# An Assignment Comprising Of Five Questions

• December 28, 2023/

## Question 1: Decision Analysis

1. a. The decision making process has mainly the following five steps highlighted below (MTaylor and Tyalor, 2004):

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Step 1: Define the expectation from the decision and with highlighting the potential outcomes of the decision that needs to be made.

Step 2: Listing out all the potential alternatives based on brainstorming

Step 3: For each of the alternatives possible, a risk and outcome analysis needs to be performed which highlights the potential outcomes and the underlying risk associated.

Step 4: For each of the alternatives, quantify the risk and outcomes in terms of cost and benefit.

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Step 5: Select the relevant decision model and then choose the relevant alternative which can enable making the appropriate decision.

b. The alternatives refer to the various strategies that the decision maker can potentially deploy and hence are within the control of the decision maker. States of nature are potential future events which may occur and are outside the control of the decision maker. The intrinsic relation between alternatives and state of nature is that for different state of nature, a different alternative or strategy would work. As a result, the decision maker needs to make a choice with regards to the optimum alternative keeping in mind the potential probabilities associated with states of nature (Lieberman, et. al., 2013).

c. The information and data are shown below:

 Sales (kg) Number of items Cost of urchase  (\$ per kg) 15 Marginal profit (\$ per kg) 15 10 10 Sale price (\$ per kg) 30 Marginal loss (\$ per kg) 5 15 20 Salvage price (\$ per kg) 10 20 40 25 20 30 10

1. Conditional profits matrix for the five alternatives and for five sales levels is highlighted below (Hillier, 2006).

 Conditional profits matrix Probability 0.1 0.2 0.4 0.2 0.1 Sales 10 15 20 25 30 Alternatives (Purchase Qty. )  kg 10 150 150 150 150 150 15 125 225 225 225 225 20 100 200 300 300 300 25 75 175 275 375 375 30 50 150 250 350 450

2. