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1 Solvng Machne Schedulng Problem Usng Partcle Sarm Optmzaton Method Hanan A. Chachan Mathematcal Dept./College of Scence/Al-Mustansryah Unversty Abstract In ths paper the problem of schedulng n jobs n a sngle machne s consdered to mnmze the total cost of sum eghted completon tme, mum eghted lateness and mum penalty earlness (,e to mnmze the multple objectve functons h W C L E )). The Partcle Sarm Optmzaton (PSO) methods are appled ( as ne local search method on a set of randomly generated problems to solve machne schedulng problem th multple objectve functons. Comparson studes are made beteen PSO and Genetc Algorthm (GA) to sho hch one s the better method n applcatons. In addton, tunng the parameters of every method has been suggested n order to mprove the applcaton of every method. A ne style of development steps has been proposed to acheve good convergence n applcaton. Snce our problem s NPhard, e propose a ne heurstc method lke partcle sarm optmzaton to fnd near optmal solutons specally hen the number of jobs exceed the ablty of some exact methods lke Branch and Bound Methods (BAB) n solvng such problems. Last, the to proposed local search methods results are compared th complete search method n solvng problem lke machnes schedulng problem. Computatonal experence found that these local search algorthms solve problem to '2000 'jobs th reasonable tme. 1. Introducton The functon to be mzed or mnmzed s called the objectve functon. A vector, x for the standard mum problem or y for the standard mnmum problem, s sad to be feasble f t satsfes the correspondng constrants. The set of feasble vectors s called the constrant set. A lnear programmng problem s sad to be feasble f the constrant set s not empty; otherse t s sad to be nfeasble. A feasble mum (resp. mnmum) problem s sad to be unbounded f the objectve functon can assume arbtrarly large postve (resp. negatve) values at feasble vectors; otherse, t s sad to be bounded. The value of a bounded feasble mum (resp, mnmum) problem s the mum (resp. mnmum) value of the objectve functon as the varables range over the constrant set. A feasble vector at hch the objectve functon acheves the value s called optmal [20]. The meanng of the Partcle Sarm Optmzaton (PSO) refers to a relatvely ne famly of algorthms that may be used to fnd optmal (or near optmal) solutons to numercal and qualtatve problems. PSO s an extremely smple algorthm that seems to be effectve for optmzng a de range of Applcatons [17]. Genetc Algorthms (GA s) are a class of optmzaton algorthms. GA s attempts to solve problems through modelng a smplfed verson of genetc process. There are many problems for hch a GA approach s useful. It s, hoever, untradtonal f assgnment s such a problem [16]. 197

