HOPX Crossover Operator for the Fixed Charge Logistic Model with Priority Based Encoding

International Journal of Electrical and Computer Engineering (IJECE)
Vol.8, No.6, December2018, pp. 5351~5358
ISSN: 2088-8708, DOI: 10.11591/ijece.v8i6.pp5351-5358  5351
Journal homepage: http://iaescore.com/journals/index.php/IJECE
HOPX Crossover Operator for the Fixed Charge Logistic
Model with Priority Based Encoding
Ahmed Lahjouji El Idrissi1
, Chakir Tajani2
Mohamed Sabbane3
1,3
Faculty of Science Meknes, Moulay Ismail University, Morocco
2
Polydisciplinary faculty of Larache, AbdelmalekEssaadi University, Morocco
Article Info ABSTRACT
Article history:
Received Mar 10, 2018
Revised Jul 25, 2018
Accepted Aug 10, 2018
In this paper, we are interested to an important Logistic problem modelised
us optimization problem. It is the fixed charge transportation problem
(FCTP) where the aim is to find the optimal solution which minimizes the
objective function containig two costs, variable costs proportional to the
amount shipped and fixed cost regardless of the quantity transported. To
solve this kind of problem, metaheuristics and evolutionary methods should
be applied. Genetic algorithms (GAs) seem to be one of such hopeful
approaches which is based both on probability operators (Crossover and
mutation) responsible for widen the solution space. The different
characteristics of those operators influence on the performance and the
quality of the genetic algorithm. In order to improve the performance of the
GA to solve the FCTP, we propose a new adapted crossover operator called
HOPX with the priority-based encoding by hybridizing the characteristics of
the two most performent operators, the Order Crossover (OX) and Position-
based crossover (PX). Numerical results are presented and discussed for
several instances showing the performance of the developed approach to
obtain optimal solution in reduced time in comparison to GAs with other
crossover operators.
Keyword:
Genetic algorithm
Logistic model
Priority based encoding
Transportation problem
Copyright © 2018Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Ahmed Lahjouji El Idrissi,
Departement of Mathematics,
Faculty of Science Meknes, Morocco.
Email: idrissila@gmail.com
1. INTRODUCTION
Logistic model are the most important problem that need to be optimized for the smooth operation
of the entire supply chain [1]. It determines the number and type of plants, warehouses and distribution
centers (DCs) to be used. It also establishes distribution channels and the quantity of products to be shipped
from suppliers for each customer. Logistic model covers a wide range of formulations from linear
deterministic models to nonlinear stochastic complexes ones.
For the first formulation called linear logistic model problem or liner transportation problem. It is a
network optimization problem introduced by Hitchcock [2] which consists to minimize the total cost in order
to transport homogeneous products from several sources to several deposits satisfying the limits of supply
and demand. This model can be find in industry, planning, communication network, scheduling,
transportation and attribution. Several search attacks the linear logistics models which can be solved by the
simplex method introduced by George Dantzig in 1947 [3]. Also, it can be solved by approximation methods
such us the method of Russell and the method of Vogel [4], [5].
The second formulation, which is the objective of this study, is the fixed charge logistic model. The
literature around this first case extension is very rich. Hirsch and Danzig in [6] was the first to formulate the
FCTP us extension of the liner logistic model. Many practical transportation and distribution problems, such
as the minimum cost network flow (transshipment) problem with a fixed-charge for logistics, can be

