Sequencing algorithms for flexible flow shop problems with unrelated parallel machines, setup times, and dual criteria

Transcription

1 Sequencing algorihms for flexible flow shop problems wih unrelaed parallel machines, seup imes, and dual crieria Jii Jungwaanaki a, Manop Reodecha a, Paveena Chaovaliwongse a, Frank Werner b,* a Deparmen of Indusrial Engineering, Faculy of Engineering, Chulalongkorn Universiy, Bangkok 0330 Thailand b Faculy of Mahemaics, Oo-von-Guericke-Universiy, P.O. Box 420, D-3906 Magdeburg, Germany * Absrac This paper considers a flexible flow shop scheduling problem, where a leas one producion sage is made up of unrelaed parallel machines. Moreover, sequence- and machine- dependen seup imes are given. The objecive is o find a schedule ha minimizes a posiively weighed convex sum of makespan and he number of ardy jobs in a saic flexible flow shop environmen. For his problem, a 0- mixed ineger program is formulaed. The problem is however a combinaorial opimizaion problem which is oo difficul o be solved opimally for large problem sizes, and hence heurisics are used o obain good soluions in a reasonable ime. The proposed consrucive heurisics for sequencing he jobs sar wih he generaion of he represenaives of he operaion ime for each operaion. Then some dispaching rules and flow shop makespan heurisics are developed. To improve he soluions obained by he consrucive algorihms, polynomial heurisic improvemen algorihms based on shif moves and pairwise inerchanges of jobs are applied. In addiion, meaheurisics are suggesed, namely simulaed annealing, abu search and geneic algorihms. The basic parameers of each meaheurisic are briefly discussed in his paper. The performance of he heurisics is compared relaive o each oher on a se of es problems wih up o 50 jobs and 20 sages and wih an opimal soluion for small-size problems. We have found ha among he consrucive algorihms he inserion based approach is superior o he ohers, whereas he proposed simulaed algorihms are beer han abu search and geneic algorihms among he ieraive meaheurisic algorihms. Keywords: Flexible flow shop; Consrucive algorihms; Simulaed Annealing; Tabu Search; Geneic algorihms * Corresponding auhor. Tel.: ; /fax.: address: frank.werner@mahemaik.uni-magdeburg.de

2 . Inroducion Producion scheduling is a decision-making process in he operaion level. I can be defined as he allocaion of available producion resources o carry ou cerain asks in an efficien manner. A frequenly occurring scheduling problem is difficul o solve due is complex naure. This paper is primarily concerned wih indusrial scheduling problems, where one firs has o assign jobs o limied resources and hen o sequence he assigned jobs on each resource over ime. I is mainly concerned wih processing indusries ha are esablished as muli-sage producion faciliies wih muliple producion unis per sage (i.e., parallel machines), e.g. a exile indusry (Karacapilidis and Pappis []), an auomobile assembly plan (Agneis e al. [2]), a prined circui board manufacure (Alisanoso, Khoo and Jiang [3], and Hsieh, Chang and Hsu [4]), and so on. In such indusries, a some sages he faciliies are duplicaed in parallel o increase he overall capaciies or o balance he capaciies of he sages, or eiher o eliminae or o reduce he impac of boleneck sages on he shop floor capaciies. Due o he mixed characer of such a producion sysem, which lies beween flow shop and parallel machines, i is known as a flexible or hybrid flow shop environmen. An ordinary flow shop model is a muli-sage producion process, where he jobs have o visi all sages in he same order sring, whereas a flexible flow shop model, a generalizaion of he classical flow shop model, is more realisic, and i assumes ha a leas one sage mus have muliple machines. Moreover, a machine can process a mos one job a a ime and a job can be processed by a mos one machine a a ime. Preempion of processing is no allowed. The problem consiss of assigning he jobs o machines a each sage and sequencing he jobs assigned o he same machine so ha some opimaliy crieria are minimized. Alhough he flexible flow shop problem has been widely sudied in he lieraure, mos of he sudies relaed o flexible flow shop problems are concenraed on problems wih idenical processors, see for insance, Gupa, Krüger, Lauff, Werner and Soskov [5], Alisanoso, Khoo and Jiang [3], Lin and Liao [6], and Wang and Hunsucker [7]. In a real world siuaion, i is common o find newer or more modern machines running side by side wih older and less efficien machines. Even 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 a longer operaing ime for he same operaion. In his paper, he flexible flow shop problem wih unrelaed parallel machines is considered, i.e., here are differen parallel machines a every sage and speeds of he machines are dependen on he jobs. Moreover, several indusries encouner seup imes which resul in even more difficul scheduling problems. In his paper, boh sequence- and machine-dependen seup ime resricions are aken ino accoun as well. A deailed survey for he flexible flow shop problem has been given by Linn and Zhang [8] and Wang [9]. Mos of he earlier lieraure has considered he simple case of only wo sages. Arhanari and Ramamurhy [0] and Salvador [] are among he firs who define he flexible flow shop problem. They propose a branch and bound mehod o ackle he problem. Such a mehod is an exac soluion echnique which guaranees opimal soluions. However, he exac algorihm presened can only be applied o very small insances. Oher exac approaches for he muli-sage flexible flow shop problem are proposed by many auhors, e.g. branch and bound algorihms are given in Brah and Hunsucker [2] and Moursli and Poche [3]. When an exac algorihm is applied o large flexible flow shop problems, such an approach can ake hours or days o derive a soluion. On he oher hand, a heurisic approach is much faser bu does no guaranee an opimum soluion. Gupa [4] proposes heurisic echniques for a simplified flexible flow shop makespan problem wih wo sages and only one machine a sage wo. Sriskandarajah and Sehi [5] develop simple heurisic algorihms for he wo-sage flexible flow shop problem. They discuss he wors and average case performance of algorihms for finding minimum makespan schedules. Their soluions are based on Johnson s rule. Guine, Solomon, Kedia and Dussauchoy [6] also propose a heurisic for he makespan problem in a wo-sage flexible flow shop based on Johnson s rule. They compare his heurisic wih he 2

