An Effective League Championship Algorithm for the Stochastic Multi- Period Portfolio Optimization Problem

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1 An Effecve League Champonhp Algorhm for he Sochac Mul- Perod Porfolo Opmzaon Problem Al Huenzadeh Kahan *1, Mohammad Eyvaz 2, Amn Abba-Pooya 3 Faculy of Indural and Syem Engneerng, Tarba Modare Unvery, Tehran, Iran. * 1 Correpondng Auhor, E-mal addre: a.kahan@modare.ac.r, Cellphone: , Offce Tel: , Addre: Nar, Jalal AleAhmad, Tehran, P.O.Box: E-mal addre: mohammad.eyvaz@modare.ac.r, Cellphone: E-mal addre: a.abbapooya@modare.ac.r, Cellphone: Abrac The mul-perod porfolo opmzaon model were nroduced o overcome he weaknee of he ngle-perod model va conderng a dynamc opmzaon yem. However, due o he nonlnear naure of he problem and rapd growh of he ze complexy wh ncreang he number of perod and cenaro, h udy devoed o developng a novel league champonhp algorhm (LCA) o maxmze he porfolo mean-varance funcon ubjec o dfferen conran. A Vecor Auo Regreon model alo developed o emae he reurn on rky ae n dfferen me perod and o mulae dfferen cenaro of he rae of reurn accordngly. Bede, we proved a vald upper bound of he objecve funcon baed on he dea of ung urrogae relaxaon of conran. Our compuaonal reul baed on ample daa colleced from S&P 500 and 10-year T. Bond ndce ndcae ha he qualy of porfolo, n erm of he meanvarance meaure, obaned by LCA 10 o 20 percen beer han hoe of he commercal ofware. Th ound promng ha our mehod can be a uable ool for olvng a varey of porfolo opmzaon problem. Keyword: porfolo opmzaon; ngle and mul-perod model; league champonhp algorhm 1 Inroducon The problem of decon-makng under uncerany for choong ae clae one of he mporan opc n fnancal area. Markowz [1] wa he fr cholar ha conduced reearch on porfolo opmzaon model. By combnng opmzaon wh probably heory, [1] model nvemen conderng uncerany. Takng accoun of he reurn on nvemen a he average of reurn and he rk of nvemen a he varance wa he echnque for mahemacal modelng. Therefore, rk condered by calculang he varance n he mean-varance model, whch ha 1

2 been uded n Yohmoo [2], Be and Hloukova [3], Lu e al. [4], Corazza and Favareo [5], o menon a few example. I worh menonng ha he above-menoned ude have he aumpon of ngle-perod porfolo opmzaon, whch doe no uffcenly model he realworld condon n nvemen. Due o he fac ha marke condon change over me and nveor decde on her wealh accordngly, ngle-perod model need o be exended o mulperod one. In oher word, wealh allocaon o ae clae n nconen fnancal marke wh hgh dvery requre employng mul-perod ochac opmzaon model. Mul-perod model for porfolo opmzaon adequaely ake accoun of uncerany for effecve parameer lke reurn of ae cla and exernal cah flow [6]. Th one of he mporan advanage of mul-perod porfolo opmzaon model over ngle-perod model. The goal of mul-perod porfolo opmzaon (MPPO) o mnmze rk and maxmze reurn. Whle maxmzng reurn of nvemen n porfolo enangled wh makng decon on he percenage of he overall porfolo value allocaed o each porfolo componen, mnmzng he rk of dfferen nvemen nrumen alo mporan a he ame me o creae or manan porfolo wh he pecfed rk-reurn characerc. MPPO echnque demonrae dynamc apec of model for achevng opmal oluon and effcen froner [6,7]. 2 Leraure Revew A prevouly mpled, ngle-perod model form he ba for mul-perod one. A recen udy by Erenlce and Kalayc [8] ha nvegaed he ue of warm nellgence for porfolo opmzaon n ngle- and mul-perod opmzaon cae. A an example of ngle-perod model, Yohmoo [2] addreed he porfolo opmzaon conderng ranacon co and effec on he porfolo. Be and Hloukova [3] propoed a cloed-form oluon for porfolo elecon problem for uncorrelaed and bounded ae. Deng e al. [9] developed a parcle warm opmzaon algorhm for cardnaly conraned porfolo opmzaon (CCPO) problem, whch ouperformed genec algorhm, mulaed annealng, and abu earch. Woodde-Orakh e al. [10] preened meaheurc baed upon genec algorhm, mulaed annealng, and abu earch for mean-varance model of Markovz conderng he dcree conran of buy-n hrehold and cardnaly. A greedy randomzed adapve earch procedure developed n [11] for CCPO. An arfcal bee colony algorhm alo propoed n [12] for CCPO. There are pleny of model propoed for mul-perod porfolo opmzaon. For nance, Bradley and Crane [13] nroduced a mul-perod bond porfolo model and a new approach for effcenly olvng problem wh decompoon algorhm of mahemacal programmng. A ochac lnear programmng formulaon of a frm' hor-erm fnancal plannng problem had been modeled under uncerany by Kallberg e al. [14]. Sochac nework opmzaon model were decrbed for nvemen plannng under uncerany, and he performance of he model n 2

3 mulaon baed on horcal daa wa nvegaed by Mulvey and Vladmrou [15]. Furhermore, a mul-perod mean-varance porfolo elecon model wh bankrupcy conran under he framework of probably heory wa developed for a ochac marke by We and Ye [16]. Berma and Pachamanova [17] preened dfferen robu formulaon for he mulperod porfolo opmzaon problem and condered ranacon co n her model. Furhermore, hey compared he performance of robu formulaon o he performance of he radonal mean-varance formulaon. Cakmak and Ozekc [18] propoed a mul-perod porfolo opmzaon model ha rebalance he porfolo accordng o me horzon and he change of marke parameer. L and Ng [19] formulaed an analycal expreon for he mulperod mean-varance effcen froner. They alo nroduced an algorhm for fndng he opmal porfolo polcy. The bankrupcy approach wa ued for execung opmal porfolo polcy by Zhu e al. [20]. Ung he downde rk creron, Pnar [21] reved he mul-perod porfolo model. The mul-perod mean-emvarance-enropy model baed on pobly heory wa formulaed by Zhang e al. [22]. Fang e al. [23] ook a fuzzy e baed heory approach o he mul-perod porfolo opmzaon problem. The mul-perod porfolo model wh dfferen rae ha been alo nroduced for borrowng and lendng n fuzzy envronmen by Sadjad e al. [24]. Zhang and Zhang [25] condered a mul-perod fuzzy porfolo elecon problem wh abolue devaon a he rk conrol of porfolo. The model ncluded ranacon co, borrowng conran, hrehold conran and cardnaly conran. Addonally, dcree approxmae eraon mehod appled o olve he opmal porfolo. Yao e al. [26] preened mul-perod mean-varance porfolo elecon problem wh a ochac nere rae, where he movemen of he nere rae follow Vacek model. In addon, dynamc programmng approach and Lagrange dualy heory were ued o overcome ncreang complexy. Gven he fac ha he MPPO problem a nonlnear complex problem wh many local opma, and me a conran for fnancal problem, heurc mehod eem o be good ool for achevng a rade-off beween he qualy and he compuaonal me. Heurc mehod uch a Tabu Search [27,28], Genec Algorhm (Chan e al. [29]), Parcle Swarm Opmzaon (Sun e al. [30]) are ju ome example. Yan e al [31] nroduced a cla of mul-perod emvarance model and appled a novel hybrd Genec Algorhm (GA) wh Parcle Swarm Opmzaon (PSO) algorhm for olvng h model. Zhang e al. [32] ued poblc meanvarance approach o exend mul-perod fuzzy porfolo elecon problem. Moreover, hey formulaed a Parcle Swarm Opmzaon algorhm for hee porfolo elecon problem. Lu e al. [33] nvegaed a mul-perod porfolo elecon problem wh bankrupcy conrol and affne recoure n fuzzy nvemen envronmen and propoed a credblc mul-perod porfolo opmzaon model wh bankrupcy conrol and affne recoure. Furhermore, a hybrd Parcle Swarm Opmzaon algorhm wa ued for olvng he model. Lu e al. [34] preened a robu mul-perod porfolo model baed on he robu heory and propec heory. To olve he model, an mproved Parcle Swarm Opmzaon algorhm wa developed. Wang e al. [35] uded MPPO problem wh reurn condered a fuzzy random varable and propoed a fuzzy 3

