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

Authors
Journal
Computers & Operations Research
0305-0548
Publisher
Elsevier
Publication Date
Volume
27
Issue
10
Identifiers
DOI: 10.1016/s0305-0548(99)00068-4
Keywords
• Inventory
• Membership Functions
• Extension Principle
• Fuzzy Inventory Without Backorder
• Fuzzy Economic Order Quantity
• Fuzzy Demand Quantity

## Abstract

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.

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