New Approach for Quantifying Process Feasibility: Convex and 1-D Quasi-Convex Regions

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1 New Approach for Quantfyng Process Feasblty: Convex and 1-D Quas-Convex Regons Maranth G. Ierapetrtou Dept. of Chemcal and Bochemcal Engneerng, Rutgers Unversty, Pscataway, NJ Uncertantes n chemcal plants come from numerous sources: nternal lke fluctuated alues of reacton constants and physcal propertes or external such as qualty and flow rates of feedstreams. Accountng for uncertanty n arous stages of plant operatons was dentfed as one of the most mportant problems n chemcal plant desgn and operatons. A new approach proposed descrbes process s feasble regon and a new metrc for e aluatng process flexblty based on the con ex hull that s nscrbed wthn the feasble regon and determnes ts olume based on Delaunay Trangulaton. The two steps n ol ed are: 1. a seres of smple optmzaton problems are sol ed to determne ponts at the boundary of the feasble regon;. g en the set of ponts at the boundary of the feasble regon, the con ex hull nscrbed wthn the feasble regon s determned. Ths s ache ed by mplementng the Quckhull algorthm, an ncremental procedure for e aluatng the con ex hull, and then by computng a Delaunay Trangulaton to determne the olume of the con ex hull pro dng a new metrc for process flexblty. Ths approach not only pro des another feasblty measure, but an accurate descrpton of the feasble space of the process. It was appled to 1-D con ex problems, and work s n progress to extend t to noncon ex systems. Introducton Producton systems typcally nvolve sgnfcant uncertanty n ther operaton. Varablty of process parameters durng operaton and plant model msmatch Žboth parametrc and structural. could gve rse to suboptmalty and even nfeasblty of the determnstc solutons. Consequently, plant flexblty has been recognzed to represent one of the mportant components n the operablty of the producton processes. A bref overvew of the approaches that exst n the lterature to deal wth the problem of feasblty and flexblty quantfcaton follows n ths secton. In a broad sense the area covers Ž. a feasblty test that requres constrant satsfacton over a specfed space of uncertan parameters, Ž. a flexblty ndex assocated wth a gven desgn that represents a quanttatve measure of the range of uncertanty space that satsfes the feasblty temperature, and Ž. the ntegraton of desgn and operatons where trade-offs between desgn cost and plant flexblty are consdered. Halemane and Grossmann Ž proposed a feasblty measure for a gven desgn based on the worst ponts for feasble operaton, whch can be mathematcally formulated as a max-mn-max optmzaton problem as follows Ž d. s max mn g T z max jg x, g I 4 h Ž d, z, x,. s0; g Ž d, z, x,. F0, Ž 1. where the functon Ž d. represents a feasblty measure, d corresponds to the vector of desgn varables, z the vector of control varables, x the vector of state varables, and the vector of uncertan parameters. If Ž d. F0, desgn d s feasble for all gt, whereas f Ž d. 0, the desgn cannot operate for at least some values of gt. The above mass-mn-max problem defnes a nondfferentable global optmzaton problem, whch, however, can be reformulated as the followng two-level optmzaton problem j AIChE Journal June 001 Vol. 47, No

2 Ž d. s max Ž d,. g T s.t. h Ž d,. s mn u z,u Ž d, z, x,. s0; g I gjž d, z, x,. Fu jg J, Ž. where the functon Ž d,. s0 defnes the boundary of the feasble regon n the space of the uncertan parameters. The desgn flexblty ndex problem as ntroduced by Swaney and Grossmann Ž 1985a. can be reformulated to represent the determnaton of the largest hyperrectangle that can be nscrbed wthn the feasble regon of the desgn. Followng ths dea, the mathematcal formulaton of the flexblty problem has the followng form F smn s.t. Ž d,. s0 Ž d,. s mn u z hž d, z, x,. s0, g I gjž d, z, x,. Fu, jg J N y N T Ž. s N y F F y q 4 G0. Ž. 3 Other approaches that exst n the lterature to quantfy the flexblty for a gven desgn nvolve the determnstc measures such as the Reslence Index Ž RI., proposed by Saboo et al. Ž 1983., the flexblty ndex proposed by Swaney and Grossmann Ž 1985a,b. and