Gams Travelling Salesman Problem Model
Essay by YurtdisiAlisver • November 15, 2016 • Term Paper • 5,755 Words (24 Pages) • 1,957 Views
1. STRATEGY & FORMULATION:
In order not to deal with excessive number of constraints, we skipped the constraint (4) directly, which is more trustable, but includes so many constraints. Because of the fact that the constraint, which is found by Desrochers and Laporte, is a stronger version of the Miller-Tucker-Zemlin(MTZ), we used constraint (6) in order to solve Travelling Salesman Problem(TSP) for the given data. As one can see in below, we defined a new decision variable u(i), which indicates the sequence in which the city i is visited.
[pic 1]
[pic 2]
[pic 3]
[pic 4]
[pic 5]
2. TSP WITH 15 CITIES:
2.a.CODE:
*****************************************************
*This is the code for TSP with 15 cities.
*****************************************************
set i /1*15/;
alias (i,j);
table
d(i,j) distance
$include "TR_15cities.txt";
binary variables
x(i,j);
positive variables
u(i)
variable
opt;
Equations
TotalDistance
AssignmentConstraint_1(j)
AssignmentConstraint_2(i)
Initialize_U
SubSetConstraint_6(i,j) ;
TotalDistance..
opt =E= SUM(i, SUM(j, d(i,j)*x(i,j) ) );
AssignmentConstraint_1(j)..
SUM(i, x(i,j) ) =E= 1 ;
AssignmentConstraint_2(i)..
SUM(j, x(i,j) ) =E= 1 ;
Initialize_U..
u('1') =E= 1;
* We initialiazed the city 1 as the first city.
SubSetConstraint_6(i,j)$(ord(i) gt 1 and ord(j) gt 1)..
u(i) - u(j) + 14 * x(i,j) + 12 * x(j,i) =L= 13;
MODEL IE413_TSP /ALL/;
option optcr = 0.0 ;
*We used the option in order to decrease the tolerance to 0.
SOLVE IE413_TSP MINIMIZING opt USING MIP;
DISPLAY opt.l, x.l, x.m, u.l;
2.b.S O L V E S U M M A R Y:
MODEL IE413_TSP OBJECTIVE opt
TYPE MIP DIRECTION MINIMIZE
SOLVER CPLEX FROM LINE 67
**** SOLVER STATUS 1 Normal Completion
**** MODEL STATUS 1 Optimal
**** OBJECTIVE VALUE 5161.0000
RESOURCE USAGE, LIMIT 1.574 1000.000
ITERATION COUNT, LIMIT 9748 2000000000
IBM ILOG CPLEX Jul 4, 2010 23.5.1 WIN 18414.18495 VS8 x86/MS Windows
Cplex 12.2.0.0, GAMS Link 34
GAMS/Cplex licensed for continuous and discrete problems.
Cplex MIP uses 1 of 8 parallel threads. Change default with option THREADS.
MIP status(101): integer optimal solution
Fixed MIP status(1): optimal
Proven optimal solution.
MIP Solution: 5161.000000 (9748 iterations, 965 nodes)
Final Solve: 5161.000000 (0 iterations)
Best possible: 5161.000000
Absolute gap: 0.000000
Relative gap: 0.000000
We reached the best possible solution by using constraint (6) in a very short time interval.
2.c.SOLUTION:
1 2 3 4 5 6
1 1.000
3 1.000
7 1.000
8 1.000
12 1.000
14 1.000
+ 7 8 9 10 11 12
4 1.000
6 1.000
9 1.000
10 1.000
13 1.000
15 1.000
+ 13 14 15
2 1.000
5 1.000
11 1.000
2.d.MAP:
[pic 6]
3. TSP WITH 81 CITIES:
3.a.CODE:
*This is the code for TSP with 81 cities.
*****************************************************
set i /1*81/;
alias (i,j);
table
d(i,j) distance
$include "TR_81cities.txt";
binary variables
x(i,j);
positive variables
u(i);
variable
opt;
Equations
TotalDistance
AssignmentConstraint_1(j)
AssignmentConstraint_2(i)
Initialize_U
SubSetConstraint_6(i,j);
TotalDistance..
opt =E= SUM(i, SUM(j, d(i,j)*x(i,j) ) );
AssignmentConstraint_1(j)..
SUM(i, x(i,j) ) =E= 1 ;
AssignmentConstraint_2(i)..
SUM(j, x(i,j) ) =E= 1 ;
Initialize_U..
u('1') =E= 1;
SubSetConstraint_6(i,j)$(ord(i) gt 1 and ord(j) gt 1)..
u(i) - u(j) + 80 * x(i,j) + 78 * x(j,i) =L= 79;
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