Hierarchical Optimization Using Hierarchical Multi-competitive Genetic

Algorithm and its Application to Multiple Vehicle Routing Problem

Shudai Ishikawa

, Ryosuke Kubota

and Keiichi Horio

Kyushu Institute of Technology, Kitakyushu, Japan

Ube National College of Technology, Ube, Japan

Keywords:

Hierarchical Optimization, Genetic Algorithm, Multiple Vehicle Routing Problem, Multiple Solution Spaces.

Abstract:

In this paper a new optimization technique which is effective for hierarchical optimization problem is pro-

posed. This technique is an extension of the multiple-competitive distributed genetic algorithm (mcDGA).

This method consists of two levels upper and lower. The solution space to be searched is determined at the

upper level, and the optimum solution in a given solution space is determined at the lower level. The migra-

tion of the individual and competition is performed at the lower layer thereby optimal solution can be found

efﬁciently. We apply the proposed hierarchical mcDGA to the mVRP to conﬁrm the effectiveness. Simulation

result shows the effectiveness of the proposed method.

1 INTRODUCTION

Some complex systems have hierarchical structure

which is divided into multiple levels and optimiza-

tion of such systems is known as hierarchical opti-

mization. For example, suppose that the system is di-

vided into two levels upper and lower. In the upper

level, some solution spaces to be searched are deter-

mined and at the lower level, the optimal solutions

are searched for each solution spaces which are deter-

mined in the upper level. Thus, optimal solutions at

the lower level are reﬂected in the upper level. At the

upper level, optimization is performed to determine

the solution space to be searched based on the result

of lower levels. By repeating this operation, the op-

timal solution of the whole system is obtained. The

example of hierarchical optimization using multiple

vehicle routing problem is shown in Figure 1. In this

problem, a product is to be delivered to a customer

by using a plurality of vehicles with minimized path

of each vehicle. As shown in Figure 1, the alloca-

tion of customer for each vehicle is optimized at the

upper level and the route is optimized at the lower

level. In this situation, the problem of the hierarchical

system, the searching space is reduced, and discov-

ery of the solution is facilitated. However, when the

optimal solution is not found at the lower level, it is

difﬁcult to ﬁnd the optimal solution for the entire sys-

tem. In addition, if much time is taken for calculation

at the lower level, computational cost will be huge in

amount.

The VehicleRoutingProblem (VRP) (Bernard and

Hubert, 1959) is a problem that minimizes the deliv-

ery path between the distribution center and the cus-

tomers. In recent years, the VRP has been attracted

much attention and studied actively. Although the

VRP is similar to the well-known Traveling Sales-

man Problem (TSP) (Flood, 1956), some constraints

are taken into account in the VRP. For instance, the

number of vehicles, the time windows (time schedul-

ing), and the capacity of vehicles and so on (Laporte,

1992). Namely, the VRP is an extension of the TSP.

In the VRP with multiple vehicles (mVRP), it is nec-

essary to minimize the total delivery path of vehicles

and therefore it is difﬁcult to ﬁnd the optimal solution

because optimization of allocation for customers of

each vehicle and the path of vehicles are performed

at the same time. To facilitate the discovery of op-

timum solution, the method of determining the cus-

tomer for each vehicle by clustering has been pro-

posed (Sofge et al., 2002), (Nallusamy et al., 2009).

In this method the discovery of the solution is facili-

tated because the solution space is reduced. However

if some constraints such as the time window or the

capacity of vehicles are added, this method cannot be

applied. It is reported that this problem is solved by

two interconnected Genetic Algorithms (GAs) (Potter

and Bossomaier, 1995). In this method, at the upper

level, a GA determines the allocation to customers of

each vehicle and at the lower level a second GA for

269

Ishikawa S., Kubota R. and Horio K..

Hierarchical Optimization Using Hierarchical Multi-competitive Genetic Algorithm and its Application to Multiple Vehicle Routing Problem.

