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.     

An optimal credit policy to increase supplier's profits with price-dependent demand functions

This paper deals with the problem of determining an optimal length of credit period from the perspective of supplier. We assume that a retailer jointly determines the unit retail price and order size to maximize profit when he/she purchases a product for which the supplier offers a trade credit. Two widely used demand functions are adopted for the study in which demands are decreasing functions of the retail price. A procedure is presented which shows how to achieve an optimal length of credit period for suppliers. The effects of credit period on the behaviour of retailers are also investigated using an example.     

A decision support system for a third-party coordinator managing supply chain with demand uncertainty

Global supply chains reduce cost but increase lead times, complexities and uncertainties. Retailers in consumer products industry are getting shorter lead time to respond to market demand. To meet this challenge, many rely on third party supply chain managers (SCMs) for economically supplying required quantities of finished products quickly. However, due to shorter ‘time to market’, the SCM has to procure raw materials and start production process based on expected demand. Since SCM absorbs financial penalties associated with under- and over-estimation of demand from retailer, finding an optimal production lot size and product customisation strategy are essential to an SCM's operation. We develop a profit maximisation model and provide a close-form solution that allows an SCM to calculate optimal production lot size. The model is used to examine profitability of postponing product customisation. Finally, the effect of demand variation on SCM's profitability is explored.     

Allocation of Returned Products among Different Recovery Options through an Opportunity Cost–Based Dynamic Approach*

In a make‐to‐order product recovery environment, we consider the allocation decision for returned products decision under stochastic demand of a firm with three options: refurbishing to resell, parts harvesting, and recycling. We formulate the problem as a multiperiod Markov decision process (MDP) and present a linear programming (LP) approximation that provides an upper bound on the optimal objective function value of the MDP model. We then present two solution approaches to the MDP using the LP solution: a static approach that uses the LP solution directly and a dynamic approach that adopts a revenue management perspective and employs bid‐price controls technique where the LP is resolved after each demand arrival. We calculate the bid prices based on the shadow price interpretation of the dual variables for the inventory constraints and accept a demand if the marginal value is higher than the bid price. Since the need for solving the LP at each demand arrival requires a very efficient solution procedure, we present a transportation problem formulation of the LP via variable redefinitions and develop a one‐pass optimal solution procedure for it. We carry out an extensive numerical analysis to compare the two approaches and find that the dynamic approach provides better performance in all of the tested scenarios. Furthermore, the solutions obtained are within 2% of the upper bound on the optimal objective function value of the MDP model.     

A newsboy problem with an emergency order under a general backorder rate function

In this paper, a generalization is presented of the newsboy problem where an emergency lot can be ordered to provide for a certain fraction of shortage. This fraction is described by a general backorder rate function which is non-increasing with respect to the unsatisfied demand. An exponential distribution for the demand during the selling season is assumed. An expression is obtained in a closed form for the optimal lot size and the maximum expected profit. A general sensitivity analysis of the optimal policy with respect to the backorder rate function and the parameters of the inventory system is developed. When the backorder rate function is described by some particular functions, its behavior is analyzed with respect to changes in the parameters. To illustrate the theoretical results, some numerical examples are also given.     

Impact of investing in quality improvement on (Q,r, L) model involving the imperfect production process

This paper investigates the impact of quality improvement on the modified lot size reorder point models involving variable lead time and partial backorders. The formulated models include the imperfect production process and an investing option of improving the process quality. The objective is simultaneously optimizing the lot size, reorder point, process quality level and lead time. We first assume that the lead time demand follows a normal distribution, then relax this assumption to consider the distribution-free case where only the mean and standard deviation of lead time demand are known. An algorithm procedure of finding the optimal solution is developed, and two numerical examples are given to illustrate the results.     

