On stability of proper efficient solutions in multiobjective fractional programming problems under fuzziness

Mathematical and Computer Modelling 45 (2007) 221–231 www.elsevier.com/locate/mcm

On stability of proper efficient solutions in multiobjective fractional programming problems under fuzziness Omar M. Saad ∗ Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, P.O. Box 11795, Cairo, Egypt Received 2 March 2006; received in revised form 19 May 2006; accepted 24 May 2006

Abstract In this paper a solution algorithm to fuzzy multiobjective fractional programming problems is suggested. These problems involve fuzzy parameters usually in the right-hand side of the constraints. In order to defuzzify the problem the concept of α-level set of a fuzzy number is given. For obtaining proper efficient solutions, Geoffrion results are extended to fuzzy multiobjective fractional programming problems. In addition, some stability notions are defined and characterized for the problem of concern. Illustrative numerical examples are presented to clarify the theory and the solution algorithm. c 2006 Elsevier Ltd. All rights reserved.

Keywords: Multiobjective fractional programming; Fuzzy numbers; α-Level set; Efficient solutions; Proper efficiency; Stability

1. Introduction Fractional programming has attracted the attention of many researchers in the past. The main reason for interest in fractional programming stems from the fact that linear fractional objective functions occur frequently as measures of performance in a variety of circumstances such as when satisfying objectives under uncertainty. In some real world decision-making situations, when formulating fractional objectives, some or all of the parameters of the optimization problem are described by fuzzy or stochastic variables. A number of reported studies have been done by the author and others in the field of linear, nonlinear, integer fractional programming and multiobjective fractional programming under fuzziness, for example, [1–9]. The chanceconstrained programming problems have been solved using an accurate linear approximation in [10]. When the variables in an optimization fractional programming problem are assumed to be integers, cutting-plane technique or the branch-and-bound method [11] can be used efficiently to find the optimal solutions to such problems. In previous work, Saad [1] presented a solution procedure for solving linear fractional programs having fuzzy parameters in the right-hand side of the constraints. These parameters have been characterized by fuzzy numbers and the concept of α-optimality has been introduced. On the other hand, Bicriterion integer nonlinear fractional programs

∗ Corresponding address: Qatar University, Department of Mathematics, College of Science, P.O. Box 2713, Doha, Qatar.

E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2006.05.008

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(BINOLFP) involving fuzzy parameters in the objective functions have been studied by Saad and Abdelkader in [2]. Moreover, a solution algorithm has been described to solve the (BINOLFP). Furthermore, a solution algorithm has been proposed by Saad and Abd-Rabo [3] for solving integer linear fractional programs involving random parameters in the right-hand side of the constraints. The suggested procedure in [3] was based upon the chanceconstrained programming technique [10] along with the branch-and-bound method [11]. Recently, Saad and Sharif developed a solution method to solve integer linear fractional programs with chance constraints and having statistically independent random parameters [4]. Pareto-optimality for multiobjective linear fractional programming problems with fuzzy parameter has been discussed by Sakawa and Yano [5]. Programming with linear fractional functions was introduced into the literature by Charnes and Cooper [6]. Results of Geoffrion for efficient and properly efficient solutions of multiobjective programming problems are extended in [7] to multiobjective fractional programming problems. Moreover, multiobjective fractional programming duality theory has been discussed in detail. In [8], Hughes introduced a geometric approach for finding interior efficient solutions in bicriterion linear fractional programming problems. The general conditions for these problems to have interior efficient solutions have been derived. It has been shown that when these conditions are satisfied, the interior efficient solutions form a hyperplane separating the points where the individual objective functions are optimized. Later on, Saad and Hughes [9] considered bicriterion integer linear fractional programs with single-scalar parameter in the objective functions. For such programs, the stability of efficient solutions in the decision space has been studied and the stability set of the first kind has been defined and characterized. Stability becomes more and more attractive in the area of multiobjective mathematical programming. Publications on this topic usually investigate the impact of parameter changes (in the right-hand side or/and in the objective function or/and in the left-hand side or/and the domination structure) on the solution in various models of vector maximization problems. According to our experiences, it is believed that the stability of the solution in fuzzy multiobjective fractional programming problems has not been treated and discussed in the literature before. Stability study allows the decision maker to take on decisions under various changes keeping the solution of the problem under consideration the same in a specified solution domain, which has great importance in management decision making well as outside of it. We list the following frequently occurring fractional objectives in which the stability of solution can be investigated under fuzzy environment: – – – – –

A multi-facility location-queueing problem, Financial planning with multiple fractional goals, An application in computational geometry leading to a convex/convex quadratic problem, Fractional semi-infinite programming problems, Applications in engineering give rise to such a problem when a lower bound for the smallest eigenvalue of an elliptic differential operator is to be determined.