DOI: 10.5220/0005043202690274

In Proceedings of the 11th International Conference on Signal Processing and Multimedia Applications (SIGMAP-2014), pages 269-274

ISBN: 978-989-758-046-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

Upper level optimization

Lower level optimization

Vehicle 1 Vehicle 2

Depot

Solution Space 1

Solution Space S

Solution Space 1

Individual1

Solution Space S

Individual N

Figure 1: Conceptual diagram in case of mVRP to be the

hierarchical structure.

each vehicle determines the shortest route of the cus-

tomers allocated to the vehicles. This method is valid

in solution search and difﬁcult to set population size.

If population size is too large, computational cost is

more whereas if population size is too small, solution

cannot be found stably at the lower level.

This proposed multi-space competitivedistributed

GA (mcDGA) (Ishikawa et al., 2011), competition is

performed among different solution spaces, and the

population size of the solution space is changed based

on the result of the competition. The individual is

added to the solution space which won the compe-

tition, and the individual is removed in loser. By re-

peating this process, the proposed method terminates

the search of the solution space less likely to solution

exists, and concentrates on the individuals for the so-

lution space more likely solution exists. Namely, it

is possible to improve the stability of ﬁnding optimal

solution and reduce the computational cost.

This proposed hierarchical mcDGA is a stable and

effective search method and we apply it to the mVRP

a supposed hierarchical optimization problem. The

proposed method performs not only the optimization

of the individual but also the optimization of the solu-

tion space. Namely, in the proposed method, reducing

the computational cost and improving the stability of

discovery of the optimal solution in the lower level

are expected. We apply the proposed method to the

mVRP for two vehicles and try to show the effective-

ness.

2 MULTIPLE-VEHICLE

ROUTING PROBLEM

The multiple vehicle problem is an extension of the

well-known vehicle routing problem, this problem

consists of determining a set of routes for some ve-

hicles. Generally, the prerequisites of the mVRP are

as follows:

• The vehicles return to the depot after they leave

the depot and visit customers.

• The positional coordinates of customers are given.

• The requirement of customers is ﬁlled by either

vehicles visit only once.

• The distance of each delivery is calculated based

on the Euclidean distance between customers.

• The total delivery distance is minimized.

In this paper, we consider two vehicles without its ca-

pacity.

3 GA AND MCDGA

3.1 GA

In nature the living things adapted by the environ-

ment, survive with high possibility and they have

much opportunity to pass their genes on to their off-

spring. The GA, developed by John Holland in the

1960s (Holland, 1962), imitates the evolution of liv-

ing things and this is one of powerful optimization

algorithms. A problem to be solved and candidates

for a solution are related to the environment and in-

dividuals respectively. The features of the GA are as

follows: (1) a population that is a set of individuals is

held (2) the individuals are evolved by genetic oper-

ators such as selection, reproduction, rearrangement

and mutation (3) an evaluation of the environment is

given to each individual and (4) the individuals whose

evaluations are high survivewith a high probability by

selection and reproduction, whereas perturbation by

rearrangement and mutation produces various kinds

of individuals. In other words, in the GA, candidates

for optimal solutions for the given problem are rep-

resented as coordinates in the solution space and the

candidates search the optimal solution by re-genesis

based on selection, reproduction, mutation and so on.

The GA achieves local and global searches in the so-

lution space by employing adequate genetic opera-

tors.

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3.2 Multi-space Competitive

Distributed GA

The mcDGA can be constructed as an extension of the

DGA (Tanese, 1989) for multi-space solution search.

Sub-populations are prepared in the corresponding

solution spaces. Each sub-population searches an

optimal solution in the corresponding solution space

in accordance with the procedure of the GA. Dur-

ing the evolution, the mean value of the evaluations

or the max value of the evaluations and so in each

sub-population is calculated and their values are com-

pared to each other. This operation is called ”com-

petition”. The sub-populations are ranked based on

the competition. The population sizes in the supe-

rior sub-populationsincrease, on the other hand, those

decreases in the inferior ones. In particular, individ-

uals are added in the sub-population with the high-

est evaluation and one individual is removed from the

other sub-population. The individual to be added and

to be removed is randomly chosen from each sub-

population. This operation is called migration. The

migration improves the search efﬁciency. The sub-

population, which won the competition has the largest

population size after the convergence and thereby it

is easy to decide the correct solution space based on

the population sizes. The GA and the DGA perform

the parameter optimization. On the other hand, the

mcDGA perform not only the parameter optimiza-

tion(the solution) but also the model selection (solu-

tion space).

