An iterative goal programming approach for solving fuzzy multiobjective integer linear programming problems

Applied Mathematics and Computation 170 (2005) 216–225

www.elsevier.com/locate/amc

An iterative goal programming approachfor solving fuzzy multiobjective integer

linear programming problems

Omar M. Saad

Department of Mathematics, College of Science, Qatar University, P.O. Box 2713, Doha, Qatar

Abstract

This paper presents an iterative goal programming approach for solving fuzzy mul-

tiobjective integer linear programming problems. These problems involve fuzzy param-

eters in the right-hand side of the constraints. The concept of a-level set of these fuzzyparameters is introduced with the definition of their membership function. A solution

algorithm is described in sequential steps to solve the formulated model. The suggested

approach in this paper is mainly based on the iterative goal programming technique

together with Gomory cuts. An illustrative numerical example is given to clarify the the-

ory and the solution algorithm.

� 2005 Elsevier Inc. All rights reserved.

Keywords: Goal programming; Multiobjective integer linear programming; Fuzzy parameters;

Gomory cuts

1. Introduction

In literature there are several approaches available to deal with multiobjec-

tive integer linear programming problems. Goal programming is one powerful

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2004.11.026

E-mail address: [email protected]

mailto:[email protected]

O.M. Saad / Appl. Math. Comput. 170 (2005) 216–225 217

approach that has been proposed for the modeling, analysis and solution of

multiobjective optimization problems. Previous results on the use of a wide

variety of goal programming models to solve these problems are reported in

[2,6–8,13–15,18].

Effective integer goal programming methods for all-integer, mixed-integer

and zero–one multiobjective problems have been introduced by Lee andMorris [10]. These methods were based on the cutting-plane algorithm,

branch-and-bound method, and implicit enumeration technique.

In Ignizio [6], a hybrid approach combining generalized goal programming

and generalized networks has been presented for the modeling of certain large-

scale multiobjective integer programming problems.

Hughes and Saad [5] discussed stability of efficient solutions in the decision

space for parametric multiobjective integer linear programming problems via

goal programming tools. The approach presented in [5] depends on Balinskialgorithm [1] together with the iterative approach suggested by Dauer and

Krueger [3] with the help of Gomory cuts [9,16]. Later on, Osman et al. [11]

investigated stability of efficient solutions for zero–one multiobjective linear

programming problems using goal programming.

More recently, Saad and Sharif [12] presented an iterative goal program-

ming approach for solving multiobjective integer linear programming prob-

lems. The definition of, and an algorithm to determine the stability set of the

first kind for these problems have been characterized. The present paper isan extension of the study introduced in [12] to cover the problem investigated

previously, but with fuzzy parameters in the right-hand side of the constraints.

To our knowledge, this problem has not yet studied before in the integer case

and under fuzzy environment.

The roots of the present paper lie in the following sections: Section 2

presents the formulation of multiobjective integer linear programming pro-

blem involving fuzzy parameters in the right-hand side of the constraints

with the associated fuzzy integer linear goal programming model. Section 3contains some basic definitions in the fuzzy theory together with the state-

ment of the nonfuzzy version of the formulated goal programming model. In

Section 4, a solution algorithm is suggested and described in sequential steps

to solve such programs. Section 5 provides a numerical example to illustrate

the theory and the solution algorithm. Finally, Section 6 contains the

conclusions.

2. Problem formulation

In this section, we begin by introducing the following multiobjective integer

linear programming problem with fuzzy parameters in the right-hand side of

the constraints:

218 O.M. Saad / Appl. Math. Comput. 170 (2005) 216–225

ðMOILPPÞ~m : max ZðxÞ; ð1:aÞsubject to

x 2 X ð~mÞ; ð1:bÞ

where

X ð~mÞ ¼ x 2 Rn=grðxÞ 6 ~mr; r ¼ 1; 2; . . . ;m; x P 0 and integerf g;

and Z:Rn ! Rk, Z(x) = (z1(x),z2(x), . . .,zk(x)) is a vector-valued criterionwith zi(x), i = 1,2, . . .,k, are real-valued linear objective functions, ~m ¼ð~m1;~m2; . . . ;~mmÞT is a vector of fuzzy parameters and Rn is the set of all ordered

n-tuples of real numbers. Furthermore, the constraints functions gr(x),

r = 1,2, . . .,m, are assumed to be linear.

Now, going back to problem ðMOILPPÞ~m (1.a) and (1.b), we can write an

associated fuzzy integer linear goal programming model ðILGPÞ~m consisting

of k goals and having ~m 2 Rm a vector of fuzzy parameters in the right-hand side

of the constraints. This model may be expressed as:

ðILGPÞ~m : Achieve : z1ðxÞ ¼ h1; ð2:aÞz2ðxÞ ¼ h2;

..

