A TABU SEARCH ALGORITHM FOR SOLVING THE EXTENDED MAXIMAL AVAILABILITY LOCATION PROBLEM

Transcription

1 A TABU SEARCH ALGORITHM FOR SOLVING THE EXTENDED MAXIMAL AVAILABILITY LOCATION PROBLEM Fernando Y. Chiyoshi Roberto D. Galvão Programa de Engenharia de Produção COPPE/Federal University of Rio de Janeiro, RJ, Brazil Rio de Janeiro, RJ, Brazil Reinaldo Morabito 1 Departamento de Engenharia de Produção Federal University of São Carlos São Carlos, SP, Brazil ABSTRACT: The objective of this study is to develop a tabu search (TS) procedure for the Extended Maximal Availability Location Problem (EMALP) and compare the results obtained with this procedure against those obtained with the Simulated Annealing (SA) procedure developed by Galvão et al. (2005) for the same problem. It is shown that in terms of quality the solutions SA outperforms TS for the smaller networks, while TS outperforms SA for the larger 200- and 250-node randomly generated networks. Comparative data related to processing times for both algorithms are also given. Keywords: Probabilistic location models; congested emergency systems; hypercube model; tabu search. 1 Corresponding author: morabito@ufscar.br 1

2 1. Introduction In a previous paper, Galvão et al. (2005) gave a unified view of the Maximal Expected Covering Location Problem (MEXCLP) of Daskin (1983) and of the Maximal Availability Location Problem (MALP) of ReVelle and Hogan (1989), identifying similarities and dissimilarities between these models and showing how they relate to each other. They also developed an extension of MALP with the hypercube model (see Larson and Odoni, 1981) embedded into it, and used simulated annealing (SA) in an attempt to enhance the local searches developed for the extended MEXCLP (see Batta et al., 1989; Chiyoshi et al., 2003) and MALP models. MALP is a probabilistic extension of its deterministic equivalent, the Maximal Covering Location Problem (MCLP), defined by Church and ReVelle (1974). In the MALP model there is a restriction that at least one server must be available within a critical distance S for any given demand area with probability greater than or equal to a given reliability. There are actually two versions of MALP: MALPI, where it is assumed that all servers have a common busy fraction, and MALPII, where different busy fractions are defined for different areas into which the region under study is divided. The Extended MALP model (EMALP), defined by Galvão et al. (2003, 2005), is an extension of MALPI which considers that each server has a different busy fraction. In this case the hypercube model is solved for each configuration of servers in order that the corresponding busy fractions are determined. A detailed review of probabilistic location problems can be found in Owen and Daskin (1998). Swersey (1994), Brotcorne, Laporte and Semet (2003), Ingolfsson et al. (2008) and Galvão and Morabito (2008) also examine some probabilistic models. The objective of this study is to develop a tabu search (TS) procedure (Glover and Laguna, 1997) for the extended MALP model and compare the results obtained with this procedure against those obtained with the SA procedure developed by Galvão et al. (2005). This procedure uses a local search heuristic based on the vertex substitution algorithm of Teitz and Bart and extended as in Chiyoshi and Galvão (2000). We did not consider applying TS to the extended MEXCLP because, as pointed out in that paper, the results produced by the SA procedure suggest that its objective function is topped by an easy to reach, plateau-like, slightly irregular surface, with many local optima very close to each other. We therefore would not expect TS to produce results that were very different from those obtained using SA. 2

3 This however is not the case for EMALP, since the coverage provided by different SA solutions are at great variance, with the better solutions produced by SA (as compared to those produced by a local search) being of considerable practical importance. As a consequence we considered that TS might further improve the results shown in the previous paper in this case. In the present study tests were conducted using both the 55, 100 and 150-node test problems used in that paper and randomly generated networks of up to 250 nodes (with 5, 10 and 15 servers in each case). It is shown that in terms of quality of the solutions SA outperforms TS for the smaller networks, while TS outperforms SA for the larger 200 and 250-node randomly generated networks. Comparative data related to processing times for both algorithms are also given. The SA algorithm was found to be faster than the TS algorithm. The SA to TS processing time ratios vary in an irregular way depending on the network size and the number of servers. On the whole it can be said that, on average, the SA algorithm is 10% to 15% faster than the TS algorithm. This difference is attributable to the nature of the TS algorithm, which requires more bookkeeping work than the SA algorithm. The paper is organized in the following manner. The tabu search procedure and the fine tuning of its parameters are described in Section 2. This is followed, in Section 3, by the description of the test problems used and the corresponding computational results that were obtained. A Conclusions section closes the paper. 2. The Tabu Search Procedure The implementation of the TS procedure followed the probabilistic model proposed by Xu et al. (1998). It works as follows: starting from an initial solution, the TS algorithm searches for better solutions using a node substitution procedure (for details of this substitution procedure see Section 4 of Galvão et al., 2005) associated with a tabu list. In each iteration of the algorithm one node in the solution is tested against all nodes not in the solution as to the effect of the substitution on the current value of the objective function. If a node not in the solution is found that improves the current objective function, the substitution is made provided that the node is not in the tabu list. However, the tabu condition is overridden if a tabu node is capable of improving the best available solution (aspiration criterion). The node leaving the solution is made tabu for a certain number of iterations. 3

