Retraction Retraction: Optimization of Fuzzy Model for Signed Distance

The research involves the estimation of minimum total cost of an inventory under both stable and imprecise environment. Demand related to unit cost is assumed here. Cost parameters and decision variables are uncertain in nature that are defuzzified by signed distance method. The KKT conditions are applied to optimize the objective function.. Numerical example is contributed to describe the comparison between crisp and fuzzy solutions.


Introduction
Decision making is one of the well known process of attaining an optimum solution in an environment with different criterions. Decision making problems are common in real life and most of the real life problems involves several objects which are considered into account. An inventory problem is a decision making problem which optimizes the objective function under various constraints [1]. Inventory control is one among the most important problems both from theoretical and practical approach. An abundant number of papers has been devoted to this area of research. The basic idea is to meet sufficient demand with stock as minimum as possible [2].
Though the literature involves many inventory control systems, a demand varying over unit price is assumed with different costs in uncertain conditions. An EOQ based inventory problem in crisp circumstances is promoted with minimum total cost without any ambiguity. Though these models are capable of providing an optimum solution under various situations, they are unfit to represent the real life scenario. This results in errors in decision making. Thus the concept of fuzzy approach is introduced instead of using usual probability theory which gives an accurate result [3].
Initially, the concept of fuzziness was first defined by [4] which receives a considerable attention from researchers of all fields like production, inventory management etc. [5] the objective function by converting a purchasing problem to an NLPP. [6] developed a non-linear fuzzy model where some of the input parameters are fuzzified with imprecise storage space. [7] attempted to approach operational research in an imprecise environment. [8] proposed an imprecise purchasing model by using trapezoidal fuzzy numbers for back orders. [ [10] considered fuzzy numbers to calculate the total price without backorders.
In this work, a fuzzy EOQ model with shortages and a varying demand is optimized. Numerical examples are illustrated to calculate both crisp and fuzzy annual total cost which leads to choose which model gives a better optimal solution.

Mathematical formulation
A model with shortages and demand dependent on unit price is assumed whose objective function is given by Equations (1)

Crisp Mathematical Model
The suggested crisp economic order quantity (EOQ) model is illustrated by Equations (3) ,(4) and (5) According to inventory management, the analytic solution of the proposed model is given by R e t r a c t e d  (4) (1 )

Fuzzified Mathematical Model
In general, holding and ordering cost are often imprecise and hence expressed in linguistic terms. With this fact, the shortage cost, batch size, unit cost and inventory level are also considered as uncertain parameters.
Equation (1) can be fuzzified as follows by Equations (6-12): where ~ indicates the fuzzification of parameters. SDM is used to calculate the optimum solution of (2).
The above mentioned fuzzy parameters are defined as follows.   Table 1 shows the calculations of Total cost for a crisp EOQ model by varying the values of β.
• As the β value increases, the unit cost also increases.
• As the β value increases, the inventory level, lot-size and also the total cost value decreases. From Table 2, it is clear that • The increment in β value rises the unit price value, whereas the inventory level, ordering quantity and inventory cost value depreciates.

Conclusion and Future Scope
It is observed from the numerical example that the total cost value obtained using signed distance method is though nearer to the crisp values, it is more accurate than that of crisp values. Hence it can be concluded that the SD method optimizes the objective function. The results indicates that the increase in the parameter value β depreciates the annual inventoryin different proposed models.
The model can be improved by comparing the calculated values of total cost by SD and GMI methods. The total cost can also be determined by assuming various costs as trapezoidal and pentagon fuzzy numbers.