Decision analysis using decision tables

Ilya and Gregor are friends who recently completed studies in IT and business respectively. They are considering starting a part-time consultancy business. To get started they need to rent some office space; however, office space is relatively expensive, and this could affect the success of their venture.

They have identified three different strategies for selecting an office. Strategy 1 is to rent a fairly expensive office in a location in close proximity to the business district where many of their potential customers are located. Although the office space is expensive, in a favourable market they would expect to attract a lot of business and hence make a good profit; however in an unfavourable market, they may not attract sufficient business to cover the cost in rent, and therefore lose money.

They estimate that in a favourable market, they would be able to obtain a net profit of $20,000 over the next two years; but if the market was unfavourable, they could lose $16,000. Strategy 2 is to rent a less expensive office in a neighbouring suburb. They estimate that under a favourable market they could get a return of $15,000, but in an unfavourable market would lose $6,000.

The third strategy is to do nothing; i.e., not set up the business. 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:

a. What would Ilya’s decision be? Why?

b. What would Gregor’s decision be? Why?

c. 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%.

d. 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.

e. Find the range of values for P for which the following decisions would be made

## Optimal Strategies for Different Types of Individuals

There lies different opportunities for acquiring profit as well as loss in the concerned business by Ilya and Gregor, which in turn depend on the set of three probable strategies which yield different results in the favourable market and in the unfavourable market. Strategy 1 involves Gregor and Ilya renting a considerably costly office in the location near their probable customers. On the other hand, under Strategy 2, they can rent a comparatively cheaper office in the neighbouring suburb region and the last strategy is the strategy of not at all opening any business venture.

As can be seen from the above, Ilya being a risk loving and optimist personnel is eager to opt for maximum risk to get greater benefits. Keeping this into account the optimum strategy in the perception of Ilya is that of Strategy 1, as here, both the risks as well as level of expected profits are maximum.

Gregor being more conservative and risk averse than that of Ilya, the optimum strategy of the same is the one with maximum profit along with minimum risk. Thus, Gregor will choose the third strategy as the loss is minimum in the concerned strategy.

- As can be seen from the concern problem, the probability of the market to be favourable for the business is 0.55 while the probability of the same to be non-favourable is 0.45. Thus, the expected pay-offs of each of the concerned strategies, in this situation, can be compared with the help of the following table:

As is evident from the above numbers, the expected profit is comparatively highest in case of the second strategy, which in turn indicates to the fact that the second strategy of getting an office in the cheaper suburbs will be chosen.

- If the probability of the presence of a favourable market is not fixed at 0.55 and instead varies between range of 0 to 1, then according to the different values of the probability of the same the expected pay-offs or returns from the concerned market can be seen from the following figure:

- i. The probability range in which the first strategy can be chosen is 0.67≤P≤1
- The probability range in which the second strategy can be chosen is 0.29≤P≤0.66

iii. The probability range in which the third strategy can be chosen is 0≤P≤0.28

From the concerned problem, the linear programming model can be constructed as below:

Min (Z) = 960(TV) + 480 (Radio) + 600(Billboards) + 120 (Newspaper)

Subject to:

TV≤10

Radio≤10

Billboards≤10

Newspaper≤10

TV≥6

Radio≥6

TV + Radio≥6

960(TV) – 600(Billboards) – 120(Newspaper)≥0

960(TV) + 480(Radios) + 600(Billboards) + 120(Newspapers)≤14000

Non-negativity constraints: TV≥0, Radios≥0, Newspaper≥0, Billboards≥0

The solution of the above problem can be seen as follows:

- Thus, the maximum number of person Jim can reach within his weekly budget of 14,000 is:

= (6*36000) + (6*26500) +(8*30000) = 615000.

- To achieve the desired result Jim should post 6 ads on TV, 6 ads on Radio and 8 ads on the billboards.

- From the simulation result it can be seen that in the concerned scenario of Max, the best case for least cost for the inventory policy is $49,000 and the worst case of highest cost for the inventory policy is $63,020. Thus, the average cost of the concerned policy which will be incurred by Max, taking into consideration, reorder point 5 and quantity 5 is of amount $56,530.
In case of the reorder point being 3 and the reordering quantity being 3, the minimum cost which has to be incurred by Max is $29,760 while the maximum cost of the same is $40,260. The average cost incurred in this respect is $37,362.**i)**Taking into consideration the reward point 7 and the reordering quantity 7, the lowest cost is $81,840 and that of the highest cost is that of $94,380, the average cost being $90,044.**ii)**- Thus, it can be asserted clearly that in case of the reordering point 3 and reordering quantity 3, the maximum cost for the inventory policy is lesser than even the minimum cost which is incurred for the same in the case of reordering point 7 and reordering quantity 7, which in turn implies that the former reordering point and quantity (3) is better than the latter (7), thereby indicating that the former is a better option for the concerned business than the latter.

