Developing Four Metaheuristic Algorithms for Multiple-Objective Management of Groundwater

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1 Journal of Soft Computng n Cvl Engneerng -4 (018) 01- Contents lsts avalable at SCCE Journal of Soft Computng n Cvl Engneerng Journal homepage: Developng Four Metaheurstc Algorthms for Multple-Objectve Management of Groundwater H.A. El-Ghandour 1 * and E. Elbeltag 1. Assocate Professor, Irrgaton & Hydraulcs Dept., Fac. of Engrg., Mansoura Unv., Mansoura 35516, Egypt.. Professor, Structural Engneerng Dept., Fac. of Engrg., Mansoura Unv., Mansoura 35516, Egypt. Correspondng author: hamdy_a@mans.edu.eg ARTICLE INFO Artcle hstory: Receved: 3 Aprl 018 Revsed: 19 June 018 Accepted: 0 June 018 Keywords: Genetc algorthms, Memetc algorthms, Partcle swarm, Shuffled frog leapng, Compromse soluton, Multple objectves optmzaton. ABSTRACT Groundwater s one of the mportant sources of freshwater and accordngly, there s a need for optmzng ts usage. In ths paper, four mult-objectve metaheurstc algorthms wth new evoluton strategy are ntroduced and compared for the optmal management of groundwater namely: Multobjectve genetc algorthms (MOGA), mult-objectve memetc algorthms (MOMA), mult-objectve partcle swarm optmzaton (MOPSO), and mult-objectve shuffled frog leapng algorthm (MOSFLA). The suggested evoluton process s based on determnng a unque soluton of the Pareto solutons called the Pareto-compromse (PC) soluton. The advantages of the current development stem from: 1) The new multple objectves evoluton strategy s nspred from the sngle objectve optmzaton, where ftness calculatons depend on trackng the PC soluton only through the search hstory; ) a comparson among the performance of the four algorthms s ntroduced. The development of each algorthm s brefly presented. A comparson study s carred out among the formulaton and the results of the four algorthms. The developed four algorthms are tested on two multple-objectve optmzaton benchmark problems. The four algorthms are then used to optmze two-objectve groundwater management problem. The results prove the ablty of the developed algorthms to accurately fnd the Pareto-optmal solutons and thus the potental applcaton on real-lfe groundwater management problems. SCCE s an open access journal under the CC BY lcense (

2 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) Introducton Groundwater s consdered as one of the freshwater sources needed to serve human actvtes, such as drnkng, ndustral, domestc and rrgaton. Due to rregular pumpng of groundwater, serous envronmental hazards have been occurred ncludng groundwater levels declne and nterference among wells. To pump the maxmum amount of groundwater wthout aqufer depleton and achevng mnmum operaton cost, powerful smulaton-optmzaton models have to be appled to obtan the best strategy. These models are adopted to deal wth both the desgn and operaton problems correspondng to remedaton, water supply and controllng of groundwater flow [1]. Optmzaton models can be classfed accordng to ther objectves nto sngle objectve and multple objectves. In sngle objectve optmzaton, a unque optmal soluton s produced related to the maxmzaton or mnmzaton of a sngle objectve [1 19]. In multple objectves optmzaton, a group of non-domnated solutons, called Pareto-front (PF), are usually produced [0 5]. Optmzaton models are classfed nto determnstc and metaheurstc. Determnstc methods nclude lnear, non-lnear and dynamc programmng [14 17, 6 33]. Whle, metaheurstc algorthms are stochastc optmzaton technques that mmc nature percepton, learnng and reasonng such as: Genetc Algorthms (GA), Memetc Algorthms (MA), Partcle Swarm (PSO) and Shuffled Frog Leapng (SFLA) [1 13, 18 5]. Determnstc optmzaton methods tend to become overly complcated and requre massve computatons when dealng wth complex optmzaton problems wth numerous constrants and decson varables, such as real-lfe water related optmzaton problems. Whle, metaheurstc optmzaton algorthms are usually used due to ther drect gettng solutons wthout the need for gradents and ntal solutons. Smulaton models are coupled wth optmzaton technques to evaluate proposed objectve functon(s). Smulaton models apply ether numercal or analytcal approaches to smulate groundwater flow and calculate the hydraulc heads of the studed aqufer. Numercal approaches nclude fnte element (FEM), fnte dfference and boundary element methods whle, analytcal approaches use analytcal element method and Fourer seres [7 8, 10]. The development made n the current paper s dfferent from prevous works n many aspects. The use of a new evoluton strategy for generatng the Pareto optmal solutons n four dfferent mult-objectves algorthms namely: MOGA, MOMA, MOPSO and MOSFLA. Usng such evoluton strategy resembles solvng sngle optmzaton problems. Performng a comparson study among the four mult-objectves optmzaton algorthms for optmzng ground water management problems. Whle, all prevous comparsons have been performed among sngle objectve optmzaton algorthms [34 35].. New evoluton strategy Multple objectves optmzaton problems are characterzed by the exstence of many optmal solutons (non-domnated) called Pareto-Front (PF). Many dfferent algorthms are used to solve

