Jointly pricing and ordering for a multi-product multi-constraint newsvendor problem with supplier quantity discounts

We present an extension to the multi-product newsvendor problem by incorporating the retailer’s pricing decision as well as considering supplier quantity discount. The objective is to maximize the expected profit of the retailer through jointly determining the ordering quantities and selling prices for the products, subject to multiple capacity constraints. We formulate the problem as a Generalized Disjunctive Programming (GDP) model and develop a Lagrangian heuristic approach for its solution. Randomly produced instances involving up to 1000 products are used to test the proposed approach. Computational results show that the Lagrangian heuristic approach can present very good solutions to all instances in reasonable time.     

Optimal order lot sizing and pricing with free shipping

Companies, especially those in e-business, are increasingly offering free shipping to buyers whose order sizes exceed the free shipping quantity. In this paper, given that the supplier offers free shipping, we determine the retailer’s optimal order lot size and the optimal retail price. We explicitly incorporate the supplier’s quantity discount, and transportation cost into the model. We analytically and numerically examine the impacts of free shipping, quantity discount and transportation cost on the retailer’s optimal lot sizing and pricing decisions. We find that free shipping can benefit the supplier, the retailer, and the end customers, and can effectively encourage the retailer to order more of the good, to the extent of ordering a few times of the optimal order lot size without free shipping. The order lot size will increase and the retail price will decrease if the supplier offers proper free shipping.     

基于M-CVaR和生产资金约束的供应链协调研究

本文针对供应商面临生产资金约束的情况,在需求随机条件下研究了由一个风险中性零售商和一个具有风险偏好的供应商组成的二级供应链的协调问题。文章在零售商享有批发价折扣的提前支付和供应商银行信贷两种融资方式下,分别建立了零售商的最优决策模型以及基于M-CVaR测度工具的供应链整体订购决策模型,并给出了采用两种不同融资模式时零售商和供应链整体的最优订购量以及供应链的协调条件,分析了供应商的风险偏好对供应链整体最优决策及协调条件的影响。最后通过算例验证了文章的主要结论。研究表明,当提前支付价格折扣大于临界值时,零售商会选择提前支付货款;提前支付模式下供应链整体的最优订购量大于银行信贷模式;随着供应商的风险偏好由风险规避向风险喜好转变,供应链整体的最优订购量逐渐增大。     

Optimal solutions for the economic lot-sizing problem with multiple suppliers and cost structures

This paper considers a multi-supplier economic lot-sizing problem in which the retailer replenishes his inventory from several suppliers. Each supplier is characterized by one of three types of order cost structures: incremental quantity discount cost structure, multiple set-ups cost structure and all-unit quantity discount cost structure. The problem is challenging due to the mix of different cost structures. For all cases of the problem where each supplier is characterized by one of the first two cost structures, some optimality properties are proposed and optimal algorithms based on dynamic programming are designed. For the case where all suppliers are characterized by all-unit quantity discount cost structures, it is hard to design a polynomial time algorithm by the analyzed optimal properties. However, it is proved that one of its special cases can be solved in polynomial time.     

Heuristics for sourcing from multiple suppliers with alternative quantity discounts

In this paper, we analyze the impact of supplier pricing schemes and supplier capacity limitations on the optimal sourcing policy for a single firm. We consider the situation where the total quantity to be procured for a single period is known by the firm and communicated to the supplier set. In response to this communication, each supplier quotes a price and a capacity limit in terms of a maximum quantity that can be supplied to the buyer. Based on this information, the buyer makes a quantity allocation decision among the suppliers and corresponding to this decision is the choice of a subset of suppliers who will receive an order. Based on industry observations, a variety of supplier pricing schemes from the constituent group of suppliers are analyzed, including linear discounts, incremental units discounts, and all units discounts. Given the complexity of the optimization problem for certain types of pricing schemes, heuristic solution methodologies are developed to identify a quantity allocation decision for the firm. Through an extensive computational comparison, we find that these heuristics generate near-optimal solutions very quickly. Data from a major office products retailer is used to illustrate the resulting sourcing strategies given different pricing schemes and capacity limitations of suppliers in this industry. We find for the case of capacity constrained suppliers, the optimal quantity allocations for two complex pricing schemes (linear discount, and incremental units discount) are such that at most one selected supplier will receive an order quantity that is less than its capacity.     

An incremental discount pricing schedule with multiple customers and single price break

This paper analyzes how a supplier can formulate the terms of a quantity discount pricing schedule. Assuming a single incremental discount system, a model is developed from which an algorithm is derived for an optimal discount schedule. The cases are investigated in which both the discount rate and the break point are unknown and either one is prescribed. A numerical example is presented to illustrate the algorithm developed.     

