Computational Strategies for Non-convex Multistage MINLP Models with Decision-Dependent Uncertainty and Gradual Uncertainty Resolution

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1 Manuscrip Click here o download Manuscrip: Paper.ps Click here o view linked References Compuaional Sraegies for Non-convex Mulisage MINLP Models wih Decision-Dependen Uncerainy and Gradual Uncerainy Resoluion Bora Tarhan, Ignacio E. Grossmann, Deparmen of Chemical Engineering, Carnegie Mellon Universiy Pisburgh, PA Vikas Goel ExxonMobil Upsream Research Company, Houson, TX 0 Absrac In many planning problems under uncerainy he uncerainies are decision-dependen and resolve gradually depending on he decisions made. In his paper, we address a generic non-convex MINLP model for such planning problems where he uncerain parameers are assumed o follow discree disribuions and he decisions are made on a discree ime horizon. In order o accoun for he decision-dependen uncerainies and gradual uncerainy resoluion, we propose a mulisage sochasic programming model in which he non-anicipaiviy consrains in he model are no prespecified bu change as a funcion of he decisions made. Furhermore, planning problems consis of several scenario subproblems where each subproblem is modeled as a nonconvex mixed-ineger nonlinear program. We propose an efficien soluion sraegy ha combines global opimizaion and ouer-approximaion in order o opimize he planning decisions. We apply his generic problem srucure and he proposed soluion algorihm o several planning problems o illusrae he efficiency of he proposed mehod. Keywords: decision making under uncerainy, decision dependen uncerainy, gradual uncerainy resoluion, mulisage sochasic programming, non-convex mixed ineger nonlinear program, global opimizaion, ouer-approximaion, oil or gas field exploraion, synhesis of process neworks 1 Inroducion Sochasic programming is a mahemaical programming framework for modeling opimizaion problems ha involve uncerainy in he daa which are represened by probabiliy disribuions. There are wo broad barhan@andrew.cmu.edu To whom all correspondence should be addressed. grossmann@cmu.edu vikas.goel@exxonmobil.com 1

2 classes of sochasic programming problems: chance consrained There are wo broad classes of sochasic programming problems: chance consrained (Charnes and Cooper, ) and problems wih recourse (Birge and Louveaux, ). In his work we focus on mulisage sochasic programming wih recourse and for he sake of simpliciy, we will refer o i hroughou he paper shorly as sochasic program. The fundamenal idea behind sochasic programming is he concep of recourse which is he abiliy of he decision-maker o ake correcive acion afer a random even has aken place over a sequence of sages. In his paper he uncerain parameers are assumed o follow discree probabiliy disribuions and he planning horizon consiss of a fixed number of decision poins. Using hese wo assumpions, he sochasic process can be represened using scenario rees. Figure 1 is a sandard represenaion of a scenario ree having one uncerain parameer (ξ ) wih wo discree values in wo ime periods, which leads o four scenarios. Uncerain parameers ξ 1 and ξ reveal a he end of firs and second ime periods respecively. 1 = 1 = 1 =1 = 1 = = = = Scenario 1 Scenario Scenario Scenario Figure 1: Scenario ree wih uncerain parameers ξ 1 and ξ. In a sandard scenario ree, each node for ime period represens a possible sae of he sysem a ha ime period. Each arc represens he possible ransiion from one sae in ime period o anoher sae in ime period + 1. A pah from he roo node o a leaf node represens a scenario. Thus, a scenario is a combinaion of possible uncerain parameers in each of he ime periods which in urn define a sage. Figure is an alernaive represenaion of he scenario ree in Figure 1, proposed by Ruszczyński (). In his represenaion each scenario is denoed by a se of unique nodes. The horizonal lines connecing nodes in ime period indicae ha hese nodes have he same informaion in ha ime period, and herefore are indisinguishable. These non-anicipaiviy consrains sugges ha decisions canno be based on knowledge ha will be revealed in he fuure. The horizonal lines reduce he ree in Figure o he one shown in Figure 1. For modeling he problem, scenario rees presened in Figure will be considered in order o explicily handle he non-anicipaiviy consrains and incorporae decision dependen uncerainies (Tarhan and Grossmann, 00). (SP) is a sandard linear sochasic program wih T ime periods and S scenarios, modeled using he alernaive scenario ree proposed by Ruszczyński (). The parameers p s represen he probabiliy of scenario s, while he variables x s represen variables for ime period in scenario s. Eq. (1.1) corresponds o he objecive of maximizing he expecaion of an economic crierion (e.g. ne presen value). Eq. (1.) =

3 = = 1 1 = = 1 = 1 = 1 = 1 = Scenario 1 Scenario Scenario Scenario Figure : Alernaive scenario ree wih uncerain parameers ξ 1 and ξ. represens he muli-period consrains for each scenario s. consrains which sae ha decisions x s and x s =1 = = Eq. (1.) represens he non-anicipaiviy mus be idenical if scenario pair (s,s ) is indisinguishable in ime period. These non-anicipaiviy consrains correspond o he horizonal lines connecing differen nodes in Figure. Eq. (1.) represens he bounds and inegraliy consrains on he variables. (SP) max p s c x s (1.1) s S T A s τ xs τ as (,s) (1.) τ x s = x s (,s,s ) N (1.) x s χs (,s) (1.) One imporan aspec of his paper is decision-dependen uncerainy. According o Jonsbraen (), uncerainy in sochasic programming problems can be divided ino wo classes: exogenous uncerainy and endogenous uncerainy. Mos previous work in he lieraure deals wih problems wih exogenous uncerainy (e.g. demands) where he opimizaion decisions canno influence he sochasic process. Reviews of previous work on problems wih exogenous uncerainy can be found in (Sahinidis, 00; Schulz, 00). Problems where sochasic processes are affeced by decisions are said o possess endogenous uncerainy. According o Goel and Grossmann (00), decisions affec he sochasic process in a leas wo ways: decisions can aler he probabiliy disribuions (ype 1), or decisions can ac o discover more accurae informaion (ype ). In his paper, we focus on he laer ype of endogenous uncerainy where he decisions ac o resolve uncerainy. Lieraure on he class of problems ha deal wih endogenous uncerainy is limied. The only papers ha we are aware of are Pflug (0); Jonsbraen e al. (); Ahmed (000); Viswanah e al. (00); Held and Woodruff (00); Goel and Grossmann (00, 00); Goel (00); Goel e al. (00); Tarhan and Grossmann (00); Boland e al. (00); Solak (00). Deailed discussion on hese papers can be found in Tarhan and Grossmann (00) and Tarhan e al. (00). Boland e al. (00) applies mulisage sochasic programming o open pi mine producion scheduling, which is modeled as a mixed-ineger linear program. They consider endogenous uncerainy (ype ) where

4 he excavaion decisions resolve uncerainy in geology immediaely. They follow a similar approach as Goel and Grossmann (00) for modeling he problem, wih he excepion of eliminaing some of he binary variables used in he general formulaion o represen condiional non-anicipaiviy consrains. Furhermore, hey solve he model in full-space wihou using any decomposiion algorihm. I is more convenien o represen he endogenous ype of uncerainy using he alernaive scenario ree in Figure where he non-anicipaiviy consrains are explicily handled as funcions of decisions. For insance, if he decisions a ime period 1 and are such ha all scenarios are indisinguishable a ime period and some are disinguishable a ime period hen he corresponding scenario ree will be as shown in Figure. This is differen from he ree in Figure where scenarios and a ime period, and all scenarios a ime period are disinguishable. Therefore, in case of endogenous uncerainy, he se of non-anicipaiviy consrains (N) in eq. (1.) is no fixed bu i is a funcion of he decisions (x s ). 1 = = 1 1 = = 1 = 1 = 1 = 1 = Scenario 1 Scenario Scenario Scenario Figure : Scenario ree wih differen uncerainy resoluion scheme. Anoher aspec ha will be addressed in his paper is gradual resoluion of endogenous uncerainy over ime. Recen publicaions in he area, Goel and Grossmann (00) and Boland e al. (00), consider ha he endogenous uncerainy resolves immediaely when he decisions o resolve he uncerainies are made. Differen from he previous work, we consider he case where such uncerainy resolves gradually depending on he decisions made. For insance he uncerainy in he yield of a process can resolve as more producion is performed. i is of endogenous ype. Since he resoluion of yield is dependen on he operaing decisions, I is assumed ha uncerainy in he yield resolves gradually depending on he operaing decisions made in each ime period. This brings some changes o he model, underlying scenario ree, and he non-anicipaiviy consrains. The lieraure abou his subjec in he conex of planning problems is also very limied. =1 = = The only lieraure ha we know is Sensland and Tjøsheim (1); Jonsbraen (); Dias (00); Tarhan and Grossmann (00). Deailed discussion on hese papers can also be found in Tarhan and Grossmann (00). Recenly, Solak (00) considers he sochasic programming models in which imes of uncerainy realizaions are dependen on he decisions made and uncerainy resolves gradually. They use a similar modeling approach presened in Tarhan and Grossmann (00). Specifically, hey consider he projec porfolio opimizaion problem ha deals wih he selecion of research and developmen projecs and deerminaion of opimal resource allocaions. They use sample

5 average approximaion mehod (Kleyweg e al., 00) for solving he problem, where he sample problems are solved hrough Lagrangian relaxaion and heurisics. Harrison (00) uses a differen approach for opimizing wo-sage decision making problems under uncerainy. Some of he uncerainy is assumed o resolve afer he observaion of he oucome of he firs sage decision. The auhor develops a new mehod called Bayesian Programming, where he necessary inegrals are approximaed using Markov Chain Mone Carlo simulaions, and simulaed annealing ype of mea-heurisic are used o opimize he decisions. The ouline of his paper is as follows. In Secion we presen he generic mahemaical model for he class of problems under consideraion. In Secion he proposed soluion approach is explained. Secion compares he resuls found by he proposed soluion approach, and discusses he resuls on synhesis of process neworks and planning of offshore oil or gas field infrasrucure problems. Mahemaical model In his chaper, only generic variable names are used in he explanaion of he model for he problems under consideraion. The generic variable names are seleced based on wheher a variable is decision, sae, or recourse variable. The decision, sae, and recourse variables are denoed by vecors d, x and u wih dimension n d, n x, and n u respecively. The firs n d n d, n x n x, n u n u elemens of hese vecors are resriced o be discree variables. Also, w and z are logic/binary variables used for defining he condiional non-anicipaiviy consrains. The planning horizon is divided ino T planning periods. The sequence of decisions is as follows: he decision variables d are implemened in he beginning of ime period are which is followed by he resoluion of uncerainy. The sae variables x are auomaically calculaed when he decisions are fixed. A he end of ime period, some recourse decisions u are implemened. (See Figure ) Period 1 Period Period +1 Period T -1 Period [ d ] [ x ] [ u ] Figure : Represenaion of planning horizon. Based on he above definiions, he proposed mulisage sochasic MINLP model is as follows: T

6 (P) max T s S p s f (d s,us ) (.1) h(d s,xs,us ) 0 T, s S (.) g(d s,x s,u s ) 0 T, s S (.) w l,s Φ(d s,x s,u s ) T, s S, l L (.) z s,s d s +1 = ds +1 z s,s Ψ(w s ) T, ( s,s ) M q, q Q (.) ] [ z s,s ( s,s, ) N C, ( s,s ) M q, q Q (.) d s = d s ( s,s, ) N I (.) d s,xs,us χs T, s S (.) z s,s,w s {True,False} ( s,s, ) N C (.) In eq. (.1), he objecive is o maximize he expeced ne presen value, which is a linear or non-linear funcion of decision and sae variables. Eq. (.) is a generic muli period linear consrain ha relaes he decision, sae, and recourse variables for every scenario. Similarly, eq. (.) is a generic non-linear consrain for every scenario. Eq. (.) relaes he binary/logic variables w wih he discree and coninuous decision, and sae variables. These w variables are used in eq. (.) o model problem specific uncerainy resoluion rules o deermine if scenario pair (s,s ) is indisinguishable or no. There are wo ypes of non-anicipaiviy consrains: iniial (N I ) and condiional (N C ). Eq. (.) is he iniial non-anicipaiviy consrains ha hold regardless of any decision aken. Noe ha he iniial non-anicipaiviy consrains include no only he firs period non-anicipaiviy consrains bu also some oher subsequen period nonanicipaiviy consrains ha mus hold because of gradual uncerainy resoluion. The condiional nonanicipaiviy consrains in he disjuncion (eq. (.)) are included ino he model if wo scenarios are indisinguishable a he end of ime period (z s,s is rue). The condiional non-anicipaiviy consrains dicae ha he decisions a he beginning of ime period + 1 in scenarios s and s mus be idenical. Oherwise hey are no imposed on he feasible space. As proved in Goel and Grossmann (00), i is enough o consider a subse of scenario pairs ha differ in only one uncerain parameer. Therefore, in eqs. (.) and (.) each such scenario pair (s,s ) will be an elemen of exacly one of he ses M q, q Q. Noe ha in eqs. (.) and (.), vecor w s conains w l,s for each resoluion level l L (e.g. in Figure L = {1,, }). Eqs. (.) (.) represen he variables properies and inegraliy requiremens. Noe ha condiional non-anicipaiviies are no imposed on recourse variables u. This is differen han Goel and Grossmann (00) where he uncerainy was assumed o resolve insananeously. In case of

