Improving a supplier's quantity discount gain from many different buyers

We consider the pricing and inventory decisions of a vendor who supplies a single product to multiple heterogeneous buyers. The problem is analyzed as a Stackelberg game in which the vendor acts as the leader by announcing its pricing policy to all the buyers in advance and the buyers act as followers by choosing their order quantity and the sassociated purchasing price independently under the vendors' pricing scheme. We propose in this paper a pricing policy for the vendor that offers price discounts based on the percentage increase from a buyers' order quantity before discount. The proposed policy is defined as a discrete all-unit quantity discount schedule with many break points. We show that: (i) the proposed policy offers a higher price discount to a buyer ordering a larger quantity and hence complies with general fair trade laws; (ii) an explicit solution is obtained for the vendors' optimal decision; and (iii) although suppliers in reality normally offer price discounts based on a buyers' unit increase in order quantity, the proposed policy is superior for the vendor when there are many different buyers. Other benefits of the proposed pricing policy are demonstrated by numerical examples.     

Approximations for multiproduct contracts with stochastic demands and business volume discounts: single supplier case

In this paper we analyze contracts for multiple products when the supplier offers business volume discounts. The contract takes the form of price discounts for total minimum dollar volume commitments over the horizon with flexibility to adjust the total purchases upwards by a fixed percentage about this minimum commitment. The optimal policy could be complex since it involves the solution of a constrained multi-period multi-product dynamic program. We suggest approximations that give us an upper bound. To develop this upper bound, we present the optimal policy for a similar contract for a single product problem. We then develop a lower bound for the multiproduct problem and show, numerically, that the gap is small. Given the approximations standard single period resource allocation algorithms can be used to solve the problem. The resulting solution methodology is fast and permits evaluation of various what-if scenarios.     

短缺量拖后率与价格折扣相关变质品VMI模型

考虑短缺量拖后率与订货商给予顾客的价格折扣正线性相关,提出一种需求指数时变的变质物品供应商管理库存模型,研究了供应链最优库存策略。数值仿真和主要参数灵敏度分析表明,订货商在缺货期间向顾客提供价格折扣有利于降低丢单损失和系统库存总成本;当需求增长因子变化时,VMI系统应保持库存控制策略不变;当变质系数和拖后率上限变大时,VMI系统则应保持订货商补货次数和价格折扣不变,同时适当降低订货商服务水平。     

Coordinating orders in supply chains through price discounts

In this paper, we examine the issue of order coordination between a supplier and multiple retailers in a decentralized multi-echelon inventory/distribution system where the supplier provides a product to multiple retailers who experience static demand and standard inventory costs. Specifically, we propose and analyze a new policy where a manufacturer, who outsources production to an Original Equipment Manufacturer, offers a price discount to retailers when they coordinate the timing of their orders with the manufacturer's order cycle. We show that our proposed policy can lead to more efficient supply chains under certain conditions, and present a straightforward method for finding the manufacturer's optimal price discount in this decentralized supply chain. A numerical experiment illustrates the managerial implications of our model as well as conditions when a manufacturer should consider adopting such a policy.     

DISCOUNTING DECISIONS IN A SUPPLIER-BUYER RELATIONSHIP WITH A LINEAR BUYER'S DEMAND

In this paper, we analyze discounting decisions for a supplier with a group of homogeneous customers. We focus on two aspects: the gaming nature of the discount problem and the demand consideration in the process. We use a general quantity discount schedule and start with the Stackelberg equilibrium of the problem. It is shown that, for the seller to gain from quantity discount, he has to set up his quantity discount schedule such that the buyer will order more than his EOQ. Both the seller and the buyer can gain significantly from quantity discount. The incentive for discount is twofold: reducing inventory related cost and attracting more demand from the customers. In addition, quantity discount schedule can be very efficient in obtaining the maximum gain the seller and the buyer can possibly obtain together.     

Dynamic pricing with ‘BOGO’ promotion in revenue management

We consider a dynamic pricing problem when a seller, facing uncertain demands, sells a single product in a finite horizon. The seller actively adopts dynamic pricing and quantity discount schemes. The proposed model is based on the assumption that each customer has random reservation prices and the purchase size depends on the posted price and discount. We particularly focus on the widely adopted promotional schemes ‘buy one get one free’ and ‘50% off’ and study the optimal strategic choices of the seller. Analytical results together with numerical experiments are presented to help us obtain managerial insights. Additional numerical results for a generalised model are provided so as to examine the effectiveness of promotional schemes.     

Quantity discounts in single-period supply contracts with asymmetric demand information

We investigate how a supplier can use a quantity discount schedule to influence the stocking decisions of a downstream buyer that faces a single period of stochastic demand. In contrast to much of the work that has been done on single-period supply contracts, we assume that there are no interactions between the supplier and the buyer after demand information is revealed and that the buyer has better information about the distribution of demand than does the supplier. We characterize the structure of the optimal discount schedule for both all-unit and incremental discounts and show that the supplier can earn larger profits with an all-unit discount.     