2 2. Multple Objectve Problems The Machne Schedulng Problems (MSP) plays a very mportant role n most manufacturng and producton systems as ell as n most nformaton processng envronment. Schedulng theory has been developed to solve problems occurrng n for nstance producton facltes. The basc schedulng problem can be descrbed as fndng for each of the tasks, hch are also called jobs, an executon nterval on one of the machnes that are able to execute t, such that all sde-constrants are met; obvously, ths should be done n such aay that the resultng soluton, hch s called a schedule, s best possble, that s, t mnmzes the gven objectve functon [8]. For many years, schedulng researchers focused on sngle regular performance measures that are non-decreasng n job completon tme. Typcally, each crteron has been studed separately, even though most real lfe schedulng problems nvolve multple crtera [4]. Hoever fe studes consdered multple crtera together. Three types of multple crtera problem can be dentfed. The frst of these types of problems nvolves dentfyng all sequence that mnmzes the frst objectve. One of these sequences that mnmze a second objectve s chosen as the optmal sequence for that problem, ths approach s called herarchcal approach [4]. The second of these multple crtera problems, hen the crtera are eghted dfferently, an objectve functons and transform the problem nto a sngle crteron schedulng problem. Ths approach s called smultaneous optmzaton along th the thrd type of multple crtera problems [4]. The thrd one of these multple crtera problems s gong to consder both crtera as equally mportance. The problem no s to fnd a sequence that does ell on both objectves. Schedulng problem s specfc case of the multple objectve (mult-crtera) schedulng problems can be formulated as follos: mnmze or mze F(s)=(f 1 (s),(f 2 (s),,f k (s)) s.t. ss here s s a soluton, S s the set of feasble soluton, k s number of objectves n problem, F(s) s the mage of s n the k-objectve space and each f (s), =1,,k represents one (mnmzaton or mzaton) objectve. In many problems, the am s to obtan the optmal arrangement of group of dscrete enttes n such a ay that the addtonal requrements and constrants (f they exst) are satsfed. If the problem s a mult objectve one, varous crtera exst to evaluate the equalty of soluton and there s an objectve (Mn. or Max.) attached to each of these crtera [4]. The lterature on multple objectve problems for sngle machne problems s summarzed by Dleepan and Sen [3], Fry et al [5], Hoogeveen [6], Lee and Varaktaraks [11] and Nagar et al [12] provde a detaled for MSP s. In ths paper e consder the problem of schedulng n jobs on a sngle machne to mnmze total cost of sum of eghted completon tme, mum eghted lateness and mum penalty earlness (.e. to mnmze the multple objectve functons h W C L E )). ( Ths MSP can be descrbed as follos: a set of n jobs N={1,2,,n} are avalable for processng at tme zero and each j requres processng durng an unnterrupted perod of gven length p j and deally should be completed at ts due-date d j. Our objectve s to fnd a sequence that mnmzes the multple objectve functons, the 198

3 sum eghted completon tme, mum eghted lateness and mum penalty h earlness. Ths problem denoted by 1 / / W C L E, our man object s to fnd the optmal and near optmal schedules that mnmze the multple objectve functons. 3. Problem Formulaton We consder the problem of schedulng n jobs on a sngle machne to mnmze the sum eghted completon tme, mum eghted lateness and mum penalty earlness (.e. to mnmze the multple objectve functons (MOF)). Let N={1,2,,n} be the set of ndependent jobs (.e. no precedence constrants are on jobs) hch are processed on a sngle machne. The jobs have to be scheduled thout preempton on the machne, here the machne can process one job at a tme (.e. the machne cannot process to jobs at the same tme). Gven a schedule (1,2,,n) then for each job j e calculate the completon tme C j p k k1 s.t. no to jobs overlap n ther executon, the earlness and tardness of job j are defned by: E {d C,0} and L j =C j -d j respectvely correspondng a job s called early f t j 1 jn j j s completed before ts due-date and tardy f t s completed after ts due-date. If a job s completed exactly at ts due-date, then t s called just-n-tme, the objectve s to fnd a: Mn F( ) mn{ S n j1 ( j) C ( j) L E h Subject to C (j) p (j), j=1,2,,n. C (j) C (j-1) + p (j), j=2,3,,n. L = { (j) L (j) }, j=1,2,,n. (p) L (j) = C (j) - d (j), j=1,2,,n. L (j) 0, j=1,2,,n. E h = {h (j) E (j) }, j=1,2,,n. E (j) = { d (j) -C (j),0}, j=1,2,,n. E (j) 0, j=1,2,,n. (j) 0, h (j) 0, j=1,2,,n. } processng order of jobs, =((1),(2),...,(n)) hch mnmzes the multple objectve functons (MOF) defned by: Mn F( ) mn{ S n j1 ( j) C ( j) L E h } (1) here (j) s postve eghted for completon tme of job j, as ell as, t s postve number (penalty for tardness of job j), S s the set of all feasble solutons, s a schedule n S. 4. Dervaton of Loer and Upper Bound 199

