chapter 5 genetic algorithm based search heursitic...

80

CHAPTER 5

GENETIC ALGORITHM BASED SEARCH HEURSITIC

5.1 INTRODUCTION

GA originated from the studies of cellular automata, conducted by

Holland (1992), and his colleagues at the University of Michigan. Holland’s

book that was published in 1975 is generally acknowledged as the beginning

of the research of GA. Until the early 1980s, the research in genetic

algorithms was mainly theoretical (Davidor 1991), with few real applications.

From the early 1980s the community of genetic algorithms has experienced

an abundance of applications which spread across a large range of disciplines.

Each and every additional application gave a new perspective to the theory.

Furthermore, in the process of improving performance, new and important

findings regarding the generality, robustness and applicability of genetic

algorithms became available. Genetic algorithm (GA) is one of the widely

used computational methods, and has been successfully implemented in a

wide variety of problem domains due to its robustness and flexibility (Berger

and Barkaoui 2003). The details about the design of GAs heuristic and a GA

based heuristic for VRPTW and MDVRPTW are presented in this chapter

5.2 GENETIC ALGORITHM BACKGROUND

The Theory of Natural Selection was proposed by Charles Darwin.

The theory states that individuals with certain favorable characteristics are

more likely to survive and consequently pass their characteristics on to their

81

offspring. Individuals with less favorable characteristics will gradually

disappear from the population. In nature, the genetic inheritance is stored in

chromosomes made of genes. The characteristics of every organism are

controlled by the genes which are passed on to the offspring when the

organisms reproduce. Occasionally a mutation causes changes in the

chromosomes. Due to natural selection, the population will gradually improve

on the average as the number of individuals having the favorable

characteristics increases.

The GA is a randomized global search algorithm that solves

problems by imitating genetic processes observed during natural evolution.

The “survival of the fittest” nature of this algorithm lends itself favorably to

being extremely robust in its search for optimality (Gen and Cheng 2000).

Fundamentally, the GA evolves a population of bit strings, or chromosomes,

where each chromosome encodes a solution to a particular problem. This

evolution takes place through the application of genetic operators which

mimic phenomena such as reproduction and mutation observed in nature.

5.3 GENETIC ALGORITHM PROCEDURE

The procedure of the traditional GA may be described as follows.

The GA starts from some randomly generated initial population which is a set

of solutions. Davis (1987) suggests that for research purposes, a good deal can

be learned by initializing a population randomly. Moving from a

randomly-created population to a well-adapted population is a good test of the

algorithm. By doing this, important features of the final solution will have

been produced by the search and recombination mechanism of the algorithm,

rather than the initialization process. To generate and to search for an optimal

solution, a function which evaluates the survivability of solutions is required

in the initialization process. This is also called the fitness function, because it

82

ranks each feasible solution in accordance to its fitness value. The fitness

function is the most critical part of the GA, as it is the one which decides how

much time it takes to find the optimal solution.

The second step, a reproductive process allows parent solutions to

be randomly selected from the population. Typically, a lower selection

pressure is indicated at the start of a search in favor of a wide exploration of

search space, while a higher selection pressure is recommended at the end to

narrow the search space (Gen and Cheng 2000). Offspring solutions are made

by the reproductive processes using a crossover operator. Offspring solutions

are produced which inherit some of the characteristics from each parent.

Then, a random mutation could be applied to the offspring with a certain

probability. Gen and Cheng (2000) proved that the mutation operator can

sometimes play a more crucial role than the crossover. Therefore, the

crossover and mutation operators need to be well-designed in accordance with

the problem on hand.

Finally, generation replacement takes place in the third step. The

evaluation of the solutions can be related to the objective function value. In

the VRPs, the total distance travelled and the level of any constraint violation

can be the fitness functions. Analogous to biological processes, offspring with

relatively good fitness levels are more likely to survive and reproduce with

the expectation that fitness levels throughout the population will improve as

they evolve. More details can be found in Reeves (1993). The overall

procedure for GA is given below:

g = number of generations

pop = size of population

for population = 1 to pop

{

83

/Initial population/

Generate initial chromosomes at random and compute the fitness values

}

for generation = 1 to g

{ for population = 1to pop

{

regenerate chromosomes from the current population

compute the fitness values

}

for population = 1 to pop-1 step 2

{

perform crossover and generate the offspring, subject to the crossover

rate

}

for population = 1 to pop

{

Perform mutation, subject to the mutation rate

}

}

Report the best objective function value and the corresponding chromosome.

5.4 DESIGN OF A GENETIC ALGORITHM BASED

HEURISTIC FOR VRPTW AND MDVRPTW

The details of a GA based model to solve the VRPTW and

MDVRPTW are described below.

84

5.4.1 Chromosome Encoding

A chromosome is represented as an integer string of length N

(number of customers). Each gene in the string is the integer node number

pre-assigned to the customer. A gene in a given chromosome indicates the

original node number assigned to a customer, while the sequence of genes in

the chromosome string represents the order in which customers are visited.

No specific genes are put in the chromosome, either to mark the depot or to

show the limits of the route, because these lead to invalid offsprings resulting

from reproduction. A solution to the problem is assigned by first decoding the

chromosomes into routes which are then assigned to and are scheduled within

vehicle planning. One chromosome representation is given below

6 4 8 3 5 7 10 9 2 1

This represents the sequence where customer 6 is visited first and

so on and customer 1 is visited last in the sequence.

For MDVRPTW problems, once the customers are assigned to the

depots using PAM, the proposed algorithm runs sequentially between Depots.

Hence the chromosome representation used for SDVRPTW is sufficient to

handle MDVRPTW problems also.

5.4.2 Initial Population

The initial population for the Genetic algorithm plays an important

role in the convergence of the GA. For a GA to converge quickly, the

population needs to have diversity as well as good solution quality. If the

initial population is generated by random methods the diversity of the

population will be good but the solution quality will be poor. If the initial

85

population is generated by means of the greedy functions, the solution quality

will be good but the diversity will be very poor. The proposed algorithm uses

the Greedy Randomized Adaptive Search Procedure (GRASP) (Resende and

Ribeiro 2003) technique for generating an initial population which is a

combination of greedy as well as random solutions. Basically the GRASP

consists of two stages, namely, the construction phase and the local search

phase. However, in this work, only the GRASP construction phase is

considered to generate the population. In the GRASP construction phase, at

each construction iteration, the choice of the next element to be added is

determined by ordering all the candidate elements in a candidate list with

Procedure grasp construct (g ( ),γ, x)

1. x = Φ

2. Initialize candidate set C;

3. While C = Φ do

4. S1 = min {g(t) | t ε C};

5. S2 = max {g(t) | t ε C};

6. RCL = { S ε C | g( S) ≥ S1 + γ (S1 - S2 };

7. Select S, at random , from the RCL;

8. x = x ∪ {S};

9. Update candidate set C;

10. end while;

11. End grasp construct;

The GRASP construction pseudo code

86

The detailed steps involved in GRASP construction procedure is

given below.

Step 1: Consider an empty set for initialization.

Step 2: Compute the distance for all the customers.

Step 3: Find out the minimum and maximum value of the distance

Step 4: Find out the range

Step 5: Choose the γ parameter (greedy value)

Step 6: Find out the width. (Range x γ )

Step 7: Choose candidate for entry in RCL i.e., RCL ={ Minimum,

Minimum + Width }

Step 8: Define rank ‘r’ for each customer in RCL.

Step 9: Calculate rank for random bias function.

Step 10: Sample and update the solution.

5.4.3 Fitness evaluation function

The fitness evaluation function assigns to each member of the

population, a value reflecting their relative superiority (or inferiority). Each

chromosome is evaluated according to its fitness function using the following

expression.

Fitness ∑

=

i

iDf

1 (5.1)

where iD is the distance of the route i

5.4.4 Selection operator

87

The selection scheme specifies the methodology employed to select

the chromosome from the current population for regeneration. There are

various selection operators available, which can be used to select the parents.

In the roulette wheel selection, parents are selected according to their fitness

value. The better the fitness, the greater is the chances of being selected. In

Random selection, the individual is chosen at random which leads to poor

solution quality. Similarly, if the ranking method is used to select the

individual, not all the individuals get the opportunity to mate, which results in

poor diversity. Typically, a lower selection pressure is desirable at the start

of the genetic search in favor of a wide exploration of the search space, while

a higher selection pressure is recommended at the end to converge efficiently

(Goldberg 1988, 1989).

