Deck 15: Introduction to Simulation Modeling

Question 1

Assume that x is a random number between 0 and 1, and that the number of units expected to be sold is uniformly distributed between 300 and 500. In this case, sales are given by the expression:

A) 300 + x
B) 500 - x
C) 300 + 200 x
D) 500 - 200 x
E) 300 + 500 x

Answer 1

300 + 200 x

Question 2

Discrete distributions are sometimes used as approximations of continuous distributions:

A) because they are more accurate
B) because they are simpler
C) when the mean and variance of the distribution are unknown
D) when the shape of the continuous distribution is not symmetric

Answer 2

because they are simpler

Question 3

When determining the most appropriate input probability distribution in a simulation model, first select the most appropriate family, and then select the most appropriate member of that family.

Answer 3

True

Question 4

A primary difference between standard spreadsheet models and simulation models is that at least one of the input variable cells in a simulation model contains random numbers.

Question 5

Each different set of values obtained for the uncertain quantities in a simulation model can considered to be:

A) the mean of the probability distribution
B) a scenario
C) a best guess
D) all of these choices

Question 6

In general, important characteristics of probability distributions include the following distinctions:

A) discrete versus continuous
B) symmetric versus skewed
C) bounded versus unbounded
D) positive (or nonnegative) versus unrestricted
E) all of these choices

Question 7

The deterministic (non-simulation) approach, using best guesses for the uncertain inputs, is:

A) better to use in complicated real world applications
B) a good estimate of what the answer will be using a simulation approach
C) generally not the appropriate model
D) the preferred approach when there is correlation between input variables

Question 8

Simulation models are particularly useful for:

A) forecasting
B) obtaining deterministic outputs
C) evaluating constraints
D) asking what-if questions

Question 9

The binomial distribution is a discrete distribution that is applied to situations where n independent and identical "trials" occur, with each trial resulting in a "success" or "failure," and where we want to generate the random number of successes in the n trials.

Question 10

Which of the following statements is correct regarding the graph of a discrete probability distribution?

A) It is a series of spikes.
B) The height of each spike is the probability of the corresponding value.
C) There is an empty space between adjacent spikes.
D) All of these statements are correct.

Question 11

A company is about to develop and then market a new product. It wants to build a simulation model for the entire process, and one key uncertain input is the development time, which is measured in an integer number of months. For each of the scenarios in the questions below, choose an "appropriate" distribution, together with its parameters, and explain your choice.
Company experts believe the development time will be from 5 to 9 months. They believe that 7 months is twice as likely as either 6 months or 8 months and that either of these latter possibilities is three times as likely as either 5 months or 9 months.

Question 12

The "building blocks" of all spreadsheet simulation models are:

A) deterministic inputs
B) random numbers between 0 and 1
C) decision variables
D) probability distributions for uncertain input variables

Question 13

Which of the following statements is/are true regarding a simulation model?

A) It makes decision-making easier.
B) It explicitly incorporates uncertainty in one or more input variables.
C) It does not require any best guesses.
D) All of these statements are true.

Question 14

The three parameters required to specify a triangular distribution are the minimum, mean, and maximum.

Question 15

One of the primary advantages of simulation models is that they enable managers to answer what-if questions about changes in systems without actually changing the systems themselves.

Question 16

Which of the following statements are true?

A) A probability distribution is symmetric (around some point) if the distribution to the left of the point is a mirror image of the distribution to the right of the point.
B) We say a distribution is skewed to the right (or positively skewed) if the "longer tail" is the right tail.
C) We say a distribution is skewed to the left (or negatively skewed) if the "longer tail" is the left tail.
D) All of these choices are correct.

Question 17

Which of the following statements are false regarding the graph of a continuous probability distribution?

A) It is characterized by a density function, a smooth curve.
B) It is a series of spikes.
C) The height of the density curve above any point is not actually a probability-that is, it is not necessarily between 0 and 1.
D) Heights above the density function indicate relative likelihoods but are not necessarily values between 0 and 1.

