A Production–Inventory Model for a Push–Pull Manufacturing System with Capacity and Service Level Constraints

We study a hybrid push–pull production system with a two‐stage manufacturing process, which builds and stocks tested components for just‐in‐time configuration of the final product when a specific customer order is received. The first production stage (fabrication) is a push process where parts are replenished, tested, and assembled into components according to product‐level build plans. The component inventory is kept in stock ready for the final assembly of the end products. The second production stage (fulfillment) is a pull‐based assemble‐to‐order process where the final assembly process is initiated when a customer order is received and no finished goods inventory is kept for end products. One important planning issue is to find the right trade‐off between capacity utilization and inventory cost reduction that strives to meet the quarter‐end peak demand. We present a nonlinear optimization model to minimize the total inventory cost subject to the service level constraints and the production capacity constraints. This results in a convex program with linear constraints. An efficient algorithm using decomposition is developed for solving the nonlinear optimization problem. Numerical results are presented to show the performance improvements achieved by the optimized solutions along with managerial insights provided.     

Inventory Rationing for Multiple Class Demand under Continuous Review

We consider how a firm should ration inventory to multiple classes in a stochastic demand environment with partial, class‐dependent backlogging where the firm incurs a fixed setup cost when ordering from its supplier. We present an infinite‐horizon, average cost criterion Markov decision problem formulation for the case with zero lead times. We provide an algorithm that determines the optimal rationing policy, and show how to find the optimal base‐stock reorder policy. Numerical studies indicate that the optimal policy is similar to that given by the equivalent deterministic problem and relies on tracking both the current inventory and the rate that backorder costs are accumulating. Our study of the case of non‐zero lead time shows that a heuristic combining the optimal, zero lead time policy with an allocation policy based on a single‐period profit management problem is effective.     

Myopic Analysis for Multi‐Echelon Inventory Systems with Batch Ordering and Nonstationary/Time‐Correlated Demands

We provide an exact myopic analysis for an N‐stage serial inventory system with batch ordering, linear ordering costs, and nonstationary demands under a finite planning horizon. We characterize the optimality conditions of the myopic nested batching newsvendor (NBN) policy and the myopic independent batching newsvendor (IBN) policy, which is a single‐stage approximation. We show that echelon reorder levels under the NBN policy are upper bounds of the counterparts under both the optimal policy and the IBN policy. In particular, we find that the IBN policy has bounded deviations from the optimal policy. We further extend our results to systems with martingale model of forecast evolution (MMFE) and advance demand information. Moreover, we provide a recursive computing procedure and optimality conditions for both heuristics which dramatically reduces computational complexity. We also find that the NBN problem under the MMFE faced by one stage has one more dimension for the forecast demand than the one faced by its downstream stage and that the NBN policy is optimal for systems with advance demand information and stationary problem data. Numerical studies demonstrate that the IBN policy outperforms on average the NBN policy over all tested instances when their optimality conditions are violated.     

Cash‐Flow Based Dynamic Inventory Management

Small‐to‐medium‐sized enterprises (SMEs), including many startup firms, need to manage interrelated flows of cash and inventories of goods. In this study, we model a firm that can finance its inventory (ordered or manufactured) with loans in order to meet random demand which in general may not be time stationary. The firm earns interest on its cash on hand and pays interest on its debt. The objective is to maximize the expected value of the firm's capital at the end of a finite planning horizon. The firm's state at the beginning of each period is characterized by the inventory level and the capital level measured in units of the product, whose sum represents the “net worth” of the firm. Our study shows that the optimal ordering policy is characterized by a pair of threshold parameters as follows. (i) If the net worth is less than the lower threshold, then the firm employs a base stock order up to the lower threshold. (ii) If the net worth is between the two thresholds, then the firm orders exactly as many units as it can afford, without borrowing. (iii) If the net worth is above the upper threshold, then the firm employs a base stock order up to the upper threshold. Further, upper and lower bounds for the threshold values are developed using two simple‐to‐compute myopic ordering policies which yield lower bounds for the value function. We also derive an upper bound for the value function by considering a sell‐back policy. Subsequently, it is shown that policies of similar structure are optimal when the loan and deposit interest rates are piecewise linear functions, when there is a maximal loan limit and when unsatisfied demand is backordered. Finally, further managerial insights are provided with extensive numerical studies.     

