elpee 0.4.5

Solving linear programming problems step-by-step

Elpee

Solving linear programming problems one step at a time.

Elpee (short for L inear P rogramming) is a Python library for students and academics to viusalize the steps and iterations required to solve constrained linear optimization problems using linear programming methods.

Demo

View how linear programming problems can be solved using the elpee-calc.streamlit.app powered by the elpee library.

Documentation

Follow the documentation to get the best use of elpee.

Install

To use elpee Python library, install it using pip:

(.venv) $ pip install elpee

Example Problem

Define the Linear Programming Problem to be solved using the All Stack Starting Method by defining the Objective Function and the Constraint Expressions.

from elpee import LinearProblem

# set up a maximization problem
problem = LinearProblem(is_maximization=True)

# define the objective function
problem.add_objective('x + y')

# add the constraints to the problem
problem.add_constraint('-x + y <= 2')
problem.add_constraint('6*x + 4*y >= 24')
problem.add_constraint('y >= 1')

Add additional configurations to apply Big-M or Dual Simplex Method

problem.use_dual_simplex() # use to configure problem to use dual simplex method
problem.use_bigM() # use to configure problem to use big M method

Apply the Solver to solve the Linear Problem. The iterations to produce the results will be printed on the command line.

from elpee import ElpeeSolver

solution = ElpeeSolver.solve(problem)

Sample Output will be as follows (using dual_simplex).

...Generating Initial Feasible Solution for
     MIN           x            y            S1           S2           S3          Sol
      P           -1.0         -1.0          0            0            0            0
      S1          -1.0         1.0           1            0            0           2.0
      S2          -6.0         -4.0          0            1            0          -24.0
      S3           0           -1.0          0            0            1           -1.0
===========================================================================================

Taking S2 = 0; Entering x as a new basic variable;

...Generating Initial Feasible Solution for
     MIN           x            y            S1           S2           S3          Sol
      P           0.0         -0.333        0.0         -0.167        0.0          4.0
      S1          0.0         1.667         1.0         -0.167        0.0          6.0
      x           1.0         0.667         -0.0        -0.167        -0.0         4.0
      S3           0           -1.0          0            0            1           -1.0
===========================================================================================

Taking S3 = 0; Entering y as a new basic variable;

Feasible Solution # 1
     MIN           x            y            S1           S2           S3          Sol
      P           0.0          0.0          0.0         -0.167       -0.333       4.333
      S1          0.0          0.0          1.0         -0.167       1.667        4.333
      x           1.0          0.0          0.0         -0.167       0.667        3.333
      y           -0.0         1.0          -0.0         -0.0         -1.0         1.0
===========================================================================================

Optimized Solution Received!

And obtain an interpretation to the solution

Minimum Value for Objective Function = 4.333

Values for Decision Variables :
      x       = 3.333
      y       = 1.0

Surplus & Slack variables
  Constraint #1 Surplus    = 4.333 units
  Constraint #2 Surplus    : Satisfied at Boundary
  Constraint #3 Surplus    : Satisfied at Boundary