Previous papers in the literature do not discuss directly proper efficiency and stability of solutions in fuzzy multiobjective fractional programming, but they could be interesting reference works and have to be cited here also [12–22]. In the present paper we consider a fuzzy multiobjective fractional programming problem with each component of the objective function having a different denominator. In this setting, use of the Charnes and Cooper transformation [6] seems inhibitive. The problem of concern involves fuzzy parameters in the right-hand side of the constraints. A solution algorithm is described in a finite number of steps to solve the model of the fuzzy multiobjective fractional programming problem. Throughout the paper we distinguish between ≤ and ≺ or ≥ and . ≡

≡

This paper is organized and consists of the following main sections. Section 2 contains the mathematical formulation of the fuzzy multiobjective fractional programming problem and surveys some basic definitions in the fuzzy set theory. In addition, a nonfuzzy version of the formulated model is stated along with the concepts of efficient and proper efficient solutions. Results of Geoffrion for efficient and proper efficient solutions to multiobjective programming problems are extended to fuzzy multiobjective fractional programming problems. Section 3 outlines the solution algorithm in finite steps. Some basic stability notions for the problem of concern are defined and characterized in Section 4. Finally, the paper is concluded in Section 5.

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

2. Problem formulation and the solution concept The problem to be considered in this paper is the following fuzzy multiobjective fractional programming problem:     f p (x) f (x) f 1 (x) f 2 (x) , ,..., (FMOFP): Max = , g(x) g1 (x) g2 (x) g p (x) ˜ = {x ∈ R n /Ax ≤ b, ˜ x ≥ o}, subject to x ∈ X (A, b) where A is an (mxn)-matrix b˜ is an m-vector of fuzzy parameters and we suppose that they are given by fuzzy numbers, estimated from the information provided by the decision maker. Moreover, gi (x)  0, (i = 1, 2, . . . , p) for all x in the feasible region of problem (FMOFP). A fuzzy number is defined differently by different authors. The most frequently used definition belongs to a trapezoidal fuzzy type as follows: Definition 1 ([23]). It is appropriate to recall that a real fuzzy number a˜ is a continuous fuzzy subset from the real line R whose membership function µa˜ (a) is defined by: (1) (2) (3) (4) (5) (6)

A continuous mapping from R to the closed interval [0,1], µa˜ (a) = 0 for all a ∈ (−∞, a1 ], µa˜ (a) is strictly increasing on [a1 , a2 ], µa˜ (a) = 1 for all a ∈ [a2 , a3 ], µa˜ (a) is strictly decreasing on [a3 , a4 ], µa˜ (a) = 0 for all a ∈ [a4 , +∞),

Fig. 1 illustrates the graph of a possible shape of a membership function of a fuzzy number a. ˜ Here, the vector of fuzzy parameters b˜ involved in problem (FMOFP) is a vector of fuzzy numbers whose membership function is µb˜ (b). In what follows, we give the definition of the α-level set or α-cut of the fuzzy vector b˜ = [b˜1 , b˜2 , . . . , b˜m ]. Definition 2 ([23]). The α-level set of the vector of fuzzy parameters b˜ in problem (FMOFP) is defined as the ordinary ˜ for which the degree of its membership function exceeds the level set α ∈ [0, 1], where: set L α (b) ˜ = {b ∈ R m /µ ˜ (b) ≥ α}. L α (b) b For a certain degree α = α ∗ in [0, 1], estimated by the decision maker, the (FMOFP) can be understood as the following nonfuzzy α-multiobjective fractional programming problem (α-MOFP):     f p (x) f 1 (x) f 2 (x) f (x) = , ,..., , (α-MOFP): Max g(x) g1 (x) g2 (x) g p (x) ˜ subject to x ∈ X (A, b) = {x ∈ R n /Ax ≤ b, x ≥ o, b ∈ L α (b)}.

