2007-2008 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

Cheng, TCE

Ji, M

European Journal of Operational Research

PolyU Institutional Repository

Parallel-machine scheduling with simple lineardeterioration to minimize total completion timeMin Ji1College of Computer Science & Information Engineering,Zhejiang Gongshang University, Hangzhou 310018, P.R. ChinaT. C. E. Cheng2Department of Logistics,The Hong Kong Polytechnic University,Kowloon, Hong Kong1Email: jimkeen@163.com2Corresponding author. Email: LGTcheng@polyu.edu.hkThis is the Pre-Published Version.AbstractWe consider the parallel-machine scheduling problem in which the processing time of a job is a simplelinear increasing function of its starting time. The objective is to minimize the total completiontime. We give a fully polynomial-time approximation scheme (FPTAS) for the case with m identicalmachines, where m is fixed. This study solves an open problem that has been posed in the literaturefor ten years.Keywords. Parallel-machine scheduling; Deteriorating jobs; FPTAS11 IntroductionFor most scheduling problems it is assumed that the job processing times are fixed parameters [14].However, such restrictive assumptions represent an oversimplified view of reality. Job processingtimes are not necessarily fixed because jobs may deteriorate while waiting to be processed. Examplescan be found in financial management, steel production, resource allocation and national defence,etc., where any delay in processing a job may result in deterioration in accomplishing the job. Fora list of applications, the reader is referred to [2, 10, 11, 13]. Such problems are generally known asthe deteriorating job scheduling problem.Work on the deteriorating job scheduling problem was initiated by Brown and Yechiali [2],and Gupta and Gupta [6]. They focused on the single-machine makespan problem under the lineardeteriorating condition. Since then, scheduling problems with time-dependent processing times havereceived increasing attention. An extensive survey of different models and problems was providedby Alidaee and Womer [1]. Cheng, Ding and Lin [5] recently presented an updated survey of theresults on scheduling problems with time-dependent processing times.The problem under consideration can be formally described as follows: There are n independentjobs J = {J1, J2, · · · , Jn}, which are simultaneously available at time t0 > 0, to be processed non-preemptively on m identical parallel machines. We assume, as in Mosheiov [12, 13], and Chen [3, 4],that the actual processing time of job Jj is pj = αjsj , where sj and αj are the starting time and thegrowth (or deterioration) rate of Jj , respectively. The assumption ”t0 > 0” is made here to avoidthe trivial case of t0 = 0 (when t0 = 0, the completion time of each job will be 0). Let C[i] denotethe completion time of the ith job in a schedule. It is easy to see that C[i+1] = s[i+1] + p[i+1] =s[i+1](1 + α[i+1]) = C[i](1 + α[i+1]). Thus, by induction, C[i] = t0∏ij=1(1 + α[j]) for every i ≥ 1 ina schedule. Since t0 is a common factor in all the completion times, it will not affect the optimalschedule under any performance criterion. Without loss of generality, we assume that t0 and all αjfor every j are integral. Our goal is to minimize the total