Shades of Grey: A Critical Review of Grey-Number Optimization

Transcription

1 Utah Stat Unvrsty CEE Faculty Publcatons Cvl and Envronmntal Engnrng 2009 Shads of Gry: A Crtcal Rvw of Gry-Numbr Optmzaton Davd E. Rosnbrg Utah Stat Unvrsty Follow ths and addtonal works at: Part of th Cvl and Envronmntal Engnrng Commons Rcommndd Ctaton Davd Rosnbrg (2009) Shads of Gry: a crtcal rvw of gry-numbr optmzaton. Engnrng Optmzaton. 41(6), do / , pp Ths Artcl s brought to you for fr and opn accss by th Cvl and Envronmntal Engnrng at DgtalCommons@USU. It has bn accptd for ncluson n CEE Faculty Publcatons by an authorzd admnstrator of DgtalCommons@USU. For mor nformaton, plas contact dylan.burns@usu.du.

2 Shads of gry: a crtcal rvw of gry-numbr optmzaton Davd E. Rosnbrg Dpartmnt of Cvl and Envronmntal Engnrng Utah Watr Rsarch Laboratory Utah Stat Unvrsty 4110 Old Man Hll Logan, UT USA davd.rosnbrg@usu.du Tlphon: 001 (435) Fax: 001 (435) Submttd to: Engnrng Optmzaton July, 2008 R-submttd: Novmbr, 2008 Accptd: Dcmbr, 2008 Manuscrpt# GENO

3 #GENO , p. 1 Shads of gry: a crtcal rvw of gry-numbr optmzaton Davd E. Rosnbrg* Dpartmnt of Cvl & Envronmntal Engnrng and Utah Watr Rsarch Laboratory, Utah Stat Unvrsty, Logan, Utah, USA Abstract -- A gry numbr s an uncrtan numbr wth fxd lowr and uppr bounds but unknown dstrbuton. Gry numbrs fnd us n optmzaton to systmatcally and proactvly ncorporat uncrtants xprssd as ntrvals plus communcat rsultng stabl, fasbl rangs for th objctv functon and dcson varabls. Ths papr crtcally rvws thr us n lnar and stochastc programs wth rcours. It summarzs gry modl formulaton and soluton algorthms. It advancs multpl countr-xampls that yld rsk-pron gry solutons that prform wors than a worst-cas analyss and do not span th stabl fasbl rang of th dcson spac. Th papr suggsts rasons for th poor prformanc and dntfs condtons for whch t typcally occurs. It also dntfs a fundamntal shortcomng of gry stochastc programmng wth rcours and suggsts nw soluton algorthms that gv mor rsk-advrs solutons. Th rvw hlps clarfy th mportant advantags, dsadvantags, and dstnctons btwn rsk-pron and rsk-advrs gry-programmng and bst/worst cas analyss. Kywords: ntrval numbr; lnar program; stochastc program wth rcours; optmzaton wth uncrtanty. Introducton Ovr th last thr dcads, a varty of tchnqus hav surfacd to optmz n th fac of uncrtanty. Tchnqus such as chanc constrants, gry numbrs, fuzzy numbrs, probablstc, possblstc, flxbl, and stochastc programs wth rcours hav bn prsntd to systmatcally and proactvly ncorporat numrcal uncrtants n optmzaton modls (Sahnds, 2004). Hr, I rvw th proactv systms analyss tchnqu of gry numbr optmzaton and suggst som modfcatons. A gry numbr (also calld an ntrval numbr) taks an unknown dstrbuton btwn ± ± fxd lowr and uppr bounds,.., w [ w, w ] or w w w, whr w - and w, ar, rspctvly, th lowr and uppr bounds for w. In optmzaton, gry numbrs fnd us to systmatcally and proactvly ncorporat uncrtants xprssd as ntrvals plus communcat rsultng stabl, fasbl rangs for th objctv functon and all dcson varabls. Gry numbr programs ar dcomposd nto two computatonally-ffcnt, ntractng dtrmnstc sub-modls that ar thn solvd squntally. Dcson makrs * Corrspondng author. Emal: davd.rosnbrg@usu.du

4 #GENO , p. 2 us th rsultng gry ntrvals for dcson varabls to slct altrnatvs wthn proscrbd bounds. Gry numbrs hav bn appld to a varty of lnar (Ishbuch and Tanaka, 1990, Huang, Batz and Patry, 1992, Huang and Moor, 1993), mxd ntgr (Huang, Batz and Patry, 1995, Huang, 1998), quadratc (Huang and Batz, 1995, L and Huang, 2007), and stochastc (Huang and Loucks, 2000, Maqsood and Huang, 2003, Maqsood, Huang and Zng, 2004, Maqsood, Huang and Yomans, 2005, Maqsood, Huang, Huang and Chn, 2005, L, Huang and N, 2006, L and Huang, 2006, L, Huang and Batz, 2006, L, Huang, N and Huang, 2006, L, Huang and N, 2007, Rosnbrg and Lund, 2008) programs wth applcatons ncludng hypothtcal numrcal xampls for sold wast managmnt (Huang, Batz and Patry, 1992, Huang, Batz and Patry, 1995, Maqsood and Huang, 2003, Maqsood, Huang and Zng, 2004, L and Huang, 2006, L, Huang, N, N and Maqsood, 2006), watr rsourcs allocaton (Huang and Loucks, 2000, Maqsood, Huang and Yomans, 2005, Maqsood, Huang, Huang and Chn, 2005, L, Huang and N, 2006), and flood dvrson plannng (L, Huang and N, 2007). Lmtd practcal xampls nclud for watr qualty managmnt n Chna (Huang, 1998), sold wast managmnt for th cty of Rgna (L and Huang, 2006), and watr systm plannng n Amman, Jordan (Rosnbrg and Lund, 2008). Apart from th practcal xampls, most gry optmzaton work has focusd on modl formulatons and soluton tchnqus for hypothtcal xampls. Thr has bn lttl ntrprtaton of soluton rsults nor comparson to rsults from othr soluton approachs such as snstvty or bst/worst cas analyss. Snstvty analyss (also calld rang-of-bass) s a ractv approach that aftr soluton xamns how or whthr th optmal soluton changs wth changs n nput paramtr valus. Snstvty can b xamnd manually (changng an ndvdual nput, rsolvng, and notng soluton changs) or by analyzng th rang-of-bass output producd by most optmzaton solvrs. Unfortunatly, rang-of-bass rsults apply only to ndvdual changs n nput paramtrs and not combnatons of paramtr changs as accommodatd by gry-numbr approachs. Th long-standng approach of bst/worst-cas analyss smply solvs a program twc for th combnatons of paramtr valus that rprsnt th most favourabl (bst cas) and last favourabl (worst cas) condtons. Rosnbrg and Lund (2008) compard a gry stochastc program wth rcours to dtrmnstc-quvalnt, robust, and bst/worst cas formulatons and found that th gry modl prformd wors than th worst-cas analyss. Ths papr furthr xplors rasons for th rsk-pron prformanc, charactrzs condtons undr whch th problm s lkly to ars, and suggsts altrnatv soluton approachs that ar mor rsk advrs. Th papr s organzd as follows. Sctons 2 and 3 rvw problm formulaton and soluton tchnqus for gry lnar programs and gry stochastc programs wth rcours. Each scton dntfs problms wth xstng soluton tchnqus and charactrzs stuatons n whch ths problms ars. Scton 4 prsnts two altrnatv gry soluton tchnqus that ar mor rsk advrs. Scton 5 dscusss and hghlghts th mportant

