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Q4. A post office is planning to build mail boxes along a wall in the mail room that is 1.8 metres high and 12 metres wide. The boxes come in two sizes: large boxes are 25cm wide and 15cm high; small boxes are 12cm wide and 10cm high. To simplify the design of the space, it has been decided that columns will contain exclusively large boxes or exclusively small boxes (i.e, no column will contain a mixture of large and small boxes). Post office officials have determined that at least half the available space should contain large boxes. Determine the design in order to have the maximum number of boxes. Just build the LP program - not need to figure out the optimal values. (9 points)

Answer 1

Solution:

Here,
Height of the wall = 1.8 meters = 1.8 x 100 = 180 cm
Width of the wall = 12 meters = 12 x 100 = 1200 cm

∴ Area of the Wall (i.e., Total Available Space) = 180 x 1200 = 216000 cm

Now, we need to maximize the total no. of boxes, so the attempt will be to fully utilize the total available space, thus, we assume that no vacant space will be left in a particular column. Moreover, there is a condition that a particular column should contain either large boxes or small boxes, not both.

Therefore,

Area of a large box column = Height of Wall x Width of Large Box

= 180 x 25

= 4500 cm

Area of a small box column = Height of Wall x Width of Small Box

= 180 x 12

= 2160 cm

Decision Variables:

Let,

x1 = # No. of Large Box Columns to be Arranged

x2 = # No. of Small Box Columns to be Arranged

Objective Function:

As an objective is to maximize the total no. of boxes, the objective function would be:

Max Z = x1 + x2

Subject to Constraints:

C1 = 4500 x1 + 2160 x2 ≤ 216000 (Total available space)

C2 = 4500 x1 ≥ 0.50 (216000) (The area allocated for the large boxes should be greater than or equal to 50% of the total space available)

∴ 4500 x1 ≥ 108000

Where, x1, x2 ≥ 0 and Integers

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