Simulerad glödgning för sökning i Python

Denna handledning visar hur du implementerar simulerad glödgning för sökning i Python, algoritmen kommer att användas för att hitta en lösning på handelsresandeproblemet. Simulerad glödgning är en lokal sökalgoritm som använder sjunkande temperatur enligt ett schema för att gå från mer slumpmässiga lösningar till mer förfinade lösningar.

En simulerad glödgningsalgoritm kan användas för att lösa verkliga problem med många permutationer eller kombinationer. Vägen till målet får inte vara viktig och det finns inga garantier för att algoritmen kommer att hitta en optimal lösning. Algoritmen kan hitta en tillfredsställande lösning snabbt och den behöver inte ta mycket minne i anspråk.

Simulerad glödgningssökning använder sjunkande temperatur enligt ett schema för att ha en högre sannolikhet att acceptera sämre lösningar i början (kunna hoppa ut från ett lokalt maximum/lokalt minimum), algoritmen är mindre benägen att kasta bort goda lösningar allt eftersom temperaturen sjunker. Simulerad glödgning börjar med en initial lösning som kan genereras slumpmässigt eller enligt vissa regler, den initiala lösningen muteras sedan i varje iteration och den bästa lösningen returneras när temperaturen är noll.

Handelsresandeproblemet

Jag kommer att hitta en tillfredsställande lösning på ett handelsresandeproblem med 13 städer (Traveling Salesman Problem). Detta problem har 479001600 ((13-1)!) permutationer och det skulle ta lång tid att testa varje permutation för att hitta den optimala lösningen. Målet är att hitta den rutt som inkluderar alla städer och som är kortast, resan börjar och slutar i samma stad.

# 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()

Resultat

Den initiala lösningen kan väljas slumpmässigt, väljas slumpmässigt med avståndsvikter eller väljas i enlighet med det kortaste avståndet mellan varje stad. Den bästa lösningen är 7293 miles, denna algoritm kan skapa en lösning som är sämre än den ursprungliga lösningen.

-- 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
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