Scheduling with regular performance measures and optional job rejection on a single machine

Transcription

1 Schedulig with regular performace measures ad optioal job rejectio o a sigle machie Baruch Mor 1, Daa Shapira 2 1 Departmet of Ecoomics ad Busiess Admiistratio, Ariel Uiversity, Israel 2 Departmet of Computer Sciece, Ariel Uiversity, Israel Correspodig Author: Baruch Mor, Departmet of Ecoomics ad Busiess Admiistratio, Ariel Uiversity, Ariel, Israel. Phoe: Fax: baruchm@ariel.ac.il

2 Abstract We address sigle machie problems with optioal job rejectio, studied recetly i Zhag et al. [21] ad Cao et al. [2]. I these papers, the authors focus o miimizig regular performace measures, i.e., fuctios that are o-decreasig i the jobs completio time, subject to the costrait that the total rejectio cost caot exceed a predefied upper boud. They also prove that the cosidered problems are ordiary NP-hard ad provide pseudo-polyomial-time Dyamic Programmig (DP) solutios. I this paper, we focus o three of these problems: makespa with release-dates; total completio times; ad total weighted completio, ad preset ehaced DP solutios demostratig both theoretical ad practical improvemets. Moreover, we provide extesive umerical studies verifyig their efficiecy. Keywords: Schedulig; Sigle machie; Regular measures; Job-rejectio; Dyamic programmig. 2

3 1. Itroductio Durig the last decade, schedulig problems with optioal job-rejectio have attracted cosiderable iterest from researchers. Ulike the traditioal approach i determiistic schedulig theory, i which all jobs must be processed, i the cosidered family of problems we are give the choice of job rejectio. I a recet ad a comprehesive survey, Shabtay et al. [20] state that i may practical cases, mostly i highly loaded make-to-order productio systems, acceptig all jobs may cause a delay i the completio of orders which i tur may lead to high ivetory ad tardiess costs. Thus, the optio of job-rejectio provides the productio maager with a ew level of freedom, ad allows him to accept (process) oly a subset of the give jobs, while rejectig (out-sourcig or refusig to process) the complemetary subset. Uquestioably, the productio maager must cosider the cost accompaied by the rejectio, as each rejected job icurs a jobdepedet pealty. The survey of Shabtay et al. [20] also presets both the theoretical ad practical sigificace of eablig job-rejectio i schedulig problems ad describes a abudace of problems studied i this area i the past few years. Studies published later iclude e.g., Domaiç ad Plaxto [3] cosider schedulig uit jobs with a commo deadlie to miimize the sum of weighted completio times ad rejectio pealties. Shabtay [19], studies the sigle machie serial batch schedulig problem with rejectio to miimize total completio time ad total rejectio cost. Li ad Zhao [14] focus o deterioratig jobs schedulig o a sigle machie with release dates, rejectio ad a fixed o-availability iterval. Differet parallel machie settigs with jobrejectio were cosidered i several studies: Ou et al. [17] work o a improved heuristic; Jiag ad Ta [12] cosider o-simultaeous machie available time; Ma ad Yua [15] examie olie schedulig with rejectio to miimize the total weighted completio time. He et al. [9, 10] ivestigate schedulig a sigle machie with parallel batchig to miimize makespa ad total rejectio cost, ad improve algorithms for sigle machie schedulig with release dates ad rejectios, respectively. Ou et al. [16] preset faster algorithms for sigle machie schedulig with release dates ad rejectio. Wag et al. [25] study bi-criteria schedulig problems ivolvig job rejectio, cotrollable processig times ad rate-modifyig activity. Zhag et al. [22] provide approximatio algorithms for precedece-costraied idetical machie schedulig with rejectio. Zou ad Miao [24] focus o the sigle machie serial batch schedulig problems with rejectio. Li ad Che [13] research schedulig with rejectio ad a deterioratig maiteace activity o a 3

