A particle swarm optimization to vehicle routing problem with fuzzy demands

Transcription

1 A partcle swarm optmzato to vehcle routg problem wth fuzzy demads Yag Peg, Ye-me Qa A partcle swarm optmzato to vehcle routg problem wth fuzzy demads Yag Peg 1,Ye-me Qa 1 School of computer ad formato egeerg, Zhejag GogShag uversty Hagzhou. Cha, Hagzhou sttute of commerce, Zhejag GogShag uversty Hagzhou. Cha, pegyag@mal.zjgsu.edu.c do: /jct.vol5.ssue6.11 I ths paper, a ovel real umber ecodg method of Partcle Swarm Optmzato (PSO) for Vehcle Routg Problem s proposed. Objectve s to solve the vehcle routg problem wth fuzzy demads (FVRP), FVRP, a fuzzy chace costraed program model s desged, based o fuzzy credblty theory. Frstly costruct a sutable mappg betwee problem soluto ad PSO partcle, ad adopted approprate procedure the method. To llustrate the effectveess ad good performace of the proposed algorthm, a umber of umercal eamples are carred out, ad the algorthm s compared wth other heurstc methods for the same problem. eywords: Partcle Swarm Optmzato, Vehcle Routg Problem, Fuzzy Possblty 1. Itroducto The vehcle routg problem (VRP), frst troduced by Datzg ad Ramser [1], s a well-ow combatoral optmzato problem the feld of servce operatos maagemet ad logstcs. The problem s cocered wth fdg effcet routes of mmum total cost, begg ad edg at a cetral depot, for a fleet of vehcles to serve a umber of customers for some commodty, such that each customer s vsted eactly oce by oe vehcle, ad satsfyg some costrats such as capacty costrats, durato costrats ad tme wdow restrctos. Hudreds of papers lterature have bee devoted to ths problem, But most of them assume that the formato s determstc, such as customer formato, vehcle formato ad so o, Actually, may logstcs systems, t s hard to descrbe the parameters of the vehcle routg problem as determstc. For stace, may evromets, the formato about demad at each customer s ofte ot precse eough. For eample, based o eperece, t ca be cocluded that demad of a customer s aroud 100 uts, betwee 30 ad 50 uts, etc. Geerally, we adopt fuzzy varables to deal wth these ucerta parameters, fuzzy vehcle routg problem (FVRP) arses wheever some elemets of the problem are ucerta, subjectve, ambguous ad vague[][3]. VRP s a typcal NP-complete problem, However, eact techques are appled oly to small- scale problem FVRP s much more dffcult tha VRP, whch s a specal eample of VRP uder ucertaty. So tellget methods have gaed wde research, as was frst preseted [3] VRP, ad Cheg ad Ge [4] used a geetc algorthm to solve the vehcle routg problem wth fuzzy due-tme. Moreover, such as Lu ad La et al. [5][6] modeled VRP wth fuzzy travel tmes by fuzzy programmg wth a possblty measuremet, ad adopted the geetc algorthm to solve the model. Zheg ad Lu [7] researched the vehcle routg problem wth fuzzy travel tme, ad preseted a chace costraed program (CCP) model wth credblty measuremet, the tegrated a fuzzy smulato ad geetc algorthm (GA) etc, ad acheved good s. Partcle swarm optmzato (PSO) algorthm s a parallel populato-based computato techque orgally developed by eedy ad Eberhart[8][9], PSO ca solve a varety of dffcult optmzato problems because of ts advatages. Although the PSO s developed for cotuous optmzato problem tally, there have bee some reported wors focused o dscrete problems recetly[10][11]. A partcle swarm optmzato method for FVRP s troduced ths paper. Frstly, the bass of FVRP s aalyzed secto. Secodly, a PSO for FVRP s proposed secto 3. At last, the umercal epermets are carred out by ths algorthm. Smulato s ad comparsos prove that 11

