Different initial solution generators in genetic algorithms for solving the probabilistic traveling salesman problem

Applied Mathematics and Computation 216 (2010) 125–137

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Different initial solution generators in genetic algorithms for solving the probabilistic traveling salesman problem Yu-Hsin Liu Department of Civil Engineering, National Chi Nan University, No. 1, University Road, Puli, Nantou Hsien 54561, Taiwan, ROC

a r t i c l e

i n f o

Keywords: Genetic algorithm Initial solution generator Probabilistic traveling salesman problem Permutation test TSPLIB

a b s t r a c t The probabilistic traveling salesman problem (PTSP) is a topic of theoretical and practical importance in the study of stochastic network problems. It provides researchers with a modeling framework for exploring the stochastic effects in routing problems. This paper proposed three initial solution generators (NN1, NN2, RAN) under a genetic algorithm (GA) framework for solving the PTSP. A set of numerical experiments based on heterogeneous and homogeneous PTSP instances were conducted to test the effectiveness and efficiency of the proposed algorithms. The results from the heterogeneous PTSP show that the average E½s values obtained by the three generators under a GA framework are similar to those obtained by the ‘‘Previous Best,” but with an average computation time saving of 50.2%. As for the homogeneous PTSP instances, NN1 is a relatively better generator among the three examined, while RAN consistently performs worse than the other two generators in terms of average E½s values. Additionally, as compared to previously reported studies, no one single algorithm consistently outperformed the others across all homogeneous PTSP instances in terms of the best E½s values. The fact that no one initial solution generator consistently performs best in terms of the E½s value obtained across all instances in heterogeneous cases, and that the performance of each examined algorithm is dependent on the number of nodes (n) and probability (p) for homogeneous cases, suggest the possibility of context-dependent phenomenon. Finally, to obtain valid results, researchers are advised to include at least a certain amount of test instances with the same combination of n and p when conducting PTSP experiments. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction The probabilistic traveling salesman problem (PTSP) is an extension of the well-known traveling salesman problem (TSP), which has been extensively studied in the field of combinatorial optimization. The goal of the TSP is to find the minimum length of a tour to all customers, given the distances between all pairs of customers, whereas the objective of the PTSP is to minimize the expected length of the a priori tour, where each customer requires a visit only with a given probability [1–3]. The main difference between the PTSP and the TSP is that in the PTSP the probability of each node being visited is between 0.0 and 1.0, while in TSP the probability of each node being visited is 1.0. Due to the fact that the element of uncertainty not only exists, but also significantly affects the system performance in many real-world transportation and logistics applications, the results from the PTSP can provide insights into research in other probabilistic combinatorial optimization problems. Moreover, the PTSP can also be used to model many real-world applications in logistical and transportation planning, such as daily pickup-delivery services with stochastic demand, job sequencing involving changeover cost, design

E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.021

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of retrieval sequences in a warehouse or in cargo terminal operations, meals on wheels in senior citizen services, trip-chaining activities, vehicle routing problem with stochastic demand, home delivery service under e-commerce, and probabilistic arc-routing problems for small-package local routing [4–9]. Early PTSP computational studies, dating from 1985, adopted heuristic approaches that were modified from the TSP (e.g., nearest neighbor, savings approach, spacefilling curve, radial sorting, 1-shift, and 2-opt exchanges) [1–3,10–13], but due to the less than satisfactory performance with regard to solution quality, researchers in recent years have turned to metaheuristic methods, such as ant colony optimization [14–18], evolutionary algorithms [19], greedy randomized adaptive search procedure (GRASP) [20,21], honey bees mating optimization [22], iterative local search algorithms [18,23–26], memetic algorithms [18], particle swarm optimization [21], scatter search [27,28], simulated annealing [29] and threshold accepting [9]. Since genetic algorithms (GA), using the conceptual framework of the population-based metaheuristic method, have been shown to yield promising outcomes for solving various complicated optimization problems over the past three decades [30–34], this study will further investigate the effects of different methods based on random generation and nearest neighbor algorithms for generating initial solutions under a GA framework for solving the PTSP. To validate the effectiveness and efficiency of the proposed algorithmic procedures, a set of heterogeneous and homogeneous PTSP test instances, as investigated in previous studies, will be used as the basis for comparison. The remainder of this paper is organized as follows. In the next section, expressions for exactly and approximately evaluating the a priori tour for the PTSP are introduced. The details of the proposed algorithmic procedure for the PTSP are then described. The results of the numerical experiments are presented and discussed in the next section, followed by concluding comments. 2. Definition and evaluation of the PTSP The PTSP is defined on a directed graph G :¼ ðV; EÞ, where V :¼ f0; v 1 ; v 2 ; . . . ; v n g is the set of nodes or vertices, and E # V  V is the set of directed edges. Node 0 represents the depot. Each non-depot node v i is associated with a presence probability pi that represents the possibility that node v i will be present in a given realization. Given a directed graph G, the PTSP is to find the a priori Hamiltonian tour with the minimal expected length in G. Solving the PTSP mainly relies on computing the expected length of an a priori tour. The computation of the expected length of a specific a priori PTSP tour s, denoted as E½s, depends on the relative location of nodes on that tour and the presence probability of each node in a given instance. Jaillet and Odoni [35] proposed an approach to exactly calculate E½s in the complexity of O(n3) for the PTSP.

E½s ¼

n nþ1 X X

( dsðiÞsðjÞ psðiÞ psðjÞ

i¼0 j¼iþ1

j1 Y

) ð1  psðkÞ Þ

ð1Þ

k¼iþ1

dij represents the distance between nodes i and j; sðiÞ denotes the node that has been assigned the ith stop in tour s and psðiÞ is the probability of node sðiÞ. sð0Þ and sðn þ 1Þ represent node 0, which is the depot. Even though (1) yields a polynomial evaluation time for the PTSP, the resulting O(n3) time for calculating E½s is still very long, especially for metaheuristic methods which need to repeatedly evaluate the objective function value E½s. In this study, the proposed algorithm needs to repeatedly compare two solutions (i.e., the new solution before and after the local search procedure, which is described in the next section) based on their values of E½s. Therefore, the depth approximate evaluation [16,28] was used to increase the computational efficiency of the proposed algorithm.

