Research on chaos PSO with associated logistics transportation scheduling under hard time windows

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1 Advanced Scence and Technology Letters Vol76 (CA 014), pp Research on chaos PSO wth assocated logstcs transportaton schedulng under hard tme wndows Yuqang Chen 1, Janlan Guo *, Xuanz Hu 3 Department of Computer Engneerng DongGuan Polytechnc Dongguan,Guangdong, Chna 1,chenyuqang@16com,,rachel0519@16com, 3, huxuanz@16com Abstract: Ths paper study the assocated logstcs transportaton schedulng problem wth hard tme wndows and construct the mathematcal model Optmze the chaos PSO problem usng the mproved adaptve nerta weght factor method to mprove the accuracy and effcency of the algorthm Comparng our method wth the GA and standard PSO algorthm, our algorthm has better performance wth solvng such problem Keywords: chaos PSO algorthm; hard tme wndow; assocated logstcs transportaton schedulng problem 1 Introducton Now, logstcs ndustry has been an mportant economc sector n our country and develops n a rapd speed Logstcs need large amount of consumpton to acheve space transfer of goods Ths consumpton whch usually wth long tme and bg dstance s the bggest cost n the total logstcs cost From studyng the logstcs transportaton schedulng problem can mprove the effcency of logstcs transportaton, decrease the logstcs transportaton cost, optmze the structure of logstcs ndustry and has a wde range of practcal sgnfcance Transportaton schedulng problem s an mportant factor n logstcs problem RVRP s a branch of the VRP(Vehcle Routng Problem) The problem of optmzng dstrbuton path has practcal sgnfcance Researchers from domestc and abroad have done a lot of wor on logstcs transportaton schedulng problem GBAlvarenga etc [1] proved that heurstc algorthm s better than precse algorthm from studyng VRPTW wth GA and partton functon Rta Macedo etc [] solved the VRP wth multple drecton and tme wndows through constructng pseudo polynomal model Yanns Marnas etc [3] studed hybrd generc PSO based VRP In domestc, Chen Jn, Ca Yanguang, L Yongsheng etc studed unon transportaton schedulng problem Qu Yuan, L Mn etc studed the transportaton problem wth multple dstrbuton centers The paper use the chaos PSO algorthm to study the assocated logstcs transportaton problem wth hard tme wndows ISSN: ASTL Copyrght 014 SERSC

2 Advanced Scence and Technology Letters Vol76 (CA 014) Problem descrpton and mathematcal model constructon In the actual stuaton, logstcs companes need to meet the customers request that goods must be dstrbuted to the customer before a tme or n a tme range That s called the logstcs transportaton problem wth hard tme wndows The logstcs transportaton problem wth hard tme wndows n ths paper s that goods must be dstrbuted to the customers n a tme range It s a fal dstrbuton f goods are dstrbuted before or after the tme range 1 Queston Descrpton The logstcs transportaton problem wth hard tme wndows s descrbed as follows: One logstcs company wth one logstcs center, l customers, demands of customer q (=1,,, l ),m dstrbuton vehcles wth the maxmum load Q (=1,,,m) for one vehcle, dstrbuton tme range [ A, B ] d j s the dstance from customer to customer j S s the tme that dstrbuton vehcle reach customer T s the tme whch s cost by dstrbuton vehcle servng customer T j s the tme cost of dstrbuton vehcle travellng from customer to custom j n s the customer number served by vehcle There are assocated constrants between customers dstrbuton tme range One dstrbuton vehcle only can serve one customer Dstrbuton vehcle starts from the dstrbuton center, dstrbute the goods to the customer and return the dstrbuton center The queston s how to reasonably arrange the dstrbuton vehcles and dstrbuton paths to lowest the cost under the condton of meetng all the customers needs Constrants are as follows: (1)The dstrbuton center s fxed and unque ()The dstrbuton process s close as the dstrbuton vehcle starts from the dstrbuton center and return the dstrbuton center (3)The dstrbuton s not n a full load stuaton and t has the maxmum dstrbuton load and maxmum dstrbuton dstance (4)Every customer s served by one vehcle and all customers dstrbuton needs must be meet (5)The maxmum load of the dstrbuton vehcle and the demands of customer s nown (6)The dstrbuton vehcle must complete dstrbuton tas n the stpulated tme range, else the dstrbuton tas s fal (7)Tme wndow for each customer s nown (8)Customers need dstrbute and fetch goods n the same tme (9)There are assocated constrants between customers dstrbuton tme range 76 Copyrght 014 SERSC

