University of Oklahoma, 100 E. Boyd Street, T-335, Norman, Oklahoma Y it )

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1 8136 Ind. Eng. Chem. Res. 2005, 44, On a New MILP Model for he Planning of Hea-Exchanger Nework Cleaning. Par III: Muliperiod Cleaning under Uncerainy wih Financial Risk Managemen Javier H. Lavaja and Miguel J. Bagajewicz* Universiy of Oklahoma, 100 E. Boyd Sree, T-335, Norman, Oklahoma This paper is a follow-up o a wo-par paper on he scheduling of nework cleaning. In he firs par (Lavaja, J. H.; Bagajewicz, M. Ind. Eng. Chem. Res. 2004, 43, ), a new mixed-ineger linear model for he planning of - cleaning in chemical plans was presened, where he ne presen cos is minimized by carefully choosing he appropriae cleaning schedules. In he second par (Lavaja, J. H.; Bagajewicz, M. Ind. Eng. Chem. Res. 2005, 44, ), hroughpu loss was added o he model, hus allowing he hroughpu o be reduced when convenien. Because of he high imporance of financial risk in almos any indusry and he presence of uncerainy in some of he parameers involved in he operaion of neworks, a sochasic version of he model was developed and applied o crude fracionaion unis. The model considers uncerainy no only in he fuure price of he naural gas expended in he furnace bu also in he acual value of he fouling raes of he crudes and in he schedule of change of feedsock during he operaion. Inroducion * Corresponding auhor. Tel.: (405) Fax: (405) bagajewicz@ou.edu. In a previous paper (Par I), 1 we inroduced a new model for opimizing he cleaning schedule of neworks. We applied his model o crude fracionaion unis. In a second par, 2 we exended he model o consider hroughpu reducion during fracions of he ime (mainly cleaning subperiods) and showed ha a cleaning schedule ha considers his hroughpu reducion can be used o increase profi. In paricular, we showed how his echnique can be used o increase he average hroughpu in unis where he boleneck for hroughpu is he furnace capaciy. In his second par, we also inroduced he exension of he model o muliperiod producion, ha is, he use of crudes wih differen fouling characerisics for differen periods of ime. In his par, we exend he model o consider uncerainy and manage financial risk. We omi covering he background of he problem because i is reviewed in Par I. In addiion, decision making in process operaions under uncerainy is a maure field. Being such a maure field, we also omi reviewing i. In paricular, he scheduling of nework cleaning has uncerain parameers, such as he fuure price of he gas (or fuel) expended in he furnace, he exac ime when a disillaion uni would change he ype of crude processed, and he fouling rae of he crude processed. The paper is organized as follows: We presen firs a brief revision of he model (for compleeness); hen he effec in he oal uni operaing coss of he crude schedule variabiliy, he variabiliy in he fouling rae for he crude processed, and he flucuaion of he fuure price of he naural gas expended in he furnace are sudied. In he case of he naural gas price, risk analysis was performed using he echnique developed by Aseeri and Bagajewicz, 3 which is based on he wosage sochasic programming formulaion. Las, risk analysis is addressed by inroducing uncerainy in he hree parameers previously described, and an example illusraing he conceps is given. Highlighs of he Previous Model. The model previously presened in Par I 1 is a rigorous mixedineger linear programming (MILP) model which minimizes he expeced ne presen value (hroughou ime horizon) of he operaing coss arising from he radeoff beween furnace exra fuel coss due o fouling and cleaning coss (which include manpower, chemicals, and mainenance). Consider a nework (HEN) of a crude disillaion uni where is recovered from disillaion column producs and pump-around sreams. We consider ha ime is discreized in inerval periods (ypically s), and each one of hese is subdivided ino a cleaning subperiod and an operaion one. Thus, he objecive is o deermine which is o be cleaned in which period given oher resricions and resource availabiliy so ha he ne presen value is maximized. The soluion should also ake ino accoun he possibiliy of changing any nework flow rae and/ or fluid for any operaion period. The clean and acual ransfer coefficien in period (U i c and U i, respecively) are relaed o he fouling facor (r i )by r i ) 1 U i - 1 U i c (1) We define a binary variable ha idenifies when and which is cleaned as follows: Y i ) { 1 if he ih is cleaned in period 0 oherwise (2) The clean and acual ransfer coefficien for each /ie050319y CCC: $ American Chemical Sociey Published on Web 09/10/2005

