Oil Refinery Maintenance as a Strategic Decision

Transcription

1 Ol Renery Mantenance as a Strategc Decson by Anar Yusov Submtted to Central Euroean Unversty Deartment o Economcs In artal ulllment o the requrements or the degree o Master o Arts Suervsor: Proessor Andrzej Banak Budaest, Hungary 008

2 Abstract In ths thess I construct a model o nteracton o renery rms whch decde on ther mantenance and roducton levels. Both o these choces are consdered strategc varables n my model as they aect the robablty o alure o renery machnery. Usng game theory tools and smulatons, I rst analyze the cometton o reneres n statc one-shot game and denty ther strateges that lead to a unque equlbrum outcome. Further, to mrove the assumtons on mantenance and roducton choces, I model a smle -erod dynamc game whch s later extended wth an ntroducton o uncertanty. Dynamc model allows answerng the queston o whether t s benecal to take u mantenance at all. In each model smulatons are used to smly the dervatons and uncover the deendence o results on arameter values. The results o statc model suggest that reneres should roduce at a low level wth a less need or mantenance. Dynamc model suggests that t s better to sk the roducton n the rst erod and take u mantenance nstead. However, ater ntroducng uncertanty the results o dynamc model deend on beles o renery rms about uture demand.

3 Acknowledgments I would lke to thank my suervsor, Proessor Andrzej Banak, or hs valuable suggestons and comments durng my research. I am also grateul to Péter Smon Vargha and László Varró rom MOL Hungaran Ol & Gas Plc. or an oortunty o research nternsh. Fnally, I thank my arents or ther never-endng suort.

4 Table o Contents Introducton... Chater : Statc Model and One-shot Games Ol Renery Process n Detal Smulaton Method Sngle Frm Analyss Duooly Analyss Smultaneous Move Cometton Sequental Move Cometton Concludng Remarks... 5 Chater : Dynamc Model and Reeated Games Dynamc Programmng Sngle Frm Analyss Duooly Analyss Smultaneous Move Cometton Sequental Move Cometton Introducng Uncertanty Concludng Remarks... 7 Conclusons... 8 Aendx Reerences v

5 Introducton Generally crude ol can not be consumed as a uel or lubrcant n ts ntal orm. Changng ts structure to make t consumable s the mddle ste o the whole rocess. Ths sgncant oston s held by renery ndustres whch lnk crude ol extracton to sales o nal rened roducts n markets. The seccty o ths oston can be seen n ts many derences rom other arts o the ol sector. One o the most mortant o these derences s that renery rms have to answer more roducton questons n decson-makng and thereore have a larger scoe or otmzaton. Reneres are constantly allng under the ressure o market demands due to the act that the basc renng rocess allows the rms to extract only a xed amount o derent roducts rom a unt o nut (crude ol). So n order to satsy the market demand, rms are comelled to aly more comlex rocesses, whch demand costly nvestments. Ths ressure ncreases wth the need or requent mantenance works on renng machnery and equment. In the majorty o cases these knds o works ncur not only costs but also demand the stoage o roducton or the erod o mantenance. Wth growng demand on etroleum roducts, renng rms have a strong ncentve to roduce at maxmum roducton rates and caacty ossbltes. Renng acltes rocess several hundred thousand barrels o crude ol er day. Takng ths nto consderaton, one understands that the suly sde o the market and the rces o nal roducts sgncantly deend on caacty ossbltes, unexected renery outages and decsons on requency o mantenance works. It s clear that choosng the rght tme or stoage s a challenge or renery rms. For more detaled normaton about renng technologes see Manzano (005).

6 An mortance o mantenance decsons can also be seen rom the ersectve o comlexty o renery systems: the sohstcaton o the renng rocess has grown snce the begnnng o technologcal revoluton due to cometton among rms and the need or requent changes n the ractons o derent rened roducts to satsy the constantly changng demand. As a result, vulnerablty o renery system and machnery to breakdowns (even n small areas) has greatly ncreased and brought u new reasons or ncreasng requency and qualty o machnery mantenance. There were numerous exlosons and res at ol reneres n derent countres durng the ast decades, majorty o whch haened due to the alure n machnery system or leakages. For examle, Brtsh Petroleum ol renery exloson n Texas (March 005) led to teen deaths, more eole were wounded. Ths event led to huge losses or socety, envronment and or the renery rm tsel. The reorts sad that the exloson haened due to a breakdown n the system. Another examle s the re n Bombay Hgh ol eld n Inda (005), when our eole were klled and the latorm tsel was comletely destroyed n the re. The reason or ths event was a leakage n a nahtha elne 3. The mantenance decsons have become crucally mortant n reventng these knds o dsasters and n saer cases huge costs to the rm. In my thess I analyze decsons o cometng renery rms on quantty suled to the market and mantenance works n game theory context n order to answer the questons o how much eort t s otmal to ut n takng u mantenance and how to combne ths choce wth an otmal choce o roducton level. Mantenance s assumed to be a strategc decson or a rm as t drectly aects the robablty o machnery alure. Choce o mantenance wll be rst resented as a choce o ercentage o the whole renng machnery to be reared or renewed n a statc model, later resented as a bnary choce n a dynamc model. Makng reasonable assumtons about the URL: htt:// 3 URL: htt://news.bbc.co.uk//h/south_asa/ stm

