e E c ex e subtour elimination constraints

Transcription

1 1 A Symmetric TSP For the symmetric traveling salesman problem we assume c ij = c ji. In graph theoretic terminology we now deal with an undirected graph G = (V, E) with V a set of nodes and E a set of edges. The problem stated in these terms becomes: stsp minimize subject to e E c ex e x(δ(v)) = 2 v V x(e) {0, 1} subtour elimination constraints where δ(v) is the set of edges incident to v. In GAMS we can solve the relaxation by ignoring the subtour elimination constraints as follows: set lt(i,j) lower triangular; lt(i,j)$(ord(i)>ord(j)) = yes; free variable z; binary variable x(i,j equations twomatch(i), obj; obj.. z =e= sum(lt(i,j), d(i,j)x(i,j) twomatch(i).. sum(lt(j,i), x(j,i)) + sum(lt(i,j),x(i,j)) =e= 2; model match /obj,twomatch/; solve match minimizing z using mip; display x.l; To find subtours we can use the following algorithm: 1

2 t := 1 tour(i) := i S := { (i, j) x ij = 1 } while S do choose (i, j) such that (i, j) S T := {(i, j)} S := S {(i, j)} K := { k ({i, j} T ) ({i, k} S {k, j} S) } R := k K δ(k) while R do T := T + R S := S R K := { k ({i, j} T ) ({i, k} S {k, j} S) } R := k K δ(k) od tour(t) := T t := t + 1 od In GAMS this could be implemented as: find and display tours set s(i,j) current solution; set tour(t,i,j) subtours; set tt(t) current subtour; set first(i,j) first (i,j) in S ; set reach(i,j) "(i,j) s connected to tour(tt)"; scalar done /0/; scalar continue; alias(i,k tt(t)$(ord(t)=1) = 1; s(i,j)=yes$(x.l(i,j) > 0.5 // initialize tt(1) = yes // initialize to current solution while(not done, pick any (i,j) from remaining solutions first(i,j)=no; loop((i,j)$(card(first)=0), first(i,j) = s(i,j this is he beginning of a new subtour 2

3 tour(tt,first) = yes; s(first) = no; continue = 1; while(continue, find (i,j) s connected to tour(tt) note that the loop over tt is just syntax: tt contains one element loop((tt,k), reach(i,j)$(tour(tt,k,j) and s(i,j)) = yes; reach(i,j)$(tour(tt,i,k) and s(i,j)) = yes; reach(i,j)$(tour(tt,k,i) and s(i,j)) = yes; reach(i,j)$(tour(tt,j,k) and s(i,j)) = yes; if (card(reach)=0, continue = 0 add them to the current subtour note that the loop over tt is just syntax: tt contains one element loop(tt, tour(tt,reach) = yes; s(reach) = no; reach(i,j) = no; if no remaining solutions, we are done if (card(s)=0, done=1 new subtour tt(t) = tt(t-1 display tour; Consider the data set from [1] which is electronically available from TSPLIB (see [4]). It consists of 42 cities in the U.S. The model solves in about 10 cycles (it depends a bit on which solver is used and what options as there are multiple solutions the solver can choose from during the process). In each cycle a handful of subtours are detected and thus increase the size of the model: in each cycle they cuts are added as constraints. We have reproduced some pictures of intermediate results: the relaxed solution with 8 subtours, the solution after cycle 1 which has 3 subtours, and finally the optimal solution. The pictures were produced using XY-pic (see [2],[5], [6]). The instructions were generated using a GAMS restart file plot42.gms which 3

4 obj equations cuts added relaxed cycle cycle cycle cycle cycle cycle cycle cycle cycle cycle Table 1: atsp42.gms results produces the L A TEX file plot42.tex. Figure 1: Relaxed solution Model stsp42.gms $eolcom // // // This model solves a Symmetric TSP using a simple algorithm // that adds cuts to exclude subtours found in the previous // solution. The data set is dantzig42 from TSPLIB. // // Reference: Dantzig, Fulkerson, Johnson, "Solution of a Large-Scale // Traveling-Salesman Problem", Operations Research, // 2, 1954, pp // 4

