Elsevier

Information Sciences

Volume 181, Issue 24, 15 December 2011, Pages 5515-5522
Information Sciences

Scheduling with general position-based learning curves

https://doi.org/10.1016/j.ins.2011.07.051Get rights and content

Abstract

Learning effect in scheduling problems has received growing attention since Biskup [3] introduced the position-based model, where the learning curve is expressed as a power function of a job position. Hurley [11] pointed out that the actual processing time of a given job drops to zero precipitously as the number of jobs increases in the standard power model. Moreover, the learning rates show considerable variation within industries or firms. The variation extends not only across firms at a given time, but also within firms over time. For instance, the learning curves usually have an initial downward concavity, and no further improvements are made after some amount of production. Beside the standard power model, learning curve is seldom discussed in scheduling. In this paper, we offer a surprising simple yet realistic learning effect model which has the flexibility to describe different learning curves easily. For instance, the standard power model, the well-known S-shaped and the plateau functions are special cases of the proposed model. We then present the optimal solution for some scheduling problems.

Introduction

Job processing times are assumed to be fixed and known in traditional scheduling problems. However, recent empirical studies in several industries have verified that unit costs decline as firms produce more of a product and gain knowledge or experience. For instance, Biskup [3] pointed out that repeated processing of similar tasks improves worker skills; workers are able to perform setup, to deal with machine operations and software, or to handle raw materials and components at a greater pace.

Gawiejnowicz [10], Donheti and Mohanty [8], Biskup [3] and Cheng and Wang [5] were among the pioneers to bring the concept of learning effect into scheduling problems. Biskup [3] introduced the job-position-based learning effect model in which the actual job processing time is a power function of its position in schedule. He showed that two single-machine scheduling problems remain polynomially solvable. Mosheiov [16] gave several examples to demonstrate that the optimal schedules of some problems with learning effect may be different from those of the classical ones without learning consideration. Lee et al. [15] studied the single machine bi-criterion scheduling problem to minimize the sum of the total completion time and the maximum tardiness. Zhao et al. [22] developed the optimal solutions of some single machine and flowshop problems in some special cases. Koulamas and Kyparisis [13] expressed the learning effect as a function of the normal processing times of jobs already processed. They showed that the shortest processing time (SPT) sequence is optimal for the single-machine makespan and the total completion time problems. In addition, they also proved that the two-machine flowshop makespan and the total completion time problems are optimally solved by the SPT rule when the job processing times are ordered. Wang [18] considered some single-machine problems with the effects of learning and deterioration. Moreover, Biskup [4] provided a comprehensive survey of scheduling problems with learning effects. Recently, Janiak and Rudek [12] considered a scheduling problem in which each job provides a different experience to the processor. They formulated the shape of the learning curve as a non-increasing k-stepwise function. Cheng et al. [6], [7] considered some single-machine scheduling problems with specific forms of learning effect. Wu and Lee [19] considered the total completion time problem in the multiple machine permutation flowshop. Yin et al. [20] provided a general model with position-dependent and time-dependent learning effects. Zhang and Yan [21] provided the optimal schedules for some single machine problems.

Among the articles in scheduling with learning effects, it is assumed in the majority of models that the learning curve is expressed as a power function of its position in schedule. Despite its frequent use, Hurley [11] pointed out that the actual processing time of a given job drops to zero precipitously as the number of jobs increases in the standard power model. Moreover, Lapre et al. [14] claimed that the traditional power form does not accommodate two often observed patterns: initial downward concavity and the plateau effect where no further improvements are made after some amount of production. In this paper, we present a new model of the learning effect which has the shaping flexibility. The flexibility of the proposed model allows us to describe different learning curves or fit more complex environments easily. As pointed out in the Biskup [4] survey paper, it is a challenge to model with the learning effect as realistic as possible. Thus, the first contribution of this paper is to provide a surprisingly simple yet realistic model. It is completely unexpected in the sense that a simple model can describe many different learning curves in the literature, such as the well-known plateau or the S-shaped phenomena, which are more realistic. Moreover, Dutton and Thomas [9] found that the learning rates show considerable variation within industries or firms after a study of more than 200 learning curves. The variation extends not only across firms at a given time, but also within firms over time. Thus, the second contribution of the paper is to provide a more flexible model.

The remainder of this note is organized as follows. The solution procedures for some single-machine problems are presented in the next section. In Section 3, the optimal solution for a two-machine flowshop problem is derived. The conclusion is given in the last section.

Section snippets

Single-machine single-criterion problems

Formulation of the proposed learning effect model in the single-machine case is as follows. There are n jobs to be processed on a single machine. Each job j has a normal processing time pj, a weight wj, and a due date dj. Due to the learning effect, the actual processing time of job j ispj[r]=pjl=0r-1al,if it is scheduled in the rth position in a sequence, where a0  = 1 and 0 < al  1 for l = 1,  , n. Note that al denotes the learning impact on processing the lth position job and l=0ral denotes the

A two-machine flowshop problem

The formulation of the proposed model in the off-line two-machine permutation flowshop case is similar to that of the single-machine case and stated as follows. There are n jobs to be processed on machines M1 and M2. Each job j consists of two operations to be processed on one machine each. Operation of job j on machine M2 can start only operation on machine M1 is completed. A machine can handle one job at a time and preemption is not allowed. The actual processing time pkj[r] of job j on

Conclusions

In this paper, we proposed a new model of the learning effect. For the single-machine case, we showed that the problem to minimize the total tardiness is polynomially solvable under certain agreeable conditions. Moreover, we provided the optimal solutions for two single-machine multiple-criteria problems. For the case of two-machine permutation flowshop, we presented polynomial-time optimal solutions for the makespan problem under the assumption of ordered processing times. For future research,

Acknowledgements

The author is grateful to the editor and the referees, whose constructive comments have led to a substantial improvement in the presentation of the paper. This work was supported by the NSC of Taiwan, ROC, under NSC 98-2221-E-035-033-MY2.

References (22)

Cited by (0)

View full text