7 behavor and choces avalable to rm, I wll rst model the roblem o one rm and ts actons assumng no cometton. Later, usng game theory I wll model and analyze the nteracton o two renery rms n artcular stuatons n order to denty ther strateges and resonses to the actons o rval rm. Numercal smulatons wll be used n all analyzed roblems both or smlcaton o dervatons and quanttatve evaluaton o results. Generally, known rom MOL exerence 4, renery rms do not otmze, but terate ther rots usng derent combnatons o strategc varables and choosng the ones yeldng the hghest value or the rots. Ths artcular way o maxmzaton can sometmes be useul; however teraton does not necessarly yeld maxmzng results and there s a scoe or otmzaton. Lttle research has been devoted to an otmal choce o mantenance and otmzaton o roducton shut down. Muehlegger (997) n hs doctoral thess models a choce o ol renery between outut and mantenance to nd out whether change n ownersh aects the robablty o outage. He addresses the roblem o unexected outages and ther ossble correlaton to the mmedate rce ncreases, whch can create an ncentve or amed (and sometmes unnecessary) shutdowns. Fudenberg and Trole (984) analyze smlar to Muehlegger`s (997) two-ste game wth caacty and entry decsons. A game o smultaneous and sequental choce o road mantenance and tollng decson or cometng road rms s analyzed n de Palma et al. (006). Even though there are major derences between road and renery sectors, ther model s able to denty otmal mantenance levels and together wth Muehlegger`s (997) model wll serve as a bass or a statc model analyzed n my thess. As a scence, game theory can be consdered young n comarson wth other branches o math and economcs. The man concet o game theory Nash Equlbrum has been ntroduced less than 60 years ago. Nevertheless t has already created a lot o derent branches n tsel 4 From a ersonal dscusson wth Mr. Györ János SCM Otmzaton Manager o MOL Hungaran Ol & Gas Comany. 3

8 (Evolutonary, Exermental Game Theory, etc.) and s wdely used n derent knds o researches. I chose game theory as a tool or ths research because t roved to be a owerul and convenent tool or analyzng and redctng actons o ndvduals and rms (as enttes controlled by ndvduals). Wth the hel o classcal game theory, I wll model the nteracton o ratonal selregardng renery rms, whch (accordng to ths theory) wll arrve at ther equlbrum choces a state whch aears as a result o best resonse o each rm to the actons o others. Ths thess s constructed as ollows: Chater develos a statc model o nteracton o renery rms wth an assumton about smultaneous choce o mantenance and roducton levels and dentes equlbrum strateges o renery rms; Chater reeats the same analyss or a dynamc model wth derent assumtons about the choces avalable to renery rms and later extends the model wth uncertanty about demand. 4

9 Chater : Statc Model and One-shot Games In the rst chater I concentrate on buldng a statc renery model ollowng Muehlegger (997) and de Palma et al. (006). I aly t to smultaneous and sequental one-shot games by ntroducng two cometng renery rms and secc demand uncton. Even though the analyss o a statc model has ts drawbacks (whch wll be dscussed later) t wll hel to understand the actons o renery rm under secc condtons. The chater begns wth the ntroducton to a more detaled renery rocess and comlcated choces whch the rms ace. It then contnues wth a ste-by-ste constructon o nteracton o renery rms.. Ol Renery Process n Detal The choce o the crude ol tye s the rst ste o renery rocess. The rce o crude ol ders manly accordng to ts densty (lght, medum or heavy), content o sulur (sweet or sour) and acds. Lght sweet crude s rocessed more easly and yelds a greater amount o lght roducts. Hence, ths tye o crude ol s more exensve due to an ncreased demand or lghter uels durng the last decades. Acd content s mortant n the sense that corroson o renery equment ncreases wth ts level 5. Intally the crude ol has to be searated by dstllaton nto derent arts, whch are later mroved wth derent renng technologes to match the qualty o market demand. Renery rms have to adjust the suly o derent tyes o roducts to seasonalty o ther consumton. Ths 5 See Manzano (005) or more detaled normaton. 5

10 adjustment s done ether through the choce o crude ol tye or technology (or both) subject to caacty constrants o the rm. Renng machnery and equment have become much more sohstcated snce the begnnngs o the renery ndustry. The reason or ths sohstcaton s technologcal nnovatons that ntroduced new ossbltes or rms to quckly adat to market demand and ncrease ecency. On the other hand, these new otons came wth greater vulnerablty o renery systems to breakdowns even n small areas. Such ncdences as leakage or machnery alure could lead to res and exlosons wth tragc consequences. Hence, the requent mantenance o renng machnery and equment works are necessary to avod the costs o alure. Mantenance decsons become strategc when a renery rm chooses the tme, requency, length and qualty o mantenance. Due to the changes n demand (esecally seasonally) and cometton these choces along wth the choce o outut determne the rots o the reneres. Otmzaton o mantenance s thus crucal or a survval o the rm. Mantenance works n realty could be taken u smultaneously wth roducton, however majorty o works demand stoage o roducton because o comlex nterconnecton o all arts o renng machnery. Wth statc model t s mossble to ut restrctons that would allow accountng or stoage o roducton. In ths sense ths model s less realstc than dynamc one; however t allows denng a unque equlbrum and greatly smles the analyss.. Smulaton Method Smulaton methods have become very oular n all branches o economcs. Even though they are used more oten n emrcal researches, here I use smulaton wth base values o arameters n order to smly burdensome dervatons and arrve at an equlbrum outcome. Smulatons are 6

11 erormed usng Wolram Mathematca sotware 6 and wth ts hel I analyze the model and varatons o equlbrum choces and rots wth all arameters. The ollowng numbers wll be used as base values or smulaton: K M F c g a b Table. Base values o arameters used or smulatons n statc model. Smulatons wth other values are erormed n Aendx ( ) or comarson. Caacty lmt and maxmum mantenance level or each rm are assumed to be equal durng smulatons; however n dervatons o otmal strateges and equlbrum, I use searate arameters or each rm n order to obtan a general soluton, whch s then smled wth assumed base values..3 Sngle Frm Analyss Even though renery rms roduce xed amount o heterogeneous outut rom one unt o crude ol deendng on technology and the tye o crude ol, here I wll assume that renery rms roduce sngle unt o homogeneous outut q rom a homogeneous nut o a unt o crude ol (that s, the technology rate s :) subject to caacty constrant K. These assumtons smly the analyss n the sense that renery rms cannot aect ther market suly and rots by choosng 6 Mathematca ver the roduct o Wolram Research, Inc. 7

