DESIGN OF PLASTICS SEPARATION SYSTEMS UNDER UNCERTAINTY BY THE SAMPLE AVERAGE APPROXIMATION METHOD

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1 DEIG OF PLATIC EPARATIO YTE UDER UCERTAITY BY THE APLE AVERAGE APPROXIATIO ETHOD Jng We and atthew J. Realff * chool of Chemcal & Bomolecular Engneerng Georga Insttute of Technology, Atlanta, GA ABTRACT Ths paper s an applcaton of the systematc desgn approach to the plastcs separaton systems synthess under uncertanty. The problem s dstngushed from the tradtonal chemcal dstllaton processes n that, frst there are many separaton mechansms can be used and a combnaton of them s a necessty and second there s a sgnfcant varaton of feed composton leadng to the desgn of a flexble system crucal. The problem s formulated as a stochastc ILP, whch s solved by the sample average approxmaton method proposed prevously by the authors. The method can handle large uncertanty space and determne approprate sample szes needed, therefore mang the systems desgn under uncertanty of realstc sze possble. ITRODUCTIO Plastcs recyclng wll play an ncreasngly mportant role n the desgn of effectve product recovery systems. The synthess of a plastcs separaton flowsheet s smlar to the tradtonal dstllaton problems n terms of the need to determne the optmal separaton sequence and unt parameters. The former s stll challengng manly due to the followng two ssues. Frst, a plastc mx can be separated by dfferent mechansms (for example, by the dfference of densty or polarty or surface tenson). Therefore there are several types of equpment to choose from. oreover, partcles n a batch are not dentcal due to both the varaton of partcle propertes (sze, densty, charge etc.) and the random performance n the separaton unts, such as the dffuson n the settlng process, contacts wth bubbles n the froth flotaton unts and enterng poston n the electrostatc separators. econd, there are many sources of uncertantes, such as feed composton, prces etc. In partcular, the large varaton of the feed composton, whch can have a sgnfcant mpact on the process feasblty and product qualty, s ntrnsc n ths problem because the feed s usually a mx of varous products n varyng fractons. For the frst ssue, a modelng approach has been developed n We and Realff (2003a-c) for many separaton mechansms, whch taes account of the dstrbuton of partcle propertes and other random factors. The approach embedded the nfluences of dstrbuton parameters nto separaton models whch have a unfed form. To approach the second challenge systematcally, the problem s formulated as a stochastc mxed nteger nonlnear program (silp). * Correspondng author: matthew.realff@chbe.gatech.edu

2 v* = mn s. t. x, y,z g j Ε [ f ( x, y, z, θ )] ( x, y, z θ ) x X, θ Θ θ, 0 z Z, y j J { 0, } m (ILP) Where y s a vector of bnary 0- varables denotng the choce of the unts or the exstence of the streams, x s a vector of desgn varables such as unt szes, z s a vector of control/state varables, whch can vary over perods/scenaros, and θ represents a vector of uncertan parameters. The constrant set J ncludes mass balances, unt desgn/operatng models, desgn/operatng specfcatons and some logcal constrants. The expected value s often approxmated through sample average by onte Carlo method (hapro et al 2000), whch transforms the stochastc problem nto a determnstc one. vˆ = mn x, z = s.t. g j ( x, y,z θ ) x X, z Z [ f ( x, y,z, θ )], 0 I ; j J (AA-ILP) The above AA-ILP s solved by the ample Average Approxmaton method (AA) proposed by We and Realff (2004). A major problem n prevous soluton technques s the lac of a method to determne the confdence of the soluton, and the nablty to consder a large number of uncertan parameters due to the computatonal complexty of the problem. Therefore, the sample sze was usually chosen to be suffcently large subject to the problem can stll be solved n a reasonable tme frame. The sample sze could be under or over estmated, resultng n a poor soluton qualty or unnecessarly long computatonal tme. The AA method determnes the sample sze based on the confdence nterval of the optmalty gaps, constructed from solutons of replcates of smaller sample sze problems and a larger sample sze problem. The values of the decson varables are found from the smaller sample sze problems and are fxed n the larger sample sze problems, therefore, the scenaros of the larger sample sze problems are decoupled, whch contrbutes to sgnfcant computatonal savngs. In the next secton, we brefly ntroduce ths algorthm and dscuss some computatonal ssues. Then the followng secton presents ts applcaton to the plastcs separaton problem. The last secton concludes the paper. THE ALGORITH ample average approxmaton method for stochastc convex ILPs The basc dea of the sample average approxmaton method s to construct confdence ntervals of the stochastc bounds. The upper bound of a stochastc program can be found relatvely easly snce any feasble soluton s an upper bound. The confdence nterval for the upper bound s smply

