Songklanakarin J. Sci. Technol. 40 (5), , Sep. - Oct Original Article. Anot Chaimanee* and Wisut Supithak

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1 Songklanakarn J. Sc. Technol. 40 (5), , Sep. - Oct Orgnal Artcle A Meetc Algorth to Mnze the Total Su of Earlness Tardness and Sequence Dependent Setup Costs for Flow Shop Schedulng Probles wth Job Dstnct Due Wndows Anot Chaanee* and Wsut Supthak Departent of Industral Engneerng, Faculty of Engneerng, Kasetsart Unversty, Chatuchak, Bangkok, Thaland Receved: 21 Deceber 2016; Revsed: 22 June 2017; Accepted: 16 July 2017 Abstract The research consders the flow shop schedulng proble under the Just-In-Te (JIT) phlosophy. There are n jobs watng to be processed through operatons of a flow shop producton syste. The objectve s to deterne the job schedule such that the total cost consstng of setup, earlness, and tardness costs, s nzed. To represent the proble, the Integer Lnear Prograng (ILP) atheatcal odel s created. A Meetc Algorth (MA) s developed to deterne the proper soluton. The evolutonary procedure, worked as the global search, s appled to seek for the good job sequences. In order to conduct the local search, an optal tng algorth s developed and nserted n the procedure to deterne the best schedule of each job sequence. Fro the nuercal experent of 360 probles, the proposed MA can provde optal solutons for 355 probles. It s obvous that the MA can provde the good soluton n a reasonable aount of te. Keywords: flow shop schedulng, earlness tardness, due wndow, optal tng algorth, Meetc Algorth 1. Introducton In the past several years, any researchers have conducted research n the producton schedulng under the Just-In-Te (JIT) phlosophy. The objectve of the JIT s to produce and delver products not before or after ther cotted due dates. Any jobs copleted early ust be held by anufacturer untl ther due dates and, hence, ncur soe costs as a result of product deteroraton, storage, and nsurance. On the other hand, those jobs copleted after ther due dates can cause any probles such as custoer penaltes, loss of sales, or potental loss of reputaton. In accordance wth Baker and Scudder (1990), an deal schedule s the one n whch all jobs are fnshed exactly on ther due dates. The ost obvous objectve of schedulng proble under the JIT *Correspondng author Eal address: anotchaanee@hotal.co polcy s to nze the devatons of job copleton tes around ther due dates. It can be seen as the nzaton proble of total su of earlness and tardness penaltes (E/T schedulng proble). The E/T schedulng proble can be dvded nto several categores accordng to the types of achne syste, due date, and characterstcs of weght penaltes. Accordng to Pnedo (2002), the E/T schedulng proble of jobs havng dfferent due dates n a sngle achne producton syste s NP hard. Lee and K (1995) studed the job schedulng proble on sngle achne wth coon due date. The objectve was to nze the total generally weghted of earlness and tardness penaltes. The slar proble but wth dstnct due dates was dscussed by Lee and Cho (1995). Szwarc and Mukhopadhyay (1995) proposed an optal tng to fnd an optal tng job startng poston for the E/T proble wth the predeterned job sequence. Sarper (1995) proposed the nzaton proble of the su of absolute devatons of job copleton tes around a coon due date for the two achnes flow shop. Mosleh et

