An Improved Flower Pollination Algorithm for Solving Integer Programming Problems

Transcription

1 Appl. Mah. Inf. Sc. Le. 3, No. 1, ( Appled Mahemacs & Informaon Scences Leers An Inernaonal Journal hp://dx.do.org/ /amsl/ An Improved Flower Pollnaon Algorhm for Solvng Ineger Programmng Problems Khall AL-Wagh* Faculy of Compuer scence & Informaon Sysem, Thamar Unversy, Thamar, Republc of Yemen *E-mal: Receved: 8 Mar. 01, Revsed: 16 Dec. 015, Acceped: 19 Dec. 015 Publshed onlne: 1 Jan. 015 Absrac: Flower pollnaon algorhm s a new naure-nspred algorhm, based on he characerscs of flowerng plans. In hs paper, a new mehod s developed based on he flower pollnaon algorhm combned wh chaos heory (IFPCH o solve neger programmng problems. IFPCH rounds he parameer values o he closes neger afer producng new soluons. Numercal smulaon resuls show ha he algorhm proved o be superor n almos all esed problems. Keywords: Flower pollnaon algorhm; mea-heurscs; opmzaon; chaos; neger programmng. 1 Inroducon The real world opmzaon problems are ofen very challengng o solve, and many applcaons have o deal wh NP-hard problems [1]. To solve such problems, opmzaon ools have o be used even hough here s no guaranee ha he opmal soluon can be obaned. In fac, for NP problems, here are no effcen algorhms a all. As a resul of hs, many problems have o be solved by ral and errors usng varous opmzaon echnques []. In addon, new algorhms have been developed o see f hey can cope wh hese challengng opmzaon problems. Among hese new algorhms, many algorhms such as parcle swarm opmzaon, cuckoo search and frefly algorhm, have ganed populary due o her hgh effcency. In hs paper, we have used IFPCH algorhm for solvng neger programmng problems. Ineger programmng s NP-hard problems [3-10]. The name lnear neger programmng s referred o he class of combnaoral consraned opmzaon problems wh neger varables, where he objecve funcon s a lnear funcon and he consrans are lnear nequales. The Lnear Ineger Programmng (also known as LIP opmzaon problem can be saed n he followng general form: Max cx (1 s..ax b, ( xz n (3 where he soluon x Z n s a vecor of n neger varables: x = (x 1, x,, x n T and he daa are raonal and are gven by he m n marx A, he 1 n marx c, and he m 1 marx b. Ths formulaon ncludes also equaly consrans, because each equaly consran can be represened by means of wo nequaly consrans lke hose ncluded n eq. (. Ineger programmng addresses he problem rased by non-neger soluons n suaons where neger values are requred. Indeed, some applcaons do allow a connuous soluon. For nsance, f he objecve s o fnd he amoun of money o be nvesed or he lengh of cables o be used, oher problems preclude : he soluon mus be dscree [3]. Anoher example, f we are consderng he producon of je arcraf and x 1 = 8. je arlners, roundng off could affec he prof or he cos by mllons of dollars. In hs case we

2 K. AL-Wagh :An Improved Flower Pollnaon Algorhm _3 need o solve he problem so ha an opmal neger soluon s guaraneed. The possbly o oban neger values s offered by neger programmng: as a pure neger lnear programmng, n whch all he varables mus assume an neger value, or as a mxedneger lnear programmng whch allows some varables o be connuous, or a 0-1 neger model, all he decson varables have neger values of zero or one[8-9]. A wde varey of real lfe problems n logscs, economcs, socal scences and polcs can be formulaed as lnear neger opmzaon problems. The combnaoral problems, lke he knapsack-capal budgeng