2 ths algorthm s a effectve method for FVRP.. Fuzzy model for VRP.1 Computato based o credblty theory Joural of Covergece Iformato Techology Volume 5, Number 6, August 010 Some basc cocepts ad s o fuzzy varables were troduced by Zheg[7] ad Lu[1], Now a tragular fuzzy varable d ( d1, d, d3) s descrbed from the deftos of possblty, ecessty ad credblty. t s easy to obta: 1, f r d d3 r Pos { d r}, f d r d (1) 3 d3 d 0, f r d3 1, f r d1 d r () Nec{ d r}, f d1 r d d d1 0, f r d 1, f r d1 d d1 r, f d1 r d ( d d1) (3) Cr{ d r} d3 r, f d r d3 ( d3 d ) 0, f r d3 We assume that servce s provded by vehcles of the same capacty. We wll deote vehcle capacty by C ad the fuzzy umber represetg demad at the th customer by d ( d 1, d, d3) equal Q Q C 1.After servg the frst customers, the avalable capacty of the vehcle wll d, Q s also a tragular fuzzy umber by usg the rules of fuzzy arthmetc, ad ( C d, C d, C d1 ) ( q1,, q,, q3, customer demad does ot eceed the remag capacty of the vehcle. ).We obta the credblty that the et Cr cr { d, 1 Q } Cr{( d1, 1 q1,, d, 1 q,, d3, 1 q3 ) 0} 0, q3, d1, *( q3, d1, 1 d d3, 1 q1, *( d *( q, d, 1 d 1, 1, 1, 1 3, 1 q q q,, 1, f, f ) ), f ) f d d 1, 1 d d 1, 1, 1 3, 1 q q q q 3, 3,, 1,, d, d, 1 3, 1 q q, 1, (4) 113

3 A partcle swarm optmzato to vehcle routg problem wth fuzzy demads Yag Peg, Ye-me Qa If the vehcle's remag capacty s greater ad the demad at the et customer s less, the the vehcle's chace'' of beg able to fsh the et customer's servce become greater. Cr deotes the magtude of our preferece to sed the vehcle to the et customer after t served curret customer. Obvously, Cr[0,1 ] ad Cr 0 shows completely that the vehcle should retur to the depot, Cotrarly Cr 1, we are absolutely certa that we wat the vehcle to serve the et customer.. The model of Fuzzy VRP We assume that:(1)each vehcle has a cotaer wth a physcal lmtato ad total loadg of each vehcle caot eceed ts capacty C;() each vehcle has mamum dstace costrats, ad total travel dstace of each vehcle caot eceed L; (3) a vehcle wll be assged for oly oe route o whch there may be more tha oe customer; (4) a customer wll be vsted by oe ad oly oe vehcle; (5) each route begs ad eds at the compay depot;(6)the demad of each customer s a tragular fuzzy umber d ( d1, d, d3),the dstace betwee customer ad customer j s c j, ad there are vehcle the depot. For coveece, the capacty of the vehcles s the same, ad the mamum dstace that each vehcle ca travel s equal, they ca be deoted respectvely as C ad L.(7) The dspatcher preferece de s descrbed as Cr *, Cr *[0,1], f the relato Cr Cr * s fulflled, the the vehcle should be set to the et customer; otherwse, the vehcle should be retured to the depot, ad sed aother vehcle to the et customer. Ad Cr * epresses the dspatcher's atttude toward rs. Parameter Cr * s subjectvely determed has a etremely great mpact o both the total legth of the plaed routes ad o the addtoal dstace covered by vehcles due to falures at some customers. Lower values of parameter Cr * epress the dspatcher's desre to use vehcle capacty the best he ca. These values shorter plaed dstaces. But lower values of parameter Cr * crease the umber of stuatos whch vehcles arrve at a customer ad are uable to servce them, thereby creasg the total dstace. We assume the decso varable =1, f arc(, j) belogs to the route operated by vehcle, j otherwse s 0; y =1, f customer s servced by vehcle, otherwse s 0. The correspodg chace costraed program (CCP) mathematcal formulato of FVRP based o credblty theory s gve as followg. j1 m c j j (5) 1 0 j0 m c ' (6) s. t. Cr( d y C) Cr*, 0,1,, (7) j 0 1, j 1,,, (8) j j 0, j 0,1,, ; 0,1,, 0 j 1, 1,,, j 0 y j, (9) (10) j 0,1,, ; 0,1,, (11) 114

