在弹性需求和物品易变质条件下数量折扣定价模型

研究在弹性需求和易变质物品条件下,当供应商和零售商独自决策时,供应商如何确定最优数量折扣问题。基于Stackelberg博奕建立了数学模型,证明了零售商的出售价格随供应商出售价格的降低而降低,从而需求量增大。当供应商给予价格折扣时,零售商的利润是增加的。在此基础上,给出了最优数量折扣的计算方法。对供应商和零售商单独决策时,供应商利用数量折扣对供应链进行协调产生的系统利润与供应商和零售商联合决策时的系统利润做了数值分析和比较。结果表明,供应商采用数量折扣的方法使供应链协调是有效的;价格折扣随价格敏感系数的增大而增大,随变质率的增大而减小。     

基于多级折扣价格的易逝品订货策略研究

基于价格折扣条件下的报童模型,研究了分销商面向折拐的需求价格曲线.采用呈等差数列下降的多级折扣价格销售产品,以期望利润最大为目标,给出了求解最优折扣次数的算法,进而得到相应的分销商最优订货量.与需求价格曲线的斜率b不变时相比较,b增大对最优订货量无显著影响;b减小对最优订货量有显著影响.     

Supplier selection with multiple criteria in volume discount environments

Supplier selection is a multi-criteria decision making problem which includes both qualitative and quantitative factors. In order to select the best suppliers it is necessary to make a trade-off between these tangible and intangible factors some of which may conflict. When business volume discounts exist, this problem becomes more complicated as, in these circumstances, buyer should decide about two problems: which suppliers are the best and how much should be purchased from each selected supplier. In this article an integrated approach of analytical hierarchy process improved by rough sets theory and multi-objective mixed integer programming is proposed to simultaneously determine the number of suppliers to employ and the order quantity allocated to these suppliers in the case of multiple sourcing, multiple products, with multiple criteria and with supplier's capacity constraints. In this context, suppliers offer price discounts on total business volume, not on the quantity or variety of products purchased from them. A solution methodology is presented to solve the multi-objective model, and the model is illustrated using two numerical examples.     

An Inventory Model with Planned Shortages and Price Dependent Demand

Previous research has yielded a procedure for a retailer to determine the optimal lot size and selling price when a supplier offers all-unit quantity discounts and demand is a decreasing function of price. In this paper, we extend that research by allowing for shortages. An algorithm is presented that determines the optimal lot size, order level, and selling price for a class of demand functions, including the constant price-elasticity and linear demand functions.     

ECONOMIC ORDER QUANTITIES WITH QUANTITY DISCOUNTS: GRANDMA DOES IT BEST

We examine a new algorithm developed by Kuzdrall and Britney [5] for locating the optimal order quantity in the presence of quantity discounts. Their algorithm, based on a model for the supplier's formulation of the price schedule, involves a regression analysis to identify the supplier's variable cost per unit and the fixed cost that the supplier seeks to recover, followed by an iterative search for the optimum. The authors describe this method as a “convenient alternative to the aimless searching of traditional approaches” [5, p. 101]. We examine the allegation of superiority of their total setup lot-sizing model over the classical method and dispute their claim of superiority.     

A Note on Determining Optimal Selling Price and Lot Size Under All-Unit Quantity Discounts

A recent study conducted by Abad [1] described a method of determining the optimal price and lot size when the supplier offers all-unit quantity discounts. The author developed a procedure using centralized and decentralized approaches. According to the author's suggestion, further study is necessary to determine if the decentralized approach always provides an optimal solution. In the present study, an attempt is made to investigate the uniformity between the centralized and decentralized approaches and to verify whether the decentralized approach always yields optimal solutions.     

Determining Optimal Selling Price and Lot Size When the Supplier Offers All-Unit Quantity Discounts*

This paper is concerned with finding the optimal price and lot size for a retailer purchasing a product for which the supplier offers all-unit quantity discounts. Demand for the product is assumed to be a decreasing function of price, and a procedure is developed for finding the optimal price and lot size for a class of demand functions. The procedure then is applied to two common demand functions: (1) the constant price-elasticity function, and (2) the linear demand function.     

Economic Inventory/Transport Lot,Sizing with Quantity and Freight Rate Discounts

Discounts are a primary marketing mechanism for inducing customers to increase the size of their purchases. Quantity discounts from suppliers and freight discounts from shippers are commonly encountered by organizations. This paper structures quantity and freight discounts into the order size decision in a deterministic economic order quantity system. Optimum lot-sizing algorithms are derived for the dual discount situations of all-units or incremental quantity discounts and all-weight or incremental freight discounts.     