5 #GENO , p. 3 advantags, dsadvantags, and dstnctons btwn rsk-pron and rsk-advrs gryprogrammng and bst/worst cas analyss. Scton 6 concluds. Gry lnar programmng Modl formulaton and soluton Early applcatons of gry lnar programmng ncorporatd gry numbrs nto th objctv functon (Ishbuch and Tanaka, 1990), constrant matrx (Huang and Moor, 1993, Tong, 1994), rght-hand sds of constrants, and all of th abov (Huang, Batz and Patry, 1992, Huang, Batz and Patry, 1995, Huang, 1996). Th procss works as follows. A lnar program wth objctv functon f, dcson varabls, objctv functon coffcnts c, constrant matrx coffcnts a j, and rght-hand-sd constrant coffcnts b j Max f = c (1a) s.t. a j b j, j (1b) 0, (1c) s turnd nto a gry lnar program by substtutng gry numbrs for ach of th nput coffcnts a ±, b ±, and c ±. Ths substtutons turn th objctv functon (f ± ) and dcson varabls ( ± ) gry and yld th gry lnar program (2): ± = c ± ± ± ± b j j Max f (2a) s.t. a ± j,, and (2b) ± 0, (2c) whr f ± s th uncrtan gry objctv functon wth lowr- and uppr bounds, rspctvly, f - and f ; smlarly for th othr dcson varabls and nput coffcnts. W solv gry lnar program (2) by dcomposng t nto two dtrmnstc sub-modls (Huang, 1996). Th two sub-modls corrspond to th uppr and lowr bounds of th gry objctv-functon and ntract. For a maxmzaton problm, solv th uppr bound sub-modl frst. Th uppr bound sub-modl corrsponds to f and uss nput coffcnts (c, a -, and b ) that maxmz th objctv functon and allow to rach thr uppr lmts. = c Max f (3a)

6 #GENO , p. 4 s.t. a j b j, j, and 0,. (3b) (3c) Modl (3) s also th bst-cas formulaton. Th lowr bound sub-modl corrsponds to f - and uss th nput coffcnts (c -, a, and b - ) that mnmz th objctv functon and forc towards thr lowr lmts. Max f = (4a) c s.t. a j b j, j, and (4b) 0,. (4c), (4d) opt Furthr, lowr bound sub-modl (4) also contans an ntracton constrant (4d) that rqurs th lowr bound soluton ( - ) to b n th soluton bass of th uppr-bound submodl ( opt ). Th ntracton constrant forcs soluton consstncy across th upprand lowr-bound sub-modls. Modl dscusson and comparsons Solutons to sub-modls (3) and (4) span maxmal, stabl, fasbl rangs for th objctv functon and dcson varabls. Ths rangs ar f ± opt = [f -, f ] and ± opt = [ -, ] whr f and ar solutons to uppr-bound sub-modl (3) and f - and - ar solutons to lowr-bound sub-modl (4). Th ntracton constrants allow dcson makrs to choos btwn thr rangs - and and b guarantd an objctv functon valu btwn f - and f. For a mnmzaton problm, th soluton ordr dscussd abov s rvrsd. Frst solv th lowr bound sub-modl (wthout ntracton constrant (4d)). Scond, solv th uppr bound sub-modl (wth an ntracton constrant, ). Furthr not that for a maxmzaton problm th bst cas formulaton s dntcal to th uppr bound sub-modl (3) whl th worst cas formulaton s smply th lowr bound sub-modl (4) wthout th ntracton constrant (4d). Bcaus th Bst/Worst cas solutons do not lmt th bass, solutons can hlp judg th systm s capablty to ralz a dsrd goal but do not ncssarly construct a st of stabl solutons for gnratng dcson altrnatvs. opt Problms For a smpl optmzaton problm

7 Max [ 3,4] [ 5,6] 1 #GENO , p (5) 1 0; 2 0 th gry lnar program formulaton and soluton algorthm gvs th gry soluton f ± opt = [5, 6]; 1 opt = [0]; and 2 opt = [1]. Hr th gry soluton s dntcal to th bst/worst cas soluton and th soluton bass for th two cass both contan 2. Howvr, whn th lowr bound of th objctv functon coffcnt for 2 changs from 5 to 2, th solutons dvrg (Tabl 1). In ths cas th gry lnar program dntfs 2 as part of th soluton bass n th uppr bound sub-modl whl skng a maxmum valu for th objctv functon, but th ntracton constrant xcluds 1 from th soluton bass for th lowrbound sub-modl. Consquntly, th objctv functon valu falls to 2. Absnt th ntracton constrant, th worst-cas analyss swtchs th soluton bass to 1 wth an mprovmnt n th objctv functon valu to 3. Th gry lnar program dntfs th maxmal, stabl fasbl rang for th objctv functon but prforms wors than th worst cas. Th lowr-bound sub-modl s mor constrand than th worst-cas submodl. Morovr, bcaus th ntracton constrant forcs 1 to stay at zro, th gry lnar program fals to rport th full stabl, fasbl rang for dcson varabl 1 (.., for unfavourabl coffcnt valus t s prfrabl to mplmnt 1 ). Smlar prformanc wors than th wors cas and falur to rport th full stabl fasbl rang for th dcson varabls s also sn whn th constrant matrx coffcnt for 2 changs from [1, 1] to [1, 4] (Tabl 1). Soluton mscharactrzaton and prformanc wors than th wors cas dntfy th gry soluton algorthm as rsk pron. Whn unfavourabl condtons ars, a dcson makr mplmntng th gry-numbr soluton could do bttr by adoptng a worst-cas or possbly othr soluton. 2 Condtons undr whch th problms ars A rtrospctv analyss of gry lnar program xampls (Huang, Batz and Patry, 1992, Huang and Moor, 1993, Huang, Batz and Patry, 1995, Tong, 1994, Huang, 1996) shows that svral of gry-numbr solutons prform wors than th worst cas (Tabl 2). Th rang of th gry objctv functon s wdr than objctv functon rang for th bst/worst cass. Howvr, ths analyss was complcatd by th facts that svral of th works ar () nfasbl as publshd (Huang, Batz and Patry, 1992, Huang, Batz and Patry, 1995, Tong, 1994), () nstad us a bst/worst cas soluton algorthm (wthout ntracton constrants) but call t a gry soluton tchnqu (Huang, Batz and Patry, 1992, Huang and Moor, 1993, Tong, 1994), or () do not prsnt nough nput data to vrfy th publshd soluton (Huang and Moor, 1993, Huang, 1998, Huang, 1996, Yh, 1996). Rworkng wth fasbl data and th gry soluton algorthm shows that svral of th xampls prform no dffrnt that bst/worst cas analyss (Huang, Batz and Patry,

