Modeling a Multi Depot K- Chinese Postman Problem with Consideration of Priorities for Servicing Arcs

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1 Copyright 2015 America Scietific Publishers All rights reserved Prited i the Uited States of America Modelig a Multi Depot K- Chiese Postma Problem with Cosideratio of Priorities for Servicig Arcs Masoud Rabbai 1a *, Setare Mohammadi 1b School of Idustrial Egieerig, College of Egieerig Uiversity of Tehra, Tehra, Ira a*mrabai@ut.ac.ir bmohammadi.9009@ut.ac.ir Abstract: This study deals with the Chiese Postma Problem i critical situatio. Critical situatio is defied by movig o arcs. Oe aim of this ivestigatio is visitatio of critical edges as soo as possible. The startig poits of the study are the high cost of activities, decreased out of priority i servicig arcs, high umbers of vehicles ad routes to service. I this respect, the problem was determied as multi depot k-chiese postma problem, a kid of arc routig problem. This Problem cosiders three weighted goals. I this study, the optimal umber of postma ad the optimal path for each of them is determied. This mathematical model was solved by a heuristic algorithm which determies possible routes for each postma the by comparig paths ad elimiatig expesive ways sets the best route ad the best umber of postma. For compariso of the curret solutio, Artificial itelligece algorithms etitled Geetic ad Particle swarm were applied. Keywords: Chiese postma problem; geetic algorithm; graph theory; particle swarm optimizatio algorithm 1 INTRODUCTION I 1962, a Chiese mathematicia [1] added aspects of the optimizatio problem. Edmods [2] offered a polyomial algorithm to solve the Chiese postma problem. Gua ad Fleischer [3] showed that for a plaar graph the aswer of Maximum Weighted Cycle Packig Problem is equal to CPP, so both of them are equivalet. Edmods ad Johso [4] foud that the DCPP (Directed Chiese Postma Problem) ca be solved i appropriate time. They offered a polyomial algorithm to solve it. Existece of Euler tour eeds eve vertices degree of the correspodig graph. So that the for each vertex etrace edges are equal to outgoig edges. Mioux [5] preseted a model which was amed Optimal Lik Test Patter Problem but could t recogize it s relatio to Maximum Weighted Cycle Packig Problem. Chiese Postma Problem allows travelig edges i both directios but DCPP allows passig them i just oe determied directio. The first exact algorithm based o brach ad boud approach for Mixed Chiese Postma Problem was preseted by Christofides, et al. [6]. O some of the Chiese postma Problems the highest goal is to miimize the maximum path. This is writte i the followig form: max w C* mi{ w ( C) C is a k-postma tour o E} The goal is fidig a tour C* with k-postma which miimizes w max. Chiese postma problem with simultaeously k differet postma is called K- max ISSN: (prit), ISSN: (olie), DOI: /aiem

