ASSIGNMENT NAME: Decision Making Techniques
STUDENT NAME: Shezan Shakeel
SUBJECT CODE: MTH 543
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Q1: Simulation: Monte Carlo Model
Investment Portfolio Model
Year opening balance Return
1
100,000
2
105,400
3
111,092
4
117,091
5
123,413
Gross
5.40%
5.40%
5.40%
5.40%
5.40%
5,400
5,692
5,999
6,323
6,664
Closing balance
105,400
111,092
117,091
123,413
130,078
5-year closing balance
Net Gain (Loss)
Net Return %
130,078
30,078
30.08%
Using the rate of return of 5.4%, the overall return for the 5 years period is obtained as
30,078 which translates to 30.08%
Using Simulation
The obtained return is treated as the expected return. Using this value and a standard
deviation of 8.3% it is possible to apply the Monte Carlo simulation
The net return calculated for 5 years is not very much different from a single fixed
observation
Returns
Variation Value
Events No Simulaltion
5,400.00
5399.90
1
5,691.60
5691.52
2
5,998.95
5999.05
3
6,322.89
6322.98
4
6,664.33
6664.38
5
6526
5910
6154
5915
6066
Other applications of the simulation techniques
One of the major advantages of simulation is it allows business to setup environment where
new ideas can be tested prior to coming up with complex business decisions. The techniques
used in the analysis allows for modification of parameters to have a glimpse of the relevant
information that is most crucial for the decision making. By applying this, business are able
to make better and less riskier choices. Some of the areas that we can apply simulation
include:
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Training: When simulation is applied during training sessions people are able to experience
complex situations and try a variety of techniques. An example of training on how to
complete tasks using software.
Improvement of processes: Simulation methods allows experimentation of processes to help
analyze business practices. A typical simulation model puts focus on unique aspect of the
business-like manufacturing or finance. Through enhancing the details of how the operations
are conducted bottlenecks can be identified and hence come up with small changes which can
have big impacts.
Prediction of Outcomes: Making use of Excel spreadsheet it’s possible to simulate the
future outcomes under different conditions. This way it’s possible to come up with more
reliable forecasts.
Risk Management: By means of data manipulation a business is able to determine the
efficient amount that needs to be invested so as to optimize the profits. One of the techniques
that are applicable in this case is the Monte Carlo simulation.
Q2: Regression
a. Devising linear regression model
From the excel output given, the intercept represents the value of the house price when all
the other variables are zero. Thereafter the values under the coefficients represent the
multiplier of the variables affecting the price.
By denoting the given variables by values x1 to x7 and the house price by y we can derive
the linear regression model below
Denoting the variables
Price
Squre Footage Bedrooms Bathrooms Car garage have a pool on a lake on golf course
y
x1
x2
x3
x4
x5
x6
x7
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INTERPRETING REGRESSION STATISTICS TABLE
Multiple R: This the correlation coefficient it tells how strong the linear relationship is. A
value of 1 means a perfect positive relationship and a value of zero means no relationship
at all. In this case is 0.97 which is close to 1 means a nearly perfect positive relationship.
𝑅 = √𝑅 2
R Square: This is R2 the coefficient of determination. It tells how many points fall on the
regression line. In this case 0.95 of the values fit the model.
𝑅2 =
𝑆𝑆𝑅(𝑆𝑢𝑚 𝑜𝑓 𝑆𝑞𝑎𝑢𝑟𝑒𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑅𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛)
𝑆𝑆𝑇(𝑆𝑢𝑚 𝑜𝑓 𝑆𝑞𝑎𝑢𝑟𝑒𝑠 𝑇𝑜𝑡𝑎𝑙)
Adjusted R Square: It explains the degree to which input variables explain the variation
of output/ predicted variable. In this case 0.82, it means 82% of the variation in the output
variable is explained by the input variable.
𝑆𝑆𝐸
𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑅 𝑆𝑞𝑢𝑎𝑟𝑒 = 1 − 𝑛 − 𝑘 − 1
𝑆𝑆𝑇
𝑛−1
SSE (Sum of Squares of Error): The total sum of the squared differences between each
observation(Y) and the predicted value (𝑌̂).
