Journal of Industrial Engineering, University of Tehran, Special Issue, 2011, PP. 69-77 69
Inventory Control of Perishable Items in a Two-Echelon Supply Chain

Fariborz Jolai *1, Elmira Gheisariha1 and Farnaz Nojavan1
1 Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran
(Received 14 November 2010, Accepted 24 April 2011)

Abstract
In this paper, we develop an inventory model for perishable items with random lifetime in a two-echelon production-distribution system. There is a manufacturer at the first stage that produces its product with a constant rate. Deterioration in this stage is modeled via a twoparameter Weibull distribution. At the second stage, the retailer places the order and receives the product instantly. The deterioration rate at this stage is a three-parameter Weibull distribution, which its initial value depends on the time the product has spent in the first stage before being transferred. The behavior of different key parameters of the model is analyzed using numerical studies.

Keywords: Inventory control, Deteriorating items, Production-distribution systems, Twoechelon supply chains

Introduction
We develop a mathematical model for a deterioration is modeled via a two-parameter two-echelon supply chain with perishable Weibull distribution similar to Covert and items. Products are produced in the first Philip [3] and at the second echelon stage and are sent to the second stage to be deterioration rate follows a three -parameter stocked. Different deterioration functions are Weibull distribution as Philip [8].
imposed for two stages. Our purpose is to Studies on integrated inventory models find the optimal policy, i.e., the optimal with perishable items when the production time, production cycle, inventory manufacturers and the retailers coordinate level and delivery cycle of the system. The their production and ordering policies have wide variety of perishable items and the received much attention from researchers in enormous number of factories and recent years. Yang and Wee [12] considered manufacturers dealing with these products as a two-echelon system with one manufacturer well as the application of two-echelon supply and several customers with constant chains for their production and distribution production rate, deterioration rate and systems is the main motivation of this study. demand. Wee et al. [11] cited two possible
Integrating production planning and flaws in the cost function of Wee and Yang’s distribution planning in a supply chain is model and give a proposal to eradicate the necessary to achieve its optimal flaws. Rau et al. [9] proposed a model performance. These systems are called similar to that of Yang and Wee [12] with Explicit Production-Distribution (EPD) the only difference that the deterioration rate systems. Among different groups of EPD is set to be exponential. Yang and Wee [13] defined by Chen [1], joint Lot-Sizing and developed a multi-lot-size production and Finished-Product Delivery Problem is the inventory model of deteriorating items with closest to our study. constant production and demand rates. Lo et
In this study we assume production period al [7] derive an optimal solution for an is equal to distribution period and items lose integrated production-inventory model with their values in both stages. The deterioration imperfect production processes and Weibull functions are stochastic and follow Weibull distribution deterioration under inflation. distribution. At the first echelon, Cheng and Wee [2] studied a production-
* Corresponding author: Tel: +98- 21- 82081067 Fax: +98- 21- 88013102 Email: [email protected]

inventory deterioration model considering pricing policy, warranty period, imperfect production and stock dependent demand. Lee and Hsu [6] study the two-warehouse inventory control model for deteriorating items with finite replenishment rate. Wang et al [10] empirically showed different deterioration rates in each echelon affect performances of individuals and integrated inventory policies. To the best of our knowledge, this is the first study considering perishable items in a two-echelon supply chain having production at the first stage and warehouse at the second stage.
The following assumptions are made in developing the mathematical model:

Demand rate is constant.
The product is produced on one production line or production machine. There is constant set up cost at the beginning of each production cycle. 3- Inventory control is continuous.
Lead-time is constant and zero
At the first stage there is only one producer.
At the second stage there is only one retailer.
Shortage is not allowed.
Deterioration of items begins at the first stage right after being produced.
The rate of deterioration and its parameters are known for both stages.
Replacing or repairing the deteriorated items is not allowed.
Our purpose is to find the optimal policy, i.e., the optimal production time, production cycle, inventory level and delivery cycle of the system.

The rest of this paper is organized as follows: In section 2 we develop a mathematical problem based on our assumptions. In Section 3 we solve the proposed model using numerical analysis and investigate the behavior of optimal solution as different parameters of the model change. Conclusion is presented in Section 4.

Mathematical Model
For describing the model and its solution we need the following notations, see figure 1:

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