## Simulation Problems

1. Virtual Reality Goggle Inventory. Galaxy Co. sells virtual reality (VR)

goggles, particularly targeting customers who like to play video

games. Galaxy procures each pair of goggles for \$150 from its

supplier and sells each pair of goggles for \$300. Monthly demand for

the VR goggles is a normal random variable with a mean of 160 units

and a standard deviation of 40 units. At the beginning of each month,

Galaxy orders enough goggles from its supplier to bring the inventory

level up to 140 goggles. If the monthly demand is less than 140, Galaxy

pays \$20 per pair of goggles that remains in inventory at the end of

the month. If the monthly demand exceeds 140, Galaxy sells only the

140 pairs of goggles in stock. Galaxy assigns a shortage cost of \$40 for

each unit of demand that is unsatisfied to represent a loss-of-goodwill

among its customers. Management would like to use a simulation

model to analyze this situation.

a. What is the average monthly profit resulting from its policy of

stocking 140 pairs of goggles at the beginning of each month?

b. What is the proportion of months in which demand is

completely satisfied?

c. Use the simulation model to compare the profitability of

monthly replenishment levels of 140 and 160 pairs of goggles.

Use a 95% confidence interval on the difference between the

average profit that each replenishment level generates to make

2. Dice Rolls. Construct a spreadsheet simulation model to simulate

1,000 rolls of a die with the six sides numbered 1, 2, 3, 4, 5, and 6.

a. Construct a histogram of the 1,000 observed dice rolls.

b. For each roll of two dice, record the sum of the dice. Construct a

histogram of the 1,000 observations of the sum of two dice.

c. For each roll of three dice, record the sum of the dice. Construct

a histogram of the 1,000 observations of the sum of three dice.

d. For each roll of four dice, record the sum of the dice. Construct a

histogram of the 1,000 observations of the sum of four dice.

e. Compare the histograms in parts (a), (b), (c), and (d). What

statistical phenomenon does this sequence of charts illustrate?

3.

Wearable Electronic Product Launch. The management of Madeira

Computing is considering the introduction of a wearable electronic

device with the functionality of a laptop computer and phone. The

fixed cost to launch this new product is \$300,000. The variable cost

for the product is expected to be between \$160 and \$240, with a most

likely value of \$200 per unit. The product will sell for \$300 per unit.

Demand for the product is expected to range from 0 to approximately

20,000 units, with 4,000 units the most likely.

a. Develop a what-if spreadsheet model computing profit for this

product in the base-case, worst-case, and best-case scenarios.

b. Model the variable cost as a uniform random variable with a

minimum of \$160 and a maximum of \$240. Model the product

demand as 1,000 times the value of a gamma random variable

with an alpha parameter of 3 and a beta parameter of 2.

Construct a simulation model to estimate the average profit and

the probability that the project will result in a loss.

c. What is your recommendation regarding whether to launch the

product?

4.

Profitability of New Product. The management of Brinkley

Corporation is interested in using simulation to estimate the profit

per unit for a new product. The selling price for the product will be

\$45 per unit. Probability distributions for the purchase cost, the labor

cost, and the transportation cost are estimated as follows:

a. Construct a simulation model to estimate the average profit per

unit. What is a 95% confidence interval around this average?

b. Management believes that the project may not be sustainable if

the profit per unit is less than \$5. Use simulation to estimate the

probability that the profit per unit will be less than \$5. What is a

95% confidence interval around this proportion?

5.

Estimating Auto Accident Costs. Statewide Auto Insurance believes

that for every trip longer than 10 minutes that a teenager drives,

there is a 1 in 1,000 chance that the drive will result in an auto

accident. Assume that the cost of an accident can be modeled with a

beta distribution with an alpha parameter of 1.5, a beta parameter of

3, a minimum value of \$500, and a maximum value of \$20,000.

Procurement

Cost (\$)

Probability Labor

Cost

(\$)

Probability Tran

C

10 0.25 20 0.10

11 0.45 22 0.25

12 0.30 24 0.35

25 0.30

Construct a simulation model to answer the following questions.

(Hint: Review Appendix 11.1 for descriptions of various types of

probability distributions to identify the appropriate way to model the

number of accidents in 500 trips.)

a. If a teenager drives 500 trips longer than 10 minutes, what is the

average cost resulting from accidents? Provide a 95% confidence

interval on this mean.

b. If a teenager drives 500 trips longer than 10 minutes, what is the

probability that the total cost from accidents will exceed \$8,000?

Provide a 95% confidence interval on this proportion.

6. Automobile Collision Claims. State Farm Insurance has developed

the following table to describe the distribution of automobile collision

claims paid during the past year.

a. Set up a table of intervals of random numbers that can be used

with the Excel VLOOKUP function to generate values for

automobile collision claim payments.

b. Construct a simulation model to estimate the average claim

payment amount and the standard deviation in the claim

payment amounts.

c. Chapter 4 describes the analytical calculation of the mean and

standard deviation of a random variable.

Let X be the discrete random variable representing the dollar

value of an automobile collision claim payment. Let,

represent possible values of X. Then, the mean

and standard deviation of X can be computed as

, and

x1, x2, … , xn (μ)

(σ)

μ = x1 × P(X = x1) + ⋯ + xn × P(X = xn)

.

Compare the values of sample mean and sample standard

deviation in part (b) to the analytical calculation of the mean

and standard deviation. How can we improve the accuracy of

the sample estimates from the simulation?

7.

Playoff Series in National Basketball Association. The Dallas

Mavericks and the Golden State Warriors are two teams in the

σ = √(x1 − μ)2 × P(X = x1) + ⋯ + (xn − μ)2 × P(X = xn)

Payment(\$) Probability

0 0.83

500 0.06

1,000 0.05

2,000 0.02

5,000 0.02

8,000 0.01

10,000 0.01

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