2 Ind. Eng. Chem. Res., Vol. 44, No. 21, subperiod can be wrien in erms of he binary variable and he fouling facor as follows, -1 U ecp i ) [ a c iky ik (1 - Y ij )] + k)0 j)k+1 U i c c Y i + c i b i -1 eop ) [ a o iky ik k)0 b i j)k+1 o o Y i + c i p)0 (1 - Y ij )] p)0 (1 - Y ip ) i, g 1 (3) + (1 - Y ip ) i, g 1 (4) where a ik, b i, and c c i are consans ha are funcions of he differen parameers. These equaions are subsiued in he equaions corresponding o he balance o render an expression for he ho oule emperaure (Th 2i ). Th 2i ) R i e d i[ k)1 (R i - 1)Th 1i - R i Tc 1i -1( a iky ik j)k+1-1( a iky ik j)k+1-1( a iky ik j)k+1 R i Tc 1i[ ed i[ k)1 R i e d i[ k)1 + (1-Y ij )) +b iy i +c i (1-Y ip )] - 1 where d i ) (A i /Fc i Cc i )(R i - 1). The expression can be easily linearized hrough sandard ricks. 1 The model minimizes he expeced ne presen value (hroughou ime horizon) of he operaing coss arising from he radeoff beween furnace exra fuel coss due o fouling and cleaning coss (which include man power, chemicals, and mainenance). NPC ) where NPC is he ne presen cos, Ef is he acual cl furnace s energy consumpion, Ef is he furnace s energy consumpion for clean condiion, C Ef is he furnace s fuel cos, C cl is he cleaning cos, η f is he furnace efficiency, and d is he discoun facor. In he second par, 2 he model for he opimizaion of he cleaning schedule for a mulipurpose/muliperiod HEN presened in Par I 1 was exended o opimize he hroughpu reducion required for he operaion of neworks when he maximum capaciy of he furnace is reached a a cerain poin of he ime horizon under consideraion, or when i is desired o operae he neworks a a higher hroughpu during a cerain ime horizon. The model gives he flexibiliy o operae he nework under igh energy condiions, by reducing he hroughpu for shor periods of ime, allowing for more cleanings and recovering he performance of he uni. The model also provides opimal cleaning schedules for he operaion of mulipurpose/muliperiod neworks, also allowing for hroughpu reducion when i is required. The model was coded in GAMS. 4 p)1 (1-Y ij )) +b iy i +c i (1-Y ip p)1 )]] (1-Y ij )) +b iy i +c i (1-Y ip p)1 )] - 1 (Ef - Ef cl ) d η f C Ef + i, } g 1 (5) d Y i C cl (6) i Effec of he Crude Schedule on he Cleaning Schedule. In his secion, he effec of he variaions in he crude schedule on he cleaning schedule is analyzed by changing he in which he swich of he crude is done in he ime horizon. The analysis was performed using he HEN shown in Figure 1 (daa from Table 1) for wo crudes, ligh and heavy, for a hroughpu of 132 Kbpd wih a furnace capaciy of 542 MMBu/h and penalies β ) 2 $/bbl and γ ) 92 $/bbl. The example is he same as he one used in Par II. 2 Table 2 shows he hree cases of crude schedule considered. The ligh crude is processed during he firs 5, 7, and 9 s of he ime horizon, cases 1, 2, and 3, respecively. Figures 2, 3, and 4 show he hroughpu profile for each case, respecively, and Tables 3, 4, and 5 show he cleaning schedule for each case, respecively. Table 6 conains he oal coss for each case, along wih he oal number of cleanings corresponding o each schedule. By comparing he resuls obained, one can noice ha swiching crude 2 s before has a negaive impac of 2% in he oal operaing coss, even hough he oal number of cleanings is he same for boh cases. Since 2% of 20 million dollars is a significan number, we conclude ha including he uncerainy of he swiching ime in he problem is imporan. Effec of he Fouling Rae on he Cleaning Schedule. Differen fouling models can be obained from experimenal laboraory sudies, 6 on-line monioring, 7 and daa reconciliaion. 