scholarly journals Mixed Graph Colorings: A Historical Review

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 385 ◽  
Author(s):  
Yuri N. Sotskov
Keyword(s):  
Historical Review ◽  
Scheduling Problem ◽  
Scheduling Problems ◽  
Graph Colorings ◽  
Mixed Graph ◽  
Criterion A ◽  
Complexity Results ◽  
Recent Developments ◽  
Time Scheduling ◽  
Makespan Criterion

This paper presents a historical review and recent developments in mixed graph colorings in the light of scheduling problems with the makespan criterion. A mixed graph contains both a set of arcs and a set of edges. Two types of colorings of the vertices of the mixed graph and one coloring of the arcs and edges of the mixed graph have been considered in the literature. The unit-time scheduling problem with the makespan criterion may be interpreted as an optimal coloring of the vertices of a mixed graph, where the number of used colors is minimum. Complexity results for optimal colorings of the mixed graph are systematized. The published algorithms for finding optimal mixed graph colorings are briefly surveyed. Two new colorings of a mixed graph are introduced.

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 A Study on the Enhanced Best Performance Algorithm for the Just-in-Time Scheduling Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Sivashan Chetty ◽  
Aderemi O. Adewumi
Keyword(s):  
Local Search ◽  
Current Problem ◽  
Initial Study ◽  
Just In Time ◽  
Scheduling Problem ◽  
Metaheuristic Algorithm ◽  
Scheduling Problems ◽  
Specific Point ◽  
Time Scheduling ◽  
Problem Setting

The Just-In-Time (JIT) scheduling problem is an important subject of study. It essentially constitutes the problem of scheduling critical business resources in an attempt to optimize given business objectives. This problem is NP-Hard in nature, hence requiring efficient solution techniques. To solve the JIT scheduling problem presented in this study, a new local search metaheuristic algorithm, namely, the enhanced Best Performance Algorithm (eBPA), is introduced. This is part of the initial study of the algorithm for scheduling problems. The current problem setting is the allocation of a large number of jobs required to be scheduled on multiple and identical machines which run in parallel. The due date of a job is characterized by a window frame of time, rather than a specific point in time. The performance of the eBPA is compared against Tabu Search (TS) and Simulated Annealing (SA). SA and TS are well-known local search metaheuristic algorithms. The results show the potential of the eBPA as a metaheuristic algorithm.

Scheduling Problems and Mixed Graph Colorings

Optimization ◽  
2002 ◽  
Vol 51 (3) ◽  
pp. 597-624 ◽  
Author(s):  
Yuri N. Sotskov ◽  
Vjacheslav S. Tanaev ◽  
Frank Werner
Keyword(s):  
Scheduling Problems ◽  
Graph Colorings ◽  
Mixed Graph

Multicriteria Flow-Shop Scheduling Problem

2010 ◽  
pp. 211-233 ◽  
Author(s):  
Ethel Mokotoff
Keyword(s):  
Combinatorial Optimization ◽  
Flow Shop ◽  
Real Life ◽  
Combinatorial Problems ◽  
Flow Shop Scheduling ◽  
Scheduling Problem ◽  
Scheduling Problems ◽  
Shop Scheduling ◽  
Recent Developments ◽  
Metaheuristic Techniques

Quality is, in real-life, a multidimensional notion. A schedule is described and valued on the basis of a number of criteria, for example: makespan, work-in-process inventories, idle times, observance of due dates, etc. An appropriate schedule cannot be obtained unless one observes the whole set of important criteria. The multidimensional nature of the scheduling problems leads us to the area of Multicriteria Optmization. Thus considering combinatorial problems with more than one criterion is more relevant in the context of real-life scheduling problems. Research in this important field has been scarce when compared to research in single-criterion scheduling. The proliferation of metaheuristic techniques has encouraged researchers to apply them to combinatorial optimization problems. The chapter presents a review regarding multicriteria flow-shop scheduling problem, focusing on Multi-Objective Combinatorial Optimization theory, including recent developments considering more than one optimization criterion, followed by a summary discussion on research directions.

scholarly journals Mixed graph colouring as scheduling multi-processor tasks with equal processing times

2021 ◽  
pp. 67-81
Author(s):  
Yuri N. Sotskov
Keyword(s):  
Optimal Schedule ◽  
Graph Colouring ◽  
Scheduling Problem ◽  
Scheduling Problems ◽  
Mixed Graph ◽  
Partially Ordered ◽  
Processing Times ◽  
Dedicated Machines ◽  
Mixed Graphs ◽  
Equal Processing Times

