Some scheduling problems with deteriorating jobs and learning effects

Single-machine scheduling problems with deteriorating jobs and learning effects

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

Scheduling with learning effects has been widely studied. However, there are situations where the learning effect might accelerate. In this paper, we propose a new model where the learning effect accelerates as time goes by. We derive the optimal solutions for the single-machine problems to minimize the makespan, total completion time, total weighted completion time, maximum lateness, maximum tardiness, and total tardiness.

The paper deals with some single-machine scheduling problems with setup time considerations where the processing time of a job is given as a function of its starting times and position in a 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 (\( \delta \geqslant 0 \)) power of job completion times, the total weighted completion time, the maximum lateness and the number of tardy jobs. We show that the makespan minimization problem, the total completion time minimization problem, and the sum of the δth power of job completion times minimization problem can be solved in polynomial time, respectively. We also show that the total weighted completion time minimization problem, the maximum lateness minimization problem and the number of tardy jobs minimization problem can be solved in polynomial time under certain conditions.

This paper considers the multi-criteria scheduling problem with total completion time, maximum lateness, and maximum earliness as the objectives. The problem is a minimisation problem; the total completion time, a MIN-SUM problem, the maximum earliness, a MIN-MAX problem, and the maximum lateness, a MIN-MAX problem. Though, the problem is NP-hard, shortest processing time (SPT) rule, yields optimal for total completion time while early due date rule (EDD) is the optimal solution for maximum lateness and maximum earliness if each criterion were to be considered separately. Two heuristics, named SOL I and SOL II were proposed and the results for each of the criteria were compared to the optimal of the sub-problems. Results of the computational experiment on small job-sizes (5 ≤ n ≤ 30) and large job-sizes (40 ≤ n ≤ 100) show that the two heuristics results are not significantly different from the optimal at 99% significant level for total completion time and maximum lateness performance measures. However, for maximum earliness, the optimal solutions are significantly better than the two proposed heuristics.

Scheduling with deterioration effects has been widely investigated in the past two decades. In realistic situations, the deterioration effect might accelerate, especially in the food manufacturing industry. In this paper we propose a scheduling model with an accelerating deterioration effect. We show that the single-machine problems under the model to minimize the makespan, total completion time, total weighted completion time, maximum lateness, maximum tardiness, and total tardiness remain polynomially solvable.

Execution times are crucial for effectiveness of tasks or jobs scheduling. It is very hard to accurately estimate execution times because they are influenced by many factors. Though there are some models for traditional machine scheduling problems, no attention has been paid on virtual machine scheduling in cloud computing. Based on cloud agent (VM administrator, scheduler or intelligent procedure) experiences, we develop integrated learning effects models to obtain accurate execution times. Based on the constructed learning effects model for single virtual machine scheduling, optimal schedule rules are proposed for minimizing makespan, the total completion time and the sum of (square) completion times. Problems with the total weighted completion time and the maximum lateness minimization are proved to be optimally solvable in polynomial time only for certain assumptions. Furthermore, we adapt the developed learning effects model to two special m-virtual machine MapReduce scenarios, for which optimal schedule rules are introduced correspondingly. Optimal solutions are demonstrated by examples of the problems under study using the constructed rules.

This paper considers a scheduling model with general learning effects and deteriorating jobs simultaneously, where the actual processing time of a job is a function depends not only on the start time of the job, but also on the total normal processing times of jobs in front of it in the sequence. It shows that the optimal schedule for the single-machine scheduling problem to minimize makespan is obtained by the longest processing time first (LPT) rule. For the problem to minimize total complete time, we use the smallest processing time first (SPT) rule as a heuristic algorithm and analyze its worst-case bound.

This paper deals with a single machine scheduling problem with general past-sequence-dependent (psd) setup time and log-linear learning in which the setup times and learning effects are job-dependent. The setup times are unique functions of the length of already processed jobs, and the learning effects show that processing times are unique decreasing functions of job positions. The goal is to find the optimal sequences that minimise objective functions such as the makespan, total completion time, total lateness, total waiting cost, total waiting time, total absolute differences in completion times, and the sum of earliness, tardiness and common due date penalty. Special cases of the resulting problems are solvable in polynomial time; however, the general problems are difficult to solve. We propose branch-and-bound (B&B) methods to derive the optimal sequences for such problems. Computational results show that the proposed methods can solve relatively large problem instances in reasonable amounts of time.

In most studies, the learning effect model applied in a scheduling problem assumes that the actual processing time of a job is a function of its scheduled position or the sum of the normal processing times of the jobs already scheduled. In this article, we introduce a learning effect model and a past-sequence-dependent (psd) setup time into a single machine group scheduling problem. The actual processing time of a job in each group is a function of the sum of the normal processing times of the jobs already scheduled and the position of the corresponding job group in the sequence. It shows that the addressed problem remains polynomially solvable for the objectives: minimisation of the makespan and the total completion time. It also shows that the smallest processing time rule provides the optimal jobs sequence in each group and the optimal group sequence for the addressed problem, respectively.

This paper considers the combination of the general sum-of-processing-time effect and position-dependent effect on a single machine. The actual processing time of a job is defined by functions of the sum of the normal processing times of the jobs processed and its position and control parameter in the sequence. We consider two monotonic effect functions: the nondecreasing function and the nonincreasing function. Our focus is the following objective functions, including the makespan, the sum of the completion time, the sum of the weighted completion time, and the maximum lateness. For the nonincreasing effect function, polynomial algorithm is presented for the makespan problem and the sum of completion time problem, respectively. The latter two objective functions can also be solved in polynomial time if the weight or due date and the normal processing time satisfy some agreeable relations. For the nondecreasing effect function, assume that the given parameter is zero. We also show that the makespan problem can remain polynomially solvable. For the sum of the total completion time problem and a 1 is the deteriorating rate of the jobs, there exists an optimal solution for a 1 ≥ M ; a V-shaped property with respect to the normal processing times is obtained for 0 < a 1 ≤ 1 . Finally, we show that the sum of the weighted completion problem and the maximum lateness problem have polynomial-time solutions for a 1 > M under some agreeable conditions, respectively.

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

This article introduces a general single-machine setup times scheduling model with learning effect and deteriorating jobs simultaneously which is a generalisation of some existing models in the literature. 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). By the effects of learning and deterioration, we mean that the actual processing time of a job depends not only on the starting time of the job but also on its scheduled position. The article shows that the problems to minimise the makespan, sum of the kth power of completion times, total lateness and sum of earliness penalties (with common due date) are polynomially solvable under the proposed model. It further shows that the problems to minimise total weighted completion time, maximum lateness, maximum tardiness, total tardiness and total weighted earliness penalties (with common due date) are polynomially solvable under certain conditions.

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.

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.