2 Chiese Postma Problem (k-cpp) [1].The CPP is a particular case of k-cpp. The CPP aims are to miimize the total cost of the tour, while k-cpp with k 2. The goal is to miimize the total cost of the tour. Ecoomically, the worst form of k-cpp i this case is just oe postma who services all other edges. Problems i the real world require employig multiple postme istead of cosiderig oly the postma. Usually, a compay is t oly oe perso to serve i differet parts but also serve several uses. Services such as the delivery of mail, distributio services, sow removal services, milk delivery systems, all require several people to service with miimal cost while time is limited. I a study coducted by Pear [7], geeral coditios ecessary to solve a k-cpp is k-cpp etworks completely udirected, k-cpp is fully directed etworks, k-cpp i hybrid etworks, eve ad symmetrical, k-cpp Widy etworks, alog with periodically symmetric. Geerally k-cpp is a NPhard problem [8]. So the ways of solvig this problem are meta-heuristic ad approximate methods [9]. Sorge [10] i his foud that the k-cpp if the parameter (k) is assumed costat would be solvable ad Guti, et al. [11] i their paper prove this. 2 PROBLEM FORMULATIONS The problem studied i the research is a multiobjective multi-depot k-chiese postma problem, which is a class of ARC routig problem. The objective of the Chiese postma problem is miimizatio of the total cost of the arc routig problem by allowig crossig each arc at least oe time. Liear programmig model of Chiese postma problem is as follows: mi CX (1) ij ij i1 j1 X X ji 0 i V (2) ij j1 j1 X ij X 1 ( i, j) A (3) ji X 0 Iteger i, j V (4) ij Equatio (1) is objective fuctio that miimizes the umber of traversed arcs. The equality of the etry ad exit umbers of each vertex is expressed i equatio (2). Equatio (3) poits out that each arc must be at least oce visited ad last equatio meas all variables must be positive iteger umbers [12]. K vehicles or postme with specified capacity leave the determied depots ad visit the arcs. It should be cosidered that vehicle s tour should ot exceed the amout of each vehicle s capacity. Now, it ca be expressed complete explaatio of problem. The aim is to miimize distace, miimize early observatio of low priority arcs i the ivestigatio ad miimize total umber of postma. 2 decimal allocated to each of the arcs. Oe of them is the distace of each arc ad the other oe the lack of priority. If the ocritical arcs visit earlier the result would be costs icreases. If the arcs are duplicated oce agai expeses will icrease. The optimum umber of postme is ot kow ad should be determied after problem solvatio. Parameters ad idexes h v D w c ij I ij { M1, M2..., M k } { e1, e2..., e } Depot umber Selected depot vertexes set The maximum acceptable distace for each Euleria path The degree of importace of the objective fuctio The cost of distace from origi i to destiatio j The low degree of priority Euler paths set traveled by the postma Depot set T D/mi(C ij ) Variables K h X M f ijt The umber of postma assiged to the ode h. 0, 1 variable of travelig arc (ij) i Euler path M f at step t 148

3 Objective fuctio ad restrictios TotalCost mi e ef e1 Z mi( w1 C X j1 Z CostPer() t IijX V 1 2 eh M f hjt M { M, M..., M } K eh ij M fijt M f 1 t1 j1 i1 w2 CostPriority( t) w3 K ) b h i1 V j1 f b 1 i X M f jit T V V t0 j1 i1 Mk 1 0 M f tij K X C D V If X M f ij( t1) i1 ij b Otherwise T V Mk T V X 1 M f jit M fijt M f M1 t1 j1 M f M1 t1 j1 M k V M f 1 j1 X M f hj0 M K T V V K h h1 CostPriority ( t) 2 CostPeriority ( t 1) CostPer () t h X (1) (2) (3) (4) (5) (6) (7) (8) (8) represets the umber of arcs exit from depot at the first step is equal to the umber of the postma. 2.1 Graph Form of Problem Small-scale graph form is examied. Graph is plotted i Fig. 1 is small scale drawig of the problem. Vertices are divided ito two categories. The Vertices which represet depots ad others represet itersectios. Note that the depots are located at the itersectios. Various depots are plotted i the form of the vertices of differet colors. Vertices that are draw blue color are ot the locatio of depots. I this problem we have 5 depots. For the black ad gray depots the umber of postma is equal to 2 while i other places just a postma offers service. For ease of display regardig the severity of the critical pathways arcs are draw distict colors. The umbers represet the time it takes passig each of the arcs. Each of the arcs must be visited by at least oe of the postma. The two of the proposed solutios of the problem is preseted. Equatio (1) is the cost of passig ocritical arc at t=0. Equatio (2) express the value of a recursive cost fuctio of ocritical arcs, so that if the arc of low priority placed first it s ocritical value (I ij) at each step is added to objective fuctio. The umber of postma allocated to each depot determied by Equatio (3). is a 0, 1 variable that oly oe of them is ozero which is represeted by Equatio (4). Equatio (5) states if i previous step passed a arc which is eded to vertex j ow this step oe arc started with j should be chose. Equatio (6) Implies the distace walked per Euleria path should ot be more tha D. The umber of outside arcs for each vertex is equal to the umber of iside arcs which is determied i equatio (7). Equatio Fig. 1. Small scale of problem A total of 7 postme must pass the arcs. Fig. 2 is oe of the proposed solutios to the problem. Fig. 3 also offers aother solutio. All proposed solutios should satisfy the problem costraits. If the capacity of postma is limited they caot travel ay differet path legths so a series of solutios proposed would be uacceptable. To remove such solutios a pealty fuctio is applied i objective fuctio. I this problem the umber of the postma at ay depot is ot fixed. If the arcs are passed several times the cost of the system icreases so the postma travels the streets surroudig. Priority routes are those which more critical arcs are passed earlier. The aim is 149