12 derent tyes o crude ol or technologes. So the choces o renery rms are lmted to the choces o homogeneous outut and mantenance 7. The cost unctons or the two varables o man nterest are gven resectvely by: c( q) c q g( m) g m In my analyss I assume that caactes o reneres aect the choces o outut through the robablty o alure uncton and do not aect the cost uncton, thus allowng roducng wth constant margnal cost. The robablty o alure or a renery rm s dened as a uncton o q (quantty roduced) and m (qualty o mantenance) wth cost F ncurred n case o alure. It s reasonable to assume that q and m ndeendently aect (even though n realty ther eects can be somehow correlated), so that the robablty uncton s reresented by a lnear uncton o two ncreasng and convex robablty unctons: ( q, m) ( q) ( m) q (), () 0 ; (), () 0 q m q m m In cases o otmal choce dervaton a smle lnear robablty uncton ( q, m) q m wll be used nstead o the general orm, wth ; = K M, where K s the caacty lmt o renery rm and M s the maxmum ossble mantenance level 8. Wth ths constructon the eect o roducton level on robablty o alure reaches 00% as roducton level aroaches caacty 7 Relaxng these assumtons wll not change the outcome o the analyss. It s ossble to redene: q j q j, where j s the number o derent tyes o roducts. The result wll be the same snce the analyss does not nvolve ugradng technologes that would change the ractons o outut yelds. 8M can be thought o as a mantenance o the whole renng machnery and equment,.e. choosng mantenance level M/ means mantanng hal o the renery system. Durng smulatons M wll be assumed to be equal to 00 and thus the choce o mantenance wll be bascally a choce o ercentage o renng machnery to be renewed or reared. 8

13 lmt. It s obvous that () decreases wth K and ncreases wth M (the more comlex s the renng machnery, the more danger there s or a alure n one o ts arts). These assumtons also assure that robablty o alure stays n the zero-one nterval wth an addtonal condton that q K m (whch s satsed or all ollowng results). M Assumng there are N rms oeratng on the market, the rce s determned rom total outut suled by all rms: (Q) and the roblem o the rm s: max [ ( Q) q c( q )] ( ( q, m )) F( q, m) g( m) q, m subject to: q K, m M and Q= q j j N () Otmal smultaneous choce o outut and qualty o mantenance can be ound rom the rst order condtons or the roblem: q [ q( Q) ( Q) c( q)] ( ( q, m)) drect eect (ncrease n due to an ncrease n outut roduced) ( q, m) ( q, m) [ Qq ( ) cq ( )] F 0 q q ndrect eect (decrease n due to an ncrease n robablty o alure) () ( q, m) ( q, m) [ ( Q) q c( q)] F g( m) 0 m m m ths eect s ostve due to decrease n robablty o alure (3) In ths art o analyss demand wll be treated as xed at hgh enough level (so that whatever s roduced by the rm s sold), hence the renery s a rce-taker. Usng assumed unctonal orms o roducton and mantenance costs and robablty o alure uncton, one can nd an otmal 9

14 choce o outut and mantenance or renery rm as unctons o xed rce and costs. Grah G8 n Aendx shows how rot uncton changes wth man varables o choce. Solvng rst order condtons smultaneously otmal choces are dened as (see A n Aendx): q ; m K ( c)( F gm ) FgM M ( K ( c) F) ( c)(4 gm K ( c)) 4 gm K ( c) An ncrease n caacty leads to an ncrease both n otmal outut and mantenance levels. Ater derentatng both results wth resect to M t s obvous that both q and m decrease wth maxmum ossble mantenance level. Usng reasonable assumton about an ncrease n otmal choce o mantenance wth an ncrease n alure (and mled necessary condtons or t), the necessary condton or q to decrease wth F s: gm K ( c)) 9 (see A n Aendx)..4 Duooly Analyss Ths art o the Chater deals wth the nteracton o two ndeendent cometng rms n a world o erect normaton. Renery rms choose quanttes that all-together dene the nal rce; hence the cometton s a la Cournot. Prce s determned accordng to the ollowng unctonal b b orm: ( q, q) a, whch relects the weak resonse o rce to suly changes q q quantty suled s large enough and strong resonse suly s on low level. Ths uncton has a constant elastcty coecent o mnus one and s dected n Grah. 9 Ths condton holds or assumed base values o arameters unless xed rce s ncredbly hgh. 0

15 Grah. Inversed demand uncton (rce uncton) Cometton o reneres wll be constructed rst under smultaneous choce o outut and mantenance by rms, ollowed by sequental move game..4. Smultaneous Move Cometton Under ths constructon o the roblem renery rms smultaneously choose ther outut and mantenance levels. The equlbrum outcome can be ound by maxmzng the rot uncton () wth resect to q and m. The resultng rst order condtons are gven n () and (3) assumng that N. Each rm s otmal choce wll deend on the choce made by a cometng rm. It can be seen rom () that otmal outut or rm s dened as a reacton uncton Solvng (3) or otmal mantenance level one obtans reacton uncton q ( q ( m ), m ). m ( q, q ( m )) or mantenance. Equlbrum choces are derved by combnng both rms` reacton unctons and solvng equatons or choce varables 0. 0 The choces o renery rms n equlbrum are exected to be symmetrcal because they are assumed to be dentcal,.e. have equal robablty o alure unctons, costs etc.

16 Wth assumed unctonal orms o robablty and cost unctons, the roblem o rm becomes: b b q m q m max [( ) ] ( )) ( ) q, m a q cq F gm q q K M K M subject to: q K and m M (4) Ater solvng an equvalent roblem or reacton unctons o rm analyss o smultaneous choce game comes down to our equatons wth our unknowns (see A n Aendx). The outut and mantenance reacton unctons o rm are resectvely: q b( K ( M m ) M q ) q ( FM K ( ac)( M m )) ( q, m) M( bq( ac)) (5) m q ( F q ( a c) b) bq (6) ( q, q) gmq ( q Smultaneous soluton o (5) and (6) leads to a reacton uncton q ) (see A n Aendx). Usng assumed values o arameters rom Table the roducts o renery rms are seen to be strategc substtutes rom the ollowng grah: Grah. Reacton curves determnng equlbrum.

