1. Jack Porter Jarrod Phillips Scott Simon Weon Kim

2. Our group provides consulting services to a petroleum company. We are to advise them on how to meet the demands of their customers for motor oil, diesel oil, and gasoline. The company has three plants and has decided not to store any excess production. From a barrel of crude oil, factory #1 can produce 20 gallons of motor oil, 10 gallons of diesel oil, and 5 gallons of gasoline. Factory #2 can produce 4 gallons, 5 gallons, and 12 gallons, respectively, of motor oil, diesel oil, and gasoline. Factory #3 can produce 4 gallons, 5 gallons, and 12 gallons of motor oil, diesel oil, and gasoline.

3. From the given specifications, we can set up a system of equations to represent the output of the factories: Motor Oil: 20f 1 + 4f 2 + 4f 3 = x 1 Diesel Oil: 10f 1 + 14f 2 + 5f 3 = x 2 Gasoline: 5f 1 + 5f 2 + 12f 3 = x 3 Where f 1, f 2, and f 3 are the number of barrels of oil provided to each factory, and x 1, x 2, and x 3 are the outputs of each product, respectively. We can then use these equations to find the amount of oil to give to each factory to meet a given demand.

4. The first demand to meet is 5000 gallons of motor oil, 8500 gallons of diesel oil, and 10000 gallons of gasoline. Solving the system of equations gives us the input for each factory in barrels of oil, as shown in the table below: Factory 1Factory 2Factory 3 Demand/Total Barrels Motor Oil20445000 Diesel Oil101458500 Gasoline551210000 Barrels of Oil (exact) 48.75331.25675.001055 Barrels of Oil (approximate) 493326751056

5. Next, we are given the situation where the demand has doubled. The solution for the new system of equations is shown below: It is observed that if demand doubles, then the barrels of oil doubles as well. Factory 1Factory 2Factory 3 Demand/Total Barrels Motor Oil204410000 Diesel Oil1014517000 Gasoline551220000 Barrels of Oil (exact) 97.5662.513502110 Barrels of Oil (approximate) 9866313502111

6. We are now given a different set of demands to meet: 2000 gallons of motor oil, 4000 gallons of diesel oil, and 4000 gallons of gasoline. Factory 1Factory 2Factory 3 Demand/Total Barrels Motor Oil20442000 Diesel Oil101454000 Gasoline55124000 Barrels of Oil (exact) 12.5187.5250450 Barrels of Oil (approximate) 13188250451

7. We now add the first and second demand to get 7000 gallons of motor oil, 12500 gallons of diesel oil, and 14000 gallons of gasoline. The solution to the added demands is simply the solution of both demands added together. Each of these solutions is said to be unique because there is only one combination of inputs that allows the company to meet the required demand. Factory 1Factory 2Factory 3 Demand/Total Barrels Motor Oil20447000 Diesel Oil1014512500 Gasoline551214000 Barrels of Oil (exact) 61.25518.759251505 Barrels of Oil (approximate) 625199251506

8. Sensitivity analysis is performed by varying coefficients in the production equations and noting how this affects the solution. We performed sensitivity analysis on our factories by varying the output of the first factory by ±3% individually for each product and recalculating the output for each variation. A system that does not change much for modest changes in parameters is said to be robust.

9. Factory 1Factory 2Factory 3Demand/Total Barrels Motor Oil20.6445000 Diesel Oil101458500 Gasoline551210000 Barrels of Oil (exact)47.06734332.36785675.235341054.67053 Barrels of Oil (approximate) 483336761057 Percentage Change-0.03% Factory 1Factory 2Factory 3Demand/Total Barrels Motor Oil 19.444 5000 Diesel Oil 10145 8500 Gasoline 5512 10000 Barrels of Oil (exact) 50.55743330.04926674.747211055.3539 Barrels of Oil (approximate) 513316751057 Percentage Change 0.03%

10. Factory 1Factory 2Factory 3Demand/Total Barrels Motor Oil20445000 Diesel Oil10.31458500 Gasoline551210000 Barrels of Oil (exact)48.92122329.90467675.489211054.3151 Barrels of Oil (approximate) 493306761055 Percentage Change-0.06% Factory 1Factory 2Factory 3Demand/Total Barrels Motor Oil20445000 Diesel Oil9.71458500 Gasoline551210000 Barrels of Oil (exact)48.57997332.58595674.514201055.68012 Barrels of Oil (approximate) 493336751057 Percentage Change0.06%