The proposed selection scheme with elitism combines the salient

features of all the three to overcome the demerits. Two tournaments are

created among the population, based on the ranking method (The first

tournament contains the better half of the population, and the second

tournament contains the worse half of the population). Then one individual is

selected at random from each of the tournaments for mating. This

incorporates the random as well as the ranking method of selecting. Elitism

refers to a method that copies the best chromosome (or a few best

chromosomes) to the new population. Elitism can rapidly increase the

performance of the GA, because it prevents the loss of the best found

solution. During regeneration, the two worst chromosomes from the current

population are replaced by the current best chromosomes. This is expected to

facilitate a better convergence of the GA.

5.4.5 Crossover Operator

88

In the crossover operation, members of the generated chromosomes

are mated at random. The crossover operator used in the proposed algorithm

preserves the commonality of the parents to the child. The best parent (P1) is

used to preserve the solution quality and the worst parent (P2) is used to

improve the diversity. The crossover used in the algorithm is explained

below:

Step 1: Randomly select cut points in P1 (best parent) and in P2 (worst

parent) with cut length = Integer (chromosome length/3)

Step 2: Exchange the cut section of the parent to the offspring i.e. the cut

section of the best parent is mapped into the worst parent and so on.

But when mapping the cut section from the worst parent to the

offspring, the relative order of genes in the chromosome of the

best parent has to be maintained.

Step 3: Delete the customers who are already in the sequence. The resulting

sequence of customers contains the customers that the offspring

needs.

This crossover helps in avoiding infeasible solutions by means of

repeating the customers and also helps in preserving the better customer order

for the offspring. The schematic representation of the proposed cross over

operation is given in Figure 5.1.

89

9 8 4 5 6 7 1 3 2 10 9 8 4 5 6 7 1 3 2 10

8 7 1 2 3 10 9 5 4 6 8 7 1 2 3 10 9 5 4 6

Figure 5.1 Cross over operation

To determine the usefulness of the proposed crossover operator,

experiment with Partially Matched Crossover operator (PMX) was performed.

The best results for various instances of MDVRPTW using PMX and the

proposed crossover operator are provided in Table 5.1. The results clearly

demonstrate the New Crossover operator is giving better solution.

9 8 4 3 2 10 1 5 6 7 8 3 1 5 6 7 9 2 4 10 C1

P1

P2

P1

P2

C2

90

Table 5.1 Comparison of PMX with Proposed Crossover Operator

Pb.No

Solution by Partial

mapped cross over

operator

Solution by

Proposed Cross

over Operator

Pr01 1083.00 929.19

Pr02 1785.38 1606.61

Pr03 2408.00 2087.49

Pr04 2958.00 2571.05

Pr05 3146.44 2927.76

Pr06 3904.00 3375.11

Pr07 1423.00 1297.52

Pr08 2394.03 2143.07

Pr09 2966.29 2751.47

Pr10 4236.30 3528.64

Pr11 1007.60 954.02

Pr12 1551.91 1427.70

Pr13 2254.23 2012.26

Pr14 2715.21 2217.18

Pr15 3109.14 2509.75

Pr16 3517.86 2901.18

Pr17 1250.00 1218.23

Pr18 2241.86 1799.43

Pr19 3000.22 2293.62

Pr20 4146.18 3080.45

91

5.4.6 Mutation operator

Mutation is used to prevent the genetic algorithm from getting

stuck in a local extreme. Mutation is applied to a single solution with a certain

probability. It makes small random changes in the solution. These random

changes will gradually add some new characteristics to the population, which

cannot be supplied by the crossover. The proposed algorithm uses the

modified edge and swap mutation operator which randomly flips an edge of

the chromosome with another and also swaps the edge. This mechanism is

explained in Figure 5.2.

9 8 4 5 6 7 1 3 2 10

Figure 5.2 Mutation operation

5.4.7 Termination Criteria

Theoretically, it requires a long enough Markov chain to guarantee

the convergence of the GA which may lead to huge computations. Now that

there is no practical rule to set suitable stopping condition and it is also

impossible for the GA to evolve with too long a time in real application, the

usual way is to set a limit to the number of generations. In this problem, the

stopping condition is the total number of generations, fixed at 150. The

convergence plot for R101 instance is given in Figure 5.2a.

9 8 4 1 3 7 6 5 2 10

Before mutation: P1

After mutation: P1

1 3

6 5

92

1240

1260

1280

1300

1320

1340

1360

1380

1400

1 25 49 73 97 121 145

No of Generations

Ob

jecti

ve f

un

cti

on

valu

e

Figure 5.2a Convergence of PGA for R101 Problem

5.5 PROPOSED GENETIC ALGORITHM FOR VRPTW

5.5.1 Notations used in Genetic Algorithm

Ps population size

Pc crossover probability

Pm mutation probability

P(k) population of kth

generation

Dk best distance at kth

generation

r real random number between 0 -1

k current generation number

Pi , Pi’ , Pi

’’ chromosome and temporary chromosome

Ng maximum generation

5.5.2 Algorithm

93

Step 0: Given the parameters required, such as population size Ps,

crossover probability Pc, mutation probability Pm, etc., set k=1 and

generate an initial population of size Ps using GRASP.

P(k)={P1(k), P2(k),…, P Ps (k)} // check feasibility

Step 1: Evaluate the fitness value of chromosomes in P(k), and set the top

two chromosomes as P* and P

** respectively.

Step 2: Set the current best distance as Dk .(Corresponding to P*)

Step 3: Set q = 0.

Step 4: Perform Selection (as given in Section 4.5)

Step 5 : Crossover

Generate ‘ r’ randomly (0 ≤ r ≤1)

If r ≤ Pc (Prob_Cross),

Do cross over (as given in Section 4.6 ) // check feasibility

Let P1′ and P2′ be the offsprings of crossovering parents P1 and P2.

Let P1= P1′ and P2 = P2′

else

No cross over

Step 6: Mutation

Generate ‘r’ randomly ((0 ≤ r ≤1)

If r ≤ Pm (Prob_Mute),

Do Mutation (as given in Section 4.7) // check feasibility

else

No mutation

Step 7: q = q+1.

If q < Ps / 2

go to step 10

else

go to step 8.

94

Step 8: Update P*, P

** and Dk in P (k).

Step 9: Adopt Elitist Strategy. Insert the top two chromosomes P*

and P**

into the current population by removing the bottom two

chromosomes (having maximum distance).

Step 10: If k > Ng

go to step 11.

else

set k = k+1 and go to step 3.

Step 11: Number of generations over. Output the current best sequence and

the corresponding performance measure. Stop.

The detailed frame work of PGA is given in Appendix

(Figure A1.1).

5.6 DESIGN OF PGA ALGORITHM CONTROL PARAMETERS

Experiments are conducted using the orthogonal array (OA)

(Belavendram (1995) and Ross 1996)) to decide the values of the parameters

to be used in the PGA. There are no interaction effects. Using the L8

orthogonal array eight experiments with two replications are conducted. The

levels considered for various GA parameters are given in Table 5.1a.

For PGA, four factors at two levels are considered. Since we are

interested in setting the levels for the main factors, the interaction effects

between the factors are not considered. For this problem degrees of freedom

required is 4. Hence, a L8 orthogonal array is selected. The factors are

assigned sequentially to the four columns of L8 orthogonal array. Eight

95

experiments with two replications are conducted as specified by the L8

orthogonal array. Table 5.2 gives the objective value (distance) obtained for

different experiments done using the L8 orthogonal array.

Table 5.2 Results of the OA Experiments

Expt. No Parameter/control factors

Objective Value Pc Pm Ps Cl

1 1 1 1 1 933.00 939.51

2 1 1 1 2 952.12 951.92

3 1 2 2 1 942.28 941.16

4 1 2 2 2 942.88 943.77

5 2 1 2 1 960.13 950.83

6 2 1 2 2 944.94 953.89

7 2 2 1 1 933.43 947.06

8 2 2 1 2 944.85 946.62

To decide the levels for the factors, the steps followed in Taguchi’s

response table method was used. Since our objective is a minimisation type,

it is decided to select the appropriate levels of the parameter having minimum

value (i.e. best quality solutions were obtained at the values

,8.0=cP ,9.0=mP 50=sP and 3/NCl = ) in the response table (Table 5.3).

