IJSER 1 INTRODUCTION. limitations with a large number of jobs and when the number of machines are more than two.

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1 Iteratioal Joural of Scietific & Egieerig Research, Volume 7, Issue, March ISSN Schedulig to Miimize Maespa o Idetical Parallel Machies Yousef Germa*, Ibrahim Badi*, Ahmed Bair*, Ali Shetwa** * Faculty of Egieerig, Misurata Uiversity, Misurata, Libya ** The college of Idustrial Techology, Misurata, Libya Abstract- Reducig productio time is a importat factor for compaies which their mai objectives are to imize profits ad miimize costs. To achieve these objectives oe must follow the scietific methods for schedulig productio time. This paper studies the Maespa Miimizatio for Idetical Parallel Machies. The problem ivolves a assigmet umber of jobs (N) to a set of idetical parallel machies (m), whe the objective is to miimize the maespa (imum completio time of the last job o the last machie of the system). I the literature the problem is deoted by ( P m ). The objective of this study is to fid the optimal schedule (solutio) for idetical parallel machies schedulig problems, by usig hypothetical situatio uder defied assumptios ad costraits. Algorithms are developed ad used i this paper to represet ad solve the problem. Iteger Liear Programmig model (ILP) is used to formulate the problem. Logest Processig Time algorithm (LPT) is used to fid (geerate) the iitial solutio, the the developed algorithm is used to improve the iitial solutio. The solutio of the algorithms are coded i MATLAB. The results demostrated that, the mathematical modelig ad the algorithms are powerful tools ad are more effective for these ids of problems, compared with traditioal methods. Keywords: Maespa, Parallel machies, Iteger Liear Programmig INTRODUTION this study the problem deals with umber of jobs (N jobs ) Ito be processed o a umber of idetical parallel machies (m machies), whe the objective is to miimize maespa (imum completio time of the last job). The problem deoted by ( P m ). Figure () shows (N) idepedet jobs o (m) parallel machies. Fig.. (m) parallel machies with (N) jobs. To solve the problem statemet of this study, the followig assumptios are used:. All machies are idetical ad are able to perform all operatios. 2. Each machie ca oly process oe tas at a time.. Permissio of a job o aother machie is ot allowed. 4. All jobs are available at time zero. There are may studies i literature dealig with parallel machies schedulig problems. Although the relatively small sized of idetical parallel machies problems ca be solved by operatioal methods such as dyamic programmig, brach ad bod method. However, these methods still have 206 limitatios with a large umber of jobs ad whe the umber of machies are more tha two. 2 LITERATUR REVIEW Koulamas ad Kyparisis proposed a modified logest processig time (MLPT) heuristic algorithm for the two uiform machie maespa miimizatio problem. The MLPT algorithm schedules the three logest jobs optimally first, followed by the remaiig jobs sequeced accordig to the LPT rule. The obtaied results showed that the MLPT rule has the tight worst-case boud of.22 [], a improvemet over the LPT boud of.28 [2]. haudhry et al preseted a spreadsheet based o geetic algorithm approach to miimize the maespa for schedulig a set of tass o idetical parallel machies ad worer assigmet to the machies. The performace of the proposed approach is compared agaist two data sets of bechmar problems available o the iteret. It was foud that the proposed approach produces optimal solutio for almost 95 percet of the problems demostratig the effectiveess of the proposed approach []. Queiroz ad Medes addressed a idetical parallel machies schedulig problem with release dates i which the total weighted tardiess has to be miimized. The obtaied results showed that the istaces of 0 jobs ad 2 machies for which the metaheuristic achieved a optimal solutio i a competitive amout of time, whe compared with the exact approach. For the cases of 5 jobs ad machies ad 20 jobs ad machies, the performace of the metaheuristic was similar [4]. A ew method has bee developed by Navid Hashemia to schedule jobs o parallel machies with availability costraits. The objective of the problem was to miimize the

