2 nd Mercosur Congress on Chemcal Engneerng 4 h Mercosur Congress on Process Sysems Engneerng HIERARCHICAL DECISIONS FOR LINEAR/NON-LINEAR DISJUNCTIVE PROLEMS Jorge M. Monagna and Aldo R. Vecche * INGAR Insuo de Desarrollo y Dseño UTN Faculad Regonal Sana Fe E-mal: mmonagna, aldovec@cerde.gov.ar Absrac. Ths arcle presens several synhess and desgn opmzaon problems n process sysem engneerng nvolvng herarchcal decsons modeled by means of nesed dsuncons. I does no exss an algorhm o solve problems formulaed h hose ype of dsuncons, herefore s needed a ransformaon of nesed dsuncons no a formulaon here he problems can be solved. Vecche and Grossmann(200) have presened a ransformaon for embedded dsuncons. Wh hs approach n mnd, he dea behnd hs arcle s o gve o he reader a beer undersandng of he seps nvolved n he ransformaon by shong several examples. Some resuls obaned n he problem soluons are also shon. Keyords: Dsuncve Programmng, Herarchcal Dscree Decsons, Process Sysem Engneerng. Inroducon Herarchcal dscree decsons are very mporan for synhess and desgn problems n process sysem engneerng. Problems nvolvng o or hree dscree decson levels here he loer levels are subeced o he prevous one are very common n he leraure. Turay and Grossmann (998) have made an srucural floshee opmzaon nvolvng complex nvesmen cos funcons. The model consss of a process synhess problem here he frs level of decson nvolves he selecon of a pece of equpmen beeen several opons, and hen he correspondng cos funcon o apply accordng o he dmenson, pressure and emperaure range. Grossmann e al. (200) presened a general dsuncve model negrang plannng and schedulng n process operaons. The model s based on a Sae Tas Neor (STN) supersrucure here a se of uns can perform a se of ass over a me horzon H. The me horzon s dvded no a number of plannng and schedulng perods. Three levels of herarchcal decsons are nvolved n he model. The frs level of decson consss on selecng he nverson on he un, he second level; he operaon of ha un n a specfc perod of me, he hrd level conemplaes he capacy expanson of he process and he schedulng decsons of he uns. oh examples presened before consss manly of o erm dsuncons, here he herarchcal decson s on he frs erm once he un s seleced. If he un s no seleced, no dscree decson exss on he second level. Monagna e al. (2004) formulaed a dsuncve model for he opmal process synhess n he producon of mulple recombnan proens. In hs problem a o level dscree decson s formulaed hrough a nesed dsuncon. The frs level corresponds o he selecon of he hos o be employed for a parcular produc. Once he hos s seleced and for some ass n he bach processes, he operaon o ll perform hose ass mus be chosen. Dsuncons for hs case are dfferen han he prevous examples because hey have several erms h a second level of decson on each of hem. The formulaon of herarchcal decsons by means of algebrac equaons/nequales s no an easy as. For dsuncve problems hese ypes of decsons are formulaed usng nesed dsuncons hch, n some sense, faclae her formulaon, due o he expressveness of he dsuncons. The modeler can easly vsualze he