Discount strategies for one-time-only sales

Inventory clearing sales, especially in the case of highly competitive, established products, are sometimes accompanied by temporary discounts in the purchase price, to encourage customers to place orders larger than usual. This paper presents a generalized model of this temporary price discount problem in which explicit recognition is given to widely used variations (i) in the length of the grace period within which prospective buyers may place the discounted order; (ii) in the size of the order needed to qualify for the discounts; and (iii) in the portion of the order eligible for the discount.     

Optimal production and pricing policies in a combined make-to-order/make-to-stock system

In this paper, we consider a production system which is capable to produce two types of products. The first type of products is make-to-order, while the second type is make-to-stock whose demand is satisfied by the on-hand inventory. The demand arrival rates of both types of products are price-sensitive. The excess demand that cannot be satisfied immediately is either backlogged or lost. The system costs include the holding costs of product inventories and shortage costs of unsatisfied demand. The objective is to maximise the total discounted profit over an infinite planning horizon by coordinating the production process and pricing decisions. By analysing the properties of objective functions, we characterise the optimal control policy by two switch curves and the optimal price is also given for different ordering and inventory levels. We also explore the monotonicity of both switch curves which will reduce the computation effort. Numerical experiments are conducted to demonstrate the use of the switch curves in managing the production system and illustrate that compared with the static pricing policy, the optimal integrated price and inventory control policy can result in a significant profit improvement in the make-to-order/make-to-stock system that is much higher than in a single-product system.     

Decentralised planning coordination with quantity discount contract in a divergent supply chain

This paper proposes to solve a supply chain planning problem with realistic features. The problem consists of planning productions, transportations and storage activities in a supply chain at a tactical level on a finite horizon. The main features considered are decentralised decision making and iteration of the planning process on a rolling horizon basis. In each planning process, the actors optimise their local planning and coordinate to achieve a good overall planning. A multi-agent system is used to model such supply chain behaviour. The study is conducted in a divergent two-echelon supply chain with one manufacturer and multiple independent retailers. Coordination is achieved using a standard contract in practice, known as the ‘quantity discount’ contract. The planning framework on the supply chain structure is detailed. Lot-sizing models integrating the quantity discount are presented for the local planning problems. Experimental tests are conducted with three major parameters: quantity discount price, quantity discount breakpoint and rolling horizon length. They are used to determine the quantity discount parameters in achieving the best supply chain profit, and to analyse the increasing profit of the actors. A decision-making tool which is able to consider realistic features of supply chain planning is therefore resulted.     

The supplier–buyer integrated production-inventory model with random yield

This paper revisits the traditional supplier–buyer integrated production-inventory model which deals with the problem of a manufacturer (supplier) supplying a product to a retailer (buyer) serving the consumer market with constant stationary demand. The product is manufactured in batches at a finite rate. The supplier's production batch is depleted by the buyer's replenishment orders at periodic intervals. The buyer's inventory is then consumed by the market demand at a fixed rate. The problem is the simultaneous computation of the manufacturer's production lot-size and the buyer's replenishment order quantity, i.e. the integrated production-inventory policy parameters. The key characteristic considered in this paper is that the manufacturing process is imperfect, and, hence, there are defective items in each production lot. As a result, each replenishment order shipped to the buyer includes defective products and the non-defective percentage in each such shipment is random. Considering the case where the supplier replenishes the buyer via equal-sized shipments, we develop an analytical expression of the total expected cost for the supplier–buyer system under consideration, with and without a considerable inspection time. We first examine the case where the inspection time is negligible, and then we present a generalisation to consider the inspection time explicitly. Our goal is to model the impact of random yield on the system performance. Our findings are useful for computing the integrated production-inventory policy parameters while considering the supply uncertainty due to an imperfect manufacturing process. Through numerical examples, we quantify the impact of supply with random yield on the system performance and illustrate its relationship with the demand and production rate.     

Optimal pricing policy for a deteriorating product by dynamic tracking control

We study the optimal selling price of a deteriorating product under a deterministic situation in a finite time horizon where the time horizon is either known or unknown. Inventory holding cost is expressed as a quadratic function of the current inventory level. Given a known time horizon, we develop a model by considering the deterioration dynamics of the product, and show its equivalence to a generalised optimal control problem of a linear quadratic form, i.e. an optimal dynamic tracking problem with constraints on the control variable. An optimal pricing policy is derived based on the maximum value principle. The control policy takes a state feedback form; it exhibits a closed-loop relationship between the optimal selling price (control variable) and the optimal inventory level (state variable). Given an unknown time horizon, an optimal pricing policy is derived through a similar approach when the initial inventory level meets certain conditions. Numerical situations are conducted to illustrate the effectiveness of the derived price control policies. Some interesting managerial insights are discussed.     

