Clonal Selection of Artificial Immune System for Solving ... · Clonal Selection of Artificial Immune System for Solving the Capacitated Vehicle Routing Problem 1Warattapop Chainate,

Clonal Selection of Artificial Immune System for Solving the Capacitated Vehicle Routing Problem

1Warattapop Chainate, 2Pupong Pongcharoen, 3Peeraya Thapatsuwan

1,3 Industrial Engineering Department, Faculty of Engineering, Naresuan University, Pitsanulok, Thailand 65000, [email protected], [email protected]

*2, Corresponding Author Industrial Engineering Department, Faculty of Engineering, Naresuan University, Pitsanulok, Thailand 65000, [email protected]

Abstract

Vehicle routing is one of the crucial logistic activities for delivering goods or services. This paper demonstrates the application of the clonal selection of Artificial Immune System (AIS), Generalized Evolutionary Walk Algorithm (GEWA) and Genetic Algorithm (GA) for solving capacitated vehicle routing problem (CVRP). The optimal parameter settings of the proposed algorithms were investigated using statistical design and analysis of experiment. Sequentially, the performances of the proposed algorithms were compared using twenty benchmarking CVRP instant datasets. The results showed that the parameters of each algorithm were statistically significant with a 95% confidence interval. It was found that the best-so-far solutions obtained from the AIS were up to 71.4% more efficient than those produced by the GEWA and GA for all problem sizes but the AIS required longer computational time.

Keywords: Vehicle Routing, Artificial Immune System, Generalized Evolutionary Walk Algorithm,

Genetic Algorithm, Parameter Setting, Experimental Design and Analysis

1. Introduction

Nowadays, the transportation of commodities, people and services has a considerable effect on most businesses and also has a large impact on the environment. In business, well-organized routing can minimize the transportation costs, develop a customer’s trust and also improve a firm’s competitive advantage. For example, Thailand’s logistics report in 2011 [1] showed that the transportation costs had the largest proportion (about 47% or 776.4 billion baht) of total logistics cost. For an environmental concern, prior-plan for routing may decrease the traffic congestion and also reduces carbon credits and emission.

Vehicle routing problem (VRP) is a generic name given to a class of logistic activities related to the search of the most efficient route for a fleet of vehicles. Each vehicle departs from a depot, serves a given of scattered customers with a known non-negative demand, and returns back to the same depot [2, 3]. The capacitated vehicle routing problem (CVRP) is additionally constrained by the vehicle’s loading capacity. The common objective of CVRP is to minimize the total cost or distance associated with the vehicle usage. The CVRP is, however, categorized as the non-deterministic polynomial hard (NP-hard) problem, which means that the computational effort required for solving this problem increases exponentially with the problem size [4].

Although the exact algorithms guarantee the best solution, these are not appropriate approaches to solve very large-size NP problems, particularly, solving within a limited time period. Therefore, approximation algorithms so called, Metaheuristics or Nature-inspired optimization methods have received more attention in the last few decades. Metaheuristics that iteratively conduct stochastic search process to find near optimal solutions in acceptable computational time can be categorized into three groups: physically-cased, socially-based and biologically-based inspiration [5].

Artificial Immune System (AIS) is the recent bio-inspired algorithms based on the principles and processes of the vertebrate immune system. Based on the variety of immunological theories, there are a number of AIS algorithms including immune network [6], negative selection [7], danger theory [8] and clonal selection [9]. Clonal selection based on a situation of ‘B’ cell response against a nonself molecule called antigen with an affinity by proliferating and producing antibody in order to kill antigenic cells [10].

AIS with Clonal Selection performs multiple directional searches using a set of cloned antibodies, conducting by affinity maturation and receptor editing processes. However, the performance of

Clonal Selection of Artificial Immune System for Solving the Capacitated Vehicle Routing Problem Warattapop Chainate, Pupong Pongcharoen, Peeraya Thapatsuwan

Journal of Next Generation Information Technology(JNIT) Volume4, Number3, May 2013 doi:10.4156/jnit.vol4.issue3.20

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conventional AIS may not be influential for some problems [11]. A probable reason is that the mutation operation may not play an important role for all problem domains. Pongcharoen et al. [12] improved the conventional AIS for solving TSP by embedding two effective mutation operations (Inversion and Shifted Operations mutations), which have been previously investigated via statistical design and analysis [13]. The AIS has also been successfully applied to solve various combinatorial optimization problems such as flow shop scheduling [14], machine loading [15] and traveling salesman [12]. However, the application of AIS has been rarely found to solve CVRP.

The objectives of this paper were to: i) investigate the suitable setting of AIS parameters for solving the CVRP through the statistical design of experiment, and ii) comparatively study the performance of the proposed AIS and other methods (GA and GEWA) for benchmarking twenty CVRP instances in terms of the quality of solutions obtained and the computational time used.

