SIMULATION OPTIMIZATION: APPLICATIONS IN RISK MANAGEMENT

Internatonal Journal of Informaton Technology & Decson Makng Vol. 7, No. 4 (2008) 571 587 c World Scentfc Publshng Company SIMULATION OPTIMIZATION: APPLICATIONS IN RISK MANAGEMENT MARCO BETTER and FRED GLOVER OptTek Systems, Inc., 1919 Seventh Street Boulder, Colorado 80302, USA better@opttek.com glover@opttek.com GARY KOCHENBERGER Unversty of Colorado Denver, 1250 14th Street Sute 215, Denver, Colorado 80202, USA Gary.kochenberger@cudenver.edu HAIBO WANG Texas A&M Internatonal Unversty Laredo, TX 78041, USA hwang@tamu.edu Smulaton optmzaton s provdng solutons to mportant practcal problems prevously beyond reach. Ths paper explores how new approaches are sgnfcantly expandng the power of smulaton optmzaton for managng rsk. Recent advances n smulaton optmzaton technology are leadng to new opportuntes to solve problems more effectvely. Specfcally, n applcatons nvolvng rsk and uncertanty, smulaton optmzaton surpasses the capabltes of other optmzaton methods not only n the qualty of solutons but also n ther nterpretablty and practcalty. In ths paper, we demonstrate the advantages of usng a smulaton optmzaton approach to tackle rsky decsons, by showcasng the methodology on two popular applcatons from the areas of fnance and busness process desgn. Keywords: Optmzaton; smulaton; portfolo selecton; rsk management. 1. Introducton Whenever uncertanty exsts, there s rsk. Uncertanty s present when there s a possblty that the outcome of a partcular event wll devate from what s expected. In some cases, we can use past experence and other nformaton to try to estmate the probablty of occurrence of dfferent events. Ths allows us to estmate a probablty dstrbuton for all possble events. Rsk can be defned as the probablty of occurrence of an event that would have a negatve effect on a goal. On the other hand, the probablty of occurrence of an event that would have a postve mpact s consdered an opportunty (see Ref. 1 for a detaled dscusson of rsks and 571

572 M. Better et al. opportuntes). Therefore, the porton of the probablty dstrbuton that represents potentally harmful, or unwanted, outcomes s the focus of rsk management. Rsk management s the process that nvolves dentfyng, selectng, and mplementng measures that can be appled to mtgate rsk n a partcular stuaton. 1 The objectve of rsk management, n ths context, s to fnd the set of actons (.e. nvestments, polces, resource confguratons, etc.) to reduce the level of rsk to acceptable levels. What consttutes an acceptable level wll depend on the stuaton, the decson makers atttude toward rsk, and the margnal rewards expected from takng on addtonal rsk. In order to help rsk managers acheve ths objectve, many technques have been developed, both qualtatve and quanttatve. Among quanttatve technques, optmzaton has a natural appeal because t s based on objectve mathematcal formulatons that usually output an optmal soluton (.e. set of decsons) for mtgatng rsk. However, tradtonal optmzaton approaches are prone to serous lmtatons. In Sec. 2 of ths paper, we brefly descrbe two promnent optmzaton technques that are frequently used n rsk management applcatons for ther ablty to handle uncertanty n the data; we then dscuss the advantages and dsadvantages of these methods. In Sec. 3, we dscuss how smulaton optmzaton can overcome the lmtatons of tradtonal optmzaton technques, and we detal some nnovatve methods that make ths a very useful, practcal, and ntutve approach for rsk management. Secton 4 llustrates the advantages of smulaton optmzaton on two practcal examples. Fnally, n Sec. 5 we summarze our results and conclusons. 2. Tradtonal Scenaro-Based Optmzaton Very few stuatons n the real world are completely devod of rsk. In fact, a person would be hard-pressed to recall a sngle decson n ther lfe that was completely rsk-free. In the world of determnstc optmzaton, we often choose to gnore uncertanty n order to come up wth a unque and objectve soluton to a problem. However, n stuatons where uncertanty s at the core of the problem as t s n rsk management a dfferent strategy s requred. In the feld of optmzaton, there are varous approaches desgned to cope wth uncertanty. 2,3 In ths context, the exact values of the parameters (e.g. the data) of the optmzaton problem are not known wth absolute certanty, but may vary to a larger or lesser extent dependng on the nature of the factors they represent. In other words, there may be many possble realzatons of the parameters, each of whch s a possble scenaro. Tradtonal scenaro-based approaches to optmzaton, such as scenaro optmzaton and robust optmzaton, are effectve n fndng a soluton that s feasble for all the scenaros consdered and mnmzng the devaton of the overall soluton from the optmal soluton for each scenaro. These approaches, however, only consder a very small subset of possble scenaros, and the sze and complexty of models they can handle are very lmted.