EAP k ½s ¼

n X

minfnþ1;iþkg X

i¼0

j¼iþ1

( dsðiÞsðjÞ psðiÞ psðjÞ

j1 Y

) ð1  psðkÞ Þ

ð2Þ

k¼iþ1

The only difference between (1) and (2) is the choice of truncation position k in (2). Eq. (2) will have the computational complexity of Oðnk2 Þ, instead of Oðn3 Þ in (1). It is easy to see that (2) becomes more accurate when k increases. A larger value of k, however, requires more computational effort for the computation of (2). Eq. (2) can perform a very good approximation Qj1 ð1  psðkÞ Þ will yield a very small value and of E½s with a smaller value of k when the value of psðkÞ gets larger, because k¼iþ1 can be omitted. Nevertheless, Eq. (2) will need a larger value of k to perform a good approximation when the value of psðkÞ is small. The detailed procedure of using the approximate evaluation shown in (2) to accelerate the algorithm is described in the next section. 3. Solution algorithms The proposed genetic algorithm consists of four components, as shown in Fig. 1. They are the initialization, including three initial solution generators – random generator (RAN), nearest neighbor algorithm – type 1 (NN1), nearest neighbor algorithm – type 2 (NN2); local search with stochastically selecting from two different methods (i.e., 1-shift and 2-opt exchanges); selection; and edge recombination (ER) crossover. When starting to solve the PTSP (Generation 0, g ¼ 0), initial solutions are created based on three different generators, which are then improved by the local search. ‘‘Selection” is then utilized to further select solutions to be mated based on their solution quality (objective function value). Pairs of solutions

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Initialization (g=0)

Selection

Random (RAN) NN1

Fitness-proportional Tournament selection

NN2

Elitism g=g+1

ER Crossover Local Search 1-shift 2-opt

No

Stopping criteria met? Yes STOP

Fig. 1. The GA procedure with three initial solution generators for solving the PTSP.

are used to generate the new solutions via ER crossover. The newly generated solutions are then improved using the local search. The solutions are allowed to evolve through successive generations until a termination criterion is met. A detailed description of the embedded components is given in the following sections. 3.1. Initialization This procedure is designed to generate m initial solutions (m = 10 in this study). Three initial solution generation methods are proposed. The detailed procedures of these three methods are described as follows. 3.1.1. Random generator (RAN) The random generator is used to randomly create a set of initial solutions. Considering a PTSP with n nodes (excluding the depot, node 0), the node, ar1 , is randomly chosen from the set of unselected nodes ðSu Þ and inserted into the second location. The set of unselected nodes ðSu Þ should then be updated by erasing node ar1 . The third location can be filled by randomly choosing a node from Su . Following the above procedure, the initial solution (tour) can be built. 3.1.2. Nearest neighbor algorithm – type 1 (NN1) Considering a PTSP with n nodes (excluding the depot, node 0), the two nearest nodes, a0 and b0 , from node 0 are first selected and randomly inserted into the second and the last locations, respectively. Basically, the nearest and second nearest nodes from the current node are selected by the probability of 0.9 and 0.1, respectively. This mechanism is used to generate different initial solutions. The procedure of building the initial solution is as follows. After selecting nodes a0 and b0 , one of the top two nearest nodes from a0 is randomly selected ða1 Þ based on the specific probability (i.e., 0.9 for the nearest node and 0.1 for the second nearest node) and inserted behind a0 . Similarly, the node ðb1 Þ is selected and inserted in front of b0 . Then, among the remaining nodes, one of the top two nearest nodes from a1 is randomly selected ða2 Þ based on the specific probability and inserted behind a1 , while node b2 is randomly selected based on the rule described previously and inserted in front of b1 . The initial solution (tour) is thus built by following the above rule, and can be expressed as follows.

0

a0

a1

a2

......

b2

b1

b0

0

3.1.3. Nearest neighbor algorithm – type 2 (NN2) Considering a PTSP with n nodes (excluding the depot, node 0), the farthest node, a0 , from node 0 is selected first and randomly inserted into a location between ðbðn þ 1Þ=2c  4Þ and ðbðn þ 1Þ=2c þ 4Þ. The reason why bðn þ 1Þ=2c  4 and bðn þ 1Þ=2c þ 4 are used rather than the middle point of the tour ðbðn þ 1Þ=2cÞ is to generate a set of different initial solutions. The nearest neighbor algorithm, which finds the unvisited node with the shortest distance, is used to build up the sequence of the tour. Basically, the nearest and second nearest nodes from the current one are selected by the probability of 0.9 and 0.1, respectively. This mechanism is used to generate different initial solutions. The procedure of building the initial solution is as follows. After selecting node a0 , one of the top two nearest nodes from a0 is randomly selected ða1 Þ based on the specific probability and inserted in front of a0 . Similarly, the node a2 is selected and inserted behind a0 . Then, among the remaining nodes, one of the top two nearest nodes from a1 is randomly selected ða3 Þ based on the specific probability and inserted in

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front of a1 , while node a4 is randomly selected based on the rule described previously and inserted behind a2 . The initial solution (tour) is thus built by following the above rule, and is expressed as follows.

