Discrete OptimizationTwo simple constant ratio approximation algorithms for minimizing the total weighted completion time on a single machine with a fixed non-availability interval
Introduction
In this paper, we address the problem of minimizing the sum of the weighted completion times on a single machine, provided that the machine is not available for processing during a given time interval and the job that cannot complete before that interval restarts from scratch when the machine becomes available again.
Our scheduling model belongs to the family of scheduling models with machine availability constraints. We refer to a recent survey by Lee [12] for a state-of-the-art review of deterministic scheduling under availability constraints. Assume that for the processing machine there is a known non-availability interval [s, t], during which the machine cannot perform the processing of any job. This non-availability interval can be due to some scheduled activity other than processing (a maintenance period, a rest period, etc.). The job that is affected by the non-availability interval is called the crossover job. There are several possible scenarios to handle the crossover job. Under the non-resumable scenario, the crossover job that cannot be completed by time s is restarted from scratch at time t. Under the resumable scenario the crossover job is interrupted at time s and resumed from the point of interruption at time t.
As an illustration of the single machine problem with a single non-availability interval under the non-resumable scenario, consider the following situation. A student uses her laptop to download files from an Internet site. She will need her laptop from 1 pm to 2 pm to do a presentation of her project, so that the laptop cannot be used for downloading during this time slot. In the case that she fails to completely download a file by 1 pm, she will start downloading that file again after 2 pm.
Formally, we are given a set of jobs that have to be processed on a single machine. The processing of job takes time units. There is a weight associated with job j, which indicates its relative importance. All values and are positive integers. The machine processes at most one job at a time. The completion time of job in a feasible schedule S is denoted by , or shortly if it is clear which schedule is referred to. It is required to minimize the total weighted completion time, i.e., the functionExtending standard scheduling notation, we denote the resulting problem under the non-resumable scenario by , and that under the resumable scenario by . A schedule is optimal if for any feasible schedule S. Both problems are NP-hard in the ordinary sense, see [1], [11].
The fact that the problems under consideration are NP-hard calls for design and analysis of approximation algorithms. Recall that for a problem of minimizing a function , where x is a collection of decision variables, a polynomial time algorithm that finds a solution such that is at most times the optimal value is called a -approximation algorithm; the value of is called a worst-case ratio bound. Given an , a family of -algorithms is called a fully polynomial time approximation scheme (FPTAS) if its running time depends polynomially on the size of the input of the problem and .
For problem with the machine non-availability interval , assume that the jobs are numbered in such a way thatWe call the sequence of jobs numbered in accordance with (1) a Smith sequence or a WSPT sequence (weighted shortest processing time). It is well-known that in an optimal schedule for the classical problem of minimizing the sum of the weighted completion times on a single permanently available machine the jobs are processed according to the WSPT sequence, see, e.g. [15].
Lee [11] proves that for problem there exists an optimal schedule in which the jobs sequenced before interval I and after that interval obey the WSPT rule. Furthermore, if the jobs are scheduled according to their numbering (1) and the crossover job restarts from scratch at time t, then for the resulting schedule the ratio can be arbitrarily large; see [11]. For problem , effective branch and bound algorithms are developed in [5], [6]. Kacem and Chu [4] study the WSPT rule and its modification and give conditions under which these heuristics behave as 3-approximation algorithms. Together with a recent 2-approximation algorithm designed by Kacem [3], this paper is the first successful attempt to develop a constant ratio approximation algorithm for the general problem .
For the unweighted case, the numbering (1) is equivalent to the SPT (shortest processing time) rule. Lee and Liman [13] by refining the analysis by Adiri et al. [1] demonstrate that for problem the schedule found by the SPT rule guarantees that , and this bound is tight. For problem , an algorithm with a worst-case performance ratio of is given by Sadfi et al. [14], and that has been recently further improved by Breit [2].
Problem in which the crossover job is dealt with according to the resumable scenario is studied by Lee [11]. He shows that this problem is NP-hard, but its counterpart with equal weights, i.e., problem , is solvable in time, since schedule is optimal. Similarly to the non-resumable scenario, for problem there exists an optimal schedule in which the jobs sequenced before and after the crossover job follow the WSPT rule. For problem , Wang et al. [16] have recently reported a 2-approximation algorithm that requires time.
The remainder of this paper is organized as follows. In Section 2, we show that a ρ-approximation algorithm for problem can be converted into a -approximation algorithm for problem . In particular, this gives a 4-approximation algorithm for the latter problem. On the other hand, we demonstrate that the worst-case performance ratio delivered by any algorithm of this type cannot be smaller than 2. In Section 3, we present a simple -approximation algorithm for problem based on an approximate solution of a linear Knapsack problem. Section 4 contains some concluding remarks.