SST = Sum of Squares of Total: The total sum of the squared differences between each
observation (Y) and the mean 𝑌̅
Standard Error: The standard error of the regression (S), also known as the standard
error of the estimate, represents the average distance that the observed values fall from
the regression line. Smaller values are better because it indicates that the observations are
closer to the fitted line.
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐸𝑟𝑟𝑜𝑟(𝑆𝐸) = √𝑀𝑆𝐸
Observations: This refers to the numbers of events in the sample. In this case are 56
INTERPRETING ANOVA TABLE
ANOVA (Analysis of Variance) consists of calculation that provides information about
levels of variability within a regression model and for tests of significance.
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degrees of
ANOVA
freedom
df
Regression k
n-k-1
Residual
n-1
Total
Calculation of ANOVA Table
sum of
squares
mean square
SS
MS=(SS/DF)
SSR
MSR=SSR/k
SSE
MSE=SSE/(n-k-1)
SST
Critical value of F
F stat
stat
F
Significance F
MSR/MSE P(F>MSR/MSE)
The difference between SST and SSE is the sum of squares explained by the regression. It
is called SSR.
𝐼𝑛 𝑡ℎ𝑖𝑠 𝑐𝑎𝑠𝑒 𝑆𝑆𝑇 − 𝑆𝑆𝐸 = 𝑆𝑆𝑅
8.44E+12 - 5.62E+13 = 4.78E+13
Each sum of squares has corresponding degrees of freedom (DF) with it. Errors are
obtained after calculating two regression parameters from the data, errors have n-2
degrees of freedom.
𝑆𝑒 = √
𝑆𝑆𝐸
𝑛−2
SSE/ (n-2) is called mean squared errors or MSE.
𝑆 2 = 𝑀𝑆𝐸 =
𝑆𝑆𝐸
𝑛−𝑘−1
An estimate of the variance of the errors in regression; n is the sample size and k is the
number of independent variables.
𝑀𝑆𝑅 =
𝑆𝑆𝑅
𝑘
Mean Square of Regression, k is the number independent variables. F statistic used to test
the significance of overall regression model.
𝐹=
𝑀𝑆𝑅
𝑀𝑆𝐸
=
6.8248𝐸+12
1.7584𝐸+11
= 3.88E+01 calculated as per values given in the above table.
To determine which of the independent variables in a regression model is significant, a
significance test on the coefficient for each variable is performed. The null hypothesis is
that the coefficient is 0 𝐻0 : 𝛽1 = 0 and the alternate hypothesis is that it is not 0
𝐻0 : 𝛽1 ≠ 0. If the p-value is lower than the level of significance then the null hypothesis
is rejected. The reliability of the linear model in predicting the house prices will be
evaluated based on the model significance. Looking at the F statistic it indicates a value
that is way below 1% and 5% hence at either 99% or 95% level of significance the model
cannot be accepted as reliable. On the other hand, the value of R square is 84.98%, this
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indicates that up to approximately 85% of the house price variations can be explained by
the changes in the seven variables (x1 to x7) listed above
b. Other predictive approaches

Logistic regression: This is a regression that is best applicable in case where the
dependent is dichotomous. It is used in describing data and give explanations in a
situation where there is one binary variable and one or more ordinal, nominal
interval or ratio level independent variable.

Random forests: These are ensemble learning techniques that apply in classifying
regression and other tasks. Their operation depends on the construction of a
multitude decision trees at the training phase and outputting the class that
represents the mode of the classes.

Decision Tree: Decision tree builds regression or classification models in the
form of a tree structure. It breaks down a dataset into smaller and smaller subsets
while at the same time an associated decision tree is incrementally developed. The
final result is a tree with decision nodes and leaf nodes.
Q3: Linear Programming Model
To obtain the optimal nutritional needs there is need for the foods to be satisfied in a way that
all the ingredients are supplied in the right quantity while at the same time minimizing the
cost.