Smulaton Optmzaton 573 2.1. Scenaro optmzaton Dembo 4 offers an approach to solvng stochastc programs based on a method for solvng determnstc scenaro subproblems and combnng the optmal scenaro solutons nto a sngle feasble decson. Imagne a stuaton n whch we want to mnmze the cost of producng a set J of fnshed goods. Each good j (j =1,...,n) has a per-unt producton cost c j assocated wth t, as well as an assocated utlzaton rate a j of resources for each fnshed good. In addton, the plant that produces the goods has a lmted amount of each resource ( =1,...,m), denoted by b. We can formulate a determnstc mathematcal program for a sngle scenaro s (the scenaro subproblem, or SP) as follows: SP: z s = mnmze Subject to n c s jx j, (1) j=1 n a s jx j = b s for =1,...,m, (2) j=1 x j 0 for j =1,...,n, (3) where c s,a s,andb s, respectvely, represent the realzaton of the cost coeffcent, the resource utlzaton, and the resource avalablty data under scenaro s. Consder, for example, a company that manufactures a certan type of Maple door. Dependng on the weather n the regon where the wood for the doors s obtaned, the costs of raw materals and transportaton wll vary. The company s also consderng whether to expand producton capacty at the faclty where doors are manufactured, so that a total of sx scenaros must be consdered. The sx possble scenaros and assocated parameters for Maple doors are shown n Table 1. The frst column corresponds to the partcular scenaro; Column 2 denotes whether the faclty s at current or expanded capacty; Column 3 shows the probablty of each capacty scenaro; Column 4 denotes the weather (dry, normal, or wet) for each scenaro; Column 5 provdes the probablty for each weather nstance; Column 6 denotes the probablty for each scenaro; Column 7 shows the cost assocated wth each Table 1. Possble scenaros for Maple doors. Scen Cap P (C) Wther P (W ) P (Scen) Cost c j Utl a j Aval b j 1 Curr 50% Dry 33% 1/6 L H L 2 Norm 33% 1/6 M L L 3 Wet 33% 1/6 H L L 4 Exp 50% Dry 33% 1/6 L H H 5 Norm 33% 1/6 M L H 6 Wet 33% 1/6 H L H

574 M. Better et al. scenaro (L = low, M = medum, H = hgh); Column 8 denotes the utlzaton rate of the capacty (L = low, H = hgh); and Column 9 denotes the expected avalablty assocated wth each scenaro. The model SP needs to be solved once for each of the sx scenaros. The scenaro optmzaton approach can be summarzed n two steps: (1) Compute the optmal soluton to each determnstc scenaro subproblem SP. (2) Solve a trackng model to fnd a sngle, feasble decson for all scenaros. The key aspect of scenaro optmzaton s the trackng model n step 2. For llustraton purposes, we ntroduce a smple form of trackng model. Let p s denote the estmated probablty for the occurrence of scenaro s. Then, a smple trackng model for our problem can be formulated as follows: Mnmze s p s ( j c s j x j z s ) 2 + s p s ( j a s j x j b s ) 2, (4) Subject to x j 0 for j =1,...,n. (5) The purpose of ths trackng model s to fnd a soluton that s feasble under all the scenaros, and penalzes solutons that dffer greatly from the optmal soluton under each scenaro. The two terms n the objectve functon are squared to ensure nonnegatvty. More sophstcated trackng models can be used for dfferent purposes. In rsk management, for nstance, we may select a trackng model that s desgned to penalze performance below a certan target level. 2.2. Robust optmzaton Robust optmzaton may be used when the parameters of the optmzaton problem are known only wthn a fnte set of values. The robust optmzaton framework gets ts name because t seeks to dentfy a robust decson.e. a soluton that performs well across many possble scenaros. In order to measure the robustness of a gven soluton, dfferent crtera may be used. Kouvels and Yu dentfy three crtera: (1) absolute robustness; (2) robust devaton; and (3) relatve robustness. 5 We llustrate the meanng and relevance of these crtera, by descrbng ther robust optmzaton approach. Consder an optmzaton problem where the objectve s to mnmze a certan performance measure such as cost. Let S denote the set of possble data scenaros over the plannng horzon of nterest. Also, let X denote the set of decson varables, and P the set of nput parameters of our decson model. Correspondngly, let P s dentfy the value of the parameters belongng to scenaro s, andletf s dentfy the set of feasble solutons to scenaro s. The optmal soluton to a specfc scenaro s

