Advances in Decision Sciences

Advances in Decision Sciences / 2008 / Article

Research Article | Open Access

Volume 2008 |Article ID 760191 | https://doi.org/10.1155/2008/760191

Mohamed El-Amine Chergui, Mustapha Moulaï, "An Exact Method for a Discrete Multiobjective Linear Fractional Optimization", Advances in Decision Sciences, vol. 2008, Article ID 760191, 12 pages, 2008. https://doi.org/10.1155/2008/760191

An Exact Method for a Discrete Multiobjective Linear Fractional Optimization

Academic Editor: Wai-Ki Ching
Received09 Jun 2007
Revised09 Jan 2008
Accepted17 Mar 2008
Published13 Apr 2008

Abstract

Integer linear fractional programming problem with multiple objective (MOILFP) is an important field of research and has not received as much attention as did multiple objective linear fractional programming. In this work, we develop a branch and cut algorithm based on continuous fractional optimization, for generating the whole integer efficient solutions of the MOILFP problem. The basic idea of the computation phase of the algorithm is to optimize one of the fractional objective functions, then generate an integer feasible solution. Using the reduced gradients of the objective functions, an efficient cut is built and a part of the feasible domain not containing efficient solutions is truncated by adding this cut. A sample problem is solved using this algorithm, and the main practical advantages of the algorithm are indicated.

1. Introduction

Fractional programming has been widely reviewed by many authors (Schaible [1], Nagih, and Plateau [2]) and there are entire books and chapters devoted to this subject (Craven [3], Stancu-Minasian [4], Horst et al. [5], and Frenk and Schaible [6]). A bibliography, with 491 entries presented by Stancu-Minasian [7], attracts our attention to the amount of work that has been done in the field in recent years. This bibliography of fractional programming is a continuation of five previous bibliographies by the author [8]. Schaible [1] has published a comprehensive review of the work in fractional programming, outlining some of its major developments. Stancu-Minasian's textbook [4] contains the state-of-the-art theory and practice of fractional programming, allowing the reader to quickly become acquainted with what has been done in the field.

The mathematical optimization problems with a goal function that is a ratio of two linear functions have many applications: in finance (corporate planning, bank balance sheet management), marine transportation, water resources, health care, and so forth. Indeed, in such situations, it is often a question of optimizing a ratio debt/equity, output/employee, actual cost/standard cost, profit/cost, inventory/sales, risk-assets/capital, student/cost, doctor/patient, and so on subject to some constraints [9]. In addition, if the constraints are linear, we obtain the linear fractional programming (LFP) problem.

Different approaches have been proposed in the literature to solve both continuous LFP and integer linear fractional programming (ILFP) problems. These can be divided in studies that have developed solution methods (e.g., [4, 1014]) and those which concentrated on applications (e.g., [4, 6]).

The multiple objective linear fractional programming (MOLFP) problem is one of the most popular models used in multiple criteria decision making. Numerous studies and applications have been reported in the literature in hundreds of books, monographs, articles, and books' chapters. For an overview of these studies and applications, see, for instance, [4, 79, 15, 16, 17, 18, 19, 20, 21, 22], and references therein.

Contrary to the multiple objective linear programming (MOLP) problem, Steuer [9] shows that the efficient solutions set in MOLFP problem is not necessarily closed; some interior points of the feasible solutions set may be efficient, while others are not, and efficient extreme solutions need not all be connected by a path of efficient edges. It becomes difficult to generate the whole efficient solutions. As the efficient set may be too difficult to determine, Kornbluth and Steuer [20] propose an algorithm for MOLFP problem that generates the set of the so-called weakly efficient solutions by means of a simplex-based algorithm. A new technique to optimize a weighted sum of the linear fractional objective functions is proposed by Costa [19]. This technique generates only one nondominated solution of the MOLFP problem associated with a given weight vector. At each stage of the technique, the nondominated domain is divided in two subdomains and each of them is analyzed in order to discard the one not containing the nondominated solutions. The process ends when the remaining domains are so little that the differences among their nondominated solutions are lower than a predefined error.

