Bi-level multi-objective programming problem with fuzzy demands: a fuzzy goal programming algorithm Ibrahim A. Baky, Mohamed H. Eid & Mohamed A. El Sayed
OPSEARCH ISSN 0030-3887 OPSEARCH DOI 10.1007/s12597-013-0145-2
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Author's personal copy OPSEARCH DOI 10.1007/s12597-013-0145-2 T H E O R E T I C A L A RT I C L E
Bi-level multi-objective programming problem with fuzzy demands: a fuzzy goal programming algorithm Ibrahim A. Baky & Mohamed H. Eid & Mohamed A. El Sayed
Accepted: 15 May 2013 # Operational Research Society of India 2013
Abstract In this paper, a fuzzy goal programming (FGP) algorithm for solving bi-level multi-objective programming problems with fuzzy demands is presented. These fuzzy demands reflect the experts' imprecise or fuzzy understandings of the nature of parameters in the problem formulation process are assumed to be characterized as fuzzy numbers. Using the level sets of fuzzy parameters, the corresponding non fuzzy bilevel programming problem is introduced. In the proposed algorithm, the membership functions for the defined fuzzy goals of all objective functions at the two levels, as well as the membership functions for the vector of decision variables controlled by FLDM are developed in the model formulation of the problem. Then FGP algorithm is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtaining the most satisfactory solution for all decision makers. Illustrative numerical example is given to demonstrate the proposed algorithm. Keywords Bi-level programming . Fuzzy sets . Fuzzy demands . Goal programming . Fuzzy goal programming . Multi-objective programming
1 Introduction Bi-level multi-objective programming problem (BL-MOPP), an apparatus for modeling decentralized decisions, consists of the objectives of the first level decision maker I. A. Baky : M. A. El Sayed (*) Department of Basic Sciences, Faculty of Engineering, Benha University, EL-Qalyoubia, Egypt e-mail:
[email protected] I. A. Baky e-mail:
[email protected] M. H. Eid Department of Engineering Mathematics and Physics, Faculty of Engineering-Shoubra, Benha University, EL-Qalyoubia, Egypt M. H. Eid e-mail:
[email protected]
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(FLDM) at the upper level and the objectives of the second level decision maker (SLDM) at the lower level. The execution of decision is sequential, from upper level to lower level; each decision maker (DM) independently controls only a set of decision variables and is interested in optimizing his/her net benefits over a common feasible region [4, 5, 13, 22]. In a hierarchical DM context, it has been realized that each DM should have a motivation to cooperate with the other, and a minimum level of satisfaction of the DM at the lower level must be considered for the overall benefit of the organization [3, 19, 22]. During the last three decades, bi-level programming problem (BLPP) [2, 3, 6, 8] as well as multi-level programming problem (MLPP) [9, 15, 16, 23] in general for hierarchical decentralized planning problems have been deeply studied and many methodologies have been developed to solve them [2, 8, 13]. The uses of the concept of membership function of fuzzy set theory to multi-level programming problems for satisfactory was first introduced by Lai [9] in 1996. Thereafter, Lai’s satisfactory solution concept was extended by Shih et al. [15] and a supervised search procedure with the use of max-min operator of Bellman and Zadeh [7] was proposed. AboSinna [1, 3] extended the fuzzy approach for multi-level programming problems of Shih et al. [15] for solving bi-level and three-level non-linear multi-objective programming problems. Sakawa et al. [18] developed interactive fuzzy programming for MLLP with fuzzy parameters in 2000. Abo-Sinna and Baky [1] present balance space approach for non-linear multi-objective bi-level programming problem. Baky [3] studied FGP algorithm for solving decentralized bi-level multi-objective programming problem. Pramanik and Roy [12] proposed a solution methodology based on fuzzy goal programming (FGP) for solving MLPP. Bi-level mathematical programming (BLMP) is identified as mathematical programming that solves decentralized planning problems with two decision makers (DMs) in two-levels or hierarchical organization [6, 15]. The basic concept of BLMP is that a first-level decision maker (FLDM) (the leader) sets his goals and/or decisions and then asks each subordinate level of the organization for their optima which are calculated in isolation; the second-level decision maker (SLDM) (the follower) decisions are then submitted and modified by the FLDM with consideration of the overall benefit of for the organization; the process is continued until a satisfactory solution is reached [6, 22]. Thus, contrary to what is stated in the previous paragraph, Pramanik and Dey [13] proposed a fuzzy goal programming model for solving BL-MOPP that didn’t follow the basic concepts of bi-level programming and neglect the upper level decisions. As this is equivalent to solve the BL-MOPP as a single level multi-objective programming problem. Hence, bi-level programming is a hierarchical optimization problem consisting of two levels, the first of which (the leader’s level) is dominant over the other (the follower’s one). The order of the play is very important, the choice of the dominant level limits or highly affects the choice or strategy of the lower level. Knowing the selection of the leader, the FGP model of BL-MOPP in which the decision variables of the FLDM appear as a membership functions to give the lower level extended feasible region to obtain a satisfactory solution. When formulating a mathematical programming problem which closely describes and represents the real-world decision situation, various factors of the real-world system should be reflected in the description of objective functions and constraints.