5 Figure 2: Cycle 1 solution // optimal solution: 699 // option optcr=0; option iterlim=100000; option reslim=10000; option limrow=0; option limcol=0; option solprint=off; set i cities / c1 Manchester, N.H. c2 Montpelier, Vt. c3 Detroit, Mich. c4 Cleveland, Ohio c5 Charleston, W.Va. c6 Louisville, Ky. c7 Indianapolis, Ind. c8 Chicago, Ill. c9 Milwaukee, Wis. c10 Minneapolis, Minn. c11 Pierre, S.D. c12 Bismarck, N.D. c13 Helena, Mont. c14 Seattle, Wash. c15 Portland, Ore. c16 Boise, Idaho c17 Salt Lake City, Utah c18 Carson City, Nevada c19 Los Angeles, Calif. c20 Phoenix, Ariz. c21 Santa Fe, N.M. c22 Denver, Colo. c23 Cheyenne, Wyo. c24 Omaha, Neb. 5

6 Figure 3: Optimal solution /; c25 Des Moines, Iowa c26 Kansas City, Mo. c27 Topeka, Kans. c28 Oklahoma City, Okla. c29 Dallas, Tex. c30 Little Rock, Ark. c31 Memphis, Tenn. c32 Jackson, Miss. c33 New Orleans, La. c34 Birmingham, Ala. c35 Atlanta, Ga. c36 Jacksonville, Fla. c37 Columbia, S.C. c38 Raleigh, N.C. c39 Richmond, Va. c40 Washington, D.C. c41 Boston, Mass. c42 Portland, Me. alias (i,j,k table d(i,j) c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 c16 c17 c18 c19 c20 c21 c2 8 c c c c c c c c

7 c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c22 c23 c24 c25 c26 c27 c28 c29 c30 c31 c32 c33 c34 c35 c36 c37 c38 c39 c40 c41 c23 5 c c c c c c c c c c c c c c c c c c c ; 7

8 set lt(i,j) lower triangular; lt(i,j)$(ord(i)>ord(j)) = yes; free variable z; binary variable x(i,j equations twomatch(i), obj; obj.. z =e= sum(lt(i,j), d(i,j)x(i,j) twomatch(k).. sum(lt(i,k), x(i,k)) + sum(lt(k,j),x(k,j)) =e= 2; model match /obj,twomatch/; solve match minimizing z using mip; display x.l; dynamic number of cuts set cycle /cycle1cycle50/; set t tours /t1t100/; set cutindex(cycle,t) equation cut(cycle,t) parameter cutcoeff(cycle,t,i,j) parameter cutlength(cycle,t) dynamic set for addressing cuts; actual cuts; coefficients in cuts; rhs for cuts; cut(cutindex).. sum((i,j), cutcoeff(cutindex,i,j)x(i,j)) =l= cutlength(cutindex)-1; model tsp /obj, twomatch, cut/; used to find and display tours set s(i,j) current solution; set tour(t,i,j) subtours; set tt(t) current subtour; set first(i,j) first (i,j) in S ; set reach(i,j) "(i,j) s connected to tour(tt)"; scalar done /0/; scalar continue; parameter results; // for reporting results( relaxed, obj )=z.l; results( relaxed, total constraints ) = match.numequ; parameter solutions(,i,j solutions( relaxed,i,j) = x.l(i,j 8

9 loop(cycle, initialization tour(t,i,j) = no; s(i,j) = no; first(i,j) = no; reach(i,j) = no; done = 0; tt(t) = no; tt(t)$(ord(t)=1) = 1; s(i,j)=yes$(x.l(i,j) > 0.5 // initialize tt(1) = yes // initialize to current solution while(not done, pick any (i,j) from remaining solutions first(i,j)=no; loop((i,j)$(card(first)=0), first(i,j) = s(i,j display first; this is he beginning of a new subtour tour(tt,first) = yes; s(first) = no; continue = 1; while(continue, find (i,j) s connected to tour(tt) note that the loop over tt is just syntax: tt contains one element reach(i,j) = no; loop((tt,k), reach(s(i,j))$tour(tt,k,j) = yes; reach(s(i,j))$tour(tt,i,k) = yes; reach(s(i,j))$tour(tt,k,i) = yes; reach(s(i,j))$tour(tt,j,k) = yes; display tour,s,reach; if reach = empty then we can stop the while loop if (card(reach)=0, continue = 0 add them to the current subtour 9

