TACTICAL PLANNING OF THE OIL SUPPLY CHAIN: OPTIMIZATION UNDER UNCERTAINTY Gabriela Ribas Idusrial Egieerig Deparme Poifical Caholic Uiversiy of Rio de Jaeiro PUC-Rio, CP38097, 22453-900 Rio de Jaeiro Brazil gabiribas@gmail.com Adriaa Leiras Idusrial Egieerig Deparme Poifical Caholic Uiversiy of Rio de Jaeiro PUC-Rio, CP38097, 22453-900 Rio de Jaeiro Brazil adriaaleiras@yahoo.com.br Silvio Hamacher Idusrial Egieerig Deparme Poifical Caholic Uiversiy of Rio de Jaeiro PUC-Rio, CP38097, 22453-900 Rio de Jaeiro Brazil hamacher@puc-rio.br ABSTRACT The oil idusry is icreasigly ieresed i improvig he plaig of heir operaios due o he dyamic aure of his busiess. Decisios made a he oil chai differ i he aciviy rage (spaial iegraio) ad plaig horizo (emral iegraio). This paper purse is o address he spaial iegraio uder uceraiy i he oil chai a he acical plaig level ad proses a mahemaical model o maximize he profi of his chai. The model is formulaed as wo-sage sochasic program, where uceraiy is icorraed i price ad demad parameers. A idusrial ale sudy from he Brazilia idusry was coduced. The Expeced Value of Perfec Iformaio (EVPI) ad he Value of he Sochasic Soluio (VSS) 1.55% ad 13.76% of he wai-ad-see soluio idicaed he beefi of icorraig uceraiy i he plaig ad demosraes he effeciveess of he prosed approach. The ceralized coordiaio (spaial iegraed) provided a beer uilizaio of he available resources. KEYWORDS. Two-sage sochasic opimizaio. Tacical plaig. Oil chai. P&G OR i he Oil & Gas area.
1. Iroducio The ucerai aure ad high ecoomic iceives of he refiig busiess are drivig forces for improvemes i he refiery plaig process. This requires a high level of decisiomakig o oly o a sigle faciliy ale bu also o a eerprise-wide ale (Chopra ad Meidl, 2004). Plaig for his kid of operaio should be carried ou cerally, hus allowig for proper iegraio amog all operaig faciliies ad, cosequely, a efficie uilizaio of available resources (Al-Qahai ad Elkamel, 2008). Decisios made a he oil chai differ maily i he rage of aciviies coordiaed i he supply chai (spaial iegraio) ad i he coordiaio of decisios across differe ime ales (emral iegraio). The udersadig of such iegraio beefis has araced aeio i he research area of supply chai plaig (Kogu ad Kulailaka, 1994; Huchzermeier ad Cohe 1996; Smih ad McCardle, 1998; Harriso ad Va Mieghem, 1999; Cacho 2002). Plaig is basically a aciviy i which producio arges are se ad marke forecass, resource availabiliy, ad iveories are cosidered. I geeral, plaig is caegorized io hree ime frames: sraegic (log erm), acical (medium erm), ad operaioal (shor erm). Sraegic plaig deermies he srucure of he supply chai. Tacical plaig is cocered wih decisios such as he assigme of producio arges o faciliies ad he rasraio from faciliies o disribuio ceers. O he oher had, operaioal plaig deermies he assigme of asks o uis a each faciliy, cosiderig resource ad ime cosrais (Maravelias ad Sug, 2009). This paper covers he acical plaig level ad addresses de problem of spaial iegraio a he oil chai. Alhough plaig i he oil idusry was radiioally developed wih well esablished deermiisic models, hese models have bee exeded o iclude uceraiies i parameers. I fac, Be-Tal ad Nemirovski (2000) sress ha opimal soluios of deermiisic models may become ifeasible eve if he omial daa is oly slighly perurbed. This idea is supred by Se ad Higle (1999), who affirmed ha uder uceraiy he deermiisic formulaio i which ucerai radom variables are replaced by heir expeced values, may o provide a soluio ha is feasible wih respec o he radom variables. Thus, uceraiies are ieviable i mahemaical modelig ad also i eforcig he plaig model o realisic