Particle Swarm Optimization for Scheduling to Minimize Tardiness Penalty and Power Cost

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1 Partcle Swarm Optmzaton for Schedulng to Mnmze Tardness Penalty and Power Cost Kue-Tang Fang and Bertrand M.T. Ln Department of Informaton and Fnance Management Insttute of Informaton Management Natonal Chao Tung Unversty, Hsnchu 300, Tawan e-mal: Abstract: Tradtonal research on machne schedulng focuses on job allocaton and sequencng to optmze certan objectve functons that are defned n job completon tmes. Wth regard to envronmental concerns, energy consumpton becomes another crtcal concern n hgh-performance systems. In ths paper, we address the schedulng problem n a multple machne system where the computng speeds of the machnes are allowed to be adjusted durng the course of executon. The CPU adjustment capablty enables the flexblty for mnmzng electrcty cost from energy savng by sacrfcng job completon tmes. The decson of the studed problem s to dspatch the jobs to the machne and to determne the job sequence and processng speed of each machne wth the objectve functon comprsed of the total weghted job tardness and the power cost. We gve a formal formulaton, propose two heurstcs, and desgn a partcle swarm optmzaton (PSO) algorthm. The experment results provdes qualty solutons. Keywords: schedulng; parallel machne; total weghted tardness; power consumpton; partcle swarm optmzaton; 1 Introducton In the tradtonal schedulng research, the processng speeds of machnes are assumed to be statc. In ths paper, we consder a schedulng problem wth parallel heterogeneous machnes, whch has the ablty to tune ther processng speeds. Hgher speeds permt a shorter processng makespan of the jobs; n the mean more power cost wll be ncurred. The schedulng polcy thus rests n the tradeoff between job completon tmes and power cost. To be more precsely, we have to (1) dspatch the jobs to the machnes; (2) sequence the jobs on each machne; (3) select the processng speed for each job. The objectve functon s a weghted sum of the total job tardness penalty and the total power consumpton. 235

2 2 Problem Statements and Lterature Revew Ths secton presents a formal defnton of the allocaton and schedulng of jobs n a cloud envronment from the aspect of parallel-machne schedulng. 2.1 Problem defnton and ILP Consder a set of n jobs {J 1,..., J n } to process on m non-homogeneous machnes {M 1,..., M m }. Each job J j has a processng load l j and a due date d j and a weght ndcatng the penalty of unt-tme tardness w j. The actual processng tme of a job J j depends on the speed selected for the machne n charge of t. Notaton: j : Index 1 j n for jobs. : Index 1 m for machnes. l : Index 1 l n for postons on each machne. s : Index 1 s µ for states (speeds) of machne M, where µ s the number of dfferent states of machne M. l j : Processng load of job J j. S s : Processng speed of machne M n state s. e s : Energy consumpton (per tme unt) of machne M n state s. b l : Startng tme of the job at l th poston on machne M t js : Executon tme of job J j when assgned to machne M n state s; t js = l j /S s. p js : Power cost of job J j when assgned to machne M at speed S s,.e. t js e s. Defne bnary decson varables x jls = 1 such that f job J j s assgned to the l th poston on machne M n state S s, then x jls = 1; x jls = 0, otherwse. Formulaton of Problem WTPC: Mnmze n n m n µ α w j T j + (1 α) t js e s x jls (1) =1 =1 j=1 l=1 s=1 236