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It should be emphasized here in the (α-MOFP) above that the vector of parameters b is treated as a vector of decision variables rather than constants. Problem (α-MOFP) can be reformulated in the following form: (P): subject to

Max F(x) = (F1 (x), F2 (x), . . . , F p (x)), x ∈ X (A, b),

(x ) where Fi (x) = f i (x) − θi∗ gi (x) and θi∗  0, θr∗ = gfrr (x ∗ ) , (i = 1, 2, . . . , p) are fixed parameters and for their ≡ specification, see [7]. ˜ we introduce the concept of α-efficient Based on Definition 2 of the α-level set of the vector of fuzzy numbers b, solution of problem (P) above as follows: ∗

Definition 3 ([5]). A point x ∗ ∈ X (A, b) is said to be an α-efficient solution of problem (P) if and only if there ˜ such that Fi (x ∗ ) ≤ Fi (x); (i = 1, 2, . . . , p) with strictly inequality holding exists no other x ∈ X (A, b), b ∈ L α (b) for at least one i, where the corresponding values of parameters br∗ (r = 1, 2, . . . , m) are called the α-level optimal parameters. Now, consider λ is a p-dimensional strictly positive fixed vector, then problem (P) can be rewritten again in a problem of scalar single-objective function (Pλ ) in the following form: (Pλ ):

Max

p X

λi Fi (x),

i=1

subject to

x ∈ X (A, b).

Let X (A, b) denote the set of feasible solutions of problem (α-MOFP) or (P) or (Pλ ). We assume that f (x)  0, ≡ g(x)  0, for all x ∈ X (A, b). We further assume that f, −g are concave functions and X (A, b) is a convex set. It follows that F is concave (see [7]). Pp Problem (Pλ ) can be solved at λi = λi∗  0 and i=1 λi∗ = 1 with the corresponding fixed parameters θi = θi∗ , (i = 1, 2, . . . , p) using any available nonlinear programming package, for example, GINO [24], to find the α-optimal solution x ∗ together with the optimal parameters, br∗ (r = 1, 2, . . . , m). It should be noted from [7] that x ∗ is an α-efficient solution to problem (α-MOFP) or problem (P) with the corresponding α-level optimal parameters br∗ (r = 1, 2, . . . , m) if there exists λ∗ ≥ 0 such that x ∗ solves problem (Pλ ) and either one of the following conditions holds: (i) λi = λi∗  0 for all (i = 1, 2, . . . , p). (ii) x ∗ is the unique maximizer of problem (Pλ ). Definition 4 ([25]). Consider the multiobjective programming problem Max φ(x) = (φ1 (x), φ2 (x), . . . , φk (x)), subject to x ∈ S ⊆ R n . We say that x 0 ∈ S is efficient if and only if there exists no x ∈ S such that φ(x 0 ) ≤ φ(x). Definition 5 ([25]). For the multiobjective programming problem in Definition 4, we say that an efficient solution x 0 is properly efficient if and only if for each i and x ∈ S, there exists a positive real number M and a j such that φ j (x 0 ) − φ j (x)  0 whenever φi (x) − φi

(x 0 )

and

φi (x) − φi (x 0 ) ≺ M((φ j (x 0 ) − φ j (x))), ≡

 0.

Before we go any further, the reader is reminded that for multiobjective linear fractional programming, when the emphasis is on finding efficient solutions, there is no general method for finding all the efficient solutions but Choo and Atkins [26] have developed an algorithm, using row parameters, for solving the bicriterion linear fractional programming problem (BLFP). Choo [27] has also shown that if x 0 is an efficient solution to (BLFP) then x 0 is properly efficient [25]. The nonnegativity of θi∗ is needed to establish part (b) of Theorem 1 below.

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Theorem 1. (a) If x ∗ is an α-optimal solution of (Pλ ), then x ∗ is properly an α-efficient for (P). (b) If f and −g are concave and x ∗ is properly an α-efficient for (P), then it is an α-optimal for (Pλ ). To prove Theorem 1 above, the reader is referred to [25]. The following theorem does not require any convexity assumptions, (see [7]). Theorem 2. The point x ∗ ∈ X (A, b) is an α-efficient solution of (α-MOFP) if and only if it is an α-efficient of (P) with F(x ∗ ) = 0. Proof. Suppose x ∗ ∈ X (A, b) is an α-efficient solution of (α-MOFP). Then by Definition 4, there is no x ∈ X (A, b) such that f i (x ∗ ) f i (x) ≤ , ∗ gi (x ) gi (x) Letting θi∗ = such that

f i (x ∗ ) gi (x ∗ )

∀i = 1, . . . , p. for i = 1, . . . , p, we see from the above inequality that there does not exist an x ∈ X (A, b)

0 ≤ f i (x) − θi∗ gi (x) = Fi (x),

∀i = 1, . . . , p.