4 mulaon-baed Parcle Swarm Opmzaon algorhm for olvng he problem. L e al. [36] condered an unceran mul-perod porfolo elecon problem wh he nfluence of ranacon co and bankrupcy. They olve he problem ung a genec algorhm wh penaly funcon. Dcouned ranacon co n a fuzzy envronmen ha been condered n mulperod porfolo elecon problem. Afer ranformng he problem no ngle-objecve equvalen, a dfferenal evoluon algorhm ha been ued o olve he problem. Th paper conder he ochac opmzaon model for mul-perod ae cla porfolo problem. Ae cla nclude cah, ock, bond and real ae. Sample daa are colleced from S&P 500 and 10-year T. Bond ndce. To exrac cenaro from horcal daa, a Vecor Auo Regreon (VAR) model fr developed o predc he reurn of rky ae. An upper bound on he opmal value of he mean-varance objecve funcon propoed baed on he urrogae relaxaon of conran va aggregaon over all cenaro. We prove ha he bound vald. To olve he problem nance of MPPO, a meaheurc algorhm developed baed on he League Champonhp Algorhm (LCA) and a penaly baed mehod ued o handle enropy conran. Our compuaon reveal ha he propoed mehodology ha a drac mpac on he qualy of he conued porfolo. I mprove he mean-varance objecve funcon from 10 o 20 percen over he reul provded by a commercal olver. Therefore, he conrbuon of h paper hreefold; Fr, a mul-perod porfolo opmzaon problem formulaed conderng dverfcaon n porfolo and uncerany n reurn of rky ae. To dverfy he porfolo conrucon, he Shannon Enropy meaure ued a an opmzaon conran. To cope wh uncerany, a cenaro baed approach followed and a VAR model developed o predc he reurn of rky ae o be ncluded n he opmzaon model. Va adopng a urrogae relaxaon echnque and mahemacal propere, an upper bound of he opmal objecve funcon value alo provded. To cope wh he nonlneary of he problem and o olve for larger complex nance a League Champonhp Algorhm propoed whch work effecve. Sacal reul how ha LCA gnfcanly mprove he reul provded by LINGO commercal olver. The remander of he paper organzed a follow. In Secon 2, we formulae a mulperod porfolo opmzaon (MPPO) problem wh nvemen rebalancng n everal dcree me pon (perod). Snce here uncerany n MPPO problem, we alo emae reurn rae for rky ae wh Vecor Auo Regreon (VAR) model baed on cenaro ree n Secon 3. Secon 4 devoed o degnng a mehod o oban an upper bound on he opmal value of he objecve funcon for our problem. In Secon 5, we gve a bref nroducon o League Champonhp Algorhm (LCA) and he applcaon of LCA on MPPO problem a hand. Secon 6 preen compuaonal expermen and reul of olvng he problem wh LINGO/Quadrac Solver and he propoed algorhm along wh he analy of he gap beween he reul of he Quadrac Solver and LCA. Fnally, he paper concluded n Secon 7. 3 Mul-Perod Porfolo Opmzaon (MPPO) Problem 4

5 The MPPO problem expree ochac opmzaon wh nvemen rebalancng n everal dcree me pon called perod. In MPPO problem, we mu pecfy plannng horzon and ae nvemen clae. Plannng horzon cover τ me perod nroduced by T {0,1,..., }. T 0 repreen he preen poon; alo T he plannng horzon. Preen poon condered a a arng pon for cenaro generaon. Decon are made a he end of each me perod. Ae nvemen clae, nroduced a he e A {1,2,..., E}, nclude broad nvemen clae, for example, ock, bond, real ae, or cah. Becaue ae clae mu rack marke, n our cae, S&P 500 ndex repreenave of ock and 10-year T. Bond for bond. Uncerany modeled hrough a fne number of cenaro each repreenng a poble realzaon of all unceran parameer. The e of all cenaro repreened by S. We wll how how we can realze dfferen cenaro ung a VAR model. The mahemacal model of MPPO problem can be preened a follow: Maxmze z Mean ( w ) (1 ) Var( w ) (1)..: x n,0 w 0 S (2) n A x n, w, n A n, rn, 1 x n, 1 S T (3) S, T, n A (4) x c (1 ) y S, T, n A, n 1 (5) n, n, n, n, n,, x y (1 ) c 1, 1, n, n, n, n A, n 1 n A, n 1 x x n, n, ln e,, n A w w S T (6) S T n A (7) The e of parameer and decon varable ued n he mahemacal model of MPPO are defned a follow: Parameer r 1 l Where l n, he rae of reurn of he ae n n me perod under n, n, cenaro w Inal wealh a me 0 0 Tranacon co ncurred n rebalancng rky ae n a he begnnng of n, me perod e Mnmum lm for dverfcaon conran a me 5