The remaining sections are organized as follows. Section 2 is a reviews of the literature related to the CVRP including its classification and the mathematical definition. In Section 3, the processes and pseudo codes of AIS including GEWA and GA for solving the CVRP are described. Section 4 demonstrates two-step sequential experiment and data analyses. Finally, the conclusions of this research are shown in section 5.

2. Vehicle Routing Problem (VRP)

Dantzig and Ramser [16] initially proposed a problem entitled “the truck dispatching problem”.

Their work was regarded as the first article originating to the well-known Vehicle Routing Problem, which has been extensively studied in the last few decades [17-21]. Nowadays, VRP and its variants have become one of the most important applications in the area of distribution management [22]. There are various types of VRP classified by their additional constraints such as loading capacity limitation [20], multiple depots [23], restricted time window [24] and fuzzy demands [25].

The classical CVRP considers the vehicle routes, in which each vehicle has the equivalent loading capacity. The problem starts with each vehicle departing from a depot and then routing to serve through geographically dispersed customers. Each customer has known demands and can only be visited by one vehicle. Finally, the problem is terminated when all vehicles have returned to the depot. On loading capacity constraint, each vehicle cannot be loaded excessively beyond the maximum loading capacity. The objective of CVRP is to minimize the traveling cost associated with the routing of vehicles. The mathematical formulation of the CVRP modified from Lin et al. [26] is described as follows:

Notation:

i, j customer ith (i=0, 2, …, N) (j=0, 2, …, N) k vehicle kth (k=1, 2, …, K) Cij cost incurred on customer i to customer j Qk loading capacity of vehicle k Di demand at customer i Xijk binary variable if Xijk = 1, vehicle k travels from customer i to customer j directly, otherwise it is equal to 0

N

i

N

j

K

kijkij XCMinimise

0 0 1

(1)

Subject to:

N

i

N

jkiijk QDX

0 0

, 1 ≤ k ≤ K (2)

N

j

N

jjikijk XX

1 1

1 , for i = 0 and k {1, 2, ...., k} (3)

K

k

N

iijkX

1 0

1

, for j = 1, ..., N, i ≠ j (4)


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K

k

N

jijkX

1 0

1

, for i = 1, ..., N, i ≠ j (5)

K

k

N

jijk KX

1 1

, for i =0 (6)

1,0ijkX , i ≠ j and i, j {0, 1,..., N} (7)

The equation (1) is the objective function aimed to minimize the total cost. Constraint (2) ensures

that the demands do not exceed the loading capacity of vehicle k. Constraint (3) ensures that every route starts and ends at the delivery depot. The constraints (4) and (5) indicate that each customer is served by only one vehicle. Constraint (6) specifies that there are maximum K routes going out of the delivery depot, and constraints (7) specify the range of the decision variables.

Various metaheuristics have been applied for solving CVRP such as genetic algorithm [3, 27, 28]. However, there has been very little research works, in which the Artificial Immune System (AIS) has been applied to solve the vehicle routing problem (VRP). Among a few, Masutti and de Castro [29] have proposed The RABNET-CVRP (Real-valued AntiBody NETwork to solve the capacitated vehicle routing problem), which is the immune method based relating to self-organizing networks. Their proposed method has been benchmarked with other self-organizing networks and the experimental results have shown that the performance of immune technique was capable of finding the better solutions than the results obtained from other methods.

Ma et al. [30] have developed an improved clonal selection with three ideas. A new process called vaccination, which is the process of learning the prior knowledge of the best antibody (individual) has been added in order to improve the quality of searching. Cloning operators have also been introduced to maintain the diversity of the population and improve the performance of algorithm. The best individual has been retained using the immune mechanism. They have shown that the quality of solution obtained from the proposed clonal selection is better than the other algorithms (SGA and IGA). However, a computation experiment has been conducted using a single instant dataset without showing a convergence graph.

3. Metaheuristics Optimization Algorithms

Metaheuristics have emerged over the last few decades because it can find near optimal solutions

for a very large-size problem within an acceptable amount of computational time. The methods are particularly popular for solving complex combinatorial optimization problems with very large-scale solution space, which full enumerative search is impractical. The following subsections briefly describe three metaheuristics that are applied to solve capacitated vehicle routing problems.

3.1. Artificial Immune System (AIS)

Immune system is one of the most intricate biological mechanisms within an organism. The immune

system is a collection of cells, tissues, and molecules that mediate resistance to infections. The responsibility of immune system is to prevent infections and to eradicate recognized infections [31]. There are many factors to cause disease. Infectious disease is a disease incident caused by pathogen’s invading. A pathogen or infectious agent is a microorganism (e.g. viruses or bacteria) that aims to successfully infect its host e.g. plant, animal or even human [32, 33]. The primary function of the immune system is to distinguish between self-cells (harmless) and nonself-cells (disease causing), and to eliminate the nonself-cells out of the organism [10].

