Transcript
OPTIMIZATION HEURISTIC SOLUTIONS, HOW GOOD CAN THEY BE? WITH EMPIRICAL APPLICATIONS IN LOCATION PROBLEMS
XIANGLI MENG
SCHOOL OF TECHNOLOGY AND BUSINESS STUDIES DALARNA UNIVERSITY BORLÄNGE, SWEDEN MARCH 2015
ISBN: 978-91-89020-94-8
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Abstract Combinatorial optimization problems, are one of the most important types of problems in operational research. Heuristic and metaheuristics algorithms are widely applied to find a good solution. However, a common problem is that these algorithms do not guarantee that the solution will coincide with the optimum and, hence, many solutions to real world OR-problems are afflicted with an uncertainty about the quality of the solution. The main aim of this thesis is to investigate the usability of statistical bounds to evaluate the quality of heuristic solutions applied to large combinatorial problems. The contributions of this thesis are both methodological and empirical. From a methodological point of view, the usefulness of statistical bounds on p-median problems is thoroughly investigated. The statistical bounds have good performance in providing informative quality assessment under appropriate parameter settings. Also, they outperform the commonly used Lagrangian bounds. It is demonstrated that the statistical bounds are shown to be comparable with the deterministic bounds in quadratic assignment problems. As to empirical research, environment pollution has become a worldwide problem, and transportation can cause a great amount of pollution. A new method for calculating and comparing the CO2-emissions of online and brick-and-mortar retailing is proposed. It leads to the conclusion that online retailing has significantly lesser CO2-emissions. Another problem is that the Swedish regional division is under revision and the border effect to public service accessibility is concerned of both residents and politicians. After analysis, it is shown that borders hinder the optimal location of public services and consequently the highest achievable economic and social utility may not be attained.
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Sammanfattning Kombinatoriska optimeringsproblem, är en av de viktigaste typerna av problem i operationsanalys (OR). Heuristiska och metaheuristiska algoritmer tillämpas allmänt för att hitta lösningar med hög kvalitet. Ett vanligt problem är dock att dessa algoritmer inte garanterar optimala lösningar och sålunda kan det finnas osäkerhet i kvaliteten på lösningar på tillämpade operationsanalytiska problem. Huvudsyftet med denna avhandling är att undersöka användbarheten av statistiska konfidensintervall för att utvärdera kvaliteten på heuristiska lösningar då de tillämpas på stora kombinatoriska optimeringsproblem. Bidragen från denna avhandling är både metodologiska och empiriska. Ur metodologisk synvinkel har nyttan av statistiska konfidensintervall för ett lokaliseringsproblem (p-median problemet) undersökts. Statistiska konfidensintervall fungerar väl för att tillhandahålla information om lösningens kvalitet vid rätt implementering av problemen. Statistiska konfidensintervall överträffar även intervallen som erhålls vid den vanligen använda Lagrange-relaxation. I avhandlingen visas även på att metoden med statistiska konfidensintervall är fungerar väl jämfört med många andra deterministiska intervall i ett mer komplexa optimeringsproblem som det kvadratiska tilldelningsproblemet. P-median problemet och de statistiska konfidensintervallen har implementerats empiriskt för att beräkna och jämföra e-handelns och traditionell handels CO2-utsläpp från transporter, vilken visar att ehandel medför betydligt mindre CO2-utsläpp. Ett annat lokaliseringsproblem som analyserats empiriskt är hur förändringar av den regionala administrativa indelningen av Sverige, vilket är en aktuell och pågående samhällsdiskussion, påverkar medborgarnas tillgänglighet till offentlig service. Analysen visar att regionala administrativa iii
gränserna lett till en suboptimal placering av offentliga tjänster. Därmed finns en risk att den samhällsekonomiska nyttan av dessa tjänster är suboptimerad.
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Acknowledgment Four score and five years, my parents brought forth on this world a new boy, conceived in their smartness and affection, and dedicated to academic research. Now I am engaged in a PhD, testing whether that statistical bounds, so effective and so distinguished, can long be applicable. I am met on a great battlefield of that academic war. I have come to dedicate a portion of that field, as a thriving area for those who gave their work that that research might flourish. Many years later as I faced this thesis, I was always to remember that distant afternoon when my supervisor Kenneth Carling, asked me why I chose for PhD. At that time, I was just a young man wanted to enjoy myself. Now, your marvelous guidance enlightened me to become a good researcher, a good runner, a fine medicine ball thrower, and most importantly, a good person. This experience of working with you, is supercalifragilisticexpialidocious. The guidance from my second supervisor, Johan Håkansson, is a box of chocolate, you never know what you are gone get. Your supervision helped me greatly in many ways and always gave me pleasant surprises. With you being there to supervise me in thesis, I feel secure, just as when you were there with me in downhill skiing. Hard work leads to good career, intelligence leads to pleasure moments, and nice virtues lead to happy life. Lars Rönnegård shows me a perfect combination of these three. Beyond that, you show me there is no limits on how nice can a man be. You taught me not only the three virtues, but also being a virtuous man in almost all aspects. Working under you is a pleasure, an experience that I will truly treasure. Being the first boss in my life, Anders Forsman, you set the bar too high. It would be really difficult for me to find another boss as patient as you, as considerate as you, or as responsible as you. I am grateful for
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everything you have done for me, as well as every signature you signed for me which cost a fortune. It is great to have you, Song William, Wei, to assist my work, and host the warm dinners with your family. In a place 20000 leagues over the sea away from China, the dinners taste like home, so is your support. I developed the statistical bounds theories further by standing on the shoulders of Pascal Rebreyend and Mengjie Han. With your excellent work, you have built up strong shoulders, and my research benefits greatly. I am glad you two were there. Catia Cialani, you are always bringing us smiles as well as the sweet Italy calories. Grazie per i dolci e le conversazioni e pasti belle. E 'davvero fortunato ad averti qui. Fate vita veramente felice e desidera che abbiate una vita felice! I deeply thank the person(s) who took Moudud Alam and Daniel Wikström into this department. It requires a smart and hardworking person to recruit another smart and hardworking person. Moudud, your quick reaction and strong memory are amazing. You set a splendid example for me not only in work but also in blodomloppet. Daniel, you warm up atmosphere, and make every big and small conversion interesting. It is so enjoyable to work with you that I cannot image how the work would be done without you. In this journey, the PhD students have been my great companions. During the hard time when I got stuck in my paper, it is always comforting to hear other students also get trapped in research. Xiaoyun Zhao, you have the nicest characteristics of people from our province, hardworking, smart, honest, and kind to others; every time I ask some favor from you, I know that I can bet on it. It is lucky to share office with you. Dao Li, even though we have different ideas on almost everything, you are always generous to send me your help. Xia Shen, it is always delightful to have you visit us, and we miss your presence here. Kristin Svenson, your stories and topics are delightful and insightful, we enjoy vi
that a lot. Majbritt Felleki, you are not talkative most of the time, but every word you shines with intelligence. Yujiao Li, thanks for hosting delicious dinners for us. Mevludin Memedi, when I work here in the nights and weekends and holidays, it is so happy seeing you coming to work also. Ola Nääs, you are always nice and smiling and seeing you is always a pleasure. It feels so warm every time I went for you for favors. Thanks for the things you have done for me. Rickard Stridbeck, I still remember the full mark I had in Statistical Inference exam. It was my happiest moment in years. You are the exact person that witnesses me changing from a master student who is talkative but not good at language into a PhD student who is talkative but not good at language. Thanks for bearing my English. Siril Yella, your corrections for my first paper greatly improve its quality a lot. I couldn’t make it without your help. I am very grateful for that. Special thanks to Professor Sune Karlsson, I always admire you, and I will never forget what you have done for me. To myself, this PHD is not the end, it is not even the beginning of the end. But it is, perhaps, the end of the beginning. From now on, I will be strong, and I will not depend on anything, or anyone, except Kenneth, Johan, Lars, Anders, William, Pascal, Catia, Daniel, Moudud, Xiaoyun, Kristin, Sune, everyone in this university and every good people in the world. If the sky in which you did not manage to fly yesterday is still there, what are you waiting for? Wish you sleep like a baby. So say we all! Xiangli Meng 2015-03-16
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Contents 1. Introduction ................................................................................................... 1 2. Combinatorial problems and data processing ................................................ 3 2.1 Combinatorial problems .......................................................................... 3 2.2. Data processing ...................................................................................... 4 3. Methodological analysis of statistical bounds ............................................... 4 3.1 Theories of statistical bounds .................................................................. 5 3.2 Experimental design ................................................................................ 8 4. Results ........................................................................................................... 8 4.1. SOET on p-median problems ................................................................. 8 4.2 SOET on Quadratic Assignment problem ............................................. 10 4.3. Data processing .................................................................................... 10 4.4. Empirical location optimization problems ............................................11 4.4.1 CO2-emissions induced by online and brick-and-mortar retailing ......11 4.4.2 On administrative borders and accessibility to public services: The case of hospitals in Sweden ......................................................................... 13 5. Future research ............................................................................................ 15 6. Paper list: ..................................................................................................... 16 7. Reference ..................................................................................................... 17
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1. Introduction Operations research or operational research, which has its origins in military planning before World War II, provides optimized solutions of complex decision-making problems in pursuit of improved efficiency. Its significant benefits have been recognized in many fields and thus been widely studied. Combinatorial optimization problems, abbreviated as combinatorial problems, are one of the most important types of problems in operational research. They aim at finding the optimal solutions in a finite solution set. In many combinatorial problems, the number of solutions is too large, rendering exhaustive search impossible. Therefore, heuristic and metaheuristics algorithms are widely applied to find a good solution. However, a common problem is that these algorithms do not guarantee that the solution will coincide with the optimum and, hence, many solutions to real world OR-problems are afflicted with an uncertainty about the quality of the solution. One common strategy is to use algorithms providing deterministic bounds, such as Lagrangian relaxation (Fisher, 2004), and Branch and bound (Land and Doig, 1960). This strategy is popular and reasonable for many problems, but it confines the choice of heuristic algorithms, and its performance (largely) depends on the choice of parameters. Another lesser but potentially useful approach is to employ statistical bounds, or statistical optimization estimation techniques (SOETs), which estimates the value of the minimum, based on a sample of heuristic solutions, and places confidence limits around it. However, the investigation of the usage for SOETs is quite limited and many questions remaining unanswered, thus the application of SOETs is hindered. The main aim of this thesis is to investigate the usability of statistical bounds to evaluate the quality of heuristic solutions applied to large combinatorial problems. The results are applied to two widely studied empirical location problems in social research. The first is to analyze the environment effect of online and brick-and-mortar retailing regarding the CO21
emissions they induce, and the second is to investigate the borders effect on accessibility to public services. To do so, the data processing issues for location models are also discussed. The contributions of this thesis are both methodological and empirical. From a methodological point of view, the usefulness of statistical bounds on p-median problems is thoroughly investigated. The statistical bounds have good performance in providing informative quality assessment under appropriate parameter settings. Also, they outperform the commonly used Lagrangian bounds. It is demonstrated that the statistical bounds are shown to be comparable with the deterministic bounds in quadratic assignment problems. The location problems need a complete accurate road network graph which is not directly derived by original data base. Suggestions regarding how to transform a large road network database into a usable graph are proposed in an example of Sweden. It leads to a graph containing 188325*1938 nodes with over 5 million residents. As to empirical research, environment pollution has become a worldwide problem, and transportation can cause a great amount of pollution. A new method for calculating and comparing the CO2-emissions of online and brick-and-mortar retailing is proposed. It leads to the conclusion that online retailing has significantly lesser CO2-emissions. Another problem is that the Swedish regional division is under revision and the border effect to public service accessibility is concerned of both residents and politicians. After analysis, it is shown that borders hinder the optimal location of public services and consequently the highest achievable economic and social utility may not be attained. The content of the research is presented as follows: Section 2 introduces combinational problems; Section 3 describes the methodology of statistical bounds; Section 4 describes all the results of this thesis; and Section 5 discusses future research.
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2. Combinatorial problems and data processing 2.1 Combinatorial problems Combinatorial problems aim at finding an optimal object from a finite set of objects. There are many different combinatorial problems, such as travelling salesman problems and scheduling problems. In this thesis, two of the important combinatory problems are considered, p-median problems and quadratic assignment problems (QAPs). The p-median problem is a widely used location problem in many areas, such as transportation and location planning, see Reese (2006) for a complete survey. It deals with the challenge of allocating P facilities to a population geographically distributed in Q demand points, so that the population’s average or total distance to its nearest service center is minimized. The distances considered are network distance and time. Like many other combinatorial problems, the p-median problem is NPhard, (Kariv and Hakimi, 1979), and further shown to be NP-complete (Garey and Johnson, 2000). The quadratic assignment problem (QAP) is a classical combinational problem in operational research. Besides location planning and transportation analysis, it fits quite a variety of situations, such as control panel design, archaeological analysis, and chemical reaction analysis (Burkard, et al., 1997). It is formulated as follows: Consider two Ndimension square matrixes 𝐴 = (𝑎𝑖𝑗 )𝑁 and 𝐵 = (𝑏𝑖𝑗 )𝑁 , find a permutation (𝑥1 , 𝑥2 , … , 𝑥𝑁 ) of integers from 1 to 𝑁 that minimises the 𝑁 objective function 𝑔 = ∑𝑁 𝑖=1 ∑𝑗=1 𝑎𝑖𝑗 𝑏𝑧𝑖 𝑧𝑗 . The QAPs are also NP-hard
and difficult to solve, especially when 𝑁 > 15 (Loiola, et al. 2007). These two combinatorial problems are classical and widely applied. SOETs have not been studied systematically for these problems. Thus this thesis investigates the usage for SOETs on heuristic solutions, for both combinatorial problems.
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2.2. Data processing The p-median problems are often studied with data from problem libraries, which have a sound organized, road network matrix. In realworld applications, the data sets usually do not provide such a complete road network. Solving p-median problems requires complete and accurate road networks with the corresponding population data in a specific area. The Euclidian distance, which is one choice, would underestimate the real distance and lead to poor location choices (Carling, et al., 2012). To address that problem, using network distance is suggested. However, the original network data sets, which researchers have obtained, usually do not fit and are unable to be used directly. The data sets need to be cleaned and transformed to a connected road network graph before being used for finding the best locations for facilities. Many challenging problems can arise in this process, such as incomplete information or isolated sub-graphs. Dealing with these problems inappropriately could produce an inaccurate graph and then lead to poor solutions. Especially for data sets on a large scale, inappropriate treatment can cause significant trouble, and could lead to seriously deviated results. Thus, the necessity of poorly addressing the problems in the data set, and the transformation process is crucial. This thesis proposes a procedure of converting the road network database to a road graph which could be used for localization problems.
3. Methodological analysis of statistical bounds Following the pioneering work of Golden and Alt (1979), on statistical bounds of travelling salesman problems and quadratic assignment problems, some researches have been carried out in the 1980s, but since then the statistical approach has received little attention. Giddings, et al., (2014) summarizes the current research and application situation of SOETs on operational problems and gives the following framework. A problem class 𝒥 is a group of problem instances 𝛪, so 𝛪 ∈ 𝒥. This class contains well-known combinatorial optimization problems, such as TSP, Knapsack, and Scheduling. A heuristic 𝐻 is a solution method with a given random number seed. The heuristic class ℋ is the collection of 4
possible heuristics. 𝑛 is the number of replicates arising from unique random number seeds. The SOETs consist of all the combination sets for 𝒥 × ℋ 𝑛 . For a complete investigation of the SOET performance, all the restrictive types 𝐼 × 𝐻 𝑛 need to be checked. The usefulness of SOET discussed for instance by Derigs (1985), presumably still applies and therefore deserves a critical examination before application, which is the focus of the theoretical analysis in this thesis. For a specific combination set of 𝐼 × 𝐻 𝑛 , the application of SOETs on p-median location problems is examined. Then the analysis is extended to the Genetic solutions of Quadratic assignment problems. Table 1 gives the statement of corresponding work. Table 1. 𝐼 × 𝐻 𝑛 combination cases analysed I
H Simulated Annealing
Vertex Substitution
Genetic Algorithm
p-median
Yes
Yes
No
QAP
No
No
Yes
3.1 Theories of statistical bounds From a statistical point of view, solving a p-median problem means identifying the smallest value of a distribution. A few notations that are used here are provided for the sake of clarity. Such notations are: 𝑧𝑝 = feasible solution of locating P facilities in N nodes. 𝐴 = the set of all feasible solutions 𝐴 = {𝑧1 , 𝑧2 , … , 𝑧(𝑁) }. 𝑃
𝑔(𝑧𝑝 ) = the value of the objective function at solution 𝑧𝑝 . 5
𝜃 = min𝐴 𝑔(𝑧𝑝 ). 𝜃̂ = an estimator of 𝜃. 𝑛 = the number of runs of the heuristic algorithm with random starting values. 𝑥̃𝑖 = the heuristic solution of ith run 𝑖 = 1,2, … , 𝑛. 𝑥̃(𝑖) = the ith order statistic of the n heuristic solutions. Since the aim of p-median problems is to identify the unknown minimum of the objective function, 𝜃, and the corresponding solution 𝑧𝑝 , from a statistical point of view, the distribution of the objective function needs to be checked before further analysis. The problems in OR-lib (Beasley, 1990) are employed for analysis. A large random sample is drawn from each problem. The distributions of most problems in the OR-library mimic the Normal distribution. Only when P is small, slight skewness is exhibited. Thus, the objective function can be regarded as approximately Normally distributed truncated in the left tail, namely the minimum 𝜃. For a good 𝜃̂, feasible solutions whose values are close to 𝜃 would be required. For 𝜃 far out in the tail, a huge subset of A is required. For many of the OR-library p-median problems, the minimum is at least 6 standard deviations away from the mean requiring a subset of a size of 1/𝛷(−6) ≈ 109 ( 𝛷 is the standard Normal distribution function) to provide expections of obtaining feasible solutions close to 𝜃. Good SOETs require a high quality sample and good techniques. Previous research shows that repeated heuristic solutions mimic a random sample in the tail, so long as the starting values are picked at random (McRoberts, 1971 and Golden and Alt, 1979). Thereby, random values in the tail can be obtained with much less computational effort, and used for estimating 𝜃. As for the estimation techniques, there are 2 6
general approaches of SOETs, the Jackknifing approach (JK), and the extreme value theory approach (EVT). The JK-estimator is derived by: 𝜃̂𝐽𝐾 = ∑
𝑀+1 (−1)(𝑚−1) ( ) 𝑧̃(𝑚) 𝑚 𝑚=1 𝑀+1
where M is the order and 𝑧̃(𝑚) is the 𝑚𝑡ℎ smallest value in the sample. Dannenbring (1977) and Nydick and Weiss (1988) suggest using the first order, i.e. 𝑀 = 1, for point estimating the minimum. The upper bounds of the JK estimators are the minimum of 𝑥̃𝑖 . The lower bound is suggested to be
[𝜃̂𝐽𝐾 − 3𝜎 ∗ (𝜃̂𝐽𝐾 )] , where 𝜎 ∗ (𝜃̂𝐽𝐾 ) is the standard
deviation of 𝜃̂𝐽𝐾 obtained from bootstrapping the n heuristic solutions (1,000 bootstrap samples are found to be sufficient). The scalar of 3 in computing the lower bound renders the confidence level to be 99.9%, under the assumption that the sampling distribution of the JK-estimator is Normal. The extreme value theory (EVT) approach assumes the heuristic solutions to be extreme values from different random samples, and they follow the Weibull distribution (Derigs, 1985). The confidence interval is derived by the characteristic of the Weibull distribution. The estimator for 𝜃 is 𝑧̃(1) , which is also the upper bound of the confidence interval. The Weibull lower bound is [𝑧̃(1) − 𝑏̂] at a confidence level of (1 − 𝑒 −𝑛 ), 𝑏̂ is the estimated shape parameter of the Weibull distribution. The Weibull approach has been commonly accepted. However, based on the extreme value theory, when the parent distribution is Normal, the extreme values follow the Gumbel distribution. Therefore the EVT approach is completed by considering the Gumbel estimators. It has the same point estimator and upper bound as the Weibull estimator, but different lower bounds. The Gumbel lower bound was derived by its 7
theoretic percentile [𝜇 − 𝜎 𝑙𝑛(−𝑙𝑛(1 − 𝛼) )] , where μ and σ are the location and shape parameters of the Gumbel distribution while α is the confidence level. 3.2 Experimental design To verify the usefulness of SOETs, full factorial experiments are conducted.
The test problems used are from QAPLIB (Burkard et al., 1997). There are 40 p-median problems in total with known 𝜃 . The experiments consist of 4 estimators, 40 complexities, 2 heuristics, 4 sample size, and 3 computing time. The four estimators considered are the 1st and the 2nd order JK-estimators, the Weibull estimator and the Gumbel estimator. Forty problems with distinct complexities are considered. Two heuristics considered are Simulated Annealing (SA, Al-Khedhairi, 2008), and Vertex Substitution (VS, Densham and Rushton 1992). They are known to be among the most commonly used heuristic approaches in solving pmedian problems (Reese, 2006), and the ideas of these two heuristics are quite different from each other. Four sample sizes considered are 3, 10, 𝑉
𝑉
𝑉
25, 100. Three computing times are 2 ∗ (100) , 20 ∗ (100) , 60 ∗ (100) seconds per replicate, where 𝑉 is the number of nodes of the test problem, and computing time refers to CPU time of one processor, on an Intel i5-2500 and 3.30GHz. To compare SOETs with deterministic bounds, Lagrangian Relaxation (LR, Daskin, 1995) is employed as benchmark, these are run with the same computing time to get the deterministic bounds. To assess the performance, the following statistics are considered, average relative biasness (
𝑏𝑖𝑎𝑠 𝜃
∗ 100%), coverage rate
(the proportion of intervals cover 𝜃 ) and the proportion of SOET intervals shorter than the length of the deterministic bounds. 4. Results 4.1. SOET on p-median problems The experiments lead to the following results regarding SOETs on pmedian problems. 8
(1) The SOETs are quite informative, given that the heuristic solutions derived are close enough to the optimum. A statistic named SR is proposed for evaluating whether this condition is satisfied. The statistical bounds will cover the optimum almost certainly if SR is smaller than the threshold 4. (2) Comparing the performances of different SOET estimators, the 2nd order JK-estimator and the Weibull estimator have better performance with smaller bias and statistical bounds covering the optimum with higher probability. When SR<4, the bounds cover the optimum almost certainly. The Gumbel estimator and the 1st order Jackknife estimator perform worst. (3) Small sample size n such as 3 leads to unstable intervals, but 10 heuristic solutions provide almost equally good statistical bounds with 100 heuristic solutions. Thus, the effect of having more than 10 heuristic processes would have small effect on the functioning of SOET. (4) Different heuristics do not affect the performance of statistical intervals. The solutions derived by Simulated Annealing are not significantly different from those derived by Vertex Substitution. The performance of point estimators and statistical bounds are almost the same as long as SR<4. (5) Under the same computing time, statistical intervals give better results than deterministic intervals derived by Lagrangian relaxation. The statistical intervals have much shorter lengths in most of the cases while covering the optimum almost certainly. (1), (2), and (4) are novel conclusions and could not trace back to similar previous research results. (3) is analogous with Brandeau and Chiu (1993), which states 𝑛 = 10 would obtain as good solutions as 𝑛 = 2000, and statistical bounds yield better lower bounds than the available analytical bounds. (5) coincides with Brandeau and Chiu (1993), and Derigs (1985). 9
4.2 SOET on Quadratic Assignment problem The SOETs work well with the p-median problems. Next, the research is generalized to another combination set in the SOET framework 𝒥 × ℋ 𝑛 , namely the Genetic algorithm on Quadratic assignment problems (QAPs). The Genetic algorithm is one of the most widely used algorithms in solving operational problems, including QAPs (Loiola, et al. 2007). It is known to be able to find good solutions consistently, while computationally affordable and exceptionally robust to different problem characteristics or implementation (Tate and Smith, 1995). It is the leading algorithm that researchers seek to solve QAPs although the quality of the solutions remains ambiguous. The functioning of different SOETs is examined with similar procedures to those in the p-median problems. The 1st, 2nd, 3rd, 4th order JKestimators are compared and the Weibull estimator, regarding biasness, coverage rate, length of interval and functioning of SR. Then the SOET is compared with deterministic bounds. The following conclusions are derived. The Jackknife estimators have better performance than the Weibull estimators, and when the number of heuristic solutions is as large as 100, higher order JK-estimators perform better than lower order ones. Compared with the deterministic bounds, the SOET lower bound performs significantly better than most deterministic lower bounds and is comparable with the best deterministic ones. One disadvantage of SOETs is that they have a certain probability of not covering the optimum which should be considered in practical usage. 4.3. Data processing The problems accounted are quite common in location models. The problems are listed in Table 2, together with their treatments. The reasons for choosing these treatments and the effect of these are given in detail in the thesis. By solving all these problems, a road network is derived with high accuracy and could be used in location model analysis. Other researchers may not have the same data set, but this approach will provide them with appropriate suggestions for deriving good network graphs. 10
Table 2 Problems encountered in data processing and their treatments Problem
Treatment
Missing crossing, Round the coordinates to the nearest multiple connectivity information of 2 meters and add virtue edges into the graph Super large graph
Remove the dead end and divide the whole network into small segments
Long time to calculate Add the Fibonacci heap into the Dijkstra distance algorithm
4.4. Empirical location optimization problems The theoretical research of SOETs can be quite useful in empirical location optimization problems. The location problems have wide empirical applications, especially in transportation and resource planning. With SOETs, it is possible to find the solutions close enough to optimal solutions (henceforth refer to derived solutions as optimal solutions). The optimal solutions would serve as a bench mark to see the effects of, for instance, logistic planning under the best scenario. Thus the thesis considers two applications regarding transportation and resource allocation issues. The first it to compare CO2-emissions induced by online and brick-and-mortar retailing, and the second to investigate administrative borders effect on accessibility to public services. The SOETs are employed to find the solution that is sufficiently close to the optimum. 4.4.1 CO2-emissions induced by online and brick-and-mortar retailing Environmental aspects are of high priority and much research is devoted to meeting the challenges of climate change and sustainability, as well as related environmental issues. The environmental impact of the retail industry on CO2-emissions should not be underestimated. The primary aim of this study is to calculate and compare the environmental impact of buying a standard electronic product online with buying the same product in a brick-and-mortar store. The focus is on consumer 11
electronics as this category of consumer products is the largest category in online retailing in Sweden and presumably is leading the way for online shopping of other consumer products in the future. As demonstrated by Carling, Håkansson, Jia (2013), and Jia, Carling, Håkansson (2013), customers tend to take the shortest route from their home to the brick-and-mortar shopping areas, and this route highly correlated with the route that minimized CO2-emissions. Thus, shopping-related trips by the shortest route can approximate CO2emissions. This scenario fits the p-median model well and therefore SOETs can be used to derive a solution sufficiently close to the optimum. By finding the environmentally best locations of brick-andmortar stores and post offices, it is possible to compare the CO2emissions result from the two retailing methods. The comparison is conducted in a representative geographical area of Sweden. The CO2emissions from both the supply and demand side are computed, and then aggregated to calculate the total carbon footprint. In addition, potential long-term general equilibrium effects of increased online retailing such as the exit and/or relocation of mortar-and-brick stores, and potential effects on consumer demography, are also analyzed. The following conclusions are derived. (1) Online shopping induces much less CO2-emissions compared with brick-and-motar stores. The estimated, yearly reduction of CO2emissions of e-tailing of consumer electronics amounts to 28 million kg in Sweden, due to the recent emergence of this form of distribution. (2) When the locations of stores are optimized, the CO2-emissions by brick-and-motar shopping will decrease, but will still be much larger than that of online shopping. (3) Online shopping will retain its environmental effect for a long time. The conclusions are shown stable for most assumptions by sensitivity check. Only one exception exists when over 80% consumers first visit 12
the brick-and-mortar store and thereafter order goods online. In this case the advantage of online retailing would be completely offset. The method of calculating and comparing the CO2 emissions is another contribution of this paper. This proposes a new and fair way to compare the emissions efficiently and fairly. The idea and the structure of the method could be used in other similar research. 4.4.2 On administrative borders and accessibility to public services: The case of hospitals in Sweden An administrative border might hinder the optimal allocation of a given set of resources by restricting the flow of goods, services, and people. Thereby resources are commonly trapped into suboptimal locations and may reduce efficiency in using these resources. A core part of EU policy has been to promote cross-border transaction of goods, services, and labor towards a common European market. There is also a growing amount of cross-border cooperation of public authorities in Europe, while public services in the EU are still normally confined by national or regional borders. As an illustration, López et al. (2009) discuss the funding of Spanish rail investments, in light of their having substantial spill-over in French and Portuguese regions bordering Spain. It studies how regional borders affect the spatial accessibility to hospitals within Sweden. Since Swedish regions are comparable in geographical size to many European countries, such as, Belgium, Denmark, Estonia, Slovenia, Switzerland, and the Netherlands, as well as provinces in Italy and Spain, and states in Germany with a self-governing of the health care, the results are informative regarding the internal borders’ of Europe effect on the accessibility of health care. To be specific, three issues are addressed. The first is the effect of borders on inhabitants’ spatial accessibility to hospitals. The second is the quality of the location of hospitals, and the resulting accessibility. The third is accessibility, in relation to population dynamics. Sweden, for several reasons, is a suitable case for a borderland study of accessibility to hospitals. Firstly, we have access to good data of the 13
national road network, and a precise geo-coding of inhabitants, hospitals, and regional borders. Secondly, hospital funding, management, and operation are confined by the regional borders. Thirdly, after 200 years of a stable regional division of the country a substantial re-organization of the regions is due. To measure the accessibility, the average travel distance/time to the nearest hospital is employed. Compared with other location models commonly used for optimizing spatial accessibility of hospitals (Daskin and Dean, 2004), the p-median model fits this scenario best in finding the best locations of hospitals. The SOETs are used in finding the best locations without borders in Sweden nationwide. With the age data, the dynamic of the results are checked. The experiments lead to the following conclusions. (1) Half of the inhabitants are unaffected by removing borders. Border inhabitants are affected and have moderate benefits. (2) Most hospitals are currently located in optimal locations. A small number of hospitals need to be relocated and significant benefits should be derived from that. (3) The results are robust to the population dynamics. These findings imply that administrative borders only marginally worsen accessibility. However, borders will hinder the optimal location of public services. As a consequence, in particular in borderlands, highest achievable economic and social utility may not be achieved. For this reason, it seems sensible that EU policy has been to promote crossborder transaction of goods, services, and labor towards a common European market. Public services have, however, been exempted from the free flow of services and largely confined by national and regional borders.
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5. Future research As shown above, the SOETs are a potentially useful method of assessing the quality of heuristic solutions. Much more research however remains to be done. Future research relates to the following three issues: (1) pmedian problems and QAPs are shown to have differences in SOET functioning. Thus, the SOETs need to be adjusted according to different problems. (2) The reason SOETs perform differently to problems has to be explained. Is it because of the characteristics of different problems? If so, what characteristics cause the difference? (3) Mathematical support of SOETs is necessary. The heuristic processes can be treated as stochastic processes, therefore the solutions might be derived by a mathematical approach and provide us with some information, such as, asymptotic distributions. As to empirical research, many issues depending on different problems need to be explored, especially when data size is currently becoming larger. Analyzing location problems with very large data sets would be important. The big data analysis techniques would hopefully be incorporated into solving these problems.
15
6. Paper list: I. Carling, K., Meng, X., (2015). On statistical bounds of heuristic
solutions to location problems. Journal of Combinatorial Optimization, 10.1007/s10878-015-9839-0. II. Carling, K., Meng, X., (2014). Confidence in heuristic solutions?. Journal of Global Optimization, to appear. III. Meng, X., (2015). Statistical bounds of genetic solutions to quadratic assignment problems. Working papers in transport, tourism, information technology and microdata analysis, 2015:02 IV. Meng, X., Rebreyend, P., (2014), On transforming a road network database to a graph for localization purpose. International Journal on Web Services
Research, to appear. V. Carling, K., Han, M., Håkansson, J., Meng, X., Rudholm, N., (2014). Measuring CO2 emissions induced by online and brick-and-mortar retailing (No. 106). HUI Research. VI. Meng, X., Carling, K., Håkansson, J., Rebreyend, P., (2014). On administrative borders and accessibility to public services: The case of hospitals in Sweden. Working papers in transport, tourism, information technology and microdata analysis, 2014:15. Papers not included: Meng, X., Carling, K., (2014). How to Decide Upon Stopping a Heuristic Algorithm in Facility-Location Problems?. In Web Information Systems Engineering–WISE 2013 Workshops (pp. 280-283). Springer Berlin Heidelberg. Meng, X., He, C., (2012). Testing Seasonal Unit Roots in Data at Any Frequency, an HEGY approach. Working papers in transport, tourism, information technology and microdata analysis, 2012:08. 16
Meng, X., (2013). Testing for Seasonal Unit Roots when Residuals Contain Serial Correlations under HEGY Test Framework. Working papers in transport, tourism, information technology and microdata analysis, 2013:03. 7. Reference Al-Khedhairi, A., (2008). Simulated annealing metaheuristic for solving p-median problem. International Journal of Contemporary Mathematical Sciences, 3:28, 1357-1365. Beasley, J.E., (1990), OR library: Distributing test problems by electronic mail, Journal of Operational Research Society, 41:11, 10671072. Burkard, R. E., Karisch, S. E., Rendl, F. (1997), QAPLIB–a quadratic assignment problem library. Journal of Global optimization, 10(4), 391403. Carling, K., Han, M., Håkansson, J. (2012). Does Euclidean distance work well when the p-median model is applied in rural areas?. Annals of Operations Research, 201(1), 83-97. Carling, K, Håkansson, J, and Jia, T (2013) Out-of-town shopping and its induced CO2-emissions, Journal of Retailing and Consumer Services, 20:4, 382-388. Daskin, M.S., (1995). Network and discrete location: models, algorithms, and applications. New York: Wiley. Daskin M S, Dean L K, (2004). Location of health care facilities. In operations research and health care (Springer, US) pp 43-76. Densham, P.J, Rushton, G., (1992). A more efficient heuristic for solving large p-median problems. Papers in Regional Science, 71, 307-329.
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Fisher, M.L., (2004), The Lagrangian relaxation method for solving integer programming problems. Management science, 50(12_supplement), 1861-1871. Garey, M.R., Johnson, D.S, (2002). Computers and intractability, 29; W.H. Freeman, New York. Giddings, A.P., Rardin, R.L, Uzsoy, R, (2014). Statistical optimum estimation techniques for combinatorial problems: a review and critique. Journal of Heuristics, 20, 329-358. Golden, B.L., Alt, F.B., 1979. Interval estimation of a global optimum for large combinatorial optimization, Operations Research 33:5, 10241049. Hakimi, S.L., (1964). Optimum locations of switching centers and the absolute centers and medians of a graph, Operations Research, 12:3, 450-459. Hakimi, S.L., (1965). Optimum Distribution of Switching Centers in a Communication Network and Some Related Graph Theoretic Problems, Operations Research, 13:3, 462-475. Han, M., Håkansson, J., Rebreyend, P. (2013). How do different densities in a network affect the optimal location of service centers? Working papers in transport, tourism, information technology and microdata analysis, ISSN 16505581; 2013:15 Jia, T, Carling, K, Håkansson, J, (2013). Trips and their CO2 emissions induced by a shopping center, Working papers in transport, tourism, information technology and microdata analysis, 2013:02.
Kariv, O., Hakimi, S.L., (1979). An algorithmic approach to network location problems. part 2: The p-median, SIAM Journal of Applied Mathematics, 37, 539-560.
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Kotz, S., Nadarajah, S., (2000). Extreme value distributions, theory and applications, Imperial College Press. Land, A.H., Doig, A.G. (1960), An automatic method of solving discrete programming problems. Econometrica: Journal of the Econometric Society, 497-520. Loiola, E.M., Abreu, N.M.M, Boaventura-Netto, P.O., Hahn.P., Querido, T., (2007). A survey for the quadratic assignment problem. European Journal of Operational Research 176.2, 657-690. McRobert, K.L., (1971). A search model for evaluating combinatorially explosive problems, Operations Research, 19, 1331-1349. Reese, J. (2006), Solution methods for the p‐median problem: An annotated bibliography. Networks, 48(3), 125-142. Tate, D.M., Smith, A.E., (1995). A genetic approach to the quadratic assignment problem. Computers & Operations Research, 22(1), 73-83.
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PAPER I This paper is published by Journal of Combinatorial Optimization. We acknowledge the journal and Springer for this publication.
J Comb Optim DOI 10.1007/s10878-015-9839-0
On statistical bounds of heuristic solutions to location problems Kenneth Carling · Xiangli Meng
© Springer Science+Business Media New York 2015
Abstract Combinatorial optimization problems such as locating facilities frequently rely on heuristics to minimize the objective function. The optimum is often sought iteratively; a criterion is therefore necessary to be able to decide when the procedure attains such an optimum. Pre-setting the number of iterations is dominant in OR applications, however, the fact that the quality of the solution cannot be ascertained by pre-setting the number of iterations makes it less preferable. A small and, almost dormant, branch of the literature suggests usage of statistical principles to estimate the minimum and its bounds as a tool to decide upon the stopping criteria and also to evaluate the quality of the solution. In the current work we have examined the functioning of statistical bounds obtained from four different estimators using simulated annealing. P-median test problems taken from Beasley’s OR-library were used for the sake of testing. Our findings show that the Weibull estimator and 2nd order Jackknife estimators are preferable and the requirement of sample size to be about 10. It should be noted that reliable statistical bounds are found to depend critically on a sample of heuristic solutions of high quality; we have therefore provided a simple statistic for checking the quality. The work finally concludes with an illustration of applying statistical bounds to the problem of locating 70 post distribution centers in a region in Sweden. Keywords p-Median problem · Simulated annealing · Jackknife · Discrete optimization · Extreme value theory
K. Carling · X. Meng (B) School of Technology and Business Studies, Dalarna university, 791 88 Falun, Sweden e-mail:
[email protected]
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1 Introduction Consider the problem of finding a solution to min f () where the complexity of the function renders analytical solutions infeasible. If a solution is found by a heuristic, how can the quality of the heuristic solution be assessed? The issue can be exemplified by the common p-median problem. The p-median problem deals with the challenge of allocating P facilities to a population geographically distributed in Q demand points such that the population’s average or total distance to its nearest service center is minimized. Hakimi (1964) considered the task of locating telephone switching centers and has further shown (Hakimi 1965) that the optimal solution of the p-median model existed at the nodes in a given network. If N is the number of nodes, then there are N possible solutions for a p-median problem. A substantial amount of research has P been devoted to finding efficient (heuristic) algorithms to solve the p-median model (see Handler and Mirchandani 1979 and Daskin 1995 as examples); bearing in mind that enumerating all the solutions is not possible as the problem size grows. In this particular work a common heuristic known as simulated annealing1 has been investigated and is based on the implementation reported by Levanova and Loresh (2004).2 The virtue of simulated annealing, as other heuristics is that the algorithm will iterate towards a good solution, not necessarily the actual optimum. The prevailing practice is to run the heuristic algorithm for a pre-specified number of iterations or until improvements in the solution become infrequent. But given a specific problem, such practice does not readily lend itself to the determination of quality of the solution. One approach to assess the quality therefore is to seek deterministic bounds for the minimum employing techniques such as Lagrangian Relaxation (see Beasley 1993). Such an approach is popular and reasonable for many problems, but the deterministic bounds depend on the chosen parameters and are available for only a limited set of heuristic algorithms. An alternative approach is to employ statistical bounds. In short, the statistical approach is to estimate the value of the minimum based on a sample of heuristic solutions and put confidence limits around it. Golden and Alt (1979) did pioneering work on statistical bounds followed by others in the 1980s, but thereafter the statistical approach has received little attention. Akyüz et al. (2012) state to the best of their knowledge that the statistical approach had not been used in location problems since 1993. However, the usefulness of statistical bounds, as discussed for instance by Derigs (1985), presumably still applies and they therefore deserve a critical examination. A few open questions relevant to statistical bounds and their implementation are as follows. How to estimate the minimum? How and according to which principle should the bounds be derived? What is the required sample size? Are they reliable? Are they computationally affordable? Does the choice of heuristic matter? How do they perform in various OR-problems and are they competitive with deterministic bounds? 1 Simulated annealing is one of the most common solution methods to the p-median problem according
to Reese’s (2006) review. 2 Simulated annealing was implemented in R (www.r-project.org) and the code is attached in the Appendix.
