Inernaonal Journal of Aled Engneerng Research ISSN 0973-4562 Volume 11, Number 4 (2016) 2839-2847 Research Inda Publcaons. h://www.rublcaon.com Cooerave Random Walk for Pe Nework Layou Omzaon Sefan Ivć* Faculy of Engneerng, Unversy of Reka, Reka, Croaa. Luka Grbčć Faculy of Engneerng, Unversy of Reka, Reka, Croaa. Snša Družea Faculy of Engneerng, Unversy of Reka, Reka, Croaa. Absrac Pe nework layou omzaon s an moran and dffcul roblem n e nework engneerng. Dfferng from he revously roosed aroaches for varous omzaon formulaons, he auhors use a mehodology for e nework omzaon conssng of nework layou omzaon couled wh e szng analyss. The mehodology s based on a novel heursc omzaon algorhm nsred by An Colony Omzaon, called Cooerave Random Walk (CRW). CRW successfully fnds an omal nework layou for a range of e dameer values, as shown on a es base nework of 40 nodes and 68 es wh one source and 12 consumer nodes. The resuls clearly show how larger e dameers allow for smler nework layous, whle smaller e szes auomacally mly more comlex and longer neworks. Alhough CRW s esed on a waer suly nework roblem, s suable for oher nework omzaons as well. Keywords: Cooerave Random Walk, Pe Nework Layou Omzaon, Sochasc omzaon mehod. Inroducon Pe nework layou omzaon s an moran roblem n waer suly nework engneerng, as well as n e nework engneerng n general. Consderng he ofen hgh nsallaon cos of e neworks, fndng he mos cos-effecve e nework confguraon s necessary for achevng a long economc lfe of a e nework. Many auhors have examned he conce of omzng e neworks, coverng varous aroaches for solvng omzaon roblems arsng from dfferen asecs of e nework desgn, manenance, conrol or oeraon. Pe neworks omzaon roblems, as well as omzaon roblems n general, are formulaed by omzaon varables, obecves and consrans. The overvew of man yes of e nework omzaons s gven n Table 1. In hs aer we deal wh he roblems of omal desgn of e neworks, whch can be, accordng o as researches, classfed as szng omzaon (.e. e dameer omzaon), nework layou omzaon, or e nework layou and szng omzaon. Unrelaed o he ye of omzaon varables, dfferen omzaon obecves and omzaon consrans are defnng he omal desgn of a e nework. Mos commonly used obecve, whch regulaes he economc effcency of a nework, s he oal cos of es. The hydraulc asec of nework desgn s manly handled by dfferen consrans: rescrbed mnmum ressure or mnmum flow, nework relably ec. Omzaon of neworks by szng he es used n a redefned fxed nework layou s he mos commonly used ye of e nework desgn omzaon. Pe dameers can be consdered as dscree omzaon varables, whch mles combnaoral omzaon roblem. Ths elemenary characersc of he omzaon roblem allows for he alcaon of wde range of omzaon mehods so as o acheve omal e szng. For solvng combnaoral roblem, where e dameers are chosen from a se of avalable values, a grea number of varous mehods have been roosed: Smulaed Annealng [2], Tabu Search [3], Genec Algorhm [4, 1], Harmony Search [5], Shuffled Comlex Evoluon [6], An Colony Omzaon [7, 8, 9], Parcle Swarm Omzaon [10], Soccer League Comeon Algorhm [11] and Cross Enroy Omzaon [12]. By allowng for he excluson of es from he base nework, he e szng roblem can be easly exended o nework layou and szng omzaon. A "zero dameer" s added o he se of avalable dameers whch can be used on every e n he base nework. The e wh "zero dameer" has no cos and here s no flow hrough ha e, hus can be consdered as excluded from he e nework. Ths rases a fundamenal roblem of ossble nework layous whch are oologcally nvald, such as a nework wh a consumer node no conneced o he source or a nework wh solaed es. The descrbed layou and szng formulaon defnes a smlar omzaon roblem as e szng, and s solved by several auhors usng dfferen omzaon echnques: An Colony Omzaon [13, 14], Genec Algorhm [15, 16] and hybrd omzaon mehods such as combnaon of Genec Algorhm and Parcle Swarm Omzaon [17, 18]. Omzng only he layou of a nework, comared wh e szng or couled e szng and nework layou, has no ye 2839