Farmer et al. [6] initially published the idea of the computational immunology named the Artificial Immune System (AIS). Based on immunology theory, many types of AIS have been proposed to solve complex computational, engineering or optimization problems [10]. Focusing on clonal selection, there are two main processes (clonal selection and affinity maturations) inspired by the biological stimulating ‘B’ cells and ‘T’ cells to produce an antibody in order to match and kill the foreign invaders [11, 34]. The pseudo code is shown in Figure 1.


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Figure 1. The AIS pseudo code for CVRP

Clonal Selection Procedure for CVRP

Clonal selection algorithm consists of six main processes: problem encoding, antibody initialization,

affinity evaluation, mutation operation and receptor editing. The main processes are described as follows.

Problem Encoding and Antibody Initialization

Firstly, a solution is encoded and called an antibody, which represents the sequence of customers to

be served by each vehicle. Each sub-antibody in the string is assigned by the integer number referring to a customer to be visited. The sequence of sub-antibodies is the order of visiting customers as shown in Fig. 2. Therefore, the length of an antibody (possible solution) is determined by the total number of customers. The process of generating antibody can be repeated until the size of population is satisfied. The population size (P) determines the number of candidate solutions in the solution space. However, increasing of the population size will increase the amount of searches and also the bulk of memory and computation required.

Figure 2. An example of antibody representing of 3 vehicles, 8 customers and a single depot

Affinity Evaluation

After generating an antibody, the next step is to measure the affinity of the antibodies through the affinity function, which is determined by the total traveled distance from all vehicles. Therefore, the shortest distance is equal to the highest affinity value. The affinity function is presented below:

N

i

N

j

K

kijkij XClueAffinityVa

0 0 1

/1 (8)

Initialize the value of AIS parameters [antibody population (P), iterations (Imax), and percentage of antibody elimination (%B)]. Generate a population of P antibodies based on the feasible fleet of vehicles For each antibody (i P), calculate affinity (i) Set current iteration (I) = 1 Do

For each antibody (i) Calculate the number of clones (Nc) and clone antibody (i) For each clone, apply inverse mutation to create a new antibody

Calculate the affinity of the new antibody from the tour obtained If affinity (new antibody) is better than the clone then clone = new antibody Else perform shift operation mutation to create a new antibody

Calculate the affinity of the new antibody from the tour obtained If affinity (new antibody) is better than the clone then clone = new antibody

antibody (i) = clone Eliminate the worst antibodies from the population based on %B Create new antibodies to replace the eliminated antibodies I = I + 1

While I ≤ Imax


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Mutation Operation

Mutation has been known to be an important part of genetic evolution of GA and also the crucial processes within the AIS algorithm. It determines the diversity of antibody populations and the amount of exploration within the search space. In the mutation process, each antibody is cloned where the number of clones is determined by its affinity value (fitness) and the size of antibody population. Each clone is then mutated using the inverse mutation operator [35]. If a mutated antibody is better than the original clone, the clone is replaced by the new mutated antibody. Otherwise, the shift operation mutation [36] is used to try and produce an antibody with a better affinity value. This process is repeated until all the antibodies are mutated. Examples on the mechanism of the inverse and shift operation mutations are demonstrated in Figure 3.

Figure 3. Inverse mutation and Shift operation mutation

Receptor Editing

After finishing the mutation process, the cloned antibodies are sorted. Next step, the receptor editing is to eliminate bad antibodies from the current population. The number of eliminating antibodies for each iteration depends on the percentage of antibody elimination (%B).

Finally, the whole process is repeated until the given number of iterations (Imax) is satisfied. Similarly to the population size, the number of iterations directly determines the amount of searches, which is related to the probability of finding the optimum solution. But higher number of iterations requires intensive numerical and long computational time and resources.

3.2. Genetic Algorithm (GA)

Genetic Algorithms (GA) were initially introduced by John Holland [37]. The idea of GA is inspired

by Darwin’s theory about evolution in which the strong survives and the weak perishes. Nowadays, GA have become one of the well-known biology-inspired metaheuristics which have drawn a great deal of attention from researchers.

The simple GA starts by encoding the problem to produce a list of genes in the form of a string. The genes are randomly combined to create a population of chromosomes representing to a possible solution. The population size (P) and the number of generations (G) are important parameters that need to be pre-assigned. Then, genetic operations including crossover and mutation are performed on chromosomes. To produce offspring, chromosomes are randomly selected from the population as parents (one for mutation and two chromosomes for crossover operations). Crossover mechanism helps in search strategy to explore the solution space whilst exploitation is conducted by the mutation mechanism. The next step is to measure the chromosomes’ fitness value of which the probability of the survival is determined. After performing the fitness evaluation process, a chromosome selection mechanism such as the roulette wheel [35], is then used to stochastically choose the same amount of chromosomes to the next generation. The GA process is repeated until a termination condition is satisfied. The mechanism of GA is demonstrated as the pseudo code in Figure 4.

