Research on Route guidance of logistic scheduling problem under fuzzy time window

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1 Advanced Scence and Technology Letters, pp Research on Route gudance of logstc schedulng problem under fuzzy tme wndow Yuqang Chen 1, Janlan Guo 2 * Department of Computer Engneerng DongGuan Polytechnc. Dongguan,Guangdong, Chna 1 chenyuqang@126.com, 2 rachel0519@126.com Abstract.Study the problem of logstc schedulng problem wth fuzzy tme wndow, construct mathematcal model, propose double objectve functon method. In the soluton process, for the feature of the double objectve functons, use the phased. In the frst phase, use the chaos PSO to get the optmal soluton. In the second phase, use smulated annealng algorthm and the prelmnary solutons got from the frst phase to solute the objectve functon. Keywords: logstc schedulng problem; chaos PSO; smulated annealng; fuzzy tme wndow 1 Introducton Modern logstcs s the product of economc globalzaton and t also s an mportant servces to promote economc globalzaton. In the recent years, Chna s logstcs ndustres are also growng rapdly. But our logstcs also have problems. Logstcs complete the space to space transfer of good wth number of power consumpton. The consumpton s long tme and long dstance. The logstcs schedulng problem wth fuzzy tme wndow s an mportant drecton. Researchers from domestc and foregn have done a lot of work. G.B.Alvarenga [1] studed the problem wth GA and phased functon. Rta Macedo [2] solute the VRP usng pseudo-polynomal. Yanns Marnaks [3] studed the hybrd genetc PSO VRP. In domestc, Cheng Jn, Ca Yanguang, L Yongsheng studed the unon schedulng problem. Qu Yuan and L Mn studed the logstc schedulng problem wth multple dstrbuton center. Ths paper s to solute the logstc schedulng problem wth fuzzy tme wndow. Studed the mproved chaos PSO and smulated annealng problem. 2 The mathematcal model of the logstc schedulng problem wth fuzzy tme wndow In the real lfe, dstrbuton process s often be dsturbed lke the customer change the dstrbuton tme, car traffc jams, accdents and other unseen crcumstances. From the pont vew of the customer, the dstrbuton s needed to be completed n the prescrpt tme. When the above crcumstances occur, customers are able to accept a certan ISSN: ASTL Copyrght 2014 SERSC

2 Advanced Scence and Technology Letters extent of ths phenomenon whch s the customer servce satsfacton problem. The greater the scope beyond the tme wndows, customer satsfacton wll be lower. To the logstcs company pont of vew, they want to fnd optmal dstrbuton path to save tme. But the companes also want good customer satsfacton to company s development. Ths needs the companes to fnd the balance pont of the cost and the customers satsfacton. From the above pont, ths paper use double objectve functons to solute the problem. One objectve functon s total cost mnmzaton and the other s servce satsfacton optmzaton. 2.1 Descrpton of the problem of logstc schedulng problem wth fuzzy tme wndow The problem can be descrbed as followng: There s a dstrbuton center wth I customers. The demand of customers are q (=1,2,...,l). There are m dstrbuton cars. There are k knds of cars (1, 2,..., k ) wth k1 represent the 1 car of the k knd. The k max load for each car s Q k (k=1,2,...,m), d j s the dstance between customer and j. T s the reachng tme of vehcle l to customer. T s the watng cost tme n servng customer. T j s the tme from customer to customer j. The tme wndows requre the vehcle arrves the customer between [ A, B ]. The servce satsfacton s L, L s between [0,1]. If the dstrbuton servce s completed n the tme wndow, L=1. The customer has a acceptaton scope between E E T, E L T. There are relatonshps between dfferent customers. Then can construct the fuzzy membershp functon lke Eq.1. 0 t E E T t E E T f ( t ) E E T t A A E E T L( t) E L T t g ( t ) B t E L T E L T B 1 A t E L T From the above functon, membershp functon s an trapezodal functon. In the scope of E E T, E L T, the servce satsfacton wll change as trapezodal, t wll decrease whle tme wndow ncrease. (1) 2.2 The mathematcal model of problem of logstc schedulng problem wth fuzzy tme wndow The mathematcal model s lke followng: Varable defntons: x jk l 1 k v e h c le l f r om to j 0 e lse ; 22 Copyrght 2014 SERSC