Smulaton Optmzaton 575 s then z s = f(xs,p s )= mn f(x, P s ). (6) X F s We assume here that f s convex. The frst crteron, absolute robustness, also known as worst-case optmzaton, seeks to fnd a soluton that s feasble for all possble scenaros and optmal for the worst possble scenaro. In other words, n a stuaton where the goal s to mnmze the cost, the optmzaton procedure wll seek the robust soluton, z R, that mnmzes the cost of the maxmum-cost scenaro. We can formulate ths as an objectve functon of the form z R =mn{max f(x, P s )}. (7) s S Varatons to ths basc framework have been proposed (see, for example, Ref. 5) to capture the rsk-averse nature of decson makers, by ntroducng hgher moments of the dstrbuton of z s n the optmzaton model, and mplementng weghts as penalty factors for nfeasblty of the robust soluton wth respect to certan scenaros. The problem wth both of these approaches, as wth most tradtonal optmzaton technques that attempt to deal wth uncertanty, s ther nablty to handle a large number of possble scenaros. Thus, they often fal to consder events that, whle unlkely, can be catastrophc. Recent approaches that use nnovatve smulaton optmzaton technques overcome these lmtatons by provdng a practcal, flexble framework for rsk management and decson makng under uncertanty. 3. Smulaton Optmzaton Smulaton optmzaton can effcently handle a much larger number of scenaros than tradtonal optmzaton approaches, as well as multple sources and types of rsk. Modern smulaton optmzaton tools are desgned to solve optmzaton problems of the form Mnmze F (x) (Objectve functon), Subject to Ax b (Constrants on nput varables), g l G(x) g u (Constrants on output measures), l x u (Bounds), where the vector x of decson varables ncludes varables that range over contnuous values and varables that only take on dscrete values (both nteger values and values wth arbtrary step szes). 6 The objectve functon F (x) s, typcally, hghly complex. Under the context of smulaton optmzaton, F (x) could represent, for example, the expected value of the probablty dstrbuton of the throughput at a factory; the ffth percentle of the dstrbuton of the net present value (NPV) of a portfolo of nvestments; a measure of the lkelhood that the cycle tme of a process wll be lower than a

576 M. Better et al. desred threshold value; etc. In general, F (x) represents an output performance measure obtaned from the smulaton, and t s a mappng from a set of values x to a real value. The constrants represented by nequalty Ax b are usually lnear (gven that nonlnearty n the model s embedded wthn the smulaton tself), and both the coeffcent matrx A and the rght-hand sde values correspondng to vector b are known. The constrants represented by nequaltes of the form g l G(x) g u mpose smple upper and/or lower bound requrements on an output functon G(x) that can be lnear or nonlnear. The values of the bounds g l and g u are known constants. All decson varables x are bounded and some may be restrcted to be dscrete, as prevously noted. Each evaluaton of F (x) andg(x) requres an executon of a smulaton of the system. By combnng smulaton and optmzaton, a powerful desgn tool results. Smulaton enables fast, nexpensve and nondsruptve examnaton and testng of a large number of scenaros pror to actually mplementng a partcular decson n the real envronment. As such, t s quckly becomng a very popular tool n ndustry for conductng detaled what-f analyss. Snce smulaton approxmates realty, t also permts the ncluson of varous sources of uncertanty and varablty nto forecasts that mpact performance. The need for optmzaton of smulaton models arses when the analyst wants to fnd a set of model specfcatons (.e. nput parameters and/or structural assumptons) that leads to optmal performance. On the one hand, the range of parameter values and the number of parameter combnatons are too large for analysts to enumerate and test all possble scenaros; hence, they need a way to gude the search for good solutons. On the other hand, wthout smulaton, many real-world problems are too complex to be modeled by tractable mathematcal formulatons that are at the core of pure optmzaton methods lke scenaro optmzaton and robust optmzaton. Ths creates a conundrum; as shown above, pure optmzaton models alone are ncapable of capturng all the complextes and dynamcs of the system; hence, one must resort to smulaton, whch cannot easly fnd the best solutons. Smulaton optmzaton resolves ths conundrum by combnng both methods. Optmzers desgned for smulaton embody the prncple of separatng the method from the model. In such a context, the optmzaton problem s defned outsde the complex system. Therefore, the evaluator (.e. the smulaton model) can change and evolve to ncorporate addtonal elements of the complex system, whle the optmzaton routnes reman the same. Hence, there s a complete separaton between the model that represents the system and the procedure that s used to solve optmzaton problems defned wthn ths model. The optmzaton procedure usually based on metaheurstc search algorthms uses the outputs from the system evaluator, whch measures the mert of the nputs that were fed nto the model. On the bass of both current and past