Question 18

A continuous probability distribution is characterized by a(n):

A) symmetric shape
B) series of spikes
C) density function
D) bounded shape

Question 19

Spreadsheet simulation modeling is quite similar to the other modeling applications in that it begins with input variables and then relates these with appropriate Excel^® formulas to produce output variables of interest.

Question 20

The binomial distribution can be well approximated by the normal distribution when the number of trials n is sufficiently small and the probability of success p is not too close to 0 or 1.

Question 21

When n is reasonably large and p isn't too close to 0 or 1, the binomial distribution can be well approximated by which of the following distributions?

A) uniform distribution
B) normal distribution
C) triangular distribution
D) none of these choices

Question 22

The "random" numbers generated by the RAND function (or by any other package's random number generator) are not really random.

Question 23

The normal distribution is often used in simulation models because it is the most common distribution in statistics and it does not allow negative values.

Question 24

In order to generate random numbers in Excel^® from a discrete distribution with a finite number of possible values and corresponding probabilities, we can use:

A) only the RAND function
B) only the VLOOKUP function
C) only the VLOOKDOWN function
D) the RAND function along with a VLOOKUP function

Question 25

A probability distribution is continuous if its possible values fall alongy some continuum.

Question 26

A probability distribution is bounded if there are values A and B such that only one possible value can be less than A or greater than B.

Question 27

A probability distribution is bounded if there are values A and B such that:

A) A and B represent the 95% confidence interval
B) A and B are the mean and standard deviation, respectively
C) A and B are the mean and variance, respectively
D) no value can be less than A or greater than B

Question 28

A random number from a binomial distribution indicates the number of successes in a certain number of identical trials.

Question 29

The uniform distribution is bounded by a minimum and a maximum, and all values between these two extremes are equally likely.

Question 30

Which of the following statements is/are true regarding the triangular distribution?

A) It is a discrete distribution with a minimum, maximum and most likely value.
B) It is more flexible and intuitive than the normal distribution.
C) It is a symmetric distribution.
D) All of these statements are true.

Question 31

Sometimes it is convenient to treat a discrete probability distribution as continuous, and vice versa.

Question 32

Which of the following statements is/are true regarding the normal distribution?

A) It is always the appropriate distribution in simulation modeling.
B) It does not permit negative values.
C) There is a 95% chance that values will be within ± 2 standard deviations of the mean.
D) All of these statements are true.

Question 33

One important special case of bounded distributions is when the only possible values are:

A) zero or one
B) uniformly distributed around the mean
C) skewed to the right
D) nonnegative

Question 34

The RAND() function in Excel^® models which of the following probability distributions?

A) Normal(0,1)
B) Uniform(0,1)
C) Normal(-1,1)
D) Uniform(-1,1).

Question 35

If we want to model the monthly return on a stock, a good option would be:

A) a symmetric distribution around 0
B) a positively skewed distribution
C) a negatively skewed distribution
D) all of these choices

Question 36

We typically choose between a symmetric and skewed distribution on the basis of realism.

Question 37

Which of the following statements are false regarding the numbers generated by the RAND function in Excel^®?

A) Approximately 10% of the numbers will be between 0.0 and 1.0.
B) Approximately 20% of the numbers will be between 0.50 and 0.70.
C) Approximately 40% of the numbers will be between 0.20 and 0.60.
D) Approximately 60% of the numbers will be between 0.15 and 0.75.
E) All of these statements are false.

Question 38

Which of the following statements are false regarding the numbers generated by the RAND function in Excel^®?

A) The numbers are random between 0 and 1.
B) The numbers are probabilistically dependent.
C) The numbers are probabilistically independent.
D) The numbers are uniformly distributed between 0 and 1.

Question 39

We can think of the uniform distribution as:

A) the "I have no idea" distribution
B) a skewed distribution
C) only modeling positive values
D) a bell curve

Question 40

The built-in functions in Excel^®, along with the RAND function, can be used to generate random numbers from many different types of probability distributions.