3 Shores Processing Time (SPT) and he Longes Processing Time (LPT) dispaching rules. They conclude ha he LPT rule gives good resuls for he wo-sage makespan problem. Gupa and Tunc [7] consider he wo-sage flow shop scheduling problem where here is one machine a sage one and he number of idenical machines in parallel a sage wo is less han he oal number of jobs. The seup and removal imes of each job a each sage are separaed from he processing imes. They propose heurisic algorihms ha are empirically esed o deermine he effeciveness in finding an opimal soluion. Sanos, Hunsucker and Deal [8] invesigae scheduling procedures which seek o minimize he makespan in a saic flow shop wih muliple processors. Their mehod is o generae an iniial permuaion schedule based on he Palmer, CDS, Gupa and Dannenbring flow shop heurisics, and hen i is followed by he applicaion of he Firs in Firs ou (FIFO) rule. To obain a near-opimal soluion, meaheurisic algorihms have also been proposed. Gourgand, Grangeon and Norre [9] presen several simulaed annealing (SA)-based algorihms for he flexible flow shop problem. A specific neighborhood is used and he auhors apply he mehods o a realisic indusrial problem. Jin, Yang and Io [20] consider he flexible flow shop wih idenical parallel machines. They propose wo approaches o generae he iniial job sequence and use a simulaed annealing algorihm o improve i. I can be seen ha a simulaed annealing algorihm has been successfully applied o various combinaorial opimizaion problems. For an exensive survey of he heory and applicaions of he SA algorihm, see Koulamas, Anony and Jaen [2]. Furhermore, Nowicki and Smunicki [22] propose a abu search (TS) algorihm for he flexible flow shop makespan problem. A geneic algorihm has been widely used in many previous works for he flow shop makespan problem, see e.g. Werner [23], Reeves [24]. Cheng, Gen and Tozawa [25] address he earliness/ardiness scheduling problem wih idenical parallel machines, and hey apply a geneic algorihm o solve his problem. Ruiz, Maroo and Alcaraz [26] use a geneic algorihm o deal wih he permuaion flow shop scheduling problem wih sequence-dependen seup imes. However, lile research has been done for flexible flow shop scheduling problems, especially for he general case wih unrelaed parallel machines and seup imes (see for insance he recen review on scheduling wih seup imes by Allahverdi, Ng, Cheng and Kovalyov [27]). In his paper, a flexible flow shop problem wih unrelaed parallel machines and seup imes is sudied. The goal is o seek a schedule which minimizes a posiively weighed convex sum of makespan and he number of ardy jobs. We will invesigae how o apply consrucive and improvemen algorihms as well as meaheurisics o solve he flexible flow shop problem wih unrelaed parallel machines approximaely. In paricular, consrucive heurisics based on dispaching rules and flow shop makespan heurisics are adaped and fas polynomial heurisic improvemen algorihms are used o improve he qualiy of he soluion. Meaheurisics, namely simulaed annealing, abu search and geneic algorihms are proposed. The res of his paper is organized as follows: The problem considered is described in Secion 2. The mahemaical model for his problem is inroduced in Secion 3. The heurisic consrucion of a schedule and ieraive algorihms are skeched in Secion 4 and Secion 5, respecively. Compuaional resuls wih he heurisics are briefly discussed in Secion 6 and conclusions are given in Secion Problem descripion The flexible flow shop sysem is defined by a se O = {,,,, k} of k processing sages. A each sage, O, here is a se M = {,, i,, m } of m unrelaed machines. The se J = {,, j,, n} of n independen jobs has o be processed on machines of he ses M,, M k. Each job j, j J, has is release dae r j 0 and a due dae d j 0. I has is fixed sandard processing ime for every sage, O. Owing o he unrelaed machines, he processing ime p of job j on machine i a sage is equal o ij ps v, where j ps is he sandard processing ime of job j a sage, and ij v is he relaive speed of job j which is processed by he machine i a sage. j ij 3