Inventory Management for Customers with Alternative Lead Times

It is common for suppliers operating in batch‐production mode to deal with patient and impatient customers. This paper considers inventory models in which a supplier provides alternative lead times to its customers: a short or a long lead time. Orders from patient customers can be taken by the supplier and included in the next production cycle, while orders from impatient customers have to be satisfied from the on‐hand inventory. We denote the action to commit one unit of on‐hand inventory to patient or impatient customers as the inventory‐commitment decision, and the initial inventory stocking as the inventory‐replenishment decision. We first characterize the optimal inventory‐commitment policy as a threshold type, and then prove that the optimal inventory‐replenishment policy is a base‐stock type. Then, we extend our analysis to models to consider cases of a multi‐cycle setting, a supply‐capacity constraint, and the on‐line charged inventory‐holding cost. We also evaluate and compare the performances of the optimal inventory‐commitment policy and the inventory‐rationing policy. Finally, to further investigate the benefits and pitfalls of introducing an alternative lead‐time choice, we use the customer‐choice model to study the demand gains and losses, known as demand‐induction and demand‐cannibalization effects, respectively.     

End‐of‐Life Inventory Problem with Phaseout Returns

We consider the service parts end‐of‐life inventory problem of a capital goods manufacturer in the final phase of its life cycle. The final phase starts as soon as the production of parts terminates and continues until the last service contract expires. Final order quantities are considered a popular tactic to sustain service fulfillment obligations and to mitigate the effect of obsolescence. In addition to the final order quantity, other sources to obtain serviceable parts are repairing returned defective items and retrieving parts from phaseout returns. Phaseout returns happen when a customer replaces an old system platform with a next‐generation one and returns the old product to the original equipment manufacturer (OEM). These returns can well serve the demand for service parts of other customers still using the old generation of the product. In this study, we study the decision‐making complications as well as cost‐saving opportunities stemming from phaseout occurrence. We use a finite‐horizon Markov decision process to characterize the structure of the optimal inventory control policy. We show that the optimal policy consists of a time‐varying threshold level for item repair. Furthermore, we study the value of phaseout information by extending the results to cases with an uncertain phaseout quantity or an uncertain schedule. Numerical analysis sheds light on the advantages of the optimal policy compared to some heuristic policies.     

Dynamic Pricing and Inventory Management with Regular and Expedited Supplies

We consider a periodic‐review inventory system with regular and expedited supply modes. The expedited supply is faster than the regular supply but incurs a higher cost. Demand for the product in each period is random and sensitive to its selling price. The firm determines its order quantity from each supply in each period as well as its selling price to maximize the expected total discounted profit over a finite or an infinite planning horizon. We show that, in each period if it is optimal to order from both supplies, the optimal inventory policy is determined by two state‐independent thresholds, one for each supply mode, and a list price is set for the product; if only the regular supply is used, the optimal policy is a state‐dependent base‐stock policy, that is, the optimal base‐stock level depends on the starting inventory level, and the optimal selling price is a markdown price that decreases with the starting inventory level. We further study the operational impact of such supply diversification and show that it increases the firm's expected profit, reduces the optimal safety‐stock levels, and lowers the optimal selling price. Thus that diversification is beneficial to both the firm and its customers. Building upon these results, we conduct a numerical study to assess and compare the respective benefit of dynamic pricing and supply diversification.     