Question 41

(A) What fraction of the random numbers are smaller than 0.5?
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(B) What fraction of the time is a random number less than 0.5 followed by another random number less than 0.5?
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(C) What fraction of the random numbers are larger than 0.8?
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(D) What do you expect the answers to (A), (B) and (C) to be before simulating? Do the answers you provided to those questions match your expectations? Explain why or why not.
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(E) Suppose your answers to (A), (B) and (C) are not close to the expected answers. What can you do to obtain answers from the simulation that are closer to the expected answers?

Question 42

If a model contains uncertain outputs, it can be very misleading to build a deterministic model by using the means of the inputs to predict an output. This is called the:

A) Law of Large Numbers
B) Flaw of Averages
C) Law of Inevitable Disappointment
D) Central Limit Theorem

Question 43

A company is about to develop and then market a new product. It wants to build a simulation model for the entire process, and one key uncertain input is the development time, which is measured in an integer number of months. For each of the scenarios in the questions below, choose an "appropriate" distribution, together with its parameters, and explain your choice.
Company experts believe the development time will fall into the range of 5 to 9 months. They believe the probabilities of the extremes (5 and 9 months) are both 10%, and the probabilities will vary linearly from those endpoints to a most likely value at 7 months.

Question 44

A discrete distribution is useful for many situations, either when the uncertain quantity is not continuous (the number of televisions demanded, for example) or when we want a discrete approximation to a continuous variable.

Question 45

@RISK introduces uncertainty explicitly into a spreadsheet model by allowing several inputs to have probability distributions and then enabling the simulation of random values from these inputs.

Question 46

Which of the following statements are false?

A) A probability distribution is discrete if it has a finite number of possible values.
B) A probability distribution is continuous if its possible values fall along some continuum.
C) An example of a discrete probability distribution is the amount of rain that falls during the month of June in Michigan.
D) None of these statements are false.

Question 47

A company is about to develop and then market a new product. It wants to build a simulation model for the entire process, and one key uncertain input is the development time, which is measured in an integer number of months. For each of the scenarios in the questions below, choose an "appropriate" distribution, together with its parameters, and explain your choice.
Company experts believe the development time will be from 5 to 9 months, but they have absolutely no idea which of these will result.

Question 48

If we want to model a random stock price, we should do so with an unbounded symmetric probability distribution.

Question 49

(A) Generate the "birthdays" of 30 different people, assuming that each person has a 1/365 chance of having a given birthday (call the days of the year 1, 2, 3, ……..,365). You can use a formula involving the DISCRETE and RAND functions to generate birthdays.
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(B) Once you have generated 30 people's birthdays, you can tell whether at least two people have the same birthday using Excel's RANK function (i.e., in the case of a tie, two numbers are given the same rank). Do you see any people with the same birthday in your sample?
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(C) Obtain at least 20 samples of the 30-person group using the F9 key. What do you estimate the probability of finding two people with the same birthday in a sample of 30 people to be?

Question 50

The flaw of averages is the reason deterministic models can be very misleading.

Question 51

A company is about to develop and then market a new product. It wants to build a simulation model for the entire process, and one key uncertain input is the development cost. For each of the scenarios in the questions below, choose an "appropriate" distribution, together with its parameters, and explain your choice.
(A) Company experts have no idea about the distribution of their development cost. All they can state is that "we are 90% sure it will be somewhere between $450,000 and $650,000."
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(B) Company experts can still make the same two statements as in (A), but now they can also state that "we believe the distribution is symmetric and its most likely value is about $550,000."
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(C) Company experts can still make the same two statements as in (A), but now they can also state that "we believe the distribution is skewed to the right, and its most likely value is about $500,000."

Question 52

Suppose that the demand for cars is normally distributed with mean of 120 and standard deviation of 20. Use @RISK simulation add-in to determine the "best" order quantity; that is, the one that has the largest expected profit. Using the statistics and/or graphs from @RISK, discuss whether this order quantity would not be considered the "best" by the car dealer.

Question 53

If you add several normally distributed random numbers, the result is normally distributed, where the mean of the sum is the sum of the individual means, and the variance of the sum is the sum of the individual variances. This result is difficult to prove mathematically, but it is easy to demonstrate with simulation. To do so, run a simulation where you add three normally distributed random numbers, each with mean 100 and standard deviation 10. Your single output variable should be the sum of these three numbers. Verify with @RISK that the distribution of this output is approximately normal with mean 300 and variance 300 (hence, standard deviation = 17.32).

