An Improvement of Genetic Algorithm for Location-routing ... · two questions, depot location and vehicle routing at the same time, calling it location-routing problem (LRP) The LRP

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5464-5471

© Research India Publications. http://www.ripublication.com

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An Improvement of Genetic Algorithm for Location-routing Problem

Apichat Buakla1 and Sirichai Tanratanawong2

School of Logistics and Supply Chain, Naresuan University, Phisanulok, Thailand.

Abstract

This paper proposed an improvement of genetic algorithm for

solving the location-routing problem (LRP). In this study,

tournament selection technique and k-mean clustering are

applied to create customer routing in each depot. The

algorithm is tested on benchmark problems set by data and

compared with former algorithm. The computation results

illustrate that the proposed algorithm can produce optimal

solutions better than comparative method. The findings prove

that the proposed algorithm can solve LRP effectively which

lead to minimize the total cost of network distribution in real

world business.

INTRODUCTION

Nowadays, the logistics costs consume a major proportion of

business operational costs. In Thailand, the average logistics

cost is 14.2% of GDP and 52.8% of total logistics cost is

transportation cost [1] which impacts on goods price increase

[2] [3] [4] and household expenses [5]. These costs can be

condensed extremely by designing an efficient distribution

network of the business supply chain. For example, in the

present the popular distribution network is transportation of

goods from factory to distribution centers or central depots.

After that, some goods which must be transported long haul use

full truckload (FTL) to carry and wait in regional depot. When

transporting goods from regional depot to each customer, the

customer demand should be less than truckload (LTL).

Therefore, the last distribution section must ship around to

many customers or milk run, which collects many goods of

demand in one truck, transports them from regional depot to

fulfil each demand and come back to the same depot as shown

in figure 1.

The regional depot location and suitable vehicle routing for

serving all customers are two important factors, which create

the potential for this distribution network. In the past, designers

found depot locations first and vehicle routing later, not solving

two problems at the same time. This procedure led to higher

distribution costs and management problems in routing.

Because when designers select depot location, first priority has

to constraint and effect in routing design later. To improve the

efficiency in supply chain distribution network, researchers

develop the mathematical model which combines and solves

two questions, depot location and vehicle routing at the same

time, calling it location-routing problem (LRP)

The LRP is one type of geographical problem similar to vehicle

routing problem [6] which consist of location data and their

attributes to both supply side (depot and vehicle data) and

demand side (customer data). In 1989, Salhi and Rand [7]

introduced LRP benefits when compared with former

procedures, after that LRP has been widely studied among

researchers. The high complexity and significance of the

problem have been attracting researchers to study this

discipline extensively. In real world problems, there are many

variants of the LRP incorporating constraints and conditions as

indicated by many review research works [8] [9] [10] e.g. the

Capacitated LRP (CLRP) that the depots and vehicles have

capacity constraints [11] [12] [13] and Two-echelon LRP

(LRP-2E) which compose of three layers (factories,

warehouses/depots and customers) [14] [15][16].

Presently researchers bring many methodologies to solve LRP.

In the earliest period, they suggested the exact method [17] [18]

to solve the problem only in small sized instances. After that,

heuristics [19] [20][21], meta-heuristics [22] [23] [24]and

hybrid methods [25] [26] [27] have been developed to work out

the disadvantages of the existing method. They can solve the

problem efficiently by using less computation time. The hybrid

method is the most recent approach for solving LRP. Due to the

significance of this problem, the researches around the world

have attracted to continually construct new methodologies to

solve this problem. Therefore, this study aims to develop a new

effective methodology for solving LRP which is necessary for

distribution network design in supply chain.

This paper proposes an improvement of genetic algorithm for

location-routing problem. The tournament selection and k-

mean clustering are applied to generate an initialization of the

population. If initial population is sufficiently diversified then it

is possible to choose the best solution for recombination and

reduction of the computational time.

The following contents include: section 2, describes the

location-routing problem (LRP). Section 3, presents the

proposed algorithm in details. The experimental results are

demonstrated in section 4. Finally, the conclusion is illustrated

and discussed in section 5.



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Potential regional depot

Opened regional depotRegional / Central DC

CustomerMilk Run 1

Regional DCFTL

Milk Run 2 FTL

Milk Run 3

FTLCentral DC

FTL

>>

>

>

Figure 1: Modern distribution network

PROBLEM DESCRIPTION

The CLRP is the basic variants of LRP [28] which can be

described as a combination of location and vehicle routing

problem with capacity constraints on depots and vehicles.

Demand side and supply side data of network are given as

follows: the number, location and demand of customers, the

number, location, and capacity of all potential depots, the

vehicle type and size. The problem must be overcome

constraints as follow: each customer demand must be served by

one single vehicle; each route must begin and end at the same

depot and not exceed vehicle capacity and the total load of the

routes assigned to a depot must not exceed depot capacity.

