Single machine scheduling with past-sequence-dependent setup times and learning effects

This paper studies a single machine scheduling problem with setup times and learning considerations. The setup times are proportional to the length of the already scheduled jobs. That is, the setup times are past-sequence-dependent. It is assumed that the learning process reflects a decrease in the process time as a function of the number of repetitions, i.e., as a function of the job position in the sequence. The following objectives are considered: the makespan, the total completion time, the total absolute differences in completion times and the sum of earliness, tardiness and common due-date penalty. Polynomial time algorithms are proposed to optimally solve the above objective functions.     

Some unrelated parallel machine scheduling problems with past-sequence-dependent setup time and learning effects

In this paper, we study an unrelated parallel machine scheduling problem with setup time and learning effects simultaneously. The setup time is proportional to the length of the already processed jobs. That is, the setup time of each job is past-sequence-dependent. The objectives are to minimize the total absolute deviation of job completion times and the total load on all machines, respectively. We show that the proposed problem is polynomially solvable. We also discuss two special cases of the problem and show that they can be optimally solved by lower order algorithms.     

Note on “Single-machine and flowshop scheduling with a general learning effect model” and “Some single-machine and m-machine flowshop scheduling problems with learning considerations”

In this note, we show that the main results in the two papers [C.C. Wu, W.C. Lee, Single-machine and flowshop scheduling with a general learning effect model, Computers and Industrial Engineering 56 (2009) 1553-1558, W.C. Lee, C.C. Wu, Some single-machine and m-machine flowshop scheduling problems with learning considerations, Information Sciences 179 (2009) 3885-3892] are incorrect.     

Solving a two-agent single-machine scheduling problem considering learning effect

Scheduling with multiple agents and learning effect has drawn much attention. In this paper, we investigate the job scheduling problem of two agents competing for the usage of a common single machine with learning effect. The objective is to minimize the total weighted completion time of both agents with the restriction that the makespan of either agent cannot exceed an upper bound. In order to solve this problem we develop several dominance properties and a lower bound based on a branch-and-bound to find the optimal algorithm, and derive genetic algorithm based procedures for finding near-optimal solutions. The performances of the proposed algorithms are evaluated and compared via computational experiments.     

Single-machine and flowshop scheduling with a general learning effect model

Learning effects in scheduling problems have received growing attention recently. Biskup [Biskup, D. (2008). A state-of-the-art review on scheduling with learning effect. European Journal of Operational Research, 188, 315–329] classified the learning effect scheduling models into two diverse approaches. The position-based learning model seems to be a realistic assumption for the case that the actual processing of the job is mainly machine driven, while the sum-of-processing-time-based learning model takes into account the experience the workers gain from producing the jobs. In this paper, we propose a learning model which considers both the machine and human learning effects simultaneously. We first show that the position-based learning and the sum-of-processing-time-based learning models in the literature are special cases of the proposed model. Moreover, we present the solution procedures for some single-machine and some flowshop problems.     

Some single-machine scheduling problems with a truncation learning effect

Scheduling with learning effects has received growing attention nowadays. A well-known learning model is called “sum-of processing-times-based learning” in which the actual processing time of a job is a non-increasing function of the jobs already processed. However, the actual processing time of a given job drops to zero precipitously when the normal job processing times are large. Motivated by this observation, we propose a truncation learning model where the actual job processing time is a function which depends not only on the processing times of the jobs already processed but also on a control parameter. The use of the truncated function is to model the phenomenon that the learning of a human activity is limited. Under the proposed learning model, we show that some single-machine scheduling problems can be solved in polynomial time. In addition, we further provide the worst-case error bounds for the problems to minimize the maximum lateness and total weighted completion time.     

Some single-machine and m-machine flowshop scheduling problems with learning considerations

Scheduling with learning effect has drawn many researchers’ attention since Biskup [D. Biskup, Single-machine scheduling with learning considerations, European Journal of Opterational Research 115 (1999) 173-178] introduced the concept of learning into the scheduling field. Biskup [D. Biskup, A state-of-the-art review on scheduling with learning effect, European Journal of Opterational Research 188 (2008) 315-329] classified the learning approaches in the literature into two main streams. He claimed that the position-based learning seems to be a realistic model for machine learning, while the sum-of-processing-time-based learning is a model for human learning. In some realistic situations, both the machine and human learning might exist simultaneously. For example, robots with neural networks are used in computers, motor vehicles, and many assembly lines. The actions of a robot are constantly modified through self-learning in processing the jobs. On the other hand, the operators in the control center learn how to give the commands efficiently through working experience. In this paper, we propose a new learning model that unifies the two main approaches. We show that some single-machine problems and some specified flowshop problems are polynomially solvable.     

