The Rayhoon Restaurant: Brayden and Behrad were roommates. They decided to open a restaurant in the small town that they were living. They bought an old home for their new restaurant which they named Rayhoon. Not knowing the taste of their customers, they decided to serve only two full course meals each night, one with pork and the other with chicken. They estimated that they would sell a maximum of 60 meals each night. Each chicken dinner requires 15 minutes to prepare, and each pork dinner takes twice as long. There is a total of 20 hours of kitchen staff labor available each day. They thought that they would sell at least 3 chicken dinners for every two pork dinners. They also believe that at least 10 percent of their customers will order pork dinners. The profit from each chicken dinner will be $12, and the profit from a pork dinner will be $16. Formulate a linear programming model for B & B, that will help them estimate the number of meals they should prepare each night and solve this model by computer. They are considering investing in some advertising to increase the maximum number of meals they serve. They estimate that if they spend $30 per day on a newspaper ad, it will increase the maximum number of meals they serve per day from 60 to 70. Should they make the investment? They are concerned about the reliability of their kitchen staff. They estimate that on some evenings they could have a staff reduction of as much as 5 hours. How would this affect their profit level? The final question they would like to explore is raising the price of the chicken dinner. Brayden has suggested a price increase that will increase profit for the chicken dinner to $14. Would this be acceptable, and how much additional profit would be realized? Ho: = .602 Ha: >.602 Checking a normal table the critical value is z= 1.645 260/400= 0.65 z= (.65.602)/(.602*(1.602)/400)= 1.96 Since 1.96>1.645 we reject the null hypothesis. There is evidence to support the proposition that the percentage of inmates serving time for drug offenses is higher than .602. Eagle Tavern Cost/Gallon Selling price profit Capacity Demand for Yodel Demand for Shotz Demand for Rainwater Decision Variables Objective Function Yodel 1.5 3 1.5 1 1 0 0 Shotz 0.9 2.5 1.6 1 0 1 0 Rainwater 0.5 1.75 1.25 1 0 0 1 # of Gallons # of Gallons # of Gallons Yodel Shotz Rainwater 400 500 100 1525 1000 400 500 100 = <= <= <= 1000 400 500 300 congressman's district Program Votes/ Dollar Job training Parks 0.03 0.08 Sanitation 0.05 Mobile library 0.03 900000 Constraints Comments Job training 900000 900000 300000 900000 900000 Decision Variables Objective Maximize Votes 900000 900000 900000 900000 1200000 900000 Parks Sanitation Mobile library Parks <= Sanitation + Mobile Library Job training >= Sanitation 3000000 spending <= <= <= <= <= >= = 3000000 all four job catagories Job training Parks Sanitation Mobile library 153000 900000 900000 900000 300000 Anna Broderick Lunch Menu Calories Constraints Minimize Cost ($) Protein Carbohydrates (per lb.) Chicken Fish Ground beef Dried beans Lettuce Potatoes Milk (2%) Iron (mg/lb.) (g/lb.) (g/lb.) 500 480 840 590 40 450 220 1300 >= 1300 <= 2100 4.2 17 0 3.1 85 0 0.25 82 0 3.2 10 30 0.4 6 0 2.25 10 70 0.2 16 22 6.4169401641 31.09565739 65.437162297 >= >= >= 4 30 60 Fat (g/lb.) 30 5 75 3 0 0 10 15 >= 15 <= 55 Chol-esterol Cost (mg/lb.) 180 90 350 0 0 0 20 35 <= 35 $/lb. 0.85 3.35 2.45 0.85 0.7 0.45 0.82 (a) Objective (b) if a serving of each of the the food items (other than milk) was limited to no more than a half pound, what effect would this have on the solution? Decision Variables Objective Chicken Fish Ground beef Dried beans Lettuce Potatoes Milk (2%) Minimize Cost ($) Answer: 2.11 0.00 0.00 0.08 1.98 0.00 0.00 0.27 <= <= <= <= <= <= 2.11 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 30 0 10 10 50 10 10 0 50 0 0 0 170 # of Nurses working 30 30 40 50 70 80 80 70 70 60 50 50 >= >= >= >= >= >= >= >= >= >= >= >= 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 Objective function <= <= <= <= <= 1200 2100 2400 3000 4000 2000 2000 2000 2000 2000 = = = = = 1200 2100 2400 3000 4000 152300 O1 O2 O3 O4 O5 Cost for recorders produced during regular working hours is $10 Cost for recorders produced during regular working hours is $15 300 600 600 600 600 <= <= <= <= <= 600 600 600 600 600 I_1 I_2 I_3 I_4 1100 1600 1800 1400 Ho: = .602 Ha: >.602 Checking a normal table the critical value is z= 1.645 260/400= 0.65 z= (.65.602)/(.602*(1.602)/400)= 1.96 Since 1.96>1.645 we reject the null hypothesis. There is evidence to support the proposition that the percentage of inmates serving time for drug offenses is higher than .602. Eagle Tavern Cost/Gallon Selling price profit Capacity Demand for Yodel Demand for Shotz Demand for Rainwater Decision Variables Objective Function Yodel 1.5 3 1.5 1 1 0 0 Shotz 0.9 2.5 1.6 1 0 1 0 Rainwater 0.5 1.75 1.25 1 0 0 1 # of Gallons # of Gallons # of Gallons Yodel Shotz Rainwater 400 500 100 1525 1000 400 500 100 = <= <= <= 1000 400 500 300 congressman's district Program Votes/ Dollar Job training Parks 0.03 0.08 Sanitation 0.05 Mobile library 0.03 900000 Constraints Comments Job training 900000 900000 300000 900000 900000 Decision Variables Objective Maximize Votes 900000 900000 900000 900000 1200000 900000 Parks Sanitation Mobile library Parks <= Sanitation + Mobile Library Job training >= Sanitation 3000000 spending <= <= <= <= <= >= = 3000000 all four job catagories Job training Parks Sanitation Mobile library 153000 900000 900000 900000 300000 Anna Broderick Lunch Menu Calories Constraints Minimize Cost ($) Protein Carbohydrates (per lb.) Chicken Fish Ground beef Dried beans Lettuce Potatoes Milk (2%) Iron (mg/lb.) (g/lb.) (g/lb.) 500 480 840 590 40 450 220 1300 >= 1300 <= 2100 4.2 17 0 3.1 85 0 0.25 82 0 3.2 10 30 0.4 6 0 2.25 10 70 0.2 16 22 6.4169401641 31.09565739 65.437162297 >= >= >= 4 30 60 Fat (g/lb.) 30 5 75 3 0 0 10 15 >= 15 <= 55 Chol-esterol Cost (mg/lb.) 180 90 350 0 0 0 20 35 <= 35 $/lb. 0.85 3.35 2.45 0.85 0.7 0.45 0.82 (a) Objective (b) if a serving of each of the the food items (other than milk) was limited to no more than a half pound, what effect would this have on the solution? Decision Variables Objective Chicken Fish Ground beef Dried beans Lettuce Potatoes Milk (2%) Minimize Cost ($) Answer: 2.11 0.00 0.00 0.08 1.98 0.00 0.00 0.27 <= <= <= <= <= <= 2.11 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 30 0 10 10 50 10 10 0 50 0 0 0 170 # of Nurses working 30 30 40 50 70 80 80 70 70 60 50 50 >= >= >= >= >= >= >= >= >= >= >= >= 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 Objective function <= <= <= <= <= 1200 2100 2400 3000 4000 2000 2000 2000 2000 2000 = = = = = 1200 2100 2400 3000 4000 152300 O1 O2 O3 O4 O5 Cost for recorders produced during regular working hours is $10 Cost for recorders produced during regular working hours is $15 300 600 600 600 600 <= <= <= <= <= 600 600 600 600 600 I_1 I_2 I_3 I_4 1100 1600 1800 1400