Alternate Approxmaton of Concave Cost Functons for Process Desgn and Supply Chan Optmzaton Problems Dego C. Cafaro * and Ignaco E. Grossmann INTEC (UNL CONICET), Güemes 3450, 3000 Santa Fe, ARGENTINA Department of Chemcal Engneerng, Carnege Mellon Unversty, Pttsburgh, PA 53, U.S.A. ABSTRACT. Ths short note presents an alternate approxmaton of concave cost functons used to reflect economes of scale n process desgn and supply chan optmzaton problems. To approxmate the orgnal concave functon, we propose a logarthmc functon that s exact and has bounded gradents at zero values n contrast to other approxmaton schemes. We llustrate the applcaton and advantages of the proposed approxmaton.. INTRODUCTION For prelmnary calculatons n chemcal process desgn and supply chan strategc plannng problems, the eupment or faclty cost (f(x)) ncreases non-lnearly wth the sze or capacty (x), as a concave functon (Crc and Floudas, 99; Begler et al., 997; Sztka et al., 003). As a result, power law expressons of the form f(x) = c x r wth exponents less than one are usually adopted for capturng the effects of economes of scale. In such optmzaton problems, one of the maor decsons s whether or not to buy/construct a certan eupment/faclty, as well as determnng ts sze or capacty, x (Begler and Grossmann, 004). A maor drawback of the typcal concave cost functon f(x) s that ts dervatve at x = 0 (a feasble value for x) s unbounded, whch causes falures n the Karush-Kuhn- Tucker condtons of the assocated nonlnear program. Common methods for dealng wth such dffcultes are: (a) approxmate the concave functon by a pecewse lnear functon (Geoffron, 977), * Correspondng author. Tel.: +54 34 455 975; fax: +54 34 455 0944. E-mal: dcafaro@f.unl.edu.ar
or (b) add a very small value ε to the varable x, thus slghtly dsplacng the curve towards the negatve values of x. Approxmaton (a) s computatonally costly and rather mprecse unless a fne dscretzaton of the doman s used. Although n prncple approxmaton (b) s reasonable, t has a number of drawbacks, especally f the exponents are small. To overcome such lmtatons, an approxmaton of logarthmc form s proposed n ths short note.. CONCAVE COST FUNCTION AND CLASSICAL APPROXIMATION Gven s the concave cost functon for economes of scale wth the form: f(x) = c x r, where varable x 0 s the sze of the eupment, f(x) s the cost of the eupment of sze x, c > 0 s a constant parameter, and 0 < r < s a real exponent. Ths functon has the property that ts dervatve wth respect to x becomes unbounded when x = 0. An approxmaton that has been used to avod computatonal falures of Non-Lnear Programmng (NLP) and Mxed-Integer Non-Lnear Programmng (MINLP) solvers s to add a small value ε to the x n the functon f(x) (Yee and Grossmann, 990; Ahmetovć and Grossmann, 0), so that: f(x) h(x) = c (x+ε) r. Although ths approxmaton yelds bounded dervatves at x = 0 and a relatvely good estmaton of f(x) when small values of ε are adopted, t has several drawbacks:. The smaller the parameter ε, the more precse the estmaton, but the larger ts dervatve at x = 0, snce: h'(x) = c (x + ε) r-, and h'(0) = c / ε -r. If such dervatves become very large, NLP solvers can lead to falures snce the Karush-Kuhn-Tucker condtons (Bazaara et al., 994; Begler, 00) cannot be satsfed due to ll condtonng.. The functon h(x) at x = 0 s not exactly eual to zero but h(x) = c ε r. If ε s not small enough, the decson not to nstall,.e. x = 0, may ncur a non-neglgble cost, partcularly f r s small. To llustrate some lmtatons wth the approxmaton h(x) wth smaller values of r, consder the smple example presented n Fgure. There are = 8 potental stes for locatng one plant (denoted by X ), and = 9 markets (represented by O ). The plant produces a sngle lud product that s suppled by dedcated ppelnes to the selected markets. The plant capacty s gven, and the fxed and varable charges for the plant nstallaton (α, β ) are ndependent on ts locaton.
