Ant Colony Algorithm Based on Information Entropy Theory to Fuzzy Vehicle Routing Problem

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1 A Coloy Algorh Bsed o Iforo Eropy Theory o Fuzzy Vehcle Roug Proble Lsheg Tg Weg Cheg Zeqg Zhg B Zhog Reserch Isue of Mechcl Egeerg, Souhes Joog Uversy, Chegdu 6003, P. R. Ch Absrc To dp he chgg of rke eed, logscs provders ke effors o reduce coss d prove cusoer servce levels o ee he cusoer ssfco s dvdul deds. I he process of cul dsrbuo, delvery vehcles ll ecouer vrous ofe ucer exerl fcors, resulg delvery es of ucery. Ad drecly ffecs he orl produco d opero. I hs pper, he rvel e bsed o fuzzy hecl odel of vehcle roug proble ke he e do s fuzzy vrble. Iforo eropy d he ph chose by he use of rdo dsurbce corol sregy o he dpve lgorh. Flly, uercl exple s gve o sho he effecveess of he lgorh. Keyords: Vehcle roug proble, Fuzzy, A coloy lgorh, Iforo eropy. Iroduco Vehcle Roug Proble (VRP) derved fro he rspor, Dzg 959, ypcl NP-hrd coborl opzo proble. Al referes o he gs so for he sudy of he rspor of perol delvery roue opzo, d quckly bece he forefro feld of operos reserch d coborl opzo sudy d ho, rcg lrge uber of cdecs o crry ou reserch. Usully grph G = (V, E) s used o descrbe he proble, G = (V, E), V={0,,2,,). E={ (, ),,, V), ode deoes he depo d oher odes deoe cusoers. Ech cusoer s ded s q, hle edge (, ) correspodg o he e or dsce or rsporo coss s C. Q deoes he cpcy of vehcle, ll vehcles fro he rehouse, coplee sk d reur o rehouse. Ech cusoer c oly be vsed oe e d he obecve fuco s usully o ze he uber of vehcles or rspor coss. A coloy lgorh s e heursc lgorh opzo ehod, suble for vehcle roug d oher coborl opzo probles. ACO s proposed lly for solvg he rvelg sles proble by he Il scholrs Dorgo []. Alog h he deepeg of reserch, he ACO hs bee roduced o elecrocs, elecoucos, d shop schedulg d oher egeerg felds. I prccl pplcos, due o he ucery of vrous exerl fcors, hch led o vehcle served e ucer, ffecg he orl opero he se e, ll led o he decle cusoer ssfco h he servces. Thus, he cery VRP of rdol heores d ehods re o loger suble for delg h ucer probles. There re lo fuzzy heory reserch hs bee used o VRP [2]-[9]. Soe doesc scholrs hve rsed he Fuzzy VRP hecl odels d lgorhs [0]-[2]. Becuse he ousdg perforce of he eropy heory delg h fuzzy ssues, lgorh bsed o foro eropy heory s used o he vehcle roug proble h fuzzy rvel e (FTTVRP) [3]. 2. Reled o he Fuzzy Theory d FTTVRP Models 2.. Fuzzy Theory Fuzzy soluos d he process s e vrble, hch c be descrbed s follos: f coplee se U, u* U, oe of he elees. If fer gve process, u becoes se of fuzzy se A, hch kes U s uverse se. Ths prculr process s ko s fuzzy. Vrble e fors rgulr fuzzy ubers, s sho Fgure. Fuzzy coverso forul s s follog:

2 u, u b u A( u) =, < u b () b 0, ohers Defuzzfy process s o he corry o fuzzy, s cler vlue W* fuzzy ses he process. I hs pper, e ke he ceer dspel defuzzfy. Is forul s follog: Servce Servce Level Level W * = Fg. : Trgle fuzzy uber = = B ( ) d B ( ) d Te(h) Te(h) b b (2) Te s o cosdered frs, fer TSP clculed for he shores ph, Fuzzy e bss o deere he level of servce d cusoer servce order. We gve prory o he eeds of odes hgh level of servce requrees The Mhecl Model of FTTVRP Zhg proposed he fuzzy vehcle roug proble bsed o vehcles rvel e (FVRP) []. FVRP s descrbed s follos: rspor eork, here s depo d servce odes for servce, deoes h 0, l, 2,,. Vehcles sr fro he depo, reured fer servced cer uber of odes. I s ko h C deoes he vehcle rspor cpcy d ech ode hs ded d. Beee he odes d expeced rvel e s fuzzy uber. Mu rsporo coss o rffc roues o ee he requrees. Ths pper cosders he shores perod of e s he obecve, kg o he ssfco of servced e cosdero. For he ske of coveece, beee y o odes,, e ssue h he rvel e % s ~ 2 3 rgulr fuzzy ubers (,, ) =. deoes he lef sde of fuzzy fgure, 2 deoes he degree of ebershp s correspod o, 3 deoes he rgh sde of fuzzy fgure. I order o cosruc he hecl odel, e defe he vrbles x k d y k. If he vehcles k rvel fro cusoer o cusoer, he he vlue of x k equls o, else equls o 0; f he sk of cusoer s copleed by he vehcle k, he he vlue of y k equls o, else equls o 0. We defe he hecl odel of he vehcle roug proble h rvel e s follog: ~ z ~ x k (3) = k dy k C, k (4) yk =, =, 2, L, (5) k xk = yk, = 0,, L, ; k (6) xk = yk, = 0,, L, ; k (7) ( k ) X = x S (8) S = ( xk ) xk R, R {, 2, L, } ; k R R Pr{ f( xy,, ) [ b, ]} β, =,2,..., (9) Forul (3) s he obecve fuco, he rvel ~ e s fuzzy fgure, hch deoes fuzzy rvel e fro o. Forul (4) s cpcy cosr, hch deoes h he sk ol cpcy c o exceed he vehcle s cpcy. Forul (5) deoes h ech cusoer could be vsed oly oe e. Forul (6) d (7) deoe he reloshp of vrble y k, y k d x k. Forul (8) s brch elo cosr. Forul (9) deoes he probbly of coplee sk ssge e dos, hch s used o esure he degree of cusoer servce e ssfco. 3. A Coloy Algorh Bsed o Iforo Eropy 3.. Theory Cocepo of Iforo Eropy Theory d chrcersc The cocep of Eropy ce fro physcs, proposed by Clusus Ger physcs 854, o descrbe he dsorder of herodyc syse. I s roduced o uber of oher dscples, Bolz eropy, foro eropy, probblsc

3 eropy, d so o. The U.S. scholrs Sho roduced he herodyc eropy o foro heory s por cocep of ucery ehods, d ofe s used o gve rough esure of ucery. I order o dscree rdo vrbles, s foro eropy. he probbly of se. p 0, S = k p l p. p deoes = = p =. Iforo eropy hs he follog properes: syery, Noegve, Addve d us A Coloy Algorh Bsed o Iforo Eropy I bsc ACO, We ssue h s he uber d d s he dsce beee cusoer d. The vsbly of Edge (, ) η =/d, hch reflecs he cusoer's spro level fro cusoer o. τ s he sregh of foro-rck o Edge (, ). Δτ k s pheoeo quy u legh lef by k o rc (, ). p k s he se rso probbly of k fro cusoer o. β ( ) η ( ) () (), α β τ η α τ k lloedk p = s lloedk (0) 0, oherse lloed k ={0,,, -} deoe he ex cusoer hch k ll choce. For he o preers α d β, hey reflec he ccuuled foro he course of ovg d he relve porce of heursc foro he s choce ph. Ech hs boo ble desged o record he depos k pssed, d o s lloed o repe he curre cycle. I bsc ACO he ph foro ou s ucery, so does he ph chooses. We roduce foro eropy o he lgorh, by corollg foro eropy vlue he ph chose d he rdo vro locl perurbos. Whe eropy reches prculr reques, sop serchg. Here s he defo of foro eropy, S () = k p()l p(). Illy, he se = foro eropy ll of phs, s soe seleced ph foro crese, he eropy decrese grdully. Eveully y led o sgo, here locl opl soluo s obed. So S S() S S() α '( ) = d β '( ) = s S 2S roduced, β '( ) s he probbly he opl ph o be ed. α '( ) s he proporo of ol coloy lloed o choose he ppropre ph sll res. Algorhs process s dced Fgure 2. Here e defe lrger eropy vlue sds crer bgger h 0 d codos s ero Pherooe Upde Sregy d Iproved Pherooe upded sregy s he key sep lgorh, rpd updg leds o sgo or eve fll o locl opl resul, hle oo slo ll o serch he opl resul. Whe s fd fesble soluo, pherooe of ll secos (, ) should be globl upded. Ilzo Ilzo Fll Fll codo codo?? No No Upde Upde globl globl Pherooe Pherooe Ccule Ccule foro foro eropy eropy d d α '( ) Vro Vro h h probbly probbly β '( ) Ccule Ccule foro foro eropy eropy Fg. : Algorh flo chr. Yes Yes Oupu Oupu ( ) ( ) τ ( ) = ρ τ ( ) + ρτ (), ρ 0, () L gb s he globl opu legh of he le prese. Q s cos. ρ s pherooe volle fcor. τ 0 s cos, τ 0 =/(l ). l s he legh of he shores le curre cycle. We se loer l of pherooe, reserve he vlue loer o prove he sbly of lgorh. 4. Nuercl Experes

