TSP

A library for solving instances of the travelling sales agent problem

Tutorial

In this tutorial we will see how to use tsp to solve instances of the Travelling Salesman Problem

Assuming we have the following distance matrix:

>>> import numpy as np
>>> distance_matrix = np.array(((0, 5, 2, 9), (5, 0, 3, 1), (2, 3, 0, 7), (9, 1, 7, 0)))

We can obtain a tour using the following:

>>> import tsp
>>> tour = tsp.run_2_opt_algorithm(
...     number_of_stops=4, 
...     distance_matrix=distance_matrix, 
...     iterations=1000, 
...     seed=0,
...    )

We can see the tour here:

>>> tour
[0, 3, 1, 2, 0]

The tsp library includes further functionality which you can read in the How To guides.

How to guide

How to obtain a basic tour

To obtain a basic tour, we use the tsp.get_tour function:

>>> import tsp
>>> tsp.get_tour(number_of_stops=5)
[0, 1, 2, 3, 4, 0]

If we pass a random seed this will also shuffle the interior stops (in a reproducible manner):

>>> tsp.get_tour(number_of_stops=5, seed=0)
[0, 3, 4, 2, 1, 0]

How to swap two stops

To swap two stops for a given tour, we use the tsp.swap_stops function:

>>> tour = [0, 1, 2, 3, 4, 5, 0]
>>> tsp.swap_stops(tour=tour, steps=(2, 4))
[0, 1, 4, 3, 2, 5, 0]

How to get the cost of a tour

To calculate the cost of a given tour, we use the tsp.get_cost function:

distance_matrix = np.array(((0, 5, 2, 9), (5, 0, 3, 1), (2, 3, 0, 7), (9, 1, 7, 0)))
tour = [0, 1, 2, 3, 0]
tsp.get_cost(tour=tour, distance_matrix=distance_matrix)

which gives:

How to plot a tour

To plot a tour we use the tsp.plot_tour function:

xs = (0, 1, 1, 2.5)
ys = (0, 5, 1, 3)
tour = [0, 1, 3, 2, 0]
tsp.plot_tour(x=xs, y=ys, tour=tour)

This gives the following image:

How to use the 2-opt algorithm

To run the full algorithm, we use the tsp.run_2_opt_algorithm function:

distance_matrix = np.array(((0, 5, 2, 9), (5, 0, 3, 1), (2, 3, 0, 7), (9, 1, 7, 0)))
tour = tsp.run_2_opt_algorithm(number_of_stops=4, distance_matrix=distance_matrix, iterations=1000, seed=0)
tour

This gives:

[0, 3, 1, 2, 0]

Explanations

This software implements the 2-opt algorithm for the travelling sales agent problem.

The TSP

As an example, if we consider three cities with the following matrix defining their distances between them:

$$ \begin{pmatrix} 0 & 4 & 1\\ 4 & 0 & 2\\ 9 & 2 & 0 \end{pmatrix} $$

Note that the distance matrix is not symmetric, it is a lot further to go from the 3rd to the 1st city (9) than to go from the 1st to the 3rd (1)

If a tour starts and finishes at the first city there are in fact 2 possibilities:

$$T \in {(0, 1, 2, 0), (0, 2, 1, 0)}$$

The cost $c(t)$ for $t\in T$ of these tour tours is taken to be the total distance travelled:

For $t=(0, 1, 2, 0)$ we have $c(t)=4 + 2 + 9=15$
For $t=(0, 2, 1, 0)$ we have $c(t)=1 + 2 + 2=5$

As the size of our problem grows the complexity of finding the optimal tour grows in complexity. In fact this problem is NP-hard, which puts it in a class of problems for which a general solution cannot be obtained efficiently for any given sized problem.

The 2-opt algorithm

One solution approach of this is the 2-opt algorithm which is what is implemented in this software.

The 2-opt algorithm is an example of a neighbourhood search algorithm which means that it iteratively improves a given solution by looking in at other candidates near it.

The 2-opt algorithm does this by randomly choosing two stops in a tour, and swapping the order between them. Essentially picking stop $n$ and stop $n + k$ and reversing the order. Thus the new candidate would visit the same stops as the original tour, until it got to stop $n$, when it would instead go to stop $n + k$ and reverse its way back to stop $n$.

Once this candidate tour is obtained the cost is evaluated and if it is good it is accepted as the new solution.

This has the effect of essentially untangling a given tour.

Alternative algorithms

There are various other algorithms for solving the TSP. They includ:

Trying all possible permutations.

Reference

List of functionality

This software implements 5 functions:

plot_tour
get_tour
swap_cities
get_cost
run_2_opt_algorithm

Bibliography

The wikipedia page on the TSP offers good background reading: https://en.wikipedia.org/wiki/Travelling_salesman_problem

The following text is a recommended text on neighbourhood search algorithms:

Aarts, Emile, Emile HL Aarts, and Jan Karel Lenstra, eds. Local search in combinatorial optimization. Princeton University Press, 2003.

drvinceknight / tsp

TSP