Name: Mathematical Formulation of LPP: Illustration 2
Uploaded: 2019-07-19
Duration: 8 min 40 s
Description: LPP can be used in many real-life scenarios. Here is one of them. Watch this video to get an idea of how a manufacturing unit plans production using LPP.

Question

A school wants to buy small and large minibuses to transport students to sports activities. It has $$ 220000$ to buy the minibuses, and $$ 1200$ to insure them. The table gives some information about different sizes of minibuses.

Size	Maximum number of people	Cost	Insurance
Small	$8$	$$ 11000$	$$ 120$
Large	$16$	$$ 24000$	$$ 80$

Determine how many of each size minibus the school should buy to maximize the number of students they can transport.

Accepted Answer

Let $x$ denotes small size and $$ 1200$ denotes large size minibus

According to the given question, the constraints are

$11000 x + 24000 y \leq 220000, 120 x + 80 y \leq 1200, x \geq 0, y \geq 0$

$\Rightarrow 11 x + 24 y \leq 220 . . . . . (1), 12 x + 8 y \leq 120 . . . . . (2), x \geq 0, y \geq 0$

To maximise $z = 8 x + 16 y$

$11 x + 24 y = 220$ $12 x + 8 y = 120$

$x = 0 \Rightarrow y = 9.166$ $x = 0 \Rightarrow y = 15$

$y = 0 \Rightarrow x = 20$ $y = 0 \Rightarrow x = 10$

Solving $11 x + 24 y = 220, 12 x + 8 y = 120$ , we get $x = 5.6, y = 6.6$

The feasible region is given by $A (0, 9.166), B (5.6, 6.6), C (10, 0)$

Objective function:

Corner points $z = 8 x + 16 y$

$A (0, 9.166)$ $z = 0 + 16 (9.166) = 146.656$

$B (5.6, 6.6)$ $z = 8 (5.6) + 16 (6.6) = 44.8 + 105.6 = 150.4$ Maximum

$C (10, 0)$ $z = 8 (10) + 0 = 80$

Therefore, $5$ small size and $7$ large size minibuses(approximately) the school should buy to maximise the number of students they can transport.

Type of investment	Interest per annum
Municipal bond	$2 %$
Bank Mutual fund	$3.2 %$
Speculative money market fund	$5 %$

Important Questions on Decisions, Decisions (Inequalities)

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