A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: III. Improving the Quality of Robust Solutions

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1 Ths s a ope access artcle publshed uder a ACS AuthorChoce Lcese, whch permts copyg ad redstrbuto of the artcle or ay adaptatos for o-commercal purposes. Artcle pubs.acs.org/iecr A Comparatve Theoretcal ad Computatoal Study o Robust Couterpart Optmzato: III. Improvg the Qualty of Robust Solutos Zuku L ad Chrstodoulos A. Floudas*, Departmet of Chemcal ad Materals Egeerg, Uversty of Alberta, Edmoto, AB T6G V4, Caada Departmet of Chemcal ad Bologcal Egeerg, Prceto Uversty, Prceto, New Jersey 08544, Uted States Dowloaded va o November 10, 018 at 15:4:38 (UTC). See for optos o how to legtmately share publshed artcles. ABSTRACT: I ths paper, we study the soluto qualty of robust optmzato problems whe they are used to approxmate probablstc costrats ad propose a ovel method to mprove the qualty. Two soluto frameworks are frst compared: (1) the tradtoal robust optmzato framework whch oly uses the a pror probablty bouds ad (3) the approxmato framework whch uses the a posteror probablty boud. We llustrate that the tradtoal robust optmzato method s computatoally effcet but ts soluto s geeral coservatve. O the other had, the a posteror probablty boud based method provdes less coservatve soluto but t s computatoally more dffcult because a ocovex optmzato problem s solved. Based o the comparatve study of the two methods, we propose a ovel teratve soluto framework whch combes the advatage of the a pror boud ad the a posteror probablty boud. The proposed method ca mprove the soluto qualty of tradtoal robust optmzato framework wthout sgfcatly creasg the computatoal effort. The effectveess of the proposed method s llustrated through umercal examples ad applcatos plag ad schedulg problems. 1. INTRODUCTION Data ucertaty wdely exsts realstc problems due to ther radom ature, measuremet errors, or other reasos. As a result, decso makg heretly volves cosderato of such ucertates sce the soluto of a optmzato problem ofte exhbts hgh sestvty to data perturbatos, ad gorg the ucertaty could lead to suboptmal or eve feasble solutos. I past decades, developg optmzato methods ad tools to facltate decso makg uder ucertaty has become oe of the most mportat topcs both the operatos research commuty ad also the process systems egeerg commuty. Robust optmzato belogs to a mportat methodology for dealg wth optmzato problems wth data ucertaty. Ths type of method eforces the costrat satsfacto for all possble realzatos of ucerta parameters sde a predefed ucertaty set. Comparg t to other methodologes that deal wth ucertaty, oe maor motvato of robust optmzato s that may applcatos the data set s a approprate oto of parameter ucertaty, especally for those cases that the parameter ucertaty s ot stochastc, or for staces where o dstrbutoal formato s avalable. Oe of the earlest papers o robust couterpart optmzato s the work of Soyster, 1 who cosdered smple perturbatos the data ad amed to fd a reformulato of the orgal lear programmg problem such that the resultg soluto would be feasble uder all possble perturbatos. The approach admts the hghest protecto ad s the most coservatve oe sce t esures feasblty agast all potetal realzatos. Thus, t s hghly desrable to provde a mechasm to allow for the trade-off betwee robustess ad performace. The work by Be-Tal ad Nemrovsk,,3 El-Ghaou et al., 4,5 ad Bertsmas ad Sm 6 vestgated the framework of robust couterpart optmzato by troducg dfferet types of ucertaty sets. Be-Tal ad Nemrovsk 3 proposed the ellpsodal set based robust couterpart formulato. El-Ghaou ad Lebret 4 troduced a robust optmzato approach for least-squares problems wth ucerta data. Bertsmas ad Sm 6 studed robust lear programmg wth coeffcet ucertaty usg a ucertaty set wth budgets. I ths robust couterpart optmzato formulato, a budget parameter (whch takes a value betwee zero ad the umber of ucerta coeffcet parameters the costrats ad s ot ecessarly teger) s troduced to cotrol the degree of coservatsm of the soluto. L et al. 7 ad Jaak et al. 8 developed the theory of the robust optmzato framework for geeral mxed-teger lear programmg problems ad cosdered both bouded ucertaty ad several kow probablty dstrbutos. The robust optmzato framework was later exteded by Verderame ad Floudas 9 ad they studed both cotuous (geeral, bouded, uform, ormal) ad dscrete (geeral, bomal, Posso) ucertaty dstrbutos ad appled the framework to operatoal plag problems. The work was further compared wth the codtoal value-at-rsk based method Verderame ad Floudas. 10 I the frst two parts of ths paper seres, 11,1 we systematcally studed the set duced robust couterpart optmzato techque for lear ad mxed teger lear optmzato problems. Dfferet ucertaty sets were extesvely studed, cludg those studed lterature ad ovel oes were troduced ths work. The relatoshp betwee dfferet represetatve ucertaty sets was dscussed, ad ther correspodg robust couterpart formulatos for both lear optmzato (LP) ad mxed teger lear optmzato (MILP) problems were derved. Probablstc guaratees Receved: May 8, 014 Revsed: Jue 18, 014 Accepted: July 3, 014 Publshed: July 3, Amerca Chemcal Socety 1311 dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

2 Idustral & Egeerg Chemstry Research o costrat satsfacto for the robust soluto of those dfferet ucertaty set duced robust couterpart optmzato models were derved, for both bouded ad ubouded ucertaty, wth ad wthout detal probablty dstrbuto formato. A key elemet applyg the robust optmzato framework s the selecto of the type ad the sze of the ucertaty set, whch s strogly related to the desred relablty of the soluto (.e., the probablty of costrat satsfacto). It s kow that chace costrat/probablstc costrat s the most drect way to eforce the relablty of the soluto of a optmzato problem, 13,18 where the relablty s expressed as a mmum requremet o the probablty of satsfyg costrats. Chace costraed optmzato problems face a lot of challeges for ther soluto. For example, eve evaluatg the dstrbuto of a sum of uformly dstrbuted depedet radom varables s very dffcult. 19 Whe the program has structural propertes that allow for a equvalet determstc formulato, a chace costraed problem ca be coverted to a determstc problem ad ca be solved drectly. 