Single-machine scheduling with past-sequence-dependent setup times and learning effects: a parametric analysis

In this article, we consider the single-machine scheduling problem with past-sequence-dependent (p-s-d) setup times and a learning effect. The setup times are proportional to the length of jobs that are already scheduled; i.e. p-s-d setup times. The learning effect reduces the actual processing time of a job because the workers are involved in doing the same job or activity repeatedly. Hence, the processing time of a job depends on its position in the sequence. In this study, we consider the total absolute difference in completion times (TADC) as the objective function. This problem is denoted as 1/LE, s _psd /TADC in Kuo and Yang (2007 Kuo, WH and Yang, DL. 2007. Single Machine Scheduling with Past-sequence-dependent Setup Times and Learning Effects. Information Processing Letters, 102: 22–26. [Crossref], [Web of Science ®] , [Google Scholar]) (‘Single Machine Scheduling with Past-sequence-dependent Setup Times and Learning Effects’, Information Processing Letters, 102, 22–26). There are two parameters a and b denoting constant learning index and normalising index, respectively. A parametric analysis of b on the 1/LE, s _psd /TADC problem for a given value of a is applied in this study. In addition, a computational algorithm is also developed to obtain the number of optimal sequences and the range of b in which each of the sequences is optimal, for a given value of a. We derive two bounds b* for the normalising constant b and a* for the learning index a. We also show that, when a?a* or b?>?b*, the optimal sequence is obtained by arranging the longest job in the first position and the rest of the jobs in short processing time order.     

The single-machine total weighted tardiness scheduling problem with position-based learning effects

This study addresses the problem of minimizing the total weighted tardiness on a single-machine with a position-based learning effect. Several dominance properties are established to develop branch and bound algorithm and a lower bound is provided to derive the optimal solution. In addition, three heuristic procedures are developed for near-optimal solutions. Computational results are also presented to evaluate the performance of the proposed algorithms.     

A single-machine scheduling problem with learning effects in intermittent batch production

This paper studies a single-machine scheduling problem with three models of learning and forgetting effects in intermittent batch production. They are the models of no transmission, partial transmission and total transmission of learning from batch to batch. The phenomena exist in many realistic production systems. The objective is to minimize the makespan. We show that the problems with the models of no transmission and partial transmission of learning from batch to batch are polynomially solvable. We also provide two polynomial time algorithms for two special cases in the problem with the total transmission model.     

Single-machine scheduling with sum-of-logarithm-processing-times-based learning considerations

Scheduling with learning effects has attracted growing attention of the scheduling research community. A recent survey classifies the learning models in scheduling into two types, namely position-based learning and sum-of-processing-times-based learning. However, the actual processing time of a given job drops to zero precipitously as the number of jobs increases in the first model and when the normal job processing times are large in the second model. Motivated by this observation, we propose a new learning model where the actual job processing time is a function of the sum of the logarithm of the processing times of the jobs already processed. The use of the logarithm function is to model the phenomenon that learning as a human activity is subject to the law of diminishing return. Under the proposed learning model, we show that the scheduling problems to minimize the makespan and total completion time can be solved in polynomial time. We further show that the problems to minimize the maximum lateness, maximum tardiness, weighted sum of completion times and total tardiness have polynomial-time solutions under some agreeable conditions on the problem parameters.     

Single-machine scheduling with learning effect and resource-dependent processing times

We consider resource allocation scheduling with learning effect in which the processing time of a job is a function of its position in a sequence and its resource allocation. The objective is to find the optimal sequence of jobs and the optimal resource allocation separately. We concentrate on two goals separately, namely, minimizing a cost function containing makespan, total completion time, total absolute differences in completion times and total resource cost; minimizing a cost function containing makespan, total waiting time, total absolute differences in waiting times and total resource cost. We analyse the problem with two different processing time functions. For each combination of these, we provide a polynomial time algorithm to find the optimal job sequence and resource allocation.     

