When do we use linear programming technique? Explain mathematical formulation of Linear programming Problem (LPP).

Linear programming is a mathematical technique used when we want to find the best outcome in a mathematical model whose requirements are represented by linear relationships. It is used in various fields such as economics, business, engineering, and military applications.

The mathematical formulation of a Linear Programming Problem (LPP) involves the following steps:

1. Decision Variables: These are the variables that decide the output. For example, in a production problem, the quantities of different products to be produced are decision variables.

2. Objective Function: This is the function that needs to be optimized (maximized or minimized). For example, in a production problem, the profit function which needs to be maximized is the objective function.

3. Constraints: These are the restrictions or limitations on the decision variables. They form a system of linear inequalities or equations. For example, in a production problem, the limitations on the availability of resources like labor, material etc. are the constraints.

Mathematically, a linear programming problem can be represented as:

Maximize (or Minimize) Z = c1x1 + c2x2 + ... + cnxn

Subject to:

a11x1 + a12x2 + ... + a1nxn ≤ b1

a21x1 + a22x2 + ... + a2nxn ≤ b2

...

am1x1 + am2x2 + ... + amnxn ≤ bm

and x1, x2, ..., xn ≥ 0

Where:

• Z is the objective function
• x1, x2, ..., xn are the decision variables
• c1, c2, ..., cn are the coefficients of the objective function
• a11, a12, ..., amn are the coefficients of the constraints
• b1, b2, ..., bm are the right-hand side values of the constraints.