Travelling Salesman Problem
Suppose a salesman wants to visit certain number of cities, say, $n$. Let $c_{ij}$ be the distance from city $i$ to city $j$. Then the problem of salesman is to select such a route that starts from his home city, passes through each city once and only once, and returns to his home city in the shortest possible distance. Such a problem is known as Travelling Salesman Problem.
Suppose $x_{ij}=1$ if the salesman goes directly from city $i$ to city $j$, and $x_{ij}=0$ otherwise. Then the objective function is to $$ \begin{equation*} \min z= \sum_{i=1}^n\sum_{j=1}^n x_{ij}c_{ij} \end{equation*} $$ subject to $$ \begin{equation*} \sum_{j=1}^n x_{ij} =1,; \text{ for } i=1,2,\ldots, n \end{equation*} $$
$$ \begin{equation*} \sum_{i=1}^n x_{ij} =1,; \text{ for } j=1,2,\ldots,n \end{equation*} $$ where $$ \begin{equation*} x_{ij}=\left{ \begin{array}{ll} 1, & \hbox{if salesman goes from $i^{th}$ city to $j^{th}$ city;} \ 0, & \hbox{otherwise.} \end{array} \right. \end{equation*} $$ With one more restriction that no city is visited twice before the tour of all cities is completed. The salesman cannot go from city $i$ to city $i$ itself. This possibility may be avoided by adopting the convention $c_{ii}=\infty$ which insures that $x_{ii}$ can never be one.
| From \ To | $A_1$ | $A_2$ | $\cdots$ | $A_j$ | $\cdots$ | $A_n$ |
|---|---|---|---|---|---|---|
| $A_1$ | $\infty$ | $c_{12}$ | $\cdots$ | $c_{1j}$ | $\cdots$ | $c_{1n}$ |
| $A_2$ | $c_{21}$ | $\infty$ | $\cdots$ | $c_{2j}$ | $\cdots$ | $c_{2n}$ |
| $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
| $A_i$ | $c_{i1}$ | $c_{i2}$ | $\cdots$ | $c_{ij}$ | $\cdots$ | $c_{in}$ |
| $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
| $A_n$ | $c_{n1}$ | $c_{n2}$ | $\cdots$ | $c_{nj}$ | $\cdots$ | $\infty$ |
Apply the usual Hungarian method to find the optimal route. (It should be in cyclic order, i.e., no city should be visited twice).
