IE415
SUMMER 2015
Homework 2
1. A decision to invest in a new facility must be made. The facility is a warehouse
costing $100K. The warehouse has an expected useful life of 35 years and a net
salvage value of $25K. Annual receipts are expected to be $17K and annual costs of
the facility are $4K for maintenance and $2K for taxes. The assumed interest rate =
6%.
a. Compute the NPV of investing in the warehouse using the best estimated cash
flows stated in the problem.
b. Assume the salvage value, annual receipts, and annual maintenance costs are
expected to vary independently and uniformly ±50% around the best estimated
values. Incorporate this into a spreadsheet simulation. Simulate 50000 times
(50000 replications). Use an initial seed of 999. What is the average NPV, and
estimated standard deviation of the NPV? Also turn in the output chart (frequency
plot) for the simulation.
c. Repeat b but include a positive rank correlation of 0.85 between annual receipts
and annual maintenance costs.
2. A uniform distribution between 0 and 1 (denoted U(0,1)) is a fundamental
distribution that is used as part of generating random observations from other
distributions. Develop a simulation in Excel to estimate the expected value of Z (i.e.,
E[Z]), where Z = max(X,Y) and X and Y are independent U(0,1) random variables.
Provide the estimate of E[Z] as well as the standard deviation of the simulation
output. Simulate 500,000 times (500,000 replications or trials) and use an initial seed
of 999. Turn in a copy of the forecast statistics table and frequency chart
3. Consider the x-y plane and a circle of radius = 1, centered at x=2, y=2. An algorithm
for generating random points within this circle is as follows:
1. Generate a random angle  that is uniformly distribute d between   , and   .
2. Generate a random distance r from the center of the circle where r ~ U (0,1).
3. Compute the coordinate s of the point
x  2  r * cos( ),
y  2  r * sin(  ).
Conduct a Monte Carlo simulation and plot such randomly generated points to
empirically demonstrate that this algorithm does not generate points that are
uniformly distributed in the circle.
i.
ii.
Turn in your plot as part of your answer.
Where is the concentration of points greater?
4. Three fair die are rolled. Use Monte Carlo simulation to estimate the probability
distribution of the maximum of the three rolled die (i.e., let X represent the maximum
when three fair die are rolled. Estimate p(X=i) for i= 1 to 6.). Simulate 500,000 times
(500,000 replications). Turn in a copy of the forecast frequency chart.
5. A function of two random variables (X and Y) needs to be evaluated. X is a lognormal
random variable with mean = 25 and standard deviation = 5. Y is a uniform random
variable with minimum = 40 and maximum = 90. The function is
𝑍=
Trial
values
1
2
3
4
5
5𝑌
+ ln(𝑌)
ln(𝑋)
Lognormal (Mean = 25,
StdDev =5)
28.4
27.6
23.2
31.7
33
Uniform(40,90)
77.6
82.8
87.1
62.6
52.8
a. Use the five realizations of X and Y to manually conduct a five trial Monte Carlo
simulation of Z. Based on these results estimate E[Z].
b. What is the sample rank correlation of the X and Y values in the table?
c. Assuming the same X and Y values in the table above were generated for a Monte
Carlo simulation such that the sample rank correlation is 1. Estimate E[Z].
6. Estimate the mean and standard deviation of completion times for the project network
below (for which a node-arc incidence matrix was completed as an in-class exercise).
The arcs have the following distribution of completion times:
1-2: Uniform(3,7)
1-5: Lognormal(Mean = 20, StdDev = 30)
1-4: Lognormal(Mean = 10, StdDev = 10)
2-3: Uniform(5,10)
2-5: Lognormal(Mean = 15, StdDev = 10)
3-4: Uniform(1,5)
3-5: Uniform(1,15)
4-5: Lognormal(Mean = 10, StdDev = 10)
2
1
5
3
4
Use Crystal Ball simulation to develop your estimates. Simulate for 50,000 trials and
use an initial seed of 999. Turn in the statistics view and frequency chart.
2
7. Use Crystal Ball simulation to estimate the value of

1

  1  0.25 cos x dx
0
Use 1,000,000 trials and turn in the statistics view for the forecast.