8,9 Bu here are cases in which he predicions can depar significanly from he real value of he fouling rae. Polley e al. 10 repored cases where he models used o predic he fouling raes underpredic he values wih up o 41% of deviaion. In his secion, he effec of he deviaion of he fouling rae from he prediced value is analyzed. The same HEN was used (Figure 1 and Table 1) for a hroughpu of 120 Kbpd, wih a furnace capaciy of 542 MMBu/h and penalies β ) 2 $/bbl and γ ) 92 $/bbl. The analysis was performed for he heavy crude for he original fouling rae repored for each in Table 1 and for wo cases where he fouling raes are all 20% below and above he original value (Table 7). Tables 8, 9, and 10 show he cleaning schedule for each case, respecively. Table 11 conains he oal coss for each case, along wih he oal number of cleanings associaed wih each schedule. By comparing he schedules, i is observed ha, in all he cases, he hoes s have he same cleaning schedule. This happens because he high fouling raes of he hoes s do no allow for differen allocaions of he cleanings, even when hese values change, wihin he range of deviaion considered. The variaions in he schedules are regisered basically among he s belonging o he cold rain, because heir prediced fouling raes are no so high, allowing for differen cleaning allocaions for differen values of he fouling raes. The resuls also show ha he oal coss are affeced by he deviaion of he prediced fouling rae from he observed value. In his example, he oal coss can be affeced from 2 up o 6%, if we consider ha i is possible o have deviaions on he order of 40% or even more. Effec of he Naural Gas Price on he Cleaning Schedule. In previous work, 11 a risk analysis was

3 8138 Ind. Eng. Chem. Res., Vol. 44, No. 21, 2005 Figure 1. Two-branch mulipurpose/muliperiod nework of Bagajewicz and Soo. Reproduced wih permission from ref American Chemical Sociey. Table 1. Daa for he Nework of Bagajewicz and Soo a No. crude ype N i,cr A (f 2 ) Fc Cpc [Bu/(h F)] Fh Cph [Bu/(h F)] U c [Bu/(f 2 h F)] U [Bu/(f 2 h F)] r [(f 2 F)/Bu] 1 ligh E-05 heavy E-05 2 ligh E-06 heavy E-05 3 ligh E-05 heavy E-05 4 ligh E-05 heavy E-05 5 ligh E heavy E ligh E-05 heavy E-05 7 ligh E heavy E ligh E heavy E ligh E heavy E ligh E heavy E ligh E heavy E ligh E heavy E ligh E heavy E ligh E heavy E Tcin ( F) ligh 70 heavy 70 Tcou ( F) ligh 679 heavy 670 fr () 0.2 η f 0.75 η c a Reproduced wih permission from ref American Chemical Sociey. Thin ( F)

4 Ind. Eng. Chem. Res., Vol. 44, No. 21, Table 2. Schedule of he Crude Type Processed in he Time Horizon (Three Differen Cases) case X cr, cr) ligh X cr, cr) heavy X cr, cr) ligh X cr, cr) heavy X cr, cr) ligh X cr, cr) heavy performed o see he effec of he naural gas price on he cleaning schedule and he oal operaing coss. The wo-sage sochasic programming echnique was used o build he model, considering he binary variable of cleaning he or no (Y (i,) ) as he only firssage decision variable, since his problem has he peculiariy ha once his variable is fixedswhich means ha he cleaning schedule is fixedshe soluion is fixed, wihou room for correcions by he second-sage variables. The scenarios were consruced by sampling naural gas prices, which are assumed o follow a cyclical average rend of seasonal variaions, based on U.S. Deparmen of Energy daa. Ten base curves were consruced around he average rend, wih increasing deviaion from i for similar s of he year as he ime horizon increases (Figure 5). Then sampling around hese base curves was performed assuming normal disribuions and equal probabiliy for all he scenarios. The model was solved for he en- nework used by Lavaja and Bagajewicz, 11 for a ime horizon of 18 s. I was assumed ha fouling has already aken place in some s a he beginning of he ime horizon. Figure 2. Throughpu profile for case 1: ligh crude processed during he firs 5 s. Figure 3. Throughpu profile for case 2: ligh crude processed during he firs 7 s.

