This tutorial will show you how to implement a simulated annealing search algorithm in Python, to find a solution to the traveling salesman problem. Simulated annealing is a local search algorithm that uses decreasing temperature according to a schedule in order to go from more random solutions to more improved solutions.

A simulated annealing algorithm can be used to solve real-world problems with a lot of permutations or combinations. The path to the goal should not be important and the algorithm is not guaranteed to find an optimal solution. It can find an satisfactory solution fast and it doesn’t need a lot of memory.

Simulated annealing search uses decreasing temperature according to a schedule to have a higher probability of accepting inferior solutions in the beginning and be able to jump out from a local maximum, as the temperature decreases the algorithm is less likely to throw away good solutions. Simulated annealing starts with an initial solution that can be generated at random or according to some rules, the initial solution will then be mutated in each iteration and the the best solution will be returned when the temperature is zero.

## Traveling Salesman Problem (TSP)

I am going to find a satisfactory solution to a traveling salesman problem with 13 cities (Traveling Salesman Problem). This problem has 479001600 ((13-1)!) permutations and it would take a long time to test every permutation in order to find the optimal solution. The goal is to find the route with the shortest total distance all cities included, starting and ending in the same city.

```
# Import libraries
import sys
import random
import copy
import numpy as np
# This class represent a state
class State:
# Create a new state
def __init__(self, route:[], distance:int=0):
self.route = route
self.distance = distance
# Compare states
def __eq__(self, other):
for i in range(len(self.route)):
if(self.route[i] != other.route[i]):
return False
return True
# Sort states
def __lt__(self, other):
return self.distance < other.distance
# Print a state
def __repr__(self):
return ('({0},{1})\n'.format(self.route, self.distance))
# Create a shallow copy
def copy(self):
return State(self.route, self.distance)
# Create a deep copy
def deepcopy(self):
return State(copy.deepcopy(self.route), copy.deepcopy(self.distance))
# Update distance
def update_distance(self, matrix, home):
# Reset distance
self.distance = 0
# Keep track of departing city
from_index = home
# Loop all cities in the current route
for i in range(len(self.route)):
self.distance += matrix[from_index][self.route[i]]
from_index = self.route[i]
# Add the distance back to home
self.distance += matrix[from_index][home]
# This class represent a city (used when we need to delete cities)
class City:
# Create a new city
def __init__(self, index:int, distance:int):
self.index = index
self.distance = distance
# Sort cities
def __lt__(self, other):
return self.distance < other.distance
# Return true with probability p
def probability(p):
return p > random.uniform(0.0, 1.0)
# Schedule function for simulated annealing
def exp_schedule(k=20, lam=0.005, limit=1000):
return lambda t: (k * np.exp(-lam * t) if t < limit else 0)
# Get the best random solution from a population
def get_random_solution(matrix:[], home:int, city_indexes:[], size:int, use_weights:bool=False):
# Create a list with city indexes
cities = city_indexes.copy()
# Remove the home city
cities.pop(home)
# Create a population
population = []
for i in range(size):
if(use_weights == True):
state = get_random_solution_with_weights(matrix, home)
else:
# Shuffle cities at random
random.shuffle(cities)
# Create a state
state = State(cities[:])
state.update_distance(matrix, home)
# Add an individual to the population
population.append(state)
# Sort population
population.sort()
# Return the best solution
return population[0]
# Get best solution by distance
def get_best_solution_by_distance(matrix:[], home:int):
# Variables
route = []
from_index = home
length = len(matrix) - 1
# Loop until route is complete
while len(route) < length:
# Get a matrix row
row = matrix[from_index]
# Create a list with cities
cities = {}
for i in range(len(row)):
cities[i] = City(i, row[i])
# Remove cities that already is assigned to the route
del cities[home]
for i in route:
del cities[i]
# Sort cities
sorted = list(cities.values())
sorted.sort()
# Add the city with the shortest distance
from_index = sorted[0].index
route.append(from_index)
# Create a new state and update the distance
state = State(route)
state.update_distance(matrix, home)
# Return a state
return state
# Get a random solution by using weights
def get_random_solution_with_weights(matrix:[], home:int):
# Variables
route = []
from_index = home
length = len(matrix) - 1
# Loop until route is complete
while len(route) < length:
# Get a matrix row
row = matrix[from_index]
# Create a list with cities
cities = {}
for i in range(len(row)):
cities[i] = City(i, row[i])
# Remove cities that already is assigned to the route
del cities[home]
for i in route:
del cities[i]
# Get the total weight
total_weight = 0
for key, city in cities.items():
total_weight += city.distance
# Add weights
weights = []
for key, city in cities.items():
weights.append(total_weight / city.distance)
# Add a city at random
from_index = random.choices(list(cities.keys()), weights=weights)[0]
route.append(from_index)
# Create a new state and update the distance
state = State(route)
state.update_distance(matrix, home)
# Return a state
return state
# Mutate a solution
def mutate(matrix:[], home:int, state:State, mutation_rate:float=0.01):
# Create a copy of the state
mutated_state = state.deepcopy()
# Loop all the states in a route
for i in range(len(mutated_state.route)):
# Check if we should do a mutation
if(random.random() < mutation_rate):
# Swap two cities
j = int(random.random() * len(state.route))
city_1 = mutated_state.route[i]
city_2 = mutated_state.route[j]
mutated_state.route[i] = city_2
mutated_state.route[j] = city_1
# Update the distance
mutated_state.update_distance(matrix, home)
# Return a mutated state
return mutated_state
# Simulated annealing
def simulated_annealing(matrix:[], home:int, initial_state:State, mutation_rate:float=0.01, schedule=exp_schedule()):
# Keep track of the best state
best_state = initial_state
# Loop a large number of times (int.max)
for t in range(sys.maxsize):
# Get a temperature
T = schedule(t)
# Return if temperature is 0
if T == 0:
return best_state
# Mutate the best state
neighbor = mutate(matrix, home, best_state, mutation_rate)
# Calculate the change in e
delta_e = best_state.distance - neighbor.distance
# Check if we should update the best state
if delta_e > 0 or probability(np.exp(delta_e / T)):
best_state = neighbor
# The main entry point for this module
def main():
# Cities to travel
cities = ['New York', 'Los Angeles', 'Chicago', 'Minneapolis', 'Denver', 'Dallas', 'Seattle', 'Boston', 'San Francisco', 'St. Louis', 'Houston', 'Phoenix', 'Salt Lake City']
city_indexes = [0,1,2,3,4,5,6,7,8,9,10,11,12]
# Index of start location
home = 2 # Chicago
# Distances in miles between cities, same indexes (i, j) as in the cities array
matrix = [[0, 2451, 713, 1018, 1631, 1374, 2408, 213, 2571, 875, 1420, 2145, 1972],
[2451, 0, 1745, 1524, 831, 1240, 959, 2596, 403, 1589, 1374, 357, 579],
[713, 1745, 0, 355, 920, 803, 1737, 851, 1858, 262, 940, 1453, 1260],
[1018, 1524, 355, 0, 700, 862, 1395, 1123, 1584, 466, 1056, 1280, 987],
[1631, 831, 920, 700, 0, 663, 1021, 1769, 949, 796, 879, 586, 371],
[1374, 1240, 803, 862, 663, 0, 1681, 1551, 1765, 547, 225, 887, 999],
[2408, 959, 1737, 1395, 1021, 1681, 0, 2493, 678, 1724, 1891, 1114, 701],
[213, 2596, 851, 1123, 1769, 1551, 2493, 0, 2699, 1038, 1605, 2300, 2099],
[2571, 403, 1858, 1584, 949, 1765, 678, 2699, 0, 1744, 1645, 653, 600],
[875, 1589, 262, 466, 796, 547, 1724, 1038, 1744, 0, 679, 1272, 1162],
[1420, 1374, 940, 1056, 879, 225, 1891, 1605, 1645, 679, 0, 1017, 1200],
[2145, 357, 1453, 1280, 586, 887, 1114, 2300, 653, 1272, 1017, 0, 504],
[1972, 579, 1260, 987, 371, 999, 701, 2099, 600, 1162, 1200, 504, 0]]
# Get the best route by distance
state = get_best_solution_by_distance(matrix, home)
print('-- Best solution by distance --')
print(cities[home], end='')
for i in range(0, len(state.route)):
print(' -> ' + cities[state.route[i]], end='')
print(' -> ' + cities[home], end='')
print('\n\nTotal distance: {0} miles'.format(state.distance))
print()
# Get the best random route
state = get_random_solution(matrix, home, city_indexes, 100)
print('-- Best random solution --')
print(cities[home], end='')
for i in range(0, len(state.route)):
print(' -> ' + cities[state.route[i]], end='')
print(' -> ' + cities[home], end='')
print('\n\nTotal distance: {0} miles'.format(state.distance))
print()
# Get a random solution with weights
state = get_random_solution(matrix, home, city_indexes, 100, use_weights=True)
print('-- Best random solution with weights --')
print(cities[home], end='')
for i in range(0, len(state.route)):
print(' -> ' + cities[state.route[i]], end='')
print(' -> ' + cities[home], end='')
print('\n\nTotal distance: {0} miles'.format(state.distance))
print()
# Run simulated annealing to find a better solution
state = get_best_solution_by_distance(matrix, home)
state = simulated_annealing(matrix, home, state, 0.1)
print('-- Simulated annealing solution --')
print(cities[home], end='')
for i in range(0, len(state.route)):
print(' -> ' + cities[state.route[i]], end='')
print(' -> ' + cities[home], end='')
print('\n\nTotal distance: {0} miles'.format(state.distance))
print()
# Tell python to run main method
if __name__ == "__main__": main()
```

