"Every real-world problem has limitations—budget, time, workforce, or materials. LPP considers these constraints to find an optimal solution." "Businesses use LPP to make crucial decisions, like how to distribute resources in production or logistics."

1. Linear Programming Problem
(LPP)
Presented by
Dr. S. Anitha
Assistant Professor
Department of S&H
SRM MCET

2. Outline of the Presentation
• What is Linear Programming?
• How do we formulate an LPP?

3. LPP
• LPP is a systematic approach to optimization.
• In mathematics, linear programming problem is also known
as optimization problem.
• It helps in finding the best possible outcome whether it’s
maximizing profit or minmizing cost or optimizing resource
allocation.

4. LPP has three major components:
• Decision Variables → Represent quantities to be determined (e.g.,
number of products to manufacture).
• Objective Function → The function to be maximized or minimized
(e.g., profit, cost).
• Constraints → Limitations on resources like materials, time, labor, or
budget.

5. Formulation of an LP Problem:
• Identify decision variables and assign symbols (e.g., X, Y, Z). These
are the quantities we want to find out.
• Express all constraints as inequalities based on decision variable.
• Define an objective function (goal) to maximize profit or minimize
cost
• Add the non-negative condition/constraint.

6. Example
A house wife wishes to mix two types of food F1 and F2 in such a way
that the vitamin contents of the mixture contain at least 8 units of
vitamin A and 11 units of vitamin B. Food F1 costs Rs.60/Kg and Food
F2 costs Rs.80/kg. Food F1 contains 3 units/kg of vitamin A and 5
units/kg of vitamin B while food F2 contains 4 units/kg of vitamin A
and 2 units/kg of vitamin B. Formulate this problem as a linear
programming problem to minimize the cost of the mixtures,
S o u r c e : h t t p : / / w w w. ma t h s . u n p . a c . z a / c o u r s e w o r k / ma t h 3 3 1 / 2 0 1 2 / l i n e a r p r o g r a mmi n g . p d f

7. Let’s follow the steps given to formulate and LP:
1. The kgs of the F1 and F2 in the mixture are our decision variables. Suppose the mixture has
X Kg of food F1 and Y Kg of food F2.
2. In this example, the constraints are the minimum requirements of the vitamins. The
minimum requirement of vitamin A is 8 units. Therefore 3X + 4Y ≥ 8. Similarly, the minimum
requirement of vitamin B is 11 units. Therefore, 5X + 2Y ≥ 11
3. The cost of purchasing 1 Kg of food F1 is Rs.60. The cost of purchasing 1 Kg of food F2 is
Rs.80. The total cost of purchasing X Kg of food F1 and Y Kg of food F2 is C = 60X +
80Y, which is the objective function.
4. The non-negativity conditions are X ≥ 0, Y ≥ 0
Therefore the mathematical formulation of the LPP is:
Minimize: C = 60X + 80Y
Subject to: 3X + 4Y ≥ 8
5X + 2Y ≥ 11
X ≥ 0 , Y ≥ 0

8. Thank you