Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 640231, 13 pages
http://dx.doi.org/10.1155/2015/640231

Research Article

Minimization of the Total Traveling Distance and Maximum Distance by Using a Transformed-Based Encoding EDA to Solve the Multiple Traveling Salesmen Problem

S. H. Chen

Department of Information Management, Cheng Shiu University, No. 840, Chengcing Road, Niaosong District, Kaohsiung 83347, Taiwan

Received 15 May 2015; Revised 7 August 2015; Accepted 18 August 2015

Academic Editor: Jason Gu

Copyright © 2015 S. H. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Estimation of distribution algorithms (EDAs) have been used to solve numerous hard problems. However, their use with in-group optimization problems has not been discussed extensively in the literature. A well-known in-group optimization problem is the multiple traveling salesmen problem (mTSP), which involves simultaneous assignment and sequencing procedures and are shown in different forms. This paper presents a new algorithm, named , which is based on self-guided genetic algorithm with a minimum loading assignment (MLA) rule. This strategy uses the transformed-based encoding approach instead of direct encoding. The solution space of the proposed method is only . We compare the proposed algorithm against the optimal direct encoding technique, the two-part encoding genetic algorithm, and, in experiments on 34 TSP instances drawn from the TSPLIB, find that its solution space is . The scale of the experiments exceeded that presented in prior studies. The results show that the proposed algorithm was superior to the two-part encoding genetic algorithm in terms of minimizing the total traveling distance. Notably, the proposed algorithm did not cause a longer traveling distance when the number of salesmen was increased from 3 to 10. The results suggest that EDA researchers should employ the MLA rule instead of direct encoding in their proposed algorithms.

1. Introduction

Estimation of distribution algorithms (EDAs) use the learning while optimizing principle [1]. Two review articles have suggested that EDAs have emerged as a prominent alternative to evolutionary algorithms [2, 3]. In contrast to genetic algorithms (GAs), which employ crossover and mutation operators to generate solutions, EDAs explicitly extract global statistical information from the previous search to build a posterior probability model of promising solutions from which new solutions are sampled [4, 5]. This crucial characteristic distinguishes EDAs from GAs [6, 7].

Numerous studies aimed at using EDAs to solve nondeterministic polynomial-time hard (NP-hard) scheduling problems have shown that EDAs are able to perform effectively in terms of the solution quality [2, 8, 9]. Ceberio et al. [2], in particular, extensively tested 13 famous permutation-based EDAs on four combinatorial optimization problems, including the quadratic assignment problem, traveling salesman problem (TSP), permutation flowshop scheduling problems (PFSPs), and linear ordering problem. Their paper provides a good basis for comparison.

Although EDAs are effective in solving various hard problems, EDA studies seldom extensively discuss a problem. To our knowledge, only one EDA, namely, that is proposed by Shim et al. [10], can solve in-group optimization problems such as the multiple traveling salesmen problem (mTSP) and parallel machine scheduling problems (PMSPs) [11]. In-group optimization problems involve assigning and sequencing procedures simultaneously. Take the mTSP, for example: a number of cities are assigned to salesmen and these cities are visited only once by a salesman, where . Thus, this appears to be an NP-hard problem.

Because only one EDA could solve in-group optimization problems, there is much room for additional research. In-group optimization problems are relevant in industry, such as in the application of the mTSP. This research developed a new EDA, named , dealt with by using a self-guided genetic algorithm (SGGA) [12] with the minimum loading assignment (MLA) rule to solve the mTSP. As opposed to direct encoding, the proposed strategy is called the transformed-based encoding approach. The solution space of the MLA is only . We compare the proposed algorithm against the optimal direct encoding technique, the two-part encoding genetic algorithm (TPGA) [13]. Notably, the solution space of the two-part encoding approach is . The proposed MLA method, consequently, is superior to the two-part encoding technique, and an improved solution quality is expected when the SGGA works with the MLA method.

This paper is organized as follows: Section 2 primarily reviews the literature on in-group optimization problems, encoding techniques, and EDAs. In Section 3, the core MLA method is presented to dispatch cities to salesmen. This assignment rule is further employed by the SGGA in Section 4. Section 5 reveals the effectiveness of the proposed algorithm, which is compared with the existing famous direct encoding methods, including the one-chromosome and two-part chromosome encoding. Finally, we draw conclusions in Section 6.

2. Background Information

The mTSP is a well-known in-group optimization problem. We review mTSP studies and their variants in Section 2.1. To solve in-group optimization problems, numerous encoding techniques could be applied in evolutionary algorithms. Solution representations fall into two classes: direct and indirect encoding methods [11], relevant studies about which are presented in Sections 2.2 and 2.3, respectively. The final section illustrates combinatorial-based EDAs.

2.1. In-Group Optimization Problems

Bektas [11] reviewed the seven types of in-group optimization problems, which we detail in Table 1. Among the variants of in-group optimization problems, the most well-known form is the mTSP because it models daily activities and exists in every enterprise [13]. The problem properties of the mTSP include assignment and sequence optimization procedures. For instance, we must optimize the traveling sequence for the route of each salesman. Both procedures directly lead to the traveling cost and time of the trip after assigning salesmen to visit places every day. A detailed definition of the mTSP can be found in [11].

Table 1: Application contexts for the in-group optimization problems.

Application context
Type of application

Routing
mTSP [13, 24, 46–48]

Print scheduling
Print press scheduling [49]

Preprint advertisement scheduling [50]

Workforce planning
Bank crew scheduling [51]

Technical crew scheduling [52]

Photographer team scheduling [53]

Interview scheduling [54]

Workload balancing [55]

Security service scheduling [56]

Transportation
planning
School bus routing [57]

Crane scheduling [58]

Local truckload pickup and delivery [59]

Vehicle routing problem [60, 61]

Mission planning
Planning of autonomous mobile robots [62–65]

Planning of unmanned air vehicles [66]

Production planning
Hot rolling scheduling [17]

Parallel machine scheduling with setup [29]

Satellite systems
Designing satellite surveying systems [67]

Although the mTSP could be solved using exact algorithms [14–16], large-sized problems are not solved efficiently. To deal with large-size instances, evolutionary algorithms (EAs) are a commonly used approach. The first crucial step of using EAs is selecting the appropriate encodings. Encoding approaches are presented in the next section.

2.2. Direct Encoding Methods

There are five major direct encodings of EAs: one-chromosome [17], two-chromosome [18, 19], two-part chromosome [13], grouping genetic algorithms (GGAs) [20–22], and matrix representation [23]. Two-part chromosome encoding, which is superior to one- and two-chromosome encoding [13] because of its smaller solution space, is depicted in Figure 1.

Figure 1: A representation of the two-part chromosome encoding for 15 cities and three salesmen.

We assume this encoding with and . There are two distinct parts. The first part of the chromosome represents the permutation of cities. The second part of the chromosome shows the number of cities assigned to each salesman so that its chromosome length is . The total sum of genes is equal to the number of cities. In [24], they examined an improved combination of crossover and mutation operators for the two-part chromosome encoding method and suggested appropriate genetic operators that could be applied in GAs.

GGAs commonly use an array of jobs for each parallel machine, and the processing order of the jobs assigned to that machine is shown [25]. Kashan et al. [26] extended the GGAs into the grouping version of the particle swarm optimization algorithm. Later, Arnaout et al. [23] proposed a matrix representation of the jobs on machines, whose size is . Each row indicates the parallel machines and the processing sequence of the jobs on it. When there are no jobs to be processed on a machine, number 0 is inserted into the blank spaces. As a result, it became apparent that GGAs memory usage was inefficient, though Liao et al. [27] found that this approach was better than the other four variants of hybrid ant colony optimization. Thus, spaces are unused if we apply this encoding technique.

In these direct encoding techniques, the optimal approach could be the two-part chromosome technique, according to Carter and Ragsdale [13]. When we have items and groups, the solution space of one-chromosome encoding requires . Two-chromosome encoding takes and the size of the two-part chromosome is .

2.3. Indirect Encoding Methods

The transformed-based encoding type separates sequencing and assignment decisions because the complex encoding may yield poor results [28]. Its encoding strategy first utilizes permutation encoding and then assigns the items into groups at every stage. Although this approach could be used to solve the PMSP [29], the separated method is also applicable in complex flowshop problems involving numerous parallel machines in the flowshop. Ruiz and Maroto [28] referred to this application as the priority rules for hybrid flowshops. Wang et al. [30] called it the earliest completion factory rule for solving the distributed permutation flowshop scheduling problem. Salhi et al. [31] selected the index of the machine that allows a job that has the shortest completion time for solving complex flowshop scheduling problems.

To achieve optimal efficiency, this study adopts the transformed-based encoding method instead of direct encoding. In addition, several EAs could apply the assignment rule and then solve the in-group optimization problem. To evaluate the performance of the algorithms examined in this study, we select the mTSP for an extensive comparison.

