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.     

Flexible Backup Supply and the Management of Lead‐Time Uncertainty

We consider replenishment decisions for a constant rate demand environment from a supplier with uncertain lead times. We study the potential use of a flexible backup supplier as an emergency response to accurate lead‐time information arriving at (or close after) the beginning of the demand interval and well after an original order with the stochastic lead‐time supplier has been placed. The emergency response decisions involve whether to order and how much from the flexible backup supplier, with the objective of minimizing the cost of meeting demand. We derive the optimal emergency‐response policy and clearly outline its implications on the optimized safety lead time of the original order placement and on the cost of meeting demand. We examine the impact on the use of the flexible backup supplier of factors like the arrival time of accurate lead‐time information and the response lead time of the backup supplier. We further study the potential benefits of the use of the flexible backup supplier in a dual role: as one of the two suppliers in a redundant supply system assigned to originally meet the demand and as an emergency response to later‐arriving lead‐time information. Our numerical studies illustrate the benefits from the use of the flexible backup supplier as an emergency response, but for reasonable purchase premiums and short lead times of flexible backup supply options, their use in a dual (regular and emergency response) role often leads to improved performance over safety lead‐time single and uncertain lead‐time supplier‐replenishment strategies. The benefits of the backup supply options are accentuated the higher the lead‐time uncertainty of the stochastic lead‐time supplier is.     

Contracting Under Vendor Managed Inventory Systems Using Holding Cost Subsidies

Vendor managed inventory systems are becoming increasingly popular. An important issue in implementing a vendor managed inventory scheme is the contracting terms that dictate the ownership of the inventory and the responsibility of inventory replenishment decisions. Thus the performance of a vendor managed system crucially depends on these terms and on how inventory‐related costs are shared in a supply chain. We consider a system where a manufacturer supplies a single product to a retailer who faces random demand in a competitive market. The retailer incurs a fixed cost per order, inventory holding cost, and a penalty cost for a stockout (unsatisfied demand is back‐ordered). Further, the manufacturer incurs a penalty cost when there is a stockout at the retailer and a fixed replenishment cost. We assume that the players are rational and act noncooperatively. We compare the performance of retailer managed inventory systems, where the retailer places orders and makes replenishment decisions, with vendor managed inventory systems, wherein the vendor or manufacturer makes inventory and replenishment decisions. Specifically, in the vendor managed inventory system, we propose and evaluate holding cost subsidy‐type contracts on inventories offered by the retailer to improve system performance. We evaluate this contract in the context of three widely used inventory systems—deterministic economic order quantity, continuous review (Q, r) policies, and periodic review policies—and show when such contracts may improve channel performance.     

Benefits of Hybrid Lateral Transshipments in Multi‐Item Inventory Systems under Periodic Replenishment

Lateral transshipments are a method of responding to shortages of stock in a network of inventory‐holding locations. Conventional reactive approaches only seek to meet immediate shortages. The study proposes hybrid transshipments which exploit economies of scale by moving additional stock between locations to prevent future shortages in addition to meeting immediate ones. The setting considered is motivated by retailers who operate networks of outlets supplying car parts via a system of periodic replenishment. It is novel in allowing non‐stationary stochastic demand and general patterns of dependence between multiple item types. The generality of our work makes it widely applicable. We develop an easy‐to‐compute quasi‐myopic heuristic for determining how hybrid transshipments should be made. We obtain simple characterizations of the heuristic and demonstrate its strong cost performance in both small and large networks in an extensive numerical study.     

Order Progress Information: Improved Dynamic Emergency Ordering Policies

Technologies such as radio‐frequency identification and global positioning systems can provide improved real‐time tracking information for products and replenishment orders along the supply chain. We call this type of visibility order progress information. In this paper, we investigate how order progress information can be used to improve inventory replenishment decisions. To this end, we examine a retailer facing a stochastic lead time for order fulfillment. We characterize a replenishment policy that is based on the classical (Q, R) policy and that allows for releasing emergency orders in response to the order progress information. We show that the optimal structure of this policy is given by a sequence of threshold values dependent on order progress information. In a numerical study we evaluate the cost savings due to this improved replenishment policy.     

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.     

A Nonatomic‐Game Approach to Dynamic Pricing under Competition

We study a revenue management problem involving competing firms. We assume the presence of a continuum of infinitesimal firms where no individual firm has any discernable influence over the evolution of the overall market condition. Under this nonatomic‐game approach, the unanimous adoption of an equilibrium pricing policy by all firms will yield a market‐condition process that in turn will elicit the said policy as one of the best individual responses. For both deterministic‐ and stochastic‐demand cases, we show the existence of equilibrium pricing policies that exhibit well‐behaving monotone trends. Our computational study reveals many useful insights, including the fact that only a reasonable number of firms are needed for our approach to produce near‐rational pricing policies.     

