Shortest path to nonpreemptive schedules of unit-time jobs on two identical parallel machines with minimum total completion time (original) (raw)
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Ideal Two-Machine Schedules of Jobs with Unit-Execution-Time Operations: An Extended Abstract
Citeseer
Ideal schedules minimize both maximum completion time and total completion time of jobs. This paper presents polynomial-time algorithms finding ideal nonpreemptive and preemptive two-machine schedules of jobs with arbitrary release dates and precedence constraints and unit-execution-time operations on identical parallel machines, in a flow shop and an open shop. Heretofore, the complexity status of these problems has been unknown.
Single-Machine Scheduling of the Indivisible Multi-Operation Jobs
South African Journal of Industrial Engineering
This paper considers a single-machine scheduling problem of multioperation jobs where each job consists of several operations processed contiguously, rather than being intermingled with the operations of different jobs. That is, the jobs are indivisible. A sequence-independent setup is required if the machine switches from one operation to another. However, no setup is necessary before the first operation of a job if this first operation is the same as the last operation of the immediately previous job. A job is complete when all of its operations have been processed. We investigate the problem for two cases. Makespan, which is the time needed to complete all jobs, is minimised in the first case; whereas the total completion time, which is the sum of the job completion times, is minimised in the second case. We show that the makespan problem is solvable in polynomial time. For the problem of minimising total completion time, we develop a mixed integer linear programming (MILP) model, which is capable of solving small and medium-sized problem instances optimally, and obtain a very small gap between the solution found and the best possible solution for the unsolved large-sized problem instances. OPSOMMING Hierdie artikel ondersoek ʼn enkel-masjien skeduleringsprobleem van meervoudige operasie take waar elke taak uit verskeie operasies bestaan wat kontinu verwerk word eerder as om gemeng te wees met die operasies van ander take. Die take is dus onverdeelbaar. ʼn Volgorde-onafhanklike opstelling word vereis as die masjien wissel van operasie na ʼn ander. Geen opstelling is egter nodig voor die eerste operasie van ʼn taak indien die eerste operasie van die nuwe taak dieselfde is as die vorige taak se laaste operasie nie. Die tyd wat dit neem om elkeen van die take te voltooi is eerstens geminimeer. Daarna is die totale voltooi tyd (die som van al die taak voltooi tye) geminimeer. Daar word gewys dat die tyd wat dit neem om elke taak te voltooi oplosbaar is in polinome tyd. Om die totale voltooi tyd te minimeer is ʼn gemengde heelgetal lineêre programmeringmodel ontwikkel wat daartoe in staat is om klein-en mediumgrootte probleme optimaal op te los. Die model behaal ook ʼn baie klein verskil tussen die geïdentifiseerde oplossing en die beste moontlike oplossing vir groot probleemgevalle. 1 INTRODUCTION Scheduling is the allocation of resources to complete a given set of tasks over time. In manufacturing systems, resources and tasks are usually referred to as machines and jobs respectively [1]. Scheduling is a complex issue, because determining the processing sequence of jobs on a machine is affected by many factors, such as processing times, setup times, due dates, precedence relations among jobs, etc. Generally these factors cannot be handled without a systematic approach. Researchers have investigated scheduling problems to satisfy the need for a systematic approach since the 1950s. In most traditional scheduling problems, it is assumed that there is only one group
An efficient algorithm for finding ideal schedules
Acta Informatica, 2012
We study the problem of scheduling unit execution time (UET) jobs with release dates and precedence constraints on two identical processors. We say that a schedule is ideal if it minimizes both maximum and total completion time simultaneously. We give an instance of the problem where the min-max completion time is exceeded in every preemptive schedule that minimizes total completion time for that instance, even if the precedence constraints form an intree. This proves that ideal schedules do not exist in general when preemptions are allowed. On the other hand, we prove that, when preemptions are not allowed, then ideal schedules do exist for general precedence constraints, and we provide an algorithm for finding ideal schedules in O(n 3 ) time, where n is the number of jobs. In finding such ideal schedules we resolve a conjecture of Baptiste and Timkovsky [1]. Further, our algorithm for finding min-max completion-time schedules requires only O(n 3 ) time, while the most efficient solution to date has required O(n 9 ) time.
On the geometry, preemptions and complexity of multiprocessor and shop scheduling
Annals of Operations Research, 2008
In this paper we study multiprocessor and open shop scheduling problems from several points of view. We explore a tight dependence of the polynomial solvability/intractability on the number of allowed preemptions. For an exhaustive interrelation, we address the geometry of problems by means of a novel graphical representation. We use the so-called preemption and machine-dependency graphs for preemptive multiprocessor and shop scheduling problems, respectively. In a natural manner, we call a scheduling problem acyclic if the corresponding graph is acyclic. There is a substantial interrelation between the structure of these graphs and the complexity of the problems. Acyclic scheduling problems are quite restrictive; at the same time, many of them still remain NP-hard. We believe that an exhaustive study of acyclic scheduling problems can lead to a better understanding and give a better insight of general scheduling problems.
A Polynomial Algorithm for Sequencing Jobs with Release Dates and Delivery Times on Uniform Machines
2021
We consider the problem of scheduling n jobs with identical processing times and given 1 release as well as delivery times on m uniform machines. The goal is to minimize the makespan, 2 i.e., the maximum full completion time of any job. This problem is well-known to have an open 3 complexity status even if the number of jobs is fixed. We present a polynomial-time algorithm for 4 the problem which is based on the earlier introduced algorithmic framework blesscmore (“branch 5 less and cut more”). We extend the analysis of the so-called behavior alternatives developed earlier 6 for the version of the problem with identical parallel machines and show how the earlier used 7 technique for identical machines can be extended to the uniform machine environment if a special 8 condition on the job parameters is imposed. The time complexity of the proposed algorithm is 9 O(γm2n log n), where γ can be either n or the maximum job delivery time qmax. This complexity 10 can even be reduced further ...
2011
SHOP PROBLEMS IN SCHEDULING By James Andro-Vasko Dr. Wolfgang Bein, Examination Committee Chair Professor of Computer Science University of Nevada, Las Vegas The shop problems in scheduling will be discussed in this thesis. The ones I’ll be discussing will be the flow shop, open shop, and job shop. The general idea of shop problems is that you’re given a set of jobs and a set of machines. Each job is predeterminely broken into parts and there are rules to how each part is executed on a machine. In this thesis, several shop problems and their algorithms will be introduced that I have researched. There are several examples and counter examples that I have constructed. Also I will discuss how an arbitrary problem that can be solved polynomially can be changed so that there are no polynomial algorithms that can solve it. Scheduling is used in computer science in the area of operating systems and it can be used in engineering. This is an important for a company when they want to run seve...