completion time, i.e.,∑nj=1Cj . Using thenotation of Chen [3], we denote the problem as the PTCT (i.e., parallel-machine total completiontime) problem.2The above defined problem may date back to Mosheiov [12], who first considered single-machinescheduling under the simple linear deteriorating assumption. The most commonly used performancemeasures were considered, such as makespan, total completion time, total weighted completion time,total weighted waiting time, total tardiness, number of tardy jobs, maximum lateness and maximumtardiness. He showed that all these models are polynomially solvable. For the multiple machinecase of the simple linear deteriorating problem, Mosheiov [13] showed its NP-hardness even forthe two-machine case, and presented an asymptotically optimal heuristic for the parallel-machinemakespan problem. Chen [3, 4] proved that the PTCT problem is NP-hard even with a fixed numberof machines. He also proposed an approximation algorithm with a parameter dependent worst-caseratio for the two-machine case. For a fixed number of machines, he pointed out that whether thereexists a polynomial time heuristic with a constant worst-case ratio remains open. More recently, Wuand Lee [15], and Ji et al. [7] extended the single-machine problem to the situation where the machinehas an availability constraint. Wu and Lee [15] studied the resumable case of the makespan problem.They showed that the linear deteriorating model can be solved as a 0-1 integer programming problem.Ji et al. [7] considered the non-resumable case to minimize the makespan and the total completiontime. They established their computational complexity, proposed approximation algorithms withtight bounds and presented computational results.In this paper we present a fully polynomial-time approximation scheme (FPTAS) for the PTCTproblem with a fixed number of machines, which greatly improves on the bound in Chen [3], notingthat the parameter dependent worst-case ratio given in Chen [3] may be infinity. Our FPTAS closelyfollows the FPTAS developed in [8, 9]. We have thus solved the open problem that has been posedin Chen [3] for ten years. The presentation of this paper is organized as follows. In Section 2 wepropose an FPTAS for the m-machine case of the PTCT problem, where m is fixed, and prove itscorrectness and establish its time complexity. Finally, we conclude the paper in Section 3.2 An FPTASAn algorithm A is called a (1+ε)-approximation algorithm for a minimization problem if it producesa solution that is at most 1+ε times as big as the optimal value, running in time that is polynomial in3the input size. A family of approximation algorithms {Aε} is a fully polynomial-time approximationscheme (FPTAS) if, for each ε > 0, the algorithm Aε is a (1 + ε)-approximation algorithm that ispolynomial in the input size and in 1/ε. From now on we assume, without loss of generality, that0 < ε ≤ 1. If ε > 1, then a 2-approximation algorithm can be taken as a (1 + ε)-approximationalgorithm.Mosheiov [12] showed that the smallest growth rate (SGR) order is optimal for the single-machinecase of the problem, which leads to the following property.Property 1 On each machine of an optimal solution for the PTCT problem, all the jobs are se-quenced in the SGR order, i.e., in nondecreasing