5 8140 Ind. Eng. Chem. Res., Vol. 44, No. 21, 2005 Figure 4. Throughpu profile for case 3: ligh crude processed during he firs 9 s. Table 3. Cleaning Schedule for Case 1: Ligh Crude Processed During he Firs Five Monhs X X X 3 6 X X X 3 7 X 1 9 X X X 3 10 X X X X X 4 13 X 1 14 X X X X X X 6 oal cleanings 35 Table 5. Cleaning Schedule for Case 3: Ligh Crude Processed During he Firs Nine Monhs X X X X X X 3 7 X X X 3 8 X X 2 9 X X 2 10 X X 2 11 X X X X X X X 3 14 X X X X X 5 oal cleanings 37 Table 4. Cleaning Schedule for Case 2: Ligh Crude Processed During he Firs Seven Monhs X X X X X X 3 7 X X 2 8 X X 2 9 X X 2 10 X X 2 11 X X X X X 2 14 X X X X X X 6 oal cleanings 35 By using 10 rend curves, 200 soluions were obained based on he 200 scenarios generaed. Figure 6 shows all he soluions obained. In his char, he cumulaive Table 6. Toal Coss and Number of Cleanings for he Three Cases in Muliperiod Operaion case NPC (MM$) oal cleanings probabiliy is ploed agains he 200 soluions, where each of he soluions conains 200 resuls sored in descending order of NPC as he cumulaive probabiliy increases. Figure 6 also depics he soluion obained by he deerminisic model using he mean values of he rend. The heurisic approach of cleaning every ha reaches 15%, 20%, and 25% fouling was also simulaed. On one hand, he resuls show ha he flucuaions around every rend do no affec he resuls significanly; ha is, all he curves obained by sampling around he same rend are very close. For example, for rend 1, he difference in he expeced ne presen cos (ENPC) beween he bes and wors soluions is only 0.6%. On he oher hand, here are wo clear groups of

6 Ind. Eng. Chem. Res., Vol. 44, No. 21, Table 7. Fouling Raes for he Three Differen Cases (Heavy Crude) hex deviaion (%) Table 8. Cleaning Schedule for Case 1: -20% of Deviaion in he Fouling Rae X 1 2 X X 2 3 X X X X X 5 5 X X X 3 6 X X X X X X X X 4 10 X X X X X X X X X X X X 6 oal cleanings 35 Table 10. Cleaning Schedule for Case 3: +20% of Deviaion in he Fouling Rae X 1 3 X X X X X 5 5 X X X X 4 6 X X X X X X X X 4 9 X X X X 4 10 X X X X X X X X X X X X 6 oal cleanings 38 Table 9. Cleaning Schedule for Case 2: 0% of Deviaion in he Fouling Rae X X X X 4 5 X X X 3 6 X X X X X X X X 4 10 X X X X X X X X X X X X 6 oal cleanings 36 soluions, depending on which rend is associaed wih he soluion. Figure 6 shows how he heurisic soluions are far away from hose prediced by modeling. Noiceably, even in he case when he same number of cleanings is used, modeling suggess beer soluions. For example, compare he deerminisic soluion obained using he overall mean rend wih he heurisic for 80% U c : boh use 28 cleanings, bu he ENPC of he heurisic sraegy is 20.5% higher (see Tables 12 and 13). Figure 6 also shows ha one group of curves and he deerminisic curve are close. This is because he rends and he overall mean are relaively close. However, for he rends ha depar significanly from he mean (especially hose ha do no cross he mean oo many imes), he resuls group in anoher se of curves. Thus, if risk is of concern, hen one should choose he soluion corresponding o a curve exhibiing lower coss a low cumulaive probabiliies, insead of he opimal, which Table 11. Toal Coss and Number of Cleanings for he Three Cases wih Differen Fouling Raes case % deviaion NPC (MM$) oal # cleanings in his case is close o he deerminisic soluion (rends 1 and 6-10). Risk Analysis on he HEN Cleaning Schedule. On he basis of he resuls of he