New Algorithms for Nonlinear Generalized Disjunctive Programming

Transcription

1 ew Agorithms for oninear Generaized Disuntive Programming Sangbum Lee and Ignaio E. Grossmann * Department of Chemia Engineering Carnegie Meon University Pittsburgh PA U.S.A. Otober 999 / Marh * To whom a orrespondene shoud be addressed: Phone: -68- Fa: e-mai:

2 ABSTRACT Generaized Disuntive Programming GDP has been introdued reenty as an aternative mode to MILP for representing disrete/ontinuous optimization probems. The basi idea of GDP onsists of representing disrete deisions in the ontinuous spae with disuntions and onstraints in the disrete spae with ogi propositions. In this paper we desribe a new onve noninear reaation of the noninear GDP probem that reies on the use of the onve hu of eah of the disuntions invoving noninear inequaities. The proposed noninear reaation is used to reformuate the GDP probem as a tight MILP probem and for deriving a branh and bound method. Properties of these methods are given and the reation of this method with the Logi Based Outer-Approimation method is estabished. umeria resuts are presented for probems in obshop sheduing synthesis of proess networs optima positioning of new produts and bath proess design. eywords: Generaized disuntive programming branh and bound mied-integer noninear programming noninear onve hu. ITRODUCTIO Mied Integer on-linear Programming MILP modes are widey used in disrete/ontinuous optimization Grossmann and ravana 997. MILP probems arise for instane in proess synthesis heat ehanger networs and reator networs in proess design optima positioning of produt and feed oation in distiation oumn in the synthesis of proess networs and in the design and sheduing of bath and ontinuous mutiprodut pants. Agorithms for soving MILP probems inude Branh and Bound BB Gupta and Ravindran 98; Borhers and Mithe 99; Stubbs and Mehrotra 999; Leyffer 999 Outer-Approimation OA Duran and Grossmann 986; uan et a. 988; Fether and Leyffer 99 Generaized Benders Deomposition GBD Geoffrion 97 Etended Cutting Pane ECP Westerund and Pettersson 99 LP/LP based branh and bound Quesada and Grossmann 99 and branh-and-ut Stubbs and Mehrotra 999. For a detaied review see Grossmann and ravana 997. Generaized Disuntive Programming GDP whih an be regarded as a generaization of disuntive programming Baas 98 has been introdued as an aternative mode to the MILP probem that uses disuntions and ogi propositions Raman and Grossmann 99.

3 Whie the MILP mode is based entirey on agebrai equations and inequaities for disrete/ontinuous optimization probem the GDP mode aows a ombination of agebrai and ogia equations whih faiitates the representation of disrete deisions. Türay and Grossmann 996 have proposed a ogi-based Outer-Approimation agorithm for soving noninear GDP probems for proess networs invoving two terms in eah disuntion. This agorithm is based on the idea of etending the Outer-Approimation agorithm by soving LP subprobems in redued spae and MILP master probems orresponding to the onve hu of the inearization of the noninear inequaities. In addition severa LP subprobems must be soved to initiaize the master probem in order to over a the terms in the disuntions. This agorithm has been impemented in LOGMIP a omputer ode deveoped by Vehietti and Grossmann 999. In this paper we address the soution of GDP probems that invove disuntions with mutipe terms. We first desribe the onve hu of a disuntion invoving onve noninear inequaities whih provide the tightest reaation of the disuntion. The equations desribing the onve hu are used as a basis to deveop a onve noninear reaation of the GDP probem. This LP reaation an be used for reformuating it as an MILP probem or for deveoping a speia purpose branh and bound method whih wi be desribed in detai. We eamine the reation of the proposed method with the one Türay and Grossmann 996 whih an hande ony disuntions with two terms and is restrited to proess networs. We desribe in this paper the basi ideas of the proposed method and emphasize its geometria interpretation. Detaied proofs an be found in Lee and Grossmann 999. The proposed methods are appied to sma anaytia eampes and to probems deaing with obshop sheduing proess networs optima positioning of new produts and design of a bath proess. GEERALIZED DISJUCTIVE PROGRAMMIG Consider the Generaized Disuntive Programming probem Raman and Grossmann 99 whih is an etension of the wor of Baas 98. In genera the GDP mode inudes Booean variabes disuntions and ogi propositions as shown in probem P