7 gradual uncerainy, alhough he decision maker observes some oupu from he sysem, he decision maker may no differeniae scenarios. For insance, le us assume here is a process which akes a raw maerial and generaes a final produc (Figure ). The demand for final produc has o be saisfied exacly and in case of shorage, he final produc can be bough from he marke. The decision a he beginning of each ime period is he amoun of raw maerial o buy and he recourse a he end of each period is he amoun of final produc o buy from he marke o saisfy he shorage. Raw Maerial Process Final Produc o buy from marke Final Produc o sell Figure : Simple process convering raw maerial o final produc. Assume here is uncerainy in he yield of he process and i requires wo years of producion o resolve he uncerainy, which has four possible values, values, θ 1, θ, θ, θ. Afer one year of producion he decision maker may observe ha he yield is on he low side (θ 1,θ ) or on he high side (θ,θ ). A his parial resoluion sage, he decision maker has o give idenical decisions for he amoun of raw maerial o purchase for he second year for each indisinguishable scenario groups (i.e. scenarios (1-) and (-)) as shown in he scenario ree in Figure. The decision maker canno disinguish he scenarios wihin he scenario groups (1-) and (-) a parial resoluion sage since he decision maker needs one more year o know he exac yield. On he oher hand, he recourse decision (amoun of final produc o buy from he marke) a he end of firs year has o be differen among some of he indisinguishable scenarios (e.g. (1-) and (-)) because he amoun of producion varies due o differen yields whereas he demand for he final produc says he same. Therefore, when wo scenarios are indisinguishable we impose only he decision variables, and no he recourse variables, in he disjuncion in eq. (.). =1 l = 1: No resoluion = l = : Parial resoluion = = 1 θ 1 θ θ θ l : Full resoluion 1 1 θ θ θ Scenarios: 1 Figure : Scenario ree represenaion of gradual uncerainy resoluion in yield. To illusrae he consrains (.)-(.), we give specific examples from he synhesis of process neworks θ θ θ θ θ θ θ θ θ

8 problem (deails in Appendix A). In he generic model (P), eq. (.) corresponds o he linear consrains such as he maerial balance consrains (A.), (A.), (A.), (A.), (A.) on every node and process. Eq. (.) corresponds o he non-linear consrains ha are used for calculaing ne presen value for every scenario (A.). Eq. (.) corresponds o consrains (A.) and (A.) which sae ha logic variable w 1,s i, be rue if and only if process i has no been expanded, or no run as a pilo plan unil ime period τ. Similarly, w,s i, will be rue if and only if process i has operaed only one ime period or run as a pilo plan unil ime period τ. Eq. (.) corresponds o consrain (A.) which sae ha a pair of scenarios ha is an elemen of M q, q Q (see Appendix A for specific subses used for he example problem) will be indisinguishable if and only if for each process ha disinguishes he scenario pair, w l,s i, holds rue. As explained in (Goel and Grossmann, 00), i is enough o consider he scenario pairs ha differ in only one uncerain parameer as shown in Table 1. Table 1: Ses ha differ only in he specified parameers in synhesis of process neworks problem. Subses of scenario pairs (M q ) Process Sep (l) M 1 M M M 1 Soluion sraegy The size of he mulisage sochasic MINLP model (P) increases quadraically wih he number of scenarios and linearly wih ime periods. Therefore, i is difficul o solve his nonconvex MINLP model in fullspace for real size problems using commercial solvers. The model (P) is composed of nonconvex mixed-ineger nonlinear subproblems. The subproblems are conneced hrough iniial (eq. (.)) and condiional (eq. (.)) non-anicipaiviy consrains. When he iniial and condiional non-anicipaiviy consrains are relaxed, each subproblem can be solved independenly. In order o ake advanage of his special problem srucure, we propose a dualiy-based branch and bound algorihm along he lines of Goel and Grossmann (00). In he following secions, he upper and lower bounding procedures used a each node of he branch and bound ree, branching scheme, and he proposed soluion algorihm are presened..1 Upper bounding procedure In he proposed algorihm, he upper bound a he roo node of he branch and bound ree is found by opimizing model ( P0 LR ) in which he logic consrains (.)-(.) and disjuncion (.) have been removed will

