Integer Programming Models for Decision Making of. Order Entry Stage in Make to Order Companies 1. INTRODUCTION

Transcription

1 Inege Pogammng Models fo Decson Makng of Ode Eny Sage n Make o Ode Companes Mahendawah ER, Rully Soelaman and Rzal Safan Depamen of Infomaon Sysems Depamen of Infomacs Engneeng Insu eknolog Sepuluh Nopembe, Suabaya 60, INDONESIA Emal: mahenda_w@s-sby.edu Absac. hs pape addesses he decson makng ssues of ode eny sage n a make o ode (MO) companes. wo mahemacal models ha pevously developed by Ebadan e al (007) ae mplemened no a pogam. he fs model (MIP) deemnes he opmal pce and delvey mes fo he new avng ode whle he second model (MIP) helps he company o deemne he bes se of supples and subconacos o fulfll he acceped odes. Impovemens ae made on he wo models o oban moe accuae esuls. he mplemenaon esuls show ha boh models can help MO company n he ode eny pocess. he mpovemens made on boh models povde moe accuae and opmal esuls compaed o he exsng models. Exenson o he fs model called IPE povdes moe accuae epesenaons on he allocaon of esouces o pocess ode dung egula and oveme as well as subconac. Exenson o he second model (IPE) povdes a moe opmal and accuae soluon by ncopoang an algohm ha only uses subconacos once he nenal capacy s ncapable o fulfll he odes. Keywods: decson makng, make-o-ode, nege pogammng.. INRODUCION Manufacung companes have dffeen saeges o sasfy cusome demands. Make-o-sock (MS) companes poduce based on foecass and sasfy demand fom sock. Whle make-o-ode (MO) companes only poduce afe ecevng odes fom he cusomes. Ode eny s an mpoan sage fo an MO company. Due o lmaon n capacy, he company can no accep all avng odes. Fuhemoe, accepng ncoec combnaon of avng odes could affec he fulfllmen of confmed odes. heefoe, an MO company should be able o choose an opmal combnaon of odes o be fulflled whch would be pofable fo hem. hee ae seveal cea ha an MO company ypcally es o acheve n fulfllng odes ncludng mnmsng pce, sho delvey me and hgh poduc qualy. Howeve, aempng o fulfl one cea ofen conflcs wh anohe whch complcae poducon plannng. Poducng hgh qualy poducs could esuls n nceasng coss and delvey me. On he ohe hand, fulflng ode n sho delvey me ofen eques addonal esouces ha evenually leads o hghe coss. Vaous auhos aemp o solve dffeen poblems elaed o dffeen pocesses n an MO company. Ebadan e al (007) popose a decson makng sucue o manage he ode eny sage n an MO company. In he decson makng sucue, hey popose seveal pocesses ha need o be conduced n ode o make an opmal decsons. Oupus fom he pocesses ha can be used fo decson makng ae: ) he opmal coss fo new odes and expeced fulflmen me of he odes and ) when he odes ae acceped by he cusomes hen he nex sep s selecng a combnaon of supples and subconacos ha could povde maeals wh mnmal pce. hese poblems ae epesened n wo mahemacal models called MIP and MIP. he am of hs pape s wofold: Fs, wo models as he man pa of he decson makng sucue poposed by Ebadan e al (007) ae mplemened. Secondly, hs pape also povdes exensons o he models poposed by Ebadan e al (007) o oban moe accuae and opmal esuls. : Coespondng Auho

