Open-shop dense schedules: properties and worst-case performance ratio

2009 ◽  
Vol 15 (1) ◽  
pp. 3-11 ◽  
Author(s):  
Rongjun Chen ◽  
Wanzhen Huang ◽  
Zhongxian Men ◽  
Guochun Tang
Keyword(s):  
Open Shop ◽  
Performance Ratio ◽  
Worst Case ◽  
Worst Case Performance Ratio

PARALLEL MACHINE SCHEDULING WITH JOB DELIVERY COORDINATION

2017 ◽  
Vol 58 (3-4) ◽  
pp. 306-313
Author(s):  
J. M. DONG ◽  
X. S. WANG ◽  
L. L. WANG ◽  
J. L. HU
Keyword(s):  
Parallel Machines ◽  
Machine Scheduling ◽  
Parallel Machine Scheduling ◽  
Parallel Machine ◽  
Performance Ratio ◽  
Worst Case ◽  
Identical Machine ◽  
Worst Case Performance Ratio ◽  
Job Delivery ◽  
Identical Machine Scheduling

We analyse a parallel (identical) machine scheduling problem with job delivery to a single customer. For this problem, each job needs to be processed on $m$ parallel machines non-pre-emptively and then transported to a customer by one vehicle with a limited physical capacity. The optimization goal is to minimize the makespan, the time at which all the jobs are processed and delivered and the vehicle returns to the machine. We present an approximation algorithm with a tight worst-case performance ratio of $7/3-1/m$ for the general case, $m\geq 3$.

scholarly journals Two Parallel Machines Scheduling with Two-Vehicle Job Delivery to Minimize Makespan

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Lisi Cao ◽  
Jianhong Hao ◽  
Dakui Jiang
Keyword(s):  
Parallel Machines ◽  
Parallel Machine Scheduling ◽  
Performance Ratio ◽  
Identical Parallel Machines ◽  
Worst Case ◽  
Worst Case Ratio ◽  
Parallel Machines Scheduling ◽  
Np Hard Problem ◽  
Worst Case Performance Ratio ◽  
Job Delivery

A problem of parallel machine scheduling with coordinated job deliveries is handled to minimize the makespan. Different jobs call for dissimilar sizes of storing space in the process of transportation. A range of jobs of one customer in the problem have priority to be processed on two identical parallel machines without preemption and then delivered to the customer by two vehicles in batches. For this NP-hard problem, we first prove that it is impossible to have a polynomial heuristic with a worst-case performance ratio bound less than 2 unless P = NP. Thereafter, we develop a polynomial heuristic for this problem, the worst-case ratio of which is bounded by 2.

scholarly journals Analysis of an Approximation Algorithm for Scheduling Independent Parallel Tasks

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Keqin Li
Keyword(s):  
Approximation Algorithm ◽  
Bin Packing ◽  
Random Variables ◽  
Packing Problem ◽  
Performance Ratio ◽  
Worst Case ◽  
Average Case ◽  
Parallel Tasks ◽  
International Audience ◽  
Worst Case Performance Ratio

International audience In this paper, we consider the problem of scheduling independent parallel tasks in parallel systems with identical processors. The problem is NP-hard, since it includes the bin packing problem as a special case when all tasks have unit execution time. We propose and analyze a simple approximation algorithm called H_m, where m is a positive integer. Algorithm H_m has a moderate asymptotic worst-case performance ratio in the range [4/3 ... 31/18] for all m≥ 6; but the algorithm has a small asymptotic worst-case performance ratio in the range [1+1/(r+1)..1+1/r], when task sizes do not exceed 1/r of the total available processors, where r>1 is an integer. Furthermore, we show that if the task sizes are independent, identically distributed (i.i.d.) uniform random variables, and task execution times are i.i.d. random variables with finite mean and variance, then the average-case performance ratio of algorithm H_m is no larger than 1.2898680..., and for an exponential distribution of task sizes, it does not exceed 1.2898305.... As demonstrated by our analytical as well as numerical results, the average-case performance ratio improves significantly when tasks request for smaller numbers of processors.

scholarly journals PREEMPTIVE SCHEDULING ON PARALLEL PROCESSORS WITH DUE DATES

2005 ◽  
Vol 22 (04) ◽  
pp. 445-462 ◽  
Author(s):  
YAKOV ZINDER ◽  
GAURAV SINGH
Keyword(s):  
Arbitrary Number ◽  
Optimal Schedule ◽  
Precedence Constraints ◽  
Parallel Processors ◽  
Performance Ratio ◽  
Preemptive Scheduling ◽  
Due Dates ◽  
Worst Case ◽  
Worst Case Performance Ratio ◽  
Priority Algorithm

The paper presents a priority algorithm for the maximum lateness problem with parallel identical processors, precedence constraints, and preemptions. The presented algorithm calculates the priority of each task by constructing a schedule for the set of its successors. The algorithm is motivated by comparison of its nonpreemptive counterpart with other algorithms for the problem with unit execution time tasks. It is shown that the presented algorithm constructs an optimal schedule for the problem with two processors and arbitrary precedence constraints, and for the problem with an arbitrary number of processors and precedence constraints in the form of an in-tree. This proof also indicates that the presented algorithm allows the worst-case performance ratio previously established for the so-called Muntz–Coffman algorithm for a particular case of the considered problem where all due dates are zero.

