An Evaluation of Sequencing Heuristics for Flexible Flowshop Scheduling Problems with Unrelated Parallel Machines and Dual Criteria

Transcription

1 An Ealuaion of Sequencing Heurisics for Flexible Flowshop Scheduling Problems wih Unrelaed Parallel Machines and Dual Crieria Jii Jungwaanaki a, Manop Reodecha a, Paeena Chaoaliwongse a, Frank Werner b,* a Deparmen of Indusrial Engineering, Faculy of Engineering, Chulalongkorn Uniersiy, Bangkok 0330 Thailand b Faculy of Mahemaics, Oo-on-Guericke-Uniersiy, P.O. Box 420, D-3906 Magdeburg, Germany Absrac This paper deals wih he heurisic soluion of flexible flowshop scheduling problems wih unrelaed parallel machines. A seup ime is necessary before saring he processing of a ob, where he seup ime depends on he preious ob. No preempion of obs is allowed. As obecie funcion, his paper considers he minimizaion of he posiiely weighed conex sum of makespan and he number of ardy obs. This paper deelo some well-known consrucie heurisics for he pure flowshop scheduling problems such as he algorihms gien by Palmer (965), Campbell, Dudek, and Smih (970), Gupa (97), and Dannenbring (977) as well as he inserion heurisic by Nawaz, Enscore and Ham (983) o he flexible flowshop enironmen. By using one of hese heurisics, he firs sage sequence is generaed. This sequence will hen be used, in conuncion wih eiher FIFO or permuaion rules, o consruc a schedule for he oerall problem. The final soluion is hen he bes schedule obained by one of he wo rules. Furhermore, some ieraie heurisics such as a geneic algorihm and a simulaed annealing approach are proposed o increase he qualiy of he consrucie soluion. Deailed compuaional resuls are presened o ealuae he efficiency of he heurisic algorihms. Deailed compuaional resuls wih he algorihms are presened Elseiser B. V. All righs resered. Keywords: flexible flowshop; heurisics; NEH; geneic algorihm; simulaed annealing. Inroducion Producion scheduling can be defined as he allocaion of aailable producion resources oer ime o perform a collecion of asks (Baker, 974). I is an imporan decision making process in he operaion leel. In a modern manufacuring enironmen, many scheduling problems occur. Mos of he scheduling problems are ery significan and hard o sole owing o he complex naure of he problems. This paper is primarily concerned wih indusrial scheduling problems, where one firs has o assign scarce resources o obs and hen o sequence he assigned obs on each resource oer ime. I is mainly concerned wih process indusries like e.g. he glass-conainer (Paul, 979), rubber (Yanney and Kuo, 989), phoographic film (Tsubone, Ohba, Takamuki, and Miyake, 993), seel (Finke and Medeiros, 2002), exile, and food indusries. * Corresponding auhor Fax address: Frank.Werner@Mahemaik.Uni-Magdeburg.DE

2 In hese indusries, producion faciliies are esablished as mulisage producion flowshop faciliies, where a producion sage may be made up of parallel producion lines, machines or any oher producion faciliy. A some sages, he faciliies (machines, lines, ec.) are duplicaed in parallel in order o increase he oerall capaciies of he shop floor, or in order o balance he capaciies of he sages, or eiher o eliminae or reduce he impac of boleneck sages on he oerall shop floor capaciies known as flexible flowshop, muliprocessor flowshop, or hybrid flowshop enironmen. A flexible flowshop enironmen is a generalizaion of he classical flowshop model. There are k sages and some sages may hae only one machine, bu a leas one sage mus hae muliple machines. The obs hae o isi he sages in he same order sring from sage one hrough sage k. A machine can process a mos one ob a a ime and a ob can be processed by a mos one machine a a ime. Preempion of processing is no allowed. The problem consiss of assigning obs o machines a each sage and sequencing he obs assigned o he same machine so ha some opimaliy crieria are minimized. Flexible flowshop models differ in he ype and number of machines a he sages. The processing ime of a ob on he differen machines of eery sage may be idenical, uniform, or unrelaed. These hree possibiliies correspond o he hree classical parallel machine models wih similar noaions. Le p i denoe he processing ime of ob on machine i a sage, ob a sage, and beween he following hree cases:. idenical parallel machines: 2 be he sandard processing ime of i is he relaie speed of machine i for ob a sage. In general, one can disinguish p i = 2. uniform parallel machines: p i = sage, and i = i for all ; for all i and ; 3. unrelaed parallel machines: p i = for all i and. i i for all i and, where i is he relaie speed of machine i a For he pas hree decades, he flexible flowshop scheduling problem has araced many researchers. Numerous research aricles hae been published on his opic (see e.g. Wang, 2005). There are wo main reasons for his, among many ohers. Firs, a flexible flowshop enironmen is a caegory of machine scheduling problems which is difficul o sole (Garey and Johnson, 979; Gupa, 988; Pinedo, 995). The flexible flowshop problem which has wo sages, wih one sage haing a leas wo machines, has already been proed o be NP-hard (Hoogeeen, Lensra, and Velman., 996). Thus, i is unlikely ha polynomial ime algorihms exis for he exac soluion of he general problem. Second, he machine scheduling problem can find many realworld applicaions. Alhough he flexible flowshop problem has been widely sudied in he lieraure, mos of he sudies relaed o flexible flowshop problems are concenraed on problems wih idenical processors, see for insance, Gupa, Krüger, Lauff, Werner and Sosko (2002), Alisanoso, Khoo, and Jiang (2003), Lin and Liao (2003) and Wang and Hunsucker (2003). In he real world siuaion, i is common o find newer or more modern machines running side by side wih older and less efficien machines. Een hough he older machines are less efficien, hey may be kep in he producion lines because of heir high replacemen coss. The older machines may perform he same operaions as he newer ones, bu would generally require longer operaing ime for he same operaion. In his paper, he flexible flowshop problem wih unrelaed parallel machines is howeer considered, ha is, here are differen parallel machines a eery sage, and speeds of he machines are dependen on he obs. Moreoer, seeral indusries encouner sequence-dependen seup imes which resul in een more difficul scheduling problems. Therefore in his paper, sequence-dependen seup ime resricions are aken ino accoun as well. The purpose of his paper is o presen and compare seeral consrucie and ieraie heurisics o sole he flexible flowshop scheduling problem wih unrelaed parallel machines and sequence-dependen seup imes. In addiion, while many papers hae sudied only he makespan crierion, we consider his scheduling problem wih wo opimizaion crieria. One reason for his consideraion is he increasing pressure of high compeiion while