3 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) 01-3 mult-objectves optmzaton problems. Grerson [36] proposed a methodology to determne a sngle Pareto-Compromse (PC) soluton located n the PF whch farly satsfes all objectves. Ths research adopts such PC soluton for the ftness calculatons and then drvng the evoluton process n MOGA, MOMA, MOPSO, and MOSFLA algorthms. The proposed new evolutonary strategy whch s based on fndng the PC soluton s adopted due to ther proven effectveness n terms of the processng tme and number of Pareto solutons obtaned n the PF, as stated by Elbeltag et al. [37]. Also, the proposed mult-objectve evolutonary strategy s remnscent of sngle-objectve optmzaton, where ts ftness assgnment s based on trackng a sngle evolvng soluton over the search hstory. In addton, knowng the PC soluton helps decson makers to choose a sngle soluton that unquely represents a mutually agreeable trade-off between all competng objectves for the problem. The suggested methodology for developng each algorthm to handle multple objectve problems depends manly on the prncple of non-domnant solutons located n more densely populated regons n the objectve space havng lower probabltes to be selected. Consequently, better dstrbutons of optmal solutons can occur n the PF. The followng steps llustrate the suggested methodology: d 1. All solutons n an evolutonary cycle are evaluated accordng to the values of ther objectves to determne a set of non-domnated solutons called PF. Each non-domnated soluton belongs to the PF can beat all other solutons n at least one objectve.. For each obtaned PF, the PC soluton s determned mathematcally as gven by Grerson [36]. 3. The dstance ( d PC, k ) between the PC and each Pareto optmal soluton (k) s calculated based on Eq. 1: PC k n f PC fk, k = 1,, 3 N (1) 1 n whch, f PC : the objectve functon value correspondng to the PC, f k : the objectve functon value correspondng to soluton (k) located n the PF, N: number of Pareto optmal solutons located n the PF, and n: number of objectve functons. 4. The ftness of each Pareto-optmal soluton (k), Eq., equals ts dstance ( d PC, k ) from the PC soluton. Ths step ensures the dsperse of the solutons along the soluton space and prevents solutons crowdes. Ftness, k = 1,, 3 N () k d PC k 5. The relatve ftness of each soluton n the PF s then calculated as follows:

