def kantorovich():
"""
Simple implementation of the compact formulation from Kantorovich for the problem
"""
N = 10 # maximum number of bars
L = 250 # bar length
m = 4 # number of requests
w = [187, 119, 74, 90] # size of each item
b = [1, 2, 2, 1] # demand for each item
# creating the model (note that the linear relaxation is solved)
model = Model(SOLVER)
x = {(i, j): model.add_var(obj=0, var_type=CONTINUOUS, name="x[%d,%d]" % (i, j)) for i in range(m) for j in range(N)}
y = {j: model.add_var(obj=1, var_type=CONTINUOUS, name="y[%d]" % j) for j in range(N)}
# constraints
for i in range(m):
model.add_constr(xsum(x[i, j] for j in range(N)) >= b[i])
for j in range(N):
model.add_constr(xsum(w[i] * x[i, j] for i in range(m)) <= L * y[j])
# additional constraint to reduce symmetry
for j in range(1, N):
model.add_constr(y[j - 1] >= y[j])
# optimizing the model and printing solution
model.optimize()
print_solution(model)
def solve_tsp_by_mip(tsp_matrix):
start = time()
matrix_of_distances = get_matrix_of_distances(tsp_matrix)
length = len(tsp_matrix)
model = Model(solver_name='gurobi')
model.verbose = 1
x = [[model.add_var(var_type=BINARY) for j in range(length)]
for i in range(length)]
y = [model.add_var() for i in range(length)]
model.objective = xsum(matrix_of_distances[i][j] * x[i][j]
for j in range(length) for i in range(length))
for i in range(length):
model += xsum(x[j][i] for j in range(length) if j != i) == 1
model += xsum(x[i][j] for j in range(length) if j != i) == 1
for i in range(1, length):
for j in [x for x in range(1, length) if x != i]:
model += y[i] - (length + 1) * x[i][j] >= y[j] - length
model.optimize(max_seconds=300)
arcs = [(i, j) for i in range(length) for j in range(length)
if x[i][j].x >= 0.99]
best_distance = calculate_total_dist_by_arcs(matrix_of_distances, arcs)
time_diff = time() - start
return arcs, time_diff, best_distance
def test_queens(solver: str):
"""MIP model n-queens"""
n = 50
announce_test("n-Queens", solver)
queens = Model('queens', MAXIMIZE, solver_name=solver)
queens.verbose = 0
x = [[
queens.add_var('x({},{})'.format(i, j), var_type=BINARY)
for j in range(n)
] for i in range(n)]
# one per row
for i in range(n):
queens += xsum(x[i][j] for j in range(n)) == 1, 'row({})'.format(i)
# one per column
for j in range(n):
queens += xsum(x[i][j] for i in range(n)) == 1, 'col({})'.format(j)
# diagonal \
for p, k in enumerate(range(2 - n, n - 2 + 1)):
queens += xsum(x[i][j] for i in range(n) for j in range(n)
if i - j == k) <= 1, 'diag1({})'.format(p)
# diagonal /
for p, k in enumerate(range(3, n + n)):
queens += xsum(x[i][j] for i in range(n) for j in range(n)
if i + j == k) <= 1, 'diag2({})'.format(p)
queens.optimize()
check_result("model status", queens.status == OptimizationStatus.OPTIMAL)
# querying problem variables and checking opt results
total_queens = 0
for v in queens.vars:
# basic integrality test
assert v.x <= 0.0001 or v.x >= 0.9999
total_queens += v.x
# solution feasibility
rows_with_queens = 0
for i in range(n):
if abs(sum(x[i][j].x for j in range(n)) - 1) <= 0.001:
rows_with_queens += 1
check_result("feasible solution",
abs(total_queens - n) <= 0.001 and rows_with_queens == n)
print('')
def solve_tsp_by_mip_with_sub_cycles_2(tsp_matrix):
start = time()
matrix_of_distances = get_matrix_of_distances(tsp_matrix)
total_length = len(tsp_matrix)
best_distance = sys.float_info.max
found_cycles = []
arcs = [(i, i + 1) for i in range(total_length - 1)]
iteration = 0
model = Model(solver_name='gurobi')
model.verbose = 0
x = [[model.add_var(var_type=BINARY) for j in range(total_length)]