the stochastc measures such as the desgn relablty proposed by Kubc and Sten Ž 1988., the stochastc flexblty ndex proposed by Pstkopoulos and Mazzuch Ž and Straub and Grossmann Ž For the case where the constrants are one-dmensonal Ž1- D. jontly quas-convex n and quas-convex n z, t was proven by Swaney and Grossmann Ž 1985a. that the pont c that defnes the soluton to Eq. les at one of the vertces of the parameter set T. Based on ths assumpton, the crtcal uncertan parameter ponts correspond to the vertces and the feasblty test problem s reformulated n the followng manner Ž d. s max Ž d, k., Ž 4. k g V where Žd, k. s the evaluaton of the functon Ž d,. at the parameter vertex k and V s the ndex set for the n p vertces for the np uncertan parameters. In a smlar fash- on for the flexblty ndex, problem 3 s reformulated n the followng way F s mn k, Ž. 5 k g V where k s the maxmum devaton along each vertex drecton k, kgv and s determned by the followng problem s.t. g h k s max, z Ž d, z, x,. F0, jg J j Ž d, z, x,. s0, g I s N q k, G0. Ž. 6 Based on the above problem reformulatons, Halemane and Grossmann Ž proposed the drect search method that explctly enumerates all the parameter set vertces. To avod the explct vertex enumeraton, Swaney and Grossmann Ž 1985a,b. proposed two algorthms, a heurstc vertex search, and an mplct enumeraton scheme. These algorthms rely on the assumpton that the crtcal ponts correspond to the vertces of the parameter set T whch s vald f the constrants are jontly 1-D quas-convex n and quas-convex n z. To crcumvent ths lmtaton, Grossmann and Floudas Ž proposed a soluton approach based on the followng deas: Ž. a They replace the nner optmzaton problem by the Karush Kuhn, Tucker optmalty condtons Ž KKT.; Ž b. They utlze the dscrete nature of the selecton of the actve constrants by ntroducng a set of bnary varables to express f a specfc constrant s actve. Based on these deas, the feasblty test and flexblty ndex problems can be reformulated as mxed-nteger optmzaton problems ether lnear or nonlnear dependng on the nature of the constrants. Grossmann and Floudas Ž proposed the actve set strategy for the soluton of the above reformulated problems based on the property that for any combnaton of n q1 bnary varables that s selected Ž z that s, for a gven set of actve constrants., all the other varables can be determned as a functon of. They proposed a procedure of systematcally dentfyng the potental canddates for the actve sets based on the sgns of the gradents g Ž d, z, x,.. Ostrovky et al. Ž z j proposed a branch and bound approach based on the evaluaton of the upper and lower bound of functon Ž d.. Although the suggested boundng problems are smpler than the orgnal feasblty test problem, they correspond to blevel optmzaton problems where global optmalty cannot be guaranteed usng local optmzaton methods Ž Mgdalas et al., Fnally, a global optmzaton approach s proposed recently by Floudas et al. Ž 000. to guarantee the determnaton of the global optmal soluton for both the feasblty and the feasblty ndex problem by generatng a relaxatonrenlargement of the feasble regon based on the convexfcaton of the orgnal problem constrants. Followng ths ntroducton, a small example s studed n the next secton to motvate the need of developng a new approach for quantfyng the range of feasblty for a gven process. The new approach and the detaled procedure that we propose s ntroduced to determne the new metrc that descrbes the feasble regon. The motvatng example s used to llustrate the basc steps of the proposed approach and the results obtaned. Several examples from the lterature are 1408 June 001 Vol. 47, No. 6 AIChE Journal

