A generalized multiobjective particle swarm optimization solver for spreadsheet models: application to water quality

Transcription

1 Hydrology Days 2006 A generalzed multobjectve partcle swarm optmzaton solver for spreadsheet models: applcaton to water qualty Alexandre M. Baltar 1 Water Resources Plannng and Management Dvson, Dept. of Cvl and Envronmental Engneerng, Colorado State Unversty, Fort Collns Darrell G. Fontane 2 Water Resources Plannng and Management Dvson, Dept. of Cvl and Envronmental Engneerng, Colorado State Unversty, Fort Collns Abstract. Ths paper presents an applcaton of an evolutonary optmzaton algorthm for multobjectve analyss of selectve wthdrawal from a thermally stratfed reservor. A multobjectve partcle swarm optmzaton (MOPSO) algorthm s used to fnd nondomnated (Pareto) solutons when mnmzng devatons from outflow water qualty targets of: () temperature; () dssolved oxygen (DO); () total dssolved solds (TDS); and (v) potental of hydrogen (ph). The decson varables are the flows through each port n the selectve wthdrawal structure. The MOPSO algorthm, mplemented as an add-n for Excel, s able to fnd nondomnated solutons for any combnaton of the four abovementoned objectves. An nteractve graphcal method was also developed to dsplay nondomnated solutons n such way that the best compromse solutons can be dentfed for dfferent relatve mportance gven to each objectve. The method allows the decson maker to explore the Pareto set and vsualze not only the best compromse soluton but also sets of solutons that provde smlar compromses. 1. Introducton Decson makng n water resources plannng and management frequently nvolves multple objectves. As greater attenton s beng gven to the envronmental and socal aspects of water resources allocaton and management the need for effectve multobjectve optmzaton approaches s ncreasng. Many of the developments n the area of multobjectve analyss n the Unted States have come from the feld of water resources (Gocoechea et al. 1982). In multobjectve optmzaton, a set of nondomnated solutons s usually produced nstead of a sngle recommended soluton. Accordng to the concept of nondomnance, also referred to Pareto optmalty, a soluton to a multobjectve problem s nondomnated, or Pareto optmal, f no objectve can be mproved wthout worsenng at least one other objectve. Tradtonal multobjectve optmzaton methods attempt to fnd the set of nondomnated solutons usng mathematcal programmng. In the case of nonlnear problems, the weghtng method and the ε-constrant method are the 1 Dept. of Cvl and Envronmental Engneerng, Colorado State Unversty, 1600 W. Plum St. 26H, Fort Collns, CO 80521, Tel: 970/ , abaltar@engr.colostate.edu 2 Dept. of Cvl and Envronmental Engneerng, Colorado State Unversty, B213 Engr. Buldng, Fort Collns, CO 80521, Tel: 970/ , fontane@engr.colostate.edu Hydrology Days