2 nd Mercosur Congress on Chemcal Engneerng 4 h Mercosur Congress on Process Sysems Engneerng dscree decsons nvolved and he mplcaons of hem relaed o he consrans o sasfy. A problem nvolvng nesed dsuncons mus be ransformed n he form of sngle no nesed dsuncons le he Generalzed Dsuncve Problem (GDP) proposed by Lee and Grossmann(2000). Once he problem s n he GDP form, he g-m or Convex-Hull relaxaons or some oher echnque can be used o solve. A ransformaon for an embedded dsuncon no he GDP form as proposed by Vecche and Grossmann (2000). The fnal formulaon resuls no a couple of sngle dsuncons and logc proposon formulas. Ths ransformaon becomes an mporan sep for dsuncve problems h herarchcal decsons. I s he purpose of hs arcle o presen some ypcal process sysem engneerng problems h herarchcal decsons and her ransformaon no he GDP form. The general obecve s o provde a beer undersandng of he seps nvolvng n he ransformaon of herarchcal decsons proposed by Vecche and Grossmann (2000) no he GDP formulaons. The resuls obaned afer he problems ransformaon and soluon are also shon. 2. Generalzed Dsuncve Programmng (GDP) A generalzed dsuncve program, here dscree decsons are modeled hrough dsuncons and logc proposons aes he follong form (Lee and Grossmann, 2000): Mn z c K s.. r(x) 0 Y g ( x ) 0, K J c γ Ω(Y) True x 0,c 0,Y + f ( x ) { rue, false } () In hs model, x R n s he connuous varables vecor and Y are oolean varables. c R are connuous varables and γ are values ha correspond o he evaluaon of alernaves. f: R n R s he erm of he obecve funcon ha depends on varables x and r: R n R q s a general se of consrans ha do no depend on dsuncons. Ths general model assumes ha f(x) and r(x) are convex funcons. A dsuncon s composed by an operaor OR and a se of erms. In each erm of he dsuncon here s a oolean varable Y, a se of convex consrans g : R n R p and a cos varable c. If he oolean varable Y s rue, hen condons g (x) 0 and c γ mus be me. Oherse, f Y s false, he correspondng consrans are gnored. I s assumed ha each erm of he dsuncons gves rse o a non-empy feasble regon. Fnally, Ω(Y) True s a se of logcal consrans generaed by usng he se of oolean varables Y. Nonlnear problems usng hs formulaon can be solved by usng he g-m or he Convex Hull relaxaons ransformng he GDP formulaon no a Mxed Ineger Nonlnear Program (MINLP) problem, and hen solvng by means of a MINLP algorhm. Anoher alernave and dependng on he dsuncon formulaon, Logc- ased Ouer Approxmaon (Turay and Grossmann, 996) can be used. 2
2 nd Mercosur Congress on Chemcal Engneerng 4 h Mercosur Congress on Process Sysems Engneerng A smlar approach can be presened for he lnear case, havng f(x), r(x), g (x) all lnear funcons, hen he g-m or Convex-Hull relaxaons for he lnear case can be appled o ransform no a Mxed Ineger Lnear Program (MILP) problem. Fnally a MILP can be solved usng a ranch and ound algorhm. Observe ha n he GDP formulaon no nesed dsuncons are nvolved meanng ha o solve he problem s necessary o ransform he embedded dsuncon no he GDP form. 3. Examples 3.. Srucural floshee h complex nvesmen cos funcons Ths example (Turay and Grossmann, 998) corresponds o a dsuncve process synhess model here he selecon of he uns, s operang condons mus be deermned. The performance and desgn equaons of he processes are ncluded based on he nvesmen and operang coss. The nvesmen cos s generally represened by smple models usng connuous lnear or nonlnear funcons. For hs parcular model he nvesmen cos models are formulaed by dsconnuous funcons of sze, pressure and emperaures: s.. Mn f(x) + c g(x) 0 D c Y h( x ) 0, ( α d η + β ) γ ( P ) δ(t ) d d d Y, x 0 γ d 0 ( P ) γ E P c 0 P P,m δ(t ) δm F T m T Tm m (2) Ω(Y ) True here f(x) represens he operaonal cos, c 0 corresponds o he nvesmen cos for un, g(x) 0 represens general lnear/nonlnear consrans ha mus be sasfed ndependen of dscree choces, Y s a oolean varable handlng he frs level of dscree decson, decdng f he un s seleced for he process or no, h(x)0 represens mass and energy balances of un,,,, and,m are oolean varables handlng he second level of dscree decsons hch selecs: he nvesmen cos funcon (c ) o apply accordng o dfferen nervals of un szes (d ); he pressure facor γ (P) accordng o he pressure (P ) nerval, he emperaure facor δ (T) relaed o he emperaure (T ) nerval, respecvely. The ransformaon proposed no he GDP form of embedded dsuncon (2) s he follong: 3