9 and he iniial non-anicipaiviy consrain (.) are dualized as follows: ( ) P LR 0 φ LR 0 ) (λ s,s d, = max T s S p s f (d s,us )+ (s,s,) N I ( [λ s,s d, d s ds )] (.1) s.. (.) (.),(.) (.) The parameers λ s,s d, represen he Lagrange mulipliers corresponding o consrain (.). In order o find he ighes upper bound generaed by model ( P0 LR ) a he roo node, we consider he Lagrangean dual problem ( P0 LD ) ( ) ha minimizes he model P LR in he space of mulipliers. ( ) P LD 0 0 φ LD 0 = min λ φ LR 0 ) (λ s,s d, A any node n, oher han he roo node in he branch and bound ree, model (P n ) is opimized o calculae upper bounds. In model (P n ), no only he iniial non-anicipaiviy consrains, bu also some condiional non-anicipaiviy consrains are added. Iniial non-anicipaiviy consrains are added o he model regardless of any decision whereas condiional ones from he relaxed disjuncion are also included based on he branching cus. The condiional non-anicipaiviy consrains ha apply in node n are included in he dynamic se NC n. The selecion of which condiional non-anicipaiviy consrain o include ino se N n C, as well as some necessary cus o be added o (P n) will be discussed. The model (P n ), no including any such cus, is given as follows: (P n ) φ n = max T s S s.. (.)-(.) (.) p s f (d s,u s ) (.) d s = d s ( s,s, ) N I N n C (.) The upper bound a node n is generaed by opimizing model ( Pn LR ) in which he non-anicipaiviy consrain (.) is dualized as follows: ( ) P LR n φ LR n = max T s S p s f (d s,u s )+ s.. (.) (.),(.) (s,s,) N I N n C ( )] [λ s,s d, d s d s Similar o he roo node, in order o find he ighes upper bound, we again consider he Lagrangean dual problem ( Pn LD ) ( ) which is minimizaion of he model P LR n in he space of mulipliers, ( ) P LD n φ LD n = min λ φ LR n ) (λ s,s d, (.) (.)

10 where λ s,s d, represens he Lagrange mulipliers corresponding o consrain (.). Boh models ( P0 LR ) ( ) and P LR n can be decomposed ino independen nonconvex MINLP subproblems for fixed values of he mulipliers. Boh models are relaxaions of he model (P) for any fixed values of he Lagrange mulipliers, and in order o obain valid upper bounds, hese nonconvex subproblems mus be globally opimized (for proof see Tarhan and Grossmann (00)). Minimizaion of he Lagrangean dual models ( P0 LD ) ( ) and P LD n in he space of mulipliers is performed by he subgradien mehod proposed by Fisher ().. Branching In general, a any node in he branch and bound ree, he soluion of he Lagrangean dual may no saisfy he dualized non-anicipaiviy consrain (.), or he condiional non-anicipaiviy consrains in relaxed disjuncion (.) inferred by decisions. In his case, new branches are generaed from he curren node by considering he violaions in he dualized non-anicipaiviy consrains or he relaxed disjuncion...1 Branching on he dualized non-anicipaiviy consrains A any node of he branch-and-bound ree, he soluion of model ( Pn LD ) may no saisfy he dualized iniial or condiional non-anicipaiviy consrains ( N I N n C). In his case, branching is performed over he consrains in violaion. If he violaing non-anicipaiviy consrain involves discree variables hen he branching sraegy divides he feasible region ino wo, where one of he regions is resriced o d s d, d s d and he oher d s d, d s d. he variables, d = ps d s +p s d s d s p s +p s d ε and d s d + ε, d s d is calculaed by he probabiliy weighed average of. In case of coninuous variables, he branches are formed by d s d ε, d + ε where d is calculaed as shown above. A special case occurs during he branching of he firs period non-anicipaiviy consrains. As hese consrains should hold for each scenario pair, we can rewrie hese equaions as one expression, d s 1 1 = ds 1 =... = d s k 1. The branching can hen be performed so ha he enire expression is resriced o d and o d (in case of coninuous variables inequaliies should ake he forms d ε and d + ε... Branching on he disjuncions When he dualized non-anicipaiviy consrains are saisfied, i is possible o coninue searching by separaing he feasible space ino wo using he disjuncion (.). The branching is performed using he condiional non-anicipaiviy consrains in he relaxed disjuncions ha are supposed o hold, bu do no do so given he values of he variables a he previous ime periods. For insance, assume he decisions saisfy he iniial non-anicipaiviy consrains ha include he firs period non-anicipaiviy consrains Then indisinguishable scenario pairs are inferred by using (.)-(.). Given he indisinguishable scenario

11 pairs, if he variable values do no saisfy he condiional non-anicipaiviy consrains in disjuncion (.), he branching is performed on z s,s as shown in Figure. 1 z s, s = 0 z s, s = 1 Figure : Branching on he disjuncion in consrain (.). The wo generaed nodes in Figure (nodes 1 and ) have differen properies. A node 1, z s,s o false/zero and a node, z s,s is fixed o rue/one. A node 1 due o fixing z s,s is fixed o false, i is required o add some cus which will guaranee he disinguishabiliy of he scenario pair (s,s ). These cus will be generaed using consrains (.)-(.). To illusrae hese cus, we will again use he synhesis of process neworks problem (Appendix A). If we fix z s,s o false, where he scenario pair (s,s ) belongs o he se M 1, hen using Table 1 and (A.) we find ha logic consrains (.) and (.) mus be included in scenario subproblems s and s, respecively. w,s 1, False w,s 1, False Differen from node 1, a node some non-anicipaiviy consrains coming from he relaxed disjuncion are added o he se N n C in model (P n) due o he fixing of z s,s o rue. Similar o node 1, cus are generaed and added o model (P n ) where z s,s soluion algorihm convergen. (For proof see Goel and Grossmann (00)). Lower bounding procedure (.) (.) is fixed o rue. These cus are necessary o make he A each node, a lower bound is found using a heurisic which convers he soluion found by he upper bound procedure o a feasible soluion. Usually, he soluion found by he upper bound generaion does no saisfy he non-anicipaiviy consrains. Feasible soluions are generaed using a rolling horizon approach (Dimiriadis e al., ). Decisions in he indisinguishable scenario pairs are found by calculaing he probabiliy weighed average of variables in such scenarios. Afer fixing hese variables and considering he resoluion of uncerainy depending on he fixed decisions, he nex period decisions are found ieraively by calculaing similarly he probabiliy weighed average of he variables in indisinguishable scenarios. This procedure is erminaed when all scenarios are disinguishable or he end of he planning horizon is reached, whichever comes firs. Then having hese decisions fixed, he MINLP model is solved in full space using

