A Genetic Algorithm for Assembly Sequence Planning

A Gene ic Algo i hm o Assembly Sequence Planning Ca melo Del Valle1, Ra ael M. Gasca1, Miguel To o1, and Edua do F. Camacho2 1Dep . Lenguajes y Sis emas In o má icos, Uni . Se illa, A da. Reina Me cedes s/n, 41012 Se illa, Spain {ca melo, gasca, m o o}@lsi.us.es 2Dep . Ingenie ía de Sis emas y Au omá ica, Uni . Se illa, Camino de los Descub imien os s/n, 41092 Se illa, Spain [email p o ec ed] Abs ac . This wo k p esen s a gene ic algo i hm o assembly sequence plan- ning. This p oblem is mo e di icul han o he sequencing p oblems ha ha e al eady been ackled wi h success using hese echniques, such as he classic T a eling Salespe son P oblem (TSP) o he Job Shop Scheduling P oblem (JSSP). I no only in ol es he a anging o asks, as in hose p oblems, bu also he selec ion o hem om a se o al e na i e ope a ions. Two amilies o gene ic ope a o s ha e been used o sea ching he whole solu ion space. The i s includes ope a o s ha sea ch o new sequences locally in a p ede e mined assembly plan, ha o pa en ch omosomes. The o he amily o ope a o s in- oduces new asks in he solu ion, eplacing o he s o main ain he alidi y o ch omosomes, and i is in ended o sea ch o sequences in o he assembly plans. Fu he mo e, some p oblem-based heu is ics ha e been used o gene a - ing he indi iduals in he popula ion. 1 In oduc ion Gene ic Algo i hms ha e been used o sol e a a ie y o op imiza ion p oblems wi h some success. Combina o ial p oblems a e a class o p oblems pa icula ly di icul o sol e, sequencing p oblems included. Many o hem ha e been s udied using e olu ion- a y echniques, such as TSP and JSSP, well known as NP-comple e p oblems [1] [2]. This pape p esen s an applica ion o gene ic algo i hms o he p oblem o selec ing and sequencing assembly ope a ions. This is a mo e di icul planning p oblem han TSP and JSSP. I in ol es no only he op imal a angemen o asks, as in hose p ob- lems, bu also he selec ion o hem om a se o al e na i e ope a ions, and aking in o accoun he cons ain s imposed o build a easible assembly plan. Assembly planning is a e y impo an p oblem in he manu ac u ing o p oduc s. I in ol es he iden i ica ion, selec ion and sequencing o assembly ope a ions, speci- ied by hei e ec s on he pa s. The iden i ica ion o assembly ope a ions has been ackled by analysing he p oduc s uc u e, ei he using in e ac i e expe sys ems [3] [4] o , h ough planne s wo king au oma ically om geome ic and ela ional models [5] and om CAD models and o he non-geome ic in o ma ion [6] [7]. The iden i ica ion o assembly ope a ions usually leads o he se o all easible as- sembly plans. The numbe o hem g ows exponen ially wi h he numbe o pa s, and depends on o he ac o s, such as how he single pa s a e in e connec ed wi hin he whole assembly. In ac , his p oblem has been p o ed o be NP-comple e [8]. An op imum assembly plan is now sough , selec ed om he se o all easible as- sembly plans. Mos app oaches used o choosing an op imal one employ di e en kind o ules in o de o elimina e assembly plans including di icul asks o awkwa d in e media e subassemblies [9] [10]. The c i e ion ollowed in his wo k is he minimiza ion o he o al assembly ime (makespan) in he execu ion o he plan. To mee his objec i e, he algo i hm akes in o accoun he in o ma ion abou each assembly ask ( obo and ool needed and es- ima ion o assembly ime) [11]. This app oach allows applying he esul s o di e en s ages o he p ocess planning, om he design o he p