2 Baltar and Fontane most commonly used technques. Both methods transform the multobjectve problem nto a sngle objectve problem whch can be solved usng nonlnear optmzaton. Wth the weghtng method, nondomnated solutons are obtaned f all weghts are postve, however not all Pareto optmal solutons can be found unless all objectve functons as well as the feasble regon are convex. Another dsadvantage of ths method s that many dfferent sets of weghts may produce the same soluton, compromsng the effcency of the method. When the weghts reflect the preferences of the Decson Maker (DM), the method gves the best-compromse soluton,.e. the soluton whch produces the hghest utlty to the DM. The ε-constrant method, on the other hand, does not requre convexty but only leads to nondomnated solutons f certan specfc condtons are satsfed (Mettnen 2001). Accordng to Coello Coello (2001), the frst hnts on the potental of evolutonary algorthms (EA) for multobjectve optmzaton occurred n the 1960s but ths research area remaned largely unexplored untl md 1980s. Ths author also hghlghted two advantages of evolutonary algorthms that make them partcularly sutable for multobjectve optmzaton, when compared to tradtonal mathematcal programmng technques: EA work smultanously wth a set of possble solutons, the so-called populaton, and several nondomnated solutons may be found n a sngle run of the algorthm; EA are less senstve to the shape or contnuty of the Pareto surface. Snce the md 1980s, a growng number of evolutonary multobjectve optmzaton algorthms have been proposed n the lterature (Fonseca and Flemng 1995). Some studes have attempted to compare dfferent algorthms. Coello Coello et al. (2004) compared four dfferent algorthms when appled to fve dfferent test problems. All test problems, however, nvolved only two objectves. Ztzler and Thele (1999) compared another fve methods wth three test problems. Some of the methods compared were later mproved. Partcle swarm optmzaton PSO (Kennedy and Eberhart 1995) s one of the newest technques wthn the famly of evolutonary optmzaton algorthms. The algorthm s based on an analogy wth the choreography of flght of a flock of brds. Due to ts fast convergence, PSO has been advocated to be especally sutable for multobjectve optmzaton (Coello Coello et al. 2004). There are many varants of the sngle objectve PSO but n most of them the movement of the partcles towards the optmum s governed by equatons smlar to the followng: v ( ) ( t ) = w v ( t) + c r ( P ( t) x ( t) ) + c r P ( t) x ( t) (1) ( t + 1 ) = x ( t) + v ( t + 1 x ) (2) Where w s an nerta coeffcent that has an mportant role balancng global (a large value of w) and local search (a small value of w), c 1 and c 2 are g 2

3 A generalzed multobjectve partcle swarm optmzaton solver for spreadsheet models: applcaton to water qualty constants (usually c 1 = c 2 = 2), r 1 and r 2 are unform random numbers n [0,1], P s the best poston vector of partcle so far, P g s the best poston vector of all partcles so far, x (t) s the current poston vector of partcle, and v (t) s the current velocty of partcle. Mendes et al. (2004) suggests an nerta coeffcent w of less than 1, whle other authors recommend to start wth larger values and decrease wth tme, for example from a value of 1.4 to 0.5 (e.g. Elbeltag et al. 2005, Jung and Karney 2006). Coello Coello et al. (2004) hghlghted the senstvty of the standard PSO algorthm to the value of w and proposed the ntroducton of a mutaton operator that assures an adequate global search whle keepng a small value of w (suggested 0.4) whch favors a refned local search. Several applcatons wth evolutonary multobjectve optmzaton have been recently reported n the water resources lterature (e.g. Long et al. 2001, Muleta and Ncklow 2005, Suen et al. 2005, Tang and Reed 2005, Kapelan et al. 2005). None of these applcatons, however, used multobjectve PSO. Jung and Karney (2006) compared the performance of sngle-objectve Genetc Algorthm and PSO approaches to optmze the selecton, szng, and placement of hydraulc devces for transent protecton. The authors studed sx dfferent cases and concluded that both algorthms produced very smlar results n most cases but the PSO found better solutons when the same populaton sze and number of teratons were appled. In ths paper, a modfed verson of the multobjectve PSO (MOPSO) proposed by Coello Coello et al. (2004) s used wth an applcaton to analyze the problem of selectve wthdrawal from thermally stratfed reservors. A graphcal procedure to ncorporate the DM s preferences s also proposed, further explorng the set of nondomnated solutons and dsplayng the bestcompromse soluton as well as famles of solutons wth smlar compromses. 2. Multobjectve Partcle Swarm Optmzaton In the MOPSO algorthm (Coello Coello et al. 2004), the performances of dfferent partcles are always compared n terms of ther domnance relatons. The man characterstc of ths algorthm s the use of an external repostory whch stores nondomnated solutons. The algorthm starts by generatng an ntal populaton. All the partcles of ths populaton are compared to each other and the nondomnated partcles are stored n the repostory. The partcles postons wll be subsequently updated usng the followng: v ( t ) = w v ( t) + c r ( P ( t) x ( t) ) + c r ( R ( t) x ( t) ) (3) Where R h s a soluton selected from the external repostory n each teraton t, and the other terms have already been defned, wth w = 0.4. The best poston vector of partcle, P, s ntally set equal to the ntal poston of partcle. In the subsequent teratons, the best poston vector s updated n the followng way: () f the current P (t) domnates the new poston x (t+1) then P (t+1) = P (t), () f the new poston x (t+1) domnates h 3

