MAT540 Week 8 Homework Chapter 4 1. Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday, and she must determine how much beer to stock. Betty stocks three brands of beer- Yodel, Shotz, and Rainwater. The cost per gallon (to the tavern owner) of each brand is as follows: Brand Cost/Gallon Yodel $1.50 Shotz 0.90 Rainwater 0.50 The tavern has a budget of $2,000 for beer for Super Bowl Sunday. Betty sells Yodel at a rate of $3.00 per gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on past football games, Betty has determined the maximum customer demand to be 400 gallons of Yodel, 500 gallons of shotz, and 300 gallons of Rainwater. The tavern has the capacity to stock 1,000 gallons of beer; Betty wants to stock up completely. Betty wants to determine the number of gallons of each brand of beer to order so as to maximize profit. a. Formulate a linear programming model for this problem. b. Solve the model by using the computer. 2. As result of a recently passed bill, a congressman's district has been allocated $3 million for programs and projects. It is up to the congressman to decide how to distribute the money. The congressman has decide to allocate the money to four ongoing programs because of their importance to his district- a job training program, a parks project, a sanitation project, and a mobile library. However, the congressman wants to distribute the money in a manner that will please the most voters, or, in other words, gain him the most votes in the upcoming election. His staff's estimates of the number of votes gained per dollar spent for the various programs are as follows Eagle Tavern Cost/Gallon Selling price profit Capacity Demand for Yodel Demand for Shotz Demand for Rainwater Yodel 1.5 3 Shotz 0.9 2.5 Rainwater 0.5 1.75 1 1 0 0 1 0 1 0 1 0 0 1 # of Gallons Yodel Decision Variables Objective Function # of Gallons Shotz # of Gallons Rainwater = <= <= <= 1000 400 500 300 congressman's district Program Votes/ Dollar Job training Parks 0.03 0.08 Sanitation 0.05 Mobile library 0.03 Constraints <= <= <= <= <= >= = spending Decision Variables Objective Maximize Votes 900000 900000 900000 900000 3000000 Job training Parks Sanitation Mobile library Comments Job training Parks Sanitation Mobile library Parks <= Sanitation + Mobile Library Job training >= Sanitation all four job catagories Anna Broderick Lunch Menu Calories Protein Carbohydrates (per lb.) Chicken Fish Ground beef Dried beans Lettuce Potatoes Milk (2%) Iron (mg/lb.) (g/lb.) (g/lb.) Fat (g/lb.) Chol-esterol Cost (mg/lb.) 4.2 3.1 0.25 3.2 0.4 2.25 0.2 17 85 82 10 6 10 16 0 0 0 30 0 70 22 30 5 75 3 0 0 10 180 90 350 0 0 0 20 >= 1300 <= 2100 Constraints 500 480 840 590 40 450 220 >= 4 >= 30 >= 60 >= 15 <= 55 $/lb. 0.85 3.35 2.45 0.85 0.7 0.45 0.82 <= 35 Minimize Cost ($) (a) Objective (b) if a serving of each of the the food items (other than milk) was limited to no more than a half pound, what effect would this have on the solution? Decision Variables Objective Chicken Fish Ground beef Dried beans Lettuce Potatoes Milk (2%) <= <= <= <= <= <= Minimize Cost ($) Answer: No feasible solution found 0.5 0.5 0.5 0.5 0.5 0.5 Jefferson County Regional HospitalScheduling Let Xi = number of nurses that begin their 8 hour shift in period I (I = 1,2,3,4, ...., 12) period 1 12:00AM -- 2:00Am period 2 2:00AM -- 4:00 AM etc DV x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 Objective Function Period 1 2 3 4 5 6 7 8 9 10 11 12 # of Nurses working >= >= >= >= >= >= >= >= >= >= >= >= Minimum # of Nurses 30 20 40 50 60 80 80 70 70 60 50 50 Videotechnics Company (Multi Production and Inventory) Let Ri = Regular production in month I, where I = 1,2,3,4,5 Oi = Overtime production in month I, where I = 1,2,3,4,5 I_i = Inventory at end of month I, where I = 1,2,3,4 R1 R2 R3 R4 R5 <= <= <= <= <= 2000 2000 2000 2000 2000 Month 1 2 3 4 5 = = = = = 1200 2100 2400 3000 4000 Objective function O1 O2 O3 O4 O5 Cost for recorders produced during regular working hours is $10 Cost for recorders produced during regular working hours is $15 <= <= <= <= <= 600 600 600 600 600 I_1 I_2 I_3 I_4