problem, warehouse locaon problem, ravellng salesman problem, decreasng coss and machnery selecon problem, nework and graph problems, such as maxmum flow problems, se coverng problems, machng problems, weghed machng problems, spannng ree problems and many schedulng problems can also be solved as lnear neger opmzaon problems [8-9]. Exac neger programmng echnques such as cung plane echnques [8-9]. The branch and he bound boh have hgh compuaonal cos, n largescale problems [8-9]. The branch and he bound algorhms have many advanages over he algorhms ha only use cung planes. One example of hese advanages s ha he algorhms can be removed early as long as a leas one negral soluon has been found and an aanable soluon can be reurned alhough s no necessarly opmal. Moreover, he soluons of he LP relaxaons can be used o provde a wors-case esmae of how far from opmaly he reurned soluon s. Fnally, he branch mehod and he bound mehod can be used o reurn mulple opmal soluons. Snce neger lnear programmng s NPcomplee, for ha reason many problems are nracable. So nsead of he neger lnear programmng, he heursc mehods mus be used. For example, Swarm nellgence meaheurscs, amongs whch an an colony opmzaon, arfcal bee colony opmzaon parcle swarm opmzaon [8-9]. Also Evoluonary algorhms, dfferenal evoluon and abu search were successfully appled no solvng neger programmng problems [8-9]. Heurscs ypcally have polynomal compuaonal complexy, bu hey do no guaranee ha he opmal soluon wll be capured. In order o solve neger programmng problems, mos of he heurscs runcae or round he real valued soluons o he neares neger values. In hs paper, Ba algorhm s appled o neger programmng problems and he performance was compared wh oher harmony search algorhms [9]. Flower pollnaon s an nrgung process n he naural world. Is evoluonary characerscs can be used o desgn new opmzaon algorhms. The algorhm obaned good resuls were dealng wh lower-dmensonal opmzaon problems, bu may become problemac for hgherdmensonal problems because of s endency o converge very fas nally. Ths paper nroduced an mproved Flower pollnaon algorhm by negrang wh chaos o mprove he relably of he global opmaly, and also enhances he qualy of he resuls. Ths paper s organzed as follows: afer nroducon, he orgnal Flower pollnaon algorhm s brefly nroduced. Than n secon 3 nroduces he meanng of chaos. In secon, he proposed algorhm s descrbed, whle he resuls are dscussed n secon 5. Fnally, conclusons are presened n secon 6. The Orgnal Flower Pollnaon Algorhm Flower Pollnaon Algorhm (FPA was founded by Yang n he year 01. Inspred by he flow pollnaon process of flowerng plans are he followng rules: Rule 1: Boc and cross-pollnaon can be consdered as a process of global pollnaon process, and pollen-carryng pollnaors move n a way ha obeys Le'vy flghs. Rule : For local pollnaon, a boc and self-pollnaon are used. Rule 3: Pollnaors such as nsecs can develop flower consancy, whch s equvalen o a reproducon probably ha s proporonal o he smlary of wo flowers nvolved. Rule : The neracon or swchng of local pollnaon and global pollnaon can

