Robust Optimization with Multiple Ranges: Theory and Application to R & D Project Selection

Transcription

1 Robust Optmzaton wth Multple Ranges: Theory and Applcaton to R & D Project Selecton Ruken Düzgün Auréle Thele July 2010 Abstract We present a robust optmzaton approach when the uncertanty n objectve coeffcents s descrbed usng multple ranges for each coeffcent. Ths settng arses when the value of the uncertan coeffcents, such as cash flows, depends on an underlyng random varable, such as the effectveness of a new drug. Tradtonal robust optmzaton wth a sngle range per coeffcent would requre very large ranges n ths case and lead to overly conservatve results. In our approach, the decson-maker lmts the number of coeffcents that fall wthn each range; he can also lmt the number of coeffcents that devate from ther nomnal value n a gven range. Modelng multple ranges requres the use of bnary varables n the uncertanty set. We show how to address ths ssue to develop tractable reformulatons and apply our approach to a R&D project selecton problem when cash flows are uncertan. Furthermore, we develop a robust rankng heurstc, where the project manager ranks the projects accordng to denstes (rato of cash flows to development costs) or Net Present Values, whle ncorporatng the budgets of uncertanty but wthout requrng any optmzaton procedure. Whle both densty-based and NPV-based rankng heurstcs perform very well n experments, the NPVbased heurstc performs better; n partcular, t fnds the truly optmal soluton more often. Keywords: robust optmzaton; multple ranges; project selecton; robust rankng. 1 Introducton Robust optmzaton addresses data uncertanty by assumng that uncertan parameters belong to a bounded, convex uncertanty set and maxmzng the mnmum value of the objectve over Department of Industral and Systems Engneerng, Lehgh Unversty, Bethlehem, PA 18015, rud207@lehgh.edu. Department of Industral and Systems Engneerng, Lehgh Unversty, Bethlehem, PA 18015, aurele.thele@lehgh.edu. Phone: Work supported n part by NSF Grant CMMI and an IBM Faculty Award. 1

2 that uncertanty set, whle ensurng feasblty for the worst-case value of the constrants. Ths approach was poneered by Soyster [27] n the 1970s; however, hs model requred that each uncertan parameter be equal to ts worst-case value, and thus was deemed too conservatve for practcal mplementaton. In the md-1990s, Ben-Tal and Nemrovsk ([5, 6, 7]), El-Ghaou and Lebret [20] and El-Ghaou et al. [21] presented tractable mathematcal reformulatons, based on ellpsodal uncertanty sets, that turned lnear programmng problems nto second-order cone problems and reduced the conservatsm of Soyster s [27] approach. Furthermore, Ben-Tal and Nemrovsk [8] studed robust optmzaton appled to conc quadratc and semdefnte programmng. Ben-Tal et. al. [4] provdes an extensve book treatment of robust optmzaton wth an emphass on ellpsodal sets. Bertsmas and Sm [12, 13] and Bertsmas et al. [11] nvestgated n the early 2000s the specal case where the uncertanty set s a polyhedron. Specfcally, the uncertanty set conssts of range forecasts (confdence ntervals) for each parameter and a constrant called a budgetof-uncertanty constrant, whch lmts the number of coeffcents that can take ther worst-case value. The approach preserves the degree of complexty of the problem (the robust counterpart of a lnear problem s lnear) and allows the decson-maker to control the degree of conservatsm of the soluton. Robust optmzaton remans the focus of substantal research efforts; recent theoretcal advances nclude the development of adjustable optmzaton (Ben-Tal et. al. [3]) and adaptable optmzaton (Bertsmas and Caramans [10]) to ncorporate nformaton revealed over tme, whle robust optmzaton has been successfully appled to a varety of areas, from nventory management (Bertsmas and Thele [15]) to revenue management (Adda and Peraks [1]) to wreless sensor networks (Ye and Ordonez [29]). The reader s referred to Bertsmas et. al. [9] for a comprehensve revew paper and to Düzgün and Thele [18] for an overvew of dynamc models n robust optmzaton. In ths paper, we focus on problem setups where the ranges taken by uncertan coeffcents depend on the realzatons of underlyng random varables. Ths problem arses for nstance n R&D project selecton, where project cash flows are uncertan but also depend on the effectveness of the underlyng compound tested by the pharmaceutcal company. Project selecton requres bnary varables, for whch ellpsodal uncertanty sets are ll-suted as they lead to nonlnear nteger problems (Bertsmas and Sm [14]); therefore, we wll focus throughout ths paper on polyhedral uncertanty sets, specfcally, sets wth range forecasts and budget-of-uncertanty constrants. The tradtonal robust optmzaton approach, wth a sngle range for each uncertan coeffcent, would requre very large ranges and thus lead to overly conservatve solutons. The mult-range robust optmzaton approach we propose allows for a more realstc descrpton of 2

3 uncertanty. Whle Metan and Thele [25] ntroduces multple ranges for product demand n a smple two-stage robust revenue management problem for a sngle product, that approach s an hybrd between robust optmzaton and stochastc programmng, where the decson-maker gans advance knowledge of the range that product demand wll fall nto. It ncorporates nether bnary varables nor budgets of uncertanty and has a sngle source of uncertanty, and focuses on the mpact of scenaro probabltes on the qualty of the optmal soluton. Benstock [16] ncorporates multple ranges to classcal mean-varance problems n portfolo management. In that framework, rsk s dscretzed by constructng uncertanty bands around estmates of the return shortfalls of the assets; observatons n the same band represent smlar levels of rsk. The user specfes rough estmates of the frequences wth whch shortfalls fall wthn each band, and constructs ntervals for the actual number of occurrences n the bands, leadng to a hstogram model. The proposed model s solved usng a cuttng-plane algorthm wth a convex master problem (called the mplementor problem) and a mxed-nteger subproblem (called the adversaral problem), whch generates cuts for the mplementor problem. The results suggest that the cuttng-plane algorthm mplemented can successfully address uncertanty models wth non-convextes for large-scale problems and s of nterest n applcatons beyond fnance. The Research and Development (R&D) project selecton problem has been studed snce the 1960s. Competton between R&D companes has ncreased the mportance of fundng projects that would best meet ther needs. Whle many methods to dentfy these projects have been nvestgated, there s no consensus on ther practcal effectveness. Martno [24] presents varous methods avalable for selectng R&D projects, n partcular rankng methods, economc models, portfolo or optmzaton models and ad-hoc methods. Early studes of the R&D project selecton problem mostly use rankng methods. The most common ones are scorng models and the analytc herarchy procedure (AHP) (see Baker and Freeland [2] for a lterature revew on these approaches.) Economc methods, whch are recommended by Martno [24], consder the cash flows nvolved wth the project, usng metrcs such as net present value (NPV), nternal rate of return (IRR) and cash flow payback. Portfolo optmzaton methods mplement mathematcal programmng to fnd the projects, from a canddate project lst, that would gve the maxmum payoff to the frm. For nstance, Chlds and Trants [17] use a real optons framework n order to examne dynamc R&D nvestment polces and valuaton of R&D programs, and Stummer and Hedenberger [28] use a mult-objectve nteger programmng model to determne all effcent (Pareto-optmal) portfolos. Data envelopment analyss (DEA) s another method for solvng R&D project selecton decsons. Lnton et. al [23] proposed ths method to splt decsons on project portfolos nto 3

