Transport

nimopt.models.transport ships from plants to warehouses over an incomplete network: a plant serves a band of nearby warehouses and not all of them. The cost parameter has one entry per arc, and the flow variable takes its members from that parameter. The model therefore has one column per arc, not one per cell of the plant-warehouse product.

minimize    Σ_{(p,w) ∈ arcs} cost[p,w] · flow[p,w]
subject to  Σ_w flow[p,w] ≤ supply[p]     for each plant p
            Σ_p flow[p,w] ≥ demand[w]     for each warehouse w
            flow[p,w] ≥ 0                 for each arc (p,w)

from nimopt.models import transport

print(transport.definition().explain())

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Output

transport  min  not built
  sets        P · W
  parameters  cost (P,W) · supply (P) · demand (W)
  variables   flow (P×W) over cost [0.0, inf]
  constraint  supply (P)  Sum(W, flow[P, W]) <= supply[P]
  constraint  demand (W)  Sum(P, flow[P, W]) >= demand[W]
  objective   min  Sum(P, W, cost[P, W] * flow[P, W])

Supply is twice the total demand of the band of a plant. No supply row binds therefore, and each warehouse buys from the cheapest plant connected to it. reference computes that sum.

from nimopt.models import transport

inputs = transport.data()
model = transport.definition().build(inputs)
solution = model.solve()
print(model.n_columns, "columns for", len(inputs["cost"][1]), "arcs")
print(solution.objective, transport.reference(inputs))

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Output

12 columns for 12 arcs
100.99601811246072 100.99601811246073

Arcs are drawn from every warehouse but the last. No plant serves the last warehouse, and it has no demand row. absent reports the row and the rule that dropped it.

from nimopt.models import transport

model = transport.definition().build(transport.data())
print(model.absent("demand"))

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Output

demand  5 of 6 rows  stated by terms
  row absent  W='w5'  term-does-not-reach (flow)