An Effective Genetic Algorithm for Capacitated Vehicle ... · Bandung, Indonesia, March 6-8, ... at the central depot, (2) ... The fitness measure for evaluating a chromosome is total

Proceedings of the International Conference on Industrial Engineering and Operations Management

Bandung, Indonesia, March 6-8, 2018

© IEOM Society International

An Effective Genetic Algorithm for Capacitated Vehicle

Routing Problem

Hadeer Awada, Raafat Elshaera†, Adel AbdElmo’ezab and Gamal Nawaraa

a Industrial Engineering Department Faculty of Engineering, Zagazig University

Zagazig, Egypt

[email protected], [email protected], [email protected]

bIndustrial Engineering Department College of Engineering

King Khaled University, KSA

[email protected]

Abstract

The capacitated vehicle routing problem (CVRP) is an NP-hard problem. Therefore, metaheuristics are

often more suitable for practical applications. In this paper, a genetic algorithm (GA) is proposed to solve the problem. The performance of the proposed algorithm is tested on different sets of benchmark instances. The computational results indicate that the algorithm has a satisfactory performance in solving the problem.

Keywords Capacitated Vehicle Routing Problem, Genetic Algorithm

1. Introduction

The capacitated vehicle routing problem (CVRP), introduced by [1], is one of the most attractive topics in operation

research, communications, manufacturing, transportation, distribution, and logistics. It is of paramount importance

to thousands of companies and organizations engaged in the delivery and collection of goods or people. The CVRP

can be formally defined as follows [2]–[4]. it consists of designing a set of routes for a fleet of identical vehicles

with overall minimum route cost which service all the demands such that (1) all vehicles should begin and terminate

at the central depot, (2) each customer should be visited exactly once, by exactly one route, (3) the total demand of

each route does not exceed capacity of vehicle, (4) the traveling distance of each vehicle cannot exceed and the

maximum traveling distance of vehicle, and (5) the split deliveries are not allowed. For details about the variants of

the vehicle routing problem and their solution procedures, we refer the reader to several surveys and taxonomies by

[5], [6] and in many books or book chapters by [7], [8].

It is known that the CVRP is an NP-hard problem in which its real-life applications are considerably large in scale

and finding the optimal solution of an instance is very hard and requires very long computational time. Therefore,

metaheuristics are often more suitable for practical applications and have been applied for CVRP to find a near

optimal solution in a reasonable amount of time, for example: tabu search [9], [10], an adaptive memory

programming method [11], simulated annealing [12], [13], variable neighborhood search [14], [15], large neighborhood search [16], [17], ant colony optimization [16], [18], particle swarm optimization [19]–[21], a genetic

algorithm [22]–[24], and a hybrid genetic algorithm [4], [25]–[27] and other metaheuristics [28]–[30]. For details

about descriptions of these algorithms, we refer the reader to the survey papers [31], [32].

The main purpose of this paper is to present an effective genetic algorithm (GA) for solving the CVRP problem. The

remainder of the paper is organized as follows: The notation and model formulation of CVRP is given in Section 2.

A brief review has been conducted on the different GA algorithms used to solve the problem in Section 3. Section 4

† Corresponding author

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mailto:[email protected]

mailto:[email protected]




presents the proposed GA algorithm and its default settings. In Section 5, the computational results are presented

and discussed. Finally, we give a brief conclusion in Section 6.

2. Notation and model formulation

The CVRP is defined as an undirected graph 𝐺 = (𝑁, 𝐸), where 𝑁 = {0, 1, … , 𝑛} is the set of nodes, 𝐸 = {(𝑖,𝑗) ∶ 𝑖, 𝑗 ∈ 𝑁, 𝑖 ≠ 𝑗} is the set of edges joining the nodes. Node 0 is the depot and the other nodes represent

the customers having a known demand 𝑑𝑖 for customer 𝑖. The travel distance between node 𝑖 and 𝑗 is defined by

𝑑𝑖𝑗 > 0 and each vehicle 𝑘 has a unique capacity of 𝑄𝑘 . In accordance with these explanations, CVRP can be

formulated [12] as given below, where 𝑋𝑖𝑗𝑘equals to 1 if vehicle 𝑘 travels from node 𝑖 to node 𝑗 and 0 otherwise.