previous hree secions, risk analysis was performed by inroducing uncerainy ino he hree parameers previously menioned: crude schedule (X cr ), fouling rae (r), and naural gas price (C Ef ). The analysis was performed using he same mulipurpose/muliperiod HEN shown in Figure 1 (daa from Table 1), along wih he same wo crudes, for a hroughpu of 120 Kbpd wih a furnace capaciy of 500 MMBu/h and penalies β ) 2 $/bbl and γ ) 92 $/bbl. For he case of uncerainy in he crude schedule, hree differen scenarios were considered, wih differen probabiliies associaed. Table 14 shows he hree scenarios of crude schedule considered, along wih heir corresponding probabiliies. The heavy crude is processed during he firs 5, 7, and 9 s of he ime horizon, scenarios 1, 2, and 3, respecively. For he uncerainy in he fouling rae, hree differen scenarios per each ype of crude were considered, wih differen probabiliies associaed. Table 15 shows he scenarios for he fouling raes, along wih heir corresponding probabiliies. In he case of he naural gas price, five scenarios were considered, wih each one conforming o a rend for he fuure price of he uiliy. Only he base rends were used based on he resuls from Lavaja and Bagajewicz; 11 flucuaions around he rends do no affec he schedule significanly. Figure 7 shows he five rends corresponding o he five scenarios and heir probabiliies.

7 8142 Ind. Eng. Chem. Res., Vol. 44, No. 21, 2005 Figure 5. Trends and 200-scenario sampling for he naural gas price. Reproduced wih permission from ref American Chemical Sociey. Figure 6. Risk curves for he case of uncerainy in he naural gas price. On he basis of all he above scenarios for each variable, we made combinaions and generaed 135 scenarios. Then, he sochasic model was generaed, applying he echnique inroduced by Aseeri and Bagajewicz. 3 This echnique consiss of solving each scenario independenly. Then each soluion is solved again wih he firs-sage variables fixed for all he scenarios o obain all he risk curves for he res of he scenarios. As a resul, 135 risk curves were obained, each one conformed by he 135 values of NPC resuling from simulaions for each scenario, fixing he deerminisic soluion obained from 1 of he 135 scenarios. From he deerminisic poin of view and based on he resuls obained in he previous hree secions, he wors-case scenarios are hose in which he fouling raes are he highes (H+20 and L+20) (or a leas he highes for he heavy crude), he schedule for he change of he crude is heavy crude he firs nine s (H9-L3), and he rends for he gas price have he highes averages. Because of he high fouling raes in hese scenarios, more cleanings are required in order o ameliorae he effecs of he fouling process. In conras, he bes-case scenarios are hose in which he fouling raes are he lowes (H-20 and L-20), he swich from heavy o ligh

8 Table 12. Cleaning Schedule for Heurisic 80% U c (ENPC ) 1112 M$) Ind. Eng. Chem. Res., Vol. 44, No. 21, hex X X X X X 4 6 X 1 7 X X X X 4 8 X X X X X 5 9 X X X X X X 6 10 X X X X X X 6 oal cleanings 28 clean./ hex Table 13. Cleaning Schedule for Deerminisic Case (ENPC ) 923 M$) hex X 1 2 X X 2 3 X X 2 4 X X 2 6 X X 2 7 X X X X 4 9 X X X X X 5 10 X X X X X 5 oal cleanings 28 clean./ hex Table 14. Three Differen Scenarios for he Schedule of he Crude Type Processed in he Time Horizon prob. scenario (%) (H5-L7) 60 X cr, cr) ligh X cr, cr) heavy (H7-L5) 30 X cr, cr) ligh X cr, cr) heavy (H9-L3) 10 X cr, cr) ligh X cr, cr) heavy