Evan L. Porteus

Selling to the Newsvendor: An Analysis of Price-Only Contracts

We consider a simple supply-chain contract in which a manufacturer sells to a retailer facing a newsvendor problem and the lone contract parameter is a wholesale price. We develop a mild restriction satisfied by many common distributions that assures that the manufacturers problem is readily amenable to analysis. The manufacturers profit and sales quantity increase with market size, but the resulting wholesale price depends on how the market grows. For the cases we consider, we identify relative variability (i.e., the coefficient of variation) as key: As relative variability decreases, the retailers price sensitivity decreases, the wholesale price increases, the decentralized system becomes more efficient (i.e., captures a greater share of potential profit), and the manufacturers share of realized profit increases. Decreasing relative variability, however, may leave the retailer severely disadvantaged as the higher wholesale price reduces his profitability. We explore factors that may lead the manufacturer to set a wholesale price below that which would maximize her profit, concentrating on retailer participation in forecasting and retailer power. As these and other considerations can result in a wholesale price below what we initially suggest, our base model represents a worst-case analysis of supply-chain performance.

Dynamic Process Improvement

This paper explores the economics of investing in gradual process improvement, a key component, with empirically supported importance, of the well known Just-in-Time and Total Quality Control philosophies. We formulate a Markov decision process, analyze it, and apply it to the problem of setup reduction and process quality improvement. Instead of a one-time investment opportunity for a large predictable technological advance, we allow many smaller investments over time, with potential process improvements of random magnitude. We use a somewhat nonstandard formulation of the immediate return, which facilitates the derivation of results. The policy that simply maximizes the immediate return, called the last chance policy, provides an upper bound on the optimal investment amount. Furthermore, if the last chance policy invests in process improvement, then so does the optimal policy. Each continues investing until a shared target state is attained. We derive fairly restrictive conditions that must be met for the policy of investing forever in process improvements to be optimal. Decreasing the uncertainty of the process making the potential improvements more predictable has a desirable effect: the total return is increased and the target state increases, so the ultimate system is more productive. Numerical examples are presented and analyzed.

Joint Inventory and Pricing Decisions for an Assortment

We seek optimal inventory levels and prices of multiple products in a given assortment in a newsvendor model (single period, stochastic demand) under price-based substitution, but not stockout-based substitution. We address a demand model involving multiplicative uncertainty, motivated by market share models often used in marketing. The pricing problem that arises is known not to be well behaved in the sense that, in its deterministic version, the objective function is not jointly quasi-concave in prices. However, we find that the objective function is still reasonably well behaved in the sense that there is a unique solution to the first-order conditions, and this solution is optimal for our problem.

Multistage Inventory Management with Expediting

After reformulating Clark and Scarfs (1960) classical serial multi-echelon model so that the lead time between adjacent echelons is one week (period), the option to expedite between each resulting echelon is added. Thus, each week requires a decision to be made at each echelon on how many units to expedite in from the next upstream echelon (to be received immediately) and how many to regular order (to be received in one week), with the remainder detained (left as is). The model can be interpreted as addressing dynamic lead time management, in which the (remaining) effective lead time for each ordered unit can be dynamically reduced by expediting and/or extended. Use of Clark and Scarfs (1960) idea of echelon stocks reduces a complex, multidimensional stocking problem to the analysis of a series of one-dimensional subproblems. What are calledtop-down base stock policies, which are readily amenable to managerial interpretation, are shown to be optimal. Myopic policies are shown to be optimal in the stationary, in1nite horizon case. The results are illustrated numerically.

Conditions for characterizing the structure of optimal strategies in infinite-horizon dynamic programs

The study of infinite-horizon nonstationary dynamic programs using the operator approach is continued. The point of view here differs slightly from that taken by others, in that Denardos local income function is not used as a starting point. Infinite-horizon values are defined as limits of finite-horizon values, as the horizons get long. Two important conditions of an earlier paper are weakened, yet the optimality equations, the optimality criterion, and the existence of optimal “structured” strategies are still obtained.

