The Price of Bounded Preemption

2021 ◽  
Vol 8 (1) ◽  
pp. 1-21
Author(s):  
Noga Alon ◽  
Yossi Azar ◽  
Mark Berlin
Keyword(s):  
Single Machine ◽  
Upper Bound ◽  
Loss Factor ◽  
Processing Time ◽  
Minimal Length ◽  
Release Time ◽  
Preemptive Scheduling ◽  
Tight Bound ◽  
Bounded Degree ◽  
Specific Scenario

In this article we provide a tight bound for the price of preemption for scheduling jobs on a single machine (or multiple machines). The input consists of a set of jobs to be scheduled and of an integer parameter k ≥ 1. Each job has a release time, deadline, length (also called processing time), and value associated with it. The goal is to feasibly schedule a subset of the jobs so that their total value is maximal; while preemption of a job is permitted, a job may be preempted no more than k times. The price of preemption is the worst possible (i.e., largest) ratio of the optimal non-bounded-preemptive scheduling to the optimal k -bounded-preemptive scheduling. Our results show that allowing at most k preemptions suffices to guarantee a Θ(min {log k +1 n , log k +1 P }) fraction of the total value achieved when the number of preemptions is unrestricted (where n is the number of the jobs and P the ratio of the maximal length to the minimal length), giving us an upper bound for the price; a specific scenario serves to prove the tightness of this bound. We further show that when no preemptions are permitted at all (i.e., k =0), the price is Θ (min { n , log P }). As part of the proof, we introduce the notion of the Bounded-Degree Ancestor-Free Sub-Forest (BAS) . We investigate the problem of computing the maximal-value BAS of a given forest and give a tight bound for the loss factor, which is Θ(log k +1 n ) as well, where n is the size of the original forest and k is the bound on the degree of the sub-forest.

An Improved Online Algorithm for the Online Preemptive Scheduling of Equal-Length Intervals on a Single Machine with Lookahead

2015 ◽  
Vol 32 (06) ◽  
pp. 1550047
Author(s):  
Wenjie Li ◽  
Jinjiang Yuan
Keyword(s):  
Lower Bound ◽  
Single Machine ◽  
Operations Research ◽  
Competitive Ratio ◽  
Processing Time ◽  
Online Algorithm ◽  
Equal Length ◽  
Preemptive Scheduling ◽  
Time Segment ◽  
Time Point

This paper studies the online preemptive scheduling of equal-length intervals on a single machine with lookahead. Let [Formula: see text] be the length (processing time) of all intervals. In the problem, at every time point [Formula: see text], online algorithms can foresee all the intervals that will arrive in the time segment [Formula: see text] for a certain [Formula: see text]. When [Formula: see text], Zheng et al. [Comput- ers & Operations Research, 2013] established a lower bound of [Formula: see text] and provided an online algorithm with a competitive ratio of 3. In this paper, we provide for this problem an improved online algorithm with a competitive ratio of 2.

scholarly journals Single-Machine Scheduling with Upper Bounded Maintenance Time under the Deteriorating Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Pengfei Xue ◽  
Yulin Zhang
Keyword(s):  
Single Machine ◽  
Upper Bound ◽  
Processing Time ◽  
Machine Scheduling ◽  
Single Machine Scheduling ◽  
Scheduling Problem ◽  
Maintenance Time ◽  
Optimal Maintenance ◽  
Special Case ◽  
Actual Processing Time

We consider a single-machine scheduling problem with upper bounded actual processing time and upper bounded maintenance time under deteriorating effect. The actual processing time of a job is a position-dependent power function. If the actual processing time of a job exceeds the upper bound, tardiness penalty of the job should be paid. And if the maintenance time exceeds the corresponding upper bound, tardiness penalty of the maintenance should also be paid. The maintenance duration studied in the paper is a position-dependent exponential function. The objective is to find jointly the optimal maintenance frequency and the optimal job sequence to minimize the total cost, which is a linear function of the makespan and the total tardiness. We show that the studied scheduling problem can be transformed as a classic assignment problem to solve. There is also shown that a special case of the scheduling problem can be optimally solved by a lower order algorithm.

