Maharashtra State BoardHSC Commerce 12th Board Exam
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Solve the following problem : Minimize Z = 4x + 2y Subject to 3x + y ≥ 27, x + y ≥ 21, x ≥ 0, y ≥ 0 - Mathematics and Statistics

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Graph

Solve the following problem :

Minimize Z = 4x + 2y Subject to 3x + y ≥ 27, x + y ≥ 21, x ≥ 0, y ≥ 0

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Solution

To find the graphical solution, construct the table as follows:

Inequation equation Double intercept form Points (x1, x2) Points (x1, x2)
3x + y ≥ 27 3x + y = 27 `x/(9) + y/(27)` = 1 A (9, 0)
B (0, 27)

3(0) + 0 ≥ 27
∴ 0 ≥ 27
∴ non-origin-side

x + y ≥ 21 x + y = 21 `x/(21) + y/(21)` = 1 C (21, 0)
D (0, 21)
(0) + 0 ≥ 21
∴ 0 ≥ 21
∴ non-origin-side
x ≥ 0 x = 0   R.H.S. of Y-axis
y ≥ 0 y = 0     above X-axis


The shaded portion CHB is the feasible region.
Whose vertices are C(21, 0), H and B(0, 27)
H is the point of intersection of lines
3x + y = 27 …(i)
x + y = 21  …(ii)
∴ By (i) – (ii), we get
3 x + y = 27
   x + y = 21
–    –     –      
2x        = 6

∴ x = `(6)/(2)`  = 3
Substituting x = 3 in (ii), we get
3 + y = 21
∴ y = 18
∴ H (3, 18)
Here, the objective function is Z = 4x + 2y
Now, we will find minimum value of Z as follows:

Feasible points The value of Z = 4x + 2y
C (21, 0) Z = 4(21) + 2(0) = 84
H (3, 18) Z = 4(3) + 2(18) = 12 + 36 = 48
B (0, 27) Z = 4(0) + 2(27) = 54

∴ Z has minimum value 48 at H (3, 18)
∴ Z is minimum, when x = 3, y = 18

Concept: Mathematical Formulation of Linear Programming Problem
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 6 Linear Programming
Miscellaneous Exercise 6 | Q 4.02 | Page 104
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