2 . LIERAURE he decson makng sucue fo ode eny sage n MO companes poposed by Ebadan e al (007) suggess pce and ode delvey me as he man cea ha need o be consdeed n ode o oban opmal soluons. he decson also depend on he flexbly of due dae.e. whehe he due dae can o canno be negoaed based on he cusome needs. hese cea ae consdeed wh he assumpon ha he poducon pocess esuls n hgh qualy poducs. he decson makng sage s dvded no wo pas based on he flexbly of due dae:. Fo fxed due dae.. Fo negoable due dae. In pacce, companes have he own cea o poze he ncomng odes such as fs come fs seve, cusome poy ec. Ebadan e al (007) dvde he ncomng odes no wo: ) hgh poy odes ha have o be fnshed n me of he due dae and ) low poy ode ha can be delayed. he decson makng sucue can be dvded no seveal man pocesses as follow: a. Calculang he avalable capacy. hs s he nal valdaon pocess o examne whehe he avalable capacy sasfy he wokload equed by odes fo each poducon esouce. hs calculaon s conduced fo boh fxed and negoable due dae. Calculaon of capacy fo low poy odes consdes anohe paamee ha epesens he capacy eseved fo hghe poy odes. b. Deemnng dffeen alenaves fo Eales Release Dae (ERD) and Ode Compleon Daes (OCD) fo each ode. A backwad mehod, whch calculae ERD and OCD sang fom he las esouces n he poducon pocess o he pevous esouces based on he scheduled delvey me, s used fo odes wh fxed due dae. Fo ode wh negoable due dae, a fowad mehod s conduced by calculang he ERD hen OCD of he fs o he nex esouces accodng o he poducon oues of each ode o deemne he delvey me fo each ode. Oveall hs s he pocess ha dffeenae he wo pas n he decson makng sucue c. Deemnng he oveall cos fo each ode usng a model called MIP. hs model ams o mnmze he oveall opeaonal coss and he possbly of laeness penaly coss fo each ode. Fom he oal cos of each ode he company can deemne he pce of each ode by addng a sandad mak-up paamee. he sandad makup paamee s an esmaon made by he company ha epesens unexpeced coss o esmaon of expeced pof. d. Deemnng he coss of supple and subconacos fo each ode usng a model called MIP. Afe each conngency ode changed o confmed odes hen he nex sep s o deemne supples and subconacos ha could povde aw maeals and wokload fo each confmed ode wh mnmum coss. A complee fomulaon of MIP and MIP can be found n he appendx. he decson makng sucue explaned pevously s he famewok n he decson makng pocess of ode eny sage n MO company. A decson makng sucue ypcally consss of npu, pocess, oupu and feedback. Each model n he sucue eques daa as he npu fo decson makng sysem. he daa s pocessed by he model o geneae oupu. he company as he use can use he esuls fom he sysem o ejec o accep he ncomng odes. hs decson povdes feedback no he sysem ha could nae anohe eaon of he pocess.. EENSIONS O MIP AND MIP In hs secon, exensons o he wo models poposed by Ebadan e al (007) ae dscussed.. Exenson o MIP In model MIP poposed by Ebadan e al (007) hee ae seveal weaknesses n epesenng he consans. hs can be seen n consans numbe o. hese consans aemp o allocae wokloads fo all ode n each esouce n each me peod ove he poducon plannng hozons. By way of hese consans, f allocaon fo each esouce pe peod exceeds he maxmum capacy fo nomal peod, hen he excess can be coveed wh oveme o subconac. Howeve, consans o sll have a weakness n locang he coec peod fo oveme o subconacng of each ode. heefoe, an addonal consan needs o be added. hs consan s shown n equaon (). Y O S 0,,,, O () Whee s he ndex fo ode ode ( =,..., I ); s esouce ndex ( =,..., R ); O s he se of odes o be pocessed n esouce ; s ndex fo me peod ( =,..., ); Y s he amoun of esouces allocaed fo ode on esouce n peod. hs ncludes esouces used n egula me, ove me and esouces obaned hough subconac. s he amoun of esouces O allocaed fo ode on esouce n peod dung oveme and S s he amoun of esouces fo ode on

