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This paper presents a fuzzy goal programming (FGP) procedure for solving bilevel multiobjective linear fractional programming (BL-MOLFP) problems. It makes an extension work of Moitra and Pal (2002) and Pal et al. (2003). In the proposed procedure, the membership functions for the defined fuzzy goals of the decision makers (DMs) objective functions at both levels as well as the membership functions for vector of fuzzy goals of the decision variables controlled by first-level decision maker are developed first in the model formulation of the problem. Then a fuzzy goal programming model to minimize the group regret of degree of satisfactions of both the decision makers is developed to achieve the highest degree (unity) of each of the defined membership function goals to the extent possible by minimizing their deviational variables and thereby obtaining the most satisfactory solution for both decision makers. The method of variable change on the under- and over-deviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem efficiently by using linear goal programming (LGP) methodology. Illustrative numerical example is given to demonstrate the procedure.

Bi-level mathematical programming (BLMP) is identified as mathematical programming that solves decentralized planning problems with two decision makers (DMs) in a two-level or hierarchical organization [

Most of the developments on BLMP problems focus on bi-level linear programming [

The use of the fuzzy set theory [

The main difficulty that arises with the FP approach of Shih et al. is that there is possibility of rejecting the solution again and again by the FLDM and reevaluation of the problem is repeatedly needed to reach the satisfactory decision, where the objectives of the DMs are overconflicting. Even inconsistency between the fuzzy goals of the objectives and the decision variables may arise. This makes the solution process a lengthy one [

To formulate the FGP Model of the BL-MOLFP problem, the fuzzy goals of the objectives are determined by determining individual optimal solution. 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 FLDM, the optimal solution of the first-level MOLFP problem is separately determined. A relaxation of the FLDM decisions is considered for avoiding decision deadlock.

The method of variable change on the under- and overdeviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem efficiently by using linear goal programming (LGP) methodology.

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

[1st Level]

[2nd Level]

In BL-MOLFP problems, if an imprecise aspiration level is assigned to each of the objectives in each level of the BL-MOLFP, then these fuzzy objectives are termed as fuzzy goals. They are characterized by their associated membership functions by defining the tolerance limits for achievement of their aspired levels.

Since the FLDM and the SLDM both are interested of minimizing their own objective functions over the same feasible region defined by the system of constraints (

Let

It may be noted that the solutions

Membership function of minimization-type objective functions.

Following Lai [

Let

Membership function of decision vectors

It may be noted that the decision maker may desire to shift the range of

Now, in a fuzzy decision environment, the achievement of the objective goals to their aspired levels to the extent possible is actually represented by the possible achievement of their respective membership values to the highest degree. Regarding this aspect of fuzzy programming problems, a goal programming approach seems to be most appropriate for the solution of the first-level multiobjective linear fractional programming problem and the bi-level multi-objective linear fractional programming problems [

In fuzzy programming approaches, the highest degree of membership function is 1. So, as in [

In conventional GP, the under- and/or overdeviational variables are included in the achievement function for minimizing them and that depend upon the type of the objective functions to be optimized. In this approach, the over-deviational variables for the fuzzy goals of objective functions,

It can be easily realized that the membership goals in (

The FGP approach to multiobjective programming problems presented by Mohamed [

Following Pal et al. [

Now, using the method of variable change as presented by Kornbluth and Steuer [

Let

Now, in making decision, minimization of

It may be noted that when a membership goal is fully achieved,

Here, on the basis of the previous discussion, it may be pointed out that any such constraint corresponding to

Therefore, under the framework of

To assess the relative importance of the fuzzy goals properly, the weighting scheme suggested by Mohamed [

The FGP model (

In this section, the FGP model of Pal et al. [

The first-level MOLFP problem is

And the FGP model of Pal et al. [

Following the above discussion, we can now construct the proposed FGP algorithm for solving the BL-MOLFP problems.

Calculate the individual minimum and maximum of each objective function in the two levels under the given constraints.

Set the goals and the upper tolerance limits for all the objective functions in the two levels.

Elicit the membership functions

Formulate the Model (

Solve the Model (

Set the maximum negative and positive tolerance values on the decision vector

Elicit the membership functions

Elicit the membership functions

Formulate the Model (

Solve the Model (

To demonstrate proposed FGP procedure, consider the following bi-level multi-objective linear fractional programming problem:

[1st Level]

[2nd Level]

subject to

Table

Coefficients

Following the procedure, the FGP model for the first-level multi-objective linear fractional programming problem is obtained as

Using the LP-ILP linear and integer programming software program, version 1 for windows, the optimal solution of the problem is

Then, following the procedure, the proposed FGP model for the bi-level multi-objective linear fractional programming problem is obtained as

The optimal satisfactory of the problem is

This paper presents a fuzzy goal programming procedure for solving bi-level multi-objective linear fractional programming (BL-MOLFP) problems. A fuzzy goal programming model to minimize the group regret of degree of satisfactions of both the decision makers is developed to achieve the highest degree (unity) of each of the defined membership function goals to the extent possible by minimizing their deviational variables and thereby obtaining the most satisfactory solution for both decision makers. The main advantage of the proposed fuzzy goal programming procedure is that the possibility of rejecting the solution again and again by the FLDM and reevaluation of the problem repeatedly, by redefining the elicited membership functions, needed to reach the satisfactory decision does not arise. A linearization process of solving BL-MOLFP problems via