Semi-online Scheduling Problems Research Articles

Minimising total weighted completion time for semi-online single machine scheduling with known arrivals and bounded processing times

This paper addresses the semi-online scheduling problem of minimising the total weighted completion time on a single machine, where a combination of information on jobs release dates and processing times is considered. In this study, jobs can only arrive at known future times and a lower bound on jobs processing times is known in advance. A new semi-online algorithm is presented and is shown to be the best possible for the considered problem. In order to make this statement, a new lower bound on the competitive ratio of any semi-online algorithm for the problem is developed and, using competitive analysis, the proposed semi-online algorithm is shown to have a competitive ratio that matches the lower bound.

Semi-Online Scheduling on Two Identical Parallel Machines with Initial-Lookahead Information

This work investigates a new semi-online scheduling problem with lookahead. We focus on job scheduling on two identical parallel machines, where deterministic online algorithms only know the information of [Formula: see text] initial jobs (i.e., the initial-lookahead information), while the following jobs still arrive one-by-one in an over-list fashion. We consider makespan minimization as the objective. The study aims at revealing the value of knowing [Formula: see text] initial jobs, which are used to improve the competitive performance of those online algorithms without such initial-lookahead information. We provide the following findings: (1) For the scenario where the [Formula: see text] initial jobs are all the largest jobs with length [Formula: see text], we prove that the classical LIST algorithm is optimal with competitive ratio [Formula: see text]; (2) For the scenario where the total length of these [Formula: see text] jobs is at least [Formula: see text], we show that any online algorithm has a competitive ratio at least 3/2, implying that the initial-lookahead knowledge is powerless since there exists a 3/2-competitive online algorithm without such information; (3) For the scenario where the total length of these [Formula: see text] jobs is at least [Formula: see text] ([Formula: see text]), we propose an online algorithm, named as LPT-LIST, with competitive ratio of [Formula: see text], implying that the initial-lookahead information indeed helps to improve the competitiveness of those online algorithms lacking such information.

Minimizing Total Weighted Completion Times for Semi-online Single Machine Scheduling

This paper addresses the semi-online scheduling problem for total weighted completion time minimization on a single machine, where a combination of information on jobs’ release dates and processing times is considered. In this study, jobs can only arrive at known future times and a lower bound on jobs’ processing times is known in advance. A new semi-online algorithm, called MCSWPT (Modified Critical Shortest Weighted Processing Time), that makes use of the available pieces of information in order to produce better schedules compared to its online peers is presented. A numerical analysis is presented, showing the impact of having the considered partial information.

A Semi-Online Algorithm for Single Machine Scheduling with Rejection

In this paper, a single-machine semi-online scheduling problem with rejection is addressed. In this model, “semi-online” implies that [Formula: see text], where [Formula: see text] and [Formula: see text] are the maximum processing time and the minimum one among all jobs, respectively, [Formula: see text]. In this setting, each job arrives online over time, and rejection is allowable. Our goal is minimizing the total penalty cost of rejected jobs plus the total completion time of processed jobs. The seminal result of this study is that we offer an algorithm with competitive ratio [Formula: see text], which matches the result of the problem without rejection.

A semi-online algorithm and its competitive analysis for parallel-machine scheduling problem with rejection

In this paper, we focus on a semi-online scheduling problem with rejection on identical parallel machines, where “semi-online” means that the ratio of the longest processing time among all jobs to the shortest one is no more than γ with γ ≥ 1. In particular, in this setting, there are a coupling of independent jobs arriving online over time with the flexibility of rejection, which implies that each job will be either accepted and scheduled on one of identical machines or rejected at the cost of penalty cost. Our objective is minimizing the total completion time of the accepted jobs plus the total penalty cost of the rejected jobs. For this problem, we design a deterministic polynomial time semi-online algorithm entitled as α Delayed Shortest Processing Time with Rejection (ADSPTR). In competitive analysis, by adopting the approach “Improved Instance Reduction”, we obtain the competitive ratio of ADSPTR is at most 1+1+γ(γ−1)−1γ.

Online scheduling on parallel machines: A survey

Scheduling is one of the most fundamental combinatorial optimization problems. Online scheduling problems have received considerable research interest in the last three decades.In this paper, we summarize new results on online over list scheduling problems on parallel machines, including problems with an objective to minimize the makespan and the $L^p$ norm of machine loads, semi-online scheduling problems on identical, uniform and hierarchial machines, scheduling with rejection and scheduling with machine costs. We also introduce several new directions of online scheduling.