0

...... a5

a3

a1

a0

a2

a4

a6

...... 0

3.2. Local search This component is used in an attempt to further enhance the solution generated via a local search procedure. As a previous study [28] showed that the diversified local search by randomly choosing two different local search methods (i.e., 1shift and 2-opt) to improve the solution generated yielded the best results in both heterogeneous and homogeneous cases, the diversified local search strategy is used in this procedure. That is, one of these two local search methods is selected to improve the candidate solution based on the specific probability of 0.5 for these two local searches. The procedures of 1-shift and 2-opt exchanges are briefly summarized, as follows. Given an a priori tour s, its 1-shift neighborhood is the set of tours obtained by moving a node at position i to position j with the intervening nodes being accordingly shifted backwards one space. The 2-opt exchange is the set of tours obtained by reversing a section of s. The approximate evaluation of the expected length of the a priori tour shown in (2) is then used to increase the computational efficiency. For a specific tour s; EAP k ½s is always less than the value of E½s because of the truncation in calculating EAP k ½s. Let sb and sa denote the a priori tour before and after a specific local search method, respectively. This means that no improvement has been found after the local search if EAP k ½sa =E½sb . Eq. (1) is used to exactly evaluate the solution after ½ s  < E½ s . If the local search yields a better E½s value than the one from the original solution (i.e., the local search if EAP a b k E½sa  < E½sb ), the new solution ðsa Þ will replace the original one ðsb Þ. If no improvement has been found after the local search, no replacement will be made. The above procedure is repeated N LS times for each solution (N LS ¼ 30 in this study). 3.3. Selection Three selection methods are used in this study: fitness-proportionate, elitism and tournament selection. Under the fitness-proportionate selection method, the probability of selecting a particular solution for reproduction is proportional to its own fitness (i.e., E½s) relative to the average fitness of the entire current generation. With this selection method, the best solution tends to produce the largest amount of offspring and hence survive to future generations. This procedure can be regarded as a ‘‘biased” roulette wheel, where each string in the current population occupies a roulette wheel slot sized in proportion to its fitness [32]. Selection can be done by simply spinning the roulette wheel, and fitter strings will have higher chances of being selected. Tournament selection, inspired by the competition in nature among individuals for the right to mate, picks two solutions using the fitness-proportionate selection from the population, and the fittest one is selected for reproduction [31,32]. Each solution can participate in an unlimited number of tournaments. The two winning solutions in the tournament are then subjected to the crossover operators, as described in the next section. Under the elitism selection strategy, the top m1 strings (m1 is determined by the analyst) of the current generation in terms of fitness value are kept and propagated to the next generation [31]. The remaining solutions in the next generation are then generated based on the tournament selection method and the crossover operators. This procedure guarantees that the best solution in the next generation is not worse than that in the current generation. 3.4. Edge recombination (ER) crossover The main purpose of this component is to create new solutions using a given pair of solutions chosen by ‘‘selection.” Based on the results from previous studies [19,36], the edge recombination (ER) crossover from genetic algorithms performed best when compared to other crossover strategies for both TSP and PTSP. Therefore, ER crossover was adopted in this study. ER crossover was proposed by Whitley et al. [37] to solve the traditional TSP. A five-node PTSP is used as an example to describe the procedure of ER crossover. Assuming that two solutions (tours) are chosen from the ‘‘selection”– (0, 4, 3, 1, 2, 0) and (0, 1, 2, 3, 4, 0), the edges connected to each node are as follows. For node 0, the first solution indicates that node 0 connects to nodes 2 and 4, and the second solution shows that node 0 connects to nodes 1 and 4. Therefore, node 0 connects to nodes 1, 2, and 4 by considering these two solutions. Similarly, node 1 connects to nodes 0, 2, 3; node 2 connects to nodes 0, 1, 3; node 3 connects to nodes 1, 2, 4; and node 4 connects to nodes 0, 3. These are the initial edge lists for each node. The operation of the ER crossover is described as follows. Assuming that node 0 is selected as the starting node for the new solution, all edges incident to node 0 must be deleted from the initial edge list. From node 0 we can go to nodes 1, 2, or 4, while nodes 1 and 2 have two active edges and node 4 has only one active edge by deleting node 0 from the initial edge list. The node with the fewest active edges, node 4, is picked as the node next to node 0 in the new solution. The edge list for the remaining nodes (nodes 1, 2, and 3) is then further updated by deleting node 4. The updated edge list is node 1 (2, 3), node 2 (1, 3), and node 3 (1, 2). From node 4, we can only go to node 3 (as node 0 is already deleted from the list).

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Therefore, node 3 is chosen to be the node next to node 4 in the new solution. The newly generated solution is then further improved by the local search. 3.5. The procedure after the first generation The newly generated solutions from the ER crossover and local search are used to update the population in terms of the objective function value. The above procedure is repeated until a termination criterion is met. To compare the effectiveness among three initial solution generators, the preset maximum number of generations ðg max Þ was used as the termination criterion (g max is set to be two times the number of nodes, i.e., g max ¼ 2n, in this study). However, if there are no solutions to be updated for the population in the current generation, the initialization is used to generate ðm  m1 Þ new solutions in the next generation, but keeping m1 high-quality solutions (m1 ¼ 2, in this study). In addition, if the previous three generations converge to the same best solution, the local search is used to improve this solution by repeating N LS2 times to exhaustively search its neighborhood (N LS2 ¼ 300, in this study). 4. Numerical experiments A set of numerical experiments was performed to achieve two objectives: to assess the performance of the proposed GA with three initial solution generators as compared to those reported in previous studies, and to examine any differential effects in terms of computational time and average E½s value among the three proposed initial solution generators (i.e., RAN, NN1, NN2) with regard to solving the heterogeneous and homogeneous PTSP instances under a GA framework. For the heterogeneous PTSP data, ninety instances, generated by Tang and Miller-Hooks [9] and later investigated by Liu [27,28], were used in this study. Three groups of problem sets categorized by different intervals of customer presence probabilities were created for each problem size (n = 50, 75, and 100). Presence probabilities of customer nodes were randomly generated from a uniform distribution on intervals (0.0, 0.2], (0.0, 0.5], (0.0, 1.0], one for each problem size. The presence probability of the depot (node 0) was assigned as 1.0. Ten different problem instances were randomly generated for each presence probability of customer nodes. For each instance, the coordinates of one depot and n customer nodes ðxi ; yi Þ were generated based on a uniform distribution from [0, 100]2. The homogeneous PTSP instances were generated from the wellknown TSPLIB instances (i.e., kroA100, eil101, ch150, d198, and rat783) and a probability of 0.1, 0.2, 0.5 and 0.75 was assigned to each customer to denote different chances of a visit. These homogeneous PTSP instances have been investigated by several studies [14–16,18,21,22], and the obtained results from these were used as the basis of comparison with the ones yielded from the three initial solution generators under the present proposed GA framework. In the experiment, the average run time and solution quality of three generators are examined. All implementations in this study were coded in Fortran and run on an Intel Pentium IV 2.8 GHz CPU personal computer with 512 MB memory, running Windows XP operating system, using the Compaq Visual Fortran 6.5 compiler. Instead of using one run for each instance, as in Tang and Miller-Hooks’ study, each problem instance (including heterogeneous and homogeneous) was tested by running each of the three algorithmic procedures (i.e., RAN, NN1, NN2) 30 times using different random number seeds and averaging the results in an attempt to enhance the robustness of the results and further conduct appropriate statistical analysis. 5. Numerical results and discussion The numerical results and statistical analysis are described in Sections 5.1 and 5.2 for heterogeneous and homogeneous PTSP instances, respectively. 5.1. Numerical results and discussion for heterogeneous PTSP instances Since the average of 10 instances for each combination of problem sizes (n) and probability intervals (p) was used as the performance measure in previous studies [9,27,28], the average statistics (i.e., E½s value and computer execution time) of 10 instances are used in Section 5.1.1 for comparison purposes. 5.1.1. Results based on the average statistics of ten instances The average E½s values and computer execution times found from three initial solution generators for the heterogeneous PTSP are reported in Table 1. The terms used in the column headings of Table 1 are defined as follows. n denotes problem size, which is the number of customer nodes; p represents the customer presence probability interval (0.0, p]; E½s denotes the average value of the expected length of the a priori PTSP tour; CPU is the average computer execution time in seconds; ‘‘RAN”, ‘‘NN1”, and ‘‘NN2” represent the average statistics by using three initial solution generators under the GA framework in this study; ‘‘Previous Best” represents the best average E½s values and the corresponding average CPU time reported in past studies [9,27,28], and this serves as the benchmark of performance efficacy for this study. In Table 1, the ‘‘Previous Best” results for the heterogeneous PTSP data were obtained by Liu [27,28], except for n ¼ 50 and p ¼ 0:5, which were obtained by Tang and Miller-Hooks [9]. The results in Table 1 show that the average E½s values obtained