 ISSN:2088-8708
Int J Elec& Comp Eng, Vol. 8, No. 6, December 2018 : 5351 - 5358
5352
formulated as fixed-charge logistic model. For instance, a fixed cost may be incurred for each shipment
between a given plant and a given customer and a facility of a plant may result in a fixed amount on
investment. The FCTP takes these fixedcharge into account, so that the TP can be considered as an FCTP
with equal fixed costs of zero for all routs.
Balainski in 1961 modified the FCTP to a linear integer problem [7]; he observed that there is an
optimal solution for the modified version of FCTP. Adlakha in [8] proposed a method which consists of two
parts; it gets the best initial solution in the first part and uses techniques to improve this solution and to check
its optimality.
The genetic algorithm (GA) is an evolutionary algorithm that represents a famous metaheuristic
proposed by Jhon Holland in 1975 [9] which is inspired by biological mechanisms such as Mendel's laws and
the theory of evolution proposed by Charles Darwin. It uses the same vocabulary as in biology and classical
genetics; so we speak of gene, chromosome, population [10]-[13].
The genetic algorithm has been used to solve many combinatorial problems including FCTP [14].
Its main advantage is that it allows a good combination between the exploitation of solutions and the
exploration of the research space. This is established as a function of the GA parameters respectively.
However, its disadvantage lies in two points; a computational time large enough to be able to converge
towards the optimal solution and the convergence that is a big problem for GAs. In addition, the different
characteristics of the genetic operators influence on the performance and the quality of the GA. For this
reason and in order to improve the performance of the GA to solve the FCTP, we propose a new adapted
crossover operator called HOPX with the priority-based encoding by hybridizing the characteristic of the two
most performent operators, the Order Crossover (OX) and Position-based crossover (PX).
This paper is organized as follows: The second section gives an overview of the formulation of the
fixed charge logistic model. The third part presents the GA as a method of resolution and these parameters
which have influenced the quality of the results. In addition to the proposed hybrid crossover operator called
HOPX. The fourth section deals with the development and implementation of the GA with the new operator
where several numerical results are presented showing his performance in comparison
with variants operators.
2. LOGISTICS MODEL DESCRIPTION AND FORMULATION
Logistic is a profound factor on the added value of each business. Examines each of this activity is
done at several levels: logistic engineering, technical publications, procurement... Suggests to each element a
separate responsibility towards the product arrived at the customer or the consumer in the complex
network [15].
Considering the following graph G = (N, A), which consists of a finite set of nodes N = {1,2, . . . ,n}
and a set of directed arcs A = {(i, j), (k, l), . . . , (s, t)}, which joins pairs of nodes in N. It is graphically
illustrated in Figure 1.
Figure 1. Transportation plan

Int J Elec& Comp Eng ISSN: 2088-8708 
HOPX Crossover Operator for the Fixed Charge Logistic Model with Priority ... (Ahmed Lahjouji El Idrissi)
5353
2.1. Fixed-charge Logistic Model
The FCTP is much more difficult to solve due to the presence of fixed costs, which cause
discontinuities in the objective function. In the fixed-charge logistic model, two types of costs are considered
simultaneously when the best course of action is selected: (1) variable costs proportional to the activity level;
and (2) fixed costs. The FCTP seeks the determination of a minimum cost transportation plan for a
homogeneous commodity from a number of plants to a number of customers. It requires the specification of
the level of supply at each plant, the amount of demand at each customer, and the transportation cost and
fixed cost from each plant to each customer. The goal is to allocate the supply available at each plant so as to
optimize a criterion while satisfying the demand at each customer. The usual objective function is to
minimize the total variable cost and fixed costs from the allocation. It is one of the simplest combinatorial
problems involving constraints. The fixedcharge logistic model with I plants and J customers can be
formulated as follows:
Min Z = ∑ ∑ (𝐶𝑖𝑗 𝑥𝑖𝑗 + 𝑓𝑖𝑗 𝑦𝑖𝑗)𝑛
𝑗=1
𝑚
𝑖=1 (1)
∑ 𝑥 𝛾
𝑛
𝑗=1 = 𝑆 𝑖 = 1,2, … . , 𝑚 (2)
∑ 𝑥𝑖𝑗
𝑚
𝑖=1 = 𝑆 𝑖 = 1,2, … . 𝑛 (3)
𝑋𝑖𝑗 ≥ 0 𝑖 = 1,2, … , 𝑚𝑎𝑛𝑑 𝑗 = 1,2, … , 𝑛
𝑦𝑖𝑗 = 0 𝑖𝑓 𝑥𝑖𝑗 = 0
𝑦𝑖𝑗 = 1 𝑖𝑓 𝑥𝑖𝑗 > 0
xij : Unknown quantity to be transported from source i to destination j;
cij : Variable transportation cost from source i to destination j;
fij : Fixed transportation cost associated with road (i,j);
yij : A binary variable which is 1 if xij> 0 and 0 if xij= 0;
Si : Amount of supply at source i;
Dj : Amount of demand at destination j.
3. PRIORITY-BASED GENETIC ALGORITHM FOR FIXED CHARGE LOGISTIC MODEL
In order to apply the genetic algorithm (GA), it is necessary to chose and adaptate the representation
method for the solution of the problem. There are several methods of representation that are used to solve
logistics models using GA, there is the matrix representation [15], representation by the number of prüfer
[16], and there is also the priority based representation which is a new and more adapted for encoding and
decoding the logistics models. It is first used by Gen and Altiparmak [17] to code and decode a two-stage
transport problem. Then, the GA consists to several steps: Initialization processes (Generation of an
population of solutions), evaluation, selection, crossover and mutation operators to create a new
population of solutions.
3.1. Encoding chromosome
For the priority based representation, the solution (chromosome) is represented by an integral chain
of length equaling the number of sources plus the number of clients. Each gene in this chromosome indicates
the identification of a node (number).
In this representation, a gene in a chromosome contains two types of information:
a) The position of a gene to represent the nodes (source / destination)
b) The value of a gene which represents the priority of the node for the construction of a
transport tree.
A chromosome consists of the priorities of sources and deposits to obtain a transport tree, its length
is equal to the total number of sources m and deposits n, ie. (m+n). Each chromosome can be reconstructed in
a random way and there is no need for a correction algorithm after the generation of a population [18].
3.2. Selection and evaluation method
Selection and evaluation are genetic processes to evaluate product solutions and compare them with
existing chromosomes in order to choose the best to preserve them in memory. Therefore; the evaluation