4 The tabu condition is imposed in order to prevent a downhill move being reversed in the next cycle (the processing of all nodes in the solution is defined as a cycle of the algorithm). It is clear that there is no need to impose the tabu condition on the node entering the solution. If no substitution that improves the current objective function is found, a node to enter the solution is selected probabilistically starting from the best nontabu node, using a node selecting probability prob (and node rejecting probability (1- prob)). If this selection fails, the second best non-tabu node is selected based on the same probability. This process is repeated until a node is eventually selected or the k-th best node is not selected (we impose a limit k on the number of nodes probabilistically tested), in which case the first best node is selected. The search process consists of three phases. The first phase is a basic initial search phase in which, starting from a random initial solution, the TS algorithm is run for a certain number of cycles and a set of best solutions is found (the elite solutions). The elite solutions are used in the next phase, the intensification phase, whose purpose is to explore more intensely the region of the objective function found to be more promising in the previous phase. In the intensification phase, the search algorithm is run using each elite solution as the initial solution, from worst to best, with all tabu restrictions removed. In this phase the elite solutions are immediately updated if a new better solution is found. The third phase is the diversification phase, in which the search is directed to the least explored region of the objective function. In this phase the algorithm is run with initial solutions drawn from the set of nodes that least frequently entered the solution in the previous phases. In order to produce data comparable to the results obtained through simulated annealing (reported in Galvão et al., 2005), the length of the search runs for the TS algorithm were set to 30 cycles, so that the computational effort to solve the hypercube model is the same in both cases. These cycles were distributed among the three phases as follows: cycles 1 to 5 for the initial search, cycles 6 to 20 for the intensification phase and cycles 21 to 30 for the diversification phase of the algorithm. The TS algorithm described above has four parameters that need to be defined: the tabu tenure of the nodes, the node selecting probability (prob), the number of candidate nodes to enter the solution (k) and the number of elite solutions to be used in the intensification phase. The setting of these parameters was made based on the method proposed by Xu et al.(1998), using the same 100-node, 10-server problem used in Galvão et al. (2005) to set up the parameters of the SA algorithm. 4

5 The fine tuning method, called by Xu et al. (1998) the tree growing and pruning method (referred to as the TG&P method in the following), starts in an initial node from which leaves are grown for different settings of the first parameter, the tabu tenure of the nodes in our case. The test problem is run and results collected for each leaf. Then Friedman s test is used to prune the leaves associated with results significantly worse than the best result. The tabu tenure settings tested by us were: (i) Nu (number of servers); (ii) random between 1 and Nu, denoted as R(1,Nu); (iii) Nu/2 and (iv) random between 1 and Nu/2, denoted R(1,Nu/2). The values of the remaining parameters were defined as follows. The node selecting probability was set to 0.30, the value reported by Xu et al. (1998) as the most commonly used value for prob. The number of elite solutions in the intensification phase of the TS is conditioned extent by the number of cycles assigned to that phase, 15 cycles in our case. Five elite solutions seemed reasonable, allowing three search cycles for each elite solution. Using these settings, the test problem was run 20 times for each setting of the tabu tenure parameter, starting from a different initial solution each time. The relevant data for Friedman s test are shown in Table 1, where Rank Sum is a statistics used in the test and Contrast refers to the significance test of the statistic of each setting with respect to the best setting. Table 1 - Test on Tabu Tenure Run Tenure Nu R(1,Nu) Nu/2 R(1,Nu/2) Rank Sum Contrast yes no * no * - best run Friedman chi-squared: 4.47 p-value: Friedman s test leads to the pruning of the leaf associated with Run 1. The next step of the TG&P method is to grow, from each remaining leaf, new leaves for alternative settings of the next parameter, the node selecting probability prob in our case. Four values were used for prob (0.3, 0.5, 0.7 and 1.0) to grow three sets of leaves from each remaining leaf (Tables 2 to 4). 5