12 an ouer approximaion algorihm (Duran and Grossmann, ), which yields a feasible soluion.. Soluion algorihm (SP-GO) Based on he mulisage sochasic MINLP model (P) and he previous explanaions abou he upper and lower bounding procedures, he proposed algorihm is presened below. P denoes he lis of open nodes each having an upper bound φ UB n found by he Lagrangean dual problem a node n, while φ LB represens he objecive value of he bes feasible soluion obained so far. Given hese variables, he major seps of he algorihm are given below. Sep 1 Iniializaion: φ LB =, φn UB =, P = {P 0 }, where P 0 is he roo node. Sep Terminaion: If P = /0, sop. The curren bes soluion is opimal. Oherwise, repea seps o. Sep Node selecion: Selec and delee node n from P based on he bes bound. Sep Bound generaion: For he roo node, generae he upper bound ( φ0 UB ) by applying a global opimizaion algorihm and obain a lower bound ( φ LB) soluions ( d, ˆ ˆx,û ) and ( d, x,ū ), respecively. (Deails of his sep (seps a-g) will be explained below.) Sep Fahoming: Delee from P all problems P wih ˆφ (P ) φ LB. Sep Branching: Branch on he dualized non-anicipaiviy consrains or disjuncions ha are violaed by he soluion ( ˆ d, ˆx,û ) of he relaxed problem (P n ). Generae wo children nodes, add hem o P. Noe ha sep 1 is he iniializaion and he algorihm ieraes beween seps and unil convergence is achieved. The deails of he sep (seps a o g) are explained below. Sep a : Se ieraion i = 0. Sep b : While i is less han or equal o max ieraion, i = i+1, repea seps b hrough h. Sep c Generae upper bound: For fixed mulipliers, use global opimizer for each MINLP subproblem o solve ( Pn LR ) o obain soluion ( d, ˆ ˆx,û ) wih objecive funcion value ˆφ. Sep d Updae upper bound: Updae he upper bound by φn UB = min { φn UB, ˆφ }. Sep e Generae lower bound: Generae a feasible soluion ( d, x,ū ) wih objecive value φ for he model (P) based on he soluion ( ˆ d, ˆx,û ) generaed a sep c. Sep f Updae lower bound: Updae he lower bound by φ LB = max { φ LB, φ }. Sep g Updae mulipliers: Updae mulipliers using subgradien mehod. 1

13 The boleneck of he proposed algorihm (SP-GO), is he upper bounding procedure (sep c) in sep where every scenario subproblem has o be globally opimized in order o have valid upper bounds during he subgradien ieraions. This is compuaionally expensive. Since i is enough o generae a valid upper bound only a he final subgradien ieraion, he proposed algorihm, SP-GO, can be modified such ha non-convex subproblems are solved using an ouer-approximaion algorihm insead of global opimizaion a he inermediae subgradien ieraions. The reason for such a modificaion is ha in pracice during he subgradien opimizaion we do no inend o find he opimal mulipliers, bu mulipliers ha improve he upper bound. This modificaion inends o improve he Lagrange mulipliers a each ieraion, wihou solving each scenario problem globally. The drawback of he approach is no geing valid bounds during hese inermediae subgradien ieraions. The following algorihm (SP-OA) combines he global opimizaion and ouer-approximaion algorihm for expediing he soluion process wihou violaing he validiy of he bounds. Noe ha he ouerapproximaion algorihm can be replaced by any algorihm ha assumes convexiy (e.g. generalized Benders decomposiion, exended cuing plane), ha will find local soluions in shor ime.. Soluion algorihm (SP-OA) In order o incorporae his idea ino he proposed algorihm (SP-GO), seps c and d need o be modified. The ype of opimizaion (global or ouer approximaion) is based on he ieraion i and maximum ieraion limi (max ieraion). If ieraion i is one or he maximum ieraion limi, every subproblem in he Lagrangean dual of he problem ( Pn LR ) is solved using global opimizaion; oherwise an ouer-approximaion algorihm ha relies on convexiy assumpion is used. The reason for using global opimizaion a he firs ieraion is for iniializing he variable values close o global opimal soluion which is aken as an inpu for he ouerapproximaion algorihm a he second ieraion. This improves he chances of finding opimal soluions during he ouer-approximaion algorihm. The reason for using global opimizaion a he las ieraion is for generaing valid upper bounds ha will be used for pruning he pars of he branch and bound ree and calculaing he dualiy gap. In sep d, he upper bound is updaed only afer global opimizaion is used for calculaing valid bounds. Figure compares he ypical profiles of he Lagrangean dual as a funcion of mulipliers. The solid lines represen a profile generaed when all he subproblems are solved using global opimizaion, whereas he dashed lines represen a profile generaed using ouer approximaion a he inermediae ieraions. In he firs and las ieraion a soluion is found on he profile generaed by global opimizaion (solid lines) and during he inermediae ieraions ouer approximaion (dashed lines) may end up in one of he local opimal soluions (x OA i ). 1

14 Resuls LD φ OA x 1 OA x OA x Figure : Graphical comparison of SP-GO and SP-OA ieraions. In his secion, we presen resuls for comparing he soluions proposed by he algorihms SP-GO and SP- OA for he synhesis of process neworks (Tarhan and Grossmann (00)) and planning of offshore oil or gas field infrasrucure problems (Tarhan e al. (00))..1 Synhesis of process neworks This secion explains briefly he synhesis of process neworks example and Secion.1.1 analyzes he soluion ime and qualiy of SP-GO and SP-OA. In he synhesis of process neworks problem, we consider he selecion and capaciy expansion of processes over a planning horizon given ha here is uncerainy in he yields of some processes. Uncerainy can be reduced hrough invesmen in pilo plans. The rade-off is ha pilo plans delay he inroducion of processes bu a a reduced uncerainy. Concave cos funcions are assumed which are he sources of non-convexiy in his problem. Figure shows he specific nework used in he synhesis of process neworks problem. The demand for he final produc A over he planning horizon is known and he company mus saisfy ha demand. The curren producion akes place only in process, which consumes an inermediae produc B from he marke. In case of producion shorage i is possible o buy final produc A from he marke a a higher cos o saisfy he demand. Invenory for boh he inermediae and final produc can be mainained. Two new echnologies (process 1 and process ) are available o produce he inermediae produc B from wo differen raw maerials C or D. These new echnologies have uncerainy in heir yields which gradually resolve over a wo year period eiher wih invesmens in pilo plans or wih plan operaion. In Figure process is already operaional wih an exising capaciy of 000 ons/year and a known yield of 0%. The only difference beween he wo differen echnologies (Process 1 and ) is he variance 1 λ