order of αj.Let the jobs be indexed according to the SGR order so that α1 ≤ α2 ≤ · · · ≤ αn. We introducevariables xj , j = 1, 2, · · · , n, where xj = k if job Jj is processed on machine k, k ∈ {1, 2, · · · ,m}. LetX be the set of all the vectors x = (x1, x2, · · · , xn) with xj = k, j = 1, 2, · · · , n, k = 1, 2, · · · ,m. Wedefine the following initial and recursive functions on X:f i0(x) = t0, i = 1, 2, · · · ,m,h0(x) = 0,fkj (x) = fkj−1(x) + αjfkj−1(x), for xj = k,f ij(x) = fij−1(x), for xj = k, i 6= k,hj(x) = hj−1(x) +m∑i=1f ij(x).Thus, the PTCT problem with m machines reduces to the following problem:Minimize hn(x) for x ∈ X.We first introduce procedure Partition(A, e, δ) proposed by Kovalyov and Kubiak [8, 9], whereA ⊆ X, e is a nonnegative integer function on X, and 0 < δ ≤ 1. This procedure partitions A intodisjoint subsets Ae1, Ae2, · · · , Aeke such that |e(x) − e(x′)| ≤ δmin{e(x), e(x′)} for any x, x′ from thesame subset Aej , j = 1, 2, · · · , ke. The following description provides the details of Partition(A, e, δ).Procedure Partition(A, e, δ)4Step 1. Arrange vectors x ∈ A in the order x(1), x(2), · · · , x(|A|) such that 0 ≤ e(x(1)) ≤ e(x(2)) ≤· · · ≤ e(x(|A|)).Step 2. Assign vectors x(1), x(2), · · · , x(i1) to set Ae1 until i1 is found such that e(x(i1)) ≤ (1 +δ)e(x(1)) and e(x(i1+1)) > (1 + δ)e(x(1)). If such i1 does not exist, then take Aeke = Ae1 = A, andstop.Assign vectors x(i1+1), x(i1+2), · · · , x(i2) to set Ae2 until i2 is found such that e(x(i2)) ≤ (1 +δ)e(x(i1+1)) and e(x(i2+1)) > (1+δ)e(x(i1+1)). If such i2 does not exist, then take Aeke = Ae2 = A−Ae1,and stop.Continue the above construction until x(|A|) is included in Aeke for some ke.Procedure Partition requires O(|A| log |A|) operations to arrange the vectors of A in nondecreas-ing order of e(x), and O(|A|) operations to provide a partition. The main properties of Partitionthat will be used in the development of our FPTAS Aε were presented in Kovalyov and Kubiak [8, 9]as follows.Property 2 |e(x)− e(x′)| ≤ δmin{e(x), e(x′)} for any x, x′ ∈ Aej, j = 1, 2, · · · , ke.Property 3 ke ≤ log e(x(|A|))/δ + 2 for 0 < δ ≤ 1 and 1 ≤ e(x(|A|)).A formal description of the FPTAS Amε for the PTCT problem with m machines is given below.Algorithm AεStep 1. (Initialization) Number the jobs in the SGR order so that α1 ≤ α2 ≤ · · · ≤ αn (Property1). Set Y0 = {(0, 0, · · · , 0)} and j = 1.Step 2. (Generation of Y1, Y2, · · · , Yn) For set Yj−1, generate Y ′j by adding k, k = 1, 2, · · · ,m,in position j of each vector from Yj−1. Calculate the following for any x ∈ Y ′j , assuming xj = k.fkj (x) = fkj−1(x) + αjfkj−1(x), (1)f ij(x) = fij−1(x), for i 6= k,hj(x) = hj−1(x) +m∑k=1fkj (x).5If j = n, then set Yn = Y ′n, and go to Step 3.If j < n, then set δ = ε/(2(n+ 1)), and perform the following computations.Call Partition(Y ′j , f ij , δ) (i = 1, 2, · · · ,m) to partition set Y ′j into disjoint subsetsY fi1 , Yf i2 , · · · , Y fikfi.Call Partition(Y ′j , hj , δ) to partition set Y ′j into disjoint subsets Y h1 , Y h2 , · · · , Y hkh .Divide set Y ′j into disjoint subsets Ya1···amb = Y f1a1 ∩ · · · ∩ Y fmam ∩ Y hb , a1 = 1, 2, · · · , kf1 ; · · ·;am = 1, 2, · · · , kfm ; b = 1, 2, · · · , kh. For each nonempty subset Ya1···amb, choose a vector x(a1···amb)such