4 There are processing resricions of he jobs as follows: () jobs are processed wihou preempions on any machine; (2) every machine can process only one operaion a a ime; (3) operaions of a job have o be realized sequenially, wihou overlapping beween he sages; (4) job spliing is no permied. Seup imes considered in his problem are classified ino wo ypes, namely a machine-dependen seup ime and a sequence-dependen seup ime. A seup ime of a job is machine-dependen if i depends on he machine o which he job is assigned. I is assumed o occur only when he job is he firs job assigned o he machine. ch ij denoes he machine-dependen seup ime (or changeover ime) of job j if job j is he firs job assigned o machine i a sage. A sequence-dependen seup ime is considered beween successive jobs. A seup ime of a job on a machine is sequence-dependen if i depends on he job jus compleed on ha machine. s denoes he ime needed o changeover from job l o job j a sage, where job l is processed lj direcly before job j on he same machine. All seup imes are known and consan. Moreover, here is given a non-negaive machine availabiliy ime for any machine of a paricular sage. The scheduling problem under consideraion has dual objecives, namely minimizing he makespan and he number of ardy jobs. Therefore, he objecive funcion o be minimized is λc max + ( λ)η T () where C max is he makespan, which is equivalen o he compleion ime of he las job o leave he sysem, η T is he oal number of ardy jobs in he schedule, and λ is he weigh (or relaive imporance) given o C max and η T (0 λ ). 3. Mahemaical model In his secion, we provide a 0- mixed ineger linear programming formulaion for he problem under consideraion. 3.. Noaions sage index, =, 2, 3,..., k m number of parallel machines a sage i machine index, i =, 2, 3,..., m j, l job index, j, l =, 2, 3,..., n r j release dae of job j d j due dae of job j s seup ime beween job j and job l a sage jl ch ij ps j v ij a i ijl seup ime of job j if job j is assigned o machine i a he firs posiion a sage sandard processing ime of job j a sage relaive speed of machine i a sage for job j ime when machine i a sage becomes available X if job j is scheduled immediaely before job l on machine i a sage, and 0 oherwise O j C j C max U j T j η T operaing ime of job j a sage compleion ime of job j a sage makespan a Boolean variable: if job j is ardy, and 0 oherwise ardiness of job j oal number of ardy jobs in he schedule 4

5 3.2. Mahemaical formulaion The problem can be formulaed as follows. minimize λc max + ( λ)η T (2) Subjec o: m n ijl i= j= 0 m n+ ijl i= l= X =,, l (3) X =,, j (4) n + l= n X,, i (5) i0 l = X ij( n+ ) =,, i (6) j= 0 X = 0,, i, j (7) n ijj n+ X = X,, i, l (8) j= 0 ijl j= ilj X ijl {0,},, i, j, l ; j = 0; l = n+ (9) m n+ ps j O j = X ijl i= l= v,, j (0) C l C j ij s jl l + O m + [( X ) ] B,, j, l; j l () i= ijl C 0,, j (2) C j l C 0 C j = r j C j l m n X i= j= ijl s jl m i= il + ch X i0l l + O,, l (3), j (4) m m ai X i0 j + chil X i0l + O j,, j (5) i= i= k Cmax C j, j (6) T j C d, j (7) k j j T j 0, j (8) U j BT j, j (9) BU j T j, j (20) n U j j= η T =, j (2) U j {0,}, j (22) In he above formulaion, he objecive is o deermine a schedule ha minimizes he makespan and he number of ardy jobs, see Equaion (2). We have X i 0 l = if job l is sequenced as he firs job on machine i a sage, and X ij( n+ ) = if job j is sequenced as he las job on machine i a sage. Consrains (3) (9) ensure ha he parial schedule on each machine a each sage is feasible. Consrain ses (3) and (4) ensure ha only one job is assigned o each sequence posiion a each sage. Consrain ses (5) and (6) ensure ha 5

6 only one job will be assigned o he firs and las posiions, respecively, on each machine a each sage. Consrain (7) assures ha afer he job has been finished a any sage, i canno be reprocessed a he same sage. Consrain (8) forces o consruc a consisen sequence a every sage. Consrain (9) specifies he decision variables X as binary variables. Consrain (0) deermines he operaing ime of every job which ijl is dependen on he machine. Consrains () (5) find he compleion ime of every job. Consrain () is a se of disjuncive consrains. I saes ha, if jobs j and l are scheduled on he same machine a a paricular sage wih job j scheduled before job l, hen job j mus complee he processing before job l can begin. This consrain se forces job l o follow job j by a leas he processing ime of job l plus he seup ime from j o l if job l is immediaely scheduled afer job j. The value of B is se o a very big consan, i.e., i is greaer han he sum of all job processing imes and seup imes. Consrain (2) ensures ha he compleion ime of every job a each sage is a non-negaive value. Consrain (3) specifies he conjuncive precedence consrains for he jobs, which says ha a job canno sar is processing a sage + before i finishes a sage. Consrain (4) applies only o sage one, saying ha a job canno sar is processing a sage one before is release dae. Consrain (5) applies only o jobs ha are assigned o he firs sequence on each machine, ha is, he job canno sar is processing before machine availabiliy. Consrain se (6) links he makespan decision variable. Consrain ses (7) and (8) deermine he correc value of he ardiness (T j ). Consrain se (7) deermines he correc value of he laeness (L j ) and (8) specifies only he posiive laeness as he ardiness (T j = max {0, C k j d }). Consrain ses (9) (22) link he decision variable of he number of ardy jobs, ha is, if ardiness is greaer han zero, he job is ardy; oherwise his job is no ardy. I is noed ha an opimal soluion can be obained by running a commercial mahemaical programming sofware, CPLEX and AMPL, wih an Inel Penium GHz CPU. We have found ha he mahemaical model can be used for solving problems wih up o seven jobs and four sages in accepable ime. 4. Consrucive algorihms Since he flexible flow shop scheduling problem is NP-hard, algorihms for finding an opimal soluion in polynomial ime are unlikely o exis. Thus, heurisic mehods are sudied o find approximae soluions. Mos researchers develop exising heurisics for he classical flexible flow shop problem wih idenical machines by using a paricular sequencing rule for he firs sage. They follow he same scheme, see Sanos, Hunsucker and Deal [8]. The algorihm is as follows: Algorihm. Sep : Sequence he jobs by using a paricular sequence rule (he firs-sage sequence). Sep 2: Assign he jobs o he machines a every sage using he job sequence from eiher he Firs-In-Firs- Ou (FIFO) rule or he Permuaion rule. Sep 3: Reurn he bes soluion. Firsly, a job sequence is deermined according o a paricular sequencing rule, and we will briefly discuss he modificaions for he problem under consideraion in he nex secion. Secondly, jobs are assigned as soon as possible o he machines a every sage using he job 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, jobs are ordered according o heir compleion imes a he previous sage. This means ha he FIFO (Firs in Firs ou) rule is used o find he job sequence for he nex sage by means of he job sequence of he previous sage. The second way is o sequence he jobs for he oher sages by using he same job sequence as for he firs sage, called he permuaion rule. Assume now ha a job sequence for he firs sage has already been deermined. Then we have o solve he problem of scheduling n jobs on unrelaed parallel machines wih sequence- and machine-dependen 6