Improving Profitability with Quantity Discounts under Fixed Demand

Quantity discount schedules have been studied, in the past, from the retailer's, and not the supplier's, point of view. These studies address the problem of determining the economic order quantities for the retailer, given a quantity discount schedule set by the supplier. In this paper, this problem is addressed from the supplier's point of view, assuming that the retailer always uses his optimal order quantity. It is shown that under certain circumstances, quantity discounts could be of benefit to the supplier (and obviously to the retailer), even when retailer's demand is insensitive to price changes. An algorithm is developed to determine the optimal pricing policy for a linear quantity discount schedule. Numerical examples are provided, and sufficient conditions when no quantity discount should be offered are derived.     

模仿创新企业进入时机和动态定价联合决策问题研究

以系统动力学建模为研究方法,通过仿真研究模拟创新企业进入市场后的市场需求变化和收益情况,特别分析了当企业的决策目标分别为产品销量最大或产品收益最大两种情景下的运营策略。模型探讨了包括消费者犹豫行为和消费者对产品打折销售的心理预期两种不同消费者行为偏好对企业运营策略的影响。结果表明：模仿创新企业以追求产品销售最大化为目标时,其最优的运营策略为尽早打折和较大的价格折扣;而当模仿创新企业追求收益最大化时,其最优的运营策略是不打折。当消费者存在产品打折的心理预期时,企业通过推后打折时间,既满足消费者的心理预期,还能获得更多的产品收益。     

Optimal ordering and pricing policy for an inventory system with order cancellations

This paper investigates the problem of ordering and pricing over a finite time planning horizon for an inventory system with advance sales and spot sales. It is assumed that the planning horizon is divided into several sales cycles each of which is divided into an advance sales period and a spot sales period. During the advance sales period, all customers are required to make reservations for their orders and will receive them at the arrival time of the replenishment orders. In the case of the spot sales periods, all customers receive their orders at the time of the purchase. In actual practice, since customers with reservations may cancel their orders before receiving them, this paper considers this phenomenon and develops a continuous time inventory model to deal with the proposed problem. This paper maximizes the total profit over a finite time planning horizon by determining the optimal advance sales price, spot sales price, order size, and replenishment frequency. Analysis of results shows that a simple algorithm can be developed to arrive at an optimal decision.     

短生命周期产品的延期交货和价格折扣模型

用斜坡型函数来描述短生命周期产品需求变化特征,用持有成本不断增加表示短生命周期产品无形变质所带来的损失,研究在允许缺货情况下,如何设置合理价格折扣来控制订单的流失,确定各周期的最佳延迟供货量、最优价格折扣、最优订货时间和订货量,从而实现企业总成本的最优。给出了一种迭代逼近的方法来寻求在有限时域内的最优订购策略,通过数值分析验证了该模型的有效性和可操作性,对相应的参数做出了灵敏度分析。该模型能够确定出企业最佳的价格折扣以挽留住更多的顾客,使其总利润达到最优。     

Rebate, returns and price protection policies in channel coordination

We study channel coordination policies for products subject to midlife price declines during their short product life cycles. Using a two-period supply chain model consisting of one supplier and one retailer, we identify policies and/or conditions under which the supply chain can be coordinated and a win-win situation can be guaranteed. We also provide algorithms to determine the win-win policy parameters. We show that if there are two purchase opportunities, then under linear or decreasing (the more you buy, the lower price you pay) pricing strategies, there may not exist a win-win policy. Our analysis shows that several findings in the single-period models prevalent in the supply chain research literature to date do not extend to the two-period model with two purchase opportunities. For example, returns policies and revenue sharing contracts are no longer equivalent. Also, quantity discounts may never lead to a win-win situation. In fact, the supplier may need to charge more for large orders in order to achieve a win-win outcome. Finally, the analytical framework of this paper can be applied to situations in which the price is constant but production costs change over time.     

Coordinated order and production policies in supply chains

This paper is focused on the coordination of order and production policy between buyers and suppliers in supply chains. When a buyer of an item decides independently, he will place orders based on his economic order quantity (EOQ). However, the buyers EOQ may not lead to an optimal policy for the supplier. Should the buyer have the more powerful position to enforce his EOQ on the supplier, then no incentive exists for him to deviate from his EOQ. To provide an incentive to order in quantities suitable to the supplier, the supplier could offer a price discount or side payment. One critical assumption made throughout the literature dealing with incentive schemes to influence buyers ordering policy is that the supplier has complete information regarding buyers cost structure. This paper provides a bargaining model with asymmetric information about the buyers cost structure. A self-selection model for the determination of an optimal set of contracts which are specifically designed for different cost structures of the buyer, assumed by the supplier, will be presented.     

The optimal production and shipment policy for the single-vendor singlebuyer integrated production-inventory problem

This paper considers the problem of a vendor (manufacturer) supplying a product to a buyer (customer). The vendor manufactures the product in batches at a finite rate and ships the output to the buyer. The buyer then consumes the product at a fixed rate. The objective is to minimize the mean total cost per unit time of manufacturing set-up, stock transfer and stock holding. Previously published work has concentrated on finding optimal solutions from within given classes of policy. We derive the structure of the globally-optimal solution, set out an algorithm for obtaining it and illustrate the process with two numerical examples.