Potental Stes for Plant Locaton Markets 00 km 3 Fgure. An llustratve example Fgure. Hypothetc soluton for the example The am of the problem s to determne the optmal locaton for the plant (denoted by the bnary y ) and the amount of product hourly suppled to every market ( ), so as to maxmze the annual benefts: b(y, ) = (pr oc ) (α y + β ) z( ) (sales ncome operaton costs plant nstallaton costs ppelne costs). Snce: (a) the product prce and operaton costs are ndependent of the plant locaton and markets suppled (pr = pr ; oc = oc ), (b) only one plant wll be selected ( y = ), (c) the plant capacty Cap, s gven ( = Cap), and (d) fxed and varable costs for the plant nstallaton are ndependent of the locaton (α = α ; β = β ), t yelds b(y, ) = (pr oc) Cap α β Cap z( ), and the only varable terms n the obectve functon are ppelne costs z( ). The ppelne flow (eual to the varable ) s proportonal to the ppelne secton,.e. = K d, where d (m) s the ppelne dameter and K has a value of 4,39 m/h (π / 4 3600 s/h.5 m/s). For smplcty, ppelne dameters are treated as contnuous varables. Ppelne nstallaton costs follow an economy of scale functon of the form: z(l,d ) = K L d 0.60, where L (km) s the dstance between and (a gven parameter) and K =,3,500 $ km - m -0.60. Thus, the MINLP model s as follows: Mn S. t. z = J I, d I J K y = 0 L d = Cap y 0.60 I = K d y {0,} I, J Dem I, J ()
By substtutng for d n the obectve functon wth the ppelne flow euaton n the constrants,.e. d = ( / K ) 0.50, we obtan: Mn S. t. z = J I 0 I, J y =, f ( = Cap y ) = y {0,} I, J ( K I Dem / K 0.30 ) L I, J 0.30 () Note that the exponents of n the non-lnear terms of the obectve functon are only 0.30. Assume that the optmal soluton s the one depcted n Fgure, where y =,, = 75 m 3 /h for =,, 3; d, = 0.03 m (8 nches) for =,, 3; whle all the other varables take a zero value. Usng the ε-approxmaton of f( ) wth a reasonable value for ε = 0.0, the cost of the selected ppelnes wll be: h(, ) = h(,3 ) = 9,440 70.7 (75+0.0) 0.30 = 30.77936 MM$ ; h(, ) = 9,440 50 (75+0.0) 0.30 =.7645 MM$; whch s ute close to the actual values: f(, ) = f(,3 ) = 9,440 70.7 75 0.30 = 30.77884 MM$; f(, ) = 9,440 50 75 0.30 =.7643 MM$. However, for all the non-selected ppelnes featurng = 0 (totalng 69 non-used arcs -) the approxmate nstallaton cost s h( ) = h(0) = 9,440 L (0+0.0) 0.30 = 3,0 L. Summng the lengths of the non-selected ppelnes (9,03 km) yelds a total of 09.794 MM$ nstead of zero! In fact, the total ppelne cost n the optmal soluton s f( ) = 83.38 MM$, whle the approxmaton wth ε = 0.0 results n the ncorrect value of h( ) = 30.77936 +.7645 + 09.7906 = 93.0457 (5 % error!). If we try a very small value for ε, say ε = 0-9, ths results n h( ) = 84.98768 ( % error). However, the dervatves of every term h( ) at = 0 ncrease to h'(0) =.844 0 L (over 9.0 0 ),.e. an unacceptably large value for NLP solvers. The new approxmaton proposed n the next secton s ntended to overcome such lmtatons, especally for concave cost functons wth r < 0.5. 3. LOGARITHMIC APPROXIMATION OF THE CONCAVE COST FUNCTION We propose the followng approxmaton functon g(x) for f(x): f(x) = c x r g(x) = k ln(bx + ), where x s the sze of the eupment, f(x) s the actual cost of the eupment of sze x, g(x) s the