11. Factory 1Factory 2Factory 3Demand/Total Barrels Motor Oil20445000 Diesel Oil101458500 Gasoline5.1551210000 Barrels of Oil (exact)48.8599331.4332674.26711054.5602 Barrels of Oil (approximate) 493366751060 Percentage Change-0.04% Factory 1Factory 2Factory 3Demand/Total Barrels Motor Oil20445000 Diesel Oil101458500 Gasoline4.8551210000 Barrels of Oil (exact)48.6406331.0676675.72961055.4377 Barrels of Oil (approximate) 493366751060 Percentage Change0.04%

12. In our next situation, factory #3 is shut down by the EPA. We need to find a way for the first two factories to meet the original demand. We can do this by using a least squares approximation. Let A be a 3 x 2 matrix representing the outputs of the first two factories and b the demand vector. A least squares approximation can be found for the input vector x with the following formula: (A T *A) -1 A T *b = x This method gives an approximation for the inconsistent solution, but because it is an approximation, the estimate can be inexact. The table in the next slide demonstrates that in this case it is impossible for the two factories to produce exactly enough gasoline without severely overproducing diesel oil and motor oil.

13. Factory 1Factory 2Demand/Total Barrels Motor Oil2045000 Diesel Oil10148500 Gasoline5510000 Barrels of Oil (exact)145.8075646.7391792.5466 Barrels of Oil (approximate)146647793

14. Factory 1Factory 2Factory 3Factory 4 Demand/Total Barrels Motor Oil204445000 Diesel Oil1014558500 Gasoline5512 10000 Barrels of Oil (exact) 48.75331.25675 - tt1055 Barrels of Oil (approximation) 49332675 – tt1056 The situation with the EPA has caused enough concern that the CEO of our company is considering buying another plant identical to the third. We can create another system of equations for this situation and solve it for our original demand:

15. Factory 1Factory 2Factory 3Paraffin Paraffin Produced147166013503157 Each factory produces a certain amount of paraffin as waste per barrel of oil used. Specifically, factory #1 produces 3 gallons per barrel of oil, while factory #2 produces 5 gallons, and factory #3 produces 2 gallons. Supposing our petroleum company finds a candle company that will purchase the waste paraffin, we can use the amount of oil used for the original demand to determine how much paraffin can be supplied to the candle company.

16. Due to aging equipment and high labor costs, the company is considering selling factory #1. They want to know how this would affect production capability. Reverting to two factories would prevent the company from meeting the initial demand. This is because the initial demand is not a linear combination of the output of the two factories. The company could again attempt to approximate production, but as shown earlier, approximation can result in extreme overproduction or underproduction.

17. With two factories producing three products, the demands that can be satisfied are severely limited. When a system of equations has more equations than variables, that system is said to be overdetermined. An overdetermined system usually has no solutions. Eliminating the first plant would produce such a system. In this situation, if the company added another plant that was identical to plant #2 or #3, the system would still be generally inconsistent since the output of the new plant would be a scalar multiple of an existing plant. To provide versatility, the output of any new factories should be linearly independent of existing ones.

18. Conversely, when a system of equations has more variables than equations, that system is said to be underdetermined. The earlier example with a 4 th factory was an underdetermined system. Underdetermined systems generally have an infinite number of solutions. As far as our petroleum company is concerned, having more factories than products would allow for increased flexibility when meeting demand, especially if the output of the additional factories were linearly independent.

19. Finally, when a system of equations has the same amount of variables as equations, it generally has a unique solution. The first few situations in this presentation were all examples of unique solutions.

20. In conclusion, it is best for the company’s production to be an underdetermined system. This allows for increased flexibility, since a problem in one factory doesn’t prohibit the company from meeting demand. A unique solution to the company’s production is also good, but in this case a factory shutdown will put production into an overdetermined situation, in which case it is impossible for the company to meet demand without overproducing any products.