Open Access Chaotic Particle Swarm Optimization Algorithm for Hub and Spoke Systems with Congestion

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1 Sed Orders for Reprits to The Ope Automatio ad Cotrol Systems Joural, 2014, 6, Ope Access Chaotic Particle Swarm Optimizatio Algorithm for Hub ad Spoke Systems with Cogestio Weiwei Wu * ad Hui Wag Najig Uiversity of Aeroautics ad Astroautics, Najig, Jiagsu, , P.R. Chia Abstract: Cosiderig the hub airports are traffic trasfer poits, the cogestio is easily happeed. The cost caused by cogestio will rise sigificatly. The hub-ad-spoke airlie etwork optimizatio model with cogestio cost is desiged. I the actual operatios of airlies, such problem is difficult to be solved by usig the classical optimizatio methods. This paper presets a Particle Swarm Optimizatio (PSO) algorithm. To improve the performace of stadard PSO algorithm ad avoid trappig ito local excellet result, a chaos PSO algorithm of traffic volume multi-path assigmet is preseted. Empirical aalysis shows that optimizatio desig with cogestio cost ca avoid the excessive cogestio pheomeo i the hub odes. The proposed algorithm ca solve the o-liear etwork optimizatio problem efficietly. Keywords: Chaos, cogestio cost, hub-ad-spoke airlie etwork, particle swarm optimizatio. 1. INTRODUCTION Airlie etwork layout has importat strategic sigificace for the log term operatio ad market competitiveess of a compay. I air trasportatio plaig, the two most commoly used trasport etwork layouts are poit-topoit (PP) ad hub-ad-spoke (HS) structure. The disadvatage of PP etworks is that they caot effectively use the ecoomies of scale, ad curretly most researches maily focus o HS airlie etwork desig [1]. The literature [2-5] respectively preset sigle allocatio or multiple allocatio mixed iteger programmig etwork desig model, i which the goal is geerally cosidered operatio cost ad costructio cost. The solutio is Lagrage relaxatio algorithm, dual-ascet algorithm, heuristic based o tabu search ad brach ad boud method. Grove ad O Kelly [6] simulates the operatio of sigle allocatio HS system, ad aalyzes that aircraft delay is greatly iflueced by the heavy flow i hub airports. Some papers [7-12] have tackled the cogestio effects restrictig the amout of flow trasitig through a hub by meas of capacity costraits. However, i the study of etwork desig, it is isufficiet that we oly cosider that the flow of hub ode caot exceed its capacity limit, because with the traffic flow close to capacity, the airport begis to appear cogestio ad lower utilizatio of resources. So we must cosider the cogestio effect of hub airport. With the icreasig traffic flow of hub airports, the cogestio is icreasig, operatio cost will show o-liear icreasig tred. The curret airlie etwork desig model ad its algorithm caot effectively solve the cogestio problem. Based o the above aalysis, this paper maily studies o the HS etwork optimizatio desig, cosiderig the cogestio cost i the objective fuctio. We ca use heuristic algorithms to hadle o-liear objective fuctio, such as eural etwork algorithm, geetic algorithm ad at algorithm. However, the shortcomigs of these methods are to solve with low efficiecy ad poor stability, which caot esure the optimizatio effect of solutio. The particle swarm algorithm has a few idividual umbers, simple calculatio, good robustess ad parallel computig advatage. I order to avoid fallig ito local optimal solutio, it ca combie with chaos ad particle swarm optimizatio [13]. Thus, this paper costructs a chaotic particle swarm optimizatio algorithm (CPSOA) of multi path traffic assigmet, the solvig result ca avoid the heavy traffic flow of ay oe hub ode exceedig its capacity restrictio, which has effective shut effect. Also, it guaratees the efficiecy for solvig large oliear etwork optimizatio problems. 2. MODEL FORMULATION: HS NETWORK DESIGN WITH CONGESTION Cosiderig hub airport is gatherig the large amout of traffic flow, cogestio occurs maily i the hub airport. Elhedhli ad Hu [14] has cosidered the costs of the cogestio effects explicitly o the objective fuctio. Usig a covex cost fuctio that icreases expoetially as more flows go through the hubs. f (u) = au b where u is the flow at a hub; a ad b are positive costats with b 1. Usig the model of the ucapacitated multiple allocatio p hub locatio problem (UMApHMP) [15], the flow through / Betham Ope

