Triangular fuzzy numbers model with cost parameters

The scope of our research work is to analyzea purchasing model in which the uncontrolled variables are fuzzified. The fuzzy cost parameters are defuzzified and an optimal solution of the ordering quantity, maximum inventory and annual cost are estimated. Numerical example and sensitivity analysis has been done to explain the mathematical model.


Introduction
Different types of impreciseness are inherent in day to day life decision making problems that are classically modeled by probabilistic approach. In such cases arises a question that how to define inventory optimization and interpret their optimal solution. In this situation, it is more convenient to approach these models using fuzzy concepts than probability concepts.
Zadeh used fuzzy numbers to approach the new products and seasonal items for an improved solution. Since then the uncertainities are shown in several articles by Chang, Yao & Lee and Yao & Chang. A systematic progress in development of inventory models from crisp parameters to fuzzy parameters has been arrived [1][2][3][4]. Park &Vujoservic improved the inventory models by handling fuzzy numbers. An EOQ model with uncertain warehouse space and order quantity was determined by Mandal&Maiti whereas Chang considered a model with fuzzy backorders.
Yao &Chang defuzzified a fuzzy purchasing model using signed distance method.Later Kazemietal improved the model using GMI method for defuzzification. Syed & Aziz developedan EOQ model without shortage with ordering cost and holding cost as imprecise variables and optimized the annual total cost [5][6][7][8][9].
In this paper a purchasing model is presented with various costsand the demand as imprecise variables. The triangular fuzzy numbers are used to represent fuzzy quantities. GMI and SD method are applied for defuzzification. The model is solved using extended Lagrange method to obtain the optimal value of the ordering quantity, inventory level and hence the total cost.The optimal values are obtained for both crisp and decision making model and the results are compared. Sensitivity analysis is done and a numerical solution is obtained [10][11][12][13][14].

3.Assumptions and Notations
Production is instantaneous; The inventory systems involve only one item.
S -Fuzzy order cost d -Fuzzy demand rate c -Fuzzy holding cost p -Fuzzy penalty cost l -Fuzzy Lot size M -Fuzzy inventory level TC -Fuzzy total cost

The KarushKuhn-Tucker Approach
An optimal solution of an NLPP has been obtained by Kuhn & Tucker followed by Karush with the derivation of necessary optimality conditions involving inequality constraints. In this research, KKT conditions are used to solve the proposed inventory model.

Mathematical Model
The formula for economic order quantity (EOQ) in scientific inventory management is given by equation Sd p l cp Defuzzification of TC is as follows.

Graded-Mean Integration method
The annual cost is determined by By the extended Lagrange method, an optimal value of ordering quantity and maximum inventory is obtained.

Numerical Examples
The triangular fuzzy numbers for S,d,c and p are as follows in table 1-4.

Sensitivity Analysis
From the numerical example, the following optimal values are obtained. By Graded men integration method, the maximum membership function corresponds to l * = 8830, M* = 8412, TC*=1990.94 By Signed distance method, the maximum membership function corresponds to l * = 8556, M* = 8130, TC*=1809.63

Conclusion
The research, we have considered a purchasing model involving cost parameters and demand rate as input variables and the lot size and maximum inventory level as decision variables. To compare the R e t r a c t e d real life situation, the major cost parameters are taken as uncertain and used triangular fuzzy numbers to define them. The solution is optimized by utilizing the extended Lagrange method along with GMI and Signed distance method for defuzzification.
The sensitivity analysis gives a better optimal solution under fuzzy situation than crisp model. It can also be concluded that the SD method is more efficient over the GMI method in optimization.