Basic Processing Time Research Articles

Online Parallel-Batch Scheduling of Learning Effect Jobs with Incompatible Job Families for Prefabricated Components

In the production scheduling of prefabricated components, we study an online [Formula: see text] parallel-batch machines scheduling model considering learning effect jobs with [Formula: see text] incompatible job families to minimize the makespan in this paper, where the capacity of batch is unbounded. Job families indicate that a job must belong to some job family and jobs of distinct job families are incapable to be executed in the same batch. The information of each job including its basic processing time [Formula: see text] and release time [Formula: see text] is unknown in advance and is revealed at the instant of its arrival. Moreover, the actual processing time of job [Formula: see text] with learning effect is [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative parameters and [Formula: see text] denotes the starting time of prefabricated job [Formula: see text], respectively. When [Formula: see text], we propose an online algorithm with a competitive ratio of [Formula: see text]. Furthermore, the performance of the online algorithm is demonstrated by numerical experiments.

Minimizing total completion time with linear deterioration: A new lower bound

We study scheduling problems in which the job processing times are linearly deteriorating. Each job has a constant (job-independent) basic processing time and a job-dependent deterioration rate. The objective function is minimum total completion time. The machine settings considered are: single machine and parallel identical machines. The complexity status of these problems has been an open question for the last three decades. We introduce a new lower bound on the optimal total completion time, which is shown numerically (through a comparison to the results obtained by a simple heuristic) to be extremely accurate for both problems.

Optimal sequence for single server scheduling incorporating a rate-modifying activity under job-dependent linear deterioration

This study considers single processor scheduling problems incorporating a rate-modifying activity (RMA) as well as processing time deterioration rates. The RMA fully restores the processing speed of a processor. As discussed in recent literature, the optimal position of an RMA for a single processor scheduling problem with the objective of makespan minimization, 1|p[j]A=α[j]S[j],rm|Cmax, where the processing time of a job (p[j]A:[j]denotesthejthposition)is determined by its start time (S[j])and position-dependent rate (α[j]), can be found easily. This study considers six variants of the problem: (P1) 1|p[j]A=p¯+α[j]S[j],rm|Cmax, (P1+)1|p[j]A=p¯+α[j]S[j],drm|Cmax, (P2) 1|p[i,j]A=pj+α[i]Sj,rm|Cmax, (P2+)1|p[i,j]A=pj+α[i]Sj,drm|Cmax, (P3) 1|pjA=αjSj,rm|Cmax, and (P3+)1|pjA=αjSj,drm|Cmax, where p¯ is the common basic processing time, pjis the basic processing time for job j, [i,j] is the job j scheduled at the ith position, and drm is the deteriorating RMA. We show that (P1) and (P1+), where the common basic processing time is considered, can be converted into the problem 1|p[j]A=α[j]S[j],rm|Cmax, so that the optimal position of the RMA can be found easily. In the other variants, the optimal sequences of jobs and positions of the RMA must be determined. In (P2) and (P2+), the basic processing time for each job (pj)is considered, whereas, in (P3) and (P3+), the deterioration rate depends not on the position of the job (α[j]) but on the job itself (αj). We show that the optimal schedules for (P2), (P2+) and (P3), (P3+) can be found by solving the assignment problems and the knapsack problems, respectively.

Adaptive neighborhood simulated annealing for sustainability-oriented single machine scheduling with deterioration effect

This research considers the realistic phenomenon of cutting tool deterioration in the scheduling of jobs and tool replacement activities to ensure sustainable machining operations. Typically, each job has a basic processing time (independent of tool deterioration) and a tool age-dependent prolongation function. Nevertheless, such a strategy may replace the cutting tools before its useful lifespan to avoid longer processing time as it leads to higher energy consumption and likelihood of tardy jobs. Given the reduced tool operating duration, the prospect of exploiting its capability to limit the processing time prolongation is not considered by prior works. In this research, a single-machine scheduling approach that determines the job processing time based on tool age and operating duration (utilization) is studied. The problem is formulated as a mixed-integer linear program (MILP) with the objective of minimizing the weighted costs of energy consumption, tooling, and job tardiness. Due to its computational complexity, a new variant of simulated annealing (SA) algorithm is proposed to solve large instances, where an adaptive strategy is utilized to determine the neighborhood size for enhanced exploration and exploitation. Computational results for an exhaustive set of instances indicate the proposed scheduling strategy to always outperform the existing method, yielding up to a 43% reduction in total cost. Besides, the proposed variant of SA effectively solves the scheduling problem and consistently dominates the traditional SA method.