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Y.-H. Liu / Applied Mathematics and Computation 216 (2010) 125–137

Table 1 Comparison of numerical results among the three initial solution generators (i.e., RAN, NN1, and NN2) under a GA framework and the previous best results obtained by the other studies for the heterogeneous PTSP. n

50

75

100

a

p

(0.0,0.2] (0.0,0.5] (0.0,1.0] (0.0,0.2] (0.0,0.5] (0.0,1.0] (0.0,0.2] (0.0,0.5] (0.0,1.0]

RAN

NN1

NN2

Previous best

E½s

CPU

E½s

CPU

E½s

CPU

E½s

CPU

224.8380 341.0946 450.8377 265.9502 403.5445 528.8862 300.9211 461.0675 626.0304

22.3 12.0 5.7 83.1 33.1 18.2 182.0 74.7 43.5

224.8316 341.3873 449.8460 265.9505 405.6467 528.1084 300.8494 462.5060 623.5687

24.2 13.2 6.3 93.2 37.2 20.1 212.3 86.2 47.7

224.8311 341.6739 451.2593 265.9773 404.1214 528.1113 300.8671 463.3903 625.4372

24.1 13.1 6.2 93.8 37.6 19.9 206.7 83.9 46.8

224.8314 341.3000 450.2215 265.9315 403.2347 527.1907 300.8495 462.2678 624.6369

45.4 72.4a 12.4 240.6 51.8 41.5 689.9 121.2 96.7

Running on a DEC AlphaServer 1200/533 computer with 1 GB RAM.

Table 2 p-Value of permutation test for the heterogeneous PTSP. n

p

50

(0.0, (0.0, (0.0, (0.0, (0.0, (0.0, (0.0, (0.0, (0.0,

75

100

a

0.2] 0.5] 1.0] 0.2] 0.5] 1.0] 0.2] 0.5] 1.0]

RAN

NN1

NN2

0.1114 – 0.0488a – – 0.4369 0.2543 – 0.0310a

0.5085 0.0003a – 1.0000 0.0000a – – 0.0011a –

– 0.0000a 0.0033a 0.3208 0.1483 0.9976 0.5406 0.0000a 0.0486a

Means that the result is considered statistically significant.

Table 3 Detailed results of three generators for problem size n ¼ 50. PTSP instances

RAN

NN1

NN2

50-0.2-01 50-0.2-02 50-0.2-03 50-0.2-04 50-0.2-05 50-0.2-06 50-0.2-07 50-0.2-08 50-0.2-09 50-0.2-10

217.8270 205.9634 212.3572 215.6972 249.5885 218.7964 245.8009 239.1634 237.7385 205.4471

217.8270 205.9634 212.2928 215.6967 249.5864 218.7964 245.8009 239.1634 237.7387 205.4504

217.8270 205.9632 212.2928 215.6967 249.5864 218.7964 245.8009 239.1634 237.7386 205.4457

50-0.5-01 50-0.5-02 50-0.5-03 50-0.5-04 50-0.5-05 50-0.5-06 50-0.5-07 50-0.5-08 50-0.5-09 50-0.5-10

313.6070 349.8370 335.2765 348.2366 327.0591 373.5820 324.3302 337.5788 352.7215 348.7170

313.7274 349.8844 334.9818 348.1619 327.0591 373.9831 324.3374 340.3564 352.5883 348.7927

313.4036 349.8121 335.8659 348.1359 327.0591 375.9827 324.3215 339.4339 354.0252 348.6991

50-1.0-01 50-1.0-02 50-1.0-03 50-1.0-04 50-1.0-05 50-1.0-06 50-1.0-07 50-1.0-08 50-1.0-09 50-1.0-10

462.5669 405.3481 468.8349 459.9634 425.0307 506.8226 482.6801 439.9868 439.5509 417.5929

459.1414 406.1062 468.6839 457.5446 424.5510 502.6295 485.0879 439.5713 437.3155 417.8290

466.6401 405.0370 466.1305 460.8704 428.3445 503.5566 482.7533 441.5305 440.1285 417.6019

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Y.-H. Liu / Applied Mathematics and Computation 216 (2010) 125–137 Table 4 p-Value of permutation test for problem size n ¼ 50.

a

PTSP instances

RAN

NN1

NN2

50-0.2-01 50-0.2-02 50-0.2-03 50-0.2-04 50-0.2-05 50-0.2-06 50-0.2-07 50-0.2-08 50-0.2-09 50-0.2-10

– 1.0000 0.2397 1.0000 1.0000 – – – – 1.0000

– 1.0000 – – – – – – 1.0000 0.4997

– – – – – – – – 1.0000 –

50-0.5-01 50-0.5-02 50-0.5-03 50-0.5-04 50-0.5-05 50-0.5-06 50-0.5-07 50-0.5-08 50-0.5-09 50-0.5-10