8 However, f the model does ot admt suffcet structure that ca be exploted, a approxmato method has to be used. The varous approxmato methods ca be dvded to samplg based methods ad aalytcal approxmato based methods. Frst, samplg based methods are desged based o the assumpto that t s possble to draw observatos from the dstrbuto of the ucertaty. Samplg based methods fall broadly to two categores: scearo approxmato ad sample average approxmato. For scearo approxmato, t draws a fte umber of samples from a gve dstrbuto, ad eforces all sampled costrats to hold. 14 Sample average approxmato refers to replacg the dstrbuto wth aother easy-to-use dstrbuto, typcally the emprcal dstrbuto determed from a sample draw from the orgal dstrbuto. 15 Whle solvg the approxmato problem represets oe aspect of complexty, the sze of the sample requred to guaratee the qualty of the approxmato s aother mportat lmtato. Secod, aalytcal approxmato methods are based o ether robust optmzato 16 or well-kow probablty equaltes. 17 Sce the type ad the sze of the ucertaty set s determed based o a tal assumpto o the costrat satsfacto ad the a pror probablty boud formulato, 1 robust optmzato provdes a safe approxmato of probablstc costrat. I cotrast to samplg based approxmato, robust optmzato based approxmato s a promsg determstc alteratve for certa classes of chace costraed problems. I addto, other forms of determstc aalytcal approxmato use probablty equaltes, such as the Markov equalty, Chebyshev s equalty, Berste s equalty, Hoeffdg s equalty, etc. Although robust optmzato has bee used wdely dfferet areas to acheve soluto robustess/relablty, the qualty of the soluto s ofte gored. I other words, whle the desred soluto feasblty (.e., desred probablty of costrat satsfacto) s met, how far s the soluto from optmalty? I ths work, we wll frst llustrate the above ssues ad the propose a teratve strategy for mprovg the robust soluto. I the proposed method, the tght a posteror probablty bouds are used to mprove the robust soluto wth a teratve framework. Compared to the sgle-step classcal robust optmzato method, the qualty of the robust soluto ca be mproved. O the other had, compared to the pure a posteror probablty boud based methods, the proposed method has the advatage that t does ot requre the global optmzato of ocovex problem. The rest of the paper s orgazed as follows. I secto, we frst preset the problem of optmzato wth probablstc guaratee o costrat satsfacto, that s, probablstcally costraed problem, ad the troduce the tradtoal robust optmzato based approxmato framework. Next, the a posteror probablty boud based approxmato framework s preseted secto 3. Both methods are studed through a umercal example. I secto 4, we preset a ovel teratve framework whch combes the advatage of the prevous two dfferet methods. The proposed method ad the tradtoal methods are studed through producto plag ad process schedulg problems secto 5, ad the paper s cocluded secto 6.. FRAMEWORK FOR ROBUST OPTIMIZATION.1. Problem Descrpto. Cosder the followg lear optmzato problem max cx s.t. ax + ax b J J (1) where x ad x ( = 1,..., ) ca be ether cotuous or teger varables, the left-had-sde (LHS) costrat coeffcets a ĩ are subect to ucertaty, ad J represets the dex subset that cotas the varable dces whose correspodg coeffcets are subect to ucertaty. The ucertates the costrat coeffcets are ormalzed by a ĩ = a + ξ a î J wth a beg the omal value ad a î beg a costat perturbato ampltude (a î > 0), {ξ } J are radom varables whch are subect to ucertaty. Wth the above defto, the th costrat problem 1 ca be rewrtte as the follows: ax + ξ ax b J Artcle () I may practcal applcatos, eforcg costrat satsfacto for all possble values of the ucerta parameters (.e., worst-case scearo) ca be too costly or eve mpossble. Probablstc costrat (also called chace costrat) provdes a compromse to avod ths stuato ad esures that the costrats are satsfed uder certa gve probablty. A probablstc verso of the above costrat s wrtte as follows so that a probablstc guaratee o costrat satsfacto s appled: Pr{ ax + ξax b} 1 ε J (3) or a upper boud o the probablty of costrat volato s appled Pr{ ax + ξax > b} ε J (4) where ε (0 < ε < 1) s the allowed degree of costrat volato. For stace, ε = 0.05 meas that the costrat must be satsfed wth a probablty larger tha 0.95 or the probablty of costrat volato must be less tha Whle ot probablstc costrats are alteratve for modelg soluto relablty, dvdual probablstc costrats are vestgated ths paper. Motvatg Example. Cosder the followg lear optmzato problem dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

3 Idustral & Egeerg Chemstry Research max 8x1 + 1x s.t. a11 x1 + a1 x 140 a1 x1 + a x 7 x1, x 0 ad assume that the LHS costrat coeffcets of the costrats are ucerta ad subect to ucertaty wth a 11 =10+ ξ 11, a 1 =0+ξ 1, a 1 =6+0.6ξ 1, ad a =8+0.8ξ ad ξ 11, ξ 1, ξ 1, ξ are depedet ucerta parameters uformly dstrbuted [ 1,1]. For the above problem, f we set the allowed volato probablty for each of the costrat as 0.05, the the probablstc costraed verso of the problem s max 8x + 1x 1 s.t. Pr{10x1 + 0 x + ( ξ11x1 + ξ1x) > 140} 0.05 Pr{6x1 + 8 x + (0.6ξ1x ξx) > 7} 0.05 x1, x 0.. Tradtoal Applcato Framework of Robust Optmzato. I set duced robust optmzato, the ucerta data s assumed to be varyg a gve ucertaty set ad the am s to choose the best soluto amog those mmuzed agast data ucertaty. For costrat, the set duced robust optmzato method ams to fd solutos that rema feasble for ay ξ the gve ucertaty set U so as to mmuze agast feasblty, that s ax + max ξ ax b ξ U J (5) The correspodg robust optmzato problem s max cx s.t. ax + max ξ ax b ξ U J (6) For dfferet ucertaty sets, the robust couterpart formulato s dstct. Furthermore, uder specfc probablty dstrbuto assumpto, the probablstc guaratee o the costrat satsfacto ca be quatfed usg the sze of the ucertaty set. I our prevous work, we have systematcally derved the robust couterpart formulatos uder dfferet ucertaty sets 11 ad also derved ther probablty bouds o costrat volato. 1 For example, f the ucertaty set s gve by a box U = { ξξ Ψ, J} (7) where Ψ s the sze of the box, the the robust optmzato couterpart costrat s ax +Ψ a x b J (8) If the ucerta parameters are subect to depedet bouded symmetrc dstrbuto, the the followg a pror probablty boud s vald Ψ = Ψ prorub prob ( ) exp volato (9) A pror probablty boud meas that f the sze of the box set s Ψ, the the soluto of the robust optmzato problem wll esure that the probablty of costrat volato s less tha or equal to the followg boud: Pr{ ax + ξ ax > b} prob ( Ψ) J prorub volato (10) I the lterature, the tradtoal way of applyg robust optmzato 7,4 to solve the probablstcally costraed problem s as follows. Frst, the relablty level ε the probablstc costrat s set, ad the type of the robust optmzato model (.e., ucertaty set) s selected by the dstrbuto of the ucertaty. Next, the sze of the ucertaty set s evaluated based o the a pror probablty bouds. For example, assumg that the box type ucertaty set s selected for applyg robust optmzato, the sze of the ucertaty set ca be determed by the followg problem m Ψ Ψ s.t.exp ε (11) Usg the sze parameter value determed from the above problem, the robust couterpart optmzato problem ca be solved ad the soluto esures that the costrat s satsfed wth the desred probablty 1 ε. max cx s.t. ax +Ψ a x b J Artcle (1) As a summary, the tradtoal framework of applyg robust optmzato to address the probablstc guaratee o costrat satsfacto s show Fgure 1. Fgure 1. Tradtoal framework of applyg robust optmzato 7,4 for probablstcally costraed optmzato problem. The robust optmzato problem provdes a safe ad coservatve approxmato of the probablstcally costraed problem. Notce that the mmum possble value for Ψ s used to fd the best possble soluto wth ths framework. I the followg, we llustrate the approxmato of the probablstc dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

4 Idustral & Egeerg Chemstry Research costraed problem usg the robust couterpart optmzato through the motvatg example. Motvatg Example (Cotued). The terval + ellpsodal ucertaty set duced robust couterpart optmzato model s appled to solve the motvatg example problem. The robust couterpart optmzato problem uder the terval + ellpsodal ucertaty set s as follows max 8x + 1x 1 s.t. 10x1 + 0x + u11 + u1 + Ω 1 z11 + 4z1 140 u x z u, u x z u x1 + 8x + 0.6u u +Ω 0.36z z 7 u x z u, u x z u x, x where Ω 1 ad Ω are parameters determg the sze of the terval + ellpsodal ucertaty set. Usg the probablty boud o costrat volato for ths type of robust couterpart optmzato model (uder the assumpto that the ucertaty s bouded ad symmetrc whch s satsfed ths example) Ω Ω exp 0.05, exp 0.05, B( J, Ω) 0.05 J we obtaed the smallest possble value: Ω 1 = Ω = The above robust optmzato problem s covex ad ca be effcetly solved usg covex olear optmzato solvers. Wth ths value, the robust couterpart optmzato problem ca be solved, the optmal obectve value s Ob* = ad the robust soluto s x = (7.77,.773). Ths soluto esures that the costrats are satsfed wth the desred probablty 0.95 o costrat satsfacto. Oce a robust soluto s obtaed, the probablty of costrat volato ca also be quatfed by a posteror probablty boud. I our prevous paper, 1 we studed those a posteror probablty bouds. If the probablstc dstrbuto formato o the ucerta parameters s kow, the the followg relatoshp holds: 1 + ξ > θ Pr{ ax ax b} exp( ( b ax) J θξ ax + l Ee [ ]) J (13) I the above dervato, θ s a arbtrary postve umber. As studed our prevous work, 1 wth the above probablty equalty, we ca evaluate the a posteror probablty boud o costrat volato as follows oce we have a set of soluto x, (.e., we have x as the soluto): Pr{ ax + ξax > b} m exp( θ( b a x ) + l E[ e ]) θ J J θξ ax (14) Notce that the above equato, a mmzato wth respect to θ (.e., oly oe varable) s performed to fd the tghtest probablty boud. For the tradtoal robust couterpart optmzato based framework, the adustable parameter defg the sze of the ucertaty set s tally selected based o the a pror probablty boud whch s a fucto of the adustable parameter. However, usually, the resultg soluto could be too coservatve, sce the actual probablty of costrat volato s much smaller tha the boud. For example, wth the robust soluto x = (7.77,.773) obtaed for the motvatg example, the probablty of costrat volato ca be evaluated usg the above a posteror probablty boud 14, ad the followg upper bouds o costrat volato for the two costrats ca be calculated: Pr{10x1 + 0 x + ( ξ11x1 + ξ1x ) > 140} Pr{6x1 + 8 x + (0.6ξ1x ξx ) > 7} whch are far less tha the desred volato probablty Ths mples that the obtaed robust soluto s coservatve ad there s room for mprovemet. 3. A POSTERIORI PROBABILITY BOUND BASED SOLUTION METHOD Whle the a posteror probablty boud ca be used to check the probablty of costrat satsfacto wth a gve soluto, t ca also be used aother way to formulate a safe approxmato of the probablstc costrat. Usg equalty 13, the followg safe approxmato of 4 s obtaed: θξ ax exp( θ( b a x) + l E[ e ]) ε J (15) because for ay feasble soluto satsfyg 15, t also satsfes the costrat 4. Costrat 15 ca be further rewrtte as θξ ax θ( b a x) + l E[ e ] l ε J (16) ad fally the followg safe approxmato of the probablstc costraed problem s obtaed: m cx x, θ θξ ax s.t. θ( b a x) + l E[ e ] l ε J Artcle (17) Note that whle for ay fxed value of θ > 0, 15 s a approxmato of the orgal probablstc costraed problem, here θ s set as a decso varables problem 17 so as to fd the tghtest possble approxmato ad to seek the best possble soluto. The a posteror probablty boud based framework addressg the probablstc costrat ca be represeted usg Fgure. It s see that the approxmato optmzato problem s costructed based o the selected a posteror boud, comparso to the selecto of ucertaty set the tradtoal robust optmzato framework. Defg ξ = ξ a î x, the the explct formulato of above approxmato problem depeds o the momet geeratg dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

5 Idustral & Egeerg Chemstry Research Fgure. Soluto framework of a posteror boud based method. Table 1. Summary o the pdf ad mgf of Some Dstrbutos dstrbuto of ξ uform U(a,b) fucto E[e θξ ]. For several kow dstrbutos, ther probablty desty fuctos ad momet geeratg fuctos are lsted Table 1, whch ca be substtuted to 17 ad the resultg problem ca be solved as a determstc optmzato problem. The above probablty equalty based approxmato framework s llustrated through the motvatg example. Motvatg Example (Cotued). Followg the dervato ths secto, the probablty equalty based safe approxmato of the probablstc costraed problem of the motvatg example s obtaed: max 8x + 1x x, θ 1 θξ x s.t. θ1(140 10x1 0 x) + l E[ e ] θξ 11x + l Ee [ ] l θξ x 1 1 θ(7 6x1 8 x) + l E[ e ] 0.8θξ x + l Ee [ ] l 0.05 x1, x 0, θ1, θ > 0 Sce the radom varable ξ 11 s subect to uform dstrbuto [ 1, 1], we have θξ e 111x1 Ee [ ] = probablty desty fucto f(ξ) θb θa 1/( b a), a ξ b e e 0, otherwse θ( b a) θ1x1 θ1x1 e θ x 1 1 momet geeratg fucto E(e θξ ) tragular θ θ ξ + 1, 1 ξ 0 e + e ξ + 1, 0 ξ 1 θ expoetal λξ 1 1 exp(λ) λe ξ 0 (1 θλ ) for λ θ 0 ξ < 0 ormal N(μ,σ) ξ 1 ( μ) θμ+ 0.5σ θ exp e πσ σ Evaluate the expectato terms the smlar way ad fally the followg problem s obtaed: max 8x + 1x x, θ 1 θx e e s.t. θ1(140 10x1 0 x) + l θ1x 1 θ1x θ1x e e + l l ε 4θ1x θ x e e θ(7 6x1 8 x) + l 1.θx1 0.8θx 0.8θx e e + l l ε 1.6θx x, x 0, θ, θ > θ1x θx1 The above problem s a ocovex optmzato problem, whch ca be solved through a determstc global optmzato approach. We solve the above problem through global optmzato solver ANTIGONE GAMS 4.. (wth relatve optmalty gap tolerace optcr = 0 ad resource lmt reslm = 10000) ad obta the followg soluto after s (wth a relatve gap 0.11% to the upper boud 9.33): Ob*= 9.9, x*= (7.3588,.7799), Artcle θ = (0.9501, 1.938) Comparg the above soluto wth the soluto from the tradtoal robust optmzato framework, t s observed that whle both solutos esure the desred probablty o costrat satsfacto, the a posteror probablty boud based method geerates a soluto whch s better tha the classcal method. Note though that the computatoal effort creases sce global optmzato s eeded. 4. ITERATIVE SOLUTION STRATEGY Comparg the prevous two methods, the followg observatos ca be made: (1) I terms of the formato eeded, the a posteror probablty boud based approxmato method eeds the exact probablty dstrbuto fucto whle the robust optmzato method oly eeds partal formato. For stace, the studed robust optmzato formulatos, the assumptos o ucertaty are oly bouded ad symmetrc so that a probablstc guaratee s vald. () I terms of the soluto complexty, the probablty equalty based approxmato problem ca be ocovex ad global optmzato s ecessary (.e., hgher computato complexty). The robust optmzato based approxmato leads to covex problem whch ca be solved very effcetly. (3) I terms of the qualty of the soluto, the a posteror probablty boud based approxmato method leads to less coservatve soluto because t s tghter tha the a pror probablty boud as llustrated prevous work. 1 The aforemetoed observatos show that there s a trade-off betwee the two dfferet types of approxmatos. To fully take advatage of both of them, a teratve soluto framework, whch s also the maor cotrbuto of ths paper, s proposed dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

6 Idustral & Egeerg Chemstry Research Artcle to combe the use of the tradtoal robust optmzato approxmato ad the a posteror probablty boud. The obectve s to mprove the qualty of robust soluto whle stll esure the probablstc guaratee of the robust soluto. At the same tme, the computatoal complexty s decreased comparg to the a posteror probablty boud based approxmato method. The proposed soluto framework s show Fgure 3, whch ca be detaled as follows. Itally, the type of the ucertaty set Notce that the proposed teratve framework, the problem of evaluatg the a posteror probablty boud usg the rghthad sde (RHS) of 14 s a optmzato problem. Sce ay feasble soluto of the RHS of 14 wll be a vald a posteror upper boud o the probablty of costrat volato, t s ot ecessary to obta the global optmal soluto here. Furthermore, sce x s a kow soluto ad take fxed value, the RHS of 14 s a sgle varable optmzato problem, whch ca be solved relatvely effcetly. Motvatg Example (Cotued). Applyg the proposed framework ad usg the terval + ellpsodal ucertaty set based robust couterpart optmzato formulato, we obta the followg results for the motvatg example as show Table for dfferet teratos (.e., k stads for terato): Fgure 3. Proposed teratve soluto framework to mprove the qualty of robust soluto. (.e., the robust optmzato formulato) s selected. Based o the desred degree of costrat satsfacto, the smallest possble sze of the ucertaty set s determed usg the specfc a pror probablty bouds (e.g.,exp( Ω /), exp( Ω / J ), etc.) of that type of robust formulato. Wth the determed sze of the ucertaty set, the robust couterpart optmzato problem s solved ad the robust soluto s obtaed. The, a upper boud o the costrat volato s evaluated usg the derved soluto ad the a posteror probablty boud. Ths probablty value s compared to the desred degree of costrat volato. If the gap betwee them s larger tha a certa predefed tolerace, the sze of the ucertaty set s adusted ad the robust optmzato problem s solved aga. The sze adustg s the mportat step the algorthm. Based o the fact that the a posteror probablty upper boud s mootocally decreasg fucto of the set sze, 1 ths adustmet ca be made heurstcally: f the probablty upper boud exceeds the desred level, the set sze should be decreased; f the boud s below the desred level, the set sze should be creased. The above procedure s repeated utl the gap betwee the desred degree of costrat volato ad the computed upper boud o costrat volato s less tha the tolerace. Fally, the soluto from the last roud robust optmzato step s reported. A pseudo code of the teratve algorthm s gve as follows: Table. Soluto Procedure k Ω 1,Ω exp( Ω posterorub /) Ob* (x 1, x ) prob volato 1 (.4477,.447) (1.38, 1.38) 3 (0.6119, ) 4 (0.9179, ) 5 (1.0709, ) 6 (1.1474, ) 7 (1.1856, ) (0.05, 0.05) (7.77,.773) (0.479, (7.745, 0.479).8009) (0.893, (7.6045, 0.893).9049) (0.656, (7.4, 0.656).859) (0.5636, 9.71 (7.335, ).836) (0.5177, 9.36 (7.93, ).84) (0.495, (7.354, ).777) ( , ) (0.0305, 0.005) (0.5486, 0.546) (0., 0.05) (0.105, 0.086) (0.06, 0.045) (0.045, 0.045) The detaled soluto procedure s explaed as follows: Step 1: Italze Ω 1 satsfy =.4477, Ω 1 volate =0,Ω satsfy =.4477, Ω volate = 0 usg the a pror boud. Set tolerace parameter δ = Step : Set Ω 1 = Ω 1 satsfy =.477, Ω = Ω satsfy =.477. Iterato 1 Step 3: Solve the robust optmzato problem ad obta soluto Ob* = 90.91, x 1 = 7.77, x =.773. Step 4: Compute the a posteror probablty boud from soluto P 1 = , P = Step 5: Sce the probablty volato upper boud s less tha 0.05 for both costrats, there s room to cotract the ucertaty set ad mprove the soluto. So decrease the sze of both ucertaty dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