6 Decon varable x Amoun of money for ae n n me perod under cenaro afer n, rebalancng Amoun of money n ae n a he end of he perod under cenaro n, before rebalancng Wealh a he begnnng of me perod under cenaro w c Amoun of ae n purchaed for rebalancng a me under cenaro n, y Amoun of ae n old for rebalancng n me perod under cenaro n, The objecve funcon of (1) n he form of mean-varance. In h funcon, Mean ( w ) he mean and Var ( w ) he varance of he oal wealh acro all cenaro n he fnal perod, namely. Parameer ndcae he relave gnfcance of mean compared o he varance componen. There are oher objecve funcon uch a mean-varance-kewne and Von Neumann Morgenern expeced uly of wealh ha can be ued, oo. Equaon (2) ae ha he nveor oal ae n me 0 are equal o he wealh a he begnnng. Equaon (3) guaranee ha he nveor oal ae n me are equal o he wealh a he begnnng of me perod under cenaro. Equaon (4) updae he wealh accumulaed a he end of he perod under cenaro for each ae n before rebalancng. Equaon (5) depc he flow balance for all ae n each me perod and cenaro. Equaon (6) calculae he amoun of money nveed n cah n me perod under cenaro afer rebalancng. Fnally, conran (7) are he enropy conran ued o dverfy he porfolo. 4 Scenaro Generaon Ung Vecor Auo Regreon Model Uncerany n MPPO problem modeled wh a cenaro ree. The mporan parameer ha hould be emaed for nex perod he reurn rae of rky ae. There are wo general mehod for modelng fuure ae reurn [37]. The fr baed on he economc parameer ncludng nere rae, nflaon, and marke ndex. Th mehod called raonal expecaon and employ concepual macroeconomc model. For example, marke ndex can be ued o generae cenaro for he nex ock reurn [30]. The econd mehod, whch called adapve expecaon, depend only on he horcal daa of he explanaory varable. Th paper doe no focu on he evaluaon of hee mehod. Due o he fac ha modelng fuure even baed on macroeconomc model dffcul, he mehod employed n h paper baed on he econd approach. A Vecor Auo Regreon (VAR) model ued o conruc cenaro ree. Le S repreen he cenaro e ha defned by S : {1,2,..., S }, and each cenaro ha a probably ha 6

7 denoed by. Scenaro generaon can have dfferen model. For nance, a cenaro may be generaed wh a bnary ree ha bfurcae from each node wo branche. The ree ha ued n our model nclude dfferen pah ha are depced n Fgure 1, whch ha 6 cenaro and 6 perod. <<Iner Fgure 1 around here>> The general equaon of VAR model a follow: r H E r E r E r k (8) q q Then we have: E ( r f ) H E r E r... E r (9) q q n whch r he vecor of rae of reurn of he rky ae group. k he vecor of random durbance wh mean zero and a known varance whch drbued ndependenly n me horzon, and q he number of lag ued n he model. Furhermore, E,..., 1 E q are me ndependen conan marce ha are predced hrough acal mehod uch a maxmum lkelhood emaon. H he vecor of nercep from auoregreon. Rae of he reurn for rky ae uch a ock and bond are modeled baed on pa reurn. Redual play he man role n modelng he rae of reurn baed on pa daa becaue hey are ued o model he durbance of reurn n me horzon. VAR model for ock and bond are emaed ung he daa from 2001 o 2013 obaned from Yahoo Fnance a: r2, r 2, r2, r3, r3, 2 z (10) r r r r r u (11) 3, 3, 1 3, 2 2, 1 2, 2 where r 2, he rae of ock reurn and r 3, he rae of bond reurn under cenaro. Rae of cah reurn for all of he perod conan and equal o 0.12 and he model are run for 4 and 7 perod. The auocorrelaon of redual checked for he model aumpon. The fr lag coeffcen are acally gnfcan a 10% level. Uncerany characerzed by ock and z for u for bond. Thee random number are generaed ung a normal drbuon, nce he drbuon of he redual normal. Equaon (10) and (11) are ued for cenaro generaon wh 2, 5, 10, 20 and 50 pah for 4 and 7 perod. Each pah from T=0 o T=τ repreen a cenaro. Table 1 and 2 how ome realzaon of he cenaro generaed wh VAR model, pecfcally, he rae of reurn for wo rky ae under 10 cenaro and 7 perod. <<Iner Table 1 around here>> 7

8 <<Iner Table 2 around here>> 5 An Upper Bound for he Opmal Value of he Mean-Varance of he MPPO Problem The purpoe of h econ o provde a mehod o oban an upper bound on he opmal value of he mean varance (objecve funcon) for he MPPO problem. Our dea for he propoed upper bound mehod rele on he concep of urrogae conran. One way o relax gh conran o negrae hem no a e of conran ha weaker. The reul a lnear programmng problem ha called he relaxed problem wh urrogae/ubue conran. Sarng wh Equaon (2)-(5) and ummng over all cenaro, hen dvdng by he number of cenaro (.e., S ), we arrve a: x n, / S w 0 (12) n x n, / S w / S (13) n x / S r x / S c (1 )/ S y / S n, (14) n, n, 1 n, 1 n, n, n, Le u defne he followng equale: x n,0 / S x n,0, /, c S c n n,, x /, S x n n,, w / S w, y /, S y n n, A a reul, we can rewre Equaon (12)-(14) a follow: x w n,0 0 n x n, w (16) n x r x / S c (1 ) y n, (17) n, n, 1 n, 1 n, n, n, (15) Lemma 1. Gven ha rn, 1 x n, 1 / S rn, 1, he followng nequaly vald. r x / S r x 2 2 n, 1 n, 1 n, 1 n, 1 Proof: Baed on he Cauchy-Schwarz nequaly, we have he fr expreon n he rgh de of Equaon (17), we have: x y x y 2 2. Sarng from 8

9 r x ( r ) ( x ) n, 1 n, 1 n, 1 n, 1 r x / S ( r ) / S ( x ) / S r x n, 1 n, 1 n, 1 n, 1 n, 1 n, 1 (18) Lemma r, 1, 1 (1 S n x n ) rn, 1 x n, 1. 9 Proof. I wa proved by Jacobon [38] ha for x 0, he nequaly Such an nequaly enure ha x max and x 2 x 2 x 2 /9 max var( x ) x / 9 hold. 2 max. Furhermore, ha been acceped ha Nx where N he number of obervaon. So we can wre S r. Now he requred reul aanable n, 1 n, 1 rn, 1 r So, he fnal form of (17) reduced o (19) a follow: x S n, 1 n, 1 x n, 1 2 S x n, (1 ) rn, 1 x n, 1 cn, (1 n, ) y n,, n, n 1 (19) 9 x 9 In a mlar way, we can rea Equaon (6) o oban he one accordng o (20). 2 S x (1 ) r x y (1 ) c 1, 1, 1 1, 1 n, n, n, 9 n, n 1 n, n 1 (20) Now we arrve a he objecve funcon. Gven ha w τ = Mean (w τ ) an upper bound on he objecve funcon, he opmal value of he objecve funcon of he followng lnear programmng problem ((21)-(25)) gve an upper bound of he opmal value of he meanvarance funcon. Max z w (21) x n,0 w 0 (22) n x n, w (23) n 2 S x n, (1 ) rn, 1 x n, 1 cn, (1 n, ) y n,, n 1 (24) 9 9

