Computing globally optimal (s,S,T) inventory policies

We consider a single-echelon inventory installation under the (s,S,T) periodic review ordering policy. Demand is stationary random and, when unsatisfied, is backordered. Under a standard cost structure, we seek to minimize total average cost in all three policy variables; namely, the reorder level s, the order-up-to level S and the review interval T. Considering time to be continuous, we first model average total cost per unit time in terms of the decision variables. We then show that the problem can be decomposed into two simpler sub-problems; namely, the determination of locally optimal solutions in s and S (for any T) and the determination of the optimal T. We establish simple bounds and properties that allow solving both these sub-problems and propose a procedure that guarantees global optimum determination in all policy variables via finite search. Computational results reveal that the usual practice of not treating the review interval as a decision variable may carry severe cost penalties. Moreover, cost differences between (s,S,T) and other standard periodic review policies, including the simple base stock policy, are rather marginal (or even zero), when all policies are globally optimized. We provide a physical interpretation of this behavior and discuss its practical implications.     

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.     

RETAIL INVENTORY CONTROL WITH LOST SALES,SERVICE CONSTRAINTS,AND FRACTIONAL LEAD TIMES

This paper describes a periodic review, fixed lead time, single-product, single-facility model with random demand, lost sales and service constraints that was developed for potential application at a Western Canadian retailer. The objective of this study was to determine optimal (s, S) policies for a large number of products and locations. To this end, we evaluate the long run average cost and service level for a fixed (s, S) policy and then used a search procedure to locate an optimal policy. The search procedure is based on an efficient updating scheme for the transition probability matrix of the underlying Markov chain, bounds on S and monotonicity assumptions on the cost and service level functions. A distinguishing feature of this model is that lead times are shorter than review periods so that the stationary analysis underlying computation of costs and service levels requires subtle analyses. We compared the computed policies to those currently in use on a test bed of 420 products and found that stores currently hold inventories that are 40% to 50% higher than those recommended by our model and estimate that implementing the proposed policies for the entire system would result in significant cost savings.     

A JOINT ECONOMIC-LOT-SIZE MODEL FOR PURCHASER AND VENDOR

In a typical purchasing situation, the issues of price, lot sizing, etc., usually are settled through negotiations between the purchaser and the vendor. Depending on the existing balance of power, the end result of such a bargaining process may be a near-optimal or optimal ordering policy for one of the parties (placing the other in a position of significant disadvantage) or, sometimes, inoptimal policies for both parties. This paper develops a joint economic-lot-size model for a special case where a vendor produces to order for a purchaser on a lot-for-lot basis under deterministic conditions. The focus of this model is the joint total relevant cost. It is shown that a jointly optimal ordering policy, together with an appropriate price adjustment, can be beneficial economically for both parties or, at the least, does not place either at a disadvantage.     

A discussion on economic ordering policy of deteriorated item for vendor and buyer

Yang and Wee (Economic ordering policy of deteriorated item for vendor and buyer: an integrated approach. Prod. Plan. & Cont., 2000, 11(5), 474–480) proposed an ordering policy for a vendor--buyer integrated model. This note examines the cost component of Yang and Wee's model and gives some insight on the derivation of the holding cost function. We have discovered that their model violates the positive holding cost characteristic and the total quantity-equality characteristic. A proposal to eradicate the problem is given.     

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.     

Point‐of‐Use Hybrid Inventory Policy for Hospitals

Modern point‐of‐use technology at hospitals has enabled new replenishment policies for medical supplies. One of these new policies, which we call the hybrid policy, is currently in use at a large U.S. Midwest hospital. The hybrid policy combines a low‐cost periodic replenishment epoch with a high‐cost continuous replenishment option to avoid costly stockouts. We study this new hybrid policy under deterministic and stochastic demand. We develop a parameter search engine using simulation to optimize the long‐run average cost per unit time and, via a computational study, we provide insights on the benefits (reduction in cost, inventory, and number of replenishments) that hospitals may obtain by using the hybrid policy instead of the commonly used periodic policies. We also use the optimal hybrid policy parameters from the deterministic analysis to propose approximate expressions for the stochastic hybrid policy parameters that can be easily used by hospital management.      

The Individually Responsible and Rational Decision Approach to Economic Lot Sizes for One Vendor and Many Purchasers: A Comment*

Joglekar and Tharthare [6] presented an alternate approach for minimizing total inventory carrying and ordering costs of a vendor and the purchaser(s). This approach permits the vendor and the purchaser(s) to rationally select their operating policies. Joglekar and Tharthare claimed that their approach is more economical than the joint lot-size approach. In this note we identify some conceptual issues in their approach and demonstrate the superiority of the joint lot-size approach with the help of an example.     