4 s. t. min Z J r g γ Ω True f { true fase} P Here R n is the vetor of ontinuous variabes and are Booean variabes. R are ontinuous variabes and γ are fied harges; f: R n R is the term for ontinuous variabes in the obetive funtion and r: R n R q are ommon onstraint sets that hod regardess of the disrete deisions. f and r are onve funtions. A disuntion is omposed of an OR operator and a number of terms. In eah term there are the Booean variabes a set of onve noninear inequaities g g i : R n R ii where I is an inde set of inequaities and a ost variabe. If is true then g and γ are enfored. Otherwise the orresponding onstraints are ignored. We assume here that eah term in the disuntions gives rise to a non-empty feasibe region. In proess synthesis probems g are heat or mass baane equations or speifiations of the proess and γ are fied harges for eah proess. J is an inde set of the terms for eah disuntion J {...m }. Finay Ω True orrespond to ogi propositions in terms of the Booean variabes. The ogi propositions Ω are epressed in Conuntive orma Form CF: Ω... P Q S s s s where P s is the set of Booean variabes whih are true and Q s is the set of Booean variabes whih are fase in ause s s S. In CF every ause that is epressed in terms of the OR operator must be true. In probem P the funtions f r and g are assumed to be onve and bounded. Aso it is assumed that probem P has a non-empty ompat feasibe region. The GDP probem P an be reformuated as the foowing MILP probem BM by repaing the Booean variabes by binary variabes y and using big-m onstraints. The ogi onstraints Ω are onverted into inear inequaities Wiiams 98 whih eads to the foowing big-m MILP;

5 J y a Ay y J y M g r s t f y Z J J {} BM.. min γ In this mode M are the big-m parameters that render the inequaities g redundant when y. The inequaities Ay a an be systematiay derived from the CF form of Ω as disussed in Raman and Grossmann 99. ote aso that the reaation of BM is obtained by treating the binary variabes as ontinuous in the range. ILLUSTRATIVE EXAMPLE Consider the foowing GDP probem with one disuntion. } { 8.. min fase true s t Z There are three terms in the disuntion and eaty one of them must be true. The feasibe region of is given by three disonneted ires as seen in Figure. The goba optima soution of is Z*.7 * fasetruefase and * By using - variabes y an be reformuated as an MILP probem BM with big-m onstraints: {} 8.. min M y y y y y y y M y M y M s t y y y Z

6 If y then the first inequaity onstraint is enfored and if y it beomes redundant assuming that M is a suffiienty arge number. If the binary variabes y are treated as ontinuous variabes in the MILP probem then for M the reaed MILP probem of has the optima soution Z*. and y* GDP PROBLEM WITH OE DISJUCTIO For simpiity we wi first assume that in probem P we have ony one disuntion i.e.. Hene eah term in the disuntion has ony one Booean variabe and the inde an be removed from P eading to P min Z f s. t. r J g P γ Ω True { true fase} J Eah term in the disuntion defines a feasibe region S J where S { γ r g }. ote that probem is a partiuar instane of probem P. In the foowing setions we derive a noninear reaation of probem P whih is tighter than the reaation of the big-m MILP probem BM. We use the proposed LP reaation as a basis for deriving an MILP reformuation and propose a speia purpose branh and bound method. We then generaize this method to probem P whih invoves mutipe disuntions >. COVEX HULL OF OLIEAR DISJUCTIO Consider the foowing disuntion that arises in probem P: g J γ where the funtions g are assumed to be bounded onve funtions over. In addition is

7 assumed to be bounded i.e. U. The disuntion means that eaty one of the Booean variabes must be true whih in turn means that g and γ. These onstraints are redundant when is fase. The onve hu of the disuntion in is given by a points that an be generated from taing the inear ombination of points in the feasibe regions S J. Figure iustrates geometriay the onve hu of the disuntion S S S. As is shown in Appendi A the onve hu of is given by the foowing set of equations: J J / J J U J J g J γ The equations in define a onve set in the spae. This property foows from the fat that a equations in are inear and the ast inequaity is onve. As proven by Hiriart- Urruty and Lemaréha 99 if g is onve and bounded over the feasibe region and then the funtion h g/ is a bounded onve funtion when h is defined as its imiting vaue. Hene the inequaities g/ are onve see aso Stubbs and Mehrotra 999. The equations in desribe the onve reaation of the disuntion in. ote in that is epressed as the sum of disaggregated variabes and is epressed as a onve ombination of γ with weight fators. The reaation in provides the tightest reaation of the disuntive feasibe region of as it orresponds to its onve hu. Aso if then and γ. impies that is true and the -th onstraint g / in is the same as the onstraint of the -th term in. Hene the -th term in the disuntion of is satisfied when equas one in is true. Finay notie that if g is a inear funtion redues to the equations proposed by Baas 98. 6

8 OLIEAR COVEX RELAXATIO PROBLEM We define a ontinuous reaation of P using as a basis the equations of the onve hu. Sine this reaation probem has no Booean variabes the ontinuous variabes are used instead and the ogi propositions are represented with the inequaities A a. The Conve Reaation Programming CRP probem for one disuntion is then given as foows: min Z s. t. L γ J r f J U J CRP J g / A a J J where for impementation the inequaity / must be reformuated as ε g / ε where ε is a sma toerane typia vaue.. ote that in CRP the number of onstraints inreases by nn m where n is the dimension of vetor. This is due to the onstraints Σ U and Σ. The number of variabes inreases by m n where m is the number of terms in the disuntion m J. Probem CRP whih an be regarded as an etension from the wor of Ceria and Soares 999 for disuntive programming orresponds to a onve noninear programming probem. This foows from the fat that the ogi inequaities are inear and the feasibe region of probem CRP is onve. Sine the obetive funtion ontains the inear summation term and f is onve the obetive funtion is onve. Therefore probem CRP is a onve LP probem. It aso foows that if the probem GDP has a bounded optima soution then the optimum of CRP is unique and orresponds to its goba minimum. Furthermore the feasibe region of CRP F C is a reaation of the feasibe region of probem P F P. Therefore sine F P F C and the obetive funtion of CRP is aso a reaation of the obetive funtion of P the soution of CRP Z L * yieds a ower bound of the optima soution of probem P Z* i.e. g 7

