Your team provides consulting services to a petroleum company. The task i
s to advise them on how to meet the demands of their customers for
motor oil, diesel oil and gasoline. They have, at the moment, three
refineries. They have decided
not to store any excess production for a variety of reasons,
including added insurance costs, environmental factors, and
deterioration of gasoline over time.
From a barrel of crude oil, each refinery produces three different
products: motor oil, diesel oil, and gasoline. Each refinery also
produces a waste product, called parrafin. The table below shows
production figures for each of the three refineries. The numbers in the
table are gallons of product or waste produced from each barrel of
crude oil. (A barrel is a unit of measurement, equal to 42 gallons,
used mainly for historical reasons.)
Suppose that the current daily demand from your distributor for
products is 1950 gallons of
motor oil, 3100 gallons of diesel oil, and 5100 gallons of gasoline.
- Set up a system of equations that describes the situation
- Solve your equations to determine the number of barrels of crude
oil each refinery should get so that the refineries, as a group, meet the
demand for the three products.
- Suppose that the demand for each product doubled
simultaneously. How would your answer to the previous problem change?
Explain your result mathematically.
- Now suppose that another distributor has come forward and says
that it would require 750 gallons of motor oil, 2000 gallons of
diesel and 2000 gallons of gasoline per day. How would you set up
production to satisfy this distributor only? That is, suppose this is
the only distributor to which the company is selling. Is there only
one way of doing this?
- Now calculate the needs of each refinery, in barrels of crude oil
per day, if both distributors are to be satisfied. How does this
compare to your two previous answers? What mathematical conclusion can
you draw? (Use the original demand from the first distributor of 1950
motor oil, 3100 gallons of diesel oil, and 5100 gallons of gasoline.)
- In real life applications, constants are rarely ever exactly equal
to their stated value; certain amounts of uncertainty are always
present. This is part of the reason for the science of statistics. In
the above model, the daily productions for the refineries would be
averages over a period of time. Your job here is to explore what
effect small changes in the parameters have on the output.
To do this, pick any 3 coefficients, one at a time, and vary them one
way and the other by 3%. For each case , note what effect this has on
the solution, as a percentage change. Can you draw any overall conclusion?
The activity just described is called a sensitivity
analysis. A model which does not change much for modest changes in
its parameters is said to be robust.
- Suppose refinery 3 is shut down temporarily by the EPA for
excessive emissions. If the demand is still 1950 gallons of
motor oil, 3100 gallons of diesel oil, and 5100 gallons of gasoline,
what would you now say about the company's ability to meet it? What
production schedule do you recommend? (Hint - set the coefficients for
refinery 3 to zero and try to solve the resulting problem.)
- The situation in the problem above has caused enough concern
that the company is considering buying another refinery, identical to
refinery 3, and using it permanently. Assuming that all 4 refineries are on
line, what production schedule do you recommend to meet the demand
given in the previous exercise? In general, what can you say about
any increased flexibility that the fourth refinery might provide?
- The company has just found a candle maker that will buy the
parrafin waste product. Assuming three refineries and the same demand of
1950 gallons of motor oil, 3100 gallons of diesel oil, and 5100
gallons of gasoline how much parrafin can be supplied?
- The company is considering selling refinery 1, due to aging
and inefficient equipment and high workmans compensation costs in the
refinery 1 is located. They would like to know what this would do to
their production capacity. Specifically, they would like examples of
demands they could and could not meet with only refineries 2 and 3 in
operation. Also, they would like you to comment on the effect of
having refineries 2, 3 and 4. Any general statements you could make here
would be helpful. (Hint - your comments should include a discussion of
the terms unique, no solution,
overdetermined, and underdetermined as they apply in
the context of the petroleum refineries.)
This document was generated using the
LaTeX2HTML translator Version 97.1 (release) (July 13th, 1997)
Copyright © 1993, 1994, 1995, 1996, 1997,
Computer Based Learning Unit, University of Leeds.
The command line arguments were:
latex2html -split +0 project_template.tex.
The translation was initiated by William W. Farr on 9/1/1998
William W. Farr