17 Reacton curves o renery rms ntersect at equlbrum ont E (at ths ont q q 386 and m m 6.37 (%) ). Mantenance reacton curve o rm s dected n Grah G9 n Aendx. The resultng choce o roducton and mantenance levels states that rms n equlbrum wll reer to roduce only at 40% o ther caacty ossbltes. Ths choce has a logcal exlanaton: by dong so rms both decrease ther need or more ntensve mantenance (only about 7% o the total mantenance ossble) and ncrease the rce. These results are suorted by addtonal smulatons erormed n Aendx, excet or smulaton o Table 6 n whch reneres choose roducton level at more than 50% o ther caacty lmt and mantan 5% o the whole system. Grahs G4-G9 n Aendx dect the varaton o equlbrum outut choce wth all arameters o the model: or ths urose exlanatory arameter vares n some range around ts base value, whle all other arameters are held constant at ther base values. The resultng curves are retty ntutve: t turns out that equlbrum choce o outut ncreases only wth an ncrease n caacty and arameters o rce uncton. Varaton o equlbrum rots wth outut and mantenance choces are shown resectvely n Grahs G30 and G3 n Aendx. These curves show the varaton wth one strategc varable whle holdng another constant (hence the maxmum onts n these grahs are not equlbrum values)..4. Sequental Move Cometton It s ossble that n a regonal market rms are not equal n tmng o takng actons and hence make ther choces one ater another. Wth sequental move settng, the cometton among rms Comarng to Table, Table 6 n Aendx assumes hgher caacty and rce, lower alure and roducton costs. Ths exlans the choce o hgher roducton level by reneres. It can be seen rom Grah G9 n Aendx that the eect o an ncrease n a s much larger n magntude than that o b. Ths s exlaned by seccty o rce uncton and ts elastcty coecent. 3

18 conssts o stages: n the rst stage rst-movng renery rm chooses ts otmal mantenance and roducton levels; ater observng the choce o ts rval, second-movng rm decdes on ts roducton and mantenance levels n the second stage. Equlbrum outcome s ound as usually by backward nducton. Assumng that renery rm moves rst, dervaton o otmal solutons or both rms starts wth analyss o second stage (rm actons). As beore, maxmzng rots wth resect to q and m and solvng rst order condtons reacton uncton q ( q ) s obtaned or rm (same as A n Aendx, but or q ). Gong back to rst stage renery rm maxmzes: b b q m q m max [( ) ] ( )) ( ) q, m a q cq F gm q q( q) K M K M subject to: q K and m M At ths stage dervatons are becomng laborous and there s a need or smulaton. Usng base values o arameters equlbrum outut and mantenance are obtaned or each rm: q 387.3, q , m 6.39 (%), m 6.37 (%). As was exected, rst-movng renery rm roduces more outut, but has a need or more mantenance. Grahs G0-G5 n Aendx show the deendence o otmal outut choces on arameter values. The derence between two rms choces decreases as caacty and mantenance cost ncrease. Prots o cometng rms are dected n Grahs G6-G n Aendx. As mantenance cost ncreases, rot o rst-movng renery rm s aroachng that o second movng rm. Ths s exlaned by the choce o rst-movng rm greater level o roducton demands greater level o mantenance and ncurs hgher mantenance costs. values. Addtonal smulatons n Aendx ( ) show the same results or derent arameter 4

19 .5 Concludng Remarks In ths Chater nteracton o renery rms has been modeled as a statc game, n whch mantenance works and roducton have been assumed to be taken u smultaneously. Ths assumton can be consdered as an unrealstc one or majorty o mantenance works whch demand stoage o roducton. Nevertheless, the model has allowed dentyng unque strateges o renery rms that dene Nash equlbrum. Both smultaneous and sequental move comettons have showed smlar results n smulatons that used base values o arameters gven n Table : t has turned out that t s better or each renery rm to roduce at about 40% o ts caacty lmt. Ths strategy has two benets low roducton (suly) mles hgher rce and decreases the need or more ntensve mantenance. Sequental move game has showed that rm chooses to roduce slghtly more than n smultaneous case, but stll stcks to the otmal strategy o roducng less than hal o caacty ossbltes. Next Chater develos ths model urther to a dynamc case whch allows makng more strct assumtons about mantenance and constructng nteracton o renery rms n a more realstc model. 5

20 Chater : Dynamc Model and Reeated Games In ths chater I develo revous model by ntroducng tme erods. Such an extenson has ts benets as well as drawbacks. The man advantage o ths addton les n the ossblty o makng more realstc assumtons. The renery model o revous chater has assumed that mantenance and roducton can be taken u at the same tme. Wth the ntroducton o dynamc model t s easble to make a reasonable assumton about mantenance beng a tme consumng event,.e. renery rms need to ut the roducton o lne n order to take u mantenance. In ths context t wll be ossble to answer the queston o whether t s otmal to take u the mantenance and sto the roducton or to contnue roducng no matter what. Dynamc model has ts dsadvantage n comarson wth statc model n ndng equlbrum. Accordng to olk theorem, reeated games have a roblem o multle equlbra 3. However the model wll be extended to two erods only and equlbrum wll stll be dened as a unque one n ths context. Hence the model allows analyzng the nteracton o cometng reneres n a more realstc constructon.. Dynamc Programmng Dynamc rogrammng s a method that allows solvng a roblem wth overlang subroblems,.e. a roblem that can be dvded nto searate arts and otmzed ste-by-ste. Here ths roblem s dened as maxmzng total rots o renery rms. A smle nte horzon model 3 Folk Theorem states that sucent condton or an outcome to be equlbrum n a reeated game s to satsy mnmax condtons,.e. the choce mnmzes maxmum ossble loss or a layer. 6