3 Appl. Mah. Inf. Sc. Le. 3, No. 1, ( be conrolled by a swch probably p[0,1], wh a slgh bas oward local pollnaon. In order o formulae updang formulas, we have o conver he aforemenoned rules no updang equaons. For example, n he global pollnaon sep, flower pollen gamees are carred by pollnaors such as nsecs, and pollen can ravel over a long dsance because nsecs can ofen fly and move n a much longer range[10].therefore, Rule 1 and flower consancy can be represened mahemacally as: 1 x x L( ( x B (1 Where x s he pollen or soluon vecor x a eraon, and B s he curren bes soluon found among all soluons a he curren generaon/eraon. Here γ s a scalng facor o conrol he sep sze. In addon, L(λ s he parameer ha corresponds o he srengh of he pollnaon, whch essenally s also he sep sze. Snce nsecs may move over a long dsance wh varous dsance seps, we can use a Le'vy flgh o mae hs characersc effcenly. Tha s, we draw L > 0 from a Levy dsrbuon: ( sn( / 1 L ~,( S S0 0 ( 1 S Here, Γ(λ s he sandard gamma funcon, and hs dsrbuon s vald for large seps s > 0. Then, o model he local pollnaon, boh Rule and Rule 3 can be represened as x x U( x 1 j k x and j x (3 Where xk are pollen from dfferen flowers of he same plan speces. Ths essenally maes he flower consancy n a lmed neghborhood. Mahemacally, f x and j x k comes from he same speces or seleced from he same populaon, hs equvalenly becomes a local random walk f we draw U from a unform dsrbuon n [0, 1].Though Flower pollnaon acves can occur a all scales, boh local and global, adjacen flower paches or flowers n he no-so-far-away neghborhood are more lkely o be pollnaed by local flower pollen han hose faraway. In order o mae hs, we can effecvely use he swch probably lke n Rule or he proxmy probably p o swch beween common global pollnaon o nensve local pollnaon. To begn wh, we can use a nave value of p = 0.5as an nally value. A prelmnary paramerc showed ha p = 0.8 mgh work beer for mos applcaons [10]. The basc seps of FP can be summarzed as he pseudo-code shown n Fgure 1. Flower pollnaon algorhm Defne Objecve funcon f (x, x = (x 1, x,..., x d Inalze a populaon of n flowers/pollen gamees wh random soluons Fnd he bes soluon Bn he nal populaon Defne a swch probably p [0, 1] Defne a soppng creron (eher a fxed number of generaons/eraons or accuracy whle ( <MaxGeneraon for = 1 : n (all n flowers n he populaon f rand <p, Draw a (d-dmensonal sep vecor L whch obeys a L evy dsrbuon Global pollnaon va 1 x ( x L B x else Draw U from a unform dsrbuon n [0,1] Do local pollnaon va 1 x x U( x x end f Evaluae new soluons If new soluons are beer, updae hem n he populaon end for Fnd he curren bes soluon B end whle Oupu he bes soluon found Fg. 1 Pseudo code of he Flower pollnaon algorhm 3 Chaos Theory Generang random sequences wh a longer perod and good conssency s very mporan for easly smulang complex phenomena, samplng, numercal analyss, decson makng and especally n heursc opmzaon [11]. Is qualy deermnes he reducon of sorage and compuaon me o acheve a desred accuracy [1]. Chaos s a deermnsc, random-lke process found n nonlnear, dynamcal sysem, whch s non-perod, non-convergng and bounded. Moreover, depends on s nal condon and parameers [1-19]. Applcaons of chaos n several dscplnes ncludng operaons research, physcs, engneerng, economcs, bology, phlosophy and compuer scence [19-]. Recenly chaos has been exended o varous opmzaon areas because can more easly escape from local mnma and mprove global convergence n comparson wh oher sochasc opmzaon algorhms [-5]. Usng chaoc j k