2 1204 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , 2018 al. (2009) studed slar proble but dstnct due dates. The author addressed the case of nzng the su of axu earlness and tardness. Yoon and Ventura (2002) proposed a procedure for nzng the ean weghted absolute devaton of job copleton tes around ther due dates when jobs are scheduled n a lot-streang flow shop. Sufen et al. (2005) dscussed the E/T schedulng proble of n jobs achnes flow shop wth uncertanty of job processng tes. Chandra et al. (2009) consdered the E/T proble under a coon due date n the flow shop producton syste. The slar proble but wth dstnct due date was dscussed by Schaller and Valente (2013) and M Hallah (2014). Accordng to the survey research on E/T schedulng probles wth job due wndow conducted by Janak et al. (2015), both coon and dstnct due wndow probles are NP-hard. Whle Yeung et al. (2004) dscussed the E/T schedulng proble wth coon due wndow, the cases of proble wth dstnct due wndows were studed by Behnaan et al. (2009), Koulaas (1996), and Wan and Yen (2002). Nonetheless, as beng known so far, there s no research conducted relevant to the E/T schedulng proble wth flow shop achne syste and job dstnct due wndows. In recent-years, a growng nuber of lteratures suggest the applcaton of Genetc Algorth (GA) as one of those powerful etaheurstc beng used to solve cobnatoral optzaton probles (Cheng el al., 1995, Reeves, 1995; Sevaux & Dauzere-Peres, 2003). Accordng to Sevaux and Dauzere-Peres (2003), the an dfference between the GA and other etaheurstcs such as Tabu Search (TS) or Sulated Annealng (SA) s that not only GA antans the populaton of soluton rather than unque current soluton but t allows the exploraton of a larger soluton space as well. Because of those benefts dscussed prevously and the splcty to represent each job as a gene of a chroosoe n the soluton representaton, the GA has been appled by any researchers to seek for the good soluton n the job sequencng probles. Soe of the are Lee and Cho (1995), Lee and K (1995), Murata et al. (1996), Reeves (1995), and Sufen et al. (2005). The Meetc Algorth (MA) can be consdered as the extenson odel of general GA. Accordng to Tavakkol-Moghadda et al. (2009), unlke tradtonal GA, the Meetc Algorth s populaton-based search approach cobnng evolutonary procedure wth local refneent strateges such as local neghborhood search. Ths paper consders the E/T schedulng proble of jobs havng dstnct due wndows n -operaton flow shop producton syste. The atheatcal odel s ntroduced to represent the proble. In order to deterne the startng te of each job when the job sequence s known, the optal tng algorth for the E/T flow shop schedulng syste s created. The Meetc Algorth based on evolutonary procedure wth the nserton of optal tng algorth s presented to deterne the good soluton to the proble. 2. Proble Characterstcs There are n jobs watng to be process on achnes flow shop producton syste. Each job has ts own earlness penalty, tardness penalty, earlest due date, and latest due date. Any jobs copleted before ther earlest due dates ncur earlness penaltes. On the other hand, those jobs copleted after ther latest due dates ncur tardness penaltes. All jobs are assued to be avalable for the producton at the begnnng of plannng horzon. The objectve s to deterne producton schedule such that the su of earlness and tardness costs of all jobs are nzed. The followng notatons are used throughout the paper. n = nuber of jobs = nuber of achnes, j = ndex of jobs ;, j 0,1,2,..., n [ ],[ j] = ndex of job postons n gven sequence ;, j = 1,2,,n k = ndex of achnes ; k 1,2,..., g = ndex of sub-schedules ; g 1,2,..., G r = ndex of job clusters ; r 1,2,..., R C, = copleton te of job on achne k k W = length of due wndow of job ; W t e e t = earlest due date of job = latest due date of job E = earlness of job ; E ax e C,,0

3 T A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , = tardness of job ; T C t,0 ax, OT 1 = the te perod of job when that job copleted after ts earlest due date ; OT ax C, e,0 1 OT 2 = the te perod of job when that job copleted before ts latest due date ; OT ax t C, 2,0, j P, k TC J J 0 J [ frst J g last J g = earlness cost of job = tardness cost of job = setup cost of job j when t edately processed after job. Here, beng process on a achne. = processng te of job on achne k = total cost = job of all n jobs = a duy job havng processng te on all achnes equal to zero = job n th poston of the gven sequence = the frst job n sub-schedule g = the last job n sub-schedule g 0, j refers to the setup cost of job j when t s the frst job r = cluster r J, r J F, r J E, r = job n cluster r ; = F, F+1,, E, E+1,, W, W+1,,T, T+1,,L = the frst job n cluster r = the last early job n cluster r J, = the last job s copleted n due wndow and condton t 0 s hold W r W C W, r J T, r = the frst tardy job n cluster r J L, r x, j = = the last job n cluster r 1 f job precedes job j 0 otherwse y 1 = the shft dstance beng calculated fro early jobs n cluster r; y 1 n e C, r FE y 2 = the shft dstance beng calculated fro jobs copleted n due wndow wth the condton t 0 for E 1,..., W are hold; t C y 2 n, r E1 W C, r frst = startng te of the job n sub-schedule G+1; SG 1 frst SG 1 C = copleton te of the last job n sub-schedule G; C R g M last G = the last cluster on sub-schedule g = large nuber frst SG 1 last G The followng exaple deonstrates the proble characterstc.