7 seup imes using his given job sequence for he firs sage. We apply a greedy algorihm which consrucs a schedule for he n jobs a a paricular sage provided ha a cerain job sequence for his sage is known (remind ha he job sequence for his paricular sage is derived eiher from he FIFO rule or from he permuaion rule), where he objecive is o minimize he flow ime and he idle ime of he machines. The idea is o balance evenly he workload in a heurisic way as much as possible. 4.. Consrucive Heurisics In order o deermine he job sequence for he firs sage by some heurisics, we remind ha he processing and seup imes for every job are dependen on he machine and he previous job, respecively. This means ha hey are no fixed, unil an assignmen of jobs o machines for he corresponding sage has been done. Thus, for applying an algorihm for fixing he job sequence for sage one, an algorihm for finding he represenaives of he machine speeds and he seup imes is necessary. The represenaives of machine speed v / ij and seup ime s / lj for sage, =, k, use he minimum, maximum and average values of he daa. Thus, he represenaive of he operaing ime of job j a sage is / he sum of he processing ime ps j v plus he represenaive of he seup ime s /. Nine combinaions of ij relaive speeds and seup imes will be used in our algorihms. The job sequence for he firs sage is hen fixed as he job sequence wih he bes funcion value obained by all combinaions of he nine differen relaive speeds and seup imes. For deermining he job sequence for he firs sage, we adap and develop several basic dispaching rules and consrucive algorihms for he flow shop makespan scheduling problem. Some of he dispaching rules are relaed o ardiness-based crieria, whereas ohers are used mainly for comparison purposes. The seps of he algorihm are as follows: Algorihm 2. Sep : Selec he represenaives of relaive speeds and seup imes for every job and every sage by using he combinaions of he minimum, maximum, and average daa values. / Sep 2: Calculae he represenaives of he operaing ime by using he erm ps j vij + s / lj. Sep 3: Use a paricular dispaching rule or consrucive algorihm o find he firs-sage sequence based on he specific represenaives of he operaing ime. Sep 4: Apply he heurisic schedule consrucion (see Algorihm ). Sep 5: Reurn he bes soluion. The Shores Processing Time (SPT) rule is a simple dispaching rule, in which he jobs are sequenced in non-decreasing order of he processing imes, whereas he Longes Processing Time (LPT) rule orders he jobs in non-increasing order of heir processing imes. The Earlies Release Dae firs (ERD) rule is equivalen o he well-known firs-in-firs-ou (FIFO) rule. The Earlies Due Dae firs (EDD) rule schedules he jobs according o non-decreasing due daes of he jobs. The Minimum Slack Time firs (MST) rule is a variaion of he EDD rule. This rule concerns he remaining slack of each job, defined as is due dae minus he processing ime required o process i. The Slack ime per Processing ime (S/P) is similar o he MST rule, bu is slack ime is divided by he processing ime required as well (Baker [28], and Pinedo and Chao [29]). The hybrid SPT and EDD (HSE) rule is developed o combine boh SPT and EDD rules. Firsly, consider he processing imes of each job and deermine he relaive processing ime compared o he maximum processing ime required. Secondly, deermine he relaive due dae compared o he maximum due dae. Nex, calculae he prioriy value of each job by using he weigh (or relaive imporance) given o C max and η T for he relaive processing ime and relaive due dae. lj 7