3 covered and compared wth the results of the prevously used approaches. The extenson of the basc framework to cover 1-D convex regons s dscussed and the basc advantages of the proposed approach are summarzed. Motvatng Example The example presented here s a modfcaton of example used by Pstkopoulos and Ierapetrtou Ž The desgn s descrbed by two parameters Ž d, d. 1 whereas z 1, z are the control varables and 1, are the uncertan parameters. The constrants of the problem are the followng f sy1.3q1.6 y1.6 z q.14 z F f s y.5 q1 z y z F0 1 1 f s.61 z y d F f s3.14 z y d F0 4 f sy z F0 5 1 f sy z F0 6 0F F 1 0F F8. The desgn examned frst corresponds to d1s3.5, ds0.0. For ths problem, frst the actve sets are dentfed based on the fact that three constrants should be actve at a tme snce there are two control varables ŽGrossmann and Floudas, Actve Set 1s f 1, f, f34 Actve Set s f 1, f, f54 Actve Set 3s f 1, f, f64 Actve Set 4s f 1, f, f44 Actve Set 5s f 1, f 3, f64 Actve Set 6s f, f 4, f54 Fgure 1. Feasble regon of Desgn ). ( d s3.5, d s 1 The results presented for the specfc example clearly motvate the need of developng a new approach for determnng the feasblty range for a gven process and a new metrc for evaluatng ts flexblty that can be used for comparson between desgns. Proposed Approach To llustrate the basc concept of the proposed approach, let s assume that the specfc desgn s descrbed by a set of nequalty constrants f Ž d, z,. j F0 assumng that the equal- ty constrants have been elmnated for ease n the presentaton. In these constrants d represents the set of desgn varables, and z the set of control varables that can be adjusted to accommodate varatons on the uncertan parameter vector. Followng the deas of Grossmann and Halemane Ž 198., one can map the feasble regon nto the uncertanty For each one of these actve sets, the feasblty functon s determned and plotted n Fgure 1. The flexblty ndex problem s then solved usng the actve set strategy approach proposed by Grossmann and Floudas Ž that results n a value of F s0.78. Ths value descrbes the rectangular whch s shown dark shaded n Fgure 1. Note that, although the actual feasble regon of the desgn s the lght shaded regon, the flexblty ndex dentfes only a small percentage of t and thus t largely underestmates desgn s feasblty. Moreover, the flexblty ndex does not provde always accurate results for comparng dfferent desgns. For example, we are consderng next the desgn that corresponds to d s.5, d s3.5. For ths desgn, the feasble 1 regon s evaluated based on the determnaton of the dfferent actve sets. The results are llustrated n Fgure. Note that ths desgn has a smaller feasble regon snce actve set 5 constrant moved to the left and thus the desgn s feasble regon s more restrcted. However, the flexblty of ths desgn s F s0.78, the same as for the desgn 1. Fgure. Feasble regon of Desgn 3.5 ). ( d s.5, d s 1 AIChE Journal June 001 Vol. 47, No

4 space by evaluatng the feasblty functon Ž d,. s mn z,u u fjž d, z,. Fu, jg J. Alternatvely, the same result can be obtaned by dentfyng all the actve constrant sets. The queston that arses at ths stage n many dfferent engneerng problems s how to descrbe and then quantfy the feasble range. The frst case that we are gong to address s the convex case, the extenson to the 1-D convex case wll be dscussed later. The man dea n the proposed approach s frst to determne ponts at the boundary of the feasble regon and then to evaluate the convex hull defned by those ponts. For the case of the convex feasble regon, the convex hull determned n ths way s guaranteed to be nscrbed wthn the feasble regon. However, for the nonconvex case, there s no guarantee that ths would be always the case. To overcome ths dffculty, an extenson of the basc proposed approach s presented later whereas the general nonconvex case would be the subject of future publcaton. For the convex case, the followng steps are consdered: Ž. 1 Solve the problems of determnng the maxmum devaton from the nomnal pont towards the vertces Ž k K. of the expected range of uncertan parameter Ž. varablty max k s.t. f Ž d, z,. F0, jg J j s N k Ž. Solve the problems of determnng the maxmum devaton from the nomnal pont by varyng one uncertan parameter Ž. at a tme max s.t. fjž d, z, 1,. F0, jg J s N js j N j Ž. 3 Determne the convex hull based n the ponts obtaned from steps Ž. 1 and Ž. applyng the Quckhull algorthm whch s descrbed below. Ž. 4 Determne the volume of the convex hull obtaned from step Ž. 3 usng Delaunay Trangulaton Žthe bascs of the Delanay Trangulaton are explaned n the sequel.. At the same step and usng the vertces of the expected range of uncertan parameter