OR-Tools Vehicle Routing with Multiple Trips, Multiple Pickup and limited capacity

I am trying to solve a routing problem as follows:

• We have many ‘tasks’ and each task contains many items to be collected by workers
• items can appear in multiple tasks (e.g. item 1 can be in both task A and B)
• We already have the distance matrix of the items
• depot is fixed
• in each trip, each worker can ONLY collect items in AT MOST 3 tasks (business domain constraint)

My question is how to use or-tools to implement a solver that:

• allows each worker to “unload” the items collected at the depot and continue to next trip
• set a constraint that limits workers to collect items in at most 3 tasks

So far I have tried:

• treat same items appearing in n tasks as n different nodes (reflected in the distance matrix, and the distance among these n nodes are set to 0)
• uses pickup and deliveries to model each task, so one item in each task will be pointed by other items in within the same task. And create a capacity constraint of 3 and set the demand of that node as 1. (e.g. task A contains [1, 2, 3, 4]. I add pickup and deliveries [1, 4], [2, 4], [3, 4]. Then create a capacity constraint of 3 for each worker, and set node 4 a demand of 1.) But adding this seems to kill the jupyter notebook kernal. (Removing this the code can run.)

Sorry for such a long question, thanks and please help!

Update: I made use of AddDisjunction and AddPickupAndDelivery and the results seem to be what I expected. I am not 100% sure if this is the answer to this problem. I am treating same items appearing in different tasks as different nodes. And add the whole set of items in each task as a disjunction set. For pickup and delivery, I didn’t duplicate the nodes, I simply make each item points to the same 1 item in that task.

The code I wrote (updated):

    # "order" is the same as a "task"
    data = {
        'distance_matrix': get_distance_matrix(locations),
        'demands': demands,
        'num_workers': number_of_order_groups,
        'max_num_orders': [num_orders_in_group] * number_of_order_groups,
        'disjunctions': disjunctions,
        'depot': 0,
    }

    manager = pywrapcp.RoutingIndexManager(len(data['distance_matrix']), data['num_workers'], data['depot'])

    routing = pywrapcp.RoutingModel(manager)

    def distance_callback(from_index, to_index):
        from_node = manager.IndexToNode(from_index)
        to_node = manager.IndexToNode(to_index)
        return data['distance_matrix'][from_node][to_node]

    transit_callback_index = routing.RegisterTransitCallback(distance_callback)

    routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index)

    def demand_callback(from_index):
        """Returns the demand of the node."""
        # Convert from routing variable Index to demands NodeIndex.
        from_node = manager.IndexToNode(from_index)
        return data['demands'][from_node]

    demand_callback_index = routing.RegisterUnaryTransitCallback(demand_callback)
    routing.AddDimensionWithVehicleCapacity(
        demand_callback_index,
        0,  # null capacity slack
        data['max_num_orders'],  # vehicle maximum capacities
        True,  # start cumul to zero
        'Capacity')

    for d in data['disjunctions']:
        routing.AddDisjunction([manager.NodeToIndex(i) for i in d], 100000000, d.shape[0])

    for d in data['disjunctions']:
        for i in d[:-1]:
            routing.AddPickupAndDelivery(manager.NodeToIndex(i), manager.NodeToIndex(d[-1]))

    # Setting first solution heuristic.
    search_parameters = pywrapcp.DefaultRoutingSearchParameters()
    search_parameters.first_solution_strategy = routing_enums_pb2.FirstSolutionStrategy.AUTOMATIC
    search_parameters.local_search_metaheuristic = routing_enums_pb2.LocalSearchMetaheuristic.AUTOMATIC

    # Solve the problem.
    solution = routing.SolveWithParameters(search_parameters)

    # Print solution on console.
    if solution:
        print_solution(data, manager, routing, solution)

    else:
        print('No solution found !')

The result I got:

Objective: 4329
Route for worker 0:
 0 Load(0) ->  49 Load(0.0) ->  64 Load(0.0) ->  48 Load(0.0) ->  50 Load(0.0) ->  62 Load(0.0) ->  46 Load(0.0) ->  47 Load(0.0) ->  63 Load(0.0) ->  67 Load(0.0) ->  51 Load(0.0) ->  52 Load(1.0) ->  66 Load(1.0) ->  65 Load(2.0) ->  68 Load(2.0) ->  69 Load(3.0) ->  0 Load(3.0)
Distance of the route: 421m
Load of the route: 3.0