3 ' v ' ± t ', α / 2 ' where, v ' s the objectve value of a -sample sze problem at any feasble soluton, v ' = ' f ' =, and ' ' = ' ( ' ) ' = ( ) 2 f v ' A wdely used lower boundng technque ( a et al., 999; Kleywegt et al 200; orn et al 998) s to tae advantage of the followng fact: E mn f ( x) x X mn E x X [ f ( x) ] Therefore, the average of the mnmums of replcated problems s an unbased estmator of the left hand sde f the batches of samples are..d. Then the confdence nterval for the lower bound s where, v v ± t,, α / 2, =, ( m v ), wth m= (m) v denotng the soluton of the m-th problem wth sample sze, and, ( ( ) ) 2 m = v v,, ( m=, 2,, ). ( ) m= Then, the confdence nterval of the optmalty gap s, ' 0, v + + ' v, t, α / 2 t ', α / 2 ' When the above boundng technques are appled to each LP and ILP of the Outer Approxmaton method (Duran and Grossmann, 986) for ILPs, the confdence nterval of the optmalty gap of each LP and ILP can be found. The sample szes are ncreased f the nterval of any optmalty gap s not suffcently small. The complete algorthm dagram can be found n We and Realff (2004). It has also been proposed that ths rule to ncrease sample szes may not be good because t s not necessary to mae the confdence nterval of a very bad soluton suffcently small. A dfferent rule to ncrease the sample szes, developed n We and Realff (2004), was to guarantee that the confdence ntervals do not overlap. The probablty of cuttng off the optmal nteger soluton was also calculated, whch can be used to adjust the sample szes and other algorthm parameters. onconvexty ssue The nonconvextes of the plastcs separaton problem manly come from the blnear terms of flow rate balances for the unts and the mxers, and also the unt models. For the LP subproblems, a global optmzaton solver, such as BARO (ahnds, ), should be used, otherwse the convergence of onte Carlo samplng can not be guaranteed and confdence nterval of the upper bound

4 s not vald. Unfortunately, only a local optmzaton LP solver, OPT, s the only one avalable to the authors. Therefore, a global soluton can not be guaranteed. The nonconvexty ssue has a more sgnfcantly mpact on the ILP part, snce the lnearzatons may cut off a sgnfcant part of the feasble regon. Approaches to generate vald lower bounds nclude soluton of a convexfed ILP (Lee and Grossmann, 200; Kesavan et al, 2004) or ILP (Tawarmalan and ahnds, 2002, 2004). In ths paper, a heurstc strategy (Vswanathan and Grossmann, 990) wth the am of reducng the effect of nonconvextes on generatng lower bounds n master problems s appled. Ths approach ntroduced slac varables n the master problem whch has the followng form mn Z T g K h ( x,y ) + f ( x, y ) ( x, y ) ( x, y ) + g( x, y ) B = α + s.t. α f y K = x X,y Y, p w p x x y y y,q p + w B 0 q q T x x, =,...,K y y p, =,...,K T x x q, =,...,K y y, =,...,K (ILP-APER) Where w p, w q are weghts that are chosen suffcently large; T ={sgn(λ )} n whch λ s the multpler assocated wth the equaton h (x,y)=0. Lagrangan decomposton The smaller LP problems have the followng form wth the dscrete varables already fxed. mn f ( x,z ) x,z = s.t. g ( x,z ) 0 =, L, x X,z Z The problem has a bloc dagonal structure. olvng such a problem as a whole s tme-consumng and sometmes ntractable f the sample sze, number of constrants and varables are very large. However, the specal structure of the problem can be exploted to decouple the scenaros and solve the problem n an teratve way untl the decson varables x solved at each scenaro converge to the same value. Ths can be done by frst splttng the decson varable x nto varables x (=,, )

