scholarly journals Branch Less, Cut More and Schedule Jobs with Release and Delivery Times on Uniform Machines

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 633
Author(s):  
Nodari Vakhania ◽  
Frank Werner
Keyword(s):  
Parallel Machines ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Delivery Time ◽  
Processing Times ◽  
Delivery Times ◽  
Uniform Machines ◽  
Complexity Status ◽  
Algorithmic Framework ◽  
Job Delivery

We consider the problem of scheduling n jobs with identical processing times and given release as well as delivery times on m uniform machines. The goal is to minimize the makespan, i.e., the maximum full completion time of any job. This problem is well-known to have an open complexity status even if the number of jobs is fixed. We present a polynomial-time algorithm for the problem which is based on the earlier introduced algorithmic framework blesscmore (“branch less and cut more”). We extend the analysis of the so-called behavior alternatives developed earlier for the version of the problem with identical parallel machines and show how the earlier used technique for identical machines can be extended to the uniform machine environment if a special condition on the job parameters is imposed. The time complexity of the proposed algorithm is O(γm2nlogn), where γ can be either n or the maximum job delivery time qmax. This complexity can even be reduced further by using a smaller number κ<n in the estimation describing the number of jobs of particular types. However, this number κ becomes only known when the algorithm has terminated.

scholarly journals A Polynomial Algorithm for Sequencing Jobs with Release and Delivery Times on Uniform Machines

2021 ◽  
Author(s):  
Nodari Vakhania ◽  
Frank Werner
Keyword(s):  
Time Complexity ◽  
Polynomial Algorithm ◽  
Delivery Time ◽  
Uniform Machine ◽  
Delivery Times ◽  
Uniform Machines ◽  
Complexity Status ◽  
Algorithmic Framework ◽  
Identical Machine ◽  
Job Delivery

The problem of sequencing $n$ equal-length non-simultaneously released jobs with delivery times on $m$ uniform machines to minimize the maximum job completion time is considered. To the best of our knowledge, the complexity status of this classical scheduling problem remained open up to the date. We establish its complexity status positively by showing that it can be solved in polynomial time. We adopt for the uniform machine environment the general algorithmic framework of the analysis of behavior alternatives developed earlier for the identical machine environment. The proposed algorithm has the time complexity $O(\gamma m^2 n\log n)$, where $\gamma$ can be any of the magnitudes $n$ or $q_{\max}$, the maximum job delivery time. In fact, $n$ can be replaced by a smaller magnitude $\kappa&lt;n$, which is the number of special types of jobs (it becomes known only upon the termination of the algorithm).

Scheduling with Discretely Compressible Processing Times to Minimize Makespan

2014 ◽  
Vol 575 ◽  
pp. 926-930
Author(s):  
Shu Xia Zhang ◽  
Yu Zhong Zhang
Keyword(s):  
Dynamic Programming ◽  
Polynomial Time ◽  
Processing Time ◽  
Parallel Machines ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Identical Parallel Machines ◽  
Processing Times ◽  
Release Times ◽  
Scheduling Model

In this paper, we address the scheduling model 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 on identical parallel machines. Jobs may have simultaneous release times. We design a pseudo-polynomial time algorithm by approach of dynamic programming and an FPTAS.

scholarly journals Scheduling Jobs with Variable Job Processing Times on Unrelated Parallel Machines

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Guang-Qian Zhang ◽  
Jian-Jun Wang ◽  
Ya-Jing Liu
Keyword(s):  
Resource Allocation ◽  
Parallel Machines ◽  
Waiting Times ◽  
Optimal Schedule ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Unrelated Parallel Machines ◽  
Scheduling Problems ◽  
Processing Times ◽  
Parallel Machines Scheduling

munrelated parallel machines scheduling problems with variable job processing times are considered, where the processing time of a job is a function of its position in a sequence, its starting time, and its resource allocation. The objective is to determine the optimal resource allocation and the optimal schedule to minimize a total cost function that dependents on the total completion (waiting) time, the total machine load, the total absolute differences in completion (waiting) times on all machines, and total resource cost. If the number of machines is a given constant number, we propose a polynomial time algorithm to solve the problem.

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 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 .

Proportionate Flow Shop Scheduling with Rejection

2017 ◽  
Vol 34 (04) ◽  
pp. 1750015 ◽  
Author(s):  
Shi-Sheng Li ◽  
De-Liang Qian ◽  
Ren-Xia Chen
Keyword(s):  
Processing Time ◽  
Flow Shop ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Flow Shop Scheduling ◽  
Total Weighted Completion Time ◽  
Processing Times ◽  
Proportionate Flow Shop ◽  
Total Penalty ◽  
Scheduling With Rejection