estmated cost, and k, b > 0 are real numbers selected to ft f(x) as closely as possble. The proposed functon has two man advantages:. The cost of x = 0 s exactly zero: g(0) = k ln(b 0 +) = k ln() = 0.. The dervatves of g(x) for all x 0 are postve (bounded) values, gven by g'(x) = bk / (bx + ). In partcular at the orgn (x = 0), g'(x) = bk. In order to fnd approprate values for b and k, a smple approach s to select two non-zero values for the varable x (0 < x < x ) and solve the followng system of non-lnear euatons: c x c x r r = k ln( b x = k ln( b x + ) + ) (3) where k and b are the two values to be determned. Snce c, r, x, x > 0, we can dvde both expressons to obtan: (x / x ) r = ln(bx + ) / ln(bx + ), whch n turn can be rearranged as: x r ln(bx + ) x r ln(bx + ) = 0. The mplct euaton for varable b can be easly solved, for nstance, usng Newton s method, to fnd the value of b and by extenson the value of k that satsfes: f(x ) = g(x ) and f(x ) = g(x ). Let us reconsder the llustratve example, where f( ) = 9,440 L 0.30, wth 0 55. Snce the frst two factors are gven, we analyze φ( ) = 0.30, and followng the procedure explaned above, we can formulate the followng system of euatons: 0.30 0.30 = k ln( b = k ln( b + ) + ) (4) Regardng the values of and, we can make the selecton based on our knowledge on the problem. Suppose that f a ppelne s nstalled, t s unlkely to supply less than lo = 50 m 3 /h, whle up = 55 m 3 /h (the plant capacty) s the maxmum possble flow. Then, we may smply set = 50 and = 55 to obtan: b = 0.5885; k =.4763, whch leads to φ() = 0.30 γ() =.4763 ln(0.5885 + ). Alternatvely, less extreme values (for nstance, = 45, = 430), may yeld better approxmatons (γ() =.64756 ln(0.095853 + )). Fgure 3 shows the comparson of the two
proposed approxmatons wth the actual functon, as well as the values of the dervatves at the orgn, whch n fact are rather small. Fgure 3. Actual concave functon (φ) and logarthmc approxmatons (γ) By determnng the absolute and relatve errors of the latter opton (see Fgure 4) t can be seen that close to the orgn the estmatons are not very accurate as expected, but n the orgn the estmaton s exact. Relatve errors are below % n the set x {0} [05 ; 55]. Fgure 4. Absolute and relatve errors of the approxmaton γ() =.64756 ln(0.095853 + ) Revstng the optmal soluton assumed for the problem (y =,, = 75 m 3 /h for =,, 3; d, = 0.03 m (8 ) for =,, 3; and all the other varables wth a zero value), the estmated cost of the selected ppelnes s: g(, ) = g(,3 ) = 9,440 70.7 [.64756 ln(0.095853 75 + )] = 30.99099 MM$; g(, ) = 9,440 50 [.64756 ln(0.095853 75 + )] =.945 MM$; whch
s close to the actual values: f(, ) = f(,3 ) = 9,440 70.7 75 0.30 = 30.77884 MM$; f(, ) = 9,440 50 75 0.30 =.7643 MM$ (only 0.6893 % error!). Perhaps even more mportant, all the ppelnes that are not selected ( = 0) feature the exact value of g( ) = g(0) = f(0) = 0. Therefore, the total ppelne nstallaton cost ( f( ) = 83.38 MM$) s very close to that one found through the proposed approxmaton, gven by: g( ) = 30.99099 +.945 = 83.8963 MM$ (0.6893 % error). Other approaches for parameter estmaton Another approach for estmatng parameters k and b n the proposed functon g(x) = k ln(bx +) s to consder a set of n representatve values for varable x (x, x,, x n ) together wth the actual values of the functon f(x) at that ponts (f, f,, f n ) and solve a least suares NLP problem of the form: Mn k, b S. t. z = ( g f) = (5) g = k ln( b x + ) =... n n Note that the values of the functon f(x) at the reference ponts can be obtaned ether from the orgnal expresson f(x) = c x r, or from the real-world costs of eupments of sze x. For the llustratve example, we propose the representatve values for and φ() presented n Table. The NLP model proposed n (5) s solved to global optmalty n 0.6 CPU s and 5 teratons usng GAMS/BARON 9.0.6 (n an AMD Phenom Dual Core Processor at.90 GHz), settng 0 and 00 as lower and upper bounds for k and b. The optmal results yeld b = 0.430; k =.48658, wth the total sum of suare errors eual to 0.059, and the error dstrbuton shown n Table. By mnmzng the suared errors, the approxmate functon matches the actual functon at lower values ( = 75). In ths way, the error at lo = 50 s bounded more tghtly. Table. Representatve values for and φ(), and approxmaton errors of γ() usng least suares 50 50 50 350 450 55 φ () = 0.30 3.3364 4.4960 5.406 5.7973 6.5 6.54708 e = γ () - φ () -0. 0. 0.0 0.04-0.045-0.5