varablty, the volume of the overall expected regon s determned. The rato between the feasble convex hull and the expected regon volumes s then evaluated to represent the new metrc for comparng desgn s feasblty Feasble Convex Hull Rato Ž FCHR. Volume of the feasble convex hull s Ž. 7 Volume of the overall expected range Note that, at step 3, the lnear functons descrbng the faces of the convex hull are also determned. As wll be llustrated by dfferent examples, ths provdes a very accurate descrpton of the feasble space for a specfc desgn or process whch s of great mportance n many dfferent engneerng problems. An example s the determnaton of the range of valdty of a reduced knetc model as descrbed by Androulaks et al. Ž Quckhull algorthm The convex hull of a set of ponts s the smallest convex set that contans the ponts. There exst a number of dfferent algorthms n the computatonal geometry lterature that are desgned to compute the convex hull for a gven set of ponts. A revew can be found n de Berg et al. Ž Recent work on convex hull has been focused on varatons of a randomzed, ncremental algorthm where ponts are consdered one at a tme. Quckhull algorthm Ž Barber et al., proceeds usng two geometrc operatons: orented hyperplane through Ž n. d ponts and sgned dstance to hyperplane. In partcular the followng steps are followed for each processed pont: Ž. a Locate the vsble facets for the pont. A facet s vsble f the pont s above the facet. The selecton s made by evaluatng the sgned dstance from the facet; Ž b. Construct a cone of new facets from the pont to the vsble facets; Ž. c Delete the vsble facets thus formng the convex hull of the new pont and the prevous processed ponts. The correctness of the algorthm has been proved by Barber et al. Ž They also provde emprcal evdence that the algorthm runs faster when the nput set of ponts contans no extreme ponts. The detaled descrpton of the algorthm, as well as the programmng mplementaton, can be found and downloaded from the Web ste of the geometry center n Mnneapols: Delaunay trangulaton The basc defnton of the Delauany Trangulaton s the graph that has a node for every Vorono cell and a straght lne connectng two nodes f the correspondng cells share an edge. The Vorono dagram of a set of ponts Ž n. s the subdvson of the plate nto Ž n. regons one for each ste such that the regon of a specfc ste contans all the ponts n the plane for whch ths ste s the closest. The regon of ths ste s called the Vorono cell. The computaton of the Delaunay Trangulaton s based on the same teratve procedure as the convex hull estmaton. In partcular, a Delaunay Trangulaton n R d can be computed from a convex hull n R dq1,as descrbed by Barber et al. Ž The same software descrbed above has the capablty of evaluatng Delaunay Trangulaton. Example re sted The example presented n the prevous secton s used here agan to llustrate the basc steps of the proposed approach and the results obtaned. The problem nvolves convex constrants so there s no need for functon convexfcaton. The 1410 June 001 Vol. 47, No. 6 AIChE Journal

5 Fgure 3. Feasble regon of Desgn 1 ( d1s3.5, ds 0.0 ). Fgure 4. Step 1: fndng ponts at the boundary tces ). The solutons to the above optmzaton problems gve rse to the square pont shown n Fgure 4. Note that these problems correspond to the vertex enumeraton subproblems ŽSwaney and Grossmann, 1985a. proposed to evaluate the flexblty ndex. Consequently, at ths step, one can utlze the results to evaluate the flexblty ndex that corresponds to the mnmum of the results Ž., where denotes dfferent vertces. For ths example, t s found that F s0.78 lmted by the flexblty pont, as shown n Fgure 4. Step. Solve the problems of maxmzng the devatons from the nomnal pont by varyng one uncertan parameter at a tme. The optmzaton problems solved at ths step have the followng form ( ver- actve sets of the problem have been dentfed and the feasble regon s descrbed by the shaded regon shown n Fgure 3. The steps of the proposed analyss are then followed. Step 1. Solve the optmzaton problems of maxmzng the devatons from the nomnal pont towards the edges. The problems solved have the followng form max s.t. fjž d, z, 1,. F0, js1,..., 6 s N s N. and max s.t. f Ž d, z,,. F0, js1,..., 6 j 1 s N 1 1 s N. In ths way the ponts llustrated wth the crcles n Fgure 5 are determned. Step 3. Quckhull algorthm s appled to dentfy the convex hull based on the ponts obtaned from steps 1 and. The output of the algorthm s the set of lnear constrants descrbng the convex hull. The constrants are gven and llus- max s.t. fjž d, z, 1,. F0, js1,..., 6 s N s N Fgure 5. Step : Fndng ponts at the boundary ( varyng one parameter at a tme ). AIChE Journal June 001 Vol. 47, No