In presenting the latest development in EDAs, it is clear that only a few can solve in-group assignment problems. Thus, this study is relevant to the investigation of in-group assignment problems.

2.4. Recently Developed Combinatorial-Based EDAs

Unlike the implicit processing of building blocks in GAs, EDAs explicitly rely on the used probability model. The building blocks are based on selection and crossover operators that do not preserve essential patterns [32]. The probability model is the core factor in affecting the performance of EDAs. The more accurate the probability model is, the more effective the algorithm will be in preventing the disruption of essential building blocks [33]. In general, a distinguishing characteristic of EDAs is their application of the probabilistic model, which is not used by GAs.

Numerous attempts at using EDAs to solve sequencing or combinatorial optimization problems have been made. For example, Chang et al. [34] proposed a hybrid framework to alternate between EDAs and genetic operators for solving the single machine scheduling problem. A position-based univariate probability model was used in the proposed algorithm. The hybrid framework is beneficial, because though EDAs efficiently improve solution quality in the first few runs, the loss of diversity rapidly increases as additional iterations are executed [7, 35, 36].

Jarboui et al. [37] proposed a hybrid approach, named EDA-VNS, that combined EDAs with the variable neighborhood search (VNS) [38] to solve PFSPs by using the minimization of the total flowtime. Their probabilistic model considered the order of the job queue and the building blocks of the jobs. This was the first attempt to take into account both first- and second-order statistical information. In addition, VNS improved as the EDA was run. Jarboui et al. [37] found that EDA-VNS was effective in small benchmarks; however, for larger problems, VNS was superior to EDA-VNS in terms of objective values and computational time. It was unclear why EDA-VNS did not outperform the VNS in large benchmarks. A new EDA in [4] also employed job permutation and similar blocks of jobs to solve lot-streaming flowshop problems. In this EDA, the definitions of job permutation and similar blocks differed from those of [37]; it also introduced a diversity measure to restart evolutionary progress when the population diversity decreased to a certain level.

In contrast to traditional EDAs, an SGGA uses a probabilistic model as the fitness function surrogate [39]. The probabilistic model guides the evolutionary direction in selecting candidate solutions for crossover and mutation operators. An SGGA could solve PFSPs. It could also be integrated with dominance properties to solve single machine scheduling problems [40]. An eSGGA was proposed for problems involving variable interactions [9].

To the best of our knowledge, the first EDA for the mTSP involved applying the one-chromosome representation [10]. Because there are pseudo cities introduced in the chromosome, every chromosome comprises genes. As a result, the dimension of their probability model , by computing the marginal probability of each city, is where is . This might be a drawback of the first EDAs, which were inherited from one-chromosome encoding, even though their performance was superior to three state-of-the-art multiobjective evolutionary algorithms. Consequently, the proposed algorithm , together with the MLA rule, may be the second EDA to solve the mTSP; it is a promising algorithm that does not use the larger probability model of EDAs.

3. Minimum Loading Assignment Rule in the mTSP

Suppose that there is a set of cities, sequenced in , that could be assigned to salesmen. These cities are not yet assigned to any salesmen. The sequence could be decoded by using an assignment rule to assign the cities to each salesman. After the assignment rule is executed, we can calculate the fitness function of each chromosome. We propose an MLA rule to perform the assignment work. The following pseudocode in Algorithm 1 illustrates this rule.

Require:

      : The position of a city in the sequence

      : The current number of cities assigned to a salesman

      : The visiting sequence of cities

(1) 1

(2) while    do

(3)     1

(4)    

(5)    

(6)    

(7) end while

(8) while    do

(9)    Select a salesman who could process the with the minimum objective value

(10) 

(11)  

(12)  

(13) end whie

Algorithm 1: Minimum loading assignment rule.

In the beginning, the first cities are assigned to salesmen and the objective values of each salesman are calculated. The objective function of the mTSP would be the total traveling distance or maximum traveling distance among salesman. The MLA rule is then applied iteratively for unassigned cities. The MLA rule assigns the first unassigned city in the sequence to a salesman when it results in the minimum objective value. This assigned city is removed from . This process continues until no cities are left in .

By using the MLA rule, a city could be assigned to a salesman who has less loading. It also ensures that this assigned city is close to the last city visited by the salesman; a faraway city would not be considered. Through the MLA rule, mTSP can be extended to the PMSP with a setup consideration or the distributed flowshop scheduling problem.