In this paper, we have proposed a technique for generating the efficient set of the MOLFP problem with integer variables by using all the decision criteria in an active way. This last problem, called MOILFP, is more difficult to solve than the MOLFP problem taking into account the integrity of variables. Indeed, finding all efficient solutions of multiobjective combinatorial optimization problems is, in general, NP-complete [23].

We should like to point out that the MOILFP problem has not received as much attention as did the multiple objective integer linear programming (MOILP) problem, what justified our interest to study this problem.

In [15], a considerable computation is necessary to obtain an optimal integer solution of an ILFP problem in the first stage, since the authors used a branch and bound method (see, e.g., [24]). In our method, we use only the Cambini and Martein's [10] method to obtain an optimal solution for the relaxed ILFP problem and an integer solution is detected by the branching process of the branch and bound method. In addition, a cutting plane is constructed taking into account all the criteria. In this manner, we are able to eliminate not only noninteger solutions of the feasible domain, but also integer solutions which are not efficient. Thus our method avoids to scan all the integer feasible solutions of the problem.

The notations and definitions used throughout this work are presented in Section 2. In Section 3, the algorithm generating all efficient solutions for the MOILFP problem is developed and the main theoretical results are proposed. An illustrative example is given in Section 4 and a computational experience is reported in Section 5. Section 6 provides some concluding remarks.

2. Problem Formulation

The purpose of this paper is to develop an exact method for solving the multiple objective integer linear fractional program (MOILFP):(𝑝){max𝑍1𝑐(𝑥)=1𝑥+𝛼1𝑑1𝑥+𝛽1max𝑍2𝑐(𝑥)=2𝑥+𝛼2𝑑2𝑥+𝛽2max𝑍𝑘𝑐(𝑥)=𝑘𝑥+𝛼𝑘𝑑𝑘𝑥+𝛽𝑘𝑥𝑆,𝑥integer(2.1) where 𝑘2;𝑐𝑖,𝑑𝑖 are 1×𝑛 vectors; 𝛼𝑖,𝛽𝑖 are scalars for each 𝑖{1,2,,𝑘}; 𝑆={𝑥𝑛𝐴𝑥𝑏,𝑥0}; 𝐴 is an 𝑚×𝑛 real matrix; and 𝑏𝑚. Throughout this article, we assume that 𝑆 is a nonempty, compact polyhedron set in 𝑛 and 𝑑𝑖𝑥+𝛽𝑖>0 over 𝑆 for all 𝑖{1,2,,𝑘}.

Many approaches for analyzing and solving the MOLFP problem use the concept of efficiency. A point 𝑥𝑛 is called an efficient solution, or Pareto-optimal solution, for MOLFP problem when 𝑥𝑆, and there exists no point 𝑦𝑆 such that 𝑍𝑖(𝑦)𝑍𝑖(𝑥), for all 𝑖{1,,𝑘} and 𝑍𝑖(𝑦)>𝑍𝑖(𝑥) for at least one 𝑖{1,,𝑘}. Otherwise, 𝑥 is not efficient and the vector 𝑍(𝑦)dominates the vector 𝑍(𝑥), where 𝑍(𝑥)=(𝑍𝑖(𝑥))𝑖=1,,𝑘.

The approach adopted in this work for detecting all integer efficient solutions of problem (𝑃) is based on solving a linear fractional programming problem, at each stage 𝑙:(𝑝1)max𝑍1𝑐(𝑥)=1𝑥+𝛼1𝑑1𝑥+𝛽1,𝑥𝑆𝑙,(2.2) with 𝑆0=𝑆 and without the integrity constraint of variables. Note that in place of 𝑍1, one can similarly consider the problem (𝑃𝑙) with another objective 𝑍𝑖 for any 𝑖{2,,𝑟}.

If the optimal solution of (𝑃𝑙) is integer, it is compared to all of the potentially efficient solutions already found and the set of efficient solutions is actualized. The growth direction of each criterion is determined by using its gradient. The method uses this information to deduce a cut able to delete integer solutions which are not efficient for the problem (𝑃) and determines a new integer solution. In the case where this optimal solution is not integer, two new linear fractional programs are created by using the branching process well known in branch and bound method. Each of them will be solved like the problem (𝑃𝑙).

