Inventory Model with Demand Dependant on Unit Cost- Input Parameters as Triangular Fuzzy Numbers

Abstract: This paper considers an inventory model in which the shortages are backlogged and the demand is dependent on unit cost. An optimum value for average total cost is calculated by considering various input costs, lot size and maximum inventory under fuzzy environment. The process of defuzzification is done by using the signed distance method. Numerical example and sensitivity analysis is given for calculating both crisp and fuzzy values of the total cost.


Introduction
The incorporation of fuzzy set theory in decision making has drawn much attention in decision making. The coupling of fuzziness with inventory has many real applications and are essential to improve the decision making problems.
The main aim of the production process is to maximize the profit with the minimum available stock and costs. The inventory costs consists of four costs namely purchase cost, ordering cost, inventory carrying cost and shortage cost.
The effect of different costs results in establishing a minimum total cost. The development of an effective inventory model is based on a proper balance between the carrying cost and ordering cost which will determine the lot-size with minimum total annual cost.
Park &Vujosevic proposed the inventory model in fuzzy sense with cost parameters as fuzzy numbers. Vujoseric et al developed an inventory model using triangular and trapezoidal fuzzy numbers and applied centroid method for defuzzification to estimate the total cost. But later it was found that the signed distance method is more precise when related to centroid method for defuzzification. Petroric.Retal calculated the total cost with backorder by using trapezoidal fuzzy numbers. Harish Nagar &Priyankasurana developed an inventory model for deteriorating items with varying demand and used pentagonal fuzzy numbers.K.Syed&L.A.Aziz proposed an inventory model without backorders using the method of signed distance fordefuzzification. Urgeletti developed EOQ model using triangular fuzzy numbers. D.Dutta and Pavankumar proposed inventory model in which the demand and various costs are taken under fuzzy environment. Jeuhan Chiang et al applied signed distance method for defuzzification of an inventory model with backorders. P.K.DeApurvaRawat prepared EOQ model without shortages using fuzzy numbers and computed the annual inventory cost.
In this research work the decision variables are computed for an inventory model in which the demand varies with respect to unit cost in both crisp and fuzzy environment. Signed distance method is employed for defuzzification with numerical example.

Model Description 3.1. Notations
We use the following notations and related parameters for crisp as well as fuzzy inventory model with backorders.
AC-Annual optimum cost B-Each time ordering lot size IL-Inventory level SC-Each time ordering cost R-Demand per cycle. p-unit cost HC-Annual holding cost per item PC-Penalty cost per unit of time

Assumptions
The following assumptions are considered: *Shortage cost is allowed *Time plan is fixed *Demand R is considered as a function of unit cost

Karush-Kuhn Tucker Method
Karush-Kuhn Tucker conditions also known as Kuhn-Tucker conditions is a derivative test for estimating the optimum solution of a non-linear programming problem. In this paper the Kuhn-Tucker conditions approach is used to optimize the impreciseinventory annual cost function by taking the costs like carrying cost, order cost and penalty cost as input parameters and the lot-size, inventory level and unit price as decision variables.

Crisp Inventory model
The crisp total cost function of the proposed model is given by 22 .
The input parameters HC,PC ,SC and also the decision quantities IL,B and p are considered as triangular fuzzy numbers which can be taken as 1

Conclusion:
In this paper, an EOQ model with shortages and a demand dependent on unit cost is considered with the various costs like the order cost, holding cost and the penalty cost are taken as triangular fuzzy numbers. The signed distance method is used to defuzzify the annual cost function and hence the crisp and fuzzy values are computed. The calculated values from the table shows that the total cost values remains same if we use the signed distance method to defuzzify the triangular fuzzy numbers. Sensitivity analysis is done to illustrate the specified model.

Future work:
In future, the work can be extended by using Trapezoidal and Pentagonal Fuzzy number to fuzzify the parameters and the result can be compared with this triangular fuzzy parameters. Also the Graded mean integration method can be applied for defuzzification and the accurate result can be concluded.