Standard Container Size Discounts and the Single-Period Inventory Problem*

This paper analyzes the problem of choosing the optimal order quantity and its associated number of standard containers making up the order for single-period inventory models under standard container size discounts. A range is determined that contains the optimal order quantity. Two algorithms are presented. The first algorithm solves the general case in which there is no restriction on the types of containers included in an order. The second algorithm solves a more restricted policy that requires the buyer to accept the order with successively smaller container sizes.     

报童问题的最优竞争比策略及其风险补偿模型

报童问题是库存管理中一个重要的模型。对报童问题在不同的目标函数和约束条件下的扩展模型已有了大量的研究。已有研究假设产品的需求分布已知,然而,在实际中完全刻画需求分布信息常常是很困难的,因此,部分信息下的报童模型研究近年越来越受到重视。在竞争比分析方法下,对报童问题进行了讨论,建立概率预期下风险补偿模型用于解决部分信息的报童问题。在确定性预期和基本概率预期下,分别为具有风险偏好的报童设计了最优订购策略,使报童可以根据自己不同的风险容忍度和未来预期选择订购策略。     

考虑汇率变化和运输成本分担的跨国供应链数量折扣契约

建立了跨国供应链的数量折扣模型,模型中考虑了汇率、国际运输成本分担比例等因素。通过分析得到了协调跨国供应链的数量折扣契约,在该契约下供应商、销售商以及供应链系统的利润都得到改善。最后通过实例验证了模型和有关命题,分析了汇率变化对契约参数及供应链各方、供应链系统利润的影响,结果表明:①汇率变化下,数量折扣率和国际运输成本分担比例具有稳定性,变化较大的契约参数是订购量;②即使实现了供应链协调,随着人民币对美元的升值,不仅使中国供应商的利润下降,也使美国销售商以及供应链系统的利润下降。根据实例计算结果与分析,从供应链管理的角度认为,应该保持人民币对美元汇率的稳定。     

An Auction Mechanism for Pricing and Capacity Allocation with Multiple Products

We consider a pricing and short‐term capacity allocation problem in the presence of buyers with orders for bundles of products. The supplier's objective is to maximize her net profit, computed as the difference between the revenue generated through sales of products and the production and inventory holding costs. The objective of each buyer is similarly profit maximization, where a buyer's profit is computed as the difference between the time‐dependent utility of the product bundle he plans to buy, expressed in monetary terms, and the price of the bundle. We assume that bundles' utilities are buyers' private information and address the problem of allocating the facility's output. We directly consider the products that constitute the supplier's output as market goods. We study the case where the supplier follows an anonymous and linear pricing strategy, with extensions that include quantity discounts and time‐dependent product and delivery prices. In this setting, the winner determination problem integrates the capacity allocation and scheduling decisions. We propose an iterative auction mechanism with non‐decreasing prices to solve this complex problem, and present a computational analysis to investigate the efficiency of the proposed method under supplier's different pricing strategies. Our analysis shows that the problem with private information can be effectively solved with the proposed auction mechanism. Furthermore, the results indicate that the auction mechanism achieves more than 80% of the system's profit, and the supplier receives a higher percentage of profit especially when the ratio of demand to available capacity is high.     

Potentially self-defeating: Group buying in a two-tier supply chain

Retailers have an incentive to cooperate in the form of group buying (GB) when a supplier provides quantity discounts, because wholesale price under GB depends on total purchasing quantity rather than individual purchasing (IP) quantity. Most previous studies on GB focus on the benefits that buyers get but ignore the supplier׳s response to GB. In this paper, we take the supplier׳s response into consideration, and present a game model with a single supplier and two symmetric competing retailers in two systems: the retailers purchase individually, and the retailers group buy. Under a general quantity discount schedule, each system has a unique sub-game perfect equilibrium. The comparison between IP and GB suggests that GB may sabotage the benefits of all members in the supply chain (i.e., the supplier, the retailers, and the consumer). Retailers may hold contradictory attitudes toward GB before and after the publishing of the discount schedule. These insights are shown to be robust for the case when more than two retailers are involved, as well as the case when the supplier enjoys economies of scale based on the order volume. We suggest that a mixed discount schedule may help prevent the potential damage of GB. In addition, with significant economies of scale, the supplier and retailers may be better off under GB. Then GB can be a favorable purchasing strategy.     

The Impact of Forecast Errors on Early Order Commitment in a Supply Chain*

Supply chain partnership involves mutual commitments among participating firms. One example is early order commitment, wherein a retailer commits to purchase a fixed‐order quantity and delivery time from a supplier before the real need takes place. This paper explores the value of practicing early order commitment in the supply chain. We investigate the complex interactions between early order commitment and forecast errors by simulating a supply chain with one capacitated supplier and multiple retailers under demand uncertainty. We found that practicing early order commitment can generate significant savings in the supply chain, but the benefits are only valid within a range of order commitment periods. Different components of forecast errors have different cost implications to the supplier and the retailers. The presence of trend in the demand increases the total supply chain cost, but makes early order commitment more appealing. The more retailers sharing the same supplier, the more valuable for the supply chain to practice early order commitment. Except in cases where little capacity cushion is available, our findings are relatively consistent in the environments where cost structure, number of retailers, capacity utilization, and capacity policy are varied.     