Table 5.3 Response Table

Pc Pm Ps Cl

Level 1 943.336 948.298 943.569 943.425

96

Leve2 947.718 942.756 947.484 947.63

Difference 4.381 -5.543 3.915 4.205

Rank 4 1 2 3

5.7 PERFORMANCE COMPARISON OF THE PGA FOR

VRPTW WITH PUBLISHED RESULTS

This section describes computational experiments carried out to

investigate the performance of the proposed GA. The algorithm is

implemented using VC++. The experiments are run on Intel P IV 2.40 GHz

with 256 MB memory.

Tables 5.4 to 5.9 show the solution generated by our proposed

genetic algorithm implementation for various types of VRPTW. Although

several other authors have worked on the VRPTW, we have confined our

comparisons to the most recent heuristics (Cordeau et al (2001) [CLM],

Tan et al (2001c) [TLZO], Li and Lim (2003) [LL], Lau et al (2003) [LST],

Braysy et al (2004) [B], Russell and Chiang (2006) [RC] and the best known

results [BKS].

From Tables 5.4 to 5.9, it is evident that the proposed algorithm

generated a better solution(new best known results) for 28 instances out of 56

Solomon’s VRPTW instances( 7 instances in R1 type problem,11 instances

in R2 type problem,4 instances in RC1 type problem and 6 instances in RC2

type problem). In C1 and C2 problem categories, our algorithm did not

perform well in distance minimization but in terms of vehicles we got the

solutions on par with the best known solutions. This is expected since the

genetic algorithm requires large differences in the fitness values of the

97

chromosomes to exploit the search space. Figure 5.3 compares average

number of vehicles used by the algorithms considered for comparison with

respect to different class of problem and Figure 5.4 compares average

distance travelled by different algorithms.

Table 5.10 compares the average number of vehicles (ANV) and

the average distance travelled (ADT) obtained by PGA and other heuristics

reported by different authors for the VRPTW.