We consider the problem of scheduling [Formula: see text] jobs with rejection on a set of [Formula: see text] machines in a proportionate flow shop system where the job processing times are machine-independent. The goal is to find a schedule to minimize the scheduling cost of all accepted jobs plus the total penalty of all rejected jobs. Two variations of the scheduling cost are considered. The first is the maximum tardiness and the second is the total weighted completion time. For the first problem, we first show that it is [Formula: see text]-hard, then we construct a pseudo-polynomial time algorithm to solve it and an [Formula: see text] time for the case where the jobs have the same processing time. For the second problem, we first show that it is [Formula: see text]-hard, then we design [Formula: see text] time algorithms for the case where the jobs have the same weight and for the case where the jobs have the same processing time.

scholarly journals Parallel Machine Scheduling with Nested Processing Set Restrictions and Job Delivery Times

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Shuguang Li
Keyword(s):  
Parallel Machines ◽  
Approximation Scheme ◽  
Parallel Machine Scheduling ◽  
Parallel Machine ◽  
List Scheduling ◽  
Polynomial Time Approximation Scheme ◽  
Time Approximation ◽  
Delivery Times ◽  
Specific Subset ◽  
Job Delivery

The problem of scheduling jobs with delivery times on parallel machines is studied, where each job can only be processed on a specific subset of the machines called its processing set. Two distinct processing sets are either nested or disjoint; that is, they do not partially overlap. All jobs are available for processing at time 0. The goal is to minimize the time by which all jobs are delivered, which is equivalent to minimizing the maximum lateness from the optimization viewpoint. A list scheduling approach is analyzed and its approximation ratio of 2 is established. In addition, a polynomial time approximation scheme is derived.

A HYBRID TWO-STAGE FLOWSHOP SCHEDULING PROBLEM

2007 ◽  
Vol 24 (01) ◽  
pp. 45-56 ◽  
Author(s):  
LONGMIN HE ◽  
SHIJIE SUN ◽  
RUNZI LUO
Keyword(s):  
Processing Time ◽  
Parallel Machines ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scheduling Problem ◽  
Flowshop Scheduling ◽  
Two Stage ◽  
Batch Processor ◽  
Flowshop Scheduling Problem ◽  
Stage 1

This paper considers a batch scheduling problem in a two-stage hybrid flowshop that consists of m dedicated parallel machines in stage 1 and a batch processor in stage 2. The processing time of a batch is defined as the largest processing time of the jobs contained in that batch. The criterion is to minimize the makespan, the time by which all operations of jobs have been processed. For such a problem, we present a polynomial time algorithm for the case with all jobs having the same processing time on the batch processor. An approximation algorithm with a competitive ratio 2 for the general case is also presented.

Single-Machine Due-Window Assignment and Scheduling with Learning Effect and Resource-Dependent Processing Times

2014 ◽  
Vol 31 (05) ◽  
pp. 1450036 ◽  
Author(s):  
Ji-Bo Wang ◽  
Ming-Zheng Wang
Keyword(s):  
Resource Allocation ◽  
Single Machine ◽  
Processing Time ◽  
Window Size ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Earliness And Tardiness ◽  
Processing Times ◽  
Due Window ◽  
Due Window Assignment

We consider a single-machine common due-window assignment scheduling problem, in which the processing time of a job is a function of its position in a sequence and its resource allocation. The window location and size, along with the associated job schedule that minimizes a certain cost function, are to be determined. This function is made up of costs associated with the window location, window size, earliness, and tardiness. For two different processing time functions, we provide a polynomial time algorithm to find the optimal job sequence and resource allocation, respectively.

scholarly journals Makespan minimization with OR-precedence constraints

2021 ◽  
Author(s):  
Felix Happach
Keyword(s):  
Scheduling Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Precedence Constraints ◽  
Np Hard ◽  
Approximation Factor ◽  
Processing Times ◽  
Special Cases ◽  
Approximation Guarantee ◽  
Unit Processing

AbstractWe consider a variant of the NP-hard problem of assigning jobs to machines to minimize the completion time of the last job. Usually, precedence constraints are given by a partial order on the set of jobs, and each job requires all its predecessors to be completed before it can start. In this paper, we consider a different type of precedence relation that has not been discussed as extensively and is called OR-precedence. In order for a job to start, we require that at least one of its predecessors is completed—in contrast to all its predecessors. Additionally, we assume that each job has a release date before which it must not start. We prove that a simple List Scheduling algorithm due to Graham (Bell Syst Tech J 45(9):1563–1581, 1966) has an approximation guarantee of 2 and show that obtaining an approximation factor of $$4/3 - \varepsilon $$ 4 / 3 - ε is NP-hard. Further, we present a polynomial-time algorithm that solves the problem to optimality if preemptions are allowed. The latter result is in contrast to classical precedence constraints where the preemptive variant is already NP-hard. Our algorithm generalizes previous results for unit processing time jobs subject to OR-precedence constraints, but without release dates. The running time of our algorithm is $$O(n^2)$$ O ( n 2 ) for arbitrary processing times and it can be reduced to O(n) for unit processing times, where n is the number of jobs. The performance guarantees presented here match the best-known ones for special cases where classical precedence constraints and OR-precedence constraints coincide.

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