4 accept, consder-further and reject sub-groups. Elat et. al [19] use a methodology based on an extended DEA that quantfes some qualtatve concepts embedded n the balanced scorecard (BSC) approach. They employ a DEA-BSC model frst to evaluate ndvdual R&D projects, and then to evaluate alternatve R&D portfolos. R&D project selecton problems nclude hgh levels of uncertanty n future cash flows; however, the most common approaches to project selecton replace uncertan parameters by ther expected values or rely on tradtonal, stochastc descrptons of randomness, although quantfyng accurately the probablty dstrbutons of future cash flows for a R&D project and the probabltes of project success s very dffcult n practce. As mentoned above, the classcal robust optmzaton approach also suffers from over-conservatsm n ths setup due to the large ranges that would be requred to mplement t. Ths makes mult-range robust optmzaton a novel theoretcal extenson of robust optmzaton wth valuable practcal applcatons. Contrbutons. Our contrbutons to the lterature are as follows. We defne the mult-range robust optmzaton framework and derve tractable reformulatons. In partcular, we show that the lnear relaxaton of the worst-case problem (whch computes the worst-case objectve for a gven strategy and requres bnary varables to model multple ranges) has nteger optmal solutons n both robust optmzaton models we consder. We apply the approach to a R&D project selecton problem. We present a robust rankng heurstc to dentfy projects to fund wthout any optmzaton and test t n numercal experments. Our computatonal results suggest that, n ths settng, rankng projects accordng to Net Present Values rather than denstes (rato of cash flows to development costs), yelds hgher-qualty solutons,.e., solutons closer to optmalty. To the best of our knowledge, we are the frst to ncorporate the dea of robust rankng to a range- and budgets-of-uncertanty-based descrpton of uncertanty. Outlne. Secton 2 ntroduces the generc mult-range robust optmzaton approach. In Secton 3, we apply our methodology to a R&D project selecton problem. Secton 4 ntroduces heurstcs based on robust rankng. Numercal results are presented n Secton 5. Fnally, Secton 6 contans concludng remarks. 4

5 2 Mult-Range Robust Optmzaton We frst provde a quck overvew of tradtonal (one-range) robust optmzaton before dscussng ts lmtatons and presentng the approach wth multple ranges. 2.1 Revew of One-Range Robust Optmzaton Let c be the objectve coeffcent vector of sze n. The general model we consder s: max c x s.t. x X, (1) where X s the constrant set of x, whch may nclude ntegralty constrants. We further assume that all decson varables are non-negatve, whch s a natural assumpton to make n the context of operatons management, where decson varables represent for nstance orderng quanttes or amounts transported; ths assumpton s partcularly justfed n the project management applcaton descrbed n Secton 3, where decson varables are bnary. We consder the case where the vector c s uncertan, whch wll correspond to uncertan project cash flows n Secton 3. We can apply the tradtonal one-range robust optmzaton approach that Bertsmas and Sm developed n [11], [13] to the uncertan parameter c. Specfcally, we model c, = 1,..., n, as an uncertan parameter n the nterval [c ĉ, c + ĉ ]. (Note that, snce decson varables are non-negatve, the worst case wll always be acheved at the low end of the range; therefore, knowledge of the hgh end of the range s not requred to mplement the approach and the confdence nterval does not have to be symmetrc.) Defne the scaled devaton y such that c = c + ĉ y for all. In lne wth Bertsmas and Sm [13], the scaled devatons are assumed to belong to the polyhedral uncertanty set: P = {y y Γ, y 1, }. The parameter Γ [0, n] s the budget of uncertanty whch specfes the maxmum number of coeffcents that can devate from ther nomnal values. If Γ = 0, the only feasble element n P s the zero vector, so that the problem reduces to ts determnstc counterpart. If Γ = n, each uncertan parameter takes ts worst case value. Takng a value of Γ between 0 and n allows the decson-maker to acheve a trade-off 5

6 between the nomnal performance of the determnstc model and the rsk protecton of the most conservatve model. Whle the setup above assumes that the project cash flows are ndependent, t s straghtforward to extend the approach to the case where cash flows are correlated by usng the mult-factor model descrbed n Bertsmas and Sm [13]. Ths extenson s left to the reader. The robust problem becomes: max mn s.t. s.t. x X. ( c + ĉ y ) x y P (2) Theorem 2.1 (One-range robust optmzaton (Bertsmas and Sm [13])). The robust counterpart of Problem (1) s: max s.t. c x Γz 0 z x X (3) z + z 0 ĉ x,, z, z 0 0. Proof. Ths s a drect applcaton of Bertsmas and Sm [13] to Problem (2) after njectng the fact that the worst case s always acheved for y 0 for all and that the decson vector x s non-negatve. Lmtatons of the One-Range Framework. Ths framework s not well-suted for cases where the uncertan parameters are drven by underlyng random varables. For nstance, n the case of drug trals, the potental revenue of a drug wll depend on the effectveness of the actve chemcal compound beng tested; f the performance of the compound s dsappontng, the resultng cash flows wll fall n a low range; f the compound s effectve n healng a wde array of patents, cash flows wll fall n a hgh range. Tryng to encompass all possble values of the cash flows nto a sngle nterval wll generate an overly large range forecast, wth an lldefned nomnal value lackng any realstc meanng f t falls between the two ntervals, as the decson-maker never beleves he wll observe such cash flows. Ths s partcularly a concern n robust optmzaton, snce t can be shown (see Bertsmas and Sm [13]) that at optmalty, the worst-case coeffcents of Problem (2) wll be equal to ether ther worst case or ther nomnal value, assumng the budget of uncertanty s nteger. Hence, t s 6