7 Idustral & Egeerg Chemstry Research sets based o the value terato 1: Ω 1 = Ω = ( )/ = Iterato Step 3: Solve the robust model ad obta Ob* = , x 1 = 7.745, x = Step 4: Compute the a posteror probablty boud P 1 = , P = Step 5: Sce the probablty volato upper boud s stll less tha 0.05 for both costrats, t s ecessary to further decrease the sze of both ucertaty sets based o the sze value terato 1 Ω 1 = Ω = ( )/ = Iterato 3 Step 3: Solve the robust model ad get Ob* = , x 1 = , x = Step 4: Compute the a posteror probablty boud P 1 = , P = Step 5: Sce the probablty volato upper boud becomes larger tha 0.05 for both costrats, ths meas the ucertaty set should be elarged to satsfy chace costrat. So we adust the ucertaty sets based o smallest sze leadg to costrat satsfacto so far (1.38 terato ) ad the sze that leads to volato ( terato 3): Ω 1 = Ω = ( )/ = Iterato 4 Step 3: Solve the robust model ad obta Ob* = , x 1 = 7.4, x =.859. Step 4: Compute the a posteror probablty boud P 1 = 0., P = Step 5: Sce the probablty volato s stll larger tha 0.05 for both costrats by usg sze , we eed to further elarge the ucertaty set to satsfy chace costrat. We adust the parameter toward the smallest sze leadg to costrat satsfacto (1.38 terato ): Ω 1 = Ω = ( )/ = Iterato 5 Step 3: Solve the robust model ad obta Ob* = 9.71, x 1 = 7.335, x =.836. Step 4: Compute the a posteror probablty boud P 1 = 0.105, P = Step 5: Sce the probablty volato s stll larger tha 0.05 for both costrats the prevous terato, we eed to further elarge the ucertaty sets: Ω 1 = Ω = ( )/ = Iterato 6 Step 3: Solve the robust model ad obta Ob* = 9.36, x 1 = 7.93, x =.84. Step 4: Compute the a posteror probablty P 1 = 0.06, P = Step 5: Sce the probablty volato s larger tha 0.05 for the frst costrats ad the secod costrat s satsfed, we keep Ω uchaged ad further crease Ω 1 as Ω 1 = ( )/ = Iterato 7 Step 3: Solve the robust model ad obta Ob* = 9.153, x 1 = 7.354, x =.777. Step 4: Compute the a posteror probablty P 1 = 0.045, P = 0.045; both are less tha 0.05 ad the gap s smaller tha δ = 0.01, so the terato stops. Step 6: Retur the fal soluto Ob* = 9.153, x 1 = 7.354, x =.777. The soluto procedure of the above teratve method s also llustrated Fgure 4, whch shows how the costrat Fgure 4. Iteratve soluto procedure for the motvatg example: (upper) a posteror probablty boud; (lower) optmal obectve value of robust soluto. satsfacto probablty coverges to the desred level ad how the qualty of the robust soluto s evetually mproved comparg to the tradtoal framework. Fally, solutos from three dfferet methods for the motvatg example are summarzed Table 3. The colums Table 3. Comparg the Dfferet Solutos for the Motvatg Example Artcle tradtoal a posteror teratve Ob* (1.43%) (0.08%) posterorub prob volato ( , ) (0.05, 0.05) (0.045,0.045) CPU tme (s) tradtoal, a posteror, ad teratve represet the tradtoal robust optmzato framework, the a posteror probablty boud based method, ad the teratve method, respectvely. The row Ob* ad prob posterorub volato represet the optmal obectve value of the robust optmzato problem ad the a posteror probablty boud based o the robust soluto obtaed. The percetage umbers represet the gaps betwee the tradtoal/ teratve method s solutos ad the a posteror method s soluto. Comparg the tradtoal robust optmzato based approxmato framework ad the proposed teratve method, we observed that t mproves the qualty of the soluto whle stll esures the degree of costrat satsfacto. Notce that the robust soluto has bee mproved from to The percetage gap to the a posteror soluto 9.9 has bee decreased from 1.43% to 0.08% as show Table 3. Comparg the pure a posteror probablty boud based approxmato dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

8 Idustral & Egeerg Chemstry Research method ad the proposed teratve method, we observed that ts computatoal complexty s decreased, sce oly a set of covex robust optmzato problem s solved. The global optmzato of the ocovex optmzato s avoded. I addto, whe the gap tolerace s defed small eough, the soluto of the proposed method wll be close to the soluto of the probablty equalty based method. Fally, we summarze the characterstcs of the three methods Table 4. Table 4. Summary of Dfferet Methodologes tradtoal a posteror teratve ucertaty formato eeded partal full full soluto complexty low hgh low soluto qualty coservatve good good 5. CASE STUDIES I ths secto, we apply the dfferet methods to solve a producto plag problem ad a process schedulg problem to compare ther performaces ad llustrate the effectveess of the proposed teratve method. All the optmzato problems are solved o a UNIX workstato wth 3.40 GHz Itel Core CPU ad 8GB memory. The related global optmzato problems are solved va ANTIGONE ad the (mxed teger) lear optmzato problems are solved usg CPLEX 1.0 GAMS 4... Resource lmt s set as s for all cases Example 1. Ths example was troduced by L et al., 1 whch addresses the problem of plag the producto, storage ad marketg of a product for a compay. It s assumed that the compay eeds to make a producto pla for the comg year, dvded to sx perods of moths each, to maxmze the sales wth a gve cost budget. The producto cost cludes the cost of raw materal, labor, mache tme, etc., ad the cost fluctuates from perod to perod. The product maufactured durg a perod ca be sold the same perod, or stored ad sold later o. Operatos beg perod 1 wth a tal stock of 500 tos of the product storage, ad the compay would lke to ed up wth the same amout of the product storage at the ed of perod 6. Ths problem ca be formulated as a lear optmzato problem as follows: max Pz (18a) s.t. Cx + Vy (18b) x1 ( y + z ) = (18c) y + x ( y + z ) = 0 =,..., 6 1 (18d) y = (18e) x U = 1,..., 6 (18f) z D = 1,..., 6 x, y, z 0 = 1,..., 6 (18g) I ths example, t s assumed that the producto costs C are subect to depedet ucertaty dstrbutos. The ucertaty s ormalzed usg 50% of the omal value C as the base perturbato ampltude. The the orgal costrat 18b ca be rewrtte as ( C ξc) x + Vy where ξ are depedet radom varables. To esure the relablty of the soluto, the mmum probablty for costrat 18b to be satsfed s set as 0.85 (.e., the upper boud o the probablty of costrat volato s set to 0.15), the the probablstc costraed verso for ths costrat s Pr{ Cx + Vy > } 0.15 I the sequel, ths example s studed uder dfferet assumptos o the ucertaty dstrbutos. (a) Uform Dstrbuto. I ths