Table 15. Three Differen Scenarios for he Fouling Raes per Type of Crude hex scenario probabiliy (%) crude ype deviaion (%) (H-20) 25 heavy (H 0) (H+20) (L-20) 25 ligh (L 0) (L+20) crude occurs a he fifh (H5-L7), and he rends for he gas price have he lowes averages. Because of he low fouling raes in hese scenarios, less cleanings are required in order o ameliorae he effecs of he fouling process. Inermediae-case scenarios would be hose in which he values of he hree parameers are inermediaes or hose in which he good-case and badcase scenarios for individual parameers compensae each oher when hey are combined. Figure 8 shows he 135 curves obained. I is observed ha all he soluions converge o a common region a high cumulaive probabiliies and low coss, bu when he cumulaive probabiliy decreases, hey separae ino wo differen groups. In one of he groups, which conains more curves and has lower coss, all he curves remain ogeher, following he rend of high cumulaive probabiliies. In conras, he second group is conformed by curves ha sar deparing from he rend a a cumulaive probabiliy equal o 0.7 o give shape o a riangular envelope ha reaches is wides par a he lowes cumulaive probabiliies, wih higher coss. Figure 9 shows only five curves belonging o each group, he five wih he lowes ENPC and he five wih he highes NPC ou of he 135 curves, plus he lowerbound curve. Table 16 shows he corresponding scenarios and heir probabiliy of occurrence for each of he 10 soluions shown in Figure 9. The firs five soluions in Figure 9 belong o he lowes-cos group and are he resul of fixing he schedule obained as deerminisic soluions of iner-

9 Figure 7. mediae-case scenarios. These soluions are almos suck o he lower-boundswhich is he ideal soluions which implies ha hey are very good sraegies. The difference in he ENPC beween he lower-bound and Soluion 1 is 0.4% (see Figure 9 s legend). The second five soluions belong o he highes-cos group and are he resul of fixing he schedule obained as deerminisic soluions of bes-case scenarios. They depar from he lower-bound in almos he whole range, and he difference in he ENPC beween Soluion 10 and he lower-bound is 6.7% (see Figure 9 s legend). Noiceably, he schedule in he change of he crude seems o rule he level of aggressiveness of he scenario since, for boh groups, he following is observed: (1) he combinaion of he individual scenarios for he fouling raes for boh crudes does no follow a paern and (2) he gas price scenarios are almos he same for boh groups, bu for all he bes soluions, he swich occurs a he sevenh (which is an inermediae scenario for ha parameer), while for all he wors soluions,

10 Ind. Eng. Chem. Res., Vol. 44, No. 21, Figure 9. Ten risk curves seleced from Figure 8: five wih he highes ENPC and five wih he lowes ENPC. Figure 10. Throughpu profile obained from he deerminisic resuls for Soluion 1. he swich occurs a he fifh (which is he bescase scenario for ha parameer). This fac implies ha, when differen parameers are considered under uncerainy simulaneously, he influence of he scenarios on he soluion migh be governed by one of he parameers srongly, even hough all he parameers considered uncerain have similar effecs when hey are considered individually, as in he previous secions. Figure 10 and Table 17 show he hroughpu profile and he cleaning schedule obained for Soluion 1, and Figure 11 and Table 18 do he same for Soluion 10. The deerminisic schedules and oal coss reaffirm wha was claimed previously: inermediae-case scenarios capure beer soluions han bes-case scenarios, by performing more cleanings, when uncerainy is considered. While his is rue in his paricular case, i is no necessarily a conclusion one would generalize. Table 17. Cleaning Schedule Obained from he Deerminisic Resuls for Soluion X X X 3 6 X X 2 7 X 1 9 X X 2 10 X X X X X X 4 13 X 1 1 X X X X X X 6 oal cleanings 34