9 Z L * Z*. The above properties of CRP an be epoited to reformuate probem P as an MILP. Aternativey we an deveop a speia branh and bound searh proedure as wi be shown ater in the paper. It shoud be noted that in probem CRP are the disaggregated variabes of the vetor of ontinuous variabes whie are weights that measure the oseness by whih eah term of the disuntion is satisfied as. Generay soving an optimization probem with probem CRP yieds a soution with frationa vaues. However when one of the beomes and the other weights are zero probem CRP beomes probem P with fied true and a the other fase in the disuntion. EXAMPLE COTIUED If we appy the CRP mode to the GDP probem the onve LP reaation probem is as foows: min Z ε[ / ε / ε ] 6 ε [ / ε / ε ] ε[ L s. t. 8 i ;. / ε 8 ε. i i i ;. / ε ] To avoid division by zero in the noninear onstraints ε is introdued as a sma toerane ε.. The optima soution of 6 is Z L. and L otie that the ower bound. is rather tight ompared to the optima soution.7. Aso the ower bound of the reaed big-m MILP probem is ower than the ower bound of CRP probem. vs... In fat the reaation gap of the CRP probem 6 is.% whie the reaation gap of reaed MILP probem is.%. By etting z / from the soution in 6 L an be epressed as a onve ombination of z 8

10 with weight as shown in Figure. Two important points are noted. i Eah z z ies at the boundary of eah feasibe region S when is nonzero. This means that a the reaed noninear onstraints in 6 are ative. The obetive funtion vaue at eah z f / γ yieds an upper bound to the oa soution of that feasibe region see Tabe. ii Eah shows how ose the optima point L is to eah feasibe region S the arger is the oser L is to S. From this information a good guess is that the goba optima soution of GDP probem is in S whih has the argest see z in Figure. Therefore when we appy a branh and bound method to GDP probem an be used as an indiator showing whih Booean variabe shoud be seeted as a branhing variabe at the net node in the searh tree. SOLUTIO METHODS Having derived the noninear onve reaation probem CRP there are two maor soution approahes one an tae. The simpest and most diret one is to reformuate CRP as the foowing MILP probem. min Z J γ s. t. r f J P J U A a {} ε g / ε J J J Again the toerane ε e.g.. is introdued in the onstraints to avoid division by zero and an additiona inequaity in terms of a vaid upper bound U has been introdued to ensure that when. Lee and Grossmann 999 proved that the ower bound predited by the reaation of probem P is greater than or equa to the ower bound predited by the reaation of the MILP ounterpart as given by probem BM. Probem P an be soved with any standard method for MILP probem disussed in the 9

11 introdution setion e.g. Branh and Bound Outer-Approimation Generaized Benders Deomposition and Etended Cutting Pane. The other aternative is to deveop a speifi branh and bound method that epoits more direty the property of the onve hu as wi be disussed in the net setion. A BRACH AD BOUD ALGORITHM WITH COVEX RELAXATIO A branh and bound agorithm with the proposed noninear onve reaation in CRP is outined in this setion for the ase of ony one disuntion. First the CRP probem of the given GDP probem is soved. The branhing rue that an be used is to seet the variabe osest to beause this orresponds to the disuntive term that is osest to being feasibe. By soving the orresponding LP subprobem this generay yieds a good upper bound as was shown in the iustrative eampe. After branhing on one term of disuntion we propose to seet the onve hu of the remaining terms of the disuntion J whih have not been eamined yet. For the ase of one disuntion this orresponds to the dihotomy: either fi S or fi onv S This means that either the soution is in the subregion S or ese somewhere in the onve hu of the remaining set of subregions S J. As wi be shown ater with the resuts this branhing rue is generay very effetive. Based on the above idea the main steps of the proposed branh and bound agorithm for are as foows see Figure : J Branh and bound agorithm for Generaized Disuntive Programming. Step. Initiaization a Set Z*. b Seet ε. Set T J. Step. CRP probem a Sove the CRP probem with the onve hu of S T. b Obtain the optima obetive vaue Z L and the optima point L.