soluios. Uceraiies ca be caegorized as shor-erm, mid-erm, or log-erm. Shor-erm uceraiies refer o uforesee facors i ieral processes such as operaioal variaios ad equipme failures (Subrahmayam e al., 1994). Aleraively, log-erm uceraiies represe exeral facors, such as supply, demad, ad price flucuaios, ha impac he plaig process over a log period of ime. Mid-erm uceraiies iclude boh shor-erm ad log-erm uceraiies (Gupa ad Maraas, 2003). Josbrae (1998) classified uceraiies as exeral (exogeous) uceraiies ad ieral (edogeous) uceraiies, accordig o he i-of-view of process operaios. As idicaed by he ame, exeral uceraiies are exered by ouside facors ha impac he process. O he oher had, ieral uceraiies arise from deficiecies i he complee kowledge of he process. Thus, mid-erm ad log-erm uceraiies ca also be classified as exeral uceraiies, whereas shor-erm uceraiies are ieral uceraiies. I regard of acical models, Liu ad Sahiidis (1996) developed a wo-sage sochasic model ad a fuzzy model for process plaig uder uceraiy. A mehod was prosed for comparig he wo approaches. Overall, he compariso favored sochasic programmig. Eudero e al. (1999) worked i he supply, rasformaio ad disribuio plaig problem ha accoued for uceraiies i demads, supply coss, ad produc prices. As he deermiisic reame for he problem provided usaisfacory resuls, hey applied he wosage eario aalysis based o a parial recourse approach. Dempser e al. (2000) formulaed he acical plaig problem for a oil cosorium as a dyamic recourse problem. A deermiisic muli-period liear model was used as basis for implemeig he sochasic programmig formulaio. Hsieh ad Chiag (2001) developed a maufacurig-o-sale plaig sysem ad adoped fuzzy heory for dealig wih demad ad cos uceraiies. Li e al. (2004) prosed a probabilisic programmig model o deal wih demad ad supply uceraiies i he acical problem. Kim e al. (2008) worked o he collaboraio amog refieries maufacurig
muliple fuel producs a differe locaios. Khor e al. (2008) reaed he problem of mediumerm plaig of a refiery operaio by usig sochasic programmig (a wo-sage model) ad sochasic robus programmig. Al-Ohma e al. (2008) have prosed a wo-sage sochasic model for muliple ime periods o opimize he supply chai of a oil compay isalled i a coury ha produces crude oil. Fially, Guyoe e al. (2009) cosidered oil uploadig ad produc disribuio problems i heir acical formulaio. Despie of hese sigifica coribuios, oly four (4) of hese ie (9) works represe acual applicaios ad mos of hem sill prese very simplified models ha exclude imra aspecs of a real refiery operaio such as cosrais for he specificaio of fial producs. Therefore, he refiery acical plaig problem uder uceraiy is sill a ope issue, which is releva for boh mahemaical modelig ad acual applicaios. I his coex, he mai coribuio of his work is o ackle a imra opic from boh heory ad pracice sad is, ivesigaig he spaial iegraio of he oil supply chai hrough a wo-sage sochasic model characerized by price ad demad uceraiies. The purse of he prese paper is, he o address he problem of iegraio ad coordiaio uder uceraiy i he oil supply chai a he acical level. A acical programmig model is prosed wih he objecive of maximizig he expeced oal profi over a give ime horizo. Uceraiies i demad for refied producs, oil prices, ad produc prices (exogeous uceraiy facors) accou for ecoomic risk a he acical level. The problem has bee formulaed as wo-sage sochasic liear program wih a fiie umber of realizaios. I his approach, decisio variables are cas io wo groups, firs ad secod sage variables (Dazig, 1955). The firs sage variables are decided prior o he acual realizaio of he radom parameers. Oce he ucerai eves have ufolded, furher operaioal adjusmes ca be made hrough values of he secod sage. The remaider of his paper is orgaized as follows. A overview of he refiig idusry is preseed i secio 2. Secio 3 preses he acical model for oil chai plaig uder uceraiy. Secio 4 offers resuls ad diussios i he coex of a case sudy usig real daa from he Brazilia oil idusry. Fially, some coclusios are draw i secio 5. 