One-Stage Optimization Problem with Chance Constraints

Transcription

1 One-Stage Optmzaton Problem wth Chance Constrants Gennady Ostrovsky *, Nadr Zyatdnov, Tatyana Lapteva Kazan State Technologcal Unversty, Karl Marx str., 65, Kazan, , Russa The one-stage optmzaton problem wth chance constrants s consdered. Ths problem s aroused by chemcal processes desgn under uncertantes. We develop methods of solvng the one-stage optmzaton problem wth chance constrants based on transformaton of a problem wth chance constrants nto a problem wth determnstc constrants. 1. Introducton We consder a problem of a chemcal process (CP) desgn n the case of the presence of uncertanty n the process models. Durng chemcal process desgn some desgn specfcatons (constrants) must be satsfed. Some of these are as follows: ) The specfcatons connected wth safety of the CP. ) Ecologcal specfcatons are the second type of specfcatons. The CP must be envronmentally bengn. One way to acheve ths s to mpose constrants on the maxmum effluent flowrates of hazardous chemcals. ) Performance targets n the CP (such as converson n a catalytc reactor) are desgn specfcatons. CP performance s estmated by some measure whch ncludes the captal and operatng costs. It can be a cost, revenue, product yeld. Our goal s to mnmze the cost of CP, whch conssts of the captal and operatng costs (that nclude energy expendtures). The satsfacton of the desgn specfcatons s complcated by the presence of an uncertanty n the process models. Therefore, n process desgn we are forced to use nexact process models. Thus durng CP desgn we must solve the followng problem: t s necessary to fnd such values of the desgn and control varables under whch some measure of the CP performance takes the mnmal or maxmal value and the desgn specfcatons are met wth some probablty. The mathematcal formulaton of ths problem s of the form mn Eθ { f( d, z, θ )} (1) d, z Pr{ g ( d, z, θ) 0} α, 1,, m, (2)

2 where d s a vector of the desgn varables, z s a vector of the control varables, θ s p -vector of uncertan parameters, f ( dzθ,, ) s a goal functon Pr{ g ( d, z, θ ) 0} ρ( θ) dθ Ω Ω { θ : g ( d, z, θ) 0, θ T}. (3) Pr{ g ( x, θ) 0} s the probablty measure of the regon Ω and α s a probablty level. Problem (1) s the problem of fndng a trade-off between the problems of mnmzaton of CP cost and mnmzaton of envronment polluton. Indeed, f we ncrease the values α (the probablty of satsfacton of the constrants) we dmnsh possble effluent flowrates of hazardous chemcals. However, one can prove that an ncrease of the values α worsens the optmal value of the obectve functon (partcularly, energy expendtures).e. ncreases the prce of CP and vce versa. Problem (1) s the one-stage optmzaton problem (OSOP) wth chance constrants. The man ssue n solvng the one-stage optmzaton problems s the calculaton of multple ntegrals that gve expected value of the obectve functon and probablty of constrants satsfacton. The use of the standard Gaussan quadrature for the calculaton of multple ntegrals s very ntensve computatonally even for small dmensonalty of vector θ of the uncertan parameters. In connecton wth ths the problem of smplfcaton of a multdmensonal ntegrals calculaton s very mportant. One can note three groups of methods n whch ths problem s consdered. The methods whch mprove the Gauss quadrature belong to the frst group (Bernardo et al, 1999,Pntarc and Kravana, 2004, Wey and Realff, 2004). The methods n whch the samplng technques (Monte-Carlo, Latn Hypercube or Hammersley Sequence Samplng - HSS) (Bernardo and Sarava, 1998, Dwekar and Kalagnanam, 1997) are used belong to the second group. Dwekar and Kalagnanam (1997) have shown that the HSS technque s more effcent than other samplng technques. Unfortunately, even the HSS technque requres several hundred approxmaton ponts to obtan a reasonable accuracy. Methods permttng to transform chance constrants nto determnstc ones belong to the thrd group. L et al. (2008) developed method permttng to transform chance constrants nto determnstc ones. It s based on the use of a monotone relatonshp between a constraned output and one of the uncertan parameters. Unfortunately, t s very dffcult to prove that ths property holds n process models of real processes. Besdes, some process models do not have ths property. We wll develop a new method based on the approxmate transformaton of the chance constrants nto determnstc ones. Ths permts to dmnsh sgnfcantly computatonal tme of solvng OSOP wth chance constrants. 2. Approxmate method of solvng the TSOP wth chance constrants Transform problem (1). Consder some regon, whch has the followng property Pr{ θ } α (4)

