# Travelling Salesman Problem

## 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.

## Formulation

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{align*} x_{ij}&= \begin{cases} 1, & \text{if salesman goes from } i^{th} \text{ city to } j^{th} \text{ city}; \\ 0, & \text{Otherwise}. \end{cases} \end{align*}

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).