MRP Planned Orders in a Multiple‐Supplier Environment with Differing Lead Times

This study examines a deterministic material requirements planning (MRP) problem where lead times at subsequent ordering moments differ. Adequate replenishment methods that can cope with lead time differences are lacking because of the order crossover phenomenon, that is, replenishment orders are not received in the sequence they are ordered. This study specifies how to handle order crossovers and recalculate planned order releases after an update of gross requirements. The optimal (s, S) policy is based on dynamic programing. The state space is kept to a minimum due to three fundamental insights. The performance of the optimal solution approach is compared with two heuristics based on relaxations and a benchmark approach in which order crossovers are ignored. A numerical analysis reveals that average cost savings up to 25% are possible if the optimal policy is used instead of the benchmark approach. The contribution of this study is threefold: (1) it generalizes theory on MRP ordering, allowing for lead time differences and order crossovers; (2) it develops new fundamental insights and an optimal solution procedure, leading to substantial cost saving; and (3) it provides good‐performing heuristics for a general and realistic replenishment problem that can replace the current replenishment methods within MRP.     

Priority Rules for Multi‐Task Due‐Date Scheduling under Varying Processing Costs

We study the scheduling of multiple tasks under varying processing costs and derive a priority rule for optimal scheduling policies. Each task has a due date, and a non‐completion penalty cost is incurred if the task is not completely processed before its due date. We assume that the task arrival process is stochastic and the processing rate is capacitated. Our work is motivated by both traditional and emerging application domains, such as construction industry and freelance consulting industry. We establish the optimality of Shorter Slack time and Longer remaining Processing time (SSLP) principle that determines the priority among active tasks. Based on the derived structural properties, we also propose an effective cost‐balancing heuristic policy and demonstrate the efficacy of the proposed policy through extensive numerical experiments. We believe our results provide operators/managers valuable insights on how to devise effective service scheduling policies under varying costs.     

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.     

Inventory‐Based Dynamic Pricing with Costly Price Adjustment

We study an average‐cost stochastic inventory control problem in which the firm can replenish inventory and adjust the price at anytime. We establish the optimality to change the price from low to high in each replenishment cycle as inventory is depleted. With costly price adjustment, scale economies of inventory replenishment are reflected in the cycle time instead of lot size—An increased fixed ordering cost leads to an extended replenishment cycle but does not necessarily increase the order quantity. A reduced marginal cost of ordering calls for an increased order quantity, as well as speeding up product selling within a cycle. We derive useful properties of the profit function that allows for reducing computational complexity of the problem. For systems requiring short replenishment cycles, the optimal solution can be easily computed by applying these properties. For systems requiring long replenishment cycles, we further consider a relaxed problem that is computational tractable. Under this relaxation, the sum of fixed ordering cost and price adjustment cost is equal to (greater than, less than) the total inventory holding cost within a replenishment cycle when the inventory holding cost is linear (convex, concave) in the stock level. Moreover, under the optimal solution, the time‐average profit is the same across all price segments when the inventory holding cost is accounted properly. Through a numerical study, we demonstrate that inventory‐based dynamic pricing can lead to significant profit improvement compared with static pricing and limited price adjustment can yield a benefit that is close to unlimited price adjustment. To be able to enjoy the benefit of dynamic pricing, however, it is important to appropriately choose inventory levels at which the price is revised.     

Analysis and Management of Periodic Review,Order‐Up‐To Level Inventory Systems with Order Crossover

In this article, we investigate the (R, S) periodic review, order‐up‐to level inventory control system with stochastic demand and variable leadtimes. Variable leadtimes can lead to order crossover, in which some orders arrive out of sequence. Most theoretical studies of order‐up‐to inventory systems under variable leadtimes assume that crossovers do not occur and, in so doing, overestimate the standard deviation of the realized leadtime distribution and prescribe policies that can inflate inventory costs. We develop a new analytic model of the expected costs associated with this system, making use of a novel approximation of the realized (reduced) leadtime standard deviation resulting from order crossovers. Extensive experimentation through simulation shows that our model closely approximates the true expected cost and can be used to find values of R and S that provide an expected cost close to the minimum cost. Taking account of, as opposed to ignoring, crossovers leads, on average, to substantial improvements in accuracy and significant cost reductions. Our results are particularly useful for managers seeking to reduce inventory costs in supply chains with variable leadtimes.     

Optimal Policies for a Two‐Product Inventory System under a Flexible Substitution Scheme

We study a two‐product inventory model that allows substitution. Both products can be used to supply demand over a selling season of N periods, with a one‐time replenishment opportunity at the beginning of the season. A substitution could be offered even when the demanded product is available. The substitution rule is flexible in the sense that the seller can choose whether or not to offer substitution and at what price or discount level, and the customer may or may not accept the offer, with the acceptance probability being a decreasing function of the substitution price. The decisions are the replenishment quantities at the beginning of the season, and the dynamic substitution‐pricing policy in each period of the season. Using a stochastic dynamic programming approach, we present a complete solution to the problem. Furthermore, we show that the objective function is concave and submodular in the inventory levels—structural properties that facilitate the solution procedure and help identify threshold policies for the optimal substitution/pricing decisions. Finally, with a state transformation, we also show that the objective function is ‐concave, which allows us to derive similar structural properties of the optimal policy for multiple‐season problems.     