4 Joural of Covergece Iformato Techology Volume 5, Number 6, August 010 j j0 0 j0 j c j y j, 0,1,, ; 0,1,, (1) L, 0,1,, (13) { 0,1}, y {0,1},, j 0,1,, ; 0,1,, (14) The objectve fucto (5) sees to mmze total plaed travel dstace. The objectve fucto (6) sees to mmze total addtoal travel dstace due to routes falure, c' ca be obta by stochastc smulato algorthm Secto 3.3. The sum of the plaed dstace ad addtoal dstace ca be obtaed by the mproved dfferetal evoluto algorthm. Chace costrat (7) assures that all customers are vsted wth vehcle capacty wth a cofdece level. Costrats (8) esure that each customer s vsted by eactly oe vehcle. Costrats (9) guaratee that the same vehcle arrves ad departs from each customer t serves. Restrctos (10) defe that at most vehcles are used. Restrctos (11) ad (1) epress the relato betwee two decso varable. Restrctos (13) are the mamum dstace costrats, L s the upper lmt o the total dstaces trasported by a vehcle ay gve secto of the route. Fally, costrats (14) defe the ature of the decso varable. 3. The mproved PSO method 3.1 Foudato of PSO The geeral prcples for the PSO algorthm are stated as follows. A partcle s treated as a pot a M-dmeso space, ad the status of a partcle s characterzed by ts posto ad velocty. Italzed wth a swarm of radom partcle, PSO s acheved through partcle flyg alog the trajectory that wll be adjusted based o the best eperece or posto of the oe partcle(called local best) ad the best eperece or posto ever foud by all partcles(called global best). The M- dmeso posto for the th terato ca be deoted as X (t)={ 1 (t), (t), m (t)},smlarly, the velocty (.e.,dstace chage),also a M-dmeso vector, for the th terato ca be descrbed as V (t)={v 1 (t),v (t),,v M (t)}, the partcle-updatg mechasm for partcle flyg(.e., search process) ca be formulated as followg. Where populato sze; L G V ( t) w( t) V ( t 1) c1r1 ( X X ( t 1)) cr( X X ( t 1) (15) X ( t) V ( t) X( t 1) (16) 1,,, P,ad P meas the total umber of the partcles a swarm, whch s called L L { 1 t 1,,, T, ad T meas the terato lmt; X,,, } represets the local best of the th partcle ecoutered after t-1 teratos, whle X,,, } represets L G G G { 1 the global best amog all the swarm of partcles acheved so far. c1 ad c are postve costats (amely, learg factors), ad r1 ad r are radom umbers betwee 0 ad 1; w(t) s the erta weght used to cotrol the mpact of the prevous veloctes o the curret velocty, fluecg the trade-off betwee the global ad local epereces. 3. Partcle represetato Oe of the ey ssues desgg a successful PSO algorthm s the represetato step,.e. fdg a sutable mappg betwee problem soluto ad PSO partcle. I ths paper, we setup a N-dmeso search space, N s the total umber of customer to be served, X ={ 1,,..., N }, 1,,..., N s a arrage of the customer s umber, deotes the th partcle s posto the populato, accordg to the restrcto of vehcle s capacty, the ecodg ca be decompose to several sectos, every secto L M G m 115

5 A partcle swarm optmzato to vehcle routg problem wth fuzzy demads Yag Peg, Ye-me Qa deote the customers ad the order served by a vehcle. For eample, there are 10 customer ad 3 vehcle, f a partcle s posto s:{5,3,7,,1,4,10,6,9,8}, ad decompose to {5,3,7,1,4,10 6,9,8}, the t maps to the soluto as follow: Vehcle 1: 5 3 7; vehcle : ; vehcle 3: I the partcles evoluto, the ecode of partcle wll be a real umber, the through a process we ca trasfer t to a teger umber, suppose: of ( t)' f ( ( t 1), v ( t 1), p, g ) { 1', ',, N'} j 1,, N } s a real umber, the sort t to = ', ',, '}, fd the de ad { j ' j '' ad substtute the orgal real umber. For eample, '' { 1 N (t)' = { 3.5,.3,3.7,7.3,.0, '' 8.1,6.,4.5,9.0,1.8}, sort t to {1.8,.0,.3, 3.5,3.7,4.5, 6.,7.3,8.1,9.0 }, fd the de ad substtute, the (t)' ca be coverted to ( t) {4,3,5,8,,,7,6,10,1} ad further decompose t to a soluto. I order to prevet a llegal chromosome eterg the et geerato great probablty, a pealty fucto s desged. R s the total dstace of vehcles traveled for the correspodg chromosome, ad let m r, f r, the m 0, ad R R Z * M,where Z s a very large teger. If r, the m 0. The ftess fucto ca be epressed as f 1/( R Z * M ). 3.3 Addtoal dstace algorthm for the FVRP Because the demad o each customer s a tragular fuzzy umber, we caot deal wth t drectly as a determstc umber by applyg other algorthms that solve the determstc vehcle routg problem. The real value of demad of a customer whe the vehcle reaches the customer ca be cosdered as a determstc umber by smulato. For each feasble plaed route that the soluto of the above model stads for, we obta a appromate estmate about addtoal dstaces ( c ' ). due to routes falure by a stochastc smulato algorthm, t s dstrbled as follows. Step 1: For each customer, estmate the addtoal dstaces by smulatg actual demads. The actual demads were geerated by followg processes: Step1.1: Radomly geerate a real umber the terval betwee the left ad rght boudares of the tragular fuzzy umber represetg demad at the customer, ad compute ts membershp u ; Step 1.: Geerate a radom umber a, a[0,1 ] ; Step 1.3: Compare a wth u, f a u, the actual demad at the customer s adopted as beg equal to ; the opposte case, f a >u, t s ot accepted that demad at the customer equals. I ths case, radom umbers ad a are geerated aga ad aga utl radom umber ad a are foud that satsfy relato a u ; Step 1.4: Chec ad repeat Step 1.1~ Step 1.3, ad termate the process whe each customer has a smulato actual demad quatty. Step : Move alog the route desged by credblty theory ad accumulate the amouts pced up from each customer, ad calculate the addtoal dstace due to routes falure terms of the ``actual'' demad. Step 3: Repeat Step 1 ad Step M tmes. Step 4: Compute the average value of addtoal dstace by M tmes smulato, ad t s regarded as the addtoal dstace. 3.4 The procedure of the proposed algorthm The procedure of the hybrd PSO ca be stated as followg: Beg Italze parameters: swarm sze, mamum of geerato, α, β, w, c1, c; Set t0, tf, λ, R ; t:=1; 116