3 Advanced Scence and Technology Letters Vol76 (CA 014) Mathematcal model Accordng the above descrpton, construct the assocated logstcs transportaton schedulng problem wth hard tme wndows as follows: The objectve functon: m n mn = dr ( 1) ( ) r+ d rn rsgn n 0 = 1 = 1 (1) Constrant functon: qr Q () n = 1 dr ( 1) ( ) r+ d rn rsgn n 0 D (3) n l (4) 0 m = 0 n = l (5) 1, n 1 sgn( n ) = 0, n < 1 (6) S + T + T = S r r r r r ( 1) ( 1) ( 1) (7) T = max{ A S,0}, = 1,,, l S B, = 1,,, l (8) (9) Copyrght 014 SERSC 77

4 Advanced Scence and Technology Letters Vol76 (CA 014) {(, j),, j { 1, l } Sr < Sr, Ts, j r (10) Formula (1) s the objectve functon whch represents the shortest dstrbuton path of all vehcle n the logstcs process Formula () represents the total dstrbuton goods are under the maxmum load of all vehcles Formula (3) represents each dstrbuton path s under the every car s maxmum dstrbuton dstance Formula (4) represents the customers n one dstrbuton path s under the total customers number Formula (5) promses every customer s served Formula (6) represents the vehcle s dstrbutng when the servng customer number s equal or bgger than 1, the vehcle s now dstrbutng when the servng customer number s less than 1 as sgn(n)=0formula (7) represents the tme of the dstrbuton vehcle reach the next customer S r equals the sum of the tme of the dstrbuton vehcle reached the current customer Sr (, the watng tme n the current customer S 1) r ( 1) and the tme cost of travellng from current customer to the next customer Tr ( 1) r Formula (8) represents the constrant of the watng tme on the current customer that determned by the reachng tme and the tme wndow of current customer Due to the hard tme wndow, f the reachng tme s before or equal the tme wndow, the watng tme equals the reachng tme mnus the current tme wndow, f the reachng tme s after the tme wndow, then the dstrbuton s fal and the watng tme s 0 Ths formula promses the dstrbuton vehcle must reach the customer before or equal the tme wndow Formula (9) promses the dstrbuton process must before or equal the tme wndow end tme Formula (8) and (9) wors together to promse the logstcs schedulng mathematcal s under the constrant of hard tme wndow Formula (10) represents the assocated relatons between customers tme range as customer must be served before customer j 3 Chaos PSO algorthm dealng wth logstcs transportaton schedulng problem wth hard tme wndow 31 Improved adaptve nerta weght factor adjustng chaos PSO algorthm In order to deal wth the problem of the logstcs problem falls nto the local optmal value to affect the general optmzaton results, the paper ntroduces an adaptve nerta weght factor adjustng method derved from adaptve weghtng factor method To the n-dmensonal partcle swarm of objectve functon, represents the represents the average partcle, the weghtng factor of partcle swarm s ω value of the partcles, the computaton method s shown as (11) Let the value of the optmal partcle s g, get the average value of the optmal partcles as Defne =, then accordng the followng three stuatons to optmze the 78 Copyrght 014 SERSC