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To address all these questions at once would be an insurmountable task, and therefore we limit the analysis to the first four questions in connection with the p-median problem. More specifically, the aim of the current work is to test by experimentation whether if statistical bounds can provide information on the optimum in p-median problems solved by simulated annealing. The remainder of the paper is organized as follows. Section 2 presents a review of the suggested methods for statistically estimating the minimum of the objective function, as well as bounds for the minimum. A few remarks on the issue are also added for the sake of completeness. Section 3 compares the different methods by applying them to incapacitated p-median test problems of a known optimum of varying complexity. Test problems available from the Beasley’s OR-library (Beasley 1990) were used. Section 4 illustrates the problem of locating post distribution centers in a region in mid-Sweden. The paper finally presents concluding remarks. 2 Statistical estimation of the minimum and its bounds From a statistical point of view, solving a p-median problem means identifying the smallest value of a distribution. The notation that is used throughout the paper is provided for the sake of clarity. Such notation is: z p = feasible solution of locating P facilities ⎧ in N nodes. ⎫ ⎨ ⎬ A = the set of all feasible solutions A = z 1 , z 2 , . . . , z N . ⎩ ⎭ P g z p = thevalue of the objective function at solution z p . θ = min A g z p . θˆ = an estimator of θ . n = the number of runs of the heuristic algorithm with random starting values. x˜i = the heuristic solution of the i th run i = 1, 2, . . . , n. x˜(i) = the i th order statistic of the n heuristic solutions. The aim is to identify the unknown minimum of the objective function, θ , and the corresponding solution z p . Since p-median problems usually are complex, most of the times one can only hope to get a solution near to θ (Levanova and Loresh 2004). Statistical bounds would add information about the solution by, ideally, providing an interval that almost certainly covers θ . Figure 1 gives an example of the distribution of feasible solutions to a p-median problem, namely the 14th problem in the OR-library (Beasley 1990). One million z p ’s are drawn at random from A and the histogram of the corresponding g(z p ) is given. This empirical distribution as well as the distributions of most of the other problems in the OR-library mimics the Normal distribution. However, this large sample is almost useless for identifying θ . The crux is the required size of the subset of A. The objective function in this p-median problem might be regarded as approximately Normal with a truncation in the left tail being the minimum θ . For a good θˆ , feasible solutions whose values approach θ would be required. For θ far out in the tail, a huge subset of A is required. For many of the OR-library p-median problems, the minimum is
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Fig. 1 Sample distribution of the 14th problem in the OR-library
at least 6 standard deviations away from the mean requiring a subset of a size of 1/Φ(−6) ≈ 109 (Φ is the standard Normal distribution function) to render hope of having feasible solutions close to θ . Work reported earlier has pointed out that if the starting values are picked at random, repeated heuristic solutions mimic a random sample in the tail (McRobert 1971 and Golden and Alt 1979). Thereby random values in the tail can be obtained with much less computational effort and used for estimating θ . The fact that a good θˆ needs not only a random sample in the tail but also a good estimation method led to the consideration of four point and interval estimators where the interval estimator gives the statistical bounds. The first two estimators follow from the extreme value theory (EVT). According to this theory, the distribution of the extreme value x˜i will be the Weibull distribution if g(z p ) follows a skewed (or a uniform distribution) (Derigs 1985). This is the assumption and the estimator conventionally used in the interest of obtaining statistical bounds for various problems in operations research. We have, however, taken large random subsets of A for the 40 problems in the OR-library. We have found that g(z p ) is typically symmetrically distributed, and only slightly skewed in instances in which P is small. Consequently, the extreme values might be better modelled by the Gumbel distribution due to their proximity to the Normal distribution (see Kotz and Nadarajah 2000 (p. 59)). Hence, the EVT approach is completed by considering the possibility that the distribution is Gumbel. Finally, to loosen the distributional assumptions of the EVT approach, two non-parametric estimators relying on the Jackknifing and bootstrapping methods are considered. As a side-remark, the computational cost of calculating statistical bounds for all four estimators is trivial compared with solving the p-median problem and we therefore ignore this cost. The Weibull point estimator is θˆW = x˜(1) , which is also the statistical upper bound. ˆ where bˆ is the estimated The lower bound, with a confidence of (1 − e−n ), is (x˜(1) − b) shape parameter of the Weibull distribution (Wilson et al. 2004). There are several ways of estimating the shape parameter, including the maximum likelihood estimation technique. We found the following simple estimator to be fast, stable, and giving good 2 )/( x˜ results: bˆ = x˜[0.63(n+1)] − (x˜(1) x˜(n) − x˜(2) (1) + x˜ (n) − 2 x˜ (2) ) where [0.63(n + 1)] is the floor of the value of the function (Derigs 1985).
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The Weibull and the Gumbel (θˆG ) estimators have the same point estimator and upper bound, but different lower bounds. The Gumbel lower bound was derived by its theoretic percentile [μ − σ ln (− ln (1 − α))] where μ and σ are the location and shape parameters of the Gumbel distribution while α is the √ confidence level. The Gumbel n 6var(x˜i ) i =1 x˜i parameters are estimated by the moments as σˆ = and μ ˆ = − π n 0.57722σˆ and are based on the details provided in Kotz and Nadarajah (2000, p. 12). In the Weibull approach the confidence level is determined by n. To render the Weibull and the Gumbel approach comparable in terms of confidence level we will let α = e−n . Finally, as briefly discussed in the literature, Jackknifing (hereafter JK) may be used for point estimation of the minimum as θˆJ K =
M+1
(−1)
(i−1)
i=1
M +1 i
x˜(i)
where M is the order (Quenouille 1956). Earlier relevant work suggested usage of the first order for point estimating the minimum (Dannenbring 1977; Nydick JR and Weiss 1988). The rationale for this suggestion is a lower mean square error of the first order JK-estimator compared with higher orders, in spite of a larger bias (Robson and Whitlock 1964). Both first and second order JK-estimators were used in the current (1) (2) work. The point-estimators are θˆJ K = 2 x˜(1) − x˜(2) and θˆJ K = 3x˜(1) − 3x˜(2) + x˜(3) . The upper bounds of both JK-estimators are identical to the other two estimators. As a lower statistical bound for the JK-estimators, we suggest to use the bootstrap method by Efron (1979). We define the lower bound as [θˆJ K − 3σ ∗ (θˆJ K )] where σ ∗ (θˆJ K ) is the standard deviation of θˆJ K obtained from bootstrapping the n heuristic solutions (we found 1000 bootstrap samples to be sufficient). With a scalar of 3 in computing the lower bound, the confidence level is 99.9 % provided that the sampling distribution of the JK-estimator is Normal. 3 Experimental evaluation of the estimators The two EVT-estimators Weibull and Gumbel (θˆW , θˆG ), together with the JKestimators θˆJ K and the accompanying bounds are justified by different arguments and there is no way to deem one superior to the others unless they are put to test. The OR-library’s 40 test problems were used as it is crucial to know the exact minimum θ of the problems used for comparison. Moreover, the problems vary substantially in (statistical) complexity, which we define here as ((μg(z p ) − θ )/σg(z p ) ) with μg(z p ) and σg(z p ) being the expectation and the standard deviation of g(z p ), as their minimum are varyingly far out in the tail of the distribution of g(z p ). The estimators’ ability to generate intervals that cover the problems’ minimum as well as the lengths of the intervals were investigated. Details concerning the calculation of the estimators and the bounds are given in Sect. 2, except for the size of n and the required number of iterations of the heuristic algorithm. Based on pre-testing the heuristic algorithm, we decided to evaluate the estimators after 1000, 10,000, and, for the more complex problems, 100,000 iterations. In the early literature on statistical
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J Comb Optim Table 1 Description of 6 problems from the OR-library Problem
θ
μg z p
σg z p
P11
7696
10,760
1195
Complexity 2.56
P2
4093
6054
506
3.88
P36
9934
13,436
735
4.77
P13
4374
6293
276
6.95
P33
4700
6711
186
10.81
P30
1989
3335
90
14.93
bounds little was said on the size of n, whereas Akyüz et al. (2012) and Luis et al. (2009) advocate n to be at least 100. We have examined n = 3, 10, 100 and have considered, if necessary by the experimental results, even larger values of n. 3.1 The complexity of the test problems In Table 1 the known minimum θ as well as the estimates of μg(z p ) and σg(z p ) are presented. The complexity of the problems where the parameters for the mean and the standard deviation are estimated on a random subset of size 1,000,000 of A. Instead of following the original order of the test problems, Table 1 gives the problems in ascending order of complexity (see also Table 5). The complexity varies between 2.56 for problem P11 to 14.93 for problem P30. All estimators require an efficient heuristic that produces random solutions in the tail of the distribution g(z p ). In the experiments random solutions x˜i are consistently generated in the tail by employing simulated annealing (SA) heuristic. SA has been found capable to provide good solutions to the problems in the OR-library (Chiyoshi and Galvão 2000). For each problem we run SA for 10,000 iterations (or 100,000 iterations for 22 test problems with a complexity of 6.95 and higher). For each problem we run SA 100 times with unique random starting values. Consider the issue of determining whether the solutions are in the tail of the distribution g(z p ). A sample of solutions obtained after running only a few iterations is expected to have solutions far from the optimum with great variation due to the starting points being selected at random. However, after many iterations the solutions are expectantly concentrated near to the optimum. Hence, a measure of the similarity of the heuristic solutions might be indicative of whether the sample is in the tail near to θ or not. A natural measure of similarity is the standard deviation of the heuristic solutions, σ (x˜i ).3 The variability in heuristic solutions is however sensitive to the metric (or scale) of the problem. One candidate for rescaling the variation in heuristic solutions is θ , but that parameter is of course unknown in practice. It is replaced by an estimator of it instead. The JK-estimators are the best point estimators of θ and the 3 The sample standard deviation is actually negatively biased. The bias is negligible unless n small (Cureton
1968). In the experiments, the bias of the sample standard deviation is of practical importance only in the case .
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J Comb Optim Iterations 1000 10000 100000
20
SR
15
10
5
0 1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
Complexity (ln) Fig. 2 S R as a function of the test problems complexity (1) second order estimator is slightly less biased than the first order. Nonetheless, θˆJ K was chosen for standardization since it has a smaller variation than the second order point estimator. In the following the statistic (hereafter referred to as SR) given by the (1) ratio of 1000σ (x˜i )/θˆJ K is considered for the purpose of checking if the solutions are in the tail of the distribution. Figure 2 shows S R as a function (regression lines imposed) of the problems’ complexity evaluated after 1000, 10,000, and 100,000 iterations when applicable. Some points that are worth noting in this light are as follows. A more complex, i.e. a more difficult, p-median problem will have a greater variation in the heuristic solutions because many of the solutions obtained are far from the minimum. As the number of iterations increases, the SR generally decreases, as more and more solutions are approaching the minimum.
3.2 The bias of point-estimation An initial check to see if the estimators can point-estimate the minimum was conducted. Bearing in mind the computational cost, it was decided to run simulated annealing 100 times for 100,000 iterations for each of the 40 test problems. Upon examining the statistical properties of the estimators, a re-sample with replacement from the sample of 100 heuristic solutions was chosen upon examining the statistical properties of the estimators. Table 2 contains a few results that were achieved. A complete list of the results can be found in the appendix Tables 6, 7 and 8. The table shows the results for the heuristic after 10,000 iterations and for n = 10 and n = 100. It was evident that the size of n = 3 typically led to the failure in the estimation of the minimum as well as in setting the statistical bounds and is hence not thoroughly discussed hereafter. For the three simpler problems the bias is about 0 meaning that the minimum is
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J Comb Optim Table 2 Bias of the estimators evaluated after 10,000 iterations Problem P11
Complexity 2.56
n 10
θ 7696
3.88
10
4.77
P13
6.95
P33
10.81
P30
14.93
10
0
0.20
0
0
0.44
0
0
0
1.46
0
0
0
1.46
9934
4
16
0
1.45
1
1
1
3.32
4374
15
23
14
5.88
15
15
15
6.19
4700
118
129
116
5.19
115
115
117
5.34
84
92
82
7.14
80
81
82
7.38
4093
100 10 100 10
SR
0
100 10
θˆW,G − θ
0
100 P36
(2) θˆJ K − θ
0
100 P2
(1) θˆJ K − θ
1989
100
well estimated by all the four approaches. For the three more complex problems all approaches are biased and over-estimate the minimum to the same degree. Hence, none of the approaches seems superior in terms of estimating the minimum. The bias persists when n = 100 is considered, indicating that there is no apparent gain in increasing n beyond 10. By increasing the number of iterations to 100,000 the bias of the 13th problem is eliminated and reduced for the two most complex problems in a similar way for all estimators (see Appendix Tables 6, 7, 8). Although it was expected to find the Weibull and Gumbel estimators of the minimum to be positively biased, it was dejecting to find the bias of the Jackknife-estimators to be significant. The practical value of the estimators lies in their ability in providing intervals containing the minimum. We therefore examine the intervals’ coverage percentage of the minimum θ .
3.3 Do the intervals cover the optimum? In Table 3 the same subset of test problems as of Table 2 are shown. The intervals obtained from the 2nd order Jackknife and the Weibull approaches are presented. To compute the coverage percentage of the Weibull approach for Problem 11 we took a sample with replacement of 10 (or 100) of the 100 heuristic solutions to the problem. Thereafter we computed the lower bound and checked if the minimum of 7696 was within the lower bound and the best solution (upper bound). The procedure was repeated 1000 times and the findings suggest the proportion of times the interval covered the minimum being 1.00 for this problem. A complete list of results and estimators of other test problems are presented in the appendix. The eighth column of the table gives the bounds of the intervals being the mean of the lower and the upper bounds in the 1000 replications. The coverage and bounds for the other estimators were computed in an identical manner. With a theoretical confidence level of almost 1
123
J Comb Optim Table 3 Bounds and coverage of Jackknife (2nd order) and Weibull after 10,000 iterations Problem P11
Complexity 2.56
n 10
θ
CoverageJK
IntervalJK
CoverageW
IntervalW
SR
7696
1.00
[7696,7696]
1.00
[7696,7696]
0.20
1.00
[7696,7696]
1.00
[7696,7696]
0.44
4093
1.00
[4079,4093]
1.00
[4083,4093]
1.46
1.00
[4093,4093]
1.00
[4081,4093]
1.46
9934
0.96
[9853,9950]
1.00
[9887,9950]
1.45
0.98
[9908,9935]
1.00
[9867,9935]
3.32
4374
0.91
[4329,4397]
0.99
[4355,4397]
5.88
0.23
[4374,4389]
1.00
[4344,4389]
6.19
0.20
[4737,4829]
0.03
[4773,4829]
5.19
0.00
[4793,4817]
0.00
[4758,4817]
5.34
0.19
[2019,2081]
0.03
[2047,2081]
7.14
0.00
[2051,2071]
0.00
[2034,2071]
7.38
100 P2
3.88
P36
4.77
P13
6.95
10 100 10 100 10 100
P33
10.81
10
4700
100 P30
14.93
10 100
1989
for all the interval estimators, regardless of whether n equals 10 or 100, the coverage percentage of the experiments should also be almost 100 per cent. This is the case for the simpler test problems, but certainly not for more complex problems. On the contrary, for the most complex problem (P30) all of the 1000 intervals exceeded the minimum for n = 100. It is reasonable to infer that x˜i will converge when the solutions come close to the optimum, and consequently the standard deviation of them would also decrease as discussed above. By dividing the standard deviation of x˜i by θˆJ(1) K a measure of similarity amongst the solutions is obtained allowing one to observe how far out in the tail and how close to the optimum the solutions are (reported in Tables 2 and 3 as column SR). The first three problems all have small SR. Correspondingly, the bias of the point-estimators is small and the coverage of the intervals is close to 1. For the last three problems being more complex, SR and the bias is large and the coverage percentage is poor. Hence, a large SR indicates that x˜i :s are not sufficiently near to the optimum for a reliable estimation of the minimum and bounds of it. Additional iterations of the heuristic might improve the situation. The number of iterations is further increased for problem P13 and the other problems with an even higher complexity. Table 4 gives again the coverage and the bounds of Weibull and the 2nd order Jackknife approach where the procedure is identical to the one described in relation to Table 3 with the exception that the number of iterations is 100,000. The problems P13 and P33 have a value of SR far below 5 after 100,000 iterations (Table 4). As a result the intervals generally cover the minimum. The large number of iterations is on the other hand possibly insufficient for the most complex problem (P30) as SR is about 5 and has intervals that often fail to cover the actual minimum.
3.4 The coverage and SR An examination of the intervals’ coverage for a few test problems is insufficient for drawing any general conclusions about the relationship between coverage and SR.
123
J Comb Optim Table 4 Bounds and coverage of Jackknife (2nd order) and Weibull after 100,000 iterations Problem
Complexity
P13
n
6.95
10
θ
CoverageJK
IntervalJK
CoverageW
IntervalW
SR
4374
1.00
[4370,4374]
1.00
[4370,4374]
0.91
1.00
[4374,4374]
1.00
[4369,4374]
0.93
4700
0.84
[4681,4714]
0.99
[4694,4714]
2.07
0.80
[4690,4707]
1.00
[4682,4707]
2.12
1989
0.49
[1982,2018]
0.44
[1995,2018]
4.71
0.28
[1990,2009]
0.96
[1984,2009]
4.83
100 P33
10.81
P30
14.93
10 100 10 100
Estimator Gumbel JK (1st) JK (2nd) Weibull
1.0
Coverage
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
10
12
14
16
SR Fig. 3 Coverage as a function of SR, n = 100
Figure 3 shows the coverage as a function of SR for all four estimators and n = 100. The first thing to note is that the coverage decreases drastically from about the nominal level of one to about 0.5 around SR being 4 and higher. For SR below 4 both the Weibull and the Gumbel estimator have excellent coverage whereas the two Jackknife estimators are less reliable with the first order Jackknife-estimator being the worst. Lastly, the functions depicted in the figure were estimated based on all the experiments and on all levels of iterations being 1000, 10,000, and (for about half of the test problems) 100,000. In estimating the functions we did not find any indication of different patterns depending on the number of iterations. Hence we think the results can be interpolated for iterations in the range of 1000–100,000, and probably also extrapolated beyond 100,000 iterations. However, running the heuristic algorithm in 100 parallel processes is computationally costly and it is therefore worthwhile to check how the estimators manage to uncover the minimum using a smaller sample of heuristic solutions. In Figs. 4 and 5 the coverage as a function of SR for the two lower levels of n is shown. It is evident
123
J Comb Optim Estimator Gumbel JK (1st) JK (2nd) Weibull
1.0
Coverage
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
10
12
14
16
SR Fig. 4 Coverage as a function of SR, n = 10 Estimator Gumbel JK (1st) JK (2nd) Weibull
1.0
Coverage
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
10
12
14
16
SR Fig. 5 Coverage as a function of SR, n = 3
from the two figures that the Gumbel estimator is quite poor particularly when its parameters are estimated on 10 or less observations. The Weibull estimator works well in the case of n = 10. The second order Jackknife estimator is the best estimator for n ≤ 10 and gives decent coverage even in the case of n = 3 as long as SR is below 4. Coverage is not the only factor of interest in deciding between n = 10 and n = 100. A greater value of n will reduce the bias and the choice might also affect the length
123
J Comb Optim
Fig. 6 Empirical distribution of objective function for the Swedish Post problem
of the interval. However, the work of Brandeau and Chiu (1993) suggests the bias reduction to be marginal. Investigation was limited to the Weibull estimator, on all the 36 test problems for which SR was below 4 upon running 10,000 or 100,000 iterations. The bias on average was 0.04 % larger for n = 10 than n = 100 with 0.2 % as a maximum; relative to the optimum, n = 10 produced slightly shorter intervals in 75 % of the test problems and slightly longer in the other 25 % problems. To sum up the findings in this section. Extensive experimentation on the 40 test problems from the OR-library reveal that, the Gumbel estimators and the first order Jackknife estimators are inferior to the alternatives. A sample of n = 10 seems sufficient for a reliable estimation of statistical bounds for the minimum, given that SR is modest (around 4 or less). If the SR is large, then no estimator provides reliable bounds. A complete list of all the results can be found in the appendix.
4 An illustrating problem Finally, application of the estimators to a location problem is practically illustrated. The problem of allocating 71 post distribution centers to 6,735 candidate nodes in Dalarna, Sweden is investigated. The landscape of the region and its population of 277,725 inhabitants, distributed in 15729 demand points, is described by Carling et al. (2012). Han et al. (2013) provide a detailed description of the road network and suggest that network distance is used as distance measure. The average distance in meters on the road network to the nearest postal center for the population was considered as the objective function in this particular case. A distribution of the objective function is also provided to further help realize the complexity of the problem. A random sample of 1,000,000 is drawn and the empirical distribution of g(z p ) is shown in Fig. 6. The distribution is slightly skewed to the right, but still approximately (2) Normal. To evaluate the complexity of the problem, θˆJ K was used to the estimate of the minimum θ to be able to evaluate the complexity of the problem, whereas the mean and variance of g(z p ) are derived by the 1 million random sample. The complexity being 5.47 corresponds to an intermediate OR-library problem.
123
J Comb Optim Variable SR JK Best sol.
12 10 8 6 4 2 0 0
50000
100000
150000
200000
250000
300000
Iterations Fig. 7 The Swedish Post problem and SR by the number of iterations. The best solution (%) and the Jackknife (1st order) point-estimate (%) relative to the best solution after 300,000 iterations
Drawing on the experimental results above, with n = 10 the heuristics algorithm was set to run until SR went well below 4 with a check after every 10,000 iterations. Figure 7 shows the evolution of SR (as well as the best heuristic solution and the Jackknife estimator of the minimum) as a function of the number of iterations. It was decided to stop the heuristic processes after 300,000 iterations where SR was equal to 3.12 and the statistical bounds were quite tight. Upon stopping the heuristic algorithm we used the sample of the 10 heuristic solutions to compute statistical bounds for the minimum using all the four estimators. The (1) (2) point estimators were θˆJ K = 2973, θˆJ K = 2972, and θˆW,G = 2975 with the latter being the upper bound. The lower bounds were 2964, 2956, 2959, and 2966 for first and second order Jackknife, Weibull, and Gumbel estimators, respectively. Hence, all estimators suggest an interval of at the most 20 m which is a tolerable error for this application. The problem was also addressed by using Lagrangian Relaxation (LR) to obtain deterministic bounds (Daskin 1995).4 LR was run for the computing time it took to run 300,000 iterations for SA on the problem. The deterministic upper and lower bounds were 3275 and 2534 with no improvement after half of the computing time. 5 Concluding remarks In the current work, the problem of knowing when a solution provided by a heuristic is close to optimal was investigated. Deterministic bounds may sometimes be applica4 Han et al. (2013) implemented LR for the test problems and, by pre-testing, found the mean of the
columns of the distance matrix divided by eight to yield good starting values for the algorithm. Han’s implementation of LR is mimicked.
123
J Comb Optim
ble and may be tight enough to shed knowledge on the problem. We have studied statistical bounds potentially being of a more general applicability. We have studied the occasionally used Weibull estimator as well as two variants of the Jackknife estimator which, to our knowledge, has never been used for obtaining bounds in location problems. Furthermore, we have given arguments for an alternative EVT estimator, namely the Gumbel estimator, and examined its performance. Derigs (1985) made a number of concluding observations upon studying statistical bounds with regard to TSP and QAP. We think that most of his conclusions are still valid, except one. Derigs stated “The Weibull approach leads to a proper approach,”. We have demonstrated that none of the estimators, including Weibull, are reliable unless the quality of the sample of heuristic solutions used for deriving the bounds is of high quality. To assess the quality we considered using SR which is the standard deviation of the n solutions divided by the Jackknife point-estimator. The experiments suggest that SR exceeding 4 causes unreliable bounds. The threshold of 4 should be understood as a conservative choice as the statistical bounds might be reliable in some cases where SR exceeds 4. The estimators performed similarly with a slight advantage for the second order Jackknife and the Weibull estimator. In fact, if one cannot afford to run many processes in parallel to get large n, the second order Jackknife is the first choice. We did address the question of the size of n. There is not much research on the size of n, but previous researchers have indicated that n needs to be at least 100. Our results suggest this size to be overly pessimistic, in fact most estimators provided reliable bounds at n equal to 10 (and the second order Jackknife provided fairly reliable bounds even at n equal to 3). We have limited our study to location problems by means of the p-median problems in the OR-library for which the optimum is known, and a real world p-median problem. It seems that g(z p ) closely follows the Normal distribution in these cases. Other combinatorial problems may imply, from a statistical perspective, objective functions of a more complicated kind such as multi-modal or skewed distributions. Moreover, our results seem to hold for all of the OR-library problems that represents a substantial variation in kind and complexity. However, we consistently used simulated annealing as the heuristic. We do not think this choice is crucial for our findings since the heuristic only serves to obtain a sample in the tail of the distribution and any heuristic meeting this requirement should work. However, it goes without saying that extending the variation in combinatorial problems and heuristic algorithms used would expand the knowledge about the value of statistical bounds in combinatorial optimization. In order words, it seems worthwhile for future research to address the auxiliary questions posed in the introduction; Does the choice of heuristic matter? How do statistical bounds perform in various OR-problems and are they competitive with deterministic bounds? As for the last question, there is some indications that statistical bounds are competitive with deterministic bounds. Derigs (1985) compared statistical bounds to deterministic bounds for 12 Travelling Salesman Problems and 15 Quadratic Assignment Problems. With regard to the latter, he found statistical bounds tighter (and thus more informative) than the deterministic bounds. Brandeau and Chiu (1993) found similar results on a subset of location problems. We are conducting additional experiments on statistical bounds and the preliminary results suggest statistical bounds to be competitive (Carling and Meng 2014).
123
J Comb Optim Acknowledgments We are grateful to participants at INFORMS Euro 2013 in Rome, two anonymous reviewers, and Siril Yella for useful comments on an earlier version. Financial support from the Swedish Retail and Wholesale Development Council is gratefully acknowledged.
Appendix Table 5 Description of the other 34 problems of the OR-library Problem
θ
μg(z p )
σg(z p )
Complexity
P1
5819
8426
877
2.97
P16
8162
11,353
1033
3.09
P6
7824
10,522
869
3.10
P26
9917
13,644
1133
3.29
P21
9138
12,906
1067
3.52
P38
11,060
15,078
1143
3.52
P31
10,086
13,960
1077
3.60
P35
10,400
14,179
1085
3.81
P7
5631
7930
598
3.84
P3
4250
6194
500
3.89
P27
8307
11,428
727
4.29
P17
6999
9819
631
4.47
P22
8579
11,699
676
4.62
P12
6634
9387
586
4.70
P39
9423
12,988
736
4.84
P32
9297
12,687
699
4.85
P4
3034
4618
320
4.95
P5
1355
2376
197
5.18
P8
4445
6604
356
6.07
P9
2734
4250
202
7.51
P18
4809
6769
248
7.92
P10
1255
2278
127
8.02
P23
4619
6586
220
8.94
P14
2968
4501
168
9.12
P28
4498
6369
188
9.95
P19
2845
4327
144
10.32
P15
1729
2896
109
10.67
P24
2961
4486
134
11.42
P37
5057
7246
188
11.65
P20
1789
3108
112
11.73
P40
5128
7329
179
12.32
P29
3033
4559
118
12.93
P25
1828
3131
95
13.64
P34
3013
4617
112
14.36
123
123
1000
10,000
1000
10,000
P7
P7
P2
P2
10,000
P38
P35
1000
P21
1000
1000
P26
10,000
10,000
P26
P35
1000
P6
P31
10,000
P6
1000
1000
P16
P31
10,000
P16
10,000
1000
P1
10,000
10,000
P1
P38
1000
P11
P21
1000
10,000
P11
Iter.
Problem
1.2
9.4
0.9
9.8
2.1
7.5
0.5
8.9
2.0
0.7
7.5
11.5
0.9
8.9
0.0
5.2
0.7
7.6
0.0
4.1
0.1
6.8
SR
1.00 0.92
−2
−9
0.97
0.92 0.84
2
−3
0.80 0.93
40
1.00
−5
−2
0.78
0.94
−1
69
0.83
40
1.00 0.97
0
0.78
−10
69
0.85
−2
33
0.81
40
1.00
0.93
−7
0
1.00
0.99
0
−10
23
141
20
198
62
333
15
338
77
16
333
416
25
355
0
151
17
222
0
81
2
184
–4
15
–1
64
22
98
6
64
6
10
74
42
8
42
0
3
5
20
0
9
1
19
0.93 1.00
−1
0
Bias
Cov.
Bias
Length
JK (2nd)
JK (1st)
Table 6 Results for the estimators in the computer experiments, n = 3
0.84
0.97
0.95
0.92
1.00
0.92
0.94
0.94
0.98
1.00
0.91
0.94
0.97
0.92
1.00
0.97
1.00
0.97
1.00
0.99
1.00
0.97
Cov.
41
246
34
345
110
636
27
589
135
29
580
726
44
621
0
264
31
387
0
142
3
336
Length
1.00
0.62
0.77
0.37
0.95
0.26
0.94
0.55
0.72
0.97
0.30
0.57
0.98
0.44
1.00
0.62
0.92
0.61
1.00
0.77
1.00
0.64
Cov.
Weibull
0
17
0
29
0
43
0
12
0
0
0
0
7
0
0
9
0
0
0
0
0
0
Length
2
32
2
80
1
146
0
113
3
0
146
128
0
123
0
23
0
38
0
3
0
35
Bias
θˆW,G
0.87
0.26
0.82
0.08
0.99
0.02
0.92
0.14
0.87
1.00
0.04
0.13
0.95
0.10
1.00
0.47
0.98
0.39
1.00
0.92
1.00
0.35
Cov.
Gumbel
2
15
2
25
14
36
3
34
11
4
29
40
6
29
0
16
4
26
0
12
1
23
Length
J Comb Optim
10,000
1000
10,000
1000
10,000
P32
P4
P4
P5
P5
1000
1000
P32
P13
10,000
P39
10,000
1000
P39
P8
10,000
P36
1000
1000
P36
P8
10,000
P12
10,000
P17
1000
1000
P17
P12
10,000
P27
1000
1000
P27
10,000
10,000
P3
P22
1000
P3
P22
Iter.
Problem
Table 6 continued
9.9
3.6
14.6
3.2
17.3
2.0
12.7
3.4
9.5
3.3
8.7
2.9
9.4
1.2
12.2
3.2
8.5
2.8
10.5
2.5
8.6
0.9
11.1
SR
0.99
192
0.28
0.96
−4
0.97 0.59
129
−2
0.96 0.69
−3
36
0.70
0.96
−8
57
0.67
0.90
0.57
0.86
152
3
192
9
0.61
−2
173
0.83
0.96
48
0.56
162
−10
0.97
−5
0.95 0.67
−4
105
0.67
114
1.00
179
57
272
14
93
21
156
114
353
120
340
108
354
24
315
100
280
63
294
75
286
10
174
195
3
127
2
39
1
60
5
163
8
190
17
199
8
61
1
182
9
111
5
119
6
15
0.90
−1
0
Bias
Cov.
Bias
Length
JK (2nd)
JK (1st)
0.61
0.99
0.79
0.98
0.86
0.97
0.86
0.99
0.85
0.95
0.78
0.93
0.82
1.00
0.92
0.98
0.80
0.99
0.86
0.98
0.85
1.00
0.96
Cov.
314
100
474
25
163
37
272
200
615
209
593
188
617
42
549
175
487
111
513
131
499
17
303
Length
0.06
0.68
0.15
0.87
0.19
0.86
0.19
0.67
0.17
0.56
0.15
0.60
0.19
0.92
0.36
0.66
0.18
0.77
0.17
0.63
0.22
0.96
0.63
Cov.
Weibull
0
3
9
1
0
1
0
7
27
3
0
7
52
3
39
10
0
5
0
4
48
0
11
Length
233
7
193
1
57
1
93
15
232
29
272
33
250
0
119
10
223
6
172
11
180
0
35
Bias
θˆW,G
0.00
0.54
0.00
0.87
0.00
0.83
0.01
0.48
0.00
0.24
0.00
0.17
0.00
0.98
0.08
0.60
0.00
1.00
1.00
0.51
0.00
1.00
0.35
Cov.
Gumbel
16
7
23
2
9
3
14
13
34
12
29
11
35
5
33
12
31
8
14
9
27
2
20
Length
J Comb Optim
123
123
10,000
100,000
1000
P9
P9
P18
1000
10,000
100,000
1000
10,000
100,000
1000
10,000
100,000
1000
10,000
100,000
1000
10,000
P10
P10
P10
P23
P23
P23
P14
P14
P14
P28
P28
P28
P19
P19
10,000
1000
P9
100,000
100,000
P13
P18
10,000
P13
P18
Iter.
Problem
Table 6 continued
6.0
10.6
1.8
5.5
9.2
2.0
7.8
12.1
1.4
6.7
10.6
2.4
8.1
16.4
0.9
4.9
8.9
2.4
5.8
14.2
0.8
5.3
SR
0.97
0.96
73
248
4
85
0.33
0.07
0.83
0.40
0.09
−1
306
0.60
0.19
0.91
0.51
45
210
0
77
0.14
−1
298
0.67
0.24
0.88
0.59
0.21
16
102
0
52
233
0.82 0.87
8
−2
0.45
0.97
−2
123
0.80
17
69
129
30
105
177
20
93
152
24
121
195
10
39
87
15
91
178
29
62
160
12
87
75
251
6
83
307
3
47
212
2
83
312
0
19
104
3
57
236
–3
10
126
0
23
Bias
Length
Bias
Cov.
JK (2nd)
JK (1st)
0.60
0.32
0.92
0.65
0.37
0.98
0.81
0.46
0.96
0.74
0.44
0.98
0.86
0.56
0.93
0.79
0.48
0.92
0.92
0.70
0.97
0.91
Cov.
120
225
52
184
308
34
162
265
42
211
340
18
67
151
27
159
310
50
109
280
22
151
Length
0.08
0.03
0.50
0.12
0.05
0.79
0.14
0.05
0.63
0.15
0.05
0.78
0.30
0.06
0.89
0.17
0.06
0.61
0.31
0.08
0.90
0.26
Cov.
Weibull
8
0
2
6
11
2
1
4
1
12
21
0
0
1
1
10
10
0
1
7
0
4
Length
88
277
10
110
347
2
67
245
5
104
341
0
24
122
3
72
274
5
22
160
0
36
Bias
θˆW,G
0.00
0.00
0.13
0.00
0.00
0.59
0.00
0.00
0.32
0.00
0.00
0.88
0.01
0.00
0.35
0.00
0.00
0.43
0.08
0.00
0.94
0.06
Cov.
Gumbel
6
12
3
9
15
3
8
14
3
12
20
1
4
8
2
9
16
2
6
14
2
10
Length
J Comb Optim
100,000
1000
P40
P40
P29
100,000
10,000
P40
P29
1000
P20
10,000
100,000
P20
P29
1000
10,000
P20
100,000
P37
1000
P24
10,000
100,000
P33
1000
10,000
P33
P37
1000
P33
P37
100,000
P15
10,000
10,000
P15
100,000
1000
P15
P24
100,000
P19
P24
Iter.
Problem
Table 6 continued
3.0
7.2
10.8
2.9
5.1
7.1
4.4
8.9
14.9
2.6
4.9
9.8
2.7
6.0
12.1
1.9
4.8
8.4
2.4
9.2
15.9
2.0
SR
22
106
345
32
155
481
4
62
263
23
150
434
12
94
296
11
125
381
1
39
177
4
0.56
0.27
0.04
0.59
0.17
0.01
0.85
0.35
0.06
0.63
0.17
0.08
0.65
0.25
0.08
0.73
0.21
0.04
0.89
0.53
0.16
0.82
35
90
143
60
107
158
32
66
124
50
104
217
32
76
157
35
93
170
15
64
120
20
24
106
345
33
157
484
5
63
262
25
151
434
13
93
297
12
124
382
2
41
178
7
Bias
Length
Bias
Cov.
JK (2nd)
JK (1st)
0.79
0.55
0.22
0.80
0.44
0.10
0.94
0.66
0.25
0.82
0.46
0.30
0.85
0.52
0.31
0.88
0.51
0.24
0.95
0.79
0.45
0.92
Cov.
61
158
250
106
187
275
55
116
217
87
181
379
56
133
274
64
163
296
27
111
210
34
Length
0.12
0.05
0.03
0.13
0.06
0.02
0.28
0.07
0.03
0.14
0.05
0.03
0.21
0.04
0.04
0.32
0.05
0.04
0.51
0.13
0.04
0.25
Cov.
Weibull
4
0
0
0
0
3
0
3
5
0
7
1
2
0
4
2
4
0
1
0
0
2
Length
29
127
379
46
180
518
12
78
292
34
174
485
20
112
332
19
147
420
4
54
205
8
Bias
θˆW,G
0.00
0.00
0.00
0.00
0.00
0.00
0.06
0.00
0.00
0.00
0.00
0.00
0.02
0.00
0.00
0.02
0.00
0.00
0.21
0.00
0.00
0.07
Cov.
Gumbel
4
8
13
6
10
15
3
6
10
5
9
19
3
6
14
4
8
14
2
6
11
3
Length
J Comb Optim
123
123
100,000
P34
P30
100,000
P34
1000
10,000
P34
10,000
1000
P25
P30
100,000
P25
P30
1000
10,000
P25
Iter.
Problem
Table 6 continued
4.3
6.5
6.8
3.8
5.9
5.9
3.8
7.8
7.8
SR
27
89
372
30
119
404
21
78
308
0.40
0.13
0.00
0.50
0.10
0.01
0.47
0.16
0.03
34
55
119
44
72
157
28
56
118
28
89
377
32
122
405
22
82
308
Bias
Length
Bias
Cov.