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 ∑ ∑ ∑ 𝑑𝑖𝑗

𝐾

𝑘=1

𝑋𝑖𝑗𝑘

𝑁

𝑗=0

𝑁

𝑖=0

(1)

Subject to:

∑ ∑ 𝑋𝑖𝑗𝑘 = 1 𝑗 𝜖 {1, … . . , 𝑛} ∶ 𝑖 ≠ 𝑗 (2)

𝑁

𝑖=0

𝐾

𝑘=1

∑ ∑ 𝑋𝑖𝑗𝑘 = 1 𝑖 𝜖 {1, … . . , 𝑛} ∶ 𝑖 ≠ 𝑗 (3)

𝑁

𝑗=0

𝐾

𝑘=1

∑ ∑ 𝑋𝑖𝑗𝑘 𝑑𝑖 ≤ 𝑄𝑘

𝑁

𝑗=0

𝑘 ∈ {1, … . , 𝐾}

𝑁

𝑖=0

(4)

∑ 𝑋𝑖𝑗𝑘

𝑁

𝑗=1

= ∑ 𝑋𝑗𝑖𝑘

𝑁

𝑗=1

≤ 1 𝑓𝑜𝑟 𝑖 = 0 𝑎𝑛𝑑 𝑘 𝜖 {1, … , 𝐾} (5)

∑ ∑ 𝑋𝑖𝑗𝑘

𝑁

𝑗=1

𝐾

𝑘=1

≤ 𝐾 𝑓𝑜𝑟 𝑖 = 0 (6)

Objective function (1) minimizes the total travelling distance. Constraint sets (2) and (3) guarantee that each

customer is served by exactly one vehicle. Constraint set (4) ensures that the total demand of the customers assigned

to a route 𝑘 does not exceed the vehicle capacity. Constraint set (5) indicates that the depot is the start and end node

for the trips of each vehicle. Constraint set (6) guarantees that there are maximum K routes for serving the

customers.

3. Genetic algorithm

Genetic algorithm (GA), proposed by [33], has been widely applied to solve hard combinatorial problems and it is

an effective search and optimization method that simulates the process of natural selection or survival of the fittest.

GA starts with generating random population of chromosomes. The chromosomes evolve through a series of

iterations, called generations. During each generation, the fitness of each chromosome in population is evaluated.

According to their fitness measure, select two parent chromosomes and can be crossed over by exchanging pieces

with each other and/or mutate randomly or be transferred unaltered to the next generation; this process is repeated

until a termination sequence (such as convergence) is reached [34].

In the next subsections, the GA and its elements, such as chromosome representation, initial population, GA

operators (such as crossover and mutation) and control parameters are described. Also, the different GA settings

mentioned in the literature concerning the problem under study are presented.

3.1. Chromosome representation

The first and most important step is to determine the chromosome representation (encoding). In CVRP literature,

chromosome representations include the permutation representation (e.g., [24], [26], [35]–[71]), direct

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representation (e.g., [72]–[74]), path representation [27] and binary representation (e.g., [55], [75]). In this study, the

permutation encoding is adopted, so the review of the GA settings in the next sections is focus only on literature that

uses this encoding.

3.2. Initialization of population

There is more than one way to generate initial population. First way is to generate it randomly (e.g., [24], [35],

[37], [39], [41], [42], [44]–[46], [49], [54], [59], [61], [66], [68], [72], [75], [76]). Second way is to use heuristics for

generating the initial population, such as the nearest addition method (NAM), the sweep algorithm (SWA), the

savings algorithm (SA), the Clarke and Wright heuristic (C&W), the Push Forward Insertion Heuristic (PFIH), the

Nearest Neighbor Heuristic (NNH) and Insertion heuristic (IH) , etc, (e.g., [26], [27], [40], [47], [48], [55]–[58], [60],

[62]–[64], [66], [67], [70]). Other way is to a combination of heuristic and randomly (e.g., [36], [38], [43], [50], [51],

[53], [65], [69], [71], [73]).