oduc and o he manu ac u - ing sys em, o he inal execu ion o he assembly plan. The es o he pape is o ganized as ollows: Sec ion 2 desc ibes he p oblem o selec ion and sequencing o assembly asks. The p oposed gene ic algo i hm is de- sc ibed in Sec ion 3, and some o he esul s ob ained a e p esen ed in Sec ion 4. Some inal ema ks a e made in he concluding sec ion. 2 Assembly Sequence Planning And/o g aphs ha e been used o ep esen he se o all easible assembly plans o a p oduc [12]. In his ep esen a ion, he O nodes co espond o sub-assemblies, he op node co esponds o he whole assembly, and he lea nodes co espond o he in- di idual pa s. Each And node co esponds o he assembly ask joining he sub- assemblies o i s wo inal nodes p oducing he sub-assembly o i s ini ial node. In he And/O g aph ep esen a ion o assembly plans, an And/O pa h whose op node is he And/O g aph op node and whose lea nodes a e he And/O g aph lea nodes is asso- cia ed o an assembly plan, and is e e ed o as an assembly ee. An impo an ad- an age o his ep esen a ion, used in his wo k, is ha he And/O g aph shows he independence o assembly asks ha can be execu ed in pa allel. Figu e 1 shows an example o his ep esen a ion. And nodes a e ep esen ed as hype a cs. The p oblem is ocused on sea ching an op imal assembly sequence, an o de ing o an assembly plan (one o he And/O ees o he And/O g aph). The e alua ion o solu ions implies a p e ious es ima ion o he imes and esou ces ( obo s, ools, ix- u es...) needed o each assembly ask in he And/O g aph. Ano he ac o consid- e ed he e, is he ime necessa y o changing he ools in he obo s, which is o he same o de as he execu ion ime o he assembly asks and he e o e canno be dis e- ga ded as in Pa s Manu ac u ing. ∆ch (M, C, C') will deno e he ime needed o in- s alling he ool C in he obo (machine) M i he ool C' was p e iously ins alled. No ice ha any change o con igu a ion in he obo s can be modeled in his way. Ano he issue is he anspo a ion o pa s and sub-assemblies, ha could a ec he o al assembly ime. Ideally, i would be supposed a well-dimensioned sys em, wi h a pe ec planning when execu ing he assembly plan, so ha , when a pa would be equi ed in a obo o execu ing an assembly ope a ion, i will be p esen he e. Bu he same canno be gua an eed o an in e media e subassembly, because i could be buil in a obo and equi ed immedia ely in ano he one o o m ano he subas- sembly. ∆mo (SA, M, M') will deno e he ime needed o anspo ing he subassembly SA om obo M o obo M'. 3 The Gene ic Algo i hm Gene ic Algo i hms (GAs) ha e been used o sol e a la ge a ie y o combina o ial op- imiza ion p oblems wi h some success [1] [2]. The na u e o he Assembly Sequence Planning p oblem en ails a g ea di icul y in applying GAs: a sequence o asks o ms a co ec solu ion i all o hem belong o an assembly plan, i.e. an assembly ee o he And/O g aph, and hey a e o de ed acco ding o he p ecedence cons ain s imposed by he plan. An assembly ask is de ined by he subassemblies used o o m a g ea e sub- assembly, and by he esul ing assembly. Thus, he p esence o a ask in a solu ion is s ongly condi ioned by he p esence o asks ela ed o hese subassemblies. The e will be li le chance o cons uc ing a new solu ion om signi ican pa s o any wo solu- ions. I will be necessa y o add (p obably no a ew) o he asks o comple e he solu- ion, and dele e some o he s. The i s issue in applying GAs is he ch omosomal encoding. A na u al way o ep- esen ing a solu ion is h ough a sequence o