A search heuristic for the sequence-dependent economic lot scheduling problem

Abstract

Almost all of the research on the economic lot scheduling problem (ELSP) has assumed that setup times are sequence-independent even though sequence-dependent problems are common in practice. Furthermore, most of the solution approaches that have been developed solve for a single optimal schedule when in practice it is more important to provide managers with a range of schedules of different length and complexity. In this paper, we develop a heuristic procedure to solve the ELSP problem with sequence-dependent setups. The heuristic provides a range of solutions from which a manager can choose, which should prove useful in an actual stochastic production environment. We show that our heuristic can outperform Dobson's heuristic when the utilization is high and the sequence-dependent setup times and costs are significant.

Introduction

The economic lot scheduling problem (ELSP) is one of the oldest and most studied problems in scheduling. The problem is to schedule more than one product for production on a single machine. The standard assumptions are that setup costs and setup times are sequence-independent, the production rates and demand rates are constant and the time horizon is infinite. Rogers (1958), in the fourth volume of Management Science, defined the problem and developed a method of computing production quantities. He applied the economic order quantity (EOQ) formula to items individually and then pointed out that because multiple products were sharing a single resource, it would usually be impossible to produce each product according to the EOQ formula due to interference – the requirement to produce two or more products at the same time on the single resource. Rogers ignored setup times and developed an iterative procedure to develop production schedules using equal time intervals, limiting production to one product per time interval and restricting production of a product to an even number of production intervals (e.g., product A is produced every fourth time interval).

Following Roger's work, researchers have used one of the two basic approaches: the basic periods or fundamental cycle (FC) approach and the cyclic schedule (CS) approach. These approaches are illustrated in Fig. 1. Bomberger (1966) developed the FC approach, defining a fundamental cycle T_f and requiring that the cycle time for any product T_i be an integer multiple k_i of the fundamental cycle. Fig. 1(a) graphically demonstrates an FC schedule for three products A, B and C. In this example, the multipliers are 1, 2 and 3, respectively. To insure feasibility, Bomberger required that the sum of setup and production times for all products be less than T_f. This feasibility restriction can significantly limit capacity utilization, especially when the multipliers are large.

A number of researchers have devised improved solution procedures to the ELSP problem using the FC approach, including Madigan (1968), Haessler (1971), Doll and Whybark (1973), Haessler and Hogue (1976) and Elmaghraby (1978), who provided an extensive review and analysis of approaches to the ELSP problem in addition to an improved solution approach.

In the CS approach, a production schedule (starting and ending times for setup and production of every product) is defined over a cycle time T_c with this schedule repeated indefinitely (Fig. 1(b)). The CS approach is more general than the FC approach in that any FC solution can be converted to a CS schedule, but the reverse is not always true. Hanssmann (1962) was the first to propose a CS solution, developing an approach to determine an optimal common cycle schedule, which is a CS schedule where every product is produced only once in the cycle. Maxwell (1964) considered the ELSP with sequence-dependent setups having setup costs proportional to the setup time. He further restricted the problem to cyclical schedules with no idle time. With these assumptions he showed that setup cost per unit time was a constant and could be ignored in determining a schedule. Adding the zero-switch rule (requiring the inventory of a product to be zero before production starts), he developed the best-product-in-best-position heuristic. This heuristic starts with a common cycle solution and then considers adding additional production runs if a cost improvement is possible.

Delporte and Thomas (1977) also used the CS approach. Starting with all production frequencies set to one, they evaluated increasingly complicated production schedules by increasing the production frequencies one-at-a-time. To determine the product that should have its production frequency increased, they looked at the setup and holding costs for all products assuming no production interference. With a set of production frequencies, they formed production sequences using rules and judgement. Given the production sequence, they determined production and idle times using a quadratic program that had the cycle time T_c fixed. This differs from the approach used in this paper, where a nonlinear program is used to evaluate production sequences including T_c as a variable.

Dobson (1992) considered the ELSP model with sequence-dependent setups. Through transformations and relaxations of the problem, Dobson developed a model that could solve for production frequencies. To achieve partial separation of the embedded lot sizing and travelling salesman problem, the model assumed the production runs could be evenly spaced, thus ignoring the inventory implications of production sequence and timing. By relaxing the integrality restrictions on the production frequencies, he was able to solve the model by subgradient optimization. The resulting production frequencies were scaled so that the smallest frequency was one, and the other frequencies were scaled to powers of two to simplify construction of a sequence.

With the production frequencies determined, a sequencing and timing heuristic procedure was used to determine the production sequence, production cycle time and production and idle times. Given the cycle time T_c, production and idle times were picked to space the lots as equally as possible while meeting the restriction of nonnegative inventory using the method in Dobson (1987).

Because the production frequencies were calculated ignoring the inventory implications of production sequence and timing, Dobson hypothesized that the optimal solution may have lower production frequencies than those determined by his relaxed model. To address this issue, Dobson used a post-hoc search procedure to evaluate other production schedules with fewer production runs.

In the following section, we propose a heuristic procedure to solve the sequence-dependent ELSP problem using the CS approach to build up the production sequence similar to Delporte and Thomas (1977) and Maxwell (1964). In our heuristic, we use a search procedure to determine the production sequence and evaluate the sequences using a nonlinear program to exactly determine the optimal schedule given a production sequence, solving for production quantities, inventory levels, production starting and ending times and the cycle time T_c. We compare our approach to Dobson's heuristic, generating problems using his experimental design (Dobson, 1992) as his four experiments cover a wide range of problems.