3.3. Fitness function

The fitness measure for evaluating a chromosome is total travel distance. The objective is to achieve the smallest

value of that measure. The total travel distance is given by Equation 1.

3.4. Selection operators

During the research, four types of selection methods are found: Roulette Wheel Selection (RWS) (e.g., [26], [27],

[35], [39], [42]–[45], [51], [52], [55], [68], [71], [75]), Ranking Selection (RNKS) (e.g., [37], [40], [46], [63], [76],

[77]), Tournament Selection (TTS) (e.g., [24], [35], [36], [38], [39], [41], [43], [47]–[54], [56], [57], [59]–[61], [63],

[65], [70], [72], [73], [76] ) and Uniform Selection (e.g., [43], [51]).

3.5. Crossover operators

In the literature, ten types of crossover operator methods are found: One-Point Crossover (1PX) (e.g., [43], [44],

[46], [59]), Two-Point Crossover (2PX) (e.g., [35], [44], [49]), Order Crossover (OX) (e.g., [26], [36], [37], [43],

[48], [50], [54], [56], [57], [60], [62], [71], [76]), Partially Mapped Crossover (PMX) (e.g., [40]–[43], [45], [46],

[52], [67]), Cyclical Crossover (CX) (e.g., [43], [75]), Route Based Crossover (RBX) (e.g., [61], [70], [72], [76]),

Sequence-Based Crossover (SBX) (e.g., [76]), Single Parent Crossover (SPO) (e.g., [36]), Genetic Vehicle

Representation Crossover (GVR) (e.g., [69]) and Best Cost-Best Route Crossover (BCBRC) (e.g., [38], [53], [64]–

[66]).

3.6. Mutation operators

Six types of mutation operator methods during research are found: Swap Mutation (SWM) (e.g., [24], [26], [35],

[37], [39]–[42], [44], [46], [49], [59], [62], [64], [65], [67]), Inversion Mutation (INVM) (e.g., [24], [45], [53], [64],

[65]), Insertion Mutation (INSM) (e.g., [41], [64], [67]), (Reallocation Mutation (RAM), Exchange Mutation (EXM)

and Reposition Mutation (RPM) (e.g., [61], [72], [73])).

3.7. Termination condition and parameter selection

The control parameters that control the execution of the GA and its operators are population size (𝑝𝑜𝑝𝑆𝑖𝑧 > 10),

crossover rate (𝐶𝑅 ∈ [0,1]), mutation rate (𝑀𝑅 ∈ [0,1]) and stop criteria. Each publication has its values which are

selected based on its own experiment. Many stop criteria can be used to termination the algorithm, but the known

one is the maximum number of generation (𝑚𝑎𝑥𝐺𝑒𝑛). Where 𝑚𝑎𝑥𝐺𝑒𝑛 equals number of explored chromosomes

divided by 𝑝𝑜𝑝𝑠𝑖𝑧.

4. The Proposed GA Characteristics

The good performance of a GA depends on the selection of a good combination of GA operators and parameters.

Based on the GA review in section 3, there are three selection methods (four versions of tournament methods), ten

crossover methods, and six mutation methods as shown in Table 1. For selecting our proposed GA, we design full

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factorial experiment with the GA operators and parameters shown in Table 1 and Table 2 respectively. Where Table

2 shows five, nine and seven levels for population size, crossover probability and mutation probability respectively.

As mention above, the stop criterion, 𝑚𝑎𝑥𝐺𝑒𝑛, is calculated based on the used population size at fixed number of

explored solutions 5000 (i.e. 𝑚𝑎𝑥𝐺𝑒𝑛 = 5000/𝑝𝑜𝑝𝑆𝑖𝑧 =). Therefore, the total number of combinations of GA

operators and parameters equals 113400 (6 × 10 × 6 × 5 × 9 × 7). The implementation model of the proposed has

been coded in C# language and the experiment is applied on the problems (A-n32-k5, A-n44-k6, A-n53-k7 and A-n60-k9) of set A which are mentioned in next section. The statistical analysis conducted in the computational results

shows that the best settings of our proposed GA is as shown in Table 3.