The Performance of MRP Purchase Lot-Size Procedures Under Actual Multiple Purchase Discount Conditions

Several heuristic procedures for purchase lot sizing in material requirements planning (MRP) systems were tested with actual data from manufacturing companies. Information provided by the companies for each purchased item included the estimated requirements and costs, the price discount structure from the vendor, and the actual company ordering policy. Simulation tests for each purchased item involved comparisons of several purchase lot-size procedures from the research literature along with the actual procedure used by the company providing the data. Results indicate that one of the heuristic lot-size procedures from the literature consistently outperformed the company policies as well as all other models tested. Another noteworthy result is that, in some cases, the actual company order policy was more cost-effective than some of the models from the research literature.     

POWER APPROXIMATIONS FOR A TWO-ECHELON INVENTORY SYSTEM USING SERVICE LEVELS

We consider a two-echelon inventory system with one warehouse and several stores. The warehouse as well as the stores are controlled by periodic review (s, S) inventory policies. We study the interrelationship between the safety stocks at the warehouse and the stores. Stockouts at the warehouse will result in supply delays to the stores and cause the lead time to be stochastic. The stores may react by increasing their safety stock. However, there is a trade-off between the safety stock at the warehouse and the safety stock at the stores. We use a service level at the warehouse to quantify the effect of warehouse stockouts on the lead time to the stores. The service level at the warehouse is considered a decision variable to find the best compromise between the various safety stocks by minimizing the overall costs. Using power approximations for the (s, S) policies, we provide an iterative procedure for adjusting the lead time distribution to the stores; this can result in substantial savings, but it doesn't guarantee the overall optimality. Numerical studies are provided to test the accuracy of approximations. The effects of the different system parameters on the inventory policy give general guidelines for use of the policies.     

The benefit of receding horizon control: Near-optimal policies for stochastic inventory control

In this paper we address the single-item, single-stocking point, non-stationary stochastic lot-sizing problem under backorder costs. It is well known that the (s, S) policy provides the optimal control for such inventory systems. However the computational difficulties and the nervousness inherent in (s, S) paved the way for the development of various near-optimal inventory control policies. We provide a systematic comparison of these policies and present their expected cost performances. We further show that when these policies are used in a receding horizon framework the cost performances improve considerably and differences among policies become insignificant.     

A Purchasing Framework for B2B Pricing Decisions and Risk‐sharing in Supply Chains*

This paper presents a common modelling structure for (i) the implementation of operational policies by individual purchasing managers of risk‐sharing agreements among supply‐chain partners, and (ii) the integration of brick and click purchasing policies in a B2B. The problem of price uncertainty created within these two environments is modelled as a stochastic repetitive‐sales problem, applicable to any probability distribution. The model identifies sufficient conditions for regenerative ordering cycles, which allows for the use of the renewal reward theorem. The end result is a two‐price purchasing policy, which may substantially ease implementation problems across a global corporation's purchasing managers world‐wide and across B2B markets.     

The Individually Responsible and Rational Decision Approach to Economic Lot Sizes for One Vendor and Many Purchasers

Banerjee's [2] joint economic lot size (JELS) model, along with related works by Monahan [12] and Lal and Staelin [10], represents one approach to minimizing the total inventory carrying and ordering costs of a vendor and his purchaser(s). Noting that JELS philosophy requires a coordinated system and that its practical implementation is problematic, we present an alternative approach to the same problem: the individually responsible and rational decision (IRRD) approach. The IRRD approach is consistent with a free enterprise system and easy to implement. In order to show that the IRRD approach is also more economical than the JELS approach, we first build a more refined JELS model for the case of one vendor and many identical purchasers. Drawing on earlier criticisms of specific JELS models, our refinement relaxes the lot-for-lot assumption commonly used by JELS scholars. To be comparable with earlier works, we retain the assumption of deterministic conditions and demonstrate the economic advantages of IRRD over JELS through a numerical example. An algebraic proof of IRRD's superiority over JELS is offered in the more general and realistic case of a vendor dealing with K nonidentical purchasers with reasonably predictable annual demand but uncertain order quantities and timings.     

AN APPLICATION OF INVENTORY THEORETICAL MODELS TO CASH COLLECTION

A problem common to all companies operating widely dispersed revenue-generating outlets is that of transferring these revenues to central accounts. Company funds held in local banks represent an unusable asset that, in view of the high opportunity cost of money, may substantially increase operating costs. A wide variety of systems can be devised to effect this transfer process. As an example of such systems, this paper examines a transfer procedure that uses depository transfer checks. It is shown that a simple (s, S) policy characterizes the optimal transfer policy.     

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.     