The Effects of Sharing Upstream Information on Product Rollover

The process of introducing new and phasing out old products is called product rollover. This paper considers a periodic‐review inventory system consisting of a manufacturer and a retailer, where the manufacturer introduces new and improved products over an infinite planning horizon using the solo‐roll strategy. We consider two scenarios: (1) the manufacturer does not share the upstream information about new‐product introduction with the retailer and (2) the manufacturer shares the information. For each scenario, we first derive the decentralized ordering policy and the system‐optimal ordering policy with given cost parameters. We then devise an optimal supply chain contract that coordinates the inventory system. We demonstrate that when the inventory system is coordinated, information sharing improves the performance of both supply chain entities. However, this may not be true if the inventory system is not coordinated. We also show that under the optimal contract, the manufacturer has no incentive to mislead the retailer about new‐product information in the information‐sharing model. When demand variability increases, information sharing adds more benefits to the coordinated supply chain. Our research provides insights about coordinating product, financial, and information flows in supply chains with product rollover.     

Managing stochastic demand in an Inventory Routing Problem with transportation procurement

We study an Inventory Routing Problem in which the supplier has a limited production capacity and the stochastic demand of the retailers is satisfied with procurement of transportation services. The aim is to minimize the total expected cost over a planning horizon, given by the sum of the inventory cost at the supplier, the inventory cost at the retailers, the penalty cost for stock-out at the retailers and the transportation cost. First, we show that a policy based just on the average demand can have a total expected cost infinitely worse than the one obtained by taking into account the overall probability distribution of the demand in the decision process. Therefore, we introduce a stochastic dynamic programming formulation of the problem that allows us to find an optimal policy in small size instances. Finally, we design and implement a matheuristic approach, integrating a rollout algorithm and an optimal solution of mixed-integer linear programming models, which is able to solve realistic size problem instances. Computational results allow us to provide managerial insights concerning the management of stochastic demand.     

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.     

Replenishment Policies for Multi‐Product Stochastic Inventory Systems with Correlated Demand and Joint‐Replenishment Costs

This study analyzes optimal replenishment policies that minimize expected discounted cost of multi‐product stochastic inventory systems. The distinguishing feature of the multi‐product inventory system that we analyze is the existence of correlated demand and joint‐replenishment costs across multiple products. Our objective is to understand the structure of the optimal policy and use this structure to construct a heuristic method that can solve problems set in real‐world sizes/dimensions. Using an MDP formulation we first compute the optimal policy. The optimal policy can only be computed for problems with a small number of product types due to the curse of dimensionality. Hence, using the insight gained from the optimal policy, we propose a class of policies that captures the impact of demand correlation on the structure of the optimal policy. We call this class (s, c, d, S)‐policies, and also develop an algorithm to compute good policies in this class, for large multi‐product problems. Finally using an exhaustive set of computational examples we show that policies in this class very closely approximate the optimal policy and can outperform policies analyzed in prior literature which assume independent demand. We have also included examples that illustrate performance under the average cost objective.     

Unsold Versus Unbought Commitment: Minimum Total Commitment Contracts with Nonzero Setup Costs

We study a minimum total commitment (MTC) contract embedded in a finite‐horizon periodic‐review inventory system. Under this contract, the buyer commits to purchase a minimum quantity of a single product from the supplier over the entire planning horizon. We consider nonstationary demand and per‐unit cost, discount factor, and nonzero setup cost. Because the formulations used in existing literature are unable to handle our setting, we develop a new formulation based on a state transformation technique using unsold commitment instead of unbought commitment as state variable. We first revisit the zero setup cost case and show that the optimal ordering policy is an unsold‐commitment‐dependent base‐stock policy. We also provide a simpler proof of the optimality of the dual base‐stock policy. We then study the nonzero setup cost case and prove a new result, that the optimal solution is an unsold‐commitment‐dependent (s, S) policy. We further propose two heuristic policies, which numerical tests show to perform very well. We also discuss two extensions to show the generality of our method's effectiveness. Finally, we use our results to examine the effect of different contract terms such as duration, lead time, and commitment on buyer's cost. We also compare total supply chain profits under periodic commitment, MTC, and no commitment.     

