System Optimization and Analysis Group Problem Set 5 Problem 1: An oil company produces three brands of oils: Regular, Multigrade and Supreme. Each brand of oil is composed of one or more of four crude stocks, each having a different viscosity index. The relevant data is: Crude Stock 1 2 3 4 Viscosity index 20 40 30 55 Cost ($/barrel) 7.1 8.5 7.7 9.00 Supply per day (barrels) 1000 1100 1200 1100 Each brand of oil must satisfy a minimum standard for viscosity index and each brand thus sells at a different price. The relevant data concerning the three brands are: Brand Regular Multigrade Supreme Minimum Viscosity Index 25 35 50 Selling Price ($/barrel) 8.50 9.00 10.00 One of three scenarios can occur: Regular Multigrade Supreme Low demand (prob. = 0.2) 1800 1300 650 Med. demand (prob. = 0.5) 2000 1500 750 High demand (prob. = 0.3) 2100 1600 800 The company cannot sell more than its demand. Also any unused crude oil is disposed off at no cost. The decision of buying crude stock is taken well in advance and thus the company does not know what scenario holds at the time of buying crude stock. However at the time of production of oil the company knows the exact demand. How much of crude stock must the company buy in order to maximize its expected profit? Problem 2: The Auto company of America (ACA) produces 4 types of cars: subcompact, compact, intermediate and luxury. ACA also produces trucks and vans. Vendor capacities limit total production capacity to at most 1,200,000 vehicles per year. Subcompacts and compacts are built together in a facility with a total annual capacity of 620,000 cars. Intermediate and luxury are built together in a facility with a total annual capacity of 400,000 cars; and the truck/ van facility has a capacity of 275,000. Profit margins and fuel efficiencies are summarized below: Type Subcompact Compact Intermediate Luxury Truck Van Profit Margin ($/vehicle) 150 225 250 500 400 200 Fuel Efficiency (MPG) 40 34 15 12 20 25 The company expects the demand level to be one of the three: high demand (with probability = 0.2), medium demand (with probability = 0.5) and low demand (with probability = 0.3). Note that the company does not know which scenario holds until the selling season arrives, long after it has taken its production decisions. The demands for these scenarios is: Type Subcompact Compact Intermediate Luxury Truck Van High Demand (in ‘000) 700 500 400 325 425 200 Medium Demand(in ‘000) 600 400 300 225 325 100 Low Demand (in ‘000) 550 350 250 175 275 50 If the company ends up making more cars than the demand, it can sell its excess stock through a discount channel and earn half the normal profit; for example surplus compact cars would yield profits of $112.5/ vehicle. The average fleet fuel efficiency must be atleast 27 MPG. What should the production plan of the company be so that its: a) Expected revenue is maximized b) Minimum revenue is maximized