1. TERM PROJECT
TEAM 5
SOPHIE MICHELOT
MICKENSON PIERRE
RYAN GARRINGER
QMB6603 QUANTITATIVE METHODS IN BUSINESS

2. SUMMARY
Introduction
Data & modeling
Formulation & results – Forecating
Formulation & results – Decision analysis
Conclusion
p.3
p.5
p.9
p.17
p.25

3. INTRODUCTION
Explain the problem undertaken its significance
The problem we are undertaking is estimating more accurate future property tax expenses for
Barron Collier Company (BCC). Currently, at BCC property tax expenses are projected simply using
last year’s data. Since the company owns several hundred properties we will use the forecasting OR
method to try and forecast more accurate short term property tax figures. Once we have conducted
a forecasting projection we will be constructing a decision analysis to help identify a
recommendation for future property investments, through a property tax expense approach.
Explain the benefits of the project to the business where data is collected.
1. Breakdown in property category and in depth look at each property owned.
2. Forecasting future values for most properties.
3. Analysis to provide recommendations for future transactions.
Breakdown in Property Category
We will be focusing our analysis in residential and commercial type properties for our project. Once
we have allocated a property to a category we then further break down the category by market
value. Since we are analyzing a significant amount of properties we will be able to identify if any
outliers are present. If we see that a property is over or under taxed by a significant margin we can
recommend the company to take a further look at property to make sure they are being tax
accurately.
Forecasting future values for most properties
We will be analyzing a lot of BCC property portfolio within the residential and commercial category
and it will be able to identify trends with individual properties and categories. Since we are
conducting a full scale analysis of the portfolio we hope we will be able to forecast values for most
of the properties within these categories.
Analysis to Provide Recommendations for Future Transactions
Once we have concluded a forecasting model we will be conducting a decision analysis. We will be
utilizing current Collier County figures to produce a decision tree and hope that we provide BCC
3

4. with a recommendation to for future purchases if the company is strictly considering property tax
expenses for the purchase of a property.
Explain the management science (operations research) method used to formulate the problem such as
(linear programming, transportation/assignment, project management, waiting line, simulation,
decision analysis, etc.).
First method used : Forecasting analysis
Time series methods : moving average (3 periods selected : 3 months, 5 months, and
12 months), exponential smoothing (2 smoothing constants selected)
Linear trend : least squares calculations
Forecast accuracy analysis : MAD and MAPD
Second method : decision analysis
Determining factors influencing real estate investments
Establishing probabilities and payoffs
Building the decision tree
Fill out the below table showing the tasks you went thru to accomplish this project, and then write the
name of the group member who worked on the task, and tome spent. –Add more rows or change the
names of specific tasks as applies to your project.
PROJECT TASKS
GROUP
MEMBER %TIME/EFFORT
ASSIGNED/WORKED
SPENT
Finding the problem
Ryan
5%
Contact w/ Industry
Ryan
5%
Data Collection
Ryan, Mickenson, Sophie
15%
Data Analysis
Ryan, Mickenson, Sophie
5%
Forecasting calculations
Mickenson, Sophie
25%
Decision analysis
Sophie
10%
Interpretation of results
Ryan, Mickenson, Sophie
10%
Report Writing
Ryan, Mickenson
20%
Power Point Presentation
Sophie
5%
4

5. DATA $ MODELING
Describe the business you worked with. Their product/service, size, number of employees, sales volume,
location.
The business we worked with was Barron Collier Companies located in Naples, Florida. They are a
private family company that has investment interest in real estate, agriculture, minerals and other
resources. Along with its major focuses in different investment fields the company provides other
services such as but not limited to: engineering, title insurance company, environmental services
property management, etc. The total number of employees is within the 50-100 range at the
corporate office.
Describe clearly the data, assumptions, terms used, and how the data is collected. Describe your
data collection methodology.
The data we collected was through BCC and the Collier County Property Appraisers/Collier County
Tax Collector website. BCC provided a selective amount from their vast portfolio, in which we used
all the residential and commercial properties for our project. Once we received the ID numbers
associated with BCC owned properties we were able to start our property tax search. On the county
websites we received the following data:
Market value- The value assessed to the property from 2008-2012. This is a summation of
land value and improved land value for the property. The property tax amount is derived
from the market value.
Property Taxes- We also found the previous five years of tax values (2008-2012) for five
different time periods (Nov.-March) per each year. The total taxes paid are determined by
the payment date. For our project we will be taking an average of the five pay periods per
each year.