JK (2nd)
JK (1st)
0.69
0.40
0.08
0.74
0.39
0.17
0.71
0.50
0.16
Cov.
59
96
207
77
126
273
49
98
206
Length
0.10
0.04
0.01
0.11
0.04
0.02
0.09
0.06
0.02
Cov.
Weibull
0
0
0
3
1
8
0
2
1
Length
34
102
399
40
135
441
27
90
336
Bias
θˆW,G
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Cov.
Gumbel
3
5
11
4
7
14
3
6
10
Length
J Comb Optim
10,000
1000
1000
10,000
10,000
1000
10,000
1000
10,000
P26
P26
P21
P38
P21
P38
P31
P31
P35
P35
10,000
1000
P6
P2
10,000
P6
1000
1000
P16
P2
10,000
P16
1000
1000
P1
10,000
10,000
P1
P7
1000
P11
P7
1000
10,000
P11
Iter.
Problem
1.5
10.0
1.0
11.3
2.9
8.2
0.8
9.9
2.3
1.1
8.3
12.4
1.2
9.3
0.0
5.7
0.9
8.2
0.0
4.9
0.2
7.5
SR
0.93
0
3
0
33
0
53
0
16
0
0
53
28
0
17
1.00
0.94
1.00
0.71
1.00
0.77
1.00
0.85
1.00
1.00
0.80
0.87
1.00
0.82
1.00
0.97
−5
0
1.00
0.93
1.00
1.00
1.00
0
2
0
0
0
3
6
55
5
90
1
180
1
181
10
0
180
202
2
209
0
57
1
68
0
7
0
62
0
2
0
29
0
72
0
7
0
0
48
22
0
6
0
–3
0
1
0
0
0
1
Bias
Length
Bias
Cov.
JK (2nd)
JK (1st)
Table 7 Results for the estimators in the computer experiments, n = 10
1.00
0.99
1.00
0.88
1.00
0.87
1.00
0.93
1.00
1.00
0.94
0.95
1.00
0.96
1.00
1.00
1.00
0.99
1.00
1.00
1.00
0.98
Cov.
14
97
12
155
4
291
2
309
23
0
310
349
4
357
0
101
2
123
0
19
0
109
Length
1.00
1.00
0.99
0.99
1.00
0.97
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Cov.
Weibull
10
66
8
89
8
185
2
179
26
0
226
220
5
197
0
66
2
99
0
25
0
77
Length
0
14
0
52
0
89
0
54
0
0
89
68
0
62
0
4
0
14
0
0
0
15
Bias
θˆW,G
0.99
0.90
0.99
0.60
1.00
0.27
1.00
0.73
1.00
1.00
0.38
0.69
1.00
0.51
1.00
0.97
1.00
0.98
1.00
1.00
1.00
0.97
Cov.
Gumbel
7
41
7
64
51
90
14
89
36
18
69
97
20
65
0
47
12
72
0
42
3
66
Length
J Comb Optim
123
123
1000
10,000
1000
10,000
P5
P8
P8
P32
P5
10,000
P32
1000
1000
P39
10,000
10,000
P39
P4
1000
P36
P4
1000
10,000
P36
10,000
P12
10,000
P17
1000
1000
P17
P12
10,000
P27
1000
1000
P27
10,000
10,000
P3
P22
1000
P3
P22
Iter.
Problem
Table 7 continued
4.2
16.3
3.8
20.0
2.3
14.0
3.7
10.3
3.6
9.7
1.5
10.0
1.6
13.6
3.4
9.8
3.3
11.6
2.8
9.4
1.3
12.5
SR
0.58 0.96
−1
1.00
0.62
1.00
107
0
29
0
0.56
0.97
−2
48
0.39
0.93
0.40
1.00
0.44
1.00
145
0
173
4
145
0
0.78
0.99
37
0.32
145
−2
0.98
−1
0.99 0.59
−1
93
0.56
92
1.00
16
176
2
53
4
82
34
160
58
194
57
203
1
155
29
154
14
156
24
167
0
77
–1
98
0
27
0
43
–1
142
–2
160
0
129
0
31
0
137
–1
86
–1
81
0
–6
0.94
−4
0
Bias
Cov.
Bias
Length
JK (2nd)
JK (1st)
0.99
0.80
1.00
0.86
1.00
0.77
1.00
0.77
0.97
0.65
0.96
0.68
1.00
0.93
1.00
0.61
0.99
0.83
1.00
0.78
1.00
0.96
Cov.
29
300
4
92
8
142
60
280
99
334
97
349
2
269
52
266
26
269
43
285
0
133
Length
1.00
0.85
1.00
0.97
1.00
0.95
1.00
0.85
0.99
0.84
1.00
0.89
1.00
1.00
1.00
0.76
1.00
0.95
1.00
0.96
1.00
1.00
Cov.
Weibull
23
153
4
55
7
92
48
183
63
213
58
201
3
173
41
162
20
175
32
168
0
83
Length
2
144
0
40
0
66
4
176
11
215
16
191
0
69
1
177
1
125
2
128
0
11
Bias
θˆW,G
0.99
0.02
1.00
0.11
1.00
0.09
0.99
0.01
0.82
0.00
0.78
0.03
1.00
0.57
1.00
0.01
1.00
1.00
0.98
0.06
1.00
0.94
Cov.
Gumbel
22
50
8
22
10
35
38
83
29
69
29
85
19
80
36
72
19
30
26
57
11
53
Length
J Comb Optim
100,000
P20
10,000
1000
P28
P40
100,000
P14
1000
10,000
P14
P40
1000
1000
P23
P14
100,000
P10
10,000
10,000
P10
100,000
1000
P10
P23
100,000
P18
P23
1000
10,000
P18
100,000
P9
P18
1000
10,000
P13
P9
100,000
P13
P9
1000
10,000
P13
Iter.
Problem
Table 7 continued
5.4
7.9
4.8
10.1
2.3
8.4
13.0
1.6
7.2
11.6
2.7
9.0
17.7
1.2
5.6
9.7
2.6
6.3
15.9
0.9
5.9
10.4
SR
142
467
5
294
0
40
196
0.11
0.00
0.78
0.00
0.96
0.45
0.06
0.34 0.93
68
0.02
1.00
0.32
0.13
0.80
0.41
−1
288
0
15
95
0
45
0.10
0.97
−1
215
0.84
0.28
1.00
0.76
0.11
6
110
0
15
185
76
105
13
102
4
53
94
12
69
105
1
19
54
5
53
116
13
32
98
2
40
94
133
456
5
289
0
38
189
–1
65
282
0
13
90
0
42
207
–1
5
106
0
14
177
Bias
Length
Bias
Cov.
JK (2nd)
JK (1st)
0.32
0.06
0.95
0.11
0.99
0.78
0.27
0.98
0.65
0.19
1.00
0.66
0.33
0.96
0.66
0.32
0.99
0.94
0.59
1.00
0.91
0.28
Cov.
130
181
24
174
8
91
160
20
118
181
3
33
93
9
91
198
24
55
167
4
68
162
Length
0.10
0.02
1.00
0.01
1.00
0.82
0.05
1.00
0.52
0.02
1.00
1.00
0.16
1.00
0.79
0.06
0.97
0.99
0.37
1.00
0.99
0.09
Cov.
Weibull
81
117
17
105
7
52
94
12
71
113
4
21
57
5
53
114
15
32
98
4
42
100
Length
160
490
7
315
1
51
217
2
83
310
0
19
107
1
56
241
1
13
130
0
23
205
Bias
θˆW,G
0.00
0.00
0.54
0.00
1.00
0.01
0.00
0.92
0.03
0.00
1.00
0.11
0.00
0.89
0.01
0.00
0.88
0.60
0.00
1.00
0.51
0.00
Cov.
Gumbel
20
32
8
37
9
19
30
7
28
48
5
11
17
8
22
33
6
15
32
6
25
38
Length
J Comb Optim
123
123
100,000
P30
100,000
P34
1000
10,000
P34
10,000
1000
P34
P30
100,000
P25
P30
1000
10,000
P25
100,000
P29
P25
1000
10,000
P29
100,000
P40
P29
Iter.
Problem
Table 7 continued
4.7
7.1
7.2
4.2
6.5
6.5
4.2
8.3
8.3
3.4
7.9
11.8
3.3
SR
24
84
367
27
114
396
18
78
300
20
98
328
28
0.31
0.02
0.02
0.30
0.06
0.07
0.33
0.04
0.00
0.24
0.07
0.01
0.48
21
36
62
24
41
87
18
24
74
17
56
99
36
22
82
364
25
110
391
17
76
296
20
95
320
26
Bias
Length
Bias
Cov.
JK (2nd)
JK (1st)
0.49
0.19
0.00
0.58
0.16
0.03
0.59
0.10
0.04
0.61
0.34
0.10
0.77
Cov.
36
62
106
42
71
151
31
42
126
30
96
169
62
Length
0.44
0.03
0.00
0.34
0.05
0.01
0.49
0.00
0.01
0.41
0.06
0.02
0.82
Cov.
Weibull
23
34
70
23
45
97
16
27
69
19
58
92
37
Length
29
92
380
32
123
414
22
83
316
24
110
349
35
Bias
θˆW,G
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
Cov.
Gumbel
7
11
28
11
16
32
6
15
21
9
17
26
12
Length
J Comb Optim
1000
10,000
1000
10,000
1000
10,000
1000
10,000
1000
10,000
1000
1000
10,000
10,000
1000
10,000
1000
10,000
1000
10,000
1000
10,000
P11
P11
P1
P1
P16
P16
P6
P6
P26
P26
P21
P38
P21
P38
P31
P31
P35
P35
P7
P7
P2
P2
Iter.
Problem
1.5
10.5
1.0
11.6
3.3
8.4
1.1
10.2
2.4
1.6
8.4
12.7
1.5
9.6
0.0
6.0
1.0
8.4
0.0
5.0
0.4
7.9
SR
1.00
0.94
−1
0
1.00
0.59
1.00
0.67
0
21
0
33
1.00
0.97
−3
0
1.00
1.00
0
0
0.67
0.74
33
1.00
−1
0.92
1.00
1.00
0
4
0
0
1.00
1.00
−1
0
1.00
1.00
0
0
0
17
0
30
0
56
0
55
0
0
56
86
0
51
0
0
0
10
0
0
0
15
0
−1
0
21
0
74
0
−3
0
0
34
−8
0
2
0
0
0
0
0
0
0
1.00
0.97
1.00
0.81
1.00
0.37
1.00
0.99
1.00
1.00
0.93
0.79
1.00
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.96
1.00
0.86
−1
0
0
Cov.
Bias
Length
Bias
Cov.
JK (2nd)
JK (1st)
Table 8 Results for the estimators in the computer experiments, n = 100
0
30
0
52
0
70
0
96
0
0
98
147
0
89
0
1
0
19
0
0
0
26
Length
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Cov.
Weibull
12
74
8
109
1
213
1
203
20
0
221
246
5
226
0
67
0
112
0
23
0
87
Length
0
1
0
27
0
41
0
5
0
0
41
16
0
13
0
0
0
1
0
0
0
2
Bias
θˆW,G
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Cov.
Gumbel
18
108
17
165
121
239
40
229
86
53
194
261
54
191
0
132
28
186
0
96
13
166
Length
J Comb Optim
123
123
1000
1000
10,000
1000
10,000
1000
10,000
1000
10,000
1000
P32
P32
P4
P4
P5
P5
P8
P8
P13
10,000
P36
10,000
1000
P36
P39
10,000
P12
P39
1000
P17
P12
10,000
P17
1000
1000
P27
10,000
10,000
P27
P22
1000
P3
P22
1000
10,000
P3
Iter.
Problem
Table 8 continued
10.7
4.3
16.7
4.0
20.2
2.4
14.4
3.8
10.6
3.7
10.0
3.3
10.5
1.8
13.9
3.7
9.9
3.6
11.8
3.0
9.7
1.9
12.8
SR
1.00
0
158
0
100
0
26
0
38
0.00
1.00
0.00
1.00
0.05
1.00
0.19
1.00
0.00
0.97
129
0.38
−1
0.94
−1
123
0.48
1.00
0.29
1.00
0.31
1.00
0.03
1.00
0.26
1.00
92
0
32
0
106
0
83
0
68
0
0
57
0
34
0
13
0
27
2
49
10
109
15
117
0
30
0
96
0
36
1
61
0
5
156
0
99
0
25
0
40
0
126
0
118
−1
87
0
34
0
107
0
84
0
69
0
0
Bias
Length
Bias
Cov.
JK (2nd)
JK (1st)
0.15
1.00
0.12
1.00
0.35
1.00
0.50
1.00
0.18
0.98
0.55
0.98
0.77
1.00
0.71
1.00
0.68
1.00
0.16
1.00
0.68
1.00
1.00
Cov.
97
1
59
0
23
0
48
5
85
18
184
27
196
0
56
0
162
0
63
3
108
0
10
Length
0.01
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Cov.
Weibull
128
22
173
3
62
5
106
49
209
69
248
63
245
2
192
37
193
19
186
33
194
0
79
Length
168
0
107
0
28
0
41
0
138
0
144
1
114
0
35
0
122
0
88
0
77
0
0
Bias
θˆW,G
0.00
1.00
0.98
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.88
1.00
1.00
1.00
1.00
1.00
0.99
1.00
1.00
1.00
1.00
1.00
1.00
Cov.
Gumbel
88
57
154
18
61
24
91
100
228
85
174
75
204
43
216
93
174
54
79
71
158
30
144
Length
J Comb Optim
100,000
1000
10,000
100,000
1000
10,000
100,000
1000
10,000
100,000
P18
P18
P10
P10
P10
P23
P23
P23
P14
P14
P14
1000
10,000
100,000
P19
P19
P19
100,000
10,000
P18
P28
1000
P9
1000
100,000
P9
10,000
10,000
P9
P28
1000
P13
P28
10,000
100,000
P13
Iter.
Problem
Table 8 continued
2.3
6.7
11.7
2.0
6.5
10.4
2.5
8.6
13.4
1.7
7.6
12.0
2.7
9.4
18.4
1.3
5.6
10.0
2.6
6.5
16.3
0.9
6.2
SR
4
60
236
1
50
289
0
36
181
0
61
268
0
4
74
0
36
191
0
6
99
0
15
0.00
0.00
0.00
0.86
0.36
0.00
1.00
0.01
0.00
1.00
0.00
0.00
1.00
0.66
0.15
0.99
0.35
0.00
1.00
0.20
0.00
1.00
0.05
2
21
22
4
41
20
0
13
37
1
21
48
0
24
46
2
23
58
0
4
32
0
7
4
58
235
1
47
290
0
36
181
0
61
270
0
1
71
0
35
192
0
6
99
0
15
Bias
Length
Bias
Cov.
JK (2nd)
JK (1st)
0.16
0.38
0.00
0.96
0.53
0.00
1.00
0.09
0.00
1.00
0.08
0.00
1.00
0.66
0.39
0.99
0.37
0.05
1.00
0.59
0.13
1.00
0.23
Cov.
3
35
38
7
70
36
1
22
64
2
36
88
0
42
78
3
40
98
0
8
55
0
12
Length
1.00
0.00
0.00
1.00
1.00
0.00
1.00
1.00
0.00
1.00
0.99
0.00
1.00
1.00
0.38
1.00
1.00
0.00
1.00
1.00
0.88
1.00
1.00
Cov.
Weibull
9
48
78
17
88
115
6
59
106
12
77
128
4
28
68
6
62
129
16
36
111
5
45
Length
4
64
240
2
57
291
0
39
188
0
65
274
0
9
83
0
41
201
0
6
104
0
15
Bias
θˆW,G
1.00
0.00
0.00
1.00
0.05
0.00
1.00
0.97
0.00
1.00
0.77
0.00
1.00
1.00
0.00
1.00
0.99
0.00
1.00
1.00
0.20
1.00
1.00
Cov.
Gumbel
19
38
81
21
42
106
24
54
79
20
76
123
11
24
39
19
57
88
18
41
92
13
69
Length
J Comb Optim
123
123
1000
100,000
1000
10,000
100,000
1000
10,000
100,000
1000
10,000
P20
P40
P40
P40
P29
P29
P29
P25
P25
100,000
P37
10,000
10,000
P37
P20
1000
P37
P20
100,000
P33
P24
100,000
P33
1000
10,000
P33
10,000
1000
P15
P24
100,000
P15
P24
1000
10,000
P15
Iter.
Problem
Table 8 continued
8.6
15.3
3.4
8.1
12.2
3.4
5.7
8.1
4.8
10.0
17.2
2.9
5.5
11.2
3.1
6.6
13.9
2.1
5.3
9.4
2.8
10.2
18.3
SR
72
281
19
96
298
24
118
428
4
48
225
20
133
396
7
84
239
4
115
332
1
25
155
0.00
0.00
0.00
0.00
0.00
0.14
0.02
0.00
0.37
0.06
0.00
0.00
0.00
0.00
0.20
0.00
0.00
0.69
0.00
0.00
0.28
0.43
0.00
14
43
5
12
68
13
52
79
4
25
54
7
26
46
5
16
71
12
14
65
1
25
26
71
278
19
95
296
23
115
423
4
48
221
19
131
396
7
84
235
4
115
328
1
23
155
Bias
Length
Bias
Cov.
JK (2nd)
JK (1st)
0.00
0.00
0.01
0.00
0.00
0.36
0.36
0.00
0.70
0.29
0.00
0.08
0.00
0.00
0.64
0.01
0.11
0.80
0.00
0.00
0.66
0.63
0.00
Cov.
24
74
8
21
115
22
89
133
7
42
92
12
45
81
10
27
120
17
24
111
2
43
45
Length
0.00
0.00
0.55
0.00
0.00
1.00
0.00
0.00
1.00
0.63
0.00
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
1.00
1.00
0.00
Cov.
Weibull
35
88
19
61
121
40
89
134
19
50
104
27
71
152
23
51
132
25
59
134
8
46
85
Length
74
289
20
98
311
26
128
443
4
53
235
21
138
403
8
86
253
7
117
345
1
30
160
Bias
θˆW,G
0.00
0.00
0.97
0.00
0.00
0.98
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.78
0.00
Cov.
Gumbel
37
54
25
51
62
37
43
68
20
34
53
35
55
116
17
38
64
22
53
78
12
34
66
Length
J Comb Optim
Iter.
100,000
1000
10,000
100,000
1000
10,000
100,000
Problem
P25
P34
P34
P34
P30
P30
P30
Table 8 continued
4.8
7.4
14.5
4.3
6.7
12.8
4.3
SR
19
80
361
26
98
373
15
0.11
0.00
0.00
0.00
0.00
0.00
0.07
10
12
18
5
33
58
9
20
81
360
26
96
368
14
Bias
Length
Bias
Cov.
JK (2nd)
JK (1st)
0.28
0.00
0.00
0.01
0.14
0.00
0.43
Cov.
18
20
31
10
56
99
15
Length
0.96
0.00
0.00
0.91
0.00
0.00
0.99
Cov.
Weibull
25
37
71
27
55
110
20
Length
20
82
364
26
105
385
16
Bias
θˆW,G
0.08
0.00
0.00
0.70
0.00
0.00
0.26
Cov.
Gumbel
17
29
75
28
36
82
15
Length
J Comb Optim
123
J Comb Optim
R-code of simulated annealing N=100 # Number of candidates p=5 # Number of facilities n<-100 # Number of SA processes ni<-10000 # Number of iterations heuristic.solution<-matrix(0, nrow=ni,ncol=n) heuristic.location<-matrix(0, nrow=p, ncol=n) Store.solution<-numeric() Best.solution<-numeric() Store.location<-matrix(0, nrow=ni, ncol=p) Best.location<-numeric() for (i in 1:n){ subset<-numeric(0) select.location<-sample(1:N,p,replace=F) objective.function<-sum(apply(Distance.matrix[,select.location],1,min)) iteration<-0;Tempearture<-400;beta<-0.5 #initial parameter setting. while (iteration
objective.function){ delta<-updated.objective.function-objective.function unif.number<-runif(1,0,1) if (unif.number=exp(-delta/Tempearture)) {count<-count+1;select.location<-
123
J Comb Optim
store.selection } } iteration<-iteration+1 Tempearture<-Tempearture*0.95 Store.solution[iteration]<-objective.function Best.solution[iteration]<-min(Store.solution[1:iteration]) Store.location[iteration,]<-select.location Best.location<-Store.location[min(which(Store.solution==Best.solution[iteration])),] } heuristic.solution[,i]<-Best.solution heuristic.location[,i]<-Best.location }
References Akyüz MH, Öncan T, Altınel IK (2012) Efficient approximate solution methods for the multi-commodity capacitated multi-facility Weber problem. Comput Oper Res 39(2):225–237 Beasley JE (1990) OR library: distributing test problems by electronic mail. J Oper Res Soc 41(11):1067– 1072 Beasley JE (1993) Lagrangian heuristics for location problems. Eur J Oper Res 65:383–399 Brandeau ML, Chiu SS (1993) Sequential location and allocation: worst case performance and statistical estimation. Locat Sci 1:289–298 Carling K, Han M, Håkansson J (2012) Does Euclidean distance work well when the p-median model is applied in rural areas? Ann Oper Res 201(1):83–97 Carling K, Meng X (2014) Confidence in heuristic solutions? Working papers in transport, tourism, information technology and microdata analysis. http://du.diva-portal.org/smash/record.jsf? pid=diva2%3A727755&dswid=-6054 Chiyoshi FY, Galvão RD (2000) A statistical analysis of simulated annealing applied to the p-median problem. Ann Oper Res 96:61–74 Cureton EE (1968) Unbiased estimation of the standard deviation. Am Stat 22(1):22 Dannenbring DG (1977) Procedures for estimating optimal solution values for large combinatorial problems. Manag Sci 23(12):1273–1283 Daskin MS (1995) Network and discrete location: models, algorithms, and applications. Wiley, New York Derigs U (1985) Using confidence limits for the global optimum in combinatorial optimization. Oper Res 33(5):1024–1049 Efron B (1979) Bootstrap methods: another look at the Jackknife. Ann Stat 7(1):1–26 Golden BL, Alt FB (1979) Interval estimation of a global optimum for large combinatorial optimization. Oper Res 33(5):1024–1049 Hakimi SL (1964) Optimum locations of switching centers and the absolute centers and medians of a graph. Oper Res 12(3):450–459 Hakimi SL (1965) Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Oper Res 13(3):462–475 Handler GY, Mirchandani PB (1979) Location on networks: theorem and algorithms. MIT Press, Cambridge Han M, Håkansson J, Rebreyend P (2013) How do different densities in a network affect the optimal location of service centers?. Working papers in transport, tourism, information technology and microdata analysis 2013:15 Kotz S, Nadarajah S (2000) Extreme value distributions, theory and applications. Imperial College Press, London
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J Comb Optim Levanova T, Loresh MA (2004) Algorithm of ant system and simulated annealing for the p-median problem. Autom Remote Control 65:431–438 Luis M, Sahli S, Nagy G (2009) Region-rejection based heuristics for the capacitated multi-source Weber problem. Comput Oper Res 36:2007–2017 McRobert KL (1971) A search model for evaluating combinatorially explosive problems. Oper Res 19:1331– 1349 Nydick RL, Weiss HJ (1988) A computational evaluation of optimal solution value estimation procedures. Comput Oper Res 5:427–440 Quenouille MH (1956) Notes on bias in estimation. Biometrika 43:353–360 Reese J (2006) Solution methods for the p-median problem: An annotated bibliography. Networks 48:125– 142 Robson DS, Whitlock JH (1964) Estimation of a truncation point. Biometrika 51:33–39 Wilson AD, King RE, Wilson JR (2004) Case study on statistically estimating minimum makespan for flow line scheduling problems. Eur J Oper Res 155:439–454
123
PAPER II This paper is accept by Journal of Global Optimization. We acknowledge the journal and Springer for this publication.
Confidence in heuristic solutions? Authors: Kenneth Carling and Xiangli Meng Abstract: Solutions to combinatorial optimization problems frequently rely on heuristics to minimize an intractable objective function. The optimum is sought iteratively and pre-setting the number of iterations dominates in operations research applications, which implies that the quality of the solution cannot be ascertained. Deterministic bounds offer a mean of ascertaining the quality, but such bounds are available for only a limited number of heuristics and the length of the corresponding interval may be difficult to control in an application. A small, almost dormant, branch of the literature suggests using statistical principles to derive statistical bounds for the optimum. We discuss alternative approaches to derive statistical bounds. We also assess their performance by testing them on 40 test p-median problems on facility location, taken from Beasley’s OR-library, for which the optimum is known. We consider three popular heuristics for solving such location problems; simulated annealing, vertex substitution, and Lagrangian relaxation where only the last offers deterministic bounds. Moreover, we illustrate statistical bounds in the location of 71 regional delivery points of the Swedish Post. We find statistical bounds reliable and much more efficient than deterministic bounds provided that the heuristic solutions are sampled close to the optimum. Statistical bounds are also found computationally affordable.
Key words: p-median problem, deterministic bounds, statistical bounds, jackknife, discrete optimization, extreme value theory
Kenneth Carling is a professor in Statistics and Xiangli Meng is a PhD-student in Micro-data analysis at the School of Technology and Business Studies, Dalarna university, SE-791 88 Falun, Sweden. Corresponding author. E-mail: [email protected]. Phone: +46-23-778509.
1
1. Introduction Consider the challenge of finding a solution to the minimum of an intractable objective function of a combinatorial optimization problem. This challenge arises frequently in operations research (OR) dealing with issues like automation, facility location, routing, and scheduling. Usually a good, but biased, solution (called heuristic solution) is sought by applying one of the many iterative (heuristic) methods available. None of the heuristics guarantees that the solution will coincide with the optimum and, hence, many solutions to real world OR-problems are plagued with an uncertainty about the quality of the solution. It goes without saying that many heuristic users find this uncertainty unappealing and try to take measures to reduce it. There are four relatively widespread measures for doing so. The first is to carefully choose the heuristic amongst them thoroughly tested on various problems of similar kind (see e.g. Taillard, 1995), the second is to set the number of iterations large or run until no improvements are encountered (see e.g. Levanova, and Loresh, 2004), the third is to use methods that provide deterministic bounds (see e.g. Beasley, J.E., 1993), and the fourth is to compute statistical bounds (see e.g. Gonsalvez, Hall, Rhee, and Siferd, 1987). We assume that the first measure (as well as reasonable data-reduction of the problem) is taken by the user and we will not discuss it further. The second measure is common practice, whereas the third measure occasionally is taken. Deterministic bounds are informative on the quality of the solution, but often at a high computational cost. The fourth measure is rarely taken by heuristic users in spite of its potential powerfulness (Derigs, 1985). One reason for heuristic users hesitating in taking the fourth measure is the lack of systematic investigation of statistical bounds. The aim of this paper is therefore to systematically investigate statistical bounds for the fundamental location-allocation problem being the p-median problem. In the investigation we vary heuristic method, sample size, estimator of the bounds, the number of iterations (computing time), and the test problems leading to variations in complexity. We present results on the reliability of the statistical bounds and the length of the interval compared with the length of intervals of deterministic bounds. The computer experiment is accompanied with an illustrating problem concerning the location of regional delivery points of the Swedish Post. 2
This paper is organized as follows: section two presents statistical optimum estimation techniques. In the third section we present the pmedian problem and discuss three solution methods used in the investigation. In section four details about the experiments are given concerning the implementation of the solution methods, factors varied in the experiment and factors kept constant, as well as outcome variables of interest. The fifth section gives the results and the sixth section illustrates the use of statistical bounds in a practical problem of locating 71 delivery points of the Swedish Post in a region. The seventh section concludes the paper. 2. Statistical optimum estimation techniques First some notations used throughout the paper: 𝑧𝑘 = the k:th feasible solution to the combinatorial problem; 𝐴 = the set of all feasible solutions, 𝐴 = {𝑧1 , 𝑧2 , … , 𝑧𝐾 }; 𝑔(𝑧𝑘 ) = the value of the objective function of solution 𝑧𝑘 (in the following breifly referred to as solution); 𝜃 = min𝐴 𝑔(𝑧𝑘 ). The challenge is to identify 𝜃 and the corresponding solution 𝑧𝑘 . The abbreviation SOET (statistical optimum estimation techniques) was recently introduced by Giddings, Rardin, and Uzsoy (2014) and it refers to techniques both for point and interval estimation of 𝜃. In short, the statistical approach is to estimate the value of the minimum based on a sample of heuristic solutions and put bounds (or confidence limits) around it. Golden and Alt (1979) did pioneering work on statistical bounds followed by others in the 1980:s, but thereafter the statistical approach has been dormant. In fact, Akyüz, Öncan, and, Altınel (2012) state that the statistical approach had not been used in location problems since 1993 to the best of their knowledge. Neither has our literature review finds any such application. There are four approaches to estimating statistically the optimum according to the review article of Giddings et al (2014). The dominating one is based on Extreme Value Theory (EVT) and another is the truncation-point approach which we will refer to as the Weibull (W) estimator and the Jackknife (JK) estimator, respectively. The Weibull estimators are usually the only estimator based on EVT approach. Recently, Carling and Meng (2014) proposes the Gumbel estimator based 3
on EVT approach but it is outperformed by Weibull estimators. Giddings et al, (2014) also mention the limiting-distribution approaches as well as the multinomial approaches. Neither of them is used nor seems promising and we will disregard them. As a point of departure in discussing the SOET:s, we note that even a very large random sample of the random quantity 𝑔(𝑧𝑘 ) is almost useless for identifying 𝜃. The crux is that a sample of the gigantic size of 109 will be hard pressed to contain a feasible solution close to 𝜃 even for combinatorial optimization problems of modest size (Meng and Carling, 2014). Fortunately, as several authors have pointed out, if the starting values are picked at random, repeated heuristic solutions mimic a random sample in the tail (see e.g. McRoberts, 1971, and Golden and Alt, 1979). Thereby random values in the tail can be obtained at a much less computational effort and used for estimating 𝜃. To discuss the estimation of 𝜃 and statistical bounds for it we need additional notations: 𝜃̂ = an point estimator of 𝜃; 𝑛 = the number of replicates of the heuristic algorithm with random starting values. The attained minimum value of the objective function in the ith will be noted as 𝑥̃𝑖 , and referred to as heuristic solution. The Weibull point estimator of 𝜃 is 𝜃̂𝑊 = 𝑥̃(1) where 𝑥̃(1) = min(𝑥̃1 , 𝑥̃2 , … , 𝑥̃𝑛 ) which is the best (smallest) heuristic solution in all the n replicates. The JK-estimator is introduced by Quenouille (1956): 𝑀+1
𝑀+1 𝜃̂𝐽𝐾 = ∑ (−1)(𝑖−1) ( ) 𝑥̃(𝑖) 𝑖 𝑖=1
where M is the order. Dannenbring (1977) and Nydick and Weiss (1988) suggest to use the first order, i.e. 𝑀 = 1 , for point estimating the minimum. The first order JK-estimator is more biased than higher order ones, but its mean square error is lower compared with higher orders as shown by Robson and Whitlock (1964). Carling and Meng (2014) however consider both the first and the second order JK-estimator and note that the second order JK-estimator performs quite well. The JK point(1) (2) estimators are 𝜃̂𝐽𝐾 = 2𝑥̃(1) − 𝑥̃(2) and 𝜃̂𝐽𝐾 = 3𝑥̃(1) − 3𝑥̃(2) + 𝑥̃(3) respectively. As for interval estimation, the upper bounds of the Weibull and the 4
Jackknife estimators are the same, i.e., 𝑥̃(1) . However their lower bounds differ. The Weibull lower bound with a confidence of 100(1 − 𝑒 −𝑛 )% is [𝑥̃(1) − 𝑏̂] where 𝑏̂ is the estimated shape parameter of the Weibull distribution (Wilson, King, and Wilson, 2004). There are several ways of estimating the shape parameter, including the maximum likelihood estimation technique. We and others (e.g. Derigs, 1985) have found the following simple estimator to be fast, stable, and giving good results: 𝑏̂ = 𝑥̃([0.63(𝑛+1)]) − (𝑥̃(1) 𝑥̃(𝑛) − 𝑥̃(2) 2 )/(𝑥̃(1) + 𝑥̃(𝑛) − 2𝑥̃(2) ) where [0.63(𝑛 + 1)] means the integer part of the value of the function. The Jackknife estimator, and extensions of it (Dannenbring, 1977), has only served the purpose of point estimating 𝜃 . However, Meng and Carling (2014) suggests a lower bound computed by means of bootstrapping the point estimator (Efron, 1979). The lower bound is [𝜃̂𝐽𝐾 − 3𝜎 ∗ (𝜃̂𝐽𝐾 )] where 𝜎 ∗ (𝜃̂𝐽𝐾 ) is the standard deviation of 𝜃̂𝐽𝐾 obtained from bootstrapping the n heuristic solutions (1,000 bootstrap samples were sufficient for estimation of 𝜎 ∗ (𝜃̂𝐽𝐾 )). By multiplying 𝜎 ∗ (𝜃̂𝐽𝐾 ) by 3, the confidence is 99.9% provided that the sampling distribution of the JKestimator is approximately Normal. Giddings et al (2014) provide a critique of the SOET:s which deserves to be re-stated briefly. Firstly, all the estimators presume a continuous distribution of the heuristic solutions for being theoretically justified. However, the discrete, feasible solutions amounts to a finite K and a strong heuristic method might produce a sample of clustered, discrete heuristic solutions. Thus, this assumption is, strictly speaking, false. 1 Secondly, clustering of heuristic solutions suggests a violation of the independence assumption up on which the probability statement of the interval hinges. Although several authors (see e.g. Wilson et al, 2004) have proposed goodness-of-fit tests for checking the assumed Weibull distribution and the assumed independence, one cannot expect the power of the tests to be high from a small sample in the extreme tail of the distribution. A failure of 1
We believe less complex problems to be more amenable to discreteness and consequent improper statistical bounds, whereas highly complex problems have a large number of service and demand points rendering the parent distribution almost continuous. Exact solutions are usually feasible for non-complex problems, while deterministic or statistical bounds are critical for complex problems.
5
rejecting the null hypotheses of the Weibull distribution and independence is perhaps a stronger indication of low power of the test than an indication of the correctness of the assumptions. A conclusion from Giddings’ et al (2014) critique is that theoretically perceived statistical properties of the SOET:s are possibly misleading and the properties’ applicability are specific to the heuristic and the combinatorial problem, due to the improper assumptions. As a consequence, the theoretical properties need to be empirically checked for various problems and heuristic methods. The contribution of this paper is that we check empirically the theoretical properties of SOET:s for the pmedian problem when it is solved by two of the most used solution methods for it. 3. The p-median problem and heuristic methods Location theory is an important part of operations research and it is concerned with the issue of locating facilities and assigning demand points to them in some desired way. Continuous problems like the original Weber problem deals with location in the plane, whereas discrete problems deals with location on networks with vertices (or nodes) connected by edges. The p-median problem is the most important of the four primary discrete location problems, with the other three being the p-center, the uncapacitated facility location, and the quadratic assignment problems (Reese, 2006). A nice virtue of operations research is the vast availability of test problems for which the optimum either is known or consistently is updated as improved solutions emerge. Consequently, the SOET:s can be checked with respect to known optima. We focus on the p-median problem because of its fundamental role in location theory and the large number of existing test problems in the OR-library (Beasley, 1990). The problem is to allocate P facilities to a demand geographically distributed in Q demand points such that the weighted average or total distance to its nearest service center is minimized for the demand points.2 Hakimi (1964) considered the task of locating telephone switching centers 2
The p-median problem is NP-hard (Kariv and Hakimi, 1979). In fact, it is also NPcomplete and therefore it is to be expected that solving a p-median problem is intractable (Garey and Johnson, 2000).
6
and showed later (Hakimi, 1965) that, in a network, the optimal solution of the p-median model exists at the nodes of the network. If V is the 𝑉 number of nodes, then there are 𝐾 = ( ) feasible solutions for a p-median 𝑝 problem for the largest test problem we consider in the computer experiment has about 𝐾 ≈ 2.5 ∗ 10164 . As enumerating all feasible solutions is not possible as the problem size grows, much research has been devoted to efficient methods to solve the p-median model (see Handler and Mirchandani, 1979 and Daskin, 1995 as examples). Reese (2006) reviews the literature for solution methods3 to the p-median problem. Lagrangian relaxation (LR) is the most used method and since it gives deterministic bounds we will use it as a benchmark for SOET:s (Beasley, 1993). In the class of heuristic methods, vertex substitution (VS) is the most frequent. VS starts with a (random) 𝑧𝑘 , say, and seeks a local improvement by examining all local adjacent nodes for the first facility, then the second and so on, until it reaches the last facility up on which an iteration is completed. After reaching the last facility, it returns to the first facility and repeats the procedure until no further improvement is encountered. The algorithm has no inherent randomness as it updates the solution according to a deterministic scheme. Randomness in the heuristic solution comes from either selecting the starting nodes, 𝑧𝑘 , at random or by randomizing the order of examination of the nodes in 𝑧𝑘 . Another class (the largest) of solution methods to the p-median problem is metaheuristics with a domination of genetic algorithms 4 and simulated annealing (SA) (Reese, 2006). SA does not apply a deterministic scheme in its search for a good heuristic solution. Hence, randomness in the heuristic solution will come both from random starting points and from inherent randomness in the algorithm. SA starts with a randomly picked 𝑧𝑘 and selects at random one facility in that 𝑧𝑘 , and evaluates a move of it to 3
Reese (2006) uses the term solution methods to indicate any approach to find (approximately) the optimum. Heuristics and meta-heuristics are a part of all solutions methods in his sense; however we will use solution method and heuristic interchangeably in what follows. 4 See for instance Michalewicz and Janikow (1991).
7
another node picked at random. The facility will be moved if it implies an improvement, but it may also be moved with a small probability in spite of no improvement. One random pick including the evaluation constitutes an iteration. SA runs until a pre-specified (usually large) number of iterations are performed. We limit the study to VS, SA, and LR. There are of course details regarding the implementation of the algorithms and we therefore implement all of them in the statistical language R and provide the code in the Appendix in the interest of making our study replicable.5 The virtue of SA is that the algorithm will yield a solution that gradually iterates towards the optimum since the inherent randomness implies that eventually all feasible solutions will be evaluated. Hence, by increasing the number of iterations better solutions are to be expected. VS and the primal iterates of LR are more susceptible to be trapped in local minima and thereby never approaching the optimum however large the number of iterations. Such a fundamental difference in the characteristics of SA and VS may very well have implication on the functioning of the SOET:s and the comparison with regard to LR. 4. The computational experiment The first factor varied in the computational experiment is the estimator. The point and the interval Weibull-estimator are defined in Section 2 except for the confidence level. The confidence level will be kept at 𝛼 = 0.9987 throughout the experiments which means that the lower bound will be calculated as [𝑥̃(1) − 𝑏̂ /𝑠] , where 𝑠 = (−𝑛/ln𝛼)1/𝑎̂ and 𝑎̂ = (𝑥̃(1) 𝑥̃(𝑛) − 𝑥̃(2) 2 )/(𝑥̃(1) + 𝑥̃(𝑛) − 2𝑥̃(2) ) being an estimator of the shape parameter of the Weibull distribution (Giddings et al, 2014). The first and the second Jackknife estimators are also defined in Section 2 where 𝜎 ∗ (𝜃̂𝐽𝐾 ) is calculated from 1,000 bootstrap samples of the n heuristic solutions. The second factor is complexity, in the context usually understood as the number of feasible solutions, of the p-median test problems. Carling and Meng (2014) proposes a different definition of complexity. They show that 5
(www.r-project.org).