thathj(x(a1···amb)) = min{hj(x) | x ∈ Ya1···amb}.Set Yj := {x(a1···amb) | a1 = 1, 2, · · · , kf1 ; · · · ; am = 1, 2, · · · , kfm ; b = 1, 2, · · · , kh, and Y f1a1 ∩ · · · ∩Y fmam ∩ Y hb 6= ∅}, and j = j + 1.Repeat Step 2.Step 3. (Solution) Select vector x0 ∈ Yn such that hn(x0) = min{hn(x) | x ∈ Yn}.Let x∗ = (x∗1, x∗2, · · · , x∗n) be an optimal solution for the PTCT problem with a fixed number ofmachines. Let L = log(max{n, 1/ε, 1 + αmax, t0}), where αmax = maxnj=1{αj}. We show the mainresult of this section in the following.Theorem 1 Algorithm Aε finds x0 ∈ X for the PTCT problem with m machines such that hn(x0) ≤(1 + ε)hn(x∗) in O(n2m+3Lm+2/εm+1).Proof. Suppose that (x∗1, · · · , x∗j , 0, · · · , 0) ∈ Ya1···amb ⊆ Y ′j for some j and a1, · · · am, b. By the defini-tion ofAε, such j always exists, for instance j = 1. AlgorithmAε may not choose (x∗1, · · · , x∗j , 0, · · · , 0)for further construction; however, for a vector x(a1···amb) chosen instead of it, we have|f ij(x∗)− f ij(x(a1···amb))| ≤ δf ij(x∗), i = 1, · · · ,m,and|hj(x∗)− hj(x(a1···amb))| ≤ δhj(x∗),6due to Property 2. Set δ1 = δ. We consider vector (x∗1, · · · , x∗j , x∗j+1, 0, · · · , 0) and x˜(a1···amb) =(x(a1···amb)1 , · · · , x(a1···amb)j , x∗j+1, 0, · · · , 0). WLOG., we assume x∗j+1 = k. It follows that|fkj+1(x∗)− fkj+1(x˜(a1···amb))|= |(fkj (x∗) + αj+1fkj (x∗))− (fkj (x(a1···amb)) + αj+1fkj (x(a1···amb)))|= |(1 + αj+1)(fkj (x∗)− fkj (x(a1···amb)))|≤ (1 + αj+1)δfkj (x∗)= δ1fkj+1(x∗),Consequently,fkj+1(x˜(a1···amb)) ≤ (1 + δ1)fkj+1(x∗),Similarly, for i 6= k, we havef ij+1(x˜(a1···amb)) ≤ (1 + δ1)f ij+1(x∗).Therefore, we obtain|hj+1(x∗)− hj+1(x˜(a1···amb))|= |(hj(x∗) +m∑i=1f ij+1(x∗))− (hj(x(a1···amb)) +m∑i=1f ij+1(x˜(a1···amb)))|≤ |hj(x∗)− hj(x(a1···amb))|+m∑i=1|f ij+1(x∗)− f ij+1(x˜(a1···amb))|≤ δ1(hj(x∗) +m∑i=1f ij+1(x∗))= δ1hj+1(x∗). (2)Consequently,hj+1(x˜(a1···amb)) ≤ (1 + δ1)hj+1(x∗).Assume that x˜(a1···amb) ∈ Yc1···cmd ⊆ Y ′j+1 and Algorithm Aε chooses x(c1···cmd) ∈ Yc1···cmd insteadof x˜(a1···amb) in the (j + 1)st iteration. We have|f ij+1(x˜(a1···amb))− f ij+1(x(c1···cmd))| ≤ δf ij+1(x˜(a1···amb)) ≤ δ(1 + δ1)f ij+1(x∗), i = 1, · · · ,m,7and|hj+1(x˜(a1···amb))− hj+1(x(c1···cmd))| ≤ δhj+1(x˜(a1···amb)) ≤ δ(1 + δ1)hj+1(x∗). (3)From (2) and (3), we obtain|hj+1(x∗)− hj+1(x(c1···cmd))|≤ |hj+1(x∗)− hj+1(x˜(a1···amb))|+ |hj+1(x˜(a1···amb))− hj+1(x(c1···cmd))|≤ (δ1 + δ(1 + δ1))hj+1(x∗)= (δ + δ1(1 + δ))hj+1(x∗). (4)Set δl = δ + δl−1(1 + δ), l = 2, 3, · · · , n− j + 1. From (4), we obtain|hj+1(x∗)− hj+1(x(c1···cmd))| ≤ δ2hj+1(x∗).Repeating the above argument for j + 2, · · · , n, we show that there exists x′ ∈ Yn such that|hn(x∗)− hn(x′)| ≤ δn−j+1hn(x∗).Sinceδn−j+1 ≤ δn∑j=0(1 + δ)j= (1 + δ)n+1 − 1=n+1∑j=1(n+ 1)n · · · (n− j + 2)j!δj=n+1∑j=1(n+ 1)n · · · (n− j + 2)j!(n+ 1)j(ε2)j≤n+1∑j=11j!(ε2)j≤n+1∑j=1(ε2)j≤ εn+1∑j=1(12)j≤ ε.Therefore, we have|hn(x∗)− hn(x′)| ≤ εhn(x∗).8Then in Step 3, vector x0 will be chosen such thathn(x0) ≤ hn(x′) ≤ (1 + ε)hn(x∗).The time complexity of Algorithm Aε can be established by noting that the most time-consumingoperation of iteration j of Step 2 is a call of procedure Partition, which requires O(|Y ′j | log |Y ′j |)time to complete. To estimate |Y ′j |, recall that |Y ′j+1| ≤ 2|Yj | ≤ 2k1fk2f · · · kmf kh. By Property 3, wehave kif ≤ 2(n + 1) log(t0(1 + αmax)n)/ε + 2 ≤ 2(n + 1)2L/ε + 2, i = 1, 2, · · · ,m, and the same forkh. Thus, |Y ′j | = O(n2(m+1)Lm+1/εm+1), and |Y ′j | log |Y ′j | = O(n2(m+1)Lm+2/εm+1). Therefore, thetime complexity of Algorithm Aε is O(n2(m+1)+1Lm+2/εm+1).3 ConclusionsThis paper studied the parallel-machine scheduling problem in which the processing time of a job isa simple linear function of its starting time to minimize the total completion time. We gave a fullypolynomial-time approximation scheme for the case with m machines, where m is fixed. Futureresearch may focus on scheduling problems with jobs of more general deterioration types. It willalso be interesting to investigate problems with other scheduling objectives.AcknowledgmentCheng was supported in part by The Hong Kong Polytechnic University under a grant from theArea of Strategic Development in China Business Services.References[1] B. Alidaee, N. K. Womer, Scheduling with time dependent processing times: Review and ex-tensions, Journal of Operational Research Society, 1999, 50: 711-720.[2] S. Brown, U. Yechiali, Scheduling deteriorating jobs on a single process, Operations Research,1990, 38: 495-498.[3] Z. L. Chen, Parallel machine scheduling with time dependent processing times, Discrete AppliedMathematics, 1996, 70: 81-93.9[4] Z. L. Chen, Erratum to ”Parallel machine scheduling with time dependent processing times”[Discrete Appl. Math. 70 (1996) 81-93], Discrete Applied Mathematics, 1997, 75: 103.[5] T. C. E. Cheng, Q. Ding, B. M. T. Lin, A concise survey of scheduling with time-dependentprocessing times, European Journal of Operational Research, 2004, 152: 1-13.[6] J. N. D. Gupta, S. K. Gupta, Single facility scheduling with nonlinear processing times, Com-puters and Industrial Engineering, 1988, 14: 387-393.[7] M. Ji, Y. He, T. C. E. Cheng, Scheduling linear deteriorating jobs with an availability constrainton a single machine, Theoretical Computer Science, 2006, 362: 115-126.[8] M. Y. Kovalyov, W. Kubiak, A fully polynomial approximation scheme for minimizing makespanof deteriorating jobs, Journal of Heuristics, 1998, 3: 287-297.[9] M. Y. Kovalyov, W. Kubiak, A fully polynomial approximation scheme for the weightedearliness-tardiness problem, Operations Research, 1999, 47: 757-761.[10] W. Kubiak, S. L. van de Velde, Scheduling deteriorating jobs to minimize makespan, NavalResearch Logistics, 1998, 45: 511-523.[11] A. S. Kunnathur, S. K. Gupta, Minimizing the makespan with late start penalties added to pro-cessing times in a single facility scheduling problem, European Journal of Operational Research,1990, 47: 56-64.[12] G. Mosheiov, Scheduling jobs under simple linear deterioration, Computers and OperationsResearch, 1994, 21: 653-659.[13] G. Mosheiov, Multi-machine scheduling with linear deterioration. INFOR, 1998, 36(4): 205-214.[14] M. Pinedo, Scheduling: Theory, Algorithms, and Systems, Prentice-Hall, Upper Saddle River,NJ, 2002.[15] C. C. Wu, W. C. Lee, Scheduling linear deteriorating jobs to minimize makespan with anavailability constraint on a single machine, Information Processing Letters, 2003, 87: 89-93.

Parallel-machine scheduling with simple linear deterioration to minimize total completion time

Abstract

Similar works

Full text

Available Versions

PolyU Institutional Repository