4. (25 marks) A microbrewery is located in a town which has become isolated after a powerful storm. Until the bridge on the highway into town can be repaired, the supply of hops, yeast, and barley is limited to 20 kg, 18 kg, and 19 kg respectively. A linear optimization model has been made in which the variables are the number of litres of light beer, lager and ale to be made until the bridge can be repaired. The profit margins per litre for the three types of beer are $2.50 for light, $2.30 for lager, and $2.80 for ale. To make a litre of light beer requires 9 grams (g) of hops, 8 g of yeast, and 6 g of barley. A litre of lager requires 7 g of hops, 8 g of yeast, and 12 g of barley. Each litre of ale requires 6 g of hops, 8 g of yeast, and 10 g of barley. There is also an overall production constraint; using 5 units for light, 8 for lager, and 4 for ale, the total cannot exceed 11,000. (a) Give the algebraic model for this situation. (b) Solve this model in LINGO or the Excel Solver, and obtain the sensitivity re- port. (c) Checking the allowable ranges, determine what would happen to the objective function value in each of the following situations (considered independently). (Assume that all proposed changes are feasible.) (i) A bakery in the town has some yeast which is not needed in the short term. They are willing to sell up to 1000 grams of yeast at a price of 20 cents per gram. Should the brewery take the deal, and if so, by how much would the profit increase? (ii) Up to 300 grams of any combination of hops, yeast and barley could be dropped from a drone. What should the drone deliver? Ignoring the cost of the drone, by how much would the profit increase? (iii) Something happens to the production process which causes the cost of making light beer to increase by 35 cents per litre. Is this in the allowable range? If so, what happens to the profit? (iv) The plant manager hikes in hip waders up to the main highway, and re- turns with 750 grams of yeast, but it is discovered that water damage from the storm has destroyed 2 kg of barley. Are these two changes allowable according to the 100% rules? If so, what happens to the profit? (v) The price per litre for all three types of beer rises by 17 cents. 4. (25 marks) A microbrewery is located in a town which has become isolated after a powerful storm. Until the bridge on the highway into town can be repaired, the supply of hops, yeast, and barley is limited to 20 kg, 18 kg, and 19 kg respectively. A linear optimization model has been made in which the variables are the number of litres of light beer, lager and ale to be made until the bridge can be repaired. The profit margins per litre for the three types of beer are $2.50 for light, $2.30 for lager, and $2.80 for ale. To make a litre of light beer requires 9 grams (g) of hops, 8 g of yeast, and 6 g of barley. A litre of lager requires 7 g of hops, 8 g of yeast, and 12 g of barley. Each litre of ale requires 6 g of hops, 8 g of yeast, and 10 g of barley. There is also an overall production constraint; using 5 units for light, 8 for lager, and 4 for ale, the total cannot exceed 11,000. (a) Give the algebraic model for this situation. (b) Solve this model in LINGO or the Excel Solver, and obtain the sensitivity re- port. (c) Checking the allowable ranges, determine what would happen to the objective function value in each of the following situations (considered independently). (Assume that all proposed changes are feasible.) (i) A bakery in the town has some yeast which is not needed in the short term. They are willing to sell up to 1000 grams of yeast at a price of 20 cents per gram. Should the brewery take the deal, and if so, by how much would the profit increase? (ii) Up to 300 grams of any combination of hops, yeast and barley could be dropped from a drone. What should the drone deliver? Ignoring the cost of the drone, by how much would the profit increase? (iii) Something happens to the production process which causes the cost of making light beer to increase by 35 cents per litre. Is this in the allowable range? If so, what happens to the profit? (iv) The plant manager hikes in hip waders up to the main highway, and re- turns with 750 grams of yeast, but it is discovered that water damage from the storm has destroyed 2 kg of barley. Are these two changes allowable according to the 100% rules? If so, what happens to the profit? (v) The price per litre for all three types of beer rises by 17 cents
