Risk, Profit, And Predictions Essay: Analysis.

Ilya and Gregor have very different approaches when it comes to risk. Whereas Ilya tends to be optimistic and likes to take risks, Gregor is much more conservative, and averse to risk-taking. They estimate that there is a 55% chance that the market is favourable, and therefore a 45% chance that it is unfavourable.

Provide answers, with justifications, for each of the following questions:

1. What would Ilya’s decision be? Why?
2. What would Gregor’s decision be? Why?
3. What would their decision be if they were to choose the alternative with the greatest expected value? Show all calculations and justify your answer. Ilya and Gregor now believe that the probability of a favourable market is not 55%.
4. Construct a plot showing how the expected value of the returns for Strategy 1 and Strategy 2 vary with the value of P (for 0 ≤ P ≤ 1), where P is the probability of a favourable market.

Questions

• What is the greatest number of people that Jim can reach weekly on his budget of $14,000 per week?
• How many ads of each type must he in order to achieve this result?

Estimation of Profit and Loss for Different Business Strategies

The profit and the loses that can be made from the business according to the estimates made by Gregor and Ilya considering three different strategies in a favourable and an unfavourable market condition is given in table 1.1.

According to the first Strategy, Gregor and Ilya would rent a fairly costly location in the district where their potential customers are located. According to the second strategy, they will be renting a location in the neighbouring suburb where the rent of the office will be cheaper than strategy 1. The third strategy was to not set up a business at all.

Table 1.1: Estimated Profit or Loss from the Market with Three Strategies

	Favourable Market	Unfavourable Market
Strategy 1	$20,000	$16,000
Strategy 2	$15,000	$6,000
Strategy 3	$0	$0

• From the point of view of Ilya, who is an optimist and is always prone to take the maximum risk, the strategy chosen by her will be Strategy 1 as in Strategy 1, despite of the risk, the profit that can be earned is maximum.

Table 1.2: Strategy chosen by Ilya
	Favourable Market	Unfavourable Market	Highest Profit / Loss
Strategy 1	$20,000	$16,000	$20,000
Strategy 2	$15,000	$6,000	$15,000
Strategy 3	$0	$0	$0

• From the point of view of Gregor, who is conservative and is always prone to maximizing the profit with minimum risk, the strategy chosen by him will be Strategy 1 as in Strategy 3, as there is no chance of loss in that strategy.

Table 1.3: Strategy chosen by Gregor
	Favourable Market	Unfavourable Market	Least Profit / Loss
Strategy 1	$20,000	$16,000	$16,000
Strategy 2	$15,000	$6,000	$6,000
Strategy 3	$0	$0	$0

• The probability that the market will be favourable is 0.55. Thus, the expected profits from each of the strategies can be evaluated in both the favourable and unfavourable markets. It can be seen that Strategy 2 will have the highest expected profit. Thus Strategy 2 will be chosen.

Table 1.4: Expected Profit / Loss
	Favourable Market	Unfavourable Market	Expected Profit / Loss
Strategy 1	$11,000	$7,200	$3,800
Strategy 2	$8,250	$2,700	$5,550
Strategy 3	$0	$0	$0

• When the probability of a favourable market is not exactly 0.55, and varies between a range of 0 and 1, the expected returns from the market are plotted in the following figure 1.1.

Figure 1.1: Expected Returns from Different Strategies

• i) The range of probability values for choosing strategy 1 is 0.67 ≤P ≤ 1

1. ii) The range of probability values for choosing strategy 2 is 0.29 ≤P ≤ 0.66

iii) The range of probability values for choosing strategy 3 is 0 ≤ P ≤ 0.28

The linear programming problem can be set up as follows:

Minimize Z = 960 TV + 480 Radio + 600 Billboards + 120 Newspapers

Subject to the constraints:

TV ≤ 10

Radio ≤ 10

Billboards ≤ 10

Newspapers ≤ 10

TV ≥ 6

Radio ≥ 6

TV + Radio ≥ 6

960 TV – 600 Billboards – 120 Newspapers ≥ 0

960 TV + 480 Radio + 600 Billboards + 120 Newspapers ≤ 14,000

And the non-negativity constraints TV ≥ 0, Radio ≥ 0, Billboards ≥ 0, Newspapers ≥ 0

• The maximum number of people to whom the advertisements can reach so that he can stay within his weekly budget is (6 * 36,000) + (6 * 26,500) + (8 * 30,000) = 6,15,000
• 6 ads can be posted to each of TV and Radio and 8 ads can be posted in billboards, so as to get the desired result.

• The simulation results show that the best case in this scenario for Max involving the least cost for inventory policy is $46,360. The worst case in this scenario for Max involving the highest cost for inventory policy is $64,640. The average cost of inventory policy that can be incurred by Max considering reorder point of 5 and reorder quantity of 5 is $56,862.
• i) When the reorder re-order point as 3 and the reordering quantity as 3, the minimum cost incurred is $33,200, the maximum cost incurred is $43,460 and the average cost that can be incurred is $38,058 on the inventory policy.

1. ii) When the reorder re-order point is 7 and the reordering quantity is 7, the minimum cost incurred is $80,860, the maximum cost incurred is $96,460 and the average cost that can be incurred is $86,876 on the inventory policy.

• It is clear that with reordering point and quantity of 3, the maximum cost incurred is even less than the minimum cost incurred in case of having a reordering point and quantity of 7. Thus, having a reordering point and quantity of 3 is a better option for the business.

