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Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : International Journal of Production Economics, Volume 65, Issue 2, 20 April 2000, Pages 201–216
The single-period problem (SPP), also known as the newsboy or newsvendor problem, is to find the order quantity which maximizes the expected profit in a single-period probabilistic demand framework. Previous extensions to the SPP include, in separate models, the simultaneous determination of the optimal price and quantity when demand is price-dependent, and the determination of the optimal order quantity when progressive discounts with preset prices are used to sell excess inventory. In this paper, we extend the SPP to the case in which demand is price-dependent and multiple discounts with prices under the control of the newsvendor are used to sell excess inventory. First, we develop two algorithms for determining the optimal number of discounts under fixed discounting cost for a given order quantity and realization of demand. Then, we identify the optimal order quantity before any demand is realized. We also analyze the joint determination of the order quantity and initial price. We illustrate the models and provide some insights using numerical examples.
The classical single-period problem (SPP) is to find a product's order quantity which maximizes the expected profit in a probabilistic demand framework. The SPP model assumes that if any inventory remains at the end of the period, one discount is used to sell it or it is disposed of . If the order quantity is smaller than the realized demand, the newsvendor, hereafter NV, forgoes some profit. If the order quantity is larger than the realized demand, the NV losses some money because he/she has to discount the remaining inventory to a price below cost. The SPP is reflective of many real-life situations and is often used to aid decision making in the fashion and sporting industries, both at the manufacturing and retail levels . The SPP can also be used in managing capacity and evaluating advanced booking of orders in service industries such as airlines, hotels, etc. . Several researchers have suggested SPP extensions in which demand is price dependent , , , , ,  and . Whitin  assumed that the expected demand is a function of price and using incremental analysis, derived the necessary optimality condition. He then provided closed-form expressions for the optimal price, which is used to find the optimal order quantity for a demand with a rectangular distribution. Mills  also assumed demand to be a random variable with an expected value that is decreasing in price and with constant variance. Mills derived the necessary optimality conditions and provided further analysis for the case of demand with rectangular distribution. Lau and Lau (LL)  introduced a model in which the NV has the option of decreasing price in order to increase demand. LL analyzed two cases for demand: (a) Case A: Demand is given by a simple homoscedastic regression model x=a−bP+ε, where a and b are constants, x is the quantity demanded, P is unit price, and ε is normally distributed. The above equation implies a normally distributed demand with an expected value which decreases linearly with unit price. (b) Case B: Demand distribution is constructed using a combination of statistical data analysis and experts’ subjective estimates. The `method of moments’ was used to fit the four-parameter beta distribution to estimate demand. For case A, LL showed that the expected profit is unimodal and thus the golden section method can be used for maximization. For case B, there is no guarantee the expected profit is unimodal. Thus, LL developed a search procedure for identifying local maximums. LL also solved the problem under the objective of maximizing the probability of achieving a target profit and considered both zero and positive shortage cost cases. For zero shortage cost and demand given by case A, LL derived closed-form solutions for the optimal order quantity and optimal price. For zero shortage cost and demand given by case B, LL developed a procedure for computing the probability of achieving a target profit and used a search procedure for finding a good solution. For positive shortage cost and demand given by cases A or B, the probability of achieving a target profit may not be unimodal. LL developed procedures for computing the probability of achieving a target profit and identifying a good solution. Polatoglu  also considered the simultaneous pricing and procurement decisions. Polatoglu identified few special cases of the demand process addressed in the literature: (i) an additive model in which the demand at price P is x(P)=μ(P)+ε, where μ(P) is the mean demand as a function of price, and ε is a random variable with a known distribution and E[ε]=0, (ii) a multiplicative model in which x(P)=μ(P)ε where E[ε]=1, (iii) a riskless model in which X(P)=μ(P). Polatoglu analyzed the SPP under general demand uncertainty to reveal the fundamental properties of the model independent of the demand pattern. Polatoglu assumed an initial inventory of I,μ(P) is a monotone decreasing function of P on (0, ∞), and a fixed ordering cost of k. For linear expected demand, (μ(P)=a−bP, where a,b>0) Polatoglu proved the unimodality of the expected profit for uniformly distributed additive demand and exponentially distributed multiplicative demand. Khouja  solved an SPP in which multiple discounts are used to sell excess inventory. In this model, retailers progressively increase the discount until all excess inventory is sold. The product is initially offered at the regular price P0. After some time, if any inventory remains the price is reduced to View the MathML source. In general, the prices are View the MathML source, where Pi>Pi+1. The amount demanded at each Pi is assumed to be a multiple View the MathML source of the demand at the regular price P0. Khouja solved the problem under two objectives: (a) maximizing the expected profit, and (b) maximizing the probability of achieving a target profit. Khouja showed that the expected profit is concave and derived the sufficient optimality condition for the order quantity. For maximizing the probability of achieving a target profit, Khouja provided closed-form expression for the optimal order quantity. Khouja  developed an algorithm for identifying the optimal order quantity for the multi-discount SPP when the supplier offers the NV an all-units quantity discount. Khouja and Mehrez  provided a solution algorithm to the multi-product multi-discount constrained SPP. The above models may not capture some actual problems facing many NVs. While NV may consider the demand–price relationship is determining the order quantity, he/she still faces the problem of what to do with excess inventory when the order quantity exceeds the realized demand. Most retailers, for example, do not use a single discount to sell excess inventory as assumed in the classical SPP. The assumption of multiple discounts proposed by Khouja  contributes a step toward solving this problem. However, Khouja's model is limited in that it assumes that the discount prices are preset and are not part of the decisions of the NV. The model also assumes that the quantity sold at each discount price is a given multiple of the quantity sold at the initial price without any assumptions about demand–price relationship. Finally, the model assumes that discounting a product does not incur a fixed cost, whereas many retailers incur a fixed discounting cost resulting from the need to advertise the discount and markdown the discounted items. In this paper, we extend the SPP to the case in which: 1. demand is price dependent, 2. multiple discount prices are used to sell excess inventory, 3. the discount prices used to sell excess inventory are under the control of the NV, and 4. there is a positive setup cost associated with discounting a product due to the costs of advertising and marking down the discounted items. The resulting problem is composed of two smaller problems. In the first problem, for a given realization of demand, a given demand–price relationship, and a given order quantity, the NV must determine the optimal discounting scheme. This problem will be referred to as the discounting problem. In the second problem, the NV must determine the order quantity which maximizes the expected profit prior to any demand being realized. This problem will be referred to as the order quantity problem. In the next section, we review the classical SSP. In Section 3, we analyze the discounting problem and develop two algorithms for determining the optimal discounting scheme under two different assumptions about the behavior of the NV. In Section 4, we solve the optimal order quantity problem. In Section 5, we analyze the joint quantity and initial pricing decisions. We conclude in Section 6.