Prins et.al.[29] defines CLRP as follows: it is a weighted and

directed graph G = (V,A,C). V is a set of nodes which

combines with a subset I of m (possible depot locations) and a

subset J of n (customers). Every arc a = (i,j) in the arc set A

have own transportation costs Ca. Each depot site i ∈ I have

capacity Wi and an opening cost Oi. Customer demand dj for

each customer j ∈ J is available. A vehicle set K of capacity Q

is given. When network is operated, each vehicle is used only

on one single route and has a fixed cost F. The number of total

vehicles used or routes performed is a decision variable.

The problem objective is to minimize the total cost of network

by discovering which depots should be opened and which

routes should be arranged. The depot cost is an opening cost Oi

while the total route cost combines with the vehicle fixed cost F

and the arc costs Ca. The mathematical model is defined below:

Mathematical model:

. . . ,i i a ak aki I a Ak K k K a I

z O y C x F x

Subject to:

1 ,akk K a j

x j J

(2)

. , ,j akj J a j

d x Q k K i V

(3)

0 , , ak aka i a i

x x k K i V

(4)

1 ,aka I

x k K

(5)

1 ***** , aka L S

x S S J k K (6)

1 , , ak ak ija i J a j

x x f i I j J k K

(7)

. . ,j ij i ij J

d f W y i I (8)

0,1 , akx a A k K (9)

0,1 iy i I (10)

0,1 , ijf i I j V (11)

The objective functions (1) comprises of all costs described

before. Each customer belongs to one route only and has only

one predecessor which is guaranteed by constraints (2). Vehicle

and depot capacity constraints are specified to inequalities (3)



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and (8). Constraints (4) and (5) confirm each route

continuousness and the origin depot of return. Constraint (6) is

sub-tour elimination constraint. Constraint (7) guarantees that if

a linking route among potential depots and customers is

opened, each customer can be assigned to only one depot.

Finally, all variables are Boolean which be indicated to

constraints (9), (10) and (11).

GENETIC ALGORITHM

Genetic algorithm (GA) is a heuristic search algorithm based

on the evolution. The GA concept is derived from genetic

transformation from older to next generation which consists of

selections, crossovers and mutations. Nowadays, GA is used to

solve optimization which represents an intelligent exploitation

of a random search problem. The original GA consists of the

following 5 components:

Step 1: generate an initialization of the populations NP vector

solutions from randomization.

Step 2: the fitness function/objective functions of the

populations are evaluated, after the initialization of

populations.

Step 3: the procedure for generating next generation of

population includes the following steps:

a) Apply proportional fitness selection (roulette

wheel), select two current populations, P1 and P2.

b) Apply one-point crossover operator to p1 and p2

with crossover rate (Pc) to obtain a child

chromosome C1 and C2.

c) Apply mutation operator to C1 and C2 with

mutation rate (Pm) to produce C1’ and C2’.

d) Add C’ to the successor population.

Step 4: replace the source population with the successor

population.

Step 5: if stopping criteria has not been met, return to Step 2.

THE PROPOSED ALGORITHM

This paper intends to improve the algorithm of the initialization

of the population by using tournament selection. The use of

former genetic algorithm to find the problem solution after

generating population step. The procedure of the proposed

algorithm includes the following steps:

Step 1: Initialization of the population; the structure of initial

population consists of the following components:

1. Generate feasible solutions for opened depot by equation

(12) then, sort the ascending order cost of each solution.

!

0k ! !

q jj j j

n nSolution for k nk n k

(12)

where 1,2,3, ,j n

n the maximum number of depots.

customerj depot

j

weightk

w

The example of general feasible solution, for instance A1

shown in figure 2.

Figure 2: The feasible solution for open depot of instance A1.

2. K-Means algorithm is applied to customer cluster in to the

depot. The number of clusters (k) is the same to the number of

depots in the solution. The k-mean algorithm consists of the

following components: (i) use the coordinates of each depots in

the solution as the central point of cluster. (ii) after that,

grouped customers by the k-means algorithm. The result of k-

means algorithm shown in figure 3.

Figure 3: The result of cluster using k-mean algorithm



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3. The tournament selection is applied to create routing

customer in each depot of solution.

Figure 3: The example of a chromosome from initial

population step.

After generating population fitness, the objective functions of

the populations are evaluated.

Step 2: The selection step, using roulette wheel for random

selectin of two current individuals, P1 and P2.

Step 3: The crossover operation is performed in this step. The

two-point crossover is applied for P1 and P2. After that, the

depot part of the child consists of the left and the right part of

first and second parent.

Figure 4: The example of chromosomes crossover.

Step 4: The mutation applied for the permutation vector VP

is to insert a customer, selected at first position 1rM , at a

new position 2rM , also chosen at random. The new

chromosome is then obtained by swapping customer at

position 1rM and 2rM .