Since 0 = f i (x ∗ ) − θi∗ gi (x ∗ ) = Fi (x ∗ ), i = 1, . . . , p, we see that there exists no x in X (A, b) such that ¯ Fi (x ∗ ) ≤ Fi (x) for i = 1, . . . , P. Therefore, x ∗ is an α-efficient of (P) with F(x ∗ ) = 0. Conversely, suppose that x ∗ is an α-efficient solution of (P) with F(x ∗ ) = 0 = f (x ∗ ) − θ ∗ g(x ∗ ). That means, by Definition 4, there exists no x ∈ X (A, b) such that 0 = Fi (x ∗ ) ≤ Fi (x) = f i (x) − θi∗ gi (x),

∀i = 1, . . . , p.

That is, there exists no x ∈ X (A, b) such that f i (x) f i (x ∗ ) = θi∗ ≤ , ∗ gi (x ) gi (x)

∀i = 1, . . . , p.

Hence, x ∗ is an α-efficient solution of (α-MOFP).



For the development that follows, we assume that there exist real numbers k  0, K  0 such that k ≺ gi (x) ≺ K for all i. Applying Definition 5 of the proper efficiency to problem (α-MOFP), we note that an α-efficient solution x ∗ of problem (α-MOFP) is properly α-efficient if there exists a real number M  0 such that for each i, we have f i (x)/gi (x) − f i (x ∗ )/gi (x ∗ ) ≺ M[ f j (x ∗ )/g j (x ∗ ) − f j (x)/g j (x)] ≡

for some j such that f j (x)/g j (x) ≺ f j (x ∗ )/g j (x ∗ ) whenever x ∈ X (A, b) and f i (x)/gi (x)  f i (x ∗ )/gi (x ∗ ). Or, rewriting these inequalities slightly differently, we say an α-efficient solution x ∗ of (α-MOFP) is properly α-efficient if there exists a real number M  0 such that for each i, we have [ f i (x)gi (x ∗ ) − f i (x ∗ )gi (x)]/gi (x ∗ ) ≤ M[ f j (x ∗ )g j (x) − f j (x)g j (x ∗ )]/g j (x ∗ ),

(1)

where M = M K /k for some j such that f j (x)g j (x ∗ )g j (x) ≺ 0

(2)

whenever x ∈ X (A, b) and f i (x)gi (x ∗ ) − f i (x ∗ )gi (x)  0.

(3)

To link proper α-efficiency of problem (α-MOFP) and (P), we prove the following theorem. Theorem 3. The point x ∗ ∈ X (A, b) is a properly α-efficient solution of (α-MOFP) if and only if it is a properly α-efficient solution of (P) with F(x ∗ ) = 0.

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Proof. Suppose x ∗ is a properly α-efficient solution of (α-MOFP). Then by Theorem 2, we know it is an α-efficient solution of (P) with F(x ∗ ) = 0. Now x ∗ is a properly α-efficient solution of (P) if there exists a positive real number M such that for each i, Fi (x) − Fi (x ∗ ) ≺ M(F j (x ∗ ) − F j (x))

(4)

≡

for some j such that F j (x) − F j (x ∗ ) ≺ 0,

(5)

whenever x ∈ X (A, b) and Fi (x) − Fi (x ∗ )  0.

(6) θi∗ gi (x)

Or [in view of the fact that Fi (x ∗ ) = 0 for all i and Fi (x) = f i (x) − i = 1, . . . , p], the result holds if and only if there exists an M  0 such for each i,

with

θi∗

= f i (x ∗ )/gi (x ∗ ) for

[ f i (x)gi (x ∗ ) − f i (x ∗ )gi (x)]/gi (x ∗ ) ≺ M[ f j (x ∗ )g j (x) − f j (x)g j (x ∗ )]/g j (x ∗ ) ≡

(7)

for some j such that f j (x)g j (x ∗ ) − f j (x ∗ )g j (x) ≺ 0,

(8)

whenever x ∈ X (A, b) and f i (x)gi (x ∗ ) − f i (x ∗ )gi (x)  0.

