有效期限內多階段訂購(T,Q) 報童模式最適決策之研究

A Study on Optimal Decision Analysis of (T,Q) Newsboy Model under Multi-stage Ordering Strategy

黃允成(Yun-Cheng Huang) ； 陳貞秀(Cheng-Hsiu Cheng)

《管理學報》 21卷3期 (2004/06) Pp. 345-362

https://doi.org/10.6504/JOM.2004.21.03.04

摘要

本文主要係針對多階段訂購之報童模式，提出一具多元限制條件下之數學規劃模型，以進行最適訂購週期與最適訂購量之決策分析探討。為符合實務運作之考量，本研究將次要市場之價格函數納入總期望利潤Tπ(T,Q) 模式中，建構含有限制條件之報童模式，並利用數值分析演算法，求解總期望利潤最大化下之最適訂購週期與最適訂購量。接著，進行各相關參數對總期望利潤函數之敏感度分析，以進一步了解相關參數對總期望利潤函數之影響，且以一數值範例闡述本研究主題的意義及對推論做一驗證，最後，列出六點結論供後續研究及實務應用之參考。

關鍵字

報童模式 ； 多階段訂購 ； 價格函數 ； 限制式

並列摘要

The researches of perishable goods are more popular and important topics, because the perishable goods has shorter expiry period than durable goods. Such as newspaper, sea foods, meats, cakes, fresh milk, fast foods, flowers, vegetables, fruits, festival cards and so on, all those merchandises have limited lifetime, before its expiry date, it can be sold by higher price than beyond the date. If the retailer orders too many quantities of perishable goods to sell, he or she will suffer big loss. On the other hand, if the retailer orders too few to sell, he or she will make fewer profits than optimal ordering quantity. The traditional newsboy model is a basic and simple mathematical model to find out the optimal ordering quantity to maximize the total expected profit. It means that if the purchasing quantity over the optimal quantity, the expected marginal loss will be great than the expected marginal profit, in the other hand, if the purchasing quantity is less than the optimal quantity, the expected marginal profit will be great than the expected loss. The assumptions of newsboy problem as follows:
1. The lifetime of goods is only one period, out of the period it must be processed as a scrap. 2.The demand of the perishable goods before expiry date is a random variable under a given price. 3. The unit cost and scrap value are given, it means that the marginal profit and marginal loss are known. 4. The ordering period is equal to the expiry period. 5. No any resources constraint. But, in real situation, many perishable goods its expiry period is usually great than the ordering period, it means that in expiry period, the retailer must order one more time. When the new goods is coming, the old ones must be handled. In general, there are two ways to handle the old goods, the one is dropping the selling price, and the other is transferring it to secondary market. Besides, when the old goods but not over the expiry date is delivered to the secondary market, the selling price in secondary market is a function of holding time in primary market. The longer it is hold in primary market, the lower price it has in secondary market, and the demand in secondary market also a random variable under any given prices. Furthermore, we thought that the resources constraints are needed to meet the real conditions. Based on above modifications, we proposed a nonlinear mathematical model with constraints. We applied the optimization technique and numerical analysis method to search for the optimal ordering period and optimal purchasing quantity to maximize the total expected profit. Sensitivity analyses are also taken to realize the effect of the value change of parameters. The parameters include the selling price in primary market, the unit cost of goods, the ordering cost per time, the holding cost per unit goods and the shortage cost per unit per time. We simulated a numerical example to demonstrate the feasibility of the proposed model. In this paper, we obtain six main conclusions as follows: 1. Given the ordering period, the total expected profit Tπ(Q\T) is a concave function in purchasing quantity. Therefore, the maximal value is existence. 2. Given all other conditions are fixed, the selling price in primary market is proportional to the total expected profit. It means that when the selling price is increasing, the total expected profit is also increased. 3. Given all other conditions are fixed, the unit cost of goods is inverse proportional to the total expected profit. It means that when the unit cost is increasing, the total expected profit is decreased. 4. Given all other conditions are fixed, the ordering cost per time is inverse proportional to the total expected profit, and the ordering cost per time is proportional to ordering period. It means that when the ordering cost per time is increasing, the total expected profit in decreased, and the optimal ordering period is increased. 5. Given all other conditions are fixed, the holding cost per unit is inverse proportional to the total expected profit, optimal ordering period and optimal purchasing quantity. It means that when the holding cost per unit is increasing, the total expected profit, optimal ordering period and optimal purchasing quantity are decreased. 6. Given all other conditions are fixed, the shortage cost per unit per time is inverse proportional to the total expected profit and optimal ordering period, however, the shortage cost per unit per time is proportional to the total purchasing quantity. It means that when the shortage cost per unit per time is increasing, the total expected profit and optimal ordering period are decreased, but the total purchasing quantity is increased.

並列關鍵字

Newsboy problem ； perishable goods ； expiry period ； optimization technique ； numerical analysis

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