Smulaton Optmzaton 577 Fg. 1. Coordnaton between the optmzaton engne and the smulaton. evaluatons, the method decdes upon a new set of nput values (Fg. 1 shows the coordnaton between the optmzaton engne and the smulaton model). Provded that a feasble soluton exsts, the optmzaton procedure deally carres out a specal search where the successvely generated nputs produce varyng evaluatons, not all of them mprovng, but whch over tme provde a hghly effcent trajectory to the globally best solutons. The process contnues untl an approprate termnaton crteron s satsfed (usually based on the user s preference for the amount of tme devoted to the search). As stated before, the uncertantes and complextes modeled by the smulaton are often such that the analyst has no dea about the shape of the response surface.e. the soluton space. There exsts no closed-form mathematcal expresson to represent the space, and there s no way to gauge whether the regon beng searched s smooth, dscontnuous, etc. Whle ths s enough to make most tradtonal optmzaton algorthms fal, metaheurstc optmzaton approaches, such as tabu search 7 and scatter search, 8 overcome ths challenge by makng use of adaptve memory technques and populaton samplng methods that allow the search to be conducted on a wde area of the soluton space, wthout gettng stuck n local optma. The metaheurstc-based smulaton optmzaton framework s also very flexble n terms of the performance measures the decson-maker wshes to evaluate. In fact, the only lmtaton s not on the sde of the optmzaton engne, but on the smulaton model s ablty to evaluate performance based on specfed values for the decson varables. In order to provde n-depth nsghts nto the use of smulaton optmzaton n the context of rsk management, we present some practcal applcatons through the use of llustratve examples. 4. Illustratve Examples 4.1. Selectng rsk-effcent project portfolos Companes n the Petroleum and Energy (P&E) Industry use project portfolo optmzaton to manage nvestments n exploraton and producton, as well as power plant acqustons. 9,10 Decson makers typcally wsh to maxmze the return on

578 M. Better et al. nvested captal, whle controllng the exposure of ther portfolo of projects to varous rsk factors that may ultmately result n fnancal losses. In ths example, we look at a P&E company that has 61 potental projects n ts nvestment funnel. For each project, the pro-forma revenues for a horzon of 10 20 perods (dependng on the project) are gven as probablty dstrbutons. To carry t out, each project requres an ntal nvestment and a certan number of busness development, engneerng and earth scences personnel. The company has a budget lmt for ts nvestment opportuntes, and a lmted number of personnel of each skll category. In addton, each project has a probablty of success (POS) factor. Ths factor has a value between 0 and 1 and affects the smulaton as follows: let us suppose that Project A has a POS = 0.6; therefore, durng the smulaton, we expect that we wll be able to obtan the revenues from Project A n 60% of the trals, whle n the remanng 40%, we wll only ncur the nvestment cost. The resultng probablty dstrbuton of results from smulatng Project A would be smlar to that shown n Fg. 2, where about 40% of the trals would have negatve returns (.e. equal to the nvestment cost), and the remanng 60% would have returns resemblng the shape of ts revenue dstrbuton (.e. equal to the smulated revenues mnus the nvestment cost). Projects may start n dfferent tme perods, but there s a restrcted wndow of opportunty of up to three years for each project. The company must select a set of projects to nvest n that wll best further ts corporate goals. Probably, the best-known model for portfolo optmzaton s rooted n the work of Nobel laureate Harry Markowtz. Called the mean-varance model, 11 t s based on the assumpton that the expected portfolo returns wll be normally dstrbuted. The model seeks to balance rsk and return n a sngle objectve functon, as follows. Fg. 2. Sample probablty dstrbuton of returns for a sngle smulated project.