(9)

Relations (7)–(9) hold by (1)–(3) with M = M. Conversely, suppose is a properly α-efficient solution of (P) with F(x ∗ ) = 0. Then by Definition 5, relations (4)–(6) hold for some M and each i and x ∈ X (A, b). From this it follows that (7)–(9) hold which are (1)–(3) with M = M.  x∗

3. Solution algorithm In this section, a solution algorithm to solve fuzzy multiobjective fractional programming problem (FMOFP) is described in a series of steps. The suggested algorithm can be summarized in the following manner: Step 0. Start with an initial level set α = α ∗ = 0. Step 1. Determine points (b1 , b2 , b3 , b4 ) for the vector of the fuzzy parameters b˜ in problem (FMOFP) to elicit a membership function µb˜ (b) satisfying assumptions (1)–(6) in Definition 1. Step 2. Convert problem (FMOFP) into its nonfuzzy version (α-MOFP). Step 3. Rewrite problem (α-MOFP) in the form of problem (Pλ ) of single-objective function. Pp Step 4. Choose λi = λi∗  0 and i=1 λi∗ = 1 with fixed values of θi = θi∗ (i = 1, 2, . . . , p) and use the GINO software package [24] to find the α-optimal solution x ∗ of problem (Pλ ). Step 5. Set α = (α ∗ + step) ∈ [0, 1] and go to Step 1. Step 6. Repeat again the above procedure until the interval [0, 1] is fully exhausted. Then, stop. Remark. It should be stated here that in the solution algorithm suggested above, a systematic variation of α-level set among the interval [0,1] will yield another α-optimal solution to problem (Pλ ) and the decision maker must determine this α-level set according to his desire. Example 1. In what follows we provide a numerical example to clarify the solution algorithm suggested above. Let f 1 (x) = 1 − x12 , f 2 (x) = 2,

g1 (x) = 1 + 2x22 ,

g2 (x) = 2 − x2 .

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So F1 (x) =

1 − x12 f 1 (x) , = g1 (x) 1 + 2x22

F2 (x) =

f 2 (x) 2 . = g2 (x) 2 − x2

Consider the following fuzzy bicriterion fractional programming problem (FBFP): Max F(x) = (F1 (x), F2 (x)), ˜ subject to x12 + x22 ≤ b, x1 , x2 ≥ 0, where b˜ is a fuzzy parameter and is characterized by the following fuzzy numbers: b˜ = (0, 1, 3, 5). Assume that the membership function of these fuzzy numbers in the following form:  0, b ≤ b1 ,   2    b − b2   , b1 ≤ b ≤ b2 , 1 −   b1 − b2 b2 ≤ b ≤ b3 , µb˜ (b) = 1,  2    b − b 3   , b3 ≤ b ≤ b4 , 1−   b − b  4 3  0, b ≥ b4 . Let α = 0.19, for example, then we get: 0.1 ≤ b ≤ 4.8. Choosing b = 1, the non-fuzzy α-bicriterion fractional programming problem (α-BFP) becomes: Max F(x) = (F1 (x), F2 (x)), subject to x12 + x22 ≤ 1, x1 , x2 ≥ 0, Observe that the point x ∗ = (0, 0) is an α-efficient solution of problem (α-BFP) since, for each feasible x and then we have: F1 (x) − F1 (x ∗ ) =

1 − x12 1 + 2x22

−1=−

x12 + 2x22 1 + 2x22

≺ 0, ≡

and F2 (x) − F2 (x ∗ ) =

x2 2 −1=  0, 2 − x2 2 − x2 ≡

and there is no other feasible point for which F(x) = (F1 (x), F2 (x)) ≥ (1, 1). We now consider the case when i = 2, j = 1 in the definition of a properly efficient solution and therefore it can be seen that x ∗ = (0, 0) is also a properly α-efficient solution. When x2 F2 (x)  F2 (x ∗ ) we have  0; that is, x2  0. 2 − x2 Then F1 (x ∗ ) − F1 (x) =

x12 + 2x22 1 + 2x22

0

and

F2 (x) − F2 (x ∗ )  0.

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Putting M=

x2 (1 + 2x22 ) (2 − x2 )(x12 + 2x22 )

 0.