Below is an example of our constructed data :
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6. Below is a sample of a property card with tax information:
6

7. Describe the objective function, decision variables, constraints-. If forecasting or decision analysis,
explain how the methods are implemented, probability figures, and so on.
Our objective function is to utilize past property tax information to help forecast future property
tax values for residential and commercial properties. We hope to take our forecasting findings
along with Collier County commercial/residential property information and preform a decision
analysis to provide recommendations to BCC for future property investments.
Forecasting
To implement forecasting we utilized several different methods including:
a. Moving Average
b. Exponential Smoothing
c. Least Square Calculations
d. MAD and MAPD
Moving Average
We decided to calculate first the moving average. We used three different time periods: 3 months, 5
months, and 12 months. The aim is to determine short-term forecast of the amount of tax expenses
due to the next payment periods. For more relevance, we decided to split our data into several
categories. We differentiated residential properties and commercial properties. Then, we built subcategories based on the value of the property. Indeed, tax expenses are directly linked to the value
of the taxed property. Here is a summary of the categories used:
Residential properties
Commercial properties
$0 – $99,999
$0 - $199,999
$100,000 & more
$200,000 - $999,999
-
$1,000,000 & more
Thus, for each category, we calculated the average of tax expenses per payment periods. The
payment periods are November, December, January, February, and March for the accounting years
2008 to 2012.
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8. Exponential Smoothing
The Second method of time series analysis we used is exponential smoothing. Here, the most recent
data are weighted more than the older one. We calculate the exponential smoothing with two
smoothing constants, 0.3 and 0.5. We used the average amount of tax expenses per pay period.
Least Square Calculations
The third method is different as we present forecasting based on linear trend line. We used least squares
calculations to determine the regression model where tax expenses are related to time.
MAD and MAPD
We close the forecasting analysis by determining the accuracy of our calculations. We calculated the
mean absolute deviation (MAD) and the mean absolute percent deviation (MAPD) for our exponential
smoothing and linear trend line.
Decision analysis
The purpose of decision analysis is to provide efficient tools to help the decision making process.
We developed a decision tree with probabilities in order to understand which action on the real
estate market BCC could make so as to maximize its profit.
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9. FORMULATION & RESULTS – FORECASTING
After having calculated the different average of tax expenses per payment periods, we start the
calculations of the 3-month moving average. Here is the formula for moving average :
MAn = (
𝑛
𝑖=1
𝐷𝑖)/n
Here the details of calculations for the average amount of tax expenses per payment periods:
This is an example of the details of calculations of moving average for commercial properties ($0–
$99,999):
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10. We use the same principle to calculate the 5-month and 12-month moving averages:
10

11. Moving average results
Property category
Nov 12 tax expenses
3-month moving
average
Nov 12 tax expenses
5-month moving
average
Commercial
$0 - $199,999
$7,005.44 ↑
$6,934.68 ↓
Commercial
$200,000 - $999,999
$14,673.05 ↑
$14,524.83 ↓
Commercial
$1,000,000 & more
$71,244.08 ↑
$70,532.34 ↑
Residential
$0 - $99,999
$1,007.93 ↑
$997.75 ↑
Residential
$100,000 & more
$5,383.44 ↑
$5,310.33 ↑
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12. Exponential smoothing
Here again we used the average amount of tax expenses per payment period. . Here is the
formula for exponential smoothing :
Fₓ₊₁ = αDₓ + ( 1 – α ) Fₓ
D corresponds to the average amount of tax expenses per payment periods and F relates to
the forecast. Here are the details of calculations :
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13. Exponential smoothing (α = 0.5) results
Property category
Nov 12 tax expenses
Exponential smoothing
Commercial
$0 - $199,999
$7,051.32 ↑
Commercial
$200,000 - $999,999
$14,712.55 ↑
Commercial
$1,000,000 & more
$71,293.78 ↑
Residential
$0 - $99,999
$1,009.56 ↑
Residential
$100,000 & more
$5,385.73 ↑
Least squares calculations
We present now other methods of forecasting based on linear trend line. We used least squares
calculations to determine the regression model where tax expenses are related to time. We
determined the different factors composing the least squares calculations :
b=
𝑥𝑦 −𝑛𝑥𝑦
𝑥 2 −𝑛𝑥 2
a = 𝑦 - b𝑥
Here are our detailed calculations :
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14. 14