8
the normal distribution gives a close fit to the distribution of 𝑔(𝑧𝑘 ) for most of the 40 test problems and defines complexity as the distance between the optimum and the estimated center of 𝑔(𝑧𝑘 ) in terms of standard deviations. With this definition of complexity the test problems ranges from 2.97 to 14.93. The third factor is the variation in heuristic methods. The implementations of these methods are explicitly given by the R-code provided in the Appendix. Our implementation of the solution methods follows closely Densham and Rushton (1992) (VS), Al-Khedhairi (2008) (SA), and Daskin (1995) (LR)6. In the early literature on statistical bounds little was said on the size of n, whereas Akyüz et al (2012) and Luis, Sahli, and Nagy (2009) advocate n to be at least 100. However, Brandeau and Chiu (1993) as well as Carling and Meng (2014) find 𝑛 = 10 to work well. We will examine as the fourth factor varied in the computational experiments, 𝑛 = 10, 25. However, running the heuristics SA and VS is computationally costly and repeating them is of course even more costly. For this reason we run the algorithms 100 times per test problem to obtain 100 heuristic solutions. Thereafter, heuristic solutions are sampled with replacement from the set of 100 solutions. LR is run only once per experimental combination as its initial values are deterministically determined by the distance matrix of the test problem. The fifth, and the last, factor is the computing time allotted to the solution methods. The time per iteration varies by the complexity of the test problems and it is therefore reasonable to assign more computing time to 𝑉 the more difficult problems. We run the algorithms for 2 ∗ (100) , 20 ∗ 𝑉
𝑉
(100) , 60 ∗ (100) seconds per replicate where, again, 𝑉 is the number of nodes of the test problem and computing time refers to CPU time of one processor on an Intel i5-2500 and 3.30GHz. The nodes vary from 100 to 900 in the test problem so the computing time of the algorithms varies between 2 seconds to 9 minutes. 6
Han (2013) implemented LR for the test problems and, by pre-testing, found the mean of the columns of the distance matrix divided by eight to yield good starting values for the algorithm. In our implementation of the algorithm, we follow Han’s approach.
9
We present results on the reliability of the SOET:s as the relative bias (𝜃̂ − 𝜃)/𝜃 , the proportion of intervals with lower and upper bound containing the optimum (hereafter referred to as coverage), the length of the intervals, and the proportion of intervals that are shorter than the length of the intervals defined by the deterministic bounds. 5. Results Our computer experiment is a full factorial experiment of 3 × 40 × 2 × 2 × 3 (estimator, complexity, heuristic, n, and computing time) resulting in 1440 experimental combinations for which bias, coverage, length of interval, and the proportion of intervals shorter than the length of the interval defined by the deterministic bounds are outcome variables. 7 We begin by checking for which experimental combinations it is meaningful to compute statistical bounds (and refer to them as used combinations in the following). Thereafter we check the bias of the three estimators in point estimating the optimum. Finally, we give results on the coverage of the intervals as well as their length. 5.1. Statistical bounds and the quality of heuristic solutions Monroe and Sielken (1984) observe that the heuristic solutions need to be close to the optimum for the computing of statistical bounds to be meaningful. Carling and Meng (2014) suggest the statistic SR, given by (1) the ratio 1000𝜎(𝑥̃𝑖 )/𝜃̂𝐽𝐾 , as a measure of concordance and as a check of the heuristic solutions being in the tail near to 𝜃. Figure 1 shows how coverage decreases with SR (for the about 75 per cent of experimental combinations with SR below 15). For SR above 15 the coverage is approaching zero (not shown). Carling and Meng (2014) found SR = 4 to be an operational threshold in the case of SA, but the threshold also seems to apply to vertex substitution in ensuring statistical bounds to correspond reasonably well to the stipulated confidence level. As a consequence of a large value of SR implying improper statistical bounds, we will only examine the outcome of the treatment combinations for which 𝑆𝑅 ≤ 4. There are 480 experimental combinations per estimator 7
The complete outcome matrix may be requested from the authors by contacting [email protected].
10
of which 247 are kept for analysis. Combinations, especially for vertex substitution, of short computing times and high complexity tend to fail the threshold check. Figure 2 illustrates the non-linear relationship with LOWESS curves between SR (in logarithms) and complexity for the three levels of computing time (Cleveland, 1979). Of the 247 combinations used in the analysis, 83 are VS (5, 38, and 40 with short, intermediate, and long computing time), and respectively 164 are simulated annealing (35, 57, and 72 with short, intermediate, and long computing time) combinations. The partial failure of vertex substitution to pass the threshold check is typically a consequence of the algorithm managing only a few iterations in the allotted computing time. For instance, the average number of iterations was 2.6 for the combinations possessing short computing time. We considered extending the allotted computing time in the experiment, but discarded this option as the allotted computing time generally was sufficient for both SA and LR.
1.0
Coverage
0.8
0.6
0.4
0.2
0.0 0
2
4
6
8
10
12
14
SR
Figure 1: Coverage as a function of SR. The Jackknife estimator is denoted by st nd the solid line (1 order) and the dashed line (2 order), and the Weibull estimator is denoted by the dotted line.
One factor in the experiment is the heuristic. We found, by formal testing with Analysis of Variance (ANOVA), this factor to be unrelated to the 11
outcomes of the experiment for used combinations. Outcomes of SA and VS are therefore pooled in the following when results related to the used combinations are presented. 6 5
ln (1+SR)
4 3 2 1 0 2
4
6
8 10 Complexity
12
14
16
Figure 2: Experimental combinations removed from the analysis (SR = 4, solid reference line). Short computing time (solid line, asterisks), intermediate computing time (dashed line, squares), and long computing time (dotted line, pluses).
5.2. The bias of point-estimation Before studying the properties of the statistical bounds, we first examine the estimators’ ability to point-estimate the minimum. Table 1 gives the bias of the estimators focusing on the case 𝑛 = 25. Theoretically the 2nd order Jackknife estimator has the smallest bias followed by the 1st order Jackknife estimator, while the Weibull point-estimate is simply equal to 𝑥̃(1) and thereby always larger than (or possibly equal to) the optimum. In practice, the three estimators are comparable in terms of bias. For a given computing time, solutions obtained by SA are much closer to the optimum compared with VS although solutions are consistently improving by time for both methods. LR, on the other hand, finds a decent solution after a short computing time, and trivially improves with added computing time. Looking specifically at the combinations for which 𝑆𝑅 ≤ 4, the bias is petite as the heuristic solutions as clustered near to the optimum. The bias 12
of LR is positive, particularly for the used combinations with a long computing time. These combinations have a high proportion of the most complex test problems. Although not shown in the table, the general conclusions regarding bias also apply to the case 𝑛 = 10. Table 1: Average relative bias (%) in the point estimation of 𝜃. 𝑛 = 25. a
st
E-C
JK 1
SA
Short
40
1.7
1.6
1.8
Interm.
40
0.4
0.4
0.4
Long
40
0.1
0.1
0.2
Short
40
22.2
21.8
23.4
Interm.
40
16.0
15.7
16.7
Long
40
13.0
12.8
13.5
Short
40
3.4
Interm.
40
3.3
Long
40
3.3
LR
Used b combinations
a)
Weibull
c
Time
VS
JK 2
nd
Method
LR
19
-0.00
-0.00
0.01
0.31
Interm.
47
0.02
0.02
0.03
1.18
Long
56
0.04
0.04
0.05
1.74
Short
b)
Note: Number of experimental combinations. SA and VS pooled due to c) similarities. Results based on one replicate due to the deterministic outcome of the algorithm.
5.3. Coverage and length of intervals Ideally, the estimators should provide statistical bounds (or intervals) that contain the unknown parameter of interest, 𝜃 at the pre-set confidence level. If the condition is fulfilled, then a short interval is desirable. We begin by examining the coverage of 𝜃 by the intervals. Table 2 gives the coverage of the three estimators for the used experimental combinations. There are two things to note in the table in addition to the fact that the coverage is near to 100 per cent. The first is that the coverage is similar for 𝑛 = 10 and 𝑛 = 25. The second is that the Weibull estimator almost always covers the optimum while the Jackknife estimators fail occasionally. From these findings we conclude that the 13
Weibull estimator is preferable and that we offer further evidence to the findings of Brandeau and Chiu (1993) indicating that a large number of replicates are redundant. Table 2: Proportion (%) of intervals containing 𝜃. Replicates
a
st
E-C
n=10
Short Interm. Long
n=25
Short Interm. Long
21 48 56 19 47
98.8 98.1 96.4 99.3 98.3
99.6 99.2 98.7 99.8 99.4
100.0 99.8 99.7 100.0 100.0
56
95.6
97.9
100.0
Note:
a)
JK 1
JK 2
nd
Time
Weibull
Experimental combinations. SA and VS pooled due to similarities.
The 1st order Jackknife estimator gives intervals of a length of about 60 per cent of the Weibull estimator, whereas the 2nd order Jackknife estimator and Weibull estimator are comparable with regard to length of intervals. However, we require the estimator to give intervals with a factual coverage corresponding to the asserted confidence level, a condition the 1st order Jackknife estimator fails to meet. Table 3: A comparison of statistical (Weibull estimator) and deterministic bounds (LR). Counts of experimental combinations. UBW < UBLR
UBW = UBLR
UBW > UBLR
LBW > LBLR
92
91
9
192
LBW < LBLR
48
1
6
55
140
92
15
247
To assess the precision offered by the Weibull intervals, it is useful to compare with the deterministic bounds given by LR. Table 3 gives a comparison of the upper bounds (UB) and lower bounds (LB) as obtained by the Weibull estimator and LR for the 247 used experimental combinations. In 6 + 1 of the combinations the intervals of the deterministic bounds are contained in the intervals 8 of the Weibull statistical bounds whereas it is the opposite in 92 + 91 of the 8
We computed the average of the Weibull upper and lower bounds based on the 1,000 Bootstrap samples.
14
combinations. The latter happens as a consequence of the Weibull pointestimator being a slightly better predictor of the optimum than the upper bound of LR, but more importantly because the statistical lower bound often is closer to the optimum than the deterministic lower bound. The latter finding conforms to what Brandeau and Chiu (1993) reported. The fact that the statistical intervals are contained in the deterministic intervals does not imply that the difference in length is of practical significance. We therefore compute the relative length of the statistical interval (average) to the deterministic interval. Figure 3 shows the relative length of the interval as a function of complexity, as the other factors were found to have no impact. Imposed is a resistant line, in order to accommodate some extreme outliers, depicting the relationship between relative length and complexity (Velleman, 1980). For all levels of complexity, the median relative length is 0.02 while the average is 0.25 being strongly affected by outliers coming from combinations with short computing time. 1.0
Relative length
0.8
0.6
0.4
0.2
0.0
2
4
6
8 Complexity
10
12
14
Figure 3: Relative length of the statistical intervals to the deterministic intervals as a function of the complexity. Resistant line imposed due to a few extreme outliers.
15
To conclude the result section, reliable statistical bounds require heuristic solutions of good quality (i.e. 𝑆𝑅 ≤ 4). Given high quality, the Weibull estimator gives proper intervals covering the optimum almost always with an interval substantially tighter than a deterministic interval. The required number of replicates is as modest as 10. Hence, statistical bounds are reliable, efficient and computationally affordable. 6. An illustrating case In this section, we illustrate statistical bounds applied to a practical location problem concerning allocating several distribution centers of the Swedish Post in one region in Sweden. In this real location problem the minimum is unknown. The problem is to allocate each of the 71 distribution centers of the Swedish Post to some of the 6,735 candidate nodes in the network of Dalarna in mid-Sweden. The landscape of the region and its population of 277,725 inhabitants, distributed on 15729 demand points, is described by Carling, Han, and Håkansson (2012). The objective is to minimize the sum over all the demand points of the distance between the demand point and the nearest postal center. The minimization is done over all possible locations of postal centers on the 6,735 candidate nodes in the network. Han, Håkansson, and Rebreyend (2013) provides detailed description of the road network and argue that network distance, rather than the Euclidian, is used as distance measure. Hence, we measure distance as meters on the road network.
Figure 4: Empirical distribution of objective function for the Swedish Post problem.
16
To appreciate the complexity of this illustrative problem, we provide the distribution of the objective function. As Carling and Meng (2014) did for the OR-lib problems, we draw a random sample of 1 million and show the empirical distribution of 𝑔(𝑧𝑘 ) in Figure 4. The distribution is slightly skewed to the right, but still approximately normal. To evaluate the (2) complexity of the problem, we use 𝜃̂𝐽𝐾 as the estimate of the minimum 𝜃, and the mean and variance of 𝑔(𝑧𝑘 ) are derived by the 1 million random sample. The complexity is 5.47 and the case is therefore comparable to the median of the 40 OR-lib problems. Drawing on the experimental results above, we set 𝑛 = 10 and run the SA heuristic algorithm until SR reaches 4 or less. Furthermore, we focus on the statistical bounds of the Weibull estimator. We checked SR in steps of 5 minutes both for SA and LR. Figure 5 shows the evolution of LR’s smallest upper and largest lower bounds as a function of time. Within 90 minutes, the lower bound of LR has stabilized at 2,534 whereas the upper bound reaches 3,275 after 125 minutes. After our decision to stop running the algorithms at 200 minutes, the LR’s solution is 3,275 meters as the inhabitants average distance to their respective closest postal centers. However, the LR-solution is imprecise as the gap between the bounds is 741 meters. The Jackknife 1st order point-estimate of the optimum is 2,973 as evaluated after 200 minutes, and this estimate is imposed in Figure 5 as a reference line to the unknown optimum.
17
Figure 5: The solutions to the Swedish Post problem by computing time. Upper and lower bounds in long and short dashed lines (LR) and short dashed lines st (SA). The embedded graph is a magnification. Jackknife (1 order) point-estimate as solid reference line.
The SR approaches 4 gradually, but slowly. It took 150 minutes of computing time of SA for the ratio to come down to 4, and the statistical upper and lower bounds are thereafter computed and depicted in Figure 5. To show clearly the evolution of the statistical bounds at 150 minutes and onwards, a magnifying graph is embedded in Figure 5. Up to termination of the algorithm, the gap between the upper and the lower statistical bound is decreasing and reaches 16 meters. Since 16 meters is a tolerable error for this application, there is no need to continue the search for a better solution and we content ourselves with a solution of 2,975 with a lower bound of 2,959 meters. We have also computed the statistical bounds using the JK-estimators with results differing only by a few meters. 7. Concluding discussion We have considered the problem of determining when a solution provided by a heuristic is close to optimal. Deterministic bounds may sometimes be 18
applicable and enough tight to shed knowledge on the problem. We have, however, studied statistical bounds potentially being of a more general applicability. We have examined the Weibull estimator as well as two variants on the Jackknife estimator. Furthermore, we have varied the number of replicates, the allotted computing time of the heuristic algorithms, the complexity of the combinatorial problem as well as the heuristic algorithms. We find statistical bounds to be reliable and much more efficient than deterministic bounds provided that the heuristic solutions are sampled close to the optimum, an issue further addressed below. Furthermore, statistical bounds may be computed based on a small number of replicates (𝑛 = 10) implying a modest computational cost up on exploiting parallel computing. We have, however, restricted the experiment to one type of combinatorial optimization problem, namely the p-median problem being the most common location problem in the OR-literature. Derigs (1985) made a number of concluding observations upon studying statistical bounds with regard to the Travelling Salesman Problem (TSP) and Quadratic Assignment Problem (QAP). It appears that most of the conclusions are valid, except one. Derigs stated “The Weibull approach leads to a proper approach,”. We have however demonstrated that none of the estimators, including Weibull, are reliable unless the quality of the sample of heuristic solutions used for deriving the bounds is of high quality. To assess the quality we suggest using SR which is the standard deviation of the n solutions divided by the Jackknife point-estimator. The experiments suggest that SR exceeding 4 causes unreliable bounds. Nevertheless, a systematic study of statistical bounds on various classes of combinatorial optimization problems is warranted before advocating a general usage of statistical bounds in the estimation of optima.
19
0.0 10
2.0
1.5
8
1.0
6
0.5
4
0.0
2
-0.5
0
10
20
30
40
2.5
5.0
7.5
10.0
0
50 SR
Figure 6: Skewness (left panel) and Relative (%) bias of average solution (right panel) as a function of SR. Combinations with SR equal to zero are removed and only the combinations for which 𝑛 = 25 are shown.
The empirically discovered threshold of SR less than four may seem mystical. We have noted however that the distribution of the sample of solutions tend to be normal until the solutions are coming close to the optimum. The left panel of Figure 6 shows skewness of the sample of solutions as a function of SR. For high values of SR, skewness is typically about zero and kurtosis is about three. Once the sample of solutions is coming close to the optimum, the skewness (and kurtosis) increases (i.e. the sample of solutions is becoming right-skewed) while the variation in solution decreases (i.e. SR becomes smaller). The right panel of figure 6 shows the relative bias of the average solution of the sample. For SR large the bias is also large. However, at the threshold the bias is only about one per cent. Hence, reliable statistical bounds seem to require a solution method yielding solutions within one per cent deviance from the optimum. The bias might be a better statistic than SR in deciding the reliability of the statistical bounds, but it requires of course knowing the (unknown) optimum. An inherent difficulty in executing unbiased computer experiments on 20
heuristic algorithms is the issue of their implementation. We have tried to render the comparison fair by running the algorithms on the same computers for the same computing time in the same R environment. We have also supplied the code which means that it is straightforward to replicate the experiment under alternative implementations including other parameter settings of the algorithms. For the current implementation of the algorithms, the wide gap between the upper and lower bounds of LR did not seem to tighten by computing time, on the contrary the bounds of the algorithm stabilized quickly in most experimental combinations. Vertex substitution was less successful than SA in fulfilling the requirement of SR below 4, but in the experimental combinations when both methods fulfilled the requirement the statistical bounds were very similar. Acknowledgement We are grateful to Mengjie Han, Johan Håkansson, Daniel Wikström, and two anonymous reviewers for comments on previous versions of the paper. Financial support from the Swedish Retail and Wholesale Development Council is gratefully acknowledged. References Akyüz, M.H., Öncan, T., Altınel, I.K., (2012). Efficient approximate solution methods for the multi-commodity capacitated multi-facility Weber problem. Computers & Operations Research 39:2, 225-237. Al-Khedhairi, A., (2008). Simulated annealing metaheuristic for solving pmedian problem. International Journal of Contemporary Mathematical Sciences, 3:28, 1357-1365. Beasley, J.E., (1990). OR library: Distributing test problems by electronic mail, Journal of Operational Research Society, 41:11, 1067-1072. Beasley, J.E., (1993). Lagrangian heuristics for location problems. European Journal of Operational Research, 65, 383-399. Brandeau, M.L., Chiu, S.S., (1993). Sequential location and allocation: worst case performance and statistical estimation. Location Science, 1:4, 289-298. 21
Carling, K., Han, M., Håkansson, J., (2012). Does Euclidean distance work well when the p-median model is applied in rural areas? Annals of Operations Research, 201:1, 83-97. Carling, K., Meng, X., (2014). On statistical bounds of heuristic solutions to location problems. Working papers in transport, tourism, information technology and microdata analysis, 2014:10. Cleveland, W.S., (1979). Robust Locally Weighted Regression and Smoothing Scatterplots, Journal of the American Statistical Association, 74, 829–836. Dannenbring, D.G., (1977). Procedures for estimating optimal solution values for large combinatorial problems, Management science, 23:12, 12731283. Daskin, M.S., (1995). Network and discrete location: models, algorithms, and applications. New York: Wiley. Densham, P.J, Rushton, G., (1992). A more efficient heuristic for solving large p-median problems. Papers in Regional Science, 71, 307-329. Derigs, U, (1985). Using confidence limits for the global optimum in combinatorial optimization. Operations research, 33:5, 1024-1049. Efron, B., (1979). Bootstrap methods: Another look at the Jackknife. Annals of statistics, 7:1, 1-26. Garey, M.R., Johnson, D.S, (2002). Computers and intractability, 29; W.H. Freeman, New York. Giddings, A.P., Rardin, R.L, Uzsoy, R, (2014). Statistical optimum estimation techniques for combinatorial problems: a review and critique. Journal of Heuristics, 20, 329-358. Golden, B.L., Alt, F.B., (1979). Interval estimation of a global optimum for large combinatorial optimization. Operations Research, 33:5, 1024-1049. Gonsalvez, D.J., Hall, N.G., Rhee, W.T., Siferd, S.P., (1987). Heuristic solutions and confidence intervals for the multicovering problem, European Journal of Operational Research, 31:1, 94-101. 22
Hakimi, S.L., (1964). Optimum locations of switching centers and the absolute centers and medians of a graph. Operations Research, 12:3, 450-459. Hakimi, S.L., (1965). Optimum Distribution of Switching Centers in a Communication Network and Some Related Graph Theoretic Problems. Operations Research, 13:3, 462-475. Han, M., (2013). Heuristic Optimization of the p-median Problem and Population Re-distribution. Dalarna Doctoral Dissertations, 2013:01. Han, M., Håkansson, J., Rebreyend, P., (2013). How do different densities in a network affect the optimal location of service centers?. Working papers in transport, tourism, information technology and microdata analysis, 2013:15. Handler, G.Y., Mirchandani, P.B., (1979). Location on networks: Theorem and algorithms. MIT Press, Cambridge, MA. Kariv, O., Hakimi, S.L., (1979). An algorithmic approach to network location problems. part 2: The p-median, SIAM Journal of Applied Mathematics, 37, 539-560. Levanova, T., and Loresh, M.A., (2004). Algorithm of ant system and simulated annealing for the p-median problem, Automation and Remote Control, 65, 431-438. Luis, M., Sahli, S., Nagy, G., (2009). Region-rejection based heuristics for the capacitated multi-source Weber problem. Computers & Operations Research, 36, 2007-2017. McRobert, K.L., (1971). A search model for evaluating combinatorially explosive problems. Operations Research, 19, 1331-1349. Meng, X., Carling, K., (2014). How to Decide Upon Stopping a Heuristic Algorithm in Facility-Location Problems?. In Web Information Systems Engineering–WISE 2013 Workshops, Lecture Notes in Computer Science, 8182, 280-283, Springer, Berlin/Heidelberg. Michalewicz, Z., Janikow, C.Z. (1991). Genetic algorithms for numerical optimization. Statistics and Computing, 1, 75-91. 23
Monroe, H.M., Sielken, R.L., (1984). Confidence limits for global optima based on heuristic solutions to difficult optimization problems: a simulation study. American Journal of Mathematical and Management Sciences, 4, 139167. Nydick, R.L., Weiss, H.J., (1988). A computational evaluation of optimal solution value estimation procedures. Computers & Operations Research, 5, 427-440. Quenouille, M.H., (1956). Notes on bias in estimation. Biometrika, 43, 353360. Reese, J., (2006). Solution methods for the p-median problem: An annotated bibliography. Networks, 48:3, 125-142. Robson, D.S., Whitlock, J.H., (1964). Estimation of a truncation point. Biometrika, 51, 33-39. Taillard, E. D., (1995). Comparison of iterative searches for the quadratic assignment problem. Location science, 3(2), 87-105. Velleman, P.F., (1980). Definition and Comparison of Robust Nonlinear Data Smoothing Algorithms. Journal of the American Statistical Association, 75, 609-615. Wilson, A.D., King, R.E., Wilson, J.R., (2004). Case study on statistically estimating minimum makespan for flow line scheduling problems. European Journal of Operational Research, 155, 439-454.
24
Appendix: Table A1 Table A1: Description of the 40 problems of the OR-library. Problem
P11 P1 P16 P6 P26 P21 P38 P31 P35 P7 P2 P3 P27 P17 P22 P12 P36 P39 P32 P4 P5 P8 P13 P9 P18 P10 P23 P14 P28 P19 P15
𝜃
𝜇𝑔(𝑧𝑝)
𝜎𝑔(𝑧𝑝)
Complexity
7696 5819 8162 7824 9917 9138 11060 10086 10400 5631 4093 4250 8307 6999 8579 6634 9934 9423 9297 3034 1355 4445 4374 2734 4809 1255 4619 2968 4498 2845 1729
10760 8426 11353 10522 13644 12906 15078 13960 14179 7930 6054 6194 11428 9819 11699 9387 13436 12988 12687 4618 2376 6604 6293 4250 6769 2278 6586 4501 6369 4327 2896
1195 877 1033 869 1133 1070 1143 1077 1085 598 506 500 727 631 676 586 735 736 699 320 197 356 276 202 248 127 220 168 188 144 109
2.56 2.97 3.09 3.10 3.29 3.52 3.52 3.60 3.81 3.84 3.88 3.89 4.29 4.47 4.62 4.70 4.77 4.84 4.85 4.95 5.18 6.07 6.95 7.51 7.92 8.02 8.94 9.12 9.95 10.32 10.67
25
P33 P24 P37 P20 P40 P29 P25 P34 P30
4700 2961 5057 1789 5128 3033 1828 3013 1989
6711 4486 7246 3108 7329 4559 3131 4617 3335
26
186 134 188 112 179 118 95 112 90
10.81 11.42 11.65 11.73 12.32 12.93 13.64 14.36 14.93
Appendix: R-code Simulated Annealing: V=100 # Number of candidates (dependent on test problem) p=5 # Number of facilities (dependent on test problem) n<-100 # Number of replicates (dependent on experimental combination) ni<-10000 # Number of replicates (dependent on experimental combination) heuristic.solution<-matrix(0, nrow=ni,ncol=n) heuristic.location<-matrix(0, nrow=p, ncol=n) Store.solution<-numeric() Best.solution<-numeric() Store.location<-matrix(0, nrow=ni, ncol=p) Best.location<-numeric() for (i in 1:n){ subset<-numeric(0) select.location<-sample(1:V,p,replace=F) objective.function<-sum(apply(Distance.matrix[,select.location],1,min)) iteration<-0;Tempearture<-400;beta<-0.5 # initial parameter setting. while (iterationobjective.function){ delta<-updated.objective.function-objective.function unif.number<-runif(1,0,1) if (unif.number=exp(-delta/Tempearture)) {count<-count+1;select.locationLower.bound){Lower.bound<-Lower.cu} if(Lower.cu=Upper.bound){counter.up<-counter.up+1} ub.cu<-c(ub.cu,Upper.cu);ub.pr<-c(ub.pr,Upper.bound) location.m<-cbind(location.m,location) violation<-sum((apply(indicator.matrix[,location],1,sum)-1)^2) if(counter==5){alpha<-alpha/2 counter<-0} update<-alpha*(Upper.bound-Lower.cu)/violation multi<-(apply(indicator.matrix[,location],1,sum)-1)*update lambda<-pmax(0,lambda-multi) iteration.time<-as.numeric(Sys.time()) run.time<-iteration.time-start.time iter=iter+1 } node<-which(ub.pr==Upper.bound)[1]+1 optimal<-sum(apply(Distance.matrix[,location.m[,node]],1,min)) c(Upper.bound,Lower.bound)
29
PAPER III
Statistical bounds of genetic solutions to quadratic assignment problems Author: Xiangli Meng Abstract: Quadratic assignment problems (QAPs) are commonly solved by heuristic methods, where the optimum is sought iteratively. Heuristics are known to provide good solutions but the quality of the solutions, i.e., the confidence interval of the solution is unknown. This paper uses statistical optimum estimation techniques (SOETs) to assess the quality of Genetic algorithm solutions for QAPs. We examine the functioning of different SOETs regarding biasness, coverage rate and length of interval, and then we compare the SOET lower bound with deterministic ones. The commonly used deterministic bounds are confined to only a few algorithms. We show that, the Jackknife estimators have better performance than Weibull estimators, and when the number of heuristic solutions is as large as 100, higher order JK-estimators perform better than lower order ones. Compared with the deterministic bounds, the SOET lower bound performs significantly better than most deterministic lower bounds and is comparable with the best deterministic ones.
Key words: quadratic assignment problem, genetic algorithm, Jack-knife, discrete optimization, extreme value theory
PhD-student in Micro-data analysis at the School of Technology and Business Studies, Dalarna University, SE-791 88 Falun, Sweden. E-mail: [email protected]. Phone: +4623-778509.
1. Introduction The combinatorial problems in operational research have been widely studied due to the significant utility in improving the efficiency of many reality problems. However, many combinatorial problems are NP-hard and enumerating all possible solutions becomes impossible when the problem size increases. Many studies have been devoted to developing efficient (meta-) heuristic algorithms to solve the problem and provide a good solution. A shortage of the heuristic solutions is that it is difficult to assess the quality of the solution, i.e., the difference between the heuristic solution and the exact optimum is unknown. One common strategy is to use algorithms providing deterministic bounds, such as Lagrangian relaxation (Fisher 2004) and Branch and bound (Land and Doig, 1960). This strategy is popular and reasonable for many problems, but it confines the choice of heuristic algorithms, and its performance (largely) depends on the choice of parameters. For many widely used algorithms such as Genetic algorithm and Simulated Annealing algorithm, the quality of their solutions remains vague relative to deterministic algorithms. An alternative strategy for assessing the quality of heuristic solutions is to use the statistical bounds, which is also referred to as statistical optimum estimation techniques (SOETs). The idea of SOETs is that parallel heuristic processes with random starting values will result in random heuristic solutions, thereby providing a random sample close to the optimum. Statistical theories such as nonparametric theory and extreme value theory are then applied on this random sample to estimate the optimum and provide the confidence intervals. Pioneering work has been done by Derigs (1985) on travelling salesman problems (TSPs) and quadratic assignment problems (QAPs). It is shown that statistical bounds are competitive with deterministic ones for both problems and has more potential in QAP than in TSP. After Derigs (1985) there is some research devoted into developing SOETs, but many questions remain unanswered, hindering the wide application of SOETs. Giddings (2014) summarizes the current research and application situation of SOETs on operational problems. A problem class 𝒥 is a group of problems instances 𝛪, so 𝛪 ∈ 𝒥. This class contains well-known combinatorial optimization problems such as TSP, Knapsack, and Scheduling. A heuristic 𝐻 is a combination of computer instructions of the solution method with a given random number 2
seed. The heuristic class ℋ is the collection of possible heuristics. 𝑛 is the number of replicates arising from unique random number seeds. The SOETs consists of all the combination sets for 𝒥 × ℋ 𝑛 . For a complete investigation of the SOET performance, all the restrictive types 𝐼 × 𝐻 𝑛 need to be checked. For a specific combination set of 𝐼 × 𝐻 𝑛 , Carling and Meng (2014a, 2014b) examine the application of SOETs on p-median problems. They compare the performance of SOETs systematically regarding different heuristics 𝐻 and number of replicates 𝑛 and give the following conclusions: (1) The SOETs are quite informative given that the heuristic solutions derived are close enough to the optimum. A statistic named SR (standard deviation ratio) is proposed for evaluating whether this condition is satisfied. The statistical bounds will cover the optimum almost certainly if SR is smaller than the threshold 4. (2) Comparing the performances of different SOET estimators, the 2nd order Jackknife estimator and the Weibull estimator have better performance in having smaller bias and providing statistical bounds covering the optimum. When SR<4, the bounds cover the optimum almost certainly. The Gumbel estimator and the 1st order Jackknife estimator perform worse. (3) Small sample size n, e.g., 𝑛 = 3 leads to unstable intervals, but 10 heuristic solutions provide almost equally good statistical bounds with 100 heuristic solutions. Thus the effect of having more than 10 heuristic processes would have small effect on the functioning of SOET. (4) Different heuristics do not affect the performance of statistical intervals. The solutions derived by Simulated Annealing are not significantly different from those derived by Vertex Substitution. The performance of point estimators and statistical bounds are almost the same as long as SR<4. (5) Under the same computing time, statistical intervals give better results than deterministic intervals derived by Lagrangian relaxation. The statistical intervals have much shorter lengths in most of the cases while 3
almost certainly covering the optimum. Conclusion (1), (2) and (4) are novel conclusions, i.e., they could not be traced back to similar research results while conclusion (3) is analogous with Brandeau and Chiu (1993) which states 𝑛 = 10 would obtain as good solutions as 𝑛 = 2000 and statistical bounds yield better lower bounds than the available analytical bounds. (5) coincides with Brandeau and Chiu (1993) and Derigs (1985). These conclusions provide us with an effective way of deriving useful statistical intervals. However, Carling and Meng have only conducted the analysis on p-median problems, and left the validity of SOETs unverified on many other operational problems in 𝒥. The focus of this paper is therefore to analyse the performance of different SOETs on another important combinatorial problem, namely the quadratic assignment problem. The quadratic assignment problem (QAP) is a classical combinational problem in operational research. It is formulated as follows. Consider two N-dimension square matrixes 𝐴 = (𝑎𝑖𝑗 )𝑁 and 𝐵 = (𝑏𝑖𝑗 )𝑁 , find a permutation (𝑥1 , 𝑥2 , … , 𝑥𝑁 ) of integers from 1 to 𝑁 that minimises the objective function: 𝑔=∑
𝑁 𝑖=1
∑
𝑁 𝑗=1
𝑎𝑖𝑗 𝑏𝑧𝑖 𝑧𝑗
The QAPs have many applications in real world for operational and economic problems, see Loiola, et al. (2007) for a detailed introduction. QAP is known to be a difficult problem to solve, especially when 𝑁 > 15. As stated above, heuristics such as Genetic algorithm (Tate and Smith 1995), Simulated Annealing (Wilhelm and Ward 1987) and Tabu search (Misevicius, 2005) are proposed for retrieving good solutions but the quality of these solutions are unknown and unable to be assessed. Deterministic lower bounds are available for only a few algorithms and rely on parameter choices, e.g., Adams et al., (2007). Derigs (1985) compares Weibull lower bound as representative of SOET with deterministic lower bounds by Branch and Bound algorithm, and concludes that Weibull bounds outperforms deterministic ones. Their research shows the potential of SOET, but the confined experimental design does not provide sufficient support for the usage of SOET, nor 4
suggestions on applications of SOET. Presumably the usefulness of SOET still applies, as Derigs argued, and it therefore deserves a critical systematic examination of QAPs, and application advice needs to be suggested, which is the focus of this paper. This paper aims at studying the usefulness of SOET with one combination in the SOET framework 𝒥 × ℋ 𝑛 , namely the Genetic algorithm on QAPs. The Genetic algorithm is one of the most widely used algorithms in solving operational problems including QAPs (Loiola, et al. 2007). It is known to be able to find good solutions consistently, while being computationally affordable and exceptionally robust to different problem characteristics or implementation (Tate and Smith 1995). It is the leading algorithm that researchers seek to solve QAPs although the quality of the solutions remains ambiguous. Thus it becomes our concern in applying SOETs to see the performance of assessing the quality of Genetic solutions. This paper is organized as follows. In Section 2 we investigate the features of QAPs. Section 3 reviews and proposes methods for statistically estimating the minimum of the objective function as well as the corresponding bounds. Section 4 presents the results and analysis. Section 5 makes a comparison between the SOET lower bounds and the deterministic ones. The last section concludes this paper. 2. Complexity measure of quadratic assignment problem First we introduce the notations that will be used throughout the paper. 𝑍𝑖 = feasible solution of a QAP with N dimensions, 𝑖 = 1,2, … , 𝑁!. 𝑍 = the set of all feasible solutions 𝑍 = {𝑍1 , 𝑍2 , … , 𝑍𝑁! }. 𝑔(𝑍𝑖 ) = the value of objective function for solution 𝑍𝑖 . 𝜃 = min𝑍 𝑔(𝑍𝑖 ). Before going into the comparison of different estimators, the characteristics of the problems need to be investigated. Here we focus on the complexity of the problems which is interpreted here as the difficulty for algorithms to reach 𝜃. In common sense, the complexity of QAPs is decided by the number of dimensions N. This makes sense since the number of possible solutions to the QAPs is 𝑁! and it determines the size of the population of solutions. Yet, the size of the solution population is not the only effect that influences the complexity of the problems; in fact the structures of matrix A and B also play an influential role. For example, 5
if one matrix contains most elements equal to 0, the complexity of the problem should be comparably smaller. Carling and Meng (2014a) propose a new way of measuring the complexity of the p-median problems in experimental cases. They find that the objective function values for the p-median problems are approximately normally distributed, therefore they propose measuring complexity of a problem by the number of standard deviations that the optimal value lies away from the mean, i.e., ((𝜇𝑔(𝑍) − 𝜃)/𝜎𝑔(𝑍) ), where 𝜇𝑔(𝑍) is the mean of 𝑔(𝑍) and 𝜎𝑔(𝑍) the standard deviation. 𝜇𝑔(𝑍) and 𝜎𝑔(𝑍) are estimated by drawing a large random sample of solutions. This method provides a good way of measuring the complexity of solving the problems since reaching to 𝜃 would grow tough when the it lies further away in the tail; hence the problem is more complex. Although this method is not practically useful since 𝜃 is unknown in reality problems, it is quite helpful in assessing the performance of SOETs in experiments. Therefore we follow this way and check the complexity of the QAPs. The test problems used are from QAPLIB (Burkard et al., 1997). The QAPLIB provides QAP test problems with various N, A and B. One important benefit of QAPLIB is that it has known 𝜃 for most problems. We choose 40 problems with N varies between 12 and 100, and then check their complexity.
Figure 1: Sample distribution of the 14th problem in the OR-library.
