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Tamil Nadu Board of Secondary EducationHSC Commerce Class 11

Solve the following linear programming problems by graphical method. Maximize Z = 20x1 + 30x2 subject to constraints 3x1 + 3x2 ≤ 36; 5x1 + 2x2 ≤ 50; 2x1 + 6x2 ≤ 60 and x1, x2 ≥ 0. - Business Mathematics and Statistics

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Question

Solve the following linear programming problems by graphical method.

Maximize Z = 20x1 + 30x2 subject to constraints 3x1 + 3x2 ≤ 36; 5x1 + 2x2 ≤ 50; 2x1 + 6x2 ≤ 60 and x1, x2 ≥ 0.

Graph
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Solution

Given that 3x1 + 3x2 ≤ 36

Let 3x1 + 3x2 = 36

x1 0 12
x2 12 0

Also given that 5x1 + 2x2 ≤ 50

Let 5x1 + 2x2 = 50

x1 0 10
x2 25 0

3x1 + 3x2 = 36

x1 + x2 = 12 ……….(1)

5x1 + 2x2 = 50 ………(2)

2x1 + 2x2 = 24 ....[(1) × 2]

−   −        −     
−3x1 = − 6

x1 = 2

Substituting x1 = 2 in (1) we get

2+ x2 = 12

x2 = 6

Also given that 2x1 + 6x2 ≤ 60

Let 2x1 + 6x2 = 60

x1 + 3x2 = 30

x1 0 30
x2 10 0

x1 + x2 = 12 …….(1)

x1 + 3x2 = 30 …….(2)
– 2x2 = – 18 ......[Equation (1) – (2)]

x2 = 9

x2 = 9 substitute in (1)

x1 + x2 = 12

x1 + 9 = 12

x1 = 12 – 9

x1 = 3

The feasible region satisfying all the given conditions is OABCD.

The co-ordinates of the comer points are

Corner points Z = 20x1 + 30x2
O(0, 0) 0
A(10, 0) 200
B(2, 6) 220
C(3, 9) 330
D(0, 10) 300

The maximum value of Z occurs at C(3, 9)

∴ The optimal solution is x1 = 3, x2 = 9 and Zmax = 330

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Linear Programming Problem (L.P.P.)
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Chapter 10: Operations Research - Exercise 10.1 [Page 244]

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Samacheer Kalvi Business Mathematics and Statistics [English] Class 11 TN Board
Chapter 10 Operations Research
Exercise 10.1 | Q 4. (v) | Page 244

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