INTEGER PROGRAMMING MODELING FOR THE CHINESE POSTMAN PROBLEMS ABSTRACT Feng Junwen School of Economcs and Management, Nanng Unversty of Scence and Technology, Nanng, 2009, Chna As far as the tradtonal Chnese Postman Problem (CPP) s concerned, based on the dscussons n the undrected and drected graphs respectvely, the correspondng nteger programmng models are proposed, some numercal examples are gven to demonstrate the utlty of the models. Furthermore, the models are extended to the case wth stochastc weghts (the correspondng problem s called Stochastc Chnese Postman Problem). Fnally, some possble generalzatons of the Chnese Postman Problem are dscussed brefly. Keywords: Chnese Postman Problem, Integer programmng, optmal model, weghted graph INTRODUCTION The research on optmal delvery route of the postman problem s frst proposed and studed by Chnas professor Guan Me gu n 960s, Now known as the Chnese postman problem. Professor Guan Me gu had gven a party pont dagram method for ths problem n960s. In973 Edmonds and Jonhson gave the mproved algorthm of the Chnese postman problem 2, whch s more effectve than the former calculaton. In 98 Professor Guan Me gu revews the research of the Chnese postman problem before that tme n the paper 3. The early dscussons of the Chnese postman problem s always based on undrected graph, In fact, due to a sngle lne or the slope of the up and down route and so on, The problem sometmes must be resorted to drected graph to study and resolve. So far, on the Chnese postman problem research s manly carred out from the perspectve of graph theory, the maorty s gven a varety of heurstc algorthms or recursve algorthm, the typcal research s [-2]. Ths artcle wll research from the perspectve of mathematcal programmng. Mathematc programmng model wth software package has the advantages of solvng convenent, easy to modfy and promoton and so on, whch even has the advantage of solvng large and complex problems. Ths paper frst establsh a correspondng explct nteger programmng model on the bass of tradtonal Chnese postman problem, and the numercal example llustrates the valdty of ths model, t also ponted out that a slght expanson of ths model can also be used to solve the problem based on a drected graph of the generalzed Chnese postman problem and the random Chnese postman problem, the modelng and numercal value s also gven accordngly. MODELING OF TRADITIONAL CHINESE POSTMAN PROBLEM The tradtonal Chnese postman problem can be summarzed as follows: A postman starts from hs post offce to go to messenger every tme, traveled every street that he or she s responsble for the delvery of the wthn range, After completon of the messenger task back to the orgnal post offce, what knd of routes he or she should choose In order to make the shortest dstance to go. The problem abstracted nto graph theory problem s that gven a connected graph G (V, E), where V = {V, V2,..., Vn} s a set of vertex, sad that the place of the street ntersecton, E s the set of edges between the vertces, sad street, E = {e = (V V);edge e between vertces V and V}, each edge e n E wth non negatve weght W ( e ) = w ( V V ), that the edge s the length of the street, the problem s to determne a crcle of G, whch over each edge at least once and makes the mnmum of the total weght (on each sde of the total)of the crcle. From the graph theory, If G does not contan the sngularty (the number of adacent edges s odd), then G has a crcle whch over each edge once and only once. So ths crcle s the crcle requred. If G contans a sngularty (the number of adacent edges s odd), then G s a crcle on each sde at least once, and wll over