8 Palmer s heurisic [30] is a makespan heurisic denoed by PAL in an effor o use Johnson s rule by proposing a slope order index o sequence he jobs on he machines based on he processing imes. The idea is o give prioriy o jobs ha have a endency of progressing from shor imes o long imes as hey move hrough he sages. Campbell, Dudek and Smih [3] develop one of he mos significan heurisic mehods for he makespan problem known as CDS algorihm. Is srengh lies in wo properies: () i uses Johnson s rule in a heurisic fashion, and (2) i generally creaes several 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 subproblems. Thus, k sequences are generaed. Since Johnson s algorihm is a wo-sage algorihm, a k- sage problem mus be collapsed ino a wo-sage problem. Gupa [32] provides an algorihm denoed by GUP, in a similar manner as algorihm PAL by using a differen slope index and scheduling he jobs according o he slope order. Dannenbring [33] develops a mehod, denoed by DAN, by using Johnson s algorihm as a foundaion. Furhermore, he CDS and PAL algorihms are also exhibied. Dannenbring consrucs only one wo-sage problem, bu he processing imes for he consruced jobs reflec he behavior of PAL s slope index. Is purpose is o provide good and quick soluions. Nawaz, Enscore and Ham [34] develop a probably bes consrucive heurisic mehod for he permuaion flow shop makespan problem, called he NEH algorihm. I is based on he idea ha a job wih a high oal operaing ime on he machines should be placed firs a an appropriae relaive order in he sequence. Thus, jobs are sored in non-increasing order of heir oal operaing ime requiremens. The final sequence is buil in a consrucive way, adding a new job a each sep and finding he bes parial soluion. For example, he NEH algorihm insers a hird job ino he previous parial soluion of wo jobs which gives he bes objecive funcion value under consideraion (he relaive posiion of he wo previous jobs in he sequence remains fixed). The algorihm repeas he process for he remaining jobs according o he iniial ordering of he oal operaing ime requiremens. Again, o apply hese algorihms o he flexible flow shop problem wih unrelaed parallel machines, he oal operaing imes for calculaing he job sequence for he firs sage are calculaed for he nine combinaions of relaive speeds of machines and seup imes Improvemen Heurisics Unlike consrucive algorihms, improvemen heurisics sar wih an already buil schedule and ry o improve i by some given procedures. Their use is necessary since he consrucive algorihms (especially some algorihms ha are adaped from pure makespan heurisics and some dispaching rules such as he SPT and LPT rules) do no consider due daes (and herefore, hey do no consider he minimizaion of he number of ardy jobs). In his secion, some fas improvemen heurisics will be invesigaed o improve he overall funcion value by concerning mainly he due dae crierion. The ieraive algorihms described in he following and in Secion 5 are based on he shif move (SM) and he pairwise inerchange (PI) neighborhoods. The SM neighborhood reposiions a chosen job. This means ha an arbirary job π r a posiion r is shifed o posiion i, while leaving all oher relaive job orders unchanged. If r < i n, i is called a righ shif and yields π = [π,, π r-, π r+,, π i, π r, π i+,, π n ]. If < i < r <n, i is called a lef shif and yields π = [π,, π i-, π r, π i,, π r-, π r+,, π n ]. For insance, assume ha randomly one soluion in he curren generaion is seleced, say [ ], and hen randomly a couple of job posiions for performing he shif is seleced, e.g. posiions 2 and 7 (in his case, i is a righ shif). The new soluion will be [ ]. However, if posiions 7 and 2 are randomly seleced (i.e. i is a lef shif), he new soluion will be [ ]. In he SM neighborhood, he curren soluion has (n ) 2 neighbors. The PI neighborhood exchanges a pair of arbirary jobs π r, and π i, where i, r n and i r. Such an operaion swaps he jobs a posiions r and i, which yields π = [π,, π r-, π i, π r+,, π i-, π r, π i+,, π n ]. For example, assume ha he curren soluion is [ ], and hen randomly he couple of job 8

9 posiions o be exchanged is seleced, e.g. posiions and 3. Thus, he new soluion will be [ ]. In he PI neighborhood, he curren soluion has n (n-)/2 neighbors. In order o find a saisfacory soluion of he due dae problem, we apply fas polynomial heurisics by applying eiher he above shif move (SM) algorihm as an improvemen mechanism based on he idea ha we will consider he jobs ha are ardy in a lef-o-righ scan and move each of hem lef and righ, or he pairwise inerchange (PI) algorihm, where a ardy job is seleced and swapped o differen job posiions lef and righ, and eiher o wo randomly deermined posiions (denoed by he number 2 ) or o all n possible posiions (denoed by he leer A ). The bes schedule among he generaed neighbors is hen aken as he resul. The algorihm is as follows: Algorihm 3. Sep : Selec he firs ardy job in he job sequence. Sep 2: Inerchange (i.e. apply 2-PI or A-PI) or shif (i.e. apply 2-SM or A-SM) he chosen job and evaluae he objecive funcion values. Sep 3: Updae he curren job sequence. Sep 4: Go o Sep unil all ardy jobs have been considered. Sep 5: Reurn he bes sequence soluion. Since every ardy job in he job sequence is considered a mos once, he complexiy of he 2-PI and 2-SM procedures is O(n) and he complexiy of he A-PI and A-SM procedures is O(n 2 ). 5. Ieraive algorihms 5.. Simulaed Annealing A simulaed annealing (SA) algorihm is an ieraive search mehod, in which an iniial soluion is repeaedly improved by making small local aleraions unil no such aleraion yields a beer soluion. I was developed by Kirkparick, Gela and Vecchi [35]. An SA algorihm simulaes ideas and mechanisms found in he physical annealing of a solid. The concep comes from he field of maerials science in which a solid is firs meled afer heaed in a high emperaure, and hen i is slowly cooled so ha i reaches a hermodynamic equilibrium. In general, an SA algorihm is a sochasic opimizaion mehod for minimizing a funcion f over a discree domain S. Saring from an iniial soluion s S, an SA algorihm generaes a new soluion s' S in he neighborhood of he iniial soluion s by using a suiable operaor. Concerning he neighborhood, we considered boh a shif move (SM) neighborhood (i.e. a job a an arbirary posiion is seleced and reinsered a some oher posiion) and a pairwise inerchange (PI) neighborhood (i.e. wo arbirary jobs are seleced and inerchanged). This new poin s objecive funcion value f(s') is hen compared o he iniial poin s value f(s) (remind ha he objecive funcion value of he full schedule generaed from he job sequence for he firs sage is aken). The change in he objecive funcion value, δ = f(s') f(s), is calculaed. If he objecive funcion value decreases (δ < 0), i is auomaically acceped and i becomes he poin from which he search will coninue. If he objecive funcion value increases (δ 0), hen higher values of he objecive funcion may also be acceped wih a probabiliy, usually deermined by a funcion, exp ( δ/t), where T R is a conrol parameer of he SA algorihm called he emperaure. The role of he emperaure T is significan in he operaion of an SA algorihm. This emperaure, which is simply a posiive number, is periodically reduced every NT ieraions, where NT denoes he epoch lengh, so ha i moves gradually from a relaively high value o near zero as he algorihm progresses according o a funcion referred o as he cooling schedule. In our ess, we invesigaed in paricular he influence of he chosen neighborhood and he cooling scheme for conrolling he emperaure. We used a geomeric (i.e. T new = α T old ) and a Lundy-Mees 9