Smulaton Optmzaton 579 Gven a vector of portfolo returns r and a covarance matrx Q of returns, we can formulate the model as follows: Maxmze r T w kw T Qw, (8) Subject to c w = b, (9) w {0, 1}, (10) where k represents a coeffcent of the frm s rsk averson, c represents the ntal nvestment n project, w s a bnary varable representng the decson whether to nvest n project, andb s the avalable budget. We wll use the mean-varance model as a base case for the purpose of comparng wth other selected models of portfolo performance. To facltate our analyss, we make use of the OptFolo r software that combnes smulaton and optmzaton nto a sngle system specfcally desgned for portfolo optmzaton. 12,13 We examne three cases, ncludng Value-at-Rsk (VaR) mnmzaton, to demonstrate the flexblty of ths method to enable a varety of decson alternatves that sgnfcantly mprove upon tradtonal mean-varance portfolo optmzaton, and llustrate the flexblty afforded by smulaton optmzaton approaches n terms of controllng rsk. The results also show the benefts of managng and effcently allocatng scarce resources lke captal, personnel, and tme. The weghted average cost of captal, or annual dscount rate, used for all cases s 12%. 4.1.1. Case 1: Mean-varance approach In ths frst case, we mplement the mean-varance portfolo selecton method of Markowtzdescrbed above. The decson s to determne partcpaton levels (0 or 1) n each project wth the objectve of maxmzng the expected NPV of the portfolo whle keepng the standard devaton of the NPV below a specfed threshold of $140 M. We denote the expected value of the NPV by µ NPV, and the standard devaton of the NPV by σ NPV. Ths case can be formulated as follows: Maxmze µ NPV (objectve functon), (11) Subject to σ NPV < $140M (requrement), (12) c x b (budget constrant), (13) p j x P j j (personnel constrants), (14) All projects must start n year 1, (15) x {0, 1} (bnary decsons). (16) The optmal portfolo has the followng performance metrcs: µ NPV = $394M, σ NPV = $107M, P(5) NPV = $176M,

580 M. Better et al. 140 NPV 120 100 80 60 40 20 0 136 182 227 273 319 364 410 456 502 547 593 639 684 Fg. 2. Dstrbuton of returns for mean-varance approach. where P (5) NPV denotes the ffth percentle of the resultng NPV probablty dstrbuton (.e. the probablty of the NPV beng lower than the P (5) value s 5%). The bound mposed on the standard devaton n Eq. (12) does not seem bndng. However, due to the bnary nature of the decson varables, no project addtons are possble wthout volatng the bound. Fgure 2 shows a graph of the probablty dstrbuton of the NPV obtaned from 1000 replcatons of ths base model. The thn lne represents the expected value. 4.1.2. Case 2: Rsk controlled by ffth percentle In the context of rsk management, statstcs such as varance or standard devaton of returns are not always easy to nterpret, and there may be other measures that are more ntutve and useful. For example, t provdes a clearer pcture of the rsk nvolved f we say there s a 5% chance that the portfolo return wll be below some value X than to say that the standard devaton s $107M. The former analyss can be easly mplemented n a smulaton optmzaton approach by mposng a requrement on the ffth percentle of the resultng dstrbuton of returns, as we descrbe here. In Case 2, the decson s to determne partcpaton levels (0, 1) n each project wth the objectve of maxmzng the expected NPV of the portfolo, whle keepng the ffth percentle of the NPV dstrbuton above the value of the ffth percentle obtaned n Case 1. In ths way, we seek to move the dstrbuton of returns further to the rght, so as to reduce the lkelhood of undesred outcomes. Ths s acheved by mposng the requrement represented by Eq. (18) n the model below. In other words, we want to fnd the portfolo that produces the maxmum average return, as long as no more than 5% of the tral observatons fall below $176M. In addton, we allow for delays n the start dates of projects, accordng to