SCHEDULING DETERIORATING JOBS ON A SINGLE MACHINE WITH RELEASE TIMES AND REJECTION

2012 ◽  
Vol 04 (02) ◽  
pp. 1250032 ◽  
Author(s):  
MING LIU ◽  
FEIFENG ZHENG ◽  
CHENGBIN CHU ◽  
YINFENG XU
Keyword(s):  
Single Machine ◽  
Processing Time ◽  
Deteriorating Jobs ◽  
Release Time ◽  
Starting Time ◽  
Release Times ◽  
Deteriorating Job ◽  
Total Penalty ◽  
Programming Algorithms ◽  
Increasing Function

This paper considers scheduling deteriorating jobs on a single machine with release times and rejection. Deteriorating job means that its actual processing time is a increasing function on its execution starting time. In this situation, jobs can be rejected by paying penalties. Each job is associated with a release time. The objective is to minimize the makespan plus the total penalty incurred by rejecting jobs. We present two dynamic programming algorithms and then design an FPTAS for the considered problem.

scholarly journals A best possible algorithm for an online scheduling problem with position-based learning effect

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ran Ma ◽  
Lu Zhang ◽  
Yuzhong Zhang
Keyword(s):  
Single Machine ◽  
Completion Time ◽  
Processing Time ◽  
Learning Effect ◽  
Online Algorithm ◽  
Online Scheduling ◽  
Release Time ◽  
Scheduling Problem ◽  
Over Time ◽  
Actual Processing Time

<p style='text-indent:20px;'>In this paper, we focus on an online scheduling problem with position-based learning effect on a single machine, where the jobs are released online over time and preemption is not allowed. The information about each job <inline-formula><tex-math id="M1">\begin{document}$ J_j $\end{document}</tex-math></inline-formula>, including the basic processing time <inline-formula><tex-math id="M2">\begin{document}$ p_j $\end{document}</tex-math></inline-formula> and the release time <inline-formula><tex-math id="M3">\begin{document}$ r_j $\end{document}</tex-math></inline-formula>, is only available when it arrives. The actual processing time <inline-formula><tex-math id="M4">\begin{document}$ p_j' $\end{document}</tex-math></inline-formula> of each job <inline-formula><tex-math id="M5">\begin{document}$ J_j $\end{document}</tex-math></inline-formula> is defined as a function related to its position <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula>, i.e., <inline-formula><tex-math id="M7">\begin{document}$ p_j' = p_j(\alpha-r\beta) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M8">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> are both nonnegative learning index. Our goal is to minimize the sum of completion time of all jobs. For this problem, we design a deterministic polynomial time online algorithm <i>Delayed Shortest Basic Processing Time</i> (DSBPT). In order to facilitate the understanding of the online algorithm, we present a relatively common and simple example to describe the execution process of the algorithm, and then by competitive analysis, we show that online algorithm DSBPT is a best possible online algorithm with a competitive ratio of 2.</p>

scholarly journals Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime

1986 ◽  
Vol 23 (03) ◽  
pp. 841-847 ◽  
Author(s):  
R. R. Weber ◽  
P. Varaiya ◽  
J. Walrand
Keyword(s):  
Processing Time ◽  
Parallel Machines ◽  
Preemptive Scheduling ◽  
Total Reward ◽  
Processing Times ◽  
Identical Machines ◽  
Scheduling Strategies

A number of jobs are to be processed using a number of identical machines which operate in parallel. The processing times of the jobs are stochastic, but have known distributions which are stochastically ordered. A reward r(t) is acquired when a job is completed at time t. The function r(t) is assumed to be convex and decreasing in t. It is shown that within the class of non-preemptive scheduling strategies the strategy SEPT maximizes the expected total reward. This strategy is one which whenever a machine becomes available starts processing the remaining job with the shortest expected processing time. In particular, for r(t) = – t, this strategy minimizes the expected flowtime.

scholarly journals Branch and Bound Method to Solve Multi Objectives Function

2016 ◽  
Vol 12 (3) ◽  
pp. 5964-5974
Author(s):  
Tahani Jabbar Kahribt ◽  
Mohammed Kadhim Al- Zuwaini
Keyword(s):  
Lower Bounds ◽  
Branch And Bound ◽  
Single Machine ◽  
Upper Bound ◽  
Completion Time ◽  
Heuristic Method ◽  
Total Weighted Completion Time ◽  
Branch And Bound Method ◽  
Special Cases ◽  
Weighted Completion Time

This paper  presents  a  branch  and  bound  algorithm  for  sequencing  a  set  of  n independent  jobs  on  a single  machine  to  minimize sum of the discounted total weighted completion time and maximum lateness,  this problems is NP-hard. Two lower bounds were proposed and heuristic method to get an upper bound. Some special cases were  proved and some dominance rules were suggested and proved, the problem solved with up to 50 jobs.

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