4 HIERARCHICAL MCDGA

In case of applying to the hierarchy problem with

mcDGA, the system is divided into upper level and

lower level. At the upper level, the allocation to cus-

tomers of each vehicle is determined. At the lower

level, the shortest route of a given allocation of cus-

tomers in each vehicle is determined. In order to per-

form efﬁcient search, the competition and the migra-

tion are conducted at the lower level. The solution

which has determined at the lower level reﬂects to the

upper level, and the new solution space will be deter-

mined based on the evaluation value of the solution

at the lower level. The conceptual diagram and the

simple ﬂow of the hierarchical mcDGA are shown in

Figure 1 and Figure 2. Generally, when GA is applied

to both the upper level and the lower level, a large

amount of computational cost is required. In addition,

the discovery rate of the optimal solution at the lower

level has a great inﬂuence on the discovery of opti-

mal solution overall. However hierarchical mcDGA

terminate the search of the solution space less likely

to solution exists and also reduce the computational

cost. Moreover the discovery rate of the optimal solu-

tion is improved by concentrating the individuals on

the solution space more likely solution exists. Since

it does not use a special method for genetic operators

and representation of an individual, it is applicable to

any hierarchical optimization problems.

5 SOLVING THE MVRP WITH

HIERARCHICAL MCDGA

In case of solving mVRP with improved mcDGA, the

representation of individuals is different from each

other in the upper level and lower level. Therefore,

different genetic operators are applied at the upper

and lower level.

Representation. At the upper level the individuals

are represented by a single genetic code with the ve-

hicle number inserted into each gene and at the lower

level the customer number inserted into each gene.

Fitness Function. Evaluation of the individuals is

performed at only the lower layer, the evaluation func-

tion F is given by the follow;

F =

max

, (1)

where, d

max

is the maximum distance in all vehicles.

This equation means the minimize The maximum dis-

tance of each vehicle.

Crossover. At the upper level a general crossover

method which is used for the combinational optimiza-

tion such as an one-point crossover and an uniform

crossover is applied because the combination of the

customer to be allocated to each vehicle. On the other

hand, at the lower level, a crossover method which

is devised for route optimization problems such as

Partially Matched Crossover (PMX) (D.E.Goldberg,

1985), Cycle Crossover (CX) (I.M.Oliver, 1987) and

Order Crossover (OX) (Davis, 1985) is applied. In

this paper, an uniform crossover and PMX are applied

at the upper level and lower level respectively.

Mutation. A normal mutation and gene sequence

inversion are applied at the upper level and lower level

respectively.

HierarchicalOptimizationUsingHierarchicalMulti-competitiveGeneticAlgorithmanditsApplicationtoMultipleVehicle

RoutingProblem

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Start

Initialization

Evaluation

Crossover

Mutation

Termination

lower GA

End

Initialization

Evaluation

Selection

Crossover

Mutation

Yes

Initialization

Evaluation

Selection

Crossover

Mutation

Selection

Initialization

Evaluation

Selection

Crossover

Mutation

Termination

upper GA

Solution space 1 Solution space 2 Solution space S

Output the solutions at upper

Output the solutions at lower

Yes

if the restrictions are satisfied

Competition

Population size adjustment

Upper GA Lower GA

Figure 2: Flow of the hierarchical mcDGA.

