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.     

Minimizing sum of completion times on a single machine with sequence-dependent family setup times

This paper presents a branch-and-bound (B&B) algorithm for minimizing the sum of completion times in a single-machine scheduling setting with sequence-dependent family setup times. The main feature of the B&B algorithm is a new lower bounding scheme that is based on a network formulation of the problem. With extensive computational tests, we demonstrate that the B&B algorithm can solve problems with up to 60 jobs and 12 families, where setup and processing times are uniformly distributed in various combinations of the [1,50] and [1,100] ranges.     

Worst-case behavior of simple sequencing rules in flow shop scheduling with general position-dependent learning effects

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 a general position-dependent learning effects. By the general position-dependent learning effects, we mean that the actual processing time of a job is defined by a general non-increasing function of its scheduled position. The objective is to minimize one of the five regular performance criteria, namely, the total completion time, the makespan, the total weighted completion time, the total weighted discounted 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 analyze the worst-case bound of our heuristic algorithms.     

Some single-machine scheduling problems with actual time and position dependent learning effects

In this paper we study some single-machine scheduling problems with learning effects where the actual processing time of a job serves as a function of the total actual processing times of the jobs already processed and of its scheduled position. We show by examples that the optimal schedules for the classical version of problems are not optimal under this actual time and position dependent learning effect model for the following objectives: makespan, sum of kth power of the completion times, total weighted completion times, maximum lateness and number of tardy jobs. But under certain conditions, we show that the shortest processing time (SPT) rule, the weighted shortest processing time (WSPT) rule, the earliest due date (EDD) rule and the modified Moore’s Algorithm can also construct an optimal schedule for the problem of minimizing these objective functions, respectively.     

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.     

Single machine scheduling with general time-dependent deterioration, position-dependent learning and past-sequence-dependent setup times

The paper deals with single machine scheduling problems with setup time considerations where the actual processing time of a job is not only a non-decreasing function of the total normal processing times of the jobs already processed, but also a non-increasing function of the job’s position in the sequence. The setup times are proportional to the length of the already processed jobs, i.e., the setup times are past-sequence-dependent (p-s-d). We consider the following objective functions: the makespan, the total completion time, the sum of the δth (δ ≥ 0) power of job completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the δ th (δ ≥ 0) power of job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

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

In this paper we consider the single machine past-sequence-dependent (p-s-d) setup times scheduling problems with general position-dependent and time-dependent learning effects. By the general position-dependent and time-dependent learning effects, we mean that 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. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times, the total lateness, the total weighted completion time, the maximum lateness, the maximum tardiness and the number of tardy jobs. We show that the problems of makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times and the total lateness can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem, the maximum lateness minimization problem, maximum tardiness minimization problem and the number of tardy jobs minimization problem can be solved in polynomial time under certain conditions.     

Scheduling deteriorating jobs with a learning effect on unrelated parallel machines

In this study we consider unrelated parallel machines scheduling problems with learning effect and deteriorating jobs, in which the actual processing time of a job is a function of joint time-dependent deterioration and position-dependent learning. The objective is to determine the jobs assigned to corresponding each machine and the corresponding optimal schedule to minimize a cost function containing total completion (waiting) time, total absolute differences in completion (waiting) times and total machine load. If the number of machines is a given constant, we show that the problems can be solved in polynomial time under the time-dependent deterioration and position-dependent learning model.     

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.     

Learning and forgetting effects on a group scheduling problem

This paper proposes to investigate learning and forgetting effects on the problem of scheduling families of jobs on a single machine to minimize total completion time of jobs. A setup time is incurred whenever the single machine transfers job processing from a family to another family. To analyze the impact of learning and forgetting on this group scheduling problem, we structure three basic models and make some comparisons through computational experiments. The three models, including no forgetting, total forgetting and partial forgetting, assume that the processing time of a job is dependent on its position in a schedule. Some scheduling rules and a lower bound are derived in order to constitute our branch-and-bound algorithm for searching an optimal sequence. In addition, an efficient and simply-structured heuristic is also built to find a near-optimal schedule.     

Note on “A note on single-machine group scheduling problems with position-based learning effect”

In a recent paper, Lee and Wu [W.-C. Lee, C.-C. Wu, A note on single-machine group scheduling problems with position-based learning effect, Appl. Math. Model. 33 (2009) 2159–2163] proposed a new group scheduling learning model where the learning effect not only depends on the job position, but also depends on the group position. They investigate the makespan and the total completion time minimization problems on a single-machine. As for the total completion time minimization problem, they assumed that the numbers of jobs in each group are the same and the group normal setup and the job normal processing times are agreeable. Under the assumption conditions, they showed that the total completion time minimization problem can be optimally solved in polynomial time solution. However, the assumption conditions for the total completion time minimization problem do not reflect actual practice in many manufacturing processes. Hence, in this note, we propose other agreeable conditions and show that the total completion time minimization problem remains polynomially solvable under the agreeable conditions.     