6 Table 2 - Test on prob for Run 2 Run prob Rank Sum Contrast yes no * no * - best run Friedman chi-squared: p-value: Table 3 - Test on prob for Run 3 Run prob Rank Sum Contrast yes no no * * - best run Friedman chi-squared: p-value: Table 4 - Test on prob for Run 4 Run prob Rank Sum Contrast yes yes no * * - best run Friedman chi-squared: p-value: < It should be noted that the first run at the second tree level duplicates the run associated with parent node (for prob=0.3). Using Friedman s test, runs 2, 3, 4 and 11 are pruned. The second pruning procedure of the TG&P method calls for the use of Wilcoxon s test for paired samples to compare pairwise leaf nodes that do not share a parent node and prune the leaf associated with the inferior result. Since run 13 produced the best result, all remaining results (except for run 12, which shares the same parent node with run 13) are compared with the result of run 13. The relevant statistics are displayed in Table 5. 6

7 Table 5 - Wilcoxon tests on prob pair (13,5) (13,6) (13,7) (13,8) (13,9) (13,10) s p-value s+ - Wilcoxon test statistic If we choose the significance level of 0.10, it is seen that run 13 produced significantly better results than all other runs, except for run 10. At this stage, out of 4x4 possible settings for two parameters with four alternative values each, we are left with three leaves (settings) for further analysis: run 10 [Tabu Tenure=Nu/2 and prob=1.0], run 12 [Tabu Tenure=R(1,Nu/2) and prob=0.7] and run 13 [Tabu Tenure=R(1,Nu/2) prob=1.0]. The next parameter analyzed was the number of the elite solutions to be used in the intensification phase. Five values were tested, from 1 to 5. After the same pruning process, two alternative settings (statistically equivalent) were left for the number of elite solutions: 2 and 4, both in leaves growing from the node associated with run 13. Based on the sample value, the final choice fell on 2 as the number of elite solutions to be used in the intensification phase. The final settings of the parameters of the TS algorithm are as follows: tabu tenure, random between 1 and Nu/2; entering node selection probability, 1.0 (in fact a deterministic selection, which makes irrelevant the number of candidate nodes); number of elite solutions, Test Problems and the Results Obtained The test problems used in this study are based on 11 networks. Networks 1, 2 an 5, with 55, 100 and 150 nodes, respectively, are those referred to in Galvão et al.(2005). The remaining eight networks are randomly generated networks of N = 100, 150, 200 and 250 nodes. For the randomly generated networks, for each network size N, two networks were constructed: the first with nodes generated randomly on an NxN square, with the nodal demands generated randomly between 1 and N; the second with nodes generated randomly on an 2Nx2N square, with the nodal demands generated randomly between 1 and 2N. For each network problems based on 5, 10 and 15 servers (a total of 33 problems) were solved 20 times by each method (SA and TS), starting from a different initial solution each time. In some cases, for a larger number of servers a tighter critical coverage 7

8 distance S was used, in order to keep the coverage not too close to 100%, so avoiding that the maximum value of the objective function coincides with the population size. The comparative data is given in terms of the best solution and the average quality of the solutions produced by the two methods in 20 runs. The average quality of the solutions is expressed in terms of the average rank associated with the solutions produced by each method. The procedure used in this study to evaluate the average rank consists in obtaining the frequency distributions of the solutions and determining the average rank of each method (SA and TS) in terms of the ranks associated with the distinct values of the objective function. This is shown in Table 6 for the 100 nodes, 5 server problem. Table 6 - Frequency Distribution of the Solutions for Problem #2 (100 Nodes/5 Servers) Cov Rank Frequencies SA TS Average Rank The comparative data for the solutions produced by SA and TS algorithms for the test problems are shown in Table 7 and Table 9; in these tables the best solution in each case in shown in bold. In terms of the best solution produced in 20 runs (Table 7), except for five server problems, the performance of the algorithms is dependent on both the size of the network and the number of servers. For the five server problems, both algorithms found the same best solutions for all 11 networks tested. For the 10 and 15 server problems, except for a tie in the 55 nodes/15 server problem, the algorithms produced different solutions. For the 55 and 100 nodes networks, the SA algorithm either matches or outperforms the TS algorithm. For the 150 nodes networks there is equivalence between the algorithms: in three out of six problems 8

