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 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.

Minimizing Total Earliness and Total Tardiness on Single Machine with Release Dates

2011 ◽  
Vol 5 ◽  
pp. 30-43
Author(s):  
Elkanah Oyetunji ◽  
Ayodeji E. Oluleye
Keyword(s):  
Single Machine ◽  
Single Machine Scheduling ◽  
Total Tardiness ◽  
Release Dates ◽  
Scheduling Problem ◽  
Scheduling Problems ◽  
Bicriteria Scheduling ◽  
Linear Composite ◽  
Effectiveness And Efficiency ◽  
Machine Scheduling Problems

This paper considers the bicriteria scheduling problem of minimizing the total earliness and the total tardiness on a single machine with release dates. In view of the fact that the problem has been characterized as NP-Hard, we propose two approximation algorithms (labeled as ETA1 and ETA2) for solving the problem. The proposed algorithms were compared with the MA heuristic selected from the literature. The two criteria (the total earliness and the total tardiness) were aggregated together into a linear composite objective function (LCOF). The performances of the algorithms were evaluated based on both effectiveness and efficiency. The algorithms were tested on a set of 1200 randomly generated single machine scheduling problems. Experimental results show that both the ETA1 and ETA2 algorithms outperformed (in terms of effectiveness and efficiency) the MA heuristic under all the considered problem sizes. Also, the ETA1 algorithm outperformed the ETA2 algorithm when the number of jobs (n) ranges between 20 and 500.

Computational Study of N-Job M-Machine Flow Shop Scheduling Problems: SPT, EDD, NEH, NEH-EDD, and Modified-NEH Algorithms

2017 ◽  
Vol 16 (04) ◽  
pp. 375-384
Author(s):  
Dwi Agustina Kurniawati ◽  
Yoga Isnaini Nugroho
Keyword(s):  
Flow Shop ◽  
Computational Study ◽  
Total Tardiness ◽  
Flow Shop Scheduling ◽  
Small Data ◽  
Data Sets ◽  
Scheduling Problems ◽  
Due Dates ◽  
Shop Scheduling ◽  
Small Data Sets

This paper discusses about the flow shop scheduling problems using shortest processing time, earliest due date (EDD), Nawaz, Enscore, and Ham (NEH), NEH-EDD, and modified-NEH methods. The objective of this research is to determine the performance of these methods in minimizing makespan and total tardiness. Processing times and due dates were randomly generated, and computational studies were performed in Microsoft Visual Basic 6.0. The experiments are performed for small and medium data sets. Efficiency index, relative error, and run time measure the performance of each method. Experimental results showed that NEH has the best performance in minimizing the makespan in both data sets; these are 53.35 time unit for small data sets and 83.803 time unit for medium data sets. NEH-EDD has the best performance in minimizing total tardiness with 9.37 time unit for small data sets and 231.02 time unit for medium data sets. Modified-NEH, as the proposed method for minimizing makespan and total tardiness at the same time, has good enough result. For minimizing the makespan, modified-NEH results in 57.15 time unit for small data sets and 88.107 time unit for medium data sets. For minimizing total tardiness, the modified-NEH results in 14.21 time unit for small data sets and 246.57 time unit for medium sets.

scholarly journals Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

Algorithms ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 246
Author(s):  
Yuri N. Sotskov ◽  
Еvangelina I. Mihova
Keyword(s):  
Graph Coloring ◽  
Precedence Constraints ◽  
Scheduling Problem ◽  
Scheduling Problems ◽  
Due Dates ◽  
Mixed Graph ◽  
Coloring Problem ◽  
Release Times ◽  
Schedule Length ◽  
Graph Coloring Problem

This article extends the scheduling problem with dedicated processors, unit-time tasks, and minimizing maximal lateness for integer due dates to the scheduling problem, where along with precedence constraints given on the set of the multiprocessor tasks, a subset of tasks must be processed simultaneously. Contrary to a classical shop-scheduling problem, several processors must fulfill a multiprocessor task. Furthermore, two types of the precedence constraints may be given on the task set . We prove that the extended scheduling problem with integer release times of the jobs to minimize schedule length may be solved as an optimal mixed graph coloring problem that consists of the assignment of a minimal number of colors (positive integers) to the vertices of the mixed graph such that, if two vertices and are joined by the edge , their colors have to be different. Further, if two vertices and are joined by the arc , the color of vertex has to be no greater than the color of vertex . We prove two theorems, which imply that most analytical results proved so far for optimal colorings of the mixed graphs , have analogous results, which are valid for the extended scheduling problems to minimize the schedule length or maximal lateness, and vice versa.

