scholarly journals Polynomial Algorithm for Constructing Pareto-Optimal Schedules for Problem 1∣rj∣Lmax,Cmax

2020 ◽  
Author(s):  
Alexander A. Lazarev ◽  
Nikolay Pravdivets
Keyword(s):  
Completion Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Single Machine Scheduling ◽  
Maximum Lateness ◽  
Pareto Optimal ◽  
Due Dates ◽  
Processing Times ◽  
Pareto Optimal Set ◽  
Maximum Completion Time

In this chapter, we consider the single machine scheduling problem with given release dates, processing times, and due dates with two objective functions. The first one is to minimize the maximum lateness, that is, maximum difference between each job due date and its actual completion time. The second one is to minimize the maximum completion time, that is, to complete all the jobs as soon as possible. The problem is NP-hard in the strong sense. We provide a polynomial time algorithm for constructing a Pareto-optimal set of schedules on criteria of maximum lateness and maximum completion time, that is, problem 1 ∣ r j ∣ L max , C max , for the subcase of the problem: d 1 ≤ d 2 ≤ … ≤ d n ; d 1 − r 1 − p 1 ≥ d 2 − r 2 − p 2 ≥ … ≥ d n − r n − p n .

Single Machine Scheduling with Discretely Compressible Processing Times

2013 ◽  
Vol 787 ◽  
pp. 1020-1024
Author(s):  
Shu Xia Zhang ◽  
Yu Zhong Zhang
Keyword(s):  
Dynamic Programming ◽  
Single Machine ◽  
Processing Time ◽  
Machine Scheduling ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Single Machine Scheduling ◽  
Scheduling Problem ◽  
Processing Times ◽  
Release Times

In this paper, we address the single machine scheduling problem with discretely compressible processing times, where processing any job with a compressed processing time incurs a corresponding compression cost. We consider the following problem: scheduling with discretely compressible processing times to minimize makespan with the constraint of total compression cost. Jobs may have different release times. We design a pseudo-polynomial time algorithm by approach of dynamic programming and an FPTAS.

scholarly journals Optimal algorithms for scheduling under time-of-use tariffs

2021 ◽  
Author(s):  
Lin Chen ◽  
Nicole Megow ◽  
Roman Rischke ◽  
Leen Stougie ◽  
José Verschae
Keyword(s):  
Polynomial Time ◽  
Single Machine ◽  
Completion Time ◽  
Optimal Algorithm ◽  
Optimal Schedule ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Single Machine Scheduling ◽  
Total Weighted Completion Time ◽  
Scheduling Problems

AbstractWe consider a natural generalization of classical scheduling problems to a setting in which using a time unit for processing a job causes some time-dependent cost, the time-of-use tariff, which must be paid in addition to the standard scheduling cost. We focus on preemptive single-machine scheduling and two classical scheduling cost functions, the sum of (weighted) completion times and the maximum completion time, that is, the makespan. While these problems are easy to solve in the classical scheduling setting, they are considerably more complex when time-of-use tariffs must be considered. We contribute optimal polynomial-time algorithms and best possible approximation algorithms. For the problem of minimizing the total (weighted) completion time on a single machine, we present a polynomial-time algorithm that computes for any given sequence of jobs an optimal schedule, i.e., the optimal set of time slots to be used for preemptively scheduling jobs according to the given sequence. This result is based on dynamic programming using a subtle analysis of the structure of optimal solutions and a potential function argument. With this algorithm, we solve the unweighted problem optimally in polynomial time. For the more general problem, in which jobs may have individual weights, we develop a polynomial-time approximation scheme (PTAS) based on a dual scheduling approach introduced for scheduling on a machine of varying speed. As the weighted problem is strongly NP-hard, our PTAS is the best possible approximation we can hope for. For preemptive scheduling to minimize the makespan, we show that there is a comparably simple optimal algorithm with polynomial running time. This is true even in a certain generalized model with unrelated machines.

scholarly journals Two-Agent Preemptive Pareto-Scheduling to Minimize Late Work and Other Criteria

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1517
Author(s):  
Ruyan He ◽  
Jinjiang Yuan
Keyword(s):  
Objective Function ◽  
Polynomial Time ◽  
Single Machine ◽  
Completion Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Pareto Optimal ◽  
Scheduling Problems ◽  
Trade Off ◽  
Late Work

In this paper, we consider three preemptive Pareto-scheduling problems with two competing agents on a single machine. In each problem, the objective function of agent A is the total completion time, the maximum lateness, or the total late work while the objective function of agent B is the total late work. For each problem, we provide a polynomial-time algorithm to characterize the trade-off curve of all Pareto-optimal points.

scholarly journals Single-Machine Scheduling with Learning Effects and Maintenance: A Methodological Note on Some Polynomial-Time Solvable Cases

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Kuo-Ching Ying ◽  
Chung-Cheng Lu ◽  
Shih-Wei Lin ◽  
Jie-Ning Chen
Keyword(s):  
Polynomial Time ◽  
Single Machine ◽  
Machine Scheduling ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Single Machine Scheduling ◽  
Learning Effects ◽  
Scheduling Problems ◽  
Maintenance Activity ◽  
Processing Times

This work addresses four single-machine scheduling problems (SMSPs) with learning effects and variable maintenance activity. The processing times of the jobs are simultaneously determined by a decreasing function of their corresponding scheduled positions and the sum of the processing times of the already processed jobs. Maintenance activity must start before a deadline and its duration increases with the starting time of the maintenance activity. This work proposes a polynomial-time algorithm for optimally solving two SMSPs to minimize the total completion time and the total tardiness with a common due date.

