# Problems

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1.

The Hoylake Rescue Squad receives an emergency call every 1, 2, 3, 4, 5, or 6 hours, according to the following probability distribution. The squad is on duty 24 hours per day, 7 days per week:

Time Between Emergency Calls (hr.)

Probability

1

.05

2

.10

3

.30

4

.30

5

.20

6

.05

1.00

1. Simulate the emergency calls for 3 days (note that this will require a "running," or cumulative, hourly clock), using the random number table.

2. Compute the average time between calls and compare this value with the expected value of the time between calls from the probability distribution. Why are the results different?

3. How many calls were made during the 3-day period? Can you logically assume that this is an average number of calls per 3-day period? If not, how could you simulate to determine such an average?

2.

The time between arrivals of cars at the Petroco Service Station is defined by the following probability distribution:

Time Between Arrivals (min.)

Probability

1

.15

2

.30

3

.40

4

.15

1.00

1. Simulate the arrival of cars at the service station for 20 arrivals and compute the average time between arrivals.

2. Simulate the arrival of cars at the service station for 1 hour , using a different stream of random numbers from those used in (a) and compute the average time between arrivals.

3. Compare the results obtained in (a) and (b).

3.

The Dynaco Manufacturing Company produces a product in a process consisting of operations of five machines. The probability distribution of the number of machines that will break down in a week follows :

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Machine Breakdowns per Week

Probability

.10

1

.10

2

.20

3

.25

4

.30

5

.05

1.00

1. Simulate the machine breakdowns per week for 20 weeks.

2. Compute the average number of machines that will break down per week.

4.

Solve Problem 19 at the end of Chapter 12 by using simulation.

5.

Simulate the decision situation described in Problem 16(a) at the end of Chapter 12 for 20 weeks, and recommend the best decision.

6.

Every time a machine breaks down at the Dynaco Manufacturing Company (Problem 3), either 1, 2, or 3 hours are required to fix it, according to the following probability distribution:

Repair Time (hr.)

Probability

1

.30

2

.50

3

.20

1.00

1. Simulate the repair time for 20 weeks and then compute the average weekly repair time.

2. If the random numbers that are used to simulate breakdowns per week are also used to simulate repair time per breakdown, will the results be affected in any way? Explain.

3. If it costs \$50 per hour to repair a machine when it breaks down (including lost productivity), determine the average weekly breakdown cost.

4. The Dynaco Company is considering a preventive maintenance program that would alter the probabilities of machine breakdowns per week as shown in the following table:

Machine Breakdowns per Week

Probability

.20

1

.30

2

.20

3

.15

4

.10

5

.05

1.00

The weekly cost of the preventive maintenance program is \$150. Using simulation, determine whether the company should institute the preventive maintenance program.

7.

Sound Warehouse in Georgetown sells CD players (with speakers ), which it orders from Fuji Electronics in Japan. Because of shipping and handling costs, each order must be for five CD players. Because of the time it takes to receive an order, the warehouse outlet places an order every time the present stock drops to five CD players. It costs \$100 to place an order. It costs the warehouse \$400 in lost sales when a customer asks for a CD player and the warehouse is out of stock. It costs \$40 to keep each CD player stored in the warehouse. If a customer cannot purchase a CD player when it is requested , the customer will not wait until one comes in but will go to a competitor. The following probability distribution for demand for CD players has been determined:

[Page 651]

Demand per Month

Probability

.04

1

.08

2

.28

3

.40

4

.16

5

.02

6

.02

1.00

The time required to receive an order once it is placed has the following probability distribution:

Time to Receive an Order (mo.)

Probability

1

.60

2

.30

3

.10

1.00

The warehouse has five CD players in stock. Orders are always received at the beginning of the week. Simulate Sound Warehouse's ordering and sales policy for 20 months, using the first column of random numbers in Table 14.3. Compute the average monthly cost.

8.

First American Bank is trying to determine whether it should install one or two drive-through teller windows . The following probability distributions for arrival intervals and service times have been developed from historical data:

Time Between Automobile Arrivals (min.)

Probability

1

.20

2

.60

3

.10

4

.10

1.00

Service Time (min.)

Probability

2

.10

3

.40

4

.20

5

.20

6

.10

1.00

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Assume that in the two-server system, an arriving car will join the shorter queue. When the queues are of equal length, there is a 5050 chance the driver will enter the queue for either window.

1. Simulate both the one- and two-teller systems. Compute the average queue length, waiting time, and percentage utilization for each system.

2. Discuss your results in (a) and suggest the degree to which they could be used to make a decision about which system to employ .

9.