We have F2 (x) − F2 (x ∗ ) ≤ M(F1 (x ∗ ) − F1 (x)). So the point x ∗ = (0, 0) is a properly α-efficient solution for problem (α-BFP) with the corresponding α-level set equals 0.19. 4. Basic stability notions for problem (FMOFP) In this section we give the definitions of the set of feasible parameters; the solvability set and the stability set of the first kind (SSK1) of problem (FMOFP) via problem (α-MOFP). Before going any further, let , p ( ) p X X E(λ∗ ) = x ∗ ∈ R n λi∗ Fi (x ∗ ) = max λi Fi (x) x∈X (A,b)

i=1

i=1

be the set of α-optimal solutions of problem (Pλ∗ ). 4.1. The set of feasible parameters Definition 6. The set of feasible parameters of problem (α-MOFP), which is denoted by U , is defined by: U = {b ∈ R m /br ∈ L α (b˜r ), (r = 1, 2, . . . , m) and X (A, b) 6= Φ}. 4.2. The solvability set Definition 7. The solvability set of problem (α-MOFP), which is denoted by V , is defined by: V = {b ∈ U/Problem (α-MOFP) has an α-efficient solution x ∗ , where x ∗ ∈ E(λ∗ )}. 4.3. The stability set of the first kind (SSK1) Definition 8. Suppose that b∗ ∈ V with the corresponding α-efficient solution x ∗ of problem (α-MOFP) such that x ∗ ∈ E(λ∗ ), then the stability set of the first kind of problem (α-MOFP), which is denoted by S(x ∗ ), is defined by:  S(x ∗ ) = b ∈ V /x ∗ is an α-efficient solution of problem (α-MOFP) . 4.4. Utilization of the Kuhn–Tucker conditions corresponding to problem (Pλ ) For the development that follows later, problem (Pλ ) can be rewritten in the following form: (Pλ ):

Max

p X

λi Fi (x),

i=1

subject to

ψr (x, br ) =

n X

ar j x j ≤ br ,

(r = 1, 2, . . . , m),

j=1

h r ≤ br ≤ Hr , x j ≥ 0,

(r = 1, 2, . . . , m),

( j = 1, 2, . . . , n).

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˜ in problem (P) has been replaced by the equivalent constraint h r ≤ br ≤ It is clear that the constraint b ∈ L α (b) Hr , (r = 1, 2, . . . , m) in problem (Pλ ), where h r and Hr , are lower and upper bounds on br , respectively. Therefore, the Kuhn–Tucker necessary optimality conditions corresponding to the maximization problem (Pλ ) will have the following form: p X

λi

m n ∂ψr (x, br ) X ∂ Fi (x) X − + ξr β j = 0, ∂x j ∂x j r =1 j=1

ξr

m m m X X ∂ψr (x, br ) X + ξr − γr + ηr = 0, ∂br r =1 r =1 r =1

i=1 m X r =1

ψr (x, br ) ≤ br , h r ≤ br , br ≤ Hr , x j ≥ 0, ξr [ψr (x, br ) − br ] = 0, γr (h r − br ) = 0, ηr (br − Hr ) = 0, β j x j = 0, ξr ≥ 0, β j ≥ 0, γr ≥ 0, ηr ≥ 0, where r ∈ I = {1, 2, . . . , m} and j ∈ J = {1, 2, . . . , n}. In addition, all the expressions of the Kuhn–Tucker conditions are evaluated at the α-optimal solution x ∗ of problem (Pλ∗ ). Furthermore, ξr , β j , γr , ηr are the Lagrange multipliers. The first two together with the last four relations of the above system of the Kuhn–Tucker conditions represent a Polytope in ξβγ η-space for which its vertices can be determined using any algorithm based upon the simplex method, for example, Balinski [28]. According to whether any of the variables ξr , γr , ηr , r ∈ I = {1, 2, . . . , m} and β j , j ∈ J = {1, 2, . . . , n} are zero or positive, then the set of parameters for which the Kuhn–Tucker necessary optimality conditions corresponding to problem (Pλ ) are utilized at x ∗ will be determined. This set is denoted here by T (x ∗ ). It should be noted that T (x ∗ ) ⊆ S(x ∗ ), since S(x ∗ ) represents the set of all parameters br , (r = 1, 2, . . . , m) for which the α-efficient solution x ∗ for one vector of parameters b∗ ∈ V rests efficient for all parameters b ∈ V . Example 2. Going back to Example 1 and in looking for stability of the proper efficient solution, the maximization problem with single-objective function corresponding to problem (α-BFP) can be written as: (Pλ ): subject to