2. Refiig Idusry Overview The oil chai covers sages from oil exploiaio o produc disribuio icludig a complex logisic ework ad several rasformaio processes ha ake place i refieries. The peroleum supply chai is illusraed i Figure 1. The aciviies ha comprise he oil chai are divided io hree major segmes: upsream, midsream, ad dowsream. The upsream segme icludes he exploiaio ad oil producio. The midsream is a iermediae segme ad cosiss of he refiig aciviy which icludes he rasraio of oil from he producio sie o refieries. The logisical asks ecessary o move he refied producs from he refiery o he cosumer is are i he dowsream segme. Peroleum may be produced i exploiaio fields of he compay iself or be supplied from ieraioal sources. The domesic oil is se by pipelie or oil akers o ermials from where he oil mees he demad of he refieries or is exred. Crude oil obaied from ieraioal sources is rasred by pipelie or oil akers o he ermials. The domesic ermials are i charge of receivig ad forwardig oils ad refied producs, whereas ieraioal ermials represe is of offer ad demad for foreig oils ad producs. The oil ermials are he coeced o refieries hrough a pipelie ework. Crude oil is covered o refied producs a refieries which ca be coeced o each oher i order o ake advaage of each refiery desig wihi he ework. A plaig model for oil refieries mus allow for he proper selecio of oil bledig ad cosider a appropriae maipulaio of iermediary sreams o obai he fial producs i he desired quaiies ad qualiies. The refied producs ca be moved alog he logisic ework by road, waer, rail, ad pipelie modes. Crude oil ad refied producs are ofe rasred o disribuio ceers hrough pipelies. From his level o producs ca be rasred eiher hrough pipelies or rucks, depedig o cosumer demad is. I some cases, producs are also rasred hrough vessels or by rai.
Figure 1. Oil supply chai (adaped from Ribas e al., 2010) As a resul of he complexiy of he oil chai plaig of he oil chai mus be aided by decisio-makig sysems, especially hose ha employ mahemaical programmig for example, RPMS - Refiery ad Perochemical Modelig Sysem (Boer ad Moore, 1979), OMEGA - Opimizaio Mehod for he Esimaio of Gasolie Aribues (Dewi e al., 1989), ad PIMS - Process Idusry Modelig Sysem (Bechel, 1993). I his way, mahemaical programmig plays a crucial role o assis he decisio-makig process i he oil supply chai. 3. Tacical Plaig of he Oil Supply Chai uder Uceraiy The acical plaig model prosed i his paper maximizes he oal reveue of he supply chai allocaig he producio of he differe producs o he various refieries i each ime period, while akig io accou iveory holdig coss ad rasraio coss. The opimizaio model is based o a eario aalysis approach, ad is liear. Followig Pogsakdi e al. (2006), may oliear feaures were simplified i order o gai compuaio speed, which allows he decisio-maker o beer explore he uceraiy issues i he model. The associaed prices, coss, ad demads are assumed o be exerally imsed i he plaig. The modelig cosiders a fixed marke, i.e. he model esures he oal fulfillme of he marke demad. I is assumed ha he physical seigs i he supply chai have already bee esablished, he cofiguraio of he chai is give ad he umber of faciliies a each sage is kow. I is also assumed a diree plaig