2 P PPP 610 The Ope Automatio ad Cotrol Systems Joural, 2014, Volume 6 Wu ad Wag b au a=10 a=1 b au b=1.5 b=1.3 b=1.1 a=0.1 b=1 u u b=1.3 a=1 Fig. (1). The Cogestio Cost Fuctios with Differet Values of a ad b. hub k is r s> r W X m rs rskm. So, the cogestio cost at hub k is f(u). f (u) = au b = a( W rs X rskm ) b s>r r m Fig. (1) plots the cogestio cost fuctios for differet values of the parameters a ad b. Cosiderig the cogestio effect o etwork desig, some improvemet o UMApHMP model is preseted. The cogestio covex cost fuctio is added ito the objective fuctio. The model is stated as follows: mi Z = W " C "#$ x "#$ + a W " X "#$ (1) st.. y k = p (2) k=1 x rskm = 1 ; r,s = 1,, k=1 m=1 " x rskm y k ; r,s,k = 1,, m=1 " x rskm y m ; r,s,m = 1,, k=1 y k {0,1} ; k = 1,, ; 0 x rskm 1 r,s,k,m = 1,, The ucapacitated HS etwork desig is defied o a graph G=(N,A), where N is a set of odes ad A is the set of routes (r,s) (with origi ode r ad destiatio ode s). (3) (4) (5) (6) w be the flow from ode r to ode s. We will geerally rs assume that w = w, r,s N. rs sr x be the fractio of flow from ode r to ode s that is rskm routed via hubs at locatios k ad m i that order. If k=m, the variable x rskm represets the oe-hub-stop service through hub k. c rskm be the trasportatio cost per uit of flow from ode r to ode s routed via hubs k ad m. k ca be the same as m. yk # y k = " $# be a biary variable defied as follows: 1 if a hub is loacated at ode k 0 otherwise The discout factor represets the ecomies of scale o the iter-hub coectio, 0 < < 1. The objective fuctio (1), Z calculates the sum of cogestio cost ad operatig cost. Costrait (2) states that the umber of hubs to be located is p. Costraits (3) assure that the flow for every pair r-s is routed via some hub pair. If oly oe hub is used, we have k=m. Costraits (4) ad (5) express whe a city is a o-hub city, there is o passegers trasferrig through it to other cities. Costraits (6) give the defiitio of decisio variables. The model is o-liear mixed iteger plaig problem. The papers [14, 16] solved it with Lagragea heuristic algorithm ad Beders algorithm. Cosiderig the practicability of itelliget optimizatio algorithm, we use chaotic particle swarm optimizatio algorithm to solve this problem. CPSO algorithm has o limit for target fuctio scale, which ca be used to solve the oliear objective fuctio model, ad the complexity of the objective fuctio has little impact o its covergece efficiecy..

3 Chaotic Particle Swarm Optimizatio Algorithm The Ope Automatio ad Cotrol Systems Joural, 2014, Volume CHAOTIC PARTICLE SWARM OPTIMIZATION ALGORITHM 3.2. Algorithm Implemetatio 3.1. Algorithm Descriptios Dedicated to computatioal itelligece research, Keedy [17] first proposed PSO techique to replace the existig evolutioary techiques, such as geetic algorithm (GA) [18]. The algorithm simulates bird foragig for purpose of cluster flight sharig mechaism to make the group behavior to achieve optimal iformatio i a group. Chaos is a kid of oliear pheomeo widely existig i ature. It is chaotic, ad its iteral structure is very delicate, ad it is sesitive to iitial coditios, i a certai rage accordig to fixed rules, which are ot repeated traversal of all state. Chaos have the characteristics of radomess, ergodicity ad regularity. The chaos properties ca be used to coduct search optimizatio, i order to itegrate ito particle swarm algorithm, amely, so-called CPSOA. For a optimizatio problem, the decisio variable X is dimesioal variable X = [x,, x ], ad the variables are called the particles. Assumed i the -dimesioal solutio space, each particle has two state descriptio, positio ad velocity, respectively, X = (X ", X ",, X " ) ad V = (V", V",, V" ). The positio Xi represets the solutio of problem. The basic idea of CPSOA ca be uderstood as: first iitialized particle swarm ad the i solutio space, suppose i t times, ad the optimal solutio of particle i is pbesti(t), ad this is idividual extremum, ad the optimal solutio for the whole particle swarm is gbest(t), which is global extremum. Whe iterate over the t+1 times, its update expressio is: V t + 1 = ωv t + c r pbest t X t +c r gbest(t) X t (7) Xi(t+1)= Xi(t)+ Vi(t+1) (8) I the formula, t represets the curret iteratio times, c1, c2 represet two learig factors, c1=c2=2, Ad two radom variables r, r (0,1). The iertia factor ω [ω"#, ω"# ], tmax is the maximum umber of iteratios. I order to prevet the particle from leavig far away out of the searchig space, the velocity of the particle is restricted i [Vmi,Vmax ], ad the locatio of the particle also stays i permitted rage, ad fially the solutio, gbest, is the global optimal. Accordig to the above metioed basic particle swarm algorithm (BPSO), although it eeds to cofirm the parameters less, realizig process is simple, yet it is easy to trap ito local optimum. Therefore chaotic dyamics is icorporated ito the above PSO, the logistic reflectio is defied as follows. k+1 = µ kj (1" kj ) k = 1, 2 j j "(0,1) j " 0.25,0.5,0.75 (9) The key poit of PSO techique is to fid a suitable expressio to make the particle correspodig to the appropriate solutio. Oe of the key problems of airlie etwork optimizatio desig is how to distribute the traffic flow i O-D path. Thus, this paper cosiders costructig a path umber dimesioal space, ad passegers eed to select a feasible path, ad each path value is the distributed flow of this path, amely, the particle correspodig vector Xi represets a solutio to the distributio of flow of each path. The path flow f" ad traffic demad q " of ay OD path (r,s) eed to meet the flow coservatio priciple, thus, we eed to make correspodig revisio i algorithm, The CPSO algorithm steps are listed as below. Step 1, (iitializatio): (1) iitialize, c1, c2, ω maximum umber of iteratios. (2) iitialize each dimesio of Xi radomly betwee [0, q " ], that is, f" = q " c (c [0,1]). I order to esure the costat value q ", it eeds to coduct ormalized processig of f". f" = q " f" f" (3) Iitialize each dimesio of V radomly betwee [1q ",q " -1] (4) Judge the fitess of all particles usig objective fuctio. (5) Select the iitial fitess values as the idividual historical optimal solutio Pi, the search for optimal solutio withi the swarm. Step 2, Repeat util a stoppig criterio or the maximal umber of iteratio is satisfied: (1) For each particle, calculate particle swarm V ad X from (7) ad (8). Whe V ad X exceed over the rage, accordig to boudary value, ad the ormalized processig of X. (2) Evaluate all the particles usig objective fuctio (1), whe the curret fitess fuctio of a certai particle is better tha its historical optimal fitess, the curret fitess is remarked as historical optimal fitess, ad the curret positio is the historical optimal locatio of this particle. (3) Chaos optimized the optimal locatio Pg of curret particle swarm usig Logistic reflectio (9), i order to obtai chaos optimizatio result. (4) Seekig for the curret optimal solutio i the swarm. If it is better tha the history optimal solutio, the update Pg. If the plurality of idividual have optimal solutios, the radomly selected oe is the curret optimal solutio. Durig the solutio process of CPSOA, paralled distribute the flow for each feasible path, ad flow distributio as the solutio of the correspodig particle will adjust the

4 612 The Ope Automatio ad Cotrol Systems Joural, 2014, Volume 6 Wu ad Wag Table cities ad their assiged umber. Num. Cities Num. Cities Num. Cities 1 Beijig 6 Hagzhou 11 Wuha 2 Chagsha 7 Kumig 12 Urumqi 3 Chegdu 8 Najig 13 Xiame 4 Guagzhou 9 Shaghai 14 Xi a 5 Haikou 10 Sheyag 15 Zhezhou Table 2. Flow distributio ratio for 15 cities i Chia. (a,b) Flow Distributio Ratio hub1 hub2 hub3 Max/mi a= (1,1.3) (1,1.5) (1,2.0) traffic o the path usig the historical experiece of idividual ad groups, thus gradually close to optimal positio, so as to obtai the optimal result. 