chemical brothers produces three chemicals: W, Y, and Z. The company begins the

Question

chemical brothers produces three chemicals: W, Y, and Z.

The company begins the production process by purchasing chemical X for cost of $650 per liter.

For an additional cost of $320 and 3 hours of labor, one liter of X can be turned into 0.4 liter of chemical Y and 0.6 liters of chemical W.

Chemical Y can either be sold or processed further. It costs $130 and 1 hour of labor to turn one liter Y into 0.6 liters of chemical Z and 0.4 liters of chemical W. For each chemical, the selling price(per liter) and the maximum amount that can be sold is given in the following table:

                               Product W      Product Y          Product Z

Selling price         $1250             $1800                $2680

Maximum sales      30                  60                         40

A maximum of 200 hours of labor is available.

How can chemical brothers maximize its profit?

Explanation / Answer

Let finally w liters of W, m liters of Y and z liters of Z be sold

w <=60

m<=30

z<=40

Total revenue = 1250w + 1800m + 2680z

let the company buy x liters of X,

so money spent = 650x

money spent to convert X into Y and W

1 liter of X into 0.4Y and 0.6W

so x liters into 0.4x of Y and 0.6x of W

let y of 0.4x Y be converted, so

now y of Y will be converted into 0.4y of W and 0.6y of Z

so final chemical compisition is

(0.4x-y) of Y 0.6x+0.4y of W and 0.6y of Z

so

w = 0.6x+0.4y <=60

m = (0.4x-y) <=30

z = 0.6y <=40

Money spent = 320x + 130y

total labour hours = 3x + y

so 3x + y <=200

total money spent = 650x + 320x + 130y = 970x + 130y

total money earned = 1250w + 1800m + 2680z = 1250(0.6x+0.4) + 1800 (0.4x-y)  + 2680(0.6y)

= 1470x + 308y

so profit = 1470x + 308y-320x - 130y = 1150x + 178y

so 1150x + 178y should be maximized, using conditions

0.6x+0.4y <=60

(0.4x-y) <=30

0.6y <=40

3x + y <=200

draw the 4 lines and shade the side of the graph that represents the inequality

then draw the 1150x + 178y line, and maximize it,

so maximum profit is

230000/3, with x = 200/3 and y = 0

so he should buy 200/3 liters of x and maximum profit is $ 76666.67

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