3 cusomers expec ordered goods o be deliered on ime. The remainder of his paper is as follows. In Secion 2, he problem considered in his paper is described and he noaions needed are inroduced. Secion 3 presens an algorihm for he heurisic consrucion of a schedule for he unrelaed parallel machine problem proided ha a ob sequence for he firs processing sage is known. Secion 4 gies consrucie heurisics for he deerminaion of he firs-sage ob sequence for he flexible flowshop problem while Secion 5 discusses some ieraie heurisics. A comparison of all heurisics is presened in Secion 6, and some conclusions are gien in Secion Problem Descripion We consider he problem of scheduling a flexible flowshop enironmen wih unrelaed parallel machines aking sequence-dependen seup imes ino accoun as well. The addressed scheduling problem can be described as follows. Consider a main queue of incoming n obs ( =, 2,, n), a k sage ( =, 2,, k) flowshop scheduling problem, each sage of which has m unrelaed parallel machines in he sysem. Each ob can adance o any one of he m machines a sage one. In general, each ob can be roued o any one of he m machines a sage. When he ob has been processed by he las sage k, using one of he m k machines, i is compleed, and a ha ime, he ob can leae he sysem. The machines are coninuously aailable from ime zero onwards and can handle a mos one ob a a ime. No preempion is allowed. Each ob, ready a ime zero, needs o be scheduled exacly on one machine of each of he k sages. Le he compleion ime of ob be C, hen C max = max { C }. {.. n} Associaed wih each ob is a due dae d 0. Le U = if due dae for ob is smaller han he compleion ime C of ob, oherwise U = 0. The oal number of ardy obs (η T ) is defined as η T = The obecie is o seek a schedule ha minimizes a posiiely weighed conex sum of makespan and he number of ardy obs. Thus, he obecie funcion alue for his research is defined by λc max + ( λ)η T, where 0 λ. λ denoes he weigh (or relaie imporance) gien o C max and η T. Before proceeding o he nex secion, some noaions used hroughou his paper are defined as follows: Indices: sage index, =, 2, 3,..., k i machine index, i =, 2, 3,..., m, l ob index,, l =, 2, 3,..., n Parameers: m d s l i number of parallel machines a sage due dae of ob seup ime from ob l o ob a sage n U = sandard processing ime of ob a sage relaie speed of machine i a sage for ob p processing ime of ob on machine i a sage ; i where p i = O operaing ime of ob a sage i 3

4 4 Variables: C C max U η T compleion ime of ob a sage he makespan, defined as max (C,, C n ), is equialen o he compleion ime of he las ob o leae he sysem a Boolean ariable equal o if ob is ardy, and 0 oherwise he oal number of ardy obs in he schedule 3. Heurisic consrucion of a schedule Since he flexible flowshop scheduling problem is NP-hard (Garey and Johnson, 979; Gupa, 988; Pinedo, 995), algorihms for finding an opimal soluion in polynomial ime are herefore unlikely o exis. Thus, heurisic mehods are sudied o find approximae soluions. Mos researchers deelop exising heurisics for he classical flexible flowshop problem wih idenical machines by using a paricular sequencing rule for he firs sage. They follow he same scheme (see Sanos, Hunsucker and Deal, 996). Firsly, a ob sequence is deermined according o a paricular sequencing rule using modified flowshop algorihms. This problem will be considered in Secion 4. Secondly, obs are assigned as soon as possible o he machines a eery sage using he ob sequence deermined for he firs sage. There are basically wo approaches for his subproblem. The firs way is ha for he oher sages, i.e. from sage wo o sage k, obs are ordered according o heir compleion imes a he preious sage. This means ha he FIFO (Firs in Firs ou) rule is used o find he ob sequence for he nex sage by means of he ob sequence of he preious sage. The second way is o sequence he obs for he oher sages by using he same ob sequence as for he firs sage. In oher words, he permuaion flexible flowshop sequencing problem deermines he order of processing obs on all machines. We will make use of boh procedures (see Algorihm 2) and discuss he modificaions for he problem under consideraion. In he res of his secion, we always assume ha a ob sequence for he firs sage has already been deermined. This secion deals wih he problem of scheduling n obs on unrelaed parallel machines wih sequence-dependen seup imes using his gien ob sequence for he firs sage. Firs, we presen a greedy algorihm which consrucs a schedule for he n obs a a paricular sage proided ha a cerain ob sequence for his sage is gien (see Algorihm ). The obecie is o minimize he flowime and minimize he idle ime of he machines. In he following algorihm, he do refers o an arbirary sage. Algorihm Inpu: ob sequence π =(π [], π [2],, π [n]) for some sage. Sep : For eery machine i, se he aailable ime of each machine o be zero: a[i] = 0, i {,, m}. Le u =. Sep 2: For = π(u) do he following: Sep 2.: Le be he sandard possessing ime of ob, r be he aailable ime (i.e. release dae of ob for he paricular sage), i be he relaie speed of he machine i for ob, and s be he seup ime on a paricular machine proided ha he preious ob on his machine is changed o ob. / Sep 2.2: For eery machine i, deermine he compleion ime Ci of ob on machine i by using he following equaion: / C i = max{ r, a[ i]} + + s ; Sep 2.3 If here is only machine i * ha gies he minimum compleion ime C * = C of ob, hen assign ob o machine i * and proceed o Sep 2.6. Sep 2.4 For eery machine i ha gies he minimum compleion ime C * of ob, calculae he idle ime w i of his machine if ob is assigned o i by using he following formula: / w i = Ci a[ i] s i i / * i

