Math question need help with how to set it up?

Sugar Spectral is a company that manufactures Halloween candy. It has two candy factories, Factory A and Factory

B, and two distributors for its product, Distributor 1 and Distributor 2. In September Distributor 1 needs 8,000

pounds of candy and Distributor 2 needs 7,000 pounds. Factory A has 10,000 pounds ready to ship and Factory B

11,000 pounds. Shipping costs per pound from factories to distributors are given below.

Distributor 1 Distributor 2

Factory A $1.00 $0.60

Factory B $1.20 $0.70

a) How much candy should Sugar Spectrals ship from each factory to each distributor if shipping costs are to be

kept to a minimum?

b) Suppose the price of shipping from Factory A to Distributor 1 doubles. How does this change the optimal

solution?Math question need help with how to set it up?
You've got 4 quantities:

A1, A2, B1, B2 representing amount from each factory to each distributor.



We'll figure the costs in units of per 100 lbs.

to get rid of the decimals.



Total cost is:

a1 * 100 + a2 * 60 + b1 * 120 + b2 * 70

and

a1 + b1 = 80

a2 + b2 = 70

Also

a1 + a2 %26lt; 100

b1 + b2 %26lt; 110

Cost now is

100 a1 + 60 a2 + 120 (80 - a1) + 70 (70 - a2)

100 a1 + 60 a2 + 9600 - 120 a1 + 4900 - 70 a2



-20 a1 -10 a2 + 14500 is the cost



a1 + a2 = 100

So make a1 80 (most D1 can handle), a2 20 (the rest of A),

b1 0, and b2 50

Cost is then

14500 - 80* 20 - 20 * 10 = 14500 - 1600 - 200 = 12700, as follows:



Returning to the original numbers:

8000 * 1.00 + 2000 * 0.60 + 0 * 1.20 + 5000 * 0.70 =

8000 + 1200 + 3500 = 12700.



If the cost from A -%26gt; D1 doubles, that of course changes everything:



Now we have

A: 2.00 0.60

B: 1.20 0.70

Now B has an advantage of 80c/lb over A to D1

and A has an advantage of 10c/lb over B to D2.

That makes it even more obvious what to do.

We ship all we can on those two routes, and the shipments need not be split.



8000 B -%26gt; D1 (cost $9600)

7000 A -%26gt; D2 (cost $4200)

Total $13800.



Going through the calculation to check:



Cost = a1 * 200 + a2 * 60 + b1 * 120 + b2 * 70

a1 + b1 = 80

a2 + b2 = 70

200 a1 + 60 a2 + 120 (80 - a1) + 70 (70 - a2)

80 a1 - 10 a2 + 9600 + 4900

80a1 - 10 a2 + 14500

To minimize that, it's clear that we should make a1 to be 0,

since it has a positive coefficient, and a2 = 70 (or 7000 lbs).

Cost then is 14500 - 10 * 70 = 14500 - 700 = 13800.

.