98

Tab

le 5

.4

PG

A s

olu

tion

an

d c

om

pari

son

wit

h t

he

oth

er h

euri

stic

s fo

r R

1 t

yp

e P

rob

lem

s

Inst

an

ce

CL

M

TL

ZO

L

L

LS

T

B

RC

B

KS

P

GA

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

R101

19

1650.8

0

20

1676.8

6

19

1650.8

0

20

1765.0

0

19

1650.8

0

19

1650.8

2

18

1607.7

0

16

1300.1

7

R102

17

1488.1

0

18

1558.5

9

17

1486.4

1

18

1548.6

1

17

1486.1

2

17

1486.1

2

17

1434.0

0

14

1223.2

4

R103

13

1299.7

8

15

1311.8

1

13

1292.6

8

14

1258.3

4

13

1293.6

9

13

1292.9

4

13

1292.6

8

12

1122.1

7

R104

10

984.0

0

12

1128.2

9

9

1007.3

1

10

1018.4

8

10

985.3

3

10

982.1

8

9

1007.2

4

10

1046.7

3

R105

14

1377.1

1

17

1496.3

7

14

1381.3

7

15

1462.6

9

14

1377.1

1

14

1377.1

1

14

1377.1

1

12

1133.2

8

R106

12

1253.2

3

14

1357.1

9

12

1269.7

2

12

1328.6

6

12

1253.2

3

12

1252.0

6

12

1251.9

8

11

1066.2

2

R107

10

1113.6

9

13

1240.8

2

10

1104.6

6

12

1160.0

8

10

1115.0

5

10

1104.6

6

10

1104.6

6

11

1069.8

7

R108

9

964.3

8

12

1091.6

9

9

986.2

5

10

1045.8

3

9

960.8

8

9

968.6

3

9

960.8

8

10

1064.2

4

R109

11

1199.6

3

15

1300.2

9

11

1208.9

6

13

1259.0

9

11

1201.7

8

11

1194.7

3

11

1194.7

3

10

1075.2

4

R110

10

1125.0

4

13

1315.5

6

10

1159.3

5

11

1127.7

0

10

1119.0

0

10

1121.0

1

10

1118.5

9

10

1034.8

8

R111

10

1108.9

0

12

1202.3

1

11

1066.3

2

11

1097.1

0

10

1101.2

0

10

1102.8

2

10

1096.7

2

11

1072.7

7

R112

10

957.0

4

12

1097.6

4

10

967.8

8

10

1021.9

5

9

1010.5

2

10

960.7

8

9

982.1

4

10

1045.5

1

99

Tab

le 5

.5

PG

A s

olu

tion

an

d c

om

pari

son

wit

h t

he

oth

er h

euri

stic

s fo

r R

2 t

yp

e P

rob

lem

s

Inst

an

ce

CL

M

TL

ZO

L

L

LS

T

B

RC

B

KS

P

GA

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

R201

4

1253.2

6

8

1329.7

4

4

1252.3

7

4

1292.5

3

4

1253.2

1

4

1258.6

4

4

1252.3

7

4

786.1

5

R202

3

1197.6

6

7

1307.0

3

4

1084.7

7

3

1158.9

8

3

1195.3

0

3

1228.5

4

3

1191.7

0

3

939.0

5

R203

3

945.5

5

6

1086.4

3

3

949.4

0

3

980.7

0

3

944.5

5

3

955.3

1

3

939.5

4

3

911.0

7

R204

2

849.6

2

6

956.3

8

2

849.0

5

3

847.7

4

2

838.5

6

2

850.7

0

2

825.5

2

2

823.4

0

R205

3

1008.5

2

5

1131.1

8

3

1032.5

5

3

1146.8

0

3

1009.1

0

3

1021.8

8

3

994.4

2

3

833.4

7

R206

3

913.1

8

5

1187.2

5

3

931.6

2

3

1007.0

0

3

913.2

4

3

912.2

6

3

906.1

4

3

816.1

7

R207

2

948.2

3

4

1016.6

3

2

905.1

3

3

869.9

4

2

906.6

7

2

928.1

5

2

893.3

3

2

832.3

4

R208

2

734.8

5

3

845.9

4

2

732.8

0

2

790.4

6

2

733.9

8

2

741.3

3

2

726.7

5

2

725.3

3

R209

3

916.4

7

5

1097.4

2

3

930.5

9

3

1020.0

6

3

916.4

8

3

945.7

1

3

909.1

6

2

843.7

5

R210

3

964.2

2

6

1136.5

4

3

1018.9

5

3

1032.6

5

3

939.9

1

3

983.6

6

3

939.3

4

3

909.1

8

R211

2

933.7

5

7

932.4

8

3

801.8

1

3

866.1

0

2

913.7

9

3

796.4

1

2

892.7

1

2

769.9

6

100

Tab

le 5

.6

PG

A s

olu

tion

an

d c

om

pari

son

wit

h t

he

oth

er h

euri

stic

s fo

r C

1 t

yp

e P

rob

lem

s

Inst

an

ce

CL

M

TL

ZO

L

L

LS

T

B

RC

B

KS

P

GA

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

C101

10

828.9

4

10

828.9

37

10

828.9

4

10

828.9

4

10

828.9

4

10

828.9

4

10

827.3

0

10

877.3

8

C102

10

828.9

4

10

868.7

98

10

828.9

4

10

834.6

4

10

828.9

4

10

828.9

4

10

827.3

0

10

874.7

5

C103

10

828.0

6

11

939.4

56

10

828.0

6

10

834.5

6

10

828.0

6

10

828.0

6

10

826.3

0

10

872.9

1

C104

10

824.7

8

10

963.7

2

10

824.7

8

10

846.3

2

10

824.7

8

10

824.7

8

10

822.9

0

10

874.5

7

C105

10

828.9

4

10

828.9

37

10

828.9

4

10

828.9

4

10

828.9

4

10

828.9

4

10

827.3

0

10

873.1

6

C106

10

828.9

4

10

828.9

37

10

828.9

4

10

828.9

4

10

828.9

4

10

828.9

4

10

827.3

0

10

869.4

7

C107

10

828.9

4

10

828.9

37

10

828.9

4

10

828.9

4

10

828.9

4

10

828.9

4

10

827.3

0

10

874.7

3

C108

10

828.9

4

10

828.9

37

10

828.9

4

10

828.9

4

10

828.9

4

10

828.9

4

10

827.3

0

10

868.5

5

C109

10

828.6

7

10

828.9

37

10

828.9

4

10

828.9

4

10

828.9

4

10

828.9

4

10

827.3

0

10

871.9

3

101

Tab

le 5

.7

PG

A s

olu

tion

an

d c

om

pari

son

wit

h t

he

oth

er h

euri

stic

s fo

r C

2 t

yp

e P

rob

lem

s

Inst

an

ce

CL

M

TL

ZO

L

L

LS

T

B

RC

B

KS

P

GA

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

C201

3

591.5

6

3

591.5

57

3

591.5

6

3

591.5

6

3

591.5

6

3

591.5

6

3

589.1

0

3

687.8

3

C202

3

591.5

6

4

683.8

64

3

591.5

6

3

619.3

6

3

591.5