15 C D rae d 1, rae d, Pilo Plan Process 1 Pilo Plan Process rae x, rae x, N1 rae x, inv,in u 1 B N rae d, rae x, inv,in u Process rae x, Figure : Schemaic represenaion of synhesis of process neworks problem. of yield disribuions. Alhough hey possess he same mean value, %, process has a higher variance han process 1 (see Table A.1). Process 1 can realize afer firs year a yield of or 1% and afer he second year,, and 1%. Similarly, yields of process afer firs year are 0 or 0% and afer second year 0, 0, 0 and 0%. I is assumed ha he probabiliy of each scenario is he same. The deailed opimizaion model and he scenarios and daa for his insance are presened in Appendix A. The size of his insance in full space wih periods and scenarios is given in Table. Alhough he problem seems o be rivial, combining scenario subproblems over years wih iniial and condiional non-anicipaiviy consrains, leads o a large scale problem. Table : Model size in full space ( scenarios). Individual Scenario Full Space Model ( Scenarios) Binary Variables 0,1 Coninuous Variables 1, Consrains 1,0.1.1 Performance analysis of SP-GO and SP-OA The resuls in his secion have been obained on a Penium-IV,.0 GHz Windows machine. Also, we employed AIMMS.. for implemening he soluion algorihm using solvers CPLEX.0, CONOPT.1, SNOPT.1, BARON.. (Sahinidis (000)), AOA (AIMMS Ouer Approximaion Module). The specific synhesis problem has been opimized using boh algorihms SP-GO and SP-OA wih % wors case gap. Wors case gap is calculaed by adding he specified gap of 1% for he global opimizer and specified gap of 1% for he branch and bound ree. The global opimizaion algorihms have been run wih 1% opimaliy gap, and boh algorihms were erminaed when he gap in he branch and bound ree 1 A

16 is less han 1%. The resuls are presened in Table. In his insance, alhough boh algorihms find close soluions, SP-GO finds a slighly beer (0.1%) feasible soluion han SP-OA (. vs..). However, while SP-GO requires 0. hours o complee he branch and bound search, SP-OA requires only. hours, a reducion of nearly 0%. In SP-OA he opimum soluion was firs found in.1 hours while SP-GO found i in 1.1 hours. One can also compare he bes upper bounds found by he wo algorihms. SP-OA reduces he bes upper bound more han SP-GO by searching more nodes in branch and bound ree, bu canno find a beer feasible soluion. Table : Performance comparison of SP-GO and SP-OA for synhesis of process neworks problem. SP-GO SP-OA Lower bound ($ ) Bes feasible soluion.. Upper bound ($ ).1.1 Wors case gap (%) Bes feasible soluion found afer (hrs) Toal CPU ime (hrs) 0... Planning of offshore oil or gas field infrasrucure In his secion, we briefly describe he specific planning of offshore oil or gas field infrasrucure problem. Secion..1 compares he resuls and he soluion imes found by he wo algorihms (SP-GO, SP-OA). We consider a field consising of a single reservoir (Figure ), where a number of wells can be drilled and exploied for oil in every reservoir during he planning horizon. The problem involves making invesmen and operaing decisions over he planning horizon. Invesmen decisions are selecion of he number, ype and capaciy of faciliies, and insallaion schedule of hese faciliies, as well as selecion of ypes of wells and drilling schedule of wells. Operaing decisions are amoun of oil producion for each ime period given he limiaions of he reservoirs. The goal is o capure he complex economic radeoffs ha arise from he invesmen and operaing decisions in order o maximize he expeced ne presen value of he projec. The deails and daa for he planning of oil or gas field infrasrucure under uncerainy are repored exensively in Tarhan and Grossmann (00). Briefly, he main uncerainies considered are in he iniial maximum oil or gas flowrae, recoverable oil or gas volume and waer breakhrough ime of he reservoir, which are represened by discree disribuions. Furhermore, i is assumed ha hese uncerainies are no immediaely realized, bu are gradually revealed as a funcion of well drilling and producion decisions. The model opimizes he invesmen and operaing decisions over he enire planning horizon. Two insances, 1 and have been opimized o show he efficiency of he proposed algorihm. In boh insances he reservoir behavior is nonlinear (because of he waer flowrae equaions) bu in insance 1, he maximum oil flowrae

17 Reservoir TLP FPSO Poenial well Figure : A Typical oil field infrasrucure. is assumed o be linear funcion of he cumulaive oil producion whereas in insance i is a nonlinear curve. In boh cases he nonlineariies give rise o non-convexiies in he model. The size of he wo insances are he same and given in Table. Table : Model size for he oil field problems. Individual Scenario Full Space Model ( Scenarios) Binary Variables 00 Ineger Variables 0 00 Coninuous Variables Consrains Performance analysis of SP-GO and SP-OA The resuls of his secion have been obained on a Penium-IV,.0 GHz Windows machine. Also, we employed AIMMS. for implemening he soluion algorihm using solvers CPLEX.1, CONOPT.1, SNOPT.1, BARON.. (Sahinidis (000)), AOA (AIMMS Ouer Approximaion Module). Table : Performance comparison of SP-GO and SP-OA for Insances 1 and. Insance 1 Insance SP-GO SP-OA SP-GO SP-OA Sochasic Programming Objecive funcion value ($ ).... Toal CPU ime (hrs).. Wors case gap (%).. 1. Table shows ha he soluion imes of boh insances using SP-GO are long for pracical purposes ( and hours). As presened in Table, in boh cases using algorihm SP-OA, which combines he global opimizaion and ouer-approximaion, reduces he soluion ime by % and % respecively, and does