case, t s assumed that the producto costs are subect to uform ucertaty, that s, ξ are radom varables that uformly dstrbutes [ 1, 1]. The tradtoal robust optmzato method s appled frst to solve the probablstcally costraed optmzato problem. Usg the terval + ellpsodal type ucertaty set, the followg robust couterpart optmzato costrats ca be formulated Cx + Vy + 0.5Cu + Ω 0.5C v u x v u The robust optmzato problem s obtaed by replacg the orgal determstc costrat 18b wth the above costrats. Usg the a pror probablty boud for the terval + ellpsodal set duced robust optmzato model, the sze of the ucertaty set s computed as Ω = The the robust optmzato problem s solved ad the correspodg obectve s The correspodg robust plag soluto s show Fgure 5. Fgure 5. Soluto of tradtoal robust optmzato method. The a posteror probablty boud based method s appled ext ad the followg costrat s appled to replace the orgal costrat 18b: 0.5θξCx θ( Cx Vy) + l E[ e ] l 0.15 For the a posteror probablty boud based method, the obectve value s It s see that ths soluto s better (hgher sales) tha that of the classcal robust optmzato method. The robust plag soluto s show Fgure 6. Artcle dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

9 Idustral & Egeerg Chemstry Research Fgure 6. Soluto of the a posteror probablty boud based approxmato model. Fally, the teratve framework s appled to solve the problem, the soluto procedure s show Table 5 ad the robust soluto s show Fgure 7. Table 5. Soluto Procedure of the Iteratve Method for Example 1 uder Uform Dstrbuto k Ω k exp( Ω posterorub /) Ob* prob volato Artcle reslm = s) s the best although global optmzato s ecessary. If we compare the classcal robust optmzato method ad the teratve robust optmzato method ad cosder the dfferece betwee ther soluto ad the best soluto (.e., the a posteror boud based soluto), the gap has bee decreased from to (a percetage of 9.93%), ad the soluto has bee greatly mproved through the teratve framework. Comparg the soluto of the a posteror probablty boud based method (Fgure 6) ad the teratve method (Fgure 7), t s see that the dfferece betwee the solutos s very small. Note that global optmzato s ot eeded the teratve framework ad oly fve covex robust couterpart optmzato problems are solved. (b) Tragular Dstrbuto. I ths case, t s assumed that the radom varables ξ are subect to symmetrc tragular dstrbuto wth support o [ 1,1]. Notce that ths type of dstrbuto s bouded ad symmetrc, so we ca stll apply the a pror probablty bouds to determe the sze of the ucertaty set ad the apply the tradtoal robust optmzato framework. Uder the terval + ellpsodal set duced robust optmzato model, the soluto wll be the same as the prevous uform dstrbuto sce the a pror probablty boud does ot deped o the dstrbuto. O the other had, the soluto of the other two methods wll chage sce they deped o the dstrbuto of the ucertaty. Specfcally, whle the a posteror probablty boud based method s appled, the soluto s after a s resource lmt reaches GAMS (wth a relatve gap of 7.5% to the upper boud ). Whe the teratve framework s appled, the fal soluto s ad the soluto procedure s show Table 7. Table 7. Soluto Procedure of Iteratve Method for Example 1 uder Tragular Dstrbuto Fgure 7. Soluto of teratve framework. I the above soluto procedure, the parameter Ω s adusted as follows: Ω = 0.5Ω 1, Ω 3 = 0.5(Ω 1 + Ω ), Ω 4 = 0.5(Ω + Ω 3 ). Notce that although the parameter value s decreased the fourth step, t s stll a coservatve soluto ( < 0.15), so we do ot adust the parameter usg Ω 5 = 0.5(Ω 3 + Ω 4 ), but rather Ω 5 = 0.5(Ω + Ω 4 ). The solutos of dfferet methods are summarzed the followg table. It s see from Table 6 that whle all the solutos Table 6. Results Summary of Example 1 uder Uform Dstrbuto tradtoal a posteror teratve Ob* (7.6%) (0.54%) posterorub prob volato CPU tme (s) satsfy the probablstc requremet o costrat satsfacto, the soluto of the a posteror probablty boud method (wth relatve optmalty gap tolerace optcr = 0 ad resource lmt k Ω k exp( Ω posterorub /) Ob* prob volato Table 8. Results Summary of Example 1 uder Tragular Dstrbuto tradtoal a posteror teratve Ob* (11.4%) (0.8%) posterorub prob volato CPU tme (s) The results of three dfferet methods are compared Table 8. Whle the tradtoal framework leads to a 11.4% dfferece to the a posteror method soluto, the teratve method s soluto oly has 0.8% dfferece. Ths shows that the teratve framework sgfcatly mprove the qualty of the robust soluto whle the relablty of the soluto s satsfed. I the above studes, t s assumed that the ucertaty dstrbuto s depedet, bouded ad symmetrc, such that we ca apply the tradtoal robust optmzato framework based o oly the a pror probablty bouds. However, whe the ucertaty dstrbuto does ot fall to ths characterstc, 1310 dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

10 Idustral & Egeerg Chemstry Research there s o bass for determg the sze of the ucertaty set. Cosequetly, the tradtoal robust optmzato framework caot be drectly appled (.e., f we stll use the a pror boud to determe the sze, the soluto wll ot esure the probablstc guaratee). O the other had, wth the proposed teratve framework, the robust optmzato ca stll be appled to solve the problem. Next, we study two cases where the dstrbuto does ot satsfy the bouded or symmetrc codto. (c) Expoetal Dstrbuto. I ths case, we assume the radom varables are subect to expoetal dstrbuto wth rate parameter λ = 1. Notce that ths dstrbuto s ubouded, so we apply the ellpsodal type ucertaty set duced robust optmzato model rather tha terval + ellpsodal type ths study. Wth the a posteror probablty boud based method, the fal soluto s Wth the teratve soluto framework, the soluto s ad the correspodg a posteror probablty upper boud of costrat volato s The soluto procedure s lsted Table 9. Notce that we do ot lst Table 9. Soluto Procedure of Iteratve Method for Example 1 uder Expoetal Dstrbuto posterorub k Ω k Ob* prob volato the a pror boud value here, sce t s ot applcable for the asymmetrc dstrbuto ths case. (d) Normal Dstrbuto. It s assumed that each ξ s subect to ormal dstrbuto N(0,0.5) ths case. Although for the case of the ormal dstrbuto, t s ot ecessary to apply a approxmato scheme to solve the probablstcally costraed problem sce aalytcal determstc equvalet problem ca be formulated ad solved, we study the robust optmzato