An Optimal Online Algorithm for Scheduling with Learning Consideration

This paper investigates a classic online scheduling problem with learning effect on a single machine. Specifically, a number of independent jobs that arrive online over time will be processed on a single machine and learning effect implies that the real processing time of job [Formula: see text] is a non-increasing function of its position [Formula: see text], i.e., [Formula: see text], where [Formula: see text] is the basic processing time of job [Formula: see text] and [Formula: see text] is the learning index. Our goal is to minimize the total completion time of all jobs. For the problem, we develop a deterministic polynomial time online algorithm called Delayed Shortest Basic Processing Time (DSBPT) and state that it is an online algorithm with a competitive ratio of 2, which matches the lower bound of the online scheduling problem we focus on.

Scheduling jobs with a V-shaped time-dependent processing time

In the field of time-dependent scheduling, a job’s processing time is specified by a function of its start time. While monotonic processing time functions are well-known in the literature, this paper introduces non-monotonic functions with a convex, piecewise-linear V-shape similar to the absolute value function. They are minimum at an ideal start time, which is the same for all given jobs. Then, the processing time equals the job’s basic processing time. Earlier or later, it increases linearly with slopes that can be asymmetric and job-specific. The objective is to sequence the given jobs on a single machine and minimize the makespan. This is motivated by production planning of moving car assembly lines, in particular, to sequence a worker’s assembly operations such that the time-dependent walking times to gather materials from the line-side is minimized. This paper characterizes the problem’s computational complexity in several angles. NP-hardness is observed even if the two slopes are the same for all jobs. A fully polynomial time approximation scheme is devised for the more generic case of agreeable ratios of basic processing time and slopes. In the most generic case with job-specific slopes, several polynomial cases are identified.

SINGLE MACHINE DUE DATE ASSIGNMENT SCHEDULING PROBLEM WITH PRECEDENCE CONSTRAINTS AND CONTROLLABLE PROCESSING TIMES IN FUZZY ENVIRONMENT

In this paper, a due date assignment scheduling problem with precedence constraints and controllable processing times in uncertain environment is investigated, in which the basic processing time of each job is assumed to be the symmetric trapezoidal fuzzy number, and the linear resource consumption function is used.The objective is to minimize the crisp possibilistic mean (or expected) value of a cost function thatincludes the costs of earliness, tardiness, makespan and resource consumption jointly by scheduling the jobs under precedence constraints and determining the due date and the resource allocation amount satisfying resource constraints for each job. First, the problem is shown to be NP-hard. Furthermore, an optimal algorithm with polynomial time for the special case of this problem is put forward. Moreover, an efficient 2-approximation algorithm is presented based on solving the relaxation of the problem. Finally, the numerical experiment is given, whose results show that our method is promising.

Scheduling deteriorating jobs with machine availability constraints to minimize the total completion time

In this study, we explored the scheduling problems with non-resumable deteriorating jobs and machine availability constraints. It is assumed that jobs have a different basic processing time and same deterioration rate, and the machines may be unavailable in some specified time intervals. The objective is to minimize the total completion time. We presented a pseudo-polynomial dynamic programming algorithm and a fully polynomial-time approximation scheme for the single machine version. For the parallel machine version, we first presented a pseudo-polynomial dynamic programming algorithm and then showed that there exists no polynomial time approximation algorithm with a constant worst-case bound unless P = NP.

