Open Access Study on Optimization of Logistics Distribution Routes Based on Opposition-based Learning Particle Swarm Optimization Algorithm

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1 Sed Orders for Reprts to 38 The Ope Automato ad Cotrol Systems Joural, 05, 7, 38-3 Ope Access Study o Optmzato of Logstcs Dstrbuto Routes Based o Opposto-based Learg Partcle Swarm Optmzato Algorthm Lu Xao-ju ad Zhag B* Departmet of Logstcs ad Iformato Maagemet, Zhuha College of Jl Uversty, Zhuha, Cha Abstract: I vew of shortcomgs of the partcle swarm optmzato algorthm such as poor late optmzato ablty ad proeess to local optmzato etc, ths paper proposes a opposto-based learg partcle swarm optmzato (OBLPSO) algorthm for the optmzato of logstcs dstrbuto routes, frstly, establshes a logstcs dstrbuto route optmzato mathematcal model, ad the solves through collaborato ad formato exchage amog partcles, troduces a opposto-based learg mechasm to mprove partcle swarm optmzato ablty ad covergece rate, ad fally coducts smulato test o the performace of OBLPSO algorthm o Matlab 0 platform. The smulato results show that OBLPSO algorthm ca be used to obta logstcs dstrbuto solutos wth short tme ad ratoal routes ad thus has certa practcal value, compared wth other logstcs dstrbuto route optmzato algorthms. Keywords: Opposto-based learg, partcle swarm optmzato, logstcs dstrbuto, route selecto.. INTRODUCTION Logstcs dstrbuto s to delver goods to specfed destato o tme accordg to customer requremets; dstrbuto route optmzato s the key of logstcs dstrbuto, ad ratoal selecto of logstcs dstrbuto routes ca save costs ad mprove ecoomc beefts; thus, logstcs dstrbuto route optmzato becomes a hot but dffcult topc study o logstcs feld []. Logstcs dstrbuto route optmzato s a multcostrat ad combatoral optmzato problem, ad belogs to No-determstc Polyomal Complete (NPC) dffculty; wth the rapd developmet of logstcs scale, maual arragemet of logstcs routes ca hardly meet moder logstcs requremets, ad dstrbuto routes are curretly arraged by computer automatcally []. Wth respect to dstrbuto route optmzato, scholars sped a lot of tme ad eergy o -depth ad extesve researches, ad put forward three types of dstrbuto route optmzato methods amely precse methods, heurstc algorthms ad swarm tellgece algorthms etc. Precse methods clude dyamc programmg method ad brach & boud method etc, could reach optmal solutos to small-scale logstcs dstrbuto route optmzato problems, but have great calculato complexty ad low solvg effcecy for moder large-scale logstcs dstrbuto route optmzato problems [3]. Heurstc algorthms clude costructo algorthm ad two-stage method, ad have mproved solvg effcecy compared wth precse methods, but ca hardly get optmal solutos to logstcs dstrbuto routes [4]. Swarm tellgece algorthms maly clude geetc algorthm, partcle *Address correspodece to ths author at the Departmet of Logstcs ad Iformato Maagemet, Zhuha College of Jl Uversty, Zhuha, Cha; Tel: ; E-mal: Zhagb8@sa.com swarm optmzato algorthm, frog leapg algorthm ad eural etwork algorthm etc, whch have advatages such as rapd search speed etc ad become ma methods for solvg logstcs dstrbuto problems [5-8]. Amog these methods, partcle swarm optmzato (PSO) algorthm has advatages cludg strog global optmzato ablty ad smple & easy mplemetato etc, ad thus s the most wdely used for logstcs dstrbuto route optmzato; however, PSO algorthm has shortcomgs such as slow late speed ad proeess to local optmzato etc, ad thus scholars put forward may logstcs dstrbuto route optmzato methods to mprove PSO algorthm ad ga a better optmal soluto to logstcs dstrbuto route [9]. However, further mprovemet s stll eeded ad ew deas shall be troduced for better logstcs dstrbuto route solutos. I 005, Tzhoosh put forward opposto-based learg (OBL) theory, ad beleved that the optmal soluto would fally be foud through costat terato by swarm tellgece algorthm f takg radom predctve value as the tal swarm; thus, radom predctve value was vtal for swarm tellgece algorthm, rapd covergece rate would