Simultaneous Capacity and Production Management of Short-Life-Cycle, Produce-to-Stock Goods Under Stochastic Demand

This paper derives the optimal simultaneous capacity and production plan for a shortlife-cycle, produce-to-stock good under stochastic demand. Capacity can be reduced as well as added, at exogenously set unit prices. In both cases studied, with and without carryover of unsold units, a target interval policy is optimal: There is a (usually different) target interval for each period such that capacity should be changed as little as possible to bring the level available into that interval. Our contribution in the case of no carry-over, is a detailed characterization of the target intervals, assuming demands increase stochastically at the beginning of the life cycle and decrease thereafter. In the case of carry-over, we establish the general result and show that capacity and inventory are economic substitutes: The target intervals decrease in the initial stock level and the optimal unconstrained base stock level decreases in the capacity level. In both cases, optimal service rates are not necessarily constant over time. A numerical example illustrates the results.

Bounds and Transformations for Discounted Finite Markov Decision Chains

This paper develops new improved bounds on the optimal return function in finite state and action, infinite horizon, discounted stationary Markov decision chains. The bounds are obtained by solving a single-constraint, bounded-variable linear program. They can be used for algorithmic termination criteria and improved tests for suboptimal decisions. We show how to implement these tests so that little additional computational effort is required. We consider several transformations that can be used to convert a process into an equivalent one that may be easier to solve. We examine whether the transformations reduce the spectral radius and/or the norm maximum row sum of the process. Gauss-Seidel iteration and Jacobi iteration are shown to be special cases of the general transformations. Gauss-Seidel iteration is given additional consideration. Another special case not only preserves equality of the row sums and sparsity but, when applicable, can dramatically reduce the norm. It reduces each element in a column by that columns smallest element. Several possible computational approaches are applied to a small numerical example.

Numerical Comparisons of Inventory Policies for Periodic Review Systems

This paper studies the numerical computation of the two parameters the reorder level s and the order up to level S of inventory policies for discrete time shortage cost systems. Our goal is to obtain approximately optimal policies with little computational effort. The paper introduces three new methods that are designed to achieve this goal. Two of the methods are shortcuts based on the method of Freeland and Porteus and one is a heuristic that makes several modifications to a standard continuous review approximation. The paper provides a fairly detailed survey of other methods for easily computing approximately optimal inventory policies. It then numerically compares all these methods on a reasonably broad range of problems. One of the shortcuts and the new heuristic method performed very well: the percentage error of their average costs was approximately 1%. Some commonly cited competing methods had percentage errors of over 10% and a commonly cited continuous review approximation had a percentage error of over 80%. To study the effect of extreme parameter choices in the test bed, the paper introduces a procedure to determine a subset of the parameter values, called the 1% contiguous test bed, for which each method performed well. The results show that, depending on the range of values that apply in a given practical situation, either i any of a large number of methods will yield good performance or ii a carefully selected method can achieve superior performance.

Responsibility Tokens in Supply Chain Management

The decentralized supply chain management scheme of Lee and Whang (1999) can be viewed as operationalizing the decentralized management scheme implicit in Clark and Scarf (1960). This paper proposes the use of what are called responsibility tokens (RTs) to further facilitate that operationalization. The proposal assumes that a management information system, presumably electronic, is established to monitor inventories and shipment quantities, and to carry out transfer payments between players. As in Lee and Whang(1999), the incentives of the system are aligned, so if each player is brilliantly self-serving, the system optimal solution will result. While the system administrator need not know how the system should be managed, the most upstream player must know how to manage the system optimally for the system optimal solution to be achieved. RTs endow the system with an attractive self-correcting property: An example illustrates that upstream players are given a mechanism and the incentive to correct for downstream overordering. The downstream players who over-order are penalized, but system performance is not degraded much. Extensions and further research are also discussed.

Chapter 12 Stochastic inventory theory

Publisher Summary Inventory theory deals with the management of stock levels of goods, with the intent of effectively meeting demands for those goods. The demands for goods are made by buyers and are met by sellers, regardless of whether monetary exchange is involved. This chapter discusses the stochastic inventory theory. The chapter introduces deterministic economic order quantity (EOQ) model and focuses on the single period newsvendor model. The critical difference in the analyses of these models is the mathematical form of the ordering/production cost function. Many properties of the solution to the newsvendor problem generalize to the case of the proportional ordering/production cost function. When that function is convex, but nonlinear, various reasonable properties of the optimal solution are preserved, but the possibility of using a trivial computation to find the optimal solution tends to fail. The problems are exacerbated when the ordering/production cost function is concave (and nonlinear).