Scheduling jobs with a general learning effect model

The paper deals with machine scheduling problems with a general learning effect. By the general learning effect, we mean that the actual processing time of a job is not only a non-increasing function of the total weighted normal processing times of the jobs already processed, but also a non-increasing function of the job’s position in the sequence, where the weight is a position-dependent weight. We show that even with the introduction of a general learning effect to job processing times, some single machine scheduling problems are still polynomially solvable under the proposed model. We also show that some special cases of the flow shop scheduling problems can be solved in polynomial time.     

Trapezoidal type inequalities for n-time differentiable functions

In this paper we consider the flow shop scheduling problems with the effects of learning and deterioration. In this model the processing times of a job is defined as a function of its starting time and position in a sequence. The scheduling objective functions are makespan and total completion time. We prove that even with the introduction of learning effect and deteriorating jobs to job processing times, some special flow shop scheduling problems remain polynomially solvable.     

A revision of machine scheduling problems with a general learning effect

In this paper, we show that the main results in a recent paper by Zhang and Yan [X. Zhang and G. Yan, Machine scheduling problems with a general learning effect, Mathematical and Computer Modelling 51 (2010) 84–90] are incorrect as an important reason is missing, that is, the processing time of a job is variable according to a general learning effect. Here we point out these wrong results by a counter-example. In addition, we give a revised model with a general learning effect. We show that some single machine scheduling problems are still polynomially solvable under the revised model. We also show that some special cases of the flowshop scheduling problems can be solved in polynomial time under the revised model.     

并行分批排序综述

并行分批排序起源于半导体芯片制造过程。在并行分批排序中,工件可成批加工,批加工机器最多可同时加工B个工件,批的加工时间为批中所有工件的最大工时。首先根据传统的机器环境和目标函数对并行分批排序已有成果进行分类介绍,主要为单机和平行机的机器环境,以及极小化最大完工时间、极小化总完工时间、极小化最大延迟、极小化误工工件数、极小化总延误和极小化最大延误的目标函数;然后梳理了由基本问题所衍生出来的具有新特点的16类新型并行分批排序,包括差异尺寸工件、多目标、工件加工时间或顺序存在限制、考虑费用和具有特殊机制等情况;最后展望未来的研究方向。     

Single-machine and two-machine flowshop scheduling problems with truncated position-based learning functions

Scheduling with learning effects has received growing attention nowadays. A well-known learning model is called ‘position-based learning’ in which the actual processing time of a job is a non-increasing function of its position to be processed. However, the actual processing time of a given job drops to zero precipitously as the number of jobs increases. Motivated by this observation, we propose two truncated learning models in single-machine scheduling problems and two-machine flowshop scheduling problems with ordered job processing times, respectively, where the actual processing time of a job is a function of its position and a control parameter. Under the proposed learning models, we show that some scheduling problems can be solved in polynomial time. In addition, we further analyse the worst-case error bounds for the problems to minimize the total weighted completion time, discounted total weighted completion time and maximum lateness.     

Worst-case analysis for flow shop scheduling problems with an exponential learning effect

A real industrial production phenomenon, referred to as learning effects, has drawn increasing attention. However, most research on this issue considers only single machine problems. Motivated by this limitation, this paper considers flow shop scheduling problems with an exponential learning effect. By the exponential learning effect, we mean that the processing time of a job is defined by an exponent function of its position in a processing permutation. The objective is to minimize one of the four regular performance criteria, namely, the total completion time, the total weighted completion time, the discounted total weighted completion time, and the sum of the quadratic job completion times. We present heuristic algorithms by using the optimal permutations for the corresponding single-machine scheduling problems. We also analyse the worst-case bound of our heuristic algorithms.     

Single machine scheduling with a time-dependent learning effect and deteriorating jobs

The paper deals with the single machine scheduling problems with a time-dependent learning effect and deteriorating jobs. By the effects of time-dependent learning and deterioration, we mean that the processing time of a job is defined by function of its starting time and total normal processing time of jobs in front of it in the sequence. It is shown that even with the introduction of a time-dependent learning effect and deteriorating jobs to job processing times, the single machine makespan minimization problem remain polynomially solvable. But for the total completion time minimization problem, the classical shortest processing time first rule or largest processing time first rule cannot give an optimal solution.     