approxmato based method here to compare the soluto qualty. The ellpsodal type ucertaty set s also used ths case to deal wth the ubouded dstrbuto. The a posteror method lead to soluto of , ad the teratve method leads to , wth the costrat volato probablty less tha as show Table 10. Table 10. Soluto Procedure of Iteratve Method for Example 1 uder Normal Dstrbuto posterorub k Ω k Ob* prob volato Example. I ths example, a process schedulg problem 11,0 s studed. Ths example volves the schedulg of a batch chemcal process related to the producto of two chemcal products usg three raw materals. The mxed teger lear optmzato model for the schedulg problem s formulated as follows, ad the readers are drected to the paper 11 for the detaled mxed teger lear optmzato formulato ad problem data max proft s.t. proft = prce d + prce (STI STF) s Sp, wv 1 I I,, s s, s Sr s s s C P ρ ρ st = st d b + b s S, (19a) (19b) s, s, 1 s, s,,, s,,, 1 Is J I J s st st s S, s, max s m max,,,,,,,, v wv b v wv I, J, d r s S s, s P Tf Ts + α wv + β b I, J,,,,,,,,,,, Ts Tf H(1 wv ),, + 1,,,, I, J, Ts Tf H(1 wv ),, + 1,,,,, I, J, Ts Tf H(1 wv ),, + 1,,,,, I,,, J, Ts Ts I, J, (19c) (19d) (19e) (19f) (19g) (19h) (19) (19),, + 1,, (19k) Tf Tf I, J,,, + 1,, (19l) Ts,, H I, J, (19m) Tf H I, J,,, (19) I ths example, we cosder the demad ucertaty oly ad the followg costrats are affected: r d 0 s S s s, P We assume depedet ucertaty dstrbutos o the product demad parameters r s ad assg a base perturbato of 0% of the omal demad data (r P1 = 50, r P = 75): r s = r s (1 + 0.ξ s ). We set the expected mmum probablty level o costrat satsfacto to 0.5 (.e., set the upper boud o costrat volato to 0.5). The the probablstc costraed verso s Pr{ r d > 0} 0.5 s S s s, Several dfferet type of ucertaty dstrbutos are cosdered to study the proposed method. P Artcle 1311 dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

11 Idustral & Egeerg Chemstry Research Artcle Fgure 8. Schedule obtaed from the a posteror boud based method. (a) Bouded Dstrbuto. We cosder a bouded symmetrc dstrbuto frst. Specfcally, we assume a uform dstrbuto o the product demad parameters r s (.e., ξ s s uformly dstrbuted [ 1,1]). Based o the tradtoal robust optmzato method, the robust couterpart optmzato costrat s formulated as follows: r d + Ψ(0. r) 0 s S s s, s s P whch s equvalet for box, ellpsodal, ad polyhedral type of ucertaty sets sce the umber of ucerta parameter each demad costrat s 1. From the desred boud o costrat volato 0.5, we get the ucertaty set sze value Sce the ucertaty s bouded, a set wth sze value 1 wll cover the whole ucertaty space. However, sze 1 stll makes the robust optmzato problem feasble. The a posteror probablty boud based method s appled ext ad the followg costrat s appled: 0. θξ s s r s θ( r d ) + l E[ e ] l 0.5 s S s s s, The resultg ocovex mxed teger olear optmzato problem s solved usg ANTIGONE (wth tolerace parameter optcr = 0.01 ad reslm = ) ad the obectve s (wth relatve optmalty gap 1.8% to the upper boud after s). The correspodg schedule s show Fgure 8. Fally, we apply the teratve soluto framework to solve the problem. Sce set sze value 1 makes the robust optmzato problem feasble, the teratve framework, we start from 0.5 for the parameter Ψ s to make the problem feasble. The soluto procedure s show Table 11. Notce that the Table 11. Soluto Procedure of Iteratve Method for Example uder Uform Dstrbuto k Ψ s exp( Ω posterorub /) Ob* prob volato 1 (1.0, 1.0) (0.6065, ) feasble (0.5, 0.5) (0.885, 0.885) (0.6646, 0.039) 3 (0.75, 0.5) (0.7548, 0.969) (0.3397, ) 4 (0.65, 0.375) (0.86, 0.931) (0.5070, 0.531) 5 (0.6875, 0.5) (0.7895, 0.969) (0.440, 0.839) 6 (0.6875, 0.15) (0.7895, 0.99) (0.440, 0.839) adustmets of the parameter values are based o the chage of the a posteror bouds. For example, we realze that for the frst demad costrat s the d terato ad the thrd terato. To move the boud close to less tha 0.5, we P set the ew parameter value Ψ P1 the fourth terato as ( )/ = 0.65, ad the resultg soluto lead to a ew probablty boud Notce the sxth step, there s o chage o the obectve soluto, so the soluto procedure s stopped. The fal optmal obectve value s , ad the correspodg schedule s show Fgure 9. Ths soluto has oly 0.54% dfferece to the a posteror soluto as show Table 1. Comparg the soluto from all the three dfferet methods show Table 1, the followg observatos ca be made. Frst, the tradtoal robust soluto framework s coservatve ad eve leads to a feasble problem. However, the teratve framework successfully addresses the same problem ad fds feasble soluto. The reaso s that the teratve framework utlzes ot oly the a pror probablty boud but also the a posteror probablty boud, thus avods the coservatve soluto. Secod, comparg the a posteror probablty boud based method ad the teratve method, t s see that the optmal solutos are very close. However, wth the teratve framework, we obta a soluto wth almost same qualty but far less computatoal efforts. Ths further valdates the effectveess of the proposed teratve method. (b) Ubouded Dstrbuto (Expoetal ad Normal). For ths schedulg example, two ubouded dstrbutos are also studed. The tradtoal framework s ot applcable ths stuato because the a pror probablty boud s based o the bouded dstrbuto assumpto. We frst study the expoetal dstrbuto wth parameter λ = 5. Box, ellpsodal, ad polyhedral type of ucertaty set lead to same robust optmzato formulatos here. The results of the dfferet methods are summarzed Table 13, ad the soluto procedure of teratve method s gve Table 14. The teratve method s soluto has oly 0.6% dfferece to the a posteror soluto ths case. Next, we study the case uder ormal dstrbuto N(0, 0.5). The results are gve Table 15, ad the soluto procedure of teratve method s summarzed Table 16. The a posteror soluto has a relatve gap of 1.33% to the upper boud after s. Iteratve method s soluto has oly 0.1% dfferece whe compared to the a posteror soluto. From the above results, t s observed that whle the tradtoal method s ot applcable to the ubouded dstrbuto cases, the teratve method apples the robust optmzato approxmato ad uses the a posteror boud to check the soluto relablty. The solutos of teratve method are cosstetly very close to the a posteror boud based method terms of the optmal obectve values whle the computatoal complexty s greatly reduced sce oly covex robust optmzato problem s solved several teratos. 131 dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