2 nd Mercosur Congress on Chemcal Engneerng 4 h Mercosur Congress on Process Sysems Engneerng Y Y x 0 h( x ) 0 d 0 c 0 (3) ( ( ( D E m F, c ( α d η + β ) Y ) * γ ( P ) (T ) c 0 δ d d d P T γ ( P ) γ m P δ (T ),,m T P T m Y ) γ ( P )can ae any value beeen bounds Y ) δ (T )can ae any value beeen bounds (4) (5) (6) Noe ha he frs sep n he ransformaon s o decompose he nesed dsuncon no o separae dsuncons [(3) and (4)]. Snce he nner dsuncons becomes a sngle ones (no nesed) observe ha from (4) o (6) an exra-erm s added o represen he fac hen Y s false, consran varables of he frs erms can ae cero value (4) or any value n he doman defned for hem (5 and 6). Leavng ransformaons [(3) - (6)] hou ha exra erms ould resul ha one of he frs erms of dsuncon (4-6) mus be rue, hch does no comply o he orgnal formulaon. Neverheless, some exra consrans mus be added, consderng ha: for each dsuncon one erm can be rue (exclusve or), he sasfacon of he dsuncon for he nvesmen cos funcons reles on he sasfacon of he frs dsuncon erm of he un selecon (Y rue), hen, he follong logc proposon mus be added o (3) and (4) n order o complee he ransformaon: Y Y To apply he algorhms presened n secon 2, (5) and (6) mus be ransformed no an algebrac formulaon gvng he follong consrans: Y m Y Y Y + ( Y m m Y Y Y m ) (7) (8) (9) (0) () (2) (3) (4) 4
2 nd Mercosur Congress on Chemcal Engneerng 4 h Mercosur Congress on Process Sysems Engneerng Equaons () - (4) expresses ha: - he un s seleced or no horough (), - f he un s seleced: one nvesmen cos funcon (2), one pressure facor (3) and one emperaure facor (4) mus be chosen and appled, - f he un s no seleced: no nvesmen cos funcon, nor pressure and nor emperaure facor s seleced by selecng he second erm of dsuncon (4), (5) and (6). In general, and for praccal purposes, he las erm of (4), (5) and (6) s no added o he formulaon because s mplcly sasfed horough (8) - (0) and dsuncon (3). Three possble soluon algorhms can be appled o hs parcular problem:. Transform he GDP formulaon no a MINLP by usng he g-m relaxaon of a dsuncve se (Lee and Grossmann, 2000), hen solve he problem by a MINLP algorhm 2. Transform he GDP formulaon no a MINLP by usng he Convex-Hull relaxaon of a dsuncve se (Lee and Grossmann, 2000), hen solve he problem by a MINLP algorhm. 3. Transform only he dsuncon (nner dsuncons) of he dsconnuous nvesmen cos funcons no MINLP formulaon, and hen solve he problem by applyng he Logc-ased Ouer Approxmaon mplemened n LogMIP (Vecche and Grossmann, 999). Ths algorhm can be appled for hs parcular problem because of form of dsuncon (3). We have solved hs example by applyng he hree algorhms menoned before. For he frs o, e have made he reformulaon no a MINLP by rng he consrans correspondng o he Convex Hull and he g- M relaxaon of he ransformed dsuncve problem, and hen solvng hem by usng DICOPT++ (MINLP solver) of he GAMS Sysem. For he hrd algorhm e have used LogMIP (Vecche and Grossmann (999), hp://.cerde.gov.ar/logmp) hch has mplemened Logc-ased Ouer Approxmaon mehod. The resuls obaned h hese algorhms are shon n Table I. Table. Resuls obaned solvng he process synhess h complex cos funcons Model Equaons Varables Dscree Opmal Relaxed Maor CPU me Varables Soluon Soluon Ieraons (sec.) Logc ased 232 32 36 24.6 ---- 2 0.38 g-m 7 78 36 24.6 592 6 2.87 Convex Hull 207 4 36 24.6 30 3 0.42 3.2. Inegraed Plannng and Schedulng The negraed plannng and schedulng model presened by Grossmann e al. (200) s formulaed by : p p r s s mn ( CO + CE ) + cs xs + c R + cs xs (5) s In(, ) In(, ) s s.. ( x,x ) a g s s s f s xs,xs,,x d b s, (6) ( ) s, (7) 5