11 8146 Ind. Eng. Chem. Res., Vol. 44, No. 21, 2005 Figure 11. Throughpu profile obained from he deerminisic resuls for Soluion 10. Table 18. Cleaning Schedule Obained from he Deerminisic Resuls for Soluion X X 2 3 X X X X 1 7 X X 2 9 X X X 3 10 X X X X 2 14 X X X X X 5 oal cleanings 32 From he previous analysis, i can be claimed ha, when risk is involved, he schedules resuling from he inermediae-case scenarios provide he bes sraegies. Conclusions The model developed considers uncerainy no only in he fuure price of he naural gas expended in he furnace bu also in he acual value of he fouling raes of he crudes and in he schedule of change of feedsock during he operaion. The resuls show how he opimal sraegies can vary when differen parameers are considered uncerain simulaneously and how he model helps deermine he bes sraegies o apply when risk is involved. In addiion, he model faciliaes he decision making coordinaion beween process and planning engineers by evaluaing upfron he bes sraegy (cleaning scheduling and hroughpu reducion) o apply under muliple scenarios ha are driven by differen fuure uncerain parameers. Lieraure Cied (1) Lavaja, J. H.; Bagajewicz, M. On a New MILP Model for he Planning of Hea-Exchanger Nework Cleaning. Ind. Eng. Chem. Res. 2004, 43, (2) Lavaja, J. H.; Bagajewicz, M. On a New MILP Model for he Planning of Hea-Exchanger Nework Cleaning. Par II: Throughpu Loss Consideraions. Ind. Eng. Chem. Res. 2005, 44, in press. (3) Aseeri, A.; Bagajewicz, M. New Measures and Procedures o Manage Financial Risk wih Applicaions o he Planning of Gas Commercializaion in Asia. Compu. Chem. Eng. 2004, 28 (12), (4) Brooke, A.; Kendrick, D.; Meeraus, A. GAMS: A User s Guide; GAMS Developmen Corporaion: Washingon, D. C., (5) Bagajewicz, M.; Soo, J. Rigorous Procedure for he Design of Convenional Amospheric Crude Fracional Unis. Par III: Tradeoff beween Complexiy and Energy Savings. Ind. Eng. Chem. Res. 2003, 42, (6) Knudsen, J. D.; Dahcheng, L.; Eber, W. A. Undersanding Hea Exchanger Fouling and is Miigaion; Bo, T. R., e al., Eds.; Begell House: New York, 1999; pp (7) Turakhia, M.; Characklis, W. G. Fouling of Hea Exchanger Surface: Measuremen and Diagnosis. Hea Transfer Eng. 1984, 5 (1-2), (8) Wilson, D. I.; Vassiliadis, V. S. Miigaion of refinery fouling by managemen of cleaning. In Undersanding Hea Exchanger Fouling and Is Miigaion, Proceedings of an Inernaional Conference, Caselvecchio Pascoli, Ialy, 1997; pp (9) Elahresh, H. A.; Shaibani, A. S.; Hassan, M. Opimizaion of Pre Train Cleaning Cycle. In 2nd Conference on Process Inegraion for Energy Saving and Polluion Reducion, 1999; pp (10) Polley, G. T.; Wilson, D. I. Evaluaion of Laboraory Crude Threshold Fouling Daa for Applicaion o Refinery Pre Trains. Appl. Therm. Eng. 2002, 22, (11) Lavaja, J. H.; Bagajewicz, M. J. Managing Financial Risk in he Planning of Hea-Exchanger Cleaning. In 14h European Symposium on Compuer-Aided Process Engineering, Received for review March 6, 2005 Revised manuscrip received July 18, 2005 Acceped Augus 1, 2005 IE050319Y