4. Proposed Algorithm: EDAMLA

This section explains the procedures of the EDA with the MLA rule. The advantages of the proposed method include preserving the salient genes of the chromosomes and exploring and exploiting optimal searching directions for genetic operators [40, 41]. The major difference between this proposed algorithm and other works is the problem type; other studies have been aimed at solving the sequencing problem, whereas we addressed the grouping and sequencing problems simultaneously. The major procedures of are shown in Algorithm 2.

Population: A set of solutions

Generations: The maximum number of generations

: Probabilistic model

: Generation index

(1) Initialize Population

(2)

(3) Initialize

(5) while    do

(6)    EvaluateFitness(Population) with minimum loading assignment rule

(7)    Selection/Elitism(Population)

(8)     BuildingProbabilityModel(Selected Chromosomes)

(9)    Self-Guided Crossover( )

(10) Self-Guided Mutation( )

(11)  

(12) end while

Algorithm 2: Main procedures of .

In Step (1), the population is initialized and the sequence of each chromosome is generated randomly. Step (3) builds the probability matrix with a matrix dimension of by , where is the problem size. Each is initialized to be , where is the total number of cities in . This initialization means that all solutions have the same likelihood of being an optimal solution. The reason for such an initialization is that we have no information about the location of promising solutions.

In Step (5), we evaluate the objective value of each solution. In addition, the MLA rule is used here (see Algorithm 1). After all cities are assigned to salesmen, the algorithm evaluates the total distance of all salesmen and the maximum distance among the salesmen. In Step (6), a binary tournament selection is used to select good solutions from the population.

Step (7) forms the probability model after the selection procedure. The calculation details are outlined in Section 4.1. In Steps (8) and (9), is employed in the self-guided crossover and mutation operators. The probability model is used as a fitness surrogate, which is shown in Sections 4.2 to 4.4. We use the two-point partial mapping crossover and swap mutation in the crossover and mutation procedures for solving the mTSP.

The proposed algorithm is explained in the following sections. We first describe the probability model of the EDA and then explain how the probabilistic model guides the crossover and mutation operators.

4.1. Formulation of the Probabilistic Model

The probability model of the EDA is defined as

where is the probability of city being in position in a promising solution. summarizes the global statistical information about promising solutions obtained from the previous search.

Let be the number of solutions in Parentset, in which city is in position and is the size of Parentset. in Line (7) is updated as follows:

where is the percentage of solutions in which city is in position . It represents the knowledge of promising solutions learned from the current generation. We use , the Laplace correction of in (2), to prevent from becoming very small [42–44]. is the historical knowledge of promising solutions. We update in an incremental manner, as suggested by [45]. balances the contribution from historical knowledge with that of the knowledge learned from the current generation.

4.2. Probabilistic Model as the Fitness Surrogate

With the probabilistic model , we define the following function to predict the quality of solution :

where is the position of city in . The following should be noted regarding this function:

(i) is the probability that city in position is a promising solution. Therefore, can measure how promising is.(ii)In general, is not an exact probability measure of the set of all the solutions of because

  is only an estimation value of the probability that is promising. This estimation is more effective and much easier to compute compared with other probabilistic models in the literature. Thus, this method is effective and reduces computational time.

is applied to select good candidate solutions during the crossover and mutation operation. In the following subsection, we drop in and for simplicity.

4.3. Crossover Operator with Probabilistic Model

With the surrogate function in (3), we preevaluate the solution quality of the new solutions generated by the crossover and mutation operators. In the normal two-point crossover procedure, two random cut-points, and , are set in the beginning, where is less than . Then, a parent solution mates with the other parent solution to yield a new offspring. However, a difference exists in the proposed algorithm.

Because of the difference, we let a parent solution mate with a set of randomly selected solutions . The size of ranges from 2 to , where is the number of tournaments. These crossover steps produce a set of offspring . The quality difference between offspring and parent solution is denoted as . is given as follows:

The larger is, the more likely that is superior to other offspring when a set of parent solutions mate with a solution . Hence, is added to the offspring population. We repeat the crossover steps to generate offspring until the offspring population is full. Both the concepts of self-guided crossover and self-guided mutation employ the same idea. The mutation procedure is shown in the next section.

4.4. Mutation Operator with Probabilistic Model

Suppose that two cities and are randomly selected and they are located in position and position , respectively. and denote city in position and city in position . After these two cities are swapped, the new probabilities of the two cities become and . The probability difference is calculated as (6), which is a partial evaluation of the probability difference because the probability sum of the other cities remains the same:

Now because the part of is always , it can be subtracted, and (6) is simplified as follows:

If is positive, it implies that one gene or both genes might move to a promising area. On the other hand, when is negative, the implication is that at least one gene moves to an inferior position.

On the basis of the probabilistic differences, it is natural to consider different choices of swapping points during the mutation procedure. A parameter is introduced for the self-guided mutation operator, which denotes the number of tournaments in comparing the probability differences among the choices in swap mutation. Basically, while implies that the mutation operator mutates the genes directly without comparing the probability differences among the different choices.

When , suppose the other alternative is that two cities and are located in position and position , respectively. The probability difference of exchanging cities and is

After and are obtained, the difference between the two alternatives is as follows:

If , the contribution of swapping cities and is better, so we swap cities and . Otherwise, cities and are swapped. Consequently, the option of a larger probability difference is selected and the corresponding two cities are swapped. By observing the probability difference , the self-guided mutation operator exploits the solution space to enhance the solution quality and prevent destroying some dominant genes in a chromosome. Moreover, the main procedure of the self-guided mutation is (9), where the time-complexity is only a constant after the probabilistic model is employed. This approach proves to work efficiently.

To conclude, the is obviously different from the previous EDAs. Firstly, the algorithm utilizes the transformed-based encoding instead of using the direct encoding used by Shim et al. [10]. Secondly, the proposed algorithm explicitly embeds the probabilistic model in the crossover and mutation operators to explore and exploit the solution space. Most important of all, the algorithm works more efficiently than previous EDAs [10] in solving the mTSP because the time-complexity is whereas the previous EDAs need time. They are the major differences to other existing EDAs.

5. Experimental Results

We conducted extensive computational experiments to evaluate the performance of in solving the mTSP. There were 34 instances selected from the well-known traveling salesman problem library, TSPLIB, and the size of these instances ranged from 48 to 400. This paper assumed that the first city of each instance was the home depot. The number of salesmen used was 3, 5, 10, and 20. Hence, there were 136 instances in the experiments. Across all the experiments, we replicated each instance 30 times.

The proposed algorithm was compared with the benchmark encoding algorithm and a classic encoding, which are the TPGA [13] and one-chromosome genetic algorithm [17], respectively. We employed the genetic operators and parameter settings of the TPGA suggested by S.-H. Chen and M.-C. Chen [24], because they used the design-of-experiments (DOE) to select significant parameters; the genetic operators are the two-point partial mapping crossover operator and swap mutation operator. This ensures a fair comparison between the proposed algorithm and benchmark encoding algorithm. One-chromosome GA utilizes the same operators of TPGA and also employs the DOE to tune the parameters as well. The crossover and mutation rate of the one-chromosome GA are 0.5 and 0.1, respectively. Finally, a standard genetic algorithm also applies the MLA rule, which is named . could show whether the performance is enhanced by the assignment rule proposed by this research.

We implemented the algorithms in Java 2 on an Amazon EC2 with a Windows 2012 server (32-core CPU). The stopping criterion is the number of examined solutions which is up to 100,000. The objective functions include minimizing the total traveling distance and maximizing the traveling distance, which are detailed in Sections 5.1 and 5.2, respectively.

5.1. Total Traveling Distance Results

This objective evaluates the total distance traveled by the salesmen. It reflects the total cost of the assignment. Figure 2 shows the main effects plot of the method comparison and the differences in the number of salesmen assigned. This figure clearly illustrates that the and are superior to the one-chromosome GA and TPGA. This indicates that the MLA rule, that is, the transformed-based method, could be a more promising approach than the direct encoding methods. The total distance increased greatly with the number of salesmen, particularly when 20 salesmen could be assigned. This implies that the request of too many salesmen would be inefficient from a managerial perspective.

Figure 2: Main effects plot on the total travelling distance of the compared algorithms.

Figure 3 depicts the interaction plot between the factor method and number of salesmen. Notably, the and did not yield a longer total traveling distance when the number of salesmen increased from two to 10. However, the TPGA suffered when the number of salesmen was increased. Thus, this figure reveals the effectiveness of the transform-based rule compared with the direct encoding method. According to this interaction plot, if a manager wants to determine how many salesmen are required, the lowest total traveling distance would be 5.

Figure 3: Intreaction plot on the total travelling distance.

