scholarly journals SORTING APPROACHES IN THE HEURISTIC BASED ON REMAINING AVAILABLE AND PROCESSING PERIODS TO MINIMIZE TOTAL WEIGHTED TARDINESS IN PROGRESSIVE IDLING-FREE 1-MACHINE PREEMPTIVE SCHEDULING

2021 ◽  
Author(s):  
Vadym V. Romanuke
Keyword(s):  
Computational Study ◽  
Computation Time ◽  
Combinatorial Problem ◽  
Total Weighted Tardiness ◽  
Release Dates ◽  
Preemptive Scheduling ◽  
Scheduling Problems ◽  
Due Dates ◽  
Weighted Tardiness ◽  
Priority Weights

Background. In preemptive job scheduling, total weighted tardiness minimization is commonly reduced to solving a combinatorial problem, which becomes practically intractable as the number of jobs and the numbers of their processing periods increase. To cope with this challenge, heuristics are used. A heuristic, in which the decisive ratio is the weighted reciprocal of the maximum of a pair of the remaining processing period and remaining available period, is closely the best one. However, the heuristic may produce schedules of a few jobs whose total weighted tardiness is enormously huge compared to the real minimum. Therefore, this heuristic needs further improvements, one of which already exists for jobs without priority weights with a sorting approach where remaining processing periods are minimized. Three other sorting approaches still can outperform it, but such exceptions are quite rare. Objective. The goal is to determine the influence of the four sorting approaches and try to select the best one in the case where jobs have their priority weights. The heuristic will be applied to tight-tardy progressive idling-free 1-machine preemptive scheduling, where the release dates are given in ascending order starting from 1 to the number of jobs, and the due dates are tightly set after the release dates. Methods. To achieve the said goal, a computational study is carried out with applying each of the four heuristic approaches to minimize total weighted tardiness. For this, two series of 4151500 scheduling problems are generated. In the solution of a scheduling problem, a sorting approach can “win” solely or “win” in a group of approaches, producing the heuristically minimal total weighted tardiness. In each series, the distributions of sole-and-group “wins” are ascertained. Results. The sole “wins” and non-whole-group “wins” are rare: the four sorting approaches produce schedules with the same total weighted tardiness in over 98.39 % of scheduling problems. Although the influence of these approaches is different, it is therefore not really significant. Each of the sorting approaches has heavy disadvantages leading sometimes to gigantic inaccuracies, although they occur rarely. When the inaccuracy occurs to be more than 30 %, this implies that 3 to 9 jobs are scheduled. Conclusions. Unlike the case when jobs do not have their priority weights, it is impossible to select the best sorting approach for the case with job priority weights. Instead, a hyper-heuristic comprising the sorting approaches (i. e., the whole group, where each sorting is applied) may be constructed. If a parallelization can be used to process two or even four sorting routines simultaneously, the computation time will not be significantly affected.

scholarly journals TIGHT-TARDY PROGRESSIVE IDLING-FREE 1-MACHINE PREEMPTIVE SCHEDULING WITH JOB PRIORITY WEIGHTS BY HEURISTIC’S EFFICIENT JOB ORDER INPUT

2021 ◽  
Author(s):  
Vadim V. Romanuke
Keyword(s):  
Job Scheduling ◽  
Computational Study ◽  
Computation Time ◽  
Equal Length ◽  
Computational Time ◽  
Release Dates ◽  
Preemptive Scheduling ◽  
Scheduling Problems ◽  
Due Dates ◽  
Priority Weights