15. Linear trend results
Property category
Nov 12 tax expenses
Linear trend
Commercial
$0 - $199,999
$5,233.39 ↓
Commercial
$200,000 - $999,999
$13,433.81 ↓
Commercial
$1,000,000 & more
$74,490.36 ↑
Residential
$0 - $99,999
$983.54 ↓
Residential
$100,000 & more
$3,834.76 ↓
Forecast accuracy
We close the forecasting analysis by determining the accuracy of our calculations. We calculated the
mean absolute deviation (MAD) and the mean absolute percent deviation (MAPD) for our
exponential smoothing and linear trend line. Here are the formulas we used to determine both MAD
and MAPD :
MAD =
MAPD =
𝐷−𝐹
𝑛
𝐷−𝐹
𝐷
Here are our detailed calculations :
15

16. Category
Commercial
$0 - $99,999
Commercial
$100,000 - $999,999
Commercial
$1,000,000 & more
Residential
$0 – $99,999
Residential
$100,000 & more
Method
MAD
MAPD
Exponential smoothing
α = 0.5
3,222.32
27.37%
Least squares
calculations
2,831.02
24.05%
Exponential smoothing
α = 0.5
616.71
3.44%
Least squares
calculations
852.62
4.75%
Exponential smoothing
α = 0.5
2,358.39
3.31%
Least squares
calculations
4,534.31
6.37%
Exponential smoothing
α = 0.5
18.82
1.77%
Least squares
calculations
18.65
1.75%
Exponential smoothing
α = 0.5
489.24
6.42%
Least squares
calculations
709.2
9.30%
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17. FORMULATION & RESULTS – DECISION ANALYSIS
D
ecision analysis helps, by developing tools like decision tree, to make decision in a situation
characterized by uncertainty. It consists in determining the decision alternatives, the states
of nature likely to occur under some probabilities, and the different payoffs resulting from
this succession of events. The aim of the decision analysis is to help BCC in its investments decisions
in the local real estate market. By building decision tree, we aim at providing efficient tools to make
the best decision at the highest profit for the company.
Investing in real estate is almost a science. A large number of factors can influence the
demand of properties for sale. We start our analysis with the principle that if the demand increases,
the prices on the real estate market will increase too, by a phenomenon of scarcity. Thus, the
situation of the market is favorable for the seller, but not for the buyer. On the contrary, if the
demand is low, the prices will tend to decrease by a phenomenon of the scarcity of demand, and an
abundance of offer. Here, the situation of the market is favorable for the buyer, and unfavorable for
the seller. Based on that, we get our two decision alternatives : should BCC invest or not invest ?
Now, we have to determine the states of nature.
According to the economic press and related sources of information, there are three major
factors influencing the demand on the real estate market. The first one in terms of importance is the
tax conditions. Thanks to the previous forecasting part, we have an accurate idea of what the tax
expenses will represent for BCC in the short-term future. If the tax expenses are likely to increase,
because of new tax rates or new fiscal laws for instance, the price and cost for the buyer are likely
to increase too. On the contrary, it can be a real advantage for the seller, who could be willing to
decrease its costs. The second factor is the mortgage conditions. It refers to the level of mortgage
rates, home loans, etc. If the mortgage rates increase, the situation becomes unfavorable for the
buyer, thus decreasing the demand. If the mortgage rates decrease, the demand will increase. The
third factor gathers the different elements defining the conditions of economic environment. We
will see these elements in a deeper analysis in the coming paragraphs.
We decided to conduct the decision analysis on one specific category defined in the
forecasting part : the residential properties with an estimated value between $0 and $99,999. We
will not tackle the commercial estate market as we consider that investing in that kind of property