Table 2 presents the results of the total traveling distance for the four algorithms. This table shows the minimum, mean, maximum, and the standard deviation (StDev). Among these 34 instances, is better than one-chromosome GA, TPGA, and out of the 17 cases when it comes to the average of the total distance. In addition, the standard deviation of one-chromosome GA, TPGA, , and is 21187, 33230, 19785, and 20041, respectively. It implies that the yields less variance than one-chromosome GA and TPGA. might be more robust in terms of the average performance and the variance.

Table 2: The total average distance of the three algorithms.

Instance
One-chromosome
TPGA

Min
Mean
Max
StDev
Min
Mean
Max
StDev
Min
Mean
Max
StDev
Min
Mean
Max
StDev

att48
39195
64157
104400
20065
47960
88032
142234
29831
42765
73320
119821
26312
42446
73416
121381
26676

berlin52
9355
12574
18389
2534
10693
16034
23586
3884
9654
14087
20647
3756
9410
14173
21140
3856

bier127
208466
253086
330415
32864
261954
332874
433197
47074
220310
253968
328622
18768
214928
255589
311903
18317

ch130
13488
17246
23859
3019
17602
24176
32313
3785
12781
15893
19262
1644
12020
15813
18983
1582

eil101
1021.6
1343.3
1832.4
233.2
1328.1
1829.7
2441.7
326.1
1024.4
1270.4
1625.4
183.7
943.9
1256.9
1624.1
197.1

eil51
511.2
738.2
1147.4
190.7
614
963.5
1450.8
260.7
497.2
771.2
1195.1
243.6
491.9
772.6
1186
245.7

eli76
721.8
1036.2
1457.1
226.1
1035.6
1460.9
2151.2
331.4
771.2
1082.3
1640.5
280.4
760
1087
1610.3
285.4

gr96
862
1355.4
2026.5
348.6
1291.9
1972.6
2968.2
519.3
911.3
1396.9
2260.3
405.3
900.2
1396.4
2257.7
421.3

kroa150
70880
89392
121337
14485
91575
124442
164472
18229
62801
78955
104886
8744
60914
77017
93631
7330

kroa200
99818
121945
164443
16620
136530
172730
223643
23919
90234
110421
145046
13832
86536
109347
141175
12517

kroB100
40570
57373
82441
11490
56782
83671
118140
17817
41476
54609
74861
10195
40070
54504
78913
10802

kroB150
69561
90636
122612
15676
98376
128704
171691
20984
62532
79664
93700
8023
63013
79136
97548
8422

kroB200
95106
119595
156774
15559
129137
170888
220721
25313
85875
111429
146582
13879
88753
111071
142722
12021

kroC100
40573
59920
90359
13754
61044
85942
128239
17762
37895
54879
77495
12239
40453
54072
78388
12654

kroD100
36251
57801
85497
12768
55413
85475
129610
19920
40864
56408
82342
12689
40074
56230
81408
12490

kroE100
42230
59632
84089
11448
59108
87720
128825
19932
41706
57851
80784
11665
39912
57714
85800
12747

lin105
30104
45726
71980
12090
41504
70560
107157
19321
30822
48377
79901
14521
30116
48700
77804
15158

lin318
212259
244247
307006
22524
271844
341953
447096
43659
182437
230828
311132
38487
183700
231180
318757
39493

mtsp100
42551
58361
87622
12581
57443
85849
123128
19325
42190
56268
81858
12583
39444
56187
83089
13184

mtsp150
84018
105130
139427
15595
110196
144405
184927
20654
76189
94674
115786
9230
76317
93618
118046
8784

mtsp51
524.7
744.9
1107.2
192.9
600.5
960.9
1436.8
265
514.1
771.5
1188
239.1
506.4
776.3
1198.2
241.1

pr124
150254
235652
346625
51913
244964
349702
507277
74210
133373
202203
298646
49945
134543
200514
296597
50491

pr136
225315
304074
428602
58108
300700
446351
634173
89691
223163
286062
383137
50634
200711
287362
390805
52720

pr144
205204
278907
414337
56813
289835
425899
587282
85838
194333
251995
327206
39986
198892
250551
323671
39774

pr152
279221
375123
530527
76952
380235
559885
790537
115996
207562
304803
409260
58957
217451
302093
403995
58259

pr226
465922
598166
796733
87652
645486
890751
1223229
146404
329781
488266
687780
83403
343841
489202
698623
85134