3 Condton (4) means that the probablty measure of the regon s equal to α. It should be noted that, generally speakng, condton (4) determnes not sngle regon but some nfnte set of regons. Desgnate ths set by Τ α. Consder the followng constrant max g ( d, z, θ ) 0, (5) where s one of the regons from the set Τ α constrant g ( d, z, θ ) 0 s met at each pont of the regon. If constrant (5) s met then the. Snce a probablty measure of the regon s at least α then the probablty of satsfacton of the constrant g ( d, z, θ ) 0 s at least α. Consequently, constrant (2) can be substtuted wth the constrants (4), (5). Hence, problem (1) can be rewrtten n the followng form f * mn E[ f( d, z, θ )] (6) d, z, Tα max g ( d, z, θ ) 0, 1,, m Pr{ θ } α, (7) Unfortunately t s very dffcult to look for the optmal form and locaton of the regon. Therefore, we wll restrct the class of possble regons T α and look for the optmal regon n the form of a multdmensonal rectangle. In connecton wth ths we wll consder two cases n whch form of regons wll be gven. Case 1: The uncertan parameters θ, 1,..., p, are ndependent, random varables havng the normal dstrbuton N [ [ ], ] 1 E θ σ where E[ θ ] s the expected value of the parameters θ. In ths case we wll suppose that the regons have a form of a multdmensonal rectangle Tα θ θ θ θ p, (8) L, U, { :, 1,..., } L, U, where θ, θ are upper and lower bounds of sdes of the multdmensonal rectangles L U. In ths case the uncertanty regon has the form (8), where the values θ, θ, L U 1,..., p, have the followng form: θ E[ θ] kσ, θ E[ θ] + kσ and k are some large enough coeffcents. In ths case the search of the optmal forms and locatons of the regons s reduced to the search of the optmal upper and lower bounds L, U, θ, θ of the sdes of the multdmensonal rectangles. Snce all the parameters θ are ndependent and have the normal dstrbuton then the probablty

4 measure of the multdmensonal rectangle s equal to multplcaton of the L, U, probablty measures of the ntervals I { θ θ θ }. Thus, we have p U, L, Pr{ θ } [ Φ( θ ) ( )] 1 Φ θ, (9) where Φ ( η) s the standard normal dstrbuton functon and followng form θ, L, θ have the U, θ ( θ E[ θ ]) σ,, 1, U U θ ( θ E[ θ ]) σ. L, L, 1 Replace Pr{ θ } n condton (7) by ts expresson from (9). Then problem (6) takes the form f mn E [ f ( d, z, θ )], (10) L, U, d, z, θ, θ max g ( d, z, θ ) 0, 1,, m, (11) [ Φ( θ ) Φ( θ )] [ Φ( θ ) Φ( θ )] α, 1,, m, U, L, U, L, 1 1 p p where the regons are defned by the formulae (8). Problem (10) s the sem-nfnte programmng problems. For ts solvng one can use the outer approxmaton algorthm (Hettch and Kortanek, 1993). For approxmate calculaton of the expected value of the functon f ( d, z, θ ) we wll use the teraton procedure based on partton of the regon T. Let at the k -th teraton the regon T consst of Q k subregons T ( k) q, q 1,..., Qk. For an approxmate calculaton of the expected value of the functon f( d, z, θ ) we wll use pecewse lnear approxmaton of the functon f ( d, z, θ ). For ths at each subregon T ( k ) q we wll replace the functon f( d, z, θ ) by ts lnear approxmaton q q p q q f ( d, z, θθ, ) f( d, z, θ ) + ( f( d, z, θ ) θ)( ) 1 θ θ. (12) q ( k) where θ T q. Then one can show that some approxmaton Eap[ f( d, z, θ ); T] of the expected value of the functon f ( d, z, θ ) can be represented n the followng form (13) Qk q p q ( k) q ap[ (,, θ); ] ( (,, ) ( (,, ) )( [ ; ] )) 1 q θ + θ θ q 1 θ q qθ E f d z T a f d z f d z E T a where a ρθ ( ) dθ [ Φ( θ ) Φ( θ )][ Φ( θ ) Φ( θ )] [ Φ( θ ) Φ( θ )], U, q L, q U, q L, q U, q L, q q ( k ) Tq p p