Alternatvely, model (5) can be solved usng -norm for the devatons between the approxmate and the actual values. Ths leads to b = 0.0964; k =.66748, yeldng the best approxmaton for the orgnal problem wth a total error of 0.5 % n the ppelne costs. 4. COMPUTATIONAL RESULTS The proposed approxmaton has been mplemented n the obectve functon of an MINLP model for optmzng the desgn and development of the shale gas supply chan (Cafaro and Grossmann, 03) wth very promsng results, partcularly when appled to the estmaton of gas and lud ppelne costs, whose economes of scale exponents (wth regards to the flud flows) are typcally 0.5 and 0.300, respectvely. Very good results are also obtaned n MINLP models of chemcal process desgn problems, lke the heat exchanger network synthess (Yee and Grossmann, 990) and the optmal desgn of process water networks (Ahmetovć and Grossmann, 0). In both cases, economes of scale functons wth larger exponents (0.60-0.70) are effectvely handled, always fndng the optmal soluton n short CPU tmes. In fact, the proposed approxmaton dffers less than 0.70 % from the actual eupment costs. In partcular, the heat exchanger network synthess problem, SYNHEAT, contrbuted by T. F. Yee to the GAMS model lbrary (McCarl, 0) s studed. By assumng that the exponent aexp s eual to 0.60 and mplementng the proposed logarthmc approxmaton, the global optmum s found n 0.7 CPUs by solvng GAMS/DICOPTx-C. On the contrary, when applyng the ε-approxmaton wth ε = 0-6, DICOPT can only fnd a suboptmal soluton that s 6.8 % worse, startng from the same ntal pont gven by default (see Table ). If the value of ε s ncreased to 0 -, an mproved suboptmal soluton s found, and the CPU tme ncreases by a factor of. When no approxmaton n the obectve functon s mplemented and the outer-approxmaton algorthm (DICOPT) s used, the optmal solutons of the MILP steps n teratons and are 3.3 0 3 and 6.3 0 3, respectvely. These are very large numbers that reflect the unbounded gradents of exchangers wth zero sze. By chance, even under these crcumstances, the MILP model fnds an nteger
soluton that s solved n the NLP step yeldng a value that s exactly the global optmum. However, t can be concluded that usng drectly the concave functons n the algorthm s not relable. On the other hand, from Table we can see that wth the proposed logarthmc approxmaton values wthn 0.60 % of the global optmum are found wth DICOPT and BARON. Table. Computatonal results for the heat exchanger network synthess problem (SYNHEAT) MINLP Solver Approxmate Soluton Actual Soluton Approxmaton Error Devaton from the Optmum CPUs no-approx DICOPT 0,70 0,70 0 % 0 % 0.7 BARON 0,70 0,70 0 % 0 % 348.6 ε-approx DICOPT 7,69 7,69.6 0-4 % 6.8 % 0.96 ε = 0-6 BARON 0,70 0,70 3. 0-4 % 0 % 30.68 ε-approx DICOPT 5,757 5,670 0.08 % 5.0 % 0.5 ε = 0 - BARON 0,57 0,70 0.08 % 0 % 7.68 log-approx DICOPT 09,53 0,70 0.58 % 0 % 0.7 BARON 09,53 0,70 0.58 % 0 % 77.07 5. CONCLUSIONS An alternate approxmaton of concave cost functons that captures economes of scale n process desgn and supply chan optmzaton problems has been presented. The proposed logarthmc expresson s very smple, fts ute well to the orgnal power law functons, and overcomes the drawbacks of large dervatves and large estmaton errors that are experenced wth small exponents usng approxmatons that add the tolerance ε. Promsng results are obtaned when applyng the approach to well-known process desgn problems and real-sze case studes related to the strategc plannng of natural gas supply chans. The proposed approxmaton s partcularly useful when large superstructures and low exponents (as for ppelne costs) are consdered.
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