Inernaonal Journal of Aled Engneerng Research ISSN 0973-4562 Volume 11, Number 4 (2016) 2839-2847 Research Inda Publcaons. h://www.rublcaon.com been horoughly researched. Fügenschuh e al. [19] combned mxed-neger lnear rogrammng, nonlnear omzaon and consran rogrammng for omzaon of gas dsrbuon nework. Anoher aroach, usng An Colony Omzaon, s resened n [20, 21]. The omzaon obecve, n all revewed aers, s o mnmze he oal cos of es, whch deends on lengh and dameer of all es n he nework. A mnmal ressure requremen a each demand node s he mos commonly used consran. Some auhors use addonal consrans such as boundng velocy n es [8, 13, 9, 22, 16] or requrng a ceran level of nework relably [13, 16, 21]. Hydraulc relably of waer dsrbuon sysems [23] s an moran consran n e nework desgn, and, as s shown n saed aers, can be relavely easly ncluded n he nework omzaon mehod. Table 1: Overvew of e nework omzaon aroaches [1] Omzaon Obecve Possble Man consrans ye varables Desgn Mnmze cos Pe layou; e Mn. level of dameers; e rehablaon servce; avalable dameers; rehablaon oons; avalable budge; LCC Oeraon Mnmze oeraonal cos Calbraon Level of servce Monorng sysem desgn Nework esng Mnmze dfference beween model and observed values Maxmze level of servce; e.g. ressure, waer qualy or relably Mnmze cos of monorng sysem Fnd crcal ses of evens ha may cause sysem o fal Pum conrols; reservor levels; sources and caacy Pe roughness; e dameer; valve sengs; leakage; demands All above Number and oson of monorng ons Fres; e falures; ower falures; conamnaon evens Mn. level of servce; number of um swches; source caacy; um caacy Sysem layou; avalable daa Sysem confguraon; budge Sysem confguraon; budge Sysem confguraon; number of smulaneous evens Hydraulc calculaon of nework flow s necessary o oban ressure a each of he nework nodes. In e nework omzaon research, he comuaon of hydraulc values s usually erformed by EPANET [24]. EPANET s a freely avalable comuer rogram ha erforms smulaons of hydraulc and waer qualy behavor whn e neworks. Some auhors have mlemened fundamenal exensons of he sandard omzaon roblem formulaon and aroach: ncludng more omzaon varables such as sloes and um locaon [22], use of mulle sources [20, 16], nework aronng for more effecve omzaon [25]. Cooerave Random Walk All revous researches ha ackled he e nework omzaon roblem wh dervaons of An Colony Omzaon (ACO) mehod ddn' really ake no accoun all nformaon rovded by nework oology. Aroach resened n hs aer, called Cooerave Random Walk (CRW), grealy reles on rncles used n varous an-based algorhms, bu addonally ulzes he oology of a nework of all ossble lnks (base nework). The base nework (Fgure 1) s a nework of all ossble es, from whch only a subse of es wll be used o consruc ossble soluons used n omzaon rocess and, fnally, he omal nework. We denoe P = { 1 n } as se of all es n he base nework, where n s number of es n base nework. A se N = { 1 n n} reresens a se of all nodes n he base nework, where n n s he number of nodes. The source and consumer nodes n he base nework are reaed dfferenly han oher nodes n he nework due o her characersc hydraulc asec as well as her ulzaon n he Cooerave Random Walk algorhm. We denoe se of source nodes as Ns N and he se of consumer nodes as Nc N. A secfc nework confguraon s defned as a se of es used o buld a layou. Ths se of es used o defne a secfc layou s denoed as L P, whch s effecvely a reresenaon of a desgn vecor for he omzaon rocess. Each (-h) e n he base nework a eraon has a robably of usage whch s used o drec he movemen of agens exlaned n secon A. Ths robably s adused durng he omzaon rocess, based on he obaned bes soluons, as descrbed n deal n secon