3.3. Generalized Evolutionary Walk Algorithms (GEWA)

Generalized Evolutionary Walk Algorithm (GEWA) was originally designed by Xin-She Yang [38]. Evolutionary walk is a random walk, but with a biased selection towards optimality. Generalized Evolutionary Walk Algorithms is based on three major components: i) global exploration by


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randomization, ii) intensive local search by random walk, and iii) the selection of the best with some elitism.

The main steps of the GEWA start from initializing a population of walkers. The population is evaluated to find the current best (g*) solution. The next step is to perform local search (random walk) or global search (randomization) which is randomly selected compared with α (see Eq. 9). During comparison loop, the best-so-far solution is iteratively updated.

Xt+1 = g* + εd (9)

where ε is drawn from a Normal distribution and d is the step length vector related to the actual

scales of independent variables. The randomization step can be achieved by Eq. 10.

Xt+1 = L + (U – L)ϵ (10) where ϵ is drawn from a uniform distribution. U and L are the upper and lower bound vectors,

respectively. The GEWA process is repeated until termination criteria are satisfied. The pseudo code of GEWA is demonstrated in Figure 5.

Figure 4. The pseudo code of the GA procedure

Figure 5. The pseudo code of the GEWA procedure

4. Experimental Design and Analysis

The experiments conducted in this work were based upon a two-step sequential experiment.

Experiment A was designed to identify appropriate settings of the AIS, GA and GEWA parameters whilst Experiment B was aimed at comparing the performance of the AIS with the GA and GEWA in terms of the quality of solutions obtained and the computational time required. All computational runs were conducted on a personal computer with Intel Core i7-2630QM 2.00 GHz CPU and 8 GB DDRIII RAM. This research considered the twenty benchmarking instances of CVRP. All instances were selected from the vehicle routing data sets [39, 40] as shown in details in Table 1.

Initialize the value of GA parameters [population size (P), number of generations (G), and probabilities of crossover (Pc) and mutation (Pm)]. Generate a population of P chromosomes For each chromosome (i P), calculate fitness (i) Set current generation (g) = 1 Do Based on Pc, randomly select two parent chromosomes for crossover operation Based on Pm, randomly select a parent chromosome for mutation operation Calculate the fitness of the offspring If offspring is better than the parent, replace the parent Randomly select the survived chromosome for next generation using roulette wheel

g = g + 1 While g ≤ G

Initialize a population of n walkers xi (i = 1, 2, …, n) Evaluate fitness Fi of n walkers and find the current best g* While (t < MaxGeneration) or (stop criterion) Discard the worst solution and replace it by (a) or (b) If (rand < α) Local search: random walk around the best Xt+1 = g* + εd (a) Else Global search: randomization Xt+1 = L + (U – L)ϵ (b) End Evaluate new solutions and find the current best g*t t = t + 1 End while


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Table 1. Benchmarking CVRP instance datasets Problem Instance

Customers Vehicles Vehicle’s Capacity

Problem Instance

Customers Vehicles Vehicle’s Capacity

A-n32-k5 31 5 100 B-n31-k5 30 5 100 A-n33-k5 32 5 100 B-n41-k6 40 6 100 A-n33-k6 32 6 100 B-n50-k7 49 7 100 A-n34-k5 33 5 100 B-n68-k9 67 9 100 A-n55-k9 54 9 100 B-n78-k10 77 10 100 A-n63-k10 62 10 100 M-n101-k10 100 10 200 A-n64-k9 63 9 100 M-n121-k7 120 7 200 A-n65-k9 64 9 100 M-n151-k12 150 12 200 A-n69-k9 68 9 100 M-n200-k16 199 16 200 A-n80-k10 79 10 100 M-n200-k17 199 17 200

Experiment A The aim of this experiment was to identify appropriate settings of the AIS, GA, GEWA parameters

for a CVRP. AIS parameters include the combination of the number of antibodies and the number of iterations (AI) as well as the percentage of eliminating antibodies (%B). GA parameters include the combination of the number of population and the number of iterations (PG), probability of crossover (Pc) and probability of mutation (Pm). GEWA parameters include the combination of the number of population and the number of iterations (PG), probability of selected procedure (α) and probability of discard the worst solution (Pd). The full factorial experimental design and the range of values considered for each of the factors is shown in Table 2. The computational runs were replicated 30 times with different random seed numbers.

The combination of PG or AI directly determines the amount of searches conducted in each algorithm. The setting of these combinations can be ideally defined as large as possible. The large amount of searches theoretically increases the probability of finding an optimal solution but requires intensive numerical and long computational time and resource. In practice, there may be limitation on computational time and resources. For benchmarking perspective, the comparison of algorithm’s performance must be unbiased and should base on the similar amount of searches. Therefore, the amount of searches considered in this work was fixed at 100,000 candidate solutions. This amount of searches was sufficient to achieve the convergence of results during pre-experimental test runs.

