Fuzzy inventory without backorder for fuzzy order quantity and fuzzy total demand quantity

Abstract

In this paper, we consider the inventory problem without backorder such that both order and the total demand quantities are triangular fuzzy numbers Q ̃ =(q 1, q 0, q 2) , and R ̃ =(r 1, r 0, r 2) , respectively, where q 1=q 0− Δ 1, q 2=q 0+ Δ 2, r 1=r 0− Δ 3, r 2=r 0+ Δ 4 such that 0< Δ 1<q 0, 0< Δ 2, 0< Δ 3<r 0, 0< Δ 4 , and r 0 is a known positive number. Under conditions 0⩽ q 1< q 0< q 2< r 1< r 0< r 2 we find the membership function μ G( Q ̃ , R ̃ ) (z) of the total fuzzy cost function G( Q ̃ , R ̃ ) and their centroid, then obtain order quantity q ∗∗ in the fuzzy sense and the estimate of the total demand quantity. Scope and purpose This paper deals with the inventory problem without backorder with total cost function F(q)=cTq/2+ar/q, q>0 . In the classical inventory (without backorder) model, both the total demand over the planning time period [0, T] and the period from ordering to arriving are fixed. In the real situation, the total demand r and order quantity q probably will be different from the values used in the total cost function. Also, r influences the values of T. In view of this circumstances, we consider the inventory problem in which both order and total demand quantities are triangular fuzzy numbers Q ̃ =(q 1, q 0, q 2) , and R ̃ =(r 1, r 0, r 2) , respectively, where q 1=q 0− Δ 1, q 2=q 0+ Δ 2, r 1=r 0− Δ 3, r 2=r 0+ Δ 4 such that 0<Δ 1< q 0, 0<Δ 2, 0<Δ 3< r 0, 0<Δ 4; where r 0 is a known number and q 0 is unknown. Letting G( Q ̃ , R ̃ )=cT Q ̃ /2+a R ̃ / Q ̃ , we use the extension principle to find the membership function μ G( Q ̃ , R ̃ ) of the fuzzy total cost function G( Q ̃ , R ̃ ) and their centroid (see Proposition 3). Therefore, given the value of q 1, q 0, q 2, r 1 and r 2, we can find an estimate of the total cost in the fuzzy sense. Finally, we make a comparison between the crisp sense and fuzzy sense by some numerical result.