B. For all es n he base nework, robably s nally se as 0 0, P (1) where φ 0 s he nal robably, whch s a arameer of roosed omzaon mehod. Lke many oher sochasc omzaon mehods, he roosed algorhm roduces mulle canddaes n each eraon (smlar o a oulaon n GA or a swarm n PSO). A canddae,.e. a ossble soluon of he omzaon roblem, s dscovered by a eam of agens ha are exlorng he base nework. On each eraon, n eams are used and each eam s ndeendenly formng a ossble soluon based on he robably of lnks n he base nework. Number of eams n s he arameer of he omzaon mehod. Usng he eam movemen algorhm descrbed n secon A, n each eraon n canddae soluons are generaed where each reresens oologcally vald soluon of e nework omzaon roblem. Toologcal valdy mles ha each consumer node s conneced o a leas one source, whch auomacally means here are no solaed (unconneced) es n nework. For each canddae a fness funcon s evaluaed accordng o he obecves and consrans of gven omzaon roblem. Proosed mehod s desgned as a mnmzaon mehod and 2840
Inernaonal Journal of Aled Engneerng Research ISSN 0973-4562 Volume 11, Number 4 (2016) 2839-2847 Research Inda Publcaons. h://www.rublcaon.com can be used for any fness funcon whch yelds a osve fness value f(l) > 0, L. whch (3) s me s chosen as a nex lnk n agen s ah. If condon (3) remans no sasfed afer checkng on all ossble lnks, he nex lnk for agens wh "dry" saus s chosen randomly from S, whle agens wh "we" saus so. Fgure 1: Base nework skech Thus, we can use hs mehod o solve several fundamenally dfferen roblems, consderng varous obecves, consrans and goal funcon formulaons. Afer evaluaon of he layous of all eams, a bes layou B for -h eraon and he overall bes layou B o are recorded. Team formaon and agen movemen: For creaon of an arbrary layou, classc ACO algorhm aroach uses a sngle agen (an) ha ados a lnear ah hrough all ossble lnks n a nework. Here, a grou of agens moves hrough he base nework and consrucs a soluon. The movemen of agens s dreced by he robably assgned o lnks (es) of he base nework, smlar o ACO where heromones regulae he selecon of an agen s ah. For varous formulaons of e nework omzaon, he connecon of consumer nodes and source nodes s requred. Thus, from every source and consumer node one agen s sen. Agens are smulaneously movng from node o node and leavng a ral on es. The k-h agen s ral s defned by he ls of used lnks (.e. enabled es) A k P. A layou L generaed by a eam of agens s a unon of es used by each agen: L n a A k k 1 (2) where n a s he number of agens n a eam whch s equal o he number of source and consumer nodes (n a = N s + N c ). Noe ha he es used by several agens are resen n layou L only once. The ral has a saus "dry" or "we" wheher he orgn of an agen s a consumer node or a source node, resecvely. The ah of an agen s deermned by sochasc decsons made on each node. To deermne along whch lnk wll agen make he nex se, a ls of all ossble lnks from curren node o neghborng nodes s creaed excludng lnks o nodes ha are already conaned n he agen s ah (Fgure 2). We denoe ls of ossble lnks as S. The decson of an agen s nex se s made by checkng he condon r (3) n where rn s a random number unformly seleced from he nerval [0, 1]. Condon (3) s checked sequenally for each lnk n a shuffled ls of all ossble lnks S. Frs lnk for Fgure 2: Agen movemen The agen movemen algorhm eraes and agens are rogressng hrough he base nework unl one of he followng condons are me:. all es n L are conneced o a leas one source node (all es have saus "we"),. number of agen ses m max reached he maxmum number of ses max, or. all agens have soed. Only he frs lsed song condon ndcaes ha he generaed layou L s oologcally vald. The eam s resared f one of he oher wo condons caused he layou dscovery rocess o so, because hey ndcae ha he creaed layou s no oologcally vald. Due o he generaly and