Table 1. Selection, Crossover and Mutation methods

used in CVRP

Table 2. GA parameters

Table 3. Proposed GA settings (Default settings)

5. Computational study

The performance of the proposed algorithm has been validated on groups of benchmark problems. The benchmark

problems consist of six sets as follow: seventy-four instances ( sets A, B and P) from [78], eleven instances (set E)

from [79], three instances (set F)from [80], five instances (set M) from [81]. All the problem instances were

downloaded from the site <http://branchandcut.org>.

The computational results of benchmark problems were obtained by running the coded algorithm on an Intel® Core

(TM) i3-4160 CPU 3.60 GHz personal computer. The results of the problems are as shown in Tables 4-9 for the

problem sets A, B, E, F, M and P respectively. In the first column of the tables the name of each problem is denoted.

The next three columns are the most important characteristics, number of nodes (n), capacity of the vehicles (Q), and number of Vehicles (K). The fifth column shows the best-known solution (BKS). Columns 6–9 present respectively

the best, the worst, the average and the standard deviation of 10 runs of the proposed algorithm. The efficiency of

the proposed GA algorithm is measured by the quality of the produced solutions. The quality is given in terms of the

percentage relative deviation from the best-known solution, %𝐷𝑒𝑣 = 100 ∗ (𝐵𝑒𝑠𝑡 − 𝐵𝐾𝑆)/𝐵𝐾𝑆 shown in last

column. The numbers in bold in column (Best) indicates that the solution obtained meets the best known one (i.e.

%𝐷𝑒𝑣 = 0.00). Whereas, a negative %𝐷𝑒𝑣 indicates that the solution obtained by the proposed GA is better than

the best known one and it is marked with (*).

It can be seen from Table 4 (Set A) that the proposed algorithm in seventeen out of the twenty-seven instances has

reached the best-known solution. For the other thirteen instances the quality of the solutions is between 0.17% and

2.33% and the average quality for the twenty-seven instances is 0.26%. For the Table 5 (Set B) the algorithm has

found the best-known solution in fifteen out of the twenty-three instances, for the rest the quality is between 0.08%

and 1.72% and the average quality for the twenty-three instances is 0.18%. From the observation of the Table 6 (Set

E), there are five best known solutions, for the rest the quality is between 0.2% and 3.44% and the average quality

for the 11 instances is 0.75. In Table 7 (Set F), our algorithm obtains one best known solution, two new solutions in

problem (F-n45-k4 and F-n135-k7) and the average quality for the 3 instances is -0.19%. For the Table 8 (Set M) the

Methods Codes

Sele

cti

on

s Roulette Wheel Selection RWS

Ranking Selection RNKS

Tournament Selection (tour size = 2) TTS-2




Cross

over

Order Crossover OX

Partially Mapped Crossover PMX

Cyclical Crossover CX

Route Based Crossover RBX

Sequence-Based Crossover SBX

Single Parent Crossover SPO

Genetic Vehicle Representation Crossover GVR

Best Cost-Best Route Crossover BCBRC

Mu

tati

on

s Swap Mutation SWM

Inversion Mutation INVM

Insertion Mutation INSM

Reallocation Mutation RAM

Exchange Mutation EXM

Reposition Mutation RPM

Parameters Value

Population size, 𝑝𝑜𝑝𝑆𝑖𝑧 10, 20, 40, 50, 100

Crossover probability, 𝐶𝑅 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7,

