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25.3.2 Traveling Salesman Problem

The TSP (Traveling Salesman Problem) is the classic combinatorial optimization problem. I have provided a very simple version of it, based on the coordinates of twelve cities in the southwestern United States. This should maybe be called the Flying Salesman Problem, since I am using the great-circle distance between cities, rather than the driving distance. Also: I assume the earth is a sphere, so I don’t use geoid distances.

The gsl_siman_solve routine finds a route which is 3490.62 Kilometers long; this is confirmed by an exhaustive search of all possible routes with the same initial city.

The full code can be found in siman/siman_tsp.c, but I include here some plots generated in the following way:

$ ./siman_tsp > tsp.output
$ grep -v "^#" tsp.output  
 | awk '{print $1, $NF}'
 | graph -y 3300 6500 -W0 -X generation -Y distance 
    -L "TSP - 12 southwest cities"
 | plot -Tps > 12-cities.eps
$ grep initial_city_coord tsp.output 
  | awk '{print $2, $3}' 
  | graph -X "longitude (- means west)" -Y "latitude" 
     -L "TSP - initial-order" -f 0.03 -S 1 0.1 
  | plot -Tps > initial-route.eps
$ grep final_city_coord tsp.output 
  | awk '{print $2, $3}' 
  | graph -X "longitude (- means west)" -Y "latitude" 
     -L "TSP - final-order" -f 0.03 -S 1 0.1 
  | plot -Tps > final-route.eps

This is the output showing the initial order of the cities; longitude is negative, since it is west and I want the plot to look like a map.

# initial coordinates of cities (longitude and latitude)
###initial_city_coord: -105.95 35.68 Santa Fe
###initial_city_coord: -112.07 33.54 Phoenix
###initial_city_coord: -106.62 35.12 Albuquerque
###initial_city_coord: -103.2 34.41 Clovis
###initial_city_coord: -107.87 37.29 Durango
###initial_city_coord: -96.77 32.79 Dallas
###initial_city_coord: -105.92 35.77 Tesuque
###initial_city_coord: -107.84 35.15 Grants
###initial_city_coord: -106.28 35.89 Los Alamos
###initial_city_coord: -106.76 32.34 Las Cruces
###initial_city_coord: -108.58 37.35 Cortez
###initial_city_coord: -108.74 35.52 Gallup
###initial_city_coord: -105.95 35.68 Santa Fe

The optimal route turns out to be:

# final coordinates of cities (longitude and latitude)
###final_city_coord: -105.95 35.68 Santa Fe
###final_city_coord: -103.2 34.41 Clovis
###final_city_coord: -96.77 32.79 Dallas
###final_city_coord: -106.76 32.34 Las Cruces
###final_city_coord: -112.07 33.54 Phoenix
###final_city_coord: -108.74 35.52 Gallup
###final_city_coord: -108.58 37.35 Cortez
###final_city_coord: -107.87 37.29 Durango
###final_city_coord: -107.84 35.15 Grants
###final_city_coord: -106.62 35.12 Albuquerque
###final_city_coord: -106.28 35.89 Los Alamos
###final_city_coord: -105.92 35.77 Tesuque
###final_city_coord: -105.95 35.68 Santa Fe

Here’s a plot of the cost function (energy) versus generation (point in the calculation at which a new temperature is set) for this problem:


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