2 Iteratioal Joural of Scietific & Egieerig Research, Volume 7, Issue, March ISSN maespa of the total productio schedule. It was cocluded that the exact algorithm is able to solve large-scale problems which are ot solvable by ay other method icludig the curret best ILP solver. Based o the results of the experimets for the ILP, it has bee demostrated that the proposed ILP ca be used to solve the small-size problems effectively. The performace of the proposed algorithm is idepedet of the patter of o-availability periods [5]. The aim of this study is to fid the optimal schedule (solutio) for idetical parallel machies schedulig. This paper preseted bactracig algorithm to fid the optimal schedule (solutio) for idetical parallel machies schedulig. Also the paper suggested a algorithm to improve the iitial solutio. PARALLEL MAHINES SHEDULLING The aim of machie schedulig is to assig jobs to the machies based o related objective fuctio to miimize operatig time ad icrease productivity [6]. Parallel machies schedulig is the tas of determiig whe each operatio has to start ad fiish o each machie ad usig available resources i efficiet maers to execute (assig) jobs or tass o machies. The parallel machies ca be idetical (uiform) or urelated. I this paper the case of study is idetical parallel machies. Idetical parallel machies are a set of machies which have the same speed factor ad they ca process all the jobs. Figure (2) shows Gatt chart for set of machies ad jobs. M M 2 Machies (N) i decreasig order waitig for processig is selected to be J J 2 ext [6]. J J 4 m x ij = j=,.,n () i= x ij {0,} i=,.,m, j=,.,n (4) 0 (5) The first equatio () i the model is the objective fuctio ( ) maespa, which should be miimized. ostrait (2) assures that the load o ay machie is equal or less tha ( ). ostrait () shows that each job must be assiged to exactly oe machie. ostrait (4) describes the type of the assigmet decisio variable ( x ij ): x if the job j is assiged to machie i ij = { 0 if the job j is ot assiged to machie i ostrait (5) shows that, the objective fuctio is iteger variable. 4 Solutio method I this paper, the solutio ca be obtaied i two steps: The LPT algorithm is used to fid (geerate) the iitial solutio, while the developed algorithm is used to improve the iitial solutio. 4- LPT algorithm The (LPT) algorithm always puts the smaller jobs towards the ed of schedule, that maes it easier to balace the machies loads. Accordig to the (LPT) algorithm, wheever oe of the (m) machies is freed, the logest job amog umber of jobs The ext job (j) will be scheduled o machie ( i ) accordig to M J 5 J 6 J 7 i = argmi the equatio: { L + p : i =,..., m } i j (6) M 4 J 8 J 9 J 0 Time uit Fig. 2. Gatt chart for set of machies ad jobs. Assume that: : the maespa (imum completio time). N : umber of jobs (iteger). m : umber of machies. (iteger). p j :processig time of job ( j ) (iteger). x ij : the assigmet (decisio) variable. The mathematical model of the problem as follows: Mi () subject to N j= p j x ij i=,.,m (2) Where: L i is the load o machie ( i ), p j the processig time of job ( j ). The maespa ( ) of ay feasible solutio is: = { i : i =,..., m} (7) The steps of LPT algorithm are as follows: Step: sort (N) jobs accordig to the o-icreasig order of their processig time. Step 2: set ( j= ). Step : assig job ( j ) to machie (i) accordig to equatio (6). Step 4: if j=n (all jobs are allocated) the go to the ext step, otherwise set j=j+ ad go to the step. LPT Step 5: calculate by usig equatio (7). 4-2 Developig the algorithm The developed algorithm, is used as a mai algorithm ad is based o two types of operatios: (costructio ad 206

