INVENTORY MODEL WITH DEMAND AS TYPE-2 FUZZY NUMBER: A FUZZY DIFFERENTIAL EQUATION APPROACH

An inventory model is formulated with type-2 fuzzy parameters under trade credit policy and solved by using Generalized Hukuhara derivative approach. Representing demand parameter of each expert's opinion is a membership function of type-1 and thus, this membership function again becomes fuzzy. The final opinion of all experts is expressed by a type-2 fuzzy variable. For this present problem, to get corresponding defuzzified values of the triangular type-2 fuzzy demand parameters, first critical value (CV)-based reduction methods are applied to reduce corresponding type-1 fuzzy variables which becomes pentagonal in form. After that $alpha$- cut of a pentagonal fuzzy number is used to construct the upper $alpha$- cut and lower $alpha$- cut of the fuzzy differential equation. Different cases are considered for fuzzy differential equation: gH-(i) differentiable and gH-(ii) differentiable systems. The objective of this paper is to find out the optimal time so as to minimize the total inventory cost. The considered problem ultimately reduces to a multi-objective problem which is solved by weighted sum method and global criteria method. Finally the model is solved by generalised reduced gradient method using LINGO (13.0) software. The proposed model and technique are lastly illustrated by providing numerical examples. Results from two methods are compared and  some sensitivity analyses both in tabular and graphical forms are presented and discussed. The effects of total cost with respect to the change of demand related parameter ($beta$),  holding cost parameter ($r$), unit purchasing cost parameter ($p$), interest earned $(i_e)$ and interest payable $(i_p)$ are discussed. We also find the solutions for type-1 and crisp demand as particular cases of type-2 fuzzy variable. This present study can be applicable in many aspects in many real life situations where type-1 fuzzy set is not sufficient to formulate the mathematical model. From the numerical studies, it is observed that under both gH-(i) and gH-(ii) cases, total cost of the system gradually reduces for the sub-cases - 1.1, 1.2 and 1.3 depending upon the positions of N(trade credit for customer) and M (trade credit for retailer) with respect to T (time period).