Table 2. Experimental factors and levels considered Methods Factors Level Values

AIS Amount of searches (AI) 4 100*1000, 200*500, 500*200, 1000*100 Percentage of eliminating antibody (%B) 5 10, 30, 50, 70, 90

GA Amount of searches (PG) 4 100*1000, 200*500, 500*200, 1000*100 Probability of crossover (Pc) 3 0.1, 0.5, 0.9 Probability of mutation (Pm) 3 0.1, 0.5, 0.9

GEWA Amount of searches (PG) 4 100*1000, 200*500, 500*200, 1000*100 Probability of selected procedure (α) 3 0.25, 0.5, 0.75 Probability of discard (Pd) 4 1/P, 0.1, 0.25, 0.5

The results were analyzed using the general linear model form of analysis of variance (ANOVA), shown in Table 3. For AIS parameters the percentages of eliminated antibodies (%B) were statistically significant with a 95% confidence level. All GA parameters were statistically significant in terms of the main effect with a 95% confidence interval. For the GEWA parameters, probability of selected procedure (α) and probability of discard the worst solution (Pd) were statistically significant with a 95% confidence level. In order to identify the appropriate setting of the factors considered, the main effect plot and the interaction plot are provided in Figures 6. It can be seen that the best settings of AIS parameters including AI and %B were 100*1000 and 50, respectively. The best settings of GA parameters that were PG, Pc and Pm were 100*1000, 0.5 and 0.1, respectively. Finally, the best settings of GEWA parameters including PG, α and Pd were 100*1000, 0.75 and 0.5, respectively.


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Table 3. Analysis of variance (ANOVA) on the experimental results Methods Source DF Sum of Squares Mean Square F ρ

AIS

AI 3 7397 2466 1.77 0.152 %B 4 84483 21121 15.14 0.000 AI*%B 12 34907 2909 2.08 0.016 Error 580 809218 1395 Total 599 936006

GA

PG 3 3686648 1228883 394.75 0.000 Pc 2 3224639 1612320 517.92 0.000 Pm 2 9099111 4549556 1461.43 0.000 PG*Pc 6 465533 77589 24.92 0.000 PG*Pm 6 717896 119649 38.43 0.000 Pc*Pm 4 1557504 389376 125.08 0.000 PG*Pc*Pm 12 1173500 97792 31.41 0.000 Error 1044 3250071 3113 Total 1079 23174902

GEWA

PG 3 43316 14439 1.73 0.159 α 2 161585 80793 9.68 0.000 Pd 3 15877024 5292341 634.10 0.000 PG*α 6 116248 19375 2.32 0.031 PG*Pd 9 294269 32697 3.92 0.000 α*Pd 6 79795 13299 1.59 0.145 PG*α*Pd 18 203690 11316 1.36 0.144 Error 1392 11617856 8346 Total 1439 28393784

Figure 6. Main effect and interaction plots results obtained from the proposed algorithms

Experiment B

This experiment was aimed to benchmark the performance of the AIS, GA and GEWA using the

best parameter settings that had previously been identified in experiment A. For each algorithm, the computational runs were replicated 30 times with different random seeds. The experimental results for 20 different problems were analyzed in terms of the average, standard deviation (Std.Dev.), maximum and minimum travelled distance, and computational time as shown in Table 4. It can be seen that the average solutions obtained from AIS outperformed those obtained from GA and GEWA for all problem sizes by up to 71.39%. Not only average solutions, but also the minimum and maximum solutions produced by AIS were 71.14% greater than the routes produced by other algorithms for all problem sizes. In term of computational time usage, GA was the quickest approach in this test whilst AIS consumed the longer computational time for finding the high-quality solution.


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Table 4. The experimental results of twenty benchmarking using AIS, GA, and GEWA