10 note that the loop over tt is just syntax: tt contains one element loop(tt, tour(tt,reach) = yes; s(reach) = no; display tour; if no remaining solutions, we are done if (card(s)=0, done=1 new subtour tt(t) = tt(t-1 // t := t + 1 cutindex(cycle,t)$(sum(tour(t,i,j),1)>0.5) = yes; check option tour:0:1:1; display tour; loop(cutindex(cycle,t), abort$(sum(tour(t,i,j),1) < 2.5) "subtour with 1 or 2 points"; if (sum(cutindex(cycle,t),1) > 1.5, cutcoeff(cycle,tour(t,i,j))$cutindex(cycle,t) = 1; cutlength(cutindex(cycle,t)) = sum(tour(t,i,j),1 option cutindex:0:1:1; option cutcoeff:0:1:1; display cutindex,cutcoeff,cutlength; solve tsp minimizing z using mip; display x.l; results(cycle, obj ) = z.l; results(cycle, cuts added ) = sum(cutindex(cycle,t), 1 results(cycle, total constraints ) = tsp.numequ; display results; else solutions(cycle,i,j) = x.l(i,j display "Optimal solution found!"; done = 1; 10

11 solutions( optimal,i,j) = x.l(i,j display results; Model plot42.gms table xy(i,) coordinates x y c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c

12 c ; display xy; file f /plot42.tex/; put f; put \documentclass{article} /; put \usepackage{amsmath} /; put \usepackage[all]{xy} /; put \begin{document} /; put \begin{figure}[h] /; put \[\begin{xy} 0;<0.5mm,0mm>: / loop(i, if(ord(i)>1, put, put (, xy(i, x ),, xy(i, y ), ){\bullet} / loop((i,j)$(solutions( relaxed,i,j)>0.5), if(ord(i)>1, put, put (, xy(i, x ),, xy(i, y ), (, xy(j, x ),, xy(j, y ), )@{-} / put \end{xy}\] / put \caption{relaxed solution} /; put \end{figure} /; put \begin{figure}[h] /; put \[\begin{xy} 0;<0.5mm,0mm>: / loop(i, if(ord(i)>1, put, put (, xy(i, x ),, xy(i, y ), ){\bullet} / loop((i,j)$(solutions( cycle1,i,j)>0.5), if(ord(i)>1, put, put (, xy(i, x ),, xy(i, y ), (, xy(j, x ),, xy(j, y ), )@{-} / put \end{xy}\] / put \caption{cycle 1 solution} /; put \end{figure} /; put \begin{figure}[h] /; put \[\begin{xy} 0;<0.5mm,0mm>: / loop(i, if(ord(i)>1, put, put (, xy(i, x ),, xy(i, y ), ){\bullet} / 12

13 loop((i,j)$(solutions( optimal,i,j)>0.5), if(ord(i)>1, put, put (, xy(i, x ),, xy(i, y ), (, xy(j, x ),, xy(j, y ), )@{-} / put \end{xy}\] / put \caption{optimal solution} /; put \end{figure} /; put \end{document} /; putclose; Model plot42.tex \documentclass{article} \usepackage{amsmath} \usepackage[all]{xy} \begin{document} \begin{figure}[h] \[\begin{xy} 0;<0.5mm,0mm>: ( , 85.00){\bullet},( , 88.00){\bullet},( , 73.00){\bullet},( , 70.00){\bullet},( , 55.00){\bullet},( , 53.00){\bullet},( , 60.00){\bullet},( , 68.00){\bullet},( , 74.00){\bullet},( 99.00, 83.00){\bullet},( 73.00, 79.00){\bullet},( 72.00, 91.00){\bullet},( 37.00, 94.00){\bullet},( 6.00, ){\bullet},( 3.00, 97.00){\bullet},( 21.00, 82.00){\bullet},( 33.00, 67.00){\bullet},( 4.00, 66.00){\bullet},( 3.00, 42.00){\bullet},( 27.00, 33.00){\bullet},( 52.00, 41.00){\bullet},( 57.00, 59.00){\bullet},( 58.00, 66.00){\bullet},( 88.00, 65.00){\bullet},( 99.00, 67.00){\bullet},( 95.00, 55.00){\bullet},( 89.00, 55.00){\bullet},( 83.00, 38.00){\bullet} 13

