Solving linear programming problems (LPPs) using Pyomo
In this tutorial, we learn how to solve linear programming problems (LPPs) using Pyomo. Pyomo stands for Python Optimization Modeling Objects. This is an algebraic modeling language.
Pyomo is a Python-based, open-source optimization modeling language with a diverse set of optimization capabilities. — http://www.pyomo.org/
In this tutorial, we consider an optimization problem and then we formulate it as a linear programming problem (LPP). After that, we learn how to write a Pyomo-based model using Python to solve the problem.
Consider a chocolate manufacturing company that produces only two types of chocolate — A and B. Both the chocolates require Milk and Choco only. To manufacture each unit of A and B, the following quantities are required:
- Each unit of A requires 1 unit of Milk and 3 units of Choco,
- Each unit of B requires 1 unit of Milk and 2 units of Choco.
The company kitchen has a total of 5 units of Milk and 12 units of Choco. On each sale, the company makes a profit of
- Rs. 6 per unit A sold,
- Rs. 5 per unit B sold.
Now, the company wishes to maximize its profit. How many units of A and B should it produce, respectively?
Now, we formulate the above optimization problem as an LPP. An LPP has mainly three components — decision variables, the objective function, and constraints.
- x be the total number of units produced by A,
- y be the total number of units produced by B.
Maximize 6x + 5y
x + y ≤ 5
3x + 2y ≤ 12
x ≥ 0, y ≥ 0
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We can use the following command to install Pyomo:
pip install pyomo
In this tutorial, we assume that Python3 and IBM ILOG CPLEX Optimization Studio or later are installed on your machine.
Now, we write down the Pyomo model for the example problem mentioned above.
Import required packages
from pyomo.environ import *
from pyomo.opt import SolverFactory
Create a model
model = ConcreteModel(name=”(My-Pyomo-Model)”)
Here, we have created a Concrete Pyomo model. Note that the argument name is optional.
Define decision variables of the model
model.x = Var(domain = NonNegativeReals, name = ‘x’)
model.y = Var(domain = NonNegativeReals, name = ‘y’)
Define the objective function of the model
model.objective = Objective(expr = 6*model.x + 5*model.y, sense = maximize)
Add constraints to the model
model.constraints = ConstraintList()model.constraints.add(model.x + model.y <= 5); # constraint-1
model.constraints.add(3*model.x + 2*model.y <= 12); # constraint-2
# to suppress unnecessary outputs, use a semicolon at the end
Note that we don’t need to include the non-negativity constraints (x≥0, y≥0) because, by default, these are already included in the definition of the decision variables.
Create a solver
solver = SolverFactory(“/opt/ibm/ILOG/CPLEX_Studio1271/cplex/bin/x86–64_linux/cplexamp”)
Here, we have used the CPLEX solver. We can use other solvers too. Note that we may need to specify the absolute solver path if it is not available as a system PATH variable.
Solve the model
sol = solver.solve(model)
Print the solution
The output of the above print statement is shown below:
- Lower bound: -inf
Upper bound: inf
Number of objectives: 1
Number of constraints: 2
Number of variables: 2
- Status: ok
Message: CPLEX 220.127.116.11\x3a optimal solution; objective 27; 2 dual simplex iterations (1 in phase I)
Termination condition: optimal
Error rc: 0
- number of solutions: 0
number of solutions displayed: 0
As we can see above, the problem has been solved to optimality and the objective function value is 27.
Retrieving decision variables’ values
print(‘x =’, model.x.value)
print(‘y =’, model.y.value)
The outputs of the above print statements are shown below:
x = 2.0000000000000004
y = 2.9999999999999996
Retrieve the objective value
print(‘The objective value is =’,model.objective())
The output of the above print statement is:
The objective value is = 27.0
Save the model in a file
The above command will create a file named ‘(My-Pyomo-Model)’.lp. The contents of this file are shown below:
\* Source Pyomo model name=‘(My-Pyomo-Model)’ *\max
ONE_VAR_CONSTANT = 1.0bounds
0 <= x1 <= +inf
0 <= x2 <= +inf
Display model information
This will display the following information:
x : Size=1, Index=None
Key : Lower : Value : Upper : Fixed : Stale : Domain
None : 0 : 2.0000000000000004 : None : False : False : NonNegativeReals
y : Size=1, Index=None
Key : Lower : Value : Upper : Fixed : Stale : Domain
None : 0 : 2.9999999999999996 : None : False : False : NonNegativeRealsObjectives:
objective : Size=1, Index=None, Active=True
Key : Active : Value
None : True : 27.0Constraints:
constraints : Size=2
Key : Lower : Body : Upper
1 : None : 5.0 : 5.0
2 : None : 12.0 : 12.0
This is the end of this tutorial. I have already written similar tutorials for solving linear programming problems (LPPs) using different algebraic modeling languages. You can find them from the links mentioned below:
Python API of DOcplex for solving linear programming problems
In this tutorial, we will learn how to write a model for linear programming problems (LPPs) using Python API and solve…
Solving Linear Programming Problems (LPPs) Using PuLP and Python
In this tutorial, we will learn how to solve linear programming problems (LPPs) using PuLP and Python. At first, we…
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