5 5 Sep 2.5 Selec a machine i * wih smalles idle ime w * of ob. i Sep 2.6 Updae he aailable ime of machine i * : a[i * ] = / C i*. / Sep 2.7 Sore he compleion ime C of ob be equal o C i*. Sep 2.8 u = u +. If u n, go o Sep 2. Oherwise, sop. The idea of he heurisic is o assign he obs o he machines eenly in order o balance he workload as much as possible heurisically. The following Algorihm 2 consrucs a for a gien ob sequence ω for he firs sage a complee schedule for he problem under consideraion. I considers boh he FIFO and he permuaion rules o fix he ob sequences for sages 2, 3,, k by means of he ob sequence for he firs sage. Algorihm 2 Inpu: ob sequence ω = (ω [], ω [2],, ω [n]) for he firs sage. Sep : Use he FIFO rule o consruc a schedule. Sep.: Se =. Sep.2: For sage, se π = ω Sep.3: Se (, r, i, s ) = (, r, i, s ). Sep.4: Use Algorihm o find he schedule for sage. + Sep.5: Se r = C (C according o Sep 2.7 in Algorihm ). Sep.6: Deermine ω + such ha compleion ime of ob ω + [] compleion ime of ob ω + [ +], {,.., n-}. Sep.7: If < k (k is he number of sages), hen = + and go o Sep.2, oherwise go o Sep 2. Sep 2: Use he permuaion rule o consruc a schedule. Sep 2.: Se ω = ω 2 =... = ω k = (ω [], ω [2],, ω [n]) and =. Sep 2.2: For sage, se π = ω. Sep 2.3: Se (, r, i, s ) = (, r, i, s ). Sep 2.4: Use Algorihm o find he schedule for sage. + Sep 2.5: Se r = C (C according o Sep 2.7 in Algorihm ). Sep 2.6: If < k, hen = + and go o Sep 2.2, oherwise go o Sep 3. Sep 3: Reurn he bes soluion. Therefore, only he ob sequence used for he firs sage is imporan for his approach. We remind ha he processing and seup imes for eery ob are dependen on he machine and he preious ob, respeciely. This means ha hey are no fixed, unil an assignmen of obs o machines for he corresponding sage has been done. Thus, for applying an algorihm for fixing he ob sequence for sage one, an algorihm for finding he represenaies of he machine speeds and he seup imes is necessary. This algorihm is as follows. The represenaies of machine speed / i and seup ime alues of he daa. Thus, he represenaie of operaing ime ime / i plus he represenaie of he seup ime s / l O / s / l use he minimum, maximum and aerage of ob a sage is he sum of he processing. Nine combinaions of relaie speeds and seup imes will be used in he suggesed algorihm. The ob sequence for he firs sage is hen fixed as he ob sequence wih he bes funcion alue obained by all combinaions of he nine differen relaie speeds and seup imes (see Algorihm 3). Algorihm 3 Inpu: Relaie speeds and seup imes. Sep : Deermine he represenaies of relaie speeds and seup imes for =,,k.

6 Sep.: a. If speed =, hen / = min{ ; i {,..., m }, {,..., n} } i b. If speed = 2, hen / = max{ ; i {,..., m }, {,..., n} } i i c. If speed = 3, hen / = aerage{ ; i {,..., m }, {,..., n} } Sep.2: a. If seup =, hen b. If seup = 2, hen c. If seup = 3, hen i l i s / = min{ ; l {,..., n}, l } l s l i s / = max{ ; l {,..., n}, l } l s l s / = aerage{ ; l {,..., n}, l } s l Sep 2: For eery speed, deermine he firs-sage ob sequence and a complee schedule. Sep 2.: Se speed =. Sep 2.2: Se seup =. Sep 2.3: For eery sage (=,,k) and ob (=,,n), deermine he represenaie of he operaing ime O / by using he following equaion: O / = Sep 2.4: Find he sequence ω of he firs sage, by using he represenaies of he operaing imes from Sep 2.3 o find he sequence of he firs sage wih some modified flowshop heurisic. Sep 2.5: Assign all obs a sage (=,,k) o some machines by using Algorihm 2. Sep 2.6: If seup < 3, hen seup = seup + and go o Sep 2.3. Sep 2.7: If speed < 3, hen speed = speed + and go o Sep 2.2 else go o Sep 3. Sep 3: Reurn bes soluion. I remains o discuss how he ob sequence for he firs sage is found in Sep 2.4. This is done in Secion 4 in deail. 4. Consrucie algorihms In his secion we presen some consrucie algorihms for deermining he ob sequence for he firs sage for he problem considered. This exends exising algorihms mosly deeloped for he flexible flowshop problem wih idenical parallel machines wihou seup ime consideraions. Using hen Algorihms 3 from Secion 3, a heurisic schedule for he problem under consideraion is found. We adap and deelop seeral consrucie algorihms for he pure flowshop scheduling problem. As flowshop heurisics, we use he algorihms gien by Palmer (965), Campbell, Dudek, and Smih (970), Gupa (97), and Dannenbring (977) and he inserion heurisic by Nawaz, Enscore, and Ham (983). Nex, we discuss he generalizaions of hese heurisics o he problem under consideraion. 4. Palmer A heurisic deeloped by Palmer (965), in an effor o use Johnson s rule, is buil around he noion of a slope index. The slope index gies a large alue o obs ha hae a endency of progressing from small o large operaing imes as hey moe hrough he sages. The sequence of operaions is gien by prioriy o obs haing he sronges endency o progress from shor imes o long imes. This means ha he ob sequence can be generaed based upon an non-increasing order of he slope indices. Le S () be he slope index for ob and O be he operaing ime of ob a sage. Palmer s slope index is calculaed as follows: k S( ) = = / i + s / l {[ k (2 )] } O 6