pr264
285262
373461
479017
37923
385908
527207
698832
78944
172047
230943
311573
33215
173869
230313
314209
36013

pr299
241958
295390
381982
33177
343357
432610
585709
60203
218292
272548
337654
28996
220328
272507
341877
27987

pr76
164907
249381
389365
73419
231932
363181
560341
107747
175830
268514
434977
84973
168656
269631
440088
88178

rat195
6740
8679
12038
1491
9573
12711
16873
2107
7200.4
8721.7
10212.1
884.8
7059.3
8739.1
10285.1
887.8

rat99
2149.2
3295.3
5008.3
913.2
3148
4989
7707
1417
2270
3672
5940
1198
2279
3651
5924
1221

rd400
82572
91903
104639
4786
101925
114910
135460
9463
53411
80713
117869
20490
53204
82324
119892
21221

st70
1023.9
1487.8
2264.9
372
1309.4
2163.9
3315.9
609.2
939.3
1594.3
2594
530.9
972.7
1606.4
2641.4
537

tsp225
13022
16756
22477
2559
18365
24630
32360
4086
13615
16592
20431
1548
13417
16775
20124
1553

Average
95930
126303
173878
21187
131434
182519
252133
33230
82825
112163
154056
19785
82557
112010
154332
20041

5.2. Maximum Traveling Distance Results

The maximum traveling distance was used as the second objective tested by the three algorithms. Thus, the algorithms minimized the loading of the salesman with the highest loading. As a result, this objective balanced the loading among the salesmen. As shown in Figure 4, both the and remained superior to the one-chromosome GA and TPGA. A primary reason for these results could be the selection of a suitable salesman during the assignment phase according to the MLA rule. Hence, following this rule reduced the maximum traveling distance.

Figure 4: Main effects plot of the maximum traveling distance for the compared algorithms.

The assignment of 20 salesmen (see Figure 4) caused the lowest maximum loading on a salesman. This is a reasonable result because the loading is distributed over many salesmen. However, the assignment of 20 salesmen also resulted in the longest total traveling distance (see Section 5.1). Hence, the two objectives present a trade-off and should be considered simultaneously. In Figure 5, it shows the interaction plot between the method and the number of salesmen. This plot indicates and perform well no matter how many salesmen are assigned. In addition, the number of salesmen yields the lower maximum traveling distance solved by the four algorithms.

Figure 5: Interaction plot of the maximum distance for the compared algorithms.

Table 3 shows the complete results for the four algorithms. The and are evidently superior to the one-chromosome GA and TPGA. The and have 20 and 14 lower mean values, respectively. This phenomenon indicates that the indirect encoding is better than the direct coding approach. The standard deviation of one-chromosome GA, TPGA, , and the is 14944, 21728, 13037, and 12940, respectively. StDev indicates that the has less variation than and TPGA. The might perform well in the minimization of the maximum traveling distance.

Table 3: Maximum distance of the three compared algorithms.

Instance
One-chromosome
TPGA

Minimum
Mean
Maximum
StDev
Min
Mean
Max
StDev
Min
Mean
Max
StDev
Min
Mean
Max
StDev

att48
13668
16178
23013
2972
13678
18440
31915
4870
13668
14650
18185
1489
13668
14687
19252
1570

berlin52
2440
3267.4
5506
913.3
2440
3671
6885
1253
2440
2850.3
4357.3
566.9
2440
2838.5
4020.1
543.1

bier127
28890
68194
127487
32040
34697
88175
160235
42537
24007
50498
105182
29541
24007
50441
107861
29014

ch130
2119
4685
8530
2088
2472
6288
11699
2968
1177
3113
6777
2002
1177
3112
6718
1962

eil101
156
349.3
621
145.4
178
438.4
811
201.2
107.2
234.3
504.2
132.3
108
236.2
505
130.5

eil51
109
173.23
301
54.08
109
195.7
352
76.63
108.89
137.98
245.88
39.32
108.89
137.24
238.07
38.33

eli76
140
261.84
451
99.51
141
318.8
576
141
124.28
189.96
369.26
79.58
124.42
190.91
350.25
79.71

gr96
181
341.1
572
120.4
204
434.3
834
182.4
169.74
249.33
479.74
92.83
169.74
253.24
461.39
97.9

kroa150
10877
24701
43153
11241
12890
33802
65152
16075
5395
15378
34924
10129
5395
15389
34281
9933

kroa200
13968
34060
63613
16286
18234
47219
85907
22693
6221
21580
48028
15050
6271
21647
48262
15043