4 In ths secton of the paper e ll derve the loer and upper bound of machne schedulng problem usng the objectve functon (19). 4.1 Dervaton of Loer Bound Consder the formulaton of the problem (p), the problem can be decomposed nto three subproblems th a smple structure. Then the loer bound of the problem (p) s the sum of the mnmum value for each subproblem. Consder the three subproblems (p 1 ),(p 2 ) and (p 3 ) as follos: Z n 1 S j1 mn{ C } (2) ( j) ( j) Subject to C (j) p (j), j=1,2,,n. C (j) C (j-1) + p (j), j=2,3,,n. (p 1 ) Z mn{l } (3) 2 S Subject to L = { (j) L (j) }, j=1,2,,n. (p 2 ) L (j) = C (j) - d (j), j=1,2,,n. L (j) 0, j=1,2,,n. (j) 0, j=1,2,,n. Z 3 mn{e S h } (4) Subject to E h = {h (j) E (j) }, j=1,2,,n. E (j) = { d (j) -C (j),0}, j=1,2,,n...(p 3 ) E (j) 0, j=1,2,,n. h (j) 0, j=1,2,,n. Its clear that for the decomposton, (p 1 ),(p 2 ) and p( 3 ) have smpler structure than (p), and thus appear easly frst to solve optmalty for (p 1 ) to get z 1 by applyng shortest eghted processng tme (SWPT) rule. Second, to solve optmalty for (p 2 ) to get z 2 by applyng Laler algorthm [10], Laler algorthm says that, the 1/ /f problem s mnmzed as follos: hle there are unassgned jobs, assgn the job that has mnmum cost hen scheduled n that last unassgned poston n that poston. Thrd, to get mnmum value z 3 for (p 3 ) by usng large mum slack tme (LST) rule, th set h (j) =1 for all j, j=1,2,,n. the LST rule [7] says that, the 1/ /E problem s solved by sequencng the jobs accordng to the SMT rule, that s n order non-ncreasng d j -p j. Last, hence LB=z 1 +z 2 +z 3 as a loer bound for the problem, snce: mn{ S n j1 ( j) C ( j) L E h } z 1 +z 2 +z 3 = LB (5) 200

5 4.2 Dervaton of Upper Bound We propose to use a smple heurstc soluton hch s obtaned by orderng the jobs n SWPT rule to provde an ntal upper bound (UB) on the MOF. Let the, =((1),(2),...,(n)) be such ordered, then: UB= n j1 h ( j) C( j) L E 5. Partcle Sarm Optmzaton (PSO) One of the mportant ne learnng methods s a Partcle Sarm Optmzaton (PSO), hch s smple n concept, has fe parameters to adjust and easy to mplement. PSO has found applcatons n a lot of areas. In general, all the applcaton areas that the other evolutonary technques are good at are good applcaton areas for PSO [17]. PSO as orgnally developed by a socal-psychologst J. Kennedy and an electrcal engneer R. Eberhart n 1995 and emerged from earler experments th algorthms that modeled the flockng behavor seen n many speces of brds. Where brds are attracted to a roostng area n smulatons they ould begn by flyng around th no partcular destnaton and n spontaneously formed flocks untl one of the brds fle over the roostng area [9]. PSO has been an ncreasngly hot topc n the area of computatonal ntellgence. It s yet another optmzaton algorthm that falls under the soft computng umbrella that covers genetc and evolutonary computng algorthms as ell [15]. The evoluton of several paradgms outlned, and an mplementaton of one of the paradgms had been dscussed. In 1999, Eberhart R.C. and Hu X. [8], arranged a ne method for the analyss of human tremor usng PSO hch s used to evolve an NN that dstngushes beteen normal subject and those th tremor. In 2004, Sh Y. [17], surveyed the research and development of PSO n fve categores: algorthms, topology, parameters, hybrd PSO algorthms, and applcatons. There are certanly other research orks on PSO hch are not ncluded due to the space lmtaton. PSO s an extremely smple concept, and can be mplemented thout complex data structure. No complex or costly mathematcal functons are used, and t doesn t requre a great amount of memory [17]. The facts of PSO has fast convergence, only a small number of control parameters, very smple computatons, good performance, and the lack of dervatve computatons made t an attractve opton for solvng the problems. 5.1 Ftness Crteron One of these stoppng crtera s the ftness functon value. The ftness value s related by the knd of the objectve functon, the PSO can be appled to mnmze or mze ths functon, n ths paper e focused n mnmzng the objectve functon n order to mprove the results. 201