18 no sacrifice he qualiy of he soluion. In fac, SP-OA obains beer feasible soluions (. vs..,. vs..) and narrower gaps beween he upper and lower bounds han he one found wih SP-GO. Similar o he resuls obained in he synhesis of process neworks problem, large reducions in soluion ime are achieved by combining local and global MINLP solvers. However, in hese problems, SP-OA also found beer feasible soluions. Conclusion We have presened in his paper a generic non-convex mulisage MINLP model wih decision dependen uncerainies. We have proposed an improvemen on he dualiy-based branch and bound algorihm for solving he large sized insances where global opimizaion and ouer-approximaion algorihms are combined. The performance of he new algorihm (SP-OA) has been compared wih he previous approach (SP-GO) for wo problems, he synhesis of process neworks and planning of offshore oil or gas field infrasrucure. In he synhesis of process neworks problem, SP-OA obained a soluion ha is 0.1% worse han SP-GO while reducing he soluion ime abou 0%. In he oil field infrasrucure problem, SP-OA improved no only he bes feasible soluion found by SP-GO by -%, bu also he soluion ime by -%. The improvemen made by SP-OA can be accouned for combining he global opimizer and ouer-approximaion in a cerain way o reduce he soluion ime. SP-OA akes advanage of he srenghs of boh algorihms, leading o large reducions in soluion ime wihou necessarily sacrificing he qualiy of he soluion. The resuls also show ha a combinaion of local and global MINLP solvers can lead o beer soluions faser han using only a global solver. The global opimizer Baron was used o generae valid bounds in longer ime, while he ouer-approximaion algorihm AOA was used o updae Lagrange mulipliers in a shorer ime wihou finding he global opimum soluions. Acknowledgemens The auhors acknowledge he financial suppor from Exxon-Mobil Upsream Research Company and parial suppor of he Naional Science Foundaion under gran CTS-0. References Ahmed, S., 000. Sraegic planning under uncerainy: Sochasic ineger programming approaches. Phd hesis, Universiy of Illinois a Urbana-Champaign. Birge, J. R., Louveaux, F.,. Inroducion o Sochasic Programming. Springer-Verlag, New York.

19 Boland, N., Dumirescu, I., Froyland, G., 00. A mulisage sochasic programming approach o open pi mine producion scheduling wih uncerain geology. Opimizaion Online, hp:// HTML/00//1.hml. Charnes, A., Cooper, W. W.,. Deerminisic equivalens for opimizing and saisficing under chance consrains. Operaions Research (1),. Dias, M., 00. Invesmen in informaion in peroleum, real opions and revelaion. In: Proceedings of he h Annual Inernaional Conference on Real Opions. Real Opions Group a Cyprus, Cyprus. Dimiriadis, A. D., Shah, N., Panelides, C. C.,. RTN-based rolling horizon algorihms for medium erm scheduling of mulipurpose plans. Compuers and Chemical Engineering 1, 1. Duran, M. A., Grossmann, I. E.,. An ouer-approximaion algorihm for a class of mixed-ineger nonlinear programs. Mahemaical Programming (), 0. Fisher, M. L.,. An applicaions oriened guide o lagrangian relaxaion. Inerfaces 1 (), 1. Goel, V., 00. Sochasic programming approaches for he opimal developmen of gas fields under uncerainy. Phd hesis, Carnegie Mellon Universiy. Goel, V., Grossmann, I. E., 00. A sochasic programming approach o planning of offshore gas field developmens under uncerainy in reserves. Compuers and Chemical Engineering (),. Goel, V., Grossmann, I. E., 00. A class of sochasic programs wih decision dependen uncerainy. Mahemaical Programming (-, Ser. B),. Goel, V., Grossmann, I. E., El-Bakry, A. S., Mulkay, E. L., 00. A novel branch and bound algorihm for opimal developmen of gas fields under uncerainy in reserves. Compuers and Chemical Engineering 0,. Harrison, K. W., 00. Two-sage decision-making under uncerainy and sochasiciy: Bayesian Programming. Advances in Waer Resources 0 (), 1. Held, H., Woodruff, D. L., 00. Heurisics for muli-sage inerdicion of sochasic neworks. Journal of Heurisics (-), 00. Jonsbraen, T.,. Opimizaion models for peroleum field exploiaion. Phd hesis, Norwegian School of Economics and Business Adminisraion. Jonsbraen, T. W., Wes, R. J., Woodruff, D. L.,. A class of sochasic programs wih decision dependen random elemens. Annals of Operaions Research,.

20 Kleyweg, A. J., Shapiro, A., Homem-de Mello, T., 00. The sample average approximaion mehod for sochasic discree opimizaion. SIAM Journal on Opimizaion 1 (), 0. Pflug, G. C., 0. Online opimizaion of simulaed markovian processes. Mahemaics of Operaions 1 (), 1. Ruszczyński, A.,. Decomposiion mehods in sochasic programming. Mahemaical Programming (1-),. Sahinidis, N. V., 000. Baron: Branch-and-reduce opimizaion navigaor. users manual. version.0. hp://archimedes.cheme.cmu.edu/baron/manuse.pdf. Sahinidis, N. V., 00. Opimizaion under uncerainy: sae-of-he-ar and opporuniies. Compuers and Chemical Engineering (-), 1. Schulz, R., 00. Sochasic programming wih ineger variables. Mahemaical Programming (1-, Ser. B), 0. Solak, S., 00. Efficien soluion procedures for mulisage sochasic formulaions of wo problem classes. Phd hesis, Georgia Insiue of Technology. Sensland, G., Tjøsheim, D., 1. Opimal decisions wih reducion of uncerainy over ime - an applicaion o oil producion. In: Lund, D., øksendal, B. (Eds.), Sochasic Models and Opion Values. pp. 1. Tarhan, B., Grossmann, I. E., 00. A mulisage sochasic programming approach wih sraegies for uncerainy reducion in he synhesis of process neworks wih uncerain yields. Compuers and Chemical Engineering,. Tarhan, B., Grossmann, I. E., Goel, V., 00. A mulisage sochasic programming approach for he planning of offshore oil or gas field infrasrucure under decision dependen uncerainy. (To appear) Indusrial and Engineering Chemisry. Viswanah, K., Peea, S., Salman, S. F., 00. Invesing in he links of a sochasic nework o minimize expeced shores pah. lengh. Purdue Universiy Economics Working Papers 1, Purdue Universiy, Deparmen of Economics, Wes Lafayee, IN. 0

