ISYE 6669 A/Q Homework #3, Due Friday September 14th
1.
Napier Chemicals produces three types of chemicals: A, B, and C. The chemicals are produced
by running two machines. Machine 1 costs $4000 per hour to run, and yields 300 gallons of
chemical A, 100 gallons of chemical B, and 100 gallons of chemical C. Machine 2 costs $1000
per hour to run, and yields 100 gallons of chemical A, 100 gallons of chemical B, and 200
gallons of chemical C. Each day, Napier must make at least 3000 gallons of chemical A, at
least 500 gallons of B, and at least 2000 gallons of C.
(a)
Formulate a linear program that Napier can use to find the cheapest way of producing
the required daily amounts of each chemical.
(b)
Solve the problem graphically. (Hint: if your formulation of part (a) has more than two
variables, you might need a different formulation in order to do part (b).)
(c)
Solve the problem using the Revised Simplex Method. Begin the algorithm with all of
your excess variables and machine variables basic. (Order them with the excess
variables first.) Show all your work. [Hint: the optimal solution should be to run
machine 1 for 2 hours each day, and machine 2 for 24 hours each day.]
(d)
At the optimal basis, answer the following questions:
(i)
(ii)
(iii)
(iv)
(v)
2.
An engineer claims that he can increase Napier’s production of chemical A by
200 gallons per day, but wants to be paid $50,000 per year if hired. Is it
profitable for Napier to hire the man? Show your work.
Leary Chemicals, one of Napier’s former competitors, has recently closed its
nearby plant. Leary’s president has offered to rent Napier a machine identical to
machine 2, for $50,000 per year. If Napier buys the machine, how many days
will it take for them to make up the $50,000 yearly rental cost?
Instead of renting a machine identical to machine 2, Napier’s president wants to
rent a third type of machine for $50,000 per year. This machine would cost
$3000 per hour to run, but would create 200 gallons of A, 300 gallons of B, and
300 gallons of C per hour. If Napier buys this machine, how many days will it
take to make up the $50,000 rental cost?
Out of these three possibilities ((i), (ii), and (iii)), which do you think is the best
use of Napier’s $50,000? Why?
In part (iii), suppose the hourly operating cost of the new machine could be
decreased. How low would the cost need to be in order for the machine to be
part of the daily optimal strategy? [Note: We didn’t talk about this in class…
but you should be able to figure it out yourself using the same type of thinking
we used to calculate variable values, shadow prices, etc. for Revised Simplex. I
want you to learn to think this way on your own! ]
Page 120, Problem #49 (airline refueling)
(a)
(b)
(c)
(d)
Formulate the problem. How many variables and constraints do you have?
Suppose you were going to solve the problem using the Revised Simplex Method. How
large a matrix would you need to invert?
Explain how you could reduce the size of your matrix from part (b) as much as possible.
Now how large a matrix would you need to invert?
Which size problem would you rather solve by hand, (b) or (c)?