3 Iteratioal Joural of Scietific & Egieerig Research, Volume 7, Issue, March ISSN bactracig). The load of machies are determied i sequece, oe after aother. Therefore, the potetial load of machie (i) with (< i m) depeds o the loads of the previous machies, the the loads o machies cotiue util assigig the last job [4]. Assume that, ( ): is the maespa of the curret feasible solutio. So if the optimal solutio has ot bee explored yet, The algorithm is always looig for the optimal solutio. Therefore the bactracig is applied wheever ay of the followig two situatios are accouted: - Whe the load of a machie is ot feasible (caot satisfy equatio(0)). 2- Whe a ew feasible solutio has bee foud for all machies (updatig maespa). the its value is ot greater tha ( ) i the case of Whe the machies has o further feasible load ( the load of machies are ot satisfy equatio (0) ). The there is o iteger processig times. Therefore: feasible solutio for the curret upper boud ad optimal UB = (8) * Where (UB ) is the upper boud for the load of all machies i maespa ( ) is equal to pervious upper boud. the feasible solutio ad still to be ivestigated. * = UB + () Ad the lower boud ( LB ) for all the other loads i the same 4- Steps of the improvemet algorithm: feasible solutio ca be foud by the followig equatio: Step : set ( i = ) ad ( t = ). N LB = 0, p j ( m ) UB (9) Step 2: use (LPT) algorithm to fid the iitial solutio (upper j= boud). Equatio (9) implies that if all machies except oe have a Step : load machie (i) accordig to the equatios () ad total load equal to the upper boud, the the remaiig load (2). is the lower boud. After fidig a ew feasible solutio, both Step 4: if machie (i) does ot satisfy iequality (0) go to step bouds (upper ad lower boud) tighte up. To esure that (6). the load o ay machie is feasible, the total load o the Step 5: if (i =m) set ( UB machie, must be betwee the lower ad upper bouds. ) ad ( ) ad go to step (). otherwise set ( i = i + ) ad go to step (). LB p j j UB (0) j= Step 6: calculate by usig equatio (). Where: ( j ) is iteger, ad j =,..,. Step 7: prit results ad stop ostructio phase Figure () shows the flow chart of the suggested algorithm. There are may feasible solutios that ca satisfy the equatio (0). For a implicit eumeratio procedure, the feasible 5 omputer program solutios must be ordered somehow ad eumerated i this order. Oe easy way to perform this tas is to order them i lexicographical order also ow as (the dictioary order) or (the alphabetic order). I the costructio phase, ad to load the machies oe by oe, the largest solutio i the lexicographical order for () jobs types is give by equatios () ad (2). That is whe the machie has o previous load. UB mi p r =, () j UB ph h h j mi = =, r j, j = 2,...,. (2) p j The feasibility of the load is checed by equatio (-0). If the costructio is successful (all machies are loaded ad all coditios are satisfied), both the upper boud ad lower boud are updated Bactracig phase The developed algorithms have bee coded i MATLAB ad executed o Itel (R) core (TM)i5 PU 2.0 GHz, RAM 4.00 GH. System type (64bit) i order to remai comparable i terms of the computatioal time. The program is desiged to solve variety of problems depedig o umber of machies (m) ad umber of jobs (N). May computatioal applicatios with differet levels of difficulties have bee carried out to evaluate the performace of the suggested algorithm. 5- ase () Fid the optimal schedule (solutio) by usig the (LPT) ad the algorithms to miimize imum completio time ) for (7) jobs their processig times ( p j ) give ( bellow: p j = [,,, 4, 4, 5, 5 ] 206

4 Iteratioal Joural of Scietific & Egieerig Research, Volume 7, Issue, March ISSN Fig. (). The flow chart of the algorithm. i=i+ t=t+ start Iput No; of jobs (N) ad their processig time. Iput No; of machies ( ) The jobs processed by () idetical parallel machies, each machie ca oly process oe job at ay time ad permissio of a job o aother machie is ot allowed. 