21 Appendices A Mahemaical Model for Synhesis of Process Neworks 1

22 Nomenclaure: Ses: D(s,s ) : Processes ha differeniae scenarios s and s DK : Sreams employed as decision variables (DK K) FK : Final produc sreams in he process nework (FK K) I IK IU IP(i) IN( j) J T J K : Processes in he process nework : Sreams employed as decision variables for inermediae produc purchase (IK DK) : Processes wih uncerain yield in he process nework : Inpu sreams o process i : Inpu sreams o node j : Nodes in he process nework : Nodes relaed o he balance of inermediae produc in he process nework : Sreams in he process nework L : Levels of gradual uncerainy resoluion (see Fugure ) ˆL(i,s,s ) : Highes uncerainy resoluion level in which scenarios (s,s ) are indisinguishable in process (ˆL L ) M q N I N C N n C ON( j) OP(i) : Scenario pairs (s,s ) ha differ in only one uncerain parameer : Se of iniial non-anicipaiviy consrains : Se of condiional non-anicipaiviy consrains : Se of condiional non-anicipaiviy consrains a node n : Oupu sreams from node j : Oupu sreams from process i Q : Subses of he scenario pairs (s,s ) S : Possible scenarios SK : Sreams employed as sae variables (SK K) T Indices: i j k : Periods in ime horizon : Process in se I : Node in se J : Sream in se K

23 l q s,s,τ : Level in se L : Elemen in se Q : Scenario in se S : Time period in se T Binary variables (or equivalen Boolean variables): d exp,s i, d oper,s i, d pilo,s i, w l,s i, z s,s : Wheher or no process i is expanded in period, scenario s : Wheher or no process i is operaed in period, scenario s : Wheher or no pilo plan for process i is buil in period, scenario s : Wheher or no he yield of process i is in level l in period, scenario s : Wheher or no scenarios s,s are indisinguishable in period Coninuous variables: d qe,s i, d rae,s k, enpv npv s : Capaciy expansion of process i in period, scenario s : Flowrae of sream k DK in period, scenario s : Expeced ne presen value : Ne presen value of projec under scenario s inv f inal,s u : Amoun of final produc o pu ino invenory a he end of period, scenario s u inv in,s u sales,s p f inal,s u x cap,s i, x rae,s k, Parameers: D FE i, FO i, : Amoun of inermediae produc o pu ino invenory a he end of period, scenario s : Amoun of sales of final produc in period, scenario s : Amoun of purchases of final produc in period, scenario s : Capaciy of process i in period, scenario s : Flowrae of sream k SK in period, scenario s : Demand for final produc in period : Fixed expansion cos for process i in period : Fixed operaing cos for process i in period L (.) (.) : Lower bounds p s PP i, : Probabiliy of scenario s : Fixed and operaing cos for pilo plan for process i in period

24 U (.) (.) : Upper bounds VE i, VO k, π f inal π in γ f inal γ in δ θ l,s i θ i : Variable expansion cos for process i in period : Variable operaing cos corresponding o d rae,s k, : Purchase price for final produc in period : Purchase price for inermediae produc in period in period : Cos of mainaining invenory of final produc in period : Cos of mainaining invenory of inermediae produc in period : Duraion of period : Yield of process i in level l in scenario s, i IU : Yield of process i, i I\IU α : Exponen in he erm for expansion cos (0 < α < 1) τ,τ,τ : Time delays In his insance, for simpliciy, we assume ha uncerainy resolves in wo seps (i.e. in hree levels, L = {1,,}). Based on he above definiions, he model for he synhesis of process neworks is as follows: Equaion (A.1) represens he expeced ne presen value which is o be maximized over he se of scenarios S. enpv = p s npv s s S The ne presen value for each scenario s is given by, npv s = T i I T k DK T i IU T γ in ( FE i, d exp,s i, +VE i, ( VO k, d rae,s k, PP i, d pilo,s i, u inv in,s T δ T ( T γ f inal d qe,s i, k SK k IK ) α ) T i I VO k, x rae,s k, δ π in d rae,s k, + π f inal FO i, d oper,s i, δ p f inal,s u ) δ (A.1) inv f inal,s u s S (A.) where α is a fracional exponen, 0 < α < 1, which gives rise o a concave cos funcion. Disjuncion (A.) represens he inpu-oupu relaionships for he processes wih uncerain yields, θ l,s i, a each period and scenario. Noe ha a each ime period and scenario, he yield of a process mus be in

25 one of he possible resoluion levels l L. l L w l,s i, x rae,s k, = θ l,s i k OP(i) ( d rae,s k, + x rae,s k, k IP(i) DK k IP(i) SK ) i IU, s S, T (A.) Eqs. (A.) and (A.) relae he Boolean variable w l,s i, o he previous ime period decision variables so ha he some se of consrains in (A.) will be valid. Assuming ha he uncerainy will resolve in wo seps, here are hree possibiliies. If we build neiher a pilo plan nor acual plan, he se of consrains in (A.) ha has zero yield (since here is no producion capaciy) will be valid which is expressed by (A.). If we build/operae he acual plan one year or run pilo plan one year, he se of consrains in (A.), ha have sill uncerain yields (a level ), will be valid. This is refleced by (A.). Finally, if he plan operaes wo or more years or a leas one year afer pilo plan operaion, he se of consrains in (A.) ha have exac yield will be valid (level ). This is capured by (A.) since a anyime a process has o be in one of he levels of uncerainy. Noe ha insead of using (A.) we could define he rule for l = bu we simplify he consrains by using (A.). Insead of specifying a rule for level l = as in eqs.(a.)-(a.), we only specify ha he process yield mus in one of he levels l L. w l,s τ i, w l,s i, l L τ=1 τ τ=1 ( ( d exp,s i,τ d oper,s i,τ ) d pilo,s i,τ d pilo,s i,τ ) l = 1, i IU, T, s S l =, i IU, T, s S (A.) (A.) w l,s i, i IU, T, s S (A.) Eq. (A.) represens he inpu-oupu relaionships for he processes wih cerain yields, θ i, a each period and scenario. ( x rae,s k, = θ i k OP(i) d rae,s k IP(i) DK k, + k IP(i) SK x rae,s k, The mass balance consrains a each node in he nework are given by, ) i I\IU, s S, T (A.) x rae,s k, = x rae,s k, j J, s S, T (A.) k ON( j) k IN( j) The balance consrain ha relaes invenory, purchase, sales and producion for final produc a