Smulaton Optmzaton 581 wndows of opportunty defned for each project. In order to acheve ths, n the smulaton model we have created copes of each project that are shfted by one, two, or three perods nto the future (accordng to the wndows of opportunty defned for each project). Mutual exclusvty clauses among these stage copes of a project ensure that only one start date s selected. For example, to represent the fact that Project A can start at tme t =0, 1, or 2, we nclude the followng mutual exclusvty clause as a constrant: Project A 0 + Project A 1 + Project A 2 1. The subscrpt followng the project name corresponds to the allowed start dates for the project, and the constrant only allows at most one of these to be chosen: Maxmze µ NPV, (17) Subject to P (5) NPV > $176M (requrement), (18) c x b (budget constrant), (19) p j x P j j (personnel constrants), (20) x 1 (mutual exclusvty), (21) m M x {0, 1} I (bnary decsons), (22) where m denotes the set of mutually exclusve projects related to project. In ths case we have replaced the standard devaton wth the ffth percentle as a measure of rsk contanment. The resultng portfolo has the followng attrbutes: µ NPV = $438M, σ NPV = $140M, P (5) NPV = $241M. By usng the ffth percentle nstead of the standard devaton as a measure of rsk, we are able to obtan an outcome that shfts the dstrbuton of returns to the rght, compared wth Case 1, as shown n Fg. 3. Ths case clearly outperforms Case 1. Although the dstrbuton of returns exhbts a wder range (remember, we are not constranng the standard devaton here), not only we do obtan sgnfcantly better fnancal performance but we also acheve a hgher personnel utlzaton rate and a more dverse portfolo. 4.1.3. Case 3: Probablty maxmzng and VaR In Case 3, the decson s to determne partcpaton levels (0, 1) n each project wth the objectve of maxmzng the probablty of meetng or exceedng the mean NPV found n Case 1. Ths objectve s expressed n Eq. (23) of the followng model: Maxmze P (NPV $394M), (23)

582 M. Better et al. 140 NPV 120 100 80 60 40 20 0 136 182 227 273 319 364 410 456 502 547 593 639 684 Fg. 3. Dstrbuton of returns for Case 2. Subject to c x b (budget constrant), (24) p j x P j j (personnel constrants), (25) x 1 (mutual exclusvty), (26) m M x {0, 1} (bnary decsons). (27) Ths case focuses on maxmzng the chance of achevng a goal and essentally combnes performance and rsk contanment nto one metrc. The probablty n (23) s not known apror, hence, we must rely on the smulaton to obtan t. The resultng optmal soluton yelds a portfolo that has the followng attrbutes: µ NPV = $440M, σ NPV = $167M, P (5) = $198M. Although ths portfolo has a performance smlar to the one n Case 2, t has a 70% chance of achevng or exceedng the NPV goal (whereas Case 1 had only a 50% chance). As can be seen n the graph of Fg. 4, we have succeeded n shftng the probablty dstrbuton even further to the rght, therefore ncreasng our chances of exceedng the returns obtaned wth the tradtonal Markowtz approach. In addton, n Cases 2 and 3, we need not make any assumpton about the dstrbuton of expected returns. As a related corollary to the last case, we can conduct an nterestng analyss that addresses VaR. In tradtonal (securtes) portfolo management, VaR s defned as the worst expected loss under normal market condtons over a specfc tme nterval and at a gven confdence level. In other words, VaR measures how much the nvestor can lose, wth probablty = α, over a certan tme horzon. 14

Smulaton Optmzaton 583 140 NPV 120 100 80 60 40 20 0 136 182 227 273 319 364 410 456 502 547 593 639 684 Fg. 4. Dstrbuton of returns for Case 3. In the case of project portfolos, we can defne VaR as the probablty that the NPV of the portfolo wll fall below a specfed value. Gong back to our present case, the manager may want to lmt the probablty of ncurrng negatve returns. In ths example, we formulate the problem n a slghtly dfferent way: we stll want to maxmze the expected return, but we lmt the probablty that we ncur a loss to α = 1% by usng the requrement shown n Eq. (29) as follows: Maxmze µ NPV, (28) Subject to P (NPV < 0) 1% (requrement), (29) c x b (budget constrant), (30) p j x P j j (personnel constrants), (31) x 1 (mutual exclusvty), (32) m M x {0, 1} (bnary decsons). (33) The portfolo performance under ths scenaro s µ NPV = $411M, σ NPV = $159M, P (5) = $195M. The results from the VaR model turn out to be slghtly nferor to the case where the probablty was maxmzed. Ths s not a surprse, snce the focus of VaR s to lmt the probablty of downsde rsk, whereas before, the goal was to maxmze the probablty of obtanng a hgh expected return. However, ths last analyss could prove valuable for a manager who wants to lmt the VaR of the