3 esouce obaned fom subconac n peod. Equaon apples o each ode n each esouce. hs consan makes sue ha oveme o subconac s allocaed n he elevan peod when s needed. he addon of hs consan povdes a moe accuae esul compaed o he MIP poposed by Ebadan e al (007).. Exenson o MIP A majo weakness s also evden n IP model poposed by Ebadan e al (007). In selecng he bes combnaon of subconac and supples, he model always allocae each ode fo subconacos ove he poducon plannng hozon. hs esuls n hgh opeaonal coss. Logcally, he model should only allocae pocess o subconacos f he esouce n house could no sasfy he odes o f povdes a moe cos effecve soluon. hus, an algohm s desgned o ncopoae hs logc o exend he MIP model. he exenson of MIP s called IPE. he algohm as can be seen n fgue woks as follow:. Addng and changng consans on he model. a. Changng he equaon n consan by addng a vaable S ha epesens esouce allocaon fo each ode n each esouce fo each peod ha wll be subconaced. hs addon s shown n equaon. ( Y NO( ) O S ) CR ( α ),,, () c. Addng a consan o ensue ha subconacng and oveme s allocaed n he elevan peod as shown n equaon () descbed pevously.. Conducng sepaae opmzaon on each pas of he objecve funcons.e. subconacng cos, supple cos, and opeaonal cos. Sa Addng a new consan : Changng consan o hs consan Opmzng he objecve funcon sepaaely. hee ae subconaco cos, supple cos and opeaonal cos No Addng hs consan : s== Yes S>Ss Yes Selecng anohe subconaco ha could povde he equed amoun of maxmum capacy (Ss) wh mnmum pce (Ps) No NO() s he se of confmed odes fom he cusomes; CR s he maxmum capacy avalable n esouce n peod fo egula me poducon; α s he pecenage of maxmum capacy fo each esouce whch s allocaed fo hghe poy odes. hs consan dffes fom he MIP poposed by Ebadan e al (007) whee he poducon s always allocaed up o he maxmum capacy of he chosen subconaco. As menon pevously, hs logc could esul n hghe cos. b. Addng a consan o lm he amoun of esouce allocaon fo subconacng whn he deemned maxmum capacy as shown n equaon (). S CS NO( ),,, () CS s he maxmum subconacng capacy of esouce n peod. Yes S == 0 s = 0 No Updang he subconaco cos objecve funcon wh he new value of s Updang he whole objecve funcon n IP model Fnsh Fgue : Flowcha of IPE.. Pefomng updae on he value of supple selecon decson vaables ( ) wh dffeen s condons: a. If based on he subconaco seleced pevously was found ha he amoun of esouce equed fom subconaco ( S ) s geae han maxmum capacy of

4 he subconaco ( S s ), hen anohe subconaco ha could povde he capacy wh mnmum pce s seleced. Based on hs selecon a new value of s s deemned. b. If hee s no need fo subconacng fo each ode hen he value of s equals o Conducng an updae on he subconacng cos of he objecve funcon usng he new value of s. 5. Conducng an updae on he ene objecve funcon of he MIP model. he dagam of he poposed algohm s shown n fgue. he algohm s caed ou wh an assumpon ha hee s a leas one subconaco ha wll be able o sasfy he equed capacy. If hs assumpon s dffcul o sasfy hen he company can lowe he maxmum allowable capacy fo subconacng of s esouces ( CR ). 4. EPERIMENS AND ANALYSIS he poposed exensons o MIP (called IPE) and MIP (called IPE) ae mplemened n MALAB 6.5 usng OMLAB Solve (Holmsom, 007). he same daa se povded by Ebadan e al (007) s used n he mplemenaon. hee dffeen expemens ae used o evaluae he mplemenaon of boh he exsng models as well as he exended models: a. Expemens on exsng models (MIP & MIP) o evaluae he soluons of he models poposed by Ebadan e al (007) b. Expemens on exended models (IPE & IPE) o evaluae he soluons of he exended models c. Expemens on IPE wh a dffeen condon o evaluae whehe he algohm could povde opmal soluons. Resuls fom hese expemens ae used o evaluae he pefomance of he exended models compaed o hose poposed by Ebadan e al (007). 4. Compason of MIP wh IPE he opmal soluon obaned fom MIP s whle ohe decson vaables ae 6 46 =, F = F 4 = 6, L = and L 4 =. he same values ae obaned fom unnng he IPE. Howeve, he esuls fom boh models dffe n ems of he allocaon of esouces fo egula, oveme and subconac. Fgue and show he esouce allocaon found fom MIP and IPE, especvely. As can be seen fom fgue, he allocaon of oveme and subconac obaned fom MIP ae naccuae. Fo example, hee s 0 hous of esouce assgned o ode a week 4 dung oveme. hs allocaon s unecessay as hs ode has been allocaed fo poducon n week. I s acually ode 4 ha needs oveme n ha peod. Smlaly, hee s also naccuae allocaon fo subconac. IPE povdes a moe accuae epesenaon fo esouce allocaon. Fgue chas he esul fom IPE whee oveme and subconac s gven only o odes ha canno be fulflled n egula me. In hs case, ode eques subconac n week 4 and 6, whle ode 4 eques oveme n he same peod. Fgue : Allocaon of esouces obaned fom MIP