10 2 S x (1 ) r x y (1 ) c n, n, 1 n, 1 n, n, n, 9 n 1 n 1 (25) 6 A League Champonhp Algorhm (LCA) Appled o MPPO Problem Th econ devoed o nroducon of he League Champonhp Algorhm (LCA) along wh applcaon o he problem of MPPO. There are ome feaure of LCA ha ha made an approprae choce for he problem of h udy. Fr, from a heorecal perpecve, LCA' concep ealy underandable and mplemenable. I alo ha few parameer, whch a feaure ha would make ue more derable. From he praccal perpecve, he conraned veron of LCA ha been evaluaed n erm of performance on 24 benchmark problem n [39]. Seven problem were quadrac and LCA demonrae ouandng performance compared o oher comparaor algorhm. Therefore, he performance of h newly propoed algorhm worh nvegang on he MPPO problem, whch of quadrac naure. 6.1 The League Champonhp Algorhm The League Champonhp Algorhm (LCA) a populaon-baed global opmzaon algorhm propoed by Huenzadeh Kahan [39,40,41,42,43], whch npred by por league. In h econ, a bref nroducon o LCA preened along wh requred adjumen o make uable for olvng MPPO problem. The mappng beween LCA and opmzaon problem elemen a follow: week repreen eraon, he playng rengh expreed a fne value, eam formaon repreen oluon, change n formaon lke he generaon of a new oluon, and he number of he eaon repreen oppng condon Rule of LCA There are ome dealzed rule of he regular champonhp envronmen o magne he arfcal champonhp modeled by LCA. Thee are: 1. The reul of a mach no predcable. 2. A eam wh hgher playng rengh ha more lkelhood of wnnng he oher eam. 3. From boh eam vewpon, he probably ha one eam bea anoher eam aumed equal. 4. The reul of a mach only wn or lo. 5. Team only concenrae on he nex mach nead of all fuure mache. 6. Any weakne n one eam he lack of a parcular rengh n ha eam. In LCA, a eam formaon (oluon) can be repreened wh a vecor of ze 1 n (n he number of varable) of real number. Each elemen relaed o one of he player and how he value of he varable of he problem. Le f ( X ( x 1, x 2,..., x n )) be an n varable numercal 10

11 n funcon ha hould be mnmzed over he decon pace pecfed a a ube of R. A eam formaon (a poenal oluon) for eam a week can be repreened by X ( x 1, x 2,..., x n )) X ( x, x,..., x ) wh f ( X ) ang he funcon value reuled from 1 2 n X Creang he League Schedule The fr ep o chedule he game n each eaon. In h paper, a ngle round-robn chedule ued. A ample league chedulng depced n Fgure 2 for a por league wh 8 eam. For he fr week (a), 1 play wh 8, 2 play wh 7, and o on. For he nex week, one eam fxed and oher eam are roaed clockwe o make a complee chedule. Aumng ha a por league ha L eam, he ngle round-robn ournamen need L (L-1)/2 mache, where (L-1) he number of mache and L/2 mache wll be held n parallel. For a por league wh S eaon, here are S (L-1) week of mache. <<Iner Fgure 2 around here>> Decdng he Oucome of a Mach To pecfy wnner or loer, ournamen elecon can be ued. Baed on he dealzed rule 1, we may wre p * f ( X ) f j * f ( X j) f p where X ( X ) he formaon of eam ( eam j) a week, f ( X )( f ( X )) he j playng rengh of eam ( eam j), p ( p ) he probably ha eam ( eam j) bea eam j ( * eam ) a week, and f a bound on he opmal value of he objecve funcon. Addonally, from dealzed rule 3 we have From (26) and (27), we oban p j j (26) p 1 (27) j p * f ( X j ) f f ( X ) f ( X ) 2f j * (28) In order o decde he wnner and loer, a unformly drbued random number n [0, 1] generaed. Two cae can occur: 1. If h random number le han or equal o 2. If h random number greaer han 11 p, eam wn and eam j loe. p, eam j wn and eam loe Seng Up a New Team Formaon In each eraon, here hould be a mechanm o move a e of oluon (populaon) by changng he confguraon of each oluon (eam). Aemen of eam weaknee and

12 rengh fr ep for eng up a new formaon. Furhermore, change n each eam formaon are baed on he reul of he eam whch we wll couner n week +1. Th ak conduced baed on SWOT analy. Le u defne he hree followng ndce: l he eam ha wll play wh eam (= 1,, L) a week +1 j he eam ha played wh eam (= 1,, L) a week k he eam ha played wh eam l a week The SWOT analy conduced accordng o he SWOT marx of Fgure 3. In order o deermne he eam formaon for playng wh eam l, f and l boh won her prevou mache, he S/T raegy (fr column n Fgure 3) for eam o focu on own rengh (or j weaknee) and rengh of l (or k weaknee). Oher cae can be nerpreed mlarly. X k <<Iner Fgure 3 around here>> Accordng he raegy adoped, we may wre he equaon for updang he oluon. X defned a he dfference beween he arrangemen of eam and eam k, focung on he rengh of eam k, whle X X focung on he weaknee of eam k. Knowng ha k each eam uch a play baed on be formaon B ( b 1, b 2,..., bn ), and he reul of mach analy, we may wre four raege for new eam formaon: If had won and l had won oo, hen he new formaon generaed by: x b y ( w r ( x x ) w r ( x x )) d 1,..., n (29) 1 d d d 1 1d d kd 1 2d d jd If had won and l had lo, hen he new formaon generaed by: x b y ( w r ( x x ) w r ( x x )) d 1,..., n (30) 1 d d d 2 1d kd d 1 2d d jd If had lo and l had won, hen he new formaon generaed by: x b y ( w r ( x x ) w r ( x x )) d 1,..., n (31) 1 d d d 1 1d d kd 2 2d jd d If had lo and l had lo oo, hen he new formaon generaed by: x b y ( w r ( x x ) w r ( x x )) d 1,..., n (32) 1 d d d 2 1d kd d 2 2d jd d In he above equaon, d he dmenon ndex. r 1d and r 2d are unform random number n [0,1]. w 1 and w 2 are coeffcen ued o cale he conrbuon of rerea or approach componen, repecvely. In equaon (29) o (32), whch ndcae wheher 1 x d y d a bnary change varable dffer from b or no. Only y 1 allow for dfference. d d 12