6 SIMULATION AND RESULT

6.1 Simulation Setup

A computation experiment has been conducted to

compare with two-level GA and conﬁrm the efﬁ-

ciency of the proposed method. Table 1 shows the

parameter of two-level GA and mcDGA used in com-

Table 1: Parameter of two-level GA and mcDGA.

two-level GA mcDGA

Upper level Population size 10

Generation 30000

Crossover rate 0.8

Mutation rate 0.3

Lower level Population size 20, 50 adaptive

Generation 500, 1000 1000

Crossover rate 0.8

Mutation rate 0.1

putation experiment. These parameters were deter-

mined experimentally. In the two-level GA, popula-

tion size set to 20 and 50, and the termination in the

lower level set to 500 and 1000. i.e. we compared the

experiments of four values for this proposed method.

The number of customers set to 20. Parameter values

for the hierarchical mcDGA are deﬁned as follows.

At the upper level, population size and terminate gen-

eration set to 10 and 30000 respectively. Crossover

rate and mutation rate set to 0.8 and 0.3 respectively.

At the lower level, initial population size set to 20

and 50 & terminate generation set to 500 and 1000.

Crossover rate and mutation rate set to 0.8 and 0.1

respectively. In the competition, the maximum eval-

uation value of the solution space is compared. The

migration is performed based on the update of the op-

timal solution and the generations. When the genera-

tion is more than 300 and the optimal solution of each

solution space is updated, the migration is performed.

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0.5

1.5

4300

4350

4400

4450

4500

Maximum distance in all vehicles

Calculation time

500

1000

500

1000

(Proposed)

Time [sec]

Distance

Population size

Number of generation

Variable

1000

Figure 3: The result of simulation. The gray bar shows the maximum distance in all vehicles. The black dots show the

calculation time.

When the generation is 300 to 500, the population size

in the solution space of the upper half of the eval-

uation value is increased and the population size in

the solution space of the lower half of the evaluation

value is decreased, i.e. the solution space searched

got narrows to half. Then, minimum and maximum

population size of each solution spaces set to 0 and

30, respectively. When the generation is more than

500, the population size of the solution space which

is increased the highest evaluation value, and the pop-

ulation size of the other solution space is decreased.

Then, maximum population size of solution space set

to 50. Namely, the solution space to be searched is

narrowed to one. Then, minimum and maximum pop-

ulation size of each solution spaces set to 0 and 50

respectively.

6.2 Result and Discussion

The trial is performed 10 times. Figure 3 and Figure

4 show the result of the experiments.

In Figure 3, the bar graph shows the distribution

and the average of the maximum distance of the vehi-

cles , and line graph shows the calculation time which

is required for one generation respectively. From Fig-

ure 3, If population size and generation are set to

20 and 500 respectively then calculation time is very

short and stability of ﬁnding the solution is poor. On

the other hand, If population size and generation are

set to 50 and 1000 respectively, then calculation time

is very long and stability of ﬁnding the solution is

good. This proposed method is able to ﬁnd the solu-

tion in short calculation time at the lower layer. Fig-

ure 4 is the best solution of the problem handled in

Vehicle 1

Vehicle 2

Depot

Figure 4: The best solution of this simulation. The black

stars and the light gray stars show the customer of vehicle 1

and vehicle 2 respectively. The black line and the light gray

line show the route of each vehicle.

this paper. The black stars and the light gray stars

show the customer of vehicle 1 and vehicle 2 respec-

tively. The black line and the light gray line show the

route of each vehicle. From this result, it is shown

that the probability of the discovery of the optimal so-

lution has improved by collecting the individual to the

solution space having high evaluation value. In other

word, the competition and the migration at the lower

level is effective in ﬁnding the optimal solution. Also

it is shown that the proposed method is able to reduce

the calculation time due to migration of individuals of

small evaluation value to search solution space. Fig-

ure 5 shows the image of competition and migration.