7 To illusrae Palmer s mehod for use in he flowshop enironmen, he example gien in Table will be uilized. Table Sandard processing imes and due dae for eery ob of a hree-sage flowshop problem Job d Since here is only one machine for eery sage, he operaing imes for eery ob are equal o is sandard processing imes ( O = ). The slope indices for he fie obs are now calculaed as follows: S() = - [2 O + 0 O 2 2 O 3 ], i.e.. S() = -2() - 0(2) + 2(20) = 8; 2. S(2) = -2(3) - 0(5) + 2(8) = 0; 3. S(3) = -2(20) - 0(0) + 2(2) = -6; 4. S(4) = -2(9) - 0(2) + 2(20) = 22; 5. S(5) = -2() - 0(8) + 2(5) = 8. Palmer s heurisic sequences he obs in non-increasing order of he slope indices. For he ob se of Table, his heurisic yields he sequence π = (4,, 2, 5, 3) for he firs sage. Palmer s heurisic yields a makespan alue of 06. Using a 0- mixed ineger programming formulaion, one can confirm ha 06 is he opimal makespan alue as well. Now, Palmer s mehod for he flexible flowshop problem wih unrelaed parallel machines and sequencedependen seup imes is deeloped as follows. Le S (, i / /, s ) be he slope index for ob a relaie speed / i and seup ime s / l, be he sandard processing ime of ob a sage, relaie speed on machine i a sage for ob, and s / l l / i 7 be he represenaie of he be he represenaie of he seup ime from ob l o ob a sage. Palmer s slope index for he flexible flowshop problem wih unrelaed parallel machines and seup imes is calculaed as follows: k / / / S(, i, sl ) = [ k (2 )]( + s / l ) = i Since he processing imes and he seup imes for eery ob are dependen on he machine and he preious ob, respeciely, he represenaie of he operaing ime of ob a sage in Palmer s mehod adapaion is he sum of he processing ime we se / i / i O / plus he seup ime s / l o be he minimum, maximum, and aerage relaie speeds, and. To consruc a schedule for he oerall problem, s / l o be he minimum, maximum, and aerage seup imes regardless for he machine and he preious ob. All hese nine combinaions of relaie speeds and seup imes will be used as has been described in Algorihm 3 and finally, he soluion wih he bes obecie funcion alue obained by all differen relaie speeds and seup imes is aken. 4.2 Campbell, Dudek, and Smih Campbell, Dudek, and Smih (970) deelop one of he mos significan heurisic mehods for flowshop problems wih makespan crierion, in he following denoed by CDS. Is srengh lies in wo properies: () i

8 uses Johnson s rule in a heurisic fashion, and (2) i generally creaes seeral schedules from which a bes schedule can be chosen. In so doing, k sub-problems are creaed and Johnson s rule is applied o each of he sub-problems. Thus, k ob sequences are generaed. Since Johnson s algorihm is a wo-sage algorihm, a k- sage problem mus be collaed ino a wo-sage problem. Le g be a couner for he k sub-problems, he operaing imes for he firs sage are denoed as a(, g), where denoes he ob and g denoes he g-h subproblem. Similarly, b(, g) denoes he second sage operaing imes of ob and sub-problem g. Gien hese noaions, he operaing imes are calculaed by he following wo formulas: and g O = a(, g) = b(, g) = k O = k g+ For each of he sub-problems, Johnson s algorihm proides a ob sequence using he alues a(, g) and b(, g). Once Johnson s sequence is creaed, he problem is hen reurned o he consideraion of all k sages. Under he CDS adapaion rule, he operaing imes for he flexible flowshop problem wih unrelaed parallel machines and seup imes are calculaed as follows: g / / / a(, g, i, sl ) = ( + s / l ) = and b(, g, / i, s / l ) = = k i k ( + s / g+ To generae he firs sage, Johnson s ordering is creaed, and he problem is hen reurned o he consideraion of k sages by calling Algorihm 2 for all nine combinaions of relaie speeds and seup imes considered in Algorihm Gupa Gupa (97) proides an algorihm which, in a similar manner as Palmer s algorihm, uses a slope index. Denoe G() as he slope index generaed by Gupa s mehod for ob. Then G() is calculaed as follows: i / l ) 8 where if k e = k if O O ( ) = e G ; g + min { O + O g } O > O g k. Afer calculaing G() for all obs, he obs are subsequenly ranked in a non-decreasing order of he slope indices. / / Under he Gupa adapaion rule, le G (, i, sl ) be he slope index for ob a relaie speed / i and seup ime s / l. Gupa s slope index for he flexible flowshop wih unrelaed parallel machines and seup imes is hen calculaed as follows: G (, / i, s / l ) = g min {( g k / g i + s e / g l ) + ( / g+ g+ i + s / g+ l )}