13 <<Iner Fgure 4 around here>> Baed on he above-menoned concep, he flowchar of LCA depced n Fgure 4. The fr ep o nalze conrol parameer and a random populaon of ndvdual (eam). Then, a league chedule generaed baed on ngle round-robn algorhm. The wnnng and long eam of each mach are deermned accordng o he ournamen elecon preened n The algorhm hen move o he nex e of oluon by ung he mach analy of SWOT marx. Th procedure connued unl a oppng creron me. 6.2 Applcaon of LCA o MPPO Problem When LCA ued o olve he mahemacal model (MPPO model), he objecve funcon and oluon repreenaon mu be defned fr. In h udy, we ued he clacal reurn-rk funcon a he objecve funcon. Due o he naure of uncerany of mul-perod problem, we need o form cenaro ree. Afer conrucon of cenaro ree, we wll nroduce he parameer requred o olve he MPPO model. A prevouly menoned, cenaro ree wa ued only for predcon of one parameer,.e., rae of reurn on rky ae ( l n, ). Probably of each cenaro ( ) aumed o be equal for each problem, for example, f he MPPO model ha 5 cenaro, equal o 0.2 for each cenaro. Addonally, he value of ranacon co ( n, ) avalable drecly from he marke. Wh l n,, and n, a parameer of he MPPO problem, we can apply LCA o fnd a oluon ha opmze he objecve funcon Soluon Repreenaon To run he algorhm, we nroduce a new decon varable denoed by k n,. I he proporon of ae wealh nveed n he ae afer rebalancng under everal cenaro over he plannng horzon, and equal o x, / w. Th percenage n each node mu be one hundred n percen for hree ued ae n h problem. If he amoun of k n, n each node were no equal o one hundred percen, we mu normalze k n, wh he equaon (33). 3 n, n, / n, n 1 k k k (33) The amoun of purchae and ell are wo oher decon varable n he mplemenaon of LCA. Therefore, hree e of decon varable are ued n each node for each ae and regardng n ae, we have 3*n decon varable n oal n each node. Therefore, he oluon (eam n LCA) repreened by a vecor comprng of hree e of varable: k,,,, n cn y n,, namely he amoun of money for ae n, he amoun of ae n purchaed, and he amoun of ae n old, repecvely (Fgure 5 ). 13

14 <<Iner Fgure 5 around here>> The number of node n h algorhm equal o: no. of node=[(number of perod -1)*number of cenaro]+1 and he oal number of varable [9*number of node], ha how problem dmenon Inroducon of Dverfcaon Conran n LCA by Penaly Funcon LCA wa orgnally nroduced for olvng unconraned connuou problem. Th algorhm can be ued for olvng porfolo dverfcaon whou conran by adopng a oluon repreenaon ha ake accoun of oher rercon. Bu when dverfcaon conran (7) are appled, we need a echnque o overcome he lmaon of LCA a well. To h purpoe, he penaly funcon mehod ued n h paper. The earch pace of conraned opmzaon problem con of wo ype of pon: feable and nfeable. Feable pon afy all conran, whle n nfeable pon, a lea one of he conran volaed. Penaly funcon echnque olved he conraned opmzaon problem hrough a ere of opmzaon problem whou conran. If he penaly oo large, mnmzng algorhm uually fall no he rap of local mnmum. On he oher hand, f he penaly oo mall, he algorhm can hardly deec feable opmal oluon. Penaly funcon commonly fall no wo man group: ac and dynamc. Sac penaly funcon ue fxed penaly value, whle dynamc penaly funcon adju penaly value n he coure of earch. The objecve funcon wh penaly funcon (F(x)) defned a follow: F( x ) f ( x ) h( k ) H ( x ) n x S R (34) where f ( x ) he objecve funcon of he problem, hk ( ) he penaly value ha adjued dynamcally, k he curren eraon of he algorhm, and H( x ) a penaly funcon ha can be defned a follow: m ( q ( x )) (35) 1 H ( x ) ( q ( x )) q ( x ) where q ( x ) max{0, g ( x )}, 1,..., m a funcon of conran volaon, ( q ( x)) perodc agnmen funcon, ( q ( x)) he power of penaly funcon, g ( x ) a pove varable and h (.), (.), (.) are dependen funcon. 14

15 7 Compuaonal Expermen and Reul In h econ, he parameer value ued for olvng he problem are preened. Numercal expermen and her analye are alo preened for demonrang he effcency of he propoed algorhm. 7.1 Parameer Seng Pror o he mplemenaon of he LCA and opmzaon olver, horcal daa of rae of reurn of rky ae were exraced from 2001 o 2013 a yearly nerval. A already aed, S&P 500 ndex and 10-year T. Bond are repreenave of ock and bond. The oher ae called cah; herefore, we have hree ae n oal n h udy. Alo, he eed perod are 4 and 7 and he eed cenaro are 2, 5, 10, 20 and 50. Moreover, he rae of reurn for rky ae predced wh VAR model and for cah, condered conan and equal o Lower bound of dverfcaon conran equal o ln(0,n), and condered equal o 0.6 n h udy. Tranacon co equal o for ock and for bond. Thee value are fxed for ock and bond n all perod and nal nvemen nended o be equal o 10 $. Furhermore, In LCA, he league ze e equal o 16, probably of ucce equal o and he number of eraon for each problem equal o w 1 and w 2 are alo generaed every me randomly beween [0,2]. Thee value are choen accordng o a plo udy on varou nance. The parameer value ha reul n he be reul were eleced for runnng numercal expermen. 7.2 Numercal Expermen The value of r n, were emaed by ung VAR model. Due o he pace lmaon, we ju repored n Table 1 and 2 he reul of r n, for 10 cenaro and 7 perod. LCA and LINGO/Quadrac Solver were run o maxmze he objecve funcon correpondng o a ere of for each mul-perod problem (QS wll be ued hereafer nead of LINGO/Quadrac Solver for he ake of brevy). In order o compare he performance of LCA wh QS n acal erm, Wlcoxon gned rank e ha been conduced wh.05 level of gnfcance. I worh nong ha he oupu of QS he ame for 10 run. The reul repored n he column 'p-value' how he probably ha he medan of he wo pared ample (objecve value of LCA and QS) are equal. Enre where he dfference gnfcan are underlned The MPPO Problem whou Dverfcaon Conran Table 3-5 conan he reul for problem e wh = 0.1,.5, and.9 each havng 10 problem nance. The fr econ of Table 3-5 repor he problem number and number of perod and cenaro. The econd par of Table 3-5 repor he objecve funcon and run me 15

