A FUZZY PROGRAMMING APPROACH TO INVENTORY CONTROL PROBLEM

Purpose of study: Main aim of this study is to deals with the problem of inventories. Their holding cost, set-up cost, and many more related to that. All the problems are flexible and having fuzzy nature. Methodology: The model takes the form of a Geometric Programming problem. Hence geometric programming algorithm is used here. Main Finding: The developed models may be used for a single item with a single constraint of limitation on storage area and multi-item inventory problems. Application of this study: This study is useful in the area of inventories. There holding cost and set-up cost etc. The originality of this study: This study may help the stockholders for storing goods and minimizing the cost of holding.


INTRODUCTION
The control of inventories of physical goods is a problem common to all enterprises in any sector of an economy. In any industry, the inventories are essential. The excess inventories are undesirable, which calls for controlling the inventories most profitably. The different types of costs (item cost, ordering cost, carrying cost, understocking cost, overstocking cost, etc.) involved inventory problems that affect the efficiency of an inventory control problem.
In inventory, holding costs, set-up costs, shortage costs, production costs may be flexible with vagueness in their values. All these parameters are variable and imprecise. Due to the imprecise nature of the parameters, the problem becomes fuzzy and unsolvable. The concept of fuzzy set theory introduced by (Zadeh, L.A. (1965)) helps in solving the problem. The most important aspect in the fuzzy set theory approach is the compensatory, competitive, or compensatory-cumcompetitive nature of the aggregate operator. Many authors (Burnwal, A.P., Mukerjee, S.N., Singh, D. This paper focuses on Economic Order Quantity (EOQ) inventory model in a fuzzy environment. Compensatory operator, Min operator and Polynomial & posynominal models of fuzzy geometric programming FGP for the inventory control problem have been devised which is solvable by GP Algorithm.

FORMULATION OF MATHEMATICAL MODEL
The mathematical model of a constrained multi-objective inventory control problem is stated as follows : And f i (x), for i=1,2,…………..,k are various costs and , for j = 1,2,……..,n are various constraints. and are either all polynomials or some polynomials and some polynomials in decision variables. Solving the optimization problem as: Minimizing each f i (x) subject to all constraints. Let us write the model (2.1) as Determine 'x' Where P=(p 1 , p 2, p 3 , ……………,p k ) to 'k' objectives function. Assuming that goals are not defined precisely, therefore, the fuzzy model of (2.2) can be given as (2.3). In a real situation, rigid goals vectors P and B do not exist. Therefore, it is more convenient to assign fuzzy (imprecise) goals by allowing some flexibility in the right-hand sides of all the objectives and constraints.
Where, the wave symbol '~' stands for fuzzification of the aspiration levels b j (j=1,…….,n) and the values b 1 , b 2 ,……. , b n of the given constraints.
Here each 'p i ' of the vector P is increased by and each b j of B is increased by . The model (2.3) has to be defuzzified which is done by using the concept of fuzzy set theory. Linear membership functions denoted by and respectively for fuzzy goals and constraints are given by and respectively. Where, The set of achievement functions in different forms such as Linear quadratic & posynomial using a compensatory operator with normalized weights and without weights as well as min operator is given as The crisp model using : The constrained FGP inventory control problem can be stated as The above models can be solved by the GP algorithm.

SAMPLE PROBLEM: GP Model of Inventory Control
Minimize the total inventory cost associated with ordering and holding costs while sacrificing certain warehouse space limitations. Let annual demand for each product is 'N' units and the rate of demand is 'k' per unit time.
If the order size of produce A is units and the stock is depleted at a constant rate 'k' then the stock of product A will be zero after units of time. If is the lot size of product B, its stock will be zero after units of time. Let lot size orders of size & respectively to arrive simultaneously. This problem can be given in any of the above models (2.4 to 2.9).

CONCLUSION
The developed models may be used for a single item with a single constraint of limitation on storage area and multi-item inventory problems under many limitations, such as their inventory levels, warehouse space, and budget limitations, etc. In multiple objective optimization cases, the objectives and constraints show some degree of dependence over each