6
Table 1: Description of the problem complexity of the QAPLIB. 𝜃 𝜇𝑔(𝑧𝑝) 𝜎𝑔(𝑧𝑝) Problem N Complexity tai64c
64
1855928
2955590.56
346815.9
3.17
bur26c
26
5426795
5941725.45
104825.6
4.91
nug16b
16
1240
1727.95
81.45
5.99
nug24
24
3488
4766.84
159.25
8.03
tai80b
80
818415043
1242911278
30709661
13.82
lipa60b
60
2520135
3272434.13
18091.48
41.58
Figure 1 gives the empirical distribution of a random sample for the problem bur26c. One million random solutions are generated and collected. The value of QAP objective function is approximately normally distributed. The distributions of the other test problems match that in Figure 1. The complexity together with the sample mean and the sample standard deviation for 6 problems are given in Table 1. The full results are given in Appendix I. The complexity of problem varies from 3 to 41. It is easy to reach an optimum which lies only 3 times standard deviation away from the mean, while rather difficult to reach an optimum which lies 41 times standard deviation away from the mean. 3. Statistical estimation of the minimum and its bounds There are two approaches of SOETs which provide estimators of the minimum and the bounds based on different statistic theories: first, the truncation points approaches, and second, the Extreme value theory approaches. Both approaches require the sample to be randomly selected. However, as Meng and Carling (2014) show, the performance of SOETs requires a randomly selected sample containing values close to 𝜃, in such case, the size of that sample would be enormously large and infeasible to retrieve. As several researchers have pointed out, if the starting values are selected at random, parallel heuristic solutions simulate a random sample in the tail (see e.g. McRoberts, 1971, and Golden and Alt, 1979). In other words, we could get a desired random sample with much less effort. We denote 𝑧̃𝑖 as the heuristic solution in the 𝑖 𝑡ℎ , 𝑖 = 1,2, … , 𝑛 heuristic process, and use them to compare the functioning of SOETs. In the truncation points approach, the most commonly used method is the Jackknife estimator introduced by Quenouille (1956): 7
𝜃̂𝐽𝐾 = ∑
𝑀+1 (−1)(𝑚−1) ( ) 𝑧̃(𝑚) 𝑚 𝑚=1 𝑀+1
where M is the order and 𝑧̃(𝑚) is the 𝑚𝑡ℎ smallest value in the sample. Dannenbring (1977) and Nydick and Weiss (1988) suggest using the first order, i.e. 𝑀 = 1, for point estimating the minimum. The upper bounds of the JK estimators are the minimum of 𝑥̃𝑖 , and the lower bound is [𝜃̂𝐽𝐾 − 3𝜎 ∗ (𝜃̂𝐽𝐾 )], where 𝜎 ∗ (𝜃̂𝐽𝐾 ) is the standard deviation of 𝜃̂𝐽𝐾 obtained from bootstrapping the n heuristic solutions (1,000 bootstrap samples are found to be sufficient). The scalar of 3 in computing the lower bound renders the confidence level to be 99.9% under the assumption that the sampling distribution of the JK-estimator is Normal. The 1st order JK-estimator is more biased than the higher order ones, but its mean square error is lower, as shown by Robson and Whitlock (1964). Carling and Meng (2014a) checked the performance of 1st and 2nd JK-estimators, finding that 2nd order JK-estimator performs better by providing a higher coverage rate at the cost of a slightly longer interval. The smaller bias of 2nd order JKestimator improves the performance of the estimator. Therefore, it is reasonable to wonder whether JK-estimators with even higher order would provide better estimation results. To check that, we extend to the 3rd and 4th JK-estimators in our experiments. The extreme value theory (EVT) approach assumes the heuristic solutions to be extreme values from different random samples, and they follow the Weibull distribution (Derigs, 1985). The confidence interval is derived from the characteristic of Weibull distribution. The estimator for 𝜃 is 𝑧̃(1) , which is also the upper bound of the confidence interval. The Weibull lower bound is [𝑧̃(1) − 𝑏̂] at a confidence level of (1 − 𝑒 −𝑛 ), 𝑏̂ is the estimated shape parameter of the Weibull distribution. Derigs (1985) provides a simple fast way of estimating parameters: 𝑏̂ = 𝑧̃[0.63(𝑛+1)] − (𝑧̃(1) 𝑧̃(𝑛) − 𝑧̃(2) 2 )/(𝑧̃(1) + 𝑧̃(𝑛) − 2𝑧̃(2) ), where [0.63(𝑛 + 1)] means the integer value of the function. As stated in the Introduction, Carling and Meng (2014a, 2014b) argue SOETs would work when 𝑥̃(𝑖) are close enough to 𝜃 . They propose a (1) statistic 𝑆𝑅 = 1000𝜎(𝑧̃𝑖 )/𝜃̂𝐽𝐾 to evaluate if that condition is satisfied. That statistic mimics a standardization of the standard deviation for different heuristic solutions. 𝑆𝑅 < 4 indicates the Weibull, and JK 8
intervals cover the optimum almost certainly. It is proved useful in pmedian problems, and we will check its functioning in QAPs. 4. Experimental evaluation of SOETs With the SOETs introduced above, we design experiments to investigate their usefulness on Genetic solutions of QAPs. The implementation of Genetic algorithm follows Tate and Smith (1995). The reproduction and mutation proportions are 25% and 75% respectively. The same 40 problems as used for complexity analysis are chosen for experiments. 100 genetic processes with 1000 iterations each are carried out for each problem. With this number of iterations, we have some problems with solutions close to optimum and some far from the optimum, this gives us diversified information for SOET performance in different situations. The first factor tested is the effect of estimators, where the Weibull estimator together with four JK-estimators are considered. The second factor considered is the effect of 𝑛, where we vary n to be 10 and 100. The third factor tested is the effect of complexity. These three factors result in a 40 × 1 × 2 × 5 experiment combination set, where 40 indicates 40 problem instances in 𝒥, 1 indicates 1 heuristic in ℋ, 2 indicates 2 sample size 𝑛. To assess the performance of estimators, we first draw a random sample of size n with replacement from the 100 solutions, and then calculate the estimators and confidence intervals. The procedure is repeated 1000 times for every combination. Then we get their average 𝑏𝑖𝑎𝑠 relative biasness ( ∗ 100%), coverage rate (the proportion of intervals 𝜃
𝑙𝑒𝑛𝑔𝑡ℎ
cover 𝜃 ), and average relative length of the interval ( 𝜃 ∗ 100% ). These three indicators are used to evaluate the performance of the estimators under different circumstances. The performance of SR statistic will also be checked. The results of the experiments are reported below with figures, and the details are provided in Appendix II. 4.1. The relative bias of estimators First we check the performance regarding the biasness. It is reasonable to expect Jackknife type estimators to have smaller bias than the Weibull estimator. Figure 1 confirms this by giving the Lowess smoothing line (Cleveland, 1979) of the relative bias for the five estimators when sample size 𝑛 is 10 and 100. But the difference between the four JK-estimators is 9
marginal for both levels of n. When the complexity of the problem is larger than 25, the biases for all the estimators increase sharply. The 2nd and 3rd order JK-estimator have smaller superiority than the 1st and 4th order ones under 𝑛 = 10. The mean difference between 1st and 3rd order JK-estimator is merely 0.23% of 𝜃. When the sample size increases to 100, the advantage of JK-estimators over the Weibull estimator still exists, but the differences between different order JK-estimators diminish. 1st and 2nd order JK-estimators reduce their relative bias by 0.3% and 0.1% of 𝜃 respectively, while there is almost no drop for both 3rd and 4th order JKestimators. Sample size 10
15 10 5 0
0
5
10
15
Relative Bias of estimators
20
20
Sample size 100
0.00
10.00
20.00 30.00 Complexity of the problem
1st JK-estimator Bias 3rd JK-estimator Bias Weibull estimator Bias
40.00
0.00
2nd JK-estimator Bias 4th JK-estimator Bias
10.00
20.00 30.00 Complexity of the problem
1st JK-estimator Bias 3rd JK-estimator Bias Weibull estimator Bias
40.00
2nd JK-estimator Bias 4th JK-estimator Bias
Figure 1. Lowess line of the biasness of the 5 estimators for sample size 10 and 100.
4.2.Interval coverage rate and relative length Next, we check the coverage rate of the 5 estimators when 𝑛 = 10 and 100. Figure 2 gives the Lowess smoothing line of the results. Table 2 gives the mean and median coverage rate together with lengths. For both levels of n, when the complexity of the problems increases, the coverage rates for all the 5 estimators decline sharply. The JK-estimators again outperform the Weibull ones with a higher coverage rate. Among the JKestimators, the 4th order performs better than the other three orders except when the complexity goes beyond 30. The 1st order JK-estimator has the worst performance among JK-estimators. Due to the deterioration of 4th order JK-estimator coverage rate, the mean difference between 1st and 4th order JK-estimators is 20% when 𝑛 = 10, and 14% when 𝑛 = 100.
10
Sample size 100
.8 .6 .4 .2 0
0
.2
.4
.6
.8
Coverage Rate of estimators
1
1
Sample size 10
0.00
10.00
20.00 30.00 Complexity of the problem
1st JK-estimator C.R. 3rd JK-estimator C.R. Weibull estimator C.R.
40.00
0.00
2nd JK-estimator C.R. 4th JK-estimator C.R.
10.00
20.00 30.00 Complexity of the problem
1st JK-estimator C.R. 3rd JK-estimator C.R. Weibull estimator C.R.
40.00
2nd JK-estimator C.R. 4th JK-estimator C.R.
Figure 2. Coverage percentage of 1st order, 2nd, 3rd and 4th order Jackknife estimator, and Weibull estimator under sample size 10 and 100. Sample size 100
6 4 2 0
0
10
20
30
40
Interval length of estimators
8
50
Sample size 10
0.00
10.00
20.00 30.00 Complexity of the problem
1st JK estimator Lengh 3rd JK estimator Length Weibull estimator Length
40.00
0.00
2nd JK estimator Length 4th JK estimator Length
10.00
20.00 30.00 Complexity of the problem
1st JK estimator Lengh 3rd JK estimator Length Weibull estimator Length
40.00
2nd JK estimator Length 4th JK estimator Length
Figure 3. Interval length of 1st order, 2nd, 3rd and 4th order Jackknife estimator, and Weibull estimator under sample size 10 and 100. Table 2: Coverage rate and relative length 5 estimators. Coverage Rate (%) Sample size 10
Relative Length (%)
Sample size 100
Sample size 10
Sample size 100
Estimator
Mean
Median
Mean
Median
Mean
Median
Mean
Median
1st JK
66
85
67
98
3.01
1.27
0.92
0.14
2nd JK
73
95
71
99
5.24
2.50
1.63
0.24
3rd JK
80
100
75
100
9.18
4.69
2.99
0.45
4th JK
86
100
81
100
16.36
8.74
5.62
0.92
Weibull
66
90
66
90
2.59
1.61
2.48
1.69
11
As for the relative length of the interval, Figure 3 shows there is no clear tendency of the relationship between the relative length of the intervals and the complexity of the problems. The Weibull intervals are the shortest, with the mean being around 2.5% of 𝜃 for both 𝑛 = 10 and 100. The lengths of the JK-estimator interval almost double when the order increases by 1 and are highly affected by sample size. The lengths when 𝑛 = 10 are almost 3 times of 𝑛 = 100. When 𝑛 = 10, 1st JK-estimator interval has a mean length 3% of 𝜃, a little higher than the Weibull ones while they have the same coverage rate. The 2nd order JK-estimator has a higher coverage rate together with a longer interval. The situation deteriorates for 3rd and 4th JK-estimator with much longer confidence interval. When 𝑛 = 100 , the Weibull estimator has almost the same performance while the JK-estimators have better performance. The 3rd order JK-estimator has a similar mean length and a much shorter median length but with 9% more average coverage rate. Thus, when the sample size is small, 1st and 2nd order JK estimators are suggested, otherwise 3rd and 4th order JK-estimators are suggested when the sample size is large. 4.3.SR performance Next, the performance of the statistic SR is checked. Based on the analysis above, we focus specifically on 2 cases, the 2nd order JK-estimator when 𝑛 = 10 and 4th order JK-estimator, when 𝑛 = 100. Figure 4 gives the scatter plot and Lowess line between the coverage rate and SR. It can be seen that for both cases, a small SR close to 0 does not guarantee a high coverage rate, while as large a SR as 60 may correspond to a coverage rate even as high as 100%. The problems with high complexities are more likely to have different heuristic solutions trapped in the same suboptimal or similar suboptimals, leading to a trivial SR. As to easy problems, a small SR does indicate a high coverage rate close to 1. Figure 5 provides the instances with SR<7 for both cases. The threshold 7 is chosen because it is the integer part of the smallest SR where the coverage rate for easy problems drops below 0.95 for both sample cases and for all 5 intervals. The size of the circle indicates the complexity of the problem. The problems with trivial coverage rate have complexities over 17. Therefore, the performance of SR is related to the complexity of the problem. For easy problems, small SR supports that the confidence interval covers 𝜃, but not for difficult problems. There is no clear pattern that can be concluded for the functioning of SR. The application of SR remains an 12
open question. Sample size 100
.8 .6 .4 .2 0
0
.2
.4
.6
.8
4th JK-estimator Coverage Rate
1
1
Sample size 10
0
20
40 SR
60
80
0
bandwidth = .8
20
40 SR
60
80
bandwidth = .8
Figure 4. Scatter plot between SR and coverage rate of 2 nd order JK-estimator for n=10 and 4th order JK-estimator for n=100.
.8 .6 .4 .2 0
0
.2
.4
.6
.8
4th order JK-estimator Coverage rate
1
Sample size 100
1
Sample size 10
0
1
2
3
4
5
0
SR
1
2 SR
3
4
Figure 5. Scatter plot between SR<7 and coverage rate of 2nd order JK-estimator (left), between SR<7 and coverage rate of 4th order JK-estimator (right). The size of the circle stands for complexity of the problem.
5. Lower bound comparison As a small part in deriving quality of solutions, it is of great concern to compare SOET with the common approach, namely the deterministic bounds, especially the lower bound. Several lower bounds are proposed. Loiola (2007) collects different lower bounds of several problems. The deterministic bounds stated are: Gilmore-Lawler bound (GLB62), from Gilmore (1962); the interior-point bound (RRD95), from Resende et al. (1995); the 1-RLT dual ascent bound (HG98), from Hahn and Grant (1998); the dual-based bound (KCCEB99), from Karisch et al. (1999); the quadratic programming bound (AB01), from Anstreicher and Brixius (2001); the SDP bound (RS03), from Sotirov and Rendl (2003); the lift13
and-project SDP bound (BV04), from Burer and Vandenbussche (2006); the Hahn-Hightower 2-RLT dual ascent bound (HH01), from Adams et al. (2007). To incorporate our results to their framework, we compare the SOET lower bound by techniques in the previous section with deterministic ones. The heuristic solutions are derived by running 100 Genetic processes with 3000 iterations each. Almost all the processes stopped improving after 2500 iterations, with very few exceptions. Then we derive the SOET lower bounds by the 4th order JK-estimator. The lower bounds are provided in Table 3. To assess the performance conveniently, we calculate the average absolute relative deviation of the |𝑙𝑜𝑤𝑒𝑟.𝑏𝑜𝑢𝑛𝑑−𝑜𝑝𝑡𝑖𝑚𝑎𝑙| lower bounds, i.e., ∗ 100%, and report them in the 𝑜𝑝𝑡𝑖𝑚𝑎𝑙 last row of Table 3.
Out of 15 problems, SOET has 2 best lower bounds while HH01 and BV04 share the rest 13 best lower bounds. SOET lower bounds perform better than the first 6 deterministic lower bounds but are surpassed by BV04 and HH01, which are acknowledged to be the best deterministic lower bounds. The average absolute bias percentage for SOET, HH01 and BV04 is not large, being 6%, 3% and 3% respectively, yet still significantly smaller than the other 6 lower bounds. Out of 15 problems, SOET lower bounds cover 14 optimums. Therefore, the SOET lower bound is competitive with the best deterministic bounds. It shows great potential in application even though it fails to cover the optimum with small probability.
14
Table 3: Statistical lower bounds and deterministic lower bounds. Proble m
Optim
GLB62
HG98
KCCEB 99
AB01
RS03
BV04
HH01
JK4
Had16
3720
3358
3558
3553
3595
3699
3672
3720
3704
Had18
5358
4776
5083
5078
5143
5317
5299
5358
5261
Had20
6922
6166
6571
6567
6677
6885
6811
6922
6810
Kra30 a
88900
68360
75853
75566
68572
77647
86678
86247
83867
Kra30 b
91420
69065
76562
76235
69021
81156
87699
87107
88601
Nug12
578
493
523
521
498
557
568
578
515
Nug15
1150
963
1039
1033
1001
1122
1141
1150
1143
Nug20
2570
2057
2179
2173
2290
2451
2506
2508
2266
Nug30
6124
4539
4793
4785
5365
5803
5934
5750
5857
Rou15
35421 0
29854 8
32394 3
32358 9
30377 7
33328 7
35020 7
34521 0
31778 2
Rou20
72552 0
55994 8
64205 8
64142 5
60782 2
66383 3
69512 3
69939 0
67944 1
Tai20a
70348 2
58067 4
61720 6
61664 4
58513 9
66373 00
67168 5
67587 0
65679 4
Tai25a
11672 56
96241 7
10067 49
10059 78
98345 6
10413 37
11128 62
10916 53
10846 65
Tai30a
18181 46
15046 88
15663 09
15653 13
15180 59
16521 86
17068 75
16862 90
22107 30
Tho30
14993 6
90578
99995
99855
12468 4
13605 9
14281 4
13670 8
14561 6
19.08
12.99
13.17
13.65
61.92
2.94
3.26
6.34
Bias.%
Source: Loiola (2007) except for last column and last row. Bold number means best lower bound.
6. Concluding discussion In this paper, we analyse the performance of SOETs on QAPs. Based on 15
the framework proposed by Giddings (2014), the paper extends the work by Derigs (1985), by systematically verifying the usefulness of SOETs and comparing with deterministic bounds, and it extends the work of Carling and Meng (2014a, 2014b) by testing on QAPs. We tested 5 estimators on 40 problems with different sample sizes. In our analysis, SOETs can be useful in providing helpful intervals covering the optimum. The JKestimators have better performance than Weibull estimators. When the sample size is small, the 2nd order JK-estimator is suggested, and when the sample size is large, the 4th order JK-estimator is suggested. The statistics SR do not provide accurate information, especially when the solutions are trapped into suboptimal for complex problems. The comparison between SOET lower bound and deterministic ones shows that the SOET performs close to the best deterministic lower bounds. Thus, it shows SOETs have great potential in accurately assessing the quality of the heuristic solutions. References Adams, W.P., Guignard, M., Hahn, P.M., Hightower, W.L., (2007), A level-2 reformulation-linearization technique bound for the quadratic assignment problem, European Journal of Operational Research 180, 983-996. Anstreicher, K.M., Brixius, N.W., (2001), Solving quadratic assignment problems using convex quadratic programming relaxations. Optimization Methods and Software, 16(1-4), 49-68. Burer, S., Vandenbussche, D, (2006), Solving lift-and-project relaxations of binary integer programs. SIAM Journal on Optimization, 16(3), 726-750. Burkard, R.E., Karisch, S.E., Rendl, F., (1997), QAPLIB – A Quadratic Assignment Problem Library, Journal of Global Optimization 10, 391-403 Carling, K., Meng, X., (2014a), On statistical bounds of heuristic solutions to location problems. Working papers in transport, tourism, information technology and microdata analysis, 2014:10. Carling, K., Meng, X., (2014b), Confidence in heuristic solutions. Working papers in transport, tourism, information technology and microdata analysis, 2014:12. 16
Cleveland, W.S., (1979), Robust Locally Weighted Regression and Smoothing Scatterplots, Journal of the American Statistical Association, 74, 829–836. Dannenbring, D.G., (1977), Procedures for estimating optimal solution values for large combinatorial problems, Management science, 23:12, 12731283. Derigs, U, (1985), Using confidence limits for the global optimum in combinatorial optimization. Operations research, 33:5, 1024-1049. Fisher, M.L., (2004), The Lagrangian relaxation method for solving integer programming problems. Management science, 50(12_supplement), 18611871. Giddings, A.P., Rardin, R.L, Uzsoy, R, (2014), Statistical optimum estimation techniques for combinatorial problems: a review and critique. Journal of Heuristics, 20, 329-358. Gilmore, P.C., (1962), Optimal and suboptimal algorithms for the quadratic assignment problem. Journal of the Society for Industrial & Applied Mathematics, 10(2), 305-313. Golden, B.L., Alt, F.B., (1979), Interval estimation of a global optimum for large combinatorial optimization, Operations Research 33:5, 1024-1049. Hahn, P., Grant, T., (1998), Lower bounds for the quadratic assignment problem based upon a dual formulation. Operations Research, 46(6), 912-922. Karisch, S.E., Cela, E., Clausen, J., Espersen, T., (1999), A dual framework for lower bounds of the quadratic assignment problem based on linearization. Computing, 63(4), 351-403. Land, A.H., Doig, A.G. (1960), An automatic method of solving discrete programming problems. Econometrica: Journal of the Econometric Society, 497-520. Loiola, E.M., Abreu, N.M.M, Boaventura-Netto, P.O., Hahn.P., Querido, T., (2007), A survey for the quadratic assignment problem. European Journal of Operational Research 176.2, 657-690. 17
McRobert, K.L., (1971), A search model for evaluating combinatorially explosive problems, Operations Research, 19, 1331-1349. Meng, X., Carling, K., (2014), How to Decide Upon Stopping a Heuristic Algorithm in Facility-Location Problems?. In Web Information Systems Engineering–WISE 2013 Workshops, Lecture Notes in Computer Science, 8182, 280-283, Springer, Berlin/Heidelberg. Misevicius, A., (2005), A tabu search algorithm for the quadratic assignment problem. Computational Optimization and Applications, 30:95-111. Nydick JR, R.L., and Weiss, H.J., (1988), A computational evaluation of optimal solution value estimation procedures, Computers & Operations Research, 5, 427-440. Resende, M.G.C., K.G. Ramakrishnan, and Z. Drezner, (1995), Computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming, Operations Research 43, 781-791. Robson, D.S., and Whitlock, J.H., (1964), Estimation of a truncation point, Biometrika, 51, 33-39. Sotirov, R., Rendl, F., (2003), Bounds for the Quadratic Assignment Problem Using the Bundle Method. Department of Mathematics, University of Klagenfurt, Austria, Tech. Rep. Tate, D.M., Smith, A.E. (1995), A genetic approach to the quadratic assignment problem. Computers & Operations Research, 22(1), 73-83. Wilhelm, M.R., Ward, T.L. (1987), Solving quadratic assignment problems by ‘simulated annealing’. IIE transactions, 19(1), 107-119.
18
Appendix I Table A1: Description of the 40 problems of the QAPLIB. Problem
n
𝜃
𝜇𝑔(𝑧𝑝 )
𝜎𝑔(𝑧𝑝 )
Complexity
els19
19
17212548
58712359.92
10526387
3.94
chr12a
12
9552
45123.58
8833.06
4.03
esc16c
16
160
249.32
20.05
4.45
chr15a
15
9896
61378.34
11444.91
4.5
had12
12
1652
1888.27
50.18
4.71
chr18b
18
1534
4601.64
618.07
4.96
bur26d
26
3821225
4211133.26
75035.14
5.2
nug14
14
1014
1363.54
65.77
5.31
had16
16
3720
4226.97
85.82
5.91
chr20b
20
2298
10708.88
1415.13
5.94
had18
18
5358
5990.01
105.86
5.97
scr20
20
110030
226272.64
19391.22
5.99
chr20a
20
2192
10707.62
1406.16
6.06
nug18
18
1930
2565.41
98.92
6.42
chr25a
25
3796
19877.53
2494.68
6.45
ste36a
36
9526
22750.27
2022.71
6.54
had20
20
6922
7764.79
125.98
6.69
tai35b
35
283315445
516046202
33413753
6.97
nug27
27
5234
7128.04
233.68
8.11
tai40b
40
637250948
1132975203
56180695
8.82
nug30
30
6124
8132.35
212.94
9.43
kra30b
30
91420
137016.96
4621.8
9.87
kra30a
30
88900
134657.56
4487.12
10.2
kra32
32
88900
137137.22
4716.35
10.23
lipa20a
20
3683
3942
24.79
10.45
sko81
81
90998
108443.68
1004.45
17.37
tai60a
60
7208572
8518524.44
70989.41
18.45
sko90
90
115534
136878.06
1143.79
18.66
wil100
100
273038
299759.06
1367.71
19.54
lipa50a
50
62093
64035.91
98.86
19.65
19
sko100c
100
147862
174507.74
1304.2
20.43
tai80a
80
13557864
15624432.79
94608.26
21.84
lipa70a
70
169755
173757.85
168.43
23.77
lipa40b
40
476581
621324.38
5245.29
27.59
20
Appendix II Table A2: Relative bias, coverage rate and relative length of 1st JK, 2nd JK and 3rd JK estimators when 𝑛 = 10. CV stands for coverage rate, RB stands for relative bias in percentage, RL stands for relative length in percentage Problem
1st JK
SR
2nd JK
3rd JK
CV
RB
RL
CV
RB
RL
CV
RB
RL
bur26d
0.28
0.98
0.00
0.01
0.99
0.00
0.02
1.00
0.00
0.04
lipa70a
0.3
0.00
0.95
0.08
0.00
0.94
0.14
0.00
0.94
0.24
had16
0.41
1.00
0.00
0.03
1.00
0.00
0.06
1.00
0.00
0.15
esc16c
0.44
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
bur26c
0.49
0.97
0.00
0.02
0.99
0.00
0.03
1.00
0.00
0.05
lipa50a
0.56
0.00
1.13
0.14
0.00
1.12
0.24
0.03
1.12
0.42
wil100
1.56
0.02
1.59
0.52
0.15
1.53
0.88
0.33
1.48
1.47
lipa60b
1.79
0.00
19.55
0.71
0.00
19.48
1.21
0.00
19.43
2.03
tai80a
1.81
0.00
4.25
0.45
0.00
4.23
0.77
0.00
4.22
1.32
had18
1.97
1.00
-0.01
0.12
1.00
0.01
0.24
1.00
0.01
0.54
tai64c
2.17
0.97
-0.02
0.27
0.99
-0.01
0.49
1.00
-0.01
0.93
had12
2.33
1.00
0.00
0.04
1.00
0.00
0.12
1.00
0.00
0.35
had20
2.35
1.00
0.00
0.10
1.00
0.01
0.24
1.00
0.00
0.58
tai60a
3.26
0.01
3.94
0.99
0.08
3.84
1.69
0.20
3.77
2.83
sko81
3.52
0.03
2.64
0.89
0.13
2.57
1.52
0.35
2.52
2.59
sko9100c
3.81
0.00
3.04
0.94
0.08
2.97
1.61
0.32
2.93
2.74
sko90
3.96
0.00
2.80
0.90
0.08
2.78
1.53
0.34
2.78
2.67
els19
5.18
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.01
lipa20a
7.89
0.88
-0.35
3.75
0.88
-0.46
6.43
0.90
-0.32
10.90
nug18
8.71
0.86
0.00
1.75
0.96
-0.11
3.02
1.00
-0.18
5.22
nug30
8.85
0.72
0.47
1.51
0.91
0.41
2.60
0.99
0.37
4.60
nug27
9.51
0.88
-0.16
2.22
0.96
-0.23
3.80
1.00
-0.22
6.59
nug24
10.22
0.93
-0.14
2.04
0.98
-0.17
3.52
1.00
-0.17
6.25
nug14
10.48
0.99
-0.10
1.31
1.00
-0.03
2.41
1.00
0.00
4.77
kra30b
11.7
0.84
0.22
2.33
0.95
0.11
4.01
0.99
0.07
7.00
kra32
11.73
0.53
1.21
2.74
0.76
0.96
4.71
0.95
0.77
8.08
nug16b
12.46
0.99
-0.06
1.65
0.99
0.15
3.41
1.00
0.20
7.52
21
kra30a
12.77
0.40
1.28
3.26
0.70
0.95
5.62
0.95
0.68
9.55
tai80b
14.51
0.43
2.60
3.23
0.66
2.38
5.55
0.91
2.21
9.61
chr18b
14.54
1.00
-0.03
0.73
1.00
0.05
1.56
1.00
0.02
3.64
scr20
14.85
0.99
-0.17
1.93
1.00
-0.03
3.52
1.00
0.05
6.92
tai35b
20.02
0.94
0.18
1.73
0.98
0.23
3.05
1.00
0.30
5.72
ste36a
25.24
0.81
1.19
4.81
0.94
0.98
8.31
1.00
0.83
14.68
tai40b
28.25
0.83
0.33
6.37
0.93
-0.09
10.85
0.98
-0.26
18.47
chr12a
31.36
1.00
-0.01
1.23
1.00
0.06
3.13
1.00
0.00
8.07
chr20b
36.85
0.36
6.58
6.92
0.72
6.52
12.03
0.95
6.57
21.72
lipa40b
47.08
0.74
-0.79
28.43
0.79
-3.25
48.22
0.82
-3.32
78.48
chr20a
60.44
0.61
3.76
15.68
0.80
2.09
26.80
0.96
0.90
45.06
chr15a
61.88
1.00
-0.09
2.18
1.00
0.14
4.61
1.00
0.05
10.83
chr25a
79.02
0.68
8.01
18.56
0.86
7.16
31.68
0.97
7.00
54.64
Table A3: Relative bias, coverage rate and relative length of 4th JK and Weibull estimator when 𝑛 = 10. CV stands for coverage rate, RB stands for relative bias in percentage, RL stands for relative length in percentage Problem
4th JK
SR
Weibull
CV
RB
RL
CV
RB
RL
bur26d
0.28
1.00
0.00
0.08
0.99
0.00
0.02
lipa70a
0.3
0.01
0.94
0.42
0.00
0.97
0.08
had16
0.41
1.00
0.00
0.34
1.00
0.00
0.04
esc16c
0.44
1.00
0.00
0.01
1.00
0.00
0.00
bur26c
0.49
1.00
0.00
0.11
1.00
0.00
0.02
lipa50a
0.56
0.17
1.11
0.74
0.01
1.17
0.15
wil100
1.56
0.60
1.45
2.45
0.04
1.72
0.48
lipa60b
1.79
0.00
19.38
3.42
0.00
19.72
0.19
tai80a
1.81
0.05
4.23
2.33
0.00
4.35
-0.15
had18
1.97
1.00
-0.01
1.19
1.00
0.00
0.20
tai64c
2.17
1.00
0.01
1.83
1.00
0.02
0.37
had12
2.33
1.00
0.02
0.94
1.00
0.00
0.15
had20
2.35
1.00
-0.01
1.36
1.00
0.00
0.26
tai60a
3.26
0.46
3.72
4.75
0.02
4.17
0.84
sko81
3.52
0.70
2.47
4.48
0.04
2.84
0.98
22
sko9100c
3.81
0.68
2.91
4.73
0.01
3.26
0.91
sko90
3.96
0.80
2.80
4.74
0.02
2.99
0.84
els19
5.18
1.00
0.00
0.05
1.00
0.00
0.00
lipa20a
7.89
0.95
-0.12
18.70
0.54
0.40
1.59
nug18
8.71
1.00
-0.21
9.18
0.93
0.40
1.73
nug30
8.85
1.00
0.39
8.31
0.85
0.78
1.70
nug27
9.51
1.00
-0.18
11.59
0.90
0.33
2.28
nug24
10.22
1.00
-0.17
11.34
0.97
0.24
2.19
nug14
10.48
1.00
0.03
9.54
1.00
0.08
1.76
kra30b
11.7
1.00
0.09
12.43
0.92
0.71
2.45
kra32
11.73
1.00
0.62
14.00
0.59
1.79
2.97
nug16b
12.46
1.00
0.18
16.11
0.96
0.01
1.89
kra30a
12.77
0.99
0.46
16.34
0.41
2.05
3.83
tai80b
14.51
0.99
2.06
16.89
0.39
3.31
3.40
chr18b
14.54
1.00
0.03
8.22
1.00
0.00
1.64
scr20
14.85
1.00
0.11
13.73
0.99
0.06
2.42
tai35b
20.02
1.00
0.39
11.11
0.98
0.47
2.33
ste36a
25.24
1.00
0.73
26.50
0.88
2.20
4.75
tai40b
28.25
1.00
-0.25
31.94
0.90
1.68
11.50
chr12a
31.36
1.00
0.14
19.43
1.00
0.00
3.43
chr20b
36.85
1.00
6.77
40.20
0.50
7.83
6.78
lipa40b
47.08
0.84
-1.87
127.79
0.33
5.04
-0.74
chr20a
60.44
1.00
0.18
76.18
0.63
7.20
15.66
chr15a
61.88
1.00
-0.01
24.74
1.00
0.03
4.81
chr25a
79.02
1.00
7.25
96.12
0.75
11.82
19.80
Table A4: Relative bias, coverage rate and relative length of 1st JK, 2nd JK and 3rd JK estimators when 𝑛 = 10. CV stands for coverage rate, RB stands for relative bias in percentage, RL stands for relative length in percentage Problem
1st JK
SR
2nd JK
3rd JK
CV
RB
RL
CV
RB
RL
CV
RB
RL
bur26d
0.31
0.00
0.94
0.03
0.00
0.94
0.05
0.00
0.94
0.09
lipa70a
0.43
0.98
0.00
0.00
0.98
0.00
0.00
0.99
0.00
0.00
23
had16
0.51
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
esc16c
0.56
0.00
1.09
0.10
0.00
1.09
0.18
0.00
1.09
0.31
bur26c
0.61
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
lipa50a
0.97
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
wil100
1.6
0.00
1.45
0.33
0.02
1.45
0.55
0.12
1.45
0.95
lipa60b
1.84
0.00
4.22
0.11
0.00
4.22
0.19
0.00
4.22
0.36
tai80a
1.85
0.00
19.35
0.49
0.00
19.37
0.85
0.00
19.39
1.54
had18
2.16
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
tai64c
2.22
1.00
0.00
0.00
1.00
0.00
0.01
1.00
0.00
0.02
had12
2.36
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
had20
2.47
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
tai60a
3.35
0.00
3.55
0.83
0.00
3.51
1.41
0.14
3.49
2.39
sko81
3.63
0.00
2.34
0.63
0.02
2.31
1.07
0.38
2.30
1.81
sko9100c
3.92
0.00
2.85
0.44
0.00
2.84
0.74
0.01
2.85
1.29
sko90
3.97
0.00
2.70
0.32
0.00
2.71
0.55
0.00
2.72
0.97
els19
8.31
1.00
0.00
0.02
1.00
0.00
0.07
1.00
0.00
0.22
lipa20a
8.75
0.99
-0.02
0.37
0.99
0.01
0.72
1.00
0.02
1.47
nug18
9.02
0.44
0.35
0.37
0.76
0.35
0.68
0.97
0.35
1.28
nug30
9.76
1.00
0.00
0.05
1.00
0.00
0.13
1.00
0.00
0.33
nug27
10.46
1.00
0.00
0.07
1.00
0.00
0.14
1.00
0.00
0.33
nug24
10.62
1.00
0.00
0.01
1.00
0.00
0.03
1.00
0.00
0.08
nug14
12.13
0.81
-0.34
3.25
0.89
-0.54
5.46
0.95
-0.54
8.99
kra30b
12.15
0.80
0.14
0.40
0.98
0.17
0.72
1.00
0.19
1.37
kra32
12.58
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
nug16b
13.12
0.85
-0.23
3.86
0.85
-0.16
6.70
0.86
0.11
11.76
kra30a
14.35
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
tai80b
14.89
0.40
1.83
1.84
0.65
1.76
3.17
0.86
1.76
5.47
chr18b
15.14
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
scr20
15.17
1.00
0.00
0.01
1.00
0.00
0.02
1.00
0.00
0.04
tai35b
20.9
0.20
0.22
0.17
0.56
0.21
0.29
0.91
0.21
0.54
ste36a
25.61
0.74
0.74
1.62
0.87
0.79
2.82
0.97
0.87
5.05
tai40b
29.2
1.00
-0.01
1.27
1.00
0.07
2.36
1.00
0.11
4.74
chr12a
31.36
1.00
-0.01
1.23
1.00
0.06
3.13
1.00
0.00
8.07
chr20b
37.55
0.00
6.36
1.11
0.00
6.41
1.94
0.07
6.47
3.60
lipa40b
50.75
1.00
-0.04
1.71
1.00
0.08
3.73
1.00
0.12
8.61
24
chr20a
64.2
0.96
-0.52
8.63
0.99
-0.49
14.91
1.00
-0.32
26.65
chr15a
64.72
1.00
0.00
0.00
1.00
0.00
0.00
1.00
0.00
0.00
chr25a
81.99
0.54
5.17
7.38
0.77
4.93
12.49
0.91
4.99
21.35
Table A5: Relative bias, coverage rate and relative length of 4th JK and Weibull estimators when 𝑛 = 10. CV stands for coverage rate, RB stands for relative bias in percentage, RL stands for relative length in percentage Problem
4th JK
SR
Weibull
CV
RB
RL
CV
RB
RL
bur26d
0.31
0.00
0.94
0.16
0.01
0.97
0.08
lipa70a
0.43
1.00
0.00
0.01
0.99
0.00
0.01
had16
0.51
1.00
0.00
0.00
1.00
0.00
0.05
esc16c
0.56
0.00
1.10
0.55
0.01
1.17
0.15
bur26c
0.61
1.00
0.00
0.00
1.00
0.00
0.02
lipa50a
0.97
1.00
0.00
0.00
1.00
0.00
0.00
wil100
1.6
0.56
1.46
1.68
0.05
1.72
0.31
lipa60b
1.84
0.00
4.22
0.69
0.00
4.35
0.31
tai80a
1.85
0.00
19.38
2.85
0.00
19.72
0.56
had18
2.16
1.00
0.00
0.00
1.00
0.00
0.21
tai64c
2.22
1.00
0.00
0.06
0.99
0.02
0.37
had12
2.36
1.00
0.00
0.00
1.00
0.00
0.27
had20
2.47
1.00
0.00
0.00
1.00
0.00
0.14
tai60a
3.35
0.62
3.48
4.10
0.03
4.18
0.86
sko81
3.63
0.55
2.30
3.09
0.03
2.83
0.82
sko9100c
3.92
0.23
2.88
2.27
0.02
3.24
0.86
sko90
3.97
0.09
2.74
1.75
0.01
2.99
0.89
els19
8.31
1.00
0.00
0.63
0.53
0.34
1.77
lipa20a
8.75
1.00
0.01
3.00
0.92
0.41
1.71
nug18
9.02
0.99
0.36
2.46
0.85
0.78
1.66
nug30
9.76
1.00
0.00
0.82
0.92
0.28
2.39
nug27
10.46
1.00
0.00
0.74
0.97
0.27
2.10
nug24
10.62
1.00
0.00
0.22
1.00
0.08
1.76
nug14
12.13
0.96
-0.46
14.96
0.56
1.80
2.64
kra30b
12.15
1.00
0.21
2.65
0.90
0.72
2.35
25
kra32
12.58
1.00
0.00
0.00
0.97
0.01
1.96
nug16b
13.12
0.92
0.37
20.99
0.41
2.09
3.56
kra30a
14.35
1.00
0.00
0.00
1.00
0.00
0.01
tai80b
14.89
0.97
1.83
9.56
0.38
3.35
4.16
chr18b
15.14
1.00
0.00
0.00
1.00
0.00
1.63
scr20
15.17
1.00
0.00
0.10
0.99
0.07
2.47
tai35b
20.9
0.99
0.21
1.02
0.98
0.47
2.27
ste36a
25.61
1.00
0.92
9.26
0.89
2.24
5.51
tai40b
29.2
1.00
0.10
9.61
0.90
1.69
6.23
chr12a
31.36
1.00
0.14
19.43
1.00
0.00
3.25
chr20b
37.55
0.45
6.53
6.86
0.48
7.98
8.14
lipa40b
50.75
1.00
0.25
19.53
0.33
5.51
5.50
chr20a
64.2
1.00
-0.14
48.59
0.65
7.33
9.11
chr15a
64.72
1.00
0.00
0.00
1.00
0.04
5.05
chr25a
81.99
0.98
5.16
37.22
0.72
12.12
17.97
26
27
PAPER IV This paper is accept by International Journal on Web Services Research, we acknowledge the journal for this publication.