16 of QS for each problem nance. The hrd econ repor he be, he wor, he average and he andard devaon of objecve funcon value obaned, and he average run me of he LCA algorhm for he problem wh dfferen perod and cenaro where he dverfcaon conran no condered. I hould be noed ha LCA wa execued 10 me for each problem nance o oban he average performance n erm of he objecve funcon and run me. Baed on he Table 3-5, he QS and LCA reul can be compared n erm of he objecve funcon obaned and he run me. Specfcally, he be, wor and average objecve funcon value obaned by he LCA are beer han QS n all problem nance. Th how he uperor performance of LCA. Bu, he run me of QS (le han one econd) le han he run me of LCA, whch range beween 1.5 and 39 mnue. The column conanng LCA me alo how ha n a fxed number of perod, he run me enve o he number of cenaro. The p-value for nance how ha he dfference acally gnfcan mo of he me (29 nance ou of 30 nance for all value of ). <<Iner Table 3-5 around here>> 7.3 Analy of he Reul of LCA and QS In h econ, he gap beween he reul of QS and LCA analyzed ung (36) The MPPO Problem wh Dverfcaon Conran Table 6, 7 and 8 repor he reul for he MMPO problem where he dverfcaon conran condered a decrbed prevouly. A can be een n all Table 6-8, reul of LCA are beer han QS n erm of objecve funcon value obaned for all of perod and cenaro. The run me of QS le han 1.5 mnue whle he run me of LCA range beween 2 and 38 mnue. A wa expeced, he run me ncreae when he number of cenaro ncreae for a fxed number of perod. In erm of objecve funcon value, he dfference beween LCA and QS gnfcan n mo of he nance (29 nance ou of 30 nance for all value of ). obj. value of LCA obj. value of QS Gap (36) obj. value of LCA Due o he nonlneare n he MPPO problem, QS oluon for he orgnal problem wh and whou he dverfcaon conran (Shannon enropy) are local. A a reul, poble ha he oluon o he problem ha olved by LCA mea-heurc algorhm be beer han he reul of QS. Fgure 6 and 7 how he gap beween he reul of QS and LCA whou dverfcaon conran for 4-perod and 7-perod cae, repecvely, under 2, 5, 10, 20 and 50 cenaro. For he cae of 4 perod (Fgure 6), gap vary beween 12 and 18 percen and here are mlar rend for dfferen value of θ. <<Iner Fgure 6 around here>> 16

17 For he cae of 7 perod (Fgure 7), he gap relavely wder and beween 9 and 21 percen for dfferen cenaro and. Moreover, Fgure 8 and 9 repor mlar reul for he problem wh dverfcaon conran. Fgure 8 how he gap for he 4-perod cae and he gap beween 8 o 14 percen for all cenaro and. The gap for he 7-perod cae beween 8 o 16 percen, a Fgure 9 demonrae. <<Iner Fgure 7 and 8 around here>> A Fgure 7-9 demonrae, all of he reul of LCA are beer han he oupu of QS n erm of objecve funcon value (beween 8 o 20 percen). <<Iner Fgure 9 around here>> In order o ee he behavor of LCA n boh cae of wh and whou dverfcaon conran, he convergence dagram lluraed n Fgure 10 for he nance 1 wh =0.9 for 1000 eraon. <<Iner Fgure 10 around here>> 8 Concluon and Fuure Reearch Th paper ued he Vecor Auo Regreon (VAR) o model he reurn of rky ae. VAR work baed on horcal daa. Alernavely, oher mehod ue economc ndcaor uch a nflaon, nere rae and marke ndexe. I mporan o noe ha he mehod of ung economc ndcaor depend on he knowledge of he economy fuure and economc varable under udy. A we know, he fuure of he marke complex o predc due o he mpac of macroeconomc varable. Therefore, baed on horcal daa from 2000 o 2013 for ock and bond, reurn of rky ae wa modeled. Scenaro made for he ock and bond were 2, 5, 10, 20 and 50 wh a perod of 4 and 7 year. Afer cenaro generaon, he upper bound model wa developed whch had a ngle cenaro and wa convenen o olve n comparon wh he problem mahemacal model. Mul-perod porfolo opmzaon model wa olved ung LINGO/Quadrac Solver (QS) and a new mea-heurc called LCA. Applyng h algorhm o mul-perod opmzaon problem wa due o nonlnear and complcaed naure of he problem. The reul of LCA for 4 and 7 perod and all cenaro were beer han he oupu of QS (beween 8 o 20 percen) n erm of he objecve funcon value obaned. The dfference n he objecve funcon obaned wa acally gnfcan n mo of nance. Th prove he effecvene of he LCA propoed for he problem. The curren reearch ough o maxmze mean-varance objecve funcon. There are oher objecve funcon uch a mean-varance-kewne and Von Neumann Morgenern expeced uly of wealh ha can be opmzed nead. Addonally, he effcency of oher recenly propoed mea-heurc algorhm uch a OIO [44,45] may be nvegaed n fuure reearche. The exenon of our model o connuou me cae alo worh analyzng, conderng he fac ha many nveor change her porfolo connuouly raher han a 17

18 dcree pon n me. Takng accoun of lqudy, whch one of he man concern for nveor n makng a porfolo decon, anoher ue o nvegae. 9 Reference 1. Markowz, H. "Porfolo elecon", The journal of fnance, 7(1), pp (1952). 2. Yohmoo, A. "The mean-varance approach o porfolo opmzaon ubjec o ranacon co", Journal of he Operaon Reearch Socey of Japan, 39(1), pp (1996). 3. Be, M. J., and Hloukova, J. "Porfolo Selecon and Tranacon Co", Compuaonal Opmzaon and Applcaon, 24(1), pp (2003). 4. Lu, S., Wang, S. Y., Qu, W., " Mean-varance-kewne model for porfolo elecon wh ranacon co", Inernaonal Journal of Syem Scence, 34(4), pp (2010). 5. Corazza, M., Favareo, D., "On he exence of oluon o he quadrac mxed-neger mean varance porfolo elecon problem", European Journal of Operaonal Reearch, 176(3), pp (2007). 6. Mulvey, J. M., and Shey, B. "Fnancal plannng va mul-age ochac opmzaon", Compuer & Operaon Reearch, 31(1), pp (2004). 7. Carno, D. R., Myer, D. H., and Zemba, W. T. "Concep, echncal ue, and ue of he Ruell-Yauda Kaa fnancal plannng model", Operaon reearch, 46(4), pp (1998). 8. Erenlce, O., and Kalayc, C. B., "A urvey of warm nellgence for porfolo opmzaon: Algorhm and applcaon", Swarm and Evoluonary Compuaon, 39, pp (2018). 9. Deng, G.-F., Ln, W.-T., and Lo, C.-C. "Markowz-baed porfolo elecon wh cardnaly conran ung mproved parcle warm opmzaon", Exper Syem wh Applcaon, 39, pp (2012). 10. Woodde-Orakh, M., Luca, C., and Bealey, J. E., "Heurc algorhm for he cardnaly conraned effcen froner", European Journal of Operaonal Reearch, 213, pp (2011). 11. Baykaoğlu, A., Yunuoglu, M. G., and Özoydan, F. B., "A GRASP baed oluon approach o olve cardnaly conraned porfolo opmzaon problem", Compuer & Indural Engneerng, 90, pp (2015). 12. Kalayc, C. B., Erenlce, O., Akyer, H., and Aygoren, H., "An arfcal bee colony algorhm wh feably enforcemen and nfeably oleraon procedure for cardnaly conraned porfolo opmzaon", Exper Syem wh Applcaon, 85, pp (2017). 13. Bradley, S. P., and Crane, D. B. "A dynamc model for bond porfolo managemen", Managemen Scence, 19(2), pp (1972). 18

