The Network Solver

Consider a cross-country trip where you want to travel the fewest miles to visit all of the capital cities in all US states except Alaska and Hawaii. Finding the optimal route is an instance of the traveling salesman problem, which is described in section Traveling Salesman Problem.

The following PROC SQL statements use the built-in data set maps.uscity to generate a list of the capital cities and their latitude and longitude:

/* Get a list of the state capital cities (with lat and long) */
proc sql;
   create table Cities as
   select unique statecode as state, city, lat, long
      from maps.uscity
      where capital='Y' and statecode not in ('AK' 'PR' 'HI');
quit;

From this list, you can generate a links data set CitiesDist that contains the distances, in miles, between each pair of cities. The distances are calculated by using the SAS function GEODIST.

/* Create a list of all the possible pairs of cities */
proc sql;
   create table CitiesDist as
   select
      a.city as city1, a.lat as lat1, a.long as long1,
      b.city as city2, b.lat as lat2, b.long as long2,
      geodist(lat1, long1, lat2, long2, 'DM') as distance
      from Cities as a, Cities as b
      where a.city < b.city;
quit;

The following PROC OPTMODEL statements find the optimal tour through each of the capital cities:

/* Find optimal tour by using the network solver */
proc optmodel;
   set<str,str> CAPPAIRS;
   set<str> CAPITALS = union {<i,j> in CAPPAIRS} {i,j};
   num distance{i in CAPITALS, j in CAPITALS: i < j};
   read data CitiesDist into CAPPAIRS=[city1 city2] distance;
   set<str,str> TOUR;
   num order{CAPITALS};

   solve with NETWORK /
      loglevel  = moderate
      links     = (weight=distance)
      tsp
      out       = (order=order tour=TOUR)
   ;

   put (sum{<i,j> in TOUR} distance[i,j]);
   /* Create tour-ordered pairs (rather than input-ordered pairs) */
   str CAPbyOrder{1..card(CAPITALS)};
   for {i in CAPITALS} CAPbyOrder[order[i]] = i;
   set TSPEDGES init
      setof{i in 2..card(CAPITALS)} <CAPbyOrder[i-1],CAPbyOrder[i]>
      union {<CAPbyOrder[card(CAPITALS)],CAPbyOrder[1]>};
   num distance2{<i,j> in TSPEDGES} =
      if i < j then distance[i,j] else distance[j,i];
   create data TSPTourNodes from [node] tsp_order=order;
   create data TSPTourLinks from [city1 city2]=TSPEDGES distance=distance2;
quit;

The progress of the procedure is shown in Output 9.7.1. The total mileage needed to optimally traverse the capital cities is miles.

The following PROC GPROJECT and PROC GMAP statements produce a graphical display of the solution:

/* Merge latitude and longitude */
proc sql;
   /* merge in the lat & long for city1 */
   create table TSPTourLinksAnno1 as
   select unique TSPTourLinks.*, cities.lat as lat1, cities.long as long1
      from TSPTourLinks left join cities
      on TSPTourLinks.city1=cities.city;
   /* merge in the lat & long for city2 */
   create table TSPTourLinksAnno2 as
   select unique TSPTourLinksAnno1.*, cities.lat as lat2, cities.long as long2
      from TSPTourLinksAnno1 left join cities
      on TSPTourLinksAnno1.city2=cities.city;
quit;

/* Create the annotated data set to draw the path on the map
    (convert lat & long degrees to radians, since the map is in radians) */
data anno_path;
   set TSPTourLinksAnno2;
   length function color $8;
   xsys='2'; ysys='2'; hsys='3'; when='a'; anno_flag=1;
   function='move';
   x=atan(1)/45 * long1;
   y=atan(1)/45 * lat1;
   output;
   function='draw';
   color="blue"; size=0.8;
   x=atan(1)/45 * long2;
   y=atan(1)/45 * lat2;
   output;
run;

/* Get a map with only the contiguous 48 states */
data states;
   set maps.states (where=(fipstate(state) not in ('HI' 'AK' 'PR')));
run;

data combined;
   set states anno_path;
run;

/* Project the map and annotate the data */
proc gproject data=combined out=combined dupok;
   id state;
run;

data states anno_path;
   set combined;
   if anno_flag=1 then output anno_path;
   else                output states;
run;

/* Get a list of the endpoints locations */
proc sql;
   create table anno_dots as
   select unique x, y from anno_path;
quit;

/* Create the final annotate data set */
data anno_dots;
   set anno_dots;
   length function color $8;
   xsys='2'; ysys='2'; when='a'; hsys='3';
   function='pie';
   rotate=360; size=0.8; style='psolid'; color="red";
   output;
   style='pempty'; color="black";
   output;
run;

/* Generate the map with GMAP */
pattern1 v=s c=cxccffcc repeat=100;
proc gmap data=states map=states anno=anno_path all;
   id state;
   choro state / levels=1 nolegend coutline=black
                 anno=anno_dots des='' name="tsp";
run;

The minimal cost tour through the capital cities is shown on the US map in Output 9.7.2.

Output 9.7.2: Optimal Traveling Salesman Tour through US Capital Cities

The data set TSPTourLinks contains the links in the optimal tour. To display the links in the order they are to be visited, you can use the following DATA step:

/* Create the directed optimal tour */
data TSPTourLinksDirected(drop=next);
   set TSPTourLinks;
   retain next;
   if _N_ ne 1 and city1 ne next then do;
      city2 = city1;
      city1 = next;
   end;
   next = city2;
run;

The data set TSPTourLinksDirected is shown in Figure 9.76.