The time between arrivals of oil tankers at a loading dock at Prudhoe Bay is given by the following probability distribution:

Time Between Ship Arrivals (days)

Probability

1

.05

2

.10

3

.20

4

.30

5

.20

6

.10

7

.05

1.00

The time required to fill a tanker with oil and prepare it for sea is given by the following probability distribution:

Time to Fill and Prepare (days)

Probability

3

.10

4

.20

5

.40

6

.30

1.00

1. Simulate the movement of tankers to and from the single loading dock for the first 20 arrivals. Compute the average time between arrivals, average waiting time to load, and average number of tankers waiting to be loaded.

2. Discuss any hesitation you might have about using your results for decision making.

10.

The Saki automobile dealer in the MinneapolisSt. Paul area orders the Saki sport compact, which gets 50 miles per gallon of gasoline, from the manufacturer in Japan. However, the dealer never knows for sure how many months it will take to receive an order once it is placed. It can take 1, 2, or 3 months, with the following probabilities:

Probability

1

.50

2

.30

3

.20

1.00

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The demand per month is given by the following distribution:

Demand per Month (cars)

Probability

1

.10

2

.30

3

.40

4

.20

1.00

The dealer orders when the number of cars on the lot gets down to a certain level. To determine the appropriate level of cars to use as an indicator of when to order, the dealer needs to know how many cars will be demanded during the time required to receive an order. Simulate the demand for 30 orders and compute the average number of cars demanded during the time required to receive an order. At what level of cars in stock should the dealer place an order?

11.

State University is playing Tech in their annual football game on Saturday. A sportswriter has scouted each team all season and accumulated the following data: State runs four basic playsa sweep, a pass, a draw, and an off tackle; Tech uses three basic defensesa wide tackle, an Oklahoma, and a blitz. The number of yards State will gain for each play against each defense is shown in the following table:

Tech Defense

State Play

Wide Tackle

Oklahoma

Blitz

Sweep

3

5

12

Pass

12

4

10

Draw

2

1

20

Off tackle

7

3

3

The probability that State will run each of its four plays is shown in the following table:

Play

Probability

Sweep

.10

Pass

.20

Draw

.20

Off tackle

.50

The probability that Tech will use each of its defenses follows:

Defense

Probability

Wide tackle

.30

Oklahoma

.50

Blitz

.20

The sportswriter estimates that State will run 40 plays during the game. The sportswriter believes that if State gains 300 or more yards, it will win; however, if Tech holds State to fewer than 300 yards, it will win. Use simulation to determine which team the sportswriter will predict to win the game.

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12.

Each semester, the students in the college of business at State University must have their course schedules approved by the college adviser. The students line up in the hallway outside the adviser's office. The students arrive at the office according to the following probability distribution:

Time Between Arrivals (min.)

Probability

4

.20

5

.30

6

.40

7

10

1.00

The time required by the adviser to examine and approve a schedule corresponds to the following probability distribution:

Schedule Approval (min.)

Probability

6

.30

7

.50

8

.20

1.00

Simulate this course approval system for 90 minutes. Compute the average queue length and the average time a student must wait. Discuss these results.

13.

A city is served by two newspapersthe Tribune and the Daily News . Each Sunday readers purchase one of the newspapers at a stand. The following matrix contains the probabilities of a customer's buying a particular newspaper in a week, given the newspaper purchased the previous Sunday:

Simulate a customer's purchase of newspapers for 20 weeks to determine the steady-state probabilities of a customer buying each newspaper in the long run.

14.

Loebuck Grocery orders milk from a dairy on a weekly basis. The manager of the store has developed the following probability distribution for demand per week (in cases):

Demand (cases)

Probability

15

.20

16

.25

17

.40

18

.15

1.00

The milk costs the grocery \$10 per case and sells for \$16 per case. The carrying cost is \$0.50 per case per week, and the shortage cost is \$1 per case per week. Simulate the ordering system for Loebuck Grocery for 20 weeks. Use a weekly order size of 16 cases of milk and compute the average weekly profit for this order size. Explain how the complete simulation for determining order size would be developed for this problem.

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15.

The Paymore Rental Car Agency rents cars in a small town. It wants to determine how many rental cars it should maintain. Based on market projections and historical data, the manager has determined probability distributions for the number of rentals per day and rental duration (in days only) as shown in the following tables:

Number of Customers/Day

Probability

.20

1

.20

2

.50

3

.10

1.00

Rental Duration (days)

Probability

1

.10

2

.30

3

.40

4

.10

5

.10

1.00

Design a simulation experiment for the car agency and simulate using a fleet of four rental cars for 10 days. Compute the probability that the agency will not have a car available upon demand. Should the agency expand its fleet? Explain how a simulation experiment could be designed to determine the optimal fleet size for the Paymore Agency.

