This post is also available as a pdf, a Jupyter Notebook, and a py file.

The travelling salesman problem is a classic geographic problem that may be framed as “Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city and returns to the origin city?”.

The same problem may be applied to community nurses: given a list of patients to see in a day, what order should the nurse visit the patients to minimise time spent travelling (to free up the most time for nursing time).

This problem is a class of problems known as ‘np-hard’, that is the problem size grows non-linearly with the number of points to visit, and soon becomes too computationally expensive to compute all possibilities. There is also no algorithm known that efficiently gives a certain optimum solution. For example with 5 points to visit there are 120 possible routes, but with 10 points there are 3,628,800 possible routes, and with 20 points to visit there are 2,432,902,008,176,640,000 routes!

Np-hard problems are ‘solved’ by using a ‘heuristic algorithm’. Heuristic algorithms follow a method designed to efficiently give a sufficiently good solution at the expense of not guaranteeing that an optimal solution is found.

In the case of the travelling salesman problem (or, in our case, a community nurse with a set of patients to visit), there are are many heuristics described. It is a favourite problem of algorithm writers!

In the code below we will use a ‘hill-climbing’ method based on reversing portions of the route (or a ‘pairwise exchange’ approach). Hill-climbing methods perform some search that then leads to them picking the best improvement in that step of the search. The search method is then repeated until no further improvement is found. Hill-climbing methods may get trapped in local optima in complicated problems – the solution found depends on the starting point. To reduce the impact of this the algorithm is repeated multiple times with different start points.

Our algorithm will follow these steps:

1) Pick a route at random (randomly order to points to visit)

2) Examine all possible pairwise exchanges in the route in (e.g. if we have a route A-B-C-D-E-F-G-H, and pairwise exchange C and F we have a new route A-B-F-E-D-C-G-H). Choose the pairwise exchange that gives the best reduction in route distance).

3) Repeat step 2) until no further improvement

4) Repeat from step 1 for a determined number of repeats (or repeat until a maximum algorithm use time is reached).

In order to run an algorithm like this we need to know the travel times or distances between points. In the example below we calculate a straight line (or ‘Euclidean’) distance based on x and y locations. x/y locations are readily available for addresses (such as Eastings and Northings in the UK). Straight line distances or travel times, while only approximate may be good enough to get reasonable solutions. More precise solutions could use travel times or distances determined from Geographical Information Systems (an example of an implementation using Open Source methods and open mapping data is described here: https://github.com/MichaelAllen1966/batch_travel_times_and_distances).

Here is a diagram of the effect of one exchange (from https://en.wikipedia.org/wiki/Travelling_salesman_problem)