PRODUCT STRUCTURE COMPLEXITY AND MULTILEVEL LOT SIZING USING ALTERNATIVE COSTING POLICIES*

This research evaluates the effect of product structure complexity on the performance of several lot-sizing procedures in a multilevel manufacturing environment. The experiment compares two different costing policies, full value added (FVA) and marginal value added (MVA), for calculating inventory holding cost. The major finding of the research is that product structure complexity has very little effect on the performance of various lot-sizing procedures. A second finding is that when product structures with varying components per parent and stocking points for a particular end item are present, the MVA costing policy emerges as the policy of choice because it favors slightly the Silver-Meal (SM)/least-total-cost (LTC) procedures over the Wagner-Whitin (WW)/LTC procedures.     

Economic ordering policy of deteriorated item for vendor and buyer: An integrated approach

This study develops an economic ordering policy of a deteriorating item with a constant production and demand rate. By considering the view of both the vendor and buyer, a mathematical model subject to single-vendor-single-buyer and multiple deliveries per order is developed. It can be shown that the integrated approach results in an impressive cost-reduction compared with an independent decision by the buyer.     

Optimal Ordering Policies for Inventory Problems With Dynamic Information Delays

Information delays exist when the most recent inventory information available to the Inventory Manager (IM) is dated. In other words, the IM observes only the inventory level that belongs to an earlier period. Such situations are not uncommon, and they arise when it takes a while to process the demand data and pass the results to the IM. We introduce dynamic information delays as a Markov process into the standard multiperiod stochastic inventory problem with backorders. We develop the concept of a reference inventory position. We show that this position along with the magnitude of the latest observed delay and the age of this observation are sufficient statistics for finding the optimal order quantities. Furthermore, we establish that the optimal ordering policy is of state‐dependent base‐stock type with respect to the reference inventory position (or state‐dependent (s, S) type if there is a fixed ordering cost). The optimal base stock and (s, S) levels depend on the magnitude of the latest observed delay and the age of this observation. Finally, we study the sensitivity of the optimal base stock and the optimal cost with respect to the sufficient statistics.     

THE TIMING OF LABOR TRANSFERS IN DUAL RESOURCE-CONSTRAINED SYSTEMS: “PUSH” VS. “PULL” RULES*

Altough the dual resource-constrained (DRC) system has been studied, the decision rule used to determine when workers are eligible for transfer largely has been ignored. Some earlier studies examined the impact of this rule [5] [12] [15] but did not include labor-transfer times in their models. Gunther [6] incorporated labour-transfer times into his model, but the model involved only one worker and two machines. No previous study has examined decision rules that initiate labor transfers based on labor needs (“pull” rules). Labor transfers always have been initiated based on lack of need (“push” rules). This study examines three “pull” variations of the “When” labor-assignment decision rule. It compares their performances to the performances of two “push” rules and a comparable machine-limited system. A nonparametric statistical test, Jonckheere's S statistic, is used to test for significance of the rankings of the rules: a robust parametric multiple-comparison statistical test, Tukey's B statistic, is used to test the differences. One “pull” and one “push” decision rule provide similar performances and top the rankings consistently. Decision rules for determining when labor should be transferred from one work area to another are valuable aids for managers. This especially is true for the ever-increasing number of managers operating in organizations that recognize the benefits of a cross-trained work force. Recently there has been much interest in cross-training workers, perhaps because one of the mechanisms used in just-in-time systems to handle unbalanced work loads is to have cross-trained workers who can be shifted as demand dictates [8]. If management is to take full advantage of a cross-trained work force, it need to know when to transfer workers.     

Impact of Retailers with Knowledge of Supplier's Inventory on Supply Chain Performance

We model a supply chain consisting of a supplier and multiple retailers facing deterministic demand. We denote some retailers as strategic in the sense that given the supplier inventory information, they will implement the optimal stocking policy by incorporating such information. On the other hand, some retailers are denoted as naïve in the sense that they ignore supply information and resort to a simplistic ordering policy. Naïve retailers learn the optimal policy over time and adjust their orders accordingly. We study the dynamics of this game and investigate the impact of such strategic and naïve retailers on the cost, ordering pattern and stocking policies of all parties. We analyze the supply chain under two scenarios: the centralized supply chain where the objective is to minimize the total supply chain cost, and the decentralized supply chain where each self‐interested player minimizes its own cost in a Stackelberg game setting. We fully characterize the optimal policies under both centralized and decentralized scenarios and show that, surprisingly, the supply chain might be better off by virtue of naïve retailers. The result is driven by the fact that strategic and naïve players’ decisions shift the positioning of inventory in the supply chain with its final impact being determined by the relative costs of different retailer‐types. Our results also offer managerial insights into how access to supply information can improve supply chain performance.