I have created a contrived example of my problem in an example here. It involves 2 water pipes which lead from a dam to a town. I want only 1 pipe to carry water over each timestamp, t.
I have created a linear optimisation program in Pyomo to determine how best to transfer water down the pipes, but it is failing to work based on one particular constraint - my constraint that only one pipe can transfer water at a time:
model.pipe_output1 = Var(model.T, domain = NonNegativeReals)
model.pipe_output2 = Var(model.T, domain = NonNegativeReals)
def bin_constraint1(model, t):
return model.pipe_output2[t] == model.pipe_output2[t] - model.pipe_output1[t]
model.bin_constraint1 = Constraint(model.T, rule = bin_constraint1)
The output is 0 for both pipes in every timestamp in this model, even though the logic makes complete sense to me. What should happen is for the water to be travelling down 1 pipe only and not the other in each timestamp.
I thought my constraint makes sense because both variables are set to be non negative reals, so if the RHS is negative, then it should be floored at 0 anyway. So it forces one variable to take a value, and the other to go to 0, i.e.:
Goal result:
pipe_output1 = [10,20,10,40,30]
pipe_output2 = [0,0,0,0,0]
What I'm getting:
pipe_output1 = [0,0,0,0,0]
pipe_output2 = [0,0,0,0,0]
As a test, I used a friend's Gurobi solver license, using the constraint below, and it worked!
def bin_constraint1(model, t):
return model.pipe_output2[t] * model.pipe_output1[t] == 0
model.bin_constraint1 = Constraint(model.T, rule = bin_constraint1)
I can't use this constraint as I don't personally have a Gurobi license. Is anyone able to help me to understand what I might be doing wrong - why does my logic fall through? Any workaround solutions?
The concept you are looking for here is how to enforce an either-or condition in an optimization model. So you tried 2 things...
Multiplying the variables together = 0. This works, but it is a non-linear operation, so you are giving up a lot by making your model non-linear, when this can easily be linearized
Your orig constraint. The algebra doesn't work. You essentially have:
a = a - b
which simplifies to b=0
One of the common ways to implement either-or in linear fashion is to use an indicator variable and big-M constraint. If these concepts are new, you may consider looking up some online tutorials or investing a few bucks in a used LP textbook.
This works, shows the use of big-M and switches flow based on price. Note the use of the boolean logic in C2 and C3
import pyomo.environ as pyo
### DATA
demand = {0: 15, 1: 20, 2:10}
M = max(demand.values()) # big-M value
p1_cost = {0: 2.0, 1: 1.0, 2: 5.0}
p2_cost = {0: 1.5, 1: 2.5, 2: 6.0}
### MODEL
m = pyo.ConcreteModel()
### SETS
m.T = pyo.Set(initialize=demand.keys())
### VARS
m.p1 = pyo.Var(m.T, domain=pyo.NonNegativeReals)
m.p2 = pyo.Var(m.T, domain=pyo.NonNegativeReals)
m.use_p1 = pyo.Var(m.T, domain=pyo.Binary) # 1 if p1 in use, 0 if not
### PARAMS
m.demand = pyo.Param(m.T, initialize=demand)
m.p1_cost = pyo.Param(m.T, initialize=p1_cost)
m.p2_cost = pyo.Param(m.T, initialize=p2_cost)
### OBJ : minimize cost
m.obj = pyo.Objective(expr=sum(m.p1[t] * m.p1_cost[t] + m.p2[t] * m.p2_cost[t] for t in m.T))
### CONSTRAINTS
# meet demand
def demand(m, t):
return m.p1[t] + m.p2[t] >= m.demand[t]
m.C1 = pyo.Constraint(m.T, rule=demand)
# only 1 pipe or other
def switch_1(m, t):