Max λ1 [(1 − x12 ) − θ1∗ (1 + 2x22 )] + λ2 [2 − θ2∗ (2 − x2 )], x12 + x22 ≤ b, 0.1 ≤ b ≤ 4.8, x1 , x2 ≥ 0

(0,0) (0,0) provided that λ1 ≥ 0, λ2 ≥ 0 and λ1 + λ2 = 1, where θ1∗ = gf11 (0,0) and θ2∗ = gf22 (0,0) . ∗ ∗ ∗ ∗ ∗ Therefore, θ1 = 1 and θ2 = 1. Taking λ1 = λ2 = 1/2 and b = 1, then the Kuhn–Tucker necessary optimality conditions of the maximization problem (Pλ ) will have the following form:

−2λ1 x1 − 2ξ x1 + β1 = 0, −4λ1 x2 + λ2 − 2ξ x2 + β2 = 0, ξ − γ + η = 0,

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ξ(x12 + x22 − b) = 0, γ (b − 4.8) = 0, η(0.1 − b) = 0, β1 x1 = 0, β2 x2 = 0, x12 + x22 ≤ b, 0.1 ≤ b ≤ 4.8, x1 , x2 ≥ 0, ξ, β1 , β2 , γ , η ≥ 0. The set of parameters T (x ∗ ) for which the above system is utilized at the α-optimal solution x ∗ = (0, 0) can be found as: T (0, 0) = {b ∈ R/0.1 ≤ b ≤ 4.8, where λ1 = 1 and λ2 = 0}. 5. Conclusions In the presented paper a solution algorithm has been proposed to solve the fuzzy multiobjective fractional programming problem (FMOFP). Moreover, results of Geoffrion for efficient and properly efficient solutions of multiobjective problems have been extended to (FMOFP). We have defined and characterized some basic stability notions for the problem under consideration. A procedure has been suggested for the determination of the stability set of the first kind for such problems. Summarizing, many aspects and general questions remain to be studied and explored in the area of fuzzy multiobjective fractional programming problems. This paper is an attempt to establish underlying results which hopefully will help others to answer some of these questions. There are however several open points for research in the area of (FMOFP), in our opinion, to be studied in future. Some of these problems are pointed out in the following: (i) An algorithm is required for solving stochastic multiobjective fractional programming problems. (ii) An algorithm is needed for solving large-scale stochastic and fuzzy multiobjective fractional programming problems. (iii) Stability of the efficient and properly efficient solutions should be investigated for fuzzy multiobjective fractional programming problems for different values of α-level sets. (iv) Finding an equivalent deterministic-crisp multiobjective programming problem to solve a stochastic fuzzy multiobjective linear fractional programming problem. References [1] O.M. Saad, On the solution of fuzzy linear fractional programs, in: The 30th Annual Conference, ISSR, vol. 30, Part (IV), Cairo University, Egypt, 1995, pp. 1–9. [2] O.M. Saad, M.F. Abdelkader, On the solution of bicriterion integer nonlinear fractional programs with fuzzy parameters in the objective functions, The Journal of Fuzzy Mathematics 10 (1) (2002) 1–7. [3] O.M. Saad, K. Abd-Rabo, On the solution of chance-constrained integer linear fractional programs, in: The 32nd Annual Conference, ISSR, vol. 32, Part (VI), Cairo University, Egypt, 1997, pp. 134–140. [4] O.M. Saad, W.H. Sharif, On the solution of integer linear fractional programs with uncertain data, Institute of Mathematics & Computer Sciences Journal 12 (2) (2001) 169–173. [5] M. Sakawa, H. Yano, J. Takahashi, Pareto optimality for multiobjective linear fractional programming problems with fuzzy parameters, Information Sciences 63 (1–2) (1992) 33–53. [6] A. Charnes, W.W. Cooper, Programming with linear fractional functions, Naval Research Logistics Quarterly 9 (1962) 181–186. [7] C. Singh, M.A. Hanson, Multiobjective fractional programming duality theory, Naval Research Logistics 38 (1991) 925–933. [8] J.B. Hughes, Interior efficient solutions in bicriterion linear fractional programming—a geometric approach, Mathematical and Computer Modeling 17 (6) (1993) 23–28. [9] O.M. Saad, J.B. Hughes, Bicriterion integer linear fractional programs with parameters in the objective functions, Journal of Information and Optimization Sciences 19 (1) (1998) 97–108.

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