Supply Performance in Multi‐Echelon Inventory Systems with Intermediate Product Demand: A Perspective on Allocation

In this article, we study the performance of multi‐echelon inventory systems with intermediate, external product demand in one or more upper echelons. This type of problem is of general interest in inventory theory and of particular importance in supply chain systems with both end‐product demand and spare parts (subassemblies) demand. The multi‐echelon inventory system considered here is a combination of assembly and serial stages with direct demand from more than one node. The aspect of multiple sources of demands leads to interesting inventory allocation problems. The demand and capacity at each node are considered stochastic in nature. A fixed supply and manufacturing lead time is used between the stages. We develop mathematical models for these multi‐echelon systems, which describe the inventory dynamics and allow simulation of the system. A simulation‐based inventory optimization approach is developed to search for the best base‐stock levels for these systems. The gradient estimation technique of perturbation analysis is used to derive sample‐path estimators. We consider four allocation schemes: lexicographic with priority to intermediate demand, lexiographic with priority to downstream demand, predetermined proportional allocation, and proportional allocation. Based on the numerical results we find that no single allocation policy is appropriate under all conditions. Depending on the combinations of variability and utilization we identify conditions under which use of certain allocation polices across the supply chain result in lower costs. Further, we determine how selection of an inappropriate allocation policy in the presence of scarce on‐hand inventory could result in downstream nodes facing acute shortages. Consequently we provide insight on why good allocation policies work well under differing sets of operating conditions.     

Equilibrium Joining Strategies and Optimal Control of a Make‐to‐Stock Queue

We consider a make‐to‐stock, finite‐capacity production system with setup cost and delay‐sensitive customers. To balance the setup and inventory related costs, the production manager adopts a two‐critical‐number control policy, where the production starts when the number of waiting customers reaches a certain level and shuts down when a certain quantity of inventory has accumulated. Once the production is set up, the unit production time follows an exponential distribution. Potential customers arrive according to a Poisson process. Customers are strategic, i.e., they make decisions on whether to stay for the product or to leave without purchase based on their utility values, which depend on the production manager's control decisions. We formulate the problem as a Stackelberg game between the production manager and the customers, where the former is the game leader. We first derive the equilibrium customer purchasing strategy and system performance. We then formulate the expected cost rate function for the production system and present a search algorithm for obtaining the optimal values of the two control variables. We further analyze the characteristics of the optimal solution numerically and compare them with the situation where the customers are non‐strategic.     

Multi‐Echelon Inventory Management under Short‐Term Take‐or‐Pay Contracts

We extend the Clark–Scarf serial multi‐echelon inventory model to include procuring production inputs under short‐term take‐or‐pay contracts at one or more stages. In each period, each such stage has the option to order/process at two different cost rates; the cheaper rate applies to units up to the contract quantity selected in the previous period. We prove that in each period and at each such stage, there are three base‐stock levels that characterize an optimal policy, two for the inventory policy and one for the contract quantity selection policy. The optimal cost function is additively separable in its state variables, leading to conquering the curse of dimensionality and the opportunity to manage the supply chain using independently acting managers. We develop conditions under which myopic policies are optimal and illustrate the results using numerical examples. We establish and use a generic one‐period result, which generalizes an important such result in the literature. Extensions to cover variants of take‐or‐pay contracts are included. Limitations are discussed.     

Multiple In‐Cycle Transshipments with Positive Delivery Times

We study a centralized inventory sharing system of two retailers that are replenished periodically. Between two replenishments, a unit can be transshipped to a stocked‐out retailer from the other. It arrives a transshipment time later, during which the stocked‐out retailer incurs backorder cost. Without transshipment, backorder cost is incurred until the next replenishment. Since the transshipment time is shorter than the time between two replenishments, transshipments can reduce the backorder cost at the stocked‐out retailer and the holding costs at the other retailer. The system is directed by a centralized inventory manager, who minimizes the long‐run average cost consisting of replenishment, holding, backorder, and transshipment costs. The transshipment policy is characterized by hold‐back inventory levels, which are nonincreasing in the remaining time until the next replenishment. The transshipment policy differs from those in the literature because we allow for multiple transshipments between replenishments, positive transshipment times, and backorder costs. We also discuss the challenges associated with positive replenishment time and develop upper and lower bounds of average cost in this case. Bounds are numerically shown to have an average gap of 1.1%. A heuristic solution is based on the upper bound and differs from the optimal cost by at most this gap.     