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function will select or refuse an individual to retain only those individuals with the best cost according to the
current population. In practice there are several types of selection applied in genetic algorithms, especially
we employ the elitist mechanism; In fact, the best solution of the current population is selected and preserved
in memory [19].
Figure 2. Priority-based chromosomes and transportation trees
3.3. Adapted mutation
Mutation operator operates by exchanging information within a chromosome. However, instead of
using this operator between two parents, we use between two segments of a parent. We have chosen a swap
mutation operator that is proposes by (Michalewicz, 1992) which permutes the values of two randomly
selected positions for each chromosome to reduce the risk of reproducing a chromosome with the same
solution [20], [21]. The procedure of swap mutation illustrated below works as follows:
Procedure of the SWAP mutation
Figure 3. Example of the SWAP mutation operator
3.4. Proposed crossover operator for FCTP
The crossover operators are genetic processes. They work on two different parents (chromosomes)
by combining the characteristics of the two chromosomes, they consist in applying procedures with a certain
probability called the crossover rate [Pc ∈ (0 , 1)] on the individuals selected to give birth to one or more
(usually two) offspring. In this context, in order to ameliorate the performance of the GA to solve the fixed
charge logistic model, we have chosen the two more peformante crossover operators (OX and PX) following
a comparative study carried out for several ones [18]. Then, we proposed an hybridization of this two
Input : One parent
Step 1 : Select tow element at random
Step 2 : Swap the element on these positions
Output : One offspring

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operators by highlighting the characteristics of each of them. The procedure of the proposed operator is
presented bellow and the Figure 4. illustrate the procedure as an example.
Procedure of the HOPX operator :
Figure 4. Example of the HOPX operator
4. COMPUTATIONAL RESULTS
The main objectives of this work are to improve the behavior of the genetic algorithm with respect
to the crossover operators because they have an influence on the quality and performance of these algorithms
and to establish to what extent the proposed algorithm solves the different types of logistic problems
compared to the already used methods.
In this section, we have applied the developed HOPX operator to Logistic model with fixed cost in
order to compare the results with those obtained using other adapted operators namely OPEX, OX, PX. In
this study, we have chosen six instances 4x5, 5x10, 10x10, 10x20, 20x30, 30x50 already used in several
articles [22], knowing that the optimal solution is known for small instances. Therefore, we have changed the
genetic algorithm parameters, such as the number of iterations to see the influence of the latter on the results
and on the quality of the solutions. This algorithm is coded in JAVA language in the NetBeans 8.0.2 IDE.
Table 1. presents the simulation results obtained from 20 times by the genetic algorithm based on
each operator used to solve the logistic problem with fixed cost. For the results displayed in this table, we
notice that the genetic algorithm with the HOPX operator allowed obtaining the optimal solution known for
small instances as GAs with other crossover operators. However, for larger instances a better optimal
solution is achieved with an improvement in computation time. Indeed, for small instances, 4*5 and 5*10, the
obtained results show that GA with all crossover operators used allow to obtain the best chromosome
satisfying the optimal solution. However, for problems with large size, with the proposed crossover operator
HOPX, we are optimizing the objective function with a slight precision in terms of optimal solution. Then,
the latter is more advantageous to the level of number of iterations with a reduced execution time. This
means that HOPX is more appropriate than other operators that have already used to solve the fixed charge
logistic model see Figure 5. and Figure 6.
Input: Tow parents;
Step 1 :Select 1/3 substring from one parent at random;
Step 2 :Select 1/4 substring and 1/4 of set of positions from same parent at random.
Step 3 : Produce a proto-child by copying the nodes on these positions into the corresponding positions
of it;
Step 4 : Delete the nodes which are already selected from the second parent;
The resulted sequence of nodes contains the nodes of the proto-child needs;
Step 5 : Place the nodes into the unfixed positions of the proto-child from left to right according to the
order of the sequence to produce one offspring;
Output: Tow offspring.

