# Math

A mathematical model of the problem.

## Variables

• An integer variable $\text{succ}_i$ per city i is needed. It represents the successor of city i in the route.

$$\forall i \in [1,C],\, \text{succ}_i = [\![1,C]\!]$$

• An integer variable $\text{dist}_i$ per city i is needed. it maintains the distance between city i and its successor in the route.

$$\forall i \in [1,C],\, \text{dist}_i = [\![1,M]\!]$$

where $M$ is the maximum value in the D matrix.

• An integer variable $totDist$ totals all distances:

$$totDist = [\![0, C\times M]\!]$$

## Constraints

• The distance from a city i to its successor should be read from D:

$$\forall i \in [1,C], \text{dist}_{i} = \text{D}_{i,\text{succ}_i}$$

• The route over cities should form an Hamiltonian path.

• the total distance has then to be maintained:

$$totDist = \sum_{i = 1}^{C} \text{dist}_i$$

## Objective

The objective is not to simply find a solution but one that minimizes $totDist$.