A problem of scheduling partially ordered unit-time tasks processed on dedicated machines is formulated as a mixed graph colouring problem, i. e., as an assignment of integers (colours) {1, 2, …, t} to the vertices (tasks) V {ν1, ν2, …, νn}, of the mixed graph G = (V, A, E) such that if vertices vp and vq are joined by an edge [νp, νq] ∈ E their colours have to be different. Further, if two vertices νp and νq are joined by an arc (νi, νj) ∈ A the colour of vertex νi has to be no greater than the colour of vertex νj. We prove that an optimal colouring of a mixed graph G = (V, A, E) is equivalent to the scheduling problem GcMPT|pi = 1|Cmax of finding an optimal schedule for partially ordered multi-processor tasks with unit (equal) processing times. Contrary to classical shop-scheduling problems, several dedicated machines are required to process an individual task in the scheduling problem GcMPT|pi = 1|Cmax. Moreover, along with precedence constraints given on the set V {ν1, ν2, …, νn}, it is required that a subset of tasks must be processed simultaneously. Due to the theorems proved in this article, most analytical results that have been proved for the scheduling problems GcMPT |pi = 1|Cmax so far, have analogous results for optimal colourings of the mixed graphs G = (V, A, E), and vice versa.

Systematische Navigation für die Reihenfolgeplanung/Systematic solution space navigation for job shop scheduling problems

2020 ◽  
Vol 110 (07-08) ◽  
pp. 563-571
Author(s):  
Edzard Weber ◽  
Eduard Schenke ◽  
Luka Dorotic ◽  
Norbert Gronau
Keyword(s):  
Job Shop ◽  
Job Shop Scheduling ◽  
Solution Space ◽  
Comparative Test ◽  
Scheduling Problem ◽  
Scheduling Problems ◽  
Job Shop Scheduling Problem ◽  
Shop Scheduling ◽  
Space Navigation ◽  
Systematic Solution

Dieser Beitrag stellt einen Algorithmus für das Job-shop-Scheduling-Problem vor, welcher den Lösungsraum indexiert und eine systematische Navigation zur Lösungssuche durchführt. Durch diese problemadäquate Aufbereitung wird der Lösungsraum nach bestimmten Kriterien vorzustrukturiert. Diese Problemrepräsentation wird formal beschrieben, sodass ihre Anwendung als Grundlage für ein navigationsorientiertes Suchverfahren dienen kann. Ein vergleichender Test mit anderen Optimierungsansätzen zeigt die Effizienz dieser Lösungsraumnavigation.   This paper presents an algorithm for the job-shop scheduling problem indexing the solution space and performing systematic navigation to find good solutions. By this problem-adequate preparation of the solution space, the solution space is pre-structured according to certain criteria. This problem representation is formally described so that its application can serve as a basis for a navigation-oriented search procedure. A comparative test with other optimization approaches shows the efficiency of this solution space navigation.

scholarly journals Realtime production scheduling and control systems in smart industry

Impact ◽  
2020 ◽  
Vol 2020 (8) ◽  
pp. 60-61
Author(s):  
Wei Weng
Keyword(s):  
Control Systems ◽  
Real Time ◽  
Production Scheduling ◽  
Production Control ◽  
Optimal Solution ◽  
Scheduling Problem ◽  
Optimal Solutions ◽  
Scheduling Problems ◽  
Smart Industry ◽  
And Control

For a production system, 'scheduling' aims to find out which machine/worker processes which job at what time to produce the best result for user-set objectives, such as minimising the total cost. Finding the optimal solution to a large scheduling problem, however, is extremely time consuming due to the high complexity. To reduce this time to one instance, Dr Wei Weng, from the Institute of Liberal Arts and Science, Kanazawa University in Japan, is leading research projects on developing online scheduling and control systems that provide near-optimal solutions in real time, even for large production systems. In her system, a large scheduling problem will be solved as distributed small problems and information of jobs and machines is collected online to provide results instantly. This will bring two big changes: 1. Large scheduling problems, for which it tends to take days to reach the optimal solution, will be solved instantly by reaching near-optimal solutions; 2. Rescheduling, which is still difficult to be made in real time by optimization algorithms, will be completed instantly in case some urgent jobs arrive or some scheduled jobs need to be changed or cancelled during production. The projects have great potential in raising efficiency of scheduling and production control in future smart industry and enabling achieving lower costs, higher productivity and better customer service.

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