The prediction of the selling prices of the house considering area as independent variable is

Selling Price = - 34301.5987 + (62.96 * Area)

The coefficient of determination for this model is (0.7952 ^ 2) = 0.6323. This indicates that 63.23 percent of the variability in the selling price can be explained by area of the house.

When area = 2000 ft², selling price = - 34301.5987 + (62.96 * 2000) = 91618.4

Figure 4.1: 10-fold cross-validation

Figure 4.2: 15-fold cross-validation

Model 2:

The prediction of the selling prices of the house considering bedrooms as independent variable is

Selling Price = 648.6487 + (35168.9189 * Number of Bedrooms)

The coefficient of determination for this model is (0.5047 ^ 2) = 0.2547. This indicates that 25.47 percent of the variability in the selling price can be explained by the number of bedrooms in the house.

When number of bedrooms = 3, selling price = 648.6487 + (35168.9189 * 3) = 106155

Figure 4.3: 10-fold Cross-Validation

Figure 4.4: 15-fold Cross-Validation

Model 3:

The prediction of the selling prices of the house considering age as independent variable is

Selling Price = 141448.2518 + (- 2256.7296 * Age)

The coefficient of determination for this model is (0.8629 ^ 2) = 0.7446. This indicates that 74.46 percent of the variability in the selling price can be explained by the age of the house.

When age of the house = 24 years, selling price = 141448.2518 + (- 2256.7296 * 24) = 87286.7.

Area can explain the highest variability in the selling prices of the house. Thus, prediction of the selling price of the house with area is the best model.

Linear Programming Problem

Figure 4.5: 10-fold cross-validation

Figure 4.6: 15-fold cross-validation

The prediction of the selling prices of the house considering area of the house and number of bedrooms in the house are the independent variables is given by:

Selling Price = -26129.5 + (76.1268 * Area) + (-12403.1 * Bedrooms)

The coefficient of determination for this model is (0.8616 ^ 2) = 0.7423. This indicates that 74.23 percent of the variability in the selling price can be explained by area and number of bedrooms in the house.

Figure 4.7: 10-fold cross-validation

Figure 4.8: 15-fold cross-validation

The prediction of the selling prices of the house considering area of the house and age of the house are the independent variables is given by:

Selling Price = 69793.9387 + (27.0743 * Area) + (-1554.9387 * Age)

The coefficient of determination for this model is (0.8798 ^ 2) = 0.774. This indicates that 77.4 percent of the variability in the selling price can be explained by area and age of the house

Figure 4.9: 10-fold cross-validation

Figure 4.10: 15-fold cross-validation

The prediction of the selling prices of the house considering number of bedrooms in the house and age of the house are the independent variables is given by:

Selling Price = 99495.77 + (12389 * Bedrooms) + (-1985.53 * Age)

The coefficient of determination for this model is (0.9309 ^ 2) = 0.8665. This indicates that 86.65 percent of the variability in the selling price can be explained by number of bedrooms and age of the house.

Figure 4.11: 10-fold cross-validation

Figure 4.12: 15-fold cross-validation

The prediction of the selling prices of the house considering area of the house, number of bedrooms in the house and age of the house are the independent variables is given by:

Selling Price = 70181.01 + (25.1505 * Area) + (-1574.39 * Age) + (1389.257 * Bedrooms)

The coefficient of determination for this model is (0.9407 ^ 2) = 0.8848. This indicates that 88.48 percent of the variability in the selling price can be explained by area, bedrooms and age of the house.

Figure 4.13: 10-fold cross-validation

Thus, prediction of selling price of the houses considering all the variables such as age, bedrooms and area of the house as the independent variables can explain the highest variability in the selling prices among the rest of the developed models. Hence, this is considered as the best model.

The multilayer perceptron model run with three independent variables area, age and number of bedrooms to predict the selling price of a house with one hidden layer shows a correlation coefficient of 0.9399 which indicates that the model is 93.96 percent accurate. Thus, the MLP model shows a more accurate prediction than the regression model.

It can be observed further, that with the increase in the number of hidden layers in the model, the accuracy of the model decreases. Thus, with one hidden layer, MLP can be considered as a better predictor model but with the increase in the number of hidden layers for an MLP model, regression model will be considered as a better predictor model as the prediction precision does not change in this model.

1. There are 17 attributes in totality, that are present in the dataset.
2. There are 7 numeric variables in the dataset.

• There are 10 categorical variables in the dataset.

1. There are 4521 examples in the dataset.
2. The class variable is y.
3. There are two values that can be taken by the class variable.

• The two values taken by the class variable are “yes” and “no”, indicating whether there will be or will not be an opening of bank account.

Logistic Regression

• Confusion Matrix
• ROC Curve
• Area under ROC Curve

Naïve Bayes Model

• Confusion Matrix
• ROC Curve
• Area under ROC Curve

Lift Chart for Logistic Regression Model

Lift Chart for Naïve Bayes Model

Conclusion

The ROC curves shows that the area under the ROC curve for logistic regression is 0.888, which is higher than the area under the ROC curve for Naïve Bayes classifier which is 0.845. Thus, Logistic is a better classifier.

The same can be concluded from the lift charts for both the classifiers, where it can be seen clearly that the lift in the Regression classifier is higher than the lift in the Naïve Bayes classifier, indicating logistic as the better classifier.