2 nd Mercosur Congress on Chemcal Engneerng 4 h Mercosur Congress on Process Sysems Engneerng y h( Q,x,x ) d CO γ z z y Q Q, QE Q Q +, D x 0 D x 0 CE α QE + β CE 0 CO 0 v v 0 Q 0, In(, ) < R R 0 η + δ T z, (23) y, (9) τ z I τ, In(, ) v,,,, (20) (2) y, (22) v,, In(, ) (24) Ω ( y,,z,v ) True (25) 2 ( v ) True Ω (26) CE, CO, Q, QE, x, R, 0 (27) y,, z, v {True, False} (28) The obecve funcon s he cos opmzaon over he me horzon ncludng operang, expanson, and resources coss and coss assocaed h saes over he plannng and schedulng perods. Global consrans le mass balances over mxers, and global consrans vald for a parcular schedulng perod (nvenory consrans) are expressed by (6) and (7) respecvely. The ouermos dsuncon represens he decson of ncludng he un n he desgn or no. If he un s ncluded hen he se of consrans ha belongs o he lef hand sde of he dsuncon s appled, oherse a subse of he sae varables assocaed o un are se o zero for all perods. If un operaes n perod,( rue), hen consrans h y CO represenng expanson and schedulng decsons are appled. If he un does no operae n perod, a subse of sae varables and he operang cos assocaed h un are se o 0 for perod. The o nner dsuncons are only appled f True. The frs represens he decson of expandng un n plannng perod or no. If un s expanded n perod, (z True), hch s also a plannng decson, consrans Q and CE are appled. If he decson s no o expand un n perod (z False), he capacy Q remans he same as n he prevous perod, and he expanson cos CE s se o zero. Un specfc schedulng decsons are represened by he second nner dsuncon. Ths nner dsuncon s only appled for schedulng perods hn he plannng perod, as denoed by he se In(,). Ths dsuncon saes ha f as s sared on un n schedulng perod, (v True), hen he bach sze s lmed by he un capacy n and he resource usage s calculaed by R. If as s no sared on un n perod, (v False), he sarng bach sze and resource usage R are se o zero. (8) 6
2 nd Mercosur Congress on Chemcal Engneerng 4 h Mercosur Congress on Process Sysems Engneerng Consrans (9)-(26) are logc proposons represenng logcal relaonshps beeen he dscree varables. For a complee reference abou he meanng of hs logcal relaonshps see Grossmann e al. (200). The ransformaon for hs lnear problem no he GDP formulaon s: y T, D y x 0 (29) h Q CE ( Q, x, x ) d D x 0, CO γ CO 0 z z Q, + QE Q Q, α QE + β CE 0, (30) (3) v 0 < Q R η + δ v 0,, In(, ) R 0 Observe ha for hs case no exra erm s needed because he no sasfacon of he frs erm of nner dsuncon s already defned hrough he negaon of he frs erm, and he logcal relaonshps beeen he ouer and nner dsuncons erms are already expressed by means of (9) o (26). As can be seen, he procedure he ransformaon of nesed dsuncons no he GDP form s: ae ou he nner dsuncons, leavng nesed dsuncons no a se of ndvdual dsuncons, ransform he logcal proposons (9)-(26) beeen he ouer and he nner dsuncons no algebrac consrans: y + (-y ) (30) y + (3) + z, (32) ( (32) ) + y, (34) ( z ) +, (35) ( v ) + In(, ) (36) I + v, In(, ), (33) Snce hs problem corresponds o a lnear one, he algorhm o solve s o ransform he lnear dsuncve problem no a MILP, by means of g-m or Convex hull relaxaons of a dsuncve se, hen applyng a ranch and ound algorhm, or a varan of le he cung plane algorhm proposed by Saaya and Grossmann (2003). A smplfcaon of hs orgnal problem has been solved correspondng o he rerof plannng of process neors proposed by Jacson and Grossman (2002). The frs level of decson s he operaon mode m of process n each me perod represened by varable Y m, hle n he second level varable W m represens 7