A comparison of different quantity discount pricing policies in a two-echelon channel with stochastic and asymmetric demand information

In this paper, we study quantity discount pricing policies in a channel of one manufacturer and one retailer. The paper assumes that the channel faces a stochastic price-sensitive demand but the retailer can privately observe the realization of an uncertain demand parameter. The problem is analyzed as a Stackelberg game in which the manufacturer declares quantity discount pricing schemes to the retailer and then the retailer follows by selecting the retail price and associated quantity. Proposed in the paper are four quantity-discount pricing policies: “regular quantity discount”; “fixed percentage discount”; “incremental volume discount” and “fixed marginal-profit-rate discount”. Optimal solutions are derived, and numerical examples are presented to illustrate the efficiency of each discount policy.     

Managing a retail-competition distribution channel with incentive policies

In reality, the multi-retailer channel of distribution has occurred in many industries, such as electronics, appliances, and apparel. The practice of the supplier providing incentives to retailers to promote sales is also quite prevalent in today’s business. This paper considers a two-retailer supply chain in which the supplier provides a cash discount to a specific retailer (a senior company owning more market share and having less pressure on the utilization of capital investment) and a credit period to another (a junior company owning less market share and having more pressure on the utilization of capital investment). The end demand for each channel has a substitute effect in terms of retail prices. The research problem originates from operations of dual-retailer channels in the automotive parts and accessories industry. The objective of the supplier is to determine the cash discount and credit period to maximize his profit. Both retailers determine pricing and ordering policies to maximize profits. This paper applies a Supplier Stackelberg framework to obtain equilibrium solutions for every entity. Numerical analysis is conducted to discuss the influence retailer competition has on the decision behaviors of the supplier and retailers.     

延期支付条件下零售商的最优售价和订货策略

在供应商给予零售商延期支付和现金折扣的优惠政策下,进一步假设产品的年需求量依赖于零售商产品售价的基础上,建立了由一个零售商和一个供应商所构成的库存决策模型,扩展了经典的经济生产批量(EPQ)模型。通过模型的分析求解,可以得出零售商在上述情况下的最优订货周期、最优售价及最优付款时间的简单判定方法。最后,通过算例,验证了模型的可行性,得出了与实际相符的结论。     

The optimal order and payment policies for deteriorating items in discount cash flows analysis under the alternatives of conditionally permissible delay in payments and cash discount

This study investigates a deteriorating inventory problem in which the supplier simultaneously offers the retailer either a conditionally permissible delay in payments or a cash discount. In the case of a conditionally permissible delay, if the retailer orders more than a predetermined quantity, then he/she has a grace period to make the full payment. Otherwise, he/she must pay the payment for goods of certain proportion first while receiving the goods and has a grace period to pay off the rest. As to a cash discount, if the retailer pays for the entire amount of the order within a certain short period, then he/she will receive a cash discount from the supplier. In additions, from a financial standpoint, all cash outflows related to the inventory control that occur at different points of time have different values. Hence, it is necessary to take account of the factor of time value of monetary when drafting the replenishment policy. In a word, this paper uses an alternate approach-discount cash flow (DCF) analysis to establish an inventory problem for deteriorating items in which the supplier provides the retailer either a conditionally permissible delay or a cash discount. We then study the necessary and sufficient conditions for finding the optimal solution. Furthermore, we establish several theoretical results to obtain the solution that provides the smallest present value of all future cash flows. Finally, several numerical examples are given to illustrate the results and obtain some managerial insights.     

Ensuring feasibility in ‘a generalized quantity discount pricing model to increase supplier's profits’

Lee and Rosenblatt examine ordering and quantity discount decisions for a single-vendor–single-buyer system from the perspective of the supplier. By imposing a constraint on the amount of price discount which may be offered to the purchaser they suggest a procedure for determining the optimal solution regarding the underlying planning problem. However, the proposed algorithm can end up with an infeasible solution. In this paper, we analyse the reason why this may happen and provide a fast computational procedure which simultaneously guarantees optimality and feasibility.     

