# Control Chart, Present Value, Linear Programming

1. A state department of tourism and recreation collects data on the number of cars with out-of-state license plates in a state park. (The group's position is that more out-of-state plates means the state's advertising programs are working.) The sample size is fixed at n=100 each day. Data from the previous 20 days indicate the following number of out-of-state license plates:

Day Out-of-state Plates

1 24

2 33

3 18

4 21

5 26

6 31

7 15

8 18

9 23

10 13

11 22

12 15

13 25

14 33

15 16

16 22

17 34

18 10

19 23

20 13

a. Calculate the overall proportion of "tourists" (cars with out-of-state plates).

b. Calculate the LCL and UCL for these data.

c. Plot the control chart. Are all points within the control limits?

d. What action do you suggest for improving this process?

2. Cartons of Plaster of Paris are supposed to weigh exactly 32 oz. Inspectors want to develop process control charts. They take ten samples of six boxes and weigh them. Based on the following data, compute the lower and upper control limits and determine whether the process is in control.

Sample Mean Range

1 33.8 1.1

2 34.6 0.3

3 34.7 0.4

4 34.1 0.7

5 34.2 0.3

6 34.3 0.4

7 33.9 0.5

8 34.1 0.8

9 34.2 0.4

10 34.4 0.3

3. Larry's boat shop wants to monitor the number of blemishes in the paint of each boat. Construct a c-chart to determine if their paint process is in control using the following data.

Sample Number Number of

Defects

1 3

2 4

3 2

4 1

5 3

6 2

7 1

8 4

9 2

10 3

4. A firm is about to undertake the manufacture of a product, and is weighing the process configuration options. There are two different job shop processes under consideration, as well as a repetitive focus. The small job shop has fixed costs of $3,000 per month, and variable costs of $10 per unit. The larger job shop has fixed costs of $12,000 per month and variable costs of $2 per unit. A repetitive focus plant has fixed costs of $60,000 and variable costs of $0.50 per unit.

a. If demand were 1,000 units per month, what would cost be under each process configuration?

b. At what level of demand does the large job shop become cheaper than the small job shop?

c. At what level of demand does the repetitive shop become cheaper than the larger job shop?

5. The local convenience store makes personal pan pizzas. Currently, their oven can produce 50 pizzas per hour. It is has a fixed cost of $2,000, and a variable cost of $0.25 per pizza. The owner is considering a bigger oven that can make 75 pizzas per hour. It has a fixed cost of $3,000, but a variable cost of $0.20 per pizza.

a. What is the break-even point?

b. If the owner expects to sell 9,000 pizza, should he get the new oven?

6. Health Care Systems of the South is about to buy an expensive piece of diagnostic equipment. The company estimates that it will generate uniform revenues of $500,000 for each of the next eight years. What is the present value of this stream of earnings, at an interest rate of 6%? What is the present value if the machine lasts only six years, not eight? If the equipment cost $2,750,000, should the company purchase it?

7. A new machine tool is expected to generate receipts as follows: $5,000 in year one; $3,000 in year two, nothing in the next year, and $2,000 in the fourth year. At an interest rate of 6%, what is the present value of these receipts? Is this a better present value than $2,500 each year over four years? Explain.

8. A toy manufacturer has three different mechanisms that can be installed in a doll that it sells. The different mechanisms have three different setup costs (overheads) and variable costs, and therefore the profit from the dolls is dependent on the volume of sales. The anticipated payoffs are as follows.

Light Demand Moderate Demand Heavy Demand

Probability 0.25 0.45 0.3

Wind-up action $325,000 $190,000 $170,000

Pneumatic action $300,000 $420,000 $400,000

Electrical action ($400,000) $240,000 $800,000

a. What is the EMV of each decision alternative?

b. Which action should be selected?

c. What is the expected value under certainty?

d. What is the expected value of perfect information?

9. (A) The Queen City Nursery manufactures bags of potting soil from compost and topsoil. Each cubic foot of compost costs 12 cents and contains 4 pounds of sand, 3 pounds of clay, and 5 pounds of humus. Each cubic foot of topsoil costs 20 cents and contains 3 pounds of sand, 6 pounds of clay, and 12 pounds of humus. Each bag of potting soil must contain at least 12 pounds of sand, 12 pounds of clay, and 10 pounds of humus. Explain how this problem meets the conditions of a linear programming problem. Plot the constraints and identify the feasible region. Graphically or with corner points find the best combination of compost and topsoil that meets the stated conditions at the lowest cost per bag. Identify the lowest cost possible.

(B) A stereo mail order center has 8,000 cubic feet available for storage of its private label loudspeakers. The ZAR-3 speakers cost $295 each and require 4 cubic feet of space; the ZAR-2ax speakers cost $110 each and require 3 cubic feet of space; and the ZAR-4 model costs $58 and requires 1 cubic foot of space. The demand for the ZAR-3 is at most 20 units per month. The wholesaler has $100,000 to spend on loudspeakers this month. Each ZAR-3 contributes $105, each ZAR-2ax contributes $50, and each ZAR-4 contributes $28. The objective is to maximize total contribution. Write out the objective and the constraints. Do Not Solve.

10. Schriever Cabinet Specialties produces wall shelves, bookends, and shadow boxes. It is necessary to plan the production schedule for next week. The wall shelves, bookends, and shadow boxes are made of oak, of which the company currently has 600 board feet. A wall shelf requires 4 board feet, bookends require 2 board feet, and a shadow box requires 3 board feet. The company has a power saw for cutting the oak boards; a wall shelf requires 30 minutes, book ends require 15 minutes, and a shadow box requires 15 minutes. The power saw is available for 32 hours next week. After cutting, the pieces are hand finished in the finishing department, which consists of four skilled and experienced craftsmen, each of whom can complete any of the products. A wall shelf requires 30 minutes of finishing, bookends require 60 minutes, and a shadow box requires 90 minutes. The finishing department is expected to operate for 80 hours next week. Wall shelves sell for $29.95 and have a unit variable cost of $17.95; bookends sell for $11.95 and have a unit variable cost of $4.95; a shadow box sells for $16.95 and has a unit variable cost of $8.95. The company has a commitment to produce 10 wall shelf units for the Languages and Literature Department. The firm normally operates to achieve maximum contribution.

a. What are the decision variables of this problem (name them)?

b. What are the constraints of this problem (name them)?

c. Write out the objective and the constraints for this problem.

d. Solve the problem using Excel Solver

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Answers 10 questions on Control Charts, p chart, X bar chart, R chart, c chart, Demand, Present Value, EMV, expected value under certainty, expected value of perfect information, Linear Programming.