21 wth erods wthout dscountng s analyzed. Prots o renery rms are assumed to be tme searable. Snce one o the choce varables s assumed to be bnary, t s mossble to use usual dynamc rogrammng methods or soluton and a smler aroach o comarson o the outcomes wll be used. For smulaton uroses the ollowng numbers wll be used as base values: K F c G a b Table. Base values o arameters used or smulatons n dynamc model. Other assumed values o arameters are used or addtonal smulatons n Aendx (. 39).. Sngle Frm Analyss In ths art o the Chater demand s treated as xed n order to determne the actons o renery rm as a rce taker. Dynamc rogrammng allows uttng restrctons on mantenance and roducton choces: mantenance decson wll be treated as a bnary choce varable wth a xed cost; mantenance n some erod s taken u, the roducton stos; renery decdes to roduce n some erod t cannot take u mantenance. Bascally the rm aces the roblem o choosng between two otons ether to roduce or take u mantenance. The cost o mantenance s xed and equal to G. Frm starts makng ts choce n rst erod wth ntal P (alure). I n some erod mantenance has been taken u, rots rom roducton n that erod are equal to zero and robablty o alure decreases by some constant amount. Assumng that rm takes actons or two erods wthout dscountng, t has two choces: ether to take u mantenance n the rst erod and roduce n the second or to roduce n 7

22 both erods 4. Snce mantenance decson s treated as a bnary varable and uts restrctons on roducton, t would be mossble to analyze the choce o renery rm by constructng a sngle rot uncton or both erods. Thus, searate rot unctons wll be analyzed and comared: m max{ } (7) where suerscrts denote erods; m denotes rot o renery rm t takes u mantenance: m G and P (alure) ; denotes rot o renery rm under roducton: q ( c) ( P(alure)) F P(alure) wth q P( alure). K I renery rm decdes to roduce n both erods, then ts roblem s: q q max{ } q ( c)( ) F( ) q, q K K q q q q q ( c)( ) F( ) K K K K The soluton s ound by rst maxmzng rots wth resect to uncton o q and dervng otmal q as a q. Ater luggng t back nto the rot uncton and maxmzng wth resect to q maxmzed rot and otmal choces o renery rm are ound (see A3 n Aendx): q K( ) F 3 ( c) q ) ( K 3 F K( K( ) ( c) 3 c) F( ) 4 Snce T= t would be meanngless to analyze cases when renery rm takes u mantenance n the second erod, because mantenance n the last erod aects nothng but stoage o roducton n that erod. 8

23 9 In case the renery decdes to take u mantenance n the rst erod the roblem becomes: ) ( ) )( ( } max{ K q F K q c q G m q Maxmzng outut and otmal rot are obtaned by solvng rst order condtons or q o ths roblem (see A4 n Aendx): ) ( ) ( c F K q G F c K F c K m ) ( ) ( 4 4 ) ( ) ( It would be logcal to conclude that the renery rm took u mantenance n the rst erod, t would roduce more n the second erod than t would wthout mantenance. Ths leads to the ollowng condton: ) ( 3 ) 3 ( c F K It can be seen that ths condton may al ntal robablty o success and are low enough 5. The roblem o renery rm now comes to comarson o rots under two derent schemes. It s obvous that the choce o renery rm between two schemes o actons n (7) manly deends on ntal robablty o alure, and cost o mantenance G. Ater luggng base values rom Table n both cases, t turns out that renery rm enjoys hgher rots ater undertakng mantenance and roducng more n second erod ( m, 0 ). Parameter values rom Tables 7,8,9 suort ths result, but 5 Ths condton s satsed or base values o arameters rom Table.

24 usng values rom Table 0 changes the outcome n avor o roducng n both erods m ( , ). Parameter values rom Table 0 decrease the need or mantenance by assumng: lower alure cost and hgher mantenance cost; lower cost o roducton and hgher caacty and rce. Grahs G-G7 n Aendx show how overall equlbrum rots under both schemes change wth all arameters. It s mortant to notce n G5 that under these base values rot o renery rm when roducng n both erods decreases to zero as ntal robablty o alure aroaches the threshold value o about 33%. Grah G3 shows that equlbrum rots are hgher under roducton n both erods when alure costs are low enough and the benets o takng u mantenance decrease..3 Duooly Analyss Demand uncton s assumed to be gven n the same orm as n revous Chater. In Duooly case each rm has a choce n orm o (7). I one rm decdes to take u mantenance n the rst erod, the other rm gets all the resdual demand. I wll analyze nteracton o renery rms agan under smultaneous and sequental move games, later ntroducng uncertanty..3. Smultaneous Move Cometton Usng assumed unctonal orms or costs and robablty uncton, each rm maxmzes sum o ts tme-searable rots. As beore, each renery decdes whether to take u mantenance n the rst erod or to sk t. In case o duooly ths means that there are 4 ossble states n whch rms 0

25 may end u: both rms always roduce; both rms take u mantenance n the rst erod; one o the rms takes u mantenance n the rst erod, the other sules the market. Otmal solutons are obtaned by backward nducton or each case. Agan, snce mantenance s a bnary choce varable t s mossble to analyze actons o renery rms by constructng a sngle roducton uncton, and hence all cases should be comared searately usng smulatons. For the case when both reneres roduce n both erods, the roblem o rm (, ) s: q q q b b q max{ } ( )( ) ( ) q a c F q, q q K K b b q q q q q ( a c)( ) F( ) q K K K K (8) where subscrt denotes rval rm and suerscrts denote erods. The soluton s symmetrcal and s obtaned usng backward nducton - startng rom otmzaton o erod outut choce and gong back to erod. When both rms decde to take u mantenance, they comete only n second erod and the roblem o rm becomes: b b q q max{ } ( )( ) ( ) G q a c F (9) q q q K K As or the searatng case (one renery roduces n both erods, another takes u mantenance n the rst erod) the roducng renery receves all the resdual demand n the rst erod and the b rce s determned as: a. The soluton s agan derved by backward nducton. The q resultng rots and outut choces are resented n the ollowng x ayo matrx:

26 Frm Frm Produce-Produce Mantan-Produce Produce-Produce 37.3 q 3.5 q q q q 39. q q 0 q Mantan-Produce q 0 q q q q 0 q q 0 q where Produce-Produce stands or roducng n both erods and Mantan-Produce - or takng u mantenance n the rst erod. Ths game may seem smlar to Prsoner`s Dlemma, however t s obvous rom ths matrx that Mantan-Produce s a domnant strategy or both rms. Otmal choces o outut levels state the strateges o renery rms. It turns out that a rm decdes to roduce n both erods, t reers to kee the roducton on low level n the rst one to comete more harshly n the second. Much hgher level o roducton n the second erod s also exlaned by the lmtaton o model to two erods: the choce o outut n the last erod can ncrease the robablty o alure n that erod, but t does not aect the uture robablty o alure snce the game s termnated at that ont. However extenson o the model to T erods s redcted to yeld the same results: rms wll roduce on low levels untl the last erod and n erod T the roducton level wll greatly ncrease.

27 It s not surrsng that renery rms stll reer to ollow the strategy o low roducton and zero mantenance both rms choose to always roduce. However n case when one o the reneres decdes to take u mantenance, the other seems to be nsenstve to the resdual demand and stll chooses to roduce on low (but slghtly hgher) level because t antcates that the rval rm wll come back wth much stronger ossbltes to roduce n the second erod. Hence, renery rm reers to be reared or the cometton n the last erod by roducng less n the rst. In current model reneres seem to be very senstve to the robablty o alure and gve u addtonal rots rom resdual demand or saer uture roducton. It s also worth notcng that the consumers are always better o when reneres take u mantenance: the total suly s hgher and the rce s lower. Even though the best oton or the consumers s when one o the reneres s always roducng and another takes u mantenance 6, renery rms arrve at an equlbrum when both take u mantenance and enjoy hgher overall rots. Addtonal smulatons erormed n Aendx (.39) show that usng arameter values rom Tables 8 and 0 makes roducton n both erods a domnant strategy or both reneres. Ths change shows that n a dynamc model choces o reneres are senstve to market rce o roduct, cost o roducton and the amount o a decrease n robablty o alure ater mantenance works ( )..3. Sequental Move Cometton Under sequental move constructon equlbrum outcome does not change: rst-movng renery rm antcates that t wll not aect the choce o the second-movng rm, snce takng u 6 Ths case s better than the one when both rms choose to take u mantenance, because n the ormer the total suly s lower and the market s comelled to suly the consumers n the rst erod wth exensve morted roducts. 3

28 mantenance s a domnant strategy or both rms. Ths concluson s resultng rom the choce o secc values o arameters rom Table. It s ossble to readjust them n such a way that takng u mantenance s no longer a domnant strategy. Addtonal smulatons n Aendx (. 39) have showed that the equlbrum outcome changes to roducton n both erods ater assumng: hgher rce and mantenance cost; lower and roducton cost. The next subsecton ntroduces uncertanty and demonstrates how equlbrum deends on beles o renery rms about demand and shows an examle when roducng n both erods becomes a domnant strategy..4 Introducng Uncertanty In a world o erect normaton choces o renery rms are based on ther knowledge about costs and demand. Ths art o the Chater extends the revous model by ntroducng uncertanty about demand and hence rce o the roduct. Frms are assumed to have dentcal beles about the uture rce: the demand wll be low wth robablty n the rst erod and wth robablty n the second. Thereore the roblems o reneres are unchanged wth the only derence n ther beles about rce. Usng robabltes that renery rms ut on low demand, rces n rst and second erod resectvely are exected to be: ( bl bl bh bh al ) ( ) ( a ) H q q q q ( bl bl bh bh al ) ( ) ( a ) H q q q q 4

29 where subscrts L and H stand or Low and Hgh resectvely and: al ah, bl bh. The results obtaned wthout uncertanty do not change : that s, renery rm beleves that demand wll be hgh or low wth the same robablty n each erod, t wll stll stck to ts revous strategy o takng u mantenance. The ollowng grah shows how maxmzed rots change wth beles (robabltes) about demand or a sngle rce-takng rm case when L ( ) H and L ( ) H : Grah 3. Otmal rots changng wth beles (robabltes o low demand) It can be seen rom Grah 3 that roducng n both erods can be otmal or a renery rm demand s lkely to be hgh n the rst erod and low n the second ( 0. 5 and 0.5). It s also obvous that takng u mantenance s always better when,.e. when demand n the rst erod s exected to be low wth a robablty at least as great as robablty o low demand n the second erod. 5

30 Alcaton o uncertanty to nteracton o renery rms can show whether t wll sgncantly aect ther choces o outut and rots. The general dervaton o rots as unctons o beles s laborous and unntutve. I use smulaton wth base values rom Table and some addtonal values stated later to smly the analyss. The ollowng examle s an alcaton o uncertanty to smultaneous move nteracton o renery rms wth the ollowng addtonal assumtons on the values o arameters: a 0, b 500, a 0, b 00, 0, The assumtons on robabltes state that H L L demand s dentely gong to be hgh n the rst erod, but n the second erod there s a 50% robablty o a low demand. The resultng ayo matrx s: H Frm Frm Produce-Produce Mantan-Produce Produce-Produce q q q q q q q 0 q Mantan-Produce 7.60 q 0 q q q q 0 q q 0 q It s obvous that n ths case roducng n both erods s a domnant strategy or each rm. Under ths constructon equlbrum outcome s also the best or consumers whch enjoy hghest total suly and lowest average rce. 6