flexbly of he agen movemen algorhm, dfferen knds of nework layous can be acheved: lnear neworks, ree-branched neworks, looed neworks and more comlex combnaons of basc layous. Probably Udae: The robably of e selecon deends on occurrence of e n eraonal and overall bes soluons and s lmed by he robably lms mn and max n order o acheve [ mn, max]. Inally, robably of e selecon s se o 0 for all es n he base nework. The nal e robably 0 and lms mn and max are consdered as arameers of he roosed omzaon mehod. The robably of -h e s udaed afer each eraon accordng o: 1 where and are he overall and he eraonal robably adusmens for -h e a he eraon, resecvely. The overall robably adusmen s calculaed as (4) 2841
Inernaonal Journal of Aled Engneerng Research ISSN 0973-4562 Volume 11, Number 4 (2016) 2839-2847 Research Inda Publcaons. h://www.rublcaon.com 1 ( max ) ( mn ) f B o oherwse, where s he overall robably adusmen facor. The adusmen α nduces ermanen growh of robably of es used n overall bes layou B o and ermanen decay of robably n oher es of base nework. We can characerze he nfluence of α on robably as ermanen due o he fac ha he overall bes soluon s no changng frequenly. Because of a relavely unchangng α, he robables of es used n he overall bes soluon converge o a maxmum value max, whle for oher es, unused n he bes layou, he robables decay o a mnmum value mn. The eraonal robably adusmen s calculaed as 1 where 2 f ( B ) ( max ) f ( Bo ) 2 f ( B ) ( mn ) f ( Bo ) f B 0 oherwse, (5) (6) s he eraonal robably adusmen facor. Snce β deends on he bes soluon of curren eraon, s more ofen changed han α and hus s nfluence on robably of e selecon s nsan bu emorary. Pe Nework Layou Omzaon The roosed Cooerave Random Walk algorhm descrbed n revous secon can easly be used along dfferen obecves n e nework omzaon roblems. We aled CRW o he mos commonly used e nework omzaon formulaon where he goal s o mnmze he overall cos of he nework whle obeyng gven hydraulc consrans. Consrans for revenng ncomlee or unconneced neworks, whch are used n oher aroaches, are no needed here due o he fac ha CRW algorhm always roduces oologcally vald neworks. Hydraulc calculaon: In he research resened n hs aer, EPANET 2.0 was used for he hydraulc calculaons. EPANET gves an nsgh o he flow n each e and ressure a each node n he sysem hroughou he nework durng a smulaon erod comrsed of mulle me ses. A nework consss rmarly of es and nodes. Pums, valves, anks or reservors can be added no he waer dsrbuon nework. EPANET calculaon s based on he laws of conservaon of mass and energy n a e nework sysem. These laws could also be consdered as a sor of hydraulc consrans [16] ha mus be me. There are hree ossble formulas used n EPANET for he calculaon of frcon head loss n a e. Those are he Hazen-Wllams formula, whch s used n hs aer, he Darcy-Wesbach formula and he Chezy-Mannng formula. Usng he Hazen-Wllams formula comonens, he head loss equaon s 1.852 4.871 1.852 h 4.727C D l Q. (7) where C sands for he Hazen-Wllams roughness coeffcen, whch s used as C = 130,, whch reresens cas ron as a maeral for all es n he waer dsrbuon nework. D s he dameer of he -h e and l s he lengh of he -h e. A slghly dfferen form of he Hazen-Wllams equaon has been nroduced n [26], made arcularly for omzng waer dsrbuon neworks: Q h l D (8) C For hs equaon o be equvalen o he Hazen-Wllams equaon used n EPANET 2.0 (7), he coeffcens mus be equal o m = 10.6668, l = 1.852 and g = 4.871. Fness funcon: The obecve, for layou-only e nework omzaon, s o mnmze he oal cos of all es n he nework. The oal cos of nework deends on he lenghs and szes of used es and, snce he e dameer s known, can be exressed as he sum of used es lenghs n he nework: O (9) l L where l s he lengh of he -h e n layou L, and O denoes he oal cos,.e. omzaon obecve value. A funconal e nework layou mus ensure