4 Baltar and Fontane P (t) then P (t+1) = x (t+1), or () f no one domnates the other then one of them s randomly selected to be the P (t+1). In MOPSO there s no such thng as the best poston vector (P g ) as n the standard PSO. There are several equally good nondomnated solutons stored n the external repostory. To update the velocty of each partcle usng Equaton (3), the algorthm has to select one of the poston vectors stored n the repostory. Ths selecton s made n such a way that nondomnated solutons located n regons more densely populated n the objectve space have lower probabltes of beng selected, therefore leadng to better dstrbutons of ponts n the Pareto front. Instead of usng the adaptve grd proposed n Coello Coello et al. (2004), the approach followed n ths study smply calculates, n the objectve space, the densty of ponts around each soluton stored n the repostory and performs a roulette wheel selecton such that the probablty of choosng one pont s nversely related to ts assocated densty. In every teraton t, the new postons of all partcles are compared among themselves and the nondomnated ones are then compared wth all solutons stored n the repostory. The repostory s then updated, addng new nondomnated solutons and elmnatng old solutons that are now domnated. The sze of the repostory s an mportant parameter to be set. Once the repostory s full and a new nondomnated soluton s found, then ths new soluton takes the place of another nondomnated soluton n the repostory whch s selected randomly usng a smlar procedure based on densty as descrbed above but now assgnng hgher probabltes of beng selected to solutons located n denser regons of the objectve space. The algorthm runs untl the maxmum number of teratons (cycles) s reached. The algorthm handles constrants n a very smple and effcent way. When comparng two dfferent solutons, wth at least one nfeasble, the algorthm does the followng: () one feasble soluton domnates other whch s nfeasble; () wth two nfeasble solutons the one wth smaller volaton of the constrants domnates the other. To mplement ths procedure when several constrants are mposed, an ndex s calculated to reflect the aggregated degree of constrant volaton. 3. An Applcaton to Water Qualty MOPSO s appled to fnd nondomnated solutons for the operaton of a selectve wthdrawal structure. The algorthm s used to determne flows from the varous ports that mnmze the square devatons from outflow water qualty targets of: () temperature, () DO, () TDS, and (v) ph. The model was mplemented n a spreadsheet format usng Mcrosoft Excel and Vsual Basc for Applcatons The select wthdrawal model The selectve wthdrawal (SELECT) model s descrbed n detal n Bohan and Grace (1973), and Fontane and Schneder (1994). Fgure 1 presents a schematc vew of the type of selectve wthdrawal structure modeled n ths applcaton. 4

5 A generalzed multobjectve partcle swarm optmzaton solver for spreadsheet models: applcaton to water qualty SELECTIVE WITHDRAWAL STRUCTURE LEVEL 1 LEVEL 2 LEVEL 3 LEVEL 4 LEVEL 5 LEVEL 5 Fgure 1. Selectve Wthdrawal Structure The SELECT model evaluates the release temperature, DO content, TDS and ph for any combnaton of flows from dfferent ports. The user has to provde, as man nputs, the levels where the ports are located, and the water qualty profles n the reservor for the four mentoned varables. Usng the temperature profle, the model computes the densty profle whch s used to determne the zone of wthdrawal of each port usng Equaton (4). V o Z = A 2 o ρ g Z ρ o Where V o s the average velocty through the port, Z s the vertcal dstance from the elevaton of the port center lne to the lower or upper lmt of the zone of wthdrawal, A o s the area of the port openng, ρ densty dfference between the port center lne and the lower or upper lmt of the zone of wthdrawal, ρ o s the densty at the elevaton of the port center lne, and g s acceleraton due to gravty. Once the zone of wthdrawal s defned, the velocty profle wthn ths zone s determned as a functon of the densty profle and the average velocty. Detals of these calculatons can be found n Bohan and Grace (1973). The water qualty characterstcs of the total outflow are then computed as a flow-weghted average throughout the depth of the reservor usng the velocty profle and the qualty profles. Fontane and Schneder (1994) used a lnear goal programmng approach to fnd the flows that meet release qualty targets as closely as possble. Ths applcaton, however, produced only one soluton for each set of weghts ntroduced by the DM to penalze devatons from the qualty targets. The SELECT model s frst used to fnd qualty characterstcs for each port (4) 5

