def build_knapsack_cqm(costs, weights, max_weight):
"""Construct a CQM for the knapsack problem.
Args:
costs (array-like):
Array of costs for the items.
weights (array-like):
Array of weights for the items.
max_weight (int):
Maximum allowable weight for the knapsack.
Returns:
Constrained quadratic model instance that represents the knapsack problem.
"""
num_items = len(costs)
print("\nBuilding a CQM for {} items.".format(str(num_items)))
cqm = ConstrainedQuadraticModel()
obj = BinaryQuadraticModel(vartype='BINARY')
constraint = QuadraticModel()
for i in range(num_items):
# Objective is to maximize the total costs
obj.add_variable(i)
obj.set_linear(i, -costs[i])
# Constraint is to keep the sum of items' weights under or equal capacity
constraint.add_variable('BINARY', i)
constraint.set_linear(i, weights[i])
cqm.set_objective(obj)
cqm.add_constraint(constraint, sense="<=", rhs=max_weight, label='capacity')
return cqm
def build_cqm(W, C, n, p, a, verbose=True):
"""Builds constrained quadratic model representing the optimization problem.
Args:
- W: Numpy matrix. Represents passenger demand. Normalized with total demand equal to 1.
- C: Numpy matrix. Represents airline leg cost.
- n: Int. Number of cities in play.
- p: Int. Number of hubs airports allowed.
- a: Float in [0.0, 1.0]. Discount allowed for hub-hub legs.
- verbose: Print to command-line for user.
Returns:
- cqm: ConstrainedQuadraticModel representing the optimization problem.
"""
if verbose:
print("\nBuilding CQM...\n")
# Initialize the CQM object
cqm = ConstrainedQuadraticModel()
# Objective: Minimize cost. min c'x+x'Qx
# See reference paper for full explanation.
M = np.sum(W, axis=0) + np.sum(W, axis=1)
Q = a * np.kron(W, C)
linear = ((M * C.T).T).flatten()
obj = BinaryQuadraticModel(linear, Q, 'BINARY')
obj.relabel_variables({
idx: (i, j)
for idx, (i, j) in enumerate(
(i, j) for i in range(n) for j in range(n))
})
cqm.set_objective(obj)
# Add constraint to make variables discrete
for v in range(n):
cqm.add_discrete([(v, i) for i in range(n)])
# Constraint: Every leg must connect to a hub.
for i in range(n):
for j in range(n):
if i != j:
c1 = BinaryQuadraticModel('BINARY')
c1.add_linear((i, j), 1)
c1.add_quadratic((i, j), (j, j), -1)
cqm.add_constraint(c1 == 0)
# Constraint: Exactly p hubs required.
linear_terms = {(i, i): 1.0 for i in range(n)}
c2 = BinaryQuadraticModel('BINARY')
c2.add_linear_from(linear_terms)
cqm.add_constraint(c2 == p, label='num hubs')
return cqm