4 1206 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , 2018 Exaple 1 Gven that fve jobs are watng to be processed through three operatons of a flow shop producton syste. The data of each job are presented n the table 1. Table 1. Data for Exaple 1. J, j Setup cost n dollars/setup ( ) Earlness cost n dollars/day ( ) j=1 j=2 j=3 j=4 j=5 Tardness cost n dollars/day ( ) Processng tes (days) P,1 P,2 P,3 Due wndows e t = = = = = = The Gantt chart n the Fgure 1 represents one possble producton schedule of the proble (ths schedule ay not be the optal). Here, the research assues that all jobs are processed wth the sae sequence on all achnes. Machne Machne Machne Fgure 1. One possble producton schedule. The cost assocated wth schedule n Fgure 1 can be calculated as follows: Setup Cost: The total setup cost ncurred fro the sequence can be calculated as 0,1 1,3 3,2 2,5 = , 4 = $13. Earlness cost: Job 1, 2, and 5 are early for 6, 2, and 3 days, respectvely. The earlness cost can be calculated as E E = 2(6)+3(2)+2(3) = $ E5 Tardness cost: Only job 4 s tardy. The tardy cost s 4T 4 = 4(1) = $4. Note that job 3 has no penalty snce t s copleted wthn the due wndow. The total cost; TC = = $ Integer Lnear Prograng The atheatcal odel based on Integer Lnear Prograng (ILP) was developed and can be shown as follows. Objectve Functon n E T, j x j nze TC, 1 n n 0 j1 j

5 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , Constrants n j1 n 0 j n j1 j x 0, j, j 1 x 1 ; j 1, 2,..., n x 1 ; 1, 2,... n, j, (1) (2) (3) C P ; 1, 2,..., n (4),1,1 j, k, k, j j, k C C M 1 x P ;, j( j) 1,..., n; k 1,..., (5) C, C, 1 P, ; 1, 2,..., n; k 2,..., (6) k k k C E OT e ; 1, 2,..., n (7), 1 C, T OT t ; 1, 2,..., n (8) 2 The objectve functon represents the su of total cost ncludng setup, earlness and tardness costs of all jobs. Constrant (1) dentfes the frst job of the sequence. Constrant (2) ensures that each job can have at ost one edately precedng job. Accordng to constrant (3), there s at ost one job can be edately processed after job. Constrant (4) guarantees that the frst job beng process on the frst achne cannot start before the te of zero. Constrant (5) ensures that any two adjacent jobs on the sae achne are processed contnuously wthout overlappng. Constrant (6) requres that any operatons of a job cannot be overlapped. Constrants (7) and (8) are to deterne earlness and tardness of job. In order to deterne the proble soluton, the Meetc Algorth s proposed. The evolutonary procedure s appled to search for the good job sequence. In order to deterne the optal schedule for each job sequence, the optal tng algorth s constructed and nserted nto the evolutonary procedure. 3. Proble Propertes Defne the ntal schedule as the schedule n whch all jobs n the gven sequence are started on each achne as soon as possble. Note that n the optal schedule, each job cannot be processed before ts startng tes n the ntal schedule. Property 1: If there s no dle te between any two consecutve jobs J [ and J [ 1] n the ntal schedule, the optal schedule can have dle te between J and [ J only when [ 1] t [ 1] e[ P[ 1],. Explanaton: Gven that, n an ntal schedule, there s no dle te between jobs optal schedule, f job t e [ 1] [ J s not early e [ ust be greater than C and job [ [ C [ 1], C[ J and [ J s not tardyc t [ 1], whch can be wrtten as shown n (9). [ 1], [ 1] J on the last achne. In the [ 1], snce C [ 1], C[, the ter t [ 1] e[ C[ 1], C[ (9)