Single-machine scheduling with a sum-of-actual-processing-time-based learning effect

In this paper we consider the single-machine scheduling problems with a sum-of-actual-processing-time-based learning effect. By the sum-of-actual-processing-time-based learning effect, we mean that the processing time of a job is defined by a function of the sum of the actual processing time of the already processed jobs. We show that even with the introduction of the sum-of-actual-processing-time-based learning effect to job processing times, the makespan minimization problem, the total completion time minimization problem, the total completion time square minimization problem, and some special cases of the total weighted completion time minimization problem and the maximum lateness minimization problem remain polynomially solvable, respectively.     

A bicriteria parallel machine scheduling with a learning effect of setup and removal times

In studies on scheduling problems, generally setup times and removal times of jobs have been neglected or by including those into processing times. However, in some production systems, setup times and removal times are very important such that they should be considered independent from processing times. Since, in general jobs are done according to automatic machine processes in production systems processing times do not differ according to process sequence. But, since human factor becomes influential when setup times and removal times are taken into consideration, setup times will be decreasing by repeating setup processes frequently. This fact is defined with learning effect in scheduling literature. In this study, a bicriteria m-identical parallel machines scheduling problem with a learning effect of setup times and removal times is considered. The objective function of the problem is minimization of the weighted sum of total completion time and total tardiness. A mathematical programming model is developed for the problem which belongs to NP-hard class. Results of computational tests show that the proposed model is effective in solving problems with up to 15 jobs and five machines. We also proposed three heuristic approaches for solving large jobs problems. According to the best of our knowledge, no work exists on the minimization of the weighted sum of total completion time and total tardiness with a learning effect of setup times and removal times.     

Some scheduling problems with time-dependent learning effect and deteriorating jobs

Although machine scheduling problems with learning and deteriorating effects consideration have received increasing attention in the recent years, most studies have seldom considered the two phenomena simultaneously. However, learning and deteriorating effects might co-exist in many realistic scheduling situations. Thus, in this article, a model which takes the effects of time-dependent learning and deterioration simultaneously is proposed and applied into some scheduling problems. Under the proposed model, the processing time of a job is determined by a function of its corresponding starting time and positional sequence in each machine. We show that some single machine and flowshop scheduling problems are polynomially solvable with the certain performance measures such as makespan, total completion time, and weighted completion time.     

Research on permutation flow shop scheduling problems with general position-dependent learning effects

Machine learning exists in many realistic scheduling situations. This study focuses on permutation flow shop scheduling problems, where the actual processing time of a job is defined by a general non-increasing function of its scheduled position, i.e., general position-dependent learning effects. The objective functions are to minimize the total completion time, the makespan, the total weighted completion time, and the total weighted discounted completion time, respectively. To solve these problems, we present approximation algorithms based on the optimal permutations for the corresponding single machine scheduling problems and analyze their worst-case error bound.     

具有CON/SLK交货期指派的一类单机排序问题

研究工件的实际加工时间既具有指数学习效应,又依赖所消耗资源的准时制排序问题.在模型中,探讨了共同交货期(CON)和松弛交货期(SLK)两种情形.管理者的目标是确定最优序、最优资源分配方案和最佳工期(共同交货期或松弛交货期)以便极小化工件的总延误、总提前、总工期和资源消耗费用的总和.对于工件的实际加工时间是资源消耗量的线性函数的排序问题,通过将其转化为指派模型,给出了时间复杂性为O(n~3)的算法,从而证明该类排序问题是多项式时间可求解的.针对工件的实际加工时间是资源消耗量的凸函数的排序问题,也给出了多项式算法.     