7 mportant for the relevance of the robust optmzaton approach and ts adopton by practtoners that the optmal values of the uncertan coeffcents correspond to values these parameters can actually take. Smlar arguments can be made n the case of demand for a new product, the sales of whch depends on the degree of popularty or market share that the product wll acheve. Such tems, wth a wde range of possble outcomes, requre a fner-graned representaton of uncertanty than the one-range model s able to provde. 2.2 The Case Wth Multple Ranges Instead of havng a sngle range of uncertanty, we now assume that we have multple ranges that the uncertan values can take values from. For notatonal smplcty, we assume that each uncertan parameter has the same number m of possble ranges, but the approach can be extended easly to the case where the number of ranges depends on the uncertan parameter. We wll analyze two cases: 1. The smple case where the (pessmstc) decson-maker assumes that each uncertan parameter takes the worst value of the range t falls nto, and the maxmum number of parameters that can fall n a gven range s bounded by a budget of uncertanty. 2. The more complex case where the decson-maker extends the setup n Case 1 to ntroduce another famly of budgets of uncertanty lmtng the number of parameters that can take ther worst-case value n a gven range. Ths allows some parameters to be equal to ther nomnal value, rather than ther worst-case value, n that range Case 1: Wthout a Budget For the Devatons Wthn The Ranges Let c k, resp. c k+ be the lower, resp. hgher, bound of range k for parameter, = 1,..., n, k = 1,..., m. The budget Γ k constrans the maxmum number of coeffcents that can fall wthn range k, k = 1,..., m. (The decson maker can also choose to ntroduce these budgets only for the lowest ranges, correspondng to the most conservatve outcomes, to lmt the conservatsm of the approach.) 7

8 The robust problem can be formulated as a mxed-nteger programmng problem (MIP): max x X mn c,y s.t. c x c k y k ck ck+ y k,, k, y k = 1, k=1 c = c k, k=1 y k Γ k, y k,, k, {0, 1},, k. (4) The tractablty of the robust optmzaton paradgm reles on the decson-maker s ablty to convert the nner mnmzaton problem nto a maxmzaton problem, of such a structure that the master maxmzaton problem (ncorporatng the outer maxmzaton problem and the new nner maxmzaton problem) can be solved effcently. Strong dualty has emerged as the tool of choce to mplement ths converson (Bertsmas and Sm [13]); however, the model of uncertanty we propose requre the use of nteger (bnary) varables, whch makes the rewrtng of a mnmzaton problem as an equvalent maxmzaton one consderably more dffcult. It s thus natural to nvestgate whether the lnear relaxaton of the nner mnmzaton problem n Problem (4) yelds bnary y varables at optmalty. Ths s the purpose of Lemma 2.2. Lemma 2.2. The lnear relaxaton of the nner mnmzaton problem: mn c,y s.t. c x c k y k ck ck+ y k,, k, y k = 1, k=1 c = c k, k=1 y k Γ k, y k,, k, {0, 1},, k, (5) has a bnary optmal vector y for any gven nteger Γ l and nonnegatve vector x. Proof. The objectve s a mnmzaton over c of c x where c = c k k=1 for all and x s non- 8

9 negatve. Hence, c k wll take the mnmum value n ts range,.e., c k = ck y k at optmalty for all, k. It follows that c = c k y k for all and the feasble set s reduced to m k=1 yk = 1,, k=1 n yk Γ k, k, and y k {0, 1},, k,. The feasble set of the lnear relaxaton has bnary extreme ponts, thus provng the lemma. Ths allows us to derve a tractable reformulaton of Problem (4). Theorem 2.3. Problem (4) has the equvalent robust lnear formulaton: max s.t. p γ k Γ k k=1 k=1 p γ k z k ck x,, k, x X γ k, z k z k 0,, k. (6) Proof. As n the proof of Lemma 2.2, we notce that, due to the non-negatvty of the vector x, the optmal objectve coeffcents n the robust optmzaton framework are always acheved at the low end of the range. Therefore, we can rewrte the group of constrants: as: c k y k c k c k+ y k, c = c = We use Lemma 2.2 to rewrte Problem (5) as: k=1 c k y k. c k, k=1 mn c,y,u s.t. k=1 y k = 1, k=1 c k y k x, y k Γ k, 0 y k, k, 1,, k, (7) whch s a lnear programmng problem wth a non-empty, bounded feasble set. We can then nvoke strong dualty to reformulate the mnmzaton as a maxmzaton problem,.e., replace the prmal formulaton by ts dual. Re-njectng yelds Problem (6). 9

10 2.2.2 Case 2: Wth a Budget For the Devatons Wthn the Ranges In practce, t s unlkely that every sngle uncertan parameter wll take the worst-case value of the range t falls n. The purpose of ths secton s to extend the robust optmzaton approach presented n Secton to the case where the manager also decdes how many parameters, at most, can take the worst-case value n the ranges they are n. As before, the uncertan coeffcents satsfy: c = c k c k,, k=1 y k ck ck+ y k y k = 1,, k=1 y k Γ k, k, y k {0, 1},, k.,, k, Because we need to defne the devaton of each parameter wthn ts gven range, we further assume that the nomnal value of parameter n range k, denoted c k, s known for all = 1,..., n and k = 1,..., m. The measure of uncertanty for parameter of range k s then defned as ĉ k = c k ck for all = 1,..., n and k = 1,..., m. Agan, because the decson varables are non-negatve, the part of the range forecast above the nomnal value wll not be used n the robust optmzaton approach and the optmal uncertan coeffcents satsfy: c = ( c k ĉ k z k ) y k, k=1 where z k s the scaled devaton of coeffcent, = 1,..., n, from ts nomnal value n range k, k = 1,..., m wth: k=1 0 z k z k Γ, 1,, k. Lemma 2.4. For any feasble x X, the worst-case objectve can be computed as a mxed-nteger 10