5 and then applyng Lagrangan Relaxaton (Fsher, 98; Gugnard and Km, 987) to the copy constrants x = x. Ths s called Lagrangan Decomposton. mn f ( x,z ) x,x,z = s.t. g ( x,z ) 0, =, L, x = x, =, L, x,x X,z Z The Lagrangan of the above problem s L = f ( x,z ) + µ g + π ( x x ) = = = Theoretcally, f the problem s convex and the varables are contnuous, the orgnal problem s equvalent (n terms of the optmal objectve value) to solvng the followng LD problem: max mn f ( x,z ) + π ( x x ) g ( x,z ) 0,x,x X,z Z (LD) π x,x = = However, a dualty gap mght exst n ths case due to the non-convextes of the problem. Therefore the soluton of LD provdes a lower bound to the orgnal problem. Any feasble soluton to the orgnal problem s an upper bound. Typcally a sub-gradent method s used to update the multplers to. CAE TUDY In ths case study, the flow stream to be processed s a mxture of TV and Computer products. The components of a TV or a computer are assumed to be fxed, respectvely. However, the fracton of TVs or Computers s uncertan, whch maes the fracton of each component n the feed stream also a random number. The total feed flow rate s fxed at 3600 g/hr (= g/s) and the fracton of TVs has a normal dstrbuton wth mean 74% and standard devaton 7%. In TVs, HIP s the domnant materal, whle n computers, AB domnates. The data are shown n Table. Feed Components Table Feed components and fractons (APC report, 2000) PE HIP AB PPO PC/AB Total: 3600 g/ hr, TV (ean: 74%; TD 7%) + Computer For TV 5% 75% 8% 2% 0% For Computer 0% 5% 57% 36% 2% The product prces are also uncertan and correlated, whch are assumed to have normal dstrbutons, wth ther respectve mean and standard devaton and the correlaton coeffcent matrx shown n Table 2.

6 Prce ($/g) Table 2 Product prce dstrbuton PE HIP AB PPO PC/AB ean T.D Correlaton Coeffcent The followng superstructure (Fgure a,b) s used, whch conssts of 0 optonal unts (4 sn-float tans (F), 2 froth flotaton tans (FF), 2 free-fall electrostatc separators (FE) and 2 drum separators (DE)). Fgure a Phase of plastcs separaton superstructure

7 Fgure b Phase 2 of plastcs separaton superstructure The problem was solved wth APL CPLEX 8. and OPT on a PC wth 2.53GHz CPU and G memory. All the objectve values reported below have the unts of mllon dollars. tartng wth (00) for the bnary varables, the optmal soluton was reached at the 2 nd teraton ( ) wth an optmal objectve value (the negatve sgn s due to the transformaton of the maxmzaton problem nto a mnmzaton problem). One sn-float tan and two free-fall electrostatc separators are selected. Lagrangan decomposton result The Lagrangan decomposton problems can n general converge wthn 2 teratons wth ntal multpler values 0.0. For example, at the frst teraton, the lower bound generated by Lagrangan decomposton s and the upper bound by the heurstc rule s At the second teraton, the lower bound s and the upper bound s nce the relatve dfference of these two values s wth 2%, the Lagrangan decomposton procedure s consdered to have converged. Therefore, despte of the nonconvextes n the nonlnear problems, the dualty gaps are neglgble. If the ntal multpler values are ncreased to 0., the number of teratons for the convergence wll ncrease to 6. As a heurstc rule to fnd an upper bound, the varables x are fxed at the maxmum of the xˆ values found n step and then the problem s solved agan. Ths heurstc rule can provde a tght upper bound and also the values of the decson varables for the larger LP.

8 Comparson between the uncertan and the average condtons The determnstc case under the average condton (feed fracton and product prces) s also solved and the optmal flowsheet s shown n Fgure 2. Average condton Feed fracton: (0.037, 0.568, 0.207, 0.82, 0.006) Purty requrement: 95% PE FE HIP.0 F AB ddlng 0 PC/AB FE7 0.9 PPO Optmal objectve value= Fgure 2 Optmal flowsheet of the determnstc case wth 95% purty The optmal objectve value s , whch s slghtly better than that of the uncertan case, and the choces of the unts are the same as n the uncertan case (Table 3). There s no dfference between the uncertan and the average condtons n terms of the choce of unts (.e., flowsheet structure) because the varaton n the feed composton has no nfluence on the unt separaton effcences. However, from Table 3 t can be seen that the capacty of the second free-fall electrostatc separator desgned under the average condton s under-desgned. Table 3 Comparson of the uncertan and average condton Under uncertanty Average condton Objectve value Choce of unts ( ) ( ) Capacty of unt Capacty of unt Capacty of unt