6 1208 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , 2018 When there s dle te between jobs condton can be wrtten as follows. J [ and J[ 1], the aount of C [ 1], C[ P[ 1], C [ 1], C[ s greater than P [ 1],. The (10) The nequalty (11) s constructed fro nequaltes (9) and (10) t [ 1] e[ P[ 1], (11) Fro the ntal schedule, gven that two consecutve jobs are placed n the sae cluster wth the followng condtons. t [ 1] e[ P[ 1], (12) S C 0 (13) [ 1], [ Property 2: In the optal schedule, jobs belong to the sae cluster ust be processed wthout nterrupton. Explanaton: Inequalty (12) and equaton (13) dentfy the property. Property 3: In the optal schedule, for each cluster, those early jobs ust be processed before those tardy jobs. Explanaton: Gven that, on the last achne, jobs J [ and J[ 1] equaton can be created by addng the ter P on both sdes of the nequalty (12). S [ [ are grouped n the sae cluster, the followng t [ 1] S[ P[ P[ 1], e[ S[ P[ (14) The nequalty (15), reduced fro nequalty (14), confrs that the early job ust be processed before the tardy job. t [ 1] C[ 1], e[ C[ (15) Property 4: In the optal schedule, for each cluster, those early jobs ust be processed before those on-te jobs. Explanaton: Snce e [ 1] t[ 1], the ter e [ 1] C[ 1], s less than nequalty (15), the result can be shown as follows whch concludes to the property. t [ 1] C[ 1],. Applyng ths relatonshp to the e [ 1] C[ 1], t[ 1] C[ 1], e[ C[ (16) Property 5: In the optal schedule, for each cluster, those on-te jobs ust be processed before those tardy jobs. Explanaton: Snce e t, the ter [ [ e C[ ], [ s less than nequalty (15), the result can be shown as follows whch concludes to the property. t C[ ], [. Applyng ths relatonshp to the t [ 1] C[ 1], e[ C[ t[ C[ (17) Property 6: In the optal schedule, f two consecutve jobs of the sae cluster are early, then E E. [ ] [ 1] Explanaton: If two consecutve early jobs are grouped n the sae cluster, the condton hold. The result s shown n the followng nequalty (18). t [ 1] e[ P[ 1], ust be

7 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , e [ 1] e[ P[ 1], (18) The followng nequalty s created by addng the ter P on both sdes of the nequalty (18). S [ [ e [ 1] S[ P[ P[ 1], e[ S[ P[ (19) The nequalty (19) can be reduced to the nequalty (20). Snce both jobs are early, the two ters on both sdes are postve and therefore the earlness of J [ s greater than the earlness of J[ 1]. e [ 1] C[ 1], e[ C[ (20) Property 7: In the optal schedule, f two consecutve jobs of the sae cluster are tardy, then T T [ 1] [. Explanaton: If two consecutve tardy jobs are grouped n the sae cluster, the condton be hold, whch results n the followng nequalty (21). t [ 1] e[ P[ 1], ust t [ 1] t[ P[ 1], (21) The followng nequalty s created by addng the ter P on both sdes of the nequalty (21). S [ [ t [ 1] S[ P[ P[ 1], t[ S[ P[ (22) The nequalty (22) can be reduced to the nequalty (23). Here, both jobs are late; the two ters on both sdes are negatve. Snce, the tardness s the absolute value of lateness, the tardness of J[ 1] s greater than J [. t [ 1] C[ 1], t[ C[ (23) 4. Optal Tng Algorth The functon of Optal Tng Algorth (OPT) s to deterne the optal startng te of each job for a gven sequence. The concept of the algorth starts fro constructng the ntal schedule and, then, dvdng jobs nto clusters before groupng several clusters nto sub-schedule. The fnal step of the algorth s to shft each job cluster to the rght sde untl the su of earlness and tardness costs of all jobs s lowest. The concept of OPT procedure ntroduced n ths study can be explaned as follows. On the last achne, any two jobs are assgned to the sae cluster when the condtons (12) and (13) are hold. Jobs n the sae cluster should be processed contnuously wthout dle te. Snce earlness and tardness of each job are deterned by coparng the job copleton te on the last achne wth ts due wndow, only startng, processng, and copleton tes of job on the last achne are consdered. Applyng the clusterng ethod to the ntal schedule n the Fgure 1, there are three clusters as shown n the Fgure 2.