6 Joural of Covergece Iformato Techology Volume 5, Number 6, August 010 Italze the partcles posto X ad the partcles velocty, partcles velocty s geerated as secto 3., due to the fuzzy demad of customer, the partcles posto X ca be geerated as followg: (1) Radomly geerate a partcles posto real umber as secto 3.; ()Choosg the frst customer of partcle, accordg to the customer demad ad the vehcle remag capacty, we ca obta Cr from Eq.(4). For a dspatcher preferece de value Cr *, f Cr Cr *, the the customer s assged to the curret vehcle; otherwse use aother vehcle to servce ths customer. (3)Delete the frst customer from the chromosome. (4)Repeat step, step 3, f all of the customers have bee assged to routes, we obtaed a feasble partcle. (5)Repeat the frst step to the forth steps for a gve populato sze NP. Evaluate each partcle s ftess accordg to Eq.(5)ad secto 3.3; Obta X g ad X p ; Repeat Compute V(t+1) accordg to Eq.(15); Obta X(t+1) accordg to V(t+1); Compute each partcle s ftess accordg to Eq.(5)ad secto 3.3; Fd ew Xg ad Xp ad updated; Carry out SA subprogram o each route of each partcle; Compute the partcle s ftess; Update Xg ad X p ; t=t+1; Utl (oe of termato codtos s satsfed) Output the optmzato s; Ed 4. Numercal Epermets The proposed PSO for FVRP has bee programmed usg Vsual C ad mplemeted o a Petum PC rug at.4ghz wth 51MB RAM. To test the computatoal performace of the method, we compare t w th the sweep algorthm troduced by Teodorovc[3]. We assume that there are 0 customers ad oe depot are geerated radomly [100,100], ad the fuzzy demads of customers are geerated radomly [0.,5], the vehcle capacty C s 8, the coordates of customers ad fuzzy demad umbers are descrbed table 1. The parameters for PSO s set as follows, mamum umber of geeratos: 100; swarm sze: 30; 0. 3 ; 0. 7 ; w 0. ; c ; c 0. 5 ; t 30 ; t 0. 1 ; 0. 9 ; The value of dspatcher preferece de Cr * 0 f vared wth the terval of 0 to 1, respectvely show the tedeces regardg the plaed dstaces, addtoal dstaces due to falures at the customers. I Tables, as dspatcher preferece de Cr * rose, a strctly rsg tedecy was oted the total average total dstaces, because of the addtoal dstace that vehcles had to mae due to falures at the customers strct crease. Whe the dspatcher preferece dees Cr * creases from 0 to 0.6, the creasg quatty of the average total dstaces s smaller tha the Cr * >0.6, whe the dspatcher preferece dees Cr * >0.6, the creasg quatty of the average total dstaces s strctly larger tha the decreasg quatty of the addtoal dstaces. 117