5 Fgue : Allocaon of esouces obaned fom IPE 4. Compason of MIP wh IPE Whle MIP and IPE focus on esouce allocaon and deemnaon of pce fo each ode, MIP and IPE aemp o selec he bes combnaon of supples and subconacos. In boh MIP and IPE, s assumed ha one of he ode (ode ) s ejeced by he cusome. he opmal soluon obaned fom MIP s and he esouce allocaon s shown n fgue 4. As one of he ode s ejeced hen hee ae changes n he poducon allocaon. As expeced fom he model, all of he odes s allocaed fo subconac whle hee s sll enough capacy o pocess hem n egula hous. he opmal soluon obaned fom IPE s.90. hs esul s 76% smalle han he opmal soluon obaned fom MIP. As shown n fgue 5, only pocessng of ode 4 on esouce a week 4 ha needs o be conduced dung oveme. hee s no need fo oveme o subconac because hee s sll enough capacy o pocess hem n egula me. heefoe hs condon could educe oal cos fo each ode. Lke IPE, IPE povdes a moe accuae epesenaon fo esouce allocaon. he esuls of IPE he ohe decson vaables ae: kl =, 4 = s Fgue 4: Allocaon of esouces obaned fom MIP 4. IPE wh Subconac he pevous expemen on IPE used he same daa se poposed by Ebadan e al (007). he opmal soluon Fgue 5: Allocaon of esouces obaned fom IPE shows no subconac s necessay n ha suaon. hs expemen s conduced o evaluae he coecness of he new algohms of IPE n a condon whee

6 subconacng s needed. heefoe some changes ae done on he daa.e. s assumed ha ode 4 s hgh poy ode ha mus be delveed on me whle ode s ejeced by he cusome. Usng hs scenao he esul s shown n fgue 6. I s clea ha n ode o delve ode 4 n me, some oveme and subconacng s necessay. Howeve, oveme and subconac ae assgned only when he wokload fo a cean esouce exceeds he lm of egula capaces. he opmal soluon obaned fom hs expemen s he esuls of he ohe decson vaables ae as follow: kl =, 4 = 4 4 = s. 5. CONCLUDING REMARKS Fgue 6: Allocaon of esouces obaned fom IPE (expemens ) hs pape pesens an mplemenaon and exenson o he wok of Ebadan e al (007) on decson makng sucue of ode eny sage n make o ode companes. he pape focused on he wo mahemacal models ha epesens he man pa of he decson makng sucue. Exenson o he fs model called IPE povdes moe accuae epesenaons on he allocaon of esouces o pocess ode dung egula and oveme as well as subconac. Exenson o he second model (IPE) povdes a moe opmal and accuae soluon by ncopoang an algohm ha only uses subconacos once he nenal capacy s ncapable o fulfll he odes. hs povdes 76% educon o he opmal soluon compaed o he old model. Fuhe wok on hs subjec should consde elaxng he assumpon ha supples and subconacos povde hgh qualy maeals. In ealy, supples and subconacos have dffeen levels of qualy. heefoe, qualy s an mpoan faco o consde n choosng supples and subconacos n addon o pce. APPENDI he mahemacal models.e. MIP and MIP poposed by Ebadan e al (007) as well as daa used n he mplemenaon ae summazed n hs secon. Moe deals explanaon on he models can be found n he pape. MIP Indces I Ode fom cusome ( =,..., I ). R fo poducon pocess n ems of machne, shop floo, human esouce, ec ( =,..., R ). Plannng hozon peod ( =,..., ). Paamees CR Maxmum capacy of esouce a peod dung egula me, ypcally n machne hous CO Maxmum capacy of esouce a peod dung oveme, ypcally n machne hous CS Maxmum capacy of esouce a peod fo subconacng, ypcally n machne hous WK Requed pocessng me of ode on esouce ; ypcally n machne hous. p Accepance pobably of ode, whch commonly epesens he poy of ode. α Pecenage of oal maxmum capacy of each esouce ha s pu asde fo hgh poy fuue avng odes a peod. dd Due dae of ode. c Poducon cos of ode on esouce a peod n egula me co Poducon cos of ode on esouce a peod n oveme

