We described a method based on integer linear programming for solving the many-visitis m-traveling salesman problem.Descreve-se um método baseado em programação linear inteira para a solução do problema do m-caixeiro viajante com múltiplas visitas

Szwarcfiter, Jayme Luiz

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--, :\;c" --,---,"'.,-.,r-~.;"7c;c ,:  "(,íi' - ,. ;' :.;0  -So'"."" = ".== - ):'-=7';,!," - AN ALGORITHM FOR THE M-A JY-VISITSm-TRAVELING SALESMAN PROBLEM -.. ; --- \"I,November 1985 Núcleo de Computação EletrônicaCaixa Postal 232420001 -Rio de Janeiro -RJ Brasil1R  5 Gó2. -jg -o  -i,  C f. ! U F. R. J.818LI"YreCAr":.;:-7:.;; :::,. Z- ij- '"" ---, :;; ;;::- ,--~'.AN ALGORITHM FOR THE MANY-VISITSrn-TRAVELING SALESMAN PROBLEM...-. .. \We described a method based on integer linear programmingfor solving the many-visits m-traveling salesman problem.RESUMO Descreve-se um método baseado em programaçã  linear inteira p ra a solução do problema do m-cai.xeiro viajante com multiplasv i s i t a s  -.,-"c :\!1,.c-,J,.: ., .'"'-- -,- -..,:Jayme Luiz SzwarcfiterUniversidade Federal do Rio.de JaneiroNúcleo de Computação Eletrônica.c .c., .,I -. ,- fi  ":11. INTRODUCTIONWe describe an integer line   pro raw in  (ILP) approachto the many-visits trave.lin  salesman problem of (2, 5]. The meth."od is based on [3] and is an application of [4]. As usual with ILPTheforrnulations of the TSP a central point is handlin  subtours.,,-'constraints described in this paper do not eliminate them necessar?-lr and instead allow the existence of certain subtours, which,are sUbsequently transforrned into proper tours of' the -sarne costa, 'The m-TSP arises naturally when considerina many-visitsthere is a distinguished city which is toI' .,type'problems. That is,it isbe visited m times, once by each salesman. In addition,useful, for instance, when solving the aircraft schedulinq problemThis problem isto find a landing schedúle(ASP) with m runway .of a set of n airplanes 50 as to minimize the time of the latestlandirig. Each plane is o.:f a certain class and the time gap redependsquired between two consecutive landin s on a same runwayThe number k of distinct classesonly on the two classes involved.the I!l > I case has beenis supposed to be fixed [2,5]. In [5],discussed, but only solutions for m=2 have been qiven. The ASPwith m runways can be solved by an a19orithm-for the many-visitsm-TSP.For convenience, we adopt the interpretation consideringthe many-visits m-TSP as a special case of the ordinary m-TSPThat is, we forrnulate an algorithrn for solving a general asyrnme!ric.m-TSP which starts by dividing the n vertices into k disjointIn theclasses C , according to a certain eq ivalence relation.vismany-visits m-TSP, each Ci would corres ond to a city to betimes. If k is fixed and the rnaxirnuw. distance of theited  .5.Lenuna I: Algorithm I constructsa subtour T* for X...Proof: First, we show by induction that in each iteration 1 there is a pair of vertices satisfying (7)-(9) . This is.c1ear for 1=1. Suppose 1<1 n+rn-1. Frorn (5) I it fol1ows we can: .c  .  e-v e: Ci' w e: Cj such that out (v) =in (w) =o, for every R. ;:,It re   \...  '4.. ..r- ,.--mains to prove that there is a choice with vFw. If i:/j this stri.y --   -=- -.' -   ".,ial. Suppose i=j and that (7) holds only for  . Then there:.: ". exists u E: C., u'lv with t(u)=2, otherwise I c.1. -.1..dicting (6) .In .the i teration that chose u 1, ' :'4=1, Xij=l, ;contraast, t.(u) =1 and' t(v)::;Q,which contradicts (8) ar (9), and the inductia? hypathesis. In ad7)-and not (8) t ereis v'dition, if v satisfies v1tw,E: c. ,1.with t(v')=O. ln this case, replace v by VI .Similarly for (9) andwe conclude that suitable v,w exist for l R. n+m-l. Each iterationdecreases .some x. .> O by 1. From (5), r x. .