Route for worker 1:
 0 Load(0) ->  178 Load(0.0) ->  163 Load(0.0) ->  179 Load(0.0) ->  136 Load(0.0) ->  137 Load(0.0) ->  160 Load(0.0) ->  170 Load(0.0) ->  143 Load(0.0) ->  183 Load(0.0) ->  145 Load(0.0) ->  144 Load(0.0) ->  181 Load(0.0) ->  169 Load(0.0) ->  132 Load(0.0) ->  165 Load(0.0) ->  167 Load(0.0) ->  182 Load(0.0) ->  138 Load(0.0) ->  140 Load(0.0) ->  166 Load(0.0) ->  133 Load(0.0) ->  168 Load(0.0) ->  172 Load(0.0) ->  161 Load(0.0) ->  171 Load(0.0) ->  142 Load(0.0) ->  162 Load(0.0) ->  164 Load(0.0) ->  139 Load(0.0) ->  175 Load(0.0) ->  159 Load(0.0) ->  177 Load(0.0) ->  134 Load(0.0) ->  173 Load(1.0) ->  135 Load(1.0) ->  141 Load(1.0) ->  146 Load(2.0) ->  176 Load(2.0) ->  180 Load(2.0) ->  184 Load(3.0) ->  0 Load(3.0)
Distance of the route: 752m
Load of the route: 3.0

Route for worker 2:
 0 Load(0) ->  34 Load(0.0) ->  24 Load(0.0) ->  21 Load(0.0) ->  29 Load(0.0) ->  2 Load(0.0) ->  19 Load(0.0) ->  25 Load(0.0) ->  8 Load(0.0) ->  5 Load(0.0) ->  20 Load(0.0) ->  9 Load(0.0) ->  11 Load(0.0) ->  13 Load(0.0) ->  1 Load(0.0) ->  10 Load(0.0) ->  14 Load(0.0) ->  7 Load(0.0) ->  3 Load(0.0) ->  27 Load(0.0) ->  4 Load(0.0) ->  189 Load(0.0) ->  31 Load(0.0) ->  32 Load(0.0) ->  15 Load(0.0) ->  6 Load(0.0) ->  23 Load(0.0) ->  33 Load(0.0) ->  22 Load(0.0) ->  12 Load(0.0) ->  28 Load(0.0) ->  26 Load(0.0) ->  16 Load(1.0) ->  190 Load(1.0) ->  30 Load(1.0) ->  35 Load(2.0) ->  191 Load(3.0) ->  0 Load(3.0)
Distance of the route: 730m
Load of the route: 3.0

Route for worker 3:
 0 Load(0) ->  109 Load(0.0) ->  110 Load(0.0) ->  148 Load(0.0) ->  111 Load(0.0) ->  112 Load(0.0) ->  147 Load(0.0) ->  149 Load(0.0) ->  150 Load(1.0) ->  113 Load(2.0) ->  157 Load(2.0) ->  158 Load(3.0) ->  0 Load(3.0)
Distance of the route: 214m
Load of the route: 3.0

Route for worker 4:
 0 Load(0) ->  117 Load(0.0) ->  129 Load(0.0) ->  127 Load(0.0) ->  76 Load(0.0) ->  123 Load(0.0) ->  71 Load(0.0) ->  122 Load(0.0) ->  115 Load(0.0) ->  119 Load(0.0) ->  125 Load(0.0) ->  74 Load(0.0) ->  73 Load(0.0) ->  72 Load(0.0) ->  130 Load(0.0) ->  116 Load(0.0) ->  120 Load(0.0) ->  124 Load(0.0) ->  70 Load(0.0) ->  75 Load(0.0) ->  118 Load(0.0) ->  128 Load(0.0) ->  77 Load(1.0) ->  126 Load(1.0) ->  131 Load(2.0) ->  121 Load(3.0) ->  0 Load(3.0)
Distance of the route: 521m
Load of the route: 3.0