some sde by more than once, If adopted the k-th edge e, let e = (V V), we add k- new edges between V and V, and the weght of the new edge s equal to the weght of the edge e, sad the new edge s the added edge of edge e. Obvously,f there are more than one add edge of edge e, we lose an even number of edges, then a total weght obtaned from graph of any sde over at least once of a crcle wll not be ncreased. So we can assume that the number of add edge of each edge add up to one. In ths way, problem of fndng the optmal delvery routes of the postman can be attrbuted to the followng graph theory problem:
Gven a connected graph G(V,E), add an edge e =(V, ) V between each edge e=(vv) E correspondng to the vertex V and V, get a connected graph G( V, E ) whose number of edges s double of G, seek EE, E E n order to make the graph G( V, E) contan no sngularty and the total weght to a mnmum.for smplcty, we () ee f e=(v V ) E, t s recorded as e, edge e E E or (), whle the correspondng add edge s,and corresponds to e,,set a 0- nteger varable x,if,the edge s from V to V or called arc, In ths way, we can e, E put an undrected graph be understood as a drected graph. Every E unquely corresponds to a set of values of x, vce versa. We can use the varables ( =,2,..., n; =,2,..., n ) to defne the constrants of optmal x, postman problem are as follows: () over each edge at least once and add edge up to at most one, values ( known as value system of E)corresponded to E need to meet: For e E, x x, (2) graph G ( V, E ) contans no sngularty, for any vertex V wth a "nto" the arc, there wll be an equvalent amount of" out" to the arc: x x, 0 The obectve of the problem s makng the mnmum total weght of G( V, E ),that s, w s mn w, x, ( ) E the weght of edge e, w, = w,so we can get the explct nteger programmng model of Chnese postman problem (CPP) as follows: mn w x ( ) E x, x 0,,2,..., n (, ) E ( ) E s. t. x x,, ( ) E x, 0or, ( ) E Ths model not only can be used to solve the Chnese postman problem, and can also determne the correspondng optmal delvery route: such as x =, means that the postman from V along the edge (.e. the street) e to V. GENERALIZED CHINESE POSTMAN PROBLE AND ITS MODELING The front secton of the postman problem assumes that the postman delvered wthn the scope of every street n the uplnk and downlnk wthout dfference, n fact that the delvery of the letter s probably not the case, such as n a case of the street of sngle lne street wth a certan slope on both sdes of the street cannot be delvered at one tme by a sngle lne and so on. Ths postman problem whch we call the generalzed Chnese postman problem the generalzed Chnese postman problem can be abstracted as a drected graph problem. Smlar to the precedng postman problem (known as tradtonal Chnese postman problem), generalzed postman problem can be descrbed as follows: Gven a connected drected graph G (V, E), each arc e has non-negatve weght w (e),we need to fnd a loop of G whch over each arc at least once and make the total weght to a mnmum. For the generalzed Chnese postman problem, the number of add arc up to one sometmes s no longer feasble, whch needs many add arc to make the correspondng connected to any one of the vertces of a graph G nto has a loop ( E loop ). Here, f number of arcs and out number of arcs to the same, n order that the G( V, E ) e=(v, V ) E, then we say that arc e s nto the arc of vertex V, whch s also out to the arc of vertex V.If the number of vertces of G (V, E) s n, t can be proved that each arc s ncreased (n-) add arc at most, each vertex can be realzed n the number of n to the arc s equal to the number of out to the arc.