4. COMPUTATIONAL EXPERIMENTS HS topologies are a importat class of etwork desig that take full advatage of ecoomies of scale o iter-hub coectios. Hub locatio is very importat ad some airports havig importat regio advatage become resources that airlie compaies use to compete. Cosiderig the rakig order of cities, passeger traffic, geographical advatages ad requiremets of the Civil Aviatio Admiistratio of Chia, we ca choose Beijig, Shaghai ad Guagzhou as hub airports. Accordig to the airlies operatioal reports, the route cost is related to segmet distace. I this computatioal example, we cosider the route cost is the route distace corrected by empirical coefficiet (airlies provided the cost of aircraft type operatig the differet routes). The passeger traffic betwee each pair of OD is obtaied from the Civil Aviatio Statistical Yearbook Suppose that a airlie has established bases i Beijig, Shaghai ad Guagzhou ad prepares to costruct HS etwork with 15 cities (See Table 1). The istace has bee solved without cogestio costs ad the usig b={1.3, 1.5, 2.0}. The flow distributio ratio of three fixed hubs are show i Table 2. I Table 2, the results of the first row do t cosider the cogestio effects. From the secod colum to the fourth colum, the percetage of the total demad is calculated i each istalled hub. The last colum is the flow imbalace ratio, that is the ratio of the largest over the lowest percetage hub flow. For istace, o row oe, the flow imbalace is take as ". =1.89. amely, the large flow imbalace would. happe whe o cogestio effects are assumed. Whe the cogestio cost is cosidered, the sortig of flow distributio of three hub airports is ot varied, but the flow imbalace ratio is decreased. For istace, b=2, the flow imbalace ratio teds to its miimum value of 1. This ca explaied by the iterferece of cogestio cost, the flow distributio of paths are very differet from that of the ucapacitated model. Figs. (2) ad (3) are respectively flow distributio results of ucapacitated HS etwork ad HS etwork with cogestio cost. Although the two situatios choose the same hub airports, their flow distributio path are differet. I Fig. (3), the umber of routes of etwork icreases, the flow of OD is o loger simply distributed with the shortest path, such as, the OD flow from Xi a to Najig, accordig to the shortest path, would select path as Xi a-beijig-shaghai-najig. Cosiderig the cogestio effect i hub airports, some passeger flow would choose Xi a-beijig-najig, some other passeger flow would choose Xi a-shaghai-najig. The OD flow from Sheyag to Shaghai, the shortest path is Sheyag-Beijig-Shaghai, cosiderig the cogestio cost i hub airports, some traffic flow would choose direct flight from Sheyag to Shaghai. Table 3 represets the flow distributio of some paths with three groups of cogestio parameters. The paths are radomly selected, ad the umber of each ode correspods with the city umber i Table 1. As a ad b chage, the distributed flow values of the same path are also differet. The last three colums show that the flow distributio value of oe path accordig to variatios i a ad b, ad 0 represets o flow distributio of this path.

5 Chaotic Particle Swarm Optimizatio Algorithm The Ope Automatio ad Cotrol Systems Joural, 2014, Volume Urumqi Sheyag Beijig Hub No-hub Truk lie Xi a Zhegzhou Chegdu Wuha Chagsha Kumig Najig Hagzhou Guagzhou Haikou Xiame Shaghai Fig. (2). Ucapacitated HS etwork (p = 3, α = 0.6, a = 0). Urumqi Beijig Sheyag Chegdu Xi a Zhegzhou Najig Wuha Chagsha Hagzhou Shaghai Hub No-hub Truk lie Kumig Xiame Guagzhou Haikou Fig. (3). HS etwork with cogestio cost (p = 3, α = 0.6, a = 1, b = 2). Table 3. Flow distributio for some paths with three groups of cogestio parameters. Path Number Some of Paths Set Flow Distributio r k m s a=1 b=1.1 a=1 b=1.5 a=1 b=

6 614 The Ope Automatio ad Cotrol Systems Joural, 2014, Volume 6 Wu ad Wag Table 3. cotd Path Number Some of Paths Set Flow Distributio r k m s a=1 b=1.1 a=1 b=1.5 a=1 b= The results show that ay hub airport ca avoid overloaded traffic flow whe cosiderig o-liear cogestio cost i the objective fuctio, ad the ubalaced flow distributio is reduced betwee hub airports. For example, As parameter a ad b icrease, the flow of some paths are chaged from zero to a distributed value, such as the 2 d ad 3 rd path, also the distributed large amout of flow of some paths are diverted, such as 1 st ad 6 th path. Therefore, some busy hub airports ca reduce cogestio i order to avoid the correspodig flight delay. CONCLUSION This paper proposes CPSO algorithm aimed to HS etwork desig cosiderig cogestio cost at hub airports. The objective fuctio is added o-liear cogestio cost, whe the flow value of hubs icreases, the cogestio cost is a covex fuctio, which would icrease expoetially, therefore, the flow of hub airports ca be diverted effectively to miimize the total