8 #GENO , p , Huang and Moor, 1993, Huang, 1996) whl othrs prform wors that th worstcas analyss (Huang, Batz and Patry, 1995, Tong, 1994). In th formr xampls, th ntracton constrants do not bnd, whras n th latr cass thy do bnd and forc a soluton that would not othrws b dsrabl. Mor gnrally, w not that prformanc wors than th worst cas and soluton mscharactrzaton ar sn whnvr th gry-numbr ntracton constrants bnd. Th gry lnar program mposs rsks and costs to mantan maxmal stabl fasbl rangs for th dcson varabls. Th cost s th shadow valu (Lagrang multplr) assocatd wth th bndng gry lnar program ntracton constrant and th rsk s, undr condtons of unfavourabl paramtr valus, th objctv functon prforms wors than th worst cas. Gry Stochastc Programmng wth Rcours Gry numbr optmzaton has also bn appld to stochastc programs wth rcours, ncludng two-stag lnar programs (Huang and Loucks, 2000, Maqsood and Huang, 2003), two-stag mxd ntgr programs (Maqsood, Huang and Zng, 2004, L and Huang, 2006, Rosnbrg and Lund, 2008), fuzzy two-stag programs (Maqsood, Huang and Yomans, 2005, L, Huang and N, 2007), and mult-stag programs (L, Huang and N, 2006), among othrs. Gry numbr stochastc programs wth rcours ncorporat uncrtants xprssd as probablty dstrbutons and as ntrvals and work as follows. Dcsons ar parttond nto two typs. Prmary-stag dcsons ar takn bfor stochastc nformaton s ralzd. Aftr th stochastc nformaton s ralzd, scondstag (rcours) dcsons ar thn mplmntd to covr th shortfalls not mt by prmary-stag dcson lvls. Snc shortfalls dffr for dffrnt stochastc ralzatons, rcours dcsons apply only to a partcular ralzaton. Stochastc ralzatons ar dscrbd by a probablty dstrbuton, whch, for a stochastc lnar program, s approxmatd by a st of dscrt lvls and lklhoods (probablts). Togthr, prmary stag dcsons plus sts of rcours dcsons for ach stochastc ralzaton consttut th dcson portfolo mx of actons to rspond to th stochastc vnts. Modl formulaton and soluton A two-stag stochastc program that has prmary dcsons of watr allocaton targts and rcours dcsons that ar shortag allocatons D to ach sctor for unmt targts gvn watr avalablty lvls q n vnts, can b xprssd as follows (Huang and Loucks, 2000): Max f p c D (6a) = b, s.t. ( D ) q, (6b)

9 #GENO , p. 7 0, ; D 0,. (6c), Hr, f s th objctv functon, b ar bnfts from watr allocaton to watr us sctor, c ar costs or pnalts n watr avalablty vnt for dlvrng a volum blow th targt, D ar actual watr dlvrs to sctor n vnt, and q and p ar, rspctvly, th watr avalablts lvls and thr assocatd probablts. Togthr, p and q dscrb a st of dscrt watr avalablty lvls and probablts that approxmat th stochastc dstrbuton of watr avalablty. Substtutng gry numbrs for ach of th nput coffcnts (b ±, c ±, and q ± ) and dcson varabls ( ± and D ± ) turns two-stag lnar program (6) nto a gry two-stag lnar program (7): ± ± ± f = b p, ± ± ± s.t. ( D ) Max c D (7a) ± ± q, (7b) ± ± 0, ; D 0, (7c), Accordng to Huang and Loucks (2000), w solv gry two-stag lnar program (7) by dcomposng t nto two dtrmnstc sub-modls. Th two sub-modls corrspond to th uppr and lowr bounds of th gry objctv-functon f ± and ntract. Wth maxmzaton, uncrtan prmary-stag dcsons ( ) ar dntfd by frst solvng th uppr-bound sub-modl. Thn, th dtrmnd prmary-stag watr allocaton targts (now calld * opt ) ar usd to solv th lowr-bound sub-modl for uppr lmts on rcours dcsons (D ). Ths ordrng dntfs th maxmal and wdst rang of systm bnfts. Dcomposton and soluton rqurs thr stps. Stp 1. St up and solv th sub-modl to dntfy th objctv functon uppr bound, f. Us paramtr valus that maxmz bnfts and mnmz th nd for rcours-stag shortags ( ± and D - ) [.., larg bnfts (b ), small pnalts (c - ), and larg watr avalablty lvls (q )]. Th program solvs for long-trm dcson lvls ( ± ) snc ths valus nflunc th objctv functon postvly or ngatvly dpndng on rcours (short-trm) dcsons. Th uppr-bound sub-modl s: f = b, s.t. ( D ) Max p c D (8a) q, (8b) 0, ; D 0, (8c), Th soluton dntfs optmal prmary-stag watr allocaton targts ( * opt) and rcours-dcson shortag lvls (D - ) that maxmz nt bnfts undr favourabl conomc condtons. Watr allocaton targt lvls ( * opt) that maxmz systm bnfts bcom nputs to th lowr-bound sub-modl.

10 #GENO , p. 8 Stp 2. St up and solv th lowr bound sub-modl to dntfy f -. Us objctv functon coffcnts and constrant valus that mnmz nt bnfts and ncras th nd for shortags (D ) [.., small bnfts (b - ), larg pnalts (c ), and small watr avalablty lvls (q - )]. Th sol dcsons ar rcours-dcson shortag lvls (D ) that mnmz bnfts wth unfavourabl conomc condtons. Th lowr-bound submodl s: * f = b opt p, s.t. q ( opt D ), Max c D (9a) * (9b) D 0,, (9c) D D, (9d) opt, Hr, ntracton constrant (9d) nforcs a stabl fasbl rang for th rcours dcsons and th modl omts non-ngatvty constrants for th prmary-stag dcsons snc th uppr-bound sub-modl fxs th watr allocaton targts ( * opt). Stp 3. Solutons to sub-modls (8) and (9) span maxmal, stabl, fasbl rangs for th objctv functon and rcours-stag dcson varabls. Ths rangs ar f ± opt = [f -, f ], *, and D ± opt = [D -, D ] whr f, *, and D - ar solutons to uppr-bound submodl (8) and f - and D ar solutons to lowr-bound sub-modl (9). Modl dscusson and comparsons As wth th gry lnar program, th bst-cas formulaton for a stochastc maxmzaton problm s th sam as th uppr bound sub-modl (8). Th worst-cas formulaton allows prmary-stag dcsons, dos not hav an ntracton constrant, and s: f = b p, s.t. ( D ) Max c D (10a) q, (10b), 0, ; D 0, (10c) Hr - ar prmary-stag watr allocaton targts dntfd undr pssmstc conomc condtons. Th prmary dffrnc btwn th two-stag bst/worst cas and gry numbr formulatons s that th prmary-stag dcson varabl valus ar fxd across th gry-numbr sub-modls (ntracton) whras thy can chang btwn th bst and worst cas modls. Also, solutons to th bst/worst cas sub-modls do not ncssarly construct a st of stabl, fasbl rangs for slctng dcson altrnatvs. For a gry two-stag mnmzaton problm, th soluton algorthm s ssntally rvrsd. Frst, solv th lowr-bound sub-modl allowng prmary-stag dcsons and wthout ntracton constrant (.., sub-modl [10]). Scond, solv th uppr-bound sub-