9 the SA outperforms TS, and is outperformed by TS in the other three. For the 200 and 250 nodes networks, the TS algorithm clearly shows a better performance: it outperforms the SA algorithm in six out of eight problems. A summary of the frequency of best solutions is shown in Table 8. Bearing in mind the caution that must be observed in looking at the figures of Table 7, given their heavy dependence on network size, it is interesting to observe that only in four out of 22 problems the absolute value of the gap between the best solutions produced in 20 runs by the SA and TS algorithms exceeded 5%. Table 7 - Comparative data related to the best solutions produced by the SA and TS algorithms Algorithm-> SA TS SA to TS Gap Network Nodes\Servers (*) ,3% 0,0% 2(*) ,3% 0,7% ,5% 12,2% ,4% 0,8% 5(*) ,7% 3,2% ,8% -1,4% ,9% 14,5% ,9% 1,0% ,5% -5,5% ,5% -0,6% ,4% -5,4% (*) - Networks solved by SA in Galvão et al. (2005). Table 8 Frequency of best solutions (10 and 15 server problems) Nodes SA TS 55 & & In terms of the average quality of the solutions (see Tables 9 and 10), the first fact to observe is that for 5 server problems the TS algorithm outperforms the SA algorithm in 9 9

10 out of 11 problems. For the 10 and 15 server problems, the behavior of the algorithms in terms of the average quality of the solutions is similar to that in terms of the best solution produced after 20 runs: the SA algorithm shows a better performance for the 55 and 100 nodes networks; there is an equivalence in performance for the 150 nodes networks; TS outperforms SA for the larger networks. Table 9 - Comparative data related to the average quality of the solutions produced by the SA and TS algorithms Algorithm-> SA TS SA to TS Gap Network Nodes\Serv ers (*) 55 1,0 8,9 3,8 1,0 10,0 3,7 0,0-1,2 0,1 2(*) 100 3,6 14,0 10,5 2,3 20,9 18,5 1,3-7,0-8, ,0 16,5 16,8 3,0 18,4 17,8 3,0-1,9-0, ,9 17,3 13,8 2,8 14,6 22,1-1,0 2,7-8,3 5(*) 150 3,4 14,0 12,4 1,9 20,3 24,0 1,5-6,4-11, ,6 19,2 22,6 2,9 16,7 18,4 2,8 2,6 4, ,4 20,2 19,0 1,6 15,5 22,1 0,8 4,7-3, ,8 26,1 15,1 3,1 13,2 24,3 0,7 12,9-9, ,7 21,8 24,2 3,2 17,8 16,8 1,5 4,0 7, ,6 22,9 23,2 3,9 17,1 16,0 2,7 5,8 7, ,9 19,0 22,5 2,5 19,4 16,6 0,4-0,4 5,9 (*) - Networks solved by SA in Galvão et al. (2005) Table 10 - Frequency of solutions with better average quality (10 and 15 server problems) Nodes SA TS 55 & & The comparative data for processing times for both algorithms are shown in Table 11. These times are derived from Pascal/Delphi based codes running on a P4/1.4 GHz computer with 256 Mb of RAM, running on a stand-alone basis. The SA algorithm was faster than the TS algorithm. The SA to TS processing times ratios vary in an irregular fashion, depending on network size and number of servers. On average the SA algorithm 10

