# Transitioning Schedules for the Economic Lot Sizing Problem

## Article excerpt

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Introduction

Consider a firm that must make on one work center many products with constant demand rates, but can produce only one product at a time. In addition, the facility requires a setup time and/or cost to change its production from one product to another. The firm achieves this by building up inventories of products between their production runs. This facility wants to minimize its relevant costs, include inventory-holding, setup, and lost order costs. This problem, called the Economic Lot Scheduling Problem (ELSP), has been researched in many forms. Maxwell (1961) first formulated this classic ELSP problem with setup costs, as well as inventory costs and setup times. Later researchers extended the problem, such as Cheng, Yan, and Yang (1998) with variable production rates, Robinson and Sahin (2001) with overtime, Sriskandarajah, and Wagneur (1999) with lot streaming. Others came up with new solution techniques, such as genetic algorithms.

Whatever the solution technique, the general solution for the ESLP is to come up with a new repeating scheduling cycle, which consists of product lot sizes and a product sequence. Individual products may be made more than once in a cycle in different lot sizes. Thereafter we will mainly refer to production times as these are identical to lot sizes but are in the same units as setup times. Dobson (1987), for example, devised a heuristic that uses time-varying lot sizes for both the zero setup cost and the non-zero setup cost cases. The classic ELSP assumes that demand rate for each product is a constant for all time, and thus gives a steady state solution. Another extensive set of research considers the situation where future demand varies. Wolsey (1995, 1997) reviews some of the work done on this problem. Bradley and Conway (2003) give a tutorial on these problems.

Quite often, many firms use the steady state scheduling solutions when demand rates are not constant but are reasonably level. In such instances, infinite scheduling is often more desirable to the changing schedules that result from finite scheduling. Constant and repetitive production times and cycle sequences need far less administration. However, even with steady state demands, changes occur such as a new product introduction, an old product deletion, a process change, or changes in customer demands. So the firm calculates a new steady state schedule. Thus the facility requires a transition from the old ELSP solution to the new cycle schedule solution. This transition must be feasible, with minimum loss of orders.

The steady state scheduling research has failed to consider this transition problem. Steady state solution approaches implicitly assumed that starting the cycle and on-hand inventories are not part of the problem. In practice, there is this starting or transition problem. This situation is not dynamic, rather part of a punctuated equilibrium. Thus there is a transition to the static solution, followed by a period of static conditions, which in turn precedes another transition to a new steady state solution, ad infinitum.

Literature Survey

Need for a Transition

Anderson (1990) stated that most of the ELSP solutions imply that demand will always be met if the initial stocks are large enough. He reported that most researchers have supposed that the initial stocks of each product are indeed large enough to sustain the optimal production cycle. His question was whether the initial stocks are large enough to cover a certain level of demand. Anderson recommended letting setup and holding costs be zero. His problem then is determining the feasibility of a production schedule, given a current level of inventory, to meet a specific static demand for several products made on one machine.

However, Anderson's problem was not finding a feasible transition sequence to enable a regular repeating cycle to start, but rather a permanent repeating cycle that can start with no transition sequence. …

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