scholarly journals Propose a Polynomial Time Algorithm for Total Completion Time Objective

2021 ◽  
Vol 6 (3) ◽  
pp. 932-943
Author(s):  
Yucel Ozturkoglu ◽  
Omer Ozturkoglu
Keyword(s):  
Single Machine ◽  
Completion Time ◽  
Polynomial Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Single Machine Scheduling ◽  
Total Completion Time ◽  
Maintenance Activity ◽  
Automated Production ◽  
Maintenance Activities

In this study, we integrate deteriorate jobs with repair&maintenance activity on a single machine scheduling subject to total completion time. This work has more than one motivation. First, jobs are assigned to machines in an automated production line. Later, to schedule the maintenance activities, if needed, to prevent machinery from breaking down later. There are some important mathematical models to solve this combination. However, due to the complexity of the problem which is Np-hard, a polynomial algorithm should be needed for solving large problems. Therefore, this article introduces several polnomial algorithms to determine the order of things best. With using these algorithms, it will be possible to determine where to assign to the schedule, taking into account the number of maintenance activities required and their optimum total completion time.

A STOCHASTIC BATCHING AND SCHEDULING PROBLEM

2001 ◽  
Vol 15 (4) ◽  
pp. 465-479 ◽  
Author(s):  
Ger Koole ◽  
Rhonda Righter
Keyword(s):  
Optimal Policy ◽  
Semiconductor Manufacturing ◽  
Processing Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Flow Time ◽  
Batch Scheduling ◽  
Scheduling Problem ◽  
Processing Times ◽  
Deterministic Problem

We consider a batch scheduling problem in which the processing time of a batch of jobs equals the maximum of the processing times of all jobs in the batch. This is the case, for example, for burn-in operations in semiconductor manufacturing and other testing operations. Processing times are assumed to be random, and we consider minimizing the makespan and the flow time. The problem is much more difficult than the corresponding deterministic problem, and the optimal policy may have many counterintuitive properties. We prove various structural properties of the optimal policy and use these to develop a polynomial-time algorithm to compute the optimal policy.

scholarly journals Single-Machine Scheduling with Rejection and an Operator Non-Availability Interval

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 668 ◽  
Author(s):  
Lili Zuo ◽  
Zhenxia Sun ◽  
Lingfa Lu ◽  
Liqi Zhang
Keyword(s):  
Polynomial Time ◽  
Single Machine ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Single Machine Scheduling ◽  
Total Weighted Completion Time ◽  
Scheduling Problems ◽  
Objective Functions ◽  
Rejection Cost ◽  
Scheduling With Rejection

In this paper, we study two scheduling problems on a single machine with rejection and an operator non-availability interval. In the operator non-availability interval, no job can be started or be completed. However, a crossover job is allowed such that it can be started before this interval and completed after this interval. Furthermore, we also assume that job rejection is allowed. That is, each job is either accepted and processed in-house, or is rejected by paying a rejection cost. Our task is to minimize the sum of the makespan (or the total weighted completion time) of accepted jobs and the total rejection cost of rejected jobs. For two scheduling problems with different objective functions, by borrowing the previous algorithms in the literature, we propose a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme (FPTAS), respectively.

scholarly journals Batch Scheduling with Proportional-Linear Deterioration and Outsourcing

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Cuixia Miao ◽  
Fanxiao Meng ◽  
Juan Zou ◽  
Binglin Jia
Keyword(s):  
Dynamic Programming Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Third Party ◽  
Batch Scheduling ◽  
Programming Algorithm ◽  
Release Dates ◽  
Linear Deterioration ◽  
Total Penalty ◽  
Maximum Completion Time

We consider the bounded parallel-batch scheduling with proportional-linear deterioration and outsourcing, in which the actual processing time is pj=αj(A+Dt) or pj=αjt. A job is either accepted and processed in batches on a single machine by manufactures themselves or outsourced to the third party with a certain penalty having to be paid. The objective is to minimize the maximum completion time of the accepted jobs and the total penalty of the outsourced jobs. For the pj=αj(A+Dt) model, when all the jobs are released at time zero, we show that the problem is NP-hard and present a pseudo-polynomial time algorithm, respectively. For the pj=αjt model, when the jobs have distinct m (<n) release dates, we provide a dynamic programming algorithm, where n is the number of jobs.

SCHEDULING TWO-MACHINE FLOW SHOPS WITH EXACT DELAYS

2007 ◽  
Vol 18 (02) ◽  
pp. 341-359 ◽  
Author(s):  
JOSEPH Y.-T. LEUNG ◽  
HAIBING LI ◽  
HAIRONG ZHAO
Keyword(s):  
Polynomial Time ◽  
Completion Time ◽  
Flow Shop ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Total Completion Time ◽  
Time Interval ◽  
Start Time ◽  
Special Cases ◽  
Efficient Approximation

We consider two-machine flow shop problems with exact delays. In this model, there are two machines, the upstream machine and the downstream machine. Each job j has two operations: the first operation has to be processed on the upstream machine and the second operation has to be processed on the downstream machine, subject to the constraint that the time interval between the completion time of the first operation and the start time of the second operation is exactly [Formula: see text]. We concentrate on the objectives of makespan and total completion time. For the makespan objective, we first show that the problem is strongly NP-hard even if there are only two possible delay values. We then show that some special cases of the problem are solvable in polynomial time. Finally, we design efficient approximation algorithms for the general case and some special cases. For the total completion time objective, we give optimal polynomial-time algorithm for a special case and an efficient approximation algorithm for another one.

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