A MIN-MAX REGRET ROBST OPTIMIZATION APPROACH FOR ARGE SCAE F FACTORIA SCENARIO DESIGN OF DATA NCERTAINTY Travat Assavapokee Department of Industra Engneerng, nversty of Houston, Houston, Texas 7704-4008, SA, travat.assavapokee@ma.uh.edu Matthew J. Reaff Department of Chemca and Bomoecuar Engneerng, Georga Insttute of Technoogy, Atanta, Georga, 3033-000, SA, matthew.reaff@chbe.gatech.edu Jane C. Ammons Department of Industra and Systems Engneerng, Georga Insttute of Technoogy, Atanta, Georga, 3033-000, SA, ane.ammons@sye.gatech.edu Ths paper presents a three-stage optmzaton agorthm for sovng two-stage robust decson makng probems under uncertanty wth mn-max regret obectve. The structure of the frst stage probem s a genera mxed nteger (bnary) near programmng mode wth a specfc mode of uncertanty that can occur n any of the parameters, and the second stage probem s a near programmng mode. Each uncertan parameter can take ts vaue from a fnte set of rea numbers wth unknown probabty dstrbuton ndependenty of other parameters settngs. Ths structure of parametrc uncertanty s referred to n ths paper as the fu-factora scenaro desgn of data uncertanty. The proposed agorthm s shown to be effcent for sovng arge-scae mn-max regret robust optmzaton probems wth ths structure. The agorthm coordnates three mathematca programmng formuatons to sove the overa optmzaton probem. The man contrbutons of ths paper are the theoretca deveopment of the three-stage optmzaton agorthm, and mprovng ts computatona performance through mode transformaton, decomposton, and pre-processng technques based on anayss of the probem structure. The proposed agorthm s apped to sove a number of robust facty ocaton probems under ths structure of parametrc uncertanty. A resuts ustrate sgnfcant mprovement n computaton tme of the proposed agorthm over exstng approaches. Keywords: Robust optmzaton, Scenaro based decson makng, Decson makng under uncertanty, Mn-max regret robust approach, Fu-factora scenaro desgn
. Introducton In ths paper we address the two-stage decson makng probem under uncertanty, where the uncertanty can appear n the vaues of any parameters of a genera mxed nteger near programmng formuaton. T T max c x + q y x, y s.t. Wy h + Tx Wy = h + Tx x x {0,}, y 0 In ths paper, ower case etters wth vector cap such as x represent vectors and the notaton x represents the th eement of the vector x. The correspondng bod upper case etters such as W denote matrces and the notaton W represents the (, ) th eement of the matrx W. In the formuaton, the vector x represents the frst-stage bnary decsons that are made before the reazaton of uncertanty and the vector y represents the second-stage contnuous decsons that are made after the uncertanty s resoved. It s worth emphaszng that many practca two-stage decson probems can often be represented by ths mathematca formuaton. For exampe n a facty ocaton probem under uncertanty n a gven network G = ( N, A), the frst-stage decson x {0,} N represents the decson of whether or not a facty w be ocated at each node n the network G. These facty ocaton decsons are consdered as ong-term decsons and have to be made before the reazaton of uncertan parameters n the mode. Once a vaues of uncertan parameters n the mode are reazed, the second-stage decson y 0 can be made. These second-stage decsons represent the fow of each matera transported on each arc n the network G. In ths paper, we consder the probem such that each uncertan parameter n the mode s restrcted to ndependenty take ts vaue from a fnte set of rea numbers wth unknown probabty dstrbuton. Ths captures stuatons where decson makers have to make the ong term decson under uncertanty when there s very mted or no hstorca nformaton about the probem and the ony avaabe nformaton s the possbe vaue of each uncertan parameter based on expert opnon. For exampe, expert opnon may gve the decson makers three possbe vaues for a gven uncertan parameter based on optmstc, average, and pessmstc estmatons. nder ths stuaton, decson makers cannot search for the frststage decsons wth the best ong run average performance, because there s a ack of knowedge about the probabty dstrbuton of the probem parameters. Instead, decson
makers are assumed to requre frst-stage decsons that perform we across a possbe nput scenaros: robust frst-stage decsons usng a devaton robustness defnton (mn-max regret robust souton) defned by Kouves and Yu (997). Two-stage mn-max regret robust optmzaton addresses optmzaton probems where some of the mode parameters are uncertan at the tme of makng the frst stage decsons. The crteron for the frst stage decsons s to mnmze the maxmum regret between the optma obectve functon vaue under perfect nformaton and the resutng obectve functon vaue under the robust decsons over a possbe reazatons of the uncertan parameters (scenaros) n the mode. The work of Kouves and Yu (997) summarzes the state-of-the-art n mn-max regret optmzaton up to 997 and provdes a comprehensve dscusson of the motvaton for the mn-max regret approach and varous aspects of appyng t n practce. Ben-Ta and Nemrovsk (998, 999, 000) address robust soutons (mnmax/max-mn obectve) by aowng the uncertanty sets for the data to be epsods, and propose effcent agorthms to sove convex optmzaton probems under data uncertanty. However, as the resutng robust formuatons nvove conc quadratc probems, such methods cannot be drecty apped to dscrete optmzaton. Mausser and aguna (998, 999a, 999b) presents the mxed nteger near programmng formuaton to fndng the maxmum regret scenaro for a gven canddate robust souton of one-stage near programmng probems under nterva data uncertanty for obectve functon coeffcents. They aso deveop an teratve agorthm for fndng the robust souton of one-stage near programmng probems under reatve robust crteron and a heurstc agorthm under the mn-max absoute regret crteron wth the smar type of uncertanty. Averbakh (000, 00) shows that poynoma sovabty s preserved for a specfc dscrete optmzaton probem (seectng p eements of mnmum tota weght out of a set of m eements wth uncertanty n weghts of the eements) when each weght can vary wthn an nterva under the mn-max regret robustness defnton. Bertsmas and Sm (003, 004) propose an approach to address data uncertanty for dscrete optmzaton and network fow probems that aows the degree of conservatsm of the souton (mn-max/max-mn obectve) to be controed. They show that the robust counterpart of an NP-hard α -approxmabe 0- dscrete optmzaton probem remans α -approxmabe. They aso propose an agorthm for robust network fows that soves the robust counterpart by sovng a poynoma number of nomna mnmum cost fow probems n a modfed network. Assavapokee et a. (007a) present a scenaro reaxaton agorthm for sovng scenaro-based mn-max regret and mnmax reatve regret robust optmzaton probems for the mxed nteger near programmng 3