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Naturally, these objective functions and constraints involve many parameters whose possible values may be assigned by the experts [20, 21, 24]. It has been observed that, in most real-world situations, for example, power markets and business management, the possible values of these parameters are often only imprecisely or ambiguously known to the experts and cannot be described by precise values. With this observation, it would be certainly more appropriate to interpret the experts understanding of the parameters as fuzzy numerical data which can be represented by means of fuzzy sets [21, 24]. An efficient methodology for solving multi-objective programming problems by using FGP approach is developed by Mohamed [11]. Thereafter, the concept of FGP approach was extended by Moitra et al. for obtaining a satisfactory solution for bi-level programming problems [23]. Recently, Baky extended FGP approach for solving a deterministic decentralized BL-MOPP [3] and MLPP [4]. In real world situation, because of incomplete or non-obtainable information, the data are often not so deterministic, therefore they usually are fuzzy. Therefore, the aim of this paper is to extend the FGP to solve bi-level multiobjective programming problem with fuzzy demands. These demands are expressed in triangular fuzzy numbers. To formulate the model of BL-MOPP with fuzzy demands, for a prescribed value of α, the model is converted to a deterministic BL-MOPP. The fuzzy goals of the objectives are determined by determining individual optimal solutions. The fuzzy goals are then characterized by the associated membership functions which are transformed into fuzzy flexible membership goals by means of introducing over- and underdeviational variables and assigning highest membership value (unity) as aspiration level to each of them. To elicit the membership functions of the decision vectors controlled by the first level DM, the optimal solution of the first level multi-objective programming (MOP) problem is separately determined. A relaxation of the decisions is considered to avoid decision deadlock. Also, illustrative numerical example is provided to demonstrate the efficiency of the proposed algorithm. The remainder of this paper is organized as follows. Section 2 provides some basic definitions. Section 3 presents the formulation of BL-MOPP with fuzzy demands. Section 4 briefly discusses the model formulation of BL-MOPP with fuzzy demands. The proposed FGP model is developed in Section 5 and the algorithm of the FGP for solving BL-MOPP with fuzzy parameters is presented in Section 6. The following section presents an illustrative numerical example in order to demonstrate the proposed algorithm. Finally, the concluding remarks are made in Section 8.
2 Definitions In this section, some basic notations of the area of fuzzy set theory have been introduced, for more details see [24]. e Definition 2.1 Let R bethe space of real numbers. A Fuzzy set Ai is a set of ordered pairs x; μ+ i ðxÞ jx ∈ R ; where μ+ i ðxÞ : →½0; 1 is called membership function of fuzzy set.
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e Definition 2.2 A convex fuzzy set, Ai , is a fuzzy set in which: ∀ x, y ∈R, ∀ 1 ∈ [0, 1] μ +i ðlx þ ð1−1Þy ≥ min μ +i ðxÞ; μ +i ðyÞ . e is called positive if its membership function is such that Definition 2.3 A fuzzy set A μ +i ðxÞ ¼ 0; ∀x ≤ 0 Definition 2.4 Triangular fuzzy number (TFN) is a convex fuzzy set which is e ¼ x; μ ~ ðxÞ where: defined as A Ai
8 ðx−aÞ > > > > < ðb−a Þ μ ðxÞ ¼ ðc−xÞ > +i > > > ðc−bÞ : 0
a≤x≤b b≤x≤c
ð2:1Þ
other wise
For convenience, TFN represented by three real parameters (a, b, c) which are (a ≤ b ≤ c) will be denoted by the triangle a, b, c (Fig. 1). e is a non-fuzzy set denoted by A e ! Definition 2.5 The α –level set of a fuzzy set A for which thedegree of its membership functions exceed or equal to a real number e ¼ fxjμ+ ðxÞ≥ αg. α ∈ [0,1],i.e. A α L U The α-level set of e a is then; e aα ¼ e aα ; e aα that is e aLα ¼ ð1−αÞa þ αb, and U L U e aα and e aα represent the lower and upper cuts respectively, aα ¼ ð1−αÞc þ αb, where, e shown in (Fig. 1).
3 Problem formulation Assume that there are two levels in a hierarchy structure with first-level decision maker (FLDM) and second-level decision maker (SLDM). Let the vector of decision variables x=(x1,x2)∈Rn be partitioned between the two planners. The first-level Fig. 1 Triangular fuzzy number
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decision maker has control over the vector x1 ∈Rn1 and the second-level decision maker has control over the vector x2 ∈Rn2 , where n = n1 + n2. Furthermore, assume that e i ðx1 ; x2 Þ : Rn1 Rn2 →Rmi ; i ¼ 1; 2 F
ð3:1Þ
are the first-level and the second-level vector objective functions, respectively. So the BLMOPP with fuzzy demands of minimization type may be formulated as follows [3, 13]: [1st Level] Min eF 1 ðx1 ; x2 Þ ¼ Min ef 11 ðx1 ; x2 Þ; ef 12 ðx1 ; x2 Þ; …; ef 1m1 ðx1 ; x2 Þ ; ð3:2Þ ︸ x1
︸ x1
where x2 solves [2nd Level] Min eF 1 ðx1 ; x2 Þ ¼ Min ef 21 ðx1 ; x2 Þ; ef 22 ðx1 ; x2 Þ; …; ef 2m2 ðx1 ; x2 Þ ; ð3:3Þ ︸ x2
︸ x1
subject to 8