horizo divided io a fiie umber of periods. The models have bee formulaed as wo-sage sochasic programs wih fixed recourse (Dazig, 1955). Uceraiies are direely represeed by SC ssible realizaio earios ad modeled as a eario ree. A eario is a pah from he roo o a leaf of he ree. The probabiliy ha he -h eario will occur is represeed by p ( p SC 0, p = 1, = 1 SC ). Based o hese assumpios, he sochasic models his paper proses ca be represeed as follows: T T Max z ( x) = c x + p q y x subjec o Ax b SC Wy h Tx, x 0, y 0 SC Firs-sage decisios are assumed o be made before he realizaio of radom variables (here-ad-ow decisios), represeed by a vecor x, while secod-sage decisios, deoed by y, are made uder complee iformaio abou he realizaio of., (1)
T The objecive fucio i Equaio (1) coais a deermiisic erm c x, which models he oil purchase decisios cocered wih oil supply by log-erm coracs. The secod T erm of Equaio (1) coais he expeced value of he secod-sage objecive p q y SC which models he sochasic operaioal profi due o he firs-sage decisio. A se of deermiisic iequaliies ( Ax b ) is used o model decisios relaed o oil purchase. Sochasic cosrais ( Wy h Tx ) are used o represe refiery operaio ad o model all operaive relaios bewee he ipus (or differe peroleum ypes) ad he oupus (or fial producs) ad he ecessary ework flows hrough he isalled rasraio ework. Uceraiy is iroduced hrough he produc prices, oil prices, ad marke demad for fial producs. I order o properly evaluae he added-value of icludig uceraiy i he problem parameers, he models ca be evaluaed usig he Expeced Value of Perfec Iformaio (EVPI) ad he Value of he Sochasic Soluio (VSS) (Birge ad Louveaux, 1997). The EVPI measures he loss of profi due o he presece of uceraiy which is also he measure of he maximum amou he decisio maker is willig o pay i order o ge accurae iformaio o he fuure. As saed by he cosrai (2), he EVPI resuls show he expeced profi differece bewee he soluio obaied by he age able o make he perfec predicio (wai-ad-see - WS) ad he oe obaied by he age ha solved he problem uder uceraiy (recourse problem - RP). EVPI = WS RP (2) A soluio based o perfec iformaio would yield opimal firs sage decisios for each realizaio of he radom parameers (Madasky, 1960). So, assumig ha he uceraiy is represeed by a fiie umber of earios ad ha ζ is a radom variable se of earios, he problem associaed wih each eario of ξ ca be defied as: x T T (, ξ ) = + max {, 0} = { :, 0} Max z x c x q y Wy h Tx y X x Ax b x I is assumed ha for all ξ here is a leas oe feasible soluio (, ) * opimal soluio o he problem (3) ad ( ) x (3) * R. Le x ( ) ξ a z x ξ ξ he opimal objecive fucio value for a eario ξ. The wai-ad-see soluio corresds o he opimal value whe he fuure realizaio of ξ is kow i.e., he decisio maker ca wai ad see he fuure before decidig. The expeced value of he wai-ad-see soluio is: ( ) * ( ) ( ) WS = E ξ max z x, ξ = Eξ z x ξ, ξ x X The recourse problem (RP) soluio is also kow as here-ad-ow decisio because he soluio he firs sage is decided wihou kowig he fuure realizaio of ξ, i.e., a he decisio ech he fuure eario is kow oly probabilisically. So, he RP corresds o he wo-sage problem defied by he model (1) ad ca be wrie as: ( ) * ( ) ( ) WS = E ξ max z x, ξ = Eξ z x ξ, ξ x X (5) The VSS, o he oher had, is defied by he differece bewee he sochasic soluio (RP) ad he average soluio of he expeced value problem (replacig he radom eves by heir meas - EEV) - cosrai (6). The VSS ca be ierpreed eiher as he beefi expeced by he age ha has ake uceraiy io accou or as he loss expeced by he age ha oped (4)