同时具有学习效应和退化效应的单机排序问题

本文给出了一种同时具有一般化学习效应和退化效应的单机排序模型。在此模型中,工件的实际加工时间既与工件所在位置又与其开工时间有关,且工件在加工之后具有一个配送时间。其中学习效应是工件所在位置的函数,退化效应是工件开工时间的函数。证明了极小化最大完工时间和极小化总完工时间问题是多项式可解的,在满足一定的条件下,极小化加权总完工时间和极小化最大延误问题也是多项式可解的。推广了一些已有文献中的结论。     

Some single-machine scheduling with sum-of-processing-time-based and job-position-based processing times

The single-machine scheduling problems with position and sum-of-processing-time based processing times are considered. The actual processing time of a job is defined by function of its scheduled position and total normal processing time of jobs in front of it in the sequence. We provide optimal solutions in polynomial time for some special cases of the makespan minimization and the total completion time minimization. We also show that an optimal schedule to be a V-shaped schedule in terms of the normal processing times of jobs for the total completion time minimization problem and the makespan minimization problem.     

Scheduling jobs with truncated exponential learning functions

In this paper we consider the single machine scheduling problem with truncated exponential learning functions. By the truncated exponential learning functions, we mean that the actual job processing time is a function which depends not only on the total normal 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. We show that even with the introduction of the proposed model to job processing times, several single machine problems remain polynomially solvable. For the following three objective functions, the total weighted completion time, the discounted total weighted completion time, the maximum lateness, we present heuristic algorithms according to the corresponding problems without truncated exponential learning functions. We also analyse the worst-case bound of our heuristic algorithms.     

Scheduling time-dependent jobs under mixed deterioration

We consider a new model of time-dependent scheduling. A set of deteriorating jobs has to be processed on a single machine which is available starting from a non-zero time. The processing times of some jobs from this set are constant, while other ones are either proportional or linear functions of the job starting times. The applied criteria of schedule optimality include the maximum completion time, the total completion time, the total weighted completion time, the maximum lateness and the number of tardy jobs. We delineate a sharp boundary between computationally easy and difficult problems, showing polynomially solvable and NP-hard cases.     

工期窗口安排与具有退化效应和维修活动的单机排序

考虑具有工件相关的退化效应和维修活动的单机排序模型,讨论了工期窗口安排问题.在这一模型中,机器在加工过程中产生退化使效率降低,工件的实际加工时间不仅与其所在排序中的位置有关并且与其本身的退化率有关;然而,维修活动能使机器的加工效率得到恢复.工期窗口的开始时间是已给定的常量,而工期窗口的结束时间是需要确定的变量.目标是得到安排维修活动的最佳时间、最佳工期窗口的大小和最优排序以便最小化流时间、提早、延误和工期窗口大小的总处罚函数.对这一问题,给出了一多项式算法.     

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.     

Single machine scheduling with exponential learning functions

In this paper we consider the single machine scheduling problem with exponential learning functions. By the exponential learning functions, we mean that the actual job processing time is a function of the total normal processing times of the jobs already processed. We prove that the shortest processing time (SPT) rule is optimal for the total lateness minimization problem. For the following three objective functions, the total weighted completion time, the discounted total weighted completion time, the maximum lateness, we present heuristic algorithms according to the corresponding problems without exponential learning functions. We also analyse the worst-case bound of our heuristic algorithms. It also shows that the problems of minimizing the total tardiness and discounted total weighted completion time are polynomially solvable under some agreeable conditions on the problem parameters.     

Precedence constrained parallel-machine scheduling of position-dependent jobs

We consider parallel-machine job scheduling problems with precedence constraints. Job processing times are variable and depend on positions of jobs in a schedule. The objective is to minimize the maximum completion time or the total weighted completion time. We specify certain conditions under which the problem can be solved by scheduling algorithms applied earlier for fixed job processing times.     

Scheduling problems with past-sequence-dependent setup times and general effects of deterioration and learning

In this paper we consider the single-machine setup times scheduling with general effects of deterioration and learning. By the general effects of deterioration and learning, we mean that the actual job processing time is a general function of the processing times of the jobs already processed and its scheduled position. The setup times are proportional to the length of the already processed jobs, i.e., the setup times are past-sequence-dependent (p-s-d). We show that the problems to minimize the makespan, the sum of the δ$\delta$th (δ>0$\delta >0$) power of job completion times, the total lateness are polynomially solvable. We also show that the total weighted completion time minimization problem, the discounted total weighted completion time minimization problem, the maximum lateness (tardiness) minimization problem, the total tardiness minimization problem can be solved in polynomial time under certain conditions.