9 AA Computatonal effcency and the confdence of the soluton The computatonal tme wth the AA method was 5 hours and 45 mnutes. The computatonal savng s apparent compared wth usng a fxed sample sze of nce at each OA teraton, the former requres solvng 3400 sngle-sze LPs/ILPs (assumng the Lagrangan decomposton converges wth 2 teratons for the frst replcaton and teraton for all other replcatons) and the latter requres solvng 8000 sngle-sze LPs/ILPs (assume the Lagrangan decomposton converges wth 2 teratons). Intutvely one would thn that such a problem wth a 0-dmensonal uncertanty space should requre very large sample szes. However, our calculaton result (Table 4) showed that the problem can be solved wth small sample szes (smaller sample sze =00, number of replcatons =6, and larger sample sze =2000) but stll acheve hgh soluton qualty. The confdence ntervals (99%) of the upper and lower bound at both teratons are less than 0.077, whch s only around 2% of the optmal objectve value. Of course, ths result s problem-specfc and can not be appled to other cases. Table 4 Case study result confdence ntervals of optmalty gaps (wth =00, =6, =2000) Iteraton (00) Iteraton2 ( ) -LP ean of UB Varance of UB L-LP ean of UB Varance of UB ean of UB gap CI of UB gap ILP ean of LB Varance of LB 4.09e-4 4.e-4 L-ILP ean of LB Varance of LB ean of LB gap CI of LB gap COCLUIO Ths paper s the frst systematc approach to solve a plastcs separaton desgn problem under uncertanty. The problem s formulated as a stochastc ILP and solved by the prevously proposed AA method. The method can not only determne the approprate sample szes based on the soluton qualty requrement or vce versa, but also reduce the computatonal tme by decouplng the scenaros usng solutons from multple problems wth smaller sample sze. However, a drawbac of the current AA algorthm s ts nablty to handle nonconvextes. Future research ncludes development of a global optmzaton algorthm for stochastc nonconvex ILPs.

10 REFERECE APC report, 2000, Amercan Plastcs Councl, Plastcs from resdental electroncs recyclng, Duran,.A. and Grossmann I. E. 986, An Outer-Approxmaton algorthm for a class of mxed nteger nonlnear programs, athematcal Programmng, 36, 307. Fsher,.L., 98, The Lagrangean relaxaton method for solvng nteger programmng problems, anagement cence, 27,. Gugnard,. and Km,., 987, Lagrangean decomposton: a model yeldng stronger Lagrangean bounds, athematcal Programmng, 39, 25. Kesavan, P., Allgor, R. J., 2004, Gatze, E P., and Barton, P. I., Outer approxmaton algorthms for separable nonconvex mxed-nteger nonlnear programs, ath Program. Kleywegt, Anton J., Alexander hapro and Tto Homem-De-ello, 200, The ample Average Approxmaton ethod for tochastc Dscrete Optmzaton, IA Journal On Optmzaton, 2, 479. Lee,. and Grossmann, I.E., 200, A global optmzaton algorthm for nonconvex generalzed dsjunctve programmng and applcatons to process systems, Computers and Chemcal Engneerng, 25, a, W.K., orton, D.P. and Wood, R.K.,999, onte Carlo boundng technques for determnng soluton qualty n stochastc programs, Operatonal Research Letters, 24, 47. orn, W. I., Pflug, Georg Ch. and Ruszczyńs, A., 998, A Branch and Bound ethod for tochastc Global Optmzaton, athematcal Programmng, 83, 425. ahnds,.v., , BARO: Branch and Reduce Optmzaton avgator, User's anual, verson 4.0, Tawarmalan, oht and ahnds,.v., 2004, Global optmzaton of mxed nteger nonlnear programs, a theoretcal and computatonal study, ath. Program., 99, 563 Tawarmalan,. and ahnds,. V., 2002, Convexfcaton and Global Optmzaton n Contnuous and xed-integer onlnear Programmng: Theory, Algorthms, oftware, and Applcatons, Kluwer Academc Publshers, Dordrecht hapro, A. and Homem-de-ello, T., 2000, On rate of convergence of onte Carlo approxmatons of stochastc programs, IA J. Optmzaton,, 70. Vswanathan, J. and Grossmann, I.E., 990, A combned penalty functon and outer approxmaton method for ILP optmzaton, Computers and Chemcal Engneerng, 4, 769.

11 We, J. and Realff,.J., 2003 (a), Desgn and optmzaton of free-fall electrostatc separators for plastcs recyclng, AIChE J., 49(2), 338. We, J. and Realff,.J., 2003 (b), Desgn and optmzaton of drum-type electrostatc separators for plastcs recyclng, submtted to Industral & Engneerng Chemstry Research. We, J. and Realff,.J., 2003 (c), A unfed probablstc approach for modelng trajectory-based solds separatons, submtted to AIChE J. We, J. and Realff,.J., 2004, The sample average approxmaton methods for stochastc ILPs, Computers and Chemcal Engneerng, 28(3), 333.