Yu-Sheng Zheng

Finding Optimal (s, S) Policies Is About As Simple As Evaluating a Single Policy

In this paper, a new algorithm for computing optimal ( s , S ) policies is derived based upon a number of new properties of the infinite horizon cost function c ( s , S ) as well as a new upper bound for optimal order-up-to levels S * and a new lower bound for optimal reorder levels s *. The algorithm is simple and easy to understand. Its computational complexity is only 2.4 times that required to evaluate a (specific) single ( s , S ) policy. The algorithm applies to both periodic review and continuous review inventory systems.

An efficient algorithm for computing an optimal ( r,Q ) policy in continuous review stochastic inventory systems

The reorder point/reorder quantity policies, also referred to as r, Q policies, are widely used in industry and extensively studied in the literature. However, for a period of almost 30 years there has been no efficient algorithm for computing optimal control parameters for such policies. In this paper, we present a surprisingly simple and efficient algorithm for the determination of an optimal r*, Q* policy. The computational complexity of the algorithm is linear in Q*. For the most prevalent case of linear holding, backlogging and stockout penalty costs in addition to fixed order costs, the algorithm requires at most 6r* + 13Q* elementary operations additions, comparisons and multiplications, and hence, no more than 13 times the amount of work required to do a single evaluation of the long-run average cost function in the point r*, Q*.

A Queueing Model to Analyze the Value of Centralized Inventory Information

Competitive pressures and technological improvements are leading many firms to consider centralized information systems to manage inventories and schedule production. We propose a simple model to explore the potential benefits of such coordination. The model represents two products competing for a single production facility. Simple Markovian behavior is assumed throughout. The key step in the analysis is the explicit solution of a queueing model with a novel priority discipline: Serve a customer from the class having the largest number of customers in the system.

Service parts logistics: a benchmark analysis

This paper summarizes the results of a benchmark study focusing on after-sales service logistics systems for technologically complex high-value products, i.e., in the computer industry. Current industrial practices and trends in service logistics operations are reported on. Specific data on costs, revenues, service measures, control policies, and distribution strategies were collected and used to define best practice performance. Analysis methodologies to support cross-sectional comparisons and causal factors are discussed.

One-Warehouse Multiretailer Systems with Centralized Stock Information

We consider a distribution system with a central warehouse and multiple retailers. The warehouse orders from an outside supplier and replenishes the retailers which in turn satisfy customer demand. The retailers are nonidentical, and their demand processes are independent compound Poisson. There are economies of scale in inventory replenishment, which is controlled by an echelon-stock, batch-transfer policy. For the special case with simple Poisson demand, we develop an exact method for computing the long-run average holding and backorder costs of the system. Based on this exact method, we provide approximations for compound Poisson demand. Numerical examples are used to illustrate the accuracy of the approximations. We also present a numerical comparison between the average costs of a heuristic, echelon-stock policy and an existing lower bound on the average costs of all feasible policies.

The joint replenishment problem with general joint cost structures

We consider inventory systems with several distinct items. Demands occur at constant, item specific rates. The items are interdependent because of jointly incurred fixed procurement costs: The joint cost structure reflects general economies of scale, merely assuming a monotonicity and concavity submodularity property. Under a power-of-two policy each item is replenished with constant reorder intervals which are power-of-two multiples of some fixed or variable base planning period. Our main results include a proof that, depending upon whether the base planning period is fixed or variable, the best among all power-of-two policies has an average cost which comes within either 6% or 2% of an easily computable lower bound for the minimum cost value. We also derive two efficient algorithms to compute an optimal power-of-two policy. The proposed algorithms generate as a by-product, a specific cost allocation of the joint cost structure to the individual items. With this specific allocation, the problem with separable costs is in fact equivalent to the original problem with nonseparable joint costs in the sense that the two problems share the same sets of optimal power-of-two policies with identical associated long-run average costs.