Heuristics for Base‐Stock Levels in Multi‐Echelon Distribution Networks

We study inventory optimization for locally controlled, continuous‐review distribution systems with stochastic customer demands. Each node follows a base‐stock policy and a first‐come, first‐served allocation policy. We develop two heuristics, the recursive optimization (RO) heuristic and the decomposition‐aggregation (DA) heuristic, to approximate the optimal base‐stock levels of all the locations in the system. The RO heuristic applies a bottom‐up approach that sequentially solves single‐variable, convex problems for each location. The DA heuristic decomposes the distribution system into multiple serial systems, solves for the base‐stock levels of these systems using the newsvendor heuristic of Shang and Song (2003), and then aggregates the serial systems back into the distribution system using a procedure we call “backorder matching.” A key advantage of the DA heuristic is that it does not require any evaluation of the cost function (a computationally costly operation that requires numerical convolution). We show that, for both RO and DA, changing some of the parameters, such as leadtime, unit backordering cost, and demand rate, of a location has an impact only on its own local base‐stock level and its upstream locations’ local base‐stock levels. An extensive numerical study shows that both heuristics perform well, with the RO heuristic providing more accurate results and the DA heuristic consuming less computation time. We show that both RO and DA are asymptotically optimal along multiple dimensions for two‐echelon distribution systems. Finally, we show that, with minor changes, both RO and DA are applicable to the balanced allocation policy.     

带学习效应的双渠道供应链库存策略研究

针对具有学习行为的双渠道供应链问题,本文研究了两种分销渠道并存下的最优库存策略。有限计划期内,分销商通过传统销售和在线销售来满足下游顾客的需求。两种分销渠道下的销售单价为时变不减线性函数,当系统中各周期的生产订购固定成本以一定的概率具有学习效应行为时,分别建立了非变质产品生产存贮问题的混合整数约束优化模型以及易变质产品存贮问题的无约束混合整数优化模型,所建立模型的目标为极大化分销商总利润函数。对于这两类模型,通过分析其最优解的性质,利用将生产订购次数松弛为连续变量的技巧证明了最优解存在的唯一性。给出了最优策略的求解方法并比较了两类模型最优利润函数值的大小。最后通过数值算例对上述模型进行了验证,数值结果表明当供应链系统中存在学习效应行为时,该系统能够获得更多的利润。     

Safety Stock,Excess Capacity or Diversification: Trade‐Offs under Supply and Demand Uncertainty

Firms mitigate uncertainty in demand and supply by carrying safety stock, planning for excess capacity and diversifying supply sources. In this study, we provide a framework to jointly optimize these three levers in a periodic review infinite horizon setting, and in particular we examine how one can reduce inventory and capacity investments through proper diversification strategies. Observing that a modified base‐stock inventory policy is optimal, we find that the capacity‐diversification problem is well behaved and characterize the optimal mix of safety stock, excess capacity and extra number of supply sources. We find that higher supply uncertainty results in higher safety stock, more excess capacity, and higher diversification. But safety stock and diversification are non‐monotonic in demand uncertainty. Our results can be extended to situations in which suppliers are heterogeneous, and can be used to develop effective heuristics.     

有限时域与采购价格上升情形下的最优订货策略

研究了有限时域下采购商面对价格上升时的订货策略问题.在分析问题的基础上提出一种新的最优采购策略,并分析了价格上升幅度对订货量的影响,以经典EOQ模型的总成本为基准,比较了本文提出的策略与文献已有策略在成本节约上的差异.本文对库存总成本的计算方法更加精确;分析表明在有限时域背景下采购商的临时订货量决定于价格上涨的幅度、在库库存以及时段长度.