4 K. AL-Wagh :An Improved Flower Pollnaon Algorhm _3 sequences n flower pollnaon algorhm can be helpful o mprove he relably of he global opmaly, and hey also enhance he qualy of he resuls. 3.1 Chaoc maps A random-based opmzaon algorhms, he mehods usng chaoc varables nsead of random varables are called chaoc opmzaon algorhms (COA [1]. In hese algorhms, due o he non-repeon and ergodcy of chaos, can carry ou overall searches a hgher speeds han sochasc searches ha depend on probables [5-9]. To acheve hs ssue, heren onedmensonal, non-nverble maps are ulzed o generae chaoc ses. We wll llusrae some of well-known one-dmensonal maps as: Logsc map The Logsc map s defned by: Y n+1 = μy n (1 Y n Y(0,1 0 < ( 3.1. The Sne map The Sne map s wren as he followng equaon: Y n+1 = μ sn(πy n Yε(0,1 0 < μ ( Ierave chaoc map The erave chaoc map wh nfne collapses s descrbed as: Y n+1 = sn ( μπ Y n μ (0,1( Crcle map The Crcle map s expressed as: Y n+1 = Y n + α ( β π sn(πy n mod 1 ( Chebyshev map The famly of Chebyshev map s wren as he followng equaon: Y n+1 = cos(kcos 1 (Y n Y ( 1,1( Snusodal map Ths map can be represened by Y n+1 = μy k sn (πy n ( Gauss map The Gauss map s represened by: 0 Y n = 0 Y n+1 = { μ Y n mod 1 Y n 0 ( Snus map Snus map s formulaed as follows: Y n+1 =.3(Y n sn(πy n ( Dyadc map Also known as he dyadc map, b shf map, x mod 1 map, Bernoull map, doublng map or saw ooh map. Dyadc map can be formulaed by a mod funcon: Y n+1 = Y n mod 1 ( Snger map Snger map can be wren as: Y n+1 = μ(7.86y n 3.31Y n Y n Y n (13 beween 0.9 and Ten map Ths map can be defned by he followng equaon: μy Y n+1 = { n Y n < 0.5 μ(1 Y n Y n 0.5 (1 The Proposed Algorhm (IFPCH for Solvng Defne Inegral In he proposed chaoc Flower pollnaon algorhm, we used chaoc maps o une he Flower pollnaon algorhm parameer and mprove he performance [11-1]. The seps of he proposed chaoc Flower pollnaon algorhm for solvng neger programmng are as follows: Sep 1 defne he objecve funcon and nalzes a populaon and fnd he bes soluon B n he nal populaon. Sep Calculae p by he seleced chaoc maps. Sep 3If (rand <p hen global pollnaon va 1 x x ( f L( ( x B else do local pollnaon va seleced chaoc map. Sep Evaluae new soluons f beer, updae hem n he populaon. Sep 5Fnd he curren bes soluon B. Sep 6Oupu he bes soluon found. 5 Numercal Resuls In hs secon, we wll carry ou numercal smulaon based on some well-known unconsraned opmzaon problems o nvesgae he performances of he proposed algorhm. The bes resuls obaned by IFPCH for es problems (1 7 are presened n Table 1. In hese problems, he nal parameers are se a n= 50 and he number of eraons s se o = 1000.The seleced

5 Appl. Mah. Inf. Sc. Le. 3, No. 1, ( chaoc map for all problems s he logsc map, accordng o he followng equaon: Y n+1 = μy n (1 Y n (15 Clearly, Y n [0,1] under he condons ha he naly 0 [0,1], where n s he eraon number and μ =.The resuls of IFPCH algorhm are conduced from 30 ndependen runs for each problem. The comparson beween he resuls deermned by he proposed approach and he sandard flower pollnaon algorhm are repored n Table 1. The resuls have demonsraed he superory of he proposed approach o fndng he opmal soluon. All he expermens were performed on a Wndows 7 Ulmae 6-b operang sysem; processor Inel Core runnng a.81 GHz; 6 GB of RAM and code was mplemened n C# Tes problem 1 The problem formulaon s: P x x x x... 1( 1 1 x D, Where D s he dmenson and x[-100,100] D. The global mnmum = Tes problem The mahemacal formulaon of he opmzaon problem can be saed as follows: x1. T P ( x x. x x1... xd.., where D s he. xd dmenson and x[-100,100] D. The global mnmum = Tes problem 3 The problem can be saed as follows: P 3 ( x (9. x1. x 11 (3. x1. x 7. The global mnmum = Tes problem The problem formulaon s: P ( x ( x 10x 5( x 1 10( x1 x The global mnmum = 0. 