运输成本对批量敏感时的供应链批量协调策略比较研究

以一个多周期、随机客户需求和单位产品运输成本对批量敏感的供应链为研究对象,建立了由生产商负责产品运输时,供应链分散决策情形下的最佳批量模型,并提出了基于批量折扣和改由批发商负责产品运输的供应链批量协调策略.研究结果表明:当由生产商负责产品运输时,单位产品运输成本对批量越不敏感,生产商的最佳生产批量越小,与批发商要求的短周期、小批量订货越接近;如果改由批发商负责产品运输,批发商的最佳订货批量更接近于生产商的最佳生产批量.最后,通过数例分析发现,对生产商来讲,当单位产品运输成本对批量不太敏感时,批量折扣政策优于改由批发商负责产品运输的策略,但当单位产品运输成本对批量较敏感时,后者优于前者.     

Joint inventory and technology selection decisions

In some manufacturing systems, the type of production technology selected dictates the lot size, or sets an upper limit on it. In this paper, we formulate a model in which the lot sizing decision has to be made at the technology selection stage. Furthermore, the cost of the technology depends on the lot size it can produce. The problem is addressed under two different assumptions about demand. First demand is assumed to be constant. A closed-form expression for the optimal lot size is derived for this case. Second, demand is assumed to be linearly decreasing. Using an accurate approximation to the total cost function, a closed-form expression for the optimal lot size is derived. We illustrate the models with numerical examples.     

静态市场下考虑重复购买的价格和质保期联合决策

本文研究静态市场下的以质保期和价格为决策变量的最优控制决策问题。考虑了初次购买和重复购买两个过程以及重复购买过程是否受价格影响两种情况。模型中的生产成本考虑了产品生产过程中的学习效应, 质保成本使用了免费维修质保策略并假设产品两次故障之间的时间服从韦伯分布。在完整建模的基础上, 应用最大化原理对基本模型进行了求解, 之后本文给出了一种基本模型的具体应用场景, 并运用Lingo11软件进行了数值试验。最后对主要研究结论进行了应用讨论。     

Optimal and Heuristic Decision Strategies for Integrated Production and Marketing Planning

In this study we examine the effects of integrating production and marketing decisions for a short- to medium-range planning horizon in a profit maximizing firm. We formulate two models for determining price, marketing expenditure, demand or production volume, and lot size for a single product with stable demand when economies of scale are present. The full integration (FI) model simultaneously determines all the decisions involved, while the partial integration (PI) model separates the lot sizing decision from the others, as happens frequently in practice. Geometric programming (GP) techniques and marginal analysis are used to compare FI and PI, and obtain important managerial implications regarding the two models.     

Coordinating Service‐Sensitive Demand and Capacity by Adaptive Decision Making: An Application of Case‐Based Decision Theory*

The subject of this article is the simultaneous choice of product price and manufacturing capacity if demand is stochastic and service‐level sensitive. In this setting, capacity as well as price have an impact on demand because several aspects of service level depend on capacity. For example, delivery time will be reduced if capacity is increased given a constant demand rate. We illustrate the relationship between service level, capacity, and demand reaction by a stylized application problem from the after‐sales services industry. The reaction of customers to variations in service level and price is represented by a kinked price‐demand‐rate function. We first derive the optimal price‐capacity combination for the resulting decision problem under full information. Subsequently, we focus on a decision maker (DM) who lacks complete knowledge of the demand function. Hence the DM is unable to anticipate the service level and consequently cannot identify the optimal solution. However, the DM will acquire additional information during the sales process and use it in subsequent revisions of the price‐capacity decision. Thus, this decision making is adaptive and based on experience. In contrast to the literature, which assumes certain repetitive procedures somewhat ad hoc, we develop an adaptive decision process based on case‐based decision theory (CBDT) for the price‐capacity problem. Finally, we show that a CBDT DM in our setting eventually finds the optimal solution, if the DM sets the price based on absorption costs and adequately adjusts the capacity with respect to the observed demand.     

WIP inventories in the simultaneous determination of optimal production and purchase lot sizes

This paper studies an integrated production and purchasing lot sizing model with work-in-process WIP inventory. In this model, the single product is made in a multiprocess manufacturing system. The raw materials are procured from outside sources and are converted gradually into the product. A solution procedure is developed to simultaneously find the optimal lot sizes for the product and its raw materials and the corresponding total relevant cost. It is shown that if the cost of WIP inventory is considered in the production lot size computation, the optimal lot sizes of the product as well as those of the raw materials could be altered significantly.     