6

3

591.5

6

3

589.1

0

3

690.5

6

C203

3

591.1

7

4

745.9

34

3

591.1

7

3

604.0

1

3

591.1

7

3

591.1

7

3

588.7

0

3

686.6

4

C204

3

590.6

3

604.9

98

3

590.6

3

644.2

3

3

590.6

3

590.6

3

590.6

0

3

693.8

8

C205

3

588.8

8

3

588.8

76

3

588.8

8

3

601.4

3

3

588.8

8

3

588.8

8

3

586.4

0

3

686.9

7

C206

3

588.4

9

3

588.4

93

3

588.4

9

3

588.8

8

3

588.4

9

3

588.4

9

3

586.0

0

3

690.6

9

C207

3

588.2

9

3

593.1

95

3

588.2

9

3

608.9

4

3

588.2

9

3

588.4

9

3

585.8

0

3

685.0

9

C208

3

588.3

2

3

590.8

73

3

588.3

2

3

591.8

3

3

588.3

2

3

588.3

2

3

585.8

0

3

692.5

5

102

Tab

le 5

.8

PG

A s

olu

tion

an

d c

om

pari

son

wit

h t

he

oth

er h

euri

stic

s fo

r R

C1 t

yp

e P

rob

lem

s

Inst

an

ce

CL

M

TL

ZO

L

L

LS

T

B

RC

B

KS

P

GA

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

RC

101

14

1708.8

0

17

1728.3

0

15

1658.6

2

15

1657.4

6

14

1697.4

3

14

1696.9

4

14

1696.9

4

12

1376.3

3

RC

102

12

1558.0

7

17

1603.5

3

13

1513.6

13

1535.7

9

12

1558.0

7

12

1557.5

6

12

1554.7

5

11

1269.5

7

RC

103

11

1262.9

8

14

1519.8

3

11

1319.9

9

12

1386.0

3

11

1262.4

3

11

1261.6

9

11

1261.6

7

11

1270.4

6

RC

104

10

1135.4

8

12

1276.0

2

10

1141.0

9

10

1213.2

5

10

1137.8

9

10

1135.5

3

10

1135.4

8

11

1223.4

6

RC

105

13

1644.4

3

17

1688.7

7

13

1637.6

2

15

1625.1

3

13

1642.5

3

13

1629.4

4

13

1629.4

4

12

1316.8

1

RC

106

11

1427.1

3

14

1491.5

8

11

1424.7

3

12

1426.0

7

11

1438.5

3

11

1425.7

7

11

1424.7

3

11

1256.9

3

RC

107

11

1231.5

3

14

1462.3

0

11

1240.6

6

11

1330.5

9

11

1233.3

0

11

1230.5

4

11

1230.4

8

11

1257.1

4

RC

108

10

1149.7

9

12

1333.1

5

10

1147.4

2

10

1175.8

8

10

1143.4

2

10

1146.6

6

10

1139.8

2

12

1317.7

3

103

Tab

le 5

.9

PG

A s

olu

tion

an

d c

om

pari

son

wit

h t

he

oth

er h

euri

stic

s fo

r R

C2 t

yp

e P

rob

lem

s

Inst

an

ce

CL

M

TL

ZO

L

L

LS

T

B

RC

B

KS

P

GA

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

RC

201

4

1406.9

4

10

1565.6

7

4

1425.2

1

4

1468.4

6

4

1412.4

5

4

1435.1

7

4

1406.9

1

4

1198.4

5

RC

202

3

1407.5

2

10

1353.2

7

3

1374.2

7

4

1222.6

9

3

1368.0

4

3

1448.1

5

3

1367.0

9

3

1020.8

2

RC

203

3

1073.3

9

6

1189.0

6

3

1088.5

3

3

1171.8

8

3

1061.1

6

3

1082.3

7

3

1049.6

2

3

882.6

7

RC

204

3

806.1

2

4

989.9

4

3

818.6

6

3

839.3

2

3

798.5

6

3

820.8

8

3

798.4

1

3

802.8

7

RC

205

4

1326.8

3

9

1465.8

4

1304.6

4

4

1338.7

0

4

1298.9

2

4

1338.2

8

4

1297.1

9

3

1046.0

6

RC

206

3

1160.9

1

5

1388.1

3

3

1159.0

3

3

1201.2

7

3

1152.1

4

3

1178.2

7

3

1146.3

2

3

1090.3

8

RC

207

3

1062.0

5

6

1304.4

8

3

1107.1

6

3

1139.4

8

3

1072.1

4

3

1076.6

2

3

1061.1

4

3

1025.1

6

RC

208

3

832.3

6

6

1003.4

3

3

862.3

4

3

985.6

0

3

829.6

9

3

834.2

7

3

828.1

4

3

1025.1

6

104

0

2

4

6

8

10

12

14

16

R1 R2 C1 C2 RC1 RC2

Problem category

Av

era

ge

nu

mb

er

of

ve

hic

les

us

ed

CLM TLZO LL LST B RC BKS PGA

Figure 5.3 Comparison of the average number of vehicles used for

different types of Solomon instances by the PGA and other

methods

0

200

400

600

800

1000

1200

1400

1600

R1 R2 C1 C2 RC1 RC2

Problem category

Av

era

ge

dis

tan

ce

tra

ve

lle

d

CLM TLZO LL LST B RC BKS PGA

Figure 5.4 Comparison of the average distance travelled for different

types of Solomon instances by the PGA and other methods

105

Tab

le 5

.10 A

ver

age

per

form

an

ce c

om

paris

on

of

the

PG

A w

ith

th

e oth

er V

RP

TW

alg

orit

hm

s

Au

thor

R1

R2

C1

C2

RC

1

RC

2

AN

V

AD

T

AN

V

AD

T

AN

V

AD

T

AN

V

AD

T

AN

V

AD

T

AN

V

AD

T

Roch

at an

d T

aill

ard (

19

95)

12.2

5 1

208.5

0

2.9

1

961.7

2 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.8

8 1

377.3

9

3.3

8 1

119.5

9

Than

gia

h (

1995a)

12.7

5 1

300.2

5

3.1

8 1

124.2

8 1

0.0

0 8

92.1

1

3.0

0 7

49.1

3 1

2.5

0 1

474.1

3

3.3

8 1

411.1

3

Potv

in a

nd B

engio

(1996

) 12.5

8 1

296.8

3

3.0

0 1

117.6

4 1

0.0

0 8

38.1

1

3.0

0 5

90.0

0 1

2.1

3 1

446.2

5

3.3

8 1

368.1

3

Chia

ng a

nd R

uss

ell

(199

7)

12.1

7 1

204.1

9

2.7

3

986.3

2 1

0.0

0 8

28.3

8

3.0

0 5

91.4

2 1

1.8

8 1

397.4

4

3.2

5 1

229.5

4

Tai

llar

d e

t al

(1997

) 12.1

7 1

209.3

5

2.8

2

980.2

7 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.5

0 1

389.2

2

3.3

8 1

117.4

4

Liu

and S

hen

(1999)

12.1

7 1

249.5

7

2.8

2 1

016.5

8 1

0.0

0 8

30.0

6

3.0

0 5

91.0

3 1

1.8

8 1

412.8

7

3.2

5 1

204.8

7

Gam

bar

del

la e

t al

(1999)