2 nd Mercosur Congress on Chemcal Engneerng 4 h Mercosur Congress on Process Sysems Engneerng he decson of mang a change m for process n perod. For hs problem Y m s equvalen o and W m s equvalen o z of he complee problem. The dsuncve problem has been solved usng LogMIP, hch auomacally reformulaes he problem no a MILP by applyng he g-m or Convex Hull relaxaons The resuls obaned h boh reformulaons are shon n Table 2. Table 2. Resuls obaned solvng he negraed plannng and schedulng problem Convex H. Equaons Varables Dscree g M g-m g-m Convex H. Convex H. varables CPU me * eraons Nodes CPU me * eraons Nodes 2 60 72 0.72 449 36 0. 22 0 * CPU Tme s expressed n seconds 3.3 Synhess and Desgn of mulproduc recombnan proens plan The problem shon n hs secon corresponds o he synhess of a bach plan for he producon of mulple recombnan proens presened by Monagna e al. (2003). The dsuncve model formulaon for hs problem s he follong: α Mn Cos M G V (37) d d d D d d β d d Q TL HT (38) h H Yh hd Sdh V d J h d D G d 0 Tdh + Tdh TL M d The obecve funcon s o mnmze he nvesmen cos of he plan, sasfyng he producon arges of he producs (,,P). α d and β d are parameers o evaluae he cos of each dmenson un V d. For he bach plan s alloed he smulaneous duplcaon of boh n-phase and ou-phases uns, represened by G d and M d, respecvely. Consran (38) seles ha for each produc, a Q quany mus be produced on he avalable me horzon HT. The embedded dsuncon (39) represens a herarchcal decson. The frs decson o mae s hch hos h H ll be seleced for he producon of, hen, dependng on he hos seleced, he subordnae decson s o deermne he operaon ha ll perform he requred separaon as J h. Dependng on he hos, he as sequence s dfferen. The number of opons for each as s represened by he se D. V d s he un volume a sage usng opon d. S dh s he sze facor correspondng o produc a sage usng hos h and opon d. s he bach of produc. Fnally, G d s he number of n-phase parallel duplcaed uns for sage, opon d. TL s he cycle me defned as he me goes beeen o successve baches of produc. I s gven by he longes processng me of all hose belongng o processng sages of produc. For he purpose of reducng he cycle me of a produc, s possble o nroduce M d ou-phase duplcaed uns. (39) 0 T dh corresponds 8
2 nd Mercosur Congress on Chemcal Engneerng 4 h Mercosur Congress on Process Sysems Engneerng o a consan me, ndependen of he dmenson of he bach o be processed; he second erm s proporonal o he bach h a consan T dh. Ths case of nesed dsuncon s dfferen from he prevous, due o: he nner and ouer dsuncon conss of several erms handled by dfferen oolean varables, he ouer dsuncon for he case of he ser J h does no have an equaon or consran ha apples hen Y h s True, follos he nner dsuncon h s equaons and consrans. The ransformaon of hs embedded dsuncon s as follos: d D hd Sdh V d G d 0 Tdh + Tdh TL M d hd d D var ables can ae J any value beeen bounds h,, h H (37) Y,h H (38) h Y h d D hd,h H, J h (39) Noe ha for hs case he ouer dsuncon dsappears, he reason s ha here no exss an equaon consran applyng hen Y h s True. Transformng (32) y (33) no an algebrac nequales leads: Yh (40) h J h d z hd Y h,h To solve hs problem here are he follong possbles: o conver he model o MINLP by he g-m or Convex Hull