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In this study, we focus on optimally determining lot-sizing policies for a deteriorating item among all the partners in a supply chain system with a single-vendor and multiple-buyers so as to minimize the average total costs. We revise Yang and Wee's [1] model using the Fourier series to precisely estimate the vendor's inventory holding costs. Also, we transform our revised model into a more concise version by applying anapproximation to the exponential terms in the objective function. In order to solve this problem, we analyze the optimality structure of our revised model and derive several interesting properties. By utilizing our theoretical results, we propose a search algorithm that can efficiently solve the optimal solution. Based on our numerical experiments, we show that the proposed algorithm outperforms the existing solution approach in the literature, especially when the number of buyers is larger in the supply chain system.

This study aims at optimally coordinating lot-sizing policies for a deteriorating item among all the partners in a supply chain system with a single-vendor and multiple-buyers so as to minimize the average total costs. The vendor (which is a producer) d i s t r i b u t e s a deteriorating item to the buyers. We assume that the replenishment cycle of each buyer, denoted by T~, must be an integer-ratio fraction of the replenishment cycle of the vendor (denoted by T). That is, Ti = T/ni and n~ E { 1 , 2 , 3 , 4 , . . . } for all i. Deterioration occurs for most products in the real world. (We note that deterioration means that a product fails to regularly implement its function.) Ghare and Schrader [2] classified the deteriorating properties of inventory into three categories: (1) direct spoilage, e.g., vegetable, fruit, and fresh food, etc.; (2) physical depletion, e.g., gasoline and alcohol, etc.; (3) deterioration such as radiation changing, negative spoiling, and loss of efficacy in inventory, e.g., electronic components and medicine. From another point of view, deterioration can also be classified by the time-value or the products' life of inventory. Raafat [3] categorized deterioration by the time-value of inventory. (1) Utility Constant: Its utility does not change significantly as time passes within its valid usage period, e.g., liquid medicine. (2) Utility Increasing: Its utility increases as time passes, e.g., some alcoholic drinks. (3) Utility Decreasing: Its utility decreases as time passes, e.g., vegetables, fruits, and fresh foods, etc. On the other hand, Nahmias [4] classified deterioration by products' lifetime of inventory. (1) Fixed Lifetime: Products' lifetime is prespecified and its lifetime is independent of the deteriorated factors; therefore, it is called time-independent deterioration. In fact, the utility of these products decreases during its lifetime, and when passing its lifetime, the product will perish completely and become of no value, e.g., milk, inventory in blood bank, and food, etc. (2) Random Lifetime: There is no specified lifetime for these products. The lifetime for these products is assumed as a random variable, and its probability distribution could be a gamma distribution, Weibull distribution, or an exponential distribution, etc. Products that keep deteriorating in some probability distribution are also the so-called timedependent deteriorating products, e.g., electronic components, chemicals, and medicine, etc. The scope of this study covers those deteriorating products being classified as utility decreasing (as regards their time-value) and also as random lifetime (as regards their lifetime). Furthermore, we assume the deterioration of inventory to be exponentially distributed. Since deterioration will incur additional costs for inventory storage, it could distort the decisionmaking scenario and mislead the decision makers' replenishment strategy if one ignores the deteriorating factor in their inventory models. However, most of the inventory models have considered the deteriorating factor as single-product or single-vendor single-buyer models, for instance, [5-9]. In the literature, the present authors have found very few articles that studied inventory models with multiple deteriorating products or single-vendor multibuyer models. Hwang and Moon [10] presented a production-inventory model that integrates the production planning of two products produced on a single facility and the raw material may be deteriorating over time with a constant rate. Kar et al. [11] proposed an inventory model for several continuously deteriorating products, sold from two shops under single management dealing with limitations on investment and total floor-space area. On the other hand, the one-warehouse multiretailer problem is one of the most representative studies in the integrated lot-sizing problems. One may refer to the following papers for further reference, namely, [12-16], etc. We note that these papers do not take into account the deteriorating factor in their mathematical models. Recently, some researchers have been working on the integrated lot-sizing models for a deteriorating item in single-vendor and multiple-buyers production-inventory systems. One may refer to [1,17-21] for reference. These inventory models share some common characteristics with the multiple-product inventory models though there still exist significant differences between them, especially in their solution approaches. In this study, we focus on solving the inventory control problem presented in Yang and Wee's [1] paper. First, we review the assumptions in Yang and Wee's model as follows. There are totally N buyers in this supply chain system. Customer demand occurs with each buyer at a constant rate. A holding cost is incurred for each unit of finished product per unit time stored, and a setup cost is charged for each order placed with the vendor and with/by each buyer. The demand rates, holding cost rates, and setup costs are stationary for the vendor and each buyer. The production rate of the production facility is finite, and it is greater than the sum of all the buyer's demands. And, no backlogging is permitted anytime in the system. Finally, the replenishment of orders is assumed to be instantaneous (though this assumption can be relaxed by adding lead times to the orders). Also, we define some notation used in Yang and Wee's model as follows. We denote T as tile length of the replenishment cycle. And, T = T1 + T2, where Tt and T2 are the length of production time and the length of nonproduction time in the replenishment cycle, respectively. We let the unit usage of raw materials per finished product be f. We set the ordering cost of raw material as Kin. The set-up cost Kp is incurred each time when the vendor starts one run of production. And, the ordering cost Kb incurs for each buyer as an order is placed. We denote di as the demand rate of buyer i and p as the production rate of the production facility at the vendor. We let the holding cost per dollar per unit time for raw material be Fro. And, let Fp and Fb be the holding cost rates of the finished product at the vendor and the buyer, respectively. We denote the unit price of raw material as Cm. Also, Cp and Cb are the unit prices of the finished product for the vendor and the buyer, respectively. And, we denote Om and 0 as the deterioration rate of the raw material and the finished product, respectively. The rest of the paper is organized as follows. In order to solve "fang and V~ree's [1] inventory control problem, we first present a revised model and conduct full theoretical analysis on the optimality structure of the optimal objective value curve in Section 2. Then, we employ our theoretical results to devise a search algorithm that solves the optimal solution for the singlevendor multibuyers system in Section 3. Next, in the first part of Section 4, we present a numerical example to demonstrate the implementation of the proposed search algorithm. Also, based on our random experiments, we show that our search algorithm outperforms Yang and Wee's heuristic in the second part of Section 4. Finally, we address our concluding remarks in Section 5.

In this study, we focus on optimally determining lot-sizing policies for a deteriorating item among all the partners in a supply chain system so as to minimize the average total costs. We revise Yang and Wee's [1] model using the Fourier series to precisely estimate the vendor's inventory holding costs. Also, we transform our revised model into a more concise version by applying an approximation to the exponential terms in the objective function. In order to solve this problem, we analyze the optimality structure of our revised model and derive several interesting properties. By utilizing our theoretical results, we propose a search algorithm that can efficiently solve the optimal solution. Based on our numerical experiments, we show that the proposed algorithm outperforms the existing solution approach in the literature, especially when the number of buyers is larger in this supply chain system.