To this aim—let 𝑥𝑙 be the first integer solution obtained after solving problem (𝑃𝑙) by using, eventually, the branching process—one defines 𝐵𝑙 as the set of indices of basic variables and 𝑁𝑙 as the set of indices of nonbasic variables of 𝑥𝑙. Let 𝛾𝑖𝑗 be the 𝑗th component of the reduced gradient vector 𝛾𝑖 defined by (2.3) for each fixed 𝑖{1,2,𝑘};𝛾𝑖=𝛽𝑖𝑐𝑖𝛼𝑖𝑑𝑖,(2.3)where 𝑐𝑖, 𝑑𝑖, 𝛼𝑖, and 𝛽𝑖 are updated values. Let us note that the gradient vector of 𝑍𝑖 and the corresponding reduced gradient vector 𝛾𝑖 for each fixed index 𝑖,𝑖{1,2,,𝑘}, have the same sign. Thus calculating 𝛾𝑖 is enough to determine the growth direction for each criterion.

In order to give the mathematical expression of the cut, we define the following sets at 𝑥𝑙: 𝐻𝑙={𝑗𝑁𝑙;𝑖1,2,,𝑘𝛾𝑖𝑗>0}{𝑗𝑁𝑙𝛾𝑖𝑗=0,𝑖1,2,,𝑘},(2.4)𝑆𝑙+1={𝑥𝑆𝑙𝑗𝐻𝑙𝑥𝑗1}.(2.5)

An efficient cut is a cut which removes only nonefficient integer solutions.

In Section 3, the approach to solve program (𝑃) is presented.

3. Methodology for Solving MOILFP

In this section, an exact method based on the branching process and using an efficient cut for generating all integer efficient solutions for problem (𝑃) is presented. First of all, the proposed method is presented in detail, the algorithm for solving the multiple objective integer linear fractional programming problems is then described. We finish the section with the theoretical results which prove the convergence of the algorithm.

3.1. Description of the Method

Starting with an optimal solution of an LFP problem, the domain of feasible integer solutions is partitioned into subdomains using the principle of branching to the search for integer solutions. As soon as an integer solution is found in a new domain, it is compared to solutions already found and hence the set of all the potentially efficient solutions is updated. An efficient cut is then added for deleting integer solutions that are not efficient. To construct this cut, the growth directions of the criteria are used. The search for the efficient solutions is made in each subdomain created. A given domain contains no efficient solutions when none criterion can grow. This last is said an explored domain. The search for the efficient solutions is stopped only if all created domains were explored domains.

First, Cambini and Martein's [10] algorithm is used for solving the following continuous linear fractional program:(𝑝){max𝑍1𝑐(𝑥)=1𝑥+𝛼1𝑑1𝑥+𝛽1,𝑥𝑆0.(3.1)

This is based on the concept of optimal level solution. A feasible point 𝑥 is an optimal level solution for the linear fractional program (𝑃0), if 𝑥 is optimal for the linear program: (𝑃𝑥𝑐){max1𝑥+𝛼1,𝑑1𝑥=𝑑1𝑥,𝑥𝑆0.(3.2)

If, in addition, 𝑥 is a vertex of the feasible solutions set 𝑆0,𝑥 is said to be a basic optimal level solution. Obviously, an optimal solution for the linear fractional program (𝑃0) is a basic optimal level solution. According to this, the algorithm generates a finite sequence of basic optimal level solutions, the first one, say 𝑥0, is an optimal solution for the linear program:𝑑min1𝑥+𝛽1,𝑥𝑆0.(3.3)

If 𝑥0 is unique, then it is also a basic optimal level solution for program (𝑃0), otherwise, solve the linear program (𝑃𝑥0) to obtain a basic optimal level solution.