3 x 5( x 5.5. Tes problem 5 The problem can be expressed as: P ( x ( x x 11 ( x1 x The global mnmum = Tes problem 6 The problem formulaon s: P ( x x x1 03.6x 18.5x1 x The global mnmum = x 5.7. Tes problem 7 The problem can be saed as follows: P7 ( x 100( x x1 (1 x1. The global mnmum = 0. x. 3 Table 1 he bes soluon of proposed algorhm and FP algorhm for solvng neger programmng problems Sandard FP Algorhm IFPCH Algorhm Sandard Sandard Tes Mean of Mean of Dmenson Success Devaon of Success Devaon of problem Ieraon Ieraon Rae Ieraon Rae Ieraon P1 P 15 17/ / / / / / / / / / / / P3 30/ / P 19/ / P5 30/ /

6 K. AL-Wagh :An Improved Flower Pollnaon Algorhm _36 P6 30/ / P7 5/ / Conclusons Ths paper nroduced an mproved Flower pollnaon algorhm by blendng wh chaos for solvng neger programmng problems. Several problems have been used o prove he effecveness of he proposed mehod. IFPCH algorhm s superor o sandard FP n erms of boh effcency and success rae. Ths mples ha IFPCH s poenally more powerful n solvng NP-hard problems. The reason for geng beer resuls han sandard flower pollnaon algorhm, usng chaos helps he algorhms o escape from local soluons. Table 1 shows he resuls of IFPCH algorhm are prvleged compared wh he resuls of Sandard flower pollnaon algorhm. References [1] L. A. Wolsey, "Ineger programmng," IIE Transacons, vol. 3, pp. -58, 000. [] G. B. Danzg, Lnear programmng and exensons: Prnceon unversy press, [3] G. L. Nemhauser and L. A. Wolsey, Ineger and combnaoral opmzaon vol. 18: Wley New York, [] E. Beale, "Ineger programmng," n Compuaonal Mahemacal Programmng, ed: Sprnger, 1985, pp. 1-. [5] C. H. Papadmrou and K. Seglz, Combnaoral opmzaon: algorhms and complexy: Courer Dover Publcaons, [6] H. Wllams, "Logc and Ineger Programmng, Inernaonal Seres n Operaons Research & Managemen Scence," ed: Sprnger, 009. [7] A. Schrjver, Theory of lnear and neger programmng: Wley. com, [8] O. Abdel-Raouf, M. Abdel-Base, and I. El-henawy."An Improved Flower Pollnaon Algorhm wh Chaos." 01. [9] O. Abdel-Raouf, M.Abdel-Base, and I. El-Henawy. "An Improved Chaoc Ba Algorhm for Solvng Ineger 01 ". Problems Programmng [10] X-S. Yang," Flower pollnaon algorhm for global opmzaon", Unconvenonal Compuaon and Naural Compuaon, Lecure Noes ncompuer Scence, Vol. 75, pp. 0-9,01. [11] L. M. Pecora and T. L. Carroll, "Synchronzaon n chaoc sysems," Physcal revew leers, vol. 6, pp. 81-8, [1] D. Yang, G. L, and G. Cheng, "On he effcency of chaos opmzaon algorhms for global opmzaon," Chaos, Solons & Fracals, vol. 3, pp , 007. [13] A. H. Gandom, G. J. Yun, X.-S. Yang, and S. Talaahar, "Chaos-enhanced acceleraed parcle swarm opmzaon," Communcaons n Nonlnear Scence and Numercal Smulaon, 01. [1] B. Alaas, "Chaoc harmony search algorhms," Appled Mahemacs and Compuaon, vol. 16, pp , 010. [15] W. Gong and S. Wang, "Chaos An Colony Opmzaon and Applcaon," n Inerne Compung for Scence and Engneerng (ICICSE, 009 Fourh Inernaonal Conference on, 009, pp [16] B. Alaas, "Chaoc bee colony algorhms for global numercal opmzaon," Exper Sysems wh Applcaons, vol. 37, pp , 010. [17] A. Gandom, X.-S. Yang, S. Talaahar, and A. Alav, "Frefly algorhm wh chaos," Communcaons n Nonlnear Scence and Numercal Smulaon, vol. 18, pp , 013. [18] J. Mngjun and T. Huanwen, "Applcaon of chaos n smulaed annealng," Chaos, Solons & Fracals, vol. 1, pp , 00. [19] L. d. S. Coelho and V. C. Maran, "Use of chaoc sequences n a bologcally nspred algorhm for engneerng desgn opmzaon," Exper Sysems wh

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