4 Here e ll gve exple o sho odels h e hve us dscussed d ho he coloy lgorh bsed o foro eropy orks. Le us cosder fuzzy rvel e vehcle roug proble sho Fg.. We ssue h here re 7 cusoers lbeled, 2,, 7 d oe depo lbeled 0. We lso ssue h he rvel es beee cusoers re ll rgulr fuzzy vrbles, he e dos d deds of cusoers re gve Tble d Tble 2. We ssue h he ulodg es re ll 0 ues d he cpces of he four vehcles re ll 0. We lso ssue h he servce level s The progr s coded C++ lguge d sulos ere perfored o persol copuer h 2700 MHz Peu 4 processor d 256MB of RAM, he rue re bou 2 s. The ol u cusoer ded Tble : Cusoer ded ble (5,6,8) (,4,9) (,3,5) (3,9,9) (4,4,6) (,2,8) (5,5,5) (3,4,9) (4,7,7) (,6,9) (4,8,9) (,2,8) (5,7,9) (2,9,9) (2,6,9) (3,8,9) (6,7,7) (,3,4) (2,5,9) (3,5,9) (7,7,8) (3,4,5) (2,5,6) (,5,7) (5,5,6) (2,6,6) (2,7,8) Tble 2: Trvel e rx (3,5,6) (6,8,8) (,6,8) (8,8,9) (,,7) (,,8) α β τ η ρ Tble 3: Bsc preer seg of ACO e rveled by he four vehcles s 4. Furherore, he he operol pl s perfored, he ru of he coloy lgorh shos h he bes operol pl s Vehcle : Vehcle 2: Vehcle 3: Vehcle 4: Coclusos Ths pper corbued o fuzzy rvel e vehcle roug proble h coloy lgorh bsed o foro eropy he follog respecs: () FTTVRP odel s proposed for fdg he opl soluos of fuzzy rvel e vehcle roug probles; (b) coloy lgorh bsed o foro eropy o solve he fuzzy rvel e vehcle roug proble s preseed, focusg o ol rvel e zo; (c) he effecveess of he coloy lgorh bsed o foro eropy s sho by soe uercl exples. Ackoledge We hk Souhes Joog Uversy for provdg he fud suppor. We grefully ckoledge he help of Professor Xo Wu ho crefully red d eded erler drfs of hs pper. Refereces [] D. Mrco, A coloes for he rvelg sles proble. Bosyses, pp.73-8, 997. [2] V.D. Albero, M. Robero, C. Nor, E.R. Adre, M.G. Luc, Te depede vehcle roug proble h ul coloy syse. Europe Jourl of Operol Reserch, Correced Proof, Avlble ole, [3] E.B. Joh, R.M. Prck, A coloy opzo echques for he vehcle roug proble. Advced Egeerg Iforcs, 8:4-48, [4] T. Duš, P. Gor, The fuzzy se heory pproch o he vehcle roug proble he ded odes s ucer. Fuzzy Ses d Syses, 82:307-37, [5] S.T. Lu, C. Ko, Solvg fuzzy rsporo probles bsed o exeso prcple. Europe Jourl of Operol Reserch, 53:66-674, [6] M. Psquer, C. Quek, M. Toh, Fuzzy lo: ovel self-orgsg fuzzy-eurl rule-bsed plo syse for uoed vehcles. Neurl Neorks, 4:099-2, 200. [7] Y.S. Zheg, B.D. Lu, Fuzzy vehcle roug odel h credbly esure d s hybrd ellge lgorh. Appled Mhecs d Copuo, 76: , [8] H.M. Sheg, J.C. Wg, H.H. Hug, D.C. Ye, Fuzzy esure o vehcle roug proble of hospl erls. Exper Syses h Applcos, 30: , [9] H. M, Z. Hu, G.J. Wg, A relble roug lgorh oble d hoc eorks usg fuzzy Per e. Globl Telecoucos Coferece Workshops, pp , [0] G.Y. Dg, J.R. Wg, J.K. Cu, Reserch of fuzzy heory coloy lgorh o VRPTW. Sscl d decso, 2:35-37, [] J.Y. Guo, J. L, A Hybrd Geec Algorh o he Vehcle Roug Proble h Fuzzy

5 Trvelg Te. Jourl of Idusrl Egeerg Mgee, 9:3-7, [2] Q. Guo, B.L. Xe, Model d lgorh of vehcle roug proble h sochsc rvel e. Jourl of syses egeerg, 8: , 2003.

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