In Figure 5, the black and white ball show the win-

ner and loser solution space respectively at the lower

level. The number in each circle shows population

HierarchicalOptimizationUsingHierarchicalMulti-competitiveGeneticAlgorithmanditsApplicationtoMultipleVehicle

RoutingProblem

273

Generation

10005003000

Winner solution space

Loser solution space

(Search is aborted)

Figure 5: Image of the competition and the migration. The

black and white ball show the winner and loser solution

space at the lower level, respectively. The number in each

circle shows the population size of each solution space. The

number of balls is equal to the number of solution space to

be searched.

size of each solution space. Let t be a calculation

time per one generation. If population size and the

terminate generation are set to 20 and 1000 respec-

tively and the competition and the migration are not

performed then total calculation time T

will be cal-

culated as below.

= 20× 10 × 1000× t, (2)

= 2.0× 10

t. (3)

On the other hand, in the proposed method, the total

calculation time T

is given by follows.

= (20×10×300+30×5×200+50×1×500)t, (4)

= 1.15× 10

t. (5)

From the above equations, the calculation time can

be signiﬁcantly reduced in the proposed method was

conﬁrmed. From above results, it was shown that the

proposed method is able to ﬁnd the solution in short

calculation time.

7 CONCLUSION

A new optimization method is proposed which is ef-

fective for hierarchical optimization problem also an

extension of the multiple-competitive distributed ge-

netic algorithm (mcDGA). This method consists of

two levels upper and lower. The solution space to be

searched is determined at the upper level, and the opti-

mum solution in a given solution space is determined

at the lower level. The migration of the individual and

competition is performed at the lower layer thereby

optimal solution can be found efﬁciently. We applied

the proposed hierarchical mcDGA to the mVRP to

conﬁrm the effectiveness and this method has shown

good discovery accuracy and short computation time.

Although the experimental validation is limited in this

paper, it is not important for our study. Because we

are aim to construct a generic optimization technique

for any problem. In the future, we will not only con-

sider timing and rules of migration but also apply the

mcDGA for other problems, (e.g. trafﬁc signal con-

trol, digital signal processing, image processing, etc).

REFERENCES

Bernard, D. G. and Hubert, R. J. (1959). The truck dispatch-

ing. Management Science, 6:80–91.

Davis, L. (1985). Applying adaptive algorithms to epistatic

domains. Proceeding of the International Joint Con-

ference on Artiﬁcial Intelligence, IEEE Computer So-

ciety Press, Los Angeles, pages 162–164.

D.E.Goldberg (1985). Alleles loci and the tsp. Proceedings

of the First International Conference on Genetic Al-

gorithms and Their Application, Lawrence Erolaum,

New Jersy, pages 154–159.

Flood, M. M. (1956). The traveling-salesman problem.

Oper. Res.

Holland, J. (1962). Outline for a logical theory of adaptive

systems. J. of the Association for Computing.

I.M.Oliver, D.J.Smith, J. H. (1987). Proceedings of the sec-

ond international conference , lawrence erolaum, new

jersy. A Study of Permutation Crossover Operator on

the TSP, pages 224–230.

Ishikawa, S., Misawa, H., Kubota, R., Tokiwa, T., Horio,

K., and Yamakawa, T. (2011). Multi-space competi-

tive dga for model selection and its application to lo-

calization of multiple signal sources. Journal of Ad-

vanced Computational Intelligence Intelligent Infor-

matics, 15(9):1320–1328.

Laporte, G. (1992). The vehicle routing problem: An

overview of exact and approximate algorithms. Eu-

ropean Journal of Operational Research, 4:61–75.

Nallusamy, R., Duraiswamy, K., Dhanalaksmi, R., and

Parthiban, P. (2009). Optimization of multiple vehicle

routing. International Journal of Engineering Science

and Technology, 1(3):129–135.

Potter, T. and Bossomaier, T. (1995). Solving vehicle rout-

ing problems with genetic algorithms. IEEE interna-

tional conference on evolutionary computing.

Sofge, D., Schultz, A., and Jong, K. D. (2002). Evolution-

ary computational approaches to solving the multiple

traveling salesman problem using a neighborhood at-

tractor schema. Lecture notes in computer science,

2279:51–60.

Tanese, R. (1989). Distributed genetic algorithm. Proc. of

the Int. Conf. on Genetic Algorithms, pages 434–439.

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