Based on the analyss above, defned a postve nteger varable x for each arc, used to represent the arc e ncreased ( e E of G (V, E), whch s x -) add arc, thus formng anther drected graph G( V, E ). Smlar to the analyss of the prevous secton, we have the followng generalzed Chnese postman problem explct nteger programmng model (GCPP): mn ( ) E w x x, x, 0,,2,..., n (, ) E ( ) E x,2,..., n ( ) E By solvng ths model, we can get the optmal delvery route of the generalzed Chnese postman problem. Several numercal examples [Example ] Consder the Chnese postman problem as shown below: Fg tradtonal Chnese postman problem (a) The undrected graph problem s equvalent to the followng drected graph: Fg 2 tradtonal Chnese postman problem (b) Accordng to the prevous dscusson of the model, the numercal example shows the correspondng nteger programmng model as follows:
mn 2( x x ) ( x x ) ( x x ) 3( x x ) 3( x x ) 6( x x ), 8 8, 8, 7 7, 8, 2 2, 8, 9 9, 8 6, 7 7, 6 2, 9 9, 2 ( x x ) ( x x ) ( x x ) ( x x ) 9( x x ) ( x x ) 9, 6 6, 9 3, 2 2, 3 9,, 9, 6 6, 3,, 3,, x2, x8, x, 2 x, 8 0 x, 2 x9, 2 x3, 2 x2, x2, 9 x2, 3 0 x2, 3 x, 3 x3, 2 x3, 0 x3, x9, x, x, 3 x, 9 x, 0 x6, x, x, 6 x, 0 x7, 6 x9, 6 x, 6 x6, 7 x6, 9 x6, 0 x6, 7 x8, 7 x7, 6 x7, 8 0 x, 8 x7, 8 x9, 8 x8, x8, 7 x8, 9 0 x2 9 x 9 x6 9 x8 9 x9 2 x9 x x 8 x, 8, x8, 7 x7, 8 x, 2 x2, x8, 9 x9, 8 x6, 7 x7, 6 x2, 9 x9, 2 x9, 6 x6, 9 x2, 3 x3, 2 x9, x, 9 x, 6 x6, x3, x, 3 x, x, x, 0or x 0,,,,,, 9, 6 9, 8 The applcaton of nteger programmng solvers QSB,and the soluton of the optmal soluton of the problem s as follows: x x x x x x, 2 2,, 8 8, 8, 7 9, 8 x x x x x x 7, 6 2, 9 3, 2 6, 9, 3 9,, 6 6,,, x x x x other x 0,the mnmum weght s 68. Assumed the post offce at the apex of vertex V, the optmal delvery routes as follows: e e e e e e e e, 2 2, 9 9, 8 8, 7 7, 6 6,, 6 6, 9 e e e e e e e e 9,,,, 3 3, 2 2,, 8 8, We should note that the optmal delvery route of ths problem s not unque, smlarly we can obtan the optmal delvery route started from any vertex. [Example 2] Consder the followng are shown n Fgure 3 of the generalzed Chnese postman problem: Fg 3 generalzed Chnese postman problem The correspondng nteger programmng model of generalzed Chnese postman problem as follows:
mn 2x 8x x 6x 7x x x x 2x 9x x 3x, 2, 3, 2,, 3 2,, 6, 7, 2 3, 7 6, 7 6, 2x x 6x 3x x x 7x 9x 2x x 8,, 9 9, 6 7, 9 7, 0 0, 9 9, 8 8,, 9 0, x3, x, 2 x, 0 x, 2 x2, x2, 0 x, 3 x3, x3, 7 0 Applcaton of the same nteger programmng x, x2, x, x6, x7, x, 3 0 x2, x8, x6, x, x, 9 0 x9, 6 x6, x6, x6, 7 0 x6, 7 x3, 7 x7, x7, 9 x7, 0 0 x9, 8 x8, x8, 0 x, 9 x0, 9 x, 9 x7, 9 x9, 6 x9, 8 0 x7, 0 x0, 9 x0, 0 x0, x8, x, 9 0 x,2,...,0 software, solvng ths model and obtan the followng optmal soluton: x 2, x, x 3, x, x, x,, 2, 3, 2,, 3 2, x, x, x, x 2, x 2, x,, 6, 7, 3, 7 6, 7 6, x, x 2, x, x, x 2, x, 8,, 9 9, 6 7, 9 7, 0 0, 9 x 2, x, x 2, x 9, 8 8,, 9 0, The mnmum weght s 9. Assumed the post offce at the apex of vertex V, the optmal delvery routes as follows: e e e e e e e e, 2 2,, 3 3,,, 3 3,, 2 e e e e e e e 2,,, 3 3, 7 7, 0 0,, 9 e e e e e e e 9, 8 8,, 9 9, 6 6,, 9 9, 8 e e e e e e e 8,, 9 9, 6 6, 7 7, 9 9, 6 6, e e e e e e e e, 3 3, 7 7, 0 0, 9 9, 6 6, 7 7, e, 3 3, It s a loop, the optmal delvery routes can be determned of the post offce n any vertex smlarly. Ths optmal delvery routes found out s based on the model of the optmal soluton wth the method smlar to the soltare game. It should be noted that f the value of x, s greater than, means that the arc to be repeated the walk. The sum of all x, should be equal to the number of street walked (ncludng