be reached f ths value was close to the optmal soluto, whle there would be log tme ad slow speed f ot; however, the effcecy of swarm tellgece algorthm would be substatally mproved f both curret soluto ad verse soluto were both searched swarm tellgece algorthm search ad the optmal soluto was selected to be predctve value accordg to results [0]. Based o ths, wth respect to shortcomgs of PSO algorthm, OBL theory s troduced to geerate a OBLPSO algorthm, ad ths algorthm s used to solve logstcs dstrbuto route problem, ad the performace of ths algorthm s tested through smulato expermet /5 05 Betham Ope

2 Study o Optmzato of Logstc The Ope Automato ad Cotrol Systems Joural, 05, Volume LOGISTICS DISTRIBUTION ROUTE OPTI- MIZATION MATHEMATICAL MODEL.. Dstrbuto Route Problem Set that a logstcs dstrbuto etwork cotas a total of M customer stes, ad thus the locatos ad goods demads of these customer stes are kow; the logstcs dstrbuto ceter has a total of K dstrbuto vehcles, each vehcle k has gve ad kow maxmum carryg capacty Pk (k=,,,k) ad starts from ad returs to the dstrbuto ceter. The objectve of logstcs dstrbuto route optmzato s to fd a optmal soluto to total costs (such as dstace ad tme etc) whle meetg the followg costrats: () The sum of goods demad o each route shall ot exceed the maxmum vehcle carryg capacty. () The locato of logstcs dstrbuto ceter s fxed ad kow. (3) Servce for oe customer ste ca oly be provded by oe vehcle. (4) Dstrbuto route legth shall ot exceed the maxmum dstace every tme a vehcle drves [0]... Logstcs Dstrbuto Route Optmzato Mathematcal Model b 0,0 deotes the dstrbuto ceter of logstcs etwork, b deotes the dstace betwee customer ad customer j,, j ad, j= 0,,, M, k deotes the umber of customers that vehcle k dstrbutes. Thus, the mathematcal model of logstcs dstrbuto route optmzato s: K k MF = b rk + b,r k k rk,0 = sg( ) () k k = Where: k = 0 sg( k ) = 0 k Costrats of logstcs dstrbuto route optmzato are: k p pk; k 0 rk = k b + b ; 0 r, k B k rk rk,0 k k = Rk Rk =, k k K Rk = {,,, M};0 k M k = () (3) Where, Rk deotes customer set of vehcle k; Bk deotes the maxmum travel dstace of vehcle k; ad r j k deotes the order of customer dstrbuto route. Accordg to formula (), the logstcs dstrbuto process has two objectves. The problem whch requres mmum dstrbuto vehcles, the shortest dstrbuto route ad delvery of goods to destato wth the specfed tme s a mult-costrat ad mult-objectve combatoral optmzato problem. As swarm tellgece optmzato algorthm has certa advatage solvg combatoral optmzato problem, ths paper adopts OBLPSO algorthm to solve formula () ad fd a logstcs dstrbuto soluto that meets all costrats. 3. OBLPSO ALGORITHM FOR LOGISTICS DIS- TRIBUTION ROUTE OPTIMIZATION 3.. Cocepts Related to OBL The basc dea of OBL algorthm s to obta curret optmal value by cosderg ad comparg curret estmated value ad verse estmated value of varable. Defto Ay real umber x [ a, b], defe ˆx= a+ b x as the verse pot of x. Defto X = [ x, x,, x D ] s a pot D- dmeso space, where x [ a, b], ad defe Xˆ = [ xˆ ˆ ˆ, x,, x D ] as the verse pot of X wth elemet of x = a + b x. Defto 3 Assume that X = [ x, x,, x D ] s a pot D-dmeso space, f() s a ftess fucto to measure ths pot, Xˆ = [ xˆ ˆ ˆ, x,, x D ] s the verse pot of x, such as f( Xˆ ) < f( X), the, replace X wth ˆX, otherwse, cotue usg pot X, ad obta a relatvely optmal soluto through ftess values of evaluato pot X ad verse pot ˆX. 3.. OBLPSO Algorthm Set the flyg speed ad locato of partcle as: v (,,, ) T = v v v ad D x (,,, ) T = x x x respectvely, D the prevous optmal locato of partcle s p = ( p, p,, pd), the optmal swarm locato s pg = ( pg, pg,, pgd), ad the locato ad speed update mode of partcle are: vd ( t + ) = wvd + crad()( pd X d ( t)) + c rad()( p X ( t)) gd d (4) x ( t+ ) = x ( t) + v ( t+ ) (5) d d d Where, c ad c are accelerato coeffcets; rad() s a radom umber [0,]; t s curret terato tmes; ad s erta weght whch s defed as follows: ter = max ( max m ) (6) ter Where, s the maxmum weght coeffcet; ad max m s the mmum weght coeffcet.