Section snippets
A heuristic based on the resumable scenario
In this section, we study a possibility of converting a schedule that is feasible for the resumable scenario into a schedule that is feasible under the non-resumable scenario.
For problem with the non-resumable scenario we introduce the associated problem with the resumable scenario. These two problems share the same data set, i.e., the values and for , and in both problems the same non-availability interval is given. Let denote an optimal
A heuristic based on the Knapsack problem
In this section, we present an algorithm for problem that for any positive behaves as a -approximation algorithm.
Given problem with the non-availability interval , for each job j, , introduce a Boolean variable in such a way thatRecall that we may only concentrate on schedules in which the jobs before interval I and those after interval I are sequenced according to the WSPT rule, i.e.,
Conclusion
In this note, we present approximation algorithms for the scheduling problem of minimizing the sum of the weighted completion times on a single machine with one non-availability interval and the non-resumable scenario. Our purpose was to demonstrate the existence of constant ratio approximation algorithm for the problem rather than to design a practical algorithm that would deliver a small worst-case ratio. Our best algorithm is a -approximation that runs in
References (16)
Improved approximation for non-preemptive single machine flow-time scheduling with an availability constraint
European Journal of Operational Research
(2007)Approximation algorithm for the weighted flow-time minimization on a single machine with a fixed non-availability interval
Computers & Industrial Engineering
(2008)- et al.
Worst-case analysis of the WSPT and MWSPT rules for single machine scheduling with one planned setup period
European Journal of Operational Research
(2008) - et al.
Efficient branch-and-bound algorithm for minimizing the weighted sum of completion times on a single machine with one availability constraint
International Journal of Production Economics
(2008) - et al.
Single-machine scheduling with an availability constraint to minimize the weighted sum of the completion times
Computers & Operations Research
(2008) - et al.
A fully polynomial approximation scheme for the single machine weighted total tardiness problem with a common due date
Theoretical Computer Science
(2006) - et al.
An improved approximation algorithm for the single machine total completion time scheduling problem with availability constraints
European Journal of Operational Research
(2005) - et al.
Single machine flow-time scheduling with a single breakdown
Acta Informatica
(1989)
Cited by (20)
Differential approximation schemes for half-product related functions and their scheduling applications
2017, Discrete Applied MathematicsSingle-machine scheduling with a variable maintenance activity
2015, Computers and Industrial EngineeringComputational performances of a simple interchange heuristic for a scheduling problem with an availability constraint
2014, Computers and Industrial EngineeringCitation Excerpt :There exist several other papers dealing with the problem in the literature. In particular, the preemptive case has been recently studied (Kacem & Chu, 2008c; Kellerer & Strusevich, 2010; Wang, sun, & Chu, 2005), as well as other variants (Kacem & Kellerer, 2011; Mellouli, Sadfi, Chu, & Kacem, 2009; Tan, Chen, & Zhang, 2011) including the weighted version of the problem (Kacem & Chu, 2008a, 2008b; Kacem, Chu, & Souissi, 2008; Kellerer, Kubzin, & Strusevich, 2009). We then present the results of the computational experiments we carried out to test MSPT-k on some randomly generated instances.
Weighted completion time minimization on a single-machine with a fixed non-availability interval: Differential approximability
2013, Discrete OptimizationCitation Excerpt :Several standard approximations have been proposed. A sample of them include the worst-case analysis of heuristic methods (see for example Adiri et al. [8]; Lee and Liman [10]; Sadfi et al. [11]; He et al. [12]; Wang et al. [13] and Breit [14]; Kacem and Chu [15]; Kacem [16]; Kellerer and Strusevich [17]). Efficient standard approximation schemes were also published in Kellerer and Strusevich [18]; Kacem and Mahjoub [19] and He et al. [12].
Parallel machines scheduling with machine maintenance for minsum criteria
2011, European Journal of Operational ResearchCitation Excerpt :Single machine scheduling problems with machine maintenance have been studied extensively. A lot of papers deal with the complexity, approximation algorithms and approximation schemes for problems with various criteria (see for example, Lee, 1996; Adiri et al., 1989; Lee and Liman, 1992; Sadfi et al., 2005; Kacem and Mahjoub, 2009; Kellerer et al., 2009; Kellerer and Strusevich, 2010). In this section, we prove the worst-case ratio of the simple LS algorithm for Pm, k∣nr − a∣Cmax.
Online scheduling to minimize modified total tardiness with an availability constraint
2009, Theoretical Computer Science