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Quantitative Decision Making

Linear Programing

Shale-Bituminous Processors

Shale Bituminous Processors (SBP) is a medium-sized oil company. Using its own patented process, it produces low- and high-sulphur crude from coal and shale. These oils are used by refiners who process them into products such as gasoline, jet fuel, industrial lubricating oils, greases and much more. Marketing director William Stanford is reviewing an order from a best customer for 10,000 gallons of low-sulphur and 5,000 gallons of high-sulphur crude. Normally such a small order would be sent directly to the Production Department, but something is bothering him. Company president T.J. Bass mentioned recently that this year was going to have record profits, with high production demand and operations running at nearly full capacity. Checking, Mr. Stanford learns that only 100 tons of coal and 150 tons of shale remain unallocated. Mr. Stanford doesn't want to turn down the order, losing its profit and alienating a good customer. But inability to produce the ordered quantities would damage SBP's credibility and reputation.
Mr. Stanford asks analyst Barry Ginocchio to determine whether or not SBP has sufficient capacity to fulfil the order and, if so, to estimate its corresponding profit. Barry checks with the Purchasing Department and learns that coal costs $20 per ton and the shale $25 per ton. The Marketing Department advises him that low-sulphur crude sells for $.50 per gallon and high sulphur crude for $.30. The Production Department tells him there are two ways to make low-sulphur and high-sulphur crude oil: gasification and pressurisation batch processes. A gasification batch requires 1 ton of coal and 2 tons of shale to yield 100 gallons of low-sulphur crude and 200 gallons of high-sulphur crude. Under pressurisation, each batch requires 2 tons of coal and 1 ton of shale to yield 150 gallons of low-sulphur crude and 100 gallons of high-sulphur crude.
Barry recently received a bachelor's degree in business from State University. From his quantitative methods course, he knows that his data can be used with linear programming to determine if the customer's order can be fulfilled with the coal and shale available and, if so, to calculate its corresponding profit.
The objective is to determine the number of gastification and pressurization batches and the number of gallons of low-sulphur and high-sulphur crude to be produced during each of the next two quarters of the coming year that will minimize costs. The quarterly orders for crude are given in the Table 1. The initial inventory of both high-sulphur and low-sulphur crude is zero Management wants to end the second quarter with 1,000 gallons of each type of crude. Assume now that 100 tons of coal and 100 tons of shale are available each quarter for producing the crudes. The variable labour costs for a gastification batch are $10 and $8 for a pressurization batch. Crude not used in a quarter can be used in future periods. The annual inventory holdings cost is 8% of the value of the crude.
Table : Crude Orders

Quarter Crude Orders (Gallons)
Low-Sulphur High -Sulphur
1 5,000 6,000
2 6,000 11,000


1. Formulate a linear program that will help the refinery manger to make a decision about the number of gasitication and pressurization batches to schedule and the quantities of low-sulphur crude to produce each quarter. Make sure that the decision variables are well defined. Find the optimal solution.

2. The economy is starting to pick up so that how it looks like the orders to be filled next year might be 10% greater than the values shown in the table. How would this effect the original solution

3. Because of a disagreement between management and the union, there will a work slowdown affecting the availability of coal and shale. If this slowdown decreases the availability to 95% of the original estimates, how would it affect management's original plans? Supposing the decrease is 20% what happens in this case?

4. An accident has occurred in the pressurization batch equipment and 20% of the production capacity will be lost during the coming year. Is this a problem? That is, how would such an accident affect the company's production schedule and projected profit? Suppose big accident occurs; yields from pressurization batches are only going o be 50% of that originally anticipated. What happens in this situation?

5. The cost of coal has unexpectedly increased to $26 per ton so that now it costs more than shale ($25). Will it mess up the company's plans? Explain.

a. The refinery manager changes his mind and requires that 2,000 gallons of each crude be in inventory at the end of the second quarter. How does this affect the solution originally obtained?
b. After determining the original solution, it is learned that 1,000 gallons of both low-sulphur and high-sulphur crude are available from a supplier at a 20% discount from their normal prices. The refinery manger might buy the crude as initial inventory at the beginning of the year. Fewer gallons would have to be produced during the year by Shale-Bituminous Processors; she thinks this would decrease cost. Should she make the purchase? Why?
c. After reviewing the original linear program, a company engineer advises the refinery manager that the limited storage capacity for the crude is not included. Tanks can store only 1,900 gallons of each of the crude at the end of the first quarter. Does this affect the original solution? If so, how?
d. An auditor questions Shale-Bituminous Processors bout their 8% holding cost figure. It might be higher or maybe even lower. Examine how the original solution changes if the holding cost is half that value; twice the value.

7. Suppose that coal and shale not used in a quarter are available for future production. How will this alter the original solution?

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Solution Summary

Word document contains context and formulation Excel file contains formulation and optimal solution along with answer, sensitivity and limits reports.