0.1139 0.6620 0.7062 0.1646 – – 0.2264 – 0.2110 0.7977

0.0180a 0.2830 – 0.5199 – 0.4707 0.0218a 0.0000a – 0.3362

– – 0.1052 – – 0.0000a – 0.0002a 0.0004a –

50-1.0-01 50-1.0-02 50-1.0-03 50-1.0-04 50-1.0-05 50-1.0-06 50-1.0-07 50-1.0-08 50-1.0-09 50-1.0-10

0.1245 0.7642 0.0579 0.0224a 0.9485 0.0663 – 0.6871 0.0747 –

– 0.2285 0.0802 – – – 0.0934 – – 0.0744

0.0146a – – 0.0457a 0.0098a 0.6094 0.9578 0.0411a 0.0000a 0.9505

Means that the result is considered statistically significant.

by three initial solution generators under the GA framework are similar to the ones obtained by the ‘‘Previous Best.” However, the computational time needed by the three generators is less than that shown for the ‘‘Previous Best” in all cases (ranging from 27.4% to 73.6% less, for an average reduction of 50.2%). This suggests that the GA solution framework proposed in this study is a promising and efficient method for solving the heterogeneous PTSP. To answer the second research question, the performance of three initial solution generators (i.e., RAN, NN1, NN2) was compared. The best average value of E½s among the three generators for each problem size with different presence probability intervals is shown in bold. As shown in Table 1, ‘‘RAN” yields the best average E½s value when p ¼ 0:5 for n ¼ 50; 75; 100, and p ¼ 0:2 for n ¼ 75. Additionally, ‘‘NN1” obtains the best average E½s value when p ¼ 1:0 for n ¼ 50; 75; 100, and p ¼ 0:2 for n ¼ 100. Finally, ‘‘NN2” yields the best average E½s value only when p ¼ 0:2 for n ¼ 50. Since the assumption of normal distribution is rarely met in minimization problems, the permutation test, rather than parametric tests, is adopted for statistical testing in this study [28,38]. A Monte Carlo method with 10,000 permutations is used to obtain the approximate p-value of the permutation test. A set of two-sample permutation tests is conducted to investigate if any statistically significant differences exist between the best average E½s value obtained and the ones obtained by the other two generators. Table 2 shows the p-values of the permutation tests, where a ¼ 0:05 is considered statistically significant. According to the results, the average E½s value obtained by ‘‘RAN” is significantly lower than those obtained by the other two generators (i.e., ‘‘NN1”, and ‘‘NN2”) for n ¼ 50; 100 when p ¼ 0:5. The average E½s value obtained by ‘‘NN1” is significantly lower than those obtained by the other two generators (i.e., ‘‘RAN”, and ‘‘NN2”) for n ¼ 50; 100 when p ¼ 1:0. Based on the results in Tables 1 and 2, no conclusions can be drawn with regard to selecting an appropriate initial solution generator for solving the PTSP. 5.1.2. Results based on the average statistics of individual instances Since clustering all instances for statistical analysis did not yield consistent findings, insights derived from phenomenological research directed the author to examine the data on a more individual basis, so as to appreciate its idiosyncratic and context-dependent nature. Specifically, rather than looking at the averaged E½s values of all ten instances, each of the instances was analyzed separately to acknowledge its possibly distinct characteristics. Tables 3, 5, and 7 show the average E½s values of each instance found from three initial solution generators for problem size n = 50, 75, and 100, respectively, while Tables 4, 6, and 8 show the associated p-values results of the permutation tests. Definitions of terms used in the column headings for Tables 3–8 are given as follows. To take ‘‘50-0.2-01” as an example in ‘‘PTSP instances”, it represents the instance with a problem size of 50, customer presence probability interval (0.0, 0.2], and instance number 01.

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Table 5 Detailed results of three generators for problem size n ¼ 75. PTSP instances

RAN

NN1

NN2

75-0.2-01 75-0.2-02 75-0.2-03 75-0.2-04 75-0.2-05 75-0.2-06 75-0.2-07 75-0.2-08 75-0.2-09 75-0.2-10 75-0.5-01 75-0.5-02 75-0.5-03 75-0.5-04 75-0.5-05 75-0.5-06 75-0.5-07 75-0.5-08 75-0.5-09 75-0.5-10 75-1.0-01 75-1.0-02 75-1.0-03 75-1.0-04 75-1.0-05 75-1.0-06 75-1.0-07 75-1.0-08 75-1.0-09 75-1.0-10

267.2378 278.1325 268.2517 271.5511 264.6652 236.2686 273.1798 251.4517 280.4570 268.3067 398.7818 424.0273 427.5079 410.0959 400.8739 378.3988 428.4649 397.6664 367.8801 401.7481 521.9330 547.6116 489.0053 579.0246 531.7827 492.2095 517.2198 560.8054 535.2565 514.0131

267.3481 278.1322 268.2517 271.5195 264.8057 236.0453 273.1889 251.4533 280.4528 268.3075 402.6016 431.6712 427.7628 410.2795 403.5626 379.5954 431.7620 395.7191 367.9123 405.6001 525.0315 532.3695 502.4464 574.6236 530.3367 489.5563 524.3398 551.0694 534.3114 516.9989

267.2163 278.1328 268.2517 271.5322 264.8237 236.4413 273.1922 251.4524 280.4213 268.3086 401.8246 430.4701 428.2333 405.6888 399.8394 379.1326 425.4925 399.6999 367.9500 402.8829 516.1022 543.8683 486.2678 579.2511 532.8435 492.7467 524.0627 554.8267 534.2880 516.8557

Table 6 p-Value of permutation test for problem size n ¼ 75.

a

PTSP instances

RAN

NN1

NN2

75-0.2-01 75-0.2-02 75-0.2-03 75-0.2-04 75-0.2-05 75-0.2-06 75-0.2-07 75-0.2-08 75-0.2-09 75-0.2-10 75-0.5-01 75-0.5-02 75-0.5-03 75-0.5-04 75-0.5-05 75-0.5-06 75-0.5-07 75-0.5-08 75-0.5-09 75-0.5-10 75-1.0-01 75-1.0-02 75-1.0-03 75-1.0-04 75-1.0-05 75-1.0-06 75-1.0-07 75-1.0-08 75-1.0-09 75-1.0-10

0.9475 1.0000 – 0.2153 – 0.3353 – – 0.4183 – – – – 0.0010a 0.2154 – 0.1383 0.0141a – – 0.0346a 0.0000a 0.2277 0.0436a 0.5666 0.2555 – 0.0340a 0.6614 –

0.0639 – – – 0.0277a – 0.5751 1.0000 0.4094 1.0000 0.0001a 0.0009a 0.7165 0.0022a 0.0017a 0.0001a 0.0094a – 0.6357 0.0233a 0.0026a – 0.0000a – – – 0.0990 – 0.9908 0.3120

– 1.0000 – 0.7288 0.0090a 0.0774 0.4930 1.0000 – 1.0000 0.0062a 0.0210a 0.5075 – – 0.0002a – 0.0000a 0.5014 0.5045 – 0.0002a – 0.0256a 0.3635 0.1472 0.0577 0.3354 – 0.3251

Means that the result is considered statistically significant.

