Combinatorics -Traveling Salesman Problem
 To model a traveling salesman problem, choose TSP from the Combinatorics menu. Our first example has 6 cities. The distances between the cities are provided by the arc length. In the dialog below we indicate that the data is to be determined by the Euclidean measure with the 6 cities located randomly in a plane. The initial worksheet is shown below. The solution is a tour through the cities starting at city 1 and ending at city 7. City 7 has the same location as city 1 and represents the completion of the tour. The computer manipulates the green colored decision variables in row 9 that indicate the next city in the tour. Row 10 holds the tour and row 11 holds the lengths of the chosen arcs. The sum of the lengths of the tour arcs is computed in cell F5. This is the objective to be minimized. Cells H4 and H5 hold the condition that a tour must visit all cities. The objective function and constraint cells are all colored yellow to indicate that they are not to be changed. This add-in only solves the traditional minimum distance TSP. The random locations of the cities are shown to the right of the distance matrix. The cells of the matrix hold formulas that compute the distances between the locations. The *** strings indicate disallowed transitions. The initial solution visits the cities in numerical order.
The Map button on the left creates a separate worksheet with a map of the solution. The button is only available when the x and y ranges in columns K and L are present. The map of the initial solution is shown below. The first city is colored gold.

The Search button calls the search dialog from the Optimize add-in. All the features of this dialog are described on the TSP page. For this small problem, we will find the solution by exhaustive enumeration.

When called from a TSP model created by the Combinatorics add-in, the search process assumes a linear model, that is, the arc lengths are computed at the beginning of the process and used for all subsequent computations. Individual solutions are not evaluated on the worksheet so the model is less flexible than models allowed when the TSP model is constructed from the Optimize add-in. Solution times are therefore much smaller than when the same problem is solved by Search command of the Optimize add-in.

The optimum solution is below. The selected cells from the distance matrix are colored.

The map of the optimum solution is created by clicking the Map button. This optimum illustrates the characteristic that the optimum tour for a Euclidean problem on a plane does not cross.

48 City Problem

A well known TSP problem involves the 48 state capitals in the continental United States. Data for the problem is available on the web. The data specifies coordinates for the cities and intercity distances are computed using the Euclidean measure. We set up and solved the problem with this add-in. The map of the published optimum obtained from the web is shown below and has a total length of 335.237.

The greedy solution starts from city 1 and the next city is the one closest to city 1. Subsequent cities are selected to be the closest non-visited city to the last city added. This is called the closest neighbor heuristic. The map of the greedy solution shown below is clearly not optimum. The total length is 405.264.

Applying the 2-change improvement procedure to the greedy solution provides a better solution, but not quite optimum. The total length is 344.055. The process evaluated 6494 solutions and required 21 seconds on the author's computer. The 2-change procedure first looks at all pairs of cities (excluding the first and last) and switches their positions in the sequence. Improving switches are immediately accepted. This search process continues until there are no pairs that can improve the solution by switching. The final solution to this process is followed by the traditional 2-opt procedure. Again we investigate all pairs of cities, but now when a pair is switched we also change the orientation of all arcs that fall between them. This tends to change larger segments of the solution. The 2-opt procedure is repeated until a pass results in no improvements. The solution tour does not cross itself. To find a better solution, we might try several random solutions combined with improvements.

Solving the TSP problems using the Search button on the worksheet provides solutions much more rapidly than solving them using the Search command from the Optimization add-in. The Combinatorics add-in solves only the traditional TSP and does not allow additional constraints or a modified objective function. The add-in makes no assumptions about the distance matrix and symmetric and asymmetric problems are solved with the same procedures. This add-in allows the visual display through maps for Euclidean problems. When distances are provided by formulas the formulas can be changed to use different distance norms.

The size of the problem that can be addressed is limited only by the size of the worksheet, memory limitations and internal restrictions of Excel. Solution times for larger problems may become prohibitive and solutions obtained through search procedures will probably not be optimum.

Operations Research Models and Methods
Internet
by Paul A. Jensen