Question 54

If we want to model the time it takes to serve a customer at a bank, we will probably choose a(n):

A) symmetric distribution
B) positively skewed distribution
C) negatively skewed distribution
D) unbounded distribution

Question 55

When we run simulation, the @RISK automatically keeps statistics such as averages and standard deviations, and can also create graphs such as histograms based on the values generated in the output cells in the simulation model.

Question 56

The triangular distribution is sometimes used in simulation models because it is more flexible and intuitive than the normal distribution.

Question 57

Obtain another set random numbers by pressing the F9 (recalculate) key. Do your results change significantly? Do the changes match your expectations? Explain your answer.

Question 58

In August 2017, a car dealer is trying to determine how many 2018 cars to order. Each car ordered in August 2017 costs $16,000. The demand for the dealer's 2018 models has the probability distribution shown in the table below. Each car sells for $21,000. If the demand for 2018 cars exceeds the number of cars ordered in August 2017, the dealer must reorder at a cost of $18,000 per car. Excess cars can be disposed of at $13,000 per car.
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(A) Use simulation to determine how many cars the dealer should order in August, 2017 to maximize his expected profit.
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(B) For the optimal order quantity, find a 95% confidence interval for the expected profit.
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(C) Why is it important to develop the confidence interval in (B)?

Question 59

It is simple to generate a uniformly distributed random number with a minimum and maximum other than 0 and 1. For example, the formula = 150+100 RAND() generates a number uniformly distributed between:

A) 100 and 150
B) 150 and 250
C) 50 and 100
D) 50 and 250

Question 60

It is common in computer simulations to estimate the mean of some distribution by the average of the simulated observations. The usual practice is then to accompany this estimate with a confidence interval, which indicates the accuracy of the estimate.

Question 61

A correlation matrix must always be symmetric, so that the correlations above the diagonal are a mirror image of those below it.

Question 62

RISKSIMTABLE is a function in @RISK for running several simulations simultaneously, one for each setting of an input or decision variable.

Question 63

Oregon State University has reached the final four in the 2016 NCAA Women's Basketball Tournament, and as a result, a sweatshirt supplier in Corvallis is trying to decide how many sweatshirts to print for the upcoming championships. The final four teams (Oregon State, University of Washington, Syracuse, and University of Connecticut) have emerged from the quarterfinal round, and there is a week left until the semifinals, which are then followed in a couple of days by the finals. Each sweatshirt costs $12 to produce and sells for $24. However, in three weeks, any leftover sweatshirts will be put on sale for half price, $12. The supplier assumes that the demand (in thousands) for his sweatshirts during the next three weeks, when interest is at its highest, follows the probability distribution shown in the table below. The residual demand, after the sweatshirts have been put on sale, also has the probability distribution shown in the table below. The supplier realizes that every sweatshirt sold, even at the sale price, yields a profit. However, he also realizes that any sweatshirts produced but not sold must be thrown away, resulting in a $12 loss per sweatshirt.
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Demand distribution at regular price Demand distribution at reduced price
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Use @RISK simulation add-in to analyze the sweatshirt sales. Do this for the discrete distributions given in the problem.

Question 64

Suppose that Ms. Smart invests 25% of her portfolio in four different stocks. The mean and standard deviation of the annual return on each stock are shown in the first table below. The correlations between the annual returns on the four stocks are shown in the second table below.
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(A) Use @RISK with 100 replications, provide a summary statistics of portfolio return; namely, minimum, maximum, mean, and standard deviation.
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(B) Use your answers to (A) to estimate the probability that Mrs. Smart's portfolio's annual return will exceed 20%.
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(C) Use your answers to (A) to estimate the probability that Mrs. Smart's portfolio will lose money during the course of a year.
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(D) Suppose that the current price of each stock is as follows: stock 1: $16; stock 2: $18; stock 3: $20; and stock 4: $22. Ms. Smart has just bought an option involving these four stocks. If the price of stock 1, six months from now are is $18 or more, the option enables Ms. Smart to buy, if she desires, one share of each stock for $20 six months from now. Otherwise the option is worthless. For example, if the stock prices six months from now are: stock 1: $18; stock 2: $20; stock 3: $21; and stock 4: $24, then Ms. Smart would exercise her option to buy stocks 3 and 4 and receive (21- 20) + (24-20) = $5 in each cash flow. How much is this option worth if the risk-free rate is 8%?

