tsp4.gms : Traveling Salesman Problem - Four
This is the third problem in a series of traveling salesman
problems. Here we revisit TSP1 and generate smarter cuts.
The first relaxation is the same as in TSP1.
Reference:
- Kalvelagen, E, Model Building with GAMS. forthcoming
Large Model of Type: MIP Includes: br17.inc
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$title Traveling Salesman Problem - Four (TSP4,SEQ=180)
$eolcom //
$Ontext
This is the third problem in a series of traveling salesman
problems. Here we revisit TSP1 and generate smarter cuts.
The first relaxation is the same as in TSP1.
Kalvelagen, E, Model Building with GAMS. forthcoming
de Wetering, A V, private communication.
$Offtext
$include br17.inc
*
* For this algorithm we can try a larger subset of 12 cities.
set i(ii) / i1*i12 /;
* options. Make sure MIP solver finds global optima.
option optcr=0;
model assign /objective, rowsum, colsum/;
solve assign using mip minimizing z;
* find and display tours
*
set t tours /t1*t17/; abort$(card(t) < card(i)) "Set t is possibly too small"
sets tour(i,j,t) subtours
from(i) contains always one element: the from city
next(j) contains always one element: the to city
visited(i) flag whether a city is already visited
tt(t) contains always one element: the current subtour;
scalar goon go on flag used to control loop;
alias(i,ix);
* initialize
from(i)$(ord(i)=1) = yes; // turn first element on
tt(t)$(ord(t)=1) = yes; // turn first element on
* subtour elimination by adding cuts
*
set cc /c1*c1000/; alias(cc,ccc); // we allow up to 1000 cuts
set curcut(cc) current cut always one element
allcuts(cc) total cuts;
parameters cutcoeff(cc, i, j)
rhs(cc)
nosubtours number of subtours;
equations cut(cc) dynamic cuts;
cut(allcuts).. sum((i,j), cutcoeff(allcuts,i,j)*x(i,j)) =l= rhs(allcuts);
model tspcut /objective, rowsum, colsum, cut/;
curcut(cc)$(ord(cc)=1) = yes;
scalar ok;
goon = 1;
loop(ccc$goon,
* initialize
from(i) = no;
tt(t) = no;
from(i)$(ord(i)=1) = yes; // turn first element on
tt(t)$(ord(t)=1) = yes; // turn first element on
tour(i,j,t)=no;
visited(i)=no;
loop(i,
next(j)=no; // find city visited after city 'from'
loop((from,j),next(j)$(x.l(from,j)>0.5) = yes);
// check x.l(from,j)=1 would be dangerous
tour(from,next,tt) = yes; // store in table
visited(from) = yes; // mark city 'from' as visited
from(j) = next(j);
if (sum(visited(next),1)>0, // if already visited...
tt(t) = tt(t-1);
loop(ix$(not visited(ix)), // find starting point of new subtour
from(j) = no; // make sure we only have one element turned on
from(ix) = yes;
)
)
);
display tour;
nosubtours = sum(t, max(0,smax(tour(i,j,t),1)));
display nosubtours;
if (nosubtours=1,
goon = 0; // done: no subtours
else // else: introduce cut
loop(t$(ord(t) <= nosubtours),
rhs(curcut)=-1;
loop(tour(i,j,t),
cutcoeff(curcut, i, j)$(x.l(i,j) > 0.5) = 1;
* not needed due to nature of assignment constraints
* cutcoeff(curcut, i, j)$(x.l(i,j) < 0.5) = -1;
rhs(curcut) = rhs(curcut) + 1;
);
allcuts(curcut) = yes; // include this cut in set
curcut(cc) = curcut(cc-1);
);
solve tspcut using mip minimizing z;
tspcut.solprint=2;tspcut.limrow=0;tspcut.limcol=0;
)
);
display x.l;
abort$(goon = 1) "Too many cuts needed"
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