asks, compa ible in o de wi h he con- s ain s imposed by he And/O g aph (o de ing and assembly plans). So, no all he asks sequences o m a legal solu ion. Figu e 2 shows how a ch omosome is decoded o p oduce a schedule. A schedule builde ans o ms he ch omosome in o a legal assem- bly schedule, aking in o accoun he p ecedence cons ain s and he sha ed esou ces o be used (machines and ools). This ansla ion is made di ec ly because o he simplici y o ha ep esen a ion. The esul could be isualized as a Gan cha , and i allows he i ness unc ion ( he makespan) o be calcula ed. No e ha , depending on he assigna ion o esou ces o asks and hei du a ions, di e en ch omosomes could be mapped in o an only schedule. I will happen when asks do no sha e he same esou ces and could be execu ed in pa allel. A B C D E A B C D A C D A B A C A D C D B E AB C D E Fig. 1. And/O g aph o p oduc ABCDE. Two amilies o gene ic ope a o s ha e been de ined o sea ching he whole solu- ion space. The i s includes ope a o s ha sea ch locally o new sequences in a p e- de e mined assembly plan, ha o pa en ch omosomes. These ope a o s, e e ed o below as Re-O de ing Tasks ope a o s, a e simila o hose used o o he sequencing p oblems, such as TSP and JSSP, in he li e a u e [1] [2], bu ob iously esul o be insu icien o ind he op imum. The o he amily o ope a o s is in ended o sea ch o sequences in o he assembly plans, and a e e e ed o as Re-Planning ope a o s. This is basically made by in oducing a new ask in a solu ion, and subs i u ing ce ain asks o o he s in o de o main ain he soundness o he ch omosomes. 3.1 Re-O de ing Tasks (ROT) Ope a o s This kind o ope a o is in ended o sea ch o new sequences in a p ede e mined as- sembly plan. Because o he imp obabili y o wo sequences o he same assembly plan coinciding in a popula ion, hey a e implemen ed as mu a ion ope a o s. They ope a e in a ch omosome by selec ing a andom ask in he sequence and a emp ing o mo e i o ano he andom posi ion. Thei p edecesso o successo asks migh be also in ol ed in he mo emen , so ha hey may keep in hei posi ions o mo e wi h he selec ed ask. Those possibili ies gi e us ou di e en gene ic ope a o s. The ansposi ion o asks is pe o med so ha he esul an indi idual is legal. 3.2 Re-Planning (RP) Ope a o s This kind o ope a o is in ended o sea ch o sequences in o he assembly plans. The esul an indi iduals will con ain new assembly asks, coming om ano he indi idual p esen in he popula ion (c osso e ope a o s) o gene a ed andomly (mu a ion op- e a o s). Some o he new asks a e equi ed in o de o comple e a co ec ch omo- ime OP-10 OP-7 OP-9 OP-4 OP-11 OP-1 OP-5 OP-11OP-9 OP-10OP-5 OP-7 OP-1 OP-4 And/O G aph Times & Resou ces o Tasks M1 M2 M3 Schedule Builde Fig. 2. The Schedule Builde . some, so ha hey subs i u e some o he s. The sequences gene a ed by hese gene ic ope a o s will main ain he posi ion o asks hey had in he pa en s, and he new ask will ill he blanks, a some compa ible o de wi h he p ecedence cons ain s. RP C osso e (RP-C) ope a o s ake wo indi iduals (pa en s) and gene a e wo child en, ying o me ge gene ic in o ma ion om he wo pa en s. Because o he na u e o Assembly Planning he e will be li le chance o cons uc ing a new solu ion om signi ican pa s o any wo solu ions. The gene a ion o child en is made by selec ing one ask in one o he pa en s, so ha hei successo asks in ha pa en a e also selec ed. The emaining asks in he new indi idual will be selec ed om he o he pa en , whene e possible, o andomly, in