The impact of dynamic pricing on the economic order decision

This paper analyzes the impact of dynamic pricing on the single product economic order decision of a monopolist retailer. Items are procured from an external supplier according to the economic order quantity (EOQ) model and are sold to customers on a single market without competition following the simple monopolist pricing problem. Coordinated decision making of optimal pricing and ordering is influenced by operating costs – including ordering and inventory holding costs – and the demand rate obtained from a price response function. The retailer is allowed to vary the selling price, either in a fixed number of discrete points in time or continuously. While constant and continuous pricing have received much attention in the literature, problems with a limited number of price changes are rather rare. This paper illustrates the benefit of dynamically changing prices to achieve operational efficiency in the EOQ model, that is to trigger high demand rates when inventories are high. We provide structural properties of the optimal time instants when the price should be changed. Taking into account costs for changes in price, it provides numerical guidance on number, timing, and size of price changes during an order cycle. Numerical examples show that the benefits of dynamic pricing in an EOQ framework can be achieved with only a few price changes and that products being unprofitable under static pricing may become profitable under dynamic pricing.     

A stochastic periodic review integrated inventory model involving defective items, backorder price discount, and variable lead time

The purpose of this article is to investigate a stochastic integrated supplier-retailer inventory problem. The model analyzed in this article explores the problem of the protection interval, the backorder price discount, the lead time, and the numbers of shipments from the supplier to the retailer in one production run as control variables to widen applications for an integrated periodic review inventory model. We consider the situation in which the supplier and the retailer establish a long-term strategic partnership and contract to jointly determine the best strategy. We assume that the protection interval demand follows a normal distribution. Our objective is to determine the optimal review period, the optimal backorder price discount, the optimal lead time, and the optimal number of shipments from the supplier to the retailer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, an algorithm of finding the optimal solution is developed. Also, the sensitivity analysis included and a numerical example is given to illustrate the results of the proposed model.     

基于多种运输渠道的经济批量问题的多项式时间算法

为了集中管理, 降低成本, 增强竞争优势, 供应商通常只负责生产,而把产品的配送业务外包给某个配送中心, 由配送中心按照零售商的需求决定发货运输的方式和时间.这样供应商, 配送中心和零售商就构成了二级供应链系统.本文研究的是运输方式为不同参数的全单位数量折扣结构时, 二级供应链系统的经济批量问题.分析了最优解的性质,并对此问题的一种特殊情形给出一个多项式时间算法.     

基于二层信用策略的两货栈库存模型

假设供应商向零售商提供信用支付期的同时,零售商也向顾客提供信用支付期,研究了两货栈的变质物品库存模型,并讨论了模型最优解的唯一性,最后给出了最优订购策略的算法步骤与数值例子.     

考虑最优广告费用及数量折扣的供应链企业间的博弈分析

主要研究由一个制造商和一个销售商组成的二级供应链系统,在随机需求下,考虑广告促销及数量折扣前后,双方如何博弈,如何决策的问题.在制造商Stackelberg策略下,对供应链协调价进行了分析,提供了制造商制定数量折扣策略的思路.     

产能限制下单向替代产品供应链生产、订货与定价研究

考虑一个产能受限的制造商和一个零售商组成的二级供应链上供需双方信息不完全对称的情景,研究具有单向替代的两种产品生产、订货及响应性定价的供应链决策问题,把问题建立成一个三阶段博弈模型。理论分析得出不同条件下制造商的最优产量及零售商的最优订货和定价决策。通过算例验证了供应链上的最优决策及其条件,灵敏度分析表明制造商产能、潜在市场需求及其方差、替代品的价格敏感系数在一定范围内增大有利于供应链系统及各成员利润增加,被替代产品的价格敏感系数增大可能会导致供应链及制造商利润下降。     

考虑合格率的单周期产品供应链决策研究

本文研究由一个供应商和一个经销商组成的关于单周期产品的供应链系统。考虑产品具有随机合格率,并且合格率的分布是供应商的私人信息。在分布函数未知情况下比较研究了供应链三种竞争协调策略:供应商具有合格率私人信息,同时给予经销商价格折扣;供应商告诉经销商合格率分布但是不提供价格折扣;供应商管理经销商库存。比较分析和实例研究说明了三种策略对供应链整体绩效具有明显的优劣和实现帕累托优化的条件。     

Ordering and pricing decisions in a two-echelon supply chain with asymmetric demand information

This article examines coordinated decisions in a decentralized supply chain that consists of one supplier and one retailer, and faces random demand of a single product with a short life cycle. We consider a setting where the retailer has accurate demand information while the supplier does not. Such a problem with asymmetric demand information can be viewed as an extension of the newsboy problem in which both the supplier and the retailer possess the same demand information. Combining the mechanism of sharing demand information and that of quantity discount and return policy enables us to develop three coordinated models in contrast with the basic and uncoordinated model. We are able to show the ordinal relationship among the retailer’s optimal order quantities in these four models under a general form of random demand, and compare the supply chain profits and conduct sensitivity analysis analytically in four models under uniform random demand. We also provide numerical results under normal random demand that bear a resemblance to those under uniform random demand.