4 4 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) 01- FtnessK Relatve Ftness k = N Ftness k1 k (3) 6. To form the next generaton, solutons from the current PF are randomly selected based on ther Relatve Ftness. Accordngly, Pareto optmal solutons wth larger values of Eucldean dstance have the hgher probablty to be selected. 3. Descrptons of the proposed metaheurstc algorthms A bref descrpton of the MOGA, MOMA, MOPSO, MOSFLA algorthms are presented n the followng subsectons Multple objectves genetc algorthms GA smulates mechansms of natural prncple of the survval of the fttest. In GA, solutons are represented usng chromosomes contanng a group of genes, havng values for the varables of an optmzaton problem. Each chromosome s, then, evaluated to determne ts ftness usng specfc objectve functon. Then, best chromosomes are exchangng ther nformaton through dfferent GA operaton (.e. selecton, crossover and mutaton), as follows, to produce offsprng chromosomes: - Selecton: Ft chromosomes are lkely to be selected to pass to next generatons. - Crossover: Parent chromones are exchangng ther genetc nformaton to produce offsprng chromosomes. - Mutaton: It allows for random change of genetc nformaton of ndvdual genes. Ths process ntroduces new genetc nformaton to the current chromosomes, ths may lead to avod stagnaton and pre-mature convergence. The GA operators are contnued for large number of generatons untl an optmal or near-optmal soluton s obtaned. The parameters that affect the performance of the GA are: number of chromosomes, number of generatons, crossover rate, mutaton rate and crossover type. The probablty of obtanng a global optmum soluton s acheved by ncreasng the numbers of both populaton and generatons but, ths substantally ncreases processng tme. In the proposed MOGA, for each generaton, the steps from (1) to (5) mentoned n the prevous secton are carred out then, step (6) s contnued to produce offsprng chromosomes through crossover and mutaton operators untl a new populaton of chromosomes s formed. Also, a replacement strategy s carred out to replace the weakest chromosome n the current generaton wth a randomly selected one located n the PF of the prevous generaton. These processes are repeated for large number of evolutonary cycles untl an optmum or near-optmal soluton s reached. Fgure 1 shows a flow chart of the proposed MOGA.

5 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) 01-5 Start Read nput data Generaton of ntal populaton of chromosomes A Evaluaton of all objectve functons and constrants Appled only n MOMA A Determnaton of nondomnated solutons located n the PF Separaton of PF n an external repostory Determnaton of the compromse soluton for the obtaned PF Calculaton of ftness and relatve ftness, Eqs. and 3, for each chromosome n the PF Applcaton of local search on each chromosome Re-evaluaton of all objectve functons and constrants The change s kept f the chromosome s performance mproves No Is max. number of swaps reached? Yes Is the termnaton condton satsfed? Yes Calculaton of fnal PF from the obtaned ones n each generaton No Wrte the fnal PF Applcaton of replacement strategy Stop Applcaton of GA operators (selecton, crossover, and mutaton) to form new populaton of chromosomes Fg. 1. Flowchart descrbng both the MOGA and MOMA algorthms.

6 6 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) Multple objectves memetc algorthm Smlar to the GA, MA chromosome s elements are called memes, not genes. In MA, chromosomes are allowed to perform local search frst, before beng nvolved n the global evolutonary search [38]. Smlar to GA, a populaton of chromosomes s, ntally, generated randomly. Then, each generated soluton s subjected to a local search to mprove ts ftness. Afterwards, the GA operators are appled, to produce new ndvduals. These new ndvduals are then performng some local search to mantan local optmalty. In ths paper, the local search s carred out through a par-wse nterchange (or n other words, swappng chromosome s memes) as shown n Fg., [38]. Fg.. Applyng local search usng par-wse nterchange (Elbeltag et al. 005). The maxmum number of swaps s lmted to 10 for each chromosome n order to mantan reasonable processng tme. After swappng, f the chromosome s ftness mproves, the change s kept; otherwse, gnore the change. As prevously mentoned, the MA parameters are the same as the parameters of the GA, n addton to the local-search mechansm. The proposed MOMA s the same as the MOGA n addton to the local search prevously mentoned. After performng each swap, a chromosome before change (ch before ) s compared wth ts mate after (ch after ) the swaps, as follows: - If the ch after domnates the ch before, then the change s kept. - If the ch before domnates the ch after, then the change s gnored and swaps are repeated for a specfed number of tmes (10 tmes). - If no domnaton occurs, then one of the two chromosomes s randomly chosen as the new chromosome. Fgure 1 shows the flow chart of the MOMA Multple objectves partcle swarm The PSO was ntally developed by Kennedy and Eberhart [39]. In sngle objectve PSO, a set of random partcles (solutons) s ntally generated to begn the evoluton process. Through such process, three values are montored by each partcle : ts current locaton [X (I)], the best poston t arrved [P (I)], and ts velocty of flyng [V (I)] where, I s the current evoluton cycle. In each evolutonary cycle, the poston of best partcle (P g ) s determned whch s consdered the best ftness for all partcles. Consequently, the velocty of each partcle s updated as gven by Sh and Eberhart [40]:

7 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) 01-7 V, (4) I V I C r P I X I C r P I X g I max I V, V V 1 max 1,...,n p Each partcle poston s updated as follows (Eq. 5): X I 1 X I V I 1 (5) n whch, V (I+1): partcle velocty n teraton (I+1); V (I): partcle velocty n teraton (I); P (I): partcle best poston n teraton (I); P g (I): the global best partcle poston n teraton (I); X (I): partcle poston n teraton (I); X (I+1): partcle poston n teraton (I+1); : constant called nerta weght; C 1 and C : learnng factors (C 1 = C = ); r 1 and r : two random numbers (rangng from 0 to 1); V max : the maxmum value of velocty change. The man PSO parameters are: the populaton sze; number of cycles; V max ; C 1 ; C ; and ω. In the suggested MOPSO, both Eqs. 4 and 5 are, also, adopted but, the two parameters P and P g need to be determned repettvely durng each cycle [41]. Intally, P represents partcle ntal poston whch updated n the subsequent teratons as gven by Baltar and Fontane [4] for each partcle : - If the current poston P domnates the new poston, then the new P s set as the current P. - If the current P s domnated by the new poston, then the new poston s set the new P. - If no domnaton occurs, then the new P s selected randomly from the two postons. In the MOPSO, there s no best partcle poston (P g ) for all partcles, but there are several equally good partcles postons (PF) n each evoluton cycle stored n the external repostory. For each evoluton cycle, the steps from (1) to (6) mentoned n the prevous secton are carred out to select one soluton from the stored ones whch s consdered as P g. The flow chart of the MOPSO s shown n Fg Multple objectves shuffled frog leapng SFLA combnes the advantages of both MA and PSO algorthms. In the sngle objectve SFLA, the evoluton process s ntalzed wth a number of memeplexes, each contans a number of frogs (ndvduals). Throughout each memeplex, two ndvduals (frogs) are determned based on ther ftness: the best ndvdual (X B ) and the worst ndvdual (X W ). Also, n each cycle of evoluton, the ndvdual wth the hghest global ftness (X G ) s determned. Then, the ndvdual wth the worst ftness s mproved n each cycle (not all frogs) through a process smlar to the PSO. Accordngly, the change n worst frog poston (D ) and ts new poston (X Wnew ) are calculated as follows [38, 43]: D = C r (X B X W ) (6) X Wnew = X Wcurrent + D, D max D - D max (7)

8 8 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) 01- n whch, C: the search acceleraton factor; r: a random number n the range [0, 1]; and D max : the maxmum allowed frog s poston change. Start Read nput data Randomly ntalze all partcles postons Evaluaton of all objectve functons and constrants Determnaton of the best poston vector of partcle (P ) Determnaton of PF and Separaton n the external repostory Determnaton of the compromse soluton for the obtaned PF Calculaton of ftness and relatve ftness, Eqs. and 3, for each partcle n the PF Determnaton of the best partcle poston, P g usng roulette wheel selecton Calculaton of velocty and updatng poston for each partcle, Eqs. 4 and 5 No Is the termnaton condton satsfed? Yes Wrte the fnal PF Stop Fg. 3. Flowchart descrbng the MOPSO algorthm.