Single-machine scheduling jobs with exponential learning functions

In a manufacturing system workers are involved in doing the same job or activity repeatedly. Hence, the workers start learning more about the job or activity. Because of the learning, the time to complete the job or activity starts decreasing, which is known as “learning effect”. In this paper, an exponential sum-of-actual-processing-time based learning effect is introduced into single-machine scheduling. By the exponential sum-of-actual-processing-time based learning effect, we mean that the processing time of a job is defined by an exponential function of the sum-of-the-actual-processing-time of the already processed jobs. Under the proposed learning model, we show that under a sufficient condition, the makespan minimization problem, the sum of the θth (θ > 0) power of completion times minimization problem, and some special cases of the total weighted completion time minimization problem and the maximum lateness minimization problem remain polynomially solvable.     

Two-agent scheduling with learning consideration

In traditional scheduling, job processing times are assumed to be known and fixed over the entire process. However, repeated processing of similar tasks improves workers’ skills. In fact, scheduling with learning effects has received considerable attention recently. On the other hand, it is assumed that there is a common objective for all the jobs. In many management situations, multiple agents pursuing different objectives compete on the usage of a common processing resource. In this paper, we studied a single-machine two-agent scheduling problem with learning effects where the objective is to minimize the total tardiness of jobs from the first agent given that no tardy job is allowed for the second agent. A branch-and-bound algorithm incorporated several properties and a lower bound is developed to search for the optimal solution. In addition, two heuristic algorithms are also proposed to search for the near-optimal solutions. A computational experiment is conducted to evaluate the performance of the proposed algorithms.     

Some scheduling problems with general position-dependent and time-dependent learning effects

In scheduling problems with learning effects, most of the research is based on specific learning functions. In this paper, we develop a general model with learning effects where the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. In particular, it is shown that some single machine scheduling problems and m-machine permutation flowshop problems are still polynomially solvable under the proposed model. These results are significant extensions of some of the existing results on learning effects in the literature.     

Notes on “some single-machine scheduling problems with general position-dependent and time-dependent learning effects”

This paper provides a continuation of the idea presented by Yin et al. [Yin et al., Some scheduling problems with general position-dependent and time-dependent learning effects, Inform. Sci. 179 (2009) 2416-2425]. For each of the following three objectives, total weighted completion time, maximum lateness and discounted total weighted completion time, this paper presents an approximation algorithm which is based on the optimal algorithm for the corresponding single-machine scheduling problem and analyzes its worst-case bound. It shows that the single-machine scheduling problems under the proposed model can be solved in polynomial time if the objective is to minimize the total lateness or minimize the sum of earliness penalties. It also shows that the problems of minimizing the total tardiness, discounted total weighted completion time and total weighted earliness penalty are polynomially solvable under some agreeable conditions on the problem parameters.     

Some scheduling problems with sum-of-processing-times-based and job-position-based learning effects

In this paper we introduce a new scheduling model with learning effects in which the actual processing time of a job is a function of the total normal processing times of the jobs already processed and of the job’s scheduled position. We show that the single-machine problems to minimize makespan and total completion time are polynomially solvable. In addition, we show that the problems to minimize total weighted completion time and maximum lateness are polynomially solvable under certain agreeable conditions. Finally, we present polynomial-time optimal solutions for some special cases of the m-machine flowshop problems to minimize makespan and total completion time.     

Scheduling problems with general effects of deterioration and learning

Scheduling with deteriorating jobs or learning effects has been widely studied recently. There are situations where both the deterioration and learning effects might exist at the same time. However, the research with the consideration of both the effects is relatively limited. Furthermore, the forms of the effects are specific functions in the literature. In this paper, we introduce a general scheduling model in the sense that the form of the function is unspecified. Under the proposed model, the actual job processing time is a general function on the processing times of the jobs already processed and its scheduled position. The optimal solutions for some single-machine problems are provided.     

A note on scheduling problems with learning effect and deteriorating jobs

In this note, we consider the machine scheduling problems with the effects of deterioration and learning. In this model, job processing times are defined by functions of their starting times and positions in the sequence. The scheduling objectives are makespan (weighted) sum of completion times and maximum lateness. It is shown that even with the introduction of deterioration and learning effect to job processing times, several single machine problems and several flow shop problems remain polynomially solvable, respectively.