0.8, 0.9

Mutation Probability, 𝑀𝑅

Number of Generations, 𝑚𝑎𝑥𝐺𝑒𝑛

0.05,0.1, 0.2, 0.3, 0.4, 0.5, 0.6

5000/𝑝𝑜𝑝𝑆𝑖𝑧

Operators / Parameters Type / Value

Selection method Ranking Selection

Crossover operator Best Cost-Best Route Crossover

Mutation operator Exchange Mutation

Crossover probability 0.8

Mutation Probability 0.1

Population size 10

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quality of solutions is between 0.1 and 17.65% and the average quality for the five instances is 6.4%. Of the twenty-

three problem instances in Table 9 (Set P), our proposed algorithm obtains thirteen best known solution, the

percentages above the best-known solutions the quality is between 0.12% and 1.47% and the average quality for the

23 instances is 0.18%.

We summarized the results given in Tables 4-9 in Table 10. The table indicates that the proposed GA algorithm has

a satisfactory performance in solving CVRP problems especially in terms of solution quality. The algorithm is

capable to solve 51 problem instances which reached optimum and finds 2 new best solution problem instances out

of 93 and it is refers to an average success rate (OA%) for both it is 55.4% and 2.2% respectively. Regarding each set, the algorithm solves the problems sets A, B, P, F, E and M with percentage success rates of optimum

achievements 62.9%, 65.2%, 56.5%, 33.3%, 45.5% and 0.00% respectively, and with percentage success rates of

number of new solutions (NNS%) 0.00%, 0.00%, 0.00%, 66.7%, 0.00% and 0.00% respectively. For more

illustration, the performance of the proposed algorithm for each set is shown in Error! Reference source not

found.1.

Table 4. Computational results for the problem set A. Table 5. Computational results for the problem set B.

Problem n Q K BKS Proposed GA

Best Worst Average SD %Dev

A-n32-k5 32 100 5 784 784 784 784 0.00 0.00

A-n33-k5 33 100 5 661 661 661 661 0.00 0.00

A-n33-k6 33 100 6 742 742 742.9 742.9 0.30 0.00

A-n34-k5 34 100 5 778 778 782 782 4.20 0.00

A-n36-k5 36 100 5 799 799 808.8 808.8 4.80 0.00

A-n37-k5 37 100 5 669 669 673.2 673.2 5.50 0.00

A-n37-k6 37 100 6 949 949 949.7 949.7 1.30 0.00

A-n38-k5 38 100 5 730 730 731.2 731.2 2.20 0.00

A-n39-k5 39 100 5 822 822 823.5 823.5 1.60 0.00

A-n39-k6 39 100 6 831 831 832.5 832.5 2.20 0.00

A-n44-k6 44 100 6 937 937 938.5 938.5 2.20 0.00

A-n45-k6 45 100 6 944 944 949.1 949.1 5.60 0.00

A-n45-k7 45 100 7 1146 1148 1163.5 1163.5 16.8 0.17

A-n46-k7 46 100 7 914 914 920.7 920.7 12.8 0.00

A-n48-k7 48 100 7 1073 1073 1095.4 1095.4 16.3 0.00

A-n53-k7 53 100 7 1010 1010 1017.8 1017.8 5.90 0.00

A-n54-k7 54 100 7 1167 1172 1184.5 1184.5 13.6 0.43

A-n55-k9 55 100 9 1073 1073 1083.6 1083.6 13.9 0.00

A-n60-k9 60 100 9 1354 1354 1373.7 1373.7 14.9 0.00

A-n61-k9 61 100 9 1034 1037 1043.7 1043.7 11.4 0.29

A-n62-k8 62 100 8 1288 1299 1324.7 1324.7 22.3 0.85

A-n63-k9 63 100 9 1616 1621 1633.7 1633.7 8.90 0.31

A-n63-k10 63 100 10 1314 1319 1327.3 1327.3 10.0 0.38

A-n64-k9 64 100 9 1401 1418 1439.1 1439.1 14.3 1.21

A-n65-k9 65 100 9 1174 1177 1182.1 1182.1 6.30 0.26

A-n69-k9 69 100 9 1159 1169 1181 1181 7.80 0.86

A-n80-k10 80 100 10 1763 1804 1833.5 1833.5 33.8 2.33

Table 6. Computational results for the problem set E. Table 7. Computational results for the problem set F.



