## Linear Programming Problem

The problem of formulating an objective function and constraints an establishing relationship between variables is called a programming problem (LPP).

If objective function and constraints are linear then the problem is called linear programming problem.

Example-

Sometimes, you come across the situation when you need to optimize your algorithm. If you can formulate your problem as a linear programming problem

then you can use linprog() function of scipy.optimize library in Python and can get optimize solution. scipy.optimize library of Python provides a variety of functions to solve optimization problems.

The simple problem which I have mentioned above I will solve using linprog() function from scipy.optimize library.

**LPP Solution in Python**

**from scipy.optimize import linprog**

**c = [4, 2] # The coefficients of x and y in objective function**

**A = [[1, -3], [-4, -1]] # The coefficients of contraints in term of a matrix**

**b = [-6, -8] # Bounds on constraints**

**x_bounds = (0, None)**

**y_bounds = (0, None)**

**res = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds])**

**print(res)**

Output:

fun: 10.461538461538463

message: ‘Optimization terminated successfully.’

nit: 2

slack: array([0., 0.])

status: 0

success: True

x: array([1.38461538, 2.46153846])

Which means for x=1.38461538 and y= 2.46153846 the objective function z will be maximum which is

z=10.461538461538463

## References

- Optimization and Root Finding, Web Link, https://docs.scipy.org/doc/scipy/reference/optimize.html
- Zoutendijk, G., 1960.
*Methods of feasible directions: a study in linear and non-linear programming*. Elsevier. - https://en.wikipedia.org/wiki/Linear_programming
- Campbell, H.G., 1965.
*An introduction to matrices, vectors, and linear programming*. Appleton-Century-Crofts.

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