7 cs Subconacng cos of ode on esouce a peod c Laeness penaly of ode pe un me wp oal wokload ha has been emaned fom he pevous plannng hozon (wokload of he odes n he pool and also he odes whch have been eleased o he shop floo bu have no ye gone hough esouce and eque opeaon on he esouce). w oal wokload of ode on esouce awang maeal wh ERD a peod,.e., npu wokload o he shop floo whch should be poduced (w =WK ) ow oal wokload of ode on esouce wh OCD a peod (.e., he oupu of wokload equed fom esouce a peod o ensue ha he ode confoms o s planned OCDs and hence mee s planned delvey dae (ow = WK ). OS () Se of odes whch mus be delveed on me. M Vey lage numbe. Decson Vaables Y Amoun of esouce assgned o ode a peod ncludng egula, oveme, and subconacng wok, ypcally n machne hous O Amoun of esouce assgned o ode a peod dung oveme, ypcally n machne hous S Amoun of esouce assgned o ode a peod whch s suppled by subconacos, L F ypcally n machne hous Laeness amoun of ode Compleon dae of ode on he las esouce Boolean vaable ha akes he value of 0 f esouce allocaon on he las esouce of ode a peod s negaves o 0. Ohewse, akes he value of f esouce allocaon on he las esouce of ode a peod s posves. Objecve Funcon Mnmze: Z = R O = = [ c ( Y O S ) + co O + cs S ] + c L, Subjec o OS () () Y O S ) CR ( α ), ( O,, () O CO, O () S CS, O,,,, (4) wp + w p Y,, O = O = (5) ow p = Y,,, k OS ( ), k = OS ( ), k = O O (6) w p Y, k k k = k =, OS( ), O, (,..., dd (7) Y, μ (, n ), M,, OS( ), (8) F + M ( ),, OS( ), (9) L ( F dd ), OS(), (0) L, OS(), dd + ( dd ) () ow p = Y, k k k = k =, OS( ), O, (,..., dd Y, O, S > 0,, O, ), (), () L, F 0 & { 0,},, OS( ), MIP Indces L Supple ( l =,..., L ) S Subkonako ( s =,..., S ) K maeal ( k =,..., K ) Paamees MAD Maxmum capacy of esouce a peod kl dung egula me, ypcally n machne hous S Maxmum wokload of ode on esouce ha s can be suppled by subconaco s a peod P Suggesed pce of subconaco s fo wokload s of ode on esouce P Suggesed pce of supple l fo aw maeal k. β kl Penaly cos of ecevng aw maeals of ode k ),

8 befoe ERD (oupu of he MIP model) pe each un of ealness β ' Penaly cos of ecevng aw maeal of ode afe ERD pe each un of laeness. Snce he laeness can cause some odes o be delveed lae hen he value of β ' mus be compued n a way ha he aw maeal s suppled on me o eale ( β >> β ' ). NO () Se of he new acceped odes. L (k) Se of supples ha can supply aw maeal k of ode L' ( k) Se of supples ha delve aw maeal k of ode befoe s ERD (MAD kl <ERD ) S () Se of subconacos ha can supply equed wokload of ode on esouce Decson Vaables Boolean vaable ha akes he value of 0 f s subconaco s s unable o supply he equed wokload of ode on esouce. Ohewse, he value equals o f subconaco s s able o supply equed wokload of ode on esouce Boolean vaable ha akes he value of 0 f kl supple l s unable o supply equed aw maeal k of ode. Ohewse, he value equals o f supple l s able o supply equed aw maeal k of ode Objecve Funcon Mnmze: Z = R s NO( ) = s S ( ) K kl NO( ) k= l L( k) K NO( ) k= l L'( k) K NO( ) k= l L'( k) R P P [ c Y MAD ERD ) O NO( ) = = s S ( ) + co O + + β ( ERD β ' ( MAD + cs s kl kl ( S s S ( ) s kl ) s kl kl ( S )] + + s s ) Subjec o () ( Y O ( S NO( ) s S ( ) s CR α ),,, ( () O CO, NO( ),, () wp + w Y,, NO( ) = s )) NO( ) = (4) ow = Y,,, OS ( ), k = NO( ) k (5) w Y, OS ( ), k = NO( ) k k k = k =, OS( ), NO( ), (,..., dd + L (6) ow = Y, k k k = k =, OS( ), NO( ), (,..., dd (7) =, OS( ), k, l L( k) kl (8) =, OS( ),, s s S ( ) (9) Y, O 0;, [0,] kl DAA In he pape Ebadan e al (007) used an assumed MO sysem wh wo esouces o pocess fou odes whn 6 weeks plannng hozon. he daa ae pesened n able A. A. whle ohe paamees ae as follow: c = 50, CR = 40, co = 50, CO = 0, cs = 50, CS = 0, α = 0. able A. DaaWK (hous) Ode Ode Ode Ode s k ), ),