5 Ur, ( k) 1 p q ( k) θ 1 L, r T 1 q θ +. E [ θ; T ] θρθ ( ) d θ I ( θρθ ( ) d θ) I where I [ Φ( θ ) Φ( θ )] [ Φ( θ ) Φ( θ )], 1 U, r L, r U, r L, r I [ Φ( θ ) Φ( θ )] [ Φ( θ ) Φ( θ )]. p U, r L, r U, r L, r p p On the bass of above consderatons we developed the teraton procedure of solvng the OSOP wth chance constrants. At each teraton we solve optmzaton problem (10) n whch the value E[ f( d, z, θ )] s replaced by the Eap[ f( d, z, θ )] (see (13)). For solvng problem (10) we can use the outer approxmatons method (Hettch and Kortanek, 1993). Thus, t s seen that the consdered method does not requre the calculaton of multple ntegrals. Case 2: The uncertan parameters θ, 1,..., p, are dependent random varables havng multvarate normal dstrbuton Np[ E[ θ], Λ ], where Λ s a covarance matrx. It s known that the random varable y where T 1 y ( θ μ) Λ ( θ μ) (where μ s p -vector wth components μ E[ θ], 1,..., p) has dstrbuton wth p degrees of freedom. Take the regons n the followng forms T 1 Tα { θ :( θ μ) Λ ( θ μ) C( α )}, 1,, m, (14) where χ 2 ( C( α)) α. In ths case, the probablty measure of the regon s equal to α. Consequently, only one regon s reduced to the followng problem f * d, z 2 χ satsfes constrant (7). Therefore, problem (6) mn E[ f( d, z, θ )], (15) max g ( d, z, θ ) 0, 1,, m. Problem (15) s the sem-nfnte programmng problems. For ts solvng one can use the outer approxmaton algorthm (Hettch and Kortanek, 1993). 3. Computatonal experment As an llustraton we consder a problem of a desgn of a chemcal process (CP) consstng from a contnuous strred tank reactor and a heat exchanger. There s the detaled descrpton of ths CP n (Halemane and Grossmann, 1983). Ths CP has two desgn varables a reactor volume V and a heat exchanger area A, two control varables the reacton temperature T 1 and the temperature T w2 of a cold water on the

6 output of the heat exchanger and fve uncertan parameters. The obectve functon f takes nto account the captal and operatng (energetc) expendtures. We solved the nomnal optmzaton problem (NOP) and the OSOP for α 0.5, 1,, m. The values of V, A and f obtaned by solvng NOP and OSOP are equal to 5.42, 5.21, 9003 and 7.40, 5.48, It s seen that n order to guarantee of satsfyng the constrants wth the gven probablty 0.5 we must ncrease the reactor volume and the heat exchanger area from 5.42, 5.21 up to 7.40, 5.48, respectvely. Ths ncreases a cost of CP by 11%. It s nterestng that f we use the straghtforward way of solvng problem (1) then the total CPU-tme requred only for the computaton of multple ntegrals used to determne the values of obectve functon (1) and constrants (2) s equal to approxmately 40 mnutes (usng the Monte Carlo method from the software package Mathematca ). At the same tme, the total CPU-tme requred to solve problem (10) wth help of our approach s equal to 3.9 s. 4. Concluson We have developed the new approach to solvng the chance constraned one-stage optmzaton problem for the case of normally dstrbuted uncertan parameters. Ths approach s based on approxmate transformaton of chance constrants nto determnstc ones and a pece-wse lnear approxmaton of the orgnal obectve functon. Partton of the uncertanty regon nto subregons s used to mprove these approxmate transformatons. The developed approach permts to solve one-stage optmzaton problems wth chance constrants n the case when the uncertan parameters are normally dstrbuted random varables wthout computatonally ntensve calculaton of multvarate ntegrals. References Bernardo F.P., Pstkopoulos E.N. and Sarava P.M., 1999, Integraton and computatonal ssues n stochastc desgn and plannng optmzaton problems, Industral and Engneerng Chemstry Research, 38, Dwekar U.M. and Kalagnanam J.R., 1997, An effcent samplng technque for optmzaton under uncertanty, AIChE Journal, 43, Halemane K.P. and Grossmann I.E., 1983, Optmal Process Desgn under Uncertanty, AIChE Journal, 29, Hettch R. and Kortanek K.O., 1993, Sem-nfnte programmng: Theory, methods and applcatons, SIAM Revew, 35, L P., Arellano-Garca H. and Wozny G., 2008, Chance constraned programmng approach to process optmzaton under uncertanty, Computers and Chemcal Engneerng, 32, Pntarc Z.N. and Kravana Z., 2004, A strategy for MINLP synthess of flexble and operable processes, Computers and Chemcal Engneerng, 28, 6-7, We J., Realff M.J., 2004, Sample average approxmaton methods for stochastc MINLPs, Computers and Chemcal Engneerng, 28, 3,