16.

A CPM/PERT project network has probabilistic activity times ( x ) as shown on each branch of the network; for example, activity 13 has a .40 probability that it will be completed in 6 weeks and a .60 probability it will be completed in 10 weeks:

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Simulate the project network 10 times and determine the critical path each time. Compute the average critical path time and the frequency at which each path is critical. How does this simulation analysis of the critical path method compare with regular CPM/PERT analysis?

17.

A robbery has just been committed at the Corner Market in the downtown area of the city. The market owner was able to activate the alarm, and the robber fled on foot . Police officers arrived a few minutes later and asked the owner, "How long ago did the robber leave?" "He left only a few minutes ago," the store owner responded. "He's probably 10 blocks away by now," one of the officers said to the other. "Not likely," said the store owner. "He was so stoned on drugs that I bet even if he has run 10 blocks, he's still only within a few blocks of here! He's probably just running in circles!"

Perform a simulation experiment that will test the store owner's hypothesis. Assume that at each corner of a city block there is an equal chance that the robber will go in any one of the four possible directions: north, south, east, or west. Simulate for five trials and then indicate in how many of the trials the robber is within 2 blocks of the store.

18.

Compcomm, Inc., is an international communications and information technology company that has seen the value of its common stock appreciate substantially in recent years . A stock analyst would like to use simulation to predict the stock prices of Compcomm for an extended period. Based on historical data, the analyst has developed the following probability distribution for the movement of Compcomm stock prices per day:

Stock Price Movement

Probability

Increase

.45

Same

.30

Decrease

.25

1.00

The analyst has also developed the following probability distributions for the amount of the increases or decreases in the stock price per day:

Probability

Stock Price Change

Increase

Decrease

1/8

.40

.12

1/4

.17

.15

3/8

.12

.18

1/2

.10

.21

5/8

.08

.14

3/4

.07

.10

7/8

.04

.05

1

.02

.05

1.00

1.00

The price of the stock is currently 62.

Develop a Monte Carlo simulation model to track the stock price of Compcomm stock and simulate for 30 days. Indicate the new stock price at the end of the 30 days. How would this model be expanded to conduct a complete simulation of 1 year's stock price movement?

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19.

The emergency room of the community hospital in Farmburg has one receptionist , one doctor, and one nurse. The emergency room opens at time zero, and patients begin to arrive some time later. Patients arrive at the emergency room according to the following probability distribution:

Time Between Arrivals (min.)

Probability

5

.06

10

.10

15

.23

20

.29

25

.18

30

.14

1.00

The attention needed by a patient who comes to the emergency room is defined by the following probability distribution:

Patient Needs to See

Probability

Doctor alone

.50

Nurse alone

.20

Both

.30

1.00

If a patient needs to see both the doctor and the nurse, he or she cannot see one before the otherthat is, the patient must wait to see both together.

The length of the patient's visit (in minutes) is defined by the following probability distributions:

Doctor

Probability

Nurse

Probability

Both

Probability

10

.22

5

.08

15

.07

15

.31

10

.24

20

.16

20

.25

15

.51

25

.21

25

.12

20

.17

30

.28

30

.10

1.00

35

.17

1.00

40

.11

1.00

Simulate the arrival of 20 patients to the emergency room and compute the probability that a patient must wait and the average waiting time. Based on this one simulation, does it appear that this system provides adequate patient care?

20.

The Western Outfitters Store specializes in denim jeans. The variable cost of the jeans varies according to several factors, including the cost of the jeans from the distributor, labor costs, handling, packaging, and so on. Price also is a random variable that varies according to competitors ' prices.

[Page 658]

Sales volume also varies each month. The probability distributions for volume, price, and variable costs each month are as follows:

Sales Volume

Probability

300

.12

400

.18

500

.20

600

.23

700

.17

800

.10

1.00

Price

Probability

\$22

.07

23

.16

24

.24

25

.25

26

.18

27

.10

1.00

Variable Cost

Probability

\$ 8

.17

9

.32

10

.29

11

.14

12

.08

1.00

Fixed costs are \$9,000 per month for the store.

Simulate 20 months of store sales and compute the probability that the store will at least break even and the average profit (or loss).

21.