for i in range(total_length)]
y = [model.add_var() for i in range(total_length)]
model.objective = xsum(matrix_of_distances[i][j] * x[i][j]
for j in range(total_length)
for i in range(total_length))
for i in range(total_length):
model += (xsum(x[i][j] for j in range(0, i)) +
xsum(x[j][i] for j in range(i + 1, total_length))) == 2
while len(found_cycles) != 1:
model.optimize(max_seconds=300)
arcs = [(i, j) for i in range(total_length)
for j in range(total_length) if x[i][j].x >= 0.99]
best_distance = calculate_total_dist_by_arcs(matrix_of_distances, arcs)
found_cycles = get_cycle(arcs)
for cycle in found_cycles:
points = {}
for arc in cycle:
points = {*points, arc[0]}
points = {*points, arc[1]}
cycle_len = len(cycle)
model += xsum(x[arc[0]][arc[1]]
for arc in permutations(points, 2)) <= cycle_len - 1
# plot_connected_tsp_points_from_arcs(tsp_matrix, arcs, '../images/mip_xql662/{}'.format(iteration))
print(iteration)
iteration += 1
time_diff = time() - start
return arcs, time_diff, best_distance
def test_tsp_mipstart(solver: str):
"""tsp related tests"""
announce_test("TSP - MIPStart", solver)
N = ['a', 'b', 'c', 'd', 'e', 'f', 'g']
n = len(N)
i0 = N[0]
A = {
('a', 'd'): 56,
('d', 'a'): 67,
('a', 'b'): 49,
('b', 'a'): 50,
('d', 'b'): 39,
('b', 'd'): 37,
('c', 'f'): 35,
('f', 'c'): 35,
('g', 'b'): 35,
('b', 'g'): 25,
('a', 'c'): 80,
('c', 'a'): 99,
('e', 'f'): 20,
('f', 'e'): 20,
('g', 'e'): 38,
('e', 'g'): 49,
('g', 'f'): 37,
('f', 'g'): 32,
('b', 'e'): 21,
('e', 'b'): 30,
('a', 'g'): 47,
('g', 'a'): 68,
('d', 'c'): 37,
('c', 'd'): 52,
('d', 'e'): 15,
('e', 'd'): 20
}
# input and output arcs per node
Aout = {n: [a for a in A if a[0] == n] for n in N}
Ain = {n: [a for a in A if a[1] == n] for n in N}
m = Model(solver_name=solver)
m.verbose = 0
x = {
a: m.add_var(name='x({},{})'.format(a[0], a[1]), var_type=BINARY)
for a in A
}
m.objective = xsum(c * x[a] for a, c in A.items())
for i in N:
m += xsum(x[a] for a in Aout[i]) == 1, 'out({})'.format(i)
m += xsum(x[a] for a in Ain[i]) == 1, 'in({})'.format(i)
# continuous variable to prevent subtours: each
# city will have a different "identifier" in the planned route
y = {i: m.add_var(name='y({})'.format(i), lb=0.0) for i in N}
# subtour elimination
for (i, j) in A:
if i0 not in [i, j]:
m.add_constr(y[i] - (n + 1) * x[(i, j)] >= y[j] - n)
route = ['a', 'g', 'f', 'c', 'd', 'e', 'b', 'a']
m.start = [(x[route[i - 1], route[i]], 1.0) for i in range(1, len(route))]
m.optimize()
check_result("mip model status", m.status == OptimizationStatus.OPTIMAL)
check_result("mip model objective",
(abs(m.objective_value - 262)) <= 0.0001)
print('')
# constraint : enter each city coming from another city
for i in range(n):
model += sum(x[j][i] for j in range(n)
if j != i) == 1, 'enter({})'.format(i)
# constraint : leave each city coming from another city
for i in range(n):
model += sum(x[i][j] for j in range(n)
if j != i) == 1, 'leave({})'.format(i)
# subtour elimination
for i in range(0, n):
for j in range(0, n):
if i == j or i == 0 or j == 0:
continue
model += \
y[i] - (n+1)*x[i][j] >= y[j] -n, 'noSub({},{})'.format(i,j)
model.optimize(maxSeconds=10)
#model.write('tsp.lp')
print('best route found has length {}'.format(model.get_objective_value()))
for i in range(n):
for j in range(n):
if x[i][j].x >= 0.98:
print('arc ({},{})'.format(i, j))
print('finished')
# constraint : leave each city coming from another city
for i in N:
model += xsum(x[a] for a in IN[i]) == 1
# no subtours of size 2
for a in A:
if (a[1], a[0]) in A.keys():
model += x[a] + x[a[1], a[0]] <= 1