the walk repeated). RANDOM CHINESE POSTMAN PROBLEM AND ITS MODELING Tradtonal and generalzed Chnese postman problem dscussed above assume that the weghts related to edge or arc are determned constants. It s often encountered that the weght s non-fxed n practce, for example, the weght consdered s the delvery tme spent on the streets, ths parameter s often not constant, the tme t takes for each delvery wll change wth the number of mal, but t generally follows a form of change, that s, the weghts are random varables wth a certan dstrbuton, then we call the correspondng problem for the random Chnese postman problem. The party graphc workng method of tradtonal Chnese postman problem and ts mproved algorthm 2 cannot solve the random problem, but by means of the nteger programmng model establshed n ths paper, and wth the applcaton of stochastc programmng theory 6-8, the problem can be solved easly. A varety of approach related to solve random problem, such as expected value method chance constraned method the optmal value dstrbuton method relevant chance constraned method, mult-stage(such as twostage)method. Ths paper only dscusses the chance constraned method, and ts core s the determnstc equvalent treatment process under the constrant condtons of probablty. Readers nterested n other method may follow the smlar treatment process of the stochastc programmng theory, whch s omtted n the paper.
Notng that the constrant of (CPP) or (GCPP) does not contan a parameter of weght, therefore solvng a queston of random weght, what need to do s ust makng the obectve functon for the correspondng determnstc equvalent treatment process. When weght s a random varable, the obectve functon s also a random varable, accordng to theory of stochastc programmng, stochastc CPP problem can be transformed nto the two determnstc equvalent model as followng: CPP( ) mn w ( ) E P( w x w) const r a nt s of ( CPP) 2 CPP( w) max ( ) E P( w x w) const r a nt s of ( CPP) The -CPP( )refers to the solvng of x, and w when s fxed, the 2-CPP(w)refers to the solvng of x, and when w s fxed. Here, we refer to as the feasble weghts of w, w s the level hoped of total weght. If the weghts are subect to normal dstrbuton and ndependent of each other, 2 w, followsn (,,, ),-CPP( )and 2-CPP(w)are equvalent to the followng two mathematcal programmng respectvely: N CPP( ) mn w 2 2 w x F,, ( ) x 0 ( ) E ( ) E const r a nt s of ( CPP) 2 N CPP( w) max 2 2 w x F,, ( ) x 0 ( ) E ( ) E const r a nt s of ( CPP) Where F(X) s the standard normal cumulatve dstrbuton functon, F - (y)s ts nverse functon. Further, t s equvalent to the followng two mathematcal programmng problems respectvely: N CPP( ) mn x F ( ) x w 2 2 ( ) E ( ) E const r a nt s of ( CPP) w ( ) E 2 N CPP( w) max F ( ) 2 2 x ( ) E const r a nt s of ( CPP) x It should be noted that F ( ) s a monotoncally ncreasng functon of.other types of random problem can also be analyzed smlarly. As the programmng above s non-lnear, we need to use the non-lnear nteger programmng tools and software or two stage method 8 to solve the problem; large problem can also be solved by means of modern ntellgent optmzaton algorthm 9. The correspondng random problem of the generalzed Chnese postman problem can also be dscussed smlarly, whch s omtted here, the nterested reader may have a try. POSSIBLE PROMOTION