8 1210 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , 2018 Machne Machne Machne {J 1,J 3 } {J 2 } {J 5,J 4 } Fgure 2. Results obtaned fro clusterng ethod. After all clusters are deterned, they ust be grouped nto sub-schedule. The clusters r and r1 wll be n the sae sub-schedule f the equaton (24) s hold. C S (24) L, r F, r1 For any cluster J,..., J, J,..., J, J J r whether or not the cluster should be shfted to the rght. J F, r, F 1, r E, r E 1, r W, r W 1, r,..., L, r, the equaton (25) s appled to deterne E F ( r), r L, r W 1 (25) If ( r ) 0, the cluster ( r ) 0, the cluster r r should not be oved because t wll only ncrease the total cost. On the other hand, f should be shfted to the rght sde. Therefore, the total cost can be reduced by the product of shft dstance E (r). Here, the shft dstance can be calculated usng the equaton (26) frst 1 last g E( r) n y, y S g C (26) 1 2, (r) and If cluster r does not have any jobs of condton e C, t, then r y (27) 2 condtons are hold. If there are no early jobs n the cluster r Suppose that all jobs belong to the cluster due dates ( t 0), then C, r, ths cluster should not be shfted to the rght. In ths case, the followng L ( r), r W 1 (28) y (29) 1 r are copleted wthn ther due wndows and ended before ther latest

9 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , ( r) 0 y 1 (30) (31) If all jobs n cluster r are tardy jobs, then L ( r) y 1, r W 1 (32) (33) y (34) 2 Step 0: Step 1: The optal tng procedure constructed n the research can be suarzed as follows: Re-ndex each job accordng to ts poston n gven sequence. Then, deterne the ntal schedule by startng each job on each achne as soon as possble. Dvde n jobs on last achne nto clusters, f nequalty (12) and equaton (13) are vald, jobs J and grouped n the sae cluster. Step 2: Group each cluster nto sub-schedule accordng to equaton (24). Step 3: Step 4: Step 5: Step 6: Set g=0. Set g=g+1, f g>g, go to step 9. Otherwse, calculate Deterne the nal h such that ( r) 0. h r1 (r) and E(r) Step 5.1: f h exsts and h=rg, go to step 4. Otherwse, consder step 5.2. Step 5.2: f h exsts and h Rg, go to step 6. Otherwse, consder step 5.3. Step 5.3: f h does not exst, then go to step 7. [ J are [ 1] for each cluster n sub-schedule g and go to step 5. The frst h clusters ust not be oved. Reove the frst h clusters fro consderaton, go to step 5 to evaluate the reanng clusters. Step 7: Consder ( r R ), deterne the sallest 1 g dstance equals to the sallest Step 8: If last frst S, update Cg g1 (r) and E(r) E(r). Go to step 8. E(r) and, then, shft clusters n the sub-schedule g to the rght at the, go to step 5. Else, f last frst S, cobne sub-schedule g wth sub-schedule Cg g1 Step 9: g+1 and go to step 4. Stop. So far, only the operatons of jobs on the last achne have been consdered. For each job, the startng te of the reanng operatons (achnes -1 to 1) can be deterned usng the followng equatons., k n S1, k,, k1 C S ; = 1,2,n-1 ; k = 1,2,-1 (35) C S ;k = 1,2,,-1 (36) n, k n, k1 Exaple 2. The calculaton of the optal tng algorth for the proble dscussed n the Exaple 1 s deonstrated n the