By taking area as the independent variable, the predicted selling prices for the house can be seen as follows:

Predicted Price = -34301.5987 + [62.96*(Area)]

For the concerned model, the coefficient of determination = (0.7952^2) = 0.6323. This in turn shows that around 62.23% of the dynamics or variability of the price of the house can be explained by the independent variable (Area) of the same.

Therefore, when Area is equal to 2000 square ft, the selling price of the same is as follows:

## Linear Programming Models for House Selling Price Prediction

Price = -34301.5987 + (62.96*2000) = 91618.4

Now, considering bedrooms as the independent variable, the predicted selling price of the concerned house is as follows:

Predicted Price = 648.6487 + [35168.9189*(Bedroom numbers)]

For this model, the determination coefficient = (0.5047^2) = 0.2547, which in turn implies that nearly 25.47% of the variations in the price of the house can be explained by the bedroom numbers present in the house.

Thus, when the number of bedrooms in the house is 3,

Price = 648.6487 + [35168.9189*3] = 106155

If age is considered to be the independent variable in the concerned problem, then,

Expected Price = 141448.2518 + [-2256.7296*(Age)]

The coefficient of determination being (0.8629^2) = 0.7446, nearly 74.46% of the selling price variability of the house can be explained by the age of the same. Thus, when the age is 24 years,

Price = 141448.2518 + [-2256.7296*24] = 87286.7.

From the above solutions, it can be asserted that the parameter area, can explain the variability of the price of the house the most, which in turn implies that using the area of the house to predict its selling price is the best possible model.

Considering the bedroom numbers as well as the area of the house as the independent variables, the predicted selling price of the same can be seen as follows:

Price = -26129.5 + [76.1268*(Area)] + [-12403.1*(Bedroom numbers)]

For this model, the determination coefficient is (0.8616^2) = 0.7423, which implies that the bedroom numbers of the house and the area of the same can explain 74.23% of all the variabilities in its selling price.

Again, considering area and age as the independent variables the predicted price for the house can be found as follows:

Price = 69793.9387 + [27.0743 * (Area)] + [-1554.9387 *(Age of the house)]

Determination coefficient in this case, is (0.8978^2) = 0.774, which in turn indicates towards the fact that 77.4% of the total variations in the selling price of the house can be explained by age and area of the same.

Predicted price of house, when bedrooms and age of the house are the independent variables is:

Price = 99495.77 + [12389 * (Number of Bedrooms)] + [-1985.53 * (Age)]

Here, the coefficient of determination is (0.9309^2) = 0.8665. Thus, 86.65% of the selling price variability can be explained by bedroom numbers and the age of the house.

On the other hand, by taking area, age and bedrooms as the independent variables, the predicted price of the concerned house can be seen as follows:

Price = 70181.01 + [25.1505 * (House Area)] + [-1574.39 * (Age)] + [1389.257 * (Bedroom Numbers)]

In this case, the determination coefficient is (0.9407^2) = 0.8848, which indicates that 88.48% of the house price variability can be explained by age, area and number of bedrooms in the concerned house.

Thus, from the above discussion and calculations it can be asserted that the model considering all the variables (age, area and number of bedrooms) explain the variability of the house selling prices the most and is thus the best model.

When the Multilayer Perception Model is run with the three concerned independent variables (age of the house, area of the house and number of bedrooms in the house) for predicting its selling price with 1 hidden layer, it shows that the correlation coefficient is of the value 0.9399. This in turn implies that there is 93.96% accuracy in the concerned model, which in turn implies that the MLP Model is much more accurate than that of the regression model in the aspect of prediction of house prices.

However, when the number of hidden layers increase, the model becomes less accurate which in turn implies that MLP Model is a more efficient one with one hidden layer. However, when there are more hidden layers, the regression model can be considered to be better than the MLP as its precision does not change. This can be shown with the help of the following tables:

- In the present dataset there are in total 17 attributes.
- In the dataset there are seven numeric

- In the dataset the number of categorical variables is 10.

- In the dataset 4521 examples are present.
- y is the class variable in this case.
- The class variable can take two values.

- The class variable can take the value “Yes” and “No”, which in turn indicates that there will be or will not be, respectively, a bank account opening.

**Logistic Regression**- Confusion Matrix:

**Conclusion**

As can be seen from the above section, for Logistic Regression, the area under the ROC Curve is 0.888 and for Naïve Bayes it is 0.845, which implies that the former is higher than the latter and thus, the former is a better classifier than the latter.

From the lift charts also, it can be seen that in case of the Regression the lift is higher than that of the Naïve Bayes classifier, which in turn also indicates that the former or the Logistic Regression is a better classifier than the Naïve Bayes classifier.

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