11 programmng problem: mn c,y k=1 ) x ( c k y k ĉ k u k s.t. u k yk,, k, u k Γ, k=1 y k = 1, k=1 y k Γ k, y k u k, k, {0, 1},, k, 0,, k. (8) Proof. Defnng u k = zk yk, we obtan: c k = c k y k ĉ k u k,, k, where 0 u k yk. The result follows from the fact that t s suboptmal to have zk u k = 0 for any, k. > 0 when The followng lemma s key to the tractablty of the robust optmzaton approach we present. Lemma 2.5. The constrant matrx of Problem (8) s totally unmodular. Proof. A matrx obtaned by a pvot operaton on a totally unmodular matrx s totally unmodular (Nemhauser and Wolsey [26]). The matrx C below s the constrant matrx of Problem (8) where the columns represent the varables [u y]. I nm I nm 1 1 nm 0 1 nm C = 0 m nm A n nm 0 n nm B m nm where I nm s the nm nm dentty matrx and matrx A n nm has the followng structure:

12 Specfcally, A s defned as: 1 f ( 1)m < j m A,j = 0 otherwse, Matrx B m nm has the followng structure: Specfcally, B has the followng structure: ) B = (I m m I m m I m m We wll do the followng operatons on C. 1) Let R j s the j th row and R j s the j th column of C. By dong the row operatons, we obtan the r-th verson of the matrx C whch s denoted by (C) r For j = nm + 2 to nm m, do R j + R nm+1 R nm+1 and call the resultng matrx (C) m. Now, for j = 1 to nm, do R j + R j+nm R j+nm. and call the resultng matrx (C) m(n+1). 2) At the end of these row/colum operatons we obtan the matrx (C) m(n+1), whch s I nm 0 nm (C) m(n+1) 1 1 nm 0 1 nm = 0 m nm A n nm 0 n nm B m nm To conclude the proof, we wll need the followng result. 12

13 Lemma 2.6. (Nemhauser and Wolsey [26], p. 544) Let A be a (0, 1, 1) matrx wth no more than two nonzero elements n each column. Then A s totally unmodular f and only f the rows of A can be parttoned nto two subsets Q 1 and Q 2 such that f a column contans two nonzero elements, the followng statements are true: a. If both nonzero elements have the same sgn, then one s n a row contaned n Q 1 and the other s n a row contaned n Q 2. b. If the two nonzero elements have opposte sgn, then both are n rows contaned n the same subset. The matrx (C) m(n+1) satsfes these condtons of total unmodularty. Snce a matrx obtaned by pvot operatons on a totally unmodular matrx s also totally unmodular, our constrant matrx C s totally unmodular. Theorem 2.7. The robust counterpart s equvalent to the followng problem, wth a lnear objectve and lnear constrants added to the orgnal feasble set: max p z k γ k Γ k Γ γ 0 k=1 k=1 s.t. π k + γ 0 ĉ k x,, k, π k + p γ k z k ck x,, k, x X γ k, γ 0, π k, zk 0, k. (9) Proof. Snce the constrant matrx of Problem (8) s totally unmodular (Lemma 2.5) and the rght-hand-sde values of the constrants are nteger, the lnear relaxaton of the problem has nteger optmal solutons. It follows from strong dualty, because the feasble set of the lnear relaxaton of Problem (8) s non-empty and bounded, that Problem (8) and the dual of ts lnear relaxaton have the same optmal objectve. Renjectng the dual yelds Problem (9). 3 Applcaton to Project Management 3.1 Problem Setup We now apply the settng descrbed n Secton 2 to an example n R&D project selecton. The manager must decde n whch projects to nvest over a fnte tme horzon. Each project has known cash requrements at each stage of ts development (for notatonal smplcty, we assume all 13

14 projects have the same number of stages; ths corresponds for nstance to the case of drug trals of small, medum and large scale leadng to possble approval by the Food and Drug Admnstraton n the Unted States), but cash flows durng and at the end of development are uncertan and depend on underlyng random varables, such as the effectveness of the actve compounds or the market response to the new product. These random varables are realzed only once (e.g., the drug compound s effectve for the dsease beng treated), so that the coeffcents for a gven project all fall n the low range or all fall n the hgh range. We allow for cash flows to be generated durng development as the company mght fle for patents or generate monetary value from the results of the ntermedary stages; the bggest cash flows, however, wll be generated at the end of the development phase. We assume that there are two uncertanty ranges for each cash flow: a project mght be successful and has hgh cash flows, or t mght be a falure and has low cash flows. Note that cash flows are non-zero, even n the low state, as the drug mght be found to be effectve on a subset of the patents and retan some market value. Because no new nformaton s revealed durng the tme horzon n ths robust optmzaton settng, we do not consder the possblty of stoppng a project after t has started, before the end of the development phase. The goal s to maxmze the worst-case cumulatve Net Present Value of the projects the manager nvests n, where the worst case s computed over the uncertanty sets descrbed n Sectons and 2.2.2, subject to constrants on the amount of money avalable at each tme perod to spend on development. We wll use the followng notaton throughout the paper. General and cost parameters. n : number of projects, T : number of tme perods, S : number of development phases for each project, B t : avalable budget for the tme perod t where t = 1,..., T, CD,s : development cost of project n phase s, r : dscount rate at each tme perod. Cash flow parameters. 14

15 CF,s l : CF l,s : CF,s l+ : ĈF l,s : CF h,s CF h,s : CF h+,s lower bound of cash flow of project n phase s f the project s unsuccessful, nomnal value of the cash flow of project n phase s f the project s unsuccessful, upper bound of cash flow of project n phase s f the project s unsuccessful, measure of uncertanty for cash flow of project n phase s n low range (= CF l,s CF,s l ), : lower bound of cash flow of project n phase s f the project s successful, nomnal value of the cash flow of project n phase s f the project s successful, : upper bound of cash flow of project n phase s f the project s successful, ĈF h,s : measure of uncertanty for cash flow of project n phase s n hgh range (= CF h,s CF,s h ), Robust optmzaton parameters and decson varables. Γ l : Γ : x,τ : y : uncertanty budget that restrcts the number of projects whose cash flows wll be n the low range, uncertanty budget that restrcts the number of projects whose cash flows devate from ther nomnal value wthn ther gven range, 1 f the project s selected to begn at tme τ, 0 otherwse, 1 f the project s n ts low range (unsuccessful), 0 otherwse. The determnstc project selecton problem where each project can be selected at most once s formulated as: max s.t. T S+1 τ=1 τ=max{1,t S+1} T x,τ 1, τ=1 [ S ] x,τ CF,s (1 + r) τ 1 (1 + r) s t CD,t τ+1 x,τ B t x,τ {0, 1},, τ. 3.2 Case 1: Robust Optmzaton Wthout a Budget for the Devaton Wthn the Ranges Frst, we consder the smple case where the manager only lmts the number of projects that wll be unsuccessful, and assumes that each cash flow wll take ts worst case wthn a gven range. t (10) 15