Problem Methods Quality of solutions obtained Time usage

(Second) Average Std. Dev. Maximum Minimum A-n32-k5

AIS 838.13 (35.48%) 31.7780 913 784 (28.79%) 68 GA 1025.63 70.4691 1159 897 26

GEWA 1299.03 102.4293 1499 1101 43 A-n33-k5

AIS 688.4 (37.65%) 21.2028 751 661 (31.93%) 33 GA 840.63 62.6244 995 727 27

GEWA 1104.17 74.6283 1218 971 47 A-n33-k6

AIS 758.97 (31.73%) 19.2147 810 742 (22.87%) 73 GA 911.77 51.7225 1056 831 27

GEWA 1111.73 67.9934 1245 962 45 A-n34-k5

AIS 808.27 (34.30%) 25.6595 900 778 (27.09%) 74 GA 971.90 58.9414 1106 874 28

GEWA 1230.17 78.7305 1399 1067 45 A-n55-k9

AIS 1171.57 (48.03%) 29.9720 1236 1119 (44.74%) 131 GA 1623.23 72.2532 1732 1462 42

GEWA 2254.33 100.9050 2470 2025 70 A-n63-k10

AIS 1476.93 (45.46%) 49.1500 1593 1397 (45.22%) 150 GA 2028.63 97.3332 2213 1801 47

GEWA 2707.77 80.7652 2848 2550 82 A-n64-k9

AIS 1554.63 (44.63%) 30.5744 1615 1510 (43.21%) 149 GA 2126.13 91.7499 2335 1939 47

GEWA 2807.80 86.0475 2971 2659 80 A-n65-k9

AIS 1352.47 (52.27%) 58.3448 1478 1270 (50.60%) 153 GA 2055.83 100.1044 2309 1861 47

GEWA 2833.80 106.9152 3036 2571 81 A-n69-k9

AIS 1334.77 (54.28%) 45.4659 1472 1250 (54.28%) 160 GA 2017.90 98.5670 2209 1826 50

GEWA 2919.70 87.3381 3057 2734 84 A-n80-k10

AIS 2068.00 (47.41%) 51.2432 2190 1984 (47.16%) 195 GA 2863.30 115.0796 3098 2696 58

GEWA 3932.47 94.2311 4215 3755 99 B-n31-k5

AIS 689.1 (20.24%) 15.72906 738 672 (11.35%) 71 GA 736.90 15.3496 759 705 27

GEWA 863.90 62.7724 961 758 44 B-n41-k6

AIS 862.43 (42.23%) 18.6948 917 831 (34.15%) 90 GA 1141.77 91.8423 1329 990 32

GEWA 1492.90 97.6038 1689 1262 55 B-n50-k7

AIS 778.70 (54.69%) 24.6691 845 747 (46.34%) 115 GA 1193.93 112.5341 1519 1034 39

GEWA 1718.70 125.1237 1928 1392 65 B-n68-k9

AIS 1412.13 (49.73%) 53.0002 1536 1335 (47.09%) 162 GA 2026.57 106.7525 2257 1868 50

GEWA 2809.23 151.3718 3038 2523 84 B-n78-k10

AIS 1432.07 (55.02%) 44.6380 1551 1365 (51.27%) 190 GA 2181.37 128.0144 2419 1922 57

GEWA 3183.87 172.0298 3477 2801 97 M-n101-k10

AIS 1156.93 (64.63%) 40.2406 1231 1089 (65.12%) 233 GA 2053.00 72.6004 2201 1914 69

GEWA 3270.83 85.2801 3417 3122 115 M-n121-k7

AIS 1516.90 (71.39%) 55.0976 1622 1424 (71.14%) 283 GA 3040.37 175.4838 3349 2626 81

GEWA 5301.80 145.5343 5608 4953 139 M-n151-k12

AIS 1865.10 (57.67%) 74.0020 2013 1741 (58.89%) 379 GA 2953.67 97.9138 3137 2806 106

GEWA 4405.97 73.1647 4522 4235 181 M-n200-k16

AIS 2825.60 (52.44%) 90.7504 2981 2596 (54.62%) 539 GA 4030.30 164.6210 4328 3771 146

GEWA 5941.40 83.4450 6077 5720 250 M-n200-k17

AIS 2825.60 (52.44%) 90.7504 2981 2596 (54.62%) 550 GA 4030.30 164.6210 4328 3771 153

GEWA 5941.40 83.4450 6077 5720 259


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Table 5 shows the percentage of margin between AIS’s best so far solution and the announced optimal solution. For the small problem size (less than 40 customers), the solutions obtained from AIS were equal to the optimal solution. However, the comparison for the larger problem size demonstrated the optimal solution is particularly better than the AIS’s solution obtained. One of the probable reasons was the fixed population size and number of generations which may not suitable to the larger size of problem.

Table 5. The comparison of AIS’s BSF solution and the optimal solution

Problem BSF

solution Optimal solution

(%Gap) Problem

BSF solution

Optimal solution (%Gap)

A-n32-k5 784 784 (0%) B-n31-k5 672 672 (0%) A-n33-k5 661 661 (0%) B-n41-k6 831 829 (0.24%) A-n33-k6 742 742 (0%) B-n50-k7 747 741 (0.81%) A-n34-k5 778 778 (0%) B-n68-k9 1335 1272 (4.95%) A-n55-k9 1119 1073 (4.29%) B-n78-k10 1365 1221 (11.79%) A-n63-k10

1397 1314 (6.32%) M-n101-k10 1089 820 (32.80%)

A-n64-k9 1510 1401 (7.78%) M-n121-k7 1424 1034 (37.72%) A-n65-k9 1270 1174 (8.18%) M-n151-k12 1741 1053* (65.34%) A-n69-k9 1250 1159 (7.85%) M-n200-k16 2596 1373* (89.08%) A-n80-k10

1984 1763 (12.54%) M-n200-k17 2596 1373* (89.08%)

* The solution is not declared as the optimal solution yet.

Figure 7 shows the comparative progress of AIS, GA and GEWA for exploring the solution space.

The total travelling costs found in the initial iteration from all three methods were similar. During progressed iteration, the travelling cost obtained from AIS iteratively reduced quicker than that produced by GA and GEWA.