A Heuristic Algorithm for the Capacitated Multiple Supplier Inventory Grouping Problem

This paper presents and solves a model for the multiple supplier inventory grouping problem, which involves the minimization of logistics costs for a firm that has multiple suppliers with capacity limitations. The costs included in the model are purchasing, transportation, ordering, and inventory holding, while the firm's objective is to determine the optimal flows and groups of commodities from each supplier. We present an algorithm, which combines subgradient optimization and a primal heuristic, to quickly solve the multiple supplier inventory grouping problem. Our algorithm is tested extensively on problems of various sizes and structures, and its performance is compared to that of OSL, a state-of-the-art integer programming code. The computational results indicate that our approach is extremely efficient for solving the multiple supplier inventory grouping problem.     

Quantity Discount‐Based Inventory Coordination: Effectiveness and Critical Environmental Factors

It is understood that quantity discounts provide a practical foundation for coordinating inventory decisions in supply chains. The primary objective of this research is to test, under a variety of environmental conditions, the effectiveness of quantity discounts as an inventory coordination mechanism between a buyer and a supplier. A comprehensive simulation experiment with anova has been designed to investigate the impacts of (1) choice of quantity discount‐based inventory coordination policies, (2) magnitude of demand variation, (3) buyer's and supplier's relative inventory cost structure, and (4) buyer's economic time‐between‐orders on the effectiveness of supply chain inventory coordination. The analytical results confirm that the quantity discount policies have managerial properties as a mediator for inventory coordination. The results also show that the performance of quantity discount‐based inventory coordination policies is influenced significantly by environmental factors, such as the magnitude of demand variation, the buyer's and the supplier's inventory cost structure, and the buyer's economic time‐between‐orders.     

Determining Order Quantity and Selling Price by Geometric Programming: Optimal Solution,Bounds, and Sensitivity*

This paper presents a geometric programming (GP) approach to finding a profit-maximizing selling price and order quantity for a retailer. Demand is treated as a nonlinear function of price with a constant elasticity. The proposed GP approach finds optimal solutions for both no-quantity discounts and continuous quantity discounts cases. This approach is superior to the traditional approaches of solving a system of nonlinear equations. Since the profit function is not concave, the traditional approaches may require an exhaustive search, especially for the continuous discounts schedule case. By applying readily available theories in GP, we easily can find global optimal solutions for both cases. More importantly, the GP approach provides lower and upper bounds on the optimal profit level and sensitivity results which are unavailable from the traditional approaches. These bounding and sensitivity results are further utilized to provide additional important managerial implications on pricing and lot-sizing policies.     

The Airline Discount Fare Allocation Problem

This paper describes a useful extension of the well-known, single-period inventory or newsboy problem. Given a fixed number of identical seats available on a scheduled airline flight, what percentage should be offered for early sale at a predetermined discount fare and what percentage reserved for later sale at a higher full fare? This two-tiered pricing strategy with early discount pricing might be appropriate in any situation in which the price sensitivity of the inventoried items decreases as the end of the period approaches. Similar to the newsboy problem solution, the decision rule that maximizes expected profit is expressed as a simple function of the percentage difference in the two fares and two carefully defined probabilities.     

Optimal Quantity Discount Design with Limited Information Sharing*

In this article we address the optimal quantity discount design problem of a supplier in a two‐stage supply chain where the supplier and the buyer share annual demand information only. The supply chain faces a constant deterministic demand that is not price sensitive and operates with fixed setup costs in both stages. We show that the supplier can actually moderate a cost‐minimizing buyer to order in quantities different than the buyer's optimal order quantity in the traditional setting and develop a multi‐breakpoint quantity discount scheme that maximizes supplier's expected net savings. The proposed multi‐breakpoint discount scheme can be easily computed from the available information and, while also maximizing the supplier's net savings, is very effective in achieving high levels of supply chain coordination efficiency in the presence of limited information.     

Single vs. multiple objective supplier selection in a make to order environment

The problem of allocation of orders for custom parts among suppliers in make to order manufacturing is formulated as a single- or multi-objective mixed integer program. Given a set of customer orders for products, the decision maker needs to decide from which supplier to purchase custom parts required for each customer order. The selection of suppliers is based on price and quality of purchased parts and reliability of on time delivery. The risk of defective or unreliable supplies is controlled by the maximum number of delivery patterns (combinations of suppliers delivery dates) for which the average defect rate or late delivery rate can be unacceptable. Furthermore, the quantity or business volume discounts offered by the suppliers are considered. Numerical examples are presented and some computational results are reported.