flow conservation constraint
Awaiting user input
I have a MILP problem where the optimal solution cannot be found (or at least takes a really really long time). The model is similar to the optimal network flow example described in netflow.py example details (gurobi.com) only with a few more decision variables and constraints:
My arcs contain connections between nodes for both directions, as the direction of the flow is not known beforehand. The cost of the transportation between the nodes is a linear function depending on the capacity of the arc which is a decision variable. The demand in each node is given, but the supply is a decision variable constrained by the availability of another commodity in the node. The capacity can be either zero or larger than 5.
In a simplified form:
nodes = [1, 2, 3, 4]
arcs = [(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(2,1),(3,1),(4,1),(3,2),(4,2),(4,3)]
flow = m.addVars(arcs, name="flow")
supply = m.addVars(len(nodes), name="supply")
capacity = m.addVars(arcs, name="capacity")
b = m.addVars(arcs, vtype='B', name="binary_bigM")
# Arc-capacity constraints
m.addConstrs((capacity[i, j] >= b[i, j] * 5 for i,j in arcs), name="larger5")
m.addConstrs((capacity[i, j] <= 100 * b[i, j] for i,j in arcs), name="BigM")
m.addConstrs((flow[i,j] <= capacity[i, j] for i, j in arcs), name="cap")
# Supply constraint (e is given for each node)
m.addConstrs((supply[i] <= e[i] for i in range(len(nodes))), name="supply")
# Flow-conservation constraints
m.addConstrs((flow.sum('*', j) + supply[idx] == flow.sum(j, '*') + demand[idx] for idx,j in enumerate(nodes)), name="flow")
# Objective
cost = quicksum([1.5*capacity[i,j] for i,j in arcs])
When I allow for only one flow direction, the optimization finds a solution fairly fast. But if I allow for both flow directions, the optimization doesn't find an optimal solution even after many hours.
Could the problem be the flow conservation constraint? Do you have any suggestions on how to formulate the problem so that the solution can be found?
Thank you!
0
-
Hi Amanda,
I'm not sure what you mean. Your example allows flow in both directions.
Could you provide a working example?
With values for \(\texttt{capacities}\), \(\texttt{e}\), and \(\texttt{demand}\).
Cheers,
David0 -
Hi David,
here is the working code:
nodes = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401]
demand = [0.0, 0.0, 0.0, 0.0, 2.45, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.85, 0.0, 0.68, 0.0, 0.0, 0.0, 0.05, 0.0, 0.0, 0.0, 0.0, 0.0, 0.07, 0.05, 0.0, 0.14, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.7200000000000002, 0.0, 0.95, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.07, 0.0, 0.05, 0.0, 0.01, 0.01, 0.0, 0.21, 0.0, 0.09, 0.0, 0.38, 0.0, 0.38, 5.109999999999999, 0.14, 0.42, 0.02, 0.3, 0.02, 0.0, 0.14, 0.08, 0.03, 0.58, 1.8800000000000001, 0.12, 0.24, 0.36, 0.15000000000000002, 0.3, 0.07, 3.38, 0.0, 0.01, 0.34, 0.02, 0.13, 1.51, 2.66, 0.0, 0.09, 0.0, 0.47, 0.13, 0.02, 0.13, 0.13, 0.44, 0.0, 0.16, 0.0, 0.0, 0.03, 0.03, 0.0, 0.0, 0.19, 0.04, 0.0, 0.01, 0.13, 0.3, 0.4, 0.2, 0.39, 0.13, 0.0, 0.03, 0.04, 0.13, 0.03, 0.09, 0.03, 0.0, 0.0, 0.0, 0.0, 0.0, 0.01, 0.06, 0.0, 0.05, 0.0, 0.0, 0.02, 0.01, 0.02, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.01, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 4.73, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.01, 0.01, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.01, 0.02, 0.01, 0.01, 0.01, 0.0, 0.02, 0.0, 0.01, 0.0, 0.0, 0.01, 0.0, 0.01, 0.01, 0.0, 0.0, 0.0, 0.01, 0.0, 0.03, 0.0, 0.01, 0.0, 0.0, 0.0, 0.0, 0.03, 0.0, 0.02, 0.01, 0.05, 0.060000000000000005, 0.01, 0.0, 0.0, 0.01, 0.0, 0.01, 0.02, 0.01, 0.0, 0.0, 1.57, 0.01, 0.0, 0.27, 0.0, 0.0, 0.0, 0.0, 0.0, 