9 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) 01-9 If a better ndvdual (soluton) s produced when applyng Eqs. 6 and 7, the worst frog s replaced by the new produced one. Otherwse, Eqs. 6 and 7 s repeatedly calculated but X B s exchanged wth the global best frog X G. If no mprovement s occurred, a new frog (soluton) s created randomly replacng the worst one. Ths process contnues for a gven number of teratons [38]. The SFLA parameters are: number of frogs; number of memeplexes; number of trals n each memeplex; number of teratons; D max ; and C. In the proposed MOSFLA, there are several solutons for (X B ) and several other solutons for (X W ) n each memeplex. Consequently, both the (X B ) and (X W ) are randomly selected from the correspondng solutons. Also, there s no global best frog (X G ) for all frogs, but there are several equally good frogs postons (PF) n each evoluton cycle stored n the external repostory. In ths case, X G s determned by applyng the steps from (1) to (6) mentoned n the prevous secton n each evoluton cycle. Fgure 4 shows the flow dagram of the MOSFLA algorthm. 4. Models mplementaton and verfcaton To facltate the mplementaton of the proposed four algorthms, each algorthm s coded by the authors usng the FORTRAN language. Accordngly, all parameters related to each algorthm are studed. The most common parameters presented n the lterature are studed here n the current development. Then, the proposed algorthms performance s compared usng two multple objectves benchmark problems (Appendx A) where the obtaned PFs are compared wth the correspondng known true PFs of the two benchmark problems. Selecton of Parameters values for each algorthm s an essental part of ths study. As such, a large number of trals are performed to obtan the most sutable parameters values that sut the two test problems and the example applcaton (presented later). In ths step, ntal parameters values are set based on relevant lterature. Then, parameters values are changed through many experments whle the results are montored. The fnal parameters values adopted for each model are presented n Table 1. Also, dfferent termnaton crtera have been expermented wth ncludng the soluton convergence and number of evolutonary cycles. Based on many trals and experments, the termnaton crteron has been set based on the number of evolutonary cycles (teraton-orented) as presented n Table 1. For the purpose of comparng the proposed algorthms performance, dfferent metrcs are used, namely: the generatonal dstance (G d ); the dstance between the PC soluton of the obtaned PF and the true PF; the processng tme; the number of Pareto-optmal solutons located on the PF and the mean square error between the obtaned compromse soluton and the nearest best alternatve Pareto optmal soluton. The values of all these metrcs are calculated and presented n Table.

10 10 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) 01- Start Read nput data Randomly ntalze all frogs postons Evaluaton of all objectve functons and constrants Determnaton of PF and t s separated n the external repostory A Local search For each memeplex, determne X B and X W Apply Eqs. (6) and (7) Determnaton of the compromse soluton for the obtaned PF Calculaton of ftness and relatve ftness, Eqs. and 3, for each frog n the PF Yes Is the new frog better than the worst? Determnaton of the global best frog (X G ) usng roulette wheel selecton No Apply Eqs. (6) and (7) wth replacng X B by X G Partton all frogs nto memeplexes Local search A Yes Is the new frog better than the worst? Shuffle the memeplexes No Generate a new frog randomly No Is the termnaton condton satsfed? Replace the worst frog Yes Wrte the fnal PF Stop Fg. 4. Flowchart descrbng the MOSFLA algorthm.

11 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) Table 1. Parameters values used n the two test problems and the example applcaton Algorthm Parameter names Test problem (1): Kta functon Test problem (): Kursawe functon MOGA - Number of chromosomes: Number of generatons: Crossover rato: Mutaton rato: Crossover type: Unform Unform MOMA MOPSO MOSFLA - Number of chromosomes: - Number of generatons: - Crossover rate: - Mutaton rate: - Crossover type: - Populaton sze: - Number of cycles: - The maxmum velocty V max : - Values of learnng factors C 1 and C : - Value of nerta weght ω: - Number of frogs: - Number of memeplexes: - Number of trals before shufflng: - Number of teratons after shufflng: - The maxmum change n frog poston D max : - The search acceleraton factor C: Unform Lnear [ ] Unform Lnear [ ] Example applcaton Unform Unform Lnear [ ] The G d metrc [44] s consdered to test the performance of the developed four models for copng wth multple objectves problems. The G d metrc s used to measure the dstance between the obtaned and the true PFs: G d N 1 N d (8) n whch, d : the Eucldean dstance between the non-domnated soluton located n the obtaned PF and the nearest one n the true PF and N s the number of non-domnated solutons located n the obtaned PF. A very small value for G d metrc means that all the generated non-domnated soluton nearly concdes wth the true PF.

