A health and fitness center operates a morning fitness program
from senior citizens. The program includes aerobic exercise, either
swimming or step exercise, followed by a healthy breakfast in the
dining room. A dietitian wants to develop a breakfast that is high
in calories, calcium, protein and fiber, which are especially
important to senior citizens, but low in fat and cholesterol. The
dietitian has selected the following food items with the individual
nutrient contributions and cost shown from which to develop a
breakfast menu.
Breakfast Food Calories Fat (g) Cholesterol (mg) Iron (mg) Calcium (mg) Protein (g) Fiber (g) Cost ($) Bran cereal (cup) 90 0 0 6 20 3 5 0.18 Dry cereal (cup) 110 2 0 4 48 4 2 0.22 Oatmeal (cup) 100 2 0 2 12 5 3 0.10 Oat bran (cup) 90 2 0 3 8 6 4 0.12 Egg 75 5 270 1 30 7 0 0.10 Bacon (slice) 35 3 8 0 0 2 0 0.09 Orange 65 0 0 1 52 1 1 0.40 2% Milk (cup) 100 4 12 0 250 9 0 0.16 Orange juice (cup) 120 0 0 0 3 1 0 0.50 Wheat toast (slice) 65 1 0 1 26 3 3 0.07 Table 2: Nutritional Content of Breakfast Foods The dietician wants the breakfast to include at least 420 calories, 5 milligrams of iron, 400 milligrams of calcium, 20 grams of protein, and 12 grams of fiber. Further, the dietician wants to limit the fat to no more than 11 grams and cholesterol to 20 milligrams.
a) Use the Solver to determine the servings of the breakfast food items that minimizes the cost of the breakfast menu while meeting all of the requirements of the dietician.
b) Starting with the optimal solution to part a, use the SolverTable add-in to see what happens to the optimal cost when the requirements for calcium vary from 300 to 500 in increments of 20 while the requirements for fat vary from 5 to 15 in increments of 2. Are the changes in optimal cost linear with respect to changes in the requirements for fat? Are the changes in optimal cost linear with respect to changes in the requirements for calcium?
3) ] A refinery blends four petroleum components into three grades of gasoline—regular, premium, and low-lead. The maximum quantities available of each component and the cost per barrel are as follows.
Component Maximum Barrels Available/day Cost/barrel 1 5,000 $ 69.00 2 2,400 $ 67.00 3 4,000 $72.00 4 1,500 $ 66.00
To ensure that each gasoline grade retains certain essential characteristics, the refinery has put limits on the percentages of the components in each blend. The limits as well as the selling prices for the various grades are as follows.
Grade Component Specifications Selling Price/barrel Regular At least 40% of 1 At most 20% of 2 At least 30% of 3 $72.00
Premium At least 40% of 3 $78.00 Low-lead At most 50% of 2 At least 10% of 1 $70.00
The refinery wants to produce at least 3,000 barrels of each grade of gasoline. Management wishes to determine the optimal mix of the four components that will maximize profit.
Formulate a linear programming model for this problem and solve it on the computer.
4) OceanEyes manufactures sunglasses. During each of the next 6 months, OceanEyes can sell up to the number of sunglasses given in the table below. Demand that is not met during a month is backlogged at a cost of $20 per unit per month. A pair of sunglasses sells for $100, requires 4 hours of labor, and uses $18 of raw material. At the beginning of month 1, OceanEyes has 8 workers and 20 units of inventory. A worker can work for up to 180 hours per month and is paid $2700 per month (regardless of how many hours he or she works). At the beginning of each month workers can be hired and/or fired. It costs $1500 to hire a worker and $1200 to fire a worker. A holding cost of $2 per pair of sunglasses is assessed against each month’s ending inventory.
Month 1 2 3 4 5 6 Demand 500 600 300 400 300 800
a) Use the Solver to determine how OceanEyes can maximize its profit for the next 6 months.
b) Modify the spreadsheet from part a so that demand that is not met during a month is lost (not backlogged). (Hint: Demand that is not met (lost) does not pay a shortage cost of $20 per unit. It is however a lost sale that is never made up.)
5) Western Family Restaurant offers a variety of low-cost meals and quick service. Other than management, the restaurant operates with two full-time employees who work 8 hours per day. The rest of the employees are part-time employees who are scheduled for 4-hour shifts that begin on the hour. On Saturdays the restaurant is open from 11:00 a.m. to 10:00 p.m. Management wants to develop a schedule for part-time employees that will minimize labor costs and still provide excellent customer service. The average wage rate for the part-time employees is $8.00 per hour. The total number of full-time and part-time employees needed varies with the time of day as shown.
Time Total Number of Employees Needed 11:00 am – Noon 9 Noon – 1:00 pm 9 1:00 pm – 2:00 pm 9 2:00 pm – 3:00 pm 3 3:00 pm – 4:00 pm 3 4:00 pm – 5:00 pm 3 5:00 pm – 6:00 pm 6 6:00 pm – 7:00 pm 12 7:00 pm – 8:00 pm 12 8:00 pm – 9:00 pm 7 9:00 pm – 10:00 pm 7 Table 5: Hourly Employee Requirements
One full-time employee comes on duty at 11:00a.m., work 4 hours, takes an hour off, and returns for another 4 hours. The other full-time employee comes to work at 1:00 pm and works the same 4-hours-on, 1-hour-off, 4-hours-on pattern. Develop a spreadsheet model for this problem. Use the Solver to determine the minimum-cost schedule for part-time employees.