5 Advanced Scence and Technology Letters Vol76 (CA 014) weghtng factor ω 1 n n = 1 = (11) Frst stuaton, s better than The partcles meet the frst stuaton are the excellent partcles n the swarm The nd partcles are close to the global optmal partcles Relatvely small weghtng factor s sutable to ths nd of partcles Ths nd of partcles can use formula (1) to adjust adaptve nerta weghtng factor ω = ω ( ω ω ) mn g (1) Second stuaton, s better than, but worse than Partcles meet the second stuaton are the general partcles whch s the majorty of all the partcles, the stablty s relatvely good and have good global and local optmzaton ablty To ths nd of partcles, we use segmentaton method At the early stage of the teratons, use the relatve bg weghtng factor to optmze At the end stage of the teratons, use the relatve small weghtng factor to optmze Accordng to the above consderaton, use Cosne functon to adjust the weghtng factor Formula (13) shows such method ω ω ω ω ( N 1) π 1+ cos( ) N 1 max = mn ( max mn ) (13) N s the teraton tme, Thrd stuaton, s worst than N max s the maxmum teraton tmes Partcles meet the thrd stuaton are relatve worse n the swarm Ths nd of partcles has worse global and local optmzaton ablty, easly to fall nto early maturty Accordng to the characters of ths nd of partcles, use formula (14) to adjust ω = r exp( r ) (14) 1 r s used to control the upper lmt of ω, In formula (34), 1 control the formula s adjustng ablty to ω r s used to Copyrght 014 SERSC 79

6 Advanced Scence and Technology Letters Vol76 (CA 014) 3 Algorthm process The dea of our method s shown as follows: Frst, use the standard PSO to ntalzaton and generated partcle swarm, get the prelmnary optmal value Then, do chaos optmzaton to the prelmnary optmal value usng chaos mappng method, get the optmal soluton The process of the algorthm s shown as Fgure (1) Begn ntalzaton Determne parameters and generate partcle swarm, nput the objectve functon and hard tme wndow constrats Compute and adjust w, use SPSO to do global search Chaos mappng Local search Early maturty? No No Normally termnaton? Yes Get the optmal dstrbuton path End Fg 1 CPSO Algorthm flowchart The steps of our algorthm: Step1 Intalzaton Intalze the partcle swarm, nput the basc parameter of the dstrbuton networ Determne the partcle swarm scale n, study factor c c 1, the maxmum and mnmum value of nertal factor w, the maxmum teraton tmes of chaos optmzaton Step Generate partcle swarm Accordng to the basc parameter, generate partcle swarm wth scale n Step3 Execute the standard PSO algorthm, preserve the partcle wth prelmnary good performance Get p g =( p g1, p g, p gm ) as p best Step4 Do chaos optmzaton to the partcles whch are preserve from step 3 p best, l get chaos sequence, do chaos mappng, acheve chaos optmzaton Fnd p * g and update the poston of the new soluton Step5 Early maturty judgment Judge f the partcle swarm fall nto early maturty 80 Copyrght 014 SERSC

7 Advanced Scence and Technology Letters Vol76 (CA 014) consstence If fall nto early maturty, select partal of the relatvely good partcle to contnue step 4 If not fall nto early maturty, go to next step The early maturty partcle swarm has two specfc characters The frst s that partcle swarm extremely accumulated The second s that partcle swarm has no change after several teratons Formula (15) s used to compute the swarm ftness varance σ of partcle swarm f f σ = = 0 f (15) Compare wth the pre-defned mnmum swarm ftness varance σ σ, the partcles extremely accumulate, fall nto early maturty mn σ mn Step6 Stop condton of judge algorthm Compute the optmal teraton tmes to l get, f the result s bgger than the pre-defned maxmum teraton tme, the p * g chaos optmzaton stops Step7 Get the global optmal soluton, as the optmal dstrbuton path Step8 Algorthm ends, f 4 Experment results and analyss The experments were dd on PC usng Matlab 70 to program One logstcs company has one logstcs center, 8 customers, Table 1 shows the locaton and dstance between logstcs center and each customer, Table shows the demand of each customer and hard tme wndow constrants The number of dstrbuton vehcle s 50, the maxmum load s 10 ton The unt transportaton cost s 1 Customer 3 needs to be served before customer 1 Table 1 Customer locaton chart Dstance Table Customer demand, servce tme and tme wndow data Customer Demands Servng tme Tme wndow [1,4] [4,6] [1,] [4,7] [3,55] [,5] [5,8] [15,4] Copyrght 014 SERSC 81