4 sigle machie. Agetis ad Mosheiov [1] cosider schedulig with job-rejectio ad positiodepedet processig times o proportioate flowshops. Gerstl ad Mosheiov [5] propose DPs ad heuristics for sigle machie schedulig problems with geeralized due-dates ad jobrejectio. Gerstl et al. [6] explore mimax schedulig problems with acceptable lead-times, ad propose extesios to positio-depedet processig times, due-widow ad job-rejectio. Zhog et al. [23] address schedulig with release times ad rejectio o two parallel machies. I this paper, we address several fudametal sigle-machie problems with optioal jobrejectio, studied i Zhag et al. [21] ad Cao et al. [2]. I these studies, the authors focus o miimizig regular performace measures, i.e., fuctios that are o-decreasig i the jobs completio time, subject to the costrait that the total rejectio cost caot exceed a predefied upper boud. The authors prove that the cosidered problems are ordiary NP-hard ad provide DP algorithms for all problems, except for the total completio time criteria. Zhag et al. [21] ad Cao et al. [2] preset FPTAS for miimizig the makespa with a release-date problem ad miimizig the total weighted completio time problem, respectively. I this paper, we focus o three of these problems: makespa with release-dates, total completio times, ad total weighted completio. We provide DP solutios demostratig both theoretical ad practical improvemets. Moreover, we have coducted extesive umerical studies to all solutios, which empirically validates the capability of our DPs to solve large-size istaces for the first two problems. For the third problem, the DP is show to be suited for medium sized orders. Our paper is costructed as follows. Sectio 2 provides the formulatio of the geeral problem. I Sectios 3, 4 ad 5 we preset the DP algorithms for the problems: miimizig the makespa with release-dates, miimizig the total completio time, ad miimizig the total weighted completio time, respectively. Sectio 6 cocludes. 4

5 2. Notatio ad formulatio A set J of jobs eeds to be processed o a sigle machie. All the jobs are available at time zero, ad preemptio is ot allowed. The scheduler is give the optio to accept (process) a subset A of the jobs ad to reject the complemetary set, R, thus J = A R ad A R =. The processig time of job j is deoted by p j, j = 1,,. For a give schedule, C j deotes the completio time of job j, j A. The rejectio cost of job j is deoted by e j, j = 1,, ad the total accepted rejectio upper boud is deoted by U. I all three problems discussed i these paper the goal is to fid a optimal schedule that miimizes a o-decreasig fuctio, of the completio times of the accepted jobs, subject to a costrait o the total rejectio cost U. For the first problem, we add a job depedet release-date deoted by r j, j J ad the schedulig criteria is the makespa. Thus, the scheduler eeds to miimize the makespa of the accepted jobs, C max = j A p j, subject to the costrait that total rejectio cost does ot exceed the upper boud U. Usig the three-field otatio, itroduced by Graham et al. [8], the first problem deoted by P1 is: P1: 1/r j, j R e j U/C max. I our secod problem, we aim to miimize the total completio time give that the total rejectio cost caot exceed U, thus the problem deoted by P2 is: P2: 1/r j, j R e j U/ C j. The last problem addressed i this paper is miimizig the total weighted completio time subject to the costrait that total rejectio cost does ot exceed the upper boud U. Let w j deote the job depedet weight of job j J ad the third problem deoted by P3 is: P3: 1/r j, j R e j U/ w j C j. 5

6 3. Problem P1: 1/r j, j R e j U/C max Zhag et al. [21] prove that the problem 1/ j R e j U/C max, is NP-hard based o the reductio from the well-kow miimizatio Kapsack problem (see Gützer ad Jugickel [7]). Thus, the exteded problem, P1, i which release dates of the jobs are itroduced, is also NP-hard. The authors the provide a O ((r + j=1 p j )) time DP algorithm for solvig the problem, provig that the problem is ordiary NP-hard. I this sectio, we provide a improved DP algorithm for Problem P1, with ruig time of O(U). Based o the theorem by Jackso [11] who showed that the problem 1/r j /C max is optimally solved by sortig the jobs i a odecreasig order of the release dates, i.e., r 1 r 2 r 1 r, the jobs are first sorted accordigly. We defie the followig state variables: Let f(j, i) deote the miimum completio time for the partial schedule of jobs 1,, j with maximum rejectio cost i. At each iteratio of the DP, the miimum completio time of jobs 1 to j, havig a upper boud i o the rejectio cost is computed, based o the completio time of jobs 1 to j 1, with a upper boud rejectio cost of either i or i e j. At each stage, oe eeds to decide whether to accept or reject job j, as follows, i. Job j must be accepted i case its rejectio cost exceeds the curret rejectio limit i. ii. Job j may be accepted i case its release time is at most or higher tha the miimum completio time of jobs 1 to j 1. I these cases, upo acceptace, the completio time should be icreased by p j or by r j + p j, i correspodece. iii. Job j may be rejected i case it miimizes the total completio time. Thus, we obtai the followig recursio formula, Dyamic programmig algorithm DP1: { mi(f(j 1, i e j), r j + p j ), f(j 1, i) < r j, e j i mi(f(j 1, i e f(j, i) = j ), f(j 1, i) + p j ), f(j 1, i) r j { r. (1) j + p j, f(j 1, i) < r j, e { f(j 1, i) + p j, f(j 1, i) r j > i j 6