5-- Usig LPT algorithm to fid the iitial schedule (solutio): Step: sort the jobs i o-icreasig order (5, 5, 4, 4,,, ) Step 2: set j = Step : assig job () to the machie (i ) accordig to equatio (6). i = argmi L + p : i =,..., m UB mi p r =, { } i j i= & t= The load o machies: = P + P5 + P7 = = 2 = P2 + P6 = 5+ = 8 Use LPT algorithm to fid = P + P4 = 4+ 4 = 8 iitial solutio UB = Step 4: j = N = 7 ( all jobs are assiged (allocated)). Step 5: calculate by usig equatio (7). Load machie (i) accordig = { i : i =, 2, } = {, 2, } to formula (-) (-2) = {,8, 8} = The machies are loaded ad the iitial solutio obtaied by LPT algorithm are illustrated i Figure (4) by Gatt chart. No 5--2 Usig the algorithm to improve the iitial solutio: Machie (i) Step : set ( i = ) ad ( t = ). satisfy Step 2: use (LPT) algorithm to fid the iitial solutio (upper boud). Yes At first iteratio UB = = The lower boud ( LB ) ca be foud by equatio (9) as follows: No N LB = 0, p j ( m ) UB i = m j= Yes 7 LB = 0, p j ( ) j= LB = { 0, 27 (2) } LB = { 0, 5} LB = 5 UB = UB - ad fid (LB) Step: load machie() accordig to the equatios () &(2). by usig equatio (-9) alculate by usig equatio (-) & prit results j UB ph h h j mi = =, rj, j = p j Therefore: 2,...,. stop 206

5 Iteratioal Joural of Scietific & Egieerig Research, Volume 7, Issue, March ISSN = mi, 2 = mi { 2.2, 2} 5 = 2 ( UB = ) ad ( t = t + ). UB = UB = UB = 0 ad t = t + 2 UB ph h t = 2 2 = mi h=, r2, j = 2 p This meas: the ew upper boud is UB = 0 ad the ext 2 iteratio is t = 2 ( p ) Ad the ew lower boud ( LB )ca be foud by equatio (9): 2 = mi, 2, j = 2 4 LB =7 With the same above procedure the ew load o machies are: (5 2) 2 = mi, 2 2 = mi { 0.25, 2} 2 = 0 4 = 0 UB 2 = 8 ph h = mi h=, r, j = = 9 p alculate by usig equatio (-7). ( ) = mi,, j = = { 0, 8, 9} = 0 = mi { 0., }, j = = 0 UB = Thus the load o UB =0 UB = 9 ad t = t + t = machie() is ( ): This meas: the ew upper boud is UB = 9 ad the ext M : = p + p2 2 + p = = 0 iteratio is t = Step 4: Ad the ew lower boud ( LB ) ca be foud by equatio (9): LB =9 LB p j j UB. (-0) 5 p j j the ew load o machies are: j= j= = 9 Where the load o machie() is: = p j j = 0 j= 2 = 9 the the machie() satisfy iequality (0) = 9 i = i + i = + i = 2 m. the the ext step is (). i =, m = i = m Step : load machie (2) accordig to the equatios ()&(2). alculate by usig equatio (7). With the same procedure for loadig machie (). The load o = {, 2, } machie (2) is( 2 ) as follows: = { 9, 9, 9} M2 : 2 = p + p2 2 + p 2 = = = 9 ( UB = ) ad ( t = t + ). Where the load o machie (2) is: 2 = p j j = UB = j= the the machie 2 satisfy iequality (0). UB = 9 UB = 8 ad t = t + t = 4 i = i + i = 2 + i = The load o machie() is ( ): M : = p + p2 2 + p = = 6 Step 4: LB p j j UB. (-0) 5 p j j j= j= Where the load o machie () is: = p j j = 6 The the machie () satisfy iequality (0). Step 5: i = m = the alculate by usig equatio (7). = { i : i =,..., m} (7) = { i : i =, 2, } = {, 2, } = { 0,, 6} = j= 206 This meas: the ew upper boud is UB = 8 ad the ext iteratio is t = 4 The the ew lower boud ( LB ) ca be foud by equatio (9): LB = the ew load o machies are: = 8 ; 2 = 8, = 8 Where the load o machies does ot satisfy iequality (0). The go to step (6) Step 6: set = UB + = 8 + = 9 The optimal solutio = 9. Figure (4) shows Gatt chart for the solutio ad the load o machies.

6 Iteratioal Joural of Scietific & Egieerig Research, Volume 7, Issue, March ISSN The solutio for case() by (LINGO, LPT ad the suggested algorithm) LINGO Fig. (4). The optimal solutio (schedule) for ase (). Improvig the iitial solutio to obtai the optimal solutio (schedule) of the problem show i figure (5). Machie Jobs Iteratio M P5, P6, P7 9 M2 P2, P M P, P4 9 LPT algorithm Machie Jobs Iteratio M P, P5, P7 M2 P2, P6 8 M P, P4 8 the suggested algorithm Machie Jobs Iteratio M P5, P6, P7 9 M2 P2, P M P, P4 9 Ad as ca be observed the solutio by LINGO ad the algorithm program is idetical ( = 9 ) with differet distributio of jobs o machies ad umber of solutio iteratios. 4-4 Problems with radom umber of machies ad jobs Number of problems with radom umber of machies ad jobs have bee geerated to esure ad demostrate effectiveess of the algorithm. Table (4-) represets the results. Fig. (5). Usig the suggested algorithm to Improve the iitial solutio. Accordig to the results i table (4-) the solutio of the problem ( ) which obtaied by LPT algorithm ( = ). TABLE 206 TABLE 2 The solutio for umber of (m) Machies ad (N) Jobs Mach Jobs Iteratio No -ies (m) (N) LINGO LPT Alg. LINGO Alg * Alg.: the suggested algorithm Table (2) represets the results for radom umber of machies ad jobs, the maespa ( ) which obtaied by LINGO ad the algorithm for problem ad 2 i the table are equal. O the other had there is a big differece, i the iteratios to reach the solutio, the algorithm is always faster tha LINGO. 6 oclusio:

7 Iteratioal Joural of Scietific & Egieerig Research, Volume 7, Issue, March ISSN I this study the ILP model itroduced to represet ad formulate the problem( P m ), U- algorithm desiged ad coded i MATLAB to solve the ILP model, also LINGO software has bee used to test ad evaluate performace of the suggested algorithm. May experimets ad problems are used with radom umber of machies (m) ad jobs (N) to demostrate effectiveess of the algorithm solutio. I this study, the ILP model ca optimally be solved by ILP solvers for small size of problems. Ad as ca be observed from tables () ad (2), LINGO ca solve the problems with small size of umber of machies ad jobs. While the suggested algorithm ca solve large sizes of parallel machies schedulig problem optimally i small umber of iteratio. The results of the study demostrated that: - Efficiecy ad performace of the algorithm is evaluated by the umber of iteratios required to fid the optimal solutio. This good performace ca be observed by varyig the umber of machies. 2- The umber of feasible solutio(s) which are geerated is equal to umber of iteratio required to fid optimal solutio mius oe (t-) ad each oe of them is better tha the previous oe. - If the umber of iteratio (t=) the iitial solutio obtaied by LPT algorithm is optimal. It was foud that computatios required by LPT will escalate cosiderably as the umber of machies icreases. However, its capability is limited i terms of solvig a large-scale problem. REFERENES [ ]. Koulamas ad G. J. Kyparisis ''A modified LPT Algorithm for the Two Uiform Parallel Machie," Europea Joural of Operatioal Research, vol.96, pp.6 68, [2] T. Gozalez, T, O.H, Ibarra ad S. Sahi '' Bouds for LPT Schedules o Uiform Processors'', SIAM Joural o omputig vol. 6, pp [] I. haudhry, S. Mahmood ad R. Ahmad ''Miimizig Maespa for Machie Schedulig ad Worer Assigmet Problem i Idetical Parallel Machie Models Usig GA'', Proc. of the World ogress o Egieerig, WE 200, Jue 0 - July 2, 200, Lodo, U.K [4] M. Queiroz ad A. Medes'' Strategies for Schedulig Jobs i a Idetical Parallel Machie Productio Eviromet, Africa Joural of Busiess Maagemet, Vol. 7(6), pp , 28 April, 20. [5] N. Hashemia'' Maespa ''Miimizatio for Parallel Machies Schedulig with Availability ostraits'', MSc dissertatio, Dept. Idustrial Egieerig, Dalhousie Ui.,200. [6] B. V Reghavedra ad A. N. Murthy Worload Balacig i Idetical Parallel Machies Schedulig Usig Geetic Algorithm, paper, ARPN Joural of Egieerig ad Applied Scieces. Vol.6, No., 20. [7] N. Hashemia'' Maespa ''Miimizatio for Parallel Machies Schedulig with Availability ostraits'', MSc dissertatio, Dept. Idustrial Egieerig, Dalhousie Ui., 200. [8]. Koulamas ad G. J. Kyparisis ''A modified LPT Algorithm for the Two Uiform Parallel Machie," Europea Joural of Operatioal Research, vol.96, pp.6 68, [9]. H. Li ad. Liao Miimizig Maespa o Parallel Machies with Machie Eligibility Restrictios, paper, The Ope Operatioal Research Joural, 2, pp. 8-24, [0] S. Kaya, Parallel Machie Schedulig Subject to Machie Availability ostraits, MSc Thesis, Bilet Ui., [] G. ND. Jatider, J. HO, ad A. Ruiz-Torres "Maespa Miimizatio o Idetical Parallel Machies Subject to Mmiimum Total Flow- Time." Joural of the hiese Istitute of Idustrial Egieers Vol. 2. No., pp , [2] M. L. Piedo, Schedulig (Theory, Algorithms ad Systems), New Yor Uiversity- USA, Fourth Editio, 20. [] K. Jihee ad Y. Harrath A Survey of Parallel Machie Schedulig Uder Availability ostraits, paper, Iteratioal Joural of omputer ad Iformatio Techology, Vol., issue 02, 204. [4] I. haudhry, S. Mahmood ad R. Ahmad ''Miimizig Maespa for Machie Schedulig ad Worer Assigmet Problem i Idetical Parallel Machie Models Usig GA'', Proc. of the World ogress o Egieerig, WE 200, Jue 0 - July 2, 200, Lodo, U.K. [5] M. L. Piedo, Plaig ad schedulig i maufacturig ad services, New Yor Uiversity- USA, 2005.