28
On transforming a road network database to a graph for localization purpose
Authors: Xiangli Meng and Pascal Rebreyend1
Abstract: The problems of finding best facility locations require complete and accurate road networks with the corresponding population data in a specific area. However the data obtained from road network databases usually do not fit in this usage. In this paper we propose a procedure of converting the road network database to a road graph which could be used for localization problems. Several challenging problems exist in the transformation process which are commonly met also in other data bases. The procedure of dealing with those challenges are proposed. The data come from the National road data base in Sweden. The graph derived is cleaned, and reduced to a suitable level for localization problems. The residential points are also processed in ordered to match the graph. The reduction of the graph is done maintaining the accuracy of distance measures in the network.
Key words: road network, graph, population, GIS.
1
Corresponding author. E-mail:[email protected]. Phone: +46-23-778921.
1
distance is deteriorated, particularly in a area with a lot natural barriers. Consequently, the usage of Euclidian distance would lead to poor locations of the model.
1. Introduction Consider the common locationallocation model (the p-median problem) which allocate P facilities to a population geographically distributed in Q demand points such that the population’s average or total distance to its nearest service facility is minimized. Solving such problems is based on computing the distance between candidate locations and all or a subset of points representing the habitat of people. The candidate locations should be connected in a graph and the corresponding distance matrix should be suitable for further calculations. A popular approach is to avoid the trouble caused by transforming the original data into a connected road network and instead use the Euclidian distance between the candidate locations and residential points. Thus, all points are assumed to be connected pairwise via straight lines. However, the Euclidian distance may be inaccurate, especially if two points are not directly connected and require detours, in which case the Euclidian distance would underestimate the real distance. Thus the quality of the
Han & al (2013) shows that using the road network to compute distances instead gives a good treatment of the problem. The road network distance gives accurate estimate of distance between 2 points. It reflects the road connections and natural barriers. In their analysis, although the complexity of the problem leads to sub-optimal solutions, the road network distance solutions still outperforms those of Euclidian ones. However, even though the usage of the road network distance is essential in solving the models, the original network data sets researchers obtained usually do not fit and are unable to be used directly. Those data sets need to be cleaned and transformed to a connected road network graph before being used for finding best locations of facilities. Many challenging problems could pop out in this process such as incomplete information or isolated sub graph. Dealing with these problems inappropriately 2
would produce inaccurate graph and then lead to bad solutions. For data sets in a large scale, the necessity of giving proper treatment of the problems in the data set and the transformation process is even more crucial.
over 34 million data points with auxiliary information like speed limits and direction. Many commonly met challenges appear in the transformation process. Proper methods need to be used for handling those challenges before getting the graph for solving the p-median models. The main challenges we encountered are as follows.
Despite the nontrivial role an accurate road network graph plays in solving the location models, few research papers give proper illustrations on how to deal with the troubles in the process of deriving it. Especially for data at a large scale for the pmedian models, there is no research investigation on how to deal with the challenges in the transformation process, which is the focus of this paper. In our case, we are interested in locating public facilities such as hospitals, public services at the scale of the country Sweden. The objective function we want to minimize is the average distance between residence’s habitants and the closest facilities nationwide. The scale of the problem is very large, making the quality for graph and distance matrix vital. The original data we get is from the National road data base (NVDB), provided by the National Swedish Road Agency (Trafikverket). It is a very detailed data set consisting of
a). Filling in the missing crossing and connectivity information properly. b). Reducing a super large graph to a manageable scale and keep the distance information as accurate as possible. c). Calculating the distance between different nodes in an affordable time. The 3 above problems are commonly met in deriving large graphs for location-allocation models. Inappropriate filling of missing information would result in incorrect graph, giving wrong distances. In our data set, cross and connection information are missing, which need to be filled properly. Wrong fill of the cross would connect 2 separated roads ‘together’. Adding connections to isolated sub graph incorrectly usually underestimates distances. 3
Reducing the large graph is necessary since our data set is based on national level, therefore the scale itself to this problem would cause trouble for further analysis. Due to the fact that pmedian problems call for the distance between any pair of nodes, the distance matrix corresponding with the graph would have very large dimensions, and it would be beyond the ability of the computers to process with heuristic algorithms or other methods. Thus it is crucial and tricky to reduce the data appropriately to an affordable level without losing too much accuracy. The commonly used Dijkstra algorithm could give the distance between any pair of nodes, but it consumes a very long time, making it inconvenient for researchers to make adjustment to it and difficult to derive large distance matrix. Thus the algorithm needs to be refined to be able to get the distance matrix in an affordable time.
country. Deriving a graph in such a large scale is quite innovative and could provide helpful techniques to researchers. The methods we proposed to deal with challenges are indicative for data at similar scales. We have tried our approach on different regions separately in Sweden, and it worked well. It shows our methods have good adaptability also for data with smaller scales. The region sizes are quite comparable to many other areas. For example, the smallest regions in Sweden are of the size of Luxembourg, the middle sized are comparable with Belgium and German states, and the largest has similar size with Hungary and Portugal. This makes our approach have wide applications and provide helpful technique when dealing with data set in many other countries and states. The method we proposed here is for the usage of the p-median models. It transforms a large detailed network data set into a connected graph and provides a detailed distance matrix which can be used for finding best locations of the models. It is also helpful in deriving usable graphs for other location-allocation models since they use the same
Sweden is about 450,000 km2 in area with 10 million population. It is fairly large in Europe and the world. The Swedish road networks are quite advanced and connect almost all the parts of the 4
type of connected graph and distance matrix in most cases, such as warehouse location problems (Beasley 1993), maximum cover problems (Kariv & Hakimi, 1979b). Our method has wider application also in circumstances other than location models. For example, the classical operational problems Travelling salesman (Reinelt, 1991), the Quadratic assignment problems (Burkard, & al 1991), and the vehicle routing problem (Laporte, 1992). The scales of those problems are becoming larger and larger and correspondingly lead to large connected graphs and distance matrixes, which makes our techniques beneficial.
version of it. The paper is organized as follows. Section 2 introduces the road data structure and our pre-processing procedure; Section 3 introduces our process in dealing with population data; Section 4 presents the graph reduction; Section 5 presents the experiments with matrix distances; Section 6 gives the conclusions. 2. Road data The road network data set we used has been provided by the National Swedish Road Agency (Trafikverket). The data comes from the National road data base (NVDB) which is operated by the Swedish Transport Agency, the Swedish Transport Administration and a few other departments. The data set has a few good characteristics. First, it is a very detailed data set that includes almost all the roads in the nation’s network. Second, the data set reflects the true road situation respecting to natural barriers and detours. Thus the distances obtained from the data set are quite accurate. Third, it includes much auxiliary information like the speed limits and latitude of the roads, making it possible to get the correct road
The outline of the paper follows our transformation process. First we introduction the basic information of the data set. Then we give the detail of filling the missing information, e.g., missing crossings and connectivity information. After getting a detailed network, we incorporate the population data into the road network. Then we reduce the size of the graph to a manageable scale. At last we derive the distance matrix in the graph with both Dijkstra algorithm and an improved 5
lengths and the travelling time required. Fourth, the data set is updated frequently. The road network data for the whole country together with its previous versions have been used for other researches as well as business usages, such as road maintenance, traffic management, navigation, etc., see Lundgren (2000) for an example. Since the location problems have different requirements for graphs, the previous transformation techniques on the NVDB data do not fit in our framework. There are few researches in location problems use the road network distance at this large scale. Although not nationwide, some researches use part of the data for the p-median and Gravity pmedian problems. See Carling etc. (2012) for an example. However the whole data set would have many more problems than data only on a single smaller area. So applying their dealing process directly are not sufficient to fulfil our purpose.
as the same road ID, the same speed limits, etc. It is represented by 1 or several connected polylines. Node: a point on an edge that indicates common vertex of 2 or more edges, or indicates the graphical character of the edge shape. Each node has a 3 dimension coordinate indicating its location and latitude. Residential point: the centre of a square habitant. It represents the residents living in the square. It is the demand point in the pmedian model. Candidate locations: a point represents the possible location for setting up a facility. It is the supply point in the p-median model. The raw data set we get is a shapefile describing the road network. Shapefiles are used in Geographical Information System (GIS) to represent spatial data. The basic entities in the shapefile are edges represented by polylines. The first task is to read all these edges and find out the connection between them. For each edge, the direction and the speed limit are provided. The raw data consists of 34,180,478 nodes. Among them, 19 are not
2.1. Terminologies and basic information of raw data. First we introduce the terminologies used throughout the paper. Edge: a small part of a road having the same parameters such 6
taken into account since they are outside the country. After removing duplicates ones, we end up with 28,061,739 different nodes and 31,406,511 edges. The average length of an edge is 21 meters with maximum 3602 meters. 26206 edges are longer than 200 meters. Only 844 of them are longer than 500 meters. Having such a high number of short edges is due to the fact that the database is used to represent the geographical information associated with a road and short edges. It aims to reflect the road directions precisely. Therefore, long edges represent straight road parts. Speed limits are kept in this step for estimating traveling time later. Table 1 provides the length of roads under different speed limits. As we can see, more than 80% of the Swedish road network has a speed limit of 70 km per hour.
Speed limit (km/hour) 60
Length (km)
(km/h) 20
(total:183 679,808) 24
30
18,958
40
4,535
50
51,104
(km/h) 70
(total:1,726 679,808) 547,866
80
23,962
90
19,581
100
8,137
110
3,977
120
753
2.2 Environmental setting. In our work, all programming has been done in C, using gcc in a Linux Environment (64 bits) on a Desktop computer with 32GB of ram with an Intel I7-3770 CPU. Parsing the shapefile has been done by using the library called shapelib. The code is available upon request. During the process, the UNIX process needs around 6GB of memory to run with the whole Swedish road network. Thus 32GB ram is not mandatory here. The computer requirement is low in our experiment. The process could also be processed in a lower speed CPU with smaller ram and produce results in acceptable time. A personal computer even a laptop is sufficient. The software environment is also quite standard regarding C language under Linux Environment. These
Table 1. Lengths of roads on different speed limits. Speed limit (km/hour) 5
Length (km)
7
settings generalize adaptability of our methods.
the
2-D grid is applied on the map. The grid structure is created, which consists of cells at the size of 500 meters by 500 meters. Each cell contains the list of nodes which belong to it. Thus, we have 1295 cells in the Xcoordinates and 3043 in the Ycoordinates. All the cells have exactly the same rectangular shape.
2.3 Data structure A key to have an efficient program in C is to have good data structures which provide a good compromise between speed and storage. In our case, all the processes from reading the road network to finalizing the reduced graph need to store information about edges and their connections. Since our main goal is to represent a graph, we have created two main structure types in C, one to represent an edge and the other to represent a node. The node structure contains information like the coordinates, ID of its edge(s). The edge information contains its length and converted travelling time.
2.4 Filling missing crossings The first challenge we encounter is the lack of direct crossing information. This is not only the problem in NVDB, but also in many other data bases. The missing crossings are due to the shortage of this kind of geographical storage methods. It identifies roads by the ID of the node. If there are no nodes happen to be on the crossings, no information of crossings will be stored. In our case, we have identified them by ourselves. Obviously, two different edges going through exactly the same node make a crossing at this node. But only a few real crossings can be detected by this approach, because in most cases two points from different edges differ by a few centimetres. The chosen approach is to, for each node, round the coordinates to the nearest multiple of 2 meters
But, we need also to have a data structure by which we can quickly find all nodes in a small given area. This is needed already from the start when we create the graph from the road network. In the process, we start with an blank graph and add nodes into it. For each node we want to add, we need to know if a node exists at or is closed to the given location. For this purpose, we have decided to use a grid to be able to find quickly all nodes in a given part of the country. A 8
(for all three axes). The choice of 2 meters comes from the general size of a one-way street as well as the height needed for a bridge or tunnel. Threshold larger than 2 meters would connect separated edges together, and threshold smaller than 2 meters will fail to identify many real crossings.
data set besides Swedish one, since many data set needs to be updated according to road changes, and it is easy to have some missing information in this process. The difference is that different data sets may vary in the percentage of missing connections, depending on quality of the data set.
After these steps, we end up with 27,922,796 different nodes and a graph since we are able to identify crossing.
Thus, the next step is to find out the different strongly connected components (we have directional edges) in the graph. The strongly connected component is a wellknown problem in graph. Therefore, we are using the Tarjan's algorithm (see Tarjan, 1972), we have identified 3057 components. The mainland is clearly recognized as the one having the highest number of nodes. 559 components have more than 10 inhabitants, 171 more than 100, 111 more than 200 persons.
2.5 Filling missing connectivity The graph we have now is representing well the road network and its structure. But when looking at the whole country we face another challenge: not all nodes are connected to each other. Although most parts of the graph are connected, there exists some parts disjoint to the rest. Those parts could not be neglected. Otherwise the some facilities of the location models will be trapped in these isolated parts. The facilities will be unable to serve residences outside this part even if they are closer to them. The residence in this part can only go to the facility inside no matter how far it is. Thus the solutions will be far away from the optimal ones. This is again a common problem in network
Others components are disconnected from the main part for two main reasons. The first is that in Sweden we have islands without any bridge to the mainland, for example the second biggest component in our case which represents the island of Gotland and its 236,235 inhabitants. It's also the case for a lot of small islands, both in the 9
archipelago and islands along the coasts as well as small ones on lakes. In most cases, communications to the mainland are done by ferry lines. It is of course possible to use directly Euclidian distance to fill in the connection, however, that largely underestimate distances and travel time. Thus we instead of using that, we use the ferry line or other transportation time and then convert it to distance.
between the mainland and the ‘islands’. The time spent on that edge is the actual transportation time regarding ferry lines or other transportations. The distances are calculated correspondingly. We can notice that in the case of Sweden, most of real islands have direct transportations to the mainland. For the isolated parts in the main lands, we simply add straight edges between the 2 closest nodes. We can also mentioned those parts of the networks will not affect so much results since virtual edges added are short (often less than 100m).
The other reason for missing connection is the wrong or inaccurate values in the database, especially regarding the altitude. This can be detected when the distance is shorter than a threshold or when the connection between the components is only possible in one direction. We choose here to have a generic solution instead of using threshold. That is because previous researches on this data set give us no information of the settings for threshold, and inappropriate threshold would lead to over or miss detection.
3. Population 3.1 Population data Another important factor in our location models is the population. The population are required to be geo-coded so that we are able to identify the habitants in each grid and find out the distance from the habitants to the facilities. We are extremely lucky to get the census population data from the Statistics Sweden (SCB) from the year 2012 for people between 20 and 64 years old. The data is well organized and do not need preprocessing.
In order to take into account the people living on these “islands”, we will add virtual edges to the graph representing ferry lines or other means of transportations. A virtual edge will be added between the closest pair of nodes 10
In the population data, the residents of Sweden are represented by a set of residential points. Each point represents the number of people living in a square with the residential point as its centre. The sizes of the square are often 500m by 500m but variations occur between big cities and the sparse populated areas. The residential points are derived after aggregating all the residents in the same squares. It means that all the persons in the same square would be assumed to have the centre of the square as the starting point, and their distances to the same facility are the same. In total we have 188,325 residential points representing 5,411,373 persons. Thus, each residential point represents on average almost 29 persons. The most populated residential point represents 2,302 persons.
population will not fit in the data frame. In that case the demand should be changed to the target people. Moreover, even for locating public facilities, the population sometimes are not exactly equal to the demand, because the demand of public facilities may vary among different types of people and different weight maybe added to different residents. For example, if we deal with the data for locating training centres, we may put smaller weight on babies; but if we deal with locating child clinics, the babies should have the highest weight. The specific methods for getting the demand of location models usually require auxiliary information of the population such as age, income, etc. Here we do not concern this problem and just propose how we connect the population data points to the graph. For the location problems with special requirements should be done in a similar way.
It should be mentioned that here we use the population to represent the demand for the facilities in location-allocation models. So we directly use the original data in our graph. This representation makes sense when it comes to locating public facilities and services like retailing centres and hospitals. But when the facilities are not targeted at the public, the
3.2 Matching residential points with nodes in the network Each residential point needs to be connected to the closest road node in order to get the its distance to the candidate locations. We look for the closest distance between the residential 11
point and the closest point of a segment. We have 497 points representing 1424 persons which are more than one kilometre to the closest road node. (resp. 6031 and 33,779 if for more than 500 meters). Approximations done here are due to the inaccuracy in data provided. Since most of the residential points provided represent the aggregated population on 500 by 500 meters squares, we may have an error of maximum 500 √2⁄2 which is about 353 meters (Euclidian distance) between where a person is living and the residential point used to represent this person. This is a basic assumption for pmedian models, and would lead to errors in the graph.
which is treated as 0 in modelling process while it is not 0 in reality. Source C comes from the residential points at the borders of some grid, which are assigned to a further facility because centre of the grid is close to that one. In our data with the grid defined before, it suggests that for Source A and B errors, the maximum error for a person should be at most 353 meters. That might be important if we locate facilities in a small but high populated area, but it is a small distance error when we consider locating facilities in a national level. Carling etc. (2012) give the mean distance in Dalarna part of Sweden for people travelling to hospitals, and it is around 40 kilometres. Thus in our case, the Source A and B error will not have big influence the quality of our graph, and will not affect the further research results. The only exception is that if researchers are locating a large number of facilities in the country and the resulting average distance to the facility is not large enough to neglect the error as large as 353 meters, then some methods should be applied to improve that. For details of those kinds of methods, see Current and Schilling (1987). As to Source C error, they will mostly
Current and Schilling (1987) point out that there are 3 types of error sources, A, B and C in this kind of aggregating processes. Source A comes from the difference of distances from a real population location inside a grid to a facility outside that grid when we “relocate” the point to the centre of the grid, i.e., the persons in the grid do not actually live in the centre of the square. Source B is the distance between the real population location and the facility in the grid where the facility is located, 12
happen to the grids that have similar distances between 2 or more facilities. In that sense the differences between 2 facilities should be within 353 meters. Consider the level of the whole country, it should not influence the results significantly.
need to keep the information needed to compute distances between people's residence and candidate locations. According Hakimi (1964), in case of the pmedian problems, locating facilities on people’s habitants gives the best solutions. Locating facilities outside these locations and into some segment will only provide sub-optimal solutions. Consequently we have information in our graph which are not used to compute distances and therefore we can remove them. In general, to compute distances in the network, we will use the Dijkstra algorithm and a modified version to compute distance from a point (typically a residential point) to all candidate locations.
4 Graph reduction At this point, we have a strongly connected graph representing our road networks. However, the graph is substantially large due to the scale of our problem and the detailed information of the networks. The problem of this large graph is that it will extensively increase the work in later calculations for finding best locations of facilities. The time and computational efforts consumed would be enormously large and beyond acceptable. This problem exists in almost all the large data sets including Swedish ones. However, reducing graph to a smaller scale would lose information, thus it is challenging to keep the useful information as much as possible during the graph reduction.
In this part, we will explain which information can be removed, how to remove them and the benefits. 4.1 Removing dead ends. In our graph, we have nodes with degree 1, i.e. nodes which are connected to only one neighbour node. Such nodes can be called dead-end nodes. If these nodes are associated with some residents, we should obviously keep them in our system. If not, we can remove this node from our graph since it will never be
Since the goal of our work is to minimize distances between where people are living and facilities are located, we only 13
used to compute the distance matrix. Obviously, the neighbour of a removed node can end up to be a new dead end. Therefore, in our algorithm, we analyse all nodes and if a node is a dead end, we recursively analyse and remove the new dead end until the graph becomes stable.
thus are denoted as degree 2. Our algorithm works as follow: For each node, we check if this node represents some population's residence. If not, we check if this node is not a crossing by checking if this node connect exactly 2 different edges. If so, we will analyse the direction of edges and remove them to disconnect this node from the graph. Then, we add in our graph a new edge between the two neighbours. The distance and traveling time of this new edge is the corresponding sum of the two removed edges. We repeat this process until not further reduction is possible.
By applying this, we can remove 9,873,764 nodes. They account for more than 35% of the whole. This is a huge reduction to our graph. Also by doing this we barely lose any information in the population or road networks. In practice, these deleted nodes mainly represent small roads on the country side, where people move away but the road information is still stored in the data base.
The speed limit information here are affected after this procedure since the two aggregated edges may have different speed limits. Therefore we drop them and keep the travelling time information. At the stage, we can remove 10,347,191 nodes.
4.2 Removing useless node of degree 2. Many nodes in the graph do not represent the structure of the road but indicate the road shape (curves, hills, etc.). We are not interested in such information but only the accurate distance by the networks between nodes as well as the traveling time. Thus these nodes can be reduced and we can get a smaller graph without losing any accuracy in distance and time. Nodes like this usually are connected to 2 neighbours,
By applying these two graph reductions, we are able to remove 20,220,955 nodes (72% of the nodes) and now, the graph has only 7,701,841 nodes. We keep all useful information regarding travelling distance and time.
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time, the closest node which is unvisited is chosen. A way to optimized computations, aside using well the C language, is to have a better data-structure for the queue used by the algorithm. The Fibonacci heap is therefore used for this purpose. We add the Fibonacci heap into the searching process of Dijkstra algorithm and greatly shorten the computing time to 1 day. For later optimization, we can even add an early stopping criterion to this algorithm and stop once distances for all nodes within a certain distance have been computed.
5. Distance matrix and optimized Dijkstra method. In this stage we derive the distance matrix with the previous graph. The first experiments we have done in this case are to compute the distance matrix between the 188,325 residential points and a set of candidate locations. As a first trial, we have established 1938 candidate locations throughout Sweden. They are related to the closest nodes in the graph. The graph itself provides us with distance between some pairs of nodes, but not all of them. The route from a node to a not adjacent one has many choices and the distance can have many true values. We aim to find the shortest. In the first part of the experiment, we were using the classical Dijkstra algorithm (Dijkstra, 1959) with a nonoptimized queue. Using such methods leads to an average of 12 days of computations.
After the matrix was derived, we make several test experiments to detect the correctness of the matrix. We pick out a few pairs of nodes and compare our distance with online distance data base. Our results are quite close to the online distances. This shows we have derived an accurate distance matrix. 6. Conclusions.
The Dijkstra algorithm is an efficient algorithm and a run of this algorithm is computing distances from (or to) a points to (or from) all others. This algorithm works by starting from the source nodes and exploring in an iterative nodes around. Each
To deal with real data leads to facing different problems which need to be checked carefully. Wrong or inaccurate data is often the reason of incorrect analysis. After deriving the clean and accurate data, another problem is 15
to reconstruct information we need based on it. In our example, we have mainly rebuilt the structure of the road network by identifying crossing based on a 2meters approximation. The approach we proposed has been used both on whole Sweden and as prototype on the province of Dalarna and in both cases results are encouraging. The missing information of the nodes has been added by linking it to the nearest neighbour and very little error generated by doing so. It should be noticed that for the nodes in Gotland, there are no bridge or tunnel connect them to the main land, and we use virtual link based on the ship speed and time required to get to the main land. In such procedure we actually underestimate the distance between the main land and Gotland in the sense that people need some time for transferring and waiting between ships and cars. However we do not have further information on the (average) time for the passengers. The graph would make more sense if we have those time data and add that to the virtual link.
not able to be handled efficiently by computers. Therefore the second goal of our approach is to reduce as maximum as possible the number of nodes. It should be mentioned that it is possible to skip the distance matrix and go directly to searching for the shortest distance between 2 nodes and just store the edges in the graph. However in our trial, that approach takes very long time for computer to process. Thus we are not quite flexible to choose good heuristic methods; neither can we apply the algorithms in an efficient way. By reducing the graph, we reduce the computational time by a huge factor, leading to more efficient algorithm and better results. In our data file, we have the speed limit for different segment of the road, therefore that factor are taken account in our process. We get a distance matrix and a time distance matrix. But the two dealing processes have quite small difference, thus we combine them together to give our process without separation. It would also be interesting to see if there are some difference in the solutions to location with the 2 matrixes.
Our approach is built to be as general as possible. The graph after identifying crossing and connecting the whole network is 16
Han, M., Håkansson, J., Rebreyend, P. (2013). How do different densities in a network affect the optimal location of service centers? Working papers in transport, tourism, information technology and microdata analysis, ISSN 1650-5581; 2013:15
Reference Beasley, J. E. (1993). Lagrangean heuristics for location problems. European Journal of Operational Research, 65(3), 383399. Burkard, R. E., Karisch, S., & Rendl, F. (1991). QAPLIB-A quadratic assignment problem library. European Journal of Operational Research, 55(1), 115119.
Huff, D.L., (1964). Defining and estimating a trade area. Journal of Marketing, 28, 34-38. Huff, D.L., (1966). A programmed solution for approximating an optimum retail location, Land Economics, 42, 293-303.
Carling, K., Han, M., Håkansson, J., (2012). Does Euclidean distance work well when the p-median model is applied in rural areas? Annals of Operations Research, 201:1, 83-97.
Kariv, O., & Hakimi, S.L., (1979a), An algorithm approach to network location problems. Part 2: The pmedian. SIAM Journal of Applied Mathematics, 37, 539-560.
Current, J., D. Schilling. (1987). Elimination of Source A and B errors in p-median location problems. Geographical Analysis 19, 95-110.
Kariv, O., Hakimi, S. L. (1979b). An algorithmic approach to network location problems. I: The p-centers. SIAM Journal on Applied Mathematics, 37(3), 513538.
Dijkstra, E.W., (1959). A note on two problems in connexion with graphs. Numerische Mathematik, 1, 269–271.
Laporte, G. (1992). The vehicle routing problem: An overview of exact and approximate algorithms. European Journal of Operational Research, 59(3), 345358.
Hakimi, S.L., (1964). Optimum locations of switching centers and the absolute centers and medians of a graph, Operations Research, 12:3, 450-459.
Lundgren, M-L (2000), The Swedish National Road Database 17
– Collaboration Enhances Quality, Proceedings of the Seventh World Congress on Intelligent Transport Systems, 6-9 November, Turin, Italy. Mladenović, N., Brimberg,J., Hansen, P., Moreno-Pérez J.A. (2007). The p-median problem: A survey of metaheuristic approaches. European Journal of Operational Research, 179 (3), pp. 927–939 Reinelt, G. (1991). TSPLIB—A traveling salesman problem library. ORSA journal on computing, 3(4), 376-384. Tarjan, R. E. (1972), "Depth-first search and linear graph algorithms", SIAM Journal on Computing 1 (2): 146–160, doi:10.1137/0201010
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PAPER V
Measuring transport related CO2 emissions induced by online and brick-and-mortar retailing Kenneth Carling, Mengjie Han, Johan Håkansson, Xiangli Meng, Niklas Rudholm Abstract We develop a method for empirically measuring the difference in transport related carbon footprint between traditional and online retailing (“etailing”) from entry point to a geographical area to consumer residence. The method only requires data on the locations of brick-and-mortar stores, online delivery points, and residences of the region’s population, and on the goods transportation networks in the studied region. Such data are readily available in most countries. The method has been evaluated using data from the Dalecarlia region in Sweden, and is shown to be robust to all assumptions made. In our empirical example, the results indicate that the average distance from consumer residence to a brick-and-mortar retailer is 48.54 km in the studied region, while the average distance to an online delivery point is 6.7 km. The results also indicate that e-tailing increases the average distance traveled from the regional entry point to the delivery point from 47.15 km for a brick-and-mortar store to 122.75 km for the online delivery points. However, as professional carriers transport the products in bulk to stores or online delivery points, which is more efficient than consumers’ transporting the products to their residences, the results indicate that consumers switching from traditional to e-tailing on average reduce their transport CO2 footprints by 84% when buying standard consumer electronics products. Keywords: E-tailing; Spatial distribution of firms and consumers; pmedian model; Emission measurement; Emission reduction JEL codes: D22, L13, L81, R12
1. Introduction
Kenneth Carling is a professor in Statistics, Mengjie Han is a PhD in Microdata Analysis, Johan Håkansson is a professor in Human Geography, Xiangli Meng is a PhD student in Microdata Analysis, and Niklas Rudholm is professor in Economics at the School of Technology and Business Studies, Dalarna University, SE-791 88 Falun, Sweden. Niklas Rudholm also works at HUI Research, Stockholm, Sweden. Corresponding author: Niklas Rudholm, e-mail:[email protected], phone: +46-70-6254627.
Environmental considerations are at the center of the agenda for politicians in many countries and much research is devoted to meet the challenges of climate change, sustainability, and related environmental issues. The environmental impact of retailing on CO2 emissions should not be underestimated. In Great Britain, the average consumer over 16 years old made 219 shopping trips and travelled a total of 926 miles for shopping in 2006 (DfT 2006). Considering that most of these trips were reportedly made by car, and that transport vehicle miles travelled is the main variable determining CO2 emissions, ways to reduce car use for shopping are sought (Cullinane 2009). In a Swedish setting, Carling et al. (2013a) studied the environmental optimality of retail locations, finding that current retail store locations were suboptimal. The suboptimal location of retailers generated on average 22% more CO2 emissions than did a case in which they were optimally located. Furthermore, in a related study, Carling et al. (2013b) used GPS data to track 250 Swedish consumers for two months. In that study, the authors compared downtown, edge-of-town, and out-of-town shopping in terms of the CO2 emissions caused by shopping trips. They concluded that downtown and edge-of-town shopping were comparable in transport CO2 emissions, but that out-of-town shopping produced approximately 60% more emissions from transportation. As traditional brick-and-mortar shopping entails substantial environmental impact, it would be pertinent to compare the CO2 emissions from transportation induced by brick-and-mortar shopping with those of online shopping that needs a physical distribution. Few recent empirical studies (e.g., Edwards et al. 2010; Wiese et al. 2012) analyze the impact of online shopping on the environment. Wiese et al. (2012) studied the CO2 effects of online versus brick-and-mortar shopping for clothing in Germany; their main finding is that, although online shopping usually induces lower CO2 emissions, the opposite is true when the distances involved are moderate. In a study of the carbon footprint of the “last-mile” deliveries of conventionally versus online-purchased goods, Edwards et al. (2010) found that neither home delivery of online purchases nor conventional shopping trips had an absolute CO2 advantage, though home delivery of 2
online-bought goods likely entailed lower CO2 emissions unless the conventional shopping trips were made by bus. In this paper, we address the issue of emissions along the entire supply chain from entry point to the studied region to consumer residence for all major suppliers of the product under study. Our study aims primarily to develop an empirical method for measuring the transportation CO2 footprint of brick-and-mortar versus e-tailing that call for a physical distribution from entry point to a region or country to consumer residence. 1 This method will then be used to calculate and compare the environmental impact of buying a standard electronics product online with buying the same product in a brick-and-mortar store in the Dalecarlia region in Sweden. In addition, the actual locations of brickand-mortar stores and online delivery points in the region will be compared with the locations that would minimize CO2 emissions. Our paper contributes to the literature in the following way. First, contrary to previous studies, the method developed makes it possible study all transport related emissions from entry into a region of interest and to the consumer residence in it. Previous studies have either been analyzing the transport related emissions within one retail chain (Wiese et al. 2012) or been focusing on the carbon footprint of the “last-mile” deliveries of conventionally versus online-purchased products (Edwards et al. 2010). Second, our method allow for simulations of how different locations of both brick-and-mortar stores and online delivery points, as well as different logistic solutions for the distribution of the goods, affects emissions. As such, our method could also be used when constructing environmentally friendly retail networks that minimize consumer travel. Third, the method also allow for simulations of changes in how attractive a consumer finds a brick-and-mortar store relative to online shopping, and the consumer’s willingness to travel to shop for the product under study.
1
Note also that this implies that the development of theory or a conceptual framework is outside the scope of this paper. The interested reader is referred to Cullinane (2009) for the outline of a conceptual framework regarding how e-tailing affects the environment.
3
We will focus on consumer electronics, as these consumer products constitute the largest e-tailing category in Sweden (HUI Research 2014), presumably leading the way to online shopping for other consumer products in the future. Consumer electronics are in the vast majority of cases imported into Sweden 2 , and pre-shipping via an entry port is required before a product reaches a consumer’s residence, regardless of whether the product is bought online or in a store. Consequently, the product’s route on the Swedish transportation network to the consumer’s residence can be identified. In brick-and-mortar shopping, the route extends from the entry port via the store to the consumer’s residence, while in online shopping, it extends from the entry port via the Swedish Post distribution points to the residence. Part of the route is covered by professional carriers, such as Swedish Post, and other parts of the route are covered by the consumer. We focus on the CO2 emissions of the complete route from regional entry point to consumer residence. The study concerns the Dalecarlia region in central Sweden containing approximately 277,000 consumers, whose residences are geo-coded. The region contains seven brick-and-mortar consumer electronic stores and 71 delivery points for online purchases. Consumers reach the stores or delivery points via a road network totaling 39,500 km. Mountains in the west and north of the region restrict the number of gateways into the region to three from the south and east, limiting the routing choices of professional carriers. The region is representative of Sweden as a whole in terms of the use of e-tailing and shares many geographical, economic, and demographic characteristics with, for example, Vermont in the USA. This paper is organized as follows. Section 2 thoroughly describes online shopping in Sweden in 2012 and 2013. Section 3 gives details of the data and the heuristic algorithm used in finding optimal locations. Section 4 presents the empirical analysis, which starts by calculating the 2
There are a few producers of consumer electronics that still manufacture their products in Sweden, but in most cases R&D and design are made in Sweden while production is located in low wage countries like China. Also, consumer electronics is the industry that has had the most rapid outsourcing of production in the Swedish economy, with textile manufacturing and rubber manufacturing as the only industries with nearly as much of the production being outsourced to low wage countries (Lennartsson and Lindholm, 2004).
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environmental damage induced by buying a standard consumer electronics product online versus in a local brick-and-mortar store. The results are also aggregated to the whole of Sweden for e-tailing in general as well as for consumer electronics products. Section 5 presents a sensitivity analysis incorporating all assumptions imposed, to arrive at the results presented in section 4. Finally, section 6 concludes the paper.
2. Online and brick-and-mortar retailing of consumer electronics in Sweden In this section, we start by describing e-tailing in Sweden for consumer electronics and in general. We then describe the delivery system from etailers to their consumers. Finally, we discuss the brick-and-mortar retailing of consumer electronics in Sweden and the Dalecarlia region. First, e-tailing is dependent on Internet access, possessed by approximately 90% of Swedish households. In addition, most workplaces have Internet access, making e-tailing available to the vast majority of the Swedish population. In the last quarter of 2012, 73% of a random sample of Swedish consumers reported having bought consumer products online in the previous three months, and 63% of the sample reported that they would buy products online in the coming three months (HUI Research, 2014).3 Moreover, 90% of respondents reported having shopped online at some time, the main cited reasons for online shopping being that it is simple, cheap, and increases the consumer’s product selection. Most online consumers use their desktop or laptop computer for online shopping, but one in five reported having used a smart phone or tablet for online shopping in 2012 (HUI Research 2014).
3
The information about Swedish online shopping comes from e-barometern 2012 and 2013. e-barometern is a yearly report on Swedish online shopping behavior produced by HUI Research (a Swedish research and consultancy firm working mainly in the retail trade industry), Posten AB (the Swedish Post), and Svensk Distanshandel (a federation of commercial enterprises in the online retail industry). The questions asked differ somewhat between years, so some statistics are from e-barometern 2012 reporting statistics for 2011 (HUI Research 2013) and others are from e-barometern 2013 reporting statistics for 2012 (HUI Research 2014).
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As stated above, we are studying the online and brick-and-mortar markets for consumer electronics. We chose electronics as the studied market because it is the largest e-tailing category in Sweden with sales of SEK 8.8 billion in 2013 (HUI Research 2014). Clothing is the next largest category with SEK 7.2 billion in sales followed by books (SEK 3.3 billion), furniture (SEK 1.2 billion), and sporting goods (SEK 1.0 billion). The fastest growing categories are sporting goods, furniture, and electronics, with annual growth rates of 28%, 19%, and 15%, respectively, in 2013. A sample of consumers was asked in a survey what products, if any, they bought online in 2013: 44% reported having bought books online, 40% clothing, 25% computers and computer accessories, and 21% other home electronics products (HUI Research 2014). There are some gender differences in e-tailing, books being the main category for women and computers and computer accessories for men (HUI Research 2013). Though the sales growth rate is impressive for sporting goods, this category is starting at a low level. As consumer electronics will continue to be one of the most important e-tailing categories for the foreseeable future, it was chosen for the present analysis.4Swedish Post delivers most e-tail packages in rural areas in northern Sweden, where over ten packages per year per household are delivered in many northern municipalities. 5 The three municipalities with the most packages delivered are Storuman, Jokkmokk, and Gällevare, all located in northern Sweden and all averaging 11.4–12.0 packages delivered per year per household. In contrast, in most municipalities in southern Sweden, particularly the three main cities, fewer than seven packages are delivered per year per household. In the municipalities of Malmö, Gothenburg, and Stockholm, 5.9–6.1 packages are delivered per household and year. The Dalecarlia region lies between the extremes of Sweden with seven to nine packages 4
This paper examines the environmental impact of transportation related to the retailing of consumer electronics, not the import or manufacturing of such products. It should, however, be noted that approximately 80% of the environmental impact of consumer electronics comes from manufacturing rather than transporting them (Weber et al. 2007). 5 The Post is note the sole provider of this type of services in Sweden, firms as DB Schenker and DHL are also active in the market. However, the Post is the market leader in the Swedish market, and also the only provider of the type of statistics reported in the section above.
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delivered per household and year by Swedish Post, with two exceptions: in the municipalities of Malung and Sälen, in the remote north of the region, over ten packages are delivered per household and year, while in Borlänge, in the center of the region and with a well-developed retail trade, fewer than 7 packages are delivered per household and year (HUI Research 2013). As such, the Dalecarlia region as a whole can be considered representative of most of Sweden, except, perhaps, for the major cities and the remote far north. In Sweden, the consumer is offered a choice of delivery points for picking up online purchases. Swedish Post, handling most e-tail packages, offers consumers a list of delivery points, the nearest the consumer’s residence being the suggested primary alternative. The opening hours of these outlets are usually 9.00–20.00. As pointed out by Cairns (2005), delivering products to intermediate points with a longer pickup time window for the consumer permits more efficient delivery, possibly reducing peak-period congestion. The vast majority (85–90%) of surveyed consumers chose to pick up products at the proposed nearest outlet, and the consumer’s preferred pickup time at the outlet was Monday to Friday after 18.00 (HUI Research 2014). Fifty percent of online shoppers reported having ever returned an online purchase, and 77% reported the experience of doing this as good or very good (HUI Research 2014). The return process usually entailed the consumer returning the package to the outlet where it was picked up, and the only product category for which consumers mention a good return system as important for their purchase decision is clothes (HUI Research 2014). It should also be noted that Swedish e-tailers are not overly exposed to foreign competition, though increased competition from abroad is expected in the future. However, 40% of surveyed consumers reported never having bought anything from a foreign e-tailer, and an additional 40% reported having bought products from foreign retailers only once per or less often per year (HUI Research 2014). Brick-and-mortar consumer electronics retailing in Sweden has a total annual turnover of approximately SEK 35 billion, but the sector’s profitability is not that impressive. In summer 2011, the Swedish brickand-mortar electronics retail chain Onoff filed for bankruptcy, and its stores were taken over by its competitor Expert. However, less than a year 7
later, Expert also filed for bankruptcy, meaning that two large, nationwide retail chains in consumer electronics have exited the market. In addition, several other chains are reporting weak profits. Meanwhile, Elgiganten, which has both brick-and-mortar stores and e-tailing for consumer electronics, is currently the best performing chain in Sweden. It is therefore conceivable that brick-and-mortar electronics retailers may leave certain local geographic markets in Sweden due to competition from etailers, increasing the potential environmental benefits of e-tailing. In 2012, there were seven brick-and mortar consumer electronics stores in Dalecarlia (see Fig. 1a), all parts of consumer electronics retail chains. Two of the chains (Elgiganten and Euronics) had three stores each, while one chain (SIBA) had only one store. Most of the stores are located in Dalecarlia’s major towns. The largest town, Borlänge, is the only town with two stores. One chain, Euronics, has a somewhat different localization pattern than do the other two chains, with two of its three stores located in smaller settlements (i.e., Malung and Svärdsjö) in the region.