12 If a are or then L is a feasibe soution to the GDP probem P. The goba optima soution is Z L and L. Set Z* Z L and * L. Eit. d Otherwise L ies outside a S m. Go to step. Step. Branh on one term a Seet with the argest in the soution of CRP probem. b Set true and fase Fi as. Sove GDP probem P with fied. d Obtain the optima obetive vaue Z U and optima point U. e Set T T \. f If Z U Z* then set Z* Z U and * U. Step. Che the remaining terms a If T is empty then eit. b Ese if T is not empty then go to step. Step. Branh on the remaining terms a Fi as remove S from onve hu. b Sove the CRP probem with the onve hu of remaining feasibe regions S T. Obtain the optima obetive vaue Z L and the optima point L. d If Z L Z U then eit. The goba optima soution is Z U U. e Ese if Z L < Z U then go to step. REMARS The above agorithm has obviousy finite onvergene sine the number of terms in the disuntion is finite. Aso sine the noninear funtions are onve the subprobems have a unique optima soution. Hene the rigorous vaidity of the bounds an be guaranteed with whih the branh and bound method is in turn guaranteed to obtain the goba optimum. Furthermore given the strength of the reaation one an aso in genera epet the enumeration of fewer nodes. An important point that is worth noting in the proposed branh and bound enumeration is the ase when the proposed branhing rue does not yied a true partition. This may arise as foows. After searhing one partiuar feasibe subregion the onve hu of the remaining feasibe subregions S generay yieds an inrease to the ower bound. However if this ower

13 bound is the same as before there is the need to verify whether partitioning has in fat taen pae. This an be done by the foowing test. If L onv S where L is the optima soution of the parent node then this is a partitionabe set see Figure a. If L onv S then the set of subregions is a non-partitionabe set see Figure b beause the point L remains feasibe in the onve hu of the subregion and hene the ower bound remains the same after branhing. This test step an be used in the proposed agorithm to aeerate the searh by avoiding repeated identia ower bounds. EXAMPLE COTIUED Figure shows the orresponding searh tree when we appy the proposed branh and bound agorithm to Eampe. At the root node the searh set is T { } and CRP probem 6 yieds a ower bound Z L.. This optima point L ies outside the feasibe region of GDP probem sine L does not satisfy any term in the disuntion see Figure. Hene this soution is infeasibe for probem. Among the weights has the argest vaue as seen in Tabe so we seet and set as true. At the first node the GDP probem is soved with fied fasetruefase. Ony is set as true and the other are set as fase. It means that we fi as and other as in probem 6. Therefore the feasibe region is restrited to S ony. Soving probem with true yieds an upper bound Z U.7. Sine S has been eamined it is removed from the searh set T {}\{} {}. At the seond node we onsider the onve hu of S and S. The CRP probem is then probem 6 without the onstraints and variabes for S. By soving this CRP probem a ower bound Z L.7 is obtained. Sine this ower bound.7 is greater than the upper bound Z U.7 the feasibe soution of S and S wi be greater than Z L.7 > Z U.7. Hene the goba optima soution is Z U.7 and the searh ends after nodes. Tabe shows the omparison among the standard BB OA GBD and ECP agorithms appied to the big-m MILP formuation and the reformuated MILP PR in 6. The standard BB finds the optima soution in nodes see Figure 6 and other agorithms show amost the same number of maor iterations in soving both the big-m and the CRP formuations. However the onve hu predits tighter ower bound than the reaation of the big-m formuation. ote that OA requires MILP and LP subprobems to sove PR ompared with MILP and LP subprobems to sove BM. GBD soves MILP and LP

14 subprobems for both formuations. ECP soves 9 and MILPs respetivey. GEERALIZATIO TO MULTIPLE DISJUCTIOS For the ase of mutipe disuntions > as in probem P the MILP reformuation of P an be readiy generaized as foows: min Z J γ s. t. r f J PR ε g J / ε U A a {} J J J where the toerane ε is aso introdued in the noninear inequaities as in 6. The dimension of the variabes in PR inreases due to the doube indies in and. Simiary as in the ase of P MILP methods suh as Branh and Bound Outer-Approimation Generaized Benders Deomposition and Etended Cutting Pane an be appied to sove probem PR. Aso the reaation of this probem yieds stronger ower bounds than the reaation of probem BM. As for the proposed branh and bound the soution proedure of the GDP probem with a number of disuntions an be easiy generaized see Lee and Grossmann 999. In this agorithm we sove the reaation of probem PR whih aows to be ontinuous between and. This reaed probem of PR orresponds to the CRP probem of GDP probem P. The goba optima soution is the best upper bound after termination of the branh and bound enumeration. Reation to ogi-based Outer-Approimation Method Türay and Grossmann 996 proposed a Logi-Based Outer-Approimation agorithm when the GDP probem P is appied to proess networs. In this ase the GDP has the foowing form

15 min Z s. t. r g B DP γ Ω True f whih in ontrast to P has ony two terms in eah disuntion to denote the eistene or non-eistene of units. In probem DP B is a matri whih fores the subset of variabes Z to zero when is fase. As shown in Appendi B an interesting point is that appying the outer-approimation method to the MILP reformuation PR redues to the ogi-based outerapproimation method by Türay and Grossmann 996. The reason is that the master probem for both methods beomes identia. UMERICAL RESULTS In this setion we present the omparison of the proposed branh and bound agorithm with standard branh and bound agorithm. Both agorithms use a depth-first searh rue. A the eampe probems were soved with GAMS Brooe et a. 997 on a MHz Pentium II PC. The GAMS/COOPT LP sover was used in both agorithms and omparisons were aso performed with GAMS/DICOPT. EXAMPLE The orresponding GDP probem taen from Grossmann and ravana 997 is given by 7. The goba optima soution is * fasetruefase * and Z*..