6 Baltar and Fontane ndvdually. It assumes that the release qualtes wll be lnear combnatons of the qualty characterstcs of each ndvdual port. It solves the lnear goal programmng problem to fnd the optmal flows from the varous ports and then uses the SELECT model agan to recalculate the characterstcs for the new set of flows. The teratve process usually converges n two or three runs of both models. Ths approach s not sutable to nvestgate trade-offs among the water qualty objectves, however. 4. Multobjectve Optmzaton Problem The MOPSO was coded n VBA as a generalzed multobjectve solver that can be mported as an add-n to Excel. The MOPSO Solver allows the user to specfy up to sx objectves to be consdered. The user specfes the cell references for the objectves, for the decson varables, and for the constrants. The solver also ncludes a scrptng tool whch allows the user to code specfc procedures that wll be called mmedately before the objectve functons are evaluated n the MOPSO algorthm. Ths tool can be used to call external programs or perform computatons that cannot be easly made n the spreadsheet. Ths consderably ncreases the flexblty of the MOPSO Solver. The MOPSO Solver Interface s presented n Fgure 2. Fgure 2. MOPSO Solver Interface 6

7 A generalzed multobjectve partcle swarm optmzaton solver for spreadsheet models: applcaton to water qualty The MOPSO Solver also ncludes a procedure to ncorporate more ntellgence n the way the partcles move from one teraton to the next. Wth Equaton (3), a partcle s poston n any teraton s updated usng a randomly defned combnaton of the partcle s ndvdual best prevous poston and the poston of a nondomnated partcle selected from the repostory. In the case of constraned problems, ths combnaton may yeld an nfeasble soluton. The MOPSO Solver has an opton that allows the evaluaton of feasble drectons for all dmensons, adjustng the velocty gven by Equaton (3) so that the partcle s not allowed to make large steps n drectons of ncreasng nfeasblty. Fgure 3 presents the temperature-do trade-off curve generated by usng the MOPSO Solver wth 100 partcles, a repostory sze of 50 solutons, and 50 teratons. The processng tme was 130 seconds n a PC AMD Athlon 64, 2.2 GHz. Ths problem has two objectves and fve decson varables. 4 DO Square Devaton [(mg/l) 2 ] Temperature Square Devaton [( o C) 2 ] Fgure 3. Temperature-Dssolved Oxygen Trade-off Curve Fgure 4 presents a plot of the Pareto surface when mnmzng the square devatons from the targets of temperature, DO, and TDS. The MOPSO processng tme was 660 seconds wth 150 partcles, repostory sze of 150 solutons, and 150 teratons. The MOPSO Solver also allows the user to defne bnary decson varables that can be appled to fnd nondomnated solutons when a restrcton on the maxmum number of ports to be used s desred. For ths problem, the user must specfy fve decson varables representng flows through each port and fve bnary varables that wll determne n each soluton whether that port wll be used. The flow n each port s the product of the flow decson varable and 7