6 Fgure 6. Step 3: Determne the feasble convex hull. trated n Fgure 6 g s y8f0 1 g sy F0 1 g s y1.3 F0 3 1 g s y7.5 q7.0f0 4 1 g s y5.71 q4.4f0 5 1 g s y5.88 q4.63f0 6 1 Step 4. The volume Žarea n ths case, snce the problem nvolves two uncertan parameters. s evaluated at ths step usng the Quckhull algorthm that utlzes Delaunay Trangulaton to determne the volume of the convex hull. The volume of the overall expected range s also calculated. For ths example, t s found that the total volume s 16 unts whereas the feasble convex hull has a volume of 10.9 unts. Ths results n a rato of 0.68 that corresponds to the percentage of desgn feasblty. The same procedure s appled for Desgn Žd1s.5, d s3.5.. As ponted out n the prevous secton, the flexblty ndex for ths desgn has the same value as the flexblty for Desgn 1 equals to F s0.78. Ths s due to the fact that the lmtng actve set s the same n both cases. However, as shown n Fgure 7, Desgn has a smaller feasble regon than Desgn 1. Ths s correctly dentfed by the proposed approach whch determnes a rato of feasblty equal to 0.64 for Desgn compared to 0.68 for Desgn 1. It s mportant to pont out that the convex hull offers a very accurate descrpton of the feasble regon for both desgns. Remark 1. It should be ponted out that the proposed rato Ž FCHR. as defned by Eq. 7 corresponds to the percentage of feasblty based on the overall expected range of uncertanty and not the true feasblty regon. As can be observed from Fgures 6 and 7, the percentage that corresponds to the feasble convex hull s much hgher n terms of the actual feasble regon. Fgure 7. Desgn : Determnaton of feasble convex hull. Remark. It should be also noted that the proposed methodology for feasble regon evaluaton and the proposed feasblty metrc s not a probablstc measure and thus does not take nto account the probablty of uncertanty realzatons. Extensons of the proposed work along the lnes proposed by Straub and Grossmann Ž and Pstkopoulos and Mazzuch Ž s a subject of present research. Remark 3. The computatonal requrements of the proposed approach nvolve the soluton of n optmzaton problems at step 1, and n optmzaton problems at step, where n s the number of uncertan parameters nvolved. However, t should be noted that one can choose to consder a subset of these problems, dentfy the correspondng ponts at the boundary of the feasble regon and proceed n the next step of evaluatng the convex hull. Examples Example 1 The frst example consdered n ths secton also nvolves a set of convex constrants that restrct the feasble regon f s q y y40f f s q y yf0 1 1 f s y4 y30f0 3 1 The feasble regon s shown shaded n Fgure 8. We assume that the nomnal pont s Ž,. sž.5, 0. 1 wth expected devatons of y 1 s7.5, q 1 s.5, q s0 and y s60. The steps of the proposed analyss are then appled to determne frst the ponts at the boundary of the feasble regon and then the feasble convex hull. At step of the procedure, the flexblty ndex s also evaluated to be equal to F s The resultng feasble convex hull has a total volume of unts, whle the expected regon corresponds to a volume of 141 June 001 Vol. 47, No. 6 AIChE Journal