Background. In setting a problem of minimizing total weighted tardiness by the heuristic based on remaining available and processing periods, there are two opposite ways to input the data: the job release dates are given in either ascending or descending order. It was recently ascertained that scheduling a few equal-length jobs is expectedly faster by ascending order, whereas scheduling 30 to 70 equal-length jobs is 1.5 % to 2.5 % faster by descending order. For the number of equal-length jobs between roughly 90 and 250, the ascending job order again results in shorter computation times. In the case when the jobs have different lengths, the significance of the job order input is much lower. On average, the descending job order input gives a tiny advantage in computation time. This advantage decreases as the number of jobs increases. Objective. The goal is to ascertain whether the job order input is significant in scheduling by using the heuristic for the case when the jobs have different lengths with job priority weights. Job order efficiency will be studied on tight-tardy progressive idling-free 1-machine preemptive scheduling. Methods. To achieve the said goal, a computational study is carried out with a purpose to estimate the averaged computation time for both ascending and descending orders of job release dates. First, the computation time for the ascending job order input is estimated for a series of job scheduling problems. Then, in each instance of this series, job lengths, priority weights, release dates, and due dates are reversed making thus the respective instance for the descending job order input, for which computation time is estimated as well. Results. The significance of the job order input is much lower than that for the case of jobs without priorities. With assigning the job priority weights, the job order input becomes further “dithered”, adding randomly scattered priority weights to randomly scattered job lengths and partially randomized due dates. On average, the descending job order input is believed to give a tiny advantage in computation time in scheduling up to 100 jobs. However, this advantage, if any (being tinier than that in the case of random job lengths without priorities), quickly vanishes as the number of jobs increases. Conclusions. It is better to compose job scheduling problems which would be closer to the case with equal-length jobs without priorities, where the saved computational time can be counted in hours. Even if the job lengths and priority weights are scattered, it is recommended to artificially “flatten” them. When artificial manipulations over job processing periods and job priority weights are impossible, it is recommended to use the descending job order input in scheduling up to 100 jobs, and either job order input in scheduling more than 100 jobs, although substantial benefits are not expected in this case.

scholarly journals A SORTING IMPROVEMENT IN THE HEURISTIC BASED ON REMAINING AVAILABLE AND PROCESSING PERIODS TO MINIMIZE TOTAL TARDINESS IN PROGRESSIVE IDLING-FREE 1-MACHINE PREEMPTIVE SCHEDULING

2021 ◽  
Author(s):  
Vadim V. Romanuke
Keyword(s):  
Negative Impact ◽  
Computational Study ◽  
Total Tardiness ◽  
Release Dates ◽  
Preemptive Scheduling ◽  
Scheduling Problem ◽  
Scheduling Problems ◽  
Due Dates ◽  
Worst Case ◽  
Priority Weights

Background. In preemptive job scheduling, which is a part of the flow-shop sequencing tasks, one of the most crucial goals is to obtain a schedule whose total tardiness would be minimal. Total tardiness minimization is commonly reduced to solving a combinatorial problem which becomes practically intractable as the number of jobs and the numbers of their processing periods increase. To cope with this challenge, heuristics are used. A heuristic, in which the decisive ratio is the reciprocal of the maximum of a pair of the remaining processing period and remaining available period, is closely the best one. However, the heuristic may produce schedules of a few jobs whose total tardiness is 25 % greater than the minimum or even worse. Therefore, this heuristic needs a corrective branch which would further try to minimize total tardiness under certain conditions. Objective. The goal is to ascertain what is to be corrected in the heuristic so that the total tardiness value could be obtained lesser. The heuristic will be applied to tight-tardy progressive idling-free 1-machine preemptive scheduling, where the release dates are given in ascending order starting from 1 to the number of jobs, and the due dates are tightly set after the release dates. In this scheduling problem, the inaccuracy of finding the minimal total tardiness has the strongest negative impact, so this is almost the worst case, which defines the accuracy limit of the heuristic and positively serves just as the principle of minimax guaranteeing decreasing losses in the worst conditions. Methods. The heuristic sorts maximal decisive ratios by release dates, where the scheduling preference is given to the earliest job. To achieve the said goal, three other sorting approaches are presented and a computational study is carried out with applying each of the four heuristic approaches to minimize total tardiness. For this, two series of 266000 and 1064000 scheduling problems are generated. Results. The earliest-job sorting ensures a heuristically minimal total tardiness value in more than 97.6 % of scheduling problems, but it fails to minimize total tardiness in no less than 2.2 % of the cases. Nevertheless, a sorting approach with minimizing remaining processing periods produces a heuristically minimal total tardiness for almost any scheduling problem. If an exception occurs, this sorting approach “loses” to the other sorting approaches very little. Moreover, the exceptions are quite rare as it has been registered just a one scheduling problem (out of 31914 cases followed by a sole “win” of a heuristic version) whose minimal total tardiness is achieved by the earliest-job sorting. Conclusions. The best heuristic version is that one which uses the sorting approach with minimizing remaining processing periods. This, however, is confirmed only for the case where jobs do not have any priorities. The case when jobs have their priority weights is to be yet analyzed.