31 .5 Concludng Remarks Ths Chater has develoed a smle but useul dynamc model o nteracton o renery rms. Dynamc aroach has allowed me to analyze the actons o renery rms under more realstc assumtons about mantenance decsons: roducton and mantenance works have been assumed to be exclusve decson varables. The results derved wth the hel o smulatons have dented the otmal strateges o renery rms. Constructon o x ayo matrx has showed that takng u mantenance s a domnant strategy under assumed values o arameters. In cases when renery decdes to roduce n both erods, t should stck to the strategy derved n the revous Chater: roduce less and enjoy hgher rces wth lower robablty o alure. The model has also demonstrated nsenstvty o renery rms to resdual demand and hgh senstvty to robablty o alure. The surrsng derence between otmal roducton choce n the rst and second erods comes rom the act that the choce game s termnated ater roducton n the second erod. Extenson o the model to more erods should not change the general result o ths outcome, because the rms wll stll roduce at an otmal low level untl the last erod and boost the roducton rght when the game termnates and robablty o alure s no longer aected. Later ntroducng uncertanty the model has showed that the equlbrum outcome deends on beles o renery rms about demand. The result has turned out retty ntutve: when demand s exected to be hgh n the rst erod and low n the second, t s otmal to roduce n both erods. Ths change o equlbrum rom mantenance to roducton s also welcomed by customers because they enjoy hgher total suly and lower average rce. 7

32 Conclusons Harsh cometton and a constantly changng demand or ol requre reneres to roduce at ull caacty ossbltes and create bad ncentves to sk roducton stoage or mantenance works. Ths, n turn, can lead to a system alure, whch creates a danger o exlosons (res) and ncurs huge costs o restoraton o renng machnery. In my thess, I have analyzed the choces o roducton and mantenance levels chosen by renery rms. Besdes denng rots, roducton and mantenance were assumed to be strategc decsons whch aect the robablty o alure. Usng classcal game theory tools and smulatons o arameters, I have dented the otmal strateges or renery rms n statc and dynamc models under derent stuatons: sngle renery rm beng a rce taker; cometton o two renery rms that make ther choces o strategc varables` levels smultaneously; cometton o two renery rms decdng on strategc varables sequentally, one ater another. Statc model n Chater has allowed otmzng reneres` otmal (smultaneous) choces o outut and mantenance levels n a convenent and smle way. In ths model, the result o nteracton o the renery rms under derent constructons (smultaneous and sequental moves) has turned out to be surrsng but ntutve: n all cases, reneres have been nvolved n a knd o tact colluson by sulyng the regonal market wth consderably less quantty o rened roducts than ossble. By dong so, renery rms have been enjoyng hgher rces and have decreased ther need or mantenance (less than 0% o total renery system demands mantenance n ths equlbrum). The advantage o the statc model les n the smlcty and the ossblty o denng a unque equlbrum. However, ths model makes t mossble to ut realstc restrctons on the choces o mantenance and roducton. The majorty o mantenance works n ol renery sector 8

33 demands roducton stoage or the erod o mantenance, manly because o comlex nterconnecton o all arts o the renery system. Introducton o dynamc model n Chater has allowed or necessary restrctons: mantenance has been assumed to be a bnary varable wth xed costs o mlementaton and xed racton o a decrease n robablty o alure; roducton and mantenance have been exclusve events. Deste ths advantage, dynamc models and reeated games come wth ther roblem o multle equlbra. Intutvely, ths means that n an nnte horzon model, both rms can end u n any o the stuatons when they are both better o. Ths dsadvantage, however, has been overcome by the ntroducton o nte horzon ( erods) model. The results o the dynamc model have showed that reneres are senstve to any ossbltes o alure and both o the cometng reneres end u takng u mantenance n the rst erod. In choosng roducton levels, rms stll ollow the otmal strategy o statc model: roducng less and enjoyng saety wth hgher rces. Later, extendng the model wth uncertanty, I have showed that decsons o reneres can swtch to roducton n both erods deendng on ther beles about demand. Both models resented n my thess are smle but normatve. Further extensons can hel to understand the nteracton and choces o ol reneres. Among ossble ones, I would suggest: analyzng n statc model the nteracton o non-dentcal reneres that der n ther caactes, market ower etc.; extendng dynamc model to more erods (nte) or nnte horzon and studyng more n-deth decsons o reneres on otmal requency o mantenance; testng both dynamc and statc models emrcally on whether the equlbrum outcome s suorted by data. 9

34 Aendx Statc Model A. Sngle Frm Choce wth Fxed Demand Usng assumed unctonal orms o robablty and cost unctons, rm maxmzes ts rot wth resect to outut and mantenance: q m q ( c) F ( ) [ c] 0 q K M K K ( c) q F gm 0 m M (A) (A) Solvng (A) and (A) or q and m resectvely brngs the roblem to: K( m M) F q M ( c) (3A) ( c) q F m (4A) gm Prot maxmzng outut and mantenance choces are ound n terms o arameters by solvng smultaneously equatons (3A) and (4A). Deendence o otmal mantenance and outut choces on derent arameters s ound by derentaton: (4 )( ) (4 gm K( c)) K m M gm F c m 0 K (4 ) (4 gm K( c)) K q gm gm F q 0 K (4 ( ))( ( )) (4 gm K( c)) M m gm K c F K c m 0 M 30