.

zkðxÞ ¼ hk;

and the constraints are given by :

grðxÞ 6 ~mr; r ¼ 1; 2; . . . ;m; ð2:bÞx P 0 and integer; ð2:cÞ

where h1,h2, . . .,hk are scalars and represent the aspiration levels associated

with the objectives z1(x),z2(x), . . .,zk(x), respectively.

3. Fuzzy concepts

The fuzzy theory has been advanced by L.A. Zadeh at the University of Cali-fornia in 1965. This theory proposes a mathematical technique for dealing with

imprecise concepts and problems that have many possible solutions. The con-

cept of fuzzy mathematical programming on a general level was first proposed

by Tanaka et al. (1974) in the framework of the fuzzy decision of Zadeh and

Bellman [17].

For the development that follows, we shall introduce some definitions con-

cerning trapeziodal fuzzy numbers and their membership functions, come from

[4], and that will be used throughout this paper. It should be noted that anequivalent approach can be used in the triangular fuzzy numbers case.


Definition 1. A real fuzzy number ~a is a fuzzy subset from the real line R with

membership function l~aðaÞ that satisfies the following assumptions:

(1) l~aðaÞ is a continuous mapping from R to the closed interval [0,1],

(2) l~aðaÞ ¼ 0 8a 2 ð1; a1�,(3) l~aðaÞ is strictly increasing and continuous on [a1,a2],(4) l~aðaÞ ¼ 1 8a 2 ½a2; a3�,(5) l~aðaÞ is strictly decreasing and continuous on [a3,a4],

(6) l~aðaÞ ¼ 0 8a 2 ½a4;þ1Þ,

where a1, a2, a3, a4 are real numbers and the fuzzy number ~a is denoted by~a ¼ ½a1; a2; a3; a4�.

Definition 2. The fuzzy number ~a is a trapezoidal number, denoted by[a1,a2,a3,a4], and its membership function l~aðaÞ is given by (see Fig. 1)

l~aðaÞ ¼

0; a 6 a1;

1 aa2a1a2

� �2

; a1 6 a 6 a2;

1; a2 6 a 6 a3;

1 aa3a4a3

� �2

; a3 6 a 6 a4;

0; otherwise:

8>>>>>>>><>>>>>>>>:

Definition 3. The a-level set of the fuzzy number ~a is defined as the ordinary

set Lað~aÞ for which the degree of their membership function exceeds the level

a 2 [0, 1]:

∼µ a (a)

1

0 a1 a2 a3 a 4 a

Fig. 1. Membership function of a fuzzy number ~a.


Lað~aÞ ¼ a 2 Rjl~aðaÞ P af g:For a certain degree a� 2 [0,1] with the corresponding a-level set of the fuzzynumbers ~mr, problem ðILGPÞ~m can be understood as the following nonfuzzy lin-ear goal programming model written as:

ðILGPÞm : Achieve : z1ðxÞ ¼ h1; ð3:aÞ

z2ðxÞ ¼ h2;

..

.

zkðxÞ ¼ hk;

subject to

grðxÞ 6 mr; r ¼ 1; 2; . . . ;m; ð3:bÞ

mr 2 Lað~mrÞ; r ¼ 1; 2; . . . ;m; ð3:cÞ

x P 0 and integer; ð3:dÞ

where Lað~mrÞ is the a-level set of the fuzzy parameters ~mr; r ¼ 1; 2; . . . ;m.We now rewrite problem (3.a) and (3.b) above in the following equivalent

form:

ðILGPÞm : Achieve : z1ðxÞ ¼ h1; ð4:aÞz2ðxÞ ¼ h2;

..

.

zkðxÞ ¼ hk;

subject to

grðxÞ 6 mr; r ¼ 1; 2; . . . ;m; ð4:bÞ

nð0Þr 6 mr 6 N ð0Þr ; r ¼ 1; 2; . . . ;m; ð4:cÞ

x P 0 and integer: ð4:dÞ

It should be noted that the constraint mr 6 Lað~mrÞ; r ¼ 1; 2; . . . ;m, has been re-

placed by the equivalent constraint nð0Þr 6 mr 6 N ð0Þr ; r ¼ 1; 2; . . . ;m, where nð0Þr

and N ð0Þr are lower and upper bounds on mr.