12 1 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) 01- Table. Comparson among algorthms results for the two test problems. Algorthm metrc Test problem (1): Kta functon Test problem (): Kursawe functon G d Dst Processng tme (second) MOGA Compromse soluton (.79, 8.11) (-16.7, -4.7) Best alternatve (.63, 8.1) (-16.69, -4.64) MSE No. of PF solutons G d Dst Processng tme (second) MOMA Compromse soluton (.78, 8.11) (-16.71, -4.7) Best alternatve (3.11, 8.09) (-16.7, -4.77) MSE No. of PF solutons G d Dst Processng tme (second) MOPSO Compromse soluton (.77, 8.11) (-16.7, -4.73) Best alternatve (.75, 8.11) (-16.7, -4.74) MSE No. of PF solutons G d Dst Processng tme (second) MOSFLA Compromse soluton (.78, 8.11) (16.71, -4.79) Best alternatve (.81, 8.11) (-16.71, -4.78) MSE No. of PF solutons The graphcal comparson between the obtaned PFs, assocated wth each proposed model, and the true PF s presented n Fg. 5. It can be seen that, the obtaned PFs seem regular and dstrbuted throughout the soluton space. The G d value s found to be , , , and for MOGA, MOMA, MOPSO, and MOSFLA respectvely. Consequently, the obtaned PFs are nearly smlar to the true PF as per the very small values obtaned of the G d. Also, the mnmum G d value obtaned s assocated wth the MOPSO algorthm. Fgure 6, on the other hand, shows the obtaned PFs for the proposed four models along wth ther correspondng true PF for the second benchmark problem. As notced n the prevous test problem, the obtaned PFs seem dstrbuted regularly all over the soluton space. The G d values are calculated as: , , , and for MOGA, MOMA, MOPSO, and MOSFLA models respectvely. Then, the obtaned PFs are nearly smlar to the true PF as per the very small dstances of G d metrc, wth the best value assocated wth the MOMA.

13 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) True MOGA True MOMA True MOPSO True MOSFLA Fg. 5. PFs produced by the four proposed algorthms and the true PF for the Kta functon. The dstance between the compromse soluton of the obtaned PF and the correspondng one of the true PF s calculated accordng to Eq. 9 and lsted n Table for each studed algorthm. The mnmum dstance reported s for the MOPSO wth the second test problem and for the MOSFLA wth the frst test problem. Dst x x y y. true obt true obt (9) n whch, x true and y true : the coordnates of the compromse soluton correspondng to the true PF and x obt and y obt : the coordnates of compromse soluton correspondng to the obtaned PF. Also, the Mean Square Error (MSE) s calculated based on the values of the crtera PC soluton and the values of the correspondng crtera solutons (Eq. 10) [36]: * 0 1 f f ;( 1,..., n) 0 f j for the * f j for each of the N Pareto-optmal MSE (1 n) (10)

14 14 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) 01- where, n: number of crtera (.e. objectve functons). The soluton that has the smallest MSE value s consdered as the best alternatve soluton of the PC soluton. True MOGA True MOMA True MOPSO True MOSFLA Fg. 6. PFs produced by the four proposed algorthms and the true PF for the Kursawe functon. All codes are runnng on a personal computer of Intel (R) Core (TM) I3 CPU and havng 4 GB Ram. The least processng tme reported wth the MOPSO for both test problems. The MOMA outperforms all other algorthms for the obtaned number of solutons located on the PF for the second test problem, whle the MOPSO outperforms other algorthms for the same metrc n the frst test problem. Generally, the results show that the MOPSO outperforms other algorthms n most of the comparson crtera as presented n Table. The results of the prevous two test problems show that, the obtaned PFs correspondng to the four models are very close to the true PFs. Also, both the rapd convergence and good results are obtaned n the two cases. Accordngly, the proposed four models havng the ablty for dealng wth problems of multple-objectve functons. 5. Example applcaton In ths secton, the four proposed models wth the new evoluton strategy are appled on a popular groundwater management problem n order to maxmze the pumpng rates and