16 Problem (4) becomes: max x s.t. mn CF,s,y s.t. T S+1 τ=1 x,τ (1 + r) τ 1 CF l,s y CF l,s [ S CF,s (1 + r) s ] :Total cash flow over tme CF l+,s y (, s) :Cash flow nterval f n low range CF,s h (1 y ) CF,s h h+ CF,s (1 y ) (, s) :Cash flow nterval f n hgh range CF,s l + CF,s h = CF,s (, s) :Cash flow s ether hgh or low y Γ l :At most Γ l projects n low range y {0, 1}, CF,s l, CF,s h, CF,s 0 (, s) t CD,t τ+1 x,τ B t t :Budget constrant at each tme perod τ=max{1,t S+1} T x,τ 1, () :Each project started at most once τ=1 x,τ {0, 1},, τ. The theoretcal results n Secton show Problem (11) can be reformulated n a tractable manner. Theorem 3.1. Problem (11) s equvalent to the mxed-nteger programmng problem: (11) max s.t. T S+1 τ=1 S τ=max{1,t S+1} T S+1 τ=1 z l + z t x,τ 1, T S+1 τ=1 x,τ CF h,s (1 + r) τ+s 1 z l Γ l CD,t τ+1 x,τ B t, x,τ (1 + r) τ 1 S (CF h,s z t,, l CF,s ) (1 + r) s,, x,τ {0, 1},, τ, z, z 0,. (12) Proof. The proof s a straghtforward applcaton of Theorem

17 3.3 Case 2: Robust Optmzaton Wth a Budget for the Devaton Wthn the Ranges Assume that cash flows for project n phase s, wth = 1,..., n, s = 1,..., S, are ether n [CF l,s ĈF l,s, CF l,s + ĈF l,s] or [CF h,s ĈF h,s, CF h,s + ĈF h,s]. In lne wth the framework n Secton 2.2.2, they can be wrtten n mathematcal terms as: CF,s = (CF l,s ĈF l,s z l,s)y l + (CF h,s ĈF h,s z h,s)y h, wth 0 z,s l, zh,s 1, s and yj {0, 1} j {l, h}. Snce the coeffcents must belong to one of the two ranges, we only ntroduce a budget-of-uncertanty constrant on the number of coeffcents that fall nto ther low range. Gven feasble bnary varables x,τ (equal to 1 f project s started at tme τ and 0 otherwse), the worst-case cash flows are gven by: mn u l,u h,y T S+1 τ=1 x,τ (1 + r) τ 1 S CF l,s y l ĈF l,s u l,s + CF h,s y h ĈF h,s u h,s (1 + r) s s.t. u l,s yl,, s, u h,s yh,, s, y l + yh = 1, y l Γ l, y j S (u l,s + u h,s) Γ, {0, 1},,, j {l, h}, u l,s, uh,s 0,, s. (13) It s a drect applcaton of Lemma 2.5 that the constrant matrx of Problem (13) s totally unmodular. 17

18 The robust optmzaton problem s gven by: max x s.t. mn u l,u h,y T S+1 τ=1 x,τ (1 + r) τ 1 S CF l,s y l ĈF l,s u l,s + CF h,s y h ĈF h,s u h,s (1 + r) s s.t. u l,s yl,, s, u h,s yh,, s, y l + yh = 1, y l Γ l, y j S (u l,s + u h,s) Γ, {0, 1}, u l,s, uh,s 0,, s, t CD,t τ+1 x,τ B t, t, τ=max{1, t S+1} T S+1 τ=1 x,τ 1,,, j {l, h}, x,τ {0, 1},, τ. (14) The followng theorem provdes a tractable reformulaton of Problem (14). straghtforward applcaton of Theorem 2.7, we state t wthout proof., Because t s a Theorem 3.2. The robust optmzaton problem (14) s equvalent to the mxed-nteger program- 18

19 mng problem: max s.t. p (z l + z h ) Γ l γ l Γ γ 0 t τ=max{1,t S+1} T S+1 τ=1 x,τ 1, S π,s l + p γ l z l S π,s h + p z h π l,s + γ 0 π h,s + γ 0 T S+1 τ=1 T S+1 τ=1 CD,t τ+1 x,τ B t, T S+1 S x,τ τ=1 T S+1 S x,τ τ=1 t,, CF l,s,, (1 + r) τ+s 1 CF h,s,, (1 + r) τ+s 1 x,τ ĈF l,s,, s, (1 + r) τ+s 1 x,τ ĈF h,s,, s, (1 + r) τ+s 1 x,τ {0, 1},, τ, π,s l, πh,s 0, s, z l, zh 0,, γ l, γ 0 0. (15) The feasble set can be decomposed as follows: The frst two groups of constrants are the same as n the determnstc model, representng the maxmum amount of money to be allocated at each tme perod and the fact that a project can be started at most once. The thrd and fourth group of constrants are the dual constrants correspondng to the prmary varables y l and yh, respectvely, and ncorporate the nformaton about the nomnal values of the cash flows. Because one of these decson varables (ether y l or yh ) wll be non-zero for each at optmalty, by complementarty slackness, one of the dual constrants wll be tght for each, thus determnng p as a functon of the nomnal cash flow for that range and the other dual varables. Ths wll brng the nomnal cash flows back nto the objectve. The ffth and sxth group of constrants are the dual constrants correspondng to the 19