Taking into account restrictions grðxÞ 6 m�r ; r ¼ 1; 2; . . . ;m, and for the pur-pose for solving the integer linear goal program (4.a)–(4.d) at mr ¼m�r ¼ N ð0Þ

r ; r ¼ 1; 2; . . . ;m, for a certain degree a = a� 2 [0, 1], we use the itera-

tive approach developed in [3] together with the Gromory cuts shown in

[9,16]. First, we solve the following integer linear optimization problem associ-ated to the first goal, viz:


P 1ðm�r Þ : minD1 ¼ d1 þ dþ

1 ; ð5:aÞsubject to

z1ðxÞ þ d1 dþ

1 ¼ h1; ð5:bÞgrðxÞ 6 m�r ; r ¼ 1; 2; . . . ;m; ð5:cÞd1 P 0; dþ

1 P 0; x P 0 and integer; ð5:dÞ

where d1 and dþ

1 are the underattainment and the overattainment, respectively,

of the first goal where d1 � dþ

1 ¼ 0.

Suppose this problem has integer optimal value D�1 ¼ d�

1 þ dþ�1 with at least

one value d�1 or dþ�

1 nonzero, i.e., d�1 � dþ�

1 ¼ 0.

Now, the attainment problem for goal 2 is equivalent to the integer optimi-

zation P 2ðm�r Þ, where


2 ; ð6:aÞsubject to

z2ðxÞ þ d2 dþ

2 ¼ h2; ð6:bÞz1ðxÞ þ d

1 dþ1 ¼ h1; ð6:cÞ

d1 þ dþ

1 ¼ D�1; ð6:dÞ

grðxÞ 6 m�r ; r ¼ 1; 2; . . . ;m; ð6:eÞdi P 0; dþ

i P 0; x P 0 and integer; i ¼ 1; 2: ð6:fÞ

Letting D�2 ¼ d�

2 þ dþ�2 denotes the integer optimal value of P 2ðm�r Þ, we can pro-

ceed to goal 3.

The general attainment problem Pjðm�r Þ for goal j is written as:

P jðm�r Þ : minDj ¼ dj þ dþ

j ; ð7:aÞsubject to

ziðxÞ þ di dþ

i ¼ hi; 1 6 i 6 j; ð7:bÞdi þ dþ

i ¼ D�i ; 1 6 i 6 j 1; ð7:cÞ

grðxÞ 6 m�r ; r ¼ 1; 2; . . . ;m; ð7:dÞdi P 0; dþ

i P 0; x P 0 and integer; 1 6 i 6 j; ð7:eÞ

where di and dþ

i are the underattainment and the overattainment, respectively,

of the ith goal level and di � dþ

i ¼ 0.The integer objective value of P jðm�r Þ, D�

j , is the maximum degree of attain-

ment for goal j subject to the maximum attainment of goals 1,2, . . ., j 1. No-

tice that D�j ¼ 0 if and only if goal j is attained.

Let x� be the optimal integer solution of the integer attainment problem

Pkðm�r Þ associated with the minimum D�k , then the solution of the integer goal

program (ILGP)m (4.a)–(4.d) is given by x� with a = a� 2 [0, 1].


4. Solution algorithm

In this section we develop a solution algorithm to solve the integer linear

goal program (4.a)–(4.d). The outline of this algorithm is as follows:

Step 0. Set a = a� = 0.Step 1. Determine the points (a1,a2,a3,a4) for each fuzzy parameter

~mr; r ¼ 1; 2; . . . ;m, in program ðILGPÞ~m (2.a)–(2.c) with the corre-

sponding membership function l~að~vÞ P a� for the vector of fuzzy

parameters ~m ¼ ð~m1;~m2; . . . ;~mmÞT.Step 2. Convert program ðILGPÞ~m (2.a)–(2.c) into the nonfuzzy integer linear

goal program ðILGPÞ~m (4.a)–(4.d).Step 3. Choose mr ¼ m�r ¼ N ð0Þ

r ; r ¼ 1; 2; . . . ;m, and solve problem P 1ðm�r Þ (5.a)–(5.d) using Gomory�s cutting-plane method [9,16] and obtain D�

1.Step 4. Set j = 2.

Step 5. Using D�1;D

�2; . . . ;D

�j1, solve P jðm�r Þ using the same method used in Step

3.

Let D�j denotes the minimum.

Step 6. If j 5 k, set j = j + 1 and go to Step 5. Otherwise, go to Step 7.

Step 7. Let x� denotes the optimal integer solution of problem Pkðm�r Þ associ-ated with the minimum D�

k .

Step 8. Set a = (a� + step) 2 [0,1] and go to Step 1.Step 9. Repeat again the above procedure until the interval [0,1] is fully

exhausted. Then stop.