10 reducion [36] (i.e. T new = T old /(+β T old )) scheme and esed he parameers of hese schemes (iniial emperaures, emperaure reducions and neighborhood srucures) Tabu Search A abu search (TS) algorihm, iniially developed by Glover [37] is an ieraive improvemen approach designed o avoid erminaing premaurely a a local opimum for solving combinaorial opimizaion problems. Similar o a simulaing annealing algorihm, a TS algorihm is based on he idea of exploring he soluion space of a problem by moving from one region of he search space o anoher in order o look for a beer soluion. The funcion ransforming a soluion ino anoher soluion is usually called a move. For any soluion s S, a subse of moves applicable o i is defined. This subse of moves generaes he neighborhood ℵ(s) of s. Saring from an iniial soluion s, he TS algorihm ieraively moves from he curren soluion s o he bes soluion s* ℵ(s) even hough s* is worse han he curren soluion s, unil some sopping crierion is saisfied. However, o escape from a local opimum, an SA algorihm acceps an inferior soluion, which may lead o beer soluions laer by using an accepance probabiliy. In conras, a TS algorihm allows he search o move o he bes soluion s* among a se of candidae moves ℵ(s) as defined by he neighborhood srucure, alhough i can move o a neighbor wih a worse objecive funcion value. Neverheless, subsequen ieraions may cause he search o move repeaedly back o he same local opimum. In order o preven cycling back o recenly visied soluions, i should be forbidden or declared abu for a cerain number of ieraions. This is accomplished by keeping he aribues of he forbidden moves in a lis, called he abu lis. The size of he abu lis, called he abu enure, mus be large enough o preven cycling, bu small enough no o forbid oo many moves. Addiionally, an aspiraion crierion is defined o deal wih he case in which a move leading o a new bes soluion is abu. If a curren abu move saisfies he aspiraion crierion, is abu saus is canceled and i becomes an allowable move. The use of he aspiraion crierion allows he TS algorihm o lif he resricions and inensify he search ino a paricular soluion region Geneic Algorihm A geneic algorihm (GA) approach is an ieraive heurisic based on Darwin s evoluionary heory abou survival of he fies and naural selecion. I belongs o he evoluionary class of arificial inelligen (AI) echniques. I was invened by Holland [38] and hen i has been applied o a large number of complex search problems. The GA approach is characerized by a parallel search of he sae space in conras o a poin-by-poin search by convenional echniques. The parallel search is achieved by keeping a se of possible soluions, called a populaion. An individual in he populaion is a sring of symbols. The GA sars wih he iniial generaion of arificial individuals which are ofen creaed randomly. Each symbol is called a gene and each sring of genes is ermed as a chromosome. The individuals in he populaion are evaluaed by a measure called he finess o describe quaniaively how well he individual masers is ask. The iniial populaion is hen evolved ino differen populaions over a number of generaions hrough he use of wo ypes of geneic operaors: () unary operaors such as muaion and inversion which change he geneic srucure of a single chromosome, and (2) a higher-order operaor, referred o as crossover which consiss of obaining new individual(s) by combining he geneic maerial from wo seleced paren chromosomes. When applying crossover, wo individuals (parens) are seleced from he populaion and new soluion(s), called he offspring, is (are) creaed. Muaion creaes a new soluion by a random change on a seleced individual. The geneic operaors are applied o randomly seleced parens o generae new offspring. Then he new populaion is seleced ou of he individuals of he curren populaion and he new generaed chromosomes. 0