Randolph College and Salem College are within 20 miles of each other, and the students at each college frequently date each other. The students at Randolph College are debating how good their dates are at Salem College. The Randolph students have sampled several hundred of their fellow students and asked them to rate their dates from 1 to 5 (in which 1 is excellent and 5 is poor) according to physical attractiveness, intelligence, and personality. Following are the resulting probability distributions for these three traits for students at Salem College:

Physical Attractiveness

Probability

1

.27

2

.35

3

.14

4

.09

5

.15

1.00

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Intelligence

Probability

1

.10

2

.16

3

.45

4

.17

5

.12

1.00

Personality

Probability

1

.15

2

.30

3

.33

4

.07

5

.15

1.00

Simulate 20 dates and compute an average overall rating of the Salem students.

22.

In Problem 21 discuss how you might assess the accuracy of the average rating for Salem College students based on only 20 simulated dates.

23.

Burlingham Mills produces denim cloth that it sells to jeans manufacturers. It is negotiating a contract with Troy Clothing Company to provide denim cloth on a weekly basis. Burlingham has established its monthly available production capacity for this contract to be between 0 and 600 yards, according to the following probability distribution:

Troy Clothing's weekly demand for denim cloth varies according to the following probability distribution:

Demand (yd.)

Probability

.03

100

.12

200

.20

300

.35

400

.20

500

.10

1.00

Simulate Troy Clothing's cloth orders for 20 weeks and determine the average weekly capacity and demand. Also determine the probability that Burlingham will have sufficient capacity to meet demand.

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24.

A baseball game consists of plays that can be described as follows.

Play

Description

An out where no runners can advance. This includes strikeouts, pop-ups, short flies, and the like.

Groundout

Each runner can advance one base.

Possible double play

Double play if there is a runner on first base and fewer than two outs. The lead runner who can be forced is out; runners not out advance one base. If there is no runner on first or there are two outs, this play is treated as a "no advance."

Long fly

A runner on third base can score.

Very long fly

Runners on second and third base advance one base.

Walk

Includes a hit batter.

Infield single

Outfield single

A runner on first base advances one base, but a runner on second or third base scores.

Long single

All runners can advance a maximum of two bases.

Double

Runners can advance a maximum of two bases.

Long double

All runners score.

Triple

Home run

Note : Singles also include a factor for errors, allowing the batter to reach first base.

Distributions for these plays for two teams , the White Sox ( visitors ) and the Yankees (home), are as follows:

Team: White Sox

Play

Probability

.03

Groundout

.39

Possible double play

.06

Long fly

.09

Very long fly

.08

Walk

.06

Infield single

.02

Outfield single

.10

Long single

.03

Double

.04

Long double

.05

Triple

.02

Home run

.03

1.00

Team: Yankees

Play

Probability

.04

Groundout

.38

Possible double play

.04

Long fly

.10

Very long fly

.06

Walk

.07

Infield single

.04

Outfield single

.10

Long single

.04

Double

.05

Long double

.03

Triple

.01

Home run

.04

1.00

Simulate a nine-inning baseball game using the preceding information. [2]

[2] This problem was adapted from R. E. Trueman, "A Computer Simulation Model of Baseball: With Particular Application to Strategy Analysis," in R. E. Machol, S. P. Ladany, and D. G. Morrison, eds., Management Science in Sports (New York: North Holland Publishing Co., 1976), 114.

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25.

Tracy McCoy is shopping for a new car. She has identified a particular sports utility vehicle she likes but has heard that it has high maintenance costs. Tracy has decided to develop a simulation model to help her estimate maintenance costs for the life of the car. Tracy estimates that the projected life of the car with the first owner (before it is sold) is uniformly distributed with a minimum of 2.0 years and a maximum of 8.0 years. Furthermore, she believes that the miles she will drive the car each year can be defined by a triangular distribution with a minimum value of 3,700 miles, a maximum value of 14,500 miles, and a most likely value of 9,000 miles. She has determined from automobile association data that the maintenance cost per mile driven for the vehicle she is interested in is normally distributed, with a mean of \$0.08 per mile and a standard deviation of \$0.02 per mile. Using Crystal Ball, develop a simulation model (using 1,000 trials) and determine the average maintenance cost for the life of the car with Tracy and the probability that the cost will be less than \$3,000.

26.

In Problem 20, assume that the sales volume for Western Outfitters Store is normally distributed, with a mean of 600 pairs of jeans and a standard deviation of 200; the price is uniformly distributed, with a minimum of \$22 and a maximum of \$28; and the variable cost is defined by a triangular distribution with a minimum value of \$6, a maximum of \$11, and a most likely value of \$9. Develop a simulation model by using Crystal Ball (with 1,000 trials) and determine the average profit and the probability that Western Outfitters will break even.

27.

