1. OPERATIONAL RESEARCH
Topic: Linear Programming Problem
Submitted To : Prof. Nilesh
Coordinators : Zeel Mathkiya (19)
Dharmik Mehta (20)
Sejal Mehta (21)
Hirni Mewada (22)
Varun Modi (23)
Siddhi Nalawade (24)

2. DEFINITION OF LINEAR
PROGRAMMING
The Mathematical Definition of LP:
“It is the analysis of problem in which a linear
function of a number of variables is to maximised
(minimised), when those variables are subject to a
number of restraints in the form of linear
inequalities”.

3. TERMINOLOGY OF LINEAR
PROGRAMMING
A typical linear program has the following
components
 An objective Function.
 Constraints or Restrictions.
 Non-negativity Restrictions.

4. TERMS USED TO DESCRIBE LINEAR
PROGRAMMING PROBLEMS
 Decision variables.
 Objective function.
 Constraints.
 Linear relationship.
 Equation and inequalities.
 Non-negative restriction.

5. FORMATION OF LPP
 Objective function
 Constraints
 Non-Negativity restrictions
 Solution
 Feasible Solution
 Optimum Feasible Solution

6. SOLVED EXAMPLE -1
 A Company manufactures 2 types of product H₁ & H₂. Both
the product pass through 2 machines M₁,M₂.The time requires
for processing each unit of product H₁,H₂.On each machine &
the available capacity of each machine is given below:
Product Machine
M₁ M₂
H₁ 3 2
H₂ 2 7
Available Capacity(hrs) 1800 1400
The availability of materials is sufficient to product 350 unit of
H₁ & 150 of H₂.Each unit of H₁ gives a profit of Rs.25,each
unit of H₂ gives profit of Rs.20.Formulate the above problem
as LPP.

7. SOLUTION
 From manufactures point of view we need to maximise the
profit.The profit depend upon the number of unit of product H₁
&H₂ produced.
Let x₁= no of unit of H₁ produce
x₂=no of unit of H₂ produce
x₁ ≥ 0  1
x₂ ≥ 0  2
3x₁ + 2x₂ ≤ 1800 3
2x₁ + 7x₂ ≤ 1400 4
Z= 25x₁ + 20x₂
LPP is formed as follows:
Maximise Z= 25x₁ + 20x₂

8. CONTI…..
 Subject to:
x₁ ≥ 0
x₂ ≥ 0
3x₁ + 2x₂ ≤ 1800
2x₁ + 7x₂ ≤ 1400

9. CONTI…...
 A Manager of hotel dreamland plans and extancison
not more than 50 groups attleast 5 must be executive
single rooms the number of executive double rooms
should be atleast 3 times the number of executive
single rooms. He charges Rs.3000 for executive
double rooms and Rs.1800 executive single rooms
per day.

10. CONTI…..
Formulate the above problume for LPP
SOLUTION →
The LPP is formulated as follows ;
Let X1 = Total No. of single executive rooms
Let X2 = Total No. of Double executive rooms
... X1 + x2 < 50
X1 > 5
x2 > 3 X1
Maximise ; Z = 1800 X1 + 3000 x2

11. The LPP is formulated as follows
Maximise ; Z = 1800 X1 + 3000 x
Subject to ; X1 + x2 < 50
X1 > 5
x 2 > 3 X1

12. GRAPHICAL METHOD
1. Arrive at a graphical solution for the following LPP.
Maximize Z = 40x1 + 35x2
Subject to : 2x1 + 3x2 < 60
4x1 + 3x2 < 96
x1 , x 2 > 0

13. Solution : Let us consider the equation
1) 2x1 + 3x2 = 60
Put x2 = 0: 2x1 = 60
x1 = 30
A = (30 , 0)
Put x1 = 0 : 3x2 = 60
x2 = 20
B = (0 , 20)

14. 2) 4x1 + 3x2 < 96
Put x2 = 0 : 4x1 = 96
x1 = 24
C = (24 , 0)
Put x1 = 0 : 3x2 = 96
x2 = 32
D = (0 , 32)

15. Y axis
40
Scale : Xaxis = 1 cm = 5 units
35 Yaxis = 1 cm = 5 units
30 D
25
20 B
15
10 p
5
C A X axis
O 5 10 15 20 25 30 35 40

16.  OBPC is the feasible region
Points x1 x2 z
O 0 0 z=0
B 0 20 z = 40(0) + 35 (20) = 700
P 18 8 z = 40(18) + 35(8) = 1000
C 24 0 z = 40(24) + 35(0) = 960
Thus, the optimal feasible solution is x1 = 18 , x2 = 8
and z = 1000

17. CONTI…..
 Find the feasible solution to following LPP
Minimize Z = 6x + 5y
Subject to = x + y > 7
x<3,y<4
x<0,y>0

18. Solution : Removing Inequality in given equation
1. x + y > 7
Put y = 0 : x = 7
Put x = 0 : y = 7
The two points are : A = (7 , 0) & B = (0 , 7)
Further,
X=3,y=4

19. Y axis
8
7 B Scale : X axis = 1cm = 1 unit
Y axis = 1cm = 1 unit
6
5
P
4
3
2
1
A
1 2 3 4 5 6 7 8 X axis
O

20. CONTI…..
 As all the 3 lines intersect each other at a common
point P( 3 , 4) it is the feasible solution to LPP
Z = 6(3) + 5(4)
= 18 + 20
= 38

21. CONCLUSION
 Linear programming is very important
mathematical technique which enables managers
to arrive at proper decisions regarding his area of
work. Thus it is very important part of operations
research.