Therefore, to develop the linear problem will first arrange the constraints
Then Carbohydrates, Protein, Fat these are the set of constraints
Carbohydrates
15𝑥 + 45𝑦 ≥ 150
Protein
60𝑥 + 15𝑦 ≥ 120
Fat
45𝑥 + 6𝑦 ≥ 180
C = Cost
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The function to be minimized
Grams of Ingredients per Serving
Ingredient
Steak
Potatoes
Daily Requirement (grams)
Carbohydrates
15
45
≥ 150
Protein
60
15
≥ 120
Fat
45
6
≥ 180
Cost per serving
$12
$6
Applying the Simplex method in excel we have
15
60
45
c= 12
-------------> 1
x
15
60
45
12
y
45
15
6
6
x
x
x
x
+
+
+
+
45
15
6
6
y ≥ 150
y ≥ 120
y ≥ 180
y
c
150
120
180
0
---------------->2 Transposing
R1
R2
R3
----->3
u
15
45
150
v
60
15
120
w
45
6
180
12
6
0
Adding Slack variables
15 u +
45 u +
-150 u -
60 v + 45 w + S1
15 v +
6 w + S2
120 v - 180 w + P
= 12
= 6
= 0
In the Tableau below R1 as row one, R2 as row two and R3 as row three.
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R1
R2
u
15
45
R3
-150
TABLEAU.01
v
w
60
45
15
6
-120
-180
R1
R2
u
0.33
45
v
1.33
15
TABLEAU.02
w
S1
1.00
0.02
6
0
R3
-150
-120
-180
R1
R2
u
0.33
43.00
v
1.33
7.00
R3
-90.00
120
R1
R2
u
0.33
1.000
v
1.33
0.163
R3
-90.00
120
u
R1 1.01
R2 1.00
v
4.04
0.16
R3 0.00
134.65
0
TABLEAU.03
w
S1
1.00
0.02
0.00
-0.13
0
S1
1
0
S2
0
1
P
0
0
C
12
6
0
0
1
0
S2
0.00
1
P
0.00
0
C
0.27
6
0
1
0
S2
0.00
1.00
P
0.00
0.00
C
0.27
4.40
0
1
48
4
TABLEAU.04
w
S1
1.00
0.02
0.00
-0.003
S2
0.00
0.023
P
0.00
0.000
C
0.27
0.10
0.00
0.00
1.00
48.00
4.00
TABLEAU.05
w
S1
S2
3.03
0.07
0.00
0.00
-0.003
0.02
P
0.00
0.00
C
0.81
0.10
0.00
1.00
57.21
3.72
2.09
The tableau is repeated until R3 have no negative values. The table represents the right
quantity of each food to serve that will minimize the cost while at the same time giving the
required nutrition.
Limitation of the Simplex approach
The methods entail understanding of many technical aspects that are not easy to be
understood by many managers who have no prior knowledge of the topic.
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Also, linear programming tasks require lots of expertise, time and are therefore cumbersome.
The method needs several steps to be adopted and must proceed in a systematic manner prior
to arriving at the correct solution.
Alternatives to the Simplex approach
Graphical solutions, instead of applying the simplex, method it is possible to obtain a solution
of the linear programme by using the graphical method. This entails graphing the constraints
and then using the objective function to arrive at the optimal solution.
Graphical methods of solving LP problems can only be used for problems with two decision
variables. Two axes representing the 2 decision variables are drawn and the lines representing
the constraints are plotted.
Graphical methods are simplest to use and should be used wherever possible. Graphical
methods can deal with any number of limitations, but as each limitation is shown as a line on
or graph, a large number of lines may make the graph difficult to read.
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REFERENCES:
https://en.wikipedia.org/wiki/Random_forest
https://statistics.laerd.com/spss-tutorials/binomial-logistic-regression-using-spssstatistics.php
https://stats.idre.ucla.edu/other/examples/ara/
https://stats.stackexchange.com/questions/215490/can-i-run-a-regression-when-bothindependent-and-dependent-variables-are-all-dic
https://en.wikibooks.org/wiki/Operations_Research/The_Simplex_Method
https://www.analyticsvidhya.com/blog/2017/02/lintroductory-guide-on-linearprogramming-explained-in-simple-english/
http://www.finance-assignment.com/limitations-of-lpp-simplex-method
https://www.saedsayad.com/decision_tree_reg.htm
http://www.jerrydallal.com/LHSP/slrout.htm
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