Product | Refinery 1 | Refinery 2 | Refinery 3 |

motor oil | 15 | 3 | 3 |

diesel oil | 10 | 14 | 5 |

gasoline | 5 | 5 | 12 |

paraffin | 3 | 5 | 2 |

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.

- 1.
- Set up a system of equations that describes the situation above.
- 2.
- 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.
- 3.
- Suppose that the demand for each product doubled simultaneously. How would your answer to the previous problem change? Explain your result mathematically.
- 4.
- 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?
- 5.
- 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 gallons of motor oil, 3100 gallons of diesel oil, and 5100 gallons of gasoline.)
- 6.
- 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. - 7.
- 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.)
- 8.
- 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?
- 9.
- 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?
- 10.
- The company is considering selling refinery 1, due to aging
and inefficient equipment and high workmans compensation costs in the
state where
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.)

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