Let’s see the code (don’t worry if you don’t understand the plot function, though hopefully you can understand enough to know how to copy and hack it to use yourself).

```
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.path import Path
import matplotlib.patches as patches
import math
def main_code():
"""Main program code called"""
# Load points to visit with co-ordinates
points_to_visit, xCo, yCo, number_of_points_to_visit = load_points()
# Create a lookup dictionary for distances between any two points_to_visit
distances_lookup = create_dictionary_of_distance(
number_of_points_to_visit, xCo, yCo)
# Run route finding algorithm multiple times
number_of_runs = 50
(final_best_distance_so_far, final_best_route_so_far,
progress_in_best_distance) = (find_shortest_route(
number_of_runs, points_to_visit, distances_lookup))
# Print results
print_results(final_best_route_so_far, xCo, yCo, points_to_visit,
final_best_distance_so_far, progress_in_best_distance)
def calculate_total_route_distance(route, distances_lookup):
"""Calculates route distances. Accesses 'distances_lookup' dictionary which
gives distance between any two points. Total distance includes return to
starting point at end of route"""
total = 0
for i in range(len(route) - 1):
total += distances_lookup[route[i], route[i + 1]]
total += distances_lookup[route[-1], route[0]]
return total
def create_dictionary_of_distance(number_of_points_to_visit, xCo, yCo):
"""Creates a dictions of distances_lookup between all combinations of
points_to_visit"""
distances_lookup = {}
for i in range(number_of_points_to_visit):
for j in range(i, number_of_points_to_visit):
x1 = xCo[i]
y1 = yCo[i]
x2 = xCo[j]
y2 = yCo[j]
distance = math.sqrt((x1 - x2) ** 2 + (y1 - y2) ** 2)
distances_lookup[i, j] = distance
distances_lookup[j, i] = distance
return distances_lookup
def load_points():
"""Usually this function will load point ids and co-ordinates from, for
example, a CSV file. Here we define a function that will generate random
points, as an example"""
number_of_points = 25
points_to_visit = list(range(0, number_of_points))
x = np.random.uniform(0, 200, number_of_points)
y = np.random.uniform(0, 200, number_of_points)
number_of_points_to_visit = len(points_to_visit)
return points_to_visit, x, y, number_of_points_to_visit
def find_shortest_route(runs, points_to_visit, distances_lookup):
"""Main algorithm code"""
final_best_route_so_far = []
final_best_distance_so_far = 1e99
progress_in_best_distance = []
for run in range(runs):
exchange_point_1 = 0
exchange_point_2 = 0
improvement_found = True
best_route_so_far = points_to_visit.copy()
np.random.shuffle(best_route_so_far)
best_distance_so_far = calculate_total_route_distance(
best_route_so_far, distances_lookup)
while improvement_found: # continue until no further improvement
improvement_found = False
# Loop through all paiwwise route section reversals
for i in range(0, len(best_route_so_far)-1):
for j in range(i, len(best_route_so_far)):
test_route = best_route_so_far.copy()
test_route = reverse_section(test_route, i, j)
test_route_distance = (calculate_total_route_distance
(test_route, distances_lookup))
if test_route_distance < best_distance_so_far:
exchange_point_1 = i
exchange_point_2 = j
improvement_found = True
best_distance_so_far = test_route_distance
if improvement_found:
best_route_so_far = reverse_section(
best_route_so_far, exchange_point_1, exchange_point_2)
if best_distance_so_far < final_best_distance_so_far:
final_best_distance_so_far = best_distance_so_far
final_best_route_so_far = best_route_so_far.copy()
progress_in_best_distance.append(final_best_distance_so_far)
return (final_best_distance_so_far, final_best_route_so_far,
progress_in_best_distance)
def plot_improvement(progress_in_best_distance):
"""Plot improvement over runs"""
plt.plot(progress_in_best_distance)
plt.xlabel('Run')
plt.ylabel('Best distance')
plt.show()
def plot_route(route, xCo, yCo, label):
"""Plot points and best route found between points"""
# Create figure
fig = plt.figure(figsize=(8, 5))
# Plot points to vist
ax1 = fig.add_subplot(121)
ax1.scatter(xCo, yCo)
texts = []
for i, txt in enumerate(label):
texts.append(ax1.text(xCo[i] + 1, yCo[i] + 1, txt))
# Plot best route found between points
verts = [None] * int(len(route) + 1)
codes = [None] * int(len(route) + 1)
for i in range(0, len(route)):
verts[i] = xCo[route[i]], yCo[route[i]]
if i == 0:
codes[i] = Path.MOVETO
else:
codes[i] = Path.LINETO
verts[len(route)] = xCo[route[0]], yCo[route[0]]
codes[len(route)] = Path.CLOSEPOLY
path = Path(verts, codes)
ax2 = fig.add_subplot(122)
patch = patches.PathPatch(path, facecolor='none', lw=0)
ax2.add_patch(patch)
ax2.set_xlim(min(xCo) - 10, max(xCo) + 10)
ax2.set_ylim(min(yCo) - 10, max(yCo) + 10)
# give the points a label
xs, ys = zip(*verts)
ax2.plot(xs, ys, 'x--', lw=2, color='black', ms=10)
texts = []
for i, txt in enumerate(label):
texts.append(ax2.text(xCo[i] + 1, yCo[i] + 1, txt))
# Display plot
plt.tight_layout(pad=4)
plt.show()
return
def print_results(final_best_route_so_far, xCo, yCo, points_to_visit,
final_best_distance_so_far, progress_in_best_distance):
"""Prints results to screeen"""
print('Best route found:')
plot_route(final_best_route_so_far, xCo, yCo, points_to_visit)
print('\nDistance = %.0f' % final_best_distance_so_far)
print('\nImprovement in distance over run:')
plot_improvement(progress_in_best_distance)
def reverse_section(orig_list, point_a, point_b):
"""Reverses section of route between points a and b (inclusive)"""
low_switch_point = min(point_a, point_b)
high_switch_point = max(point_a, point_b)
high_switch_point += 1 # Include high switch point in reversed section
section_1 = orig_list[0:low_switch_point]
section_2 = list(reversed(orig_list[low_switch_point:high_switch_point]))
section_3 = orig_list[high_switch_point:]
new_list = section_1 + section_2 + section_3
return (new_list)
# RUN MAIN CODE
# -------------
main_code()
```

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