16 . } { 7.. min fase true s.t. Z As seen in Tabe the soution of probem 7 with a standard branh and bound agorithm appied to the big-m MILP formuation Grossmann and ravana 997 predits a ower bound of. and requires nodes for the termination. In ontrast by appying the proposed speiaized branh and bound method to the GDP in 7 ony nodes are required whih is argey due to the improved ower bound of.96 that is predited. EXAMPLE : JOBSHOP SCHEDULIG PROBLEM The net eampe is sheduing probem with obs and stages Raman and Grossmann 99. The obetive of the foowing GDP mode is to minimize the maespan:. } { T T 8 T.. min fase true T s t T Z The optima soution has a maespan of Z* hours * fasetruefase and *. This optima shedue is shown in Figure 7. When the mode in 8 is onverted into a

17 big-m MILP BM a standard BB soves this probem in nodes. In ontrast the proposed BB soves the GDP in 8 in ony nodes. In this ase the ower bound from the inear onve hu 8.6 is not muh tighter than that of big-m reaation 8.. However the proposed agorithm redues the number of nodes signifianty. EXAMPLE : PROCESS ETWOR SUPERSTRUCTURE This eampe was originay proposed by Duran and Grossmann 986 as an MILP probem. The mode is given as a GDP probem by Türay and Grossmann

18 ; } { ] [ ep ep 9 7 /. ep /. ep ep..8.. min fase true a s t a Z T T

19 Logi Propositions: Speifiations: 6 The optima soution is Z* 68. * fasetruefasetruefasetruefasetrue and * Figure 8 shows the superstruture of the proess and Figure 9 shows its optima onfiguration. The Booean variabes denote the eistene or non-eistene of proess -8. As seen in Tabe the proposed agorithm appied to the GDP in 9- finds the optima soution in ony nodes. Using the big-m formuation reported by Duran and Grossmann nodes were required with the standard branh and bound method and nodes with the branh and ut by Stubbs and Mehrotra 999. As seen in Figure we first onsider the onve hu of eah of the eight disuntions and the weight of the first term in eah disuntion is shown. The reaed optimum obetive obtained at the root node 6.8 is quite ose to the optima soution 68. of GDP probem. Sine the seond term in eah disuntion sets a subset of the variabes to zero we onsider ony the weight of the first term in eah disuntion. The eight weights shown in Figure orrespond to eah Booean variabe. At the root node has the argest frationa vaue. At the first node we fi the first term of the seond disuntion set as true and onsider the onve hu in eah of the remaining seven 7 8

20 disuntions 678. After fiing the optima soution at the first node has ony one frationa weight. So is seeted and fied as at the seond node. The soution of the seond node yieds the upper bound After batraing the goba optima soution 68. is obtained at the third node. At the fourth node a ower bound 7. is obtained with frationa. Sine this ower bound is greater than the urrent upper bound of 68. the searh stops. Tabe shows the omparison with other agorithms when the probem 9- is reformuated as the MILP probem PR with the onve hu representation for the disuntions. ote that the proposed BB agorithm and the standard BB yied the same ower bound 6.8 sine they start by soving the same reaation probem of PR. The differene in the number of nodes ies in the branhing rues. The OA method requires maor iterations and the first reaed soution is ower than that of BB method. OA GBD and ECP start with initia guess []. ote that in the GBD and OA methods one maor iteration onsists of one LP subprobem and one MILP master probem. Again the proposed agorithm yieds the tightest ower bound and requires the fewest number of subprobems. For omparison the ogi-based OA method by Türay and Grossmann 996 yieds the ower bound 67.9 with initia LP subprobems LPS and finds the optima soution 68. in one maor iteration requiring a tota of LP and MILP s. The proposed method requires LP s. EXAMPLE : OPTIMAL POSITIOIG OF A EW PRODUCT The fifth eampe probem onsists in determining the optima positioning of a new produt in a mutiattribute spae Duran and Grossmann 986. Here we onsider a maret with a set of eisting produts M and a set of onsumers. The eisting produts an be oated in a mutiattribute spae of dimension with oordinates δ M. Eah onsumer is haraterized by an idea point z i and a set of weights w i i both representing onsumer s onept of an idea produt. A region whih defines oseness to the idea point for eah onsumer an be determined in terms of the eisting produts. Based on these riteria a onsumer is assumed to seet a produt osest to the idea point. The obetive is to design the optima oation of the produt to maimize the profit. The revenue of the new produt from saes to onsumer i is given i and f is the ost of reahing 9