# computing farthest point for each point
F = []
G = nx.DiGraph()
for ((i, j), d) in A.items():
G.add_edge(i, j, weight=d)
for i in N:
P, D = nx.dijkstra_predecessor_and_distance(G, source=i)
DS = list(D.items())
DS.sort(key=lambda x: x[1])
F.append((i, DS[-1][0]))
model.lazy_constrs_generator = SubTourCutGenerator(F)
model.optimize()
print(model.status)
print('best route found has length {}'.format(model.objective_value))
arcs = [a for a in A.keys() if x[a].x >= 0.99]
print('optimal route : {}'.format(arcs))
def cg():
"""
Simple column generation implementation for a Cutting Stock Problem
"""
L = 250 # bar length
m = 4 # number of requests
w = [187, 119, 74, 90] # size of each item
b = [1, 2, 2, 1] # demand for each item
# creating models and auxiliary lists
master = Model()
lambdas = []
constraints = []
# creating an initial pattern (which cut one item per bar)
# to provide the restricted master problem with a feasible solution
for i in range(m):
lambdas.append(master.add_var(obj=1, name='lambda_%d' %
(len(lambdas) + 1)))
# creating constraints
for i in range(m):
constraints.append(master.add_constr(lambdas[i] >= b[i],
name='i_%d' % (i + 1)))
# creating the pricing problem
pricing = Model(SOLVER)
# creating pricing variables
a = []
for i in range(m):
a.append(pricing.add_var(obj=0, var_type=INTEGER, name='a_%d' % (i + 1)))
# creating pricing constraint
pricing.add_constr(xsum(w[i] * a[i] for i in range(m)) <= L, 'bar_length')
pricing.write('pricing.lp')
new_vars = True
while new_vars:
##########
# STEP 1: solving restricted master problem
##########
master.optimize()
# printing dual values
print_solution(master)
print('pi = ', end='')
print([constraints[i].pi for i in range(m)])
print('')
##########
# STEP 2: updating pricing objective with dual values from master
##########
pricing.objective = 1
for i in range(m):
a[i].obj = -constraints[i].pi
# solving pricing problem
pricing.optimize()
# printing pricing solution
z_val = pricing.objective_value
print('Pricing:')
print(' z = {z_val}'.format(**locals()))
print(' a = ', end='')
print([v.x for v in pricing.vars])
print('')
##########
# STEP 3: adding the new columns
##########
# checking if columns with negative reduced cost were produced and
# adding them into the restricted master problem
if 1 + pricing.objective_value < - EPS:
coeffs = [a[i].x for i in range(m)]
column = Column(constraints, coeffs)
lambdas.append(master.add_var(obj=1, column=column, name='lambda_%d' % (len(lambdas) + 1)))
print('new pattern = {coeffs}'.format(**locals()))
# if no column with negative reduced cost was produced, then linear
# relaxation of the restricted master problem is solved
else:
new_vars = False
pricing.write('pricing.lp')
print_solution(master)
# constraint : enter each city coming from another city
for i in N:
model += xsum(x[a] for a in OUT[i]) == 1
# constraint : leave each city coming from another city
for i in N:
model += xsum(x[a] for a in IN[i]) == 1
# subtour elimination
for (i, j) in [a for a in A.keys() if n0 not in [a[0], a[1]]]:
model += \
y[i] - (n+1)*x[(i, j)] >= y[j]-n, 'noSub({},{})'.format(i, j)
print('model has {} variables, {} of which are integral and {} rows'.format(
model.num_cols, model.num_int, model.num_rows))
print("Adding SOSs")
for i in N:
sosOut = [(x[(i, j)], A[(i, j)]) for (i, j) in OUT[i]]
sosIn = [(x[(i, j)], A[(i, j)]) for (i, j) in IN[i]]
model.add_sos(sosOut, 1)
model.max_nodes = 1000
st = model.optimize(max_seconds=120)
print('best route found has length {}, best possible (obj bound is) {} st: {}'.