corresponds to specific projects between BCC and external partners, and not necessarily to a pure
answer to market opportunity, as it is more the case with residential properties.
Establishing probabilities and payoffs
There are three decisions opened to BCC : invest, sell, or do nothing. We will refer to these decisions
in the decision tree as Invest, Sell, Status quo.
The states of nature are the conditions of the market, the conditions of mortgage, and the
conditions of tax expenses. We will refer to these states of nature in the decision tree as : Good/Bad
market conditions, Good/Bad mortgage conditions, and Tax increase/Tax decrease. We will first
describe how to establish the probabilities for each state of nature.
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18. Market conditions
We established earlier that the market conditions are determined by several factors. We
select four of them to forecast the fluctuation of the demand on the real estate market. The first
factor is the prices evolution of residential properties in Collier County. After a strong decrease
from 2007 to 2009, the prices increase slowly, and we thus bet on an increase of prices (source :
movoto.com). This situation is favorable for the seller, and unfavorable for the buyer. The second
factor is the demographic evolution. The number of inhabitants in Collier County is going to
increase at regular pace (source : US census bureau). This will have a positive impact on demand,
and thus contribute to an increase of price. The situation is favorable for the seller, and unfavorable
for the buyer. The third factor is the vacancy rate. The actual rate amounts 32.5% (source :
marconews.com). We bet on a decrease on this rate, which is quite high today. This will have an
impact on the scarcity of properties, thus increasing prices. The situation is favorable for the seller,
and unfavorable for the buyer. The fourth factor is the housing starts rate. The actual rate is 52.9%
(source : metrostudyreport.com). Analysts predict that this rate is going to increase. This means
that the number of available and new houses will increase. This situation is more favorable for the
buyer than for the seller. To conclude, we estimate that the market conditions will sustain the offer
of properties. We define the probabilities of this state of nature as follows :
Market conditions
Good
Bad
Probability
0.6
0.4
Mortgage conditions
To establish the probabilities regarding mortgage conditions, we look at the mortgage rates. After a
certain decrease, the rate slightly increases. Nonetheless, we bet on a stabilization of the rate for the
coming months. We define the probabilities of this state of nature as follows :
Mortgage conditions
Good
Bad
Probability
0.5
0.5
Tax conditions
To determine the probability of increase or decrease of the tax expenses, we use our results from
the forecasting part. The forecasts for the tax expenses linked to the residential properties with an
estimated land value between $0 and $99,999, based on the least squares calculations methods,
follows this scheme :
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19. 1
2
3
4
5
6
7
8
9
10
Tax expenses
983.5417941
980.6746253
977.8074566
974.9402878
972.073119
969.2059502
966.3387814
963.4716127
960.6044439
957.7372751
9706.395346
1100
1080
1060
1040
1020
Actual
1000
980
Linear trend
960
940
920
Nov 07
Jan 08
Mar 08
Dec 08
Feb 09
Nov 09
Jan 10
Mar 10
Dec 10
Feb 11
Nov 11
Jan 12
Mar 12
Period
For the coming two years, the average tax expenses will amount approximately $9,706. The linear
trend seems to forecast a decrease in tax expenses.
We define the probabilities of this state of nature as follows :
Taxes
Increase
Decrease
Probability
0.3
0.7
We can now give a first overview of the decision tree with the different probabilities defined.
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20. 20