19 14. Kallberg, J. G., and Zemba, W. T. "Comparon of alernave uly funcon n porfolo elecon problem", Managemen Scence, 29(11), pp (1983). 15. Mulvey, J. M., and Vladmrou, H. "Sochac nework opmzaon model for nvemen plannng", Annal of Operaon Reearch, 20(1), pp (1989). 16. We, S. Z., and Ye, Z. X. "Mul-perod opmzaon porfolo wh bankrupcy conrol n ochac marke", Appled Mahemac and Compuaon, 186(1), pp (2007). 17. Berma, D., and Pachamanova, D. "Robu mulperod porfolo managemen n he preence of ranacon co", Compuer & Operaon Reearch, 35(1), pp (2008). 18. Çakmak, U., and Özekc, S. "Porfolo opmzaon n ochac marke", Mahemacal Mehod of Operaon Reearch, 63(1), pp (2006). 19. L, D., and Ng, W. L. "Opmal Dynamc Porfolo Selecon: Mulperod Mean Varance Formulaon", Mahemacal Fnance, 10(3), pp (2000). 20. Zhu, S. S., L, D., and Wang, S. Y. "Rk conrol over bankrupcy n dynamc porfolo elecon: A generalzed mean-varance formulaon", Auomac Conrol, IEEE Tranacon on, 49(3), pp (2004). 21. Pınar, M. Ç. "Robu cenaro opmzaon baed on downde-rk meaure for mulperod porfolo elecon", OR Specrum, 29(2), pp (2007). 22. Zhang, W. G., Lu, Y. J., and Xu, W. J. "A poblc mean-emvarance-enropy model for mul-perod porfolo elecon wh ranacon co", European Journal of Operaonal Reearch, 222(2), pp (2012). 23. Fang, Y., La, K. K., and Wang, S. Y. "Porfolo rebalancng model wh ranacon co baed on fuzzy decon heory", European Journal of Operaonal Reearch, 175(2), pp (2006). 24. Sadjad, S. J., Seyedhoen, S. M., and Haanlou, K. "Fuzzy mul perod porfolo elecon wh dfferen rae for borrowng and lendng", Appled Sof Compung, 11(4), pp (2011). 25. Zhang, P., and Zhang, W. G. "Mulperod mean abolue devaon fuzzy porfolo elecon model wh rk conrol and cardnaly conran", Fuzzy Se and Syem, 255, pp (2014). 26. Yao, H., L, Z., and L, D. "Mul-perod mean-varance porfolo elecon wh ochac nere rae and unconrollable lably", European Journal of Operaonal Reearch, 252(3), pp (2016). 27. Berger, A. J., Glover, F., and Mulvey, J. M. "Solvng global opmzaon problem n longerm fnancal plannng", Sac and Operaon Reearch Techncal Repor, Prnceon Unvery (1995). 28. Berger, A. J., and Mulvey, J. M. "Inegrave rk managemen for ndvdual nveor", Worldwde Ae and Lably Modelng (1996). 19

20 29. Chan, M. C., Wong, C. C., Cheung, B. K. S., and Tang, G. N. "Genec algorhm n mulage ae allocaon yem", In Syem, Man and Cybernec, 2002 IEEE Inernaonal Conference on, Vol. 3, pp. 6 pp., IEEE (2002, Ocober). 30. Sun, J., Fang, W., Wu, X., La, C. H., and Xu, W. "Solvng he mul-age porfolo opmzaon problem wh a novel parcle warm opmzaon", Exper Syem wh Applcaon, 38(6), pp (2011). 31. Yan, W., Mao, R., and L, S. "Mul-perod em-varance porfolo elecon: Model and numercal oluon", Appled Mahemac and Compuaon, 194(1), pp (2007). 32. Zhang, X., Zhang, W., and Xao, W. "Mul-perod porfolo opmzaon under pobly meaure", Economc Modellng, 35, pp (2013). 33. Lu, Y. J., Zhang, W. G., and Zhang, Q. "Credblc mul-perod porfolo opmzaon model wh bankrupcy conrol and affne recoure", Appled Sof Compung, 38, pp (2016). 34. Lu, J., Jn, X., Wang, T., and Yuan, Y. "Robu mul-perod porfolo model baed on propec heory and ALMV-PSO algorhm", Exper Syem wh Applcaon, 42(20), pp (2015). 35. Wang, B., L, Y., Waada, J., "Mul-perod porfolo elecon wh dynamc rk/expecedreurn level under fuzzy random uncerany", Informaon Scence, , pp. 1-18, (2017). 36. L, B., Zhu, Y., Sun, Y., Aw, G., and Teo, K. L., "Mul-perod porfolo elecon problem under unceran envronmen wh bankrupcy conran", Appled Mahemacal Modellng, 56, pp (2018). 37. Zhao, Y., and Zemba, W. T. A, "ochac programmng model ung an endogenouly deermned wor cae rk meaure for dynamc ae allocaon", Mahemacal Programmng, 89(2), pp (2001). 38. Jacobon, H. I. "The maxmum varance of rerced unmodal drbuon", The Annal of Mahemacal Sac, 40(5), pp (1969). 39. Huenzadeh Kahan, A. "An effcen algorhm for conraned global opmzaon and applcaon o mechancal engneerng degn: League champonhp algorhm (LCA)", Compuer-Aded Degn, 43(12), pp (2011). 40. Huenzadeh Kahan, A. "League champonhp algorhm: a new algorhm for numercal funcon opmzaon", In 2009 Inernaonal Conference of Sof Compung and Paern Recognon, pp , IEEE (2009, December). 41. Huenzadeh Kahan, A. "League Champonhp Algorhm (LCA): An algorhm for global opmzaon npred by por champonhp", Appled Sof Compung, 16, pp (2014). 42. Almorad, M. R., Huenzadeh Kahan, A. A league champonhp algorhm equpped wh nework rucure and backward Q-learnng for exracng ock radng rule, Appled Sof Compung, 68, pp (2018). 20