o de o comple e a legal ch omosome. This is done in di e en ways, depending on he dis ance o he sea ch o a p edecesso ask o he selec ed ask in he And/O g aph. RP Mu a ion (RP-M) akes an indi idual and modi ies i by changing a andom sub- ee o he assembly plan o ano he , selec ed andomly and acco ding o he cons ain s imposed by he And/O g aph. The posi ions o he new asks in he sequence will be he same ha held he subs i u ed asks. 3.3 A heu is ic o gene a ing solu ions An es ima ion o he ime needed o he execu ion o each ask and hei successo s in he And/O g aph ha e been used in o de o gene a e he solu ions in he ini ial popula ion. An op imis ic way o doing his es ima ion is done by he heu is ic unc- ion h , de ined by he equa ions below: 12 ( ) ( ) max(min( ( , )), min ( ( , ))) ij ij TO T O h T du T h c T T h c T T ∈∈ =+ (1) ()( ) () (,) ()max ,(),(), (),(),() ii i mo ii h c T T h T T M T C T sa T M T M T τ =+ ∆ (2) () () 1 ,(), i () (, , ) max 0, ( , , ) i ( ) ch MCT C M MT TMC TMC M MT ττ !∆= ∀ =#≠ ∀ ∃ (3) () ( ) () () () 12 122 ,, maxmin ,,, ,min ,,, ij ij TO T O TMC TTMC TT MC τττ ∈∈ %& =∋( )∗ (4) ()()( ) 2,, , , , () () ii i TT MC T MC h T h T ττ =−− (5) This calcula ions o h (T) ake in o accoun no only he du a ion du (T) o each ask T and he p ecedence cons ain s de ined in he And/O g aph o he asks, bu also he delays co esponding o he change o ools (con igu a ions) in he machines and o he anspo a ion o in e media e sub-assemblies be ween di e en machines. A solu ion is buil by o de ing he asks o a ee o he And/O g aph. In o de o cons uc a ee, an And node is selec ed andomly om he se o al e na i e ask o each O node, so ha he p obabili y o selec ion is in e se o he alue o h o he asks. The sequence o asks is o med selec ing consecu i ely a ask which i s imme- dia e p edecesso ha e been in oduced in he sequence. When a se o candida e asks may be selec ed, a p obabili y di ec ed p opo ional o h is used o a andom selec- ion, so ha he asks (and i s successo s) wi h mo e es ima ed ime will ha e mo e chance o being selec ed. 4 Expe imen s and Resul s A hypo he ical p oduc has been used in o de o e alua e he gene ic ope a o s de- sc ibed in he p e ious sec ion. Tha p oduc is o med by 30 pa s, and he numbe o connec ions among hem is he minimum. The p oduc includes in i s And/O g aph a ious al e na i e asks o each O node, and con ains 396 O nodes and 764 And nodes. The e a e abou 1021 possible indi iduals. No e ha he numbe o di e en schedules depends no only on he numbe o sequences, bu also on he dis ibu ion o sha ed esou ces (and hei numbe ) and du a ions among all asks. Thus, a ious indi iduals could be ans o med in an unique schedule. The alues co esponding o he highe pa o igu es 3 and 4 ep esen he a e - age o 25 ials. The lowe pa ep esen s he bes esul in all ials. Mo eo e , all alues ep esen he a e age o 10 di e en dis ibu ions o du a ions and sha ed e- sou ces among he asks. They show he bes solu ion ound un il he numbe o e alua ions indica ed. The g aphics include also he alue o he op imum solu ion (OPT) and he pe o mance o a andom algo i hm (RND). Figu es 3 and 4 show he ope a ion o he speci ic gene ic ope a o s. The high di - icul y o me ging gene ic in o ma ion om wo any indi iduals could explain he ela i ely poo esul s ob ained by RP-C in compa ison wi h hose expec ed om ypical c osso e ope a o s. In ac , RP-M ob ains sligh ly be e esul s, maybe be- cause i p ese es mo e gene ic in o ma ion in he indi iduals. Mo