Pyomo for Budget Optimization

Pyomo

Pyomo is a Python-based open-source optimization modeling language that enables users to define and solve mathematical optimization problems. It provides a flexible and expressive way to formulate optimization models using Python syntax.

Key Features

Modeling Language: Pyomo allows users to define variables, objectives, and constraints using a high-level, declarative syntax.
Solver Integration: Pyomo can interface with a variety of optimization solvers, both open-source and commercial, to solve the defined models.
Extensibility: Pyomo is designed to be extensible, allowing users to create custom components and integrate with other Python libraries.

Solvers

Pyomo supports a wide range of solvers, including:

GLPK (GNU Linear Programming Kit) - Open-source solver for linear programming (LP) and mixed-integer programming (MIP). Based on simplex and interior-point methods.
IPOPT - Open-source solver for large-scale nonlinear programming (NLP) problems. Based on interior-point methods.

Example usage

Question (Modeling & Formulation)

A small workshop produces two products, Product X and Product Y. Each unit of X yields a profit of $40, and each unit of Y yields $30. At most 40 units of X can be sold. Producing one unit of X or Y requires 1 hour of Labor A. A total of 80 hours of labor A are available. X requires 2 hours and Y requires 1 hour of Labor B. A total of 100 hours of labor B are available. Formulate a linear program to determine the production quantities that maximize profit.

Decision Variables: X, Y
Objective: $40x + 30y = profit$
Constraints
- $x \leq 40$
- $x + y \leq 80$
- $2x + y \leq 100$

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import pyomo.environ as pyo
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model = pyo.ConcreteModel()
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# define decision vars
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model.x = pyo.Var(domain=pyo.NonNegativeReals)
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model.y = pyo.Var(domain=pyo.NonNegativeReals)
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# define objective function
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model.profit = pyo.Objective(expr = 40*model.x + 30*model.y, sense=pyo.maximize)
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# define constraints
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model.constr1 = pyo.Constraint(expr = model.x + model.y <= 80)
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model.constr2 = pyo.Constraint(expr = 2*model.x + model.y <= 100)
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# print model
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model.pprint()
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# Solve
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results = pyo.SolverFactory('glpk').solve(model) #glpk is for linear programming, based on simplex method
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# Print results
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print("profit= ", model.profit())
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model.display()

You can also use matrices to write the constraints.

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import pyomo.environ as pyo
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c = [40, 30]
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A = [[1,0],
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    [1,1],
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    [2,1]]
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b = (40,80,100)
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model = pyo.ConcreteModel()
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model.x = pyo.Var(range(2), domain=pyo.NonNegativeReals)
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model.profit = pyo.Objective(expr = sum(c[i]*model.x[i] for i in range(2)), sense=pyo.maximize)
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model.Constraint_list = pyo.ConstraintList()
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for i in range(3):
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    model.Constraint_list.add(expr = sum(A[i][j]*model.x[j] for j in range(2)) <= b[i])

Output:

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2 Var Declarations
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    x : Size=1, Index=None
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        Key  : Lower : Value : Upper : Fixed : Stale : Domain
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        None :     0 :  None :  None : False :  True : NonNegativeReals
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    y : Size=1, Index=None
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        Key  : Lower : Value : Upper : Fixed : Stale : Domain
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        None :     0 :  None :  None : False :  True : NonNegativeReals
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1 Objective Declarations
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    profit : Size=1, Index=None, Active=True
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        Key  : Active : Sense    : Expression
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        None :   True : maximize : 40*x + 30*y
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3 Constraint Declarations
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    constr1 : Size=1, Index=None, Active=True
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        Key  : Lower : Body  : Upper : Active
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        None :  -Inf : x + y : 100.0 :   True
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    constr2 : Size=1, Index=None, Active=True
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        Key  : Lower : Body    : Upper : Active
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        None :  -Inf : 2*x + y : 150.0 :   True
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    constr3 : Size=1, Index=None, Active=True
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        Key  : Lower : Body : Upper : Active
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        None :  -Inf :    x :  40.0 :   True
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6 Declarations: x y profit constr1 constr2 constr3
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profit=  3400.0
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Model unknown
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  Variables:
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    x : Size=1, Index=None
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        Key  : Lower : Value : Upper : Fixed : Stale : Domain
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        None :     0 :  40.0 :  None : False : False : NonNegativeReals
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    y : Size=1, Index=None
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        Key  : Lower : Value : Upper : Fixed : Stale : Domain
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        None :     0 :  60.0 :  None : False : False : NonNegativeReals
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  Objectives:
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    profit : Size=1, Index=None, Active=True
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        Key  : Active : Value
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        None :   True : 3400.0
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  Constraints:
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    constr1 : Size=1
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        Key  : Lower : Body  : Upper
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        None :  None : 100.0 : 100.0
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    constr2 : Size=1
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        Key  : Lower : Body  : Upper
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        None :  None : 140.0 : 150.0
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    constr3 : Size=1
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        Key  : Lower : Body : Upper
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        None :  None : 40.0 :  40.0