facility.md

Facility location problem

We consider a capacitated facility location problem as an example for adjustable robust optimization. The goal is to decide on a number of facilities to build first and then determine from which facilities customers' demand should be met.

Nominal problem

# Define parameters
N = 4
M = 5
cost_facility = [100, 140, 80, 150]
cost_transport = [[2.0, 3.0, 1.5, 4.0, 3.3],
                  [1.5, 0.4, 1.0, 2.2, 3.5],
                  [3.4, 1.4, 2.1, 3.3, 1.6],
                  [2.6, 2.1, 2.0, 1.4, 1.3]]
demand = [45, 30, 60, 55, 25]
max_dem = [100, 140, 80, 150]

# Construct model
m = pe.ConcreteModel()
# Define variables
m.x = pe.Var(range(N), within=pe.Binary)
m.y = pe.Var(range(N), range(M), within=pe.NonNegativeReals)
# Add objective
expr = 0
for i in range(N):
    for j in range(M):
        expr += cost_transport[i][j]*m.y[i, j]
    expr += cost_facility[i]*m.x[i]
m.obj = pe.Objective(expr=expr, sense=pe.minimize)

# Add constraints
def sum_y_rule(m, j):
    return sum(m.y[i, j] for i in range(N)) == m.demand[j]
m.sum_y = pe.Constraint(range(M), rule=sum_y_rule)

def max_demand_rule(m, i):
    lhs = sum(m.y[i, j] for j in range(M))
    return lhs <= max_dem[i]*m.x[i]
m.max_dem = pe.Constraint(range(N), rule=max_demand_rule)

Robust problem

# Construct model
m = pe.ConcreteModel()
# Define variables
m.x = pe.Var(range(N), within=pe.Binary)
# Construct uncertainty set
m.uncset = ro.UncSet()
m.uncset.cons = pe.ConstraintList()
# Define uncertain parameter
m.demand = ro.UncParam(range(M), nominal=demand, uncset=m.uncset)
# Define adjustable variables
m.y = ro.AdjustableVar(range(N), range(M), bounds=(0, None), uncparams=[m.demand])
# Define box uncertainty set
for i in range(M):
    m.uncset.cons.add(expr=pe.inequality(0.9*demand[i], m.demand[i], 1.1*demand[i]))
# Add objective
expr = 0
for i in range(N):
    for j in range(M):
        expr += cost_transport[i][j]*m.y[i, j]
    expr += cost_facility[i]*m.x[i]
m.obj = pe.Objective(expr=expr, sense=pe.minimize)

# Add constraints
def sum_y_rule(m, j):
    return sum(m.y[i, j] for i in range(N)) == m.demand[j]
m.sum_y = pe.Constraint(range(M), rule=sum_y_rule)

def max_demand_rule(m, i):
    lhs = sum(m.y[i, j] for j in range(M))
    return lhs <= max_dem[i]*m.x[i]
m.max_dem = pe.Constraint(range(N), rule=max_demand_rule)

Running the example

The facility location problem is available as part of ROmodel and can be used as follows:

import romodel.examples as ex
import pyomo.environ as pe

m = ex.Facility()

solver = pe.SolverFactory('romodel.reformulation')
solver.solve(m)