11 #GENO , p. 9 modl usng th prmary-stag dcson valus fxd from th lowr-bound sub-modl soluton and wth an ntracton constrant on rcours-stag dcsons ( D D, ). opt, Problms and condtons undr whch thy ars Rtrospctv analyss comparng gry stochastc program xampl solutons (Huang and Loucks, 2000, Maqsood and Huang, 2003, Maqsood, Huang and Zng, 2004, Maqsood, Huang and Yomans, 2005, L, Huang and N, 2006, L and Huang, 2006, L, Huang and N, 2007, Rosnbrg and Lund, 2008) to thr bst/worst cas countrparts show that gry solutons always prform wors than thr worst cas countrparts (Tabl 3). Hr, th gry wdths for th gry objctv functons (f f - ) ar much wdr than th wdths assocatd wth th bst/worst cas sub-modls. Ths rsults dntfy gry stochastc solutons as vry rsk pron subjct to larg, undsrabl consquncs undr unfavourabl condtons that dcson makrs could mprov upon wth a dffrnt soluton approach such as solutons rcommndd by a worst-cas analyss. Prformanc s sgnfcantly wors than th worst cas bcaus th gry-soluton mthod chooss optmstc prmary-stag dcson valus to maxmz systm bnfts undr bstcas condtons. Furthr, th gry-soluton mthod fxs ths optmstc prmary-stag dcsons across th uppr- and lowr-bound sub-modls. For unfavourabl condtons, th gry-numbr approach must mplmnt th sam program of optmstc dcson valus to mantan fasbl rangs for dcsons across sub-modls. Ths sub-modl ntracton thn rqurs th gry-numbr approach to countract th fxd and optmstc program of prmary-stag dcsons wth many addtonal and mor costly rcours dcsons. Th worst-cas analyss s not smlarly constrand. Undr unfavourabl condtons, th worst-cas bass for prmary-stag dcsons can xclud, scal back, or dntfy mor approprat prmary-stag dcson targts. Morovr, fxng prmary-stag dcsons across gry-numbr sub-modls undrmns on of th tnants of gry numbr programmng: to dntfy th stabl, fasbl rang for th dcson varabls. Exstng gry-soluton tchnqus (Huang and Loucks, 2000) do not dntfy a rang for th most mportant prmary-stag plannng dcsons; thy only dntfy a gry rang for th lss mportant rcours-stag opratonal dcsons. I now propos som promsng gry-soluton tchnqus that () narrow th gry wdth of th objctv functon, and () also dntfy a stabl, fasbl rang for prmary-stag dcsons. Altrnatv gry-soluton tchnqus Hrn, I dvlop two altrnatv gry-soluton tchnqus for stochastc lnar programs, provd ratocnatons for ach tchnqu, and prsnt and dscuss soluton rsults for numrous xampls. Th frst tchnqu s trmd rsk advrs and sks to rduc th gry-wdth of th objctv functon by dntfyng a sngl st of prmary-stag dcsons and stabl, fasbl rangs for rcours dcsons. Th scond tchnqu mposs

12 #GENO , p. 10 ntracton constrants on both th prmary- and rcours-stag dcsons and dntfs stabl, fasbl rangs for both sts of dcsons. Ths approach s trmd an ntractng prmary-stag gry soluton tchnqu. Both soluton approachs guarant objctv functon valus qual or bttr than th worst-cas valu and work as follows. Rsk advrs tchnqu Th xstng rsk-pron gry-soluton tchnqu (Huang and Loucks, 2000) dntfs prmary-stag solutons by solvng th bst-cas (uppr-bound for a maxmzaton problm) sub-modl frst. Ths approach gvs a wd-rangng objctv functon valu bcaus sgnfcant (and costly) rcours dcsons ar rqurd should unfavourabl condtons (rprsntd by worst-cas paramtr valus) ars. Rvrsng th soluton procss to frst solv th worst-cas (lowr-bound for a maxmzaton problm) submodl can dntfy a mor approprat st of prmary-stag allocaton targts and rduc th nd for costly rcours dcsons. Soluton procss For a maxmzaton problm, th rsk advrs soluton procss works as follows. Stp 1. St up and solv worst-cas sub-modl (10) to dntfy th objctv functon lowr bound (f - ), prmary-stag allocaton targts undr unfavourabl paramtr valus ( - ), and uppr bounds on rcours dcsons (D ). Prmarystag watr allocaton targt lvls dntfd for pssmstc paramtr valus ( - opt) bcom nputs to th uppr-bound sub-modl. Stp 2. St up and solv an uppr-bound sub-modl to dntfy f. Hr, th sol dcsons ar rcours-dcson shortag lvls (D - ) that maxmz bnfts undr optmstc paramtr valus. Ths uppr-bound sub-modl s: f = b opt p, s.t. ( opt D ) Max c D (11a) q, (11b) D 0,, D D, opt, (11c) (11d) Intracton constrant (11d) nforcs a stabl fasbl rang for th rcours dcsons and th modl omts a non-ngatvty constrant for th prmary-stag dcsons snc watr allocaton targts ( - opt ) ar fxd n th lowr-bound submodl. D - ar th sol dcson varabls n lnar sub-modl (11), so w can formulat an analytcal soluton rul for D -. Ths rul s: for ach vnt, mnmz shortags (.., maxmz dlvry ncrass D D - ) to sctor wth th hghst

13 #GENO , p. 11 watr shortag cost (c - ) subjct to ncrasd watr avalablty (q q - ) and D - wthn th non-ngatvty (11c) and ntracton (11d) constrants. For sctors wth lowr shortag costs, maxmz dlvry ncrass subjct to ncrasd watr avalablty mnus dlvry ncrass to sctors wth hghr-shortag costs. Mathmatcally, ths rcursv soluton rul s: j1 ( q q ) ( D D ), D, j Dj Dj = Mnmum k k j,. (11) k = 1 Hr, j and k ar th watr us sctors rankd by shortag costs, c j -, so that c 1 - c 2 - c I -. Ths rul s obtand by subtractng (10b) from (11c), lmnatng th common - opt trms, sparatng shortag dcsons D j and D j - for th j th watr us sctor from shortag dcsons for th othr watr us sctors, brngng ths trms to on sd, and combnng wth constrant (11d). Furthr, snc th soluton to a constrand lnar optmzaton problm falls on th boundary of th fasbl soluton spac, th bndng nqualty constrant bcoms an qualty. Rarrangng (11) gvs th analytcal dcson rul for uppr-bound shortag dcsons D - as: j1 ( q q ) ( D D ), D, j Dj = Dj - Mnmum k k j, k 1. (11f) = Stp 3. Solutons to sub-modls (10) and (11) span stabl, fasbl rangs for th objctv functon and dcson varabls. Ths rangs ar f ± opt = [f -, f ], -, and D ± opt = [D -, D ] whr f -, -, and D ar solutons to worst-cas sub-modl (10) and f and D - ar solutons to th uppr-bound sub-modl (11). Ratocnaton Th proof that th rsk-advrs tchnqu gvs a narrowr objctv functon wdth wth a fasbl soluton and objctv functon valu qual or bttr than th worst-cas valu s straghtforward. Th proof nvolvs rntrprtng a pror thorm and proof mad by Huang t al. (1995) and thn showng soluton fasblty and rangs. In thr Thorm 2, Huang t al. (1995, p ) show that solvng th uppr-bound sub-modl frst (for a maxmzaton problm) s ncssary to dntfy th two xtrm bounds for gvn systm condton varatons (p. 601). Ths ordrng gnrats gry solutons of good qualty. Hr, qualty rfrs to th gry dgr or wdth of th gry dcson varabls and objctv functon valus (dffrncs btwn thr uppr and lowr bounds)(huang, Batz and Patry, 1995, Dfnton 13, p. 597) whl good mans ths wdths ar maxmal and larg. Huang t al. (1995, p. 598) also show th convrs solvng th lowr-bound sub-modl frst (for a maxmzaton problm) s unabl to gnrat gry solutons wth good qualty.