Figure 7. The progress graphs of each algorithms from large problem

5. Conclusions

This paper describes successful development and application of a tool incorporating an Artificial

Immune System (AIS), a Genetic Algorithm (GA) and Generalized Evolutionary Walk Algorithm (GEWA) for solving vehicle routing problem (CVRP). A two-step sequential experiment was designed and conducted to identify the best parameter configuration of the algorithms for solving twenty CVRP instances. An analysis of variance showed that the AIS parameters amount of searches (AI) and the percentage of eliminating antibodies (%B) were statistically significant with a 95% confidence interval. A main effect plot found that values of 100*1000 and 50%, performed best. All GA parameters and the GEWA parameters, probability of selected procedure (α) and probability of discard (Pd) were statistically significant in terms of the main effect with a 95% confidence interval. The best settings of GA and GEWA parameters PG, Pc, Pm, α and Pd were 100*1000, 0.5, 0.1, 0.75 and 0.5, respectively.


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In the sequential experiment, the average best-so-far solutions obtained from the AIS were the most 70% better than those produced by the GA and GEWA for all problem sizes. The convergence graph indicated that the best-so-far result obtained from AIS was decreased quicker than those from the GA and GEWA. However, the high-quality solution obtained from AIS required a longer computational time usage than the other approaches.

6. Acknowledgements

The authors would like to acknowledge Assist.Prof.Dr.Sutatip Pongcharoen and Dr.Gareth Ross for

many valuable comments and useful suggestions for preparing the manuscript.

7. References [1] NESDB, Thailand's Logistics Report 2011, Office of the National Economic and Social

Development Board (NESDB), Thailand, 2012. [2] Jean-Francois Cordeau, Michel Gendreau, Gilbert Laporte, Jean-Yves Potvin, Federic Semet, "A

guide to vehicle routing heuristics", Journal of the Operational Research Society, vol. 53, pp. 512-522, 2002.

[3] Christian Prins, "A simple and effective evolutionary algorithm for the vehicle routing problem", Computers & Operations Research, vol. 31, no. 12, pp. 1985-2002, 2004.

[4] Gilbert Laporte, "What you should know about the VRP", Naval Research Logistics, vol. 54, no. 8, pp. 811-819, 2007.

[5] Orhan Engin, Alper Döyen, "A new approach to solve hybrid flow shop scheduling problems by artificial immune system", Future Generation Computer Systems, vol. 20, no. 6, pp. 1083-1095, 2004.

[6] J. Doyne Farmer, Norman H. Packard, Alan S. Perelson, "The immune system, adaptation, and machine learning", Physica D, vol. 2, pp. 187-204, 1986.

[7] Stephanie Forrest, Alan S. Perelson, Lawrence Allen, Rajesh Cherukuri, "Self-nonself discrimination in a computer", In Proceedings of the 1994 IEEE Symposium on Research in Security and Privacy, 1994.

[8] Polly Matzinger, "The danger model: a renewed sense of self", Science, vol. 296, pp. 301-305, 2002.

[9] Leandro N. de Castro, Fernando J. Von Zuben, "An evolutionary immune network for data clustering", In Proceedings of the 6th Brazilian Symposium on Neural Network (2000), Rio de Janeiro, Brazil, 2000.

[10] Leandro N. de Castro, Jon Timmis, "Artificial immune system: a novel paradigm to pattern recognition", In Proceeding of SOCO, University of Paisley, UK, 2002.

[11] Simon M. Garrett, "How do we evaluate artificial immune systems?.", Evolutionary Computation, vol. 13, no. 2, pp. 145-177, 2005.

[12] Pupong Pongcharoen, Warattapop Chainate, Sutatip Pongcharoen, "Improving Artificial Immune System Performance: Inductive Bias and Alternative Mutations", in Artificial Immune Systems, Bentley, P.J., Lee, D., Jung, S., Editors., Springer, pp. 220-231, 2008.

[13] Pupong Pongcharoen, Warattapop Chainate, Peeraya Thapatsuwan, "Exploration of genetic parameters and operators through travelling salesman problem", ScienceAsia, vol. 33, no. 2, pp. 215-222, 2007.

[14] Akhilesh Kumar, Anuj Prakash, Ravi Shankar, M. K. Tiwari, "Psycho-Clonal algorithm based approach to solve continuous flow shop scheduling problem", Expert Systems with Applications, vol. 31, no. 3, pp. 504-514, 2006.

[15] Nitesh Khilwani, Anoop Prakash, Ravi Shankar, M. K. Tiwari, "Fast clonal algorithm", Engineering Applications of Artificial Intelligence, vol. 21, no. 1, pp. 106-128, 2008.

[16] George B. Dantzig, John H. Ramser, "The truck dispatching problem", Management Science, vol. 6, no. 1, pp. 80-91, 1959.