7 The boudary coditios are: f(j, 0) = { C j 1 + p j, r j + p j, C j 1 r j, 0 j, as i = 0 implies that o job ca be rejected; C j 1 < r j f(0, i) = 0, 0 i mi( j=1 e j, U), implies that o jobs are cosidered. The optimal solutio is give by f(, U). Theorem 1: The computatioal complexity of DP1 is O( mi( j=1 e j, U)). Proof: Usig the formula i (1), the dyamic programmig is calculated for every job j, 1 j, ad every rejectio cost i mi( j=1 e j, U), thus implyig a O( mi( j=1 e j, U)) processig time. Recostructig the solutio is doe by backtrackig, startig at f(, mi( j=1 e j, U)) ad edig at f(0,0), for a additio of O( + mi( O( mi( j=1 e j, U)) processig time. j=1 e j, U)) operatios, for a total of Example 1: Cosider the followig istace of the problem with = 10 ad U = 93, ad the jobs are sorted i o-decreasig order of the release dates ad reumbered. The job processig times are, p = (47, 41, 20, 42, 31, 15, 12, 21, 18, 24). The job release dates are, r = (18, 70, 81, 102, 144, 302, 316, 354, 359, 365). The job rejectio costs are, e = (44, 14, 20, 28, 16, 29, 46, 32, 38, 1). Applyig DP1, we obtai the followig optimal solutio: The set of rejected jobs is, R = (J 5, J 8, J 9, J 10 ). The set of accepted jobs is, A = (J 1, J 2, J 3, J 4, J 6, J 7 ). The optimal makespa is C max = 329, ad the solutio valid sice j R e j = = U. 7

8 Numerical study: We performed umerical tests i order to measure ruig times of DP1. We coded all the experimets i this paper i C++ ad executed them o a Itel (R) Core i5-6200u 2.30 GHz 4.0 GB RAM platform. We geerated radom istaces havig = 500, 1000, 1500 ad 2000 jobs. The job processig times ad the job rejectio costs were geerated uiformly i the iterval [1, 50]. Let p max = max{p j j J } deote the maximal processig time ad e max = max{e j j J } deote the latest release date, thus p max = e max = 50. The release-dates were geerated uiformly i the iterval [0, 0.80p max ] to reflect full spectrum of zero to early the total sum of processig times. To avoid trivial solutios, i.e. all the jobs are either rejected or accepted, the total rejectio cost upper boud (U), was geerated uiformly i the iterval [0.20r max, 0.30r max ]. Actually, the geerated U-values guaratee approximately equal umber of rejected ad accepted jobs. For each set of ad U, 20 istaces were costructed ad solved. Table 1 presets the average ad worst case ruig times i millisecods. The umber of jobs,, is give i the first colum. The cosidered itervals i which U was chose uiformly, is give i the secod colum. The third ad fourth colums preset the average ad worst case ruig times, respectively. The results idicate that DP1 is extremely efficiet ad ca solve large-size problems. I particular, the worst-case ruig time for problems of 2000 jobs did ot exceed 0.58 secods. U Average ruig time [msec] Worst case ruig time [msec] 500 [5000, 7500] [10000, 15000] [15000, 22500] [20000, 30000] Table 1: Average ad worst case ruig times of DP1 algorithm for Problem P1. 8