6 5.2 PSO Algorthm v k d The PSO algorthm depends on ts mplementaton n the follong to relatons: k1 k1 k1 k1 k1 k1 v c * r *( p d x d ) c * r ( p gd x d ) (5a) d k k1 k xd xd vd (5b) here c 1 and c 2 are postve constants, r 1 and r 2 are random functons n the range [0,1], x =(x 1,x 2,,x d ) represents the th partcle; p =(p 1,p 2,,p d ) represents the best prevous poston (the poston gvng the best ftness value) of the th partcle; the symbol g represents the ndex of the best partcle among all the partcles n the populaton, v=(v 1,v 2,,v d ) represents the rate of the poston change (velocty) for partcle [2]. The orgnal procedure for mplementng PSO s as follos: 1. Intalze a populaton of partcles th random postons and veloctes on d- dmensons n the problem space. 2. PSO operaton ncludes: a. For each partcle, evaluate the desred optmzaton ftness functon n d varables. b. Compare partcle's ftness evaluaton th ts pbest. If current value s better than pbest, then set pbest equal to the current value, and p equals to the current locaton x. c. Identfy the partcle n the neghborhood th the best success so far, and assgn t ndex to the varable g. d. Change the velocty and poston of the partcle accordng to equaton (5a) and (5b). 3. Loop to step (2) untl a crteron s met. Lke the other evolutonary algorthms, a PSO algorthm s a populaton based on search algorthm th random ntalzaton, and there s an nteracton among populaton members. Unlke the other evolutonary algorthms, n PSO, each partcle fles through the soluton space, and has the ablty to remember ts prevous best poston, survves from generaton to another. The flo chart of PSO algorthm s shon n fgure (1) [20]

7 Start Intalze the partcle populaton Evaluate the ftness of each partcle ftness<p d Yes Rene p d and poston No p d <p gd Yes Rene p gd No v d = *v d + c 1 *r 1 *(p d -x d )+c 2 * r 2 * (p gd -x d ) x d = x d + v d No crteron end? Yes End Fgure (1) Flochart of PSO Algorthm [16]. 5.3 The Parameters of PSO [18],[13] A number of factors ll affect the performance of the PSO. These factors are called PSO parameters, these parameters are: 1. Number of partcles n the sarm affects the run-tme sgnfcantly, thus a balance beteen varety (more partcles) and speed (less partcles) must be sought. 2. Maxmum velocty (v ) parameter. Ths parameter lmts the mum jump that a partcle can make n one step. 3. The role of the nerta eght, n equaton (5a), s consdered crtcal for the PSO s convergence behavor. The nerta eght s employed to control the mpact of the prevous hstory of veloctes on the current one. 4. The parameters c 1 and c 2, n equaton (5a), are not crtcal for PSO s convergence. Hoever, proper fne-tunng may result n faster convergence and allevaton of local mnma, c 1 than a socal parameter c 2 but th c 1 + c 2 = The parameters r 1 and r 2 are used to mantan the dversty of the populaton, and they are unformly dstrbuted n the range [0,1]. 6. Genetc Algorthms (GA s) Genetc Algorthms (GA s) are search algorthms based on the mechancs of natural selecton and natural genetcs. They combne survval of the fttest among strng structures th a structured yet randomzed nformaton exchange to form a 203