3 Advanced Scence and Technology Letters y k l 1 Po nt s c ompet ed by k v eh c l e l 0 e lse A, B s the max and mn of tme wndow of customer. E E T, E L T s the max and mn of the fuzzy tme wndow of customer satsfacton d j the dstance from customer to customer j. (1, 2,..., k ) car knd set. k k1 s the 1 car of knk k. f k s the constant usng cost of vehcle of knd K. c s the dstrbuton cost of vehcle of knd K. k s the servce satsfacton. S s the reachng tme of vehcle to customer. T s the watng tme of servng customer. T j s the cost tme n the dstrbuton process from customer to customer j. Objectve functons: K m I I I K m k 0 k l jk l j k (2) k l l j 0 k 1 l 1 m n Z f x x d c ; Constrant functons: I K I m m a x L L x T jk l 1 k 1 j 0 1 (3) K I m x 1 j (4) k 1 j 0 1 I q y Q k l k (5) 0 0 x T t q t T (6) j j j L x T K I m jk l (7) k 1 j 0 1 S S, T (, j ),, j (1,2,... l ) (8) j l Eq.2 and Eq.3 are two objectve functons. Eq.2 s the logstcs dstrbuton mn cost Z whch s total of constant cost and dstrbuton cost. Eq.3 s the max functon of Copyrght 2014 SERSC 23

4 Advanced Scence and Technology Letters customer s servce satsfacton. Eq.4 used to constran each customer can only be served once wth one vehcle. Eq.5 used to constran the total dstrbuton weght of good cannot exceed the maxmum load. Eq.6 s that one vehcle goes to serve customer j after servng customer. The relatonshp meets T t q t T. Eq.7 s used to constran the customer s j j servce satsfacton of the servce whch must exceeds. Eq.8 s the relatonshp between customers whch customer s early to be served than customer j. 3 Phased method to solve problem of logstc schedulng problem wth fuzzy tme wndow The model of fuzzy tme s complex than the normal model as t has two objectve functons. In the real lfe, logstcs companes have tme wndow constrant threshold for maturty customers. Based on a customer delvery tolerance, can only make optmzaton n the threshold range. Thus can optmze the dstrbuton route, save dstrbuton tme, decrease the dstrbuton cost wth the consderaton of dstrbuton sequence and tme whch meets the customers requrement. Accordng to the above realty, we can use two phased soluton method to solve the problem. The frst stage s to solve the problem of gettng the lowest cost of the customers whose servce satsfacton s bgger than. Specfc solvng process: of Under the condton of customer servce satsfacton s bgger than s known, then can compute the tme wndow [ A, B ] through Eq.9.. As the value 1 1 A B 1, 2... (9) f ( a ) g ( a ) Then can get the mathematcal model as below. K m I I I K m k 0 k l jk l j k (10) k l l j 0 k 1 l 1 m n Z f x x d c K I m x 1 j (11) k 1 j 0 1 I q y Q k l k (12) 0 0 x T t q t T (13) j j j 24 Copyrght 2014 SERSC

5 Advanced Scence and Technology Letters (14) A T B Eq.11 s the tme wndow constrant under the condton of the customer servce satsfacton. The second stage s makng optmzaton to the prelmnary results got from the frst stage accordng to the customer servce satsfacton. Specfc solvng process: After the soluton of the frst stage, we get the prelmnary solutons. Basng the prelmnary solutons, make dstrbuton tme optmzaton on each dstrbuton route. The goal s to mprovng customer servce satsfacton. Mathematcal model of the second stage: I K I m m a x L L x T jk l 1 k 1 j 0 1 (15) L x T K I m jk l (16) k 1 j x T t q t T (17) j j j In the stages, the frst stage solvng the prelmnary solutons s the key. Only save the excellent partcles and get prelmnary optmal soluton n the frst stage, that can get more accurate results n the second stage. Due to the two stage solvng, n order to solve for more accurate results, ths paper uses chaos PSO to optmzaton and uses smulated annealng n the second stage. The characterze of the smulated annealng s that t less demand the prelmnary solutons. 4 The soluton of the frst stage 4.1 Chaos PSO optmzaton by Logstc functon Logstc functon s also called S-functon or compresson functon whose mathematcal formula s lke bellowng: b y a 1 dx e Accordng to the characterstcs of nonlnear and everywhere derbable of the Logstc functon, ths paper uses Eq.19 to make adjustment to the nerta weght factor. (18) Copyrght 2014 SERSC 25