eo e , ROT ope a- o s imp o e mo e quickly a i s . A las , hei pe o mance is condi ioned by he as- sembly plans gene a ed in he ini ial popula ion. A las cu e is gene a ed in Fig. 6. I shows he esul s ob ained by a GA wi h all e e ed ope a o s wo king oge he (ALL). A qui e imp o emen can be obse ed. This e lec s he combina ion o he wo e ec s: ROT ope a o s op imize assembly plans ha ha e been gene a ed, and RP ope a o s ob ain new assembly plans. The in luence o ha ing an ini ial popula ion ha ha e been gene a ed andomly o using any o he in o med me hod can be seen compa ing he wo igu es. Figu e 3 shows how he GA wo ks when s a ing om a andom ini ial popula ion, and igu e 4 when he ini ial popula ion is gene a ed using he heu is ic h p esen ed in sec ion 3.3. The cu e RND in igu e 4 co esponds o he use o h in a p obabilis ic algo- i hm ha con inuously gene a es new solu ions, which subs i u es he andom algo- i hm used in igu e 3. We can see ha he p obabilis ic algo i hm imp o es he an- dom one, and he a e age bes solu ions a e nea he solu ions ob ained by some o he gene ic ope a o s wo king alone. The esul s a e be e in gene al in igu e 4, bu he imp o emen s a e di e en o each ope a o . Fo example, RP-C ope a o s wo k- ing alone ob ain simila esul s, RP-M imp o es sligh ly, and ROT ope a o s im- p o es mo e, because o he in luence o he ini ial popula ion in his beha io . The GA wi h all he ope a o s wo king oge he also imp o es he solu ion, bu only a li - le o he bes ial. The use o mo e heu is ic me hods, such as he heu is ic p esen ed, bu also du ing he sea ch, is expec ed o imp o e he GA, so ha a na u al ex ension o his wo k mus be in ha sense. 5 Conclusions A gene ic algo i hm has been p esen ed o sol ing he assembly sequence planning p oblem, a much mo e di icul p oblem han o he sequencing p oblems ha ha e 75 80 85 90 95 100 105 110 0 5000 10000 15000 20000 25000 30000 E alua ions Makespan RND ROT RP-C RP-M ALL OPT Fig. 3. Resul s o andom ini ial popula ions. 75 80 85 90 95 100 105 110 0 5000 10000 15000 20000 25000 30000 E alua ions Makespan RND ROT RP-C RP-M ALL OPT Fig. 4. Resul s o heu is ic ini ial popula ions. al eady been ackled using simila echniques, such as TSP and JSSP, because i in- ol es also he selec ion o assembly ope a ions ha will o m he assembly plan om a se o al e na i es. Two amilies o gene ic ope a o s ha e been used in o de o sea ch h ough he whole solu ion space. RP ope a o s ha e been p oposed o gene - a e new assembly plans (wi h di e en assembly asks) om o he s. On he o he hand, he e is he o de ing o assembly asks in he sequence o ob ain a good sched- ule. ROT mu a ion ope a o s ha e been used o his goal. Al hough he gene ic in o ma ion used by hese ope a o s could seem insu icien , he combined ope a ion o hem has been qui e sa is ac o y. The beha io o he GA algo i hm has been imp o ed by using a ini ial popula ion ha ha e been gene a ed using a heu is ic co esponding o an es ima ion o he ime needed o execu ing each ask and i s successo s in he And/O g aph o he p oduc o be assembled. Re e ences 1. T. S a kwea he , S. McDaniel, K. Ma hias, D. Whi ley, C. Whi ley (1991). A Compa ison o Gene ic Sequencing Ope a o s. P oceedings o he Fo h In l. Con . on Gene ic Algo i hms, ICGA-91, pp. 69-76. Mo gan Kau mann. 2. G. Syswe da (1990). Schedule Op imiza ion Using Gene ic Algo i hms. In L. Da is, ed.. The Handbook o Gene ic Algo i hms, pp. 332-349. 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