14 #GENO , p. 12 Frst, w not that larg-rangng gry wdths that comprs th xtrm bounds and forc th objctv functon to prform wors than th wors-cas soluton whn ntracton constrants bnd ar nthr good nor dsrabl outcoms. Undr unfavorabl paramtr condtons, dcson makrs may rgrt that thy could hav don bttr had thy adoptd a worst-cas or possbly othr soluton. W thrfor rntrprt Huang t al. s (1995) dfntons of good and dsrabl to allow as accptabl frst solvng th lowr-bound sub-modl to gnrat gry solutons of ndtrmnat qualty. Scond, w show that fasbl solutons xst for th uppr-bound sub-modl (11). Ths proof s straghtforward. By dfnton of th gry-numbr paramtr, q q -, V and xamnng soluton xprsson (11f), w not 0 D - D, V,, whch s compatbl wth constrants (11c) and (11d) and gvs fasbl solutons for D -. Intally, larg ncrasd watr avalablts (q q - > D ) forc th scond argumnt of th Mnmum functon n (11f) to domnat and st shortags to zro for sctors wth hgh shortag costs. Subsquntly, ncrasd allocatons to sctors wth hghr shortag costs wll balanc th ncrasd watr avalablty so that th frst argumnt of th Mnmum functon wll fall to a mnmum of zro. Ths mnmum only allows D - to rach an uppr-lmt of D and mantans th fasbl rang of soluton valus. Thrd, w show th objctv functon valu (f ) for th uppr-bound sub-modl (11) wll always b gratr than or qual to th objctv functon valu (f - ) for th lowr-bound (and worst-cas) sub-modl (10). By dfntons of th gry-numbr paramtrs b b -, V; c - c, V,; and from ntracton constrant (11d) whr D - D, V,; w hav th objctv functon valu ordrng: f = b pcd b pcd = f. Th wdth of th rsk advrs,, objctv functon rang s f f = ( b b ) p ( cd cd ) n Equaton (11f) gvs,. Substtutng j1 ( b b ) p( c c ) D pc jmn ( q q ) ( Dk Dk ), f f = Dk, j k = 1 whch shows th rsk advrs objctv functon wdth dpnds only on ncrasd bnfts (b b - ), dcrasd costs (c c - ), and ncrasd watr avalablty (q q - ) multpld by lowr-bound costs for sctors wth th most costly shortags. Fourth, w not (as do Huang t al. (1995, p. 601)) that solvng th lowr-bound (worstcas) sub-modl frst wll gnrat th worst-cas soluton but that th assocatd upprbound soluton wll lkly not rach th bst-cas objctv functon valu. Howvr, ths bhavor s not rqurd for th rsk-advrs soluton approach. (Such bhavor wll occur only whn solutons to th bst- and worst-cas sub-modls comprs th sam soluton bass and th uppr-bound sub-modl ntracton constrant dos not bnd). Fnally, combnng rsults from Ponts #3 (fasbl objctv functon valu worst-cas objctv functon valu) and #4 (uppr-bound objctv functon valu bst-cas objctv functon valu) gvs a rsk-advrs gry objctv functon rang (f f - ) that s narrowr than rang obtand by th xstng rsk-pron soluton mthod.

15 #GENO , p. 13 Exampl rsults Rsolvng ach stochastc program xampl wth th rsk-advrs gry-soluton tchnqu shows that th tchnqu gvs an objctv functon valu rang that s narrowr than both th rsk-pron gry-numbr and bst/worst cas mthods (Tabl 3). On bound of th objctv functon corrsponds to th worst-cas (lowr-bound for a maxmzaton problm; uppr-bound for a mnmzaton problm) whl th othr bound falls nsd th bst-cas soluton (uppr-bound lss than th bst-cas for a maxmzaton problm; lowr-bound gratr than th bst-cas for a mnmzaton problm). Th rsk-advrs tchnqu dntfs prmary-stag dcsons and stabl, fasbl rang of rcours-stag dcsons that mnmz dvatons of th objctv functon valu. Furthr, th objctv functon avods rsk-pron prformanc wors than th worst-cas soluton. Howvr, th rsk advrs soluton approach (lk th rsk-pron approach) fxs prmary-stag dcson valus across th sub-modls; w corrct ths falng wth an ntractng prmary-stag gry soluton approach. Intractng prmary-stag tchnqu Th rsk-pron and rsk-advrs gry soluton tchnqus fx prmary-stag dcson valus across th uppr- and lowr-bound sub-modls and fal to dntfy a stabl, fasbl rang for all dcson varabls. Hr, w ntroduc ntracton constrants for prmary-stag dcsons to dntfy th stabl, fasbl rangs for ths varabls. Soluton procss For a maxmzaton problm, frst solv th worst-cas sub-modl (10) to dntfy th objctv functon lowr bound (f - ), lowr bound on prmary-stag allocaton targts undr unfavourabl paramtr valus ( - ), and uppr bounds on rcours dcsons (D ). Scond, solv an uppr-bound sub-modl to dntfy th objctv functon uppr bound (f ), uppr bound on prmary-stag allocaton targts undr favourabl paramtr valus ( ), and lowr bounds on rcours dcsons (D - ). f = b, s.t. ( D ) Max p c D (12a) q, (12b) 0, ; D 0,, (12c) D D opt,, (12d), (12) opt Hr, dcson varabls nclud both prmary- and rcours-stags ( and D - ) wth an ntracton constrant (12) rqurng prmary-stag dcsons undr favorabl condtons to b abov th lvls dntfd n th worst-cas sub-modl ( - opt ). W can drv an

16 #GENO , p. 14 analytcal soluton for th rcours-stag shortag dcsons D - as was don for th rskadvrs approach. Excpt, hr, prmary-stag dcsons from th two sub-modls ar not ncssarly dntcal and may not cancl. Thus, D j = D j - Mnmum j1 ( q q ) ( ) ( Dk Dk ), Dj, j,. (12f) k = 1 Togthr, solutons to sub-modls (10) and (12) span stabl, fasbl rangs for th objctv functon and both prmary- and rcours-stag dcson varabls. Ths rangs ar f ± opt = [f -, f ], ± opt = [ -, ], and D ± opt = [D -, D ] whr f -, -, and D ar solutons to worst-cas sub-modl (10) and f,, and D - ar solutons to th upprbound sub-modl (12). Ratocnaton Th mathmatcal proof that th ntractng prmary stag soluton tchnqu gvs a fasbl soluton, objctv functon valu qual or bttr than th worst-cas valu, and objctv functon wdth that s qual or wdr than th rsk-advrs tchnqu follows th ratocnaton provdd for th rsk-advrs tchnqu. Hr, w smply add and account for ntracton constrants on th prmary-stag varabls. Frst, w agan rntrprt th Thorm 2 and proof mad by Huang t al. (1995) to allow that frst solvng th lowr-bound sub-modl wll gnrat gry solutons of ndtrmnat but accptabl qualty. Scond, w show that fasbl solutons xst for uppr-bound sub-modl (12). Ths proof s straghtforward. Subtractng (10b) from (12b), combnng and sparatng trms, gvs q q ( ) ( D D ) 0,. Ths xprsson s compatbl wth th pror gry-numbr paramtr dfnton for q and ntracton constrants (12d) and (12), and gvs fasbl solutons for uppr-bound allocaton targts ( ) and lowr-bound shortags (D - ). Togthr, ncrass n and dcrass n D - cannot xcd ncrasd watr avalablts sn whn movng from unfavorabl to favorabl paramtr condtons. But th xprsson stll allows for a wd rang of and D -. At worst, = - and (12f) rducs to (11f). In ths cas fasblty condtons shown n th ratocnaton for th rsk-avrs tchnqu smlarly apply. Thrd, w show f f - from uppr-bound sub-modl (12) and lowr-bound (and worstcas) sub-modl (10). Ths proof s also straghtforward. By pror dfntons of th grynumbr paramtrs and ntracton constrants on rcours dcsons (D - D, V,, [Eq. 12d]) and on prmary-stag dcsons ( -, V, [Eq. 12]), w smply hav: f = b pcd b pcd = f.,, Fourth, w show that th objctv functon wdth for th ntractng prmary stag soluton tchnqu (f ps f - ps) s gratr than or qual to th wdth for th rsk-advrs tchnqu (f ra f - ra). Snc both soluton tchnqus us th sam lowr-bound (worst cas) sub-modl (10), w nd only xamn th uppr-bound objctv functon valus