In Problem 21, assume that the students at Randolph College have redefined the probability distributions of their dates at Salem College as follows: Physical attractiveness is uniformly distributed from 1 to 5; intelligence is defined by a triangular distribution with a minimum rating of 1, a maximum of 5, and a most likely of 2; and personality is defined by a triangular distribution with a minimum of 1, a maximum of 5, and a most likely rating of 3. Develop a simulation model by using Crystal Ball and determine the average date rating (for 1,000 trials). Also compute the probability that the rating will be "better" than 3.0.

28.

In Problem 23, assume that production capacity at Burlingham Mills for the Troy Clothing Company contract is normally distributed, with a mean of 320 yards per month and a standard deviation of 120 yards, and that Troy Clothing's demand is uniformly distributed between 0 and 500 yards. Develop a simulation model by using Crystal Ball and determine the average monthly shortage or surplus for denim cloth (for 1,000 trials). Also determine the probability that Burlingham will always have sufficient production capacity.

29.

Erin Jones has \$100,000 and, to diversify, she wants to invest equal amounts of \$50,000 each in two mutual funds selected from a list of four possible mutual funds. She wants to invest for a 3-year period. She has used historical data from the four funds plus data from the market to determine the mean and standard deviation (normally distributed) of the annual return for each fund, as follows:

Return ( r )

Fund

µ

s

1. Internet

.20

.09

2. Index

.12

.04

3. Entertainment

.16

.10

4. Growth

.14

.06

The possible combinations of two investment funds are (1,2), (1,3), (1,4), (2,3), (2,4), and (3,4).

1. Use Crystal Ball to simulate each of the investment combinations to determine the expected return in 3 years. (Note that the formula for the future value, FV , of a current investment, P , with return, r , for n years in the future is FV n = P r (1 + r ) n .) Indicate which investment combination has the highest expected return.

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2. Erin wants to reduce her risk as much as possible. She knows that if she invests her \$100,000 in a CD at the bank, she is guaranteed a return of \$20,000 after 3 years. Using the frequency charts for the simulation runs in Crystal Ball, determine which combination of investments would result in the greatest probability of receiving a return of \$120,000 or greater.

30.

In Chapter 16, the formula for the optimal order quantity of an item, Q , given its demand, D , order cost, C o , and the cost of holding, or carrying, an item in inventory, C c , is as follows:

The total inventory cost formula is

Order cost, C o , and carrying cost, C c , are generally values that the company is often able to determine with certainty because they are internal costs, whereas demand, D , is usually not known with certainty because it is external to the company. However, in the order quantity formula given here, demand is treated as if it were certain. To consider the uncertainty of demand, it must be simulated.

Using Crystal Ball, simulate the preceding formulas for Q and TC to determine their average values for an item, with C o = \$150, C c = \$0.75, and demand, D , that is normally distributed with a mean of 10,000 and a standard deviation of 4,000.

31.

The Management Science Association (MSA) has arranged to hold its annual conference at the Riverside Hotel in Orlando next year. Based on historical data, the MSA believes the number of rooms it will need for its members attending the conference is normally distributed, with a mean of 800 and a standard deviation of 270. The MSA can reserve rooms now (1 year prior to the conference) for \$80; however, for any rooms not reserved now, the cost will be at the hotel's regular room rate of \$120. The MSA guarantees the room rate of \$80 to its members. If its members reserve fewer than the number of rooms it reserves , MSA must pay the hotel for the difference, at the \$80 room rate. If MSA does not reserve enough rooms, it must pay the extra costthat is, \$40 per room.

1. Using Crystal Ball, determine whether the MSA should reserve 600, 700, 800, 900, or 1,000 rooms in advance to realize the lowest total cost.

2. Can you determine a more exact value for the number of rooms to reserve to minimize cost?

32.

In Chapter 8, Figure 8.6 shows a simplified project network for building a house, as follows:

[Page 663]

There are four paths through this network:

Path A: 123467

Path B: 1234567

Path C: 12467

Path D: 124567

The time parameters (in weeks) defining a triangular probability distribution for each activity are provided as follows:

Time Parameters

Activity

Minimum

Likeliest

Maximum

12

1

3

5

23

1

2

4

24

.5

1

2

34

45

1

2

3

46

1

3

6

56

1

2

4

67

1

2

4

1. Using Crystal Ball, simulate each path in the network and identify the longest path (i.e., the critical path).

2. Observing the simulation run frequency chart for path A, determine the probability that this path will exceed the critical path time. What does this tell you about the simulation results for a project network versus an analytical result?

Introduction to Management Science (10th Edition)
ISBN: 0136064361
EAN: 2147483647
Year: 2006
Pages: 358
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