F-n45-k4 45 2010 4 724 721 725 721.8 1.70 -0.41*

F-n72-k4 72 30000 4 237 237 241 237.8 1.70 0.00

F-n135-k7 135 2210 7 1162 1160 1230 1189.3 20.6 -0.17*

Table 8. Computational results for the problem set M.



M-n101-k10 101 200 10 820 842 897 866 16.9 2.68

M-n121-k7 121 200 7 1034 1035 1188 1129 64.2 0.10

M-n151-k12 151 200 12 1053 1095 1175 1137.5 26.0 7.88

M-n200-k16 200 200 16 1274 1319 1394 1348.6 20.1 3.53

M-n200-k17 200 200 17 1373 1500 1665 1603.1 49.2 17.65



B-n31-k5 31 100 5 672 672 680 673.5 2.70 0.00

B-n34-k5 34 100 5 788 788 789 788.7 0.50 0.00

B-n35-k5 35 100 5 955 955 955 955 0.00 0.00

B-n38-k6 38 100 6 805 805 805 805 0.00 0.00

B-n39-k5 39 100 5 549 549 550 549.1 0.30 0.00

B-n41-k6 41 100 6 829 829 829 829 0.00 0.00

B-n43-k6 43 100 6 742 742 743 742.1 0.30 0.00

B-n44-k7 44 100 7 909 909 909 909 0.00 0.00

B-n45-k5 45 100 5 751 751 751 751 0.00 0.00

B-n45-k6 45 100 6 678 680 687 683.2 3.40 0.29

B-n50-k7 50 100 7 741 741 744 741.7 1.10 0.00

B-n50-k8 50 100 8 1312 1315 1337 1323.5 6.20 0.23

B-n51-k7 51 100 7 1032 1032 1033 1032.1 0.30 0.00

B-n52-k7 52 100 7 747 724 754 747.9 2.20 0.00

B-n56-k7 56 100 7 707 707 724 712.1 4.40 0.00

B-n57-k7 57 100 7 1153 1153 1153 1153 0.00 0.00

B-n57-k9 57 100 9 1598 1612 1649 1636 10.30 0.88

B-n63-k10 63 100 10 1496 1504 1552 1528.3 17.40 0.53

B-n64-k9 64 100 9 861 861 878 868.2 8.50 0.00

B-n66-k9 66 100 9 1316 1321 1335 1327.2 4.00 0.38

B-n67-k10 67 100 10 1032 1033 1052 1040.8 6.30 0.10

B-n68-k9 68 100 9 1272 1273 1293 1286.6 7.60 0.08

B-n78-k10 78 100 10 1221 1242 1281 1259.9 14.8 1.72



E-n22-k4 22 6000 4 375 375 375 375 0.00 0.00

E-n23-k3 23 4500 3 569 569 569 569 0.00 0.00

E-n30-k3 30 4500 3 534 534 537 534.3 0.90 0.00

E-n33-k4 33 8000 4 835 835 835 835 0.00 0.00

E-n51-k5 51 160 5 521 521 539 528.9 4.50 0.00

E-n76-k7 76 220 7 682 691 723 706.4 9.40 1.32

E-n76-k8 76 180 8 735 738 760 749.6 7.10 0.41

E-n76-k10 76 140 10 830 838 855 844.4 6.00 0.96

E-n76-k14 76 100 14 1021 1023 1058 1038.2 10.3 0.20

E-n101-k8 101 200 8 817 843 895 873.9 18.3 3.44

E-n101-k14 101 112 14 1077 1087 1144 1125.4 16.8 1.87

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Table 9. Computational results for the problem set P.

Table 10. Summary of computational study.

OA: Number of optimum achievements; NP: Number of problems;

NNS: Number of new solutions, NDS: Number of deviated solutions

Figure 1. Performance of the proposed GA algorithm for each data set.



