A fuzzy approach for bi-level integer non-linear programming problem

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Abstract

Bi-level programming, a tool for modeling decentralized decisions, consists of the objective of the leader at its first level and that is of the follower at the second level. Integer programming deals with the mathematical programming problems in which some or all the variables are to be integer. This paper studies a bi-level integer non-linear programming problem with linear or non-linear constraints, and in which the non-linear objective function at each level are to maximized. The bi-level integer non-linear programming (BLI-NLP) problem can be thought as a static version of the Stackelberg strategy, which is used leader-follower game in which a Stackelberg strategy is used by the leader, or the higher-level decision-maker (HLDM), given the rational reaction of the follower, or the lower-level decision-maker (LLDM). This paper proposes a two-planner integer model and a solution method for solving this problem. This method uses the concept of tolerance membership function and the branch and bound technique to develop a fuzzy Max–Min decision model for generating Pareto optimal solution for this problem; an illustrative numerical example is given to demonstrate the obtained results.

Introduction

A bi-level programming problem is formulated for a problem in which two decision-maker make decisions successively. For example, in a decentralized firm, top management, an executive board, or headquarters makes a decision such as a budget of the firm, and then each division determines a production plane in the full knowledge of the budged. We can also site the Stackelberg duo-poly: two firms supply homogeneous goods to a market, and consequently the predominant firm first its level of supply, and then the over firm determines that of itself after it realizes that of the predominant firm. Research on multi-level mathematical programming to solve organizational planning and decision-making problems has been conducted widely. The research and application have concentrated mainly on bi-level programming (see [1], [2], [3], [4], [5], [6], [7], [8]). Integer programming problems deals with linear or non-linear programming problems in which some or all of the variables assume to be integer. An integer-programming problem is said to be mixed integer or pure integer depending on whether some or all the variables are restricted to be integer values. Although there are three main techniques have been developed for solving integer-programming problems, none of these techniques is very reliable from the computational standpoint. Nevertheless, the branch and bound technique is more successful computationally than are the cutting-plane technique or the implicit enumeration technique (see [9], [12], [13]). For this reason most commercial codes are based on the use of the branch and bound technique. This paper proposes a two-planner integer model for solving the BLI-NLP problem. This model uses the concept of tolerance membership function and the branch and bound technique to develop a fuzzy Max–Min decision model for generating Pareto optimal solution for this problem. The HLDM specifies his/her objective functions and decision variable with possible tolerance, which are to be described by non-linear membership function of fuzzy set theory and solve it by branch and bound technique. Then, the LLDM uses this preferences information for the HLDM and solves his/her problem subject to the HLDM restrictions. Finally the LLDM presents his/her solution to the HLDM, if the HLDM agree to the proposed solution, a Pareto optimal solution is reached. If the HLDM rejects the proposed solution, the HLDM and the LLDM must update and change former goals and decisions until a Pareto optimal solution is reached.

Section snippets

Problem formulation and solution concept

Let xiRni (i=1,2) be a vector variables indicating the first decision level’s choice and the second decision level’s choice, ni1 (i=1,2).

Let Fi:RniRNi (i=1,2) be the first level objective function, and the second level objective function, respectively. Let the HLDM and LLDM have N1 and N2 objective function, respectively.

So the BLI-NLP problem may be formulated as follows:

[Upper level]Maxx1F1(x1,x2),where x2 solves;

[Lower level]MaxF2(x1,x2),subject toG=(x1,x2)gi(x1,x2)0,i=1,2,,m,x1,x20,

Fuzzy decision models for BLI-NLP problem

To solve the BLI-NLP by adopting, the leader-follower Stakelberg and the well-known fuzzy decision model of Sakawa [10], [11]. One first gets the satisfactory integer solution that is acceptable to HLDM, and then give the HLDM decision variable and goal with some leeway to the LLDM for him/her to seek the optimal solution, and to arrive at the solution which is closest to the optimal solution of the HLDM. This due to, the LLDM who should not only optimize his/her objective function but also try

Numerical example for BLI-NLP problem

To demonstrate the solution method for BLI-NLP problem, let us consider the following example:

[Upper level]Maxx1F1(x1,x2)=Maxx1x12+x22,where x2, solves

[Lower level]MaxF2(x1,x2)=Max(x1-1)2+x22subject to(x1,x2)G=(x1,x2)2x1+x28,x1+2x26,x1,x20,andintegers.

First, the HLDM solves his/her problem as follows:

  • 1.

    Find individual optimal solution by solving Eqs. (4), (5), we get: (F1,F¯1)=(16,0).

  • 2.

    By using (6), the HLDM build the membership function μ(F1(x)) then solve the mixed-integer Tchebycheff

Summary and concluding remarks

This paper has proposed a bi-level integer non-linear programming problem with linear or non-linear constraints, and in which the non-linear objective function at each level are to maximized. The bi-level integer non-linear programming (BLI-NLP) problem can be thought as a static version of the Stackelberg strategy, which is used leader-follower game in which a Stackelberg strategy is used by the leader given the rational reaction of the follower. This paper has proposed a two-planner integer

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