20 prmary varables u l s and uh s, respectvely, and ncorporate the nformaton about the uncertanty on the cash flows n each range. At most one of these decson varables (ether u l s or uh s ) wll be non-zero for each at optmalty; f t s non-zero, by complementarty slackness, one of the dual constrants wll be tght for each, thus determnng ether π l s or π h s as a functon of the uncertanty n that range and the other dual varables. (Otherwse the πs l and πh s varables wll be at zero.) Ths wll brng the cash flow uncertanty, through the half-range of the confdence ntervals, nto the objectve when needed. The other constrants are sgn constrants or bnary constrants. The robust formulaton (15) has n (3+T +2S)+2 decson varables and T +n(3+2s) constrants n addton to sgn and bnary constrants; therefore, the sze of the mxed-nteger programmng problem ncreases lnearly wth each of the parameters n, T, S (number of projects, length of tme horzon, number of development stages) when the others are kept constant. 4 Robust Rankng Heurstc Whle Problem (15) provdes an exact formulaton of the robust optmzaton problem for project management, we focus n ths secton on developng optmzaton-free heurstcs to provde a feasble soluton to the robust problem, whch would gve practtoners more nsghts nto the strategy they mplement and the mpact of the cash flow parameters. We are motvated by the fact that, when there s only one tme perod and one development phase (T = 1 and S = 1), the project selecton problem has the structure of a knapsack problem, for whch a well-known heurstc s to rank tems by decreasng order of densty (value to weght rato) and fll the knapsack untl the next tem n the lst does not ft (see, for nstance, Kellerer et. al. [22])). In partcular, we provde a robust rankng procedure to rank the projects wth uncertan cash flows; to the best of our knowledge, we are the frst to present such a rankng procedure n the context of robust optmzaton. We wll consder two ways to rank the projects: (a) accordng to decreasng densty, (b) accordng to decreasng Net Present Value. Method (a) s motvated by ts popularty to solve the generc knapsack problem; Method (b) s motvated by ts superor performance n the numercal experments provded n Secton 5 and the wdespread use of Net Present Value to select projects n practce. Once projects are ranked, we apply the greedy multple-knapsack heurstc descrbed n Kellerer et. al. [22] to generate a canddate soluton. Specfcally, we proceed down the ranked lst of projects and assgn project j to knapsacks t,..., t + S 1, wth t the smallest nteger such that the project development costs ft n all of these knapsacks capacty. 20

21 4.1 Case 1: Rankng for the Projects Wthout a Budget for the Devaton Wthn the Ranges Recall that, f there s no budget for the devaton wthn the ranges, the cash flows always take ther worst case wthn the range, and that the range (hgh or low) s only selected once,.e., the range does not change wth the development phase. The hgh-level dea s to () compute two rankngs, one usng the low range of the cash flows and the other usng the hgh range, () use the low-range rankng untl the budget of uncertanty has been used up, and then () use the hgh-range rankng. Rankng procedure. Step 1 Compute the followng parameters for all projects. Method (a): Denstes Method (b): Net Present Values a h = S a l = S CF h,s (1 + r) s CD,s CF l,s (1 + r) s CD,s S ( ) a h = h CF,s CD,s + (1 + r) s S ( ) a l = l CF,s CD,s + (1 + r) s For ether method, compute two rankngs: n decreasng order of a h, and n decreasng order of a l. Step 2 Add Γ l projects to your rankng lst correspondng to the projects wth the largest Γ l values of a l. Then proceed to Step 3. Step 3 Contnue untl all projects are ranked by choosng the unranked projects accordng to the largest values of a h, dscardng projects that have already been selected n Step 2. 21

22 4.2 Case 2: Rankng for the Projects Wth a Budget for the Devaton Wthn the Ranges In ths case, the cumulatve cash flow of a project can take four possble values (four cash flow measures): low value of the low range, nomnal value of the low range, low value of the hgh range, nomnal value of the hgh range. The hgh-level dea s to () compute four rankngs, one for each of the possble cash flow measures, () use the low value of the low range rankng untl one of the two budgets of uncertanty has been used up, () use ether the nomnal value of the low range rankng or the low value of the hgh range rankng (dependng on whch budget s not yet zero) untl the other budget of uncertanty has been used up, and (v) complete the procedure usng the nomnal value of the hgh range rankng. Rankng procedure. Step 1 Compute the four followng parameters for all projects. Method (a): Denstes A h = S a h = S Method (b): Net Present Values CF h,s (1 + r) s, A l CD = S CF l,s,s (1 + r) s, CD,s CF h,s (1 + r) s, a l CD = S CF l,s,s (1 + r) s. CD,s S ( A h = CD,s + CF h ),s S ( (1 + r) s, A l = CD,s + CF l ),s (1 + r) s, S ( ) a h = h CF,s S ( CD,s + (1 + r) s, a l = CD,s + ) l CF,s (1 + r) s. Usng ether method, create four rankngs, rankng projects n decreasng order of each of the parameters A h, Al, ah and a l. Step 2 Choose the projects correspondng to the largest mn(γ l, Γ) values n the rankng based on the a l parameters. Step 3 If Γ l > Γ (all cash flows wll now take ther nomnal value as we have used up the Γ budget, but the manager stll expects Γ l Γ projects to have cash flows n the low range), 22

23 add Γ l Γ projects to the ranked lst by usng the rankng based on the A l parameters, skppng the projects that have already been selected n Step 2. Step 4 If Γ Γ l > 0, add Γ Γ l projects to the ranked lst by usng the rankng based on the a h parameters, skppng the projects that have already been selected n Steps 2 and 3. Step 5 Contnue untl all projects are ranked by usng the rankng based on the A h parameters, skppng the projects that have already been selected n Steps 2, 3 and 4. 5 Numercal Example In ths secton, we nvestgate the practcal performance of our robust optmzaton models and heurstcs on an example. We focus on the case where T = 1 and S = 1, for whch the mathematcal formulaton wthout uncertanty becomes a well-known knapsack problem. Furthermore, we consder two uncertanty ranges: hgh (ndcated by the superscrpt h n relevant parameters) and low (ndcated by the superscrpt l). We have two man goals n ths experment: 1. Test whether the robust optmzaton framework does protect aganst downsde rsk as advertsed. 2. Test the performance of the heurstcs, (a) compared to the optmal soluton, (b) compared to each other. 5.1 Setup We frst provde the robust optmzaton formulatons for clarty. As ths s a specal case of Secton 3, the results are stated wthout proof. Case 1: Wthout a budget of uncertanty for the devatons wthn the ranges. The robust optmzaton problem becomes: max s.t. CF h x z l Γ l (1 + r) CD x B, z h (CF z l + z CF l ) 1 + r x, x {0, 1},, (16) z l, z 0,. 23