A joint optimal pricing and order quantity model under parameter uncertainty and its practical implementation

We consider a robust optimization model of determining a joint optimal bundle of price and order quantity for a retailer in a two-stage supply chain under uncertainty of parameters in demand and purchase cost functions. Demand is modeled as a decreasing power function of product price, and unit purchase cost is modeled as a decreasing power function of order quantity and demand. While the general form of the power functions are given, it is assumed that parameters defining the two power functions involve a certain degree of uncertainty and their possible values can be characterized by ellipsoids. We show that the robust optimization problem can be transformed into an equivalent convex optimization which can be solved efficiently and effectively using interior-point methods. In addition, we propose a practical implementation of the model, where the stochastic characteristics of parameters are obtained from regression analysis on past sales and production data, and ellipsoidal representations of the parameter uncertainties are obtained based on a combined use of genetic algorithm and Monte Carlo simulation. An illustrative example is provided to demonstrate the model and its implementation.     

A joint optimal pricing and order quantity model under parameter uncertainty and its practical implementation

We consider a robust optimization model of determining a joint optimal bundle of price and order quantity for a retailer in a two-stage supply chain under uncertainty of parameters in demand and purchase cost functions. Demand is modeled as a decreasing power function of product price, and unit purchase cost is modeled as a decreasing power function of order quantity and demand. While the general form of the power functions are given, it is assumed that parameters defining the two power functions involve a certain degree of uncertainty and their possible values can be characterized by ellipsoids. We show that the robust optimization problem can be transformed into an equivalent convex optimization which can be solved efficiently and effectively using interior-point methods. In addition, we propose a practical implementation of the model, where the stochastic characteristics of parameters are obtained from regression analysis on past sales and production data, and ellipsoidal representations of the parameter uncertainties are obtained based on a combined use of genetic algorithm and Monte Carlo simulation. An illustrative example is provided to demonstrate the model and its implementation.     

Combined Pricing and Portfolio Option Procurement

In this paper, we study a single‐product periodic‐review inventory system that faces random and price‐dependent demand. The firm can purchase the product either from option contracts or from the spot market. Different option contracts are offered by a set of suppliers with a two‐part fee structure: a unit reservation cost and a unit exercising cost. The spot market price is random and its realization may affect the subsequent option contract prices. The firm decides the reservation quantity from each supplier and the product selling price at the beginning of each period and the number of options to exercise (inventory replenishment) at the end of the period to maximize the total expected profit over its planning horizon. We show that the optimal inventory replenishment policy is order‐up‐to type with a sequence of decreasing thresholds. We also investigate the optimal option‐reservation policy and the optimal pricing strategy. The optimal reservation quantities and selling price are shown to be both decreasing in the starting inventory level when demand function is additive. Building upon the analytical results, we conduct a numerical study to unveil additional managerial insights. Among other things, we quantify the values of the option contracts and dynamic pricing to the firm and show that they are more significant when the market demand becomes more volatile.     

不确定需求下的冷链分销商最优决策

针对新鲜产品供应链 (冷链) 中分销商, 引入保鲜努力刻画分销商投入的保鲜工作, 考虑新鲜产品到达市场的数量与质量, 分析冷链分销商的最优决策问题.与前人研究相比, 本文使用加法形式需求函数刻画市场需求, 并使用相关企业数据验证加法形式需求函数合理性, 完善了冷链分销商决策问题的相关研究.研究发现, 当保鲜努力外生时, 分销商的最优订货数量和最优零售价格只与保鲜努力有关;当零售价格外生时, 分销商的最优保鲜努力与产品实际单位成本关于保鲜努力函数的一阶最优条件有关;当订货数量外生时, 分销商制定的零售价格随保鲜努力增加而增大, 所获得的利润也随保鲜努力递增, 但边际利润随保鲜努力递减.最后, 将所得结论应用于中国500强民营企业卓尔控股有限公司旗下的小雪冷链 (武汉) 物流有限公司, 为该公司提供相应的保鲜努力投入策略.     