8 search algorthm th some of the nnovatve flar of human search [2]. GA s an teratve procedure, hch mantans a constant sze populaton of canddate soluton. Durng each teraton step (Generaton) the structures n the current populaton are evaluated, and, on the basc of those evaluatons, a ne populaton of canddate solutons formed. The basc GA cycle shon n fgure (2) [1]. Old Populaton Ne Populaton Evaluaton Selecton Mutaton Matng Crossover Fgure (2) Basc cycle of GA [1]. An abstract ve of the GA s: Generaton=0; Intalze G(P); {G=Generaton ; P=Populaton} Evaluate G(P); Whle (GA has not converged or termnated) Generaton = Generaton + 1; Select G(P) from G(P-1); Crossover G(P); Mutate G(P); Evaluate G(P); End (Whle) Termnate the GA [20]. 7. Implementaton of Evolvng Methods n MSP Obvously the problems ncludng more than to crtera are more dffcult. So there s a need for local search methods to treat a large sze nstances problem. Ths s the man am of the present paper. Effectvely, evolvng methods or can be called Local Search methods lke PSO and GA have demonstrated ther ablty to solve mult objectve problems to fnd near optmal soluton to the problem (p). In ths secton, e are gong to descrbe the to methods of local search In ths secton e ll mplement to Frst, s the PSO as the man ne method, and the second, s GA as comparatve method to compare the results obtaned from the to methods n order to fnd hch s better. Before e dscuss each of method, e have to talk about the common bascs beteen the to methods, these bascs are: 1. Problem Defnton The most promnent member of the rch set of combnatoral optmzaton problems s undoubtedly the Machne Schedulng Problem (MSP). In ths problem, n jobs ant to be executed n a sngle machne n some arrangement 204

9 hch gves mnmum objectve functon (19). Obvously, a sngle machne schedulng problem sequencng example of NP-complete, the ork area to be explored gros exponentally accordng th number of jobs, and so does. In general, f n jobs ere must be arranged n a sngle machne, then the general complexty s n!. 2. Problem Representaton The chromosome representaton should be an nteger vector. In ths partcular approach e accept schedule representaton hch s descrbed as a lst of jobs. For example of (10) jobs numbered from 1 to 10. Table (1) shos the 1p 10, 1d 100, 1 10 and 1h 10 of the MSP all generated randomly. Table (1) Example of (10) jobs generated randomly for the MSP p d h Intal Populaton For the ntalzaton process e can ether use some heurstcs startng from dfferent jobs, or e can ntalze the populaton by a random sample of permutaton of {1,2,,n}. 7.1 Use of GA n MSP No e ll dscuss the Use of GA frst snce t has been used before n MSP for many tmes. 1. Genetc Operators Selecton Operator Ths method uses the roulette heel selecton method. The strng th lo ftness has a hgher probablty of contrbutng one or more offsprng to the next generaton. Crossover Operator The strength of genetc algorthms arses from the structured nformaton exchange of crossover combnatons of hghly ft ndvduals. So hat e need s a crossover-lke operator that ould explot mportant smlartes beteen chromosomes. For that purpose the crossover used n ths algorthm s the Order Crossover (OX) [21], gven to parents, bulds offsprng by choosng a subsequence of a tour from one parent and preservng the relatve order of jobs from the other parent. For example, f the parents are: v 1 = ( ) v 2 = ( ) The resultng offsprng s: o 1 = ( ) o 2 = ( ) Mutaton Operator After the ne generaton has been determned, the chromosomes are subjected to a lo rate mutaton process. For ths example apples to 205

10 mutaton operators to ntroduce genetc dversty nto the evolvng populaton of permutaton. The frst operator s a smple to pont mutaton, hch randomly selects to elements n the chromosome and sap them ( ) becomes ( ). The second operator s a shuffle mutaton, hch shunts the permutatons forard by a random number of places; thus ( ) shuffled forard sx places becomes ( ). 2. Genetc Parameters For MSP, from our experence, the follong parameters are preferred to be used: populaton sze (pop_sze=20), probablty of crossover (Pc=0.7), probablty of mutaton Pm =0.1 and some hundreds of number of generatons. 7.2 Use of PSO n MSP For MSP, from our experence, the follong parameters are preferred to be used: Number of Partcles (N_Par=20,30), Maxmum velocty (v =Number of Jobs (J)), Mnmum velocty (v mn =1), Inerta Weght ([0.4,0.9]), Frst acceleraton parameter (c 1 [0.5,2]), Second acceleraton parameter (c 2 =c 1 ), Dversty of the populaton Mantenance (random r 1,r 2 [0,1]) and some hundreds of generatons. 8. Expermental Results of PSO and GA Implementaton n MSP For ths problem, a smulaton has been constructed n order to apply the PSO and GA, hen usng the parameters of PSO and GA mentoned above, the value of MOF, tme and number of teratons for best value of MOF results are shoed, n table (2) and table (3) hch are obtaned hen applyng PSO and GA methods respectvely, from number of jobs=3 10, th 1000 generatons, for 5 experments for each number of jobs, usng the follong abbrevatons: 1. J: Number of Jobs. 2. Values of MOF: Ex: Experment number. Max: Maxmum value of MOF of experment. Opt: Optmal value of MOF of experment usng complete search. UB: Upper Bound value of MOF of experment usng complete search. BV: Best Value of MOF of experment. ABV: Average of Best Values of MOF for all experments. 3. Values of Tme: CT: Complete Tme of fnsh experment. BT: Best Tme of best value of MOF of experment. ABT: Average of Best Tme of MOF of all experments. 4. NI: Number of Iteraton of best value of MOF of experment. Note: the shaded cell represents the mnmum best value of MOF, the mnmum tme, the teraton of best value of MOF n all experments. J 3 Table (2) Applyng PSO method on MSP for J= Ex Value of MOF Tme Max Opt UB BV ABV CT BT ABT NI