Parallel-machine scheduling with past-sequence-dependent delivery times and learning effect

This paper considers identical parallel-machine scheduling problem with past-sequence-dependent (psd) delivery times and learning effect. In electronic manufacturing industry, an electronic component may be exposed to certain electromagnetic field and requires an extra time for eliminating adverse effect after the main processing. The extra time is modeled as past-sequence-dependent delivery time in the literature, which is proportional to the waiting time in the system. It is also observed that the learning process reflects a decrease in the processing time as a function of the number of repetitions, i.e., as a function of the job position in the sequence. In practice, one often has to deal with the scheduling problems with psd delivery times and learning effect. Identical parallel-machine setting is considered because the occurrence of resources in parallel is common in the real world. In this paper, three objectives are the minimization of the total absolute deviation of job completion times, the total load on all machines and the total completion time. We develop polynomial algorithms to optimally solve these problems.     

基于加工时间之和学习效应下的单机成组排序问题

研究具有加工时间之和学习效应下的一个新型成组排序问题,工件的学习效应是之前工件加工时间之和的函数,组学习效应是成组加工所在的位置的函数. 考虑最大完工时间和总完工时间两个问题,证明了这两个问题都是多项式时间可解的,并提出了相应的多项式时间算法.     

Single-machine group scheduling with time-dependent learning effect and position-based setup time learning effect

As to learning effect, it may be more appropriate to assume that position-based learning takes place during machine setups only, while sum-of-processing-time-based learning occurs in considering the experience that workers have gained from producing jobs. Thus, in this paper, we consider sum-of-processing-time-based learning on job processing time and position-based learning on setup time in single-machine group scheduling problems. The objectives are to minimize the makespan and the total completion time, respectively. We provide two polynomial time algorithms to solve the makespan minimization problems. On the other hand, we also provide two polynomial time algorithms to solve the total completion time minimization problems under certain conditions.     

Single-machine scheduling with learning considerations

The focus of this work is to analyze learning in single-machine scheduling problems. It is surprising that the well-known learning effect has never been considered in connection with scheduling problems. It is shown in this paper that even with the introduction of learning to job processing times two important types of single-machine problems remain polynomially solvable.     

Scheduling with a position-weighted learning effect

In general, human learning takes time to build up, which results from a worker gaining experience from repeating similar operations over time. In the early stage of processing a given set of similar jobs, a worker is not familiar with the operations, so his learning effect on the jobs scheduled early is not apparent. On the other hand, when the worker has gained experience in processing the jobs his learning improves. So a worker’s learning effect on a job depends not only on the total processing time of the jobs that he has processed but also on the job position. In this paper we introduce a position-weighted learning effect model for scheduling problems. We provide optimal solutions for the single-machine problems to minimize the makespan and the total completion time, and an optimal solution for the single-machine problem to minimize the total tardiness under an agreeable situation. We also consider two special cases of the flowshop problem.     

Single machine scheduling with truncated job-dependent learning effect

In this paper we consider the single machine scheduling problem with truncated job-dependent learning effect. By the truncated job-dependent learning effect, we mean that the actual job processing time is a function which depends not only on the job-dependent learning effect (i.e., the learning in the production process of some jobs to be faster than that of others) but also on a control parameter. The objectives are to minimize the makespan, the total completion time, the total absolute deviation of completion time, the earliness, tardiness and common (slack) due-date penalty, respectively. Several polynomial time algorithms are proposed to optimally solve the problems with the above objective functions.     

具有指数和位置学习效应的机器排序问题

本文考虑指数学习效应和位置学习效应同时发生的新的排序模型.工件的实际加工时间不仅依赖于已经加工过工件正常加工时间之和的指数函数,而且依赖于该工件所在的位置.单机排序情形下,对于最大完工时间和总完工时间最小化问题给出多项式时间算法.此外某些特殊情况下,总权完工时间和最大延迟最小化问题也给出了多项时间算法.流水机排序情形,对最大完工时间和总完工时间最小化问题在某些特殊情形下给出多项时间算法.     

Single-machine scheduling simultaneous with position-based and sum-of-processing-times-based learning considerations under group technology assumption

This paper considers the problems of scheduling with the effect of learning on a single-machine under group technology assumption. We propose a new learning model where the job actual processing time is linear combinations of the scheduled position of the job and the sum of the normal processing time of jobs already processed. We show that the makespan minimization problem is polynomially solvable. We also prove that the total completion time minimization problem with the group availability assumption remains polynomially solvable under agreeable conditions.