Semi-online scheduling on two identical machines with a common due date to maximize total early work

In this paper, we consider four semi-online scheduling problems with the goal of early work maximization, which shares the same essence with late work minimization when the optimal offline solutions are determined, but differs from it when the approximation or online solutions are constructed. First, we prove a tight bound 65 for scheduling jobs with a common due date when assuming that the total processing time of all jobs, or, the optimal criterion value is known in advance. Then, we show that if both the total and maximal processing time is known, the bound can be reduced to 109 and it is still tight. Finally, we prove that if only know the maximal processing time, the upper and lower bounds are 1.1375 and 1.1231, respectively.

An improved semi-online algorithm for scheduling on a single machine with unexpected breakdown

We consider a semi-online scheduling problem on a single machine with an unexpected breakdown period. In the problem, each job has a processing time and a subsequent delivery time. All these data are known in the beginning. The scheduler has to determine a sequence S of these jobs for processing. After S is given, a machine unavailability period may occur where its starting time and length are not known in advance. The objective is to minimize the time by which all jobs are delivered. We present an online algorithm with a competitive ratio of $$\frac{\sqrt{5}+1}{2}\approx 1.618$$, which improves the previously known algorithm with $$\frac{2+\sqrt{2}}{2}\approx 1.707$$-competitive.

Semi-online scheduling with bounded job sizes on two uniform machines

In this paper, we investigate a semi-online scheduling problem on two uniform machines with the speed ratio s. It is assumed that all jobs have their processing times between p and tp (p>0, t≥1). The objective is to minimize the makespan. We give the competitive ratio of LS algorithm which is a piecewise function on t≥1 and s≥1. It shows that LS is an optimal algorithm for most regions on s and t. We further present two optimal algorithms. The algorithm H1 with competitive ratio of s is optimal for 1.325≤s≤1+52 and s<t≤s2−11+s−s2. The algorithm H2 with competitive ratio of s is optimal for 1.206≤s≤1.5 and s≤t≤min⁡{2s−1,2(s2−1)1+s−s2}, and it is also optimal for 1≤s≤1+174 and max⁡{2s−1,−s+9s2+8s2s}≤t≤2s with competitive ratio of 1+t2.

Online and semi-online scheduling to minimize makespan on single machine with an availability constraint

Online and semi-online scheduling problems on a single machine with an availability constraint are considered in this paper. The machine has one unavailable interval in which jobs cannot be processed. Preemption is not allowed. Jobs arrive over time. The objective is to minimize makespan. First we discuss the online version of the problem. After giving its lower bound, we prove that Earliest Release Date (ERD) algorithm is an optimal algorithm. Then we study some semi-online problems in which the largest processing time, the total processing time, the largest release date, or the optimal makespan is known in advance. For these semi-online problems, we give their lower bounds, design semi-online algorithms and prove their competitive ratios, respectively.

Optimal algorithm for semi-online scheduling on two machines under GoS levels

Recently, Liu et al. (J Combin Optim 21:138–149, 2011) studied the semi-online scheduling problem on two machines under a grade of service provision. As the sum of jobs’ processing times \(\varSigma \) is known in advance and the processing times are bounded by an interval \([1,\alpha ]\) where \(1< \alpha <2\), they presented an algorithm which is \(\frac{1+\alpha }{2}\)-competitive when \(\varSigma \ge \frac{2\alpha }{\alpha -1}\). In this paper, we give a modified algorithm which is shown to be optimal for arbitrary \(\alpha \) and \(\varSigma \).

Semi-online scheduling with combined information on two identical machines in parallel

This paper is concerned with a semi-online scheduling problem with combined information on two identical parallel machines to minimize the makespan, where all the jobs have processing times in the interval $$[1,\,t]$$[1,t] $$(t\ge 1)$$(t?1) and the jobs arrive in non-increasing order of their processing times. The objective is to minimize the makespan. For $$t\ge 1$$t?1, we obtain a lower bound $$\max _{N=1,2,3,\ldots }\left\{ \min \{\frac{4N+3}{4N+2}\,,\frac{Nt+N+1}{2N+1}\}\right\} $$maxN=1,2,3,?min{4N+34N+2,Nt+N+12N+1} and show that the competitive ratio of the $$LS$$LS algorithm achieves the lower bound.

Semi-Online Scheduling on Two Machines withGoSLevels and Partial Information of Processing Time

This paper investigates semi-online scheduling problems on two parallel machines under a grade of service (GoS) provision subject to minimize the makespan. We consider three different semi-online versions with knowing the total processing time of the jobs with higher GoS level, knowing the total processing time of the jobs with lower GoS level, or knowing both in advance. Respectively, for the three semi-online versions, we develop algorithms with competitive ratios of 3/2, 20/13, and 4/3 which are shown to be optimal.