relaxaon, and hen applyng a MINLP algorhm Ths problem has been solved by ransformng he GDP formulaon by means of g_m and Convex Hull relaxaons, he ransformaon has been ren n GAMS (rooe e al., 992) and solved by means of DICOPT++ ncluded n he sysem solvers. The resuls obaned are he follong: Table 3. Resuls obaned solvng he synhess of mulproduc recombnan proens plan Model Equaons Varables Dscree Opmal Relaxed CPU me Ieraons Varables Soluon Soluon (sec.) g-m 530 475 94 3,53,944 477,439 5 5 Convex Hull 230 360 94 3,53,944 3,087,54 4 20 (4) 4. Conclusons We have presened hree process sysem engneerng examples nvolvng herarchcal decsons. Those decsons have been formulaed by means of dsuncons, hch faclaes her comprehenson and he vsualzaon of he consrans nvolved n he problem. There s no algorhm ha can solve a problem h nesed dsuncons, herefore s needed a ransformaon of hs ype of dsuncons n a form ha can be reaed by he soluon mehods avalable for hs ype of 9
2 nd Mercosur Congress on Chemcal Engneerng 4 h Mercosur Congress on Process Sysems Engneerng problems. One possble ransformaon s o ranslae he embedded dsuncon no he GDP formulaon here all of hem are n he form of sngle dsuncons (no embedded) of several erms. A general procedure o mae hs ranslaon s he follong: a) Tae ou he nner dsuncons such ha hey become sngle ones. b) Add a erm, hen needed, o he nner dsuncons o allo he possbly ha f none of he oher erms s sasfed because he paren erm s no sasfed. c) Add a logc proposon for he oolean varables handlng he ouer dsuncon represenng he fac ha only one of s erms can be rue. d) Add equvalen logc proposons beeen he oolean varable ha handles he paren erm and he varables handlng he chld dsuncon erms. esdes o hs general procedure, some consderaons mus be aen no accoun, dependng on he problem o be ransformed: The exra erm added n sep b) may no be needed for praccal purposes, snce s covered by he equvalence proposon added n c). The varables nvolved n he nner dsuncon consrans ll ae he value gven by oher consrans, or ll ae a value n he doman defned for he varable. Afer proposng he ransformaons, he problems have been solved horough several algorhms, and he resuls obaned n he soluon have been shon. References rooe, A., Kendrc, D., Meeraus, A. A. (992). GAMS - A User's Gude (Release 2.25). The Scenfc Press. San Francsco, CA. Grossmann I.E., van den Heever S.A. and Harunos I (200), Dscree Opmzaon Mehods and her Role n he Inegraon of Plannng and Schedulng. Proceedngs of Chemcal Process Conrol (CPC-6). Jacson J. and Grossmann I.E. (2002), Hgh Level Opmzaon Model for he rerof plannng of process neors,i&ec Research 4,3762-3770. Lee S. and Grossmann I.E. (2000), Ne algorhm for Nonlnear Generalzed Dsuncve Programmng. Compuers and Chemcal Engneerng,, 24, 9, 225-242. Monagna e. al.(2004), Synhess of oechnologcal Processes usng Generalzed Dsuncve Programmng. Ind. Eng. Chem. Res., 43, 4220-4232. Saaya, N.S. and Grossmann I.E.(2003), A cung plane mehod for solvng lnear generalzed dsuncve programmng problems. Proceedngs of PSE2003, 032. Turay M. and Grossmann I.E.(996), Logc-ased Algorhms for he Opmal Synhess of Process Neors. Comp. Chem. Eng., 20, 8, pp. 959-978. Turay M. and Grossmann I.E.(998), Srucural floshee opmzaon h complex nvesmen cos funcons. Comp. Chem. Eng., 22, 4/5, pp. 673-686. Vecche A. and Grossmann I.E. (999), LOGMIP: A Dsuncve 0- Nonlnear Opmzer for Process Sysem Models. Comp. Chem. Eng., 23, 555-565. Vecche A. and Grossmann I.E.(2000), Modelng ssues and mplemenaon of language for dsuncve programmng. Comp. Chem. Eng., 24, 9, 243-255. 0