return m.p1[t] <= m.use_p1[t] * M
m.C2 = pyo.Constraint(m.T, rule=switch_1)
def switch_2(m, t):
return m.p2[t] <= (1 - m.use_p1[t]) * M
m.C3 = pyo.Constraint(m.T, rule=switch_2)
### CHECK IT
m.pprint()
### SOLVE IT
opt = pyo.SolverFactory('cbc')
res = opt.solve(m)
if pyo.check_optimal_termination(res):
m.display()
1 Set Declarations
T : Size=1, Index=None, Ordered=Insertion
Key : Dimen : Domain : Size : Members
None : 1 : Any : 3 : {0, 1, 2}
3 Param Declarations
demand : Size=3, Index=T, Domain=Any, Default=None, Mutable=False
Key : Value
0 : 15
1 : 20
2 : 10
p1_cost : Size=3, Index=T, Domain=Any, Default=None, Mutable=False
Key : Value
0 : 2.0
1 : 1.0
2 : 5.0
p2_cost : Size=3, Index=T, Domain=Any, Default=None, Mutable=False
Key : Value
0 : 1.5
1 : 2.5
2 : 6.0
3 Var Declarations
p1 : Size=3, Index=T
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : None : None : False : True : NonNegativeReals
1 : 0 : None : None : False : True : NonNegativeReals
2 : 0 : None : None : False : True : NonNegativeReals
p2 : Size=3, Index=T
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : None : None : False : True : NonNegativeReals
1 : 0 : None : None : False : True : NonNegativeReals
2 : 0 : None : None : False : True : NonNegativeReals
use_p1 : Size=3, Index=T
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : None : 1 : False : True : Binary
1 : 0 : None : 1 : False : True : Binary
2 : 0 : None : 1 : False : True : Binary
1 Objective Declarations
obj : Size=1, Index=None, Active=True
Key : Active : Sense : Expression
None : True : minimize : 2.0*p1[0] + 1.5*p2[0] + p1[1] + 2.5*p2[1] + 5.0*p1[2] + 6.0*p2[2]
3 Constraint Declarations
C1 : Size=3, Index=T, Active=True
Key : Lower : Body : Upper : Active
0 : 15.0 : p1[0] + p2[0] : +Inf : True
1 : 20.0 : p1[1] + p2[1] : +Inf : True
2 : 10.0 : p1[2] + p2[2] : +Inf : True
C2 : Size=3, Index=T, Active=True
Key : Lower : Body : Upper : Active
0 : -Inf : p1[0] - 20*use_p1[0] : 0.0 : True
1 : -Inf : p1[1] - 20*use_p1[1] : 0.0 : True
2 : -Inf : p1[2] - 20*use_p1[2] : 0.0 : True
C3 : Size=3, Index=T, Active=True
Key : Lower : Body : Upper : Active
0 : -Inf : p2[0] - (1 - use_p1[0])*20 : 0.0 : True
1 : -Inf : p2[1] - (1 - use_p1[1])*20 : 0.0 : True
2 : -Inf : p2[2] - (1 - use_p1[2])*20 : 0.0 : True
11 Declarations: T p1 p2 use_p1 demand p1_cost p2_cost obj C1 C2 C3
Model unknown
Variables:
p1 : Size=3, Index=T
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : 0.0 : None : False : False : NonNegativeReals
1 : 0 : 20.0 : None : False : False : NonNegativeReals
2 : 0 : 10.0 : None : False : False : NonNegativeReals
p2 : Size=3, Index=T
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : 15.0 : None : False : False : NonNegativeReals
1 : 0 : 0.0 : None : False : False : NonNegativeReals
2 : 0 : 0.0 : None : False : False : NonNegativeReals
use_p1 : Size=3, Index=T
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : 0.0 : 1 : False : False : Binary
1 : 0 : 1.0 : 1 : False : False : Binary
2 : 0 : 1.0 : 1 : False : False : Binary
Objectives:
obj : Size=1, Index=None, Active=True
Key : Active : Value
None : True : 92.5
Constraints:
C1 : Size=3
Key : Lower : Body : Upper
0 : 15.0 : 15.0 : None
1 : 20.0 : 20.0 : None
2 : 10.0 : 10.0 : None
C2 : Size=3
Key : Lower : Body : Upper
0 : None : 0.0 : 0.0
1 : None : 0.0 : 0.0
2 : None : -10.0 : 0.0
C3 : Size=3
Key : Lower : Body : Upper
0 : None : -5.0 : 0.0
1 : None : 0.0 : 0.0
2 : None : 0.0 : 0.0