15 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) mnmze the operaton costs [4]. Also, a comparatve study among the algorthms results s carred out. Both the plan of the studed aqufer and ts cross-sectonal elevaton along wth the requred data are shown n Fg. 7. The aqufer s formed of sand and gravel wth homogeneous and sotropc porous medum. The hydraulc conductvty (K) equals 50 m/day and t s assumed that the aqufer s subjected to unform ranfall (W) of m/day. The objectve of ths problem s to determne the pumpng rates from the pre-specfed system of the ten wells, achevng hydraulc constrants. The mathematcal formulaton of the optmzaton problem s as follows [6, 4]: f N W max. Q P( h (11) 1 ) 1 f N W b 1 b b3 mn. a1 yd aq d h a3q d h Ph PQ (1) 1 Subjected to: N W 1 Q Q req. (13) h h, mn, = 1,, 3,.N W (14) Q mn Q Q max, = 1,, 3,... N W (15) 1 f Q 0 k = 1,, 3,...N W (16) 0 f Q 0 h mn h f h h mn P( h) = 1,, 3, N W (17) 0 f h h mn Qreq. Q f Q Qreq. P( Q) = 1,, 3, N W (18) 0 f Q Q req. where, Q : well pumpng rate; N W : number of wells; P (h) : head volaton penalty; h, mn : mnmum head allowable at well (= 0); Q mn and Q max : well mnmum and maxmum allowable pumpng rates (equals 0 and 7000 m 3 /day respectvely); a 1, a, and a 3 : cost coeffcents for drllng, pumpng equpment and operaton of well respectvely (a 1 = 41 $/m, a = 0.1 $/m 4, a 3 = 0.03 $/m 4 ); y: the penalty term; b 1, b, and b 3 : constants ndcatng economes of scale (b 1 = 0.99 and b = b 3 =1); Q req : the water demand requred (= m 3 /day); d : the depth of well ; and P (Q) : penalty term for the mnmum pumpng volaton.

16 16 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) 01- Z Rver, h=0 m N (1) () (3) (4) (5) (6) Elevaton (7) (8)(9) (10) Areal Recharge y 100 x Swamp, h=0 m 4500 m Z 1000 Rver 980 Sand & Gravel Slate Bedrock Swamp (a) (b) Fg. 7. The plan vew and sectonal elevaton of the studed aqufer (El-Ghandour and Elbeltag 014). In ths applcaton, a smulaton model, based on FEM, s lnked wth each proposed optmzaton algorthm to predct the hydraulc head at each well locaton shown n the plan, Fg. 7. FEM s used to dscretze the studed plan to number of lnear rectangular elements and the twodmensonal flow equaton, Eq. 19 s solved over each element [5]. K K W x y N W 1 Q x x y y (19) n whch, K: the hydraulc conductvty of aqufer; : the potental (= h /); h: the hydraulc head; W: the unform ranfall or unform evaporaton; Q : the rate of njecton or pumpng for the well ; δ(z): the Drac delta functon whch equals 1 f z equal zero otherwse equals zero; and N W : number of feld wells. In the smulaton model, Eq. 19 s wrtten as follows: [A] [φ] = [F] (0)

17 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) n whch, A: the conductance/stffness matrx, φ: the unknown vector of potentals and F: the load vector whch contans the external source or snk of water and flux concentraton. In order to make a comparson among the obtaned PFs of the developed four models, n addton to the metrcs descrbed before, the Spacng (S P ) metrc [44] s consdered, Eq. 1. The S P s used to measure the spread of the Pareto optmal solutons over the whole soluton space. S 1 N d P N 1 d 1 (1) n whch, d : the mnmum dstance between Pareto-soluton and all other solutons located n the PF, d : the arthmetc mean of d, and N: the number of solutons located n the PF. MOGA Compromse soluton MOMA Compromse soluton ( & ) ( & MOPSO Compromse soluton MOSFLA Compromse soluton ( & ) ( & ) Fg. 8. PFs for the example applcaton produced by the developed four algorthms A small value of S P means that solutons located n the PF are equdstantly spaced. Fgure 8 llustrates the obtaned PFs assocated wth each developed model. As presented n Table 3, the S P metrc s computed for each algorthm as 31.6, 84.6, 19.3, and for MOGA, MOMA, MOPSO, and MOSFLA respectvely. The S P value for MOPSO model s the mnmum then, the performance of ths model s the best compared wth the other models. Accordngly, the obtaned PF from MOPSO model s more regular and dstrbuted throughout the soluton space.

