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 deteriorating jobs and learning effects

Although scheduling with deteriorating jobs and learning effect has been widely investigated, scheduling research has seldom considered the two phenomena simultaneously. However, job deterioration and learning co-exist in many realistic scheduling situations. In this paper, we introduce a new scheduling model in which both job deterioration and learning exist simultaneously. The actual processing time of a job depends not only on the processing times of the jobs already processed but also on its scheduled position. For the single-machine case, we derive polynomial-time optimal solutions for the problems to minimize makespan and total completion time. In addition, we show that the problems to minimize total weighted completion time and maximum lateness are polynomially solvable under certain agreeable conditions. For the case of an m-machine permutation flowshop, we present polynomial-time optimal solutions for some special cases of the problems to minimize makespan and total completion time.     

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.     

Single-machine scheduling with truncated sum-of-processing-times-based learning effect including proportional delivery times

The single-machine scheduling problem with truncated sum-of-processing-times-based learning effect and past-sequence-dependent job delivery times is considered. Each job’s delivery time depends on its waiting time of processing. For some regular objective functions, it is proved that the problems can be solved by the smallest processing time first rule. For some special cases of the total weighted completion time and the maximum lateness objective functions, the thesis shows that the problems can be solved in polynomial time.     

Worst-case and numerical analysis of heuristic algorithms for flowshop scheduling problems with a time-dependent learning effect

In this paper, we investigate a time-dependent learning effect in a flowshop scheduling problem. We assume that the time-dependent learning effect of a job was a function of the total normal processing time of jobs scheduled before the job. The following objective functions are explored: the makespan, the total flowtime, the sum of weighted completion times, the sum of the kth power of completion times, and the maximum lateness. Some heuristic algorithms with worst-case analysis for the objective functions are given. Moreover, a polynomial algorithm is proposed for the special case with identical processing time on each machine and that with an increasing series of dominating machines, respectively. Finally, the computational results to evaluate the performance of the heuristics are provided.     

Stochastic scheduling problems with general position-based learning effects and stochastic breakdowns

The focus of this study is to analyze position-based learning effects in single-machine stochastic scheduling problems. The optimal permutation policies for the stochastic scheduling problems with and without machine breakdowns are examined, where the performance measures are the expectation and variance of the makespan, the expected total completion time, the expected total weighted completion time, the expected weighted sum of the discounted completion times, the maximum lateness and the maximum tardiness.     

Single-machine scheduling problems with deteriorating jobs and learning effects

In this paper, we introduce a new scheduling model in which deteriorating jobs and learning effect are both considered simultaneously. By deterioration and the learning effect, we mean that the actual processing time of a job depends not only on the processing time of the jobs already processed but also on its scheduled position. For the single-machine case, we show that the problems of makespan, total completion time and the sum of the quadratic job completion times remain polynomially solvable, respectively. In addition,we show that the problems to minimize total weighted completion time and maximum lateness are polynomially solvable under certain conditions.     

A note on single-machine scheduling with general learning effect and past-sequence-dependent setup time

In this paper, we study a scheduling model with the consideration of both the learning effect and the setup time. Under the proposed model, the learning effect is a general function of the processing time of jobs already processed and its scheduled position, and the setup time is past-sequence-dependent. We then derive the optimal sequences for two single-machine problems, which are the makespan and the total completion time. Moreover, we showed that the weighted completion time, the maximum lateness, the maximum tardiness, and the total tardiness problems remain polynomially solvable under agreeable conditions.     

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.     

Several flow shop scheduling problems with truncated position-based learning effect

The concept of truncated position-based learning process plays a key role in production environments. However, it is relatively unexplored in the flow shop setting. In this paper, we consider the flow shop scheduling with truncated position-based learning effect, i.e., the actual processing time of a job is a function of its position and a control parameter in a processing permutation. The objective is to minimize one of the six regular performance criteria, namely, the total completion time, the makespan, the total weighted completion time, the discounted total weighted completion time, the sum of the quadratic job completion times, and the maximum lateness. We present heuristic algorithms and analyze the worst-case bound of these heuristic algorithms. We also provide the computational results to evaluate the performance of the heuristics.     

Two-machine flowshop scheduling with three-operation jobs subject to a fixed job sequence

This paper studies the problem of scheduling three-operation jobs in a two-machine flowshop subject to a predetermined job processing sequence. Each job has two preassigned operations, which are to be performed on their respective dedicated machines, and a flexible operation, which may be processed on either of the two machines subject to the processing order as specified. Five standard objective functions, including the makespan, the maximum lateness, the total weighted completion time, the total weighted tardiness, and the weighted number of tardy jobs are considered. We show that the studied problem for either of the five considered objective functions is ordinary NP-hard, even if the processing times of the preassigned operations are zero for all jobs. A pseudo-polynomial time dynamic programming framework, coupled with brief numerical experiments, is then developed for all the addressed performance metrics with different run times.     