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Y.-H. Liu / Applied Mathematics and Computation 216 (2010) 125–137 Table 7 Detailed results of three generators for problem size n ¼ 100. PTSP instances

RAN

NN1

NN2

100-0.2-01 100-0.2-02 100-0.2-03 100-0.2-04 100-0.2-05 100-0.2-06 100-0.2-07 100-0.2-08 100-0.2-09 100-0.2-10 100-0.5-01 100-0.5-02 100-0.5-03 100-0.5-04 100-0.5-05 100-0.5-06 100-0.5-07 100-0.5-08 100-0.5-09 100-0.5-10 100-1.0-01 100-1.0-02 100-1.0-03 100-1.0-04 100-1.0-05 100-1.0-06 100-1.0-07 100-1.0-08 100-1.0-09 100-1.0-10

277.0664 298.5900 309.5734 298.7631 319.7291 301.1276 301.5002 294.6052 304.4292 303.8268 463.7327 464.8867 463.5933 445.4410 473.2588 486.3018 459.8960 454.7699 461.0018 437.7925 619.6353 638.5145 633.0711 639.1166 586.2283 661.6444 581.3484 645.2671 641.4221 614.0557

277.0661 298.5857 309.5748 298.7583 319.8163 301.1351 301.4911 293.8812 304.3633 303.8222 465.0026 460.5846 469.4656 444.4016 475.6817 486.3916 465.8255 454.6410 462.8787 440.1873 612.9643 637.5530 625.0363 640.9636 590.0275 662.6780 578.1414 637.5529 636.9095 613.8605

277.0671 298.5863 309.5751 298.7612 319.7032 301.1197 301.5848 293.9797 304.4732 303.8209 464.9629 468.9344 464.3700 451.6357 474.5723 489.0410 462.1778 451.4503 465.4602 441.2983 619.4330 632.2117 634.5632 639.2510 588.7062 662.1465 583.2056 632.1414 644.4703 618.2426

Table 8 p-Value of permutation test for problem size n ¼ 100.

a

PTSP instances

RAN

NN1

NN2

100-0.2-01 100-0.2-02 100-0.2-03 100-0.2-04 100-0.2-05 100-0.2-06 100-0.2-07 100-0.2-08 100-0.2-09 100-0.2-10 100-0.5-01 100-0.5-02 100-0.5-03 100-0.5-04 100-0.5-05 100-0.5-06 100-0.5-07 100-0.5-08 100-0.5-09 100-0.5-10 100-1.0-01 100-1.0-02 100-1.0-03 100-1.0-04 100-1.0-05 100-1.0-06 100-1.0-07 100-1.0-08 100-1.0-09 100-1.0-10

1.0000 0.3132 – 0.2497 0.4538 0.2767 0.8955 0.1197 0.0008a 0.1179 – 0.0000a – 0.3749 – – – 0.0019a – – 0.0013a 0.0577 0.0008a – – – 0.1925 0.0000a 0.3369 0.9500

– – 1.0000 – 0.0115a 0.0252a – – – 1.0000 0.4080 – 0.0004a – 0.3105 0.9440 0.0084a 0.0034a 0.3554 0.0233a – 0.0684 – 0.4514 0.2491 0.7040 – 0.0497a – –

1.0000 1.0000 1.0000 1.0000 – – 0.2734 0.4976 0.0004a – 0.4411 0.0000a 0.5237 0.0000a 0.4839 0.0795 0.3698 – 0.1096 0.0003a 0.0023a – 0.0007a 0.9468 0.2873 0.8674 0.0588 – 0.1066 0.1345

Means that the result is considered statistically significant.

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In Tables 4, 6 and 8, the values marked with a are those which are significantly different from the best average values among these three initial solution generators. As shown in Tables 4, 6 and 8, even though none of these three initial solution generators (i.e., RAN, NN1, and NN2) consistently performed best in terms of the E½s value obtained across all instances, it seems to be somewhat instance-dependent for the same combination of n and p. For example, for n ¼ 75 and p ¼ 0:5, the value obtained by RAN (398.7818) is significantly better than the values obtained by NN1 (402.6016) and NN2 (401.8246) for instance number 01; the value obtained by NN2 (405.6888) is significantly better than the values obtained by RAN (410.0959) and NN1 (410.2795) for instance number 04; and the value obtained by NN1 (395.7191) is significantly better than the values obtained by RAN (397.6664) and NN2 (399.6999) for instance number 08. Similar outcomes are found in other combinations of n and p (i.e., for n ¼ 100 and p ¼ 0:5). The fact that some instances under the same combination of n and p have smaller E½s values obtained by a specific generator while others have higher E½s values yielded by the same generator indicates that the possibility of finding the optimal solution based on the same generator may fluctuate across instances. This also suggests that the effectiveness of finding the optimal solution might be mediated by some factors, such as the characteristics of the instance (e.g., the relative location and presence probability of each node) in solving the PTSP. As such, algorithms which can dynamically select the initial solution generator according to individual instance patterns might be one approach for effectively solving the PTSP. With the rapid advances in pattern recognition technologies, or the development of an index system to categorize instances, effective and valid differentiation of individual instance patterns will be accomplished. Another implication of the current study is the inadequacy of PTSP studies that rely on a single-instance numerical experiment. Specifically, one PTSP test instance for a specific combination of n and p, as used in previous studies [16,20], assumes that the same combination of n and p will yield similar results by the same method. As can be clearly seen from the results of this study, this assumption is not upheld. The numerical experiment based on one PTSP test instance may be subject to biased sampling of the PTSP instances, which may inadvertently result in over- or under-estimation of the performance of the proposed algorithms for PTSP. As such, to obtain more valid results, researchers are advised to include at least a certain amount of test instances with the same combination of n and p when conducting PTSP numerical experiments. 5.2. Numerical results and discussion for homogeneous PTSP instances The results obtained from the three initial solution generators for the homogeneous PTSP are reported in Table 9, while Table 10 shows the associated p-value results of the permutation tests. Definitions of terms used in the column headings for Table 9 are given as follows. To take ‘‘kroA100-0.1” as an example in ‘‘PTSP instances”, it represents the instance kroA100 from TSPLIB with a problem size of 100, while customer presence probability 0.1. E½s and rE½s denote the average and standard deviation of the expected length of 30 runs for a given homogeneous PTSP instance, respectively. CPU is the 30-run average CPU running time (in seconds) for each instance. Best represents the best value found among 30 independent runs by the algorithmic procedures using the specific initial solution generator chosen in this study for a given instance. According to the results shown in Tables 9 and 10, the average E½s value obtained by ‘‘NN1” is lower than the ones obtained by the other two initial solution generators (i.e., ‘‘RAN” and ‘‘NN2”) for most of the tested cases (16 out of 20). Of the four cases when ‘‘NN1” does not yield the best average E½s value, three of them are not statistically significantly different