17 #GENO , p. 15 and show f ps f ra. Hr, not that th uppr-bound rsk-advrs sub-modl soluton (11) s part of th soluton spac to th uppr-bound ntractng prmary-stag sub-modl (12) ([ * opt, D - ] ra є [, D - ] ps ) by vrtu of ntracton constrant (12). Furthr, th uppr-bound rsk-advrs sub-modl s mor constrand than th ntractng prmarystag modl (th latr has -, V, [Eq. 12] and th formr has = * opt, V). f f = b p c D D. Thrfor, f ps f ra. Th ncras s ps ra ( opt ) ( ps ra ) Substtutng n (11f) and (12f) and notng that th sol dffrnc btwn D - ps and D - (whch rprsnts dcrasd watr avalablty from ra s th trm ( opt ) ncrasd prmary-stag watr allocaton targts), gvs f ps fra b pc ( opt ). W can also obtan th sam xprsson by formulatng th Lagrangan for sub-modl (12), spcfyng th Kuhn-Tuckr condtons, and substtutng to lmnat th Lagrang multplr assocatd wth constrant (12b). Ths xprsson says that th ntractng prmary-stag objctv functon valu wll ncras abov th rsk advrs valu whnvr uppr-bound bnfts xcd xpctd lowr-bound shortag costs.. Should bnfts not xcd xpctd shortag costs, constrant (12) wll bnd so that - opt = 0 wth no ncras. Fnally, w not agan that th assocatd uppr-bound objctv functon valu f ps wll lkly not rach th bst-cas objctv functon valu. Howvr, ths bhavor s not rqurd of th ntractng prmary-stag soluton approach., Exampl rsults Rsolvng ach of th stochastc program xampls usng th ntractng prmary-stag gry-soluton mthod shows that th approach gnrats solutons whos objctv functon wdths ar wdr than th rsk-advrs solutons but narrowr than th rsk-pron or bst/worst cas solutons (Tabl 3). Fgur 1 llustrats and compars ths objctv functon wdths for th watr rsourcs allocaton problm posd by Huang and Loucks (2000). Th prmary-stag ntracton soluton prforms no wors than th rsk-advrs approach (both mthods us worst-cas sub-modl (10) to solv for th lowr bound of th objctv functon), but shows an mprovmnt ovr th rsk-advrs approach for favourabl paramtr valus. Ths mprovmnt narly approachs th larg, optmstc uppr-bound objctv functon valu sn for th bst-cas and rsk-pron soluton mthods. Th ntractng prmary-stag mthod avods th ptfall of th rsk-pron approach (prformanc wors than th worst-cas), allows flxblty to choos prmarystag dcson valus wthn th dntfd rang, and mprovs objctv functon prformanc for favourabl paramtr valus compard to th rsk-advrs soluton approach.

18 #GENO , p. 16 Dscusson Th Bst/Worst-cas formulatons solv a lnar program twc usng th most favourabl (bst cas) and last-favourabl (worst cas) paramtr valus. Solutons from th two sub-modls can hlp judg th systm s capablty to ralz a dsrd goal but do not ncssarly construct a st of stabl rangs for gnratng dcson altrnatvs. Whn th soluton bass for th bst cas dffrs from th soluton bass for th worst cas, thr can b confuson about how to oprat th systm n th fac of uncrtan paramtr nputs. Gry lnar programs dntfy maxmal, stabl, fasbl rangs for dcson varabls by frst solvng th bst-cas (uppr-bound for a maxmzaton problm) sub-modl. Thy thn solv th lowr-bound sub-modl and ntroduc ntracton constrants to rqur lowr-bound solutons b lss than or qual to uppr-bound solutons. Ths ntracton dntfs stabl, fasbl rangs for dcson varabls and smultanously communcats that dcson varabls can b chosn wthn th proscrbd rangs whl assurng that th objctv functon valu wll vary only wthn th assocatd spcfd rang. Whn th rang of uncrtanty for nput paramtrs s small and th ntracton constrants do not bnd, th gry lnar program and bst/worst-cas formulaton solutons ar dntcal. In ths cas, th soluton bass for th bst and worst cass ar also th sam. Howvr, whn th rang of uncrtanty for nput paramtrs s sgnfcant and th ntracton constrants bnd, th gry lnar program objctv functon valu wll b wors than th worst cas. Th gry lnar program wll also fal to dntfy part of th soluton bass that s prfrabl undr unfavourabl paramtr valus. Thr ar rsks and costs to mpos a maxmal, stabl, fasbl rang of solutons. Th cost s th shadow valu (Lagrang multplr) assocatd wth th bndng ntracton constrant and th rsk s, undr unfavourabl paramtr valus, prformanc wors than th wors cas. In ths stuatons, dcson makrs wll lkly prfr to adopt a worst-cas or othr mor rskadvrs soluton. Two smpl numrcal problms and rtrospctv analyss of gry lnar program xampls from th ltratur dmonstrat ths problms. Ths problms ar magnfd for gry stochastc programs that hav prmary- and rcours-stag dcsons and ncorporat uncrtants xprssd as probablty dstrbutons and as ntrvals. Exstng gry-soluton mthods whch w trm rsk-pron dntfy maxmal, stabl, fasbl rangs for th objctv functon and rcours-stag dcson varabls by solvng th bst-cas sub-modl frst. Thy thn us dntfd prmary-stag dcson valus as nputs to th lowr-bound sub-modl. Fxng th prmary-stag dcson valus across th sub-modls, rsk-pron gry-soluton mthods fal to dntfy stabl, fasbl rangs for prmary-stag dcsons and oftn rqur sgnfcant and costly rcours-stag dcsons for unfavourabl paramtr valus. Ths rqurmnt rsults n wd-rangng and rsk-pron objctv functon valus that prform wors than th worst cas. Agan, undr unfavorabl paramtr condtons, dcson makrs could do bttr by adoptng a worst-cas or othr mor rsk-advrs soluton. To narrow th wdth of objctv functon dvatons and guarant prformanc at or bttr than th worst cas, a rsk-advrs gry-soluton mthod solvs th worst-cas submodl frst, thn uss th dntfd prmary-stag dcson valus to solv th uppr-