for deermiisic modelig usig he average sochasic parameers ( [ ] E ξ = ξ ). VSS = RP EEV (6) I order o quaify he VSS, firs i is ecessary o calculae he expeced value soluio (EV) which is defied by he soluio of he problem o he expeced eario (expeced value of ξ ). Le ξ E [ ξ ] * = ad x ( ) ξ he opimal soluio o EV, so: ( ) EV = max z x, ξ (7) x The by fixig he firs sage variables from he EV problem, he expecaio of EV (EEV) ca be obaied by allowig he opimizaio problem o choose he secod sage variables wih respec o differe realizaios: * ( ( ), ) EEV = Eξ z x ξ ξ The model formulaio prosed i his work is preseed i he ex secio. 3.1. Tacical plaig model This secio preses he sochasic formulaio for acical plaig of oil refieries. This formulaio is adaped from he model prosed by Ribas e al. (2010) by excludig all elemes relaed o ivesme decisios which mus oly be cosidered i a sraegic plaig model. The prosed liear programmig model aims o maximize he expeced profi of he oil chai ad cosider he followig facors ha affec he domesic supply: cofiguraio of refiig park; refiery operaios ad rasraio coss; imr of oil; imr ad exr of refied producs; requiremes for refied producs defied by regulaory orgaizaios; producio of crude oil; domesic cosumpio of refied producs; ad prices of oil ad refied. The model decisios o oil refiig deermies he oil bledig o each refiery, he producio level a each process ui, ad he operaioal mode for each ui a each period o mee he demad ad respec he qualiy sadards o he refied producs. A operaioal mode is characerized by a se of operaio paers o prioriize he producio of a specific produc se. Wih respec o he logisic ework, he model mus defie he miimum cos flow combiaio for he refiery supply ad he refied producs disribuio. Defiiios of parameers, ses, ad variables are provided i Tables 1 ad 2 which are followed by mahemaical formulaio. Table 1. Parameers of he acical model Parameers Operaioal cos OC r, u Oil field producio Trasraio capaciy CT a Ow cosumpio Trasraio cos TC a Miimum prorio Disillaio ui yield YDU r, u, c, o, Maximum prorio Process ui yield Y P U r, u, c, p i, Oil price - ieral disribuio Miimum capaciy UCL r, u Sochasic parameers Maximum capaciy UCU Probabiliy of eario r, u Sulfur quaiy - ery produc SIO pi Domesic produc demad Maximum sulphur SPOU Produc price - domesic marke Viosiy bledig idex B I p Produc exr price Miimum viosiy VPOL Produc imr price Oil imr price (8) FP i 1, o CP r, u, PRPL r, u, pi, c PRPU r, u, pi, c OPBR r, o P PD b, PPBR b, PPE i, PPI i, OPI i, o
Se of odes (i1, i2) Ses Table 2. Ses ad variables of he acical model Se of process uis (u, u') U Oil purchase Se of operaioal modes (c) Se of producs (pi, ) PR Bledig Se of oils (o) O Disillaio ui load Se of rasr modes (m) M Oher process ui load Se of rasr arcs (a) AT Oil imr Time periods { = 1,..., NT } N Produc exr Se of acical earios ( ) SC Produc imr I C Firs sage variables Secod sage variables Variables Refiery (r) R I Trasred flow - eerig he refiery Naural gas producers (g) NG I Oil flow Ieraioal odes (i) IN I Trasred flow - leavig he refiery Termials (r) TR I Produc flow Bases (b) B I Sock level of oil o a refiery r Oil Field (of) OF I Trasraio arcs available for rasraio of from i1 o i2 ATA AT by he mode m Model Formulaio Maximize TM = Refiig balace ( OPBRr, oqocfr, o ) r R o O N,, ( b ), b, + ( PPEi ), pexpi, PPI i, pimp i, N i I P ( OPIi ), o oimpi, o N i I o O OCr, u qdur, u, o, c OCr, u qpu r, u, pi, c N r R u U o O c C N r R u U pi P c C pa, TCa oa, o TCa a AT N P a AT N o O + P PPBR PD SC b B P N qocf r, o b r, pi, qdu r, u, c, o qpu r, u, c, pi oimp i, o pexp i, pimp i, ir r, o a, o or r, p a, vo r, o ( ) 1, r, o + r, o = r, u, c, o + r, o,,, u U c C qocf vo qdu vo r R o O N SC qdur, u, c, oydu r, u, c, o, + qpur, u, c, piypu r, u, c, pi, + br, pi, u U c C o O u U c C pi P pi P r, r,, pi r, u, c, r, u, c, pi r, u, pi P u U c C u U c C pi P + ir = b + qpu + qpu CP + or Refiig operaio cosrais r, r R, PR, N, SC (9) (10) (11)