24 The project densty parameters (Method (a)) are gven by: A h = CF h, A (1 + r) CD l = CF l (1 + r) CD. The project Net Present Value parameters (Method (b)) are gven by: A h = CD + CF h 1 + r, Al = CD + CF l 1 + r. Case 2: Wth a budget of uncertanty for the devatons wthn the ranges. The robust optmzaton problem becomes: max s.t. p (z l + z h ) Γ l γ l Γ γ 0 CD x B, π l + p γ l z l CF l 1 + r x,, π h + p z h CF h 1 + r x,, (17) π l + γ 0 ĈF l 1 + r x,, π h + γ 0 ĈF h 1 + r x,, x {0, 1},, π l, πh, zl, zh 0, γ l, γ 0 0. The project densty parameters (Heurstc (a)) are gven by: A h = CF h (1 + r) CD, A l = CF l (1 + r) CD, a h = CF h, a (1 + r) CD l = CF l. (1 + r) CD The Net Present Value parameters (Heurstc (b)) are gven by: A h = CD + CF h 1 + r, Al = CD + CF l 1 + r, a h = CD + CF h 1 + r, al = CD + CF l 1 + r. 24

25 5.2 Numercal Results We tested our formulatons and heurstcs for 4 data sets. Data Sets 1 and 2 have 10 projects whle Data Sets 3 and 4 have 20 projects. In all cases, development costs (CD ) were generated usng a Unform dstrbuton n [80 120], nomnal values of low cash flows (CF l ) were generated usng Unform dstrbuton n ( ) CD, and nomnal values of hgh cash flows (CF h ) generated usng Unform dstrbuton n (2 3.5) CD. For all, the devaton parameters ĈF l, ĈF h were selected as 0.2 CF l, 0.2 CF h respectvely. Budget for development costs was set to 500 n all cases. In addton, Data Sets 3 and 4 were also solved for a value of the budget equal to 1,000. The same dstrbutons were used to compute the actual objectve usng random cash flows once the optmzaton problem had been solved. The probablty of the cash flows beng n the low range was taken equal to 0.5. Optmal soluton. We solved Problem (15) for each data set and for each (Γ, Γ l ) combnaton. Fgure 1 shows the hstogram of revenues for Data Set 1 and the determnstc model, where parameter values are taken equal to ther expected values, here (CF h + CF l )/2 for all (red lne wth square markers) as well as two robust models: (Γ, Γ l ) = (2, 1) and (Γ, Γ l ) = (3, 4) (blue lne wth lozenge markers and green lne wth trangle markers, respectvely). These budgets were chosen to have Γ > Γ l n one case and Γ < Γ l n the other. Ths hstogram was generated usng 1,000 scenaros. Fgure 1 suggests that robust optmzaton s more conservatve than ts nomnal counterpart (lmts upsde potental) but decreases the downsde rsk. Fgures 2 and 3 show the number of teratons versus budget of uncertanty Γ l for fve dfferent Γ values, for Data Sets 1 and 3, respectvely. Recall that Data Set 1 has 10 projects and Data Set 3 has 20. (Our observatons reman vald for other values of Γ, but the correspondng graphs were omtted for graph readablty.) We observe that, for each Γ value, the number of teratons n the robust optmzaton models does not dffer substantally from the number of teratons n the determnstc model when Γ l s close to ts bounds (Γ l = 0 or Γ l = 10), whch means that most projects are n the same uncertanty range.) When projects are more evenly assgned to low and hgh ranges (mddle values of Γ l ), the number of teratons ncreases, sometmes substantally (see Fgure 3, where the top curve corresponds to Γ = 10). Snce robust optmzaton maxmzes the worst-case cash flow over the uncertanty set, t s natural to evaluate how well t protects aganst downsde rsk. To do that, we compute the frst and ffth percentle of the dstrbuton of the random objectve where we have njected the optmal soluton, for Data Set 1 and all (Γ, Γ l ) combnatons, usng 1,000 scenaros. These results 25

26 Fgure 1: Hstogram of Revenues. Fgure 2: Number of Iteratons versus Budget of Uncertantes for Data Set 1, Budget=500. are shown n Tables 1 and 2, respectvely. Table 3 shows the expected value of the objectve for reference. We see that robust optmzaton does ndeed protect aganst downsde rsk, as evdenced n the ncrease n the values for the frst and ffth percentle, wth modest performance degradaton (decrease n average objectve value). It s mportant to note that the optmal soluton wll not change once Γ or Γ l ncreases past 26

27 Fgure 3: Number of Iteratons versus Budget of Uncertantes for Data Set 3, Budget=1000. the number of projects beng funded, whch we wll denote x. If p s the (estmated) probablty of project cash flows fallng n the low range, a decson-maker nterested n protectng hs cumulatve cash flow aganst adverse events wll select Γ l p x; however, x cannot be determned before the robust optmzaton problem has been solved (and depends somewhat on Γ and Γ l, although the dependence s mnmal n our experments: the manager nvests n 4 or 5 n all data sets wth budget equal to 500, and 10 or 11 projects out of 20 n Data Sets 3 and 4 when the budget s equal to 1,000). Therefore, we recommend that the decson-maker compute Tables 1, 2 and 3 for hs own project selecton problem, and choose an approprate (Γ, Γ l ) par based on the tradeoff between downsde rsk (measured ether by frst or ffth percentle) and performance (measured by average objectve) that he wshes to acheve. Also note that several (Γ, Γ l ) pars have the same optmal soluton, due to the use of bnary varables, and that what the manager ultmately needs to determne s the strategy he wll mplement, rather than a specfc (Γ, Γ l ) par, whch would only be used to compute the correspondng optmal strategy anyway. In the case of Data Set 1, we recommend to nvest n projects 1, 3, 6, 8, 10; ths strategy s optmal for (Γ, Γ l ) pars (3, 4), (4, 4), (0, 3) and (Γ, 3) for any Γ 5. Ths choce maxmzes both frst and ffth percentles over all possble (Γ, Γ l ) combnatons, achevng the bggest shft of the cumulatve cash flow dstrbuton to the rght. 27

28 Γ Γ l Table 1: Frst percentle values for each (Γ, Γ l ) par wth Data Set 1. Γ Γ l Table 2: Ffth percentle values for each (Γ, Γ l ) par wth Data Set 1. Γ Γ l Table 3: Expected revenue for each (Γ, Γ l ) par wth Data Set 1. Heurstcs. Table 4 compares the objectve functon values of Method (a) (rankng accordng to denstes) and Method (b) (rankng accordng to NPV) wth the optmal objectve functon value. For ths comparson, the development budget was taken equal to 500 n all data sets. # Opt. ndcates the number of tmes the heurstcs gves the same objectve functon value as the optmal soluton when all possble (Γ, Γ l ) pars are enumerated. We see that Method (b) generally performs better n terms of the number of tmes t fnds the optmal value: t performs as well as Heurstc (a) for Data Set 2 and performs much better for the other three data sets. Hghest performance s acheved for Data Set 3, where Heurstc (a) never found the optmal soluton whle Heurstc (b) had a success rato of 76%. We now evaluate the optmal soluton and the heurstc solutons. We generated 100 scenaros for the cash flows when the probablty of fallng nto the low range s 0.5. We mplemented the 28