21 oations of the new produt within an attribute spae. This eampe was formuated as an MILP probem by Duran and Grossmann 986 and an be epressed as the foowing GDP probem. s. t f.6 R p i T w i a b a [] b [.89] min { min Z. z... i i i.7 i p i..9 w i δ. i R i... z f i i i... i... i. } i... [ ] { true fase} i... The data for eisting produts onsumers attributes and the GDP probem are shown in Appendi C. The optima soution is Z* -8.6 * 687 true and * The ower bound obtained from the reaation of mode PR is muh oser to the optima soution -8.6 than that of the big-m formuation BM by Duran and Grossmann see Tabe 6. The tightness of the ower bound substantiay redues the number of nodes in the BB agorithm 89 vs.. Aso mode PR is soved in maor iterations by the OA agorithm ompared with 9 of mode BM. EXAMPLE 6: DESIG OF A MULTI-PRODUCT BATCH PLAT The ast eampe is a bath pant design with mutipe units in parae and intermediate storage tans Ravemar 99; Vehietti and Grossmann 999. This probem onsists of determining

22 the voume of the equipment the number of units in parae and the voume and oation of the intermediate storage tans. The obetive is to minimize the investment ost. The GDP mode is as foows: } { {6} ; {} } {. og ; og og og og og og og og... og og og og og ep ; og ; og.. ep..6 ep min E D C B A I J z J fase true J I i B b B J V v V V v V J n m J v I i b b I i b b b v I i b b b v I i b b J z n J z m e Q H J I i m b P e J I i n b S v s t v v m n Z z z U i i L i U L U L i i i i i i i i i i z z z z z z z z I i i i i i i i i J J µ φ µ µ φ φ The disuntions orrespond to the storage tan voume and the bath size. The obetive funtion is noninear and onve and the onve hu is inear beause the onstraints in the disuntions are inear. The data and the optima soution are shown in Appendi C. The optima struture whih has a ost of 688 is shown in Figure. The reaation gap is.% PR vs. 6.% BM. The proposed BB signifianty redues the number of nodes by 8% 9 vs. 7 ompared to the standard BB see Tabe 6. When the OA agorithm is appied to both mode BM and PR it taes more CPU time to sove mode PR than BM.9 vs..7 se.

23 COCLUSIO A nove soution agorithm has been proposed for GDP probems whih orrespond to disrete/ontinuous optimization probems that invove disuntions with noninear inequaities and ogi propositions. A new noninear reaation of the GDP probem and its properties have been presented. The proposed reaation probem CRP of the GDP probem is based on the onve hu of eah noninear disuntion and is used for the reformuation of the GDP probem as the MILP probem PR whih an be soved with MILP agorithms suh as BB OA GBD and ECP. The reation of probem PR with the ogi based outer-approimation agorithm by Türay and Grossmann was estabished. A speia purpose branh and bound agorithm for the GDP probem was aso proposed based on the CRP probem. The numeria resuts of si GDP probems showed that the proposed branh and bound agorithm whih maes use of the reaation CRP requires fewer nodes and ess CPU time than the standard branh and bound method whih maes use of the big-m reaation. These GDP probems were aso reformuated as the MILP probem PR and soved by eisting MILP agorithms. Anowedgments-The authors woud ie to anowedge finania support from the SF Grant CTS-97 and partia support from Eastman Chemia Company. REFERECES Baas E. Disuntive Programming and a Hierarhy of Reaations for Disrete Optimization Probems. SIAM J. Ag. Dis. Meth Borhers B. and J.E. Mithe An Improved Branh and Bound Agorithm for Mied Integer oninear Programming. Computers and Operations Researh Brooe A. D. endri A. Meeraus and R. Raman GAMS Language Guide Reease. Version 9. GAMS Deveopment Corporation 997. Ceria S. and J. Soares Conve Programming for Disuntive Optimization. Mathematia Programming Duran M.A. and I.E. Grossmann An Outer-Approimation Agorithm for a Cass of Mied- Integer oninear Programs. Mathematia Programming Fether R. and S. Leyffer Soving Mied oninear Programs by Outer Approimation.