19 #GENO , p. 17 bound sub-modl. Idntfyng prmary-stag dcson lvls frst for unfavourabl paramtr valus mnmzs th cost of and nd for rcours-stag dcsons, but also rducs potntal bnfts undr favourabl paramtr condtons. Lk th rsk-pron approach, th rsk-advrs mthod also fxs prmary-stag dcson varabl valus across sub-modls and fals to dntfy a stabl, fasbl rang for ths dcson varabls. A thrd soluton approach uss ntracton to dntfy stabl, fasbl rangs for th objctv functon, prmary-stag, and rcours-stag dcson varabls. Th ntractng prmary-stag gry soluton mthod solvs th worst-cas sub-modl frst to dntfy lowr-bounds on th objctv functon and prmary-stag dcson varabls. Thn t solvs th uppr-bound sub-modl and uss an ntracton constrant on prmary-stag dcsons to dntfy th uppr bounds on th objctv functon and prmary-stag dcson varabls. Togthr, solutons form stabl, fasbl rangs for slctng dcson altrnatvs. Bcaus ntracton dntfs a rang for prmary-stag dcson valus, th ntractng prmary-stag gry soluton mthod s bttr abl to adapt to favourabl paramtr condtons and typcally gvs an objctv functon rang that s wdr than th rsk-advrs approach and narly approachs th bst-cas soluton valu. Tabl 4 summarzs and compars th four mthods to solv stochastc programs wth rcours that ncorporat uncrtants xprssd as ntrvals. Th choc of soluton mthod dpnds on th modlr s ams, partcularly hs/hr tolranc for objctv functon dvatons. If larg dvatons and prformanc wors than th wors cas ar accptabl should unfavourabl condtons ars, thn us th xstng rsk-pron gry soluton approach. Frst solv th bst-cas (uppr-bound for a maxmzaton problm) sub-modl and us prmary-stag dcson valus dntfd for optmstc condtons. Howvr, f objctv functon valu dvatons ar to b rducd and a soluton guarantd to b at or bttr than th worst-cas, nstad us th rsk-advrs or ntractng prmary-stag gry soluton approachs. In ths cas, frst solv th worst-cas (lowr-bound for a maxmzaton problm) sub-modl and us th prmary-stag dcson valus dntfd for pssmstc condtons. Algorthmcally, rsk tolranc bols down to a choc of frst solvng th bst- or worst-cas sub-modl. Conclusons A gry numbr xprsss uncrtanty as an ntrval btwn fxd lowr and uppr bounds. Gry numbrs fnd us n optmzaton to proactvly ncorporat uncrtants xprssd as ntrvals and dntfy maxmal, stabl, fasbl rangs for th objctv functon and dcson varabls. Ths rangs ar dntfd by ntroducng ntracton constrants to lmt dcson varabl valus for unfavourabl condtons basd on dcson varabl lvls frst dntfd for favourabl condtons. Rangs for dcson varabls can thn b usd to slct dcson altrnatvs wthn proscrbd bounds. Gry numbr programs rprsnt an mprovmnt ovr bst/worst cas analyss bcaus th lattr approach, lackng ntracton constrants, oftn offrs solutons wth dffrnt bass for favourabl and unfavourabl paramtr valus. Howvr, th ntracton

20 #GENO , p. 18 constrants also lmt gry solutons and gry programs oftn fal to dntfy part of th fasbl soluton spac, partcularly n th fac of unfavourabl paramtr valus. Morovr, ntracton constrants oftn lad th gry-numbr objctv functon valu to prform wors than th worst-cas analyss. Ths soluton mscharactrzaton and rskpron prformanc wors that th wors cas occurs whnvr th ntracton constrants bnd. Th papr shows ths mscharactrzaton and rsk-pron prformanc for numrous lnar and stochastc programmng xampls. Furthr, th xstng gry-soluton approach for stochastc programs wth rcours fxs prmary-stag dcson varabl valus across sub-modls and fals to dntfy a stabl, fasbl rang for ths mportant plannng dcson varabls. Two altrnatv gry-soluton algorthms ar prsntd to ovrcom ths problms. A rsk-advrs gry-soluton tchnqu solvs th worst-cas sub-modl frst, rducs dvatons n th objctv functon valu, and guarants an objctv functon valu no wors than th worst cas. An ntractng prmary-stag tchnqu ntroducs ntracton constrants on prmary-stag dcsons, dntfs a stabl, fasbl rang for ths dcson varabls, guarants an objctv functon valu no wors than th worst cas, yt offrs a rang that s wdr and an mprovmnt ovr th rsk-advrs tchnqu. Ths soluton bhavors ar ratocnatd, dmonstratd, and vrfd for numrous stochastc programmng xampls. Ultmatly, a modlr s or dcson makr s choc of soluton mthod to nclud uncrtants xprssd as ntrvals dpnds on thr rsk prfrncs partcularly thr tolranc for objctv functon dvatons. If wd dvatons ar accptabl wth prformanc wors than th worst cas possbl undr unfavourabl paramtr valus, thn us xstng gry-soluton tchnqus. Howvr, f wd dvatons ar to b avodd such as n rsk-advrs dcson-makng, thn th altrnatv soluton approachs may b prfrabl. Should th goal b only to charactrz systm prformanc across favourabl and unfavourabl condtons wthout nd to nforc soluton stablty across ths dffrnt nvronmnts, thn Bst/Worst cas analyss may b usd. Ths tradoffs and dstnctons hghlght th mportant advantags, dsadvantags, and dffrncs btwn rsk-pron and rsk-advrs gry-numbr programmng and bst/worst cas analyss. Acknowldgmnts Th author thanks Marclo Olvars and two anonymous rvwrs for thr commnts on arlr drafts. Rfrncs [1] Sahnds, N. V. (2004). "Optmzaton undr uncrtanty: stat-of-th-art and opportunts." Computrs & Chmcal Engnrng, 28(6-7), [2] Ishbuch, H., and Tanaka, H. (1990). "Multobjctv Programmng n Optmzaton of th Intrval Objctv Functon." Europan Journal of Opratonal Rsarch, 48(2),

21 #GENO , p. 19 [3] Huang, G. H., Batz, B. W., and Patry, G. G. (1992). "A Gray Lnar-Programmng Approach for Muncpal Sold-Wast Managmnt Plannng undr Uncrtanty." Cvl Engnrng Systms, 9(4), [4] Huang, G. H., and Moor, R. D. (1993). "Gray Lnar-Programmng, Its Solvng Approach, and Its Applcaton." Intrnatonal Journal of Systms Scnc, 24(1), [5] Huang, G. H., Batz, B. W., and Patry, G. G. (1995). "Gry Intgr Programmng - an Applcaton to Wast Managmnt Plannng undr Uncrtanty." Europan Journal of Opratonal Rsarch, 83(3), [6] Huang, G. H. (1998). "A hybrd nxact-stochastc watr managmnt modl." Europan Journal of Opratonal Rsarch, 107(1), [7] Huang, G. H., and Batz, B. W. (1995). "Gry Quadratc-Programmng and Its Applcaton to Muncpal Sold-Wast Managmnt Plannng undr Uncrtanty." Engnrng Optmzaton, 23(3), [8] L, Y. P., and Huang, G. H. (2007). "Fuzzy two-stag quadratc programmng for plannng sold wast managmnt undr uncrtanty." Intrnatonal Journal of Systms Scnc, 38(3), [9] Huang, G. H., and Loucks, D. P. (2000). "An nxact two-stag stochastc programmng modl for watr rsourcs managmnt undr uncrtanty." Cvl Engnrng and Envronmntal Systms, 17(2), [10] Maqsood, M., and Huang, G. H. (2003). "A two-stag ntrval-stochastc programmng modl for wast managmnt undr uncrtanty." Journal of th Ar & Wast Managmnt Assocaton, 53(5), [11] Maqsood, I., Huang, G. H., and Zng, G. M. (2004). "An nxact two-stag mxd ntgr lnar programmng modl for wast managmnt undr uncrtanty." Cvl Engnrng and Envronmntal Systms, 21(3), [12] Maqsood, M., Huang, G. H., and Yomans, J. S. (2005). "An ntrval-paramtr fuzzy two-stag stochastc program for watr rsourcs managmnt undr uncrtanty." Europan Journal of Opratonal Rsarch, 167(1), [13] Maqsood, I., Huang, G. H., Huang, Y. F., and Chn, B. (2005). "ITOM: an ntrvalparamtr two-stag optmzaton modl for stochastc plannng of watr rsourcs systms." Stochastc Envronmntal Rsarch and Rsk Assssmnt, 19(2), [14] L, Y. P., Huang, G. H., and N, S. L. (2006). "An ntrval-paramtr mult-stag stochastc programmng modl for watr rsourcs managmnt undr uncrtanty." Advancs n Watr Rsourcs, 29(5), [15] L, Y. P., and Huang, G. H. (2006). "An nxact two-stag mxd ntgr lnar programmng mthod for sold wast managmnt n th Cty of Rgna." Journal of Envronmntal Managmnt, 81(3), [16] L, Y. P., Huang, G. H., and Batz, B. W. (2006). "Envronmntal managmnt undr uncrtanty - An ntrnal-paramtr two-stag chanc-constrand mxd ntgr lnar programmng mthod." Envronmntal Engnrng Scnc, 23(5), [17] L, Y. P., Huang, G. H., N, S. L., and Huang, Y. F. (2006). "IFTSIP: ntrval fuzzy two-stag stochastc mxd-ntgr lnar programmng: a cas study for nvronmntal managmnt and plannng." Cvl Engnrng and Envronmntal Systms, 23(2),

22 #GENO , p. 20 [18] L, Y. P., Huang, G. H., and N, S. L. (2007). "Mxd ntrval-fuzzy two-stag ntgr programmng and ts applcaton to flood-dvrson plannng." Engnrng Optmzaton, 39(2), [19] Rosnbrg, D. E., and Lund, J. R. (2008). "Modlng ntgratd watr utlty dcsons wth rcours and uncrtants." Watr Rsourcs Managmnt, do: /s [20] L, Y. P., Huang, G. H., N, S. L., N,. H., and Maqsood, I. (2006). "An ntrvalparamtr two-stag stochastc ntgr programmng modl for nvronmntal systms plannng undr uncrtanty." Engnrng Optmzaton, 38(4), [21] Tong, S. C. (1994). "Intrval Numbr and Fuzzy Numbr Lnar Programmngs." Fuzzy Sts and Systms, 66(3), [22] Huang, G. (1996). "IPWM: An ntrval paramtr watr qualty managmnt modl." Engnrng Optmzaton, 26(2), [23] Yh, S.-C. (1996). "Gry programmng and ts applcatons to watr rsourcs managmnt."ph.d., Cornll Unvrsty, Ithaca, NY.

23 #GENO , p. 21 Tabl 1. Comparson of gry lnar program and bst/worst cas solutons for smpl optmzaton programs Soluton Elmnt Max [3, 4] 1 [5, 6] 2 Gry Lnar Program Max [3, 4] 1 [2, 6] <= <= 1 Gry Bst/Worst Lnar Cas Program Bst/Worst Cas Max [3, 4] 1 [5, 6] 2 1 [1, 4] 2 <= 1 Gry Lnar Program Bst/Worst Cas f [5, 6] <5, 6> [2, 6] <3, 6> [1.25, 6] <3, 6> 1 [0] <0, 0> [0] <1, 0> [0] <1, 0> 2 [1] <1, 1> [1] <0, 1> [0.25, 1] <0, 1>

24 #GENO , p. 22 Author Huang, Batz, and Patry Huang and Moor Tong Huang, Batz, and Patry Yh Huang Huang Programmng Yar Approach 1992 Gry lnar program 1993 Gry lnar program 1994 Intrval lnar program 1995 Gry ntgr program 1996 Gry lnar program 1996 Gry lnar program 1998 Gry lnar chanc constrant program Tabl 2. Comparson of gry numbr and worst-cas solutons for lnar program xampls Objctv Functon Soluton Mthod Applcaton Drcton Gry Soluton Worst Cas Vrfcaton / Problms Bst/Worst cas Numrcal xampl Max [765, 1931] 765 Infasbl as publshd. Landfll capacty too SWM xampl Mn [238, 517]* 517 small for optmal soluton publshd. Instad, us [2.36, 3.24]10 6 tons. Bst/Worst cas Numrcal xampl Max [764, 1930]* WR xampl Mn [8.13, 27.6] Bst/Worst cas Intractng Bst/Worst cas Intractng Bst/Worst cas; Full and partal grylzaton Intractng Bst/Worst cas Intractng Bst/Worst cas Chckn fd numrcal xampl 764 Infasbl as publshd for uppr bound of numrcal xampl. Corrct as publshd n Huang t al (1992). Not nough nput data publshd to vrfy WR soluton. Mn [242.2, 420]* 415 Infasbl as publshd. Total forag constrant msntrprtd. Should b 1 2 <= 1000; 1 2 <= Uppr bound submodl prforms wors than worst cas. SWM xampl Mn [385, 708]* 702 Error n spcfcaton of wast gnraton rats for cts 2 & 3. Uppr bound submodl prforms wors than worst cas. Rsrvor capacty xampl Numrcal xampl WQ xampl WQ x., p=0.10 WQ x., p=0.05 WQ x., p=0.01 Mn [4, 12] NA Not nough nput data publshd to vrfy soluton or calculat worst cas. Max Max [8.2, 15.4] [15.4, 20.0] Max [20.1, 22.8] [17.9, 21.2] [15.4, 20.0] * dnots author's calculaton dos not vrfy aganst publshd gry-numbr soluton for rasons dscrbd n vrfcaton column 8.2 Lowr bound s sam as worst cas. Not nough nput data to vrfy WQ xampl. NA Not nough nput data publshd to vrfy soluton or calculat worst cas.

25 Tabl 3. Comparson of gry numbr and bst/worst cas solutons for stochastc lnar program xampls #GENO , p. 23

26 #GENO , p. 24 Author Huang and Loucks Maqsood and Huang Maqsood, Huang, and Zng Maqsood, Huang, and Yomans L, Huang, N, N, and Maqsood L, Huang, and N L, Huang, and N Yar Programmng Approach 2000 Two-stag stochastc program 2003 Two-stag stochastc program 2004 Two-stag mxd ntgr program 2005 Fuzzy twostag stochastc program 2006 Two-stag mxd ntgr program 2006 Mult-stag stochastc program 2007 Fuzzy twostag mxd ntgr program Applcaton WR numrcal xampl SWM xampl SWM xampl WR numrcal xampl SWM xampl WR numrcal xampl Flood dvrson xampl Drcton Bst / Worst Cass Objctv Functon Rangs Gry Numbr Soluton Mthods Exstng (rsk pron) Rsk Advrs Intractng Prmary Stag Max <346, 592> [260, 592] [346, 462] [346, 560] Mn <0.147, 0.255> [0.147, 0.260]* [0.15, 0. 26] [0.149, 0.255] Mn <249, 432> [249, 478]* [302, 432] [272, 432] Max <203, 571> [154, 571]* [203, 462] [203, 538] Mn <119, 278> [119, 283]* [124, 278] NA Max <1435, 2605> [1240, 2605]* [1435, 2404] [1435, 2606] Mn <1899, 2215> [1899, 2634]* [2083, 2215] NA Rosnbrg and Lund 2008 Two-stag mxd ntgr program Watr supply plannng Mn <-15, 112> [-15, 281] [4, 112] [4.8, 112] * dnots author's calculaton dos not vrfy aganst publshd rsk pron soluton for rasons dscrbd n th txt

27 #GENO , p. 25 Tabl 4. Comparson of soluton mthods to stochastc programs wth rcours Dcson Varabls Soluton Mthod Sub-modls Prmary Stag Rcours Stag Nots Bst/Worst cass (8) and (10) <-, > - <D, D > Solutons do not ncssarly construct stabl, fasbl rangs for slctng dcson altrnatvs. Exstng rsk pron (8) thn (9) * - [D, D ] Wd-rangng objctv functon prforms wors than worst-cas. Prmary-stag dcsons fxd across submodls. Rsk advrs (10) thn (11) - - [D, D ] Mnmzs objctv functon dvatons. Objctv functon prforms no wors than worst-cas. Prmarystag dcsons fxd across sub-modls. Intractng prmary stag (10) thn (12) [-, ] - [D, D ] Intracton constrants dntfy rang of prmary-stag dcsons. Objctv functon prforms no wors than worst-cas and bttr than rsk-advrs tchnqu. Gry-numbr