PRPL qpu qpu PRPU qpu r, u, pi, c r, u, pi, c r, u, pi, c r, u, pi, c r, u, pi, c pi P pi P r R, u U, pi PR, c C, N, SC,,, r, u r, u, o, c + r, u, pi, c r, u o O c C pi P c C UCL qdu qpu UCU r R u U N SC (12) (13) Eviromeal legislaio requiremes,, qpu,,, SIO YPU,,,, qpu,,, YPU,,,, SPOU u U pi P c C u U c C pi P ( ) r u c pi pi r u c pi r u c pi r u c pi r R, PR, N, SC b BI + qdu YDU + qpu YPU BI r, pi, pi r, u, c, o r, u, c, o, r, u, c, pi r, u, c, pi, pi P u U c C o O u U c C pi P VPOL br, pi, + qdur, u, c, oydu r, u, c, o, + qpu r, u, c, piypu r, u, c, pi, pi P u U c C o O u U c C pi P,,, Logisic balace p + or = PD + p + ir a, i1, i1, a, i1, ( a, i2, m) ATA ( a, i2, m) ATA o + FP = qocf + o a, o i1, o i1, o a, o ( a, i2, m) ATA ( a, i2, m) ATA Logisic capaciy cosrais r R PR N SC i1 R U TR U OF, PR, N, SC i1 R U BU TR U NG U OF, o O, N, pa, + oa, o CTa a AT, N, SC P o O qocf b dfr pfr oimp pexp pimp ir o or p SC,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, + r o r pi r u c o r u c i o i i r a o r a, R (19) The objecive fucio (9) maximizes he expeced acical margi. This margi icludes he reveue from he produc sales ad he produc exrs, mius he raw maerial coss, he oil ad produc imrs, he refiig operaio coss, ad he rasraio coss. The oil purchase ( qocf r, o ) represes he firs sage decisios. The secod-sage decisios are he, amou of produc ad oil rasred ( p ad i o a, o a, o (14) (15) (16) (17) (18) ), he amou of oil imred ( oimp ), ad he amou of imred produc ad exred produc ( pimp ad pexp ). Equaio (10) ad (11) represe he oil balace ad he produc balace, respecively. For hem, he sum of he ery flows mus be equal o he sum of he oupu flows. Equaio (12) esablishes he prorio bewee he ery flows (pi) ad he oal process ui (u) loadig. The maximum ad miimum capaciies of he process ui u i period are limied by equaio (13). i Equaios (14) ad (15) limi he sulfur coe ( SIO ) ad he viosiy ( BI pi pi, i ) of he fial producs. Fial produc properies mus be wihi a rage esablished by eviromeal regulaios. Propery calculaios yield a se of oliear cosrais (Moro ad Pio, 2004) where he oliear erms arise from he muliplicaio bewee he producs properies ad heir
volumes. These erms ca be liearized by esimaig he properies of iermediae producs. A he acical level i is ssible o esimae he sulfur coe SIO pi, ad he viosiy BI pi, of he iermediae producs wih sufficie accuracy, makig he cosrai ha corols he fial producs properies liear. The acical model corols oly hese wo properies because hey are he oes ha affec mos acical decisios such as oil purchase ad oil bledig. Logisic balace cosrais (equaios 16 ad 17) deermie ha he sum of he ipu flows mus be equal o he sum of he oupu flows for each ode (i), produc () or oil (o), period of ime () ad eario ( ). ATA represes he se of rasraio arcs (a) for a produc () from a origi ode (i1) o a desiaio ode (i2) by a rasraio mode (m). Equaio (18) limis he maximum volume rasred by he rasraio arc (a) i he period. Fially, cosrais (19) defie he o-egaiviy of he variables. 