Question 65

Oregon State University has reached the final four in the 2016 NCAA Women's Basketball Tournament, and as a result, a sweatshirt supplier in Corvallis is trying to decide how many sweatshirts to print for the upcoming championships. The final four teams (Oregon State, University of Washington, Syracuse, and University of Connecticut) have emerged from the quarterfinal round, and there is a week left until the semifinals, which are then followed in a couple of days by the finals. Each sweatshirt costs $12 to produce and sells for $24. However, in three weeks, any leftover sweatshirts will be put on sale for half price, $12. The supplier assumes that the demand (in thousands) for his sweatshirts during the next three weeks, when interest is at its highest, follows the probability distribution shown in the table below. The residual demand, after the sweatshirts have been put on sale, also has the probability distribution shown in the table below. The supplier realizes that every sweatshirt sold, even at the sale price, yields a profit. However, he also realizes that any sweatshirts produced but not sold must be thrown away, resulting in a $12 loss per sweatshirt.
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Demand distribution at regular price Demand distribution at reduced price
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Use simulation to analyze the supplier's problem. Determine how many sweatshirts he should produce to maximize the expected profit.

Question 66

It is usually not too difficult to predict the shape of the output distribution from the shape(s) of the input distribution(s).

Question 67

Oregon State University has reached the final four in the 2016 NCAA Women's Basketball Tournament, and as a result, a sweatshirt supplier in Corvallis is trying to decide how many sweatshirts to print for the upcoming championships. The final four teams (Oregon State, University of Washington, Syracuse, and University of Connecticut) have emerged from the quarterfinal round, and there is a week left until the semifinals, which are then followed in a couple of days by the finals. Each sweatshirt costs $12 to produce and sells for $24. However, in three weeks, any leftover sweatshirts will be put on sale for half price, $12. The supplier assumes that the demand (in thousands) for his sweatshirts during the next three weeks, when interest is at its highest, follows the probability distribution shown in the table below. The residual demand, after the sweatshirts have been put on sale, also has the probability distribution shown in the table below. The supplier realizes that every sweatshirt sold, even at the sale price, yields a profit. However, he also realizes that any sweatshirts produced but not sold must be thrown away, resulting in a $12 loss per sweatshirt.
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Demand distribution at regular price Demand distribution at reduced price
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(A) Assume that the weight of each can in a six-pack has a 0.8 correlation with the weight of the other cans in the six-pack. What mean fill quantity (within 0.05 ounce) maximizes expected profit per six-pack?
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(B) If the weights of the cans in the six-pack are probabilistically independent, what mean fill quantity (within 0.05 ounce) will maximize expected profit per six-pack?
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(C) How can you explain the difference in the answers for (A) and (B)?

Question 68

Which of the following statements are true?

A) The @RISK contains a number of functions such as RISKNORMAL and RISKDISCRETE that make it easy to generate observations from the most important probability distributions.
B) You can specify any cell or range of cells in your simulation model as output cells.
When you run the simulation, @RISK automatically keeps summary measures (averages, standard deviation, percentiles, and others) from the values generated in these output cells across the replications.
C) @RISK has a special function, RISKSIMTABLE, which allows you to run the same simulation several times, using a different value of some key input variable each time.
D) All of these statements are true.

Question 69

When we maximize or minimize the value of a decision variable by running several simulations simultaneously, we have found an optimal solution to the problem and attitude toward risk becomes irrelevant.