14 ,( 85.00, 25.00){\bullet},( , 35.00){\bullet},( , 37.00){\bullet},( , 24.00){\bullet},( , 13.00){\bullet},( , 30.00){\bullet},( , 32.00){\bullet},( , 18.00){\bullet},( , 36.00){\bullet},( , 45.00){\bullet},( , 54.00){\bullet},( , 61.00){\bullet},( , 82.00){\bullet},( , 87.00){\bullet},( , 88.00( , , 70.00( , , 55.00( , , 53.00( , , 60.00( , , 68.00( , , 74.00( , , 74.00( , 73.00, 79.00( 99.00, 72.00, 91.00( 99.00, 72.00, 91.00( 73.00, 6.00, ( 37.00, 3.00, 97.00( 6.00, 21.00, 82.00( 3.00, 33.00, 67.00( 37.00, 4.00, 66.00( 21.00, 3.00, 42.00( 4.00, 27.00, 33.00( 3.00, 52.00, 41.00( 27.00, 57.00, 59.00( 52.00, 58.00, 66.00( 33.00, 58.00, 66.00( 57.00, 99.00, 67.00( 88.00, 95.00, 55.00( 99.00, 89.00, 55.00( 88.00, 89.00, 55.00( 95.00, 85.00, 25.00( 83.00, , 35.00( 83.00, , 37.00( , , 13.00( 85.00, , 13.00( , , 30.00( , , 30.00( , , 18.00( , , 36.00( , , 36.00( , , 54.00( , , 61.00( , , 61.00( , , 82.00( , , 87.00( , , 87.00( , \end{xy}\] 14

15 \caption{relaxed solution} \end{figure} \begin{figure}[h] \[\begin{xy} 0;<0.5mm,0mm>: ( , 85.00){\bullet},( , 88.00){\bullet},( , 73.00){\bullet},( , 70.00){\bullet},( , 55.00){\bullet},( , 53.00){\bullet},( , 60.00){\bullet},( , 68.00){\bullet},( , 74.00){\bullet},( 99.00, 83.00){\bullet},( 73.00, 79.00){\bullet},( 72.00, 91.00){\bullet},( 37.00, 94.00){\bullet},( 6.00, ){\bullet},( 3.00, 97.00){\bullet},( 21.00, 82.00){\bullet},( 33.00, 67.00){\bullet},( 4.00, 66.00){\bullet},( 3.00, 42.00){\bullet},( 27.00, 33.00){\bullet},( 52.00, 41.00){\bullet},( 57.00, 59.00){\bullet},( 58.00, 66.00){\bullet},( 88.00, 65.00){\bullet},( 99.00, 67.00){\bullet},( 95.00, 55.00){\bullet},( 89.00, 55.00){\bullet},( 83.00, 38.00){\bullet},( 85.00, 25.00){\bullet},( , 35.00){\bullet},( , 37.00){\bullet},( , 24.00){\bullet},( , 13.00){\bullet},( , 30.00){\bullet},( , 32.00){\bullet},( , 18.00){\bullet},( , 36.00){\bullet},( , 45.00){\bullet},( , 54.00){\bullet},( , 61.00){\bullet},( , 82.00){\bullet},( , 87.00){\bullet},( , 88.00( , 85.00)@{-},( , 70.00( , 73.00)@{-},( , 55.00( , 70.00)@{-},( , 53.00( , 73.00)@{-},( , 60.00( , 53.00)@{-},( , 68.00( , 60.00)@{-},( , 74.00( , 68.00)@{-},( 99.00, 83.00( , 74.00)@{-},( 72.00, 91.00( 73.00, 79.00)@{-},( 37.00, 94.00( 72.00, 91.00)@{-},( 6.00, ( 37.00, 94.00)@{-} 15