suffcen ressure head a all consumer nodes, whch can be defned as a consran: N (10) where mn, c and mn are calculaed ressure and mnmal allowable ressure, resecvely, for every node n he se of consumer nodes N c. The mnmal allowable ressure consran (10) s no handled by he CRW algorhm drecly. A enaly mehod s used o nclude handlng of a consran whn a fness funcon. A enaly value s defned wh resec o consran (10): mn n mn a b f C mn (11) 1 0 oherwse where a s he enaly um, b s he enaly facor, and C denoes oal enaly,.e. consran value. Fnally, he fness funcon, as a funcon of layou L, s defned as f ( L) O C (12) and s used n hs form o evaluae all layous roduced by he Cooerave Random Walk mehod. A mnmzaon of 2842
Inernaonal Journal of Aled Engneerng Research ISSN 0973-4562 Volume 11, Number 4 (2016) 2839-2847 Research Inda Publcaons. h://www.rublcaon.com (12) s requred o oban he omal layou for he descrbed omzaon roblem. A Pe Dameer Deenden Omal Nework Layou The Cooerave Random Walk algorhm descrbed n he revous secon s used o deermne a relaon beween omal e nework layou and e dameer. The rocedure o deermne hs relaon s ncluded he resened nework layou omzaon mehod, combned wh he roo fndng algorhm. I s an erave rocess (Fgure 3) n whch he e dameer s gradually decreased and omal layou s found for each exsng e dameer range. For a gven e dameer, an omal nework layou s deermned by use of CRW algorhm. Usng he omal layou, a crcal value of e dameer, for whch a leas one demand node has ressure equal o he mnmum requred ressure, s found usng a smle roo fndng mehod. Ths can be wren as fndng crcal dameer D cr ha solves he nonlnear equaon: mn Nc mn ( D ) 0 cr (13) and reresens a reservor wh he consan waer level. The head of he source node s 40 m ( 1 = 40m H20). Twelve consumers are rovded n he base nework, wh her ouflow demand shown n Table 3. Mnmal ressure needed a all consumer nodes s 10 m. Ths es s desgned o observe, analyze and comare omal e nework layous for varyng e dameer. There are many ossble layous whch ensure ha all consumer nodes are conneced wh he source node bu no all of hem sasfy he hydraulc consrans. In all hs combnaons, he ask of hs mehod s o fnd he shores e nework layou for a gven e dameer ha rovdes he consumers wh he needed ressure and dscharge. The man loo of he rocedure gradually lowers he e dameer by fndng he crcal e dameer for each layou. The e nework layou omzaon roblem s dfferen for every gven e dameer, so rovdes a comleely dfferen convergence of Cooerave Random Walk and a dfferen fnal soluon. Thus, he e dameer loo rocedure s also used as a benchmark rocedure for CRW mehod. where ressure values on consumer nodes N c deend on he e dameer. For solvng (13) regula fals mehod s used. Fgure 3: Flow dagram of algorhm for deermnng omal layous for dfferen e dameers Tes Cooerave Random Walk s beer sued for layou-only omzaon of e neworks, bu here s sgnfcanly more research avalable for smulaneous szng and layou omzaon. Resuls resened n [19, 21] are dealng wh layou only e nework omzaon, bu does no offer all nformaon o reroduce he resened omzaon cases. Thus, we desgned a new es case, whch s suable for esng he roosed CRW mehod. The es case s based on he base nework layou shown n Fgure 4. The base nework s conssed of 40 nodes and 68 es whose lenghs are gven n Table 2. For smlcy, he nework s consdered o be deally fla so all nodes and es are a he same hegh level. The source s locaed a node 1, Fgure 4: Base nework for es case Cooerave Random Walk mehod arameers are deermned uon relmnary esng of he algorhm on varous base neworks. For he above descrbed es, he omzaon s done wh n = 50 (number of eams) n max = 200 eraons. The robably adusmen = 0.035 and = 0.13 are used. Inal robably s se o 0 = 0.5 and s bounded wh mn = 0.03 and max = 0.90. Based on he comlexy of he 2843