cost. Comparig with other algorithms of airlie etwork desig, CPSO ca quickly get the traffic flow distributio of multi paths uder specified hubs. Give each OD demad, CPSO algorithm ca get the results of the flow distributio. Before optimizatio, this algorithm must be determied the set of paths for each OD pair by the exhaustive procedure. Usig the method of the mutative scale chaos mutatio, which ca be very good to accelerate the covergece speed ad avoid trappig ito local optimizatio. Through example demostratio ad compariso aalysis of Chiese airlie trasport data, the results show that the algorithm has a good shut effect o effectively solvig hub ode cogestio problem, ad it tries to alleviate the flight delay degree caused by hub ode flow icrease. ABOUT THE AUTHORS First Author Weiwei Wu, Uiversity associate professor of Najig Uiversity of Aeroautics ad Astroautics, Ph.D. The author s major is Maagemet Sciece ad Egieerig. Secod Author Hui Wag, master degree i trasportatio egieerig, studyig i Najig Uiversity of Aeroautics ad Astroautics. The author s major is Trasportatio Egieerig. CONFLICT OF INTEREST The author cofirms that this article cotet has o coflict of iterest. ACKNOWLEDGEMENTS This work was fiacially supported by the Natural Sciece Foudatio of Chia (No , ). REFERENCES [1] J. Zhu. Air trasportatio plaig, Northwester Polytechical Uiversity Press, 2009, pp [2] M.E. O Kelly, A quadratic iteger program for the locatio of iteractig hub facilities, Europea Joural of Operatios Research, vol. 32, pp , [3] T. Ayki, O a quadratic iteger program for the locatio of iteractig hub facilities, Europea Joural of Operatioal Research, vol. 46, pp ,1990 [4] J.F. Campbell, Hub locatio ad the p-hub media problem. Operatios Research, vol. 44, pp , [5] M.E. O Kelly, D. Brya, D.S. Kapov, ad J.S. Kapov, Hub etwork desig with sigle ad multiple allocatio: A computatioal study, Locatio Sciece, vol. 4, o. 3, pp , [6] G.P. Grove, ad M.E. O'Kelly, Hub etworks ad simulated schedule delay, Papers of the Regioal Sciece Associatio vol. 59, pp , 1986.

7 Chaotic Particle Swarm Optimizatio Algorithm The Ope Automatio ad Cotrol Systems Joural, 2014, Volume [7] H. Yama, ad G. Carello. Solvig the hub locatio problem with modular lik capacities, Computers & Operatios Research, vol. 32, pp , [8] V. Mariao, ad D. Serra, Locatio models for airlie hubs behavig as m/d/c queues, Computers & Operatios Research, vol. 30, pp , [9] T. Ayki, Lagragia relaxatio based approaches to capacitated hub-ad-spoke etwork desig problem, Europea Joural of Operatioal Research, vol 79, pp ,1994. [10] J. Ebery, M. Krishamoorthy, A. Erst, ad N. Bolad, The capacitated multiple allocatio hub locatio problems: formulatios ad algorithms, Europea Joural of Operatioal Research, vol. 120, pp , [11] J.F. Campbell, G. Stiehr, A.T. Erst, ad M. Krishamoorthy, Solvig hub arc locatio problems o a cluster of workstatios, Parallel Computig, vol. 29, pp , [12] I.R. Marti, ad J.J. Salazar-Goalez, Solvig a capacitated hub locatio problem, Europea Joural of Operatioal Research, vol. 184, pp , [13] T.M. Qi, M. Kai, ad W.X. Jiag, F.B. Jiag, Z.J. Gag, OFDM resource allocatio algorithm based o chaos particle swarm optimizatio, Cotrol ad Decisio, vol. 27, o. 7, pp , [14] S. Elhedhli, ad F.X. Hu, Hub-ad-spoke etwork desig with cogestio, Computers & Operatios Research, vol. 32, pp , [15] J.F. Campbell, Locatio ad allocatio for distributio systems with trasshipmets ad trasportatio ecoomies of scale, Aals of Operatios Research, vol. 40, pp , [16] R.S. de Camargo, G. Mirada Jr., ad R.P.M. Ferreira, ad H.P. Lua, Multiple allocatio hub-ad-spoke etwork desig uder hub cogestio, Computers & Operatios Research, vol. 36, pp , [17] J. Keedy, ad R.C. Eberhart, Particle swarm optimizatio. I: Proceedigs of IEEE Iteratioal Coferece o Neural Networks, Piscataway, NJ, 1995, pp [18] D.E. Goldberg, Geetic Algorithms i Search, Optimizatio, ad Machie Learig, MA: Addiso Wesley, Received: November 11, 2014 Revised: Jauary 07, 2015 Accepted: Jauary 21, 2015 Wu ad Wag; Licesee Betham Ope. This is a ope access article licesed uder the terms of the Creative Commos Attributio No-Commercial Licese ( which permits urestricted, o-commercial use, distributio ad reproductio i ay medium, provided the work is properly cited.