21. Now let’s determine the different payoffs. We start with the following assumptions. Let’s
consider that BCC owns one property of the category chosen (40-$99,999), and that it wants to
analyze the payoffs of the different decisions taken after two years. Thus, BCC has three decision
choices : invest and buy a new property with the same characteristics that the one it already owns,
sell this property, or do nothing. Notice that the property currently owned yields an average rent of
$886 per month. To determine the payoffs, we attribute an increase or decrease in dollar terms to
each states of nature that occurs. The following table describes the calculations of payoffs :
Decision
Invest
Market conditions
Good
Average price of the
property : $20,000
If the market conditions
are good, we estimate
that the demand is high,
thus the prices are high
too. We add $6,000 to
the starting price.
Mortgage
conditions
Good
If the mortgage
conditions are good,
we estimate that the
demand is high, thus
the prices are high
too. We add $2,000
to the starting price.
Tax variation
Payoff
Increase
The cost of purchase a
new
property
is
estimated to $38,000.
However, BCC will get
$21,264 ($886x12) of
rent revenue from its
current possession, plus
another $21,264 from its
new acquisition. BCC will
also have to pay for the
tax of its current
property ($10,00). Thus,
the final payoff will be
$42,528 - $38,000 $10,000
If the tax increases,
we estimate that
the amount of tax
expenses for the
two coming years
will be $10,000.
= $38,000
=$28,000
=$26,000
=-$5,472
Invest
Invest
Invest
Invest
Good
=$26,000
Good
=$26,000
Good
=$26,000
Bad
If the market conditions
are bad, we estimate
that the demand is low,
Good
=$28,000
Bad
If the mortgage
conditions are bad,
we estimate that the
demand is low, thus
the prices are lower.
We substract $2,000
to the starting price.
Decrease
If the tax decreases,
we estimate that
the amount of tax
expenses for the
two coming years
will be $9,706.
= $37,706
Increase
=$34,000
=$24,000
Bad
=$24,000
Decrease
=$33,706
Good
=$16,000
Increase
=$26,000
21
$42,528
$9,706
-
$37,706
-
=-$4,884
$42,528 - $34,000 $10,000
=-$1,472
$42,528 - $33,706 $9,706
=$884
$42,528 – $26,000 $10,000
=$6,528

22. Invest
Bad
Good
=$16,000
Increase
=$26,000
$42,528 – $26,000 $10,000
=$6,528
=$14,000
Bad
=$14,000
Good
=$16,000
Decrease
=$25,706
$42,528
$9,706
Invest
Bad
=$14,000
Bad
=$12,000
Increase
=$22,000
$42,528
$10,000
Invest
Bad
=$14,000
Bad
=$12,000
Decrease
=$21,706
$42,528
$9,706
Sell
Good
=$26,000
Good
=$28,000
Increase
By selling its property,
BCC gives up its rent
revenues. Thus, the only
got comes from the sale
of the property.
If the market conditions
are bad, we estimate
that the demand is low,
thus the prices are low
too.
We
substract
$6,000 to the starting
price.
Invest
-
$25,706
-
=$7,116
-
$22,000
-
=$10,528
-
$21,706
-
=$11,116
Sell
Sell
Sell
Sell
Sell
Sell
Sell
Good
=$26,000
Good
=$28,000
Good
=$26,000
Good
=$26,000
Bad
=$14,000
Bad
=$14,000
Bad
=$14,000
Bad
=$14,000
Bad
=$24,000
Bad
=$24,000
Good
=$16,000
Good
=$16,000
Bad
=$12,000
Bad
=$12,000
22
An increase in tax
expenses
can
discourage
the
demand and thus
decrease the prices.
We estimate an
impact of $2,000 on
the selling price.
=$26,000
Decrease
=$26,000
=$30,000
A decrease in tax
expenses
can
encourage
the
demand and thus
increase the prices.
We estimate an
impact of $2,000 on
the selling price.
=$30,000
Increase
=$22,000
Decrease
=$26,000
Increase
=$14,000
Decrease
=$18,000
Increase
=$10,000
Decrease
=$14,000
=$22,000
=$26,000
=$14,000
=$18,000
=$10,000
=$14,000