Figure 5: The example of chromosomes mutation.

Following that, the clockwise search process is applied to

move the customer in each vehicle and repeat until all

customers are moved. When clockwise search process is

terminated, all novel solutions which show better results than

the previous solution will be transferred to the next stage. The

figure 6 shows the clockwise process in 31car of depot 3.

Figure 6: Shows clockwise search process for 31car .

Step 5: Replace the source population with the successor

population.

Step 6: If stopping criteria has not been met, return to Step 2.



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Figure 7: The flowchart of the proposed algorithm.

EXPERIMENTAL RESULTS

The performance of the proposed algorithm is implemented in

Matlab language and tested on the 12 instances from Prins et al.

benchmark data set. [30] for the problem on vehicle and depot

capacities. The description of instances which can be classified by

number of depot and vehicles are shown in Table 1.

Table 1: The description of instance

Instance #

Depot

Capacity of

Depot

Capacity of

Vehicle

#

Customer

A1 5 140 70 20

A2 5 300 150 20

A3 5 70,140 150 20

A4 5 70,140 70 20

A5 5 350,420 70 50

A6 5 350,420 150 50

A7 5 350 70 50

A8 5 350 150 50

A9 5 300 150 50

A10 5 350 70 50

A11 5 350,420 70 50

A12 5 350,420 150 50

The computational results of the proposed model are

summarized in comparison with GRASP algorithm results from

Prins [30] who created this benchmark dataset in Table 2 and

shown thoroughly in Table 3. The comparisons demonstrate

percentage of average different result which separated in total

distribution network cost, number of opened depot and used

vehicle, the setup costs of depot (Cd) and the routing costs (Cr).

Table 2: The result comparison with former algorithm

Instance Cost #dep #veh Cd Cr

A1-A4 Average diff.(%) 0.05 - - - 0.09

Better - - - - -

A5-A12 Average diff.(%) -0.57 - - -1.60 -0.51

Better 7/8 - - 1/8 6/8

Tested instances Average diff.(%) -0.36 - - -1.06 -0.31

Better 7/12 - - 1/12 6/12

The minus numbers indicate that the proposed algorithm

produces less number of vehicle used, more effective decisions

in depot location and routing design especially in large sized

instance. Number of instances which are founded better by

results proposed by method are compared with number of

tested instances. The proposed model can reduce total network

cost by 0.36% in 7 instances form 12 tested instances. Figure 8-

11 illustrates the solution for the instance A1, A3, A6 and A9

respectively. The results demonstrate that the proposed

algorithm provides better depot location and vehicle routing

compared with the former algorithm as shown in the table.



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Table 3: The detail result from former algorithm and proposed algorithm

Instance GRASP Purposed % Difference

Cost #dep #veh Cd Cr Cost #dep #veh Cd Cr Cost #dep #veh Cd Cr

A1 55021 3 5 25549 29472 55131 3 5 25549 29582 0.20 - - - 0.37

A2 39104 2 3 15497 23607 39104 2 3 15497 23607 - - - - -

A3 48908 3 5 24196 24712 48908 3 5 24196 24712 - - - - -

A4 37542 2 3 13911 23631 37542 2 3 13911 23631 - - - - -

A5 90632 3 12 29319 59467 90160 3 12 25442 64718 -0.52 - - -13.22 8.83

A6 64761 2 6 15385 49376 63256 2 6 15385 47871 -2.32 - - - -3.05

A7 88786 3 12 29319 64270 88715 3 12 29319 59396 -0.08 - - - -7.58

A8 68042 3 6 29319 38723 67893 3 6 29319 38574 -0.22 - - - -0.38

A9 84055 3 12 19785 64270 84181 3 12 19875 64396 0.15 - - 0.45 0.20

A10 52059 3 6 18763 33296 51992 3 6 18763 33229 -0.13 - - - -0.20

A11 87380 2 12 18961 68419 86203 2 12 18961 67242 -1.35 - - - -1.72

A12 61890 2 6 18961 42929 61830 2 6 18961 42869 -0.10 - - - -0.14

Figure 8: Illustration of the solution for the problem instance

A1 with 20 customers.









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CONCLUSION

This paper proposed an improvement of genetic algorithm for

location routing problem (LRP). In order to improve the

initialized population of genetic algorithm, tournament

selection technique and k-mean custering are applied to create

customer routing in each depot. The testing on 12 instances

shows that the proposed algorithm provides the better results

when comparing with former methodology. The proposed

algorithm can reduce total distribution network cost when

comparing with the former algorithm. The experimental results

confirm that the proposed algorithm is the new alternative

methodology for solving the LRP which can minimize total

cost of distribution network in real world business.

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Documents

An Improvement of Genetic Algorithm for Location-routing ... · two questions, depot location and vehicle routing at the same time, calling it location-routing problem (LRP) The LRP