kroB100
7492
15510
27325
6333
8623
19895
35162
8710
6699
10503
20761
4546
6699
10480
19748
4563

kroB150
11170
24615
43288
11044
13835
33480
66963
15396
5750
15271
36168
9535
5750
15220
32456
9289

kroB200
14455
33215
60208
15652
17701
46330
89048
22545
6698
21760
50222
15170
6697
21740
51004
14895

kroC100
7686
15738
27000
6322
8456
20534
39549
9018
5750
10031
20252
4830
5750
9858
19638
4549

kroD100
8035
15495
26648
6166
8676
20215
37334
8865
6357
10374
20636
4679
6357
10395
19507
4606

kroE100
8321
15860
27766
6208
8896
20891
36552
8846
7038
10946
20938
4631
7038
11058
21363
4766

lin105
6687
11982
21810
4386
7334
15608
27895
6384
6375
8802
15930
3060
6375
8816
15539
3017

lin318
26565
65155
117892
32195
34679
91301
170309
45253
10175
44867
106246
33540
10266
45772
107495
34536

mtsp100
7805
15625
28178
6230
8670
20291
36571
8799
6357
10519
22419
4920
6357
10380
20549
4554

mtsp150
12427
28944
51754
13383
14710
38711
71653
18645
5352
18495
42674
12571
5306
18188
39861
12141

mtsp51
110
171.37
278
53.43
109
194.65
367
73.71
108.89
136.71
214.08
36.13
108.89
138.23
238.87
38.52

pr124
31977
65199
119940
26367
38626
89080
179180
40189
22594
38131
77415
19053
22594
38281
75941
18940

pr136
38778
81809
146217
34977
47037
110777
209589
51156
25731
53626
113915
30275
25731
53285
120554
29799

pr144
37745
79364
147984
33520
46297
108344
196336
49131
24313
49939
115229
29812
24313
49548
107708
28633

pr152
48531
101819
178569
41889
59013
142063
271591
63667
31727
60579
131923
34854
31727
61139
133314
34801

pr226
75797
169009
297280
76223
91773
243451
454610
114227
34845
101101
248381
73633
34845
100943
250000
71767

pr264
42743
101884
188189
46735
57515
149315
278195
69900
16339
46435
114410
32724
16524
46883
117359
33250

pr299
33418
78941
144137
38053
42242
113477
208744
53805
14293
51391
114725
36170
14344
51988
118493
36699

pr76
37970
61320
102253
20609
38692
77148
135311
30936
37971
48125
82033
13526
37971
48825
83361
14041

rat195
1032
2319.6
4174
1040.4
1271
3201
6023
1518
604.2
1595.2
3553
990.7
605
1593.6
3354.1
984.7

rat99
466
838.9
1491
299.2
503
1069.9
2007
442
432.4
625.3
1215.9
230.8
432.9
625.8
1120.4
227.1

rd400
9050
23770
43172
12268
12312
33286
62312
17169
2970
16509
39917
13220
2903
16603
39024
13265

st70
209
368
624
120.3
221
448.8
817
178.6
206.4
275.81
481.14
86.41
207.4
276.8
492.14
88.51

tsp225
1971
4479
7780
2066
2521
6188
10987
2891
998
3068
6843
2045
998
3130
7060
2091

Average
15970
34284
61388
14944
19258
47185
87984
21728
9797
21823
47810
13037
9805
21885
47857
12940

6. Conclusions

This study solves an in-group optimization problem that is rarely solved by EDAs. A new EDA , an EDA combined with the MLA rule, was proposed. Because the MLA rule was classified as transform-based encoding, the proposed algorithm was compared with the TPGA, the most favorable direct encoding strategy thus far. We evaluated these algorithms by solving the mTSP problem for 33 instances drawn from the TSPLIB. The scale of the experiments was larger than those of other mTSP studies. Our experimental results showed that the with the MLA rule outperformed the TPGA for both the objectives of total traveling and maximum traveling distance. Thus, the proposed algorithm is capable of efficiently solving the mTSP problem. In addition, the MLA rule was effective and could be applied with some GAs originally designed for permutation-type problems. As a result, this study provides insight for researchers investigating scheduling problems and advances the research on in-group optimization problems.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank anonymous reviewers who enhance the quality of this paper and the Minister of Science and Technology with Grant nos. MOST 102-2221-E-230-019-MY2 and MOST 101-2221-E-343-002.

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