12.0

0 1

217.7

3

2.7

3

967.0

0 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.6

3 1

382.4

2

3.2

5 1

129.1

9

Than

gia

h e

t al

(1999

) 12.3

0 1

227.4

2

3.0

0 1

005.0

0 1

0.0

0 8

30.8

9

3.0

0 6

40.8

6 1

2.0

0 1

391.1

3

3.4

0 1

173.3

8

Hom

ber

ger

and G

ehri

ng (

1999)

11.9

2 1

228.0

6

2.7

3

969.9

5 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.6

3 1

392.5

7

3.2

5 1

144.4

3

Bak

er e

t al

(2000

) 12.4

1 1

200.5

4

3.0

0

936.5

1 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

2.0

0 1

383.2

1

3.3

8 1

116.5

1

Cord

eau e

t al

(2001

) 12.0

8 1

210.1

4

2.7

3

969.5

7 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.5

0 1

389.7

8

3.2

5 1

134.5

2

Tan

et

al (

2001

c)

14.4

2 1

314.7

9

5.6

4 1

093.3

7 1

0.1

1 8

60.6

2

3.2

5 6

23.4

7 1

4.6

3 1

512.9

4

7.0

0 1

282.4

7

Pro

pose

d G

A

11.4

2 1

104.5

3

2.6

4

835.4

4 1

0.0

0 8

73.0

5

3.0

0 6

89.2

8 1

1.3

8 1

286.0

5

3.1

3 1

011.4

5

106

Tab

le 5

.10 (

Con

tin

ued

)

Au

thor

R1

R2

C1

C2

RC

1

RC

2

AN

V

AD

T

AN

V

AD

T

AN

V

AD

T

AN

V

AD

T

AN

V

AD

T

AN

V

AD

T

Rouss

eau e

t al

(2002)

12.0

8

1210.2

1

3.0

0

941.0

8 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.6

3 1

382.7

8

3.3

8 1105.2

2

Li

and L

im (

2003

) 12.0

8

1215.1

4

2.9

1

953.5

5 1

0.0

0 8

28.3

8

3.0

0 5

89.9

0 1

1.7

5 1

385.4

7

3.2

5 1142.4

8

Lau

et

al (

2003)

13.0

0

1257.7

9

3.0

0 1

001.1

8 1

0.0

0 8

23.1

3

3.0

0 6

06.2

8 1

2.2

5 1

418.7

8

3.3

8 1170.9

3

Bra

ysy

et

al (

2003)

12.1

7

1208.5

7

2.7

3

971.4

4 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.7

5 1

372.9

3

3.2

5 1154.0

4

Bra

ysy

(2003)

11.9

2

1222.1

2

2.7

3

975.1

2 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.5

0 1

389.5

8

3.2

5 1128.3

8

Ber

ger

et

al

(2003)

11.9

2

1221.1

0

2.7

3

975.4

3 1

0.0

0 8

28.4

8

3.0

0 5

89.9

3 1

1.5

0 1

389.8

9

3.2

5 1159.3

7

Ben

t an

d H

ente

nry

ck (

20

04)

12.1

7

1203.8

4

2.7

3

980.3

1 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.6

3 1

379.0

3

3.2

5 1158.9

1

Bra

ysy

et

al (

2004)

Met

hod A

12.0

0

1222.5

5

2.7

3

968.7

7 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.5

0 1

400.9

1

3.2

5 1139.5

1

Bra

ysy

et

al (

2004)

Met

hod B

12.0

0

1212.8

9

2.7

3

960.4

4 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.5

0 1

389.2

0

3.2

5 1124.1

4

Ber

ger

and B

arkao

ui

(20

04)

11.9

2

1221.1

0

2.7

3

975.4

3 1

0.0

0 8

28.4

8

3.0

0 5

89.9

3 1

1.5

0 1

389.8

9

3.2

5 1159.3

7

Hom

ber

ger

and G

ehri

ng (

2005)

11.9

2

1212.7

3

2.7

3

955.0

3 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.5

0 1

386.4

4

3.2

5 1108.5

2

Ibar

aki

et a

l (2

005)

11.9

2

1217.4

0

2.7

3

959.1

1 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.5

0 1

391.0

3

3.2

5 1122.7

9

Russ

ell

and C

hia

ng (

200

6)

12.0

8

1207.8

2

2.8

2

965.6

9 1

0.0

0 8

28.3

8

3.0

0 5

89.8

8 1

1.5

0 1

385.5

2

3.2

5 1151.7

5

Pis

inger

and R

opke

(200

7)

11.9

2

1212.3

9

2.7

3

957.7

2 1

0.0

0 8

28.3

8

3.0

0 5

89.8

6 1

1.5

0 1

385.7

8

3.2

5 1123.4

9

Pro

pose

d G

A

11.4

2

1104.5

3

2.6

4

835.4

4 1

0.0

0 8

73.0

5

3.0

0 6

89.2

8 1

1.3

8 1

286.0

5

3.1

3 1011.4

5

107

5.7.1 Instance wise comparison

To measure the relative performance of the proposed algorithm, the

percentage of improvement in the number of vehicles used and the distance

travelled is computed using the following two indices

100*

−=

b

i

p

i

b

i

iV

VVPIV (5.2)

where

PIVi - Percentage of improvement in the number of vehicles for

instance i

Vib - Number of vehicles used in the best known solution for

instance i

Vip - Number of vehicles used in the proposed algorithm for

instance i

100*

−=

b

i

p

i

b

i

iD

DDPID (5.3)

where

PIDi - Percentage of improvement in distance travelled for

instance i

Dib - Distance travelled in the best known solution for instance i

Dip - Distance travelled in the proposed algorithm for instance i

The PIV and PID values for R1, R2, C1, C2, RC1 and RC2 are

given in Tables 5.11 to 5.22. The overall average percentage of improvement

considering all 56 instances is found to be 1.31% with a maximum of 33.33%

percentage improvement in terms of the number of vehicles used while the

108

overall average percentage improvement in terms of distance travelled is

found to be 2.01%, with a maximum improvement of 37.23%.

5.7.2 Problem type wise comparison

The data used for comparison comprises different problem types

like R1, R2, C1, C2, RC1 and RC2. To study the effect of the proposed

algorithm on these types of problems, the average values of PIV and PID are

used as defined below:

p

n

i

i

n

PIV

APIV

p

p

∑== 1 (5.4)

where

APIVp - Average PIV for problem type p

np - the number of instances considered in problem type p

p

n

i

i

pn

PID

APID

p

∑== 1 (5.5)

where

APIDp - Average PID for problem type p

np - the number of instances considered in problem type p

The values of PIV and PID for various instances with respect to

algorithm considered for comparison along with APIV and APID values are

given in Table 5.11 to 5.22. Figure 5.5 compares average percentage

improvement in vehicles used by PGA with other heuristic and Figure 5.6

compares average percentage improvement in distance travelled by PGA with

other heuristic considered for comparison.

109

Table 5.11 Percentage of improvement in the number of vehicles used

for R1 problems using the PGA over earlier results

Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

R101 15.79 20.00 15.79 20.00 15.79 15.79 11.11

R102 17.65 22.22 17.65 22.22 17.65 17.65 17.65

R103 7.69 20.00 7.69 14.29 7.69 7.69 7.69

R104 0.00 16.67 -11.11 0.00 0.00 0.00 -11.11

R105 14.29 29.41 14.29 20.00 14.29 14.29 14.29

R106 8.33 21.43 8.33 8.33 8.33 8.33 8.33

R107 -10.00 15.38 -10.00 8.33 -10.00 -10.00 -10.00

R108 -11.11 16.67 -11.11 0.00 -11.11 -11.11 -11.11

R109 9.09 33.33 9.09 23.08 9.09 9.09 9.09

R110 0.00 23.08 0.00 9.09 0.00 0.00 0.00

R111 -10.00 8.33 0.00 0.00 -10.00 -10.00 -10.00

R112 0.00 16.67 0.00 0.00 -11.11 0.00 -11.11

Average 3.48 20.27 3.39 10.45 2.55 3.48 1.24

110


for R2 problems using the PGA over earlier results

Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

R201 0.00 50.00 0.00 0.00 0.00 0.00 0.00

R202 0.00 57.14 25.00 0.00 0.00 0.00 0.00

R203 0.00 50.00 0.00 0.00 0.00 0.00 0.00

R204 0.00 66.67 0.00 33.33 0.00 0.00 0.00

R205 0.00 40.00 0.00 0.00 0.00 0.00 0.00

R206 0.00 40.00 0.00 0.00 0.00 0.00 0.00

R207 0.00 50.00 0.00 33.33 0.00 0.00 0.00

R208 0.00 33.33 0.00 0.00 0.00 0.00 0.00

R209 33.33 60.00 33.33 33.33 33.33 33.33 33.33

R210 0.00 50.00 0.00 0.00 0.00 0.00 0.00

R211 0.00 71.43 33.33 33.33 0.00 33.33 0.00

Average 3.03 51.69 5.83 12.12 3.03 6.06 3.03


for C1 problems using the PGA over earlier results

Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

C101 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C102 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C103 0.00 9.09 0.00 0.00 0.00 0.00 0.00

