Linear Programming in Python

What is Linear Programming?

Linear programming widely known as linear optimization is a popular mathematical programming technique that is used to obtain the best outcome which can either maximize or minimize a linear objective function subject (such as maximum profit or minimum cost) in a mathematical model for a set of linear equality and/or inequality constraints.

Also, one connection related here is that linear programming in Python always comes in pairs:

• an original primal problem, and
• an associated dual problem.

When the primal problem involves more maximization, then the dual problem will eventually involve minimization. When the primal problem involves more minimization, then the dual problem will eventually involve maximization. A standard form of linear programming in Python and associated methods are applied to transform all the other forms of linear programming in Python problems as a standard form.

Uses of Linear Programming

The uses of Linear programming in Python can be defined as when an organization implements linear programming it considers two main where the first is minimization and the other is maximization. Minimization concerning linear programming in Python means to minimize the total cost of production whereas Maximization on the other hand means to maximize the company or organization’s profit. Hence, linear programming in Python with its graphical method helps to find the optimum solution.

Other areas where linear programming in Python can be implemented are:

Food and agriculture industry

Farmers are implementing such linear programming techniques to understand know what crops can be grown on the land and follow better farming practices.

Transportation optimization

The transportation industry works on linear programming in Python for cost and time efficiency. It helps them run and optimize the buses and train routes.

Manufacturing units

The manufacturing industry implements linear programming as the principal optimization tool to generate more profit for the organization.

Basic Terminologies of Linear Programming

Listed below are the basic terminologies of Linear Programming in Python that one must be aware of:

Objective Function: The objective function in linear programming in Python can be defined either to maximize or minimize which is the main aim of the problems. As seen in below sample equation, we want to minimize the value of 'C' where 'C' is the objective function.

Decision Variables: We can describe the decision variables as the variables that are used for deciding the output. These decision variables are the unknowns of the mathematical programming model. As the equation below, the variables a and b (known as the decision variables) for which we determine the value for minimizing the value of C.

Constraints: The constraints are the restrictions on the decision variables. As for the equation, $C = 3b + 5a$, the limitations that we have on the decision variables are the constraints of Linear programming in Python.

Non–negativity restrictions: While working in linear programming in Python, the values for the decision variables are always calculated as greater than or equal to 0.

Feasible solution: When a solution meets all the constraints for the decision variables it's represented as a feasible solution. At least one feasible solution is possible.

Infeasible solution: When a solution when all constraints are not satisfied or no optimal solution is found.

Linear Programming in Python

While working with linear programming in Python, the basic method for solving is the simplex method, which consists of several variants. While you can slowly implement the interior-point method.

The Simplex Method can be defined as an approach in which you manually use slack variables, tableaus, and pivot variables as a medium for finding the optimal solution to an optimization problem.

The Interior Point is described as the method where the approximation of the constraints is considered as a set of boundaries covering a defined region. The approximations are then utilized for problems consisting of constraints, which are discontinuous or troublesome but have the possibility of modifying them so that they can be handled.

Several well-known Python tools used for solving linear programming in Python are libraries like:

• SciPy Optimization and Root Finding.
• PuLP.
• Pyomo.
• CVXOPT.

Examples for Understanding

Case 1: Production Example for Linear programming in Python.

Problem statemnet:

A factory produces two goods namely 'S' and 'T'. Both labor and material are required for producing the products 'S' and 'T'. When each product is sold, revenue is generated. Per unit material and labor is as below:

Code:

Output:

Explanation:

As seen above, the optimal plan comes out as 2.5, 5 which tells the organization for producing 2.5 units of Product S and 5 units of Product T. The maximized revenue that could be generated is 27.5. Here, the linprog function is a black box where Python first transforms the problem as a standard form. It evaluates each inequality constraint it generates using one slack variable.

Case 2: Investment Example for Linear programming in Python.

Problem Statement:

Code:

Output:

Explanation:

As can be seen above, with the help of linear programming in Python, we get the best investment strategy. It states that the mutual fund should buy $24927 of the annuity at the beginning of the first year. And the bank account balance then shall be $75072. Then in the second year, the mutual should invest in the corporate bond of $50,000 and keep investing in an annuity where its balance in the bank account for $4658.82. In the third year, at the very beginning, the MF must borrow $20,000 and invest in the annuity. Finally, at the end of the third year, the mutual fund can then get payouts from both the annuity and corporate bond and accordingly repay its loan as well. The mutual fund would own 141018.24, which makes its total net rate for three years 41.02%.