9 4. Problem P2: 1/ j R e j U/ C j Zhag et al. [21] proved that the problem P2, is NP-hard, based o reductio from the eve odd partitio problem (Garey ad Johso [4]), but did ot preset a solutio procedure for the problem. I the followig, we provide a DP algorithm for Problem P2 ad thus prove that it is ordiary NP-hard. It is well-kow that the Shortest Processig Time first (SPT) rule, i.e., p 1 p 2 p is optimal for 1// C j, (see Piedo [18]). Thus, we start the DP by sortig the jobs i SPT order. Let f(j, i) deote the total completio time for the partial schedule of jobs 1,, j ad maximum rejectio cost i. Similar to DP1, at each iteratio of the DP, oe eeds to decide whether to accept job j, ad thus to icrease the total completio time, or rather to reject job j, i case its rejectio cost does ot exceed the curret rejectio cost i. Thus, the formal recursio fuctio is, Dyamic programmig algorithm DP2: f(j, i) = { mi (f(j 1, i) + C j, f(j 1, i e j )), e j i f(j 1, i) + C j, e j > i. (2) The boudary coditios are: j f(j, 0) = k=1 C k, 0 j. f(0, i) = 0, 0 i mi( j=1 e j, U). The optimal solutio is give by f(, U). Theorem 2: The computatioal complexity of DP2 is O( mi( j=1 e j, U)). Proof: See the proof for Theorem 1 i Sectio 3. Example 2: Cosider the followig istace of the problem with = 10 ad U = 66, ad the jobs are sorted i SPT order ad reumbered. The job processig times are, p = (15, 18, 23, 24, 28, 33, 36, 38, 46, 47). 9

10 The job rejectio costs are, e = (21, 46, 7, 10, 15, 32, 33, 10, 46, 29). Applyig DP2, we obtai the followig optimal solutio: The set of rejected jobs is, R = (J 1, J 3, J 4, J 5, J 8 ). The set of accepted jobs is, A = (J 2, J 6, J 7, J 9, J 10 ). The optimal makespa is C j = 469, ad the solutio valid sice j R e j = = U. Numerical study: We adapted the scheme plaed for Problem P1 to suit Problem P2 by elimiatig the jobs release-dates. Agai, to avoid trivial solutios the total rejectio cost upper boud (U), was geerated uiformly i the iterval [0.10r max, 0.15r max ]. Table 2, havig the same structure as Table 1, presets the average ad worst case ruig times. The worst-case ruig time for istaces of 2000 jobs did ot exceed 193 msec, demostratig that DP2 is extremely efficiet ad ca be used to solve real life-size problems. U Average ruig time [msec] Worst case ruig time [msec] 500 [2500, 3750] [5000, 7500] [7500, 11250] [10000, 15000] Table 2: Average ad worst case ruig times of DP2 algorithm for Problem P2. 10

11 5. Problem P3: 1/ j R e j U/ w j C j I this sectio we address Problem P3, studied i Cao et al. [2]. The authors prove that the cosidered problem is NP-hard, ad preset a DP solutio with computatioal complexity of O( 3 2 p max w max ) time, where p max = max{p j j J } ad w max = max{w j j J }. Therefore, they suggest a FPTAS algorithm. The suggested DP rus i O( j=1 j=1 p j U). As p j p max, our solutio is at least faster by a factor of. We start our DP by sortig the jobs i WSPT (Weighted Shortest Processig Time) first order, i.e., o icreasig order of p 1 w 1 p 2 w 2 p w. Let f(j, t, i) deote the total weighted completio time for the partial schedule of jobs 1,, j, havig completio time t ad maximum rejectio cost i. Similar to DP1, at each iteratio of the DP, oe eeds to decide whether to accept job j, ad thus icrease the total weighted completio time, or rather reject job j, i case job j s rejectio cost does ot exceed the curret rejectio limit i. The formal recursio fuctio is as follows. Dyamic programmig algorithm DP3 f(x) =, p j > t ad e j > i f(j 1, t p j, i) + w j t, f(j 1, t, i e j ), { mi ( f(j 1, t p j, i) + w j t, f(j 1, t, i e j )), The boudary coditios are: p j t ad e j > i p j > t ad e j i. (3) p j t ad e j i f(0,0, i) = 0 for all i U, as if the set of jobs is empty, their total completio time is 0, ad so is their cost. f(0, t, i) = if t 0 ad i U, as if the set of jobs is empty, their total completio time must be 0. j j f(j, t, 0) = k=1 w k C k, ad t = k=1 p k. I case the rejectio upper boud is 0, all jobs must be processed. The optimal solutio is give by mi{f(, t, i) 0 t j=1 p j, 0 i e j j J }. 11