8 Baltar and Fontane ts assocated bnary varable. The user can then nclude a constrant establshng a lmt to the number of ports used, by makng the sum of the bnary varables to be equal to ths lmt. Ths s a very dffcult problem to solve, and nonlnear mathematcal programmng optmzaton methods wll have dffculty fndng Pareto optmal solutons. Temp. Sq. Dev. [( o C) 2 ] DO Square Devaton [(mg/l) 2 ] Fgure 4. Temperature-DO-TDS Trade-off Surface TDS Square Devaton [(mg/l) 2 ] 4.1. Procedure to vsualze and explore the Pareto set A procedure was developed to ncorporate the preferences of the DM as well as to vsualze and explore the Pareto set. Frst, all Pareto solutons are normalzed usng a compromse programmng approach wth a Eucldan norm (L-2 norm). All objectves are placed as vertces equally spaced n a crcumference of dameter 1. Let N be the number of objectves. The frst objectve s arbtrarly assgned to coordnates [0.5,1.0]. The x and y coordnates of the followng objectves are gven by the followng equatons: 2π π = N θ (8) 2 π xob ( k) = xob ( k 1) + cosθ cos + π k ( 2 k 3) θ 2 (9) π yob ( k ) = yob ( k 1) + cosθ sn + π k ( 2 k 3) θ 2 (10) Where N s the number of objectves and k = 2,3,..N. The normalzed metrcs are calculated for each objectve as follows: CP (, k ) 2 [ Best( k ) R(, k) ] [ Best( k ) Worst( k )] 2 = (11) Where CP(,k) s the [0,1] normalzed metrc of partcle for objectve k, and R(,k) s the value of objectve k of partcle n the repostory. 8

9 A generalzed multobjectve partcle swarm optmzaton solver for spreadsheet models: applcaton to water qualty The normalzed metrc for each objectve s then used to calculate a new coordnate measured n the dameter correspondng to that objectve. The new coordnates are gven by: π xcp (, k) = xob ( k) + CP(, k) cos + π k ( 2 k 2) θ 2 (12) π ycp (, k ) = yob ( k) + CP(, k) sn + π k ( 2 k 2) θ 2 (13) Each partcle n the repostory now has a (x,y) coordnate for each of the N objectves. The partcle s then plotted n the centrod defned by these N ponts. A weghted average of the normalzed metrcs s calculated based on weghts gven by the DM. The DM can change the weghts and automatcally see the best-compromse Pareto soluton, as well as other Pareto solutons wth a smlar compromse. Fgure 5 presents the MOPSO Solver nterface wth the temperature-do-tds-ph trade-off graph for the case of equal weghts. Fgure 6 presents the same trade-off graph for two other sets of weghts, one set wth equal weghts for temperature and DO and no mportance to TDS and ph, and another consderng only ph. Temp DO ph TDS Fgure 5. MOPSO Solver Interface: Temperature-DO-TDS-pH Trade-off 9

10 Baltar and Fontane Temp DO ph TDS Temp DO ph TDS Fgure 6. Temperature-DO-TDS-pH Trade-off Graphs 5. Conclusons The multobjectve partcle swarm optmzaton algorthm, appled n a model for operaton of selectve wthdrawal structures, proved to be able to fnd well-dstrbuted nondomnated solutons n the objectve space. Compared to the goal programmng approach of Fontane and Schneder (1994) that only produced sngle compromse solutons based upon an a pror set of weghts, a sngle run of the MOPSO defned the trade-off regons among the four water qualty objectves. Once defned, these trade-off regons can be explored to fnd a number of compromse solutons based upon a posteror set of weghts. The graphcal procedure proposed n ths paper can help decson makers to explore Pareto sets when three or more objectves are consdered. The bestcompromse soluton may be dentfed as well as subsets of the Pareto set that provde smlar compromses. 10