24 Mathematia Programming Fippo O.E. and A.H.G. Rinnoy an Deomposition in Genera Mathematia Programming. Mathematia Programming Geoffrion A.M. Generaized Benders Deomposition. Journa of Optimization Theory and Appiation Grossmann I.E. and Z. ravana Mied-Integer oninear Programming: A Survey of Agorithms and Appiations Large-Sae Optimization with Appiations Part II: Optima Design and Contro eds. Bieger et a. Springer-Verag Gupta O.. and V. Ravindran Branh and Bound Eperiments in Conve oninear Integer Programming. Management Siene Hiriart-Urruty J. and C. Lemaréha Conve Anaysis and Minimization Agorithms Vo.. Springer-Verag 99. Lee S. and I.E. Grossmann Generaized Disuntive Programming: oninear Conve Hu Reaation and Agorithms submitted to Mathematia Programming 999. Leyffer S. Integrating SQP and branh-and-bound for Mied Integer oninear Programming submitted to Computationa Optimization and Appiations 999. emhauser G.L. and L.A. Wosey Integer and Combinatoria Optimization. John Wiey & Sons In Quesada I. and I.E. Grossmann An LP/LP Based Branh and Bound Agorithm for Conve MILP Optimization Probems. Computers Chem. Engng. 6/ Raman R. and I.E. Grossmann Reation Between MILP Modeing and Logia Inferene for Chemia Proess Synthesis. Computers Chem. Engng Raman R. and I.E. Grossmann Modeing and Computationa Tehniques for Logi Based Integer Programming. Computers Chem. Engng Ravemar E. Optimization modes for design and operation of hemia bath proesses. Ph.D. Thesis. 99. Stubbs R. and S. Mehrotra A Branh-and-Cut Method for - Mied Conve Programming. Mathematia Programming Türay M. and I.E. Grossmann Logi-based MILP Agorithms for the Optima Synthesis of Proess etwors. Computers Chem. Engng Vehietti A and I.E. Grossmann LOGMIP : A Disuntive - oninear Optimizer for Proess

25 Systems Modes. Computers Chem. Engng Westerund T. and F. Petterson An Etended Cutting Pane Method for Soving Conve MILP Probems. Computers Chem. Engng. 9 sup. S-S6 99. Wiiams H.P. Mode buiding in mathematia programming John Wiey & Sons In. 98. uan X S. Zhang L. Piboeau and S. Domeneh Une Methode d optimization onineare en Variabes Mites pour a Coneption de Poredes. Rairo Reherhe Operationnee 988. APPEDIX A Proof of Conve Hu Theorem. The onve hu of the disuntion in is given by J J g U J / J J J γ J A. Proof. The onve hu of the disuntion in an be epressed as a onve ombination of mutipiers that mutipy the onstraints in the disuntion. J g J γ J J The equation A. an be inearized by setting whih eads to γ J A. A. A. A. The inequaity A. is generay nononve due to the biinearity that is introdued by the produt. We an onveify however the inequaity by defining the new variabe. From the onveity ondition of the foowing equations hod.

26 J J J J γ A.6 A.7 Furthermore rewriting A. in terms of and for g / J A.8 From Hiriart-Urruty and Lemaréha 99 the above inequaity is onve. From the assumption is bounded by J U A.9 where U is an upper bound for eah. Hene the onve hu is given by the equations and inequaities in A.. APPEDIX B Reation to ogi-based Outer-Approimation Method for probem DP In the agorithm by Türay and Grossmann 996 whih addresses the soution of probem DP the LP subprobem for fied vaues of the Booean variabes at iteration is given by min Z s. t. g B r γ f for for true fase FX - DP The outer-approimation master probem is given by the foowing disuntive probem in whih the noninear onstraints are inearized at the optima soutions of probem FX-DP

27 6 fase true L true True B g g L for r r f f s t Z L L T T T Ω } { }... { MP..... min α γ α α The inde L orresponds to the iteration ounter whie L is the set of those iterations in whih the eft term of the -th disuntion in DP is ative thus yieding a inear approimation for the inequaity g. Probem MP an be transformed into the foowing MILP probem by using the onve hu of eah disuntion with inearized onstraints see equation : y L true a Ay y y y g g g g L for r r f f s t y Z L U U L T T Z T T T Z {} }... { ] [ MP DP min α α α γ where is the vetor of variabes whih are non-zero when is fase whie Z is the vetor of variabes that taes a vaue of zero. This partition of the ontinuous variabes is performed aording to the definition of the matri B in DP. Sine eah disuntion must have at east one inearization severa LP subprobems must be soved initiay. The fewest number of suh LP subprobems an be determined from a set overing probem Türay and Grossmann 996. For the ase of two terms in eah disuntion in probem DP probem PR redues to

28 7 a A U U U g r s t f Z Z Z {} PRT / /.. min γ where [ Z ]. ote that the above onstraints have been simpified beause Z sine the orresponding variabes Z tae a vaue of zero in the seond term of the disuntion. For fied in PRT we have is fase if is true if with whih probem PRT beomes: a A fase for U fase for true for U true for g fase for true for r s t f Z Z Z Z PRT FX -.. min γ It is ear that for fied the LP subprobem FX-PRT is identia to probem FX-DP. Rather than inearizing the origina onstraints as in probem MP we inearize the noninear onve hu formuation in PRT to define the master probem of the outer-approimation agorithm. Then the orresponding MILP master probem is given as foows

29 8 L true a A U U U g g g g L for r r f f s t Z L Z T T Z T Z T T Z {} }... { PRT M min α α γ ote that if we et / and treat as binary variabe y then the inearized onstraints M-PRT and the onve hu of the inear disuntion in probem DP-MP are the same. Aso the partition of in non-zero and zero variabes is used in the same way as in DP- MP. Hene the MILP probem in M-PRT is identia to the MILP master probem DP-MP of Türay and Grossmann 996. Thus we an onude that for the ase of probem DP appying the outer-approimation method to the MILP reformuation PR redues to the ogibased outer-approimation method by Türay and Grossmann 996. APPEDIX C Data for GDP eampe probems and 6 EXAMPLE i Idea points z i Attribute weights w i