## Output

The initial solution can be selected at random, selected at random with distance weights or selected by using the shortest distance in each step. The best solution is 7293 miles, this algorithm can produce a solution that is worse than the initial solution.

```
-- Best solution by distance --
Chicago -> St. Louis -> Minneapolis -> Denver -> Salt Lake City -> Phoenix -> Los Angeles -> San Francisco -> Seattle -> Dallas -> Houston -> New York -> Boston -> Chicago
Total distance: 8131 miles
-- Best random solution --
Chicago -> St. Louis -> New York -> Boston -> Salt Lake City -> Phoenix -> Los Angeles -> Denver -> Dallas -> Minneapolis -> Seattle -> San Francisco -> Houston -> Chicago
Total distance: 11324 miles
-- Best random solution with weights --
Chicago -> St. Louis -> Dallas -> Houston -> New York -> Boston -> Minneapolis -> Denver -> Salt Lake City -> Phoenix -> Los Angeles -> San Francisco -> Seattle -> Chicago
Total distance: 8540 miles
-- Simulated annealing solution --
Chicago -> St. Louis -> Minneapolis -> Denver -> Salt Lake City -> Seattle -> San Francisco -> Los Angeles -> Phoenix -> Dallas -> Houston -> New York -> Boston -> Chicago
Total distance: 7534 miles
```