11 The applicaion of he GA approach requires he represenaion of a soluion, he choice of geneic operaors (crossover and muaion), an evaluaion funcion, a selecion mechanism and he deerminaion of geneic parameers (populaion size as well as crossover and muaion raes). For he represenaion, consideraion of a job permuaion is sraighforward and widely used in many previous works on he GA approach for he flow shop problem, see e.g. Werner (984). Thus, in our GA approach, we apply a permuaion-based code (or job code) using inegers as he chromosome coding scheme. For insance, one chromosome of an example wih nine jobs can be coded as he job sequence [ ]. As crossover operaor, we esed a parially mapped crossover (PMX) and a hybrid order and posiion-based crossover (OPX). The muaions are based again eiher on a pairwise inerchange of wo jobs or on a shif of one chosen job. The PMX (parially mapped crossover) mehod may be he mos popular crossover operaor when operaing wih permuaions. Firsly, choose wo parens P and P2, e.g. P = [ ] and P2 = [ ], and wo cuing sies along he sring are randomly chosen, e.g. 3 and 7. The subsrings defined by he wo cupoins are called he mapping secions. Secondly, exchange he wo subsrings beween he parens o produce proochildren, and hen hey will be [ ] and [ ]. I is clear ha proochildren will ofen lead o infeasible soluions. Then, one needs o deermine he mapping relaionship beween he wo mapping secions and finally, we legalize he offspring using his mapping relaionship. In he firs proochild, we can map he wo infeasible genes 2 and 8 ouside he mapping secion, by using he mapping swaps, for insance, 2 in he firs proochild s mapping secion can be mapped o 5 in he second proochild s mapping secion corresponding o he posiion. I does however no finish, because 5 is in he firs proochild s mapping secion as well. Again, 5 in he firs proochild can be mapped o 7 in a similar way. A las, 2 in he firs proochild can be swapped o 7. Similarly, 8 in he firs proochild can be mapped o 4. Consequenly, he firs offspring is [ ]. Then, he second offspring is analogously creaed as [ ]. The OPX (combined order and posiion-based crossover) mehod may be a good crossover choice, in which i creaes feasible soluions like PMX and combines he characerisics of OX and PBX as well. We will creae he firs offspring based on OX, whereas he second offspring is characerized by PBX. Again wo parens P and P2 are randomly seleced, and consider he same example as for PMX above. Then, randomly selec a subsring from he firs paren, e.g. [ ]. Copy he subsring ino he firs proochild corresponding o he firs paren posiion, e.g. [ _ ]. Then, delee all he symbols from he second paren which are already in he subsring and place is symbols ino he unfixed posiions in he firs proochild from lef o righ according o he second paren order, e.g. [ ]. To creae he second offspring, he second proochild is creaed by copying he symbols from he second paren, where he jobs are he same as he symbols in he subsring in he corresponding posiion, e.g. [ _ 4]. Then, place he symbols form he firs paren ino he unfixed posiions in he second proochild from lef o righ according o he order of he firs paren regarding he subsring symbols o produce he second offspring, [ ]. In our ess, we invesigaed he influence of he choice of he iniial generaion, he choice of he populaion size, differen crossover and muaion operaors and he choice of probabiliies for applying crossover and muaion Choice of an iniial soluion To improve he qualiy of he soluion finally obained, we also invesigaed he influence of he choice of an appropriae iniial soluion for he SA and TS algorihms, and an iniial populaion for he GA algorihm by using he heurisic consrucive and improvemen algorihms. To his end, we use one or several consrucive algorihm(s) SPT, LPT, ERD, EDD, MST, S/P, HSE, PAL, CDS, GUP, DAN and NEH as well as he oher seleced polynomial improvemen heurisics as iniial soluion(s), respecively (for he GA algorihm, he remaining iniial soluions are sill randomly generaed).

12 Table : Average overall performance of he consrucive and polynomial improvemen heurisics λ Problem size CA 2-SM A-SM 2-PI A-PI a Sum b Sum Sum Sum Sum Sum Sum Sum a average absolue deviaion for λ = 0, b average percenage deviaion for λ>0 6. Compuaional Resuls Firsly, consrucive algorihms according o Algorihm 2 and differen fas polynomial improvemen heurisics according o Algorihm 3 are sudied. The consrucive algorihms (denoed by he leer CA ) are simple dispaching rules such as he SPT, LPT, ERD, EDD, MST, S/P, and HSE rules, and some flow shop makespan heurisics adaped such as he algorihms PAL, CDS, GUP, DAN, and NEH. Then, we addiionally applied he fas polynomial improvemen heurisics based on he four cases saed above in Secion 4.3 (denoed by 2-SM, A-SM, 2-PI, and A-PI, respecively). We used problems wih 0 jobs 5 sages, 30 jobs 0 sages, and 50 jobs 20 sages. For all problem sizes, we esed insances wih λ {0, 0.00, 0.005, , 0., 0.5, } in he objecive funcion. Ten differen insances for each problem size have been run. For our experimens wih problems wih unrelaed parallel machines, we generaed he sandard processing imes, relaive machine speeds, seup imes, release daes and due daes as follows. The sandard processing imes are generaed uniformly from he inerval [0,00]. The relaive speeds are disribued uniformly in he inerval [0.7,.3]. The seup imes, boh sequence- and machine-dependen seup imes, are generaed uniformly from he inerval [0, 50], whereas he release daes are generaed uniformly from he inerval beween 0 and half of heir oal sandard processing ime mean. The due dae of a job is se in a way ha is similar o he approach presened by Rajendaran and Ziegler [39] and is as follows: 2

13 k d j = r j + ps j + oal of mean seup ime of a job on all sages + = (n ) (mean processing ime of a job on one machine) U(0,) (23) The resuls for he consrucive algorihms and fas polynomial improvemen heurisics are given in Table. We give he average (absolue for λ = 0 resp. percenage for λ > 0) deviaion of a paricular algorihm from he bes soluion in hese ess for he hree problem sizes n k. From hese resuls i is obvious ha he fas polynomial improvemen heurisics can improve he qualiy of he consrucive algorihms by abou percen. In addiion, we have found ha for he problem size 0 jobs 5 sages he all-shif-move (A-SM) heurisic is slighly beer han he ohers, whereas he allpairwise-inerchange-based (A-PI) improvemen heurisic is he bes algorihm oherwise. However, in general he all-pairwise-inerchange algorihm should be seleced as he improvemen algorihm. Consequenly, in his paper we use in he following only he all-pairwise-inerchange-based improvemen heurisics. However, when comparing beween he 2-SM and 2-PI algorihms whose CPU ime is smaller han he CPU ime of boh he A-SM and A-PI algorihms, we have found ha he 2-SM algorihm cerainly become beer han he 2-PI algorihm. Table 2: Average performance of he consrucive algorihms of Group I & II Group I Group II λ Problem size SPT LPT ERD EDD MST S/P HSE PAL CDS GUP DAN NEH a Sum b Sum Sum Sum Sum Sum Sum Sum a average absolue deviaion for λ = 0, b average percenage deviaion for λ>0 3