3 30 The Ope Automato ad Cotrol Systems Joural, 05, Volume 7 Xao-ju ad B Table. Specfc partcle codg. Customer Ste Zx Zy Accordg to partcle decodg mode, stuato of each vehcle each customer ste s show Table. Table. Specfc partcle decodg. Customer Ste t(zx) 3 3 Zy A partcle of swarm has growg formato demad wth ts cotuous evoluto ad growth, ad mere relace o p ad p g extreme values could ot meet such demad. Thus, the ablty of other partcles the swarm to explore ew areas shall be stregtheed. Based o the above aalyss, OBLPSO algorthm s put forward. O the premse of trackg p ad p g each terato of radom partcle, aother ew extreme values shall be tracked as well, amely the worst soluto pp of partcle tself ad the worst soluto pp g occurred curretly the etre swarm. Thus, ths paper troduces the verse partcle PP g of the worst global pp g terato mechasm as a ew learg factor, ad the specfc terato mechasm s as follows: vd ( t + ) = wvd + crad()( pd X d ( t)) + (7) c rad()( p X ( t)) + c rad()( PP g X ( t)) gd d 3 d The expermet proves that PP g does ot always gude partcle swarm to the optmal value,.e. some stuatos would affect global learg ablty of partcle. Thus, mprovemet s made based o verse pot PP g. a + b ( m pp + m p), frad () < p = 0, otherwse g g PP Where, m ad m are radom umbers [0,], m +m =, ad p s the possblty of chagg to OBL algorthm ad s defed as follows: max max m (8) p= p t( p p )/ T (9) Where, p ad max p m are scopes of possblty p; ad t s curret terato tmes. OBLPSO algorthm ot oly adds partcle dversty to avod local optmzato problem, but also arrows the search space to mprove partcle search speed ad avod over learg pheomeo Partcle Codg Mode Durg logstcs dstrbuto route optmzato, the correspodece of partcle locato to optmzato soluto s the most crucal. Ths paper bulds a -dmeso space for dstrbuto to customer stes, each of whch s correspodg to servce vehcles ad executo order, ad thus partcle dvdes -dmeso vector Z to two -dmeso vectors Z x ad Z y, whch deote vehcle umber ad route order respectvely Partcle Decodg Mode () Frstly, take the teger of Z x of partcle to get correspodg vehcle umber j dstrbuted to customer. () The, determe route order of vehcle j accordg to sze order of elemet Z y. Assume that a dstrbuto ceter has 7 customers ad 3 vehcles, the code Z of partcle s show Table. The, travel routes of vehcles correspodg to partcle are (0 deotes the dstrbuto ceter): vehcle : 040; vehcle : 0570; ad vehcle 3: Solvg Steps of Logstcs Dstrbuto Route Optmzato () Set parameters for OBLPSO algorthm partcle swarm algorthm. () Italze partcle swarm, ad each partcle deotes a potetal feasble logstcs dstrbuto route. (3) Decode each partcle, calculate the shortest route of each dstrbuto soluto ad take t as the partcle ftess fucto value. (4) Compare each partcle locato wth prevous optmal values of dvduals ad the swarm, ad update prevous optmal values for dvduals ad the swarm accordg to results. (5) Geerate a value [0,] radomly, calculate the possblty p through formula (9), ad go to step (6) f ths ra-

4 Study o Optmzato of Logstc The Ope Automato ad Cotrol Systems Joural, 05, Volume 7 3 dom value s greater tha p, otherwse, eter OBL specfcally as follows: ) Calculate the verse pot of ths partcle by OBL mechasm based o crossover factor; ) Calculate the adaptve values of partcle ad verse partcle whch s based o crossover factor, compare whether the adaptve value of verse partcle s superor to that of partcle, select the optmal partcle to replace X, otherwse, cotue usg pot X. (6) Update the locato ad speed of each partcle accordg to formula (5) ad (6). (7) If the ed codto s met, take the optmal partcle locato as the optmal soluto, otherwse, go to step (3) ad cotue to optmze. (8) Get the optmal logstcs dstrbuto route soluto based o the optmal soluto. 4. SIMULATION EXPERIMENT 4.. Classc Fucto Test Two types of classc stadard multmodal fuctos are selected for the test expermet, ad compare the test results wth those obtaed by PSO algorthm. These two types of classc test fuctos are specfcally as follows: () Sphere fucto (a) Covergece Curve of Sphere Fucto f( x) = x (0) = () Grewak fucto x ( ) = / 4000 ( x ) cos( ) + = = f x () Fg. () shows ftess log value evoluto curves of test fuctos (otes: ths paper takes the base-0 logarthm of ftess values of fuctos for the coveece of dsplay ad observato of evoluto curves). I the fgure, the sold le s the covergece curve of PSO algorthm, whle the dotted le s the covergece curve of OBLPSO algorthm. As ca be kow from Fg. (), OBLPSO algorthm has a sgfcatly superor covergece rate to PSO algorthm, ad thus avods the shortcomg of PSO amely local optmzato problem. Ths s because OBLPSO cotas OBL, whch makes OBLPSO better tha tradtoal PSO algorthm terms of global search ablty, covergece precso ad speed. 4.. Smulato Test of Logstcs Dstrbuto Route Optmzato A compay has oe logstcs dstrbuto ceter ad fve goods trasport vehcles (carryg capacty of each vehcle s t), ad eeds to delver goods to eght customer stes. Coordates ad goods demad of each customer ste are show table 3 (0 deotes the dstrbuto ceter ad -8 are customer stes). (b) Covergece Curve of Grewak Fucto Ftess Fg. (). Comparso of covergece performace betwee OBLPSO ad PSO algorthm. Table 3. Coordates ad goods demads of customers. Customer No. Coordates Goods Demad 0 (40,40) (0,0).30 (5,50).38 3 (30,40).77 4 (40,60).40 5 (38,0).8 6 (50,45). 7 (70,0).8 8 (60,70).0 Parameters of OBLPSO algorthm are: swarm scale N=0, the maxmum evoluto algebra T=500, c=c=.5, c3=.0. Solve logstcs dstrbuto route optmzato problem Table 3 by PSO algorthm ad OBLPSO algorthm respectvely, ad results are show (Fgs. ad 3).