7 followng m 1.84 y f sp q Hy yk m D yp y F0.0 c m 1.84 y f syp y Hq qk m D qp y F0.0 c ˆ f s m Hy W F0.0 3 f sc yc max F0.0 4 f syc q r c max F Fgure 8. Example 1: feasble convex hull. 800 unts, that results n a feasble rato of FCHRs0.19. Note that the nscrbed convex hull descrbes the feasble regon much more accurately than the flexblty ndex that corresponds to only 3% of the expected range of varablty compared to 19% of the feasble convex hull, as shown n Fgure 8. Note that the percentage n terms of the actual feasble regon s much hgher snce only the lght shaded areas ŽFg- ure 8. are excluded from the feasble convex hull. The value of FCHR s, however, stll small because t reflects the rato of the feasble convex hull regon when compared not to the actual feasble regon that s unknown, but the expected range of uncertanty whch can be much larger. The exact lnear constrants descrbng the convex hull are g s q9.98 q4.94s g s y3. q1.31s0.0 1 ˆ The desgn varables are the drver power W, the pump head H, the ppe dameter D, and the control valve sze c max, the control varable s the valve coeffcent c, and the uncer- tan parameters are the desred pressure s P, and the 1 flow rate s m. The expected devatons consdered n ths example are q s, y s6, q s00, y s Two desgns are examned and evaluated n terms of ther feasblty Desgn 1 that corresponds to the followng set of desgn varables ˆ max W s31., H s1.3, Ds0.076, c s and Desgn that corresponds to the followng desgn values, ˆ max W s33.74, H s1.406, Ds0.0764, c s The proposed approach s appled to both desgns n order to evaluate the feasble convex hull. Fgures 9 and 10 show the results. It s found that the feasblty rato s FCHRs0.97 for desgn 1 and 0.9 for desgn compared to flexblty ndex that g s y4.0 y30.0s g s y.75 y9.38s g s q0.1 y104.74s g s q5.3 y49.3 s g s y8.4 q15.4 s g s y5.9 q8.06s Example The second example corresponds to the pump and ppe example descrbed by Swaney and Grossmann Ž 1985a.. The set of constrants represent the energy balance, the outlet pressure tolerance, the pump drver power lmt, and the constrant valve range. By elmnatng the outlet pressure P usng the energy balance, the reduced set of nequaltes s the Fgure 9. Example : feasble convex hull for desgn 1. AIChE Journal June 001 Vol. 47, No

8 Fgure 10. Example : feasble convex hull for desgn. s for desgn 1 and for desgn. Consequently, by evaluatng the feasble convex hull, we can better compare the feasblty of dfferent desgns snce, as shown n Fgures 9 and 10, desgn 1 has actually larger feasble regon than desgn whch could not be captured by the flexblty ndex. The major advantage of the proposed approach compared to the flexblty ndex s that t s not lmted by one drecton, but can capture the overall desgn behavor under parameter varablty. However, t should be ponted out that snce the feasble regon s 1-DQC and not jontly convex on 1,, there s no guarantee that the convex hull defned by the ponts found through steps 1 and would be nscrbed wthn the feasble regon. To overcome ths dffculty the approach descrbed n the followng secton s modfed as suggested. Example 3 The example consdered here s a modfcaton of the example of Pstkopoulos and Ierapetrtou Ž wth three uncertan parameters. The constrants descrbed the feasble regon of the desgn Ž d, d. have the followng form 1 f sy zy q0.5 q.0 q d y3 d y8f0 1 1 f sy zy r3y y r3q d q8r3f0 f s zq y y d q y4f where z s the control varable, 1,, 3 are the uncertan parameters wth nomnal value of 1 N s N s 3 N s and expected devatons 1 s 1 s 1 s. All the constrants are jontly convex on Ž. and Ž z.. The desgn examned here corresponds to Ž d, d. sž 3,1. 1. The proposed approach s appled for ths desgn and the followng ponts are frst dentfed at the boundary of the feasble regon. Step 1. At the frst step, the ponts towards the vertces of Fgure 11. Example 3: feasble convex hull for 3-D feasble regon. the expected uncertanty range: p1: Ž,,. sž.18,.18,.18. p: Ž,,. sž 3.000, 3.000, p3: Ž,,. sž.5, 1.748,.5. p4: Ž,,. sž 1.637,.363,.363. p5: Ž,,. sž.486, 1.514, p6: Ž,,. sž 1.514, 1.514,.455. p7: Ž,,. sž 0.000, 4.000, p8: Ž,,. sž 0.000, 0.000, Step. Varyng only one uncertan parameter at the tme we got the followng ponts: p9: Ž,,. sž.6,.000,.000. p10: Ž,,. sž 0.000,.000,.000. p11: Ž,,. sž.000,.000,.31. p1: Ž,,. sž.000,.000, p13: Ž,,. sž.000, 3.646,.000. p14: Ž,,. sž.000, 0.667, Step 3. Then the quckhull algorthm s appled to determne the feasble convex hull of the ponts obtaned at steps 1 and. The resultng convex hull s llustrated n Fgure 11. Step 4. Usng Delaunay Trangulaton the volume of the convex hull s determned to be equal to 1.6 that corresponds to 19% of the total expected range of uncertanty. To compare wth the results of the flexblty analyss the flexblty of the desgn s also determned F s The volume of the correspondng rectangular s approxmately that corresponds to only 0.1% of the expected uncertanty range Ž see Fgure June 001 Vol. 47, No. 6 AIChE Journal