The paper based on tradtonal Chnese postman problem establshed the explct nteger programmng model of the problem, and t has been extended to the generalzed Chnese postman problem based on the weghted drected graph and random Chnese postman problem wth uncertan weghts. Other possble promoton s the Chnese postman problem consdered the weght of the fuzzy type or grey type or coarse type or belef functon type, what we need to do s makng a correspondng determnstc converson of the obectve functon based on the uncertanty connotaton. Ths s the charm of the mathematcal model establshed by the paper, nterested readers can have a try. Ths paper dscusses the problem of sngle obectve, consderng the mult-obectve travelng salesman problem, the flexble and moble and easy to modfy and deformaton characterstcs of mathematcal programmng can be used to solve the problem of possble expanson easly. REFERENCES Operatons Research Teachng Materals Edtoral. Operatonal Research (revsed edton) (2008). Beng: Tsnghua Unversty Press. 2 Feng Tan, Ma Zhongfan (987). Graphs and Network Flow Theory. Beng: Scence Press. 3 Chen Wenlan, Da Shugu. Travelng Survey on salesman problem algorthm study. Journal of Chuzhou College, 2006, 03 Wang Maozh Guo Ke. The performance of the ant algorthm to solve TSP problem analyss and mprovement. Journal of Chengdu Unversty of Technology: Natural Scence Edton, 2009, Ren Xaokang, Generaton Based on search algorthm for travel salesman problem. Journal of Jamus Unversty: Natural Scence Edton, 200, 3 6 Zhong Yanhua, Yu Xaomn. Travel salesman problem of quantum algorthm. Computer engneerng and desgn, 200, 6 7 L Luhua. DNA algorthm of travelng salesman problem. Master degree theses of master of Xnang unversty, 2006 8 Gao Jngwe Zhang X L Feng, Zhao Hu. Genetc algorthm for solvng TSP problems. The computer age. 200, 02 9 LI Daun, Zhang Janwen, Guan Yunlan, Zhao Baogu. An Insert crossover operator for the travelng salesman problem. Computer engneerng and applcaton. 2003, 33 0 zhang Chunxa, Wang Ru. Algorthm desgn based on genetc algorthm to solve TSP problem. Journal of Anyang nsttute of technology, 2007, 0 L Sucheng Lu Guang. An mproved heurstc algorthm TSP problem. Journal of management engneerng. 200, 02 2 Huang Housheng. New ways to solve the travelng salesman problem research. Master degree theses of master of Tann unversty. 200 3 Wu Wencheng, Xao Jan. Chna travelng salesman problem based on ant colony algorthm satsfed soluton. Computer and modern. 2002, 08, Nng Abng, Ma Lang. The mnmum rato of travelng salesman problem (MRTSP) compettve decson algorthm. Computer engneerng and applcaton. 200, Nng Abng, Ma lang. Compettve decson algorthm based on the quck lower bound estmaton for e bottleneck travelng salesman problem Journal of Shangha unversty of scence and technology, 200, 03 6 Lu Ha Hao Zhfeng, Ln zhyong. Improved genetc crossover operator to solve TSP problem. Journal of south Chna unversty of technology (natural scence edton). 2002 7 Lng Guoxan. Improved genetc crossover operator to solve the TSP. Journal of Guangx nsttute of technology. 2003, 0 8 Malek M. Smulated Annealng and Tabu Search Algorthms for the Travellng Salesman Problem. Annals of Operatons Research, 2, 989, 9~ 8 9 Greetha, S. and Nar K,P.K. On Stochastc Spannng Tree Problem, Networks, 993 Vol.23, No.8: 67-679 20 Feng Junwen, Explct nteger programmng model for weghted mnmum spannng tree problem, system engneerng and electroncs, 998, 2 Yh-Long Chang and Robert S. Sullven. Quanttatve Systems for Busness Plus, Verson 2.0, New York: Prentce-Hall, Inc., 99 22 Bodng Lu. Theory and Practce of Uncertan Programmng (Second Edton), Beng: UTLAB, 2007 23 Bodng Lu. Uncertan Theory (Thrd Edton), Beng: UTLAB, 2007 2 Wang Lng. Intellgent optmzaton algorthm and ts applcatons. Beng: Tsnghua unversty press, 200.