[17] Lawrence Bodin, Bruce L. Golden, Arjang A. Assad, Michael O. Ball, "The state of art in the routing and scheduling of vehicles and crews", Computers & Operations Research, vol. 10, pp. 63-212, 1983.


177

[18] Michel Gendreau, Gilbert Laporte, Jean-Yves Potvin, "Vehicle routing: Modern heuristics", in Local search in combinatorial optimization, Aarts, E.H.L., Lenstra, J.K., Editors., Wiley, Chichester, pp. 311-336, 1997.

[19] Michel Gendreau, Gilbert Laporte, Jean-Yves Potvin, "Metaheuristics for the capacitated VRP", in The Vehicle Routing Problem, Toth, P., Vigo, D., Editors., Society for Industrial and Applied Mathematics (SIAM), pp. 129-154, 2002.

[20] Paolo Toth, Daniele Vigo, The Vehicle Routing Progblem, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001.

[21] Francisco B. Pereira, Jorge Tavares, Bio-inspired Algorithms for the Vehicle Routing Problem, Springer, Berlin, Heidelberg, 2009.

[22] Jean-François Cordeau, Gilbert Laporte, Martin W. P. Savelsbergh, Daniele Vigo, "Chapter 6 Vehicle Routing", in Handbooks in Operations Research and Management Science, Cynthia, B., Gilbert, L., Editors., Elsevier, pp. 367-428, 2007.

[23] Liao Wer, He Zhenggang, Song Jinyu, "The modeling and algorithm for a multi-depot vehicle routing problem based on the difference of customer demands", International Journal of Advancements in Computing Technology (IJACT), vol. 5, no. 4, pp. 778-785, 2013.

[24] Jean-François Cordeau, Gilbert Laporte, Anne Mercier, "A unified tabu search heuristic for vehicle routing problems with time windows", Journal of the Operational Research Society, vol. 52, no. 8, pp. 928-936, 2001.

[25] Yang Peng, Ye-mei Qian, "A particle swarm optimization to vehicle routing problem with fuzzy demands", Journal of Convergence Information Technology (JCIT), vol. 5, no. 6, pp. 112-119, 2010.

[26] Shih-Wei Lin, Zne-Jung Lee, Kuo-Ching Ying, Chou-Yuan Lee, "Applying hybrid meta-heuristics for capacitated vehicle routing problem", Expert Systems with Applications, vol. 36, no. 2, Part 1, pp. 1505-1512, 2009.

[27] Barrie M. Baker, M. A. Ayechew, "A genetic algorithm for the vehicle routing problem", Computers & Operations Research, vol. 7, pp. 301-317, 2003.

[28] Yannis Marinakis, Magdalene Marinaki, "A hybrid genetic – Particle Swarm Optimization Algorithm for the vehicle routing problem", Expert Systems with Applications, vol. 37, no. 2, pp. 1446-1455, 2010.

[29] Thiago Masutti, Leandro N. de Castro, "A Neuro-Immune Algorithm to Solve the Capacitated Vehicle Routing Problem", in Artificial Immune Systems, Bentley, P., Lee, D., Jung, S., Editors., Springer, pp. 210-219, 2008.

[30] Jia Ma, Liqun Gao, Gang Shi, "An improved immune clonal selection algorithm and its applications for VRP", In Proceedings of the IEEE International Conference on Automation and Logistics, Shenyang, China, 2009.

[31] Abul K. Abbas, Andrew H. Lichtman, Basic Immunology: Functions and Disorders of the Immune System, Suanders Elsevier, Philadelphia, 2004.

[32] Abul K. Abbas, Andrew H. Lichtman, Jordan S. Pober, Cellular and Molecular Immunology, W.B. Saunders, Philadelphia, 2000.

[33] Peter Delves, Seamus Martin, Dennis Burton, Ivan Roitt, Roitt's Essential Immunology, Wiley-Blackwell, London, 2006.

[34] Frank M. Burnet, The Clonal Selection Theory of Acquired Immunity, Vanderbilt University Press, 1959.

[35] David E. Goldberg, Genetic Algorithms in Search, Optimisation and Machine Learning, Addison-Wesley, Reading, MA, 1989.

[36] Tadahiko Murata, Hisao Ishibuchi, "Performance evaluation of genetic algorithms for flowshop scheduling problems", In Proceedings of the First IEEE Conference on Evolutionary Computation, 1994.

[37] John H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Michigan, 1975.

[38] Xin-She Yang, Nature-Inspired Metaheuristic Algorithm, Luniver Press, 2008. [39] P. Augerat, J. Belenguer, E. Benavent, A. Corbern, D. Naddef, G. Rinalddi, Computational results

with a branch and cut code for capacitated vehicle routing problem, University Joseph Fourier, 1995.


178

[40] Nicos Christofides, Aristide Mingozzi, Paolo Toth, "The vehicle routing problem", in Combinatorial Optimization, Christofides, N., Mingozzi, A., Toth, P., Sandi, C., Editors., Wiley-Interscience, New York, pp. 315-338, 1979.


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