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It should be noted that the genetic algorithm with the developed crossover operator is also applied to
the linear logistic model which also shows its performance with respect to the GAs with the other
mentioned operators.
Table 1. Best and average results by different operator for the test problems of Fixed-charge Logistic Model
Problem Parameter OPEX OX PX HOPX
Size m x n Popsize Maxgen Best Avrg Best Avrg Best Avrg Best Avrg
4x5 20 300 9291 9291 9291 9291 9291 9291 9291 9291
30 500 9291 9291 9291 9291 9291 9291 9291 9291
5x10 20 300 12718 12751 12718 12751 12718 12751 12718 12718
30 500 12718 12734 12718 12734 12718 12734 12718 12718
10x10 20 500 13987 14074 13987 14139 14065 14133 13987 14047
30 700 13934 14074 13987 14074 13934 14065 13934 13987
10x20 20 500 22258 22484 22376 22531 22198 22834 22150 22258
30 700 22095 22484 22095 22198 22095 22532 22095 22198
20x30 20 500 32840 34119 33192 34581 32683 34414 32492 33142
30 700 32526 33917 33917 33236 32492 33234 32471 32936
30x50 20 700 55611 56399 56399 56705 55450 56007 55269 55450
30 1000 55143 55912 55912 55912 55106 55407 54114 55106
Figure 5. Average computation time for different operators for fixed charge Logistic Models
Table 2. gives the best chromosome for the different instances which represent the solution for the
fixed charge logistic problem.
Table 2. Best chromosome for the test problems
Instance Best chromosome
4x5 1 6 8 2 4 3 9 7 5
5x10 11 1 7 3 13 14 2 5 6 12 4 10 9 15 8
10x10 8 5 14 12 19 2 1 6 15 20 9 10 7 11 3 16 4 13 18 17
10x20 24 5 21 10 9 17 13 19 12 6 29 1 28 3 18 20 15 4 7 11
25 23 26 27 16 30 22 2 8 14
20x30 5 42 13 31 33 30 23 41 48 9 27 50 11 29 40 16 20 25
3 22 28 43 32 44 38 6 26 37 49 2 24 36 7 18 35 14 8
1 46 34 17 10 15 4 12 21 45 39 47 19
30x50 59 50 66 47 8 29 2 51 74 56 35 65 27 5 28 68 25 21
19 40 38 76 53 46 71 42 14 55 61 7 41 22 6 13 72 36
15 11 33 79 57 4 49 30 43 73 60 52 44 58 26 48 64
23 9 75 20 67 24 70 32 3 62 39 10 45 54 12 17 78 16
1 69 63 77 18 34 37 80 31

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Figure 6. Optimum solution with different crossover operators OPEX, OX, PX, HOPX with different
instances for fixed charge Logistic Models
5. CONCLUSION
In this work, we are interested in solving logistics models which are classified as NP-complet
combinatorial problem called fixed charge transportation problem (FCTP). Its resolution by exact methods is
very difficult, consequently the metaheuristics can be exploited such us genetic algorithms. Then, we are
interested in improving the performance of GAs through the crossover operator that has a great effect. After
presenting a comparison with three crossover operators (OX, OPEX, and PX) adapted to our problem with
priority baseb encoding, we proposed a new hybrid operator with the two operators OX and PX that we
called (HOPX). Numerical results was developed supporting the conclusion that the proposed HOPX
operator is more efficient than the other presentedoperators either for the optimal solution level or the
execution time, especially for larger instances.

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