Optimal Policies for Selling New and Remanufactured Products

Because of environmental and economic reasons, an increasing number of original equipment manufacturers (OEMs) nowadays sell both new and remanufactured products. When both products are available, customers will buy the one that gives them a higher (and non‐negative) utility. Thus, if the firm does not price the products properly, then product cannibalization may arise and its revenue may be adversely impacted. In this paper, we study the pricing problem of a firm that sells both new and remanufactured products over a finite planning horizon. Customer demand processes for both new and remanufactured products are random and price‐sensitive, and product returns (also called cores) are random and remanufactured upon receipt. We characterize the optimal pricing and manufacturing policies that maximize the expected total discounted profit. If new products are made‐to‐order (MTO), we show that when the inventory level of remanufactured product increases, the optimal price of remanufactured product decreases while the price difference between new and remanufactured products increases; however, the optimal selling price of new product may increase or decrease. If new products are made to stock (MTS), then the optimal manufacturing policy is of a base‐stock policy with the base‐stock level decreasing in the remanufactured product inventory level. To understand the potential benefit in implementing an MTO system, we study the difference between the value functions of the MTO and MTS systems, and develop lower and upper bounds for it. Finally, we study several extensions of the base model and show that most of our results extend to those more general settings.     

Discount Pricing Policies and the Coordination of Decentralized Distribution Systems*

A challenge of supply chain management is to align the objectives, and hence coordinate the activities, of independent supply chain members. In this study, we approach this problem in a simple way by extending traditional quantity discounts that are based solely on buyers' individual order size to discount policies that are based on both buyers' individual order size and their annual volume. We show that discount policies are able to achieve nearly optimal system profit and, hence, provide effective coordination, for a decentralized two‐echelon distribution system, whereby a supplier sells a product to a group of heterogeneous and independent retailers each facing a downward‐sloping demand curve of its retail price. When buyers are heterogeneous, a critical issue of coordination is to motivate different customers to increase their demand and lot size according to their potential so as to improve profits. We show that market heterogeneity presents an effective discriminating factor for the supplier to segment customers in the design of a coordination mechanism.     

A DECISION THEORETIC APPROACH TO PRICING

The problem of price determination and revision is considered as a case of decision making under uncertainty in which profit is to be maximized. Typically price is a simple function of cost which in turn determines the quantity which will be demanded. This paper proposes that maximum profits could be realized in the long run, if a quantity corresponding to the lowest cost per unit of product under the attendant circumstances was fixed and a price established at which demand would exactly equal the fixed quantity. It further suggests a theoretical approach to the determination of this price based on decision theory. The decision theoretic approach considers the set of possible price levels at which demand will equal the fixed quantity of product as the state of nature. The set of acts consist of the establishment of the product price at each of the possible levels. After an initial price is established, empirical information can then be utilized according to some optimal decision rule for subsequent price revisions.     

Hedging through index-based price contracts in commodity-based supply chains

This paper studies the optimal hedging strategy of risk-neutral firms in supply chain settings. We model a retailer procuring goods through index-based price contracts from two commodity processors. The processors’ input commodity prices are random and correlated. The retailer faces price-sensitive demand curves; therefore, it controls product demand through retail pricing in the final product market. We characterize the optimal contracting terms for the processors and show that they prefer to hedge part of their exposure to the commodity price risk. The optimal contract for processor comprises a processing margin independent of the commodity price volatility and a hedge ratio that is a function of the commodity price volatility and the products substitution factor. We uncover a new rationale for hedging in settings where downstream firms have pricing power; both processors and the retailer benefit from the retailer’s pricing power when their margins are linked to input prices; an index-based price contract is a means to link the processors’ and the retailer’s margins. We further investigate how different market parameters affect the optimal hedge ratios and extend our model to settings with random market sizes and asymmetric substitution for final products.