9 able A. Daa fo wp (hous) 0 0 able A. Daa fo c Ode Ode Ode Ode able A.4 Daa fo p Ode Ode Ode Ode able A.5 Daa fo ERD (hous) Ode Ode Ode Ode 4 able A.6 Daa fo dd (hous) Ode Ode Ode Ode able A.8 Daa fo Ode Ode Ode Ode able A.9 Daa fo Ode Ode Ode Ode β β ' able A.7 Daa foocd (weeks) Ode Ode 5 Ode Ode able A.0 Daa fo P s Ode Ode Ode Ode able A. Daa fo P kl Supple Maeal Maeal Maeal Supple able A. Daa fo MAD kl (weeks) Ode Ode Maeal Maeal Maeal Maeal Maeal Maeal 4 Supple Ode Ode 4 Maeal Maeal Maeal Maeal Maeal Maeal

10 able A. Daa fo S s (hous/weeks) Peode Ode Peode Ode Peode Ode REFERENCES Eason, F.F., Moode, D.R. (999) Pcng and lead me decson fo make-o-ode fms wh conngen odes, Euopean Jounal of Opeaonal Reseach, Vol. 6, Issue, Ebadan, M., Rabban, M., Jola, F., oab, A., avakkol-moghaddam, R. (007) A new decson-makng sucue fo he ode eny sage n make-o-ode envonmens, Inenaonal Jounal of Poducon Economcs, Vol., Issue, Ghaehgozl, A.H., avakkol-moghaddam, R., Rabban, M., Zaepou, N. (007) Accepance/Rejecon of Incomng Odes by a Fuzzy Analycal Heachy Pocess n Make-o-Ode Envonmens. Poceedngs of he 8h WSEAS Inenaonal Confeence on Fuzzy Sysems, Vancouve, Canada, Holmsom, K., Goan, A.O.,Edvall, M.M (007). Use's Gude Fo omlab 5.9, omlab Opmzaon. AUHOR BIOGRAPHIES Mahendawah ER s a Lecue n Depamen of Infomaon Sysems, Faculy of Infomaon echnology, Insu eknolog Sepuluh Nopembe, Indonesa. She eceved a Docoal Degee n Manufacung Engneeng and Opeaons Managemen fom Nongham Unvesy, Uned Kngdom n 004. He eachng and eseach neess nclude supply chan managemen, nenaonal opeaon, poduc vaey managemen, modelng and smulaon. He emal addess s <mahenda_w@ssby.edu>

11 Rully Soelaman s a Lecue n Depamen of Infomacs, Faculy of Infomaon echnology, Insu eknolog Sepuluh Nopembe, Indonesa. He eceved a Mase Degee n Compue Scence fom Faculy of Compue Scence, Unvesy of Indonesa, Indonesa n 00. Hs eachng and eseach neess nclude nellgence sysem and opmzaon. Hs emal addess s <ully@s.s.ac.d> Rzal Safan eceved a bachelo degee fom Depamen of Infomaon Sysems, Faculy of Infomaon echnology, Insu eknolog Sepuluh Nopembe, Indonesa n 008. He has pevously conduced a eseach on Open Souce Busness Ecosysem n Indonesa appoved by Depamen of Communcaon and Infomaon echnology of he Republc of Indonesa. He can be conaced a <jolee@yahoo.com>