9 9 wih e = if if k / / k ( + s ) > ( + ) / l s / k l i i k / / k ( + s ) ( + ) / l s / k l i i 4.4 Dannenbring Like Palmer s rule, Dannenbring (977) deelo a mehod by using Johnson s algorihm as a foundaion. Furhermore, algorihms of CDS and Palmer are also exhibied. Dannenbring consrucs only one wo-sage problem, bu he operaing imes for he obs reflec he behaior of Palmer s slope index. Denoe a() and b() as he operaing imes for he consruced wo-sage problem. The calculaions of a() and b() are as follows: and k a( ) = ( k + )( = k b( ) = = Afer calculaing a() and b() for all obs, he obs are subsequenly ranked by applying Johnson s algorihm o generae he ob sequence for sage one. Under he Dannenbring adapaion rule, he operaing imes for he flexible flowshop problem wih unrelaed parallel machines and seup imes are calculaed as follows: and 4.5 NEH Heurisic a(, / i, s / l O O k ) = ( k + )( + s / = i ) / l k / / / b(, i, sl ) = ( + s l ) /. = i Nawaz, Enscore and Ham (983) deelop he probably bes consrucie heurisic mehod for he permuaion flowshop problem, called NEH algorihm. I is based on he idea ha a ob wih a high oal operaing ime on he machines should be placed firs a an appropriae relaie order in he sequence. Thus, obs are sored in nonincreasing order of heir oal operaing ime requiremens. The final sequence is buil in a consrucie way, adding a new ob a each sep and finding he bes parial soluion.. The oal operaing imes for ob is calculaed using he formula k P = O, =,, n. = 2. The n obs are sored in non-increasing order of heir oal operaing ime P on he machines. Then he firs wo obs (hose wih larges P ) are aken and he wo possible schedules conaining hem are ealuaed. The sequence wih beer obecie funcion alue is aken for furher consideraion. 3. Take eery remaining ob in he sored lis calculaed in Sep 2 and find he bes schedule by placing i a all possible posiions in he sequence of obs ha are already scheduled. For example, if (, 2, 3) is he curren sequence of scheduled obs and ob r is he remaining ob wih larges P r in he sored lis, hen ob r could be placed a four posiions: (r,, 2, 3), (, r, 2, 3), (, 2, r, 3) or (, 2, 3, r ). The sequence wih bes obecie funcion alue among he four considered is seleced for furher exension. )

10 To apply he NEH algorihm o he flexible flowshop problem wih unrelaed parallel machines, he oal operaing imes for calculaing he ob sequence for he firs sage are calculaed according o Sep 2.3 of Algorihm 3 for he nine combinaions of relaie speeds of machines and seup imes. 4.6 Illusraie Example In order o illusrae he applicaion of he proposed procedures, le n=5, k=3 and m =m 2 =m 3 =3. Table 2 shows he sandard processing imes and he due dae of eery ob. Tables 3a -3c and Tables 4a - 4c show he marix of seup ime for eery sage and he relaie speed for eery machine. Table 2 Sandard processing imes and due dae of eery ob of he hree-sage flexible flowshop Job d Table 3a The marix of seup imes from ob o ob l a sage o from Table 3b The marix of seup imes from ob o ob l a sage 2 o from Table 3c The marix of seup imes from ob o ob l a sage 3 o from

11 Table 4a The relaie speed of eery machine ( i ) a sage ob machine i Table 4b The relaie speed of eery machine ( i 2 ) a sage 2 ob machine I Table 4c The relaie speed of eery machine ( i 3 ) a sage 3 ob machine i For his example, he opimal soluions for eery λ {0, 0.5, } by using e.g. a 0- mixed ineger programming code are shown in Table 5. All proposed heurisics generae an opimal schedule for he λ alues considered. Table 5 Resuls of he calculaion of an opimal schedule λ Schedule Value Cmax η T 0 sage : 4 2 // // sage 2: 3 5 // 4 // 2 sage 3: // // sage : 4 // 5 // sage 2: 3 4 // 5 // 2 sage 3: 5 // 4 // 3 2 sage : 4 // 5 // 3 2 sage 2: 5 // 3 4 // 2 sage 3: 5 // 4 // Ieraie algorihms In his secion, we consider wo ieraie algorihms for he heurisic soluion of he problem considered, namely a geneic algorihm and a simulaed annnealing algorihm. All hese ieraie algorihms work wih he ob sequences for he firs sage. If a new ob sequence has been generaed, he schedule consrucion procedure described in Secion 3 is applied and he obecie funcion alue of his schedule is used for ealuaing he ob sequence for he firs sage.

12 2 5. Geneic Algorihm A geneic algorihm is an ieraie heurisic based on Darwin s eoluionary heory abou surial of he fies and naural selecion. I belongs o he eoluionary class of arificial inelligen (AI) echniques. Holland (975) proposes some basic principles of naural eoluion as a mehodology o sole decision-making problems. For he classical flow shop problem, he firs geneic algorihm has been gien by Werner (984). A geneic algorihm is characerized by a parallel search of he sae space in conras o a poin-by-poin search by conenional opimizaion echniques. The parallel search is achieed by keeping a se of possible soluions under consideraion, called a populaion. An indiidual in he populaion is a sring of symbols and i is an absrac represenaion of a soluion. The algorihm sars wih an iniial generaion of arificial indiiduals which are creaed randomly. Each symbol is called a gene and each sring of genes is ermed as a chromosome. The indiiduals in he populaion are ealuaed by some finess measure o describe quaniaiely how well he indiidual masers is ask. The populaion of chromosomes eoles from some generaion o he nex hrough he use of wo ypes of geneic operaors: () unary operaors such as muaion and inersion which change he geneic srucure of a single chromosome, and (2) a higher-order operaor, referred o as crossoer which consiss of obaining new indiidual(s) by combining he geneic maerial from wo seleced paren chromosomes. When applying crossoer, wo indiiduals (parens) are seleced from he populaion and new soluion(s), called offspring, is(are) creaed. Muaion creaes a new soluion by a random change on a seleced indiidual. The geneic operaors (crossoer and muaion) are applied o randomly seleced parens o generae new offspring. Then he new populaion is seleced ou of he indiiduals of he curren populaion and he new generaed chromosomes (Gen and Cheng, 997). A basic GA is shown below. Geneic Algorihm Begin N p = Populaion size. P c = Crossoer probabiliy. P m = Muaion probabiliy. P Consruc (N p ) // Consruc he iniial populaion For = o N p Ealuae Finess(p[]); EndFor Repea For = o P c N p (x, y) Selec_Cross (P); // Selec wo parens proporional o heir finess using roulee wheel. Crossed[] Cross_Oer (x, y); Ealuae Finess(Crossed[]); EndFor For = o P m N p (x) Selec_Muae (P); // Selec one paren proporional o is finess using roulee wheel. Muaed[] Muaion (x); Ealuae Finess(Muaed[]); EndFor P Selec_for_Nex_Gen // Selec from offspring and parens he new parens for he nex generaion. Unil (Sop-Crierion) Reurn (indiidual wih highes finess alue) End