3. Data and method In this paper, we identify the shortest route and transport mode the product follows on its way from regional entry port, via the retailer, to consumer residence, and calculate the emissions induced by this transport. To do so, we draw on data from the Dalecarlia region in central Sweden, and impose several identifying assumptions (labeled using Roman numerals) that are scrutinized by means of sensitivity analysis in section 5. Note that the method developed here could be used to measure emissions in any setting where the entry points into the studied region, emissions per kilometer for the transport method used, location of the final destination, and available transport network are known. The method may therefore also have important uses outside the retail sector. There are several reasons for choosing the Dalecarlia region when evaluating the measurement method developed. To perform a thorough investigation of how the various assumptions and data requirements affect the model output, we need access to data as detailed as possible regarding the location of people’s residences, the region’s road network, and all other necessary measurements. 8
First, in Dalecarlia, the population’s residences are geo-coded in 250 × 250-m squares, meaning that the actual residential location may err by 175 m at most.6 Fig. 2a shows the geographical distribution of the population: considering that consumer electronics is a broad category of products appealing to almost everyone, it is reasonable to regard anyone in the population irrespective of disposable income, gender, and age as a potential buyer of such products. 7 The population totals 277,000 people whose residency is represented by 15,729 squares whose center coordinates are known (census data from Statistics Sweden as of 2002). The population density is high in the southeast part of the region, along the two main rivers, and around a lake in the center of the region. The western and the northern parts of the region are sparsely populated. Second, previous work has carefully examined the road network in the region, so potential pitfalls encountered in working with these large databases are known (Carling, Han, and Håkansson 2012; Han, Håkansson, and Rebreyend 2013; Carling, Han, Håkansson, and Rebreyend 2014). Fig. 1b depicts the road network of Dalecarlia (actually the national roads only, as showing the many local streets and private roads in the dataset would clutter the map). The network was constructed using the national road database (NVDB), a digital database representing the Swedish road network and maintained by the National Transport Administration (Trafikverket). The database contains national roads, local streets, and private roads (both government subsidized and unsubsidized); the version used here was extracted in 2010, representing the network of that time. Furthermore, attributes of the road segments, such as their position, length, and nominal speed limits, are also given (for details, see Han, Håkansson, and Rebreyend 2013). A very realistic travel distance for a potential consumer can be derived by calculating the distance along the road system from the home to any point where either a brick-and-mortar store or a Swedish Post delivery point is located.
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In a 250 × 250-m square, the longest distance from the center point to the edge is √1252 + 1252 = 175. 7 We are using individuals rather than households as the unit of analysis since this is the type of data we have access to from Statistics Sweden. However, the example product in our paper is a computer, and we believe that in most cases such a product is today bought and used by an individual rather than shared within a household.
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Fig. 1 about here. Third, an in-depth study of consumer shopping trip behavior was conducted in Borlänge, a centrally located city in the region (Carling, Håkansson, and Jia, 2013b; Jia, Carling, and Håkansson 2013). Some 250 volunteer car owners were tracked for two months using GPS. Typical travel behavior for trips to a store selling durable goods was to drive the shortest route from the home to store, implying the lowest possible CO2 emissions. Consequently, we approximated shopping-related trips using the shortest route in the following analysis. Fourth, there are only three gateways8 into the region, meaning that it is relatively straightforward to obtain information about how consumer electronics products arrive there and are then distributed to consumer residences, irrespective of whether the purchase is made online or at a brick-and-mortar store. 9 Fig. 1a also shows the current location of the seven existing brick-and-mortar consumer electronic stores and the 71 delivery points for products purchased online. Altogether, the region’s road network is represented by 1,964,801 segments joined in about 1.5 million nodes. This means that a consumer can follow a myriad of potential routes to get to the store or delivery point. Based on previous work, we stipulate that the consumer takes the shortest route (Jia et al. 2013). However, identifying the shortest route given this vast number of alternatives is challenging in itself. We follow the convention of using the algorithm proposed by Dijkstra (1959) to find the 8
Mountains in the west and north of the region limit the number of gateways into the region to three from the south and east, limiting the routing choices of professional carriers (cf. Figure 1b). Although there are two airports in the region, neither of them is used or is suitable for freight shipments. The brick-and-mortar retailers have confirmed that all their shipments are by truck. However, Swedish Post might occasionally use train for partial shipments of the products. In such cases, our approach overestimates the CO 2 emissions induced by online shopping, as we assume truck transport. 9 The important point for us is to identify the point where the distribution network starts to differ between online and brick-and-mortar stores. This would also be the case if there were local production in the studied region, so this would not in principle affect the method developed, except that in such cases we would have to identify where in the Dalecarlia region the distribution network from producer to online or brick-andmortar retailers started to differ.
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shortest distance between all node pairs in the road system, an effort that is very time-consuming but done only once. The algorithm, in its naïve form, specifies a starting node, identifies all its adjacent nodes. Thereafter, it seeks second-order nodes adjacent to the starting node and identifies the distance to them via the adjacent nodes. Then the third order nodes adjacent to the starting node are identified and the distance to the staring node via the nodes adjacent to the staring node and the second order nodes adjacent to the starting node are identified. This process continues until all node pairs of interest have been assigned a distance. In other words, the algorithm starts with nearby nodes and calculates stepwise the distance between nodes farther and farther apart. Finally, a (non-symmetric) matrix of road distance between all node pairs is obtained in which the rows of the matrix refer to the nodes of the residences and the columns to the nodes of the stores or delivery points. Zhan and Noon (1998) confirmed that the algorithm, used successfully, identifies the shortest route in a network. Though road distance is not the same as CO2 emissions, we nevertheless assume a perfect correlation between the two. We do this despite being aware that other factors, such as speed, time, acceleration, deceleration, road and weather conditions, and driver and vehicle types, are being ignored. Stead (1999), based on data from the 1989–1991 National Travel Survey, suggested using road distance as a proxy for vehicle emissions because of the ease of collecting and computing it. Previous work in Dalecarlia indicates that, while intersections and arterial roads imply higher emissions, emissions crucially depend on road distance (Carling, Håkansson, and Jia, 2013b; Jia, Carling, and Håkansson 2013). It is an approximation to replace CO2 emissions with road distance, though it is a fairly good one, as we can demonstrate in the sensitivity analysis presented in section 5. To calculate the CO2 emissions we assume the following. First, the consumer drives a gasoline-powered Toyota Avensis 1.8 with CO2 emissions of 0.15 kg per km 10 , making the trip solely to pick up a 10
This emission rate is according to the EU norm for testing car emissions and refers to driving on a mixture of urban and non-urban roads. In 2012, newly registered cars in Sweden emitted 0.14 kg per km of CO2, whereas the existing car fleet in Sweden emitted somewhat more CO2.
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consumer electronics product (e.g., a computer or a small stereo) and return to his or her residence. The product is sold in a 0.3 × 0.6 × 0.6 m3 box weighing up till 10 kg. The product is transported by a professional carrier using a Scania truck and a trailer with a standard loading volume of 100 m3 respecting the Swedish restriction of 24 tons of load per vehicle. The Scania truck runs on diesel, emits 1.08 kg per km of CO2 (according to the producer; see www.scania.com), and is loaded to 60% of its capacity with identical products, such that the consumer’s product constitutes one of 600 in the load and is responsible for approximately 0.002 kg per km of CO2. Emissions when on- and offloading the product and when moving it indoors are neglected, and emissions from transporting the product to the region’s boundary from the manufacturer are assumed to be the same irrespective of its being purchased online or in a store and are thus set to zero in the calculations. Moreover, we stipulate that each person in Dalecarlia is equally likely to purchase the product, i.e., that there is no geographical variation in the likelihood of purchase. The online-purchased products are assumed to first arrive at the region’s six Swedish Post distribution centers via the shortest route upon entering the region through gateway B from Stockholm where parcels are sorted (see Fig. 1b). They are then transported to the 71 delivery points, again via the shortest routes.11 For a product purchased in a store, we assume that the product arrives at the store from the boundary of the region via the shortest route. The companies were not particularly willing to disclose their logistics solutions. We do know that these firms only have one distribution center within Sweden, and that this is not located within the region under study. We therefore assume that the product enters through one of the three gateways such that the gateway implies the shortest distance to the store (see Fig. 1b). This assumption is conservative, as it might underestimate the product’s actual transporting distance to the store if the retailer’s logistics solution does not use the shortest route. 11
This assumption minimizes emissions, it could be that some other type of logistics distribution system such as a spoke and hub system is used. We have not been able to get any information of the precise nature of the logistics system used, however, reloading is costly and it thus seems unlikely that, for example, a spoke and hub system are at work within the region.
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The current locations of stores and delivery points, shown in Fig. 1a, are presumably suboptimal and potentially subject to reconsideration. We therefore use the p-median model to find the best possible store locations from an environmental perspective. Hakimi (1964) developed the pmedian model to find the optimal location of switching centers in a network, which is a discrete location problem on a map with spatially distributed demand points (Hakimi 1965; Daskin 1995). In the p-median model, the demand points are assumed to be assigned to the nearest facilities. The distance is weighted by the mass of the demand points, in this case, the number of residents at a point. The goal is to locate p centers or facilities such that the average individual distance is minimized. Consequently, it is impossible to find more environmentally friendly retail outlet locations than the solution to the p-median model under our assumptions that consumers take the shortest routes and choose the nearest stores or online delivery points and that road distance and CO2 emissions are perfectly correlated. The p-median problem is non-deterministic polynomial-time (NP) hard and, unless the combinatorial problem is modest, it is impossible to find an exact solution. Instead, an approximate solution is sought using a heuristic algorithm. In this paper, we use simulated annealing (SA) because it generally provides good solutions to p-median problems, is flexible enough to be controlled, and has worked well on other p-median problems in similar contexts (Chiyoshi and Galvao 2000). Han et al. (2013) give details of SA implementation. The algorithm starts with a configuration of p facilities picked at random. One facility is picked at random and is examined to determine whether the average distance is reduced by moving the facility to any of its neighboring nodes. If so, this configuration is accepted as an improvement and the previous step of randomly selecting a facility and searching its neighborhood is repeated. If not, the original configuration is kept with a preset probability and a poorer configuration is selected with one minus this probability. This gradual movement away from the original configuration continues until the average distance is near the minimum. We use the Carling and Meng (2014) approach to obtain confidence intervals for the minimum distance, to ensure that we are only meters away from the best possible solution. 13
For clarity, we end this section by gathering together all identifying assumptions discussed above and on which the results build. Three assumptions are related to the measurement method as such: (i) the road distance and CO2 emissions are perfectly correlated; (ii) the number of brick-and-mortar stores is fixed during the studied period; and (iii) the consumer population is stable during the studied period. These three assumptions, along with knowledge of the locations of brick-and-mortar stores, online delivery points, and residences of the population of the region, and of the transportation networks used to transport goods in the studied region, are all the methodological assumptions and data required to use the model. However, the model also requires assumptions about human behavior, which can of course be altered in infinite ways. In this paper, we will test robustness to seven additional assumptions regarding consumer behavior and three additional assumptions regarding producer behavior. There are several additional assumptions about consumer behavior. (iv) Online-purchased products are picked up at the delivery point nearest the consumer’s residence, as confirmed by the surveys (HUI Research 2013 and 2014) cited in section 2. (v) Consumers in Dalecarlia take the shortest route from their residence to the brick-and-mortar store or online delivery point, as suggested by a previous study (Jia et al. 2013). (vi) Consumers always pick up the product by car and drive a car emitting 0.15 kg per km. According to the National Transport Administration, new cars in Sweden emitted on average 0.138 kg per km in 2012. Although precise figures are lacking, the older fleet of cars would typically have higher emissions, making 0.138 kg per km an underestimation of the overall average emissions. (vii) The region’s consumers are equally likely to purchase a given product. (viii) The consumers either purchase the product on visiting a brick-and-mortar store or purchase it online. (ix) The consumers are indifferent to whether they shop in a store or online. (x) The consumers shopping at a brick-and-mortar store choose the nearest one. There are three assumptions about producer behavior. (xi) The truck is loaded to 60% of its capacity. (xii) Online-purchased products arrive at the delivery point by first going via the shortest route to one of the six distribution centers and then via the shortest route from the distribution center to the delivery point. This is essentially how Swedish Post 14
described their logistics solution to us, although they were unwilling to go into detail. (xiii) A product destined for a brick-and-mortar outlet arrives at the store via the shortest route from the nearest of the three gateways into the region. The sensitivity to all these assumptions will be scrutinized in section 5.
4. Empirical analysis of CO2 emissions induced by consumers shopping To set the scene, consider a stereotypical consumer electronics product, such as a desktop computer or small stereo. 12 Such a physical product needs to be transported to the consumer’s residence, typically by car, inducing marginal freight trips for delivery to the consumer and causing additional environmental damage. On the other hand, it is a marginal product for delivery by the professional carrier, as its volume and weight are marginal to standard trucks. Of course, some consumer electronics products (e.g., books and DVDs) are tiny and easily transported by consumers walking, biking, or riding a bus from the store. However, in Sweden these products would also typically be delivered by ordinary mail to the consumer’s residential mailbox. 13 Hence, we believe that the environmental impact of the transport of these tiny products can be abstracted from. Also note that when it comes to high value consumer electronics products as a computer or a stereo, the consumer is likely to choose to pick up the product at a delivery point where the likelihood of theft is negligible.
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It should be noted that some low-end TV sets and other electronics products have at times been sold at some of the largest retail food outlets (e.g., ICA Maxi and Coop Forum) in Dalarna, but due to the low profit margins on consumer electronics the sections containing these type of products have in most cases decreased in size or been removed from these stores. The sales of such products are thus considered to be limited and are excluded from our analysis, but it should be noted that if there are additional brick-and-mortar stores selling the type of consumer electronics products being considered in this paper, this would reduce the difference in emissions between online and brick-and-mortar stores. 13 Of course, at some point online retailing could expand to the point at which Swedish Post would be required to add additional delivery trips. Although online retailing is expanding, the analysis of such effects is outside the scope of the present paper.
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Table 1 shows the consumer’s travel distance on the road network from home to the nearest store and back. The average distance to the 7 current brick-and-mortar stores in Dalecarlia is 48.5 km, with considerable variation between consumers. For 5% of consumers, the nearest store is within walking distance (under 2.6 km), while for another 5%, the nearest store is over 162 km from home. Obviously, the postal delivery points are much more conveniently located, approximately 25% of consumers having to travel under 2.1 km to the nearest delivery point, with an average of 6.7 km for consumers overall. Assuming that CO2 emissions approximately coincide with distance travelled, the average consumer induces only 14% CO2 emissions when buying the product online rather than at a store. Table 1 also shows the hypothetical situation when stores and delivery points are optimally located according to the p-median model. A first observation is that the postal delivery points are currently nearly optimally located, as the mean distance differs by under 0.7 km between the current and hypothetical locations. Note also that, comparing the current with the optimal online delivery points, the travel distance to the current locations is less than the optimal one for consumers living in urban areas in the region, while the opposite is true for consumers in rural areas. The brick-and-mortar stores could, from the environmental and consumer perspectives, be better located. Optimally locating the 7 stores would reduce the average consumer’s trip from 48.5 to 28.8 km a 41% reduction. Optimally locating the brick-and-mortar stores would generally most benefit the quartile of consumers today living farthest from a store, but optimal locations would reduce travel distance for all percentiles. Table 1 about here. The consumer’s trip to pick up the product represents a substantial part of the transport effort; the other part is transporting the product to the pickup point, whether store or postal delivery point. Table 2 shows the distance the product travels from entry into the region to the store or delivery point. The values in the table are calculated assuming travel via the shortest route and derived assigning equal weight to all outlets. The average distance from regional boundary to store is 47 km, whereas the average distance is 123 km to the delivery point. Three unsurprising things can be noted from 16
Table 2. First, products purchased online must travel farther to the pickup point than do ones sold in stores. Second, professional carriers usually carry products farther than do consumers (cf. Table 1). Third, optimally locating stores from the consumer perspective would mean longer-distance transport to the stores for the professional carriers (averaging 62 km). Table 2 about here. However, a consumer carrying a product in a car induces much higher CO2 emissions per travelled kilometer than does a professional carrier bringing many product units on the same trip. Hence, the values in Tables 1 and 2 cannot simply be added. Following Wiese et al. (2012), we started by calculating total CO2 emissions from traditional brick-and-mortar stores and then turned to CO2 emissions from e-tailers. Wiese et al. (2012) analyzed one German clothing retailer, comparing two selected brick-andmortar stores with an e-tailing system. The retail chain provided information about distances from the central warehouse to the two stores, type of transportation used, the quantity delivered to the stores, and the delivery frequency, making it possible to calculate the supply chain’s environmental impact. The demand side environmental impact was investigated using a consumer survey administered to customers of the chain’s brick-and-mortar stores. The questionnaire provided information about customer postal code, type of customer transport, and number of products bought at the store. We instead use information about the location of all individual residences, brick-and-mortar electronics stores, and Swedish Post delivery points in Dalecarlia.14 In addition, we know the layout (i.e., the different types of roads and the speed limits) of the road network connecting the brick-andmortar stores and the outlet depots to the individual household residences. We believe that the total environmental impact of online and brick-andmortar retailing can be calculated with more precision than previously. 14
The Swedish Post is the market leader in the delivery of goods bought online, and we therefore use the locations of the online delivery points that the Post uses in our analysis. There are also other firms active in the market, and the main competitors to the Post in the Swedish market are DB Schenker and DHL. It should, however, be noted that the delivery points in the Dalecarlia region for products purchased online are in the majority of cases co-located for the three main firms (the Post, DB Schenker and DHL).
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Table 3 about here. Table 3 shows the average total CO2 emissions per purchase of a standard consumer electronics product (e.g., a desktop computer or small stereo). Purchasing the product in a brick-and-mortar store induces on average 7.4 kg of CO2 emissions. This is substantially more than in the case of etailing, where the average is 1.2 kg of CO2, implying 84% lower emissions. Many consumers (about 50% according to Table 1) live near a delivery point and may prefer to pick up the product on foot, rather than by the car as assumed above (iv). The fourth and fifth columns in Table 3 show the resulting emissions if every consumer within 2 km of an outlet walks to pick up the product. This behavior is probably not that common if a desktop computer or small stereo is assumed to be the product. However, other small electronics may conveniently be carried while walking, in which case the difference in induced emissions would be greater (1.0/7.4 meaning 86% lower emissions). As mentioned in the third section, several brick-and-mortar stores were recently closed due to bankruptcy. Such unplanned closures will lead to brick-and-mortar stores being poorly located relative to consumers, so there is room for the brick-and-mortar stores to be better located. Table 4 again shows the average total CO2 emissions per purchase of the standard consumer electronics product, but assuming stores and delivery points to be located so as to minimize average CO2 emissions per purchase. In this case, seven optimally located brick-and-mortar stores would still lead to four-times-higher CO2 emissions per product than would the online alternative. It is clear that e-tailing is environmentally preferable to brickand-mortar retailing, even if it were possible to locate the brick-andmortar stores optimally from an environmental perspective. Table 4 about here. What does this effect of e-tailing in terms of reduced CO2 emissions amount to at a national level? Consumer electronics retailing totals SEK 44 billion annually, of which approximately SEK 8.8 billion constituted online purchases in 2013 (HUI Research 2014). Consumer electronics constitutes almost 25% of e-tailing, so when Swedish Post delivers eight products purchased online per household per year in Sweden, two of these 18
packages can be expected to contain consumer electronics. 15 Statistics Sweden estimated the number of households in Sweden in 2011 at approximately 2.24 million. Consequently, approximately 4.5 million consumer electronics packages were delivered in Sweden due to e-tailing. If we assume that consumer electronics items purchased in brick-andmortar stores are comparable to those bought online, then consumers took home approximately 22.5 million packages from consumer electronics stores. Before 2005, when e-tailing was nearly nonexistent in Sweden, these 27 million packages would have induced 27 ∗ 7.4 = 200 million kg of CO2. Today, they instead induce 22.5 ∗ 7.4 + 4.5 ∗ 1.2 = 172 million kg of CO2 thanks to the availability of e-tailing. In the unlikely event of brick-and-mortar stores being completely replaced by e-tailing, the emissions reduction would be substantial at 27 ∗ 1.2 = 32 million kg of CO2. Such an exercise in aggregation should, of course, be considered only indicative, but nevertheless illustrates that further growth in e-tailing might have more than a trivial impact on the environment.
5. Robustness of the measurement method To estimate the average CO2 emissions per purchased consumer electronics product, several identifying assumptions were imposed. 16 Here 15
A fraction of the packages are probably delivered directly to the consumer’s residence thereby inducing even less CO2-emissions. It is hard to say how large the fraction is. However, as an indication, the consumers report that at least 70% of them prefer to have a cell phone delivered to the delivery point rather than directly to their residence (HUI Research, 2014). 16 It has been suggested that we should also try to numerically calculate the impact of returns on our results. Unfortunately, we do not have any reliable numbers on how common returns are when it comes to consumer electronics products. However, note that returns will only affect our results if there is a difference in how common returns are when the product is bought online as opposed to in a brick-or-mortar store, or if the returned product is only transported part of the way for one or the other of the two retailing solutions being compared. Otherwise, the impact of a return is similar to one additional purchase, the only difference being that the product is now transported from the consumers residence and to either the brick-and-mortar store or the online delivery point, and back through the logistics chain. Arbitrarily assuming that 10% of the purchases are returned when buying online and 3% when buying in a brick and mortar store, and that the products are delivered at least back to the point of entry in the region in both cases, the impact on emissions can be calculated. In that case, simply multiply the emissions in table 3 for online purchases with 1.10 and the ones from the mortar and brick stores with 1.03.
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we look at the sensitivity of the results to each of these assumptions. We begin with the method-related assumptions, and then investigate the consumer behavior assumptions and finally the producer behavior assumptions. Assumption (i): The first assumption concerns the relationship between CO2 emissions and road distance. Carling et al. (2013b) found that emissions peaked at intersections and on arterial streets in urban areas due to non-constant velocity. The CO2 emissions of travelling to a delivery point could be underestimated, as such travel would usually occur in urban areas where constant speed is difficult to maintain. In towns and near intersections, the speed limit is usually 50 km per h or lower. To check assumption (i), we elaborate on the CO2 emissions for travelling on urban roads and streets by assigning higher emissions to road segments with speed limits of 50 km per h and below. On these segments, we increase the CO2 emissions of cars by 50% and trucks by 100%, as the latter are even more sensitive to varying driving speed. Considerable transport effort related to shopping occurs on urban roads with speed limits of 50 km per h and below. On average, consumers in Dalecarlia patronizing online delivery points travel on such roads for 66.3% of the distance travelled, while 36.0% of such consumers travel exclusively on them. Trucks and consumers travelling to brick-and-mortar stores as well as trucks travelling to online delivery points travel more on inter-urban roads and are therefore less exposed to urban roads inducing speed fluctuations. Nonetheless, their exposure to urban roads is nontrivial, calling assumption (i) into question. Table 5 compares products purchased in brick-and-mortar stores and online when CO2 emissions are stipulated to be higher on urban roads in the region. As seen in the table, this stipulation increases emissions in urban areas, making the online solution somewhat less attractive than the brick-and-mortar one relative to the baseline results. However, the differences are too small to significantly change our results, so we deem our original measurements robust to the assumption that distance equals emissions. Table 5 about here. 20
Assumptions (iii) and (vii)17: Assumption (iii) was that the population of the studied region remained stable during the studied period, while assumption (vii) was that all residents of the region were equally likely to purchase the product. Age is an important part of the consumer profile that we cannot access, so we may have to allow for heterogeneity between age groups. Age is highly correlated to income, for example, but can also be used to model geographical redistribution likely to represent future demographic changes in the region, i.e., assumption (iii). This is because people born into older cohorts largely live in rural areas, whereas people born into younger cohorts are more concentrated in urban areas. Due to this spatially skewed age distribution, there is an ongoing process of birth deficits and population decrease in rural areas and the opposite in many urban areas (e.g., Håkansson 2000). Table 6 shows results comparable to those in Table 3, but weighted by age. Elderly consumers (≥65 years old) have a weight of 0.5, young consumers (≤15 years) a weight of 1.5, and those in between a weight of 1. Note that these changes can be seen as altering both the population composition and likelihood of purchasing, testing both assumptions (iii) and (vii) at once. Although young consumers are now considered three times more likely to purchase electronics than are old consumers, the values in Table 6 are almost identical to those in Table 3, so we conclude that the results are insensitive to these assumptions. Table 6 about here. Assumption (iv): One assumption regarding consumer behavior (iv) is that online-purchased products are picked up at the nearest delivery point. This assumption has been confirmed in most cases in Sweden via the surveys cited in section 2 (HUI Research 2013 and 2014), in which 85–90% of surveyed consumers selected the outlet nearest their residence. Assumption (v): This assumption, that consumers in the Dalecarlia region take the shortest routes from their residences to the brick-and-mortar stores or online delivery points,18 was supported by a study cited in section 17
Assumption (ii) will be tested together with assumptions (ix) and (x), below. The method suggested can also be used if more consumers than in the studied Swedish region travel from work to the delivery points, with the added data requirement that we then also need to know where the consumer works and the 18
21
3 (Jia et al. 2013). Researchers compared actual travelling routes with the shortest routes to a shopping center, finding that only 5 of 500 investigated shopping trips did not take the shortest routes. Assumption (vi): The calculations presented above assumed that the consumer drives a car emitting 0.15 kg per km of CO2, roughly equaling the emissions of a Toyota Avensis. According to the National Transport Administration, new cars in Sweden emitted on average 0.138 kg per km in 2012, while the older fleet of cars typically had higher emissions, making 0.138 kg per km an underestimation of the overall average emissions. What is important here is that the total emissions for each purchase are calculated as follows: Total emissions = (consumer’s car emissions per km × km driven by consumer) + (distributer’s truck emissions per km × km driven by distributer)
(1)
As can be seen from equation (1), the car’s emissions can be changed at will and the total emissions recalculated, since this is only a scale factor for the total emissions of car travel. Note also that the same holds if we want to investigate how a change in truck emissions or choice of travel route (i.e., distance traveled) affects total emissions. Assumption (viii)19: We assume that consumers made the purchase either at the store or online. According to Cullinane (2009), however, if people browse online and shop in brick-and-mortar stores, some shopping journeys can be saved, but if they browse in the stores and shop online, additional travel will likely be incurred. Moreover, the RAC Foundation (2006) reports that almost 80% of surveyed consumers travel to brick-andmortar stores to compare products. We accordingly repeated the analysis, but stipulated that each online purchase was preceded 80% of the time by a trip to a brick-and-mortar store to physically assess the product and its substitutes. Under this behavioral assumption, we find that online shopping would induce 7.89 kg of CO2 on average, comparable to the exclusively brick-and-mortar store case. The environmental benefits of additional distance traveled to pick up the package. However, as demonstrated by Jia et al. (2013), such behavior is unimportant in our empirical setting. 19 Assumption (vii) was investigated together with assumption (iii), above.
22
online shopping would be completely offset if as many as 80% of consumers behaved in this way; in fact, more detailed analysis indicated that if 71% or more of consumers behaved in this way, the environmental benefits of online shopping would be offset. It should be noted that in Sweden in 2013, only 6% of consumers buying consumer electronics online reported first visiting a brick-and-mortar store and then purchasing the product online, while 32% reported first researching what product to buy online and then purchasing the product from a brick-and-mortar store (HUI Research 2014). Assumptions (ix), (x), and (ii): Table 7 shows the results of simulations in which certain customer behavior assumptions are imposed. The identifying assumptions (ix) and (x) concern how attractive a consumer finds a brick-and-mortar store relative to online shopping and the consumer’s propensity to travel to shop for consumer electronics. In this, we are applying the idea of a gravity model as proposed in an operational research setting by Drezner and Drezner (2007), which in turn draws on work in the marketing literature (particularly Huff 1964). Drezner and Drezner (2007) specify the probability that a consumer residing at q will patronize a facility located at p as
𝐴𝑝 𝑒 ∑𝑝∈ 𝐴𝑝 𝑒
𝑝 𝑝
, where 𝐴𝑝 is the
attractiveness of the facility, is the parameter of the exponential distance decay function,20 and 𝑝 is the shortest distance between residence and facility. We adapt this probability to the context such that the probability (𝑝) = of patronizing brick-and-mortar store p is 𝐴𝑝 𝑒 ∑𝑝∈ 𝐴𝑝 𝑒
𝑝 𝑝 +𝐴
𝑒
, where 𝐴 = 1 is the normed attractiveness of
online shopping and is the shortest distance to the nearest delivery point for online-purchased products. To understand this specification, consider a consumer who can choose between one brick-and-mortar store and one delivery point for online-purchased products and who lives equidistant from the two outlets. The attractiveness parameter for the brick-and-mortar store then describes how much more likely the consumer is to choose the brick-and-mortar over the online alternative. For example,
20
The exponential function and the inverse distance function dominate the literature, as discussed by Drezner (2006).
23
𝐴𝑝 = 2 means that the consumer would patronize the brick-and-mortar store two times out of three.21 In the analysis, we consider three values of = 1.0,0.11,0.035, the first referring to a situation in which the consumer is very likely to choose the nearest store or delivery point, the second being the estimated parameter value based on Californian visitors to shopping malls (Drezner 2006), and the third being the estimated value based on Swedes’ self-reported trips to buy durable goods (Carling et al. 2012). The values of can be converted into average distances travelled to a store of 1, 9, or 30 km. Furthermore, we let 𝐴𝑝 = 1.0,2.0,5.0 represent the brick-and-mortar stores, including the case of consumers indifferent to whether they see the product in the store or online (𝐴𝑝 = 1.0) and that of a consumer who finds it much more attractive to see and touch the product physically (𝐴𝑝 = 5.0). Table 7 shows how the market share of the brick-and-mortar stores increases due to their attractiveness when the market share is computed as the expected number (implied by the model) of consumers patronizing any brick-andmortar store divided by the number of consumers. Focusing on the case in which consumers are willing to consider travelling to stores other than the nearest one ( = 0.035), we note that the market share of brick-andmortar stores increases from 55% if consumers find them as attractive as online shopping (𝐴𝑝 = 1.0) to 83% if consumers find them much more attractive than online shopping (𝐴𝑝 = 5.0). Considering that = 0.035 is the most likely estimate in Sweden and that brick-and-mortar stores currently sell approximately 80% of all purchased consumer electronics, one may conjecture from Table 7 that Swedish consumers currently regard brick-and-mortar shopping as about two to five times more attractive than online shopping, on average. The last column of the table gives the average CO2 emissions per consumer and purchase. In calculating the emissions, we take into account that the consumer will shop at various brick-and-mortar stores and sometimes shop online. The formula is ∑𝑝=1 (𝑝) ∗ ( 𝑝 ∗ + ̃𝑝 ∗ ) + (1 − ∑𝑝=1 (𝑝)) ∗ ( ∗ + ̃ ∗ ), where and are the 21
One argument for a high attractiveness of the brick-and-mortar stores can be colocation of retailing giving the consumer access to several stores for the product in question in a limited geographical area.
24
CO2 emissions per kilometer driven by consumer cars and delivery trucks, respectively, ̃𝑝 is the road distance the truck travels to store p, and ̃ is the road distance the truck travels to the online delivery point. The formula therefore gives the consumer’s expected CO2 emissions for repeated purchases. An increased likelihood to travel for shopping implies a higher market share for brick-and-mortar stores, which in turn leads to a dramatic increase in CO2 emissions. Consider, for example, the case when brick-and-mortar and online shopping are equally attractive to consumers, i.e., 𝐴𝑝 = 1.0. If consumers are unwilling to travel ( = 1), they will almost always shop online and pick up their purchases at the nearest delivery points, as that implies the least travelling with resulting low CO2 emissions of 1.23 kg. If they are likely to travel ( = 0.035), then they will sometimes shop online, sometimes at stores near their residences, and sometimes at stores far from their residences. As a result, their travelling will on average be extensive, resulting in high CO2 emissions of 5.95 kg. Table 7 about here. Some of the results presented in Table 7 are illustrated in Fig. 2, which indicates the geographical areas dominated by brick-and-mortar shopping. The left panel presents the case in which = 0.11 and 𝐴𝑝 = 1, showing that most of the region, except for the centermost areas surrounding the brick-and-mortar stores, is served by e-tailing. In the right panel, consumers supposedly are likely to travel for shopping ( = 0.035) and find brick-and-mortar stores more attractive 𝐴𝑝 = 2than online shopping (𝐴𝑝 = 2), so the more densely populated areas of the region are served chiefly by brick-and-mortar shopping. Fig. 2 about here. We also elaborate on the closure of brick-and-mortar stores to check the sensitivity of assumption (ii) by stepwise removing, one at a time, the store with the smallest market share. For example, Table 8 presents the situation after closing the two stores attracting the smallest shares of consumers. Although store closure leads to a smaller market share for brick-and-mortar shopping, the general pattern found in Table 7 remains. 25
Table 8 about here. Assumption (xi): The truck is assumed to be loaded to 60% of its capacity, though the loading could be lower or higher. We therefore check the sensitivity to this assumption by stipulating that the truck is loaded to 30% of its capacity, which might be the case if the truck typically returns empty from the delivery points. We also consider an 80% loading, corresponding to efficient distribution and a good solution to the travelling salesman problem, in which the truck finds an efficient route to pass all scheduled delivery points. Table 9 shows that varying the loadings only modestly affects the CO2 emissions induced by selling a standard electronics product at a brick-and-mortar store. The assessment of the onlinepurchased product’s emissions is somewhat more sensitive to the stipulated loading, but the difference in emissions between brick-andmortar- and online-purchased products remains large. Table 9 about here. Assumption (xii): Online-purchased products arrive at the delivery point by first going via the shortest route to one of the six distribution centers and then via the shortest route from the distribution center to the delivery point. This is essentially how Swedish Post described their logistics solution to us, although they were unwilling to go into detail. Assumption (xiii): A product sold at a brick-and-mortar outlet comes to the store via the shortest route from the nearest of the three gateways into the region (xiii). Assumptions (xii) and (xiii) may be flawed and could in that case lead to underestimated CO2 emissions. From our analysis, we know the distances traveled via the shortest routes from points of entry into the region to consumer residences, and use these to calculate total emissions in accordance with equation (1). If interested, one could use equation (1) to introduce longer transportation routes for both the consumer and/or retailer distribution networks and recalculate the total emissions. The equation could for instance be used if one suspected that consumers often used multi-purpose trips when shopping, in which case we would introduce only shorter routes specifically reflecting the marginal transport effort related to shopping. However, multi-purpose shopping trips are not 26
that common in Sweden (Jia et al. 2013) and are only relevant when comparing online and brick-and-mortar shopping if behavior differs systematically between the two types of shopping.
6. Discussion Retailing creates an environmental impact that should not be underestimated. In Great Britain, the average consumer made 219 shopping trips and travelled a total of 926 miles for retail purposes in 2006 (DfT 2006). Meanwhile, in a Swedish setting, Carling et al. (2013a) reported that the current location of retailers in the Dalecarlia region of Sweden was suboptimal, and that suboptimal retailer locations generated on average 22% more CO2 emissions than did optimal locations. An empirical literature (e.g., Wiese et al. 2012; Edwards et al. 2010) analyzes the environmental impact of online shopping. However, this literature has focused on the emissions induced by consumers traveling to and from brick-and-mortar stores or online delivery points, and has not compared any but the “last-mile” environmental impacts of online versus brick-and-mortar retailing. This paper sought to develop a method for empirically measuring the CO2 footprint of brick-and-mortar retailing versus e-tailing from entry point to a region (e.g., country, county, and municipality) to the consumer’s residence. The method developed was then used to calculate and compare the environmental impacts of buying a standard electronics product online and in a brick-and-mortar store in the Dalecarlia region in Sweden. The method developed only requires knowledge of the road network of the studied region, the location of the residences of the population (measured as precisely as possible), and the locations of the brick-and-mortar outlets and e-tailer delivery points. The method also requires several assumptions that need scrutiny to determine whether the method is robust to changes in the underlying assumptions. This was done thoroughly in this study, and the results indicate that the method developed is very robust to changes in the underlying assumptions. The results indicate that e-tailing results in a substantial reduction in CO2 emissions from consumer travel. The average distance from a consumer 27
residence to a brick-and-mortar electronics retailer is 48.54 km in the Dalecarlia region, while the average distance to an online delivery point is only 6.7 km. As such, making the purchase online will lead to only 14% of the consumer travel emissions that would have resulted from purchasing the product in a brick-and-mortar store. It should also be noted that the online delivery points in the Dalecarlia region are well located relative to consumer residences. The actual delivery point locations differ from those that would minimize CO2 emissions caused by consumer travel by under 0.7 km. The results also indicate that e-tailing causes the distance traveled from regional entry point to delivery point (i.e., brick-and-mortar store or online delivery point) to increase. On average, the product travels 47.15 km to the brick-and-mortar store versus 122.75 km to the online delivery point. However, one must recall that a product carried in a consumer car induces much higher CO2 emissions than does the same product delivered by a professional carrier transporting many units simultaneously. As such, we have also calculated the total CO2 emissions from regional entry point to consumer residence for the two options, i.e., e-tailing or brick-and-mortar stores. The results indicate that purchasing the product in a brick-andmortar store on average causes 7.4 kg of CO2 to be emitted along the whole chain from regional entry point to consumer residence, while purchasing the same product online only induces on average 1.2 kg of CO2 emissions. As such, consumers in the Dalecarlia region who switch from buying the product in a store to buying the same product online on average reduce their transport CO2 emissions by approximately 84%. This is a case study and more cases are needed. It would be interesting to see how the results in this study would hold for other cases, especially in more densely populated areas in Sweden, as well as in other countries where other shopping behavior could be observed, and where the distribution of the goods is done differently. It would also be of interest to further investigate how good the transportation work done by professional carriers is from a CO2 emissions perspective, and redo the analysis in this paper taking the sub-optimality of the carrier routes into account. In the meanwhile, have we outlined a method to follow the products from entering a region to the consumer residence that seems to be a fruitful way 28
to compare transport related CO2 emissions induced by brick-and-mortar retailing with emissions from online shopping.
Acknowledgments The authors would like to thank Sven-Olov Daunfeldt, Oana Mihaescu, Pascal Rebreyend and participants at the 8th HUI Workshop in Retailing (Tammsvik, Sweden, January 16-17, 2014) and the 21th EIRASS Conference on Recent Advances in Retailing and Services Science (Bucharest, Romania, July 7-19, 2014) for valuable comments and suggestions. This study was financed by a Dalarna University internal grant, and the funding source had no involvement in study design, data collection and analysis, or the decision to submit the article for publication.