=n+m-l. Usin g (4), wel.J l.Jconclude that the process terminates after n+m-l iterations of thegeneral. step. Each one adds the edge (v,w) to E(T*) . At termination, in(u)=out(u)=l, for a 11 u. The lemrna follows.We now proceed to transform. T* into a pr per tour ToWith thispurpose we describe twooperations, SPLIT and JOIN, onthe cycles of D+. Let C(V), be the class of D-containing vertex v.whereC(rl= C(SI) . Then SPLIT removes from Z the edges (rp'sl) andThe effect of(Z,rl'sl(Sq'rl ' while adding to it (rp'rl) and (Sq'sl) .this operation is therefore ofsplit Z into the cycles Z'=rl'...'rp'rland Z"=51'...'Sq'Sl.Conversely, let Z. and Z " be disjoint cycles r I' ...,rp'r 1 and+Sl'...'Sq'Sl of D , respectively such that C(rl)=C(sl).p,q>l and the non consecutive vertices rl'sl are such that..,.-,.  ;;- ;c,,-."""C-(Z,v,w}in the cycle Z. In addition, C(v}=C(w}=Ck. That is, SPLIT..andis correct. Now, suppose there is a cycle z' having no rootsuch that there is no vertex outside Z' belonging to the same cl sas any of Z'. Let Cq ,...,Cq be the classes of D represented in1 : .e.Z'. Then S = {ql'...'q!D} c {l,...,k-l} is such that r x. .=-l. .. S l.J., l. , J E:.r ICil, contradicti g (6) .Then no such Z' can exist and JOINl. S-- -  - , c ,l\ ,-'c--         ,-  -. . ,.r -. -.-is correct. Hence the,algorithm constructs a digraph T which is ob.- -   ,=- -  -- ::;;'--"-tained from T* by a se uence of m-IT*I+p SPLIT followed by p JOIN,1 ; ,operations, where-p isc.the number of cycles of T* havi  no root.That is, Tis formedby vertex disjoint cycles, ea.ch of them con , ;: +taining exactly one root.and spqnning a 11 vertices of D .,;In addition, by lenuna 2 cost(T)=cost(T*) o There,fore, T is a touroAlgorithms 1 and 2 can be implernented in O(n) andtime, .respectively. The time required by the JOIN operations ofbethe latter dominates its complexity. However, the rnethod canimproved by adopting a systematic way of selecting the pair ofcycles to be merged by JOIN, as below. We assume  hat all requiredSPLIT operations have already been performed.In the initial  , construct a bipartite graph G havingas vertices the cy cles Z. of T* and the classes c. of D , denoted..J .l.G(Zj) and G(Ci)' respectively. G has an edCJe (G(Zj) , G(Ci))V(z J.)nCi   and there is a label v E V(Z.)nC. attached to it. G has.J l.no otheredges. Start by examining each G(Zj) , mark it if G(Zj) isand unmark it otherwise. Next remove all verticeswhen"adjacent to  (6) , ofthe degreeterrninate the alg undisrithm if all vertices G(Z.) are marked. Otherwise choose someJ .marked G(Zp) and let G(Ci) and G(Zq) be two vertices of G attances one and two from G(Zp) , respectively. Denote by Vp andVq.  ---~.10.-....[1] J.Blazewiocz and K.Eqker, A Linear time all?orithm for restric    ted bin packing and scheduling problerns.  . Res. Letters  nc: :!:: -:7:':.): (1983) 80-83i;: :  : :;,...:.::   c . I) c"[2l1 ,:.:S.S.Cosrnadakis and C.H.Papadirnitriou, The traveling salesrnan..i;.::..: ..i.:,;;;problernwith rnany.visits to few cities.      Q(1984),c'. .:; 99 - 108 ..,.'; ' :. I: ,i" ;   ::.t:"[3j' G.Dantzig, D.R.Fu1kerson ànd S.Johnson, Solution of a largef:: sca1e trave1ing sa1esman prob1em.  .   !:. {1954) 393-410;[4] H.W.Lenstra, Jr., Integer progranuning with a fixed number of[5] H.N.Psaraftis, A DynaIl1,ic programming ap!>roach for sequencinggroups of identical jobs. .  .  .  .  .   (1980)1347-1359.

An algorithm for the many-visits m-traveling salesman problem

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  Previous issue date: 1985-11-30Descreve-se um método baseado em programação linear inteira para a solução do problema do m-caixeiro viajante com múltiplas visitas.We described a method based on integer linear programming for solving the many-visitis m-traveling salesman problem

An algorithm for the many-visits m-traveling salesman problem

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