Table 9 Detailed results of three generators for homogeneous PTSP instances. PTSP instances

kroA100-0.1 kroA100-0.2 kroA100-0.5 kroA100-0.75 eil101-0.1 eil101-0.2 eil101-0.5 eil101-0.75 ch150-0.1 ch150-0.2 ch150-0.5 ch150-0.75 d198-0.1 d198-0.2 d198-0.5 d198-0.75 rat783-0.1 rat783-0.2 rat783-0.5 rat783-0.75

RAN

NN1

NN2

E½s

rE½s

CPU

Best

E½s

rE½s

CPU

Best

E½s

rE½s

CPU

Best

9051.62 11791.14 17538.28 21163.38 197.77 286.13 467.16 577.65 2485.84 3476.01 5371.90 6346.84 7457.83 9394.61 12811.92 14843.98 3388.34 4946.03 8423.48 9768.27

17.78 83.05 687.89 267.20 0.86 2.15 5.84 8.24 19.01 30.89 117.09 98.97 25.53 76.29 87.49 121.41 47.58 173.86 925.32 1601.34

30.9 22.5 16.9 15.7 29.3 21.7 16.4 15.3 78.2 61.6 51.8 49.0 162.2 134.3 113.3 106.8 4534.3 3423.0 2634.4 2707.3

9034.97 11715.14 16569.72 20001.49 197.34 283.77 459.16 566.79 2479.96 3439.42 5134.73 6142.50 7440.58 9319.35 12687.59 14592.21 3326.46 4766.55 7352.30 8643.55

9050.40 11779.23 17083.73 20441.99 197.63 285.25 464.55 575.18 2483.27 3466.70 5240.72 6129.57 7450.99 9356.27 12771.49 14741.85 3343.96 4784.57 7422.13 8719.57

19.45 69.51 528.71 755.09 0.16 0.80 5.04 7.13 4.95 32.88 121.18 88.28 13.18 31.18 125.54 122.45 20.36 36.39 54.96 50.30

34.6 25.0 17.9 16.2 32.5 23.9 17.4 15.6 85.8 68.5 55.5 51.8 183.4 162.4 135.3 127.3 4294.6 2871.5 1605.5 1393.8

9034.97 11715.14 16569.72 19366.01 197.34 284.43 459.57 561.01 2479.96 3418.95 5012.20 5898.65 7437.99 9324.12 12580.28 14479.19 3300.56 4725.28 7311.82 8626.38

9037.65 11768.50 17374.28 20947.65 197.68 286.14 465.38 573.99 2484.19 3481.98 5235.60 6233.15 7455.27 9370.77 12811.52 14853.27 3375.05 4854.68 7499.49 8750.91

7.26 105.60 527.97 448.38 0.73 1.79 5.51 7.25 11.31 61.80 115.26 135.97 13.21 28.28 133.12 158.88 34.55 54.42 127.06 88.06

33.6 23.8 17.7 16.0 31.9 23.4 17.4 16.0 87.1 68.4 55.8 51.9 183.5 162.8 137.7 129.8 4512.1 3014.1 1751.9 1553.1

9034.97 11715.14 16610.07 19791.83 197.34 283.72 460.07 563.44 2479.96 3419.36 5016.08 6005.01 7438.47 9313.35 12559.34 14588.90 3323.81 4738.52 7364.89 8607.46

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Y.-H. Liu / Applied Mathematics and Computation 216 (2010) 125–137 Table 10 p-Value of permutation test for homogeneous PTSP instances.

a

PTSP instances

RAN

NN1

NN2

kroA100-0.1 kroA100-0.2 kroA100-0.5 kroA100-0.75 eil101-0.1 eil101-0.2 eil101-0.5 eil101-0.75 ch150-0.1 ch150-0.2 ch150-0.5 ch150-0.75 d198-0.1 d198-0.2 d198-0.5 d198-0.75 rat783-0.1 rat783-0.2 rat783-0.5 rat783-0.75

0.0001a 0.3608 0.0067a 0.0000a 0.3541 0.0376a 0.0706 0.0741 0.5624 0.2695 0.0000a 0.0000a 0.2018 0.0010a 0.1505 0.0015a 0.0000a 0.0000a 0.0000a 0.0000a

0.0006a 0.6564 – – – – – 0.5287 – – 0.8605 – – – – – – – – –

– – 0.0359a 0.0029a 0.6339 0.0149a 0.5475 – 0.7741 0.2387 – 0.0008a 0.2159 0.0643 0.2418 0.0041a 0.0000a 0.0000a 0.0017a 0.0946

Means that the result is considered statistically significant.