4 to improve simultaeously both objective fuctios. A small size of problem alog with drawig correspodig graph ad the optimal solutio represeted i Fig. 4 ad Table 1. Fig. 2. Proposed solutio oe phase, solvig techiques are distictive which have bee described idividually. The first solvig method results obtaied i a loger time seem to be more accurate. To solve problems i differet aspects 10 steps would be followed. 1. Determie the parameters of the problem. 2. Specify the umber of Euleria possible cycles for each depot. 3. Cycles elimiatio over more tha capacity. 4. Determie the maximum umber of postma for each depot demostrated by K i with respect to the umber of Euleria possible cycles. 5. Oe reductio uit i the total umber of the postma. 6. Collect the problem set solutios. This set cotais all possible combiatios. If this set is empty, go to Step9. 7. Determie the optimal cycle for each of the postma. 8. Update the output value of the problem the go to step The output value of our problem is reported. Fig. 5 represets the steps take to achieve optimal solutio. Table 1. Aswer of graphical problem Solutio Depot oe: Postma oe: (1,3),(3,4),(4,1) Fig. 3. Proposed solutio two Depot Two: Postma two: (2,3),(3,1),(1,2) Depot Two: Postma three: (2,6),(6,4),(4,5),(5,2) The total distace walked: 33 Cost1=99 Cost2=131 Total Cost=115 Fig. 4. Graphical form of problem 3 METHODOLOGY To solve this problem, a optimizatio algorithm with the approach of Euleria graphical tours has bee developed which is determied i Fig. 5. The problem with the help of Geetic algorithm (GA) ad PSO has bee coducted so the results have bee compared. At the begiig of all solvig methods the total umber of possible Euleria paths for each depot couted. The coutig has iteded Euleria path legth restrictios. At this stage the umber of Euler cycles may has determied. I this Usig Geetic algorithm to solve the problem, we should follow the steps below. 1. Set parameters of the problem 2. Determiatio accordig to the Euleria possible cycle paths for each depot. 3. Omit Euleria cycles over more tha allowable legth. 4. Specificatio the maximum umber of postma for each depot represeted by K i which is equal to the umber of Euleria possible cycles obtaied i the previous step. 5. Allocatio of postma set. If N 1=0 go to step6 else the total umber of allocated postma is equal to K 1+K 2+ K -N Solvig optimizatio problems usig PSO algorithms. The solutio set at this stage cotais combiatios emerged with the help of crossover ad mutatio o populatio set i additio, the best respose i the populatio. If the aswer set was empty at this stage N 1=N 1-1 go to Step5. 6. Hold ito cosideratio the best result obtaied usig Geetic algorithm as the best respose. 7. Omit N 1 Euleria possible path which 150