13 The applicaion of a geneic algorihm o any problem requires he represenaion of a soluion for he problem as a sring of symbols, a choice of geneic operaors, an ealuaion funcion, a selecion mechanism, and he deerminaion of geneic parameers (populaion size and probabiliies of crossoer and muaion). Encoding scheme The sring coding is used for making up a chromosome direcly. Thus, he digis of he chromosome will be equal o he numbers of obs (n digis). The gene codes are posiie ineger numbers represening he numbers of obs. Thus, he mehod is named as geneic algorihm based on he ob code. When encoding he soluions, he concep of random key encoding which is he mos prominen and successful example of such an encoding proposed by Bean (994) is used. The sequence of he firs sage is represened by he numbers of genes from he lef side o he righ side as shown in Figure Fig.. Illusraion of a chromosome. Crossoer A subsequence preseraion crossoer operaor is used since he offspring inheri a subsequence from he firs paren will ry o coner he soluion. Moreoer, he adanage of his mehod guaranees he generaion of feasible sequences afer he crossoer. This mehod randomly akes wo parens from he populaion and creaes wo offspring. The firs offspring is creaed by duplicaing a subsequence from he firs paren ino an offspring and hen compleing he offspring by insering he remaining obs deried from he second paren by making a lef-o-righ scan. The second offspring is creaed by fixing a sequence of obs from he second paren whose obs are he same as he random subsequence of he firs paren and insering he remaining obs deried from he firs paren by making a lef-o-righ scan. The crossoer operaor is depiced in Figure 2. 3 Fig. 2. Illusraion of he subsequence crossoer operaor. Muaion This operaor randomly chooses a chromosome and wo of is genes as a candidae o be muaed, and hen swa he genes on such a chromosome. I is called reciprocal exchange muaion as illusraed in Figure 3. Fig. 3. Illusraion of he reciprocal exchange muaion.

14 Ealuaion policy During each generaion, chromosomes are ealuaed using some measure of finess. In mos opimizaion applicaions, he finess funcion is consruced based on he original obecie funcion. The finess alue of each chromosome is a key measure o guide he direcion of search in GA. Due o he minimizaion problem, he finess alue mus be in inerse proporion o he obecie funcion alue so ha a fier chromosome has a larger finess alue. finess ( z ) =, z =,2,..., populaion _ size ; f ( ) z where finess( z ) is he finess alue and f( z ) is he obecie funcion alue of he z-h chromosome for he complee schedule generaed from he corresponding ob sequence for he firs sage using Algorihms -2. According o his research obecie, he obecie is o minimize a posiiely weighed conex sum of makespan and number of ardy obs. Thus, he finess alue of a chromosome, finess( z ) is gien by 4 finess ( ) = z λc max ( ) + ( λ ) U ( ) + z n = z, z =,2,..., populaion _ size ; where C max ( z ) is he makespan of he z-h chromosome (resp. of he resuling complee schedule),u ( z ) is a Boolean ariable for ob of he z-h chromosome which is equal o if ob is ardy, and 0 oherwise, and λ denoes he weigh (or relaie imporance) gien o makespan and number of ardy obs. The larges alue of he finess funcion is he lowes alue of he posiiely weighed conex sum of makespan and number of ardy obs. In he denominaor alue one is added in order o preen a diision by zero when he weigh λ and he number of ardy obs are zero. Selecion policy An eliis policy and enlarged sampling space echnique are used. Boh parens and he offspring hae he same chance of compleing for surial. Figure 4 illusraes he selecion based on an enlarged sampling space. Then Holland s proporionae selecion or roulee wheel selecion is employed o reproduce he nex generaion based on he curren enlarged populaion. The idea is o deermine a selecion probabiliy (also called surial probabiliy) for each chromosome proporional o is finess alue. For chromosome z wih finess finess( z ), is selecion probabiliy prob( z ) is calculaed as follows: finess( z ) prob ( z ) = populaion _ size+ offspring _ size finess( ) z= z Fig. 4. Illusraion of he selecion performed on an enlarged sampling space.