0.02, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.1, 0.01, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.01, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.01, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.01, 0.01, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.01, 0.0, 0.01, 0.0, 0.0, 0.0, 0.0, 0.01, 0.02, 0.0, 0.0, 0.0, 0.02, 0.0, 0.0, 0.01, 0.0, 0.0, 0.01, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.060000000000000005, 0.0, 0.0, 0.04, 0.0, 0.0, 0.0, 0.0, 0.0, 0.08, 1.07, 0.0, 0.0, 0.01, 0.0, 0.06, 0.0, 0.48, 0.0, 0.0, 0.0, 0.0, 0.01, 0.0, 0.0, 0.0, 0.09, 0.01, 0.0, 0.01, 0.0, 0.0, 0.0, 0.9, 0.0, 0.0, 0.02, 0.03, 0.02, 0.0, 0.0, 0.0, 0.01, 0.01, 0.05, 0.03, 0.02, 0.11, 2.92, 0.0, 0.0, 0.0, 0.0, 0.07, 0.01, 0.01, 0.03, 0.01, 0.0, 3.26, 0.14, 0.0, 0.0, 0.0, 0.0, 0.01, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.01, 0.0, 0.0, 0.04, 0.0, 0.0, 0.0, 0.0, 0.02, 0.01]
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arcs = [(258, 264), (267, 271), (191, 194), (269, 273), (247, 255), (184, 185), (245, 251), (160, 174), (376, 388), (293, 302), (246, 252), (259, 263), (199, 201), (164, 177), (326, 340), (257, 260), (294, 303), (224, 241), (159, 172), (158, 161), (268, 272), (195, 198), (270, 275), (114, 118), (161, 196), (161, 175), (281, 286), (153, 157), (102, 127), (292, 295), (70, 89), (71, 73), (280, 285), (358, 362), (167, 322), (216, 217), (51, 54), (359, 368), (90, 118), (282, 287), (100, 105), (66, 69), (281, 282), (101, 102), (374, 391), (65, 69), (282, 283), (135, 138), (89, 90), (79, 88), (223, 238), (203, 204), (158, 196), (165, 175), (69, 70), (71, 72), (65, 70), (64, 75), (304, 309), (50, 58), (103, 162), (47, 62), (283, 284), (116, 119), (72, 73), (80, 81), (305, 314), (72, 75), (72, 81), (163, 169), (103, 125), (162, 176), (158, 175), (81, 87), (158, 166), (99, 115), (377, 380), (192, 193), (175, 196), (66, 90), (73, 75), (90, 114), (100, 106), (68, 77), (90, 94), (17, 26), (103, 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m = gp.Model('test')
flow = m.addVars(arcs, name="flow")
supply = m.addVars(len(nodes), name="supply")
capacity = m.addVars(arcs, name="capacity")
b = m.addVars(arcs, vtype='B', name="binary_bigM")
# Arc-capacity constraints
m.addConstrs((capacity[i, j] >= b[i, j] * 5 for i,j in arcs), name="larger5")
m.addConstrs((capacity[i, j] <= 100 * b[i, j] for i,j in arcs), name="BigM")
m.addConstrs((flow[i,j] <= capacity[i, j] for i, j in arcs), name="cap")
# Supply constraint (e is given for each node)
m.addConstrs((supply[i] <= e[i] for i in nodes), name="supply")
m.addConstrs((flow.sum('*', i) + supply[i] == flow.sum(i, '*') + demand[i] for i in nodes), name="flow_conservation")
cost = quicksum([60*capacity[i,j]*dist[idx] for idx,(i,j) in enumerate(arcs)]) + quicksum(supply)*0.01298
# Optimierung
m.setObjective(cost, GRB.MINIMIZE)
m.Params.timelimit = 10000.
m.setParam(GRB.Param.Threads, 10)
m.optimize()With these parameters the optimization has not been able to find the optimal solution even after many hours.
But if I only allow one flow direction, the optimum solution can be found fairly fast. That is the reason why I suspected a problem with the flow conservation constraint in the version with both flow directions.
I need to allow for both flow directions. Do you have any suggestions on how to enhance the formulation?
Thank you,
Amanda0 -
Hi Amanda,
Thanks for the updated code.This is interesting, I think we need to find a reformulation as the number of unexplored nodes in that log doesn't inspire much confidence in actually finding the optimal solution anytime soon.
Are there any more details on the formulation you are using? Is there a paper?
Are there any valid inequalities given the problem description?