10 1212 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , 2018 Exaple 2 In the Exaple 1, the gven job sequence s J 1 J 3 J 2 J 5 J 4. The optal startng te of each job n ths sequence can be deterned usng the optal tng algorth explaned prevously as follows. Step 0: The ntal schedule s shown n Fgure 1. Note that by re-ndexng jobs accordng to ther postons, jobs J 1, J 3, J 2, J and 5 J can be called 4 J[ 1], J[2], J[3], J[4], and J [5], respectvely. Step 1: Apply nequalty (12) and equaton (13) to dvde fve jobs nto clusters. The job ebers of each cluster are shown as follows: Cluster 1: Cluster 2: Cluster 3: J [, J 1] J [3] J [, J 4] [2] [5] Step 2: Group each cluster nto sub-schedule accordng to equaton (24) Sub-schedule 1: Cluster 1, Cluster 2 Sub-schedule 2: Cluster 3 Step 3: Set g=0. Step 4: Set g=0+1, consder the frst sub-schedule. 6,1,1 1 ( 1) [ 1] 2, E(1) n ( 2) 3] 3, E(2) n 2,,1 1 [ Step 5: ( 1) 0, (1) (2) 0. Step 5.3: The nal h does not exst, go to step 7. Step 7: Shft cluster 1 and 2 by n (1), E(2) 1 E, go to step 8 Step 8: C last S frst 42, cobne sub-schedule 1 wth sub-schedule 2. The second sub-schedule s now coposed of clusters 1 2 1, 2, and 3. Go to step 4. Step 4: Set g = 1+1 = 2, consder the second sub-schedule. ( 1) 1] [2] ( 2) 3], [ [ 3 1 E ( 1) n 5,,, E ( 2) n1,, 1 ( 3) 4] [5], [ 2 E ( 3) n 3,, Step 5: ( 1) 0. Step 5.2: The nal h=1, go to step 6 Step 6: The frst cluster does not ove. Delete the frst cluster fro the consderaton and go to step 5. Step 5: ( 2) 0, ( 2) (3) 0. Step 5.3: The nal h does not exst, go to step 7. Step 7: Shft cluster 2 and 3 by n (2), E(3) 1 E, go to step 8 Step 8: C last S frst (63 ), update (2), (3), E (2), E (3). Go to step ( 2) 0, E ( 2) n,3, 3 ( 3) [ 4] [5] E ( 3) n 2,,, 2

11 Step 5: ( 2) 0 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , Step 5.2: The nal h=2, go to step 6. Step 6: The second cluster does not ove. Delete the second cluster fro the consderaton and go to step 5. Step 5: ( 3) 0. Step 5.1: The nal h=rg=3, go to step 4. Step 4: Set g=2+1=3, g>g (3>2), go to step 9. Step 9: Stop. The result obtaned fro the procedures s shown n the Fgure 3 below. Machne Machne Machne Fgure 3. Results fro the optal tng algorth. Fro the Fgure 3, jobs 1 and 5 are early whle job 4 s tardy. The total cost assocated wth fnal schedule can be calculated as (setup cost = $13) + (earlness cost = $14) + (tardness cost = $8) = $ Meetc Algorth The Meetc Algorth s proposed to deterne the good soluton to the proble n a reasonable aount of te. In ths research, the evolutonary procedure s appled to deterne the good job sequence, whch can be consdered as the global search. For the local search, the optal tng algorth presented n the prevous secton s nserted n the evolutonary procedure. The functon of OPT s to deterne the best startng poston of each job for the gven job sequence. The Meetc procedure s llustrated n the Fgure 4.