The solution of the program (𝑃0) obtained in a finite sequence of optimal level solutions is optimal if and only if 𝛾1𝑗0 for all 𝑗𝐽10, where𝐽10={𝑗𝑁0𝑑1𝑗>0}.(3.4)Otherwise, there exists an index 𝑗𝐽10 for which 𝛾1𝑗>0. The nonbasic variable 𝑥𝑟,𝑟𝐽10, which must enters the basis is indicated by the index 𝑟 such that𝑐1𝑟𝑑1𝑟=max{𝑐1𝑗𝑑1𝑗,𝑗𝐽10}.(3.5)

The original format of the objectives fractional functions and original structure of the constraints is maintained and the iterations are carried out in an augmented simplex table which includes 𝑚+3𝑘 rows. The first 𝑚 rows correspond to the original constraints, the 𝑚+3(𝑖1)+1 and 𝑚+3(𝑖1)+2 rows correspond to the numerator and denominator of the objective fractional function 𝑍𝑖,𝑖{1,2,,𝑘}, of program (𝑃), respectively, and the 𝑚+3𝑖 row corresponds to the 𝛾𝑖𝑙 vector at step 𝑙.

At each stage of the algorithm, all the rows are modified as usual through the pivot operation when the nonbasic variable 𝑥𝑟,𝑟𝐽10, enters the basis, except the 𝑚+3𝑖 rows, for 𝑖{1,2,,𝑘}, which are modified using the 𝛾𝑖𝑙 formula (2.3).

Each program (𝑃𝑙) corresponds to node 𝑙 in a structured tree. A node 𝑙 of the tree is fathomed if the corresponding program (𝑃𝑙) is not feasible or 𝐻𝑙= (explored domain).

If the optimal solution 𝑥𝑙 of program (𝑃𝑙) is not integer, let 𝑥𝑗 be one component of 𝑥𝑙 such that 𝑥𝑗=𝛼𝑗, where 𝛼𝑗 is a fractional number. The node 𝑙 of the tree is then separated in two nodes which are imposed by the additional constraints 𝑥𝑗𝛼𝑗 and 𝑥𝑗𝛼𝑗+1, where 𝛼𝑗 indicates the greatest integer less than 𝛼𝑗. In each node, the linear fractional program obtained must be solved, until an integer feasible solution is found. In presence of an integer feasible solution, the efficient cut 𝑗𝐻𝑙𝑥𝑗1 is added to the program and the new program is solved using the dual simplex method. The method terminates when all the created nodes are fathomed.

3.2. Algorithm

The algorithm generating the set of all integer efficient solutions of program (𝑃) is presented in the following steps. The nodes in the tree structure are treated according to the backtracking principle.

Step 1. Initialization: 𝑙=0, create the first node with the program (𝑃0). Eff=; (integer-efficient set of program (𝑃)

Step 2. General step: as long as a nonfathomed node exists in the tree, do: choose the node not yet fathomed, having the greatest number 𝑙, solve the corresponding linear fractional program (𝑃𝑙) using the dual simplex method and the Cambini and Martein's method. (Initially, for solving program (𝑃0), only the Cambini and Martein's method is used).
If program (𝑃𝑙) has no feasible solutions, then the corresponding node is fathomed.
Else, let 𝑥𝑙 be an optimal solution. If 𝑥𝑙 is not integer, go to Step 3, else go to Step 4.

Step 3. Branching process (partition of the problem into mutually disjoint and jointly exhaustive sub-problems): choose one coordinate 𝑥𝑗 of 𝑥𝑙 such that 𝑥𝑗=𝛼𝑗, with 𝛼𝑗 fractional number, and separate the actual node 𝑙 of the tree in two nodes 𝑘, 𝑘𝑙+1, and , 𝑙+1, 𝑘.
In the current simplex table, the constraint 𝑥𝑗𝛼𝑗 is added and a new domain is considered in node 𝑘 and similarly, the constraint 𝑥𝑗𝛼𝑗+1 is added to obtain another domain in node . (Each created program must be solved using the same process until an integer feasible solution is found), go to Step 2.

Step 4. Update the set Eff: if 𝑍(𝑥𝑙) is not dominated by 𝑍(𝑥) for all 𝑥Eff, then Eff=Eff{𝑥𝑙}. If there exists 𝑥Eff such that 𝑍(𝑥𝑙) dominates 𝑍(𝑥), then Eff=Eff{𝑥}{𝑥𝑙}.
Construct the efficient cut: determine the sets 𝑁𝑙 and 𝐻𝑙.
If 𝐻𝑙=, then the corresponding node is fathomed. Go to Step 2.
Else, add the efficient cut 𝑗𝐻𝑙𝑥𝑗1 to the program (𝑃𝑙). Go to Step 2.