Breakfast Food Calories Fat (g) Cholesterol (mg) Iron (mg) Calcium (mg) Protein (g) Fiber (g) Cost ($) Bran cereal (cup) 90 0 0 6 20 3 5 0.18 Dry cereal (cup) 110 2 0 4 48 4 2 0.22 Oatmeal (cup) 100 2 0 2 12 5 3 0.10 Oat bran (cup) 90 2 0 3 8 6 4 0.12 Egg 75 5 270 1 30 7 0 0.10 Bacon (slice) 35 3 8 0 0 2 0 0.09 Orange 65 0 0 1 52 1 1 0.40 2% Milk (cup) 100 4 12 0 250 9 0 0.16 Orange juice (cup) 120 0 0 0 3 1 0 0.50 Wheat toast (slice) 65 1 0 1 26 3 3 0.07 Table 2: Nutritional Content of Breakfast Foods The dietician wants the breakfast to include at least 420 calories, 5 milligrams of iron, 400 milligrams of calcium, 20 grams of protein, and 12 grams of fiber. Further, the dietician wants to limit the fat to no more than 11 grams and cholesterol to 20 milligrams.
a) Use the Solver to determine the servings of the breakfast food items that minimizes the cost of the breakfast menu while meeting all of the requirements of the dietician.
b) Starting with the optimal solution to part a, use the SolverTable add-in to see what happens to the optimal cost when the requirements for calcium vary from 300 to 500 in increments of 20 while the requirements for fat vary from 5 to 15 in increments of 2. Are the changes in optimal cost linear with respect to changes in the requirements for fat? Are the changes in optimal cost linear with respect to changes in the requirements for calcium?
3) ] A refinery blends four petroleum components into three grades of gasoline—regular, premium, and low-lead. The maximum quantities available of each component and the cost per barrel are as follows.
Component Maximum Barrels Available/day Cost/barrel 1 5,000 $ 69.00 2 2,400 $ 67.00 3 4,000 $72.00 4 1,500 $ 66.00
To ensure that each gasoline grade retains certain essential characteristics, the refinery has put limits on the percentages of the components in each blend. The limits as well as the selling prices for the various grades are as follows.
Grade Component Specifications Selling Price/barrel Regular At least 40% of 1 At most 20% of 2 At least 30% of 3 $72.00
Premium At least 40% of 3 $78.00 Low-lead At most 50% of 2 At least 10% of 1 $70.00
The refinery wants to produce at least 3,000 barrels of each grade of gasoline. Management wishes to determine the optimal mix of the four components that will maximize profit.
Formulate a linear programming model for this problem and solve it on the computer.
4) OceanEyes manufactures sunglasses. During each of the next 6 months, OceanEyes can sell up to the number of sunglasses given in the table below. Demand that is not met during a month is backlogged at a cost of $20 per unit per month. A pair of sunglasses sells for $100, requires 4 hours of labor, and uses $18 of raw material. At the beginning of month 1, OceanEyes has 8 workers and 20 units of inventory. A worker can work for up to 180 hours per month and is paid $2700 per month (regardless of how many hours he or she works). At the beginning of each month workers can be hired and/or fired. It costs $1500 to hire a worker and $1200 to fire a worker. A holding cost of $2 per pair of sunglasses is assessed against each month’s ending inventory.
Month 1 2 3 4 5 6 Demand 500 600 300 400 300 800
a) Use the Solver to determine how OceanEyes can maximize its profit for the next 6 months.
b) Modify the spreadsheet from part a so that demand that is not met during a month is lost (not backlogged). (Hint: Demand that is not met (lost) does not pay a shortage cost of $20 per unit. It is however a lost sale that is never made up.)
5) Western Family Restaurant offers a variety of low-cost meals and quick service. Other than management, the restaurant operates with two full-time employees who work 8 hours per day. The rest of the employees are part-time employees who are scheduled for 4-hour shifts that begin on the hour. On Saturdays the restaurant is open from 11:00 a.m. to 10:00 p.m. Management wants to develop a schedule for part-time employees that will minimize labor costs and still provide excellent customer service. The average wage rate for the part-time employees is $8.00 per hour. The total number of full-time and part-time employees needed varies with the time of day as shown.
Time Total Number of Employees Needed 11:00 am – Noon 9 Noon – 1:00 pm 9 1:00 pm – 2:00 pm 9 2:00 pm – 3:00 pm 3 3:00 pm – 4:00 pm 3 4:00 pm – 5:00 pm 3 5:00 pm – 6:00 pm 6 6:00 pm – 7:00 pm 12 7:00 pm – 8:00 pm 12 8:00 pm – 9:00 pm 7 9:00 pm – 10:00 pm 7 Table 5: Hourly Employee Requirements
One full-time employee comes on duty at 11:00a.m., work 4 hours, takes an hour off, and returns for another 4 hours. The other full-time employee comes to work at 1:00 pm and works the same 4-hours-on, 1-hour-off, 4-hours-on pattern. Develop a spreadsheet model for this problem. Use the Solver to determine the minimum-cost schedule for part-time employees.
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