11 A generalzed multobjectve partcle swarm optmzaton solver for spreadsheet models: applcaton to water qualty The MOPSO Solver was found to be very flexble allowng the user to easly conduct many types of analyses. The mplementaton of the MOPSO Solver as an add-n EXCEL greatly enhances the potental applcablty of the approach. Acknowledgments. The frst author of ths paper s a sponsored by CNPq, Brazl. References Bohan, J.P. and Grace, J.L. (1973). Selectve Wthdrawal from Man-Made Lakes. Techncal Report H U.S. Army Engneer Waterways Experment Staton, Vcksburg, MS. Coello Coello, C.A. (2001). A Short Tutoral on Evolutonary Multobjectve Optmzaton. Sprnger-Verlag, Lect. Notes Comp. Sc. 1993, Berln, Germany, Coello Coello, C.A., Puldo, G.T., and Lechuga, M.S. (2004). Handlng Multple Objectves wth Partcle Swarm Optmzaton. IEEE Transactons on Evolutonary Computaton, IEEE, Pscataway, NJ, 8 (3) Elbeltag, E., Hegazy, T., and Grerson, D. (2005). Comparson Among Fve Evolutonary-based Optmzaton Algorthms. Adv. Engneerng Informatcs, Elsever, 19 (1) Fonseca, C.M. and Flemng, P.J. (1995). An Overvew of Evolutonary Algorthms n Multobjectve Optmzaton. Evolut. Comptn., MIT Press, Cambrdge, MA, 3 (1) Fontane, D.G. and Schneder, M.L. (1994). Descrpton of the Applcaton of Goal Programmng for Selectve Wthdrawal Structure Applcaton. Proc. 21 st Annual ASCE Water Resources Plng. and Mgmt. Dvson Conf. (Fontane, D.G. and Tuvel, H.N.,eds.), Denver, CO. Gocoechea, A., Hansen, D.R., and Ducksten, L. (1982). Multobjectve Decson Analyss wth Engneerng and Busness Applcatons. J. Wley & Sons, New York, NY, 519 pp. Jung, B.S. and Karney, B.W. (2006). Hydraulc Optmzaton of Transent Protecton Devces Usng GA and PSO Approaches. J. Water Resour. Plng. and Mgmt., ASCE, Reston, VA, 132 (1) Kapelan, Z.S., Savc, D.A., and Walters, G.A. (2005). Optmal Samplng Desgn Methodologes for Water Dstrbuton Model Calbraton. J. Hydraulc Engrg., ASCE, Reston, VA, 131 (3) Kennedy, J. and Eberhart, R. (1995). Partcle Swarm Optmzaton. Proc. 4 th IEEE Int. Conf. on Neural Networks, IEEE, Pscataway, NJ, Long, S., Khu, S., and Chan, W. (2001). Dervaton of Pareto Front wth Genetc Algorthm and Neural Network. J. Hydrologc Engrg., ASCE, Reston, VA, 6 (1) McMahon, T.A. and Men, R.G. (1988). Rver and Reservor Yeld. Water Resources Publcatons, Lttleton, CO, 368 pp. Mendes, R., Kennedy, J., and Neves, J. (2004). The Fully Informed Partcle Swarm: Smpler, Maybe Better. IEEE Transactons on Evolutonary Computaton, IEEE, Pscataway, NJ, 8 (3)

12 Baltar and Fontane Mettnen, K. (2001). Some Methods for Nonlnear Mult-objectve Optmzaton. Sprnger-Verlag, Lecture Notes n Computer Scence 1993, Berln, Germany, Muleta, M.K. and Ncklow, J.W. (2005). Decson Support for Watershed Management Usng Evolutonary Algorthms. J. Water Resour. Plng. and Mgmt., ASCE, Reston, VA, 131 (1) Suen, J., Eheart, J.W., and Herrcks, E.E. (2005). Integratng Ecologcal Flow Regmes n Water Resources Management Usng Multobjectve Analyss. Proc. ASCE/EWRI World Water and Envmtl. Resour. Cong., ASCE, Reston, VA. Tang, Y. and Reed, P.M. (2005). Multobjectve Tools and Strateges for Calbratng Integrated Models. Proc. ASCE/EWRI World Water and Envmtl. Resour. Cong., ASCE, Reston, VA. Ztzler, E. and Thele, L. (1999). Multobjectve Evolutonary Algorthms: A Comparatve Case Study and the Strength Pareto Approach. IEEE Transactons on Evolutonary Computaton, IEEE, Pscataway, NJ, 3 (4)