format(model.objective_value, model.objective_bound, st))
arcs = [(a) for a in A.keys() if x[a].x >= 0.99]
print('optimal route : {}'.format(arcs))
if cut.violation > 0.001:
model.add_cut(cut)
# number of queens
n = 60
queens = Model('queens', MAXIMIZE)
x = [[
queens.add_var('x({},{})'.format(i, j), var_type=BINARY) for j in range(n)
] for i in range(n)]
# one per row
for i in range(n):
queens += xsum(x[i][j] for j in range(n)) == 1, 'row({})'.format(i)
# one per column
for j in range(n):
queens += xsum(x[i][j] for i in range(n)) == 1, 'col({})'.format(j)
queens.cuts_generator = DiagonalCutGenerator()
queens.cuts_generator.lazy_constraints = True
queens.optimize()
stdout.write('\n')
for i, v in enumerate(queens.vars):
stdout.write('O ' if v.x >= 0.99 else '. ')
if i % n == n - 1:
stdout.write('\n')
"""0/1 Knapsack example"""
from mip.model import Model, xsum, maximize
from mip.constants import BINARY
p = [10, 13, 18, 31, 7, 15]
w = [11, 15, 20, 35, 10, 33]
c = 47
n = len(w)
m = Model('knapsack')
x = [m.add_var(var_type=BINARY) for i in range(n)]
m.objective = maximize(xsum(p[i] * x[i] for i in range(n)))
m += xsum(w[i] * x[i] for i in range(n)) <= c
m.optimize()
selected = [i for i in range(n) if x[i].x >= 0.99]
print('selected items: {}'.format(selected))
class UnitCommitment:
def __init__(self, generators, demand):
self.generators = generators
self.demand = demand
self.period = range(1, len(self.demand) + 1)
self.model = Model(name='UnitCommitment')
self.p, self.u = {}, {}
def build_model(self, fixed=False, u_fixed=None):
for t in self.period:
for g in self.generators.index:
if fixed:
self.u[t, g] = self.model.add_var(lb=u_fixed[t, g],
ub=u_fixed[t, g])
else:
self.u[t, g] = self.model.add_var(var_type=BINARY)
self.p[t, g] = self.model.add_var()
# Max/min power
for t in self.period:
for g in self.generators.index:
self.model.add_constr(
self.p[t, g] <=
self.generators.loc[g, 'p_max'] * self.u[t, g],
name=f'pmax_constr[{t},{g}]')
self.model.add_constr(
self.p[t, g] >=
self.generators.loc[g, 'p_min'] * self.u[t, g],
name=f'pmin_constr[{t},{g}]')
# Power balance
for t in self.period:
self.model.add_constr(xsum(
self.p[t, g]
for g in self.generators.index) == self.demand.loc[t,
'demand'],
name=f'power_bal_constr[{t}]')
# Min on
for t in self.period[1:]:
for g in self.generators.index:
min_on_time = min(t + self.generators.loc[g, 'min_on'] - 1,
len(self.period))
for tau in range(t, min_on_time + 1):
self.model.add_constr(
self.u[tau, g] >= self.u[t, g] - self.u[t - 1, g],
name=f'min_on_constr[{t},{g}]')
# Min off
for t in self.period[1:]:
for g in self.generators.index:
min_off_time = min(t + self.generators.loc[g, 'min_off'] - 1,
len(self.period))
for tau in range(t, min_off_time + 1):
self.model.add_constr(
1 - self.u[tau, g] >= self.u[t - 1, g] - self.u[t, g],
name=f'min_off_constr[{t},{g}]')
# Objective function
self.model.objective = minimize(
xsum(
xsum(self.p[t, g] * self.generators.loc[g, 'c_var'] +
self.u[t, g] * self.generators.loc[g, 'c_fix']
for g in self.generators.index) for t in self.period))
def optimize(self):
self.model.optimize()
return self.u, self.p, self.model.objective.x, self.model.status.name
def get_prices(self):
u_fixed = {(t, g): self.u[t, g].x
for g in self.generators.index for t in self.period}
self.model.clear()
self.build_model(fixed=True, u_fixed=u_fixed)
self.optimize()
return [
self.model.constr_by_name(f'power_bal_constr[{t}]').pi
for t in self.period
]
C, U = S.C, [i for i in range(S.u_max + 1)]
m = Model(sense=MINIMIZE)
x = [[m.add_var('x({},{})'.format(i, c), var_type=BINARY) for c in U]
for i in N]
z = m.add_var('z')
m.objective = minimize(z)
for i in N:
m += xsum(x[i][c] for c in U) == r[i]
for i, j, c1, c2 in product(N, N, U, U):
if i != j and c1 <= c2 < c1 + d[i][j]:
m += x[i][c1] + x[j][c2] <= 1
for i, c1, c2 in product(N, U, U):
if c1 < c2 < c1 + d[i][i]:
m += x[i][c1] + x[i][c2] <= 1
for i, c in product(N, U):
m += z >= (c + 1) * x[i][c]
m.start = [(x[i][c], 1.0) for i in N for c in C[i]]
m.optimize(max_seconds=100)
C = [[c for c in U if x[i][c] >= 0.99] for i in N]
print(C)