3 b,l+1 b,l = m n µ x jls = 1, 1 j n; (2) =1 l=1 s=1 µ n x jls 1, 1 m, 1 l n; (3) j=1 s=1 µ n x jls t js, 1 m, 1 l n; (4) j=1 s=1 T j b,l + t js x jls (1 x jls )M d j, 1 m, 1 s µ, 1 j, l n; (5) T j 0, 1 j n; (6) x jls {0, 1}, 1 m, 1 s µ, 1 j, l n. (7) 2.2 Lterature revew The objectve functon consdered n ths paper s the mnmzaton of total weghted tardness. For the sngle-machne schedulng problem, Lawler (1977) desgned a pseudopolynomal algorthm for solvng the 1 T j problem. The complexty status remaned open untl Du and Leung (1990) gave an NP-hardness proof to confrm the ntractablty of the 1 T j problem. The weghted verson of ths problem, 1 w j T j, s known to be NP-hard n the strong sense (Lenstra, Rnnooy Kan, and Brucker 1977). In the case of parallel machnes, Lenstra et al. (1977) showed the problem wth two machnes to be ordnary NP-hard. For the case wth a fxed number of machnes, Garey and Johnson (1978) gave an proof of ordnary NP-hard. The case wth an arbtrary number of machnes turns to be NP-hard n the strong sense (Garey and Johnson, 1978). Hence, the complexty of the total weghted tardness of parallel machnes s brefly NP-hard n the strong sense. The studed problem WTPC s concerned about (a) assgnng the jobs to the machne, (b) determnng the processng sequence of job on each machne, and (3) selectng the processng speed for each job such that the weghted sum of tardness penalty and power cost s mnmzed. From the complexty of the schedulng problem wth the objectve functon of total tardness, problem WTPC s obvously strongly NP-hard. 3 Heurstcs Ths secton ntroduces two constructve heurstcs, based upon the earlest due date (EDD) rule, and the weghted shortest processng tme (WSPT) rule, respectvely. The jobs are then dspatched to the machnes by lst schedulng (Graham, 1966), whch assgns the frst unscheduled job to the machne wth the shortest processng tme untl all jobs are assgned. 237

4 The EDD rule, proposed by Jackson (1955), guarantees the optmalty n the mnmzaton of the maxmum lateness on a sngle machne. It has been wdely adopted to deal wth due-date related objectves. Ths frst heurstc algorthm s outlned n the followng: Due-date-based heurstc H EDD Step 1. Sort the jobs n non-decreasng order of due dates. Step 2. Remove the frst job, say J j, from the lst and assgn t to the shortest machne. Step 3. Select the machne frequency that wll mnmze the contrbutons that J j wll make to the objectve functon. Step 4. Execute job J j usng the frequency selected n Step 3. Update the completon tme of the machne. Step 5. Repeat Steps 2-4 untl all jobs are processed. To solve the sngle-machne schedulng problem of mnmzng the total weghted completon tme, Smth (1956) proposed the WSPT rule that sequences the jobs n nondecreasng order of the rato between job processng tme and job weght. Ths stmulates the adopton of the WSPT rule for dealng wth the problem of mnmzng the total weghted tardness, especally when most jobs tend to be tardy. WSPT-based heurstc H W SP T Step 1.Sort the jobs n non-decreasng order of the rato between processng load and weght, l j /w j. Step 2-5. The same as proposed n due-date-based heurstc. 4 Partcle Swarm Optmzaton The general framework of the PSO algorthm s descrbed below. Let x t poston of partcle at teraton t, and v t denote current the velocty of partcle at teraton t. The velocty and poston of soluton the at next teraton are computed as follows: v t+1 = ωv t + c 1 rand 1 (pbest x t ) + c 2 rand 2 (gbest x t ) (8) x t+1 = x t + v t+1 (9) where: 238

5 ω: The nerta weght. c j : The acceleraton coeffcents for j = 1, 2. pbest : The best poston ever vsted n the course of partcle ; gbest: The poston of best partcle n the swarm. PSO Algorthm Step 1. Set partcle dmenson equal to the sze of jobs n {j j } J. Step 2. Randomly generate partcle poston x and velocty v for ntalzaton. Step 3. For each partcle, calculate ts ftness value. Update pbest and gbest functon. Then, use Eq. (8) and Eq. (9) to calculate next velocty and postons. Step 4. Repeat from 3, untl the stop crtera s satsfed. 5 Computatonal Experments Ths secton presents a computatonal study for examnng the performances of the proposed heurstcs and PSO algorthms. The test nstances were generated as follows. The processng load l j of each job was randomly generated by the unform dstrbuton U(1, 100). The tardness penalty weghts w j of the jobs were drawn from U(1, 10). The generaton of due date follows the scheme proposed by Hall and Posner (2000). We set a parameter R to adjust the perod of due dates. The mean value of due dates s equal to a half of the average total processng load on each machne dvded by the average machne processng speed, namely D = 1 2 j l j m n. All the proposed algorthms were coded n Vsual Studo 2008 and executed on a personal computer equpped wth an Intel Core 2 Quad Q GHz CPU and 2GB of RAM. The executon of the PSO algorthms was termnated once ether 1,000 teratons were reached or the specfed convergence status was encountered. The performance of the PSO algorthm subject to dfferent values of R are dsplayed n Table 1. We lst the s S s best objectve values and the parameter settngs of (c 1, c 2, ω). 6 Conclusons In ths paper, we formulated a schedulng problem wth heterogeneous unrelated parallel machnes to mnmze the weghted sum of tardness penalty and power consumpton 239 objectve. Snce the studed schedulng problem s NP-hard, we desgned two heurstcs

6 Table 1: the best parameter settng R obj (c 1, c 2, ω) obj (c 1, c 2, ω) obj (c 1, c 2, ω) obj (c 1, c 2, ω) (2,1,0.2) (2,1,0.2) (1,1,0.6) (1,2,0.2) (2,1,0.2) (2,1,0.2) (2,1,0.2) (1,2,0.2) (2,1,0.2) (2,1,0.2) (1,2,0.2) (2,1,0.2) (1,1,0.6) (1,2,0.2) (1,1,0.6) (1,2,0.2) (1,2,0.4) (2,1,0.2) (2,1,0.2) (1,2,0.2) (1,1,0.6) (2,1,0.2) (1,2,0.2) (2,1,0.2) (1,1,0.6) (2,1,0.2) (2,1,0.2) (2,1,0.2) (2,1,0.2) (2,1,0.2) (1,2,0.2) (2,1,0.2) (2,1,0.2) (1,2,0.2) (1,2,0.2) (1,2,0.2) (2,1,0.2) (2,1,0.2) (2,1,0.2) (2,1,0.2) algorthms and a PSO algorthms to produce approxmate solutons. Statstcs showed that the PSO algorthm can provde qualty solutons. For further research, t could be nterestng to extend the proposed approach to dealng wth other performance crteron. We can also ntroduce the concept of CPU frequency adjustment to the tradtonal machne schedulng problems. The desgn of new PSO features, e.g. dynamcally adjustng acceleraton coeffcents as the course of teratons contnues, s another research drecton. References [1] Du, J. and Leung, J.Y Mnmzng total tardness on one machne s NP-hard. Mathematcs of Operatons Research, Vol.15(3), pp [2] Garey, M.R. and Johnson, D.S Strong NP-completeness results: Motvaton, examples, and mplcatons. Journal of the AssocmUon for Computng Machnery, Vol. 25(3), pp [3] Jackson, J.R Schedulng a producton lne to mnmze maxmum tardness. Research Report 43, Management Scence Research Project, UCLA. [4] Kennedy, J. and Eberhart, R.C Partcle swarm optmzaton. Proceedngs of IEEE Internatonal Conference on Neural Networks, pp [5] Lawler, E.L A pseudopolynomal algorthm for sequencng jobs to mnmze total tardness. Annals of Dscrete Mathematcs, Vol. 1, pp [6] Lenstra, J.K., Rnnooy Kan, A.H.G. and Brucker, P Complexty of machne schedulng problems. Annals of Dscrete Mathematcs, Vol.1, pp [7] Smth, W.E Varous optmzers for sngle-stage producton. Naval Research Logstcs Quarterly, Vol. 3, pp

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