Question 70

Oregon State University has reached the final four in the 2016 NCAA Women's Basketball Tournament, and as a result, a sweatshirt supplier in Corvallis is trying to decide how many sweatshirts to print for the upcoming championships. The final four teams (Oregon State, University of Washington, Syracuse, and University of Connecticut) have emerged from the quarterfinal round, and there is a week left until the semifinals, which are then followed in a couple of days by the finals. Each sweatshirt costs $12 to produce and sells for $24. However, in three weeks, any leftover sweatshirts will be put on sale for half price, $12. The supplier assumes that the demand (in thousands) for his sweatshirts during the next three weeks, when interest is at its highest, follows the probability distribution shown in the table below. The residual demand, after the sweatshirts have been put on sale, also has the probability distribution shown in the table below. The supplier realizes that every sweatshirt sold, even at the sale price, yields a profit. However, he also realizes that any sweatshirts produced but not sold must be thrown away, resulting in a $12 loss per sweatshirt.
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Demand distribution at regular price Demand distribution at reduced price
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Use @RISK simulation add-in to analyze the sweatshirt sales. Do this for normal distributions, where we assume that the regular demand is normally distributed with mean 10,000 and standard deviation 1500, and that the demand at the reduced price is normally distributed with mean 5,000 and standard deviation 1500.

Question 71

Analysts often plan a simulation so that the confidence interval for the mean of some important output will be sufficiently narrow. The reasoning is that narrow confidence intervals imply more precision about the estimated mean of the output variable.

Question 72

Data tables in spreadsheet simulations are useful for taking a "prototype" simulation and replicating its key results a desired number of times.

Question 73

Many companies have used simulation to determine which of several possible investment projects they should choose. This is often referred to as

A) risk analysis
B) @RISK investment
C) unbounded risk
D) risk management

Question 74

Different random numbers generated by the computer are probabilistically dependent. This implies that when we generate a random number in a particular cell, it has some effect on the values of any other random numbers generated in the spreadsheet.

Question 75

A correlation matrix must always have 1's along its diagonal (because a variable is always perfectly correlated with itself) and the correlations between variables elsewhere.

Question 76

Correlation between two random input variables might not change the mean of an output, but it can definitely affect the variability and shape of an output distribution.

Question 77

(A) What are the appropriate probability distributions to model the number of faculty members showing up in each lot?
(B) Given the current situation, estimate the probability that on a peak day, at least one faculty member with a sticker will be unable to find a parking space. Assume that the number who shows up at each lot is independent of the number who shows up at the other two lots.
(C) Suppose that faculty members are allowed to park in any lot. Does this help solve the problem? Why or why not?
(D) Suppose that the numbers of faculty who show up at the three lots are correlated, with each correlation equal to 0.80. Does your answer to (C) change? Why or why not?

Question 78

Suppose you run a simulation model several times with different order quantities. What can we infer about the the quantity that maximizes the output, the company's profit?

A) This quantity is the optimal order quantity.
B) This quantity might be the optimal order quantity, but we also need to consider the company's attitude toward risk.
C) This is not necessarily the optimal order quantity, because it may have produced the largest profit by luck.
D) We cannot infer anything about the quantity.

Question 79

A common guideline in constructing confidence intervals for the mean is to place upper and lower bounds one standard error on either side of the average to obtain an approximate 95% confidence interval.

Question 80

Suppose you are going to invest equal amounts in three stocks. The annual return from each stock is normally distributed with a mean of 0.01 (1%) and a standard deviation of 0.06. The annual return on your portfolio, the output variable of interest, is the average of the three stock returns. Run @RISK, using 1000 iterations, on each of the scenarios described in the questions below, and report few results from the summary report sheets.
(A) The three stock returns are highly correlated. The correlation between each pair is 0.9
(B) The three stock returns are practically independent. The correlation between each pair is 0.1
(C) The first two stocks are moderately correlated. The correlation between their returns is 0.4. The third stock's return is negatively correlated with the other two. The correlation between its return and each of the first two is -0.8.
(D) Compare the portfolio distributions from @RISK for the three scenarios in (A), (B) and (C). What do you conclude?
(E) You might think of a fourth scenario, where the correlation between each pair of returns is a large negative number such as -0.80. But explain intuitively why this makes no sense. Try running a simulation with these negative correlations to see what happens.