scholarly journals Mathematical models for a batch scheduling problem to minimize earliness and tardiness

2018 ◽  
Vol 11 (3) ◽  
pp. 390 ◽  
Author(s):  
Basar Ogun ◽  
Çigdem Alabas-Uslu
Keyword(s):  
Mathematical Models ◽  
Real World ◽  
Lean Manufacturing ◽  
Programming Model ◽  
Computational Study ◽  
Batch Scheduling ◽  
Scheduling Problem ◽  
Scheduling Problems ◽  
Earliness And Tardiness ◽  
Customer Orders

Purpose: Today’s manufacturing facilities are challenged by highly customized products and just in time manufacturing and delivery of these products. In this study, a batch scheduling problem is addressed to provide on-time completion of customer orders in the environment of lean manufacturing. The problem is to optimize partitioning of product components into batches and scheduling of the resulting batches where each customer order is received as a set of products made of various components.Design/methodology/approach: Three different mathematical models for minimization of total earliness and tardiness of customer orders are developed to provide on-time completion of customer orders and also, to avoid from inventory of final products. The first model is a non-linear integer programming model while the second is a linearized version of the first. Finally, to solve larger sized instances of the problem, an alternative linear integer model is presented.Findings: Computational study using a suit set of test instances showed that the alternative linear integer model is able to solve all test instances in varying sizes within quite shorter computer times comparing to the other two models. It was also showed that the alternative model can solve moderate sized real-world problems.Originality/value: The problem under study differentiates from existing batch scheduling problems in the literature since it includes new circumstances which may arise in real-world applications. This research, also, contributes the literature of batch scheduling problem by presenting new optimization models.

scholarly journals JOB ORDER INPUT FOR EFFICIENT EXACT MINIMIZATION OF TOTAL TARDINESS IN TIGHT-TARDY PROGRESSIVE SINGLE MACHINE SCHEDULING WITH IDLING-FREE PREEMPTIONS

2020 ◽  
Vol 1 (1) ◽  
pp. 19-36
Author(s):  
V.V. Romanuke ◽  
Keyword(s):  
Linear Programming ◽  
Single Machine ◽  
Job Scheduling ◽  
Computation Time ◽  
Single Machine Scheduling ◽  
Total Tardiness ◽  
Release Dates ◽  
Scheduling Problem ◽  
Speed Up ◽  
Job Scheduling Problem

Abstract. A schedule ensuring the exactly minimal total tardiness can be found with the respective integer linear programming problem. An open question is whether the exact schedule computation time changes if the job release dates are input into the model in reverse order. The goal is to ascertain whether the job order in tight-tardy progressive single machine scheduling with idling-free preemptions influences the speed of computing the exact solution. The Boolean linear programming model provided for finding schedules with the minimal total tardiness is used. To achieve the said goal, a computational study is carried out with the purpose of estimating the averaged computation time for both ascending and descending orders of job release dates. Instances of the job scheduling problem are generated so that schedules which can be obtained trivially, without the exact model, are excluded. As in the case of equal-length jobs, it has been ascertained that the job order really influences the speed of computing schedules whose total tardiness is minimal. Scheduling two to five jobs is executed on average faster by the descending job order input, where 1 to 3 % speed-up is expected. Further increment of the number of jobs to be scheduled cannot guarantee any speed-up even on average. This result is similar to that in the case of equal-length jobs, but there is no regularity in such an efficient job order input. Without any assurance for a single job scheduling problem, the efficient exact minimization of total tardiness by the descending job order input must be treated as on average only.

scholarly journals The minimization of exact total weighted completion time in the preemptive scheduling problem by subsequent length-equal job importance growth

2018 ◽  
Keyword(s):  
Completion Time ◽  
Optimal Schedule ◽  
Total Weighted Completion Time ◽  
Preemptive Scheduling ◽  
Scheduling Problem ◽  
Exact Model ◽  
Priority Weights ◽  
Weighted Completion Time

For the preemptive scheduling problem in case of subsequent job importance growth, it is studied whether the optimal schedule might be found faster within an exact model. It is ascertained that when the number of jobs up to six (except for the case of four jobs) and there is no randomness in problem forming, a little advantage of weight-descending job order exists only on average. As the number of jobs increases, the advantage of either weight-descending or weight-ascending job order becomes more certain. When priority weights are formed randomly, weight-descending job order is expected to be faster than weight-ascending.

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