8 Advanced Scence and Technology Letters Vol76 (CA 014) The basc parameters used are as follows: the number of populatons of partcle w swarm 50, c1=1, c=1, adaptve weghtng factor max =08, w mn =03, the maxmum teraton tme 150 tmes The optmal dstrbuton paths got from the smulaton experments: Dstrbuton path 1: Dstrbuton path : Dstrbuton path 3: The optmal total cost s =90 To valdate the performance of our algorthm, use GA and standard PSO to do 30 tmes smulaton experments The number of populatons s 50 and teraton tme s 100 for both GA and standard PSO The results are shown as fgure and table 3 Table 3 Comparson of the three algorthms Algorthm Longest path Shortest path Average Optmal soluton tme GA PSO Our algorthm Fg Comparson of the three algorthms From the above analyss, the algorthm proposed n ths paper has good performance n dealng wth assocated logstcs transportaton problem wth hard tme wndow Our algorthm effectvely avods the partcles to fall nto maturty, mprove the global and local searchng ablty Compared wth GA and standard PSO, our algorthm has better performance n precse, accuracy, teraton effcency and optmal soluton 8 Copyrght 014 SERSC

9 Advanced Scence and Technology Letters Vol76 (CA 014) 5 Concluson Ths paper ntroduced the soluton method assocated logstcs transportaton problem wth hard tme wndow, constructed mathematcal model Introduced an adaptve weghtng factor adjustng chaos PSO algorthm whch derved from adaptve weghtng adjustng factor method The method can adjust the adaptve nerta weghtng factor accordng dfferent partcles, can effectvely mprove the global and local searchng ablty Fnally do smulaton experment usng Matlab Experment results show that our algorthm has better performance n dealng wth assocated logstcs transportaton problem wth hard tme wndow Acnowledgments The wor descrbed n ths paper was supported by Natonal Natural Scence Foundaton of Chna (No ), also supported by Guangdong Provncal of Scence and Technology Foundaton (No 01B ), also supported by the excellent young teachers program project of Guangdong Provnce(No:Yq01301), and also supported by the project of Chnese Insttute of logstcs (014CSLKT3-05) Reference 1 Yanns Marnas, Magdalene Marna A hybrd genetc-partcle Swarm Optmzaton Algorthm for the vehcle [J] Expert System wth Applcatons 37(010), pp: G B Alvarenga, GRMateus, Gde Tom, A genetc and set parttonng two-phase approach for the vehcle routng problem wth tme wndows [J] Computers&Operatons Research 34(007): Rta Macedo, Claudo Alves, J M Valero de Carvaho, Francos Clautaux, Sad Hanaf, Solvng the vehcle routng problem wth tme wndows and multple routes exactly usng a pseudo-polynomal model [J] European Journal of Operatonal Research, 14(011), pp: Eberhart R C, Sh Y, Partcle swarm optmzaton: developments, applcatons and resources ProcCongress on Evolutonary Computaton 001[C]Pscataway, NJ: IEEE Press, 001, 9(33), Ayed S, Imtaz S, Sabah A-M Partcle swarm optmzaton for tas assgnment problem[j]mcroprocessors and Mcrosystems, 00, (6): Dumas J Soums E, Passenger flow model for arlne networs Transportaton Scence, 008, 4(): Anly S, Tzur M, Shppng multple tems by capactated vehcles: ail optmal dynamc programmng approach[j] Transportaton Scence, 005, 39(): Agarwal R, Ergun O, Shp schedulng and networ desgn for cargo mutng n lner shppng[j] Transportaton Scence, 008, 4(): Schaefer A J, Johnson EL, Kleywegt AJ, Arlne crew schedulng under uncertanty[j] Transportaton Scence, 005, 39(3): Copyrght 014 SERSC 83