5. An illustrative example

Consider the following integer linear goal program involving fuzzy param-eters ð~m1;~m2;~m3Þ in the right-hand side of the constraints:

ðILGPÞ~m : goal 1 : Achieve 2x1 þ x2 ¼ h1;

goal 2 : Achieve x1 þ 2x2 ¼ h2;

subject to

x1 þ x2 6 ~m1;

x1 þ x2 6 ~m2;

6x1 þ 2x2 6 ~m3;

x1 P 0; x2 P 0 and integers:

Assume that the membership function corresponding to the fuzzy parame-

ters is in the form:


l~aðaÞ ¼

0; a 6 a1;

1 aa2a1a2

� �2

; a1 6 a 6 a2;

1; a2 6 a 6 a3;

1 aa3a4a3

� �2

; a3 6 a 6 a4;

0; a P a4:

8>>>>>>><>>>>>>>:

where ~a corresponds to each ~mi; i ¼ 1; 2; 3. In addition, we assume also that the

fuzzy numbers are given by the following values:

~m1 ¼ ð2; 4; 6; 8Þ; ~m2 ¼ ð0; 3; 5; 7Þ; ~m3 ¼ ð18; 20; 22; 24Þ:Setting a = a� = 0, then we get:

2 6 ~m1 6 8; 0 6 ~m2 6 7; 18 6 ~m3 6 24:

By choosing m� ¼ ðm�1; m�2; m�3Þ ¼ ð8; 7; 24Þ, then the aspiration levels of the goals

have been found h1 = 10 and h2 = 15, respectively.The integer optimization problem associated with the first goal is:


1 ;

subject to

2x1 þ x2 þ d1 dþ

1 ¼ 10;

x1 þ x2 6 8;

x1 þ x2 6 7;

6x1 þ 2x2 6 24;

d1 P 0; dþ

1 P 0; x1 P 0; x2 P 0 and integers:

The maximum degree of attainment of problem P 1ðm�r Þ is D�1 ¼ 0 with the

optimal integer solution

x1 ¼ ð2; 6Þ and d1 ¼ 0; dþ

1 ¼ 0:

The attainment problem for goal 2 is equivalent to the integer optimization

problem P 2ðm�r Þ whereP 2ðm�r Þ : minD2 ¼ d

2 þ dþ2 ;

subject to

x1 þ 2x2 þ d2 dþ

2 ¼ 15;

2x1 þ x2 þ d1 dþ

1 ¼ 10;

d1 þ dþ

1 ¼ 0;

x1 þ x2 6 8;

x1 þ x2 6 7;

6x1 þ 2x2 6 24;

di P 0; dþ

i P 0; x1 P 0; x2 P 0 and integers; i ¼ 1; 2:


The maximum degree of attainment of goal 2 is D�2 ¼ 1 with the optimal

integer solution

x2 ¼ ð2; 6Þ and d2 ¼ 1; dþ

2 ¼ 0:

Therefore, the optimal integer solution of the original integer linear goal pro-

gram is:

x� ¼ ð2; 6Þ;

D�1 ¼ 0 with d�

1 ¼ 0; dþ�1 ¼ 0;


2 ¼ 1; dþ�2 ¼ 0

with the corresponding used Gomory cut: x2 6 7.

On the other hand, setting a = a� = 1, we get:

4 6 ~m1 6 6; 3 6 ~m2 6 5; 20 6 ~m3 6 22:

Choosing m� ¼ ðm�1; m�2; m�3Þ ¼ ð6; 5; 22Þ, then the optimal integer solution of the

original program has been found:

x� ¼ ð2; 4Þ;


1 ¼ 0; dþ�1 ¼ 0;


2 ¼ 1; dþ�2 ¼ 0

with the corresponding used Gomory cut: 3x1 + x2 6 10.

Remark. It should be noted that a systematic variation of a 2 [0, 1] will yield anew optimal integer solution to the integer linear goal program ðILGPÞ~m.

6. Conclusions

In this paper, an iterative goal programming approach has been proposed

for solving multiobjective integer linear programming problems. These prob-

lems involve fuzzy parameters in the right-hand side of the constraints. Thealgorithm presented in this paper has the advantage of dealing with many inte-

ger linear goal programs by varying the parameters in the right-hand side of the

constraints. Certainly, there are many other points for future research in the

area of fuzzy integer linear goal programs and should be studied. Some of these

points are:

(i) A study is needed to solve fuzzy integer linear goal programs with different

a-level sets of the fuzzy parameters, i.e., the solutions obtained fordifferent values of a should be compared and saved in any step of the

algorithm.


(ii) A solution algorithm is required to treat integer linear goal programs with

fuzzy aspiration levels of the objectives.

(iii) An algorithm should be carried out to solve integer linear goal programs

involving fuzzy parameters in the left-hand side of the constraints.

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An iterative goal programming approach for solving fuzzy multiobjective integer linear programming problems