P-n16-k8 16 35 8 450 450 450 450 0.00 0.00

P-n19-k2 19 160 2 212 212 212 212 0.00 0.00

P-n20-k2 20 160 2 216 216 216 216 0.00 0.00

P-n21-k2 21 160 2 211 211 211 211 0.00 0.00

P-n22-k2 22 160 2 216 216 216 216 0.00 0.00

P-n22-k8 22 3000 8 603 603 603 603 0.00 0.00

P-n23-k8 23 40 8 529 529 529 529 0.00 0.00

P-n40-k5 40 140 5 458 458 469 461.7 4.20 0.00

P-n45-k5 45 150 5 510 510 523 516.4 4.90 0.00

P-n50-k7 50 150 7 554 554 582 563.4 8.60 0.00

P-n50-k8 50 120 8 631 632 663 644.3 10.2 0.16

P-n50-k10 50 100 10 696 697 712 703.3 4.90 0.14

P-n51-k10 51 80 10 741 741 776 755.1 10.2 0.00

P-n55-k7 55 170 7 568 570 591 581.9 6.40 0.35

P-n55-k10 55 115 10 694 695 709 704.3 4.40 0.14

P-n55-k15 55 70 15 989 989 989 989 0.00 0.00

P-n60-k10 60 120 10 744 748 768 756.1 6.10 0.54

P-n60-k15 60 80 15 968 974 1001 981 8.60 0.62

P-n65-k10 65 130 10 792 792 820 805.1 6.90 0.00

P-n70-k10 70 135 10 827 828 849 840 6.70 0.12

P-n76-k4 76 350 4 593 595 622 604 8.60 0.34

P-n76-k5 76 280 5 627 629 652 637 7.60 0.32

P-n101-k4 101 400 4 681 691 740 709.6 14.8 1.47

Algorithm Problem

Set OA NNS NDS NP OA% NNS% NDS%

Proposed

GA

A 17 0 10 27 62.9 0.00 37.0

B 15 0 8 23 65.2 0.00 34.8

P 13 0 10 23 56.5 0.00 43.5

F 1 2 0 3 33.3 66.7 0.00

E 5 0 6 11 45.5 0.00 54.5

M 0 0 5 5 0 0.00 100

Overall 51 2 39 92

Average 55.4 2.2 42.4

0%

20%

40%

60%

80%

100%

A B P F E M

Instances

Percentage ofdeviation solution

Percentage successrate of new solutions

percentage of successrate of optimumachievements

379




To study the impact of stop criterion (maximum number of generation, 𝑚𝑎𝑥𝐺𝑒𝑛) on the performance of proposed

GA, the following values, 𝑚𝑎𝑥𝐺𝑒𝑛 = {200, 500, 1000, 2000, 5000, 10000, 20000, 50000}, are applied on the

benchmark problems. The average percentage deviation and the standard deviation of ten runs on each set of

problems are computed as mention in Table 11. For more illustration, the 𝑚𝑎𝑥𝐺𝑒𝑛 and its corresponding grand

average of the percentage deviation of all data set are shown in Figure 2. The vertical line represents the standard

error of the average. The figure reveals that the proposed algorithm can keep finding better results as more

generations are explored.