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References Cairns, S., (2005), Delivering supermarket shopping: more or less traffic?, Transport Reviews: A Transnational Transdisciplinary Journal, 25, 51-84. Carling, K., Han, M., and Håkansson, J., (2012), Does Euclidean distance work well when the p-median model is applied in rural areas?, Annals of Operations Research, 2012, 201:1, 83-97. Carling, K., Han, M., Håkansson, J., and Rebreyend, P., (2012), An empirical test of the gravity p-median model, Working papers in transport, tourism, information technology and microdata analysis, 2012:10. Carling, K., Han, M., Håkansson, J. and Rebreyend, P., (2014), Distance measure and the p-median problem in rural areas, Annals of Operations Research, Online July 27. Carling, K., Håkansson, J. and Jia, T., (2013), Out-of-town shopping and its induced CO2-emissions, Journal of Retailing and Consumer Services, 20:4, 382388. Carling, K., Håkansson, J. and Rudholm, N., (2013), Optimal retail location and CO2-emissions, Applied Economic Letters, 20:14, 1357-1361. Carling, K., and Meng, X., (2014), On statistical bounds of heuristic solutions to location problems, Working papers in transport, tourism, information technology and microdata analysis, 2014:10. Chiyoshi, F.Y. and Galvão, R.D., (2000), A statistical analysis of simulated annealing applied to the p-median problem, Annals of Operations Research 96, 61-74. Cullinane, S., (2009), From bricks to clicks: The impact of online retailing on transport and the environment, Transport Reviews, 29, 759-776. Daskin, M., (1995), Network and discrete location, Wiley, New York. DfT., (2006), National Transport Survey: 2006,London, TSO. Dijkstra, E.W., (1959), A note on two problems in connexion with graphs, Numerische Mathematik, 1, 269–271. 30
Drezner, T., (2006). Derived attractiveness of shopping malls, IMA Journal of Management Mathematics, 17, 349-358. Drezner T. and Drezner Z., (2007), The gravity p-median model, European Journal of Operational Research, 179, 1239-1251. Edwards, J.B., McKinnon, A.C. and S.L. Cullinane (2010) Comparative analysis of the carbon footprints of conventional and online retailing: A “last mile” perspective. International Journal of Physical Distribution & Logistics Management, 40, 103-123. Hakimi, S.L., (1964), Optimum locations of switching centers and the absolute centers and medians of graph, Operations Research 12:3, 450-459. Hakimi, S.L., (1965), Optimum distribution of switching centers in a communications network and some related graph theoretic problems, Operations Research 13, 462-475. Han, M., Håkansson, J. and Rebreyend, P., (2013), How do different densities in a network affect the optimal location of service centers?, Working papers in transport, tourism, information technology and microdata analysis, 2013:15. Huff, D.L., (1964), Defining and estimating a trade area, Journal of Marketing, 28, 34–38. HUI Research, (2013), e-barometern, Årsrapport 2012, http://www.hui.se/statistik-rapporter/index-och-barometrar/e-barometern. HUI Research, (2014), e-barometern, Årsrapport 2013, http://www.hui.se/statistik-rapporter/index-och-barometrar/e-barometern. Håkansson, J., (2000), Impact of migration, natural population change and age composition on the redistribution of the population in Sweden 1970 – 1996, CYBERGEO, 123. Jia, T., Carling, K. and Håkansson, J., (2013), Trips and their CO2 emissions induced by a shopping center, Journal of Transport Geography, 33, 135-145. Lennartsson, D. and Lindholm, P., (2004) Utflyttning av produktion inom den svenska industrin/Outsourcing of production in Swedish industry. Statistics Sweden (in Swedish). 31
RAC Foundation, (2006), Motoring Towards 2050: Shopping and Transport Policy, RAC, London. Stead, D., (1999), Relationships between transport emissions and travel patterns in Britain, Transport Policy, 6, 247–258. Weber, C.L., Matthews, H.S., Corbett, J.J. and Williams, E.D., (2007), Carbon emissions embodied in importation, transport and retail of electronics in the U.S.: A growing global issue, in: Proceedings of the 2007 IEEE Symposium on Electronics and the Environment, 174-179. Wiese, A.., Toporowski, W. and Zielke, S., (2012), Transport-related CO2 effects of online and brick-and-mortar shopping: A comparison and sensitivity analysis of clothing retailing, Transportation Research D, 17, 473-477. Zhan, F.B. and Noon. C.E., (1998), Shortest path algorithms: an evaluation using real road networks, Transportation Science, 32:1, 65-73.
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Table 1. Consumers’ return travel distance on the road network from home to nearest brick-and-mortar store and online delivery point (in km), showing current and p-median optimal locations. Percentile: 5 25 50 75 95 Mean St. dev.
Current location Brick-andOnline mortar delivery stores points 2.64 0.80 7.12 2.08 25.86 3.64 77.96 7.88 162.10 22.58 48.54 6.70 55.72 8.46
Optimal location Brick-andOnline mortar delivery stores points 1.64 0.92 5.20 2.12 16.46 3.70 40.12 7.56 88.80 17.68 28.76 5.98 36.96 7.14
Table 2. Distance products travel on the road network to brick-and-mortar stores and online delivery points (in km). Percentile: 5 25 50 75 95 Mean St. dev.
Current location Brick-andOnline mortar delivery stores points 22.25 14.23 75.75 39.86 104.89 52.63 168.53 253.17 47.15 122.75 40.61 70.54
33
Optimal location Brick-andOnline mortar delivery stores points 29.15 18.00 78.39 51.20 107.97 115.45 192.93 272.25 62.28 130.81 47.24 76.49
Table 3. CO2 emissions (in kg) induced by transporting a typical product from the regional boundary to consumer’s home via current outlets. Brick-andOnline Brick-andOnline mortar delivery mortar (incl. (incl. a a stores points walking) walking) 5 0.48 0.29 0.08 0.06 25 1.13 0.51 1.13 0.18 50 3.96 0.78 3.96 0.41 75 11.79 1.42 11.79 1.42 95 24.58 3.65 24.58 3.65 Mean 7.44 1.22 7.40 1.05 St. dev. 8.41 1.30 8.44 1.39 a It is assumed that all consumers within 2 km of the outlet walk to pick up the product and while doing so produce no CO2 emissions. Percentile:
Table 4. CO2 emissions (in kg) induced by transporting a typical product from the regional boundary to the consumer’s home via outlets that are environmentally optimally located. Brick-andOnline Brick-andOnline mortar delivery mortar (incl. (incl. a a stores points walking) walking) 5 0.56 0.28 0.10 0.06 25 1.39 0.52 1.39 0.18 50 3.13 0.77 3.13 0.41 75 6.50 1.29 6.50 1.29 95 13.44 3.04 13.44 3.04 Mean 4.77 1.12 4.73 0.95 St. dev. 5.31 1.13 5.35 1.23 a It is assumed that all consumers within 2 km of the outlet walk to pick up the product and while doing so produce no CO2 emissions. Percentile:
34
Table 5. CO2 emissions (in kg) induced by transporting a typical product from the regional boundary to consumer’s home via current outlets; greater CO2 emissions assumed in urban areas. Percentile: 5 25 50 75 95 Mean St. dev.
Baseline Brick-andOnline mortar delivery stores points 0.48 0.29 1.13 0.51 3.96 0.78 11.79 1.42 24.58 3.65 7.44 1.22 8.41 1.30
Higher urban emissions Brick-andOnline mortar delivery stores points 0.75 0.43 1.87 0.79 4.99 1.21 13.26 1.95 26.92 4.38 8.67 1.64 9.05 1.45
Table 6. CO2 emissions (in kg) induced by transporting a typical product from the regional boundary to consumer’s home via current outlets; consumers weighted by age. Brick-andOnline Brick-andOnline mortar delivery mortar (incl. (incl. a a stores points walking) walking) 5 0.48 0.29 0.08 0.06 25 1.14 0.50 1.14 0.18 50 3.98 0.77 3.98 0.41 75 11.87 1.40 11.87 1.40 95 24.58 3.61 24.58 3.61 Mean 7.46 1.21 7.42 1.04 St. dev. 8.40 1.29 8.44 1.38 a It is assumed that all consumers within 2 km of the outlet walk to pick up the product and while doing so produce no CO2 emissions. Percentile:
35
Table 7. Market share of brick-and-mortar stores and average CO2 emissions (in kg) induced by transporting a typical product from the regional boundary to consumer’s home via current outlets; seven brickand-mortar stores. Market share (%) CO2 emissions 𝐴𝑝 1
2
5
1 0.11 0.035 1 0.11 0.035 1 0.11 0.035
11.53 30.56 55.43 16.87 41.70 69.06 24.86 55.10 82.68
1.23 1.85 5.95 1.24 2.16 7.44 1.27 2.66 9.20
Table 8. Market share of brick-and-mortar stores and average CO2 emissions (in kg) induced by transporting a typical product from the regional boundary to consumer’s home via current outlets; 5 brick-andmortar stores. Market share (%) CO2 emissions 𝐴𝑝 1
2
5
1 0.11 0.035 1 0.11 0.035 1 0.11 0.035
8.60 25.42 49.42 12.78 34.47 62.45 19.32 45.38 76.40
36
1.23 1.79 5.51 1.24 2.07 7.08 1.27 2.53 9.17
Table 9. CO2 emissions (in kg) induced by transporting a typical product from the regional boundary to consumer’s home via current outlets; trucks loaded to two capacity levels. Percentile: 5 25 50 75 95 Mean St. dev.
30% loading Brick-andOnline mortar delivery stores points 0.58 0.40 1.21 0.69 4.04 1.01 11.88 1.65 24.84 3.93 7.54 1.43 8.46 1.34
37
80% loading Brick-andOnline mortar delivery stores points 0.46 0.25 1.12 0.46 3.94 0.72 11.77 1.35 24.51 3.60 7.41 1.16 8.40 1.29
Figure 1. a) Consumer residences and current locations of brick-andmortar stores; b) national road network and current locations of postal distribution centers and delivery points.
38
Figure 2. Market areas when (a) = 0.11 and 𝐴𝑝 = 1 and when (b) = 0.035 and 𝐴𝑝 = 2 for current brick-and-mortar stores. 39
PAPER VI
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On administrative borders and accessibility to public services: The case of hospitals in Sweden. Authors1: Xiangli Meng, Kenneth Carling, Johan Håkansson2, Pascal Rebreyend Abstract: An administrative border might hinder the optimal allocation of a given set of resources by restricting the flow of goods, services, and people. In this paper we address the question: Do administrative borders lead to poor accessibility to public service such as hospitals? In answering the question, we have examined the case of Sweden and its regional borders. We have used detailed data on the Swedish road network, its hospitals, and its geo-coded population. We have assessed the population’s spatial accessibility to Swedish hospitals by computing the inhabitants’ distance to the nearest hospital. We have also elaborated several scenarios ranging from strongly confining regional borders to no confinements of borders and recomputed the accessibility. Our findings imply that administrative borders are only marginally worsening the accessibility. Key words: hospitals, optimal location, network distance, travel time, location model
1. Introduction
1 2
School of Technology and Business Studies, Dalarna University, Borlänge, Sweden Corresponding author: e-mail: [email protected]
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A national, regional or any other administrative border might be considered a barrier to the free flow of goods, services, and people, and thereby hindering the optimal allocation of a given set of resources. As a consequence, in particular in borderlands, the highest achievable economic and social utility may not be attained. Van Houtum (2000) gives an extensive review of the study of borders with an emphasis on the EU and its internal borders. In spite, or maybe because, of the globalization process, the recent upsurge of research on borders is discussed by Andersson et al (2002). While not all borderland studies view a border as a barrier, it is widely held that borders reduce trade and are a demarcation of the labor market. In fact, a core part of the EU policy has been to promote cross-border transaction of goods, services, and labor towards a common European market. There are also a growing number of cross-border cooperation of public authorities in Europe. However, it is still too early to regard such cooperation as defining new territorial entities and joint regional policies (e.g. Perkmann, 2007; Popescu, 2008; Harguindéguy and Bray, 2009). Public services in the EU are still normally confined by national or regional borders. As an illustration, López et al (2009) discuss the funding of Spanish rail investments in light of them having substantial spill-overs in French and Portuguese regions bordering Spain. Similar to transport infrastructure, health care is often under public control in the EU. In this paper, we examine how regional borders affect the spatial accessibility to hospitals within Sweden. Since Swedish regions are comparable in geographical size to many European countries such as Belgium, Denmark, Estonia, Slovenia, Switzerland, and the Netherlands as well as provinces in Italy and Spain and states in Germany with a selfgoverning of the health care, we believe the results will be informative of the internal borders’ of Europe effect on the accessibility of health care. To be specific, we address three issues. The first is the effect of borders on inhabitants’ spatial accessibility to hospitals. The second is the quality of the location of hospitals and the resulting accessibility. The third is accessibility in relation to population dynamics. Sweden, for several reasons, is a suitable case for a borderland study of accessibility to hospitals. Firstly, we have access to good data of the national road network and a precise geo-coding of the inhabitants, the hospitals, and the regional borders. Secondly, hospital funding, management, and operation are confined by the regional borders. Thirdly, after 200 years of a stable regional division of the country a substantial re42
organization of the regions is due. The paper is organized as follows: In Section 2, the institutional settings of the Swedish health care and the regional re-organization are discussed jointly with a short review on location models and their application in analyzing populations’ spatial access to health care. Section 3 presents data, defines the distance measures, and provides some descriptive statistics of key variables. Furthermore, a sketch of how health care is organized in Sweden is given jointly with maps of Sweden that put the location model into the empirical context. In Section 4 the experimental design leading to a ‘what-if’ analysis and the optimization method are described. Results are presented in Section 5, and the paper ends with a concluding discussion in Section 6. 2. Swedish health care, accessibility, and location models Health care in Sweden is organized and tax funded at a regional level because it is the regions’ primary responsibility. The health care is politically controlled and the population can respond to its management by democratic channels such as elections and (less often) referendums. The regional division of Sweden has remained stable for more than 200 years, but it is currently subject to a major revision. The primary reason for the revision is that many regions as a consequence of population dynamics and historical decisions are locked up in suboptimal solutions within the region. Therefore it is difficult to operate health care efficiently which leads to long queues and high production costs (see e.g. McKee and Healy, 2002). Health care service depends to a large extent on face-to-face activities and hence the spatial accessibility for the population is a key concern. Central to the supply of health care is the hospitals. Drawing on efficiency arguments, the trend in Sweden and elsewhere (Hope 2011) has been a concentration of hospitals in fewer locations with a possible consequent decrease in spatial accessibility for the population. The concentration seems to go hand in hand with urbanization, but it is counteracted by suburbanization, counter urbanization and urban sprawl from the 1960s. The net outcome on the accessibility for the population is unclear due to these counteracting forces. Nonetheless, the concentration of health care has led to a growing number of people questioning its management. For instance in the Swedish region Västerbotten, a recent referendum 43
regarding a political proposal of further concentration of health care was enforced in September 8th, 2013. In the referendum, about 90% of the voters rejected the proposal. The direction of the regional revision of Sweden is clear; the number of regions shall decrease from the present 21 regions to about 6 to 8 regions. The reason behind the revision is that larger regions imply greater populations, which allows greater potential to organize health care efficiently. As for spatial accessibility, such revision would reduce the presumed and negative border effect, but not necessarily lessen the sub optimality of solutions within the regions. Because of this, some political parties, and most notably the health minister, have argued for hospitals to be organized and managed on a national level. A key fact in the debate on administrative level of health care in Sweden ought to be spatial accessibility for the population under the alternatives, a fact that up to now is missing. Furthermore, there is no international study on the potential impact of a national administrative revision on the population’s spatial accessibility to the hospitals to the best of our knowledge. There are, however, many studies that measure and describe a population’s spatial accessibility to health care usually in a confined area (e.g. Higgs, 2004; Perry and Gesler, 2000; Shi et al, 2012; Tanser et al, 2006). These studies did not provide, as a benchmark, the best possible spatial accessibility. To do so, an analytic procedure that, for instance, minimizes the average distance to the health care is necessary. To address such a general location problem the p-median model is commonly used (see e.g. Hakimi, 1964; Reese, 2006). The p-median model intends to find an optimal solution for the location of supply points that minimizes the average distance to the population’s nearest supply point. This model has been applied to solve location problems of hospitals (see e.g. Daskin and Dean, 2004; Wang, 2012). Unfortunately, the p-median problem is NPhard forcing most applications to address rather small problems of limited spatial reach. The largest p-median problem solved that we are aware of is synthetically generated data consisting of 89,600 nodes (Avella et al, 2012). Avella’s et al (2012) problem is modest relative to a problem of optimizing spatial accessibility on a national level assuming geo-coded data with high geographical resolution. It is, therefore, an open question whether it is possible to derive the benchmark of the best possible spatial accessibility for the population on a national level. We shall attempt to do so using about 5,400,000 inhabitants 44
and their residence geocoded in about 190,000 squares each of which is 500 by 500 meters. The inhabitant will be assumed to patronize the nearest of Sweden’s 73 hospitals by travelling along the shortest route on Sweden’s very extensive road network of about 680,000 kilometers. The p-median model is not the only location model relevant for optimizing spatial accessibility of hospitals. In a literature review by Daskin and Dean (2004) and more recently by Wang (2012), several location-allocation models for finding optimal location of health care facilities were described and summarized. The location models optimize facility locations according to different objectives. One common location model is the location set covering problem (LSCP) which minimizes the number of facilities covering the whole demand (Toregas and ReVelle, 1972). Relative to the p-median model, the LSCP model would lead to a change in the number of hospitals compared with the present situation and thereby indicating merging of current hospitals or adding of new hospitals. Another commonly used model was developed by Church and ReVelle (1974) who go in another direction by maximizing the demand covered within a desired distance or time threshold (maximum covering location problem, MCLP). Relative to the p-median model, the MCLP model put little weight on inhabitants in remote areas implying a drastic deterioration in accessibility for them. Yet another model is the center model described by Wang (2012) with the objective of minimizing the maximum distance to the nearest facility. The center model is perhaps best suited for emergency service planning as it, compared with the p-median model, gives heavy weight to the remote inhabitants and downplays the huge demand of densely populated areas. To locate health care facilities of different hierarchical levels such as hospitals with specialized care and local health centers, the hierarchical type models have been proposed (Michael et al, 2002; Narula, 1986). Hierarchical location models locate p hospitals for health care with services on different levels simultaneously. Hierarchical location models are computationally very heavy which makes them most suitable for solving problems where the number of facilities and nodes for possible location is small. Although the alternative location models are interesting, we will focus on the best possible spatial accessibility in the sense of minimizing the average distance to the nearest hospital for the population. In other words, the p-median model will be used. Furthermore, we will only consider 45
homogenous hospitals meaning that hierarchical location models are unwarranted. 3. Data and descriptive statistics Sweden is about 450,000 km2. Figure 1 depicts the country’s 21 regions. The size of the regions ranges from 3,000 km2 (the island Gotland) to the northernmost region Norrbotten of 97,000 km2 with an average regional size of 21,000 km2. To put the geographical size of the regions of Sweden in the European perspective, it may be noted that the smallest regions are of the size of Luxembourg, the middle sized are comparable with Belgium and German states, and the largest are comparable with Hungary and Portugal. We have access to high quality, geo-coded data of the Swedish inhabitants as of 2008. They are geo-coded in squares of 500 by 500 meters. All inhabitants within a certain square are geo-coded to the center (point) of the corresponding square where the center is taken to be the demand point in the ensuing location analysis. The inhabitants are distributed in 188,325 squares making up approximately 10 percent of the country’s area. The population used in the analysis is all the inhabitants in the age of 20 to 64 years and it amounts to 5,411,573.3 Figure 1a shows the distribution of the population. The population is asymmetrically distributed in the country due to natural conditions such as climate, variation in altitude, quality of the soil, access to water and so forth. The great part of the population lives in the southern part of the country and along the coast of the northern part. While the population density of Sweden (20 inh./km2) is very low compared with other European countries, the variation in population density between the regions is substantial. The western part of northern Sweden is very sparsely populated with a population density below one inhabitant per square kilometer, whereas many regions in the southern parts have a population density of about 50 inh./km2 with an extreme of 350 inh./km2. A hospital is a complex producer of health care and consequently its definition is nontrivial as discussed by Mckee and Healy (2002). For this study we have accepted a conventional classification of health care in Sweden used for hospital ranking in Sweden 2010 (Sveriges bästa sjukhus, 3
The restriction to the working population is a consequence of the data having been gathered for labor market related studies.
46
2010). This classification identifies 73 hospitals in Sweden.4 The hospitals are located in 69 of the 1,938 settlements5 (depicted in Figure 1b). Two settlements being the two largest cities in the country – Stockholm and Gothenburg – contain three hospitals each. In the search for optimal location of hospitals each of the 1,938 settlements are considered as a candidate for locating a hospital.
Figure 1a-c: Distribution of the population (a), settlements (b), and regions, current hospitals, and major national roads (c).
Figure 1c illustrates the locations of the 73 hospitals in Sweden. The number of hospitals in Sweden is low compared with other European countries. There is about 0.75 hospitals per 100,000 inhabitants in Sweden. The overall average for Europe is 2.6 hospitals per 100,000 4
It goes without saying that a petite part of the health care is highly specialized and not offered everywhere. The national government funds and exercises the power to decide the location of such health care, but we shall abstract from it due to its rarity. 5 Only settlements with more than 200 inhabitants according to the census of 1995 are considered in the location analysis.
47
inhabitants with a range from 1 (the Netherlands) to 6 (Finland) (Hope 2011). In spite of Sweden’s dissimilarity to other European countries in this respect, the expenditure on health care in Sweden is similar to other European countries of about 10 per cent of the GDP. The population size of the regions is about 300,000 inhabitants and consequently it is expected to be three hospitals per region. In fact, this is the case with three exceptions being the markedly more populated regions surrounding the cities Stockholm, Gothenburg, and Malmo. These regions have 6-9 hospitals and a population exceeding 1,300,000 inhabitants.
Figure 2: National roads and their speed limit.
As mentioned before, the inhabitants may travel between the residence and the hospital along some 680,000 kilometers of roads. National roads maintained by the state are the most important roads in the road network and they make up 15 per cent of Sweden’s road network. We have retrieved the road network information from the national road data base (NVDB). In Figure 2 the national roads are visualized. There are 31,000,000 road segments stored in NVDB. Each segment is stored along 48
with other attributes such as speed limit. The speed limit varies between 5 and 120 km/h with 80 percent of the road segments having a speed limit of 70 km/h. From Figure 2 it may be noticed that national roads with a speed limit below 80 km/h dominate in the rural areas while national roads with higher speed limits connect the larger towns by a sparse network. Within urban areas the speed limit is usually 50km/h or lower. We have processed the data into a country wide road network to enable both the computing of travel distance and travel time between the 188,325 demand points and the 1,938 candidate nodes for hospital location (Meng and Rebreyend, 2014). While there is some latitude for the inhabitants to select the hospital to patronize within the region, it is safe to assume that the chosen hospital is that nearest to the residence and that the shortest route to the hospital is taken. This means that the shortest route between the hospitals and the demand points needs to be identified. To do so, we have used the algorithm originally proposed by Dijkstra (1959). At the onset, the algorithm identifies and set all nodes (i.e. settlements and demand points) as unvisited and assigns them infinity as distance. The algorithm begins with a starting node. This node is marked as visited and receives the distance 0. The distance of all its neighbors is then updated. The algorithm is thereafter iterating on all unvisited nodes. At each step the unvisited node with the lowest current distance from the starting node is picked. The node is marked as visited (and then its distance is the lowest distance to the starting node) and the distance of each of its neighbors to the starting node is updated if needed. The algorithm can stop at this stage if the node is the destination node. In our case, we continue the algorithm until all nodes are marked as visited since we need distances from one point to all the others. The resulting Origin-Destination (OD) matrix was created on a Dell Optiplex 9010 with an Intel Core I7-3770 (3.4 GHz) 32 Gb of RAM and a Linux operation system. It took 12.5 hours to generate the matrix. The final OD matrix is of the dimension 1,938 by 188,227 representing the candidate nodes of locating hospitals and the demand points in Sweden. 98 demand points were lost in the generation of the OD matrix due to residences without access to the road network. 4. Experimental design As stated in the introduction, we intend to address three issues. The first one is the effect of borders on inhabitants’ spatial accessibility to hospitals. 49
The second is the accessibility to hospitals without restrictions of borders and where hospitals are optimally located. The third is accessibility in relation to population dynamics. In addressing the first issue, we first compute the population’s distance to the nearest hospital along the shortest route. In this computation, the inhabitants may only patronize a hospital in their residential region. In the alternative scenario, the inhabitants may patronize hospitals in any region in which case boundaries implied by the borders are removed. Thus, we also compute the distance when the inhabitant may patronize the nearest hospital of any region. The second issue to be addressed is location of the current 73 hospitals. Are they located in a way that yields the best possible accessibility subject to the restriction of the 73 hospitals in the country? To answer the question we identify the optimum of the 73 hospitals where, by optimality, it is meant a location of the hospitals such that the population’s distance to the nearest hospital (irrespective of regional borders) is minimized. To find the optimal location of hospitals we use the p-median model. It can be stated as: 1 𝑀 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑅 ∑𝑁 𝑖=1 ∑𝑗=1 ℎ𝑖 𝑖𝑗 𝑥𝑖𝑗 s.t. ∑𝑀 𝑗=1 𝑥𝑖𝑗 = 1, 𝑖 = 1,2, … , 𝑁 𝑥𝑖𝑗 ≤ 𝑦𝑗 𝑀 ∑𝑗=1 𝑦𝑗 = 𝑝 where R is the number of inhabitants, I is the set of demand nodes indexed by i, J is the set of M candidate locations (i.e. settlements) indexed by j, ℎ𝑖 is the number of inhabitants in demand point i, 𝑖𝑗 is the distance of the shortest route between demand point i and candidate location j, and p is the number of hospitals to be located. Furthermore, 𝑥𝑖𝑗 equals one if the demand point i is assigned to a hospital at location j and zero otherwise, whereas 𝑦𝑗 equals one if a hospital is located at point j and zero otherwise. The distance is measured both as travel distance in meters and travel time in seconds in the road network. Often Euclidian distance is used as a distance measure, but it has been found to be unreliable (Bach, 1981; Carling et al, 2012; 2014). The p-median model assigns the inhabitants to the nearest hospital without considering the maximum capacity of a hospital. In this case, this might lead to absurdly large hospitals in Stockholm and Gothenburg since their large and concentrated populations are represented by one single 50
settlement each. To overcome the problem in the implementation of the pmedian model, we comply with the current situation by assigning three hospitals in the same candidate location in Stockholm and Gothenburg. To solve the p-median problem is a nontrivial task as the problem is NPcomplete (see Kariv & Hakimi, 1979) implying that enumeration of all possible solutions is infeasible. Much research has been devoted to develop efficient (heuristic) algorithms to solve the p-median model (see Daskin, 1995; Handler and Mirchandani, 1979). We solve the problem by using a common heuristic solution method known as simulated annealing (SA) (see Lenanova and Loresh, 2004). Alternative solution methods for the p-median model are extensively discussed by Reese (2006). The virtue of simulated annealing as other heuristic methods is that the algorithm will iterate towards a good solution being not necessarily the optimum as a stopping point must be given. As a consequence it is unknown whether the solution is close to the optimal solution or not. However, it has been shown that statistical confidence intervals derived by Weibull estimator may be used for estimating the uncertainty of the solution with regard to the optimum (Carling and Meng, 2014). We run SA until the confidence intervals are very tight – in matters of travel time it amounts to some seconds. As far as the third issue is concerned, identifying an optimal location of hospitals is done at a specific point of time. Is it likely that this optimum be robust to population dynamics? To address this question the population is divided by age and the optimal location of hospitals is identified for both the younger part of the population (20-39) and the older part (50-64). The dissimilarity of the two solutions is thereafter examined. In sum, the experiments related to the aim of the paper examine the current situation to a number of counterfactual scenarios with regional borders removed, national (and optimal) allocation of hospitals, and redistribution of the population. 5. Results 5.1 The effect of removing regional borders Table 1 shows the average and the median distance to the nearest of the current 73 hospitals. The inhabitants have on average 17.9 kilometers to their nearest hospital within the region while the median distance is 11.3 51
kilometers. The time it takes to travel the distance in the road network, assuming attained velocity to be the speed limit, is on average 15 minutes and 18 seconds while the median value is 11:06 minutes. If the population was free to patronize hospitals irrespective of regional borders, the distance would decrease somewhat. For instance, the inhabitants would on average have the distance to a hospital shortened by 0.6 kilometers or by 25 seconds. The resulting improvement in accessibility would be about 3 percent. The majority of the inhabitants would be unaffected by the removal of regional borders, a fact that follows from the median distance being (almost) identical in the current and the counterfactual situation. Table 1: The inhabitants distance to the nearest hospital within the region as well as within Sweden.
Unsurprisingly a fraction of the population living close to the regional borders would benefit from them being removed. To examine the size of this fraction of the population, we have computed each inhabitant’s Measure
Within the region Mean Median
Within Sweden Mean Median
Distance (km)
17.9
11.3
Time (min)
15:18
11:06
17.3 14:53
11.3 11:08
shortening of the distance to a hospital as a consequence of the removal of regional borders. Figure 3 gives the shortening in distance (in percent) to the nearest hospital. The figure shows that a majority of the inhabitants (55 per cent) would be unaffected as their nearest hospital already is located in their region of their residence. However, 45 percent of the inhabitants would be better off by having the opportunity of patronizing a hospital in a neighboring region. This opportunity would be of marginal importance though, as the shortening in distance is at the most of some 10 per cent. As a result, the removal of regional borders has little effect on improving the accessibility to hospitals in Sweden. Most inhabitants would be unaffected, but those affected would be subject to a modest improvement in accessibility.
52
8 Shortening of distance (%)
7 6
With distance measure in kilometres
5
with distance measure in minutes
4 3 2
1 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 Percentile of the population Figure 3: Rate of shortening in distance (in percent) to the nearest hospital due to removal of regional borders for percentiles of the population.
5.2 The effect of optimal location of hospitals The present spatial accessibility to hospitals is rather poor as the average distance between inhabitants and hospitals is 17.9 kilometers. Is this a result of the current 73 hospitals being poorly located with regard to the population? Table 2: The inhabitants distance to nearest current and optimally located hospital. a Current location Optimal location Shortening (%) Measure Mean Median Mean Median Mean Median Distance (km) 17.3 11.3 16.2 10.4 6.4 8.0 Time (min) 14:53 11:08 13:54 10:00 6.6 10.3 Note: a) The 99% confidence intervals for the mean values are (16.16-16.20 km) and (13:51-13:54 min).
Table 2 gives the average and the median distance to all current 73 hospitals for the population being unrestricted by regional borders (cf Table 1). It also shows the inhabitants’ distance to the nearest of 73 optimally located hospitals. The location of the 73 optimally located hospitals is depicted in Figure 4a. Finally, the table gives the resulting shortening of the distance as a consequence of hospitals being optimally located. The shortening in distance is of a modest 5-10 percent with the median indicating greater relative improvement for the inhabitants already 53
closest to the hospitals. Table 3: Relocation towards optimality. Number of hospitals, inhabitants affected and their distance to a hospital. Mean distance nearest Hospitals Inhabitants hospital Shortening Measure relocated Affected current optimal (%) Distance 17 1,323,599 25.9 21.6 16.7 (km) Time (min) 22 1,747,908 19:47 16:44 15.4
The scenario that the health care would be under national control with a resulting relocation of the 73 hospitals towards an optimal location is not far-fetched. Who would be affected if the scenario were to be realized? Table 3 gives some answers to this question. First of all, most of the current hospitals are optimally located. Only 17 (22 if optimized with respect to time) of the current hospitals would require relocation for attaining a maximum accessibility for the population under 73 hospitals (see Table 3). Secondly, a substantial proportion of the population, 24 per cent (31 per cent if optimized with respect to time), would be affected by the relocation towards optimal accessibility. Thirdly, the inhabitants affected by the relocation would have improvement in accessibility to hospitals with about 16 per cent.
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Figure 4a-b: Current and optimal 73 hospitals as well as inhabitants with improved and worsened accessibility as a consequence of optimal hospital configuration (a) and optimally located hospitals for inhabitants of 20-39 years and 50-64 years, respectively (b).
Relocation towards optimality would result in some inhabitants be closer to a hospital than presently and some inhabitants would be further away. To illustrate the underlying gross affect, inhabitants with improved and inhabitants with worsened accessibility are separated (Table 4). The magnitude of the improvement and the worsening is similar, but the number of inhabitants positively affected is about twice the ones negatively affected. Figure 4a visualizes the locations of the positively and negatively affected inhabitants (if optimized with respect to travel distance). In general, the relocation towards optimality implies a slight relocation from one town to the neighboring one. To draw conclusions regarding the location towards optimality issue, the current location of the 73 hospitals are not far from an optimal solution with regard to the population’s accessibility to hospitals. An optimal configuration of hospitals seems to be an exercise of carefully fine-tuning the location within the regions. 55
Table 4: Number of Inhabitants with affected accessibility by relocation of hospitals towards optimality, their distance to the nearest hospital, and change in distance. Measur e Distanc e (km) Time (min)
Improved accessibility Mean distance to a hospital Inhabitant Curren Optima Differenc s t l e 846,519 32.1 18.7 -13.4 1,163,453
23:16
14:34
-8:42
Worsened accessibility Mean distance to a hospital Inhabitant Curren Optima Differenc s t l e 477,083 15.0 26.7 11.7 547,450
12:15
21:02
8:47
5.3 Robustness of optimal location to population dynamics What effect do population dynamics have on the optimal locations? How much will a change in the spatial distribution of the population affect the accessibility to the optimal hospitals where the optimum is identified for a particular population at hand? We identify the optimum for groups of inhabitants. The first group is inhabitants aged between 50-64 years and the other group are those between 20-39 years. For these two sets of optimally located hospitals, we compute the accessibility for the inhabitants of 20-39 years. Figure 4b shows the location of the 73 hospitals optimized with respect to travel time and the two groups. The configurations for the two groups are similar and there are 58 hospitals coincide with each other. The figure indicates that the younger population would require more hospitals around Stockholm and Gothenburg at the cost of fewer hospitals in the northwestern part of the country. The requirement is however not very critical. The younger population has today 13:31 minutes on average to the nearest hospital. An optimal location of hospitals for them would only reduce the time to 12:25 minutes. How much worse off would the younger population be if they had to accept a configuration of hospitals optimized for the older? The answer is less than 1 per cent or 5 seconds since their travel time would increase to 12.30 minutes. Thus, an optimal location of hospitals seems to be robust to a long-term spatial redistribution of a population. 5.4 Miscellaneous results Returning to the issue of a reformation of the regional division in Sweden, what affect may it have on the spatial accessibility to health care? It is clear that the removal of borders is inconsequential. However, there is scope for some improvement by optimizing the location of hospitals. Is 56
such improvement likely to follow from merging neighboring regions? Figure 5 shows two parts of Sweden. To the left panel the region surrounding Gothenburg known as Västra Götaland (the dark gray area) is shown, but hereafter simply referred to as the Gothenburg region. To the right the Stockholm region as well as neighboring regions (the light gray area) is shown. The Gothenburg region is a forerunner in the regional reformation. In 1998 three regions near to Gothenburg were merged into the Gothenburg region and as a consequence the hospitals of the three independent regions came under the power of one region. Stockholm and the neighboring regions depicted in Figure 5 are candidates for being merged into a single region (hereafter the Stockholm region).
Figure 5: The factual Gothenburg region (dark gray area) and the hypothetical Stockholm region (light gray area).
If the reformation of the regional borders would have any effect on the interregional and suboptimal location of hospitals, then the hospital location in the Gothenburg region ought to be better than the Stockholm region which is not formed yet. This is checked by letting all regions be looked up with the current location of hospitals except the Gothenburg and 57
the Stockholm region where hospitals may be relocated to the optimum within the region. The population is free to patronize hospitals in any region. Recall from Table 1 that the average travel time was 14:53 minutes for the population to the current hospitals. If the hospitals in the Stockholm region were optimally located, the average travel time would decrease to 14:39 minutes. Yet, if the hospitals in the Gothenburg region were optimally located, the average travel time would similarly decrease to 14:38 minutes. Hence, there is no reason to expect that the formation of an extended Stockholm region would generate a better location of the hospitals in such a not yet formed region than today. The various experiments, so far, have indicated that any regional reformation will have little impact on the spatial accessibility to hospitals. One may wonder: how is the ongoing trend of concentration to fewer hospitals in Sweden as elsewhere affecting spatial accessibility? To address this question we have considered two scenarios. Out of the 73 hospitals in Sweden 48 of them are labelled emergency hospitals with a slight higher level of specialized care. The first scenario is that the 25 nonemergency hospitals would close and the country be left with current 49 emergency hospitals to serve the population. The average travel time would as a result increase by 26 per cent. The second scenario is that Sweden had twice as many hospitals as today, thereby being more similar to other European countries in terms of health care. In this scenario, the average travel time would decrease by almost a half (39 per cent). As a conclusion, the key to a spatially accessible health care is the number of hospitals. 6. Conclusion A national, regional or any other administrative border might be considered as barriers to the free flow of goods, services, and people. These barriers hinder the optimal allocation of a given set of resources. As a consequence, in particular in borderlands, the highest achievable economic and social utility may not be attained. For this reason, it seems sensible that the EU policy has been to promote cross-border transaction of goods, services, and labor towards a common European market. Public services have, however, been exempted from the free flow of services and largely confined by national and regional borders. The present EU policy 58
is, however, addressing the confinement of public services. So it is interesting to ask: Do the Europeans suffer from a poor accessibility to public services due to internal borders? In this paper we have attempted to address this question by studying the effect of administrative borders within Sweden on the population’s spatial accessibility by considering one prominent public service which is hospital service. We have elaborated several scenarios ranging from strongly confining regional borders to no confinements of borders as well as long-term population redistribution. Our findings imply that the borders are only marginally worsening the accessibility. Instead, the key to good spatial accessibility to hospital service is the number of hospitals. However, it is more likely that this number is under further decrease due to the ongoing concentration of hospitals. While we believe that the case of Sweden can be extrapolated to a European setting, it would be interesting to replicate the study on a European level. Acknowledgements We are grateful to Hasan Fleyeh for comments on an earlier version of this paper. References Anderson J, O'Dowd L, Wilson TM, 2002 “Introduction: Why Study Borders Now?” Regional & Federal Studies 12(4) 1-12. Avella P, Boccia M, Salerno S, Vasilyev I, 2012, “An aggregation heuristic for large scale p-median problem” Computers & Operations Research 397 1625-1632. Bach L, 1981, “The problem of aggregation and distance for analyses of accessibility and access opportunity in location-allocation models” Environment and Planning A: 138 955-978. Carling K, Han M, Håkansson J, 2012, “Does Euclidean distance work well when the p-median model is applied in rural areas” Annals of Operations Research, 201(1) 83-97. 59
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