9 lem. Snce then constrants consdered here are jontly quasconvex n z and 1-D quas-convex n Ž., t s proven that the lnearzaton would always result n underestmaton of the feasble regon Ž see Appendx A.. In the followng secton an example nvolvng 1-D quas-convex functons s consdered to llustrate the results of the proposed approach for ths case. Example 4 The example presented n ths secton corresponds to an example that exhbts a 1-D convex feasble regon. To llustrate the proposed deas, the example s smple and descrbed by the followng constrants f sy q q F f s Ž 6q. y350f0 1 f s q y60f f s y10 y50f Fgure 1. Example 3: Flexblty range and overall expected range of uncertanty. 1DQC Convex Feasble Regon Frst, t should be ponted out that only 1-D convex regons are examned here and not general nonconvex regons whch s the subject of the second part of ths work. For the case of the 1-D convex regon, the proposed approach s smlar to the approach for the convex case wth the dfference that after evaluatng the ponts at the boundary towards the vertces as descrbed at step 1, the nonconvex constrants are lnearzed at these ponts. It should be noted that only the nonconvex constrants that are actve at these ponts are lnearzed and not all of the nonlnear constrants n the prob- Note that the constrants satsfy propertes Ž. and Ž. 3 proved by Swaney and Grossmann Ž 1985a. to be suffcent n resultng n the 1DQC feasble regon Ž as also shown n Fgure 13.. The proposed approach s appled for ths example and the feasble convex hull s determned Ž Fgure 13.. Note that, for comparson purposes, the approach s appled for both the orgnal nonconvex constrants, as well as the convexfed ones. As suggested n the prevous secton, the convexfcaton s obtaned after the determnaton of the ponts at the boundary. For ths example, the pont used to convexfy the nonconvex constrant f s the pont Ž P1. shown n Fgure 14 dentfed from the optmzaton problem of maxmzng the devaton from the nomnal pont towards the vertex that corresponds to the maxmum postve devatons of 1,. The convexfcaton results n the followng lnear constrant shown n Fgure 14 wth a dashed lne ˆf s y51.385q4.17 F0.0 1 Fgure 13. Example 4: feasblty convex hull for 1DQC feasble regon. Fgure 14. Example 4: convexfcaton of constrant f. AIChE Journal June 001 Vol. 47, No

10 plotng the nherent dstrbuted nature of the ntal steps of the proposed approach, and Ž. c the extenson of the proposed work to ncorporate the probablty of occurrence of uncertan parameter realzatons. Acknowledgments The author gratefully acknowledges fnancal support from the Natonal Scence Foundaton under the NSF CAREER program CTS Specal thanks to the anonymous revewer that brought to our attenton smlar work done n the area of electrcal engneerng. Fgure 15. Example 4: feasble convex hull for 1DQC feasble regon usng convexfed constrant. Usng ths constrant nstead of constrant f results n the determnaton of pont Ž Pn. to be used on feasble convex hull evaluaton nstead of pont P Ž Fgure 14.. Snce the dfference s very small n ths example, we llustrate the result by focusng Fgure 15 at the area of pont Ž P. whch s the only dfference between the feasble convex hull of the orgnal problem and the feasble convex hull of the convexfed problem. Dscusson and Future Drectons A novel approach for evaluatng the range of feasblty of a gven process s proposed n ths artcle. The basc dea s to determne the convex hull that can be nscrbed wthn the feasble regon. The convex hulls were computed usng the Quckhull Algorthm whch s an ncremental algorthm for evaluatng the convex hull gven a set of ponts. The ponts used to determne the feasble convex hull are ponts at the boundary of the feasble regon that are obtaned from the soluton of a seres of small optmzaton subproblems. The outcome of the applcaton of the Quckhull Algorthm s not only the