12 1214 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , Representaton and ntalzaton Fgure 4. MA procedure. In the representaton, a chroosoe can be consdered as a sequence of jobs. Each gene s an nteger nuber represented a job n the sequence. For the llustraton, a chroosoe of [ ] represents the producton sequence of J1J3J2J5J4. Note that all achnes have the sae producton sequence and the optal tng algorth can be appled to each producton sequence n order to generate the best schedule. The chroosoes n the ntal populaton are generated untl the nuber of chroosoes equals to the ntal populaton sze. 5.2 Crossover Procedure (Unfor Order Based Crossover) Accordng to Lee and Cho (1995), the unfor order based crossover s consdered to be best ft for job sequencng probles. Therefore, ths ethod s selected as the crossover operator n the research. 5.3 Mutaton Procedure (Swappng Mutaton) Each offsprng created fro the crossover operator s evaluated to see f the utaton should occur. The swappng utaton s used here. The ethod s to, frst, randoly select two genes fro a chroosoe and, then, exchange ther postons. 5.4 Evaluaton The purpose of evaluaton s to deterne qualty and ftness value of each chroosoe. The total cost of each chroosoe (TC), represented the chroosoe qualty, can be calculated accordng to the objectve functon entoned n the sesson 2.1. The chroosoe ftness value (f) s deterned usng the equaton 37. Ths value can be consdered as the probablty that each chroosoe wll be selected as a eber of the next generaton. Here, those good chroosoes wth low total costs have greater chances to be selected. 5.5 Selecton 1 f (37) TC Slar to the work of Cheng et al. (1995), two selecton operatons, eltst and roulette wheel, are appled to perfor the reproducton step of evolutonary procedures. The eltst s pleented to preserve the best chroosoe n the enlarge populaton (parents + off-sprngs) of current generaton for the populaton of next generaton. The roulette wheel s, then, appled to select the reanng chroosoes to be the ebers of the next generaton n such a way that a ftter chroosoe has greater chance to be selected. 6. Coputatonal Results Ths secton s to evaluate the perforance of Meetc Algorth dscussed prevously. The soluton obtaned fro the Meetc Algorth s copared wth the optal soluton yelded fro Integer Lnear Prograng (sall sze probles) and the Branch and Bound ethod (sall and large sze probles). All three approaches are coded wth the MATLAB R2014b and

13 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , copled wth the Intel(R) Core(TM) 7 CPU processor 3.07 GHz RAM 7.88 GB. In the Branch and Bound (B&B), the partal job schedules are created by applyng the optal tng algorth to partal job sequences. The reanng partal job sequences after applyng the concept of Branch and Bound to the Exaple 1, usng the soluton obtaned fro the MA as an upper bound, are deonstrated n the Fgure 5. The devaton percentage value (%Dev) appled to evaluate the MA perforance can be calculated as: ( TC (% Dev) MA TC TC Opt Opt ) 100 (38), where TC s the total cost obtaned fro the MA. MA TC Opt s the optal total cost obtaned fro the ILP (sall sze probles) and the B&B (sall and large sze probles). Fgure 5. Branch and Bound structure for Exaple 1.

14 1216 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , 2018 Table 2 presents the detals of proble setup. To deterne the proper probabltes of crossover (pc) and utaton (p), forty tral probles were evaluated. The experent suggests that the cobnaton of pc=0.8 and p=0.1 should be selected. In coparson to the other, ths cobnaton yelds the best result for thrty four out of forty probles. Three stoppng crtera are appled n the MA. The frst crteron stops the search when total cost of the best chroosoe n a generaton equals to zero. The second crteron ternates the MA when the total cost reducton percentage s saller than 0.01 after 200 consecutve generatons. The last crteron fnshes the search when the nuber of generatons reaches 1,000. Table 2. Detals of proble setup. Characterstcs Values Nuber of jobs (n) 5, 10, 12 Nuber of achnes () 3, 5 Processng te (P,k) Dscrete unfor [1,10] Due dates Dscrete unfor [ n ( P, k ),(1.5 P, k )/ ) ] Devaton of Due dates Dscrete unfor [1,3] Earlest due dates (e ) Due dates - Devaton of Due dates Latest due dates (t ) Due dates + Devaton of Due dates Earlness cost ( ) Dscrete unfor [1,5] Tardness cost ( ),,, Setup cost (, j ) Dscrete unfor [0,5] k 2.0 n 1 k1 In the experent, the nfluences of nuber of jobs (3 levels), nuber of achnes (2 levels), and rato of tardness to earlness penaltes (4 levels) on the MA perforance are evaluated. Note that there are totally twenty four treatents wth ffteen replcatons n each treatent. The suary results fro twenty four treatent cobnatons are shown n table 3. Table 3. Average devaton percentage and average coputatonal te of each treatent (15 replcatons of each treatent). Treatents ( n,, / ) Nuber of optal soluton found Average Devaton Percentage Value fro 15 probles Average Coputatonal Te (sec.) MA Branch and Bound ILP (5,3,0.5) (5,3,1.0) (5,3,1.5) (5,3,2.0) (5,5,0.5) (5,5,1.0) (5,5,1.5) (5,5,2.0) (10,3,0.5) N/A (10,3,1.0) N/A (10,3,1.5) N/A (10,3,2.0) N/A (10,5,0.5) N/A (10,5,1.0) N/A (10,5,1.5) N/A (10,5,2.0) N/A (12,3,0.5) N/A (12,3,1.0) N/A (12,3,1.5) N/A (12,3,2.0) N/A (12,5,0.5) N/A (12,5,1.0) N/A (12,5,1.5) N/A (12,5,2.0) N/A *Note that the devaton percentage value deonstrates the percentage of dfference between the soluton obtaned fro the MA and the optal solutons yelded fro ILP and B&B.

