Peter L. Jackson

Modelling the evolution of demand forecasts with application to safety stock analysis in production distribution systems

In this paper, we propose a general probabilistic model for modeling the evolution of demand forecasts, referred to as the Martingale Model of Forecast Evolution (MMFE). We combine the MMFE with a linear programming model of production and distribution planning implemented in a rolling horizon fashion. The resulting simulation methodology is used to analyze safety stock levels for a multi-product/multi-plant production/distribution system with seasonal stochastic demand. In the context of this application we demonstrate the importance of good forecasting.

The Joint Replenishment Problem with a Powers-of-Two Restriction

Abstract In this paper we consider the joint replenishment problem in the light of recent work by the second and third authors concerning the selection of realistic and consistent reorder intervals in production/ distribution systems. After stating a general dynamic programming formulation of the joint replenishment problem, we present its usual statement which assumes constant reorder intervals. We then restrict the problem further by assuming the constant reorder intervals are powers-of-two multiples of some base planning interval. We present an algorithm based on concepts we developed that solves the joint replenishment problem with the powers-of-two restriction. Like other algorithms proposed for this problem, it is a simple sorting algorithm. Finally, we establish that the algorithm yields a solution whose average annual cost is within 6% of the general problems long-run minimum average annual cost.

A review of the stochastic lot scheduling problem

Abstract This paper surveys the current research literature on the stochastic lot scheduling problem which deals with scheduling production of multiple products with random demand on a single facility with limited production capacity and significant change-overs between products. The deterministic version of this problem has received significant coverage in the literature; however, the stochastic problem has been addressed only recently. Furthermore, a range of distinctly different analytical methods have been applied to this problem. This paper provides a unifying framework for discussing these approaches and offers some explanation and clarification of the different analytical methods for this problem. After discussing some of the modeling and managerial implications of this problem, a detailed review of both continuous and discrete time control strategies is given, and areas for further research are outlined.

Risk pooling in a two-period, two-echelon inventory stocking and allocation problem

The object of this article is to investigate the risk-pooling effect of depot stock in two-echelon distribution system in which the depot serves n retailers in parallel, and to develop computationally tractable optimization procedures for such systems. The depot manager has complete information about stock levels and there are two opportunities to allocate stock to the retailers within each order cycle. We identify first- and second-order aspects to the risk-pooling effect. In particular, the second-order effect is the property that the minimum stock available to any retailer after the second allocation converges in probability to a constant as the number of retailers in the system increases, assuming independence of the demands. This property is exploited in the development of efficient procedures to determine near-optimal values of the policy parameters.

Optimizing Service Parts Inventory in a Multiechelon, Multi-Item Supply Chain with Time-Based Customer Service-Level Agreements

In the realm of service parts management, customer relationships are often established through service agreements that extend over months or years. These agreements typically apply to a piece of equipment that the customer has purchased, and they specify the type and timing of service that will be provided. If a customer operates in multiple locations, service agreements may cover several pieces of equipment at several locations. In this paper, we describe a continuous-review inventory model for a multi-item, multiechelon service parts distribution system in which time-based service-level requirements exist. Our goal is to determine base-stock levels for all items at all locations so that the service-level requirements are met at minimum investment. We derive exact time-based fill-rate expressions for each item within its distribution channel, as well as approximate expressions for the gradients of these fill-rate functions. Using these results, we develop an intelligent greedy algorithm that can be used to find near-optimal solutions to large-scale problems quickly, as well as a Lagrangian-based approach that provides both near-optimal solutions and good lower bounds with increased computational effort. We demonstrate the effectiveness and scalability of these algorithms on three example problems.

The warehouse scheduling problem: formulation and algorithms

The Warehouse Scheduling Problem is a deterministic multi-item inventory problem with a restriction on warehouse floor space available. We formulate a mixed integer nonlinear programming problem for the objective of minimizing long run inventory holding and order costs per unit of time. We integrate algorithms for staggering orders, described in companion papers, with a heuristic to choose the order sequences. The result is called Sequenced Staggering. We describe a new algorithm to generate order frequencies, called the powers-of-two-factor-of-three technique, as a generalization of Roundys roundoff technique for powers-of-two policies. We report on a computational study of four hybrid algorithms for solving the warehouse scheduling problem, including the competing algorithm of Gallego, Queyranne, and Simchi-Levi. Based on these results, we recommend the combination of powers-of-two frequencies with Sequenced Staggering.

TECHNICAL NOTE---Exact Analysis of a Lost Sales Model Under Stuttering Poisson Demand

We investigate the (S-1, S) inventory policy under stuttering Poisson demand and generally distributed lead time when the excess demand is lost. We correct results presented in Feeney and Sherbrookes seminal paper [Feeney, G. J., C. C. Sherbrooke. 1966. The (S-1, S) inventory policy under compound Poisson demand. Management Sci.12(5) 391--411] and note that the stationary distribution of units on order for the general compound Poisson demand case is still an open question.

Time-variant lot sizing models for the warehouse scheduling problem

We consider the problem of scheduling the delivery of n products into a warehouse with limited space under the assumptions of continuous demands at constant rates, infinite horizon, and no backorders. The delivery schedule is described by a cyclic schedule with time-varying lot sizes. The order frequencies and the order sequence are assumed to be given. We formulate a linear program that determines delivery times relative to the cycle length to minimize the relative maximum space used and show that the optimal solution is characterized by filling the warehouse at each order. We bound the optimal solution by using a worst-case analysis and give conditions under which the linear program has the same optimal solution as a quadratic program that minimizes the holding cost. Under general conditions, we derive a bound on the cost penalty that results when using the optimal solution of the linear program as a solution to the quadratic program. Finally, we complete a solution to the nonlinear lot-sizing model by ...

Revised Dantzig-Wolfe decomposition for staircase-structured linear programming

Staircase structured linear programs arise naturally in the study of engineering economic systems. One general approach to solving such LPs is the technique of nested decomposition of the primal or dual problem. The research described in this paper proposes a revised decomposition algorithm that incorporates knowledge of the structure of the staircase basis in forming the decomposed linear programs. Column proposals from the revised subproblems are shown to achieve maximum penetration against the master problem basis. The proposed algorithm resorts to the regular Dantzig-Wolfe subproblem to test for optimality. The algorithm is shown to be finite and is compared to the Abrahamson-Wittrock algorithm. Computational results indicate substantial improvement over the Dantzig-Wolfe algorithm in most cases. A numerical example of the algorithm is provide in the appendix.

Estimating Performance Measures in Repetitive Manufacturing Environments via Stochastic Cyclic Scheduling

In this paper we review the cyclic scheduling paradigm for repetitive manufacturing environments. System performance is measured by system throughput and job flow time. For a given cyclic job sequence and operation precedence structure, we are interested in the steady-state behavior of these measures when processing times are stochastic. We present an iterative approximation scheme based on Clark’s method to estimate the two-moment behavior of these performance measures. Computational tests demonstrate that the approach is a useful alternative to simulation.