The following theorems show that the algorithm generates all integer efficient solutions of program (𝑃) in a finite number of stages. Theorem 3.1. Suppose that 𝐻𝑙 at the current integer solution 𝑥𝑙. If 𝑥 is an integer efficient solution in domain 𝑆𝑙{𝑥𝑙}, then 𝑥𝑆𝑙+1.Proof. Let 𝑥 be an integer solution in domain 𝑆𝑙{𝑥𝑙} such that 𝑥𝑆𝑙+1, then 𝑗𝐻𝑙𝑥𝑗=0, that implies 𝑥𝑗=0 for all index 𝑗𝐻𝑙.
From the simplex table corresponding to the optimal solution 𝑥𝑙, the criteria are evaluated by
𝑍𝑖(𝑥)=𝑗𝑁𝑙𝑐𝑖𝑗𝑥𝑗+𝛼𝑖𝑗𝑁𝑙𝑑𝑖𝑗𝑥𝑗+𝛽𝑖for𝑖1,,𝑘,(3.6)where 𝛼𝑖𝛽𝑖=𝑍𝑖𝑥𝑙.(3.7) Then we can write 𝑍𝑖(𝑥)=𝑗𝑁𝑙𝐻𝑙𝑐𝑖𝑗𝑥𝑗+𝛼𝑖𝑗𝑁𝑙𝐻𝑙𝑑𝑖𝑗𝑥𝑗+𝛽𝑖,𝑖1,,𝑘.(3.8)
In the other hand, 𝛾𝑖𝑗=𝛽𝑖𝑐𝑖𝑗𝛼𝑖𝑑𝑖𝑗0, for all index 𝑗𝑁𝑙𝐻𝑙 and 𝛾𝑖𝑗=𝛽𝑖𝑐𝑖𝑗𝛼𝑖𝑑𝑖𝑗<0 for at least one criterion, implies that 𝑐𝑖𝑗𝛼𝑖𝑑𝑖𝑗/𝛽𝑖 for all 𝑗𝑁𝑙𝐻𝑙 because 𝛽𝑖=𝑑𝑖𝑥𝑙+𝛽𝑖>0 for all criterion 𝑖{1,,𝑘}. The decision variables being nonnegative, we obtain 𝑐𝑖𝑗𝑥𝑗(𝛼𝑖𝑑𝑖𝑗/𝛽𝑖)𝑥𝑗 for all 𝑗𝑁𝑙𝐻𝑙 and hence 𝑗𝑁𝑙𝐻𝑙𝑐𝑖𝑗𝑥𝑗𝑗𝑁𝑙𝐻𝑙𝛼𝑖𝑑𝑖𝑗𝛽𝑖𝑥𝑗𝑗𝑁𝑙𝐻𝑙𝑐𝑖𝑗𝑥𝑗+𝛼𝑖𝑗𝑁𝑙𝐻𝑙𝛼𝑖𝑑𝑖𝑗𝛽𝑖𝑥𝑗+𝛼𝑖.(3.9)
For any criterion 𝑍𝑖, 𝑖{1,,𝑘}, the following inequality is obtained 𝑍𝑖(𝑥)=𝑗𝑁𝑙𝐻𝑙𝑐𝑖𝑗𝑥𝑗+𝛼𝑖𝑗𝑁𝑙𝐻𝑙𝑑𝑖𝑗𝑥𝑗+𝛽𝑖𝑍𝑖(𝑥)𝑗𝑁𝑙𝐻𝑙𝛼𝑖𝑑𝑖𝑗/𝛽𝑖𝑥𝑗+𝛼𝑖𝑗𝑁𝑙𝐻𝑙𝑑𝑖𝑗𝑥𝑗+𝛽𝑖𝑍𝑖((𝑥)𝛼𝑖/𝛽𝑖)𝑗𝑁𝑙𝐻𝑙𝑑𝑖𝑗𝑥𝑗+𝛽𝑖𝑗𝑁𝑙𝐻𝑙𝑑𝑖𝑗𝑥𝑗+𝛽𝑖𝑍𝑖(𝑥)𝛼𝑖𝛽𝑖𝑍𝑖(𝑥)𝑍𝑖(𝑥𝑙).(3.10)
Consequently, 𝑍𝑖(𝑥)𝑍𝑖(𝑥𝑙) for all 𝑖{1,,𝑘} and 𝑍𝑖(𝑥)<𝑍𝑖(𝑥𝑙) for at least one index. Hence 𝑍(𝑥𝑙) dominates 𝑍(𝑥) and the solution 𝑥 is not efficient.
Corollary 3.2. Suppose that 𝐻𝑙 at the current integer solution 𝑥𝑙. Then the constraint 𝑗𝐻𝑙𝑥𝑗1 is an efficient cut.Proof. By the above theorem, no efficient solution is deleted when the constraint 𝑗𝐻𝑙𝑥𝑗1 is added. We can say that this is an efficient valid constraint. In the other hand, 𝑥𝑙 does not satisfy this constraint since 𝑥𝑗=0, for all 𝑗𝑁𝑙. We conclude that the constraint is an efficient cut.Proposition 3.3. If 𝐻𝑙= at the current integer solution 𝑥𝑙, then 𝑆𝑙{𝑥𝑙} is an explored domain.Proof. 𝐻𝑙= means that 𝑥𝑙 is an optimal integer solution for all criterion, hence 𝑥𝑙 is an ideal point in the domain 𝑆𝑙 and 𝑆𝑙{𝑥𝑙} does not contain efficient solutions.Theorem 3.4. The described algorithm terminates in a finite number of iterations and generates all the efficient solutions of program (𝑃).Proof. The set 𝑆 of feasible solutions of problem (𝑃), being compact, contains a finite number of integer solutions. Each time an optimal integer solution 𝑥𝑙 is calculated, the efficient cut is added. Thus according to the above theorem and corollary, at least the solution 𝑥𝑙 is eliminated when one studies any subproblem (𝑃𝑘),𝑘>𝑙, but no efficient solution is deleted.