15 A. Chaanee & W. Supthak / Songklanakarn J. Sc. Technol. 40 (5), , Fro the study result, the MA yelds optal soluton for 355 out of 360 probles. The treatents of (10, 5, 0.5), (10, 5, 1.0),and (12, 3, 0.5) provde the optal soluton for 14 out of 15 replcatons and the treatent of (12, 5, 2.0) found the optal soluton for 13 out of 15 replcatons. On average, the axu devaton percentage, occurrng n the treatent of (12, 3, 0.5), s 0.48 percent. The axu coputatonal te of the MA and the Branch and Bound ethod are and 19, seconds, respectvely. It s obvous that when the proble sze s gettng larger, the coputatonal te of Branch and Bound ncreases draatcally. Ths result ephaszes that the MA heurstc s approprate to be appled to those edu and large sze probles. 7. Conclusons The research addresses the flow shop schedulng proble wth jobs havng dfferent due wndows under the just n te phlosophy. The objectve s to nze total cost coposng of setup, earlness, and tardness costs. The atheatcal odel s developed to represent the proble. The Meetc Algorth wth the nserton of optal tng algorth has been created to deterne the good soluton n a reasonable aount of te. Accordng to the ethod proposed, the functon of evolutonary procedure s to search for the good producton sequences. The optal tng algorth s, then, appled to deterne the optal schedule of each producton sequence. For perforance evaluaton, the solutons obtaned fro MA heurstc s copared wth the optal solutons yelded fro the Branch and Bound ethod. Fro the study result of 360 probles, the MA heurstc provdes the optal solutons for 355 probles. On average, the axu coputatonal te of the MA and the Branch and Bound are and 19, seconds, respectvely. Ths result ephaszes the beneft of applyng MA heurstc to solve the proble of edu and large szes. Acknowledgeents Ths research receved the fundng of Kasetsart Unversty Scholarshp for Doctoral Student, whch s granted by the Kasetsart Unversty. References Baker, K. R., & Scudder, G. D. (1990). Sequencng wth Earlness and Tardness Penaltes: A Revew. Operatons Research, 38(1), Behnaan, J., Zandeh, M., & Fate Gho, S. M. T. (2009). Due wndow schedulng wth sequence-dependent setup on parallel achnes usng three hybrd eta-heurstc algorths. The nternatonal Journal of Advanced Manufacturng and Technology, 44, Chandra, P., Mehta, P., & Trupat, D. (2009). Perutaton flow shop schedulng wth earlness and tardness penaltes. Internatonal Journal of Producton Research, 47(20), Cheng, R., Gen, M., & Tozawa, T. (1995). Mnax earlness/tardness schedulng n dentcal parallel achne syste usng genetc algorths. Coputers & Industral Engneerng, 30(1-4), Janak, A., Janak, W. A., Toasz, K., & Toasz, K. (2015). A survey on schedulng probles wth due wndows. European Journal of Operatonal Research, 242, Koulaas, C. (1996). Sngle-achne schedulng wth te wndows and earlness/tardness penaltes. European Journal of Operatonal Research, 91,

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