14 Table 3: Average performance of he polynomial improvemen algorihms of Group III & IV Group III Group IV λ Problem size ISPT ILPT IERD IEDD IMST IS/P IHSE IPAL ICDS IGUP IDAN INEH a Sum b Sum Sum Sum Sum Sum Sum Sum a average absolue deviaion for λ = 0, b average percenage deviaion for λ Nex, we presen he resuls for he consrucive and improvemen algorihms ha are separaed ino four main groups. The firs heurisic group includes he simple dispaching rules such as SPT, LPT, ERD, EDD, MST, S/P, and HSE. The second heurisic group includes he flow shop makespan heurisics adaped such as PAL, CDS, GUP, DAN, and NEH. The hird and fourh heurisic groups are generaed from he firs wo groups of heurisics where he soluions are improved by he seleced polynomial improvemen algorihm based on all-pairwise-inerchange improvemen heurisics, and hey are denoed by he firs leer I in fron of he leers describing he heurisics of he firs wo groups. The resuls for he consrucive and fas polynomial improvemen algorihms are given in Table 2 and Table 3, respecively (he bes varian in each group is given in ialic underlined while he overall bes varian is given in bold face underlined). From hese resuls, i is obvious ha he algorihms in he fourh heurisic group improved he pure makespan heurisics form he second heurisic group (i.e., PAL, CDS, GUP, DAN, and NEH), and hey are beer han he dispaching rules in he firs heurisic group (i.e., SPT, LPT, ERD, EDD, MST, S/P, and HSE) as well as he hird heurisic group improved from hem. Among he simple dispaching rules (heurisic Group I), he SPT, LPT, ERD, and HSE rules are good dispaching rules. However, in general he HSE rule ouperforms he oher dispaching rules for λ < 0.0, and he LPT rule is beer han he oher rules oherwise. Among he adaped flow shop makespan heurisics 4

15 in heurisic Group II, he NEH algorihm is clearly he bes algorihm among all sudied consrucive and improvemen heurisics (bu in fac, his algorihm akes he convex combinaion of boh crieria ino accoun when selecing parial sequences). The CDS algorihm is cerainly he algorihm on he second rank (bu i is subsanially worse han he NEH algorihm even if he makespan porion in he objecive funcion value is dominan, i.e. for large λ values). When we apply a fas pairwise inerchange algorihm (denoed by he leer I firs) o he dispaching rules and adaped makespan heurisics, we have found ha he qualiy of he soluion can be improved by abou percen excep for he NEH rule. I can be noed ha he NEH rule is no improved by using he improvemen heurisics of algorihm INEH because boh algorihms use a very similar sraegy. However, he improvemen of he heurisics from he adaped pure makespan heurisics in he heurisic Group IV is beer han he improvemen of he heurisics derived from he dispaching rules in he heurisic Group III. These resuls are similar o he conclusions of Jungwaanaki, Reodecha, Chaovaliwongse and Werner [40] whose experimens compared he resuls for small problem sizes wih he opimal soluions. Table 4: Tesed Parameers for he SA, TS, and GA algorihms Algorihms Parameers Levels SA Iniial emperaure 2, 4, 6, 8, 0 hrough 00, in seps of 0 Neighborhood srucures PI, SM Cooling schedules 4 (geomeric reducion wih α {0.8, 0.85, 0.9, 0.95}) 5 3 (LM reducion wih β: 0.0 hrough 0.09, in seps of 0.0) 4 23 (LM reducion wih β: 0. hrough.0, in seps of 0.) TS Number of neighbors 0 hrough 50, in seps of 0 Neighborhood srucures PI, SM Sizes of abu lis 5, 0, 5, and 20 GA Populaion sizes 0, 30, 50, 70 Crossover ypes PMX, OPX Muaion ypes PI, SM Crossover raes 0. hrough 0.9, in seps of 0. Muaion raes 0. hrough 0.9, in seps of 0. Thirdly, we have sudied he SA, TS, and GA algorihms wih a random iniial soluion (or populaion). The purpose of his sudy is o deermine he favorable parameers of he heurisics. From he preliminary ess, we se he ime limi equal o one second for he problems wih en jobs, en seconds for he problems wih 30 jobs, and 30 seconds for he problems wih 50 jobs. Again, for all ess we considered insances wih λ {0, 0.00, 0.005, 0.0, 0.05, 0., 0.5, and }. Based on he preliminary ess, he esed parameers are shown in Table 4. From he full facorial experimen, we analyzed our resuls by means of a muli-facor Analysis of Variance (ANOVA) echnique using a 5% significance level. We give he average (absolue resp. percenage) deviaion of a paricular ieraive algorihm from he bes soluion obained by he ieraive algorihms. For he SA algorihm, we have found ha for he neighborhood srucure and he cooling schedule, here are saisically significan differences, whereas here are slighly saisically significan differences in he iniial emperaure. For he iniial emperaure, we observed ha a lower iniial emperaure is effecive for he problem. However, no all he problem cases should use he same small iniial emperaure. For he problems wih λ < 0.5, he use of an iniial emperaure of wo and of en for he oher problems can be recommended. I is clear ha shif moves are beer han pairwise inerchange moves for λ > 0, especially for he problems whose makespan objecive is dominan in comparison wih he ardiness objecive, whereas he pairwise inerchange moves are slighly beer han or nearly as good as he shif moves for he oher values. Consequenly, he neighborhood srucures should be based on pairwise inerchanges for λ = 0 and on shifs of jobs oherwise, or i can also be recommended o use shif moves for 5