4. An Illustrative Example

The following program (𝑃) is given as an example of multiple objective linear fractional programming (MOLFP) in Kornbluth and Steuer [20]:max𝑍1𝑥(𝑥)=14𝑥2,+3max𝑍2(𝑥)=𝑥1+4𝑥2,+1max𝑍3(𝑥)=𝑥1+𝑥2,subjectto𝑥1+4𝑥20,2𝑥1𝑥28,𝑥10,𝑥20,andintegers.(4.1)

Using the described algorithm, program (𝑃0) is first resolved and the optimal solution is given in the simplex Table 1:


BRhsx3x4

x28/72/71/7
x132/71/74/7

c14/71/74/7
d113/72/71/7
γ13/78/7

c24/71/74/7
d215/72/71/7
γ21/78/7

c324/71/73/7

Since 𝛾1𝑗0 for all 𝑗𝐽10, 𝐽10=𝑁0={3,4}, then the obtained solution (32/7,8/7) is optimal for program (𝑃0), but not integer. Therefore, two branches are possible.

(1)𝑥151/7𝑥34/7𝑥43/7. This is not possible and (1) is fathomed.(2)𝑥141/7𝑥34/7𝑥44/7.

This constraint is added and the dual simplex method is applied. The integer optimal solution of program (𝑃2) is obtained in Table 2.


BRhsx3x5

x211/41/4
x1401
x411/47/4

c1001
d121/41/4
γ102

c2001
d221/41/4
γ202

c331/43/4

𝛾1𝑗0 for all 𝑗𝐽12, 𝐽12=𝑁2={3,5}, then the current solution (4,1) is optimal. Eff={(4,1)}, 𝐻2={5}, and 𝑆3={𝑥𝑆2|𝑥51}.

The constraint 𝑥51 is added to the current simplex table and after pivoting, we obtain Table 3.


BRhsx2x6

x3341
x1301
x4212
x5101

c1101
d1310
γ113

c2101
d2110
γ211

c3311

𝐽13= then, (3,0) is an optimal integer solution and Eff={(4,1),(3,0)}.

Proceeding in this manner, we obtain Table 4.