scholarly journals Infinity Substitute in Finding Exact Minimum of Total Weighted Tardiness in Tight-Tardy Progressive 1-machine Scheduling by Idling-free Preemptions

2021 ◽  
Vol 9 (4) ◽  
pp. 820-837
Author(s):  
Vadim Romanuke
Keyword(s):  
Positive Integer ◽  
Computation Time ◽  
Total Weighted Tardiness ◽  
Decision Variable ◽  
Scheduling Problems ◽  
Problem Model ◽  
Weighted Tardiness ◽  
Integer Linear Programming Problem ◽  
Priority Weights ◽  
Open Question

A job schedule ensuring the exact minimum of total weighted tardiness can be found with the respective integer linear programming problem model, in which the infinity showing that the respective states are either forbidden or impossible is substituted with a sufficiently great positive integer. An open question is whether the substitute can be selected so that the computation time would be decreased. Thus, it is ascertained that, whichever job lengths and its priority weights are, substituting the infinity with just “a sufficiently great positive integer” is never recommended. To decrease the computation time on average, it is strongly recommended to select the infinity substitute as multiple maximum over finite decision variable weights in the exact model. It is sufficient to take 2 to 5 such maxima as the infinity substitute. However, the shortened computation time is not guaranteed for solving a single or few scheduling problems. It is only an expected benefit, which builds up as a few hundred scheduling problems are solved at least.

scholarly journals Bi-criteria parallel batch machine scheduling to minimize total weighted tardiness and electricity cost

2020 ◽  
Vol 90 (9) ◽  
pp. 1345-1381
Author(s):  
Jens Rocholl ◽  
Lars Mönch ◽  
John Fowler
Keyword(s):  
Genetic Algorithms ◽  
Pareto Front ◽  
Total Weighted Tardiness ◽  
Mixed Integer ◽  
Due Dates ◽  
Semiconductor Wafer ◽  
Weighted Tardiness ◽  
Electricity Cost ◽  
Problem Instances ◽  
Special Case

Abstract A bi-criteria scheduling problem for parallel identical batch processing machines in semiconductor wafer fabrication facilities is studied. Only jobs belonging to the same family can be batched together. The performance measures are the total weighted tardiness and the electricity cost where a time-of-use (TOU) tariff is assumed. Unequal ready times of the jobs and non-identical job sizes are considered. A mixed integer linear program (MILP) is formulated. We analyze the special case where all jobs have the same size, all due dates are zero, and the jobs are available at time zero. Properties of Pareto-optimal schedules for this special case are stated. They lead to a more tractable MILP. We design three heuristics based on grouping genetic algorithms that are embedded into a non-dominated sorting genetic algorithm II framework. Three solution representations are studied that allow for choosing start times of the batches to take into account the energy consumption. We discuss a heuristic that improves a given near-to-optimal Pareto front. Computational experiments are conducted based on randomly generated problem instances. The $$ \varepsilon $$ ε -constraint method is used for both MILP formulations to determine the true Pareto front. For large-sized problem instances, we apply the genetic algorithms (GAs). Some of the GAs provide high-quality solutions.

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