Question

Two different types of books have to be stacked in the shelf of a library. The first type of book weighs 1 kg and has a thickness of 6 cm. The second type of book weighs 1.5 kg and has a thickness of 4 cm. The shelf is 96 cm long and can support a maximum weight of 21 kg.

How should both the types of books be placed in the shelf to include the maximum number of books? Formulate a Linear Programming Problem and solve it graphically.

Answer 1

Let x be number of books of first type and y be number of books of second type.

Objective function Max Z = x + y

Constraints are

1x + 1.5y ≤ 21

6x + 4y ≤ 96

And x ≥ 0, y ≥ 0

Now, x + 1.5y ≤ 21 and 6x + 4y ≤ 96

2x + 3y ≤ 42, 3x + 2y ≤ 48

Convert inequalities into equations

2x + 3y = 42   ...(i)

x	0	21
y	14	0

3x + 2y = 48   ...(ii)

x	0	16
y	24	0

Solve equations (i) and (ii)

2x + 3y = 42 × 2

3x + 2y = 48 × 3

4x + 6y = 84

9x + 6y = 144
 –  –          –         
     – 5x =  – 60

 x = 12

Put the value of x in equation (i),

2 × 12 + 3y = 42

3y = 42 – 24 = 18

y = 6

Coordinates of point B(12, 6).

Corner points	Value of Z = x + y
A(0, 14)	ZA = 0 + 14 = 14
B(12, 6)	ZB = 12 + 6 = 18
C(6, 0)	ZC = 16 + 0 = 16

The maximum value of Z is 18, which occurs at B(12, 6).

The maximum number of books which can be placed on the self is 18. Placing 12 books of first type and 6 books of the second type.