16 ,( 3.00, 97.00( 6.00, 21.00, 82.00( 3.00, 33.00, 67.00( 21.00, 4.00, 66.00( 33.00, 3.00, 42.00( 4.00, 27.00, 33.00( 3.00, 52.00, 41.00( 27.00, 57.00, 59.00( 52.00, 58.00, 66.00( 73.00, 58.00, 66.00( 57.00, 99.00, 67.00( 99.00, 99.00, 67.00( 88.00, 89.00, 55.00( 88.00, 89.00, 55.00( 95.00, 83.00, 38.00( 95.00, 85.00, 25.00( 83.00, , 35.00( 85.00, , 37.00( , , 24.00( , , 13.00( , , 30.00( , , 32.00( , , 18.00( , , 36.00( , , 45.00( , , 54.00( , , 61.00( , , 61.00( , , 82.00( , , 87.00( , , 87.00( , \end{xy}\] \caption{cycle 1 solution} \end{figure} \begin{figure}[h] \[\begin{xy} 0;<0.5mm,0mm>: ( , 85.00){\bullet},( , 88.00){\bullet},( , 73.00){\bullet},( , 70.00){\bullet},( , 55.00){\bullet},( , 53.00){\bullet},( , 60.00){\bullet},( , 68.00){\bullet},( , 74.00){\bullet},( 99.00, 83.00){\bullet},( 73.00, 79.00){\bullet},( 72.00, 91.00){\bullet},( 37.00, 94.00){\bullet},( 6.00, ){\bullet},( 3.00, 97.00){\bullet},( 21.00, 82.00){\bullet},( 33.00, 67.00){\bullet},( 4.00, 66.00){\bullet},( 3.00, 42.00){\bullet},( 27.00, 33.00){\bullet},( 52.00, 41.00){\bullet} 16

17 ,( 57.00, 59.00){\bullet},( 58.00, 66.00){\bullet},( 88.00, 65.00){\bullet},( 99.00, 67.00){\bullet},( 95.00, 55.00){\bullet},( 89.00, 55.00){\bullet},( 83.00, 38.00){\bullet},( 85.00, 25.00){\bullet},( , 35.00){\bullet},( , 37.00){\bullet},( , 24.00){\bullet},( , 13.00){\bullet},( , 30.00){\bullet},( , 32.00){\bullet},( , 18.00){\bullet},( , 36.00){\bullet},( , 45.00){\bullet},( , 54.00){\bullet},( , 61.00){\bullet},( , 82.00){\bullet},( , 87.00){\bullet},( , 88.00( , , 73.00( , , 70.00( , , 55.00( , , 53.00( , , 60.00( , , 68.00( , , 74.00( , 99.00, 83.00( , 73.00, 79.00( 99.00, 72.00, 91.00( 73.00, 37.00, 94.00( 72.00, 6.00, ( 37.00, 3.00, 97.00( 6.00, 21.00, 82.00( 3.00, 33.00, 67.00( 21.00, 4.00, 66.00( 33.00, 3.00, 42.00( 4.00, 27.00, 33.00( 3.00, 52.00, 41.00( 27.00, 57.00, 59.00( 52.00, 58.00, 66.00( 57.00, 88.00, 65.00( 58.00, 99.00, 67.00( 88.00, 95.00, 55.00( 99.00, 89.00, 55.00( 95.00, 83.00, 38.00( 89.00, 85.00, 25.00( 83.00, , 35.00( 85.00, , 37.00( , , 24.00( , , 13.00( , , 30.00( , , 32.00( , , 18.00( , , 36.00( , 17

18 ,( , 45.00( , , 54.00( , , 61.00( , , 82.00( , , 87.00( , , 87.00( , \end{xy}\] \caption{optimal solution} \end{figure} \end{document} References [1] Dantzig, G., Fulkerson, R., Johnson, S., Solution of a large-scale Traveling- Salesman Problem, Operations Research, 2, 1954, pp [2] Goossens, M., Rahtz, S., Mittelbach, F., The L A TEX Graphics Companion, Addison-Wesley, [3] Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.), The Traveling Salesman Problem. A Guided Tour of Combinatorial Optimization, Wiley, [4] Reinelt, G., TSPLIB A Traveling Salesman Problem Library, ORSA Journal on Computing, 3 (4), pp [5] Rose, K.H., Moore, R., XY-pic Reference Manual, [6] Rose, K.H., XY-pic User s Guide,