4. Numerical Example A idusrial ale sudy usig real daa from he Brazilia idusry was used o evaluae he performace of he prosed model i opimizig large-ale problems. The refiig sysem icludes 3 refieries (amed R1, R2, ad R3) ad represes a geeral sysem ha ca be foud i may idusrial sies aroud he world. The refieries are coordiaed cerally, he feedsock oil supply is shared, ad he refieries collaborae o mee a give marke demad. The refieries are supplied by 8 groups of aioal oils produced i 2 exploiaio fields, ad 1 group of foreig oils. The refieries process up o 50 iermediae producs o produce 10 fial producs associaed o he local marke demad. The logisic ework icludes 2 domesic ad 4 ieraioal ermials, 2 disribuio bases, ad 73 rasraio arcs relaive o road, waer, rail, ad pipelie modes. The ime horizo i he acical level covers 6 mohly periods. R1 is a small ad low complex refiery which focuses o he producio of lubricas, asphal, ad fuel oils. This refiery is supplied by 1 group of oil (group A). R2 ca also be cosidered a low complexiy refiery ha aims a he producio of solves ad fuels ad processes 3 groups of oils (groups C, D, ad E). Fially, R3 is a medium complexiy refiery ad processes 7 groups of oils (groups A, B, C, E, F, G, ad H) wih he focus o he producio of aphha, bu also has sigifica producio of je fuel, diesel, ad gasolie. The mehod used o creae he earios of he sochasic model was based o daa collecio ad direc corasiio of primary (daa obaied from he oil Brazilia oil idusry) ad secodary research (hisorical ecoomic daa available olie). Developig mehodologies for eario geeraio is beyod he ope of his paper, ad he ieresed reader ca refer o he work by Kouweberg (2001), for example. As i is esseial o es he prosed models, he eario geeraio wih he associaed probabiliies was arbiraed i cosisecy wih he real problem ad validaed wih expers of he oil idusry. Table 3 shows he probabiliy of each ssible realizaio of he sochasic uceraiy. The demad for refied producs, oil prices, ad produc prices are mid-erm uceraiies which are cosidered i he acical plaig. Table 3. Probabiliies of he sochasic parameers Model Sochasic Parameer Realizaios Probabiliy High 25% Demad Base 50% Low 25% Tacical Price High 25% Base 50% Low 25% Each sochasic parameer a he acical level (price ad demad) has hree ssible realizaios (high, medium, ad low). Assumig ha he radom variables are idepede hese wo parameers were combied o creae he ie earios preseed i Figure 2. For a give sochasic parameer, i is assumed complee depedece for all producs - for example, oe eario of high demad for oe produc implies i high demad o he oher producs. Similar
paer is preseed o oil ad produc prices. The base case (eario 5) used daa from he curre plaig sysem of Brazilia refieries. This sysem addresses oly a deermiisic case which was used o geerae he base case. The oher earios were cosruced based o he experise of employees of he idusry uder sudy. Demad Price Scearios Probabiliies Millio $ 1,800 1,600 1,400 1,200 1,000 Figure 2. Sceario ree High 1 6,25% High Base 2 12,5% Low 3 6,25% High 4 12,5% Base Base 5 25,0% Low 6 12,5% High 7 6,25% Low Base 8 12,5% Low 9 6,25% 4.1. Compuaioal resuls ad diussio The model was implemeed i he Advaced Iegraed Mulidimesioal Modelig Sofware AIMMS ad solved usig he CPLEX 12.1. Table 4 summarizes he model saisics: Table 4. Model Saisics #Variables #Cosrais #No zeros Solvig ime (s) E[margi] (millio $) 96,899 119,105 218,286 0.78 707.9 As show i Figure 3a, he bes resuls for he model were foud i he earios wih high prices. This fidig idicaes he model s sesiiviy o he ucerai parameers ad ha he prices uceraiy had a greaer impac o oal profi ha he demads uceraiy had. 800 600 400 200 1,480 693 1,503 712 1,522 0-106 -91-80 -200 1 2 3 4 5 6 7 8 9 Scearios 727 1,500 1,350 1,200 1,050 900 750 600 450 300 150 0 1 2 3 4 5 6 Periods R1/ A R2/C R3/F R3/H Figure 3a. Tacical margi soluios by earios Figure 3b. Tacical oil purchase decisios Figure 3. Tacical model soluios Oil purchase (housad m 3 / moh) The acical oil purchase decisios for he 6 periods of plaig (), defied by he firs sage variable qocf r, o, are preseed a Figure 3b. The leged represes he refiery/ group of oil allocaed o he refiery. Refiery R3 is ressible by 83.90% of he oal oil purchases