11 is 10% to 15% faster than the TS algorithm. This difference may be attributed to the nature of TS algorithm, which requires more bookkeeping work than the SA algorithm. In fact, for the SA algorithm, in addition to the use of the Metropolis criterion for deciding a downhill move, there is only the ratio of the number of accepted moves to the number of rejected moves to keep track of. In relation to the TS algorithm, there is the management of the tabu list and the collection of long-term statistics for the diversification phase. Table 11 - Comparative data for average processing times per run for The SA and TS algorithms (seconds) Algorithm-> SA TS SA to TS Ratio # Nodes\Servers ,97 75,18 391,34 7,65 90,06 478,87 0,78 0,83 0, ,41 154,53 681,62 18,65 176,70 808,38 0,83 0,87 0, ,75 307, ,32 38,83 331, ,58 0,87 0,93 0, ,98 482, ,74 61,81 517, ,95 0,87 0,93 0, ,40 695, ,25 96,02 785, ,43 0,87 0,89 0,87 Both the SA and TS algorithms allow downhill moves to escape from local maxima. The SA algorithm applies the Metropolis criterion for deciding on downhill moves, assigning lower probabilities for larger variations in the objective function. In each iteration the decision is made on a single candidate node entering the solution: the node is either accepted or rejected. The Metropolis criterion is adjusted so that the ratio of accepted to rejected moves is close to 1:1, following Metropolis objective of making the algorithm follow the behavior of the objective function. As a result of this adjustment only about half the iterations result in a new point of the objective function. For the TS algorithm, as implemented in this work, when a better substitute is not found for a given node, a downhill move is always selected. Thus the number of points of the objective function visited by TS algorithm is about twice those visited by SA algorithm. It is therefore rather unexpected that the SA algorithm has a better performance than the TS algorithm for the 55 and 100 node networks. It appears that the sample produced by SA algorithm, although smaller, is more representative of the objective function than that produced by TS algorithm. We may then ask why this advantage disappears (for the 150 nodes network) or is even reversed in favor of TS algorithm for the 11

12 Cycles larger networks. If we look at the heuristic procedures as sampling procedures, we may conjecture that this has to do with the sample size. Chart 1 provides some evidence in that direction. This Chart displays the number of cycles (for each of the 20 runs) in which the best solutions were found in three 10 server problems, two solved through SA (100 and 200 nodes networks) and one through TS (250 nodes network). In each curve the number of cycles to the best solution was sorted in ascending order. The data associated with the solutions obtained by SA method show that for the 100 nodes network the limit of the search to 30 cycles apparently did not interfere on the search process. For the 200 nodes network, on the other hand, the data show that the limit of 30 cycles may have prevented the SA algorithm from finding better solutions. The data for the TS method are included only for illustrative purposes since they are not strictly comparable to the SA data, due to the fact that the TS method is divided into phases; for example, the presence of the diversification phase of the TS algorithm has no counterpart in the SA algorithm. It is worth noticing, however, that in two of the 20 runs the best solutions of the TS algorithm were found in the diversification phase. Chart 1 Cycles to best solution for three 10 server problems SA100 SA200 TS Run The TS algorithm has the additional feature that it uses the tabu condition to prevent the reversal of a downhill move in the next few cycles. Since the decision for 12

13 substitution is made for each node in the solution, this would only be achieved with a tabu tenure for a solution leaving node not less than Nu. In our case the fine-tuning of the TS algorithm lead to a tabu tenure randomly chosen from 1 to Nu/2. This would allow the algorithm to reverse a downhill move in the next cycle, so that the most distinctive feature of the TS algorithm was not at work in our model. It may be assumed that the use of the tabu condition must have affected the search path of the TS algorithm, but its effect on the corresponding performance is hard to assess. 4. Conclusions Our study did not produce evidence of a clear dominance of either method over the other. Given an appropriate search length (like the 30 cycles for the 55 and 100 nodes networks) the SA algorithm may produce better results than the TS algorithm for the same run length. On the other hand, when the length is insufficient (like the 30 cycles for 200 and 250 nodes networks), the SA algorithm is outperformed by TS algorithm for the same run length. This is equivalent to saying that, in order to produce comparable performances, the SA algorithm seems to require longer runs than the TS algorithm. This result could be anticipated due to the fact that the SA algorithm visits about half the number of points visited by the TS algorithm. In relation to computing effort, for a fixed search length the TS algorithm is more expensive than the SA algorithm (although not overwhelmingly so). The computing time difference in favor of the SA algorithm may be attributed to the difference in the bookkeeping activities required by the algorithms. The only requirement of the SA algorithm is the number of accepted and rejected moves in order to adjust the temperature of the algorithm so as to keep the ratio of these moves close to 1:1. The TS algorithm, on the other hand, requires the collection to two types of statistics, one related to the management of the tabu list and the other to the frequency of the use of the nodes (for the diversification phase of the algorithm). References 1. R. Batta, J.M. Dolan and N.N. Krishnamurthy. (1989). The maximal expected covering location problem: Revisited, Transportation Science 23,

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