Answer to Question #255930 in Operations Research for aquero

Question

As part of a campaign to promote its Summer Annual Clearance Sale, Cassy clothing cc. decided to buy television advertising time on Etu television. Their television advertising budget was N$102 000-00. Morning time costs N$3000-00 per minute, afternoon time costs N$1000-00 per minute, and evening (prime) time costs N$12 000-00 per minute. Because of previous commitments, Etu television could not offer Cassy clothing more than 6 minutes of prime time and a total of 25 minutes of advertising time over the two weeks in which the commercials run in the morning would be seen by 200 000 people, those run in the afternoon would be seen by 100 000 people and those run in the evening would be seen by 600 000 people. How much morning, afternoon, and evening advertising time should Cassy clothing buy to maximize exposure of its products? 

Answer 1

To maximize exposure of its products;

Max Z= $200,000x_{1}+100,000x_{2}+600,000x_{3}$

Subject to:

$3,000x_{1}+1000x_{2}+12,000x_{3}\le102,000$

$x_{1}+x_{2}+x_{3}\leq25$

$x_{3}\leq6$

$x_{1},x_{2},x_{3}\ge0$

Introducing slack variables;

Max Z= $200,000x+100,000y+600,000z+0S_{1}+0S_{2}+0S_{3}$

Subject to:

$3,000x_{1}+1000_{2}+12,000_{3}+S_{1}=102,000$

$x_{1}+x_{2}+x_{3}+S_{2}=25$

$x_{3}=6$

$x_{1},x_{2},x_{3},S_{1},S_{2},S_{3}\ge0$

Negative minimum Z$_{j}$-$C_{j}$ is -600000 and its column index is 3. So, the entering variable is $x_{3}$

Minimum ratio is 6 and its row index is 3. So, the leaving basis variable is $S_{3}$ .

$\therefore$ the pivot element is 1

Entering =$x_{3}$ , Departing =$S_{3}$ , Key Element =1

R$_{3}$ (new)=R$_{3}$ (old); R$_{1}$ (new)=R$_{1}$ (old) - 12000R$_{3}$ (new) and R$_{2}$ (new)=R$_{2}$ (old) - R$_{3}$ (new)

Negative minimum Z$_{j}$ -C$_{j}$ is -200000 and its column index is 1. So, the entering variable is $x_{1}$ .

Minimum ratio is 10 and its row index is 1. So, the leaving basis variable is $S_{1}$ .

∴ The pivot element is 3000.

Entering =$x_{3}$ , Departing =$S_{1}$ , Key Element =3000

R$_{1}$ (new)=R$_{1}$ (old)$\div$3000 ; R$_{2}$ (new)=R$_{2}$ (old) - R$_{1}$ (new) and R$_{3}$ (new)=R$_{3}$ (old)

 Negative minimum Z$_{j}$ -C$_{j}$ is -200000 and its column index is 6. So, the entering variable is $S_{3}$ .

Minimum ratio is 3 and its row index is 2. So, the leaving basis variable is $S_{2}$ .

∴ The pivot element is 3.

Entering =$S_{3}$ , Departing =$S_{2}$ , Key Element =3

R$_{2}$ (new)=R$_{2}$ (old)$\div$3 ; R$_{1}$ (new)=R$_{1}$ (old) $\div$ 4R$_{2}$(new) and R$_{3}$ (new)=R$_{3}$ (old)-R$_{2}$ (new)

Since all Z$_{j}$ -C$_{j}$$\ge0$

Hence, optimal solution is arrived with value of variables as :

x$_{1}$ =22, x$_{2}$=0, x$_{3}$=3

Max Z= $200,000x_{1}+100,000x_{2}+600,000x_{3}$

Max Z=200000(22)+100000(0)+600,000(3)=6,200,000

Therefore Cassy clothing cc. should buy 22 minutes of morning time ,0 minutes of afternoon time and 3 minutes of prime time in order to maximize exposure of its products.