5 3 The Ope Automato ad Cotrol Systems Joural, 05, Volume 7 Xao-ju ad B of trackg of both global optmzato ad the verse partcle of the worst global partcle durg terato of OBPSO algorthm ad rapd covergece rate, whch avods effectvely local optmzato, prevets premature covergece ad mproves the search effcecy. CONFLICT OF INTEREST The authors cofrm that ths artcle cotet has o coflct of terest. Fg. (). The optmal logstcs dstrbuto route of PSO algorthm. ACKNOWLEDGEMENTS Declared oe. REFERENCES Fg. (3). The optmal logstcs dstrbuto route of OBLPSO algorthm. From Fg. (), there are logstcs dstrbuto routes by PSO algorthm: route s 0570 wth total dstrbuto route legth of 35.36km ad route s wth total dstrbuto route legth of 56.98km, ad thus the total route legth s 9.34km. From Fg. (3), there are logstcs dstrbuto routes by OBLPSO algorthm: route s wth total dstrbuto route legth of 5.73km ad route s wth total dstrbuto route legth of 3.09km, ad thus the total route legth s 48.8km. By comparg results of Fg. () ad Fg. (3), logstcs dstrbuto route soluto obtaed by OBLPSO algorthm s superor to that obtaed by PSO algorthm, maly because [] Y. Y. Shu, ad K. C. Ch, A geetc algorthm that adaptvely mutates ad ever revsts, IEEE Trasactos o Evolutoary Computato, vol. 3, o., pp , 009. [] L.Y. Tseg, ad Y.T. L, A hybrd geetc local search algorthm for the permutato flow shop schedulg problem, Europea Joural of Operatoal Research, vol. 98, o., pp. 84-9, 009. [3] X. Zhag, H. Lu, D. L, ad R. Yag, Study o VRP express dstrbuto based o geetc algorthm, Logstcs Techology, o. 3, pp. 0-05, 03. [4] X. Hu, N. Yu, ad Q. Dg, Sequetal decso methods for dsrupto maagemet dstrbuto, Joural of Idustral Egeerg ad Egeerg Maagemet, vol. 5, o., pp , 0. [5] B. B. L, ad L. Wag, A hybrd quatum-spred geetc algorthm for mult-objectve flow shop schedulg, IEEE Trasactos o Systems, Ma ad Cybemetcs, vol. 37, o. 3, pp , 007. [6] J. Che, Study o routg optmzato for physcal dstrbuto based o at coloy algorthm, Computer Smulato, vol., o., pp. 68-7, 0. [7] Z. L, Mxed at coloy algorthm solvg the VRP problem, Joural of Wuha Uversty of Techology (Trasportato Scece & Egeerg), vol. 30, o., pp , 006. [8] F. Zhag, ad S. Wag, Study o data aalyss oreted route programmg algorthm logstcs system, Logstcs Techology, o. 6, pp , 03. [9] L. L, ad W. Jag, Study o applcato of at coloy geetc optmzato algorthm choosg logstcs dstrbuto routes, Joural of Harb Uversty of Commerce (Natural Sceces Edto), vol. 5, o. 6, pp , 009. [0] T. Wag, ad Y. Wu, Study o optmzato of logstcs dstrbuto route based o chaotc PSO, Computer Egeerg ad Applcatos, vol. 47, o. 9, pp. 8-, 0. Receved: February 6, 05 Revsed: Aprl 9, 05 Accepted: May 0, 05 Xao-ju ad B; Lcesee Betham Ope. Ths s a ope access artcle lcesed uder the terms of the Creatve Commos Attrbuto No-Commercal Lcese ( lceses/by-c/3.0/) whch permts urestrcted, o-commercal use, dstrbuto ad reproducto ay medum, provded the work s properly cted.