Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 1490712, 10 pages

http://dx.doi.org/10.1155/2016/1490712

## Optimal Replenishment Policy for Weibull-Distributed Deteriorating Items with Trapezoidal Demand Rate and Partial Backlogging

School of Management, Shanghai University, Shanghai 200444, China

Received 2 December 2015; Accepted 27 March 2016

Academic Editor: Alireza Amirteimoori

Copyright © 2016 Lianxia Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An inventory model for Weibull-distributed deteriorating items is considered so as to minimize the total cost per unit time in this paper. The model starts with shortage, allowed partial backlogging, and trapezoidal demand rate. By analyzing the model, an efficient solution procedure is proposed to determine the optimal replenishment and the optimal order quantity and the average total costs are also obtained. Finally, numerical examples are provided to illustrate the theoretical results and a sensitivity analysis of the major parameters with respect to the stability of optimal solution is also carried out.

#### 1. Introduction

Deteriorating is a general phenomenon for many products, in that fruits or vegetables are spoiled directly while alcohol undergoes physical depletion over time, and electronic products deteriorate through a gradual loss of potential utility as time passed. Based on the above situation, the effect of deteriorating for items can not be disregarded in many inventory systems. An inventory model with a constant decaying rate for items is proposed by Ghare and Schrader in 1963 [1], while Covert and Philip [2] extended Ghare and Schrader’s model to an EOQ model with a two-parameter Weibull-distributed deteriorating rate. Since then, there are many researchers who focused on such topic; Wee et al. [3] considered an inventory model for deteriorating items with quantity discount, pricing, and partial backlogging, and more related articles can be referred to like Yang et al. [4], Shah et al. [5], Yang et al. [6], and so forth.

However, it is impossible for deteriorating items that the demand rate increases continuously during their full life cycle. Based on such reasons, Hill [7] proposed an inventory model with ramp-type demand rate, which extended to allow shortage proposed by Mandal and Pal [8]. Further, Wu [9] extended Mandal and Pal’s model to have Weibull-distributed deterioration and time-dependent backlogging. Skouri et al. [10] considered a model with general ramp-type demand rate, partial backlogging, and Weibull deterioration rate. Tan and Weng [11] considered a discrete inventory model for deteriorating items with partial backlogging. Pal et al. [12] considered a production inventory model for deteriorating items with ramp-type demand rate under the effect of inflation and shortages under fuzziness. There are many other related literatures about such inventory model, such as Ahmed et al. [13], Agrawal et al. [14], Singh and Pattnayak [15], Wu et al. [16], and Rabbani et al. [17].

While, for some short life cycle products, the demand rate may increase up to a certain level in the early stage of marketing, then reach a stabilized period, and finally decrease to zero and reach the end of their life cycle, Cheng and Wang [18] proposed an inventory model for deteriorating items with trapezoidal type demand rate. Further, Cheng et al. [19] extended the model to be of time-dependent deteriorating items with trapezoidal type demand rate and partial backlogging. Uthayakumar and Rameswari [20] studied a model for defective items with trapezoidal type demand rate to determine the optimal product reliability. Lin [21] and Lin et al. [22] considered the inventory model which was proposed by Cheng and Wang [18]. Zhao [23] studied an inventory model for Weibull-distributed deterioration items with trapezoidal type demand rate and partial backlogging in which the model begins with no-shortage. While, in real situation, customers will respond differently in case of shortages, either to wait until replenishment or to look for alternative products, in this paper, we study an inventory model beginning with shortage, Weibull-distributed deterioration items, trapezoidal type demand rate, and time-dependent partial backlogging. By analyzing the studied model, we propose an optimal replenishment strategy for it and also carry out a sensitivity analysis of the main parameters.

The remainder of the paper is organized as follows. Notations and assumptions are described in Section 2. Section 3 constructs and analyzes the inventory model starting with shortage. Some numerical examples to illustrate the solution procedure are provided; then sensitivity analysis of the major parameters is also carried out in Section 4. Finally, a conclusion is presented.

#### 2. Notations and Assumptions

The fundamental notations and assumptions used in this paper are given as follows.(i) is the inventory level at time , , where is the length of inventory cycle.(ii) denotes the replenishment time, and denotes the optimal .(iii) is the maximum inventory level for each ordering cycle.(iv) denotes the optimal ordering quantity.(v) denotes the fixed cost per order.(vi) is the cost of each deteriorated item, is the unit inventory holding cost per unit time, is the unit shortage cost per unit time, and is the unit lost sales cost.(vii), , denotes the average total cost per unit time under different conditions, respectively.(viii) denotes the average total cost per unit time.(ix)The demand rate, , which is positive and consecutive, is assumed to be a trapezoidal type function of time; that is, where is time point changing from the increasing demand function to the constant demand , and is time point changing from the constant demand to the decreasing demand function .(x)The replenishment rate is infinite; that is, replenishment is instantaneous.(xi)The time to deterioration of the item is distributed as Weibull ; that is, the deterioration rate is , where is the scale parameter, , and is the shape parameter, (xii)Shortage is allowed and let be fraction of shortages backlogged, where is the waiting time up to the next replenishment and is a tiny positive constant; that is, .

#### 3. Inventory Model Starting with Shortage

The system of model begins with shortage and until the shortage achieves its maximum; then replenishment occurs at time . All of the shortage demand during is partial backlogged (Figure 1). At , the total number of backlogged items will be satisfied by the replenishment and the inventory level up to , while the deteriorating occurs during the time of . According to the notations and assumptions mentioned above, the model is described as shown in Figure 1:with boundary condition , , .