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)

OPERATIONAL RESEARCH

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”.

TERMINOLOGY OF LINEAR PROGRAMMING

A typical linear program has the following components

An objective Function.Constraints or Restrictions.Non-negativity Restrictions.

TERMS USED TO DESCRIBE LINEAR PROGRAMMING PROBLEMS

Decision variables.Objective function.Constraints.Linear relationship.Equation and inequalities.Non-negative restriction.

FORMATION OF LPPObjective functionConstraintsNon-Negativity restrictionsSolutionFeasible SolutionOptimum Feasible Solution

SOLVED EXAMPLE -1A 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.

SOLUTIONFrom 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

CONTI…..Subject to:

x₁ ≥ 0 x₂ ≥ 0 3x₁ + 2x₂ ≤ 1800 2x₁ + 7x₂ ≤ 1400

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.

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

The LPP is formulated as follows

Maximise ; Z = 1800 X1 + 3000 x

Subject to ; X1 + x2 < 50

X1 > 5

x2 > 3 X1

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 , x2 > 0

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)

2) 4x1 + 3x2 < 96

Put x2 = 0 : 4x1 = 96

x1 = 24

C = (24 , 0)Put x1 = 0 : 3x2 = 96

x2 = 32

D = (0 , 32)

Y axis

Scale : Xaxis = 1 cm = 5 units Yaxis = 1 cm = 5 units D

B

p

C A X axis O

5 10 15 20 25 30 35 40

40

35

30

25

20

15

10

5

OBPC is the feasible regionPoints x1 x2 z

O 0 0 z = 0B 0 20 z = 40(0) + 35 (20) = 700P 18 8 z = 40(18) + 35(8) = 1000C 24 0 z = 40(24) + 35(0) = 960

Thus, the optimal feasible solution is x1 = 18 , x2 = 8 and z = 1000

CONTI…..Find the feasible solution to following LPPMinimize Z = 6x + 5ySubject to = x + y > 7

x < 3 , y < 4x < 0 , y > 0

Solution : Removing Inequality in given equation1. x + y > 7Put y = 0 : x = 7Put x = 0 : y = 7The two points are : A = (7 , 0) & B = (0 , 7)Further,X = 3 , y = 4

Y axis

B Scale : X axis = 1cm = 1 unit Y axis = 1cm = 1 unit

P

A X axis O

1 2 3 4 5 6 7 8

8

7 6

5 4

3

2

1

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

CONCLUSIONLinear 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.