584 M. Better et al. Fg. 5. Process flowchart for Level 1 patents. selected portfolo. As shown here, for ths partcular set of projects, a very good portfolo n fnancal terms can stll be selected wth that objectve n mnd. 4.2. Rsk management n busness process desgn Very common measures of process performance are the cycle tme, a.k.a turnaround tme, the throughput, and the operatonal cost. For our present example, we consder a process manager at a hosptal emergency room (ER). Typcally, emergency patents that arrve at the ER present dfferent levels of crtcalty. In our example, we consder two levels: Level 1 patents are very crtcal and requre mmedate treatment; Level 2 patents are not as crtcal and must undergo an assessment by a trage nurse before beng assgned to an ER room. Usually, a patent wll have a choce n the care provder he or she prefers; hence, there s an nherent rsk of lost busness related to the qualty of servce provded. For nstance, f patents must spend a very long tme n the ER, there s a rsk that patents wll prefer to seek care at another faclty on subsequent, follow-up vsts. Fgure 5 shows a hgh-level flowchart of the process for Level 1 patents (the process for Level 2 patents dffers only n that the Fll out regstraton actvty s done before the Transfer to room actvty, durng the trage assessment). The current operaton conssts of 7 nurses, 3 physcans, 4 patent care techncans (PCTs), 4 admnstratve clerks and 20 ER rooms. Ths current confguraton has a total operatng cost (.e. wages, supples, ER rooms costs, etc.) of $52,600 per 100 h of operaton, and the average tme a Level 1 patent spends at the ER (.e. cycle tme) has been estmated at 1.98 h. Hosptal management has just ssued the new operatonal budget for the comng year and has allocated to the ER a maxmum operatonal cost of $40,000 per 100 h of operaton. The manager has three weeks to make changes to the process to ensure complance wth the budget, wthout deteroratng the current servce levels. The process manager has stated hs goals as: to mnmze the average cycle tme for Level 1 patents, whle ensurng that the operatonal cost s below $40,000 per 100 h of operaton. Snce arrval tmes and servce tmes n the process are stochastc, we use a smulaton model to smulate the ER operaton, and OptQuest to fnd the best confguraton of resources n order to acheve the manager s goals. We obtan the

Smulaton Optmzaton 585 Fg. 6. Redesgned process for Level 1 patents. followng results: operatonal cost = $36, 200 (three nurses, three physcans, one PCT, two clerks, twelve ER rooms); average cycle tme = 2.08 h. The confguraton of resources above (shown n parentheses) results n the lowest possble cycle tme gven the new operatonal budget for the ER. Obvously, n order to mprove the servce level at the current operatonal budget, t s necessary to redesgn the process tself. The new process proposed for Level 1 patents s depcted n Fg. 6. In the proposed process, the Receve treatment and Fll out regstraton actvtes are now done n parallel, nstead of n sequence. The smulaton of ths new process, wth the resource confguraton found earler, reduces the average cycle tme from 2.08 to 1.98 h. When desgnng busness processes, however, t s very mportant to set goals that are not subject to the Law of Averages that s, the goals should be set so that they accommodate a large percentage of the demand for the product or servce that the process s ntended to delver. In the above example, t s possble that the probablty dstrbuton of cycle tme turns out to be hghly skewed to the rght, so that even f the average cycle s under 2 h, there could be a very large number of patents who wll spend a much longer tme n the ER. It would be better to restate the manager s performance goals as follows: ensure that at least 95%ofLevel1patentswllspendnomorethan2hntheERwhle ensurng that the operatonal cost s under $40,000. Ths gves a clear dea of the servce level to whch the process manager aspres. If we re-optmze the confguraton of resources wth the new goal, we obtan the followng results: operatonal cost = $31, 800 (four nurses, two physcans, two PCTs, two clerks, nne ER rooms); average cycle tme = 1.94 h; 95th percentle of cycle tme = 1.99 h.