Table 11. Effect of number of explored solutions on the %Dev

Problem

set

Number of generations, 𝑚𝑎𝑥𝐺𝑒𝑛

Size 200

500

1000

2000

5000

10000

20000

50000

Avg SD

Avg SD

Avg SD

Avg SD

Avg SD

Avg SD

Avg SD

Avg SD

A 27 3.17 3.71 1.53 1.89 0.89 1.21 0.64 1.06 0.57 1.01 0.39 0.69 0.34 0.63 0.26 0.52

B 23 1.94 1.89 1.08 1.47 0.78 1.23 0.48 0.79 0.37 0.68 0.31 0.57 0.23 0.46 0.18 0.40

P 23 2.95 3.02 1.69 2.14 0.96 1.48 0.72 1.41 0.49 0.61 0.51 0.86 0.22 0.35 0.18 0.34

F 3 1.44 2.51 1.28 2.09 0.92 1.97 0.81 1.77 0.58 1.38 0.06 0.51 0.04 0.47 -0.19 0.21

E 11 6.78 7.68 4.34 6.04 3.35 4.78 2.33 2.83 1.59 2.27 1.44 2.17 1.02 1.50 0.75 1.09

M 5 37.7 15.3 27.1 14.6 21.5 15.8 16.5 13.6 13.5 9.14 8.88 9.16 10.6 8.53 6.37 6.90

Overall 92 5.05 7.99 3.17 5.84 2.29 4.69 1.68 3.61 1.33 2.96 0.98 1.94 0.91 2.35 0.59 1.41

Figure 2. Impact of number of generation, 𝑚𝑎𝑥𝐺𝑒𝑛 on the performance of proposed GA.

6. Conclusions and future research

In this paper, the capacitated vehicle routing problem (CVRP) is addressed using a genetic algorithm (GA). Due the

quality of the performance of any genetic algorithm is depends on the good selection of its operators and parameters,

based on a review on CVRP literature solved using GA, three selection methods, ten crossover methods, and six

mutation methods are found. For selecting our proposed GA, a full factorial experiment with the GA operators and

parameters is designed. The performance of the proposed algorithm is tested on different sets of benchmark

instances. The computational results indicate that the algorithm is able to solve CVRP instances with a satisfactory

performance and proved to be very stable and efficient based on the comparisons performed with the best-known

solutions. Future researches will be focus on implementing the proposed GA on stochastic routing problems, like Vehicle Routing Problem with Stochastic Demand (VRPSD).

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383




Biographies

Gamal Nawara is an emeritus professor at the Industrial Engineering Department, Faculty of Engineering, Zagazig

University; Zagazig, Sharkia, Egypt. He received his B.Sc., from University of Ain Shams, Egypt 1963 in

Mechanical Engineering and Ph.D., from University of Leipzig, Germany 1969 in Industrial Engineering. Prof.

Nawara has several distinguished activities in the field of industrial engineering. He is planner, evaluator and

coaching projects, project manager, and trainer in several projects. In the last decade he has focused on

Development of Small and Medium Enterprises. He is also a member in number of Supreme council of Egyptian

universities. Prof. Nawara has more than 50 articles in different industrial engineering topics.

Raafat Elshaer is an Associate professor in Industrial Engineering Department, Faculty of Engineering, Zagazig

University; Zagazig, Sharkia, Egypt. He received his B.S. degree in Production Engineering from Faculty of Engineering, Helwan University in 1996, M.Sc. degree in Industrial Engineering from Faculty of Engineering,

Zagazig University in 2004, and Ph.D. in Industrial Engineering from Faculty of Engineering, Zagazig University in

2009 as a joint program between Zagazig University and Rutgers University, USA. He has published journal and

conference papers. His research interests include optimization, scheduling, project management, earned value

management and others.

Adel AbdElmoez is an Assistant professor in Industrial Engineering Department, Faculty of Engineering, Zagazig

University; Zagazig, Sharkia, Egypt. Now, He is an Assistant professor in Industrial Engineering Department,

College of Engineering, King Khaled University, Abha, Kingdom of Saudia Arabia. He received his B.Sc. degree in

Construction Engineering from Faculty of Engineering, Zagazig University in 1985, M.Sc. degree in Systems

Engineering from Faculty of Engineering, Zagazig University in 1992, and Ph.D. in Industrial Engineering from

Faculty of Engineering, Zagazig University in 1998. He has published journal and conference papers. His field of

interests include optimization, scheduling, project management, safety engineering, and human factors engineering.

Hadeer Awad is a teaching assistant in Industrial Engineering Department, Faculty of Engineering, Zagazig

University; Zagazig, Sharkia, Egypt. She received her B.S. degree in Industrial Engineering from Faculty of

Engineering, Zagazig University in 2014. She is currently working in designing and developing genetic algorithms for solving different variants of Vehicle Routing Problems.

384

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