Near-Optimal Pricing and Replenishment Strategies for a Retail/Distribution System

This paper integrates pricing and replenishment decisions for the following prototypical two-echelon distribution system with deterministic demands. A supplier distributes a single product to multiple retailers, who in turn sell it to consumers. The retailers serve geographically dispersed, heterogeneous markets. The demand in each retail market arrives continuously at a constant rate, which is a general decreasing function of the retail price in the market. The supplier replenishes its inventory through orders (purchases, production runs) from a source with ample capacity. The retailers replenish their inventories from the supplier. We develop efficient algorithms to determine optimal pricing and replenishment strategies for the following three channel structures. The first is the vertically integrated channel, where the system-wide pricing and replenishment strategies are determined by a central planner whose objective is to maximize the system-wide profits. The second structure is that of a vertically integrated channel in which pricing and operational decisions are made sequentially by separate functional departments. The third channel structure is decentralized, i.e., the supplier and the retailers are independent, profit-maximizing firms with the supplier acting as a Stackelberg game leader. We apply our algorithms to a set of numerical examples to quantify the supply chain inefficiencies due to functional segregation or uncoordinated decision making in a decentralized channel. We also gain insight into systematic differences in the associated pricing and operational patterns.

Ending Inventory Valuation in Multiperiod Production Scheduling

When making lot-sizing decisions, managers often use a model horizon T that is much smaller than any reasonable estimate of the firms future horizon. This is done because forecast accuracy deteriorates rapidly for longer horizons, while computational burden increases. However, what is optimal over the short horizon may be suboptimal over the long run, resulting in errors known as end-effects. A common end-effect in lot-sizing models is to set end-of-horizon inventory to zero. This policy can result in excessive setup costs or stock-outs in the long run. We present a method to mitigate end-effects in lot sizing by including a valuation term VIT for end-of-horizon inventory IT, in the objective function of the short-horizon model. We develop this concept within the classical EOQ modeling framework, and then apply it to the dynamic lot-sizing problem DLSP. If demand in each period of the DLSP equals the long-run average demand rate, then our procedure induces an optimal ordering policy over the short horizon that coincides with the long-run optimal ordering policy. We test our procedure empirically against the Wagner-Whitin algorithm and the Silver Meal heuristic, under several demand patterns, within a rolling horizon framework. With few exceptions, our approach significantly outperforms the other approaches tested, for modest to long model horizons. We discuss applicability to more general lot-sizing problems.

Inventory policies with quantized ordering

This article studies (nQ, r) inventory policies, under which the order quantity is restricted to be an integer multiple of a base lot size Q. Both Q and r are decision variables. Assuming the one-period expected holding and backorder cost function is unimodal, we develop an efficient algorithm to compute the optimal Q and r. The algorithm is facilitated by simple observations about the cost function and by tight upper bounds on the optimal Q. The total number of elementary operations required by the algorithm is linear in these upper bounds. By using the algorithm, we compare the performance of the optimal (nQ, r) policy with that of the optimal (s, S) policy through a numerical study, and our results show that the difference between them is small. Further analysis of the model shows that the cost performance of an (nQ, r) policy is insensitive to the choice of Q. These results establish that (nQ, r) models are potentially useful in many settings where quantized ordering is beneficial.

Optimal Control Policy for Stochastic Inventory Systems with Markovian Discount Opportunities

In this paper, we study a single-item continuous-review inventory system with Poisson demand. In addition to the standard cost structure of a fixed setup cost and a quasiconvex expected inventory holding and shortage cost, special opportunities for placing orders at a discounted setup cost occur according to a Poisson process that is independent of the demand process. This model has been studied as a subproblem of multi-item/location inventory systems where there are economies-of-scale in joint replenishment. For the single-item model, the literature proposes the ( s , c , S ) policy, under which an order is placed to increase the inventory position to S either when the inventory position drops to s , or when the inventory position is at or below c and a discount opportunity occurs. We prove that the ( s , c , S ) policy is optimal for the model, develop an efficient algorithm for computing optimal control parameters s *, c *, S*, and carry out a parametric analysis showing the effects of changes in problem parameters on the optimal control parameters and the minimum cost.