C104 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C105 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C106 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C107 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C108 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C109 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Average 0.00 1.01 0.00 0.00 0.00 0.00 0.00

111


for C2 problems using the PGA over earlier results

Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

C201 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C202 0.00 25.00 0.00 0.00 0.00 0.00 0.00

C203 0.00 25.00 0.00 0.00 0.00 0.00 0.00

C204 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C205 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C206 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C207 0.00 0.00 0.00 0.00 0.00 0.00 0.00

C208 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Average 0.00 6.25 0.00 0.00 0.00 0.00 0.00


for RC1 problems using the PGA over earlier results

Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

RC101 14.29 29.41 20.00 20.00 14.29 14.29 14.29

RC102 8.33 35.29 15.38 15.38 8.33 8.33 8.33

RC103 0.00 21.43 0.00 8.33 0.00 0.00 0.00

RC104 -10.00 8.33 -10.00 -10.00 -10.00 -10.00 -10.00

RC105 7.69 29.41 7.69 20.00 7.69 7.69 7.69

RC106 0.00 21.43 0.00 8.33 0.00 0.00 0.00

RC107 0.00 21.43 0.00 0.00 0.00 0.00 0.00

RC108 -20.00 0.00 -20.00 -20.00 -20.00 -20.00 -20.00

Average 0.04 20.84 1.63 5.26 0.04 0.04 0.04

112


for RC2 problems using the PGA over earlier results

Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

RC201 0.00 60.00 0.00 0.00 0.00 0.00 0.00

RC202 0.00 70.00 0.00 25.00 0.00 0.00 0.00

RC203 0.00 50.00 0.00 0.00 0.00 0.00 0.00

RC204 0.00 25.00 0.00 0.00 0.00 0.00 0.00

RC205 25.00 66.67 25.00 25.00 25.00 25.00 25.00

RC206 0.00 40.00 0.00 0.00 0.00 0.00 0.00

RC207 0.00 50.00 0.00 0.00 0.00 0.00 0.00

RC208 0.00 50.00 0.00 0.00 0.00 0.00 0.00

Average 3.13 51.46 3.13 6.25 3.13 3.13 3.13

Table 5.17 Percentage of improvement in the distance travelled for R1

problems using the PGA over earlier results

Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

R101 21.24 22.46 21.24 26.34 21.24 21.24 19.13

R102 17.80 21.52 17.71 21.01 17.69 17.69 14.70

R103 13.66 14.46 13.19 10.82 13.26 13.21 13.19

R104 -6.38 7.23 -3.91 -2.77 -6.23 -6.57 -3.92

R105 17.71 24.26 17.96 22.52 17.71 17.71 17.71

R106 14.92 21.44 16.03 19.75 14.92 14.84 14.84

R107 3.93 13.78 3.15 7.78 4.05 3.15 3.15

R108 -10.35 2.51 -7.91 -1.76 -10.76 -9.87 -10.76

R109 10.37 17.31 11.06 14.60 10.53 10.00 10.00

R110 8.01 21.34 10.74 8.23 7.52 7.68 7.48

R111 3.26 10.77 -0.60 2.22 2.58 2.72 2.18

R112 -9.24 4.75 -8.02 -2.31 -3.46 -8.82 -6.45

Average 7.08 15.15 7.55 10.54 7.42 6.92 6.77

113

Table 5.18 Percentage of improvement in the distance travelled for R2


Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

R201 37.27 40.88 37.23 39.18 37.27 37.54 37.23

R202 21.59 28.15 13.43 18.98 21.44 23.56 21.20

R203 3.65 16.14 4.04 7.10 3.54 4.63 3.03

R204 3.09 13.90 3.02 2.87 1.81 3.21 0.26

R205 17.36 26.32 19.28 27.32 17.40 18.44 16.19

R206 10.62 31.26 12.39 18.95 10.63 10.53 9.93

R207 12.22 18.13 8.04 4.32 8.20 10.32 6.83

R208 1.30 14.26 1.02 8.24 1.18 2.16 0.20

R209 7.93 23.12 9.33 17.28 7.94 10.78 7.19

R210 5.71 20.00 10.77 11.96 3.27 7.57 3.21

R211 17.54 17.43 3.97 11.10 15.74 3.32 13.75

Average 12.57 22.69 11.14 15.21 11.67 12.01 10.82

Table 5.19 Percentage of improvement in the distance travelled for C1


Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

C101 -5.84 -5.84 -5.84 -5.84 -5.84 -5.84 -6.05

C102 -5.53 -0.69 -5.53 -4.81 -5.53 -5.53 -5.74

C103 -5.42 7.08 -5.42 -4.6 -5.42 -5.42 -5.64

C104 -6.04 9.25 -6.04 -3.34 -6.04 -6.04 -6.28

C105 -5.33 -5.33 -5.33 -5.33 -5.33 -5.33 -5.54

C106 -4.89 -4.89 -4.89 -4.89 -4.89 -4.89 -5.1

C107 -5.52 -5.52 -5.52 -5.52 -5.52 -5.52 -5.73

C108 -4.78 -4.78 -4.78 -4.78 -4.78 -4.78 -4.99

C109 -5.22 -5.19 -5.19 -5.19 -5.19 -5.19 -5.39

Average -5.40 -1.77 -5.39 -4.92 -5.39 -5.39 -5.61

114

Table 5.20 Percentage of improvement in the distance travelled for C2


Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

C201 -16.27 -16.27 -16.27 -16.27 -16.27 -16.27 -16.76

C202 -16.74 -0.98 -16.74 -11.5 -16.74 -16.74 -17.22

C203 -16.15 7.95 -16.15 -13.68 -16.15 -16.15 -16.64

C204 -17.49 -14.69 -17.49 -7.71 -17.49 -17.49 -17.49

C205 -16.66 -16.66 -16.66 -14.22 -16.66 -16.66 -17.15

C206 -17.37 -17.37 -17.37 -17.29 -17.37 -17.37 -17.87

C207 -16.45 -15.49 -16.45 -12.51 -16.45 -16.41 -16.95

C208 -17.72 -17.21 -17.72 -17.02 -17.72 -17.72 -18.22

Average -16.86 -11.34 -16.86 -13.78 -16.86 -16.85 -17.29

Table 5.21 Percentage of improvement in the distance travelled for

RC1 problems using the PGA over earlier results

Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

RC101 19.46 20.37 17.02 16.96 18.92 18.89 18.89

RC102 18.52 20.83 16.12 17.33 18.52 18.49 18.34

RC103 -0.59 16.41 3.75 8.34 -0.64 -0.70 -0.70

RC104 -7.75 4.12 -7.22 -0.84 -7.52 -7.74 -7.75

RC105 19.92 22.03 19.59 18.97 19.83 19.19 19.19

RC106 11.93 15.73 11.78 11.86 12.62 11.84 11.78

RC107 -2.08 14.03 -1.33 5.52 -1.93 -2.16 -2.17

RC108 -14.61 1.16 -14.84 -12.06 -15.24 -14.92 -15.61

Average 5.60 14.34 5.61 8.26 5.57 5.36 5.25

115

Table 5.22 Percentage of improvement in the distance travelled for

RC2 problems using the PGA over earlier results

Instance

Imp.%

Over

CLM

Imp.%

Over

TLZO

Imp.%

Over

LL

Imp.%

Over

LST

Imp.%

Over

B

Imp.%

Over

RC

Imp.%

Over

BKS

RC201 14.82 23.45 15.91 18.39 15.15 16.49 14.82

RC202 27.47 24.57 25.72 16.51 25.38 29.51 25.33

RC203 17.77 25.77 18.91 24.68 16.82 18.45 15.91

RC204 0.40 18.90 1.93 4.34 -0.54 2.19 -0.56

RC205 21.16 28.64 19.82 21.86 19.47 21.84 19.36

RC206 6.08 21.45 5.92 9.23 5.36 7.46 4.88

RC207 3.47 21.41 7.41 10.03 4.38 4.78 3.39

RC208 -23.16 -2.17 -18.88 -4.01 -23.56 -22.88 -23.79

Average 8.50 20.25 9.59 12.63 7.81 9.73 7.42

5.7.3 Overall comparison

The cumulative value of the number of vehicles and the cumulative

distance travelled from literature is compared with the cumulative values of

the proposed algorithm in Table 5.23. CNV and CDT indicate the cumulative

values of the number of vehicles and the total distance travelled respectively,

for 56 Solomon instances. The relative improvement over literature results

compared in terms of CNV vary from 1.24% to 24.57% while in terms of

CDT, it ranges from 5.70% to 13.84%.

116

0

10

20

30

40

50

60

R1 R2 C1 C2 RC1 RC2

Problem category

Av

era

ge

im

pro

ve

me

nt

in

ve

hic

les

us

ed

(%

)

CLM TLZO LL LST B RC BKS

Figure 5.5 Average Percentage of improvement in the number of

vehicles used by the PGA with other heuristics

-20

-15

-10

-5

0

5

10

15

20

25

R1 R2 C1 C2 RC1 RC2

Problem category

Av

era

ge

im

pro

ve

me

nt

in

dis

tan

ce

tra

ve

lle

d (

%)

CLM TLZO LL LST B RC BKS

Figure 5.6 Average Percentage of improvement in the distance

travelled by the PGA with other heuristics

117

Table 5.23 Comparison of Cumulative Number of Vehicles (CNV) and

Cumulative Distance Travelled (CDT)

Author(s) CNV CDT

Rochat and Taillard (1995) 415 57231

chiang and Russell (1997) 411 58502

Taillard et al (1997) 410 57522

Liu and Shen (1999) 412 59317

Gambardella et al (1999) 407 57516

Homberger and Gehring (1999) 406 57876

Cordeau et al (2001) 407 57556

Tan et al (2001c) 525 62902

Rousseau et al (2002) 412 56953

Li and Lim (2003) 401 57469

Berger et.al (2003) 405 57952

Braysy (2003 ) 405 57710

Lau et.al (2003) 428 59164

Berger and Barkaoui (2004) 405 57952

Bent and Hentenryck (2004) 405 57273

Braysy et.al (2004 ) 406 57401

Homberger and Gehring (2005) 405 57309

Ibaraki et.al (2005) 405 57444

Russell and Chiang (2006) 408 57590

Pisinger and Ropke (2007) 405 57360

Proposed GA 396 54196

Maximum 525 62902

% Improvement with respect to Maximum 24.57% 13.84%

Minimum 401 57469

% Improvement with respect to Minimum 1.25% 5.70%

118

5.8 PERFORMANCE COMPARISON OF THE PGA FOR

MDVRPTW WITH THE PUBLISHED RESULTS

In order to compare the solution provided by the PGA with other

solution methodologies available earlier, the PGA is tested on the

MDVRPTW instances provided by Cordeau et al (2001). This consists of 20

instances with different customer sizes as well as time window tightness.

Depending upon the time window tightness the 20 instances are divided into

two groups, PR1 and PR2 problems. The performance of the PGA is

compared with the following earlier reported literature results

• Cordeau et al (2001) [TS1]

• Cordeau et al (2004) [TS2]

• Polacek et al (2004) [VNS]

• Chiu et al (2006) Method A [ECUTS]

• Chiu et al (2006) Method B [DGH]

Results obtained in terms of the number of vehicles used and the

distance travelled by the PGA with respect to TS1, TS2, VNS, ECUTS and

DGH are provided in Tables 5.24 and 5.25.

Figure 5.7 gives the comparison of the average number of vehicles

used for PR1 and PR2 problems and Figure 5.8 compares the average distance

travelled for PR1 and PR2 problems by the PGA and other algorithms under

comparison.

It can be observed from Table 5.26 that the average improvement

percentage in terms of vehicles used for PR1 problems by the PGA over the

earlier reported results of TS1, TS2 and VNS is about 18.7. With

119

ECUTS 11.93 percentage of improvement is observed. When compared to

DGH and BKS an improvement of 0.41 percentage is achieved.

It is evident from Table 5.27 for PR2 problems, that the PGA

performs equally well with respect to TS1, TS2, VNS and DGH. But when

compared to ECUTS and BKS, the PGA performance is 0.435% inferior.

From Tables 5.28 and 5.29, it is clearly seen that the PGA

outperforms with respect to the distance travelled. The maximum average

improvement of 34.67% is achieved when compared to ECUTS with a

minimum of 6 percent with respect to TS2 for the PR1 problem. For the PR2

problem a maximum average improvement of 21.74% is obtained when

compared to the DGH and a minimum of 0.54% with BKS. Graphical

comparison of APIV and APID with respect to APIV and APID is shown in

Figure 5.9 and Figure 5.10 respectively

The cumulative values of the number of vehicles and the distance

travelled for MDVRPTW from literature is compared with the cumulative

values of the proposed algorithm in Table 5.30.

