Ms. Jones installs and demonstrates smoke detectors. There are two types of smok

Question

Ms. Jones installs and demonstrates smoke detectors. There are two types of smoke detectors, Type A and Type B. Type A requires 1 hour to install and 1/3 hour to demonstrate. Type B requires 1.5 hours to install and 1/5 hour to demonstrate. Company rules require Ms. Jones to work a minimum of 20 hours as an installer and a maximum of 15 hours a week as a demonstrator. She earns $8 per hour as an installer and $5 per hour as a demonstrator. How many of each type of alarm should she install and demonstrate to maximize her income?

Explanation / Answer

Solution:

Let x be total number unit of type A smoke detector

Let y be total number unit of type B smoke detector

As per given condition

Total number of hours require for Mr. Jones to work as installer

x + 1.5y >=20

Similerly as demonstrator

(1/3) x + (1/5)y <= 15

She has to maximize objective function p = 8x +5y

using simplex method

Maximize p = 8x + 5y subject to
x + 1.5y >= 20
(1/3)x +(1/5) y <= 15

x y s1 s2 p

1 1.5 -1 0 0 20

0.33 0.2 0 1 0 15

-8 -5 0 0 1 0

Tableau #2

x y s1 s2 p

0.67 1 -0.67 0 0 13

0.2 0 0.13 1 0 12

-4.7 0 -3.3 0 1 67

Tableau #3

x y s1 s2 p

1 1.5 -1 0 0 20

0 -0.3 0.33 1 0 8.3

0 7 -8 0 1 160

Tableau #4

x y s1 s2 p

1 0.6 0 3 0 45

0 -0.9 1 3 0 25

0 -0.2 0 24 1 360

Tableau #5

x y s1 s2 p

1.7 1 0 5 0 75

1.5 0 1 7.5 0 92

0.33 0 0 25 1 370

Answer:

Optimal Solution: p = 370; x = 0, y = 75

therefore she has to install x = 0 uint of type and y = 75 unit of type B to maximize her income

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