Semi-online scheduling with two GoS levels and unit processing time

In this paper, we consider two semi-online scheduling problems of parallel machines with two GoS (grade of service) levels and unit processing time, in which the objective is to minimize the makespan. Assume that the GoS levels of the first s machines are 2 and that of the last (m−s) machines are 1 where m is the number of machines. The first problem is the lookahead version where an online algorithm is able to foresee the information of the next k jobs. The second problem is the buffer version where a buffer is available for storing at most g jobs. For the both versions, we prove that the lookahead ability or the buffer is useless for designing algorithm when k<m−s or g<min{s,m−s}. We also show that no online algorithm has a competitive ratio less than m2m(m−s)+s2 for any constant k or g. Moreover, we present two optimal online algorithms for k=m2−1s+s−m and g=m−mm(m−s)+s2. At last, for the case with only two machines, we prove that the two algorithms can get their best possible competitive ratio when k=1 and g=1.

Semi-online scheduling problems on a small number of machines

We consider the semi-online parallel machine scheduling problem of minimizing the makespan given a priori information: the total processing time, the largest processing time, the combination of the previous two or the optimal makespan. We propose a new algorithm that can be applied to the problem with the known total or largest processing time and prove that it has improved competitive ratios for the cases with a small number of machines. Improved lower bounds of the competitive ratio are also provided by presenting adversary lower bound examples.

Semi-online scheduling problems on two uniform machines under a grade of service provision

This paper studies the semi-online scheduling on two uniform machines under a grade of service (GoS) provision, where one machine is available for all jobs and the other one is only available for partial jobs. The objective is to minimize the makespan. We consider three variants, where the optimal makespan, the total size of jobs, and the largest job size are known in advance respectively, and design optimal algorithms for all of them.

Several semi-online scheduling problems on two identical machines with combined information

In this paper we consider several semi-online scheduling problems on two identical machines with combined information. The objective of each problem is to minimize the makespan. The first problem is semi-online scheduling with known optimal solution value and maximum job size. We obtain a lower bound 65 and design an optimal algorithm with a competitive ratio 65. The second problem is semi-online scheduling with a buffer of size k, where k(k≥1) is a finite positive integer, and known maximum job size. We obtain a lower bound 65 and design an algorithm with a competitive ratio 54. The third problem is semi-online scheduling with a buffer of size 1 and jobs arriving in decreasing order of their processing times. We obtain a lower bound 76, which matches an upper bound in the literature. The last problem is semi-online scheduling with a buffer of size 1 and all the job processing times being bounded in the interval [1,t](t≥1). We obtain a lower bound max{min{43,t+26},min{54,t+14},min{76,t+23}}, where the lower bound 43 for t≥6 matches an upper bound in the literature, and design an algorithm with a competitive ratio max{t+23,87} for 1≤t≤32, which is optimal for 107≤t≤32.

Algorithms better than LPT for semi-online scheduling with decreasing processing times

We consider the semi-online multiprocessor scheduling problem with m identical, parallel machines to minimize the makespan, where the jobs arrive in decreasing order of processing times. The famous Longest Processing Time (LPT) algorithm by Graham (1966) [4] for the classical offline multiprocessor scheduling problem schedules the jobs in decreasing order of processing times and has a worst-case bound of 4/3−1/(3m). So far, no algorithm with a better competitive ratio than the LPT algorithm has been given for the semi-online scheduling problem with decreasing processing times. In this note, we present a 5/4-competitive algorithm for m≥3 and an algorithm that is the best possible for m=3, i.e. an algorithm with competitive ratio (1+37)/6.

Semi-online scheduling on two identical machines with rejection

In this paper we consider two semi-online scheduling problems with rejection on two identical machines. A sequence of independent jobs are given and each job is characterized by its size (processing time) and its penalty, in the sense that, jobs arrive one by one and can be either rejected by paying a certain penalty or assigned to some machine. No preemption is allowed. The objective is to minimize the sum of the makespan of schedule, which is yielded by all accepted jobs and the total penalties of all rejected ones. In the first problem one can reassign several scheduled jobs in rejection tache, in the second a buffer with length k is available in rejection tache. Two optimal algorithms both with competitive ratio $\frac{3}{2}$ are presented.

Semi-online scheduling for jobs with release times

In this paper we consider three semi-online scheduling problems for jobs with release times on m identical parallel machines. The worst case performance ratios of the LS algorithm are analyzed. The objective function is to minimize the maximum completion time of all machines, i.e. the makespan. If the job list has a non-decreasing release times, then $2-\frac{1}{m}$ is the tight bound of the worst case performance ratio of the LS algorithm. If the job list has non-increasing processing times, we show that $2-\frac{1}{2m}$ is an upper bound of the worst case performance ratio of the LS algorithm. Furthermore if the job list has non-decreasing release times and the job list has non-increasing processing times we prove that the LS algorithm has worst case performance ratio not greater than $\frac{3}{2} -\frac{1}{2m}$ .