18 18 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) 01- Also, the processng tme of MOPSO model ( sec) s found to be the mnmum compared wth the processng tme of the other models. Also, the hghest number of Pareto optmal solutons obtaned was wth the MOPSO. From the vsual nspecton of Fg. 8, the PFs obtaned usng MOGA and MOMA models are more unformly dstrbuted compared wth the other two models. The Pareto compromse soluton for each obtaned PF s, also, determned as shown n Fg. 8. It s notced that, the devaton among the values of compromse solutons s small. In general, the results showed the effectveness of usng the suggested evoluton strategy n each model to drve the fnal PF for the example applcaton. Consequently, the proposed four models can be used to deal wth groundwater real lfe applcatons havng multple objectves. Table 3. Comparson among algorthms results for the example applcaton. metrc Algorthm S p Processng tme (second) Compromse soluton Best alternatve MOGA ( , ( , ,0) ) MOMA (4579.4, ( , ) ) MOPSO (4558., (4565.6, ) ) MOSFLA (4467.6, ( , ) ) MSE No. of PF solutons In general, the MOPSO outperforms all other algorthms n most of the comparson crtera, whch confrm wth some prevous studes. Also, dentfyng a unque PC soluton helps the decson makers to determne a unque soluton that satsfes all objectves farly. 6. Lmtatons and suggested mprovements The four developed metaheurstc models have been appled on a ground water example applcaton and worked effectvely. Also, the models were expermented wth two bench mark problems wth dfferent shapes of PFs and performed well. The developed models used a unque soluton that satsfes farly all competng crtera of the problem "PC soluton" to drve the ftness. As such, trackng a sngle pont resembles optmzng a sngle-objectve. Also, t helps the decson maker by presentng a unque PC soluton especally n complex objectve spaces. Despte ther perceved benefts, the developed model stll has some lmtatons wth a number of possble mprovements, ncludng: - For the groundwater management problem, consderng other objectves n addton to the total pumpng and total cost; - studyng dfferent aqufers shapes other than rectangular or near rectangular shapes.; and - Compare the performance of the proposed mult-objectve optmzaton strategy wth other strateges used n the lterature

19 H.A. El-Ghandour and E. Elbeltag/ Journal of Soft Computng n Cvl Engneerng -4 (018) Conclusons Ths research appled new evoluton strategy of trackng a sngle compromse soluton to modfy the GA, MA, PSO, and SFLA algorthms to tackle multple objectves problems. Two multple objectves standard test problems wth known PFs are used to check the ablty of the proposed models wth the proposed evoluton strategy to arrve at the correct PFs. The proposed four algorthms are then appled on a popular groundwater management example for maxmzng pumpng rates and mnmzng operaton costs. The suggested four models proved ther ablty to obtan the close true PFs for the two benchmark problems n addton to the groundwater applcaton. Accordngly, they can effectvely be used to deal wth real-lfe groundwater management applcaton. The performance of the dfferent mult-objectve algorthms s studed and compared. The MOPSO outperformed all other algorthms based on the used performance metrcs n terms of regularty and dstrbuton of PF throughout the soluton space and smallest processng tme when compared wth other mult-objectve optmzaton algorthms. In ths applcaton, a fnte element-based smulaton model, wth a new dea to reduce the run tme, s lnked wth each proposed model for evaluatng the potental soluton. Appendx A A.1 Benchmark problem (1) The frst benchmark problem s called the Kta problem wth two objectves (Eq. A.1), two unknowns, and three constrants (Eq. 9) [44]: f x, y) y x p, f ( x, y) 0.5x y 1 p, 0 x, y 7 (A.1) 1( Subjected to: p 1 x y 0, p x y 0, p3 5x y 30 0 (A.) 6 1p1 p p3 f p1, p, p3 0 P 0 Otherwse (A.3) where, P s the penalty term for constrants volatons. In ths study, dfferent penalty functons are expermented wth (lnear, exponental, etc.) and the best results were assocated wth the proposed lnear penalty functon (Eq. A.3). A.. Benchmark problem () The second benchmark problem s called the Kursawe problem wth two objectves and three unknowns [44].

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