Question: Dinoco produces three types of oil products: domestic oil, petrol for car and aviation fuel. Each product must meet the minimum average octane regulation. The average octane levels must be at least 5 for domestic oil, 10 for petrol and 8 for aviation fuel. To produce these products, Dinoco purchases two types of crude oil; Arabian Night (at $120 per barrel) and Brent (at $90 per barrel). Each day, at most 8,000 barrels of each type of crude oil can be purchased. Crude Oils must be distilled to be used for the production of Dinoco’s three products. Each day, at most 12,000 barrels of crude oil can be distilled. It costs $20 to distil a barrel of oil (whether it is Brent or Arabian Night). The results of distillation are as follows:

• Each barrel of Arabian Night yields 0.6 barrels of naphtha, 0.3 barrel of distilled 1, and 0.1 barrel of distilled 2.

• Each barrel of Brent yields 0.4 barrels of naphtha, 0.4 barrel of distilled 1, and 0.2 barrel of distilled 2.

Dinoco’s three products are produced by mixing naphtha, distilled 1, and distilled 2. Due to the regulation, however, distilled naphtha cannot be used for aviation oil. The octane level for types of distilled are 12, 10, and 4 for naphtha, distilled 1 and distilled 2, respectively. All domestic oil produced can be sold at $200 per barrel; all petrol produced sold at $300 per barrel, and all aviation fuel sold at $280 per barrel. For company’s product mix consideration, at least 2,500 barrels of each product must be produced daily.

1. Formulate an LP to maximise Dinoco’s daily profit. You must list any assumption(s) you have made in your formulation. And, use Management Scientist to find the optimal solution.
2. Suppose you were the person at Dinoco who just solved the above LP problem. The chairman of Dinoco just dropped you an email asking the following questions:

1. Among the three products, which one is most profitable?
2. Dinoco is thinking about expanding the capacity of distillation (currently 12,000 barrel per day). Could you give me a recommendation on how much of capacity we should expand and at what price?
3. (Confidential) Based on the report from our lobbyists at Washington DC, it is quite likely the regulation of "no naphtha for aviation fuel" will be lifted (i.e. removed) by next month. What will be the impact on our profit?
4. We are going to relax our product mix constraint just a bit. Starting from next week, at least 2450 barrels of each product must be produced daily. Will there be any change in our optimal solution?
5. What is the maximum price we can pay for each crude oil and what is the minimum selling price of each product to keep the current optimal solution still optimal?

Interpret the results from the solution output of Management Scientist and answer the chairman’s questions. For each question, explain whether you have to re-formulate and re-solve the LP problem to answer the question. (20 marks)

(c) Sir Humphrey, the chairman of Brit-oil, is a long time friend and personal rival of the chairman of Dinoco. It just happens that Sir Humphrey is visiting Dinoco while you are preparing the answers to the questions in Question 3(c). The chairman of Dinoco brings Sir Humphrey to you, and introduces you to him. "Humphrey, he/she is the one who is doing the optimisation for me!"

The response from Sir Humphrey, however, is not exactly an encouraging one. He is saying to the chairmen of Dinoco, "Don’t believe this non-sense of LP mumbo-jumbo. I have a daily capacity of 20,000 barrels for refinery. My LP guy told me that we should produce only 16,000 barrels of oil to maximise profit. It means I have to idle 20% of my capacity to maximise my profit. How can it be true? Why shouldn’t I produce more when I can? Don’t believe any of these guys." How would respond to Sir Humphrey’s comment? Explain to Sir Humphrey how his LP guy would be right. (10 marks)

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Solution: The downloadable solution consists of 9 pages
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