15 5 Terminaion condiion In he implemenaion of GA, he search procedure is erminaed when he bes obecie funcion alue found so far is no updaed for a predeermined number of generaions. I can also be erminaed when he number of generaion exceeds he predeermined number of generaions. We apply GA wih a crossoer rae and a muaion rae boh equal o 0.2. The maximum number of generaions is se equal o 000 in adance. Moreoer, GA is erminaed when wihin 00 generaions no improemen of he bes funcion alue has been obained. For he problem daa gien in Tables 2-4, he geneic algorihm also produced opimal soluions for all alues of λ considerd. 5.2 Simulaed Annealing Simulaed annealing (SA) is an enhanced ersion of local opimizaion or an ieraie search mehod, in which an iniial soluion is repeaedly improed by making small local aleraions unil no such aleraion yields a beer soluion. Annealing refers o he process which occurs when physical subsances, such as meals, are raised o a high energy leel (meled) and hen gradually cooled unil some solid sae is reached. The goal of his process is o reach he lowes energy sae. In his process, physical subsances usually moe from higher energy saes o lower ones if he cooling process is sufficienly slow, so a minimizaion naurally occurs. Due o naural ariabiliy, howeer, here is some probabiliy a each sage of he cooling process ha a ransiion o a higher energy sae will occur. As he energy sae naurally declines, he probabiliy of moing o a higher energy sae decreases. Then he simulaed annealing algorihm is as follows, where RANDOM(0,) denoes a random number aken from a random ariable uniformly disribued in he ineral (0,). As menioned earlier, SA aem o aoid an enrapmen in a local opimum by someimes acceping a moe ha deerioraes he alue of he obecie funcion f. Saring from an iniial soluion randomly generaed, SA generaes a new soluion S' in he neighborhood of he iniial soluion S. This new poin's obecie funcion alue is hen compared o he iniial poin's alue (remind ha he obecie funcion alue of he full schedule generaed from he ob sequence for he firs sage is aken). The change in he obecie funcion alue, = f(s') f(s), is calculaed in order o deermine wheher he new alue is smaller. For he case of minimizaion, if he obecie funcion alue decreases ( < 0), i is auomaically acceped and i becomes he poin from which he search will coninue. The algorihm will hen proceed wih anoher ieraion. If he obecie funcion alue increases ( 0), hen higher alues of he obecie funcion may also be acceped wih a probabiliy, usually deermined by a funcion, exp (- /T), where T is a conrol parameer called he emperaure. This emperaure, which is simply a posiie number, is periodically reduced by a emperaure scheme, so ha i moes gradually from a relaiely high alue o near zero as he mehod progresses. Thus, a he sar of SA mos deerioraing moes are acceped, bu a he end only amelioraing ones are likely o be acceped. The mehod conerges o a local opimum as he emperaure approaches zero, bu by occasionally acceping poins wih higher alues of he obecie funcion, he SA algorihm is able o escape local opima. During he algorihm, T is reduced eery NT ieraions according o T i+ = T i - RT T 0, where NT is a prese parameer which esablishes he number of ieraions beween emperaure reducions, called he epoch lengh, index i refers o he ih epoch and RT is he emperaure reducion facor. We decided o ake a linear reducion scheme (Winson and Venkaaramanan, 2003) which, for a consan number of epochs, reduces he emperaure slower han e.g. a geomeric cooling scheme. Neighborhood search In search algorihms, he neighborhood of a soluion describes which soluions can be generaed from he curren one by a specific aleraion. As an example, an inserion or pairwise inerchange neighborhood is commonly used when soluions are described by permuaions. For a ob sequence for he firs sage, a ob a some posiion is seleced and reinsered a some oher posiion in he inserion neighborhood, while wo obs are seleced and inerchanged in he laer neighborhood. We use he pairwise inerchange neighborhood, where a ob sequence has n ( n ) neighbors. 2

16 6 Simulaed annealing algorihm Begin T = T 0 ; // T 0 is he iniial emperaure S = S 0 ; // S 0 is he iniial soluion BesS = S 0 ; // BesS is he bes soluion seen so far BesF =f(s 0 ) // BesF is he bes obecie funcion alue seen so far NT // Number of ieraions wih consan emperaure Repea For l = o NT do S' = PERTURB (S); {generae a neighbor S of S} If f(s') < BesF hen BesF = f(s'); BesS = S'; = f(s') f(s); If < 0 hen S = S'; Else if (exp (- /T) > RANDOM(0,)) hen S = S'; Endif If no updae of he bes soluion wihin he las 30 ieraions hen l = NT; EndFor; UPDATE (T); Unil (Sop-Crierion) Reurn (ob sequence wih smalles funcion alue) End Terminaion condiion In SA, he emperaure is reduced o a smaller emperaure when he bes obecie funcion alue found so far is no updaed for a predeermined number of ieraions and i is also reduced when NT ieraions hae been performed (i.e. a he end of an epoch lengh). The procedure is erminaed when he emperaure becomes equal o or less han zero. We apply SA wih an iniial emperaure equal o T 0 =2000. The number of ieraions beween emperaure reducions is NT =00 and he emperaure reducion facor RT=0.. If no updae of he bes soluion is made, he emperaure is reduced afer 0.3 x NT=30 ieraions. For he problem daa gien in Tables 2-4, he simulaed annealing algorihm also produced opimal soluions for all alues of λ considerd. 6. Experimens 6. Small Size Problems All mehods presened aboe produce an opimal soluion for he example problem. In pracice, howeer, his will no always be he case. This secion proides experimens designed o sudy he effecieness of he consrucie algorihms such as Palmer (PAL), CDS, Gupa (GUP), Dannnenbring (DAN), and he NEH adapaion, and he ieraie algorihms such as he geneic algorihm (GA) and simulaed annealing (SA). The small sample consiss of 80 flexible flowshop problems. The number of obs ranges from hree o fie, he number of sages ranges form wo o fie, and he number of unrelaed parallel machines a each sage ranges from wo o fie. The sandard processing imes of he operaions are uniformly disribued ineger alues from one o fify ime unis, while he relaie speed for eery machine a each sage for eery ob aries beween 0.7 and.3, and he seup imes from he preious ob o he nex ob a each sage range from zero o en. In order o ealuae he performance of each of he heurisics, we use he relaie error for λ>0 and he absolue error for λ=0. If he posiiely weighed conex sum of he obecies obained by a paricular heurisic is denoed by f heu and he opimal posiiely weighed conex sum of he obecies is denoed by f op, hen he error E is calculaed as:

17 7 E = f heu f f heu f op op f op ; when λ > 0 ; oherwise Table 6 shows he mean error E for eery heurisic. The resuls show ha he geneic algorihm is he bes one for eery λ. Table 7 shows he percenages of he paricular alues of ardy obs obained for eery heurisic. Table 8 shows he percenage of insances for which he error is in he corresponding ineral. The geneic algorihm finds he opimal alue in abou 45% of all problems, simulaed annealing in abou 40%, and he NEH algorihm in abou 30%, while he oher heurisics in abou 25% of he problems. Table 6 Mean error E for eery heurisic Mean error E (ob unis) Mean error E (%) λ = 0 λ = 0.5 λ = PAL CDS GUP DAN NEH GA SA Table 7 Percenage of number of ardy obs for eery heurisic η T PAL CDS GUP DAN NEH GA SA Large Size Problems For a comparison of he proposed heurisics, some es problems are randomly generaed he deails of which are as follows. () A manufacuring enironmen is defined as a number of obs number of sages scenario in which he number of obs can be 5, 0, 30, or 50, and he number of sages can be 5, 0, or 20. Thus, wele manufacuring enironmens are considered for he experimens. (2) For he obecie funcion, he weigh λ for each problem can be 0, 0.25, 0.5, 0.75, or. Thus, he oal number of scenarios o be esed is equal o sixy differen problem ypes. (3) The number of unrelaed parallel machines a each sage is randomly se beween one and hree, bu a leas one sage mus hae muliple machines. (4) The sandard processing imes of a ob are inegers uniformly disribued in he ineral (, 00). The relaie speeds ary beween 0.7 and.3, and seup imes ary beween zero and en ime unis. The compuaional experimens are conduced for 50 es problems in each scenario by using an Inel Penium GHz CPU. While an opimal soluion for he small problem se is obainable using e.g. 0- mixed ineger programming, for he large size problems opimal soluions are ery difficul o find. Therefore, his paper compares he performance of all proposed heurisics by using he mean (relaie or absolue) deiaion (MD) which is calculaed as follows:

18 8 MD = 50 f ( ( 50 heu f heu f f bes f bes bes ) ) ; when λ > 0, ; oherwise where f heu is he funcion alue obained by a paricular of he fie consrucie or wo ieraie algorihms, and f bes is he bes funcion alue obained by all heurisics. The resuls for all algorihms gien in Table 9 hrough Table 3 indicae ha, among he consrucie algorihms he NEH algorihm can obain 279, 242, 249, 254, and 245 bes soluions for λ {0.00, 0.25, 0.50, 0.75 and.00}, respeciely, and he oal mean relaie deiaions (ΣMD and Σ %MD, respeciely) from he bes funcion alue are unis as well as 9.85%, 8.65%, 7.72%, and 8.34%, respeciely. The oher consrucie algorihms obain a number of bes soluions subsanially smaller han he NEH algorihm, and hae oal mean deiaions from he bes funcion alue considerably higher han he NEH algorihm. We can conclude ha he NEH algorihm clearly ouperforms he oher consrucie algorihms. Disregarding he NEH algorihm, he resuls show ha he performance of he CDS algorihm is beer han he performance of he algorihms by Palmer, Gupa and Dannenbring. Comparing he performance of consrucie and ieraie algorihms reeals ha he performance of he ieraie algorihms is subsanially beer han he performance of consrucie algorihms bu he CPU imes of he consrucie algorihms are faser han he CPU imes of he ieraie algorihms. Among he ieraie algorihms, he performance of he geneic algorihm is slighly beer han he performance of simulaed annealing. The compuaional imes of he geneic algorihm and he simulaed annealing algorihm are comparable. The resuls for he differen alues of λ are similar, bu he superioriy of he geneic algorihm oer simulaed annealing and he NEH algorihm is significan in paricular for λ=0 (see Table 9). 7. Conclusions This paper deelo and compares some consrucie and ieraie heurisics for flexible flowshop scheduling problems wih unrelaed parallel machines, where a sequence-dependen seup ime is necessary before saring he processing of a ob. As obecie funcion, his paper considers he minimizaion of he posiiely weighed conex sum of makespan and he number of ardy obs. All he heurisics suggesed in his paper generae a ob sequence for he firs sage by using generalized consrucie heurisics for pure flowshop scheduling problems such as Palmer (965), Campbell, Dudek, and Smih (970), Gupa (97), and Dannenbring (977) as well as he inserion heurisic by Nawaz, Enscore and Ham (983) o he flexible flowshop enironmen or ieraie algorihms such as a geneic or a simulaed annealing algorihm. The resuls show ha among he consrucie algorihms, a ob inserion based algorihm ouperforms he oher consrucie algorihms, while among he ieraie algorihms he geneic algorihm ouperforms he simulaed annealing algorihm. In fuure research, heurisics ha use differen sage sequencing orders will be sudied o increase he soluion qualiy furher. In addiion, he procedure for fixing he processing and seup imes as well as he relaie speeds in he consrucie algorihms can be refined furher. The use of adanced echniques, e.g. he deelopmen of a abu search, an colony or an ieraed local search algorihm, can also be worhwhile. Acknowledgemen The fourh auhor has been suppored by INTAS (proec ).

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20 20 Table 8 Percenage of numbers how ofen a paricular error E has been obained by eery heurisic E (%) PAL CDS GUP DAN NEH GA SA λ=0.5 λ=.0 λ=0.5 λ=.0 λ=0.5 λ=.0 λ=0.5 λ=.0 λ=0.5 λ=.0 λ=0.5 λ=.0 λ=0.5 λ=.0 E = < E < E < E < E < E < E < E < E < E < E < E < E < E < E < E < E < E < E