Replenishment Strategies for Micro‐Retailers in Developing Countries

When formal distribution channels are absent in developing countries, micro‐retailers travel a long distance to replenish their stocks directly from suppliers. This “informal” replenishment strategy is inefficient due to high imputed travel costs involved in the replenishment process. To improve efficiency, one “hybrid” replenishment strategy has emerged under which one of the micro‐retailers in a neighborhood, while continuing its own retail business, also operates as a wholesaler to serve other micro‐retailers in the neighborhood. A major obstacle for the transition from the informal strategy to the hybrid strategy is that the micro‐retailers are reluctant to source from a wholesaler who also competes with them in the retail market. Thus, it is not clear when the micro‐retailers adopt the hybrid strategy instead of the informal strategy. Meanwhile, the micro‐retailers may prefer the “formal” strategy: a traditional replenishment strategy under which one of the micro‐retailers relinquishes its retail business and operates purely as a wholesaler. We examine a situation when competing micro‐retailers contemplate with the three potential aforementioned replenishment strategies. Our equilibrium analysis of the two‐store model reveals that the dominant strategy is: (a) the hybrid strategy when the travel cost is high; (b) the formal strategy when the travel cost is medium; and (c) the informal strategy when the travel cost is low. This key insight is shown continue to hold when we extend the two‐store model to incorporate other issues including: quantity discounts from the supplier, variable operating costs, price competition, local monopolies, and different decision sequences. One additional finding is that the transition from the informal strategy to the hybrid strategy could benefit consumers if the micro‐retailers receive quantity discounts from the supplier. Furthermore, when there are more than two stores, we show that the formal strategy is never dominant. Moreover, when the number of stores increases, the hybrid strategy becomes more preferable to the informal strategy.     

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.     

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.     

On Optimal Expediting Policy for Supply Systems with Uncertain Lead‐Times

We examine the role of expediting in dealing with lead‐time uncertainties associated with global supply chains of “functional products” (high volume, low demand uncertainty goods). In our developed stylized model, a retailer sources from a supplier with uncertain lead‐time to meet his stable and known demand, and the supply lead‐time is composed of two random duration stages. At the completion time of the first stage, the retailer has the option to expedite a portion of the replenishment order via an alternative faster supply mode. We characterize the optimal expediting policy in terms of if and how much of the order to expedite and explore comparative statics on the optimal policy to better understand the effects of changes in the cost parameters and lead‐time properties. We also study how the expediting option affects the retailer's decisions on the replenishment order (time and size of order placement). We observe that with the expediting option the retailer places larger orders closer to the start of the selling season, thus having this option serve as a substitute for the safety lead‐time and allowing him to take increased advantages of economies of scale. Finally we extend the basic model by looking at correlated lead‐time stages and more than two random lead‐time stages.     

Managing Demand Risk in Tactical Supply Chain Planning for a Global Consumer Electronics Company

We consider the problem of managing demand risk in tactical supply chain planning for a particular global consumer electronics company. The company follows a deterministic replenishment‐and‐planning process despite considerable demand uncertainty. As a possible way to formally address uncertainty, we provide two risk measures, “demand‐at‐risk” (DaR) and “inventory‐at‐risk” (IaR) and two linear programming models to help manage demand uncertainty. The first model is deterministic and can be used to allocate the replenishment schedule from the plants among the customers as per the existing process. The other model is stochastic and can be used to determine the “ideal” replenishment request from the plants under demand uncertainty. The gap between the output of the two models as regards requested replenishment and the values of the risk measures can be used by the company to reallocate capacity among different products and to thus manage demand/inventory risk.     

Flexible Servers in Understaffed Tandem Lines

We study the dynamic assignment of cross‐trained servers to stations in understaffed lines with finite buffers. Our objective is to maximize the production rate. We identify optimal server assignment policies for systems with three stations, two servers, different flexibility structures, and either deterministic service times and arbitrary buffers or exponential service times and small buffers. We use these policies to develop server assignment heuristics for Markovian systems with larger buffer sizes that appear to yield near‐optimal throughput. In the deterministic setting, we prove that the best possible production rate with full server flexibility and infinite buffers can be attained with partial flexibility and zero buffers, and we identify the critical skills required to achieve this goal. We then present numerical results showing that these critical skills, employed with an effective server assignment policy, also yield near‐optimal throughput in the Markovian setting, even for small buffer sizes. Thus, our results suggest that partial flexibility is sufficient for near‐optimal performance, and that flexibility structures that are effective for deterministic and infinite‐buffered systems are also likely to perform well for finite‐buffered stochastic systems.     

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.