11 It s mportant to note that the optmal and mum value of MOF for each experment obtaned by usng complete search method. The complete search, of course, dffcult to be appled for jobs more than 10 jobs. For ths reason the results of the optmal and mum value of MOF are not mentoned n the tables ncluded jobs more than 10 jobs. Table (3) Applyng GA method on MSP for J= J 3 4 Ex Value of MOF Tme Max Opt UB BV ABV CT BT ABT BNI

12 In table (4) a comparson has been made beteen the results of applyng PSO (P) obtaned from table (2) and the results of applyng GA (G) obtaned from table (3) for value of MOF, Tme and number of teratons from number of jobs=3 10. Table (4) Comparson results beteen GA and PSO methods on MSP for J= J Ex Value of MOF Tme NI BV ABV CT BT ABT UB G P G P G P G P G P G P

13 The average values of MOF, tme and number of teratons for best value of MOF results are shoed, n table (5) and table (6) hch are obtaned hen applyng PSO and GA methods respectvely, from chosen number of jobs=20(10)100,100(100)1000 and 2000 th 1000 generatons, for 10 experments for each number of jobs. Table (5) Applyng PSO method on MSP for chosen J= J Value of MOF Tme NI AUB MBV ABV MBT ABT ACT MNI ANI

14 J Table (6) Applyng GA method on MSP for chosen J= Value of MOF Tme NI AUB MBV ABV MBT ABT ACT MNI ANI In table (7) a comparson has been made beteen the results of applyng PSO obtaned from table (5) and the results of applyng GA obtaned from table (6) for value of MOF, Tme and number of teratons from chosen number of jobs=20(10)100,100(100)1000 and Table (7) Comparson results beteen GA and PSO methods on MSP for chosen J= Value of MOF Tme NI J MBV ABV MBT ABT ACT MNI ANI G P G P G P G P G P G P G P

15 Fgure (3) descrbes comparson chart hch s shos the relaton beteen value of MOF and number of teratons hen applyng PSO and GA on MSP conssts of 10 jobs. Fgure (3) comparson chart of applyng PSO and GA on MSP conssts of 10 jobs. Fgure (4) descrbes comparson chart hch s shos the relaton beteen value of MOF and number of teratons hen applyng PSO and GA on MSP conssts of 150 jobs. Fgure (4) comparson chart of applyng PSO and GA on MSP conssts of 150 jobs. The PSO and GA methods ere tested by a programmng them usng verson 6 of Delph Language, and runnng on Pentum IV at 1.4 GHz, th Ram 128 MB computer. 9. Conclusons In ths paper, near optmal approaches have been developed for one machne schedulng problem to mnmze a multple objectve functon for the h 1 / / W C L E, ths problem s consdered to be NP-hard. 1. The local search methods that are used to solve all of the large problems n ths paper, the results sho the robustness and flexblty of local search heurstcs. 211

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