7 A partcle swarm optmzato to vehcle routg problem wth fuzzy demads Yag Peg, Ye-me Qa Table 1. Basc data of customer the epermets o. y fuzzy demads (0.4,.1,3.) (0.,1.7,.5) (0.5,.3,4.1) (1.0,3.,4.7) (0.7,.7,4.1) (0.3,1.9,3.) (0.4,1.5,3.1) (0.3,0.9,1.) (0.5,1.8,.6) (0.8,1.8,.6) (0.6,1.5,.0) (0.7,1.4,.1) (0.3,0.9,1.) (0.,0.7,1.0) (0.3,0.9,1.3) (0.4,1.0,1.7) (0.5,1.6,.) (0.,0.9,1.5) (0.5,1.5,.4) (0.4,1.0,1.5) Table also compares the average values, the best values ad the worst values obtaed wth the sweep algorthm troduced by Teodorovc[3] at varous Cr * levels. Ths table cofrms the superorty of the developed PSO algorthm. It s clear that the s obtaed by the proposed PSO are superor to the s obtaed by the sweep algorthm the average values. The epermets show that lower values of parameter Cr * epress our desre to use vehcle capacty the best we ca. These values correspod to routes wth shorter dstaces. O the other had, lower values of parameter Cr * crease the umber of cases whch vehcles arrve at a customer ad are uable to servce t, thereby creasg the total addtoal dstace they cover due to the falure''. Hgher values of parameter Cr * are characterzed by less utlzato of vehcle capacty. Cr * Average Best Table. Comparsos of computato s Improved PSO Worst umber of vehcles Average Sweep algorthm Best Worst umber of vehcles Therefore, whe the operatoal pla s performed, we suggest the dspatcher preferece de should be 0.6 appromate. 5. Cocluso Ths paper cotrbuted to the vehcle routg problem wth fuzzy demads the followg respects: (1) a effectve partcle swarm optmzato method ecoded by real umber s proposed for the FVRP, 118

8 Joural of Covergece Iformato Techology Volume 5, Number 6, August 010 focusg o total traveled dstace mmzato;;()a chace costraed program mathematc model of FVRP was proposed based o credblty theory; (c) the dspatcher preferece de greatly flueced the legth of the plaed routes ad the addtoal dstaces covered by vehcles due to falures at the customers, ad the propostoal ``best'' value of parameter Cr * was obtaed by the proposed hybrd algorthm. Further research wor may clude FVPR wth other costrats, such as wth tme wdows, or wth reverse logstcs etc. 6. Refereces [1] Datzg, G., Ramser, J.: The Truc Dspatchg Problem[J]. Maagemet Scece. Vol.6,pp.80-91,1959. [] M. Gedreau, G. Laporate, R. Segu, Stochastc vehcle routg, Europea Joural of Operatoal Research. vol.88,pp.3-1,1996. [3] Teodorovc D, Pavovc G. The fuzzy set theory approach to the vehcle routg p roblem whe demad at odes s ucerta [J ]. Fuzzy Sets ad Systems.vol.8,pp ,1996. [4] R.Cheg, M. Ge, Vehcle routg problem wth fuzzy due-tme usg geetc algorthm, Japaese Joural of Fuzzy Theory ad Systems.vol.7,pp ,1995. [5] Lu B., La.., Stochastc programmg models for vehcle routg problems, Asa Iformato Scece Lfe. Vol.1,pp.13-8,00. [6] La.., Lu B., Peg J., Vehcle routg problem wth fuzzy travel tmes ad ts geetc algorthm, Techcal Report,003. [7] Zheg Y, L u B. Fuzzy vehcle routg model wth credblty measure ad ts hybrd tellget algorthm [ J ]. Appled Mathematcs ad Computato.vol. 176, pp , 006. [8] Eberhart, R., eedy, J., A New Optmzer Usg Partcle Swarm Theory. Proceedg of the Sth Iteratoal Symposum o Mcro Mache ad Huma Scece, pp.39-43, [9] eedy, J., Eberhart, R.. Partcle Swarm Optmzato.Proceedg of the 1995 IEEE Iteratoal Coferece o Neural Networ, pp , [10] Sh,Y.,Eberhart, R.. Emprcal Study of Partcle Swarm Optmzato. Proceedgs of Cogress o Evolutoary Computato, vol.41, pp , [11] Salma,A.,Ahmad,I.,Mada,S. A. Partcle swarm optmzato for tas assgmet problem. Mcroprocessors ad Mcrosystems, vol.6, pp , 00. [1] Lu B. Ucertaty theory. Sprger-Verlag, Berl,