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Arc capacity constraints
We begin with a straightforward set of constraints. The sum of the flow variables on an arc must be less than or equal to the capacity of that arc:
# Arc-capacity constraints
m.addConstrs(
(flow.sum('*', i, j) <= capacity[i, j] for i, j in arcs), "cap")
Note that this one statement uses several of the concepts that were introduced earlier in this section.
The first concept used here is the sum method on flow,
which is used to create a linear expression over a subset of the
variables in the tupledict. In particular, it is summing over
all commodities (the '*' in the first field) associated with an
edge between a pair of cities i and j.
The second concept used here is a generator expression, which iterates over all arcs in the network. Specifically, this portion of the statement...
for i,j in arcsindicates that we are iterating over every edge in
arcs. In
each iteration, i and j will be populated using the
corresponding values from a tuple in arcs. In a particular
iteration, flow.sum('*',i,j) will be computed using those
specific values, as will capacity[i,j].
The third thing to note is that we're passing the result as an
argument to addConstrs. This method will create a set of
Gurobi constraints, one for each iteration of the generator
expression.
The final thing to note is that the last argument gives the base for
the constraint name. The addConstrs method will automatically
append the corresponding indices for each constraint. Thus, for
example, the name of the constraint that limits flow from Denver to
Boston will be cap[Denver,Boston].
Note that if you prefer to do your own looping, you could obtain the equivalent behavior with the following loop:
for i,j in arcs:
m.addConstr(sum(flow[h,i,j] for h in commodities) <= capacity[i,j],
"cap[%s,%s]" % (i, j))
Next: Flow conservation constraints Up: Python Dictionary Example Previous: Building a multi-dimensional array
