Question

1. TMA manufactures 37-in. high definition LCD televisions in two separate locations, Locations I and II. The output at Location I is at most 6000 televisions/month, whereas the output at Location II is at most 5000 televisions/month. TMA is the main supplier of televisions to the Pulsar Corporation, its holding company, which has priority in having all its requirements met. In a certain month, Pulsar placed orders for 3000 and 4000 televisions to be shipped to two of its factories located in City A and City B, respectively. The shipping costs (in dollars) per television from the two TMA plants to the two Pulsar factories are as follows.

	To Pulsar Factories
From TMA	City A	City B
Location I	$7	$3
Location II	$8	$11

TMA will ship x televisions from Location I to city A and y televisions from Location I to city B. Find a shipping schedule that meets the requirements of both companies while keeping costs to a minimum.

(x,y) = ( , )

What is the minimum cost?

2. A veterinarian has been asked to prepare a diet, x ounces of Brand A and y ounces of Brand B, for a group of dogs to be used in a nutrition study at the School of Animal Science. It has been stipulated that each serving should be no larger than 8 oz and must contain at least 29 units of Nutrient I and 20 units of Nutrient II. The vet has decided that the diet may be prepared from two brands of dog food: Brand A and Brand B. Each ounce of Brand A contains 3 units of Nutrient I and 4 units of Nutrient II. Each ounce of Brand B contains 5 units of Nutrient I and 2 units of Nutrient II. Brand A costs 3 cents/ounce, and Brand B costs 7cents/ounce. Determine how many ounces of each brand of dog food should be used per serving to meet the given requirements at a minimum cost.

(x,y) = ( , )

What is the minimum cost? ( Round your answer to the nearest cent.)

3. A farmer plans to plant two crops, A and B. The cost of cultivating Crop A is $40/acre whereas the cost of cultivating Crop B is $60/acre. The farmer has a maximum of $7400 available for land cultivation. Each acre of Crop A requires 20 labor-hours, and each acre of Crop B requires 25 labor-hours. The farmer has a maximum of 3300 labor-hours available. If she expects to make a profit of $170/acre on Crop A and $180/acre on Crop B, how many acres of each crop, x and y, respectively, should she plant in order to maximize her profit?

(x,y) = (?,?)

What is the optimal profit?

Answer 1

1 Answer:

Given Data

Given Maximum TV output :

at location 1 = 6000Tv's

at location 2 = 5000Tv's

Order of TVs for city A = 3000TV's

Order of TVs for city B = 4000TV's

Number of TV's to be shipped from location 2 to city A

= (3000-x) ;and

Location 2 to city B

= (4000 - y)

According to the table ; total shipping cost of TV's is given by

x = 7x + 3y + 8( 3000 - x ) + 11(4000 - y )

= 7x + 3y +24000 - 8x + 44000-11y

= - x -8y +68,000

= 68,000 - x - 8 y

Problem becomes : -

min z = 68,000 - x - 8y

subject to

( max production at location 1 is 6000 TV's)

(3000 - x) + (4000 - y) 5,000

(max output at location 2 is 5000TV's)

7000 - (x+y) 5,000

2000 (x+y)

0 x 3000 ;

0 y 4000

[ Order of city A is 3000TV's & order of city B is 4000 TV's]

Hence , precisely we need to

min z =  68,000 - x - 8 y ( in $ )

Subject to

We need to minimize z

Note that z will minimize if we maximize x and y and y have larger negative coefficient than x

Maximizing y will minimize x more .

We first maximize y subject to our constraints affecting y namely :-

2000 x +y 6000 ; (*)

and

0 y 4000

Maximum value of can be 4000

We will take y = 4000

and put it in (*) which gives

2000 x + 4000 6000

- 2000   x 2000

Moreover ; we have constraints affecting x given by:-

( and with max value of if give =new of constraint )

(A) 0 x 3000

We can have maximum value of x as 2000 according to above constraints &

Req and

Minimizing shipping cost

Shipping cost = =  68,000 - x - 8 y

= 68,000 - 1( 2,000) - 8 (4000)

= 68,000 - 2,000 - 32,000

= 34,000 $

Shipping schedule to meet requirements of both companies keeping cost minimum is given by :-

Location 1 should ship TV's

to city A and TV's to city B and

location 2 should ship ( 3000 - ) = 3,000 - 2,000

= 1,000 TV's to city A

and (4000 - ) = ( 4000 - 4000)

= 0TV's

i.e no TV to city B

This will minimize shipping cost to   = 34,000 $