Scheduling controllable processing time jobs in a deteriorating environment

In many real-life applications, job processing times are a function of the waiting time prior to their execution. In the most general setting, each job comprises of a basic processing time, which is independent of its start time, and a start time-dependent deterioration function. Some common examples of deteriorating systems include fire fighting, pollution containment, and medical treatments. To date, research has focused on scheduling models where the basic processing time of jobs is constant. However, job processing times are often controllable through the allocation of a limited non-renewable resource. We study a single-machine setting that combines these two models under the assumptions of general linear deterioration and convex resource functions. We develop a polynomial time solution for minimizing the makespan. For the total flowtime criterion, we compute the optimal resource allocation policy for a given job instance and show that the sequencing problem is at least as hard as the case with non-controllable jobs. We follow by discussing the properties of several special cases.

Single-machine scheduling with effects of exponential learning and general deterioration

The purpose of this study is to explore the single-machine scheduling with the effects of exponential learning and general deterioration. By the effects of exponential learning and general deterioration, we meant that job processing time is decided by the functions of their starting time and positions in the sequence. Results showed that with the introduction of learning effect and deteriorating jobs to job processing time, single-machine makespan, and sum of completion time (square) minimization problems remained polynomially solvable, respectively. But for the following objective functions: the weighted sum of completion time and the maximum lateness, this paper proved that the weighted smallest basic processing time first (WSPT) rule and the earliest due date first (EDD) rule constructed the optimal sequence under some special cases, respectively.

Single-machine scheduling with deteriorating jobs

In this article, we study a single-machine scheduling problem in which the processing time of a job is a nonlinear function of its basic processing time and starting time. The objectives are to minimise the makespan, the sum of weighted completion times and the sum of the kth powers of completion times. We show that the makespan minimisation problem can be solved in polynomial time. However, the total completion time and the sum of the kth powers of completion times minimisation problems can be solved in polynomial time in some cases. Besides, some useful properties are also provided for the sum of weighted completion times problem under certain conditions.

Minimizing sum of completion times for batch scheduling of jobs with deteriorating processing times

We consider the problem of scheduling n jobs on m ⩾ 1 parallel and identical machines, where the jobs are processed in batches and the processing time of each job is a step function of its waiting time; i.e., the time between the start of the processing of the batch to which the job belongs and the start of the processing of the job. For each job i, if its waiting time is less than a given threshold D, then it requires a basic processing time p i = a i ; otherwise, it requires an extended processing time p i = a i + b i . The objective is to minimize the sum of completion times; i.e., ∑ i = 1 n C i , where C i is the completion time of job i. We first show that the problem is NP-hard in the strong sense even if there is a single machine and b i = b for all 1 ⩽ i ⩽ n. We then show that the problem is solvable in polynomial time if a i = a for all 1 ⩽ i ⩽ n. Our algorithm runs in O( n 2) time. Finally, we give an approximation algorithm for the special case where b i ⩽ D for all 1 ⩽ i ⩽ n, and show that it has a performance guarantee of 2.

Batch scheduling of step deteriorating jobs

In this paper we consider the problem of scheduling n jobs on a single machine, where the jobs are processed in batches and the processing time of each job is a step function depending on its waiting time, which is the time between the start of the processing of the batch to which the job belongs and the start of the processing of the job. For job i, if its waiting time is less than a given threshold value D, then it requires a basic processing time a i ; otherwise, it requires an extended processing time a i +b i . The objective is to minimize the completion time of the last job. We first show that the problem is NP-hard in the strong sense even if all b i are equal, it is NP-hard even if b i =a i for all i, and it is non-approximable in polynomial time with a constant performance guarantee Δ<3/2, unless $\mathcal {P}=\mathcal{NP}$ . We then present O(nlog?n) and O(n 3F?1log?n/F F ) algorithms for the case where all a i are equal and for the case where there are F, F?2, distinct values of a i , respectively. We further propose an O(n 2log?n) approximation algorithm with a performance guarantee $\Delta\le1+\lfloor\frac{m^{*}}{2}\rfloor/m^{*}\le3/2$ for the general problem, where m * is the number of batches in an optimal schedule. All the above results apply or can be easily modified for the corresponding open-end bin packing problem.