6 Advanced Scence and Technology Letters 2 ( )( 1) m a x m a x m n 1 e N d N m a x (19), s the max and mn value of, m ax m n m ax s the current teraton tme, computed usng the Eq.20. N s the max teraton tmes, N s the weght factor to the partcle, parameter d s d c, (0,1), [0, n ] (20) Constant c s determned accordng to dfferent crcumstance. s the chaos varable n the th chaos teraton. n s the number of populatons. 4.2 The soluton process of the chaos PSO optmzed by Logstc functon Accordng to the objectve functon and constrant functons n Eq.9-Eq.14, use chaos PSO optmzed by Logstc functon to solve the mathematcal model n Secton.2. The algorthm process to get prelmnary solutons n phase 1 s lke bellowng. Step1 Intalzaton. Intalze the partcle swarm, nput basc data to the network. Determne the swarm scope n, learn factor c c 1 2,, the max chaos teraton tmes. m ax m n, the max and mn value of Step 2. Compute the nerta weght factor through Eq.18,Eq.19. Determne the customer servce satsfacton. Step 3. Generate partcle swarm. Basng the basc parameters to generate the partcle swarm wth scope n. Step 4. Execute the standard PSO algorthm, save the partcle wth prelmnary good performance. Get p g whch s also called p. best Step 5. Make chaos optmzaton to the prelmnary partcles p from step 3 to best get chaos sequence and make chaos mappng. Fnd the poston to update the optmal soluton ' p. g Step 6. Premature judgment. If the partcles fall n premature then let part of the partcles wth good performance go back step 4. If not fall n premature then contnue the algorthm. There are two features for the premature partcles. The frst s that the partcles are extremely gathered. The second s that partcle swarms do not change after number of teratons. Use r e x p ( r. ) 1 2 to compute the adaptve varance of 2 of the partcle swarm. Make comparson wth the predefned mnmum populaton adaptve varance gathered and fall n premature. 2 mn 2 2. If, then the partcle s extremely mn 26 Copyrght 2014 SERSC

7 Advanced Scence and Technology Letters Step 7. Algorthm stop condton. If the teraton tmes to get the optmal soluton ' p s bgger than the predefned max teraton tme, then the algorthm stops. g Step 8. Get the global optmal soluton whch s the optmal dstrbuton route. Step 9. Algorthm stops. 5 Solve the second phase usng smulated annealng algorthm 5.1 The mechansm of the smulated annealng algorthm Basng on the theory of the smulated annealng algorthm, the probablty of sold trendng to balance at a temperature T s e E / k T. E s the sold nternal energy change,k s constant.the smulated annealng algorthm frstly fnd the ntal soluton and control parameter t. Then get the update soluton, then compute the objectve functon mnus, then update soluton or abandon the soluton. Loop the above sequence to the end of the algorthm to fnd the optmal soluton. The control parameter wll decrease wth the ncrease of the teraton. The bggest feature of smulated annealng algorthm s less dependent on the ntal solutons. Ths provdes convenent to the soluton n the second stage. The procedures of smulated annealng algorthm [6] : (1)Frstly get the basc parameters lke the ntal soluton x, ntal temperature, o current temperature, max teraton tme, etc. (2)If the current temperature reaches the nsde cycle temperature, then go to (3), else, do feld operatons, select a temporary soluton x j, compute E E ( x ) E ( x ), f E j s less or equal 0 or e x p ( E / t ) ra n d (0,1), then let x j j x, repeat step (2). j (3)Update the teraton tme k=k+1, accordng the temperature control functon to update the current temperature. The temperature control functon s y ( t ) t k k. If 1 meets the algorthm stop condton, then go to step (4), else go back to step (2). (4)Output results, algorthm ends. j 5.2 Algorthm process Accordng to the mathematcal model, use the optmal solutons got from the frst stage as the ntal solutons. The second stage of the smulated annealng algorthm s lke bellowng. Frstly nput the external parameters lke car model, customer nformaton, etc. Then ntalze the objectve functon. Use the prelmnary optmal soluton O, control parameter t. Conduct feld operatons, generate a neghborng soluton O, compute the Copyrght 2014 SERSC 27