30 Eisting produts δ EXAMPLE 6 m og M n og v og V v og V J; b i og B i I; J i i produts; stages; H horizon time 6 h Q i prodution rate of produt i: A B C 8 D 6 E S i size fator for produt i at stage i \ 6 A B

31 C D E P i proessing time of produt i at stage i \ 6 A B C D E Optima Soution: * {fasetruefasefasefase} 6 V M V storage Cost Z* 688

32 LIST OF TABLES Tabe. Disaggregated variabes and oa optima points of eampe. Tabe. Comparison of the resuts for eampe. Tabe. Comparison of the resuts for eampe. Tabe. Comparison of branh and bound methods for eampe. Tabe. Comparison of agorithms for formuation PR of eampe. Tabe 6. Comparison of formuations BM and PR for GDP eampes. LIST OF FIGURES Figure. Feasibe region of eampe. Figure. Conve hu of feasibe region. Figure. The proposed branh and bound agorithm for. Figure. Partitionabe and on-partitionabe sets. Figure. The proposed branh and bound tree: eampe. Figure 6. Standard branh and bound tree: eampe. Figure 7. The optima shedue for eampe. Figure 8. Superstruture for eampe. Figure 9. The optima struture of eampe. Figure. The proposed branh and bound method for eampe. Figure. The optima pant struture of eampe 6.

33 Feasibe Region Tabe. Disaggregated variabes and oa optima points of eampe. Loa optima vaue z / f / γ Loa optima point S S S Formuation Opt. Soution Tabe. Comparison of the resuts for eampe. Lower Bound a BM.7. PR.7. Method Maor Iter. /odes Maor Iter. /odes Standard BB Proposed BB OA b GBD ECP a LP reaation. b OA begins with LP reaation. GBD and ECP begin with initia guess y. Tabe. Comparison of the resuts for eampe. Method o. of LP Subprobems Lower Bound Standard BB-formuation BM. Proposed BB-formuation PR.96 Tabe. Comparison of branh and bound methods for eampe. Method Standard BB Branh & Cut Proposed BB Formuation odes 7 6 Cuts - BM Reaed Opt Formuation odes - - PR Reaed Opt Optima Soution Tabe. Comparison of agorithms for formuation PR of eampe. Method* Standard BB Proposed BB OA Maor GBD Maor ECP Logi-based OA* o. of nodes / Iteration odes odes Iter. 8 Iter. 7 Iter. subprobem Maor Iter. Reaed Optimum *A methods eept ogi-based OA sove the reformuated MILP probem PR.

34 Tabe 6. Comparison of formuations BM and PR for GDP eampes. Probem GDP OA BB umber Goba Maor It. CPU se odes Lower Bound Opt. BM PR BM PR * *with ogi propositions

35 Figure. Feasibe region of eampe. Conve hu onv S z S Weight Conve ombination of z z v / S L z z S Goba optimum.9.77 Z*.7 Loa Optima point Conve hu optimum Z L

36 Figure. Conve hu of feasibe region. S S onv S S S S S a disuntive feasibe region b onve hu

37 Figure. The proposed branh and bound agorithm for. START Set Z* seet ε. Set T { m} Sove CRP with onvs T. A are or? Seet S with osest to one. Sove P with fied S Z U f U T T \ if Z U Z* then Z* Z U * U Goba Optima Soution Z* f L * L T is empty? Sove CRP with onvs T. Z L f L Z L Z U? Goba Optima Soution Z* f U * U ED 6

38 Figure. Partitionabe and on-partitionabe sets. S L L S S S S a Partitionabe set S S S S S L L b on-partitionabe set 7

39 Figure. The proposed branh and bound tree: eampe. Root ode Conve hu of as Z. [.6.9.9] First ode Fi Z.7 [ ] [.9.77] Z L. Branh on Seond ode Conve hu ofs S Z.7 [.7.6] Z U.7 Ba-tra Z L.7 > Z U Stop 8

40 Figure 6. Standard branh and bound tree: eampe. Z L. [.9.97] Fi Fi Z L.8 [.97.] Z [ ] Fi Fi Z U.8 [ ] Optima Soution Z*.7 * [ ] : Branhing Var. Bod : Fied Var. 9

41 Figure 7. The optima shedue for eampe. Stage Stage Stage B C A time

42 Figure 8. Superstruture for eampe

43 Figure 9. The optima struture of eampe

44 Figure. The proposed branh and bound method for eampe. Z L 6.8 [..69.] Fi Fi Z L 6.9 [.] Fi Fi Z L 7. > Z U [.] Stop Z U 7.79 [] Feasibe Soution Z U 68. Z* [] Optima Soution : Branhing Var. Bod : Fied Var.

45 Figure. The optima pant struture of eampe 6. S 6