SCHEDULING WITH DISCRETELY COMPRESSIBLE RELEASE DATES TO MINIMIZE MAKESPAN

2010 ◽  
Vol 27 (04) ◽  
pp. 493-501
Author(s):  
SHU-XIA ZHANG ◽  
ZHI-GANG CAO ◽  
YU-ZHONG ZHANG
Keyword(s):  
Performance Ratio ◽  
Release Dates ◽  
Release Date ◽  
Worst Case ◽  
Scheduling Model ◽  
Processing Order ◽  
Time Scheduling ◽  
First Time ◽  
Worst Case Performance Ratio ◽  
Np Hardness

In this paper, we address the scheduling model with discretely compressible release dates, where processing any job with a compressed release date incurs a corresponding compression cost. We consider the following problem for the first time: scheduling with discretely compressible release dates to minimize the sum of makespan plus total compression cost. We show its NP-hardness, and design an approximation algorithm with worst-case performance ratio 2, which is the best possible if the processing order of the jobs is pre-specified.

SINGLE MACHINE SCHEDULING WITH JOB DELIVERY TO MINIMIZE MAKESPAN

2008 ◽  
Vol 25 (01) ◽  
pp. 1-10 ◽  
Author(s):  
LINGFA LU ◽  
JINJIANG YUAN
Keyword(s):  
Single Machine ◽  
Machine Scheduling ◽  
Single Machine Scheduling ◽  
Performance Ratio ◽  
Extended Version ◽  
Worst Case ◽  
Transportation Time ◽  
Transportation Times ◽  
Worst Case Performance Ratio ◽  
Job Delivery

In the single machine scheduling problem with job delivery to minimize makespan, jobs are processed on a single machine and delivered by a capacitated vehicle to their respective customers. We first consider the special case with a single customer, that is, all jobs have the same transportation time. Chang and Lee (2004) proved that this case is strongly NP-hard. They also provided a heuristic with the worst-case performance ratio [Formula: see text], and pointed out that no heuristic can have a worst-case performance ratio less than [Formula: see text] unless P = NP. In this paper, we provide a new heuristic which has the best possible worst-case performance ratio [Formula: see text]. We also consider an extended version in which the jobs have non-identical transportation times and the transportation time of a delivery batch is defined as the maximum transportation time of the jobs contained in it. We provide a heuristic with the worst-case performance ratio 2 for the extended version, and show that this bound is tight.

APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

2013 ◽  
Vol 23 (06) ◽  
pp. 461-477 ◽  
Author(s):  
MINATI DE ◽  
GAUTAM K. DAS ◽  
PAZ CARMI ◽  
SUBHAS C. NANDY
Keyword(s):  
Approximation Algorithms ◽  
Simple Algorithm ◽  
Constant Factor ◽  
Performance Ratio ◽  
Approximation Result ◽  
Worst Case ◽  
Approximation Factor ◽  
Minimum Number ◽  
Unit Disks ◽  
Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

GENERALIZED FIRST-FIT ALGORITHMS IN TWO AND THREE DIMENSIONS

1990 ◽  
Vol 01 (02) ◽  
pp. 131-150 ◽  
Author(s):  
KEQIN LI ◽  
KAM-HOI CHENG
Keyword(s):  
Bin Packing ◽  
Three Dimensional ◽  
Packing Problem ◽  
Three Dimensions ◽  
Performance Ratio ◽  
Dimensional Case ◽  
Performance Bound ◽  
Worst Case ◽  
Packing Problems ◽  
The One

We investigate the two and three dimensional bin packing problems, i.e., packing a list of rectangles (boxes) into unit square (cube) bins so that the number of bins used is a minimum. A simple on-line packing algorithm for the one dimensional bin packing problem, the First-Fit algorithm, is generalized to two and three dimensions. We first give an algorithm for the two dimensional case and show that its asymptotic worse case performance ratio is [Formula: see text]. The algorithm is then generalized to the three dimensional case and its performance ratio [Formula: see text]. The second algorithm takes a parameter and we prove that by choosing the parameter properly, it has an asymptotic worst case performance bound which can be made as close as desired to 1.72=2.89 and 1.73=4.913 respectively in two and three dimensions.

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