Given: The manager of a paper company receives an order for low, medium, and high grade paper. The company consists of two factories which are each capable of producing each style of paper, but in differing quantities per day. Pertinent data are given below. Note that if you decide to run Factory 1 for one complete day, it will spew out 8 tons of low grade paper, 1 ton of medium grade paper, and 2 tons of high grade paper. That’s just what it does. It is not necessary that your solution be based on “whole” days – a part of a day is acceptable.
Factory outputs, costs to run, and order quantities are as follow:
Required amount to 
Factory 1 
Factory 2 

Low grade 
16 tons 
8 tons/day 
2 tons/day 
Medium grade 
5 tons 
1 ton/day 
1 ton/day 
High grade 
20 tons 
2 tons/day 
7 tons/day 
Daily cost to run plant 
$1000/day 
$2000/day 
Required:
 Solve the problem to fill the order at a minimum cost.
 Solve the problem to fill the order while running the two factories for as few total days as possible – i.e. get the order filled while running the two factories for a combined minimum total number of days. For example, I don’t care if factory 1 works 11.2 days and factory 2 works 6.8 days, for a total of 18 days, but I want that total number of days in the factories (18) minimized.
 Solve the problem to fill the order in a minimum time – i.e. to get the order finished and out the door and to the customer as quickly as possible.
Hint:
Dr. Lowery,
I have a question about web problem 5.18 that is due tomorrow. It appears to me that numbers 2) and 3) under what is required are asking the same thing. This is what they say:
2. Solve the problem to fill the order while running the two factories for as few total days as possible.
3. Solve the problem to fill the order in a minimum time.
Aren’t “the fewest days” and “minimum time” the same thing?
Joe:
Sorry I am just getting back to you. Too many emails.
Not really.
The optimum answer for cost might be A = 10 days, B = 30 days. (Note A+B = 40 days, cost = $100)
The optimum answer for minimum total time = A+B might be A = 14 days, B = 20 days. (A+B now = 34 days, cost = $120)
In the first case the customer would not get his order until 30 days.
In the second case he would not get his order until 20 days.
Do you see a way to get the customer his order quicker than 20 days?
L^3