Case 6.1.4 Frozen Goal Programming Problem

6.1.1.1. Zimmermann’s FGP Formulation
Consider the following: Subject to:

(11) where, represents the underdeviational variables in the goal constraint for the fuzzy multiobjective goal programming problem with aspiration level one, .
6.1.2. Proposed Weighted Fuzzy Goal Programming Formulation
Gupta and Bhattacharjee [12, 13] proposed two methods of solving a Fuzzy Goal Programming (FGP) problem by applying weighting method where the relative weights represent the objective functions relative importance. This method involves one additional goal constraint by introducing only under deviation variables to the fuzzy operator (resp., ).
Method 1:
For FGPP the Weighted Fuzzy Goal Programming Problem (WFGPP) method 1 can be written as:
Find 푥, ,
Subject to

(12)
Three different modes of weights are considered: ; (13) ; (14) . (15)
Method 2:
The WFGP method 2 can be written as:
Find 푥, Subject to:

. \ (16)
Here, consider three different modes of weights defined as: ; ; , where, 푝i represents the tolerances, and 푤i represents the weights; i = 1, 2,…,k.
7. CASE STUDY

In this section, the researchers consider an inventory problem model where the problem is to maximize the profit to holding cost ratio and minimize the total cost to total produced quantities ratio with the goals to both the objectives being imprecise in nature. It is further assumed by the researcher that there is only one period in the cycle time. Two constraints are considered in this modeling formulation: warehouse space constraint and budget