5 Fig. 7. A example of Crossover operator of GA chromosomes Fig. 8. A example of Mutatio operator of GA chromosomes Fig. 5. Represeted algorithm does t exist i best result obtaied i pervious step from the collectio the go to step The best Result should be reported. The first step i usig geetic algorithms is creatig iitial populatio. The iitial populatio i the form of a chromosome is defied as show i Fig. 6. Each oe represets ot selectio of a path ad each zero represet selectio of that path. This chromosome idicates Euleria paths will be removed from the solutio set. The parameters used i geetic algorithm for our problem are umber of iteratios, populatio size, crossover probability, ad mutatio probability. All geetic algorithm codes were writte by MATLAB R 2013a program. May differet ways is possible for processig crossover, at this crossover rate is used. Therefore, as much as the populatio size, a ew solutio is created by paret which are selected radomly. Mutatio operator is used to avoid local optimum. Fig. 6. A example of GA chromosomes Fig. 7 ad 8 shows Crossover ad Mutatio operators have bee implemeted. Usig Particle Swarm Optimizatio algorithm to solve the problem, we should follow the steps below. Step 1-5 of previous algorithm. 6. Solvig optimizatio problems usig PSO algorithms. If the aswer set was empty at this stage N 1=N 1-1 go to Step5. 6. Hold ito cosideratio the best result obtaied usig PSO algorithm as the best respose. 7. Omit N 1 Euleria possible path which does t exist i best result obtaied i pervious step from the collectio the go to step4. 7. The best Result should be reported. Various versios of the PSO algorithm are used but at this research the geeral equatio movemet of the particles is show below. All geetic Particle Swarm Optimizatio codes were writte by MATLAB R 2013a program., 1 11 gbest i i i i best i v t wv t c r x t x t c r x t x t i i i x t x t v t i v [ t 1] is the coefficiet of iertia for particle i i ext movemet. Velocity shows the desire of particle to cotiue the previous path. c & c Uif [1, 2], wuif [0.4, 0.9] ad 1 1 r & r Uif [0,1]. Icrease i the amout of w 1 1 ehaces exploratio of algorithm ad its decreasig, ehaces exploitatio ability of algorithm. 4 NUMERICAL RESULTS The problem i differet dimesios with the help of the metioed algorithms has bee ivestigated. Parameter assumed values preseted i Table

6 Arc legth ad Arcs Priority comes from uiform distributio. The weights cosidered for each goal is represeted i colum7. The umerical results are preseted i Table 3.The Problem is solved i three dimesios by three solvig method ad its umerical results are reported i Table 3. As ca be observed i Fig. 9 the first the algorithm ivestigates more solutios so it takes more time ad after more iteratios it fids the best solutio i compressio with meta-heuristics so its ru time is loger tha both of GA ad PSO but this algorithm does ot fall i local trap so its respose ca cofirm the aswers of two other algorithms. Show that the GA ad PSO algorithms did t fall i local trap.i geeral compressio PSO reports better solutios tha GA i each step. As ca be see i Fig. 9 after elimiatio of o-optimal routes all results coverge o oe had. This figure represets the cost results related to a etwork with 10 vertexes, 2 depots ad 20 arcs. The results of geetic ad PSO algorithms is ear aother therefore, the graph i the lower part of the figure is magified of this area. This figure icludes the first two compoets of cost so the third goal of miimizig total umber of postme is ot cosidered. Fig. 9. Comparative results Numerical results of GA for a small size problem reported by Tables 4, 5 ad 6. Table 4 represets the distributio of parameters. Table 5 reports the compoets of cost ad total cost of GA i 12 iteratios. Step by step the amout of Total cost decreases till the umber of the postma has bee miimized. The routs passed by the postme assiged to depots are observed i Table 6. Table 2. Parameters amout Algorithm Depots Vertexes Arcs Legth Priority Max distace Weights Heuristic uif (1,7) uif(1,10) ,0.3,0.4 3 Fuel Cost (Cij) uif(1,7) uif(1,10) ,0.3, uif(1,7) uif(1,10) ,0.3,0.4 3 GA uif(1,7) uif(1,10) ,0.3, uif(1,7) uif(1,10) ,0.3, uif(1,7) uif(1,10) ,0.3,0.4 3 PSO uif(1,7) uif(1,10) ,0.3, uif(1,7) uif(1,10) ,0.3, uif(1,7) uif(1,10) ,0.3,0.4 3 Table 3. The results related to Table 2 Parameters Algorithm Cost1 Cost2 Cost3 Total Cost Heuristic Cotiued i Next Page 152