12 Theorem 3: The computatioal complexity of DP3 is O( j=1 p j mi( e j, U The proof is similar to that of Theorem 1 i Sectio 3. j=1 )). At first sight, oe could have used formula (2) ad replaced the additive C j with w j C j. However, the followig example ituitively explais the mai differece betwee DP2 ad DP3. Cosider for example 5 jobs havig processig times 24, 44, 34, 25 ad 47, rejectio costs 19, 19, 36, 40 ad 34, job weights 16, 15, 11, 8 ad 5, ad rejectio upper boud U = 55, respectively. The jobs p j w j are 1.5, 2.9, 3.1, 3.1 ad 9.4, thus the jobs are give i WSPT order. Whe job J 1 is chose, its charge is w 1 p 1 = =384. Alteratively, if Job J 3 was the first accepted job, it would have icurred a cost of w 3 p 3 = =374, which is less tha the cost of job J 1. At a superficial glace it seems cheaper to process job J 3 rather tha job J 1. However whe cosiderig the succeedig jobs ad takig ito accout their relative high rejectio costs, 40 ad 34, it is better to process jobs J 4 ad J 5, tha reject them. Thus, as the processig time of job J 1 is less tha the processig time of job J 3, addig the costs of jobs J 4 ad J 5 results i: w 1 p 1 + w 4 (p 1 + p 4 ) + w 5 (p 1 + p 4 + p 5 ) < w 3 p 3 + w 4 (p 3 + p 4 ) + w 5 (p 3 + p 4 + p 5 ), which is ot as expected. The fact that jobs may be rejected ecessitates the computatio of the cost of each job at every time uit, as if it was its completio time, which justifies the additio of aother state variable t. Example 3: Cosider the followig istace of the problem with = 10, U = 88, ad the jobs are reumbered accordig to their WLPT order. The job processig times are, p = (5, 20, 28, 13, 33, 35, 16, 35, 41, 48). The job weights are, w = (8, 20, 24, 9, 18, 18, 8, 17, 19, 1). The job rejectio costs are, e = (36, 23, 6, 31, 3, 40, 22, 10, 32, 21). Applyig DP3, we obtai the followig optimal solutio, The set of rejected jobs is, R = (J 2, J 3, J 5, J 8, J 9 ), 12

13 The set of rejected jobs is, A = (J 1, J 4, J 6, J 7, J 10 ), w j C j = 1825, ad j R e j = = U Numerical study: We adapted the scheme plaed for Problem P2 to suit Problem P3 by addig job-depedet weights, which were geerated uiformly i the iterval [1, 25]. To guaratee solutios with approximately equal umbers of rejected ad accepted jobs the total rejectio cost upper boud (U), was geerated uiformly i the iterval [0.15r max, 0.20r max ]. Table 3, presets the average ad worst case ruig times i the same format as Table 1. As ca be see, the empirical experimet results cofirm that DP3 is efficiet ad ca be used to solve medium size istaces, the worst-case ruig time for problems of 40 jobs did ot exceed 93 msec. U Average ruig time [msec] Worst case ruig time [msec] 10 [75, 100] [150, 200] [225, 300] [300, 400] Table 3: Average ad worst case ruig times of DP3 algorithm for Problem P3 6. Coclusios I this paper, we focused o miimizig regular performace measures with optioal jobrejectio. We itroduced improved time complexity DP algorithms to well-kow DPs i schedulig theory. Moreover, extesive umerical studies were preseted, that support our achievemets, ad suggest that the ehaced DPs, for miimizig the makespa ad miimizig the total completio time, are extremely fast ad efficiet, ad thus suitable for solvig large size real life problems. 13

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