Table 11 Comparison of the proposed algorithm with other metaheuristic methods from the literature for the homogeneous PTSP instances. PTSP instances

ACO [15]

pACS [14]

HybMSPSO [20]

HBMO [21]

ILS-EE [17]

MAGX-EE [17]

ACS-EE [17]

pACS+1-shift [17,38]

RAN

NN1

NN2

kroA100-0.1 kroA100-0.2 kroA100-0.5 kroA100-0.75 eil101-0.1 eil101-0.2 eil101-0.5 eil101-0.75 ch150-0.1 ch150-0.2 ch150-0.5 ch150-0.75 d198-0.1 d198-0.2 d198-0.5 d198-0.75 rat783-0.1 rat783-0.2 rat783-0.5 rat783-0.75

– – – – – – 460.563 564.036 – – – – – – – – – – – –

9039.40 11720.60 16605.40 – 199.70 286.70 460.70 – 2493.60 3444.20 5051.30 – 7556.10 9489.20 12613.30 – 3368.90 4781.20 7261.40 –

9074.94 11726.50 16581.64 20281.91 200.03 284.93 455.65 579.24 2510.11 3444.61 5016.85 5988.34 7504.94 9415.08 12527.56 14876.35 3616.44 4775.14 7094.87 8217.31

9071.72 – 16581.64 – 200.03 – 455.65 – 2509.98 – 5016.82 – 7490.09 – 12492.62 – 3616.44 – 7085.48 –

9042 11718 16576 – 197 – – – 2484 3423 5005 – 7444 9323 12424 – 3250 4556 6857 –

9041 11717 16569 – 197 – – – 2483 3420 5005 – 7443 9318 12421 – 3246 4553 6846 –

9036 11716 16569 – 197 – – – 2481 3432 5004 – 7445 9322 12417 – 3251 4570 6854 –

9034 11714 16569 – 196 – – – 2479 3417 5033 – 7437 9312 12464 – 3341 4826 7789 –

9034.97 11715.14 16569.72 20001.49 197.34 283.77 459.16 566.79 2479.96 3439.42 5134.73 6142.50 7440.58 9319.35 12687.59 14592.21 3326.46 4766.55 7352.30 8643.55

9034.97 11715.14 16569.72 19366.01 197.34 284.43 459.57 561.01 2479.96 3418.95 5012.20 5898.65 7437.99 9324.12 12580.28 14479.19 3300.56 4725.28 7311.82 8626.38

9034.97 11715.14 16610.07 19791.83 197.34 283.72 460.07 563.44 2479.96 3419.36 5016.08 6005.01 7438.47 9313.35 12559.34 14588.90 3323.81 4738.52 7364.89 8607.46

– Represents that no results were given in the specific algorithm.

from the lowest average E½s value obtained by ‘‘NN2” (i.e., kroA100-0.1, kroA100-0.2, eil101-0.75, and ch150-0.5). This suggests that ‘‘NN1” is a relatively better generator with the current parameter settings (stopping criteria, number of initial solutions generated, etc.) under the GA framework for homogeneous PTSP instances when the problem size is more than 100. Unlike the results shown in Section 5.1, the random initial solution generator (RAN) performed worse than the other two generators for all homogeneous PTSP instances. One possible reason is that a greater number of generations is needed for the ‘‘RAN” algorithm to converge for relatively large PTSP instances. With the stopping condition used in this study (i.e., two times the number of nodes), the outcomes from the two generators whose initial solutions were systematically built, NN1 and NN2, are better than that yielded from the random generator. In Table 11, the best tours found by the three generators (i.e., RAN, NN1, and NN2) under the currently proposed GA framework are further compared to the results obtained by ACO [16], pACS [14], hybrid multi-swarm particle swarm optimization (HybMSPSO) [21], honey bees mating optimization (HBMO) [22], estimation-based metaheuristics (i.e., ILS-EE for estimation-based iterative local search, MAGX-EE for estimation-based memetic algorithms, ACS-EE for estimation-based ant colony optimization) [18], and pACS+1-shift [15,18]. As the computational time is dependent on the compiler and the

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computer hardware used, comparison of computational efficiency among these studies is meaningless unless the identical computer equipment and configuration can be secured. Thus, comparisons are focused on the quality of the best tour obtained by different algorithms. As shown in Table 11, in terms of the best E½s value found, no one single algorithm consistently outperformed the others across all homogeneous PTSP instances. To elaborate, for the instances of kroA100 with probability equal to 0.1, 0.2, and 0.5, ACS-EE, pACS+1-shift, RAN and NN1 all obtained relatively high-quality solutions in terms of the expected tour length. When the number of nodes was less than 200 (i.e., kroA100, eil101, ch150, and d198) and these was a high probability (i.e., 0.75), NN1 obtained the a priori tour with minimum expected values. On the other hand, HybMSPSO and HBMO yielded the best value for the eil101-0.5 instance. For the instances with 150 and 198 nodes (i.e., ch150 and d198), pACS+1-shift and NN2 obtained relatively high-quality solutions when the probability was less than or equal to 0.2. For the rat783 instance with probability value 0.1, 0.2 and 0.5, the estimation-based metaheuristics consistently obtained the best results among all the listed algorithms, while HybMSPSO yielded the best outcome for the instance with a probability equal to 0.75. The fact that no algorithm was better than the others in all test cases again echoed what was found in Section 5.1.2, that the effectiveness of finding the optimal PTSP solution might be affected by the context (in this case the probability and number of nodes). 6. Concluding comments In this paper, three different initial solution generators under a GA framework are proposed and developed to solve the PTSP. For the heterogeneous PTSP instances, the numerical results with regard to capability of finding similar results to the ‘‘Previous Best” not only support the effectiveness of the three proposed generators in finding the average E½s values, but further substantiate their efficacy in terms of efficiency. Specifically, the computational time needed by the three generators is much less in all test cases (ranging from 27.4% to 73.6% computation time saving, with an average of 50.2%) compared to the earlier reported data. As for the homogeneous PTSP instances, NN1 is relatively better among the three generators in terms of average E½s values, while the RAN consistently performs worse than the other two generators in terms of average E½s values (probably due to the time required for it to reach convergence, which cannot be attained under the current settings). Additionally, the fact that no one single algorithm consistently outperformed the others across all homogeneous PTSP instances in terms of the best E½s values suggests the possibility of context-dependent phenomenon. Finally, to obtain more valid results, researchers are advised to include at least a certain amount of test instances with the same combination of n and p when conducting PTSP numerical experiments. Acknowledgements This work was supported primarily by three research grants funded by the National Science Council of Taiwan (NSC 972410-H-260-012, 98-2410-H-260-052, and 98-2918-I-260-001). Special thanks are extended to anonymous referees for their constructive and informative suggestions for the refinement of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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