Inernaonal Journal of Aled Engneerng Research ISSN 0973-4562 Volume 11, Number 4 (2016) 2839-2847 Research Inda Publcaons. h://www.rublcaon.com base nework and osons of source and consumer nodes, he maxmum numbers of agen s ses s se as m max = 4. dameers and oal lengh of he es n nework are shown n Fgure 5. Table 2: Base nework e lenghs Pe ID Lengh [m] Pe ID Lengh [m] Pe ID Lengh [m] Pe ID Lengh [m] 1 437.86 18 526.11 35 371.11 52 250.65 2 440.32 19 522.24 36 599.94 53 372.13 3 489.97 20 267.59 37 402.33 54 439.17 4 598.17 21 494.56 38 442.53 55 238.13 5 455.19 22 363.79 39 516.20 56 897.18 6 452.96 23 476.03 40 297.67 57 335.82 7 400.55 24 463.58 41 267.19 58 645.83 8 477.30 25 486.13 42 507.59 59 358.63 9 405.48 26 350.10 43 395.45 60 402.64 10 312.74 27 501.30 44 439.96 61 407.08 11 437.51 28 373.95 45 244.94 62 297.89 12 394.00 29 538.35 46 348.24 63 411.29 13 316.81 30 346.56 47 306.40 64 505.65 14 329.41 31 358.76 48 431.91 65 372.98 15 460.20 32 399.13 49 315.35 66 284.79 16 430.80 33 342.48 50 427.99 67 741.79 17 428.59 34 446.06 51 258.05 68 626.14 Fgure 5: Omal layous for dfferen e dameers Table 3: Consumer nodes ouflow demand Node ID Ouflow demand [m³/s] 5 105 10 110 11 150 18 100 19 150 20 140 25 120 31 110 32 135 37 165 38 140 40 105 The rocedure for deermnng dameer deenden omal e nework layou was conduced for he descrbed es case. To fnd he omal nework layou for each e dameer, a 100 Cooerave Random Walk omzaons are conduced and he bes resul s chosen as he omal layou. Alhough hs s a somewha exensve comuaon, gves grea nsurance ha he omal layou s found. Omal e nework layous are found for sx dfferen crcal e dameers. The layous, ogeher wh he crcal As execed, lowerng he e dameer leads o longer and more comlex e neworks. Nework branchng s more frequen o aroraely dsrbue he flow whch leads o lesser hydraulc losses n es and consequenally lower ressure n he whole nework. The black crcle n layous shown n Fgure 5 marks he node wh he mnmal ressure n he nework. Ths s he crcal node for whch (D cr ) = mn. I s neresng o observe, when lowerng he e dameer, addonal branchng occurs n he branch whch conaned he crcal node n he revous omal layou. Dealed llusraon and comarson of he convergence of Cooerave Random Walk s dslayed n Table 4. Each row of he able shows convergence nformaon of e nework layou omzaon for gven e dameer. The convergence lo for all 100 runs of layou omzaon s shown, where average fness value over eraons s emhaszed (red lne). The dsrbuon of fnal fness value, whch s also he lengh of es n he nework, s shown n he fourh column of he able. Based on he convergence los, can be concluded ha Cooerave Random Walk successfully mnmzes he fness for all sx e dameers. Ou of oal 600 layou omzaon runs, no a sngle one had a maor ssue wh fness mnmzaon. To gve a more recse udgmen, he dsrbuon of fnal fness value for all 100 runs should also be observed. The 2844
Inernaonal Journal of Aled Engneerng Research ISSN 0973-4562 Volume 11, Number 4 (2016) 2839-2847 Research Inda Publcaons. h://www.rublcaon.com roosed mehod s behavng dfferenly when omzng he same layou wh dfferen e dameer. The combnaon of e dameer and hydraulc consrans makes sgnfcan nfluence on he dffculy of he layou omzaon roblem whch leads o remaure convergence,.e. convergence o subomal layou. Achevemen of he omal layou, deendng on he observed e dameer, vares from 7% o 88%. Due o he fac ha he roosed algorhm s a sochasc mehod where reeaably s ofen an ssue, we beleve ha hese resuls gve us grea confdence ha Cooerave Random Walk s an arorae and successful mehod for e nework layou omzaon. Table 4: Resuls D cr Omal layou lengh [m] 555.4 6409.27 Convergence Layou occurrence 550.7 6632.93 500.7 6718.78 478.6 6849.71 478.4 6901.79 2845