A note on machine scheduling with sum-of-logarithm-processing-time-based and position-based learning effects

Recently, Biskup [2] classifies the learning effect models in scheduling environments into two types: position-based and sum-of-processing-time-based. In this paper, we study scheduling problem with sum-of-logarithm-processing-time-based and position-based learning effects. We show that the single machine scheduling problems to minimize the makespan and the total completion time can both be solved by the smallest (normal) processing time first (SPT) rule. We also show that the problems to minimize the maximum lateness, the total weighted completion times and the total tardiness have polynomial-time solutions under agreeable WSPT rule and agreeable EDD rule. In addition, we show that m-machine permutation flowshop problems are still polynomially solvable under the proposed learning model.     

具有线性恶化加工时间的调度问题

讨论了工件具有线性恶化加工时间的调度问题.在这类问题中,工件的恶化函数为线性函数.对单机调度问题中目标函数为极小化最大完工时间加权完工时间和,最大延误以及最大费用等问题分别给出了最优算法.对两台机器极小化最大完工时间的Flowshop问题,证明了利用Johnson规则可以得到最优调度.对于一般情况,如果同一工件的工序的加工时间均相等,则Flowshop问题可以转化为单机问题.     

Two-agent singe-machine scheduling with release times to minimize the total weighted completion time

In many management situations multiple agents pursuing different objectives compete on the usage of common processing resources. In this paper we study a two-agent single-machine scheduling problem with release times where the objective is to minimize the total weighted completion time of the jobs of one agent with the constraint that the maximum lateness of the jobs of the other agent does not exceed a given limit. We propose a branch-and-bound algorithm to solve the problem, and a primary and a secondary simulated annealing algorithm to find near-optimal solutions. We conduct computational experiments to test the effectiveness of the algorithms. The computational results show that the branch-and-bound algorithm can solve most of the problem instances with up to 24 jobs in a reasonable amount of time and the primary simulated annealing algorithm performs well with an average percentage error of less than 0.5% for all the tested cases.     

Single machine scheduling with interfering job sets

We consider two single machine bicriteria scheduling problems in which jobs belong to either of two different disjoint sets, each set having its own performance measure. The problem has been referred to as interfering job sets in the scheduling literature and also been called multi-agent scheduling where each agent's objective function is to be minimized. In the first problem (P1) we look at minimizing total completion time and number of tardy jobs for the two sets of jobs and present a forward SPT-EDD heuristic that attempts to generate the set of non-dominated solutions. The complexity of this specific problem is NP-hard; however some pseudo-polynomial algorithms have been suggested by earlier researchers and they have been used to compare the results from the proposed heuristic. In the second problem (P2) we look at minimizing total weighted completion time and maximum lateness. This is an established NP-hard problem for which we propose a forward WSPT-EDD heuristic that attempts to generate the set of supported points and compare our solution quality with MIP formulations. For both of these problems, we assume that all jobs are available at time zero and the jobs are not allowed to be preempted.     

A Multiple-Criterion Model for Machine Scheduling

We consider a scheduling problem involving a single processor being utilized by two or more customers. Traditionally, such scenarios are modeled by assuming that each customer has the same criterion. In practice, this assumption may not hold. Instead of using a single criterion, we examine the implications of minimizing an aggregate scheduling objective function in which jobs belonging to different customers are evaluated based on their individual criteria. We examine three basic scheduling criteria: minimizing makespan, minimizing maximum lateness, and minimizing total weighted completion time. Although determining a minimum-cost schedule according to any one of these criteria is polynomially solvable, we demonstrate that when minimizing a mix of these criteria, the problem becomes NP-hard.     

Relocation scheduling subject to fixed processing sequences

This study addresses a relocation scheduling problem that corresponds to resource-constrained scheduling on two parallel dedicated machines where the processing sequences of jobs assigned to the machines are given and fixed. Subject to the resource constraints, the problem is to determine the starting times of all jobs for each of the six considered regular performance measures, namely, the makespan, total weighted completion time, maximum lateness, total weighted tardiness, weighted number of tardy jobs, and number of tardy jobs. By virtue of the proposed dynamic programming framework, the studied problem for the minimization of makespan, total weighted completion time, or maximum lateness can be solved in \(O(n_1n_2(n_1+n_2))\) time, where \(n_1\) and \(n_2\) are the numbers of jobs on the two machines. The simplified case with a common job processing time can be solved in \(O(n_1n_2)\) time. For the objective function of total weighted tardiness or weighted number of tardy jobs, this problem is proved to be NP-hard in the ordinary sense, and the case with a common job processing length is solvable in \(O(n_1n_2\min \{n_1,n_2\})\) time. The studied problem for the minimization of number of tardy jobs is solvable in \(O(n^2_1n^2_2(n_1+n_2)^2)\) time. The solvability of the common-processing-time problems can be generalized to the m-machine cases, where \(m\ge 3\).     

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.     

A Lagrangian approach to single-machine scheduling problems with two competing agents

In this paper, we develop branch-and-bound algorithms for several hard, two-agent scheduling problems, i.e., problems in which each agent has an objective function which depends on the completion times of its jobs only. Our bounding approach is based on the fact that, for all problems considered, the Lagrangian dual gives a good bound and can be solved exactly in strongly polynomial time. The problems addressed here consist in minimizing the total weighted completion time of the jobs of agent A, subject to a bound on the cost function of agent B, which may be: (i) total weighted completion time, (ii) maximum lateness, (iii) maximum completion time. An extensive computational experience shows the effectiveness of the approach.