BRhsx1x10

x6310
x3014/3
x4811
x5410
x7201
x9138
x8011
x2011

c1410
d1311
γ114

c2410
d2111
γ254

c3001

𝐽14= then, (0,0) is an optimal integer solution, Eff={(4,1),(3,0),(0,0)}, 𝑁4={1,10}, 𝐻4={10}, and 𝑆5={𝑥𝑆4|𝑥101}.

The constraint 𝑥101 is added and we obtain Table 5.


Rhsx11x10

x6211
x3111/3
x4712
x5311
x7201
x9235
x8110
x2110
x1111

c1311
d1210
γ114

c2311
d2210
γ254

c3001

The dual is not feasible, then the corresponding node is fathomed.

The algorithm terminates since all nodes are fathomed and the integer efficient solutions set of program (𝑃) is Eff={(4,1),(3,0),(2,0),(1,0),(0,0)}.

5. Computational Results

The computer program was coded in MATLAB 7.0 and run on a 3.40 GHz DELL pentium 4, 1.00 GB RAM. The used software was developed by the authors and was tested on randomly generated problems. We show the results of the computational experiment in Table 6.


(n,m,α), (15,10,33) (20,10,25) (25,5,17) (25,10,17)

Efficient Mean 99,50 204,50 200,45 98,00
Solutions Max 215 324 397 228
Min 4 23 67 17

CPU Mean 38,52 185,96 400,48 306,74
(second) Max 54,39 337,08 673,17 367,11
Min 23,094 143,56 193,87 177,09

Simplex Mean 75837,7 126488,50 521763,45 255726,55
Iterations Max 101207 365750 719105 306665
Min 55851 2717 369414 145324

Efficient Mean 885,3 2341,50 2607,15 979,00
Cuts (EC) Max 1152 2593 3909 1268
Min 451 679 839 440

(EC)/(Cuts) Mean 0,91 0,97 0,84 0,81

The method was tested with 𝑚=5 and 10 constraints with 𝑟=4 objectives functions and 𝑛 variables, 𝑛{15,20,25} randomly generated. The coefficients are uncorrelated integers uniformly distributed in the interval [1,100] for constraints and [1,80] for objective functions about the first two types of treated problems. For the latter two types of problems tested, the integer values of the matrix constraints vary in the interval [1,50] and for those of criteria in [1,30]. The right-hand side value is set to 𝛼% of the sum of the coefficients (integer part) of each constraint, where 𝛼{17,25,33}. With each instance (𝑛,𝑚,𝛼), a series of 20 problems is solved and the whole efficient solution set was generated for all these problems.

The method being exact, it was expected that the iteration number of the simplex method is very large taking into account the fact that, for this type of problems, the number of efficient solutions increases quickly with the data size. In addition, we should like to point out that the ratio EC/cuts tends toward the value one, showing that the number of efficient cuts introduced into the method is very large compared to the full number of cuts and indicating that this type of cuts has a positive impact on the research of the whole of efficient solutions.

6. Conclusion

In this paper, an exact method for generating all efficient solutions for multiple objective integer linear fractional programming problems is presented The method does not require any nonlinear optimization. A linear fractional program is solved using the Cambini and Martein's algorithm in the original format and then by using the well-known concept of branching in integer linear programming, integer solutions are generated. The proposed efficient cut exploits all the criteria in the simplex table, and only the parts of the feasible solutions domain containing efficient solutions are explored. Also it is easy to implement the proposed cut since to obtain integer solution 𝑥𝑘+1 from 𝑥𝑘, one has just to append the cut in the simplex table corresponding to 𝑥𝑘 and carry out pivoting iterations as in an ordinary linear fractional programming problem. The described method solves MOILFP problems in the general case. However, in order to make the algorithm more powerful, the tree structure of the algorithm can be exploited for construction of a parallel algorithm. For large scale problems, the number of efficient solutions can be very high so that it becomes unrealistic to generate them all. In this case, one can choose only the increasing directions of criteria which satisfy a desirable augmentation. This can be made by building the sets 𝐻𝑙 in an interactive way at each step of the algorithm.

Acknowledgments

The authors are grateful to anonymous referees for their substantive comments that improved the content and presentation of the paper. This research has been completely supported by the laboratory LAID3 of the High Education Algerian Ministry.

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Copyright © 2008 Mohamed El-Amine Chergui and Mustapha Moulaï. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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