21 43. Huenzadeh Kahan, A., Abba-Pooya, A., Karmyan, S. A Rg-Baed Formulaon and a League Champonhp Algorhm for Helcoper Roung n Offhore Tranporaon, Proceedng of he 2nd Inernaonal Conference on Daa Engneerng and Communcaon Technology: ICDECT 2017, Volume 828 of Advance n Inellgen Syem and Compung. 44. Huenzadeh Kahan, A. "A new meaheurc for opmzaon: opc npred opmzaon (OIO)", Compuer & Operaon Reearch, 55, pp (2015). 45. Huenzadeh Kahan, A. An effecve algorhm for conraned opmzaon baed on opc npred opmzaon (OIO), Compuer-Aded Degn, 63, pp (2015). Dr. Al Huenzadeh Kahan hold degree n Indural Engneerng from Amrkabr Unvery of Technology (PolyTechnque of Tehran), Iran. He worked a a podocoral reearch fellow a he deparmen of Indural Engneerng and Managemen Syem wh he fnancal uppor of Iran Naonal Ele foundaon. Dr. Kahan currenly an aan profeor n he Deparmen of Indural and Syem Engneerng, Tarba Modare Unvery and ha been acve n appled opmzaon reearch feld nce H reearch focue on modelng and olvng hard combnaoral opmzaon problem n area uch a logc and upply nework, revenue managemen and prcng, reource chedulng, groupng problem, fnancal engneerng, ec. A oluon mehodologe for real world engneerng degn problem, he ha nroduced everal nellgen opmzaon procedure, whch npre from naural phenomena, uch a League Champonhp Algorhm (LCA), Opc Inpred Opmzaon (OIO), Fnd-Fx-Fnh-Explo-Analyze (F3EA) meaheurc algorhm and Groupng Evoluon Sraege (GES). Dr. Kahan ha publhed over 70 peer-revewed journal and conference paper, and ha erved a a referee for everal ouandng journal uch a: IEEE Tranacon on Evoluonary Compuaon, Omega, Compuer & Operaon Reearch, Journal of he Operaonal Reearch Socey, Compuer & Indural Engneerng, Inernaonal Journal of Producon Reearch, Informaon Scence, Appled Sof Compung, Ecologcal Informac, Engneerng Opmzaon, Opmal Conrol and Applcaon ec. He ha receved everal award from Iran Naonal Ele Foundaon and n 2016 he wa honored by he Academy of Scence of Iran a he ouandng young cen of Indural Engneerng. Mohammad Eyvaz receved h BSc degree n Indural Engneerng from he Unvery of Tabrz n He receved h MSc degree n Fnancal Engneerng from Tarba Modare Unvery n H reearch nere porfolo opmzaon. Amn Abba-Pooya receved h BSc degree and MSc degree boh n Indural Engneerng from Yazd Unvery and Tarba Modare Unvery, repecvely. H reearch nere appled operaon reearch. Th nclude developmen of mahemacal and meaheurc mehod, daa analy, and applcaon of acal mehod. He ha worked on real-world projec n he area of Vehcle Roung Problem and varan (uch a helcoper 21

22 roung, pckup and delvery, and mlk-run logc), Relably Engneerng, and Spare Par Managemen and Opmzaon. 22

23 Fgure Capon Fgure 1 - Scenaro generaon for 6 perod and 6 pah Fgure 2 - A ample ngle round-robn chedulng algorhm Fgure 3. SWOT marx for eng up a new formaon Fgure 4 - The flowchar of LCA Fgure 5 - Soluon repreenaon for he MPPO problem Fgure 6 - Gap of oluon of LCA and QS oupu whou dverfcaon conran for 4 perod Fgure 7 - Gap of oluon of LCA and QS oupu whou dverfcaon conran for 7 perod Fgure 8 - Gap of oluon of LCA and QS oupu wh dverfcaon conran for 4 perod Fgure 9 - Gap of oluon of LCA and QS oupu wh dverfcaon conran for 7 perod Fgure 10 Convergence dagram for nance 1 whou (a) and wh (b) dverfcaon conran Table Capon Table 1 Sample cenaro realzed for he ock rae of reurn (10 cenaro - 7 perod) Table 2 - Sample cenaro realzed for he bond rae of reurn (10 cenaro - 7 perod) Table 3 - Reul of oluon of LCA and QS whou dverfcaon conran ( =0.1) Table 4 - Reul of oluon of LCA and QS whou dverfcaon conran ( =0.5) Table 5 - Reul of oluon of LCA and QS whou dverfcaon conran ( =0.9) Table 6 - Reul of oluon of LCA and QS wh dverfcaon conran ( =0.1) Table 7 - Reul of oluon of LCA and QS wh dverfcaon conran ( =0.5) Table 8 - Reul of oluon of LCA and QS wh dverfcaon conran ( =0.9) 23

24 Fgure Fgure 1 - Scenaro generaon for 6 perod and 6 pah Fgure 2 - A ample ngle round-robn chedulng algorhm Adop S/T raegy Adop S/O raegy Adop W/T raegy Adop W/O raegy S won, l won. Focu on own rengh (or weaknee of j) W won, l lo. Focu on own rengh (or weaknee of j) lo, l won. Focu on own weaknee (or rengh of j) lo, l lo. Focu on own weaknee (or rengh of j) O T rengh of l (or weaknee of k) weaknee of l (or rengh of k) rengh of l (or weaknee of k) Fgure 3. SWOT marx for eng up a new formaon weaknee of l (or rengh of k) 24

25 Fgure 4 - The flowchar of LCA k 1, k 2, k 3, c1, c 2, c3, y 1, y 2, y 3, Fgure 5 - Soluon repreenaon for he MPPO problem 25

26 Gap Gap nu=0.1 nu=0.5 nu=0.9 20% 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% Scenaro Fgure 6 - Gap of oluon of LCA and QS oupu whou dverfcaon conran for 4 perod 25% nu=0.1 nu=0.5 nu=0.9 20% 15% 10% 5% 0% Scenaro Fgure 7 - Gap of oluon of LCA and QS oupu whou dverfcaon conran for 7 perod 26

27 Gap Gap nu=0.1 nu=0.5 nu=0.9 16% 14% 12% 10% 8% 6% 4% 2% 0% Scenaro Fgure 8 - Gap of oluon of LCA and QS oupu wh dverfcaon conran for 4 perod nu=0.1 nu=0.5 nu=0.9 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% Scenaro Fgure 9 - Gap of oluon of LCA and QS oupu wh dverfcaon conran for 7 perod 27

28 a) whou dverfcaon conran ( =0.9) b) wh dverfcaon conran ( =0.9) Fgure 10 Convergence dagram for nance 1 whou (a) and wh (b) dverfcaon conran 28

29 Perod Perod Table Table 9 Sample cenaro realzed for he ock rae of reurn (10 cenaro - 7 perod) Scenaro Table 10 - Sample cenaro realzed for he bond rae of reurn (10 cenaro - 7 perod) Scenaro

30 Table 11 - Reul of oluon of LCA and QS whou dverfcaon conran ( =0.1) Problem QS LCA Perod Scenaro Inance Objecve Tme () Be Wor Average S. Dev. Avg. Tme() p-value p< p< p< <p< <p< p< p< p< p< <p<0.05 Problem Inance Perod Scenaro Table 12 - Reul of oluon of LCA and QS whou dverfcaon conran ( =0.5) QS LCA Objecve Tme () Be Wor Average S. Dev. Avg. Tme() p-value <p< <p< <p< <p< <p< p< p< p< p< p<

31 Problem Inance Perod Scenaro Table 13 - Reul of oluon of LCA and QS whou dverfcaon conran ( =0.9) QS LCA Objecve Tme() Be Wor Average S. Dev. Avg. Tme() p-value <p< <p< <p< <p< <p< p< p< p< p< <p<0.05 Problem Inance Perod Scenaro Table 14 - Reul of oluon of LCA and QS wh dverfcaon conran ( =0.1) QS LCA Objecve Tme() Be Wor Average S. Dev. Avg. Tme() p-value p< p< p< <p< <p< p< p< p< p< <p<