Inernaonal Journal of Aled Engneerng Research ISSN 0973-4562 Volume 11, Number 4 (2016) 2839-2847 Research Inda Publcaons. h://www.rublcaon.com 470.7 7019.64 Concluson The resened echnque erforms full e nework omzaon couled wh e szng range analyss, by emloyng a hydraulc model of he nework and a novel heursc omzaon algorhm named Cooerave Random Walk. Alhough he CRW mehod n rncle serves as a e nework layou omzer, he es used for assessmen of he roosed echnque s erformance showed ha can be successfully used for hybrd analyss of omal e dameer and nework layou. The mehod had no roblem omzng a nework of consderable sze (40 nodes and 68 connecons) wh a sasfyng success rae and reeaably. In order o furher advance he roosed e nework omzaon mehodology, he e nework cos should be made o nclude he full e cos, whch s e-szedeenden. Even he resuls of he es llusrae how usng smaller es asks for longer and more comlex e neworks n order o fulfll he demand ouflow and ressure. Ths means ha smaller es mly longer neworks whch may reduce he savngs nally execed o be acheved by usng smaller es. Whereas he rovded es nework ncludes only one source, he CRW mehod self s enrely suable for omzng e neworks wh many sources. Furhermore, CRW can be aled o he oher e nework omzaon formulaons descrbed n he Inroducon. Fnally, here are oher areas of nework omzaon where CRW should be useful, such as logscs omzaon, raffc omzaon, ec. References [1] M. Van Dk, S. Van Vuuren, and J. Van Zyl, Omsng waer dsrbuon sysems usng a weghed enaly n a genec algorhm, Waer SA, vol. 34, no. 5,. 537-548, 2008. [2] M. d. C. Cunha and J. Sousa, Waer dsrbuon nework desgn omzaon: smulaed annealng aroach, Journal of Waer Resources Plannng and Managemen, vol. 125, no. 4,. 215-221, 1999. [3] M. da Concecao Cunha and L. Rbero, Tabu search algorhms for waer nework omzaon, Euroean Journal of Oeraonal Research, vol. 157, no. 3,. 746-758, 2004. [4] G. C. Dandy, A. R. Smson, and L. J. Murhy, An mroved genec algorhm for e nework omzaon, Waer Resources Research, vol. 32, no. 2,. 449-458, 1996. [5] Z. W. Geem, Parcle-swarm harmony search for waer nework desgn, Engneerng Omzaon, vol. 41, no. 4,. 297-311, 2009. [6] S.-Y. Long and M. Aquzzaman, Omal desgn of waer dsrbuon nework usng shuffled comlex evoluon, Journal of The Insuon of Engneers, Sngaore, vol. 44, no. 1,. 93-107, 2004. [7] H. R. Maer, A. R. Smson, A. C. Zecchn, W. K. Foong, K. Y. Phang, H. Y. Seah, and C. L. Tan, An colony omzaon for desgn of waer dsrbuon sysems, Journal of waer resources lannng and managemen, vol. 129, no. 3,. 200-209, 2003. [8] M. H. Afshar, A new ranson rule for an colony omzaon algorhms: alcaon o e nework omzaon roblems, Engneerng Omzaon, vol. 37, no. 5,. 525-540, 2005. [9] L. Baker and M. R.-S. Keedwell, An colony omsaon for large-scale waer dsrbuon nework omsaon, n AISB 2008 Convenon Communcaon, Ineracon and Socal Inellgence, vol. 1,. 44, 2008. [10] I. Monalvo, J. Izquerdo, R. Pérez-García, and M. Herrera, Imroved erformance of so wh selfadave arameers for comung he omal desgn of waer suly sysems, Engneerng alcaons of arfcal nellgence, vol. 23, no. 5,. 727-735, 2010. [11] N. Moosavan and B. K. Roodsar, Soccer league comeon algorhm: a novel mea-heursc algorhm for omal desgn of waer dsrbuon neworks, Swarm and Evoluonary Comuaon, vol. 17,. 14-24, 2014. [12] A. Shbu and M. J. Reddy, Cross enroy omzaon for omal desgn of waer dsrbuon neworks, Inernaonal Journal of Comuer Informaon Sysems and Indusral Managemen Alcaons, vol. 5,. 308-316, 2013. [13] M. Afshar, Alcaon of a max-mn an sysem o on layou and sze omzaon of e neworks, Engneerng omzaon, vol. 38, no. 03,. 299-317, 2006. [14] M. H. Afshar, Layou and sze omzaon of reelke e neworks by ncremenal soluon buldng ans, Canadan Journal of Cvl Engneerng, vol. 35, no. 2,. 129-139, 2008. [15] S. H. Saleh and T. T. Tanymboh, Couled oology and e sze omzaon of waer dsrbuon 2846
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