7 GA PSO Table 4. Parameters Parameters Number of Depots Number of Vertexes Number of arcs Arc legth Arc Priority Maximum distace Cij uif(1,7) uif(1,10) 17 3 Table 5. Results reported by GA Algorithm Total Cost GA GA Cost1 GA Cost2 GA Cost Table 6. Routs passed by the postme The Postma Path Traveled Path Traveled by Postma Path Traveled by Postma Path Traveled by Postma

8 Costs Costs Rabbai ad Mohammadi, Advaces i Idustrial Egieerig ad Maagemet, Vol. 4, No. 2 (2015), As ca be see eve oe arc did t traveled more tha oe time. Graphical form of Table 5 is represeted i Fig. 10 ad Compressioal results of solvig methods are represeted i Fig Costs i 12 Iteratio other algorithms is estimated i Table 7. It is recommeded for future research to implemet model preseted at this study ad other relevat models for projects related to emergecy maagemet. It is recommeded that for power distributio etworks by this viewpoit modelig ivestigatio be coducted. Table 7. Gap of results Gap of Compressio with Heuristic GA Total Cost GA GA Cost2 GA Cost1 GA Cost PSO 0 Fig. 10. Compoets of Cost ad Total Cost obtaied by GA Comprestio of Three Methods Solutios Iteratio Number Huristic GA PSO Fig. 11. Compoets of total cost obtaied by GA, PSO ad Heuristic Algorithm 5 CONCLUSIONS This study ivestigated a multi depot Chiese postma problem which is oe of arc routig problems. The first aim is to miimize the mileage by total postme. The secod goal is travelig the critical arcs faster tha the rest of the arcs by postme. The third objective is miimizig the total umber of the postme. The legth of path of the survey by the postma is limited. This issue has bee evaluated by three separate algorithms to optimize routes were idetified. The aswers obtaied by GA ad PSO at each stage are approximately the same but PSO solutios are a little better tha GA which is determied i Table 7. If we bechmark the iitial algorithm the gap both Refereces [1] M. K. Kwa, Graphic programmig usig odd or eve poits. Chiese Math., vol. 1 1, pp [2] J. Edmods, Paths, trees, ad flowers. Caadia Joural of mathematics, vol. 17, o. 3, pp [3] M. Gua ad H. Fleischer, O the Miimum Weighted Cycle-coverig Problem for Plaar Graphs. Departmet of Combiatorics ad Optimizatio, Uiversity of Waterloo. [4] J. Edmods ad E.L. Johso, Matchig, Euler tours ad the Chiese postma. Mathematical programmig, vol. 5, o. 1, pp [5] M. Mioux, Optimal lik test patters i etworks ad the Chiese postma problem. Cahiers du Cetre d'études de recherche opératioelle, vol. 34, pp [6] N. Christofides, et al., A optimal method for the mixed postma problem. System Modellig ad Optimizatio, vol. 59, pp [7] W. L. Pear, Solvable cases of the k- perso Chiese postma problem. Operatios Research Letters, vol. 16, o. 4, pp [8] C. Thomasse, O the complexity of fidig a miimum cycle cover of a graph. SIAM Joural o Computig, vol. 26, o. 3, pp

9 [9] A. Osterhues ad F. Mariak, O variats of the k-chiese Postma Problem. Operatios Research ad Wirtschaftsiformatik, Uiversity Dortmud, vol. 30. [10] M. Sorge, Some algorithmic challeges i arc routig. Talk at NII Shoa Semiar, o. 18. [11] G. Guti, G. Muciaccia ad A. Yeo, Parameterized complexity of k-chiese postma problem. Theoretical Computer Sciece,vol 513, pp [12] İ. Z.Akyurt, T. Keskiturk ad Ç. Kalkaci, Usig Geetic Algorithm For Witer Maiteace Operatios: Multi Depot K- Chiese Postma Problem. Emergig Markets Joural, vol. 5, o. 1, pp

10 156 Rabbai ad Mohammadi, Advaces i Idustrial Egieerig ad Maagemet, Vol. 4, No. 2 (2015),