120

Tab

le 5

.24 P

GA

solu

tion

an

d c

om

pari

son

wit

h t

he

oth

er h

euris

tics

for

PR

1 t

yp

e P

rob

lem

s

Inst

an

ce

TS

1

TS

2

VN

S

EC

UT

S

DG

H

BK

S

PG

A

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

Pr0

1

8

1083.9

8

8

1074.1

2

8

1074.1

2

8

1526.1

0

6

1540.8

0

6

1074.1

2

5

929.1

9

Pr0

2

12

1763.0

7

12

1762.2

1

12

1762.2

1

11

2466.2

0

9

2308.7

0

9

1762.2

1

9

1606.6

1

Pr0

3

16

2408.4

2

16

2373.6

5

16

2373.6

5

15

3656.2

0

15

3222.2

0

15

2373.6

5

13

2087.4

9

Pr0

4

20

2958.2

3

20

2852.2

9

20

2815.4

8

19

4146.1

0

18

3813.6

0

18

2815.4

8

17

2571.0

5

Pr0

5

24

3134.0

4

24

3029.6

5

24

2993.9

4

23

4255.6

0

23

4454.9

0

23

2993.9

4

23

2927.7

6

Pr0

6

28

3904.0

7

28

3627.1

8

28

3629.7

2

28

5452.3

0

28

5048.9

0

28

3627.1

8

27

3375.1

1

Pr0

7

12

1423.3

5

12

1418.2

2

12

1418.2

2

10

1852.9

0

8

1938.1

0

8

1418.2

2

8

1297.5

2

Pr0

8

18

2150.2

2

18

2102.6

1

18

2096.7

3

15

2965.8

0

14

2930.7

0

14

2096.7

3

14

2143.0

7

Pr0

9

24

2833.8

0

24

2737.8

2

24

2730.5

4

21

4021.5

0

20

4013.5

0

20

2730.5

4

19

2751.4

7

Pr1

0

30

3717.2

2

30

3505.2

7

30

3499.5

6

28

5288.4

0

20

5187.9

0

20

3499.5

6

28

3528.6

4

121

Tab

le 5

.25 P

GA

solu

tion

an

d c

om

pari

son

wit

h t

he

oth

er h

euris

tics

for

PR

2 t

yp

e P

rob

lem

s

Inst

an

ce

TS

1

TS

2

VN

S

EC

UT

S

DG

H

BK

S

PG

A

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

NV

D

T

Pr1

1

4

1031.4

9

4

1005.7

3

4

1005.7

3

4

1302.5

0

4

1265.8

0

4

1005.7

3

4

954.0

2

Pr1

2

8

1500.4

8

8

1478.5

1

8

1472.7

6

8

1723.6

0

8

1767.3

0

8

1472.7

6

8

1427.7

0

Pr1

3

12

2020.5

8

12

2011.2

4

12

2001.8

3

12

2374.1

0

12

2554.8

0

12

2001.8

3

12

2012.2

6

Pr1

4

16

2247.7

2

16

2202.0

8

16

2215.5

1

16

2616.4

0

16

2740.8

0

16

2202.0

8

16

2217.1

8

Pr1

5

20

2509.7

5

20

2494.5

7

20

2465.2

5

20

2963.1

0

20

3162.6

0

20

2465.2

5

20

2509.7

5

Pr1

6

24

2943.9

0

24

2901.0

2

24

2896.0

3

23

3534.6

0

24

3641.9

0

23

2896.0

3

24

2901.1

8

Pr1

7

6

1250.0

9

6

1236.2

4

6

1236.2

4

6

1485.4

0

6

1602.0

0

6

1236.2

4

6

1218.2

3

Pr1

8

12

1809.3

5

12

1792.6

1

12

1796.2

1

12

2050.8

0

12

2371.2

0

12

1792.6

1

12

1799.4

3

Pr1

9

18

2310.9

2

18

2285.1

18

2292.4

5

18

2807.9

0

18

2857.0

0

18

2285.1

0

18

2293.6

2

Pr2

0

24

3131.9

0

24

3079.1

6

24

3076.3

7

24

3874.2

0

24

4077.7

0

24

3076.3

7

24

3080.4

5

122

0

5

10

15

20

25

PR1 PR2

Problem category

Avera

ge n

um

ber

of

veh

icle

s

used

TS1 TS2 VNS ECUTS DGH BKS PGA

Figure 5.7 Comparison of the average number of vehicles used for

different type of Cordeau instances by the PGA and other

methods

0

500

1000

1500

2000

2500

3000

3500

4000

PR1 PR2

Problem category

Avera

ge d

ista

nce t

ravell

ed

TS1 TS2 VNS ECUTS DGH BKS PGA

Figure 5.8 Comparison of the average distance travelled for different

type of Cordeau instances by the PGA and other methods

123


for PR1 problems using the PGA over earlier results

Instance

Imp.%

Over

TS1

Imp.%

Over

TS2

Imp.%

Over

VNS

Imp.%

Over

ECUTS

Imp.%

Over

DGH

Imp.%

Over

BKS

Pr01 37.5 37.5 37.5 37.5 16.67 16.67

Pr02 25.00 25.00 25.00 18.18 0.00 0.00

Pr03 18.75 18.75 18.75 13.33 13.33 13.33

Pr04 15.00 15.00 15.00 10.53 5.56 5.56

Pr05 4.17 4.17 4.17 0.00 0.00 0.00

Pr06 3.57 3.57 3.57 3.57 3.57 3.57

Pr07 33.33 33.33 33.33 20.00 0.00 0.00

Pr08 22.22 22.22 22.22 6.67 0.00 0.00

Pr09 20.83 20.83 20.83 9.52 5.00 5.00

Pr10 6.67 6.67 6.67 0.00 -40 -40

Average 18.7 18.70 18.70 11.93 0.41 0.41


for PR2 problems using the PGA over earlier results

Instance

Imp.%

Over

TS1

Imp.%

Over

TS2

Imp.%

Over

VNS

Imp.%

Over

ECUTS

Imp.%

Over

DGH

Imp.%

Over

BKS

Pr11 0.00 0.00 0.00 0.00 0.00 0.00

Pr12 0.00 0.00 0.00 0.00 0.00 0.00

Pr13 0.00 0.00 0.00 0.00 0.00 0.00

Pr14 0.00 0.00 0.00 0.00 0.00 0.00

Pr15 0.00 0.00 0.00 0.00 0.00 0.00

Pr16 0.00 0.00 0.00 -4.35 0.00 -4.35

Pr17 0.00 0.00 0.00 0.00 0.00 0.00

Pr18 0.00 0.00 0.00 0.00 0.00 0.00

Pr19 0.00 0.00 0.00 0.00 0.00 0.00

Pr20 0.00 0.00 0.00 0.00 0.00 0.00

Average 0.00 0.00 0.00 -0.44 0.00 -0.44

124

Table 5.28 Percentage of improvement in the distance travelled for PR1


Instance

Imp.%

Over

TS1

Imp.%

Over

TS2

Imp.%

Over

VNS

Imp.%

Over

ECUTS

Imp.%

Over

DGH

Imp.%

Over

BKS

Pr01 14.28 13.49 13.49 39.11 39.69 13.49

Pr02 8.87 8.83 8.83 34.85 30.41 8.83

Pr03 13.33 12.06 12.06 42.91 35.22 12.06

Pr04 13.09 9.86 8.68 37.99 32.58 8.68

Pr05 6.58 3.36 2.21 31.20 34.28 2.21

Pr06 13.55 6.95 7.01 38.10 33.15 6.95

Pr07 8.84 8.51 8.51 29.97 33.05 8.51

Pr08 0.33 -1.92 -2.21 27.74 26.88 -2.21

Pr09 2.91 -0.50 -0.77 31.58 31.44 -0.77

Pr10 5.07 -0.67 -0.83 33.28 31.98 -0.83

Average 8.69 6.00 5.70 34.67 32.87 5.69

Table 5.29 Percentage of improvement in the distance travelled for PR2


Instance

Imp.%

Over

TS1

Imp.%

Over

TS2

Imp.%

Over

VNS

Imp.%

Over

ECUTS

Imp.%

Over

DGH

Imp.%

Over

BKS

Pr11 7.51 5.14 5.14 26.75 24.63 5.14

Pr12 4.85 3.44 3.06 17.17 19.22 3.06

Pr13 0.41 -0.05 -0.52 15.24 21.24 -0.52

Pr14 1.36 -0.69 -0.08 15.26 19.10 -0.69

Pr15 0.00 -0.61 -1.81 15.30 20.64 -1.81

Pr16 1.45 -0.01 -0.18 17.92 20.34 -0.18

Pr17 2.55 1.46 1.46 17.99 23.96 1.46

Pr18 0.55 -0.38 -0.18 12.26 24.11 -0.38

Pr19 0.75 -0.37 -0.05 18.32 19.72 -0.37

Pr20 1.64 -0.04 -0.13 20.49 24.46 -0.13

Average 2.11 0.79 0.67 17.67 21.74 0.56

125

-5

0

5

10

15

20

PR1 PR2

Problem category

Avera

ge i

mp

rovem

en

t in

veh

icle

s u

sed

(%

)

TS1 TS2 VNS ECUTS DGH BKS

Figure 5.9 Average Percentage of improvement in the number of

vehicles used by the PGA with other heuristic

0

5

10

15

20

25

30

35

40

PR1 PR2

Problem category

Avera

ge i

mp

rovem

en

t in

dis

tan

ce t

ravell

ed

(%)

TS1 TS2 VNS ECUTS DGH BKS

Figure 5.10 Average Percentage of improvement in the distance

travelled by the PGA with other heuristic

126

Table 5.30 Comparison of the Cumulative Number of Vehicles (CNV)

and the Cumulative Distance Travelled (CDT)

Authors CNV CDT

Cordeau et al (2001) 336 46132.60

Cordeau et al (2004) 336 44969.30

Polacek et al (2004) 336 44852.60

Chiu et al (2006) Method A 321 60363.70

Chiu et al (2006) Method B 305 60500.40

Proposed GA 307 43631.70

Maximum 336 46132.60

% Improvement with respect to Maximum 8.63% 5.42%

Minimum 305 60500.40

% Improvement with respect to Minimum -0.66% 27.88%

5.9 SUMMARY

This chapter presents the genetic algorithm approach to the solve

single depot and multi-depot vehicle routing problem with time windows,

with the objective of minimizing the number of vehicles required and the total

distance travelled. The performance of the proposed genetic algorithm has

been evaluated, using the benchmark suite provided by Solomon for VRPTW

and the Cordeau instances for MDVRPTW. The experimental results have

shown that the algorithm is effective and efficient in reducing the number of

vehicles and is also competitive in terms of distance minimization.

chapter 5 genetic algorithm based search heursitic...

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