23. Status quo
Status quo
Status quo
Status quo
Status quo
Status quo
Status quo
Status quo
Good
Good
Increase
By doing nothing, BCC
gives up to act on the
real estate market.
By doing nothing,
BCC gives up to act
on the real estate
BCC chooses to
keep its current
property and thus
Thus, any changes in market. Thus, any has to pay taxes.
the market will not changes in the =-$10,000
directly affect BCC.
market will not
directly affect BCC.
Good
Good
Decrease
=-$9,706
Good
Bad
Increase
=-$10,000
Good
Bad
Decrease
=-$9,707
Bad
Good
Increase
=-$10,000
Bad
Good
Decrease
=-$9,706
Bad
Bad
Increase
=-$10,000
Bad
Bad
Decrease
=-$9,706
With
its
current
property, BCC earns
$21,264 of rent revenue
over the period.
=$11,264
=$11,558
=$11,264
=$11,558
=$11,264
=$11,558
=$11,264
=$11,558
Here is an overview of the decision tree with probabilities, payoffs, and expected values : (next
page)
The expected value of the decision analysis is $22,000, which corresponds to the decision to sell the
property. This result refers to an analysis of revenue for the coming two years. The result could be
different for a longer period, but we can assume that it could also be less accurate as the conditions of
the market would have evolved. However, this analysis represents a good tool to help the decision
making.
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24. 24

25. CONCLUSION
1. Commercial Properties
After calculating the 3-months, 5-months, and 12-months moving average of the commercial
properties, we found out that the all three moving average forecasts smooth out the variability in
the actual average. However the 3-months moving average period was the closest to the real
average. This is because the 5-months and 12-months moving average consider older data than the
3-months moving average. Moving average does not react well to variations that occur for a reason,
therefore, it will be best to use the moving average for short-term forecasting rather than
forecasting that is too far into the future. We also found out that there is a trend in the movement of
the taxes throughout the five years. Therefore, we can confidently predict that what happened in
the past will happen again.
The second calculation we performed was the exponential smoothing. We found out that alpha =
0.5 was actually closer to the actual average than alpha = 0.3. And also, we found out that there is a
linear trend in the forecast and actual average. This information is crucial to the client, because they
can predict that what happens in the past will eventually happen in the future. For example if tax
increases in a month last year, in the current year, one can expect taxes to increase throughout the
same period.
The last method we use is the linear trend line method. This method clearly shows that there is a
linear trend which closely reflects the actual data.
And finally, we determined the forecast accuracy of our data by computing the MAD and MAPD for
all our forecasts. For the exponential smoothing forecast, alpha = 0.5 has a MAPD of 27.37 %, while
alpha = 0.3 has an MAPD of 29.90%, and the linear trend has an MAPD of 24.05%. The smallest
MAPD of those three forecasts is the linear trend with an MAPD of 24.05%, therefore we would
recommend that BCC uses the linear trend method to forecast future expenses for the short term.
2. Residential Properties
We performed a moving average forecast for the residential properties and the same results as the
commercial results. The 3-months moving average was the closest to the actual average, which
means that the moving average will work best if it is performed in the short run as opposed to the
long run.
The exponential smoothing also yields the same result as the results for the commercial properties.
We found out that the forecast using alpha = 0.5 was closer to the actual average than the forecast
using alpha = 0.3, and also that there is a trend in the results meaning that what happened in one
period the previous year will also happen in the same period during the upcoming year.
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26. The last forecasting method we used was the linear trend method. We found out that there is a
linear trend in the data which clearly reflects the actual data.
And finally, we also calculated the MAD and the MAPD of all our forecasts. We found out that the
MAPD for our exponential smoothing was 8.25 for alpha = 0.3 and 5.61 for alpha = 0.5, and the
MAPD for the linear trend line was 8.15%. The lowest MAPD of all the forecasts is the exponential
smoothing alpha = 0.5 with an MAPD of 5.61%. Therefore we would recommend BCC to use
exponential smoothing with alpha = 0.5 to forecast its future tax expenses for the short-term
period.
To validate the results, we would compare each one of them to the actual data and determine what
percentage did we deviate from the actual data. Currently, BCC uses the previous year’s data to
forecast future expenses, but in our methodology, we use data from the previous five years. In the
future, instead of using the data of previous year, BCC could use economic conditions and attempt
to predict their future expenses instead of just looking at the trends, and they need to analyze the
effects of certain conditions on the total value of the properties. If we had to redo this project, we
would definitely pick another topic. Forecasting and decision analysis are two extremely
complicated methods to use and comprehend. We definitely benefited greatly from doing this
project. It helped us understand the different methods of forecasting as well as how to apply
decision analysis to a real-world situation.
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