Route for worker 5:
 0 Load(0) ->  95 Load(0.0) ->  99 Load(0.0) ->  96 Load(0.0) ->  92 Load(0.0) ->  98 Load(0.0) ->  88 Load(0.0) ->  97 Load(0.0) ->  107 Load(0.0) ->  94 Load(0.0) ->  55 Load(0.0) ->  106 Load(0.0) ->  83 Load(0.0) ->  102 Load(0.0) ->  93 Load(0.0) ->  81 Load(0.0) ->  87 Load(0.0) ->  79 Load(0.0) ->  80 Load(0.0) ->  90 Load(0.0) ->  58 Load(0.0) ->  57 Load(0.0) ->  86 Load(0.0) ->  154 Load(0.0) ->  101 Load(0.0) ->  85 Load(0.0) ->  84 Load(0.0) ->  105 Load(0.0) ->  91 Load(0.0) ->  153 Load(0.0) ->  155 Load(0.0) ->  56 Load(0.0) ->  100 Load(0.0) ->  104 Load(0.0) ->  82 Load(0.0) ->  54 Load(0.0) ->  151 Load(0.0) ->  59 Load(1.0) ->  89 Load(1.0) ->  103 Load(1.0) ->  152 Load(1.0) ->  108 Load(2.0) ->  156 Load(3.0) ->  0 Load(3.0)
Distance of the route: 721m
Load of the route: 3.0

Route for worker 6:
 0 Load(0) ->  41 Load(0.0) ->  114 Load(1.0) ->  39 Load(1.0) ->  40 Load(1.0) ->  43 Load(1.0) ->  38 Load(1.0) ->  42 Load(1.0) ->  44 Load(2.0) ->  185 Load(2.0) ->  186 Load(3.0) ->  0 Load(3.0)
Distance of the route: 369m
Load of the route: 3.0

Route for worker 7:
 0 Load(0) ->  78 Load(1.0) ->  60 Load(1.0) ->  61 Load(2.0) ->  187 Load(2.0) ->  188 Load(3.0) ->  0 Load(3.0)
Distance of the route: 231m
Load of the route: 3.0

Route for worker 8:
 0 Load(0) ->  174 Load(1.0) ->  36 Load(1.0) ->  37 Load(2.0) ->  17 Load(2.0) ->  18 Load(3.0) ->  0 Load(3.0)
Distance of the route: 198m
Load of the route: 3.0

Route for worker 9:
 0 Load(0) ->  192 Load(1.0) ->  53 Load(2.0) ->  45 Load(3.0) ->  0 Load(3.0)
Distance of the route: 172m
Load of the route: 3.0

Total distance of all routes: 4329m
Total load of all routes: 30.0

Answer

Posting my final solution for anybody who is working on a similar setting problem:

1. allows each vehicle to “unload” the items collected at the depot and continue to next trip
2. set a constraint that limits workers to collect items in at most 3 tasks

For 1, I simply set the number of vehicles to be equal the number of tasks. The solution found should contains a lot of vehicles with no items. Our business use case allows workers to move onto next route once he/she has finished the current route so there’s no need to mimic “loading/unloading” in ortools.

For 2, since each task has multiple items. I used a list to represent items for a particular task, e.g.:

items = [1, 2, 3, 4]

Then all the items are represented by a list of list, and 0 is the depot, e.g.:

items = [[0], [1, 2, 3, 4], [5, 6, 7], [8], [9, 10]] # we have 4 tasks here
demands = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] # all items have 0 demand for now

The key is to created a dummy node for each task and set its demand to 1:

items = [[0], [1, 2, 3, 4, 11], [5, 6, 7, 12], [8, 13], [9, 10, 14]] # we have 4 tasks here
demands = [0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1] # dummy nodes have 1 demand

add capacity constraint:

# define demand callback function (demand is the cost of a node)
def demand_callback(from_index):
    """Returns the demand of the node."""
    # Convert from routing variable Index to demands NodeIndex.
    from_node = manager.IndexToNode(from_index)
    return data['demands'][from_node]

demand_callback_index = routing.RegisterUnaryTransitCallback(demand_callback)

# associate demand to max_num_orders
routing.AddDimensionWithVehicleCapacity(
    demand_callback_index,
    0,  # null capacity slack
    data['max_num_orders'],  # worker maximum capacities, 3 in our case
    True,  # start cumul to zero
    'Capacity')

Then create pickup and delivery constraints, where the pickup nodes are the REAL items and the delivery nodes are the DUMMIES:

# [1:] because we don't care about the depot
for d in data['items'][1:]:
    for i in d[:-1]:
        pickup_index = manager.NodeToIndex(i)
        delivery_index = manager.NodeToIndex(d[-1])
        routing.AddPickupAndDelivery(pickup_index, delivery_index)
        routing.solver().Add(routing.VehicleVar(pickup_index) == routing.VehicleVar(delivery_index))

AddDisjunction is no longer required as my problem always has a feasible solution without dropping any node. You might add disjunctions if your problem might requires dropping nodes to get a solution.

That’s it.

If your solver get stuck finding the solution. Try to change your first solution strategy to PATH_CHEAPEST_ARC as this strategy always (I think) gives you a solution.