# Mathematics 2: Linear optimization Examples

### Jean pants

A plant produces two kinds of jean pants : Jean A and Jean B.

A made jean pants passes through three machines:

Machine M1: cut the fabric. It operates at most 5 hours,
Machine M2: sew the pants. It operates at most 4 hours,
Machine M3: install rivets. It operates at most 2 hours.

• Jean A:
It costs 50.00 $. The Machine M1 spends 5 mn to cut its fabric, the Machine M2 spends 6 mn to sew the pants, and the Machine M3 spends 8 mn to install special rivets. • Jean B: It costs 30.00$.
The Machine M1 spends 3 mn to cut its fabric, the Machine M2 spends 5 mn to sew the pants, and the Machine M3 spends 2 mn to install rivets.

How many jean pants the factory should produce to maximize its profit?

Solution

Set the unknown:

x is the number of jean pants of kind A
y is the number of jean pants of kind B

The operation time (in mn) for the three machines:

M1: 5 x + 3 y ≤ 5 x 60 = 300
M2: 6 x + 5 y ≤ 4 x 60 = 240
M3: 8 x + 2 y ≤ 2 x 60 = 120

The equivalent equations are:

M1: y = - (5/3) x + 100
M2: y = - (6/5) x + 48
M3: y = - 4 x + 60

Their graph is:

Function objective Z = 50 x + 30 y Maximum

 Vertex Z ($) O(0, 0) 0 A(0, 48) 1440 B(4.3, 42.86) → B(4, 42) 1505 C(15,0) 750 To obtain the maximum profit which is 1505$, the factory would produce 4 jean pants of kind A and 42 jean pants of kind B.

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