The major challenge facing producers of perishable or decaying products is to plan for the production and inventory levels of their products to meet random demands over the sale cycle dictated by the products’ limited life. Generally, such products are made by a processing system consisting of a series of processing stages. We assume that the system comprises n−1n−1 serial processing facilities, so it includes n stocking echelons, which hold various kinds of inventory items in the form of raw materials (echelon 1), different grades of work-in-process (echelons 2 to n−1n−1) and the finished product (echelon n ). The last echelon n is the retailer. Since there are usually more than one retailer for the product, echelon n contains m≥1m≥1 retailers. A diagrammatic representation of such a processing system is given in Fig. 1.
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Fig. 1.
A serial production–inventory system comprising n stocking echelons and m retailers.
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Because of the limited product life, there are no opportunities for the producer to reorder more raw materials from the supplier, neither is there time for the producer to modify the initial inventory levels of the different grades of work-in-process at the stocking echelons. Since the demand for the product is random and the product has a limited life, if the producer over-produces, he will have to liquidate all excess inventory held at the stocking echelons at the end of the sale cycle. This will reduce his profit over the sale cycle. However, the liquidation costs will be lower for work-in-process inventory items held at lower stocking echelons as they have been added less value. On the other hand, if the producer under-produces, he can convert some of the work-in-process inventory into the final product to meet the excess demand within the sale cycle. However, the profit over the sale cycle is lower again in this case because extra costs will be incurred to expedite the orders, and a higher production cost will result from converting the partially decayed work-in-process inventory into the final product (e.g., more energy and additives are needed). Therefore, the optimal inventory level at each echelon needs to be determined at the start of each sale cycle with the objective of maximizing the expected profit over the sale cycle.
Practical examples of such a model is the production of oriental health and herbal food, such as readily served swallow nest soup. The raw material is typically a protein-based solution, which has a short life of 5–7 days that defines the duration of the sale cycle. Ingredients such as herbs, vitamins, nutrients, scents, colouring, additives, etc. are added at different processing stages organized in a serial configuration. Each processing stage provides a unique treatment, which takes a short time compared with the cycle duration, and produces a different grade of work-in-process. The grade of the work-in-process increases with the echelon level. So it will be more profitable, and it will take a shorter time, to convert the work-in-process inventory at higher echelons to the final product, if demand for the product exceeds supply within the sale cycle. However, since the work-in-process at higher echelons has a higher added value, it will be more costly to liquidate any such excess inventory at the end of the sale cycle.
Generally, there are two kinds of inventory policies used for production and inventory planning. One is produce-in-advance (PIA) and the other is produce-to-order (PTO). Under PTO, if a shortage of the finished product occurs, the excess demand can be partially satisfied by converting some of the available raw materials and work-in-process into products. Since customers are usually unwilling to wait for a long time, the fraction of the work-in-process converted to meet the excess demand is smaller than the unsatisfied demand that is carried over to a lower echelon. Thus, adopting the PTO inventory policy will result in a loss of the customers who are unwilling to wait. Moreover, as far as production is concerned, the unit cost of a finished product under PTO is higher than that under PIA, because for the latter there is no need for rush orders, and production can be planned and implemented well in advance. On the other hand, if the PIA inventory policy is used, liquidation loss will be incurred if supply exceeds demand because of the wasted processing costs and the disposal of unused inventory. Therefore, as observed by Eynan and Rosenblatt (1995), a composite of PIA and PTO is the dominating policy.
Whitin (1953) was the first to present such a single-period problem with stochastic demand and two echelons. Hariga (1998) considered a model composed of n echelons. Subsequently, many researchers have generalized his model to different forms (see, for example, Bivin, 2003, Wong et al., 2009 and Banerjee, 2005).
In Section 2 we introduce the notation used throughout the paper and study a serial mixed PTO and PIA inventory model comprising n echelons and m retailers. We formulate the general composite model under two kinds of demand distribution, namely uniformly distributed demands and normally distributed demands, and show that the problem can be reduced to a general constrained optimization problem in Section 3. Finally, we conclude the paper in Section 4.