586 M. Better et al. Through the use of smulaton optmzaton, we have obtaned a process desgn that comples wth the new budget requrements as well as wth an mproved servce level. If mplemented correctly, we can be sure that 95% of the crtcal patents n the ER wll be ether released or admtted nto the hosptal for further treatment n less than 2 h. Thus, the rsk of havng unsatsfed patents s mnmzed. 5. Results and Conclusons Practcally every real-world stuaton nvolves uncertanty and rsk, creatng a need for optmzaton methods that can handle uncertanty n model data and nput parameters. We have brefly descrbed two popular methods, scenaro optmzaton and robust optmzaton, that seek to overcome lmtatons of classcal optmzaton approaches for dealng wth uncertanty and that undertake to fnd hgh-qualty solutons that are feasble under as many scenaros as possble. However, these methods are unable to handle problems nvolvng moderately large numbers of decson varables and constrants, or nvolvng sgnfcant degrees of uncertanty and complexty. In these cases, smulaton optmzaton s becomng the method of choce. The combnaton of smulaton and optmzaton affords all the flexblty of the smulaton engne n terms of defnng a varety of performance measures and rsk profles, as desred by the decson maker. In addton, as we demonstrate through two practcal examples, modern optmzaton engnes can enforce requrements on one or more outputs from the smulaton, a feature that scenaro-based methods cannot handle. Ths affords the user dfferent alternatves for controllng rsk, whle ensurng that the performance of the system s optmzed. The combnaton of smulaton and optmzaton creates a tool for decson makng that s fast (a smulaton runs at a small fracton of real tme, and the optmzaton gudes the search for good solutons wthout the need to enumerate all possbltes), nexpensve, and nondsruptve (solutons can be evaluated wthout the need to stop the normal operaton of the busness, as opposed to plot projects, whch can also be qute expensve). Fnally, smulaton optmzaton produces results that can be conveyed and grasped n an ntutve manner, provdng the user wth an especally useful and easy-to-use tool for dentfyng mproved busness decsons under rsk and uncertanty. References 1. D. Vose, Rsk Analyss: A Quanttatve Gude (John Wley and Sons, Chchester, 2000). 2. M. Fukushma, How to deal wth uncertanty n optmzaton some recent attempts, Int. J. Inform. Tech. Decs. Makng 5(4) (2006) 623 637. 3. H. Eskandar and L. Rabelo, Handlng uncertanty n the analytc herarchy process: A stochastc approach, Int. J. Inform. Tech. Decs. Makng 6(1) (2007) 177 189. 4. R. Dembo, Scenaro optmzaton, Ann. Oper. Res. 30 (1991) 63 80.

Smulaton Optmzaton 587 5. P. Kouvels and G. Yu, Robust Dscrete Optmzaton and Its Applcatons (Kluwer, Dordrecht, Netherlands, 1997), pp. 8 29. 6. J. P. Kelly, Smulaton optmzaton s evolvng, INFORMS J. Comput. 14(3) (2002) 223 225. 7. F. Glover and M. Laguna, Tabu Search (Kluwer, Norwell, MA, 1997). 8. F. Glover, M. Laguna and R. Martí, Fundamentals of scatter search and path relnkng, Control Cybernet. 29(3) (2000) 653 684. 9. W. J. Haskett, Optmal apprasal well locaton through effcent uncertanty reducton and value of nformaton technques, n Proc. Socety of Petroleum Engneers Annual Techncal Conference and Exhbton (Denver, CO, 2003). 10. W. J. Haskett, M. Better and J. Aprl, Practcal optmzaton: Dealng wth the realtes of decson management, n Proc. Socety of Petroleum Engneers Annual Techncal Conference and Exhbton (Houston, TX, 2004). 11. H. Markowtz, Portfolo selecton, J. Fnance 7(1) (1952) 77 91. 12. J. Aprl, F. Glover and J. P. Kelly, Portfolo optmzaton for captal nvestment projects, n Proc. 2002 Wnter Smulaton Conference, eds.s.chck,t.sanchez, D. Ferrn and D. Morrce (2002), pp. 1546 1554. 13. J. Aprl, F. Glover and J. P. Kelly, Optfolo A smulaton optmzaton system for project portfolo plannng, n Proc. 2003 Wnter Smulaton Conference, eds.s.chck, T. Sanchez, D. Ferrn and D. Morrce (2003), pp. 301 309. 14. S. Bennnga and Z. Wener, Value-at-rsk (VaR), Math. Educ. Res. 7(4) (1998) 1 7.