Discrete OptimizationMinimizing the total completion time in a single-machine scheduling problem with a time-dependent learning effect
Introduction
A scheduling problem is very important in a manufacturing system. Hence, numerous scheduling problems have been studied for many years. For most of them, the processing time of a job is independent of its position in a scheduling sequence. However, in many realistic situations, because the firms and employees perform the same task repeatedly, they learn how to perform more efficiently. Therefore, the actual processing time of a job is shorter if it is scheduled later, rather than earlier in the sequence. This phenomenon is known as the “learning effect” in the literature. Although different types of learning effects have been studied extensively in various areas (see Nadler and Smith, 1963, Yelle, 1979), it has rarely been studied in the context of scheduling.
Biskup (1999) was the first to investigate the learning effect in scheduling problems. He assumed that the time needed to perform an operation decreases by the number of repetitions, i.e., the processing time of a job is a function of the job position in a sequence. In this job-independent learning effect model, the actual processing time of job j if scheduled in position r, is given bywhere pj is the normal (sequence-independent) processing time of job j, a ⩽ 0 is a constant learning index and n is the total number of jobs. Under this assumption, the learning effect of a job only depends on the number of jobs that are scheduled in front of it. It implies the learning rates of all jobs are all the same. There are several studies in this context. Biskup showed that single-machine scheduling problems with a learning effect still remain polynomially solvable if the objective is to minimize the deviation from a common due date or to minimize the sum of flow times. Mosheiov, 2001a, Mosheiov, 2001b applied similar solution techniques to some other single-machine scheduling problems, and minimum total flow time on parallel identical machines. Lee et al. (2004) considered the learning effect in a bi-criterion single-machine scheduling problem and proposed a heuristic algorithm to search for optimal or near-optimal solutions. Lee and Wu (2004) proposed a heuristic algorithm to solve a total completion time minimization problem in a two-machine flowshop with a learning effect.
However, in some other practical situations, different jobs usually have different processing times due to the number of operations. Thus, the firms and employees will learn more if they perform a job with longer processing time. Therefore, Mosheiov and Sidney (2003) further considered the learning in the production process of some jobs to be faster than those of others, i.e., the learning is job-dependent. They showed that the makespan and the total flow time minimization problems on a single machine, a due-date assignment problem, and total flow time minimization on unrelated parallel machines remain polynomially solvable. In this job-dependent learning effect model, the actual processing time of job j if scheduled in position r, is given bywhere aj is a negative job-dependent parameter. It proposes a concept that different jobs have different learning effects. The job-dependent learning effect model is more general than the job-independent one.
On the other hand, Biskup and Simons (2004) further considered both autonomous and induced learning effects in a common due date scheduling problem. They focus on a scheduling problem where the processing times decrease according to a learning rate, which can be influenced by an initial cost-inducing investment. They derived some structural properties of the problem and presented a polynomially bound solution procedure to search an optimal solution.
In this paper, we will introduce another viewpoint of a learning effect, i.e., the more time you practice, the better performance you obtain. Thus, we propose a time-dependent learning effect model and incorporate it into a single-machine scheduling problem. The objective is to minimize the total completion time of all jobs.
Section snippets
Notations and assumptions
There are n jobs to be processed on a single machine. Each of them is available at time zero. Let pj be the normal (sequence-independent) processing time of job j (Jj, j = 1, 2, …, n) in a sequence and p[k] be the normal processing time of a job if scheduled in the kth position in a sequence. The normal processing time of a job is incurred if the job is scheduled first in a sequence. The processing times of the following jobs are smaller than their normal processing times because of the learning
Minimum total completion time
In this section, we study a single-machine scheduling problem with a time-dependent learning effect. For a given schedule q, let Ci = Ci(q) represent the completion time of Ji. The total completion time is denoted by . For convenience, we denote the time-dependent learning effect mentioned in the previous section by LEt. Thus, using the conventional notation, the problem of total completion time minimization on a single machine is denoted by .
The classical scheduling problem
Conclusions
In this study, we introduce a time-dependent learning effect into a single-machine scheduling problem. The time-dependent learning effect of a job is assumed to be a function of total processing time of jobs scheduled in front of it. It is very practical in many realistic situations. Furthermore, we show that it remains polynomially solvable for the objective, i.e., minimizing the total completion time on a single machine. In addition, we show that the SPT-sequence is still the optimal sequence
Acknowledgements
The authors would like to thank the anonymous referees for their helpful comments and suggestions. This research was supported in part by the National Science Council of Taiwan, Republic of China, under grant number NSC-93-2213-E-150-019.
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