preseed i Figure 4. Two groups of oils are allocaed o R3. This refiery processes he eire amou available of he oil F ad complees he maximum capaciy level wih oil (H). I addiio 13.76% of oal oil quaiy is allocaed o he oil family C a refiery R2. Fially, he acical model aribues he las 2.34% of oil o he family A for he lubrica producio a R1. I his umerical sudy, he EVPI reached a maximum of 1.55% of he wai-ad-see (WS) soluio 11.16 millio $. The EVPI resul shows he differece bewee he soluio of he problem i which he oil purchase decisios are sed uil ha he uceraiy is ufolded (WS) ad he soluio of he sochasic problem (recourse problem - RP). The lower he EVPI, he beer he sochasic models accommodae uceraiies as he sochasic objecive fucio value was o so far from he resul obaied by he WS soluio. So, he resul idicaes he beefi of icorraig uceraiy i he differe model parameers of he oil chai. However, sice acquirig perfec iformaio is o viable, he value of he sochasic soluio (VSS) ca be cosidered as a more realisic resul. The VSS resul, 98.95 millio $ (13.76% of he WS soluio), shows ha he sochasic model provided a good soluio as a expressive gai was obaied by he iclusio of uceraiy io he problem. 5. Coclusios The purse of his paper was o diuss he problem of spaial iegraio ad coordiaio i he acical plaig level of he oil chai. A sochasic mahemaical programmig model was developed o improve he acical plaig of oil refieries cosiderig uceraiies i demad ad prices. The model was applied o a acual refiig sysem i Brazil. The values obaied for he Expeced Value of Perfec Iformaio (EVPI) ad he Value of he Sochasic Soluio (VSS) 1.55% ad 13.76% of he wai-ad-see soluio respecively idicaed he beefi of icorraig uceraiy i he domia radom parameers of he acical plaig. The opimizaio resuls are suiable o he real plaig aciviies of he oil chai. The ceralized coordiaio (spaial iegraed) ad he shared feedsock oil supply provided a beer uilizaio of he available resources i meeig a give marke demad. Ackowledgemes The auhors would like o hak he Brazilia Federal Agecy for Supr ad Evaluaio of Graduae Educaio (CAPES). Refereces Al-Qahai, K. ad Elkamel, A. (2008), Mulisie faciliy ework iegraio desig ad coordiaio: A applicaio o he refiig idusry, Compuers ad Chemical Egieerig, 32, 2189 202. Al-Ohma W.B.E, Lababidi, H.M.S, Alaiqi, I.M., ad Al-Shayji, K. (2008), Supply chai opimizaio of peroleum orgaizaio uder uceraiy i marke demads ad prices, Europea Joural of Operaioal Research, 189, 3, 822 840. Bechel, PIMS (Process Idusry Modelig Sysem) User s maual. Versio 6.0. Houso TX: Bechel Corp., Houso 1993. Be-Tal, A. ad Nemirovski, A. (2000), Robus soluios of liear programmig problems coamiaed wih ucerai daa, Mahemaical Programmig, 88, 411 424. Birge, J. ad Louveaux, F., Iroducio o Sochasic Programmig, Spriger-Verlag, New York, 1997. Boer ad Moore, RPMS (Refiery ad Perochemical Modelig Sysem): a sysem deripio Boer ad Moore Maageme Sciece, Houso NY, 1979. Cacho G., Supply chai coordiaio wih coracs, I Graves, S., T. de Kok (Eds.) Hadbooks i Operaios Research ad Maageme Sciece, 11: Supply Chai Maageme: Desig Coordiaio ad Operaio Norh-Hollad, 2002. Chopra, S. ad Meidl, P., Supply chai maageme: sraegy, plaig, ad operaios (2d ed.), Pearso Educaio New Jersey, 2004. Dazig, G. (1955), Liear Programmig Uder Uceraiy, Maageme Sciece, 50, 12 Suppleme 1764-1769.
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