Single machine scheduling problems with deteriorating jobs

This paper considers the single machine scheduling problems with deteriorating jobs, i.e. jobs whose processing times are a decreasing linear function of their starting time. It is assumed that jobs have the different basic processing time and same decreasing rate. Based on the analysis of problems, the optimal algorithms are presented for the problems to minimize the sum of earliness penalties subject to no tardy jobs, to minimize the resource consumption with makespan constraints and to minimize the makespan with the total resource consumption constraints.

Scheduling linearly deteriorating jobs on parallel machines: A simulated annealing approach

Scheduling deteriorating jobs on parallel machines is an NP-hard problem, for which heuristics would be the first solution option. Two variants of linearly deteriorating jobs are considered. The first is that with simple linear deterioration, i.e. where there is a deterioration rate only, which is meaningful only if the jobs are assumed to be available at a positive time t0. In the second variant, there is a basic processing time and a deterioration rate and all jobs are available at time t = 0. In both cases, we seek to minimize the makespan. Starting from simple heuristics, both steepest descent search and simulated annealing are designed and implemented to arrive at optimal or near-optimal solutions. Computational results for randomly generated problem instances with different job/machine combinations are presented.

V-Shaped Policies for Scheduling Deteriorating Jobs

A set of N jobs has to be processed on a single machine. Jobs have the same basic processing time, but the actual processing time of each job grows linearly with its starting time. A (possibly) different rate of growth is associated with each job. We show that the optimal sequence to minimize flow time is V-shaped: Jobs are arranged in descending order of growth rate if they are placed before the minimal growth rate job, and in ascending order if placed after it. Efficient (0(N log N)) asymptotically optimal heuristics are developed. Their average performance is shown to be extremely good: The average relative error over a set of 20-job problems is on the order of 10−5.

Reading rates of stutterers and nonstutterers during silent and oral reading

The purpose of this study was to investigate the differences between the reading rates of stutterers and nonstutterers during both silent and oral reading of fluently spoken words. Ten adult stutterers and 14 nonstutterers were instructed to read two lists of words over four trials each. The first trial of either the first or the second list was a silent reading trial. All the other trials were oral reading trials. The word lists consisted of nouns with word lengths from two to five syllables. The dependent variables were the silent and oral reading times per word and per syllable. The major findings were that stutterers, as compared to nonstutterers, required longer reading times per word and per syllable during both silent and oral reading. These results indicate that stutterers differ from nonstutterers in the basic processing time for verbal material.

The Psychological Refractory Period and the Effect of Stimulus Discriminability

The 'psychological refractory period' has come to be the accepted label for response-delay phenomena in a double-stimulation situation. In the classic arrangement the two stimuli S1 and S2, requiring the two responses R1 and R2, are presented at interstimulus intervals of less than 500 msec. With decreasing interstimulus intervals the latency of R2 is known to increase steadily. The early studies on the refractory period were primarily concerned with the probability aspects of the various parameters involved. These parametric studies have produced a wealth of results that can hardly be encompassed by any one theory. Among the various theories that have been proposed over the years, three are most likely to survive: singlechannel, expectancy, and response-conflict theory. Single-channel theory, first proposed by Craik (1948), explains the delay in R2 in terms of a limited capacity of the central processing system (single-channel). The single channel can process only one stimulus at a time. Under double or sequential stimulation, signals are either queued or lost if the pace of their temporal presentation exceeds the basic processing time of the channel. According to this assumption, a delay in R2 should occur if the interstimulus interval falls below a critical value. In contrast to the single-channel notion, Poulton (1950) attributed delays in R2 at short interstimulus intervals to the fact that the subject

A measure of pronunciability of CVC trigrams

All 2100 CVC trigrams were scaled for pronunciability by measuring pronunciation latency (PLat). The resulting distribution of PLat scores was extremely leptokurtic and positively skewed. Scores ranged from a minimum of .531 sec to a maximum of 1.726 sec. Average PLat was .81 sec. The relationship of PLat to Archer meaningfulness was linear; however, the degree of relationship was slight (r=−.37). This finding is interpreted as indicating that PLat is relatively free of bias from such other stimulus attributes as meaningfulness. As such, P Lat is viewed as reflecting a basic processing time for such stimulus materials.