Cost minimization solved example of LPP

1. LINEAR PROGRAMMING PROBLEM
(LPP)
TOPIC: COST MINIMIZATION

2. INTRODUCTION
 Linear programming is a mathematical technique used to find the best possible solution in allocating
limited resources (constraints) to achieve maximum profit or minimum cost by modelling linear
relationships.
 Model formulation steps :
• Define the decision variables
• Construct the objective function
• Formulate the constraints
• Find the feasible solution
• Calculate the value of the objective function at each of the vertices of feasible solution area to
determine which of them has the maximum or minimum values

3. MODEL COMPONENTS
 Decision variables – mathematical symbols representing levels of activity of a firm.
 Objective function – a linear mathematical relationship describing an objective of the firm, in terms of
decision variables - this function is to be maximized or minimized.
 Constraints – requirements or restrictions placed on the firm by the operating environment, stated in
linear relationships of the decision variables.
 Parameters – numerical coefficients and constants used in the objective function and constraints.

4. LP MODEL FORMULATION
 Two brands of fertilisers available: SuperGro & CropQuick
 Field requires at least 16 kgs of nitrogen and 24 kgs of phosphate
 SuperGro costs ₹6 per bag and CropQuick costs ₹3 per bag
 SuperGro has 2 kgs of nitrogen & 4 kgs of phosphate
 CropQuick has 4 kgs of nitrogen & 3 kgs of phosphate
 Problem: How much of each brand to buy to minimize total cost of fertiliser ?
Brand Nitrogen Phosphate
SuperGro 2 4
CropQuick 4 3

5.  Decision Variables
x = No. of bags of SuperGro
y = No. of bags of CropQuick
 Objective Function
Minimize, z = 6x + 3y
 Constraints
2x + 4y >= 16 (nitrogen)
4x + 3y >= 24 (phosphate)
x, y >= 0 (non-negativity constraint)

6. Graph of Constraints
X 0 6
Y 8 0
X 0 8
Y 4 0
4x + 3y= 242x + 4y= 16

7. Feasible Solution Area

8. Optimum Solution
Points Z
A (0,8) 24
B (4.8,1.6) 33.6
C (8,0) 48
z = 6x + 3y
So, 8 kgs of CropQuick should be bought to
minimise total cost of fertiliser.

9. Thank You