8 Advanced Scence and Technology Letters energy dfference E E ( x ) E ( x ), f E j equals or less than 0 or j j e x p ( E / t ) ra n d (0,1) j, then let O=O. Generate the new soluton from the prelmnary soluton, compute the objectve functons dfference, update or abandon the new soluton. Repeat the above process wth the decreasng of control parameter t. Otherwse return the feld operaton steps. Update the teratons k=k+1, update the current temperature wth reference to the temperature control functon whch s lke y ( t ) t k k. If meet the algorthm end condton, the algorthm termnates and get the 1 optmal soluton. There are two crtera to determne algorthm convergence. The frst s max teraton tmes. The second s when the temperature s lower than a certan value or temperature does not change several tmes. e x p ( E / t ) ra n d (0,1) (21) j 6 Experment results and analyss The experments are dd on a pc wth Intel(R)Core2 CPU2.66GHz memory 2.0G, Matlab7. The specfc content s lke followng. A logstc company has a dstrbuton center. The company has four knd of cars and 8 customers. Suppose the upper and low lmt of the tme wndow and the customer acceptable fuzzy tme wndow s known. The customer 1 s requred to be served before customer 3. The experments are to determne the car type, dstrbuton routes and dstrbuton tme to mnmze the total cost. The detals are shown as below. Table 1. Vehcle nformaton Car type Number Load Constant cost Drvng cost A B C D E F Table 2. Customer dstance Dstance Copyrght 2014 SERSC

9 Advanced Scence and Technology Letters Dstance Experment results: Use the algorthm proposed n ths paper to do smulated analyss of the above case. In order to reflect customer satsfacton s mpact on results, we use constant customer satsfacton.we analyze the crcumstance when equals 1, not less than 0.9 and not less than 0.8. The experment results are lsted n table 4-6. Table 4. The expermental results when Dstrbuton Car type Route 1 C Route 2 C Route 3 A Dstrbuton cost Table 5. The expermental results when Dstrbuton Car type Route 1 C Route 2 C Route 3 A Route 5 E Dstrbuton cost Table 6. The expermental results when Dstrbuton Car type Route 1 A Route 2 C Route 3 E Route 4 E Dstrbuton cost From the above experments data we can conclude, when as the hard tme wndow problem. When = =1, the model s same 0.9, the dstrbuton routes change and total dstrbuton cost decrease. When 0.8 also can promse completng dstrbuton and the dstrbuton cost decrease reference to the above two crcumstances. So, control the customer satsfacton n a reasonable range can obvously decrease the dstrbuton cost of the logstcs company. Copyrght 2014 SERSC 29

10 Advanced Scence and Technology Letters 7 Summary The paper ntroduces the soluton method of the logstcs schedulng problem wth fuzzy tme wndow. Construct mathematcal model, ntroduce customer satsfacton as the second objectve functon. Use two stage method to solve the mathematcal model. In the frst stage, use chaos PSO algorthm. In the second stage, use the smulated annealng algorthm. Fnally, do smulaton analyss whch shows that t s reasonable to use two stage method to solve the problem. The concluson s that control the customer satsfacton n a reasonable range can obvously decrease the dstrbuton cost of the logstcs company. Acknowledgment.The work 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. 2012B ), also supported by the excellent young teachers program project of Guangdong Provnce(No:Yq ), and also supported by the project of Chnese Insttute of logstcs (2014CSLKT3-205). References 1. Yanns Marnaks,Magdalene Marnak : Expert System wth Applcatons Vol.37(2010),pp G.B.Alvarenga,G.R.Mateus,G.de Tom : Computers&Operatons Research Vol.34(2007), pp Rta Macedo,Claudo Alves,J.M.Valero de Carvaho,Francos Clautaux,Sad Hanaf. : European Journal of Operatonal Research Vol.214(2011), pp Zhou Y, LI Ha long, Wang Ru : Journal of Jln Unversty,Scence Edton Vol.02(2010),pp Ca Yanguang, Sh Ka : Computer Integrated Manufacturng Systems Vol.12(11)(2006), pp Zhou Y, LI Ha long, Wang Ru : Journal of Jln Unversty,Scence Edton Vol.02(2010),pp Suresh S, Sujt P B, Rao A K : Composte Structures Vol. 81(4)(2007),p Awad E1-Gohary,A1-Ruzaza A S : Chaos,Soltons and Fractals Vol. 34(2)(2007), pp Copyrght 2014 SERSC