1. 问题模型
TSP问题是旅行商问题的简写，问题非常简单：从原点出发经过所有需求点并回到原点，使得途经的距离最短。
ortools可以使用RoutingModel来进行求解。
2. 求解代码
首先建立RoutingModel模型：
routing = pywrapcp.RoutingModel(tsp_size, num_routes, depot)
为SetArcCostEvaluatorOfAllVehicles方法传入一个计算距离的回调函数，然后使用默认的参数pywrapcp.RoutingModel.DefaultSearchParameters()，就可以使用SolveWithParameters进行求解了。注意使用这个参数的话，给出的是一个可行初始解，而不是最优解。
代码如下
from ortools.constraint_solver import pywrapcp
from ortools.constraint_solver import routing_enums_pb2
# Cities
city_names = ["New York", "Los Angeles", "Chicago", "Minneapolis", "Denver", "Dallas", "Seattle",
"Boston", "San Francisco", "St. Louis", "Houston", "Phoenix", "Salt Lake City"]
# Distance matrix
dist_matrix = [
[ 0, 2451, 713, 1018, 1631, 1374, 2408, 213, 2571, 875, 1420, 2145, 1972], # New York
[2451, 0, 1745, 1524, 831, 1240, 959, 2596, 403, 1589, 1374, 357, 579], # Los Angeles
[ 713, 1745, 0, 355, 920, 803, 1737, 851, 1858, 262, 940, 1453, 1260], # Chicago
[1018, 1524, 355, 0, 700, 862, 1395, 1123, 1584, 466, 1056, 1280, 987], # Minneapolis
[1631, 831, 920, 700, 0, 663, 1021, 1769, 949, 796, 879, 586, 371], # Denver
[1374, 1240, 803, 862, 663, 0, 1681, 1551, 1765, 547, 225, 887, 999], # Dallas
[2408, 959, 1737, 1395, 1021, 1681, 0, 2493, 678, 1724, 1891, 1114, 701], # Seattle
[ 213, 2596, 851, 1123, 1769, 1551, 2493, 0, 2699, 1038, 1605, 2300, 2099], # Boston
[2571, 403, 1858, 1584, 949, 1765, 678, 2699, 0, 1744, 1645, 653, 600], # San Francisco
[ 875, 1589, 262, 466, 796, 547, 1724, 1038, 1744, 0, 679, 1272, 1162], # St. Louis
[1420, 1374, 940, 1056, 879, 225, 1891, 1605, 1645, 679, 0, 1017, 1200], # Houston
[2145, 357, 1453, 1280, 586, 887, 1114, 2300, 653, 1272, 1017, 0, 504], # Phoenix
[1972, 579, 1260, 987, 371, 999, 701, 2099, 600, 1162, 1200, 504, 0]] # Salt Lake City
tsp_size = len(city_names)
num_routes = 1
depot = 0
routing = pywrapcp.RoutingModel(tsp_size, num_routes, depot)
search_parameters = pywrapcp.RoutingModel.DefaultSearchParameters()
dist = lambda from_node,to_node:int(dist_matrix[from_node][to_node])
routing.SetArcCostEvaluatorOfAllVehicles(dist)
# Solve the problem.
assignment = routing.SolveWithParameters(search_parameters)
if assignment:
# Solution distance.
print("Total distance: " + str(assignment.ObjectiveValue()) + " miles\n")
# Display the solution.
# Only one route here; otherwise iterate from 0 to routing.vehicles() - 1
route_number = 0
index = routing.Start(route_number) # Index of the variable for the starting node.
route = ''
while not routing.IsEnd(index):