computaton of the convex hull descrbed by a set of lnear constrants but also ts volume. Ths s acheved based on the deas of Delaunay Trangulaton. A new metrc was ntroduced that corresponds to the rato of the feasble convex hull volume dvded by the overall expected range of uncertanty. In all examples consdered we found that the proposed approach results n much better descrpton of the feasble space. Moreover, the new metrc ntroduced to represent the feasblty of a gven process was found to descrbe accurately the feasble space and thus results n correct comparson between dfferent desgns andror processes. Although the examples addressed n ths work are of small scale, the results obtaned are very promsng regardng the applcablty of the proposed work n large-scale problems. Dfferent extensons are currently under nvestgaton regardng Ž. a the applcablty of the work to nonconvex case, and Ž b. the development of effcent mplementaton scheme ex- Lterature Cted Androulaks, I. P., A. Srdeshapande, and M. G. Ierapetrtou, Determnaton of the Range of Valdty of Reduced Knetc Mechansms, n press AIChE J., Ž Barber, C. B., D. P. Dobkn, and H. Huhdanpaa, The Quckhull Algorthm for Convex Hulls, ACM Trans. Math. Soft.,, 469 Ž de Berg, M., M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computatonal Geometry. Algorthms and Applcatons, Spreger Ž Floudas, C. A., Z. Gumus, and M. G. Ierapetrtou, Global Optmzaton for Feasblty Test and Flexblty Index Problems, Ind. Eng. Chem. Res., Ž Grossmann, I. E., and K. P. Halemane, Decomposton Strategy for Desgnng Flexble Chemcal Plants, AIChE J., 8, 686 Ž Grossmann, I. E., and C. A. Floudas, Actve Constrant Strategy for Flexblty Analyss n Chemcal Processes, Comput. Chem. Eng., 11, 675 Ž Halemane, K. P., and I. E. Grossmann, Optmal Process Desgn under Uncertanty, AIChE J., 9, 45 Ž Kubc, W. L., and F. P. Sten, A Theory of Desgn Relablty Usng Probablty and Fuzzy Sets, AIChE J., 34, 583 Ž Mgdalas, A., P. M. Pardalos, and P. Varbrand, Multle el Optmzaton: Algorthms and Applcatons, Kluwer Academc Publshers, Dordrecht, The Netherlands Ž Ostrovsky, G. M., Y. M. Voln, E. I. Bart, and M. M. Senyavn, Flexblty Analyss and Optmzaton of Chemcal Plants wth Uncertan Parameters, Comput. Chem. Eng., 18, 755 Ž Pstkopoulos, E. N., and M. G. Ierapetrtou, A Novel Approach for Optmal Process Desgn Under Uncertanty, Comput. Chem. Eng., 19, 1089 Ž Pstkopoulos, E. N., and T. A. Mazzuch, A Novel Flexblty Analyss Approach for Processes wth Stochastc Parameters, Comput. Chem. Eng., 14, 991 Ž Saboo, A. K., M. Morar, and D. C. Woodcock, Desgn of Reslent Processng Plants: VIII. A Reslence Index for Heat Exchanger Networks, Chem. Eng. Sc., 40, 1553 Ž Straub, D. A., and I. E. Grossmann, Desgn Optmzaton of Stochastc Flexblty, Comput. Chem. Eng., 17, 339 Ž Swaney, R. E., and I. E. Grossmann, An Index for Operatonal Flexblty n Chemcal Process Desgn Part I: Formulaton and Theory, AIChE J., 36, 139 Ž 1985a.. Swaney, R. E., and I. E. Grossmann, An Index for Operatonal Flexblty n Chemcal Process Desgn Part II: Computatonal Algorthms, AIChE J., 31, 631 Ž 1985b.. Appendx: Proof of Feasble Regon Underestmaton Proposton If f Ž d, z,. j are jontly quas-convex n z and 1-D quas- convex n, then the lnearzaton around and extreme pont underestmates the feasble regon. Proof As llustrated n Fgure A1, we need to bascally to prove that the pont 1 belongs n the feasble regon, that s, 1416 June 001 Vol. 47, No. 6 AIChE Journal

11 Also, we assume that the nomnal pont s a feasble pont Ž N. 1 and thus d, F0. Snce belongs to the lne wth slope Ž.Ž. 1 N fj d, z, there exst a scalar such that s q Ž. 1y. Based on theorem proved by Swaney and Grossmann Ž 1985a., f the constrant functons f Ž d, z,. j are jontly quas-convex n z and one-dmensonal quas-convex n, then the feasblty functon Ž d,. s 1-D quas-convex n, Ž d, 1. Fmax Ž d,., Ž d, N. 4. Fgure A1. Underestmaton of feasble regon. Ž. Ž N.4 Ž 1. Snce max d,, d, s 0, d, F 0 and the proposton has been proved. Ž 1. d, F0. Snce and are at the boundary Žd,. s Žd,. s0 as proved by Swaney and Grossmann Ž 1985a.. Manuscrpt rece ed June 7, 000, and re son rece ed No. 6,000. AIChE Journal June 001 Vol. 47, No