Any heuristics you can use to come up with some lower/upper bounds on the capacities? E.g. minimise (maximise) flow through then use this as a lower (upper) bound on the capacity.I have experimented with the latter and it seemed to work really fast and the bounds seem pretty harmless. Although I am not 100% sure it makes sense as the final solution is not within your bounds:
Best objective 5.479714128767e+09, best bound 5.479714128767e+09, gap 0.0000%
Cheers,
David0 -
Hi David,
thanks for your fast response.
Only that I don't just allow discrete diameters , but a continuous range. (In my case: capacity = diameter, as I assume them to be linearly dependent on each other).
I am not sure I understand your idea. With my Big-M constraint I already give a lower and upper bound for the capacity, as it can take the value 0, or something in the range of 5 to 100. This is a more or less realistic range for the problem. I guess it also makes sense to maximise the flow through each pipeline, but how can I model that and why might it help to find the final solution?
Cheers,
Amanda0 -
Hi Amanda,
Sorry for the delay and thanks for the link to the paper.
It seems that making capacity continuous is harder. Could you maybe try a discrete approach?I guess it also makes sense to maximise the flow through each pipeline, but how can I model that and why might it help to find the final solution?
I just thought it would be a quick way of obtaining an upper bound on the capacity that is feasible with respect to the supply-demand constraints.
Similarly minimising the flow to provide a lower bound. These will most likely tend to be 100 in maximisation or 0 in minimisation, and since you have some demands, it leads to some small deviations from these: min UB is 94.21, max LB is 5.622 using this approach. i.e. certain amounts of flow must go through in a certain edge to ensure demands are met.
I am really not sure of the correctness of this, probably needs some more thought. As suggested by the different objective function value, this may well be wrong. What is slightly promising is that the bounds in the final formulation are not very drastic.Below is what I meant if you decide to try it out.
Cheers,
David# ...your code...
m.setObjective(
gp.quicksum([flow[i, j] for idx, (i, j) in enumerate(arcs)]),
GRB.MAXIMIZE,
)
m.Params.timelimit = 60.0
m.optimize()
for i, j in arcs:
capacity[i, j].UB = flow[i, j].X
m.update()
m.setObjective(
gp.quicksum([flow[i, j] for idx, (i, j) in enumerate(arcs)]),
GRB.MINIMIZE,
)
m.optimize()
for i, j in arcs:
capacity[i, j].LB = flow[i, j].X
# reset objective to original
m.setObjective(cost, GRB.MINIMIZE)
m.write("finalflow.lp")
m.optimize()0 -
Hi David,
I think the problem with this approach is that any minimized flow value with
0 < flow[i, j].X < 5
leads to a lower bound on the capacity
capacity[i, j].LB == 5
since capacity values between 0 and 5 are not allowed. This leads to a higher objective function value (with the original cost function) because there exist better solutions with more capacity arcs with the value 0. I tried only setting the lower bound with minimized flow values higher 5, but then again I do not find the optimal solution.
I also tried a discrete approach, but also this approach didn't lead to the optimum.
Kind regards
Amanda0 -
Hi Amanda,
Thanks for double-checking my logic. Indeed, you are right.
It would be good to (somehow) find out which of these flow variables have to be positive in order to satisfy demand, but since you also have variable supply, this is pretty hard.I would've thought a discrete approach works much better. With capacity for example being \([0, 5, 10, \dots, 100]\), so you can change capacity to be an integer variable \([0, 1,\dots,20]\) and multiply it by 5. (Also removing the constraints to enforce that it must be between 0 and 5).
Another approach would be solving a relaxed version see model.feasRelax() to initialise capacity values and iteratively (and maybe heuristically?) come up with increasing capacity values until you hit a feasible solution.There are a few more avenues you can investigate:
- Parameter tuning for this model (I have tried grbtune to no avail).
- Finding a good heuristic to warm-start the solver.
- See if you can come up with some valid inequalities, or a reformulation altogether.
Cheers,
David0 -
Additionally, as this seems pretty tricky, you may want to consider posting your question to https://or.stackexchange.com/ to have a few more eyes on this.
Cheers,
David0 -
I would've thought a discrete approach works much better. With capacity for example being , so you can change capacity to be an integer variable and multiply it by 5. (Also removing the constraints to enforce that it must be between 0 and 5).
I have just tried this, even if you restrict the capacity to \([0,5]\) it is still super hard to solve.
I think you have to down the reformulation or heuristics way, unfortunately.
Cheers,
David0
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