# Convert variable indices to node indices in the displayed route.
route += str(city_names[routing.IndexToNode(index)]) + ' -> '
index = assignment.Value(routing.NextVar(index))
route += str(city_names[routing.IndexToNode(index)])
print("Route:\n\n" + route)
输出为：
Total distance: 7293 miles
Route:
New York -> Boston -> Chicago -> Minneapolis -> Denver -> Salt Lake City -> Seattle -> San Francisco -> Los Angeles -> Phoenix -> Houston -> Dallas -> St. Louis -> New York
可以手动设定初始可行解的求解算法：
search_parameters.first_solution_strategy = (routing_enums_pb2.FirstSolutionStrategy.PATH_CHEAPEST_ARC)
清单如下：
| Option | Description |
|---|---|
| AUTOMATIC | Lets the solver detect which strategy to use according to the model being solved. |
| PATH_CHEAPEST_ARC | Starting from a route “start” node, connect it to the node which produces the cheapest route segment, then extend the route by iterating on the last node added to the route. |
| PATH_MOST_CONSTRAINED_ARC | Similar to PATH_CHEAPEST_ARC, but arcs are evaluated with a comparison-based selector which will favor the most constrained arc first. To assign a selector to the routing model, use the method ArcIsMoreConstrainedThanArc(). |
| EVALUATOR_STRATEGY | Similar to PATH_CHEAPEST_ARC, except that arc costs are evaluated using the function passed to SetFirstSolutionEvaluator(). |
| SAVINGS | Savings algorithm (Clarke & Wright). Reference: Clarke, G. & Wright, J.W.: “Scheduling of Vehicles from a Central Depot to a Number of Delivery Points”, Operations Research, Vol. 12, 1964, pp. 568-581. |
| SWEEP | Sweep algorithm (Wren & Holliday). Reference: Anthony Wren & Alan Holliday: Computer Scheduling of Vehicles from One or More Depots to a Number of Delivery Points Operational Research Quarterly (1970-1977), Vol. 23, No. 3 (Sep., 1972), pp. 333-344. |
| CHRISTOFIDES | Christofides algorithm (actually a variant of the Christofides algorithm using a maximal matching instead of a maximum matching, which does not guarantee the 3/2 factor of the approximation on a metric travelling salesman). Works on generic vehicle routing models by extending a route until no nodes can be inserted on it. Reference: Nicos Christofides, Worst-case analysis of a new heuristic for the travelling salesman problem, Report 388, Graduate School of Industrial Administration, CMU, 1976. |
| ALL_UNPERFORMED | Make all nodes inactive. Only finds a solution if nodes are optional (are element of a disjunction constraint with a finite penalty cost). |
| BEST_INSERTION | Iteratively build a solution by inserting the cheapest node at its cheapest position; the cost of insertion is based on the global cost function of the routing model. As of 2/2012, only works on models with optional nodes (with finite penalty costs). |
| PARALLEL_CHEAPEST_INSERTION | Iteratively build a solution by inserting the cheapest node at its cheapest position; the cost of insertion is based on the the arc cost function. Is faster than BEST_INSERTION. |
| LOCAL_CHEAPEST_INSERTION | Iteratively build a solution by inserting each node at its cheapest position; the cost of insertion is based on the the arc cost function. Differs from |
| GLOBAL_CHEAPEST_ARC | Iteratively connect two nodes which produce the cheapest route segment. |
| LOCAL_CHEAPEST_ARC | Select the first node with an unbound successor and connect it to the node which produces the cheapest route segment. |
| FIRST_UNBOUND_MIN_VALUE | Select the first node with an unbound successor and connect it to the first available node. This is equivalent to the CHOOSE_FIRST_UNBOUND strategy combined with ASSIGN_MIN_VALUE (cf. constraint_solver.h). |
返回值的清单如下：
| Value | Description |
|---|---|
| 0 | ROUTING_NOT_SOLVED: Problem not solved yet. |
| 1 | ROUTING_SUCCESS: Problem solved successfully. |
| 2 | ROUTING_FAIL: No solution found to the problem. |
| 3 | ROUTING_FAIL_TIMEOUT: Time limit reached before finding a solution. |
| 4 | ROUTING_INVALID: Model, model parameters, or flags are not valid. |
3. 第二阶段算法：local search
使用local search可以在初始可行解的基础上进行优化，如下：
search_parameters.local_search_metaheuristic = (
routing_enums_pb2.LocalSearchMetaheuristic.GUIDED_LOCAL_SEARCH)
search_parameters.time_limit_ms = 30000 # 30s
可使用的清单如下：
| Option | Description |
|---|---|
| AUTOMATIC | Lets the solver select the metaheuristic. |
| GREEDY_DESCENT | Accepts improving (cost-reducing) local search neighbors until a local minimum is reached. |
| GUIDED_LOCAL_SEARCH | Uses guided local search to escape local minima (cf. http://en.wikipedia.org/wiki/Guided_Local_Search); this is generally the most efficient metaheuristic for vehicle routing. |
| SIMULATED_ANNEALING | Uses simulated annealing to escape local minima (cf. http://en.wikipedia.org/wiki/Simulated_annealing). |
| TABU_SEARCH | Uses tabu search to escape local minima (cf. http://en.wikipedia.org/wiki/Tabu_search). |
| OBJECTIVE_TABU_SEARCH | Uses tabu search on the objective value of solution to escape local minima |