29 Method (a) vs Optmal Method (b) vs Optmal % Obj. Df. # Opt. % Obj. Df. # Opt. Data Set / /121 Data Set / /121 Data Set / /441 Data Set / /441 Table 4: Optmal objectve functon value versus heurstc results. optmal and heurstc solutons of each data set for these scenaros (agan, wth a budget of 500 n all cases, to allow for easy comparson) and computed mean and standard devaton of the objectve (cumulatve dscounted cash flows). Table 5 shows the average percentage (absolute) dfference n mean and standard devaton of the smulaton results over all (Γ, Γ l ) combnatons for method (a) and method (b). We see that usng the heurstcs rather than the optmal soluton does not sgnfcantly change the objectve average, but does change the standard devaton more sgnfcantly. There was no sgn pattern n the mean dfference or standard devaton dfference, whch s why we only show absolute values. Method (a) vs Optmal Method (b) vs Optmal % Df. % Df. % Df. % Df. (Mean) (St. Dev.) (Mean) (St. Dev.) Data Set Data Set Data Set Data Set Table 5: Dfference between smulated optmal solutons versus heurstc solutons. 6 Conclusons We have presented an approach to robust optmzaton wth multple ranges for each uncertan coeffcent, derved tractable exact reformulatons and studed an applcaton to R&D project selecton. We have also provded a robust rankng heurstc and tested two possble rankng crtera: (a) accordng to project denstes, and (b) accordng to project Net Present Values. Numercal experments suggest that, whle both heurstcs exhbt good performance, Heurstc (b) performs better. The mult-range approach gves more flexblty to the decson-maker to specfy how many coeffcents can fall n each of the ranges and thus allows for a fner descrpton of uncertanty wthn the robust optmzaton framework. 29

30 References [1] Elode Adda and Georga Peraks. A robust optmzaton approach to dynamc prcng and nventory control wth no backorders. Math. Program., Ser. B, 107(97-129), [2] Norman Baker and James Freeland. Recent advances n R&D beneft measurement and project selecton methods. Management Scence, 21(10), [3] A. Ben-Tal, A. Goryashko, E. Gusltzer, and A. Nemrovsk. Adjustable robust solutons of uncertan lnear programs. Math. Program., Ser. A, 99: , [4] Aharon Ben-Tal, Laurent El-Ghaou, and Arkad Nemrovsk. Robust optmzaton. Prnceton Seres n Appled Mathematcs. Prnceton Unversty Press, Prnceton, NJ, [5] Aharon Ben-Tal and Arkad Nemrovsk. Robust convex optmzaton. Mathematcs of Operatons Research, 23(4): , [6] Aharon Ben-Tal and Arkad Nemrovsk. Robust solutons of uncertan lnear programs. Operatons Research Letters, 25:1 13, [7] Aharon Ben-Tal and Arkad Nemrovsk. Robust solutons of lnear programmng problems contamnated wth uncertan data. Mathematcal Programmng, 88: , [8] Aharon Ben-Tal and Arkad Nemrovsk. Robust optmzaton: Methodology and applcatons. Mathematcal Programmng, 92: , [9] Dmtrs Bertsmas, Davd Brown, and Constantne Caramans. Theory and applcatons of robust optmzaton. Techncal report, Unversty of Texas at Austn, Austn, TX, [10] Dmtrs Bertsmas and Constantne Caramans. Fnte adaptablty n multstage lnear optmzaton. IEEE Transactons on Automatc Control, to appear, [11] Dmtrs Bertsmas, Dessslava Pachamanova, and Melvyn Sm. Robust lnear optmzaton under general norms. Operatons Research Letters, 32: , [12] Dmtrs Bertsmas and Melvyn Sm. Robust dscrete optmzaton and network flows. Mathematcal Programmng B, 98:49 71, [13] Dmtrs Bertsmas and Melvyn Sm. The prce of robustness. Operatons Research, 52(1):35 53,

31 [14] Dmtrs Bertsmas and Melvyn Sm. Robust dscrete optmzaton under ellpsodal uncertanty sets. Techncal report, Massachusetts Insttute of Technology, Cambrdge, MA, [15] Dmtrs Bertsmas and Auréle Thele. A robust optmzaton approach to nventory theory. Operatons Research, 54(1):150168, [16] Danel Benstock. Hstogram models for robust portfolo optmzaton. Journal of Computatonal Fnance, 11(1):1 64, [17] Paul D. Chlds and Alexander J. Trants. Dynamc R&D nvestment polces. Management Scence, 45(10): , [18] Ruken Düzgün and Auréle Thele. Encyclopeda of Operatons Research and Management Scence, volume to appear, chapter Dynamc Models for Robust Optmzaton. Wley, New York, NY, [19] Harel Elat, Boaz Golany, and Avraham Shtub. Constructng and evaluatng balanced portfolos of R&D projects wth nteractons: A dea based methodology. European Journal of Operatonal Research, 172: , [20] Laurent El-Ghaou and Hervé Lebret. Robust solutons to least-square problems to uncertan data. SIAM Journal on Matrx Analyss and Applcatons, 18(4): , [21] Laurent El-Ghaou, Francos Oustry, and Hervé Lebret. Robust solutons to uncertan semdefnte programs. SIAM Journal on Optmzaton, 9:33 52, [22] Hans Kellerer, Ulrch Pferschy, and Davd Psnger. Knapsack Problems. Sprnger-Verlag, Berln, Hedelberg, Germany, [23] Jonathan D. Lnton, Steven T. Walsh, and Joseph Morabto. Analyss, rankng and selecton of r&d projects n a portfolo. R&D Management, 32(2): , [24] Joseph P. Martno. Research and Development Project Selecton. Wley Seres n Engneerng & Technology Management, New York, [25] Gokhan Metan and Auréle Thele. Scenaro desgn n robust nventory management. Techncal report, Lehgh Unversty, Bethlehem, PA, [26] George L. Nemhauser and Laurence A. Wolsey. Integer and Combnatoral Optmzaton. Wley-Interscence Seres n Dcrete Mathematcs and Optmzaton, USA,