(MIP) formuaton. Assavapokee et a. (007b) present a mn-max regret robust optmzaton agorthm for two-stage decson makng probems under nterva data uncertanty n parameter vectors ch,, h and parameter matrxes T and T of the MIP formuaton. They aso demonstrate a counter-exampe ustratng the nsuffcency of the robust souton obtaned by ony consderng scenaros generated by a combnatons of upper and ower bounds of each uncertan parameter for the nterva data uncertanty case. Ths resut s aso true for the fu factora scenaro desgn of data uncertanty case. As of our best knowedge, there exsts no agorthm for sovng two-stage mn-max regret robust decson makng probems under nterva data uncertanty n parameter vector q and parameter matrxes W and W of the genera MIP formuaton. Because the proposed agorthm n ths paper can sove the smar robust two-stage probems under fu factora scenaro desgn of data uncertanty n any parameter of the genera MIP formuaton, the proposed agorthm can aso be used as one aternatve to approxmatey sove the devaton robust two-stage probems under nterva data uncertanty n parameter vector q and parameter matrxes W and W of the genera MIP formuaton by repacng the possbe range of each uncertan parameter wth a fnte set of rea vaues. Tradtonay, a mn-max regret robust souton can be obtaned by sovng a scenarobased extensve form mode of the probem, whch s aso a mxed nteger near programmng mode. The extensve form mode s expaned n deta n secton. The sze of ths extensve form mode grows rapdy wth the number of scenaros used to represent uncertanty, as does the requred computaton tme to fnd optma soutons. nder fufactora scenaro desgn, the number of scenaros grows exponentay wth the number of uncertan parameters and the number of possbe vaues for each uncertan parameter. For exampe, a probem wth 0 uncertan parameters each wth 3 possbe vaues has over 3.4 bon scenaros. Sovng the extensve form mode drecty obvousy s not the effcent way for sovng ths type of robust decson probems even wth the use of Benders decomposton technque (J.F. Benders, 96). For exampe, f the probem contans 3.4 bon scenaros and a sub-probem of Benders decomposton can be soved wthn 0.0 second, the agorthm woud requre approxmatey one year of computaton tme per teraton to generate a set of requred cuts. Because of the faure of the extensve form mode and the Benders decomposton agorthm for sovng a arge scae probem of ths type, the three-stage optmzaton agorthm s proposed n ths paper for sovng ths type of robust optmzaton probems 4
under mn-max regret obectve. The agorthm s desgned expcty to effcenty hande a combnatora szed set of possbe scenaros. The agorthm sequentay soves and updates a reaxaton probem unt both feasbty and optmaty condtons of the overa probem are satsfed. The feasbty and optmaty verfcaton steps nvove the use of b-eve programmng, whch coordnates a Stackeberg game (Von Stackeberg, 943) between the decson envronment and decson makers, whch s expaned n deta n secton. The agorthm s proven to termnate at an optma mn-max regret robust souton f one exsts n a fnte number of teratons. Pre-processng procedures, probem transformaton steps, and decomposton technques are aso derved to mprove the computatona tractabty of the agorthm. In the foowng secton, we present the theoretca methodoogy of the proposed agorthm. The performance of the proposed agorthm s demonstrated through a number of facty ocaton probems n secton 3. A resuts ustrate mpressve computatona performance of the agorthm on exampes of practca sze.. Methodoogy Ths secton begns by revewng key concepts of scenaro based mn-max regret robust optmzaton. The methodoogy of the proposed agorthm s then summarzed and expaned n deta, and each of ts three stages s specfed. The secton concudes wth the proof that the proposed agorthm aways termnates at the mn-max regret robust optma souton f one exsts n a fnte number of teratons. We address the probem where the basc components of the mode s uncertanty are represented by a fnte set of a possbe scenaros of nput parameters, referred as the scenaro set Ω. The probem contans two types of decson varabes. The frst stage varabes mode bnary choce decsons, whch have to be made before the reazaton of uncertanty. The second stage decsons are contnuous recourse decsons, whch can be made after the reazaton of uncertanty. et vector x denote bnary choce decson varabes and et vector y denote contnuous recourse decson varabes and et c, q, h, h, W, W, T, and T denote mode parameters settng for each scenaro Ω. If the reazaton of mode parameters s known to be scenaro a pror, the optma choce for the decson varabes x and y can be obtaned by sovng the foowng mode (). 5
T T O = max cx + qy x, y s.t. W y T x h W y T x = h x {0,} and 0 x y When parameters uncertanty exsts, the search for the mn-max regret robust souton comprses fndng bnary choce decsons, x, such that the functon max( O Z ( x)) s Z x = q y W y h + T x W y = h + T x + c x. It s aso mnmzed where ( ) max { T, } T y 0 worth mentonng that the vaue of O Z ( x) s nonnegatve for a scenaros Ω. In the case when the scenaro set Ω s a fnte set, the optma choce of decson varabes x under mn-max regret obectve can be obtaned by sovng the foowng mode (). mn δ, xy, s.t. δ T T δ O qy cx W T y x h Ω y T x = h W y 0 x {0,} x Ths mode () s referred to as the extensve form mode of the probem. If an optma souton for the mode () exsts, the resutng bnary souton s the optma settng of the frst stage decson varabes x. nfortunatey, the sze of the extensve form mode can become unmanageaby arge as does the requred computaton tme to fnd the optma settng of x. Because of the faure of the extensve form mode and the Benders decomposton agorthm for sovng a arge scae probem of ths type, a new agorthm whch can effectvey overcome these mtatons of these approaches s proposed n the foowng subsecton. A key nsght of ths agorthm s to recognze that, for a fntey arge set of a possbe scenaros, t s possbe to dentfy a smaer subset of mportant scenaros that actuay need to be consdered as part of the teraton scheme to sove the overa optmzaton probem wth the use of b-eve programmng. Ω () () Mn-Max Regret Robust Agorthm for Fu-Factora Scenaro Desgn We start ths secton by defnng some addtona notatons whch are extensvey used n ths secton of the paper. The uncertan parameters n the mode () can be cassfed nto 6
eght maor types. These uncertan parameters can be combned nto a random vector ξ = ( cqh,,, h, T, T, W, W ). In ths paper, we assume that each component of ξ can ndependenty take ts vaues from a fnte set of rea numbers wth unknown probabty dstrbuton. Ths means that for any eement p of the vector ξ, p can take any vaue from the set of { p(), p(),..., p ( p) } such that p() < p() <... < p( p) where the notaton p denotes the number of possbe vaues for the parameter p. For smpcty, notatons p = p() and p = p w be used to represent the ower bound and upper bound vaues of the parameter ( p) p ξ. The scenaro set Ω s generated by a possbe vaues of the parameter vector ξ. et us defne ξ( ) as the specfc settng of the parameter vector ξ under scenaro Ω and Ξ= { ξ ( ) Ω } as the support of the random vector ξ. As descrbed beow, we propose a three-stage optmzaton agorthm for sovng the mn-max regret robust optmzaton probem under scenaro set Ω that utzes the creatve dea based on the foowng nequaty where Ω Ω. = max{ O Z( x)} mn max{ O Z( x)} mn max{ O Z( x)} = Ω x Ω x Ω In the consdered probem, we woud ke to sove the mdde probem ( mn max{ O Z ( x)} ), whch s ntractabe because Ω s extremey arge. Instead, we x Ω successvey sove the eft and rght probems for and. The eft probem s soved by utzng a reformuaton as a tractabe B-eve mode. The rght probem s soved by utzng the Benders decomposton based on the fact that Ω s reatvey sma compared to Ω. The proposed agorthm proceeds as foows. Three-Stage Agorthm Step 0: (Intazaton) Choose a subset Ω Ω and set =, and = 0. et x opt denote the ncumbent souton. Determne the vaue of ε 0, whch s a pre-specfed toerance. Step : (Sovng the Reaxaton Probem and Optmaty Check) Sove the mode () to obtan O Ω. If the mode () s nfeasbe for any scenaro n the scenaro set Ω, the agorthm s termnated; the probem s -posed. 7
Otherwse the optma obectve functon vaue to the mode () for scenaro s desgnated as O. By usng the nformaton on O Ω, Appy the Benders decomposton agorthm expaned n deta n Secton. to sove the smaer verson of the mode () by consderng ony the scenaro set Ω nstead of Ω. Ths smaer verson of the mode () s referred to as the reaxed mode () n ths paper. If the reaxed mode () s nfeasbe, the agorthm s termnated wth the confrmaton that no robust souton exsts for the probem. Otherwse, set x x whch s an optma souton from the reaxed mode () and set = Ω = max{ O Z( x )} Ω If { - } ε, x opt whch s the optma obectve functon vaue of the reaxed mode (). s the gobay ε-optma robust souton and the agorthm s termnated. Otherwse the agorthm proceeds to Step. Step : (Feasbty Check) Sove the B-eve- mode descrbed n deta n Secton. by usng the x Ω nformaton from Step. If the optma obectve functon vaue of the Beve- mode s nonnegatve (feasbe case), proceed to Step 3. Otherwse (nfeasbe case), Ω Ω { } where s the nfeasbe scenaro for x Ω generated by the B-eve- mode n ths teraton and return to Step. Step 3: (Generate the Scenaro wth Maxmum Regret Vaue for x Ω and Optmaty Check) Sove the B-eve- mode specfed n deta n Secton.3 by usng the x Ω nformaton from Step. et arg m ax{ O Z ( x )} and = max{ O Z ( x )} generated by the Ω Ω B-eve- mode respectvey n ths teraton. If <, then set xopt = x Ω and set =. If { - } ε, x opt s the gobay ε-optma robust souton and the agorthm s termnated. Otherwse, Ω Ω { } and return to Step. We defne the agorthm Steps,, and 3 as the frst, second, and thrd stages of the agorthm respectvey. Fgure ustrates a schematc structure of ths agorthm. Each of the three stages of the agorthm s detaed n the foowng subsectons. Ω Ω 8
O ptm aty check and G enerate x arg m n m ax{ O Z ( x )} Ω Frst Stage x Ω = mn max{ O Z ( x )} x Ω x Ω and Second Stage and Thrd Stage O ptm aty check and G enerate arg m ax{ O Z ( x Ω )} Ω = mn{max{ O Z ( x )}, } Ω (B-eve- mode). Ω Feasbty check (B-eve- mode) x Ω and Fgure : Schematc Structure of the Agorthm... The Frst Stage Agorthm = The purposes of the frst stage are () to fnd arg mn{ max{ } xω O Z( x)} mn max{ O Z( x)} x Ω x Ω, () to fnd, and (3) to determne f the agorthm has dscovered an optma robust souton for the probem. The frst stage utzes two man optmzaton modes: the mode () and the reaxed mode (). The mode () s used to cacuate O for a scenaros Ω Ω. If the mode () s nfeasbe for any scenaro Ω, the agorthm s termnated wth the concuson that there exsts no robust souton to the probem. Otherwse, once a requred vaues of O Ω are obtaned, the reaxed mode () s soved. In ths paper, we recommend sovng the reaxed mode () by utzng the Benders decomposton technques as foow. The reaxed mode () has the foowng structure. T T mn ( max ( O cx qy)) xy, Ω s.t. W y x h T W y T x = h Ω y 0 x x {0,} Ths mode can aso be rewrtten as T mn f( x) where f( x) = max( O Q( x) cx) and x Ω { 0,} x T Q ( x) = max q y y 0 s.t. W y h + T x ( π,, x ) W y = h + T x ( π ),, x 9
where the symbos n parenthess next to the constrants denote to the correspondng dua varabes. The resuts from the foowng two emmas are used to generate the master probem and sub probems of the Benders decomposton for the reaxed mode (). emma : f ( x ) s a convex functon on x. Proof: f x O Q x c x s a convex functon on x because of the foowng reasons. () Q ( x) T ( ) = max( ( ) ) Ω and c T x are concave functons on x ; () ( )concave functon s a convex functon; (3) Summaton of convex functons s aso a convex functon; and (4) Maxmum functon of convex functons s aso a convex functon. T ( ) T ( ) T f( x ( )) where T T T emma : ( c() π, (), x() π ( ), (), x() ( ) ) T () argmax{ O c x() Q( x())} Ω, f ( x ( )) s sub-dfferenta of the functon f at x() and ( π, ( ), x( ), π, ( ), x( ) ) s the optma souton of the dua probem n the cacuaton of Q ( x) when = ( ) and x = x ( ). Proof: From duaty theory: T T Q() ( x( )) = ( π, (), x ()) ( h () + T x( )) + ( π (), (), x ()) ( h () + T() x( )) T T and Q() ( x) ( π, (), x ()) ( h () + T x) + ( π (), (), x ()) ( h () + T() x) for arbtrary x. Thus, Q () ( x ) Q () ( x ( )) ( T T π, ( ), x( ) ) T ( x x ( )) + ( π (), ( ), x( ) ) T() ( x x ( )) and ( ) T T O() Q() x O() Q () ( x ( )) (( π, (), x() ) T + ( () π, (), x() ) T() )( x x ( )). T c x = c x() c ( x x()) and f ( x) = max( O Q ( x) c x), T T T From () () () Ω T f ( x) O() Q() ( x) c() x T T T T () () () () π, (), x() T π (), (), x() T() ( ) O Q ( x ( )) c x ( ) + c ( ) ( ) ( x x ( )) T From () argmax{ O c x() Q ( x())}, f ( x) Ω f ( x ( )) + c ( ) ( ) ( x x ( )) T T T ( () π, (), x() T π (), (), x() T() ) Based on the resuts of the emma and, we brefy state the genera Benders decomposton agorthm as t appes to the reaxed mode (). Benders Decomposton Agorthm for the Reaxed Mode (): Step 0: Set ower and upper bounds b = and ub = + respectvey. Set the teraton counter k = 0. et 0 Y ncudes a cuts generated from a prevous teratons of the 0
proposed three-stage agorthm. A these cuts are vad because the proposed agorthm aways add more scenaros to the set Ω and ths causes the feasbe regon of the reaxed mode () to shrnk from one teraton to the next. et x denote the ncumbent souton. Step : Sove the master probem b = mn θ θ, x T s.t. θ a x+ b =,,..., k k ( θ, x) Y If the master probem s nfeasbe, stop and report that the reaxed mode () s nfeasbe. Otherwse, update k = k + and et x( k) be an optma souton of the master probem. Step : For each Ω, sove the foowng sub probem: T Q ( x( k)) = max q y y 0 s.t. W y h + T x( k) ( π,, x( k) ) W y = h + T x( k) ( π ),, x( k) where the symbos n parenthess next to the constrants denote to the correspondng dua varabes. If the sub probem s nfeasbe for any scenaro Ω, go to Step 5. Otherwse, usng the sub probem obectve vaues, compute the obectve functon vaue T f ( xk ( )) = O( k) c( k) xk ( ) Q( k) ( xk ( )) correspondng to the current feasbe souton x( k) T where ( k) argmax{ O c x( k) Q ( x( k))}. If ub > f ( xk ( )), update the upper bound Ω ub = f ( x( k)) and the ncumbent souton x = xk ( ). Step 3: If ub b λ, where λ 0 s a pre-specfed toerance, stop and return x optma souton and ub as the optma obectve vaue; otherwse proceed to Step 4. T Step 4: For the scenaro ( k) argmax{ O c x( k) Q ( x( k))}, et ( π, π Ω be the optma dua soutons for the sub probem correspondng to x( k), ( k), x( k) as the, ( k), x( k) and ( k) soved n T T Step. Compute the cut coeffcents ( T T ak = c ( k) + ( π, ( k), x( k) ) T + ( π ( k), ( k), x( k) ) T ( k) ), T and b = a x( k) + f( x( k)), and go to Step. k k Step 5: et ˆ Ω be a scenaro such that the sub probem s nfeasbe. Sove the foowng optmzaton probem where 0 and represent the vector wth a eements equa to zero and one respectvey. )
mn( h + T x( k)) v + ( h + T x( k)) v v, v T T s.t. Wˆv+ W ˆv 0 0 v, v T T ˆ ˆ ˆ ˆ et v and v be the optma souton of ths optmzaton probem. Set k = k and k k T T Y Y { x ( h + T x) v + ( h + T x) v 0} and go to Step. ˆ ˆ ˆ ˆ If the reaxed mode () s nfeasbe, the agorthm s termnated wth the concuson that there exsts no robust souton to the probem. Otherwse, ts resuts are the canddate robust decson, x x, and the ower bound on mn-max regret vaue, max{ = O Z ( x )} Ω = obtaned from the reaxed mode (). The optmaty condton s then checked. The optmaty condton w be satsfed when ε, where ε 0 s pre-specfed toerance. If the optmaty condton s satsfed, the agorthm s termnated wth the ε - optma robust souton. Otherwse the souton x Ω and the vaue of second stage... The Second Stage Agorthm The man purpose of the second stage agorthm s to dentfy a scenaro admts no feasbe souton to Z ( x ) for Ω Ω are forwarded to the Ω whch =. To acheve ths goa, the agorthm soves a b-eve programmng probem referred to as the B-eve- mode by foowng two man steps. In the frst step, the agorthm starts by pre-processng mode parameters. At ths pont, some mode parameters vaues n the orgna B-eve- mode are predetermned at ther optma settng by foowng some smpe preprocessng rues. In the second step, the B-eve- mode s transformed from ts orgna form nto a snge-eve mxed nteger near programmng structure. Next we descrbe the key concepts of each agorthm step and the structure of the B-eve- mode. One can fnd a mode parameters settng or a scenaro souton to Z ( x ) for Ω Ω whch admts no feasbe = by sovng the foowng b-eve programmng probem referred to as the B-eve- mode. The foowng mode (3) demonstrates the genera structure of the B-eve- mode.
mn ξ s.t. max yss,,, s, s.t. δ ξ Ξ δ δ Wy+ s = h + TxΩ W y+ s = h + T xω Wy+ s = h Tx Ω δ s, δ s, δ s, y 0 (3) In the B-eve- mode, the eader s obectve s to make the probem nfeasbe by controng the parameters settngs. The foower s obectve s to make the probem feasbe by controng the contnuous decson varabes, under the fxed parameters settng from the eader probem, when the settng of bnary decson varabes s fxed at x Ω. In the mode (3), δ represents a scaar decson varabe and 0 and represent the vector wth a eements equa to zero and one respectvey. The current form of the mode (3) has a nonnear beve structure wth a set of constrants restrctng the possbe vaues of the decson vectors ξ = ( cqh,,, h, T, T, W, W ). Because the structure of the foower probem of the mode (3) s a near program and t affects the eader s decsons ony through ts obectve functon, we can smpy repace ths foower probem wth expct representatons of ts optmaty condtons. These expct representatons ncude the foower s prma constrants, the foower s dua constrants and the foower s strong duaty constrant. Furthermore, from the speca structure of the mode (3), a eements n decson varabe matrxes T, Wand vector h can be predetermned to ether one of ther bounds even before sovng the mode (3). For each eement of the decson matrx W n the mode (3), the optma settng of ths decson varabe s the upper bound of ts possbe vaues. The correctness of these smpe rues s obvous based on the fact that y 0. Smary, for each eement of the decson vector h and matrx T, the optma settng of ths decson varabe n the mode (3) s the ower bound of ts possbe vaues. emma 3: The mode (3) has at east one optma souton T, h, W, T, h, and W n whch each eement of these vectors takes on a vaue at one of ts bounds. Proof: Because the optma settng of each eement of T, h, and W aready takes ts vaue from one of ts bounds. We ony need to prove ths emma for each eement of T, h, and W. Each of these varabes T, h, and W appears n ony two constrans n the mode 3
W y + s = h + T x Ω and W y + s = h T x Ω (3):. It s aso easy to see that s = s and mn{ s, s} = s s /. Ths fact mpes that the optma settng of y whch maxmzes mn{ s, s } w aso mnmze s s / and vse versa under the fxed settng of ξ. Because s s /= h + TxΩ Wy, the optma settng of T, h, andw w maxmze mn h + T xω W y y χ ( x Ω ) where χ( x Ω ) = { y 0 Wy h+ Tx Ω, Wy h + T x Ω }. In ths form, t s easy to see that the optma settng of varabes T, h andw w take on one of ther bounds. et us defne the notatons and E to represent sets of row ndces assocatng wth essthan-or-equa-to and equaty constrants n the mode () respectvey. et us aso defne the notatons w, w + and w E to represent dua varabes of the foower probem n the mode (3). Even though there are sx sets of foower s constrants n the mode (3), ony three sets of dua varabes are requred. Because of the structure of dua constrants of the foower probem n the mode (3), dua varabes assocated wth the frst three sets of the foower s constrants are exacty the same as those assocated wth the ast three sets. After repacng the foower probem wth expct representatons of ts optmaty condtons, we encounter wth a number of nonnear terms n the mode ncudng: W y, Ww +, Ww, hw +, h w, Tw +, and T w. By utzng the resut form emma 3, we can repace + these nonnear terms wth the new set of varabes WY, WW, WW, HW + HW,, TW +, and TW wth the use of bnary varabes. We ntroduce bnary varabes,, and,, and bt bh bw whch take the vaue of zero or one f varabes T h W respectvey take the ower or the upper bound vaues. The foowng three sets of constrants (4), (5), and (6) w be used to reate these new varabes wth nonnear terms n the mode. In these constrants, the notatons varabes y, w + and w y, w + and w constructng these bounds of the prma and dua varabes. represent the upper bound vaue of the respectvey. Teraky (996) descrbes some technques on 4
T = T + ( T T ) bt + + + Tw TW Tw Tw TW Tw + + + TW Tw ( T + T )( w )( bt ) E, + + + TW Tw + ( T + T )( w )( bt ) TW Tw ( T + T )( w )( bt ) TW Tw + ( T + T )( w )( bt ) bt {0,} (4) h = h + ( h h) bh + + + hw HW hw hw HW hw + + + HW hw ( h + h )( w )( bh) + + + HW hw + ( h + h )( w )( bh) HW hw ( h + h )( w )( bh ) HW hw + ( h + h )( w )( bh) bh {0,} E (5) W = W + ( W W ) bw W y WY W y WY W y W W y bw WY W y W W y bw W w WW W w W w WW W w E, WW ( )( )( ) WW W w W W w bw WW W w ( W W )( w )( bw ) ( + )( )( ) + ( + )( )( ) + + + + + + W w W + W w bw + + + + ( + )( )( ) + + ( + )( )( ) WW W w W W w bw bw {0,} (6) After appyng pre-processng rues, the foower probem transformaton, and the resut from emma 3, the mode (3) can be transformed from a b-eve nonnear structure to a snge-eve mxed nteger near structure presented n the mode (7). The tabe n the mode (7) s used to dentfy some addtona constrants and condtons for addng these constrants 5
to the mode (7). These resuts greaty smpfy the souton methodoogy of the B-eve- mode. If the optma settng of the decson varabe δ s negatve, the agorthm w add scenaro, whch s generated by the optma settng of h, h, T, T, W, and W from the mode (7) and any feasbe combnaton of c and q, to the scenaro set Ω and return to the frst stage agorthm. Otherwse the agorthm w forward the souton x Ω and the vaue of to the thrd stage agorthm. mn s.t. δ W y + s = h + T x Ω WY + s = h + T x E Ω WY + s = h T x E Ω δ s, δ s E, δ s E W w + WW WW 0 E ( + ) ( + ) w + w + w = E + + δ = h + TxΩ w + HW HW + ( TW TW) xω E w 0, w 0 E, w 0 E, y 0 + Condton to Add the Constrants Constrant Reference Constrant Index Set Aways (4) For a E, For a Aways (5) For a E Aways (6) For a E, For a.3. The Thrd Stage Agorthm The man purpose of the thrd stage s to dentfy a scenaro { arg max O Z( xω )} The mathematca mode utzed by ths stage to perform ths task s aso a b-eve program referred to as the B-eve- mode. The eader s obectve s to fnd the settng of the decson vector ξ = ( cqh,,, h, T, T, W, W ) and decson vector ( x, y) that resut n the maxmum regret vaue possbe, max{ O Z( xω )} Ω Ω (7), for the canddate robust souton x Ω. The foower s obectve s to set the decson vector y to correcty cacuate the vaue of Z ( x ) under the fxed settng of decson vector ξ estabshed by the eader. The genera Ω structure of the B-eve- mode s represented n the foowng mode (8).. 6
The souton methodoogy for sovng the mode (8) can be structured nto two man steps. These two man steps ncude () the parameter pre-processng step and () the mode transformaton step. Each of these steps s descrbed n deta n the foowng subsectons. max { q y + c x q y c x )} x x {0,}, y 0, ξ s.t. T T T T ξ Ξ W y h + Tx W y = h + Tx T m ax q y y 0 s.t. W y h + T x W y = h + T x Ω Ω Ω (8).3.. Parameter Pre-Processng Step From the structure of the mode (8), many eements of decson vector ξ can be predetermned to attan ther optma settng at one of ther bounds. In many cases, smpe rues exst n dentfyng the optma vaues of these eements of decson vector ξ when the nformaton on x Ω s gven even before sovng the mode (8). The foowng secton descrbes these smpe pre-processng rues for eements of vector c and matrx T n the vector ξ. Pre-Processng Step for c The eements of decson vector c represent the parameters correspondng to coeffcents of bnary decson varabes n the obectve functon of the mode (). Each eement c of vector c s represented n the obectve functon of the mode (8) as: ( cx cx Ω ). From any gven vaue of x Ω, the vaue of c can be predetermned by the foowng smpe rues. If x Ω s, the optma settng of c s c = c. Otherwse the optma settng of c s c = c. Pre-Processng Step for T The eements of decson vector T represent the coeffcents of the bnary decson varabes ocated n the ess-than-or-equa-to constrants of the mode (). Each eement T of matrx T s represented n the constrant of the mode (8) as: W y h + T x + T x and W y h + T xω + T x k k k. From any gven k Ωk k 7
vaue of x Ω, the vaue of T can be predetermned at T = T f x Ω = 0. In the case when x Ω =, the optma settng of T satsfes the foowng set of constrants ustrated n (9) where the new varabe TX repaces the nonnear term T x n the mode (8). The nsght of ths set of constrants (9) s that f the vaue of x s set to be zero by the mode, the optma settng of T s T and TX = 0. Otherwse the optma settng of T can not be predetermned and TX = T. TX T + T ( x ) 0 TX + T T ( x ) 0 T x TX T x T T T + x ( T T ) (9).3.. Probem Transformaton Step In order to sove the mode (8) effcenty, the foowng two man tasks have to be accompshed. Frst, a modeng technque s requred to propery mode the constrant ξ Ξ. Second, an effcent transformaton method s requred to transform the orgna formuaton of the mode (8) nto a computatonay effcent formuaton. The foowng two subsectons descrbe technques and methodooges for performng these two tasks..3... Modeng Technque for the Constrant ξ Ξ Consder a varabe p whch ony takes ts vaue from p dstnct rea vaues, p (), p (),, p ( p). Ths constrant on the varabe p can be formuated n the mathematca programmng mode as: p = p() b, p = p b =, b 0 =,..., p = and { b, b,..., b } s SOS. A Speca Ordered Set of type One (SOS) s defned to be a set of varabes for whch not more than one member from the set may be non-zero. When these nonnegatve varabes, b =,..., p, are defned as SOS, there are ony p branches requred n the searchng tree for these varabes..3... Fna Transformaton Steps for the B-eve- Mode Because the structure of the foower probem n the mode (8) s a near program and t affects the eader s decsons ony through ts obectve functon, the fna transformaton steps start by repacng the foower probem wth expct representatons of ts optmaty condtons. These expct representatons ncude the foower s prma constrants, the foower s dua constrants and the foower s strong duaty constrant. The mode (0) p 8
ustrates the formuaton of the mode (8) after ths frst transformaton where decson varabes w and w E represent the dua varabe assocated wth foower s constrants. The mode (0) s a snge-eve mxed nteger nonnear optmzaton probem. By appyng resuts from parameter pre-processng steps and modeng technque prevousy dscussed, the fna transformaton steps are competed and are summarzed beow. max{ q y + c x q y c x } s.t. Ω ξ Ξ W y h + T x W y h + T x Ω W y = h + T x E W y = h + T x E Ω W w + W w q E h + T x w + h + T x w = q y w 0, y 0, y 0, x {0,} Ω Ω E (0) Fna Transformaton Steps Parameter c : By appyng the preprocessng rue, each varabe c can be fxed at Parameter T : By appyng prevous resuts, f the parameter T can be preprocessed, then fx ts vaue at the approprate vaue of c. T. Otherwse, frst add a decson varabe TX and a set of constrants ustrated n (9) to repace the nonnear term T x n the mode (0), then add a set of varabes and constrants ustrated n () to repace part of the constrant ξ Ξ for T n the mode (0), and fnay add a varabe TW and a set of varabes and constrants ustrated n () to repace the nonnear term T w n the mode (0), where w and w represent the upper bound and the ower bound of dua varabe respectvey. w T T () T = T bt, bt =, bt 0 s {,,..., T } and { bt } s SOS ( s) ( s) ( s) ( s) ( s) s s= s= T TW = T ( s) ZTW ( s) s= w bt ( s) ZTW ( s) w bt ( s), ZTW ( s) w w ( bt ( s) ), ZTW ( s) w w ( bt ( s) ) s {,..., T } () 9
Parameter T : We frst add a decson varabe TX and a set of constrants ustrated n (3) to repace the nonnear term T x n the mode (0), then add a set of varabes and constrants ustrated n (4) to repace part of the constrant ξ Ξ for T n the mode (0), and fnay add a varabe TW and a set of varabes and constrants ustrated n (5) to repace the nonnear term T w n the mode (0), where w and w represent the upper bound and the ower bound of varabe w respectvey. T TX = T ( s) ZTX ( s) s= 0,,, {,..., } ZTX ( s) ZTX ( s) bt ( s) ZTX ( s) x ZTX ( s) bt ( s) + x s T (3) T T (4) T = T bt, bt =, bt 0 s {,,..., T } and { bt } s SOS ( s) ( s) ( s) ( s) ( s) s s= s= T TW = T ( s) ZTW ( s) s= w bt s ( ) ZTW s ( ) w bt s ( ), ZTW s ( ) w w( bt s ( )), ZTW s ( ) w w( bt s ( )) s {,..., T } (5) Parameter h and h : We frst add a set of varabes and constrants ustrated n (6) and (7) to repace part of the constrant ξ Ξ for h and h respectvey n the mode (0). We then add varabes HW, HWand a set of varabes and constrants n (8) and (9) to repace the nonnear terms hw and hw respectvey n the mode (0). h h (6) h = h bh, bh =, bh 0 s {,,..., h } and { bh } s SOS ( s) ( s) ( s) ( s) ( s) s s= s= h h (7) h = h bh, bh =, bh 0 s {,,..., h } and { bh } s SOS ( s) ( s) ( s) ( s) ( s) s s= s= h HW = h ( s) ZHW ( s) s= (8) w bh () s ZHW () s w bh() s, ZHW () s w w( bh() s), ZHW () s w w ( bh() s) s {,..., h } h HW = h ( s) ZHW ( s) s= w bh ( s) ZHW( s) w bh ( s), ZHW( s) w w( bh ( s) ), ZHW( s) w w( bh ( s) ) s {,..., h } (9) 0
Parameter q : We frst add a set of varabes and constrants ustrated n (0) to repace part of the constrant ξ Ξ for q n the mode (0). We then add decson varabes QY and a set of varabes and constrants n () to repace the nonnear terms QY, qy and qy respectvey n the mode (0) where y r and y r represent the upper bound and the ower bound of varabe y r respectvey for r = and. q q (0) q = q bq, bq =, bq 0 s {,,..., q } and { bq } s SOS ( s) ( s) ( s) ( s) ( s) s s= s= q QY = q ZQY ( s) ( s) s= s () () s s (), () s ( s ()), () s ( s ()) {,..., } q = ( s) ( s) s= ( s) ( s) ( s), ( s) y y bq( s) ZQY ( s) y y bq( s) s q y bq ZQY y bq ZQY y y bq ZQY y y bq s q QY q ZQY y bq ZQY y bq ZQY () ( ), ( ) {,..., } Parameter W and W : We frst add a set of varabes and constrants ustrated n () and (3) to repace part of the constrant ξ Ξ for W and W respectvey n the mode (0). We then add a set of varabes and constrants ustrated n (4) and (5) together wth varabes WY, WY, WY, WY, WW, and WW to repace the nonnear terms W y, W y W y, W y, W w, and Ww n the mode (0). W W () W = W bw, bw =, bw 0 s {,,..., W } and { bw } s SOS ( s) ( s) ( s) ( s) ( s) s s= s= W W (3) W = W bw, bw =, bw 0 s {,,..., W } and { bw } s SOS ( s) ( s) ( s) ( s) ( s) s s= s= W = ( s) ( s) s= () s () s () s, () s ( () s ), () s ( () s ) {,..., } W = ( s) ( s) s= s () ZWY () s ybw() s ZWY () s y y bw() s ZWY () s y y bw () s s W W = ( s) ( s) s= () s () s () s, () s w ( bw() s ), ZWW () s w w ( bw() s ) s {,..., W } WY W ZWY y bw ZWY y bw ZWY y y bw ZWY y y bw s W WY W ZWY y bw, ( ), ( ) {,..., } WW W ZWW wbw ZWW wbw ZWW w (4)
W = ( s) ( s) s= () s () s () s, () s ( () s ), () s ( () s ) {,..., } W = ( s) ( s) s= () s ZWY() s ybw() s ZWY() s y y bw() s ZWY() s y y bw() s s W W = ( s) ( s) s= ( s) ( s) ( s), ( s) w bw() s ZWW () s w w bw() s s W WY W ZWY y bw ZWY y bw ZWY y y bw ZWY y y bw s W WY W ZWY y bw WW W ZWW wbw ZWW wbw ZWW w, ( ), ( ) {,..., } ( ), ( ) {,..., } (5) By appyng these transformaton steps, the mode (0) can now be transformed nto ts fna formuaton as a snge eve mxed nteger near programmng probem as shown n the mode (6). The tabe n the mode (6) s used to dentfy some addtona constrants and condtons for addng these constrants to the mode (6). = max{ QY + c x QY c x } Ω s.t. WY h + TX + T x Ind = Ind = 0 WY = h + TX WY h + T x + T x Ω Ω Ind = Ind = 0 WY = h + T x E WW Ω E E + WW q HW + HW + TW x + T w x + TW x = QY y 0, y 0, w 0, x {0,} Ω Ω Ω E Ind = Ind = 0 E Condton to Add the Constrants Constrant Reference Constrant Index Set Ind = (9), (), and () For a, For a Aways (3), (4), and (5) For a E, For a Aways (6) and (8) For a Aways (7) and (9) For a E Aways (0) and () For a Aways () and (4) For a, For a Aways (3) and (5) For a E, For a Where Ind f T vaue cannot be predetermned. = 0 otherwse (6) T Preprocessed vaue of T f T can be preprocessed = 0 Otherwse
The B-eve- mode can now be soved as a mxed nteger near program by sovng the mode (6). The optma obectve functon vaue of the mode (6),, s used to update the vaue of by settng to mn{, }. The optmaty condton s then checked. If the optmaty condton s not satsfed, add scenaro whch s the optma settngs of ξ = ( cqh,,, h, T, T, W, W ) from the mode (6) to the scenaro set Ω and return to the frst stage agorthm. Otherwse, the agorthm s termnated wth an ε optma robust souton whch s the dscrete soute on wth the maxmum regret of from the mode (6). In fact, we do not have to sove the mode (6) to optmaty n each teraton to generate the scenaro. We can stop the optmzaton process for the mode (6) as soon as the feasbe souton wth the obectve vaue arger than the current vaue of has been found. We can use ths feasbe settng of ξ = ( cqh,,, h, T, T, W, W ) to generate the scenaro n the current teraton. The foowng emma 4 provdes the mportant resut that the agorthm aways termnates at an ε-optma robust souton n a fnte number of agorthm steps. emma 4: The three-stage agorthm termnates n a fnte number of steps. When the agorthm has termnated wth ε 0, t has ether detected nfeasbty or has found an ε- optma robust souton to the orgna probem. Proof: The resut foows form the defnton of case the agorthm enumerates a scenaros Ω and Ω s fnte. and and the fact that n the worst The foowng secton demonstrates appcatons of the proposed three-stage agorthm for sovng mn-max regret robust optmzaton probems under fu-factora scenaro desgn of data uncertanty. A resuts ustrate the promsng capabty of the proposed agorthm for sovng the robust optmzaton of ths type wth an extremey arge number of possbe scenaros. 3. Appcatons of the Three-Stage Agorthm In ths secton, we descrbe numerca experments usng the proposed agorthm for sovng a number of two-stage facty ocaton probems under uncertanty. We consder the suppy chan n whch suppers send matera to factores that suppy warehouses that suppy markets as shown n Fgure (Chopra and Mend). ocaton, capacty aocaton, and transportaton decsons have to be made n order to mnmze the overa suppy chan cost. Mutpe warehouses may be used to satsfy demand at a market and mutpe factores may be used to repensh warehouses. It s aso assumed that unts have been appropratey 3
adusted such that one unt of nput from a suppy source produces one unt of the fnshed product. In addton, each factory and each warehouse cannot operate at more than ts capacty and a near penaty cost s ncurred for each unt of unsatsfed demands. The mode requres the foowng notatons, parameters and decson varabes: Suppers Factores Warehouses Markets Fgure : Stages n the Consdered Suppy Chan Network (Chopra and Mend). m : Number of markets W e : Potenta warehouse capacty at ste e n : Number of potenta factory ocaton f : Fxed cost of ocatng a pant at ste : Number of suppers f e : Fxed cost of ocatng a warehouse at ste e t : Number of potenta warehouse ocatons c h : Cost of shppng one unt from supper h to factory D : Annua demand from customer c e : Cost of shppng one unt from factory to warehouse e K : Potenta capacty of factory ste c 3e : Cost of shppng one unt from warehouse e to market S h : Suppy capacty at supper h p : Penaty cost per unt of unsatsfed demand at market y : = f pant s opened at ste ; : = 0 otherwse z e : = f warehouse s opened at ste e; : = 0 otherwse x h : = Transportaton quantty from supper h to pant x 3e : = Transportaton quantty from warehouse e to market x e s : = Transportaton quantty from pant to warehouse e : = Quantty of unsatsfed Demand at market In the determnstc case, the overa probem can be modeed as the mxed nteger near programmng probem presented n the foowng mode. When some parameters n the mode are uncertan, the goa becomes to dentfy robust factory and warehouse ocatons under devaton robustness defnton. Transportaton decsons are treated as recourse decsons whch w be made after the reazaton of uncertanty. 4
mn n t n n t t m m f y + f z + c x + c x + c x + p s e e h h e e 3e 3e = e= h= = = e= e= = = n t s.t. x S h {,..., }, x x = 0 {,..., n} h h h e = h= e= t n m xe K y e 3e e= = = m {,..., n}, x x = 0 e {,..., t} x W z e {,..., t}, x + s = D {,..., m} 3e e e 3e = e= x 0 h, x 0 e, x 0 e, s 0 h e 3e t, y {0,}, and z {0,} e e We appy the proposed agorthm to 0 dfferent expermenta settngs of the robust facty ocaton probems. Each expermenta settng n ths case study contans dfferent sets of uncertan parameters and dfferent sets of possbe ocatons whch resut n dfferent number of possbe scenaros. The number of possbe scenaros n ths case study vares from 64 up to 3 40 scenaros. The key uncertan parameters n these probems are the suppy quantty at the supper, the potenta capacty at the factory, the potenta capacty at the warehouse, and the unt penaty cost for not meetng demand at the market. et us defne notatons, n, t, and m to represent the number of suppers, factores, warehouses, and markets wth uncertan parameters respectvey n the probem. It s assumed that each uncertan parameter n the mode can ndependenty take ts vaues from r possbe rea vaues. The foowng Tabe descrbes these twenty settngs of the case study. Tabe : Twenty Settngs of Numerca Probems n the Case Study Sze of Extensve Form Mode Probem n t m n t m r Ω #Constrants #Contnuous #Bnary Varabes Varabes S 8 8 8 8 0 64 3.4 x 0 3.3 x 0 4 6 S 8 8 8 8 0 4 56.5 x 0 4 4.5 x 0 4 6 S3 8 8 8 8 0 4 56.5 x 0 4 4.5 x 0 4 6 S4 8 8 8 8 0 6 04 5.0 x 0 4.80 x 0 5 6 S5 8 8 8 8 0 4 4 04 5.0 x 0 4.80 x 0 5 6 S6 8 8 8 8 0 6 04 5.0 x 0 4.80 x 0 5 6 S7 8 8 8 8 0 4 6 4096.0 x 0 5 7. x 0 5 6 S8 8 8 8 8 0 6 4 4096.0 x 0 5 7. x 0 5 6 S9 8 8 8 8 0 6 6 4 8.03 x 0 5.88 x 0 6 6 M 8 8 8 8 0 6 6 6 8.8 x 0 7 4.6 x 0 7 6 M 6 6 6 6 6 6 6 6 3 3 4.04 x 0 3.7 x 0 3 M3 6 6 6 6 6 6 6 6 3 3 4.04 x 0 3.7 x 0 3 M4 6 6 6 6 6 6 6 6 3 3 4.04 x 0 3.7 x 0 3 M5 6 6 6 6 6 6 6 6 3 3 4.04 x 0 3.7 x 0 3 M6 6 6 6 6 6 6 6 6 3 3 4.04 x 0 3.7 x 0 3 8 8 8 8 8 8 8 8 3 3 3 9.08 x 0 6 3.6 x 0 7 6 8 8 8 8 8 8 8 8 3 3 3 9.08 x 0 6 3.6 x 0 7 6 3 8 8 8 8 8 8 8 8 3 3 3 9.08 x 0 6 3.6 x 0 7 6 4 8 8 8 8 8 8 8 8 3 3 3 9.08 x 0 6 3.6 x 0 7 6 5 0 0 0 0 0 0 0 0 3 3 40 7.4 x 0 0 3.40 x 0 0 5
A case study settngs are soved by the proposed agorthm wth ε = 0 and extensve form mode (EFM) both wthout and wth Benders decomposton (BEFM) on a Wndows XP-based Pentum(R) 4 CP 3.60GHz persona computer wth.00 GB RAM usng a C++ program and CPEX 0 for the optmzaton process. MS-Access s used for the case study nput and output database. In ths case study, we appy the proposed agorthm to these twenty expermenta probems by usng two dfferent setups of nta scenaros. For the frst setup, the nta scenaro set conssts of ony one scenaro. For the second setup, the nta scenaro set conssts of a combnatons of upper and ower bounds for each man type of uncertan parameters. For exampe, there are three man types of uncertan parameters n the probems S to S9, the nta scenaro set of these probems conssts of 3 = 8 scenaros for the second setup. Tabe ustrates the computaton tme (n seconds) and performance comparson among these methodooges over a 0 settngs. If the agorthm fas to obtan an optma robust souton wthn 4 hours or fas to sove the probem due to nsuffcency of memory, the computaton tme of -- s reported n the tabe. Because the probems consdered n ths case study are aways feasbe for any settng of ocaton decsons, the Stage of the proposed agorthm can be omtted. Tabe : Performance Comparson between Proposed Agorthm and Tradtona Methods Proposed Agorthm Probem EFM BEFM Inta Scenaros Setup Inta Scenaros Setup Souton Tme Souton Tme Stage Tme Stage3 Tme Tota Tme #Iteraton ( Ω ) Stage Tme Stage3 Tme Tota Tme #Iteraton ( Ω ) S 80 56.6 9.4.7. 4 (5) 5.6 0.9 6.5 (9) S 495 78.9 9.9.7.6 4 (5) 5.8. 6.9 (9) S3 93 86.4 9.4.4.8 3 (4) 5.7 6.7 (9) S4 7540 743. 0.6.7 3.3 4 (5) 5.8.3 7. (9) S5 38630 80.4 9.6.8.4 3 (4) 5.9 0.9 6.8 (9) S6 39757 805. 9.5.8.3 4 (5).8. 4 3 () S7 -- 538.8 9.9.6.5 3 (4) 6.. 7.3 (9) S8 -- 689.5 0.8.7 3.5 4 (5). 4. 3 () S9 -- 645.6.4.9 4.3 4 (5) 3. 5. 3 () M -- -- 57. 7. 64.3 9 (0) 45. 5.4 50.6 5 (3) M -- -- 3.5 33.7 47. 8 (9) 40 5.7 55.7 0 (6) M3 -- -- 65.9 469.8 535.7 7 (8) 39. 30.5 349.7 (7) M4 -- -- 57. 4.5 8.6 5 (6).9 9 30.9 5 () M5 -- -- 9.3 70.4 793.7 (3) 9.3 0. 30.4 8 (4) M6 -- -- 37.5 4. 79.6 4 (5) 55.6 78.7 34.3 6 (0) -- -- 550 6553 703 4 (43) 49.9 4394.8 4544.7 7 (33) -- --. 948 69. 9 (0) 69.8 069.6 39.4 0 (6) 3 -- -- 548.5 77. 65.6 35 (36) 58.4 36. 94.6 (7) 4 -- -- 95 86 3768 49 (50) 0.9 85.8 836.7 5 (3) 5 -- -- 5338 33765 8503 87 (88) 7559 9038 46597 75 (9) 6
A resuts from these expermenta runs ustrate sgnfcant mprovements n computaton tme of the proposed agorthm over the extensve form mode both wth and wthout Benders decomposton. These resuts demonstrate the promsng capabty of the proposed agorthm for sovng practca mn-max regret robust optmzaton probems under fu factora scenaro desgn of data uncertanty wth extremey arge number of possbe scenaros. In addton, these numerca resuts ustrate the mpact of the quaty of the nta scenaros setup on the requred computaton tme of the proposed agorthm. Decson makers are hghy recommended to perform thorough anayss of the probem n order to construct the good nta set of scenaros before appyng the proposed agorthm. 4. Summary Ths paper deveops a mn-max regret robust optmzaton agorthm for deang wth uncertanty n mode parameter vaues of mxed nteger near programmng probems when each uncertan mode parameter ndependenty takes ts vaue from a fnte set of rea numbers wth unknown ont probabty dstrbuton. Ths type of parametrc uncertanty s referred to as a fu-factora scenaro desgn. The agorthm conssts of three stages and coordnates three mathematca programmng formuatons to sove the overa optmzaton probem effcenty. The agorthm utzes pre-processng steps, decomposton technques, and probem transformaton procedures to mprove ts computatona tractabty. The agorthm s proven to ether termnate at an optma robust souton, or dentfy the nonexstence of the robust souton, n a fnte number of teratons. The proposed agorthm has been apped to sove a number of robust facty ocaton probems under uncertanty. A resuts ustrate the outstandng performance of the proposed agorthm for sovng the robust optmzaton probems of ths type over the tradtona methodooges. 5. References Assavapokee, T., Reaff, M., Ammons, J., and Hong, I.: A Scenaro Reaxaton Agorthm for Fnte Scenaro Based Mn-Max Regret and Mn-Max Reatve Regret Robust Optmzaton. Computers & Operatons Research, (To appear) (007a). Assavapokee, T., Reaff, M., and Ammons, J.: A Mn-Max Regret Robust Optmzaton Approach for Interva Data ncertanty. Journa of Optmzaton Theory and Appcatons, (To appear) (007b). Averbakh, I.: On the Compexty of a Cass of Combnatora Optmzaton Probems wth ncertanty, Mathematca Programmng, 90, 63-7 (00). Averbakh, I.: Mnmax Regret Soutons for Mnmax Optmzaton Probems wth ncertanty, Operatons Research etters, 7/, 57-65 (000). 7