35 4 ( ( )) (4 gm K( c)) M q gk M FK c q 0 M 4 gm K( c) F m M m 0 4 gm ( ) K c F (5A) ( )) ( c)(4 gm K( c)) M q gm K c q 0 gm K( c)) F Last condton s necessary or the otmal outut to decrease wth F and s sucent or otmal mantenance choce to ncrease wth F. A. Smultaneous Choce Game n Duooly Frst order condtons or the roblem are: b b q[ ac ] q m b q q F ( ) [ ac ] 0 q K M q K K (7A) m b b q[ ac F] q q gm M 0 (8A) The rst one leads to the outut reacton uncton o rm : q b( K ( M m ) M q ) q ( FM K ( ac)( M m )) ( q, m) M( bq( ac)) FOC or mantenance level denes otmal choce o mantenance as: m ( q, q) gmq (9A) q ( F q ( a c) b) bq (0A) Reacton uncton o outut: q ( q ) s obtaned by solvng (9A) and (0A) smultaneously or otmal choce 3

36 q ( q) ( bq( ac))( q(4 gm K( ac)) bk) q ( b K q ( K ( a c)( F gm ) FgM ) b( FK gm ( K q ) K q ( a c))) (A) Equlbrum outcome s ound by solvng the same roblem or rm n a smlar way and combnng the reacton unctons or outut o both rms. The soluton s burdensome and has no ntutve value wthout usng smulatons. Dynamc Model A3. Sngle Frm Choce wth Fxed Demand When the rm decdes to roduce n both erods, the soluton s ound by backward nducton. Maxmzng rot uncton wth resect to q, the rm obtans: ( q ) q ( c)( q K ) F K 0 q K( ) q F ( c) Pluggng t back to rot uncton and maxmzng wth resect to q : ( q ) [( c)( K( ) 3q K ) 3F] 0 q K( ) F 3 ( c) Maxmzed rot s obtaned by solvng or otmal uncton. q usng q and nally - usng both n rot For the case when a rm decdes to take u mantenance rst, otmal outut choce n the second erod s ound rom FOC: ( m q ) q ( c)( K ) F K The nal result s obtaned by luggng otmal 0 q K( ) F ( c) q back nto rot uncton. 3

37 Statc Model Smultaneous Moves G4. Varaton o equlbrum outut wth caacty G5. Varaton o equlbrum outut wth maxmum mantenance level G6. Varaton o equlbrum outut wth alure cost G7. Varaton o equlbrum outut wth mantenance Cost G8. Varaton o equlbrum outut wth roducton cost G9. Varaton o equlbrum outut wth rce uncton 33

38 Statc Model - Sequental Moves G0. Varaton o equlbrum oututs wth caacty G. Varaton o equlbrum oututs wth maxmum mantenance level G. Varaton o equlbrum oututs wth alure cost G3. Varaton o equlbrum oututs wth mantenance cost G4. Varaton o equlbrum oututs wth roducton cost G5. Varaton o equlbrum oututs wth rce uncton 34

39 Statc Model - Sequental Moves G6. Varaton o equlbrum rots wth caacty G7. Varaton o equlbrum rots wth maxmum mantenance level G8. Varaton o equlbrum rots wth alure cost G9. Varaton o equlbrum rots wth mantenance cost G0. Varaton o equlbrum rots wth roducton cost G. Varaton o equlbrum rots wth rce uncton 35

40 Dynamc Model Sngle Frm G. Varaton o equlbrum rots wth caacty G3. Varaton o equlbrum rots wth alure cost G4. Varaton o equlbrum rots wth roducton costs G5. Varaton o equlbrum rots wth ntal robablty o alure G6. Varaton o equlbrum rots wth a decrease n G7. Varaton o equlbrum rots wth xed rce robablty o alure due to mantenance 36

41 Mscellaneous G8. Varaton o rot wth outut and mantenance G9. Mantenance reacton curve n smultaneous move game n sngle rm case G30. Varaton o equlbrum rot wth outut choce G3. Varaton o equlbrum rot wth mantenance n smultaneous move game choce n smultaneous move game Addtonal Smulatons Statc Model K M F c g a b Table 3 37

42 K M F c g a b Table 4 K M F c g a b Table 5 K M F c g a b Table 6 Tye o Game Smultaneous Moves Sequental Moves Frm Values Table 3 Table 4 Table 5 Table 6 Frms & Frst-mover (Frm ) Second-mover (Frm ) q q 08.7 m m 4.58 (%) q q 9.69 m m 5.79 (%) q q m m 8.47 (%) 46.6 q q 80.7 m m 4. (%) q.98 m 4.74 (%) 45. q 9.94 m 5.8 (%) 987. q m 8.47 (%) 46.6 q m 4. (%) q m 4.50 (%) 49.8 q 9.6 m 5.79 (%) q m 8.46 (%) 46.3 q 80.7 m 4. (%)

43 Addtonal Smulatons Dynamc Model K F c G a b Table 7 K F c G a b Table 8 K F c G a b Table 9 K F c G a b Table 0 Values Choce F Produce-Produce F Produce-Produce F Mantan-Produce F Produce-Produce F Mantan-Produce F Mantan-Produce Table Table Table ; ; ; Table ;

44 Reerences De Palma, A., Klan, M., and Lndsey, R. (006) Mantenance, servce qualty and congeston rcng wth cometng roads, workng aer: URL: htt:// ; Fudenberg, D. and Trole, J. (984) The at cat eect, the uy-dog loy and the lean and hungry look, Amercan Economc Revew: Paers and Proceedngs; Manzano, F. (005) Suly Chan Practces n the Petroleum Downstream, Masters Thess submtted to the deartment o Engneerng Systems Dvson o Massachusetts Insttute o Technology; Muehlegger, E. J. (005) Essays on gasolne rce skes, envronmental regulaton o gasolne content, and ncentves or renery oeraton, Doctoral Thess submtted to the deartment o Economcs o B.A. Wllams College; Welln, P (008) Comuter Smulatons Wth Mathematca And Java book, ublsher: Srnger-Verlag New York Inc. BBC News: Four de n Inda renery re, last vew date 03/06/08 URL: htt://news.bbc.co.uk//h/south_asa/ stm Poular Mechancs: What Went Wrong: Ol Renery Dsaster, last vew date 03/06/08 URL: htt:// 40