12 Idustral & Egeerg Chemstry Research Artcle Fgure 9. Schedule obtaed from the teratve method. Table 1. Results Summary of Example uder Uform Dstrbuto tradtoal a posteror teratve Ob* feasble (0.54%) posterorub prob volato (0.5, 0.558) (0.44, 0.839) CPU tme (s) Table 13. Results Summary of Example uder Expoetal Dstrbuto tradtoal a posteror teratve Ob* ot applcable (0.6%) posterorub prob volato (0.5, 0.167) (0.406, 0.187) CPU tme (s) Table 14. Soluto Procedure of Iteratve Method for Example uder Expoetal Dstrbuto posterorub k Ψ s Ob* prob volato 1 (1.0, 1.0) feasble (0.5, 0.5) (0.5578, ) 3 (0.75, 0.5) (0.397, 0.40) 4 (0.8, 0.) (0.1991, 0.394) 5 (0.7, 0.3) (0.873, 0.094) 6 (0.6, 0.4) (0.406, 0.187) Table 15. Results Summary of Example uder Normal Dstrbuto tradtoal a posteror teratve Ob* ot applcable (0.1%) posterorub prob volato (0.5, 0.547) (0.4868, 0.58) CPU tme (s) Table 16. Soluto Procedure of Iteratve Method for Example uder Normal Dstrbuto posterorub k Ψ s Ob* prob volato 1 (1.0, 1.0) feasble (0.5, 0.5) (0.6065, 0.357) 3 (0.75, 0.5) (0.347, 0.307) 4 (0.8, 0.) (0.780, 0.343) 5 (0.7, 0.3) (0.3753, 0.901) 6 (0.6, 0.4) (0.4868, 0.58) CONCLUSION The tradtoal robust optmzato framework ca be used to approxmate probablstc costrats ad provde safe soluto. However, the soluto ca be coservatve. Whe a detaled probablty dstrbuto o ucertaty s avalable, the a posteror probablty boud based method leads to less coservatve approxmato, but the trade-off s that the resultg ocovex problem eeds to be solved va a determstc global optmzato approach. A ovel soluto framework combg the robust optmzato approxmato ad the a posteror probablty boud evaluato s proposed to mprove the soluto qualty of tradtoal robust optmzato framework wthout sgfcat computato effort. The effectveess of the proposed method has bee llustrated through a motvatg example, as well as plag ad schedulg problems. Furthermore, whle the tradtoal robust optmzato method requres formato o certa probablty dstrbuto o the ucertaty such that the a pror probablty boud s vald, the proposed teratve framework exteds the applcato to geeral dstrbutos sce we ca always use the a posteror probablty boud to esure the costrat s satsfed wth desred probablty. Fally, t s worth metog that the probablty bouds used ths work are derved based o the assumpto of depedece o the ucerta parameters. Oe of the future research drectos wll be to vestgate the correlato betwee ucerta parameters. AUTHOR INFORMATION Correspodg Author *E-mal: floudas@tta.prceto.edu. Tel.: Fax: Notes The authors declare o competg facal terest. ACKNOWLEDGMENTS The authors gratefully ackowledge facal support from the Natoal Scece Foudato (CMMI ) ad the Natoal Isttute of Health (5R01LM009338). REFERENCES (1) Soyster, A. L. Covex programmg wth set-clusve costrats ad applcatos to exact lear programmg. Operatos Research 1973, 1, () Be-Tal, A.; Nemrovsk, A. Robust solutos of ucerta lear programs. Operatos Research Letters 1999, 5 (1), dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,

13 Idustral & Egeerg Chemstry Research Artcle (3) Be-Tal, A.; Nemrovsk, A. Robust solutos of Lear Programmg problems cotamated wth ucerta data. Mathematcal Programmg 000, 88, (4) El-Ghaou, L.; Lebret, H. Robust solutos to least-squares problems wth ucerta data. Sam Joural o Matrx Aalyss ad Applcatos 1997, 18 (4), (5) El-Ghaou, L.; Oustry, F.; Lebret, H. Robust solutos to ucerta semdefte programs. SIAM J. Optm. 1998, 9, (6) Bertsmas, D.; Sm, M. The prce of robustess. Operatos Research 004, 5 (1), (7) L, X.; Jaak, S. L.; Floudas, C. A. A ew robust optmzato approach for schedulg uder ucertaty: I. bouded ucertaty. Comput. Chem. Eg. 004, 8, (8) Jaak, S. L.; L, X.; Floudas, C. A. A ew robust optmzato approach for schedulg uder ucertaty: II. Ucertaty wth kow probablty dstrbuto. Comput. Chem. Eg. 007, 31, (9) Verderame, P. M.; Floudas, C. A. Operatoal Plag of Large- Scale Idustral Batch Plats uder Demad Due Date ad Amout Ucertaty. I. Robust Optmzato Framework. Id. Eg. Chem. Res. 009, 48 (15), (10) Verderame, P. M.; Floudas, C. A. Operatoal Plag of Large- Scale Idustral Batch Plats uder Demad Due Date ad Amout Ucertaty: II. Codtoal Value-at-Rsk Framework. Id. Eg. Chem. Res. 010, 49 (1), (11) L, Z.; Dg, R.; Floudas, C. A. A Comparatve Theoretcal ad Computatoal Study o Robust Couterpart Optmzato: I. Robust Lear ad Robust Mxed Iteger Lear Optmzato. Id. Eg. Chem. Res. 011, 50, (1) L, Z.; Tag, Q.; Floudas, C. A. A Comparatve Theoretcal ad Computatoal Study o Robust Couterpart Optmzato: II. Probablstc Guaratees o Costrat Satsfacto. Id. Eg. Chem. Res. 01, 51, (13) Chares, A.; Cooper, W. W.; Symods, G. H. Cost Horzos ad Certaty Equvalets - a Approach to Stochastc-Programmg of Heatg Ol. Maagemet Scece 1958, 4 (3), (14) Nemrovsk, A.; Shapro, A., Scearo approxmatos of chace costrats. I Probablstc ad radomzed methods for desg uder ucertaty; Calafore, G.; Dabbee, F., Eds.; Sprger: Lodo, 005; pp (15) Luedtke, J.; Ahmed, S. A Sample Approxmato Approach for Optmzato wth Probablstc Costrats. SIAM Joural o Optmzato 008, 19 (), (16) Nemrovsk, A.; Shapro, A. Covex approxmatos of chace costraed programs. SIAM Joural o Optmzato 006, 17 (4), (17) Pter, J. Determstc approxmatos of probablty equaltes. ZOR - Method ad Models of Operatos Research 1989, 33, (18) Prekopa, A. Stochastc Programmg; Kluwer: Dordrecht, (19) Mtra, S. K. Probablty Dstrbuto of Sum of Uformly Dstrbuted Radom Varables. Sam Joural o Appled Mathematcs 1971, 0 (), (0) Ierapetrtou, M. G.; Floudas, C. A. Effectve cotuous-tme formulato for short-term schedulg.. Cotuous ad semcotuous processes. Id. Eg. Chem. Res. 1998, 37 (11), (1) Ierapetrtou, M. G.; Floudas, C. A. Short-term schedulg: New mathematcal models vs algorthmc mprovemets. Comput. Chem. Eg. 1998,, S419 S46. () Ierapetrtou, M. G.; Hee, T. S.; Floudas, C. A. Effectve cotuous-tme formulato for short-term schedulg. 3. Multple termedate due dates. Id. Eg. Chem. Res. 1999, 38 (9), (3) Mseer, R.; Floudas, C. A. ANTIGONE: Algorthms for contuous/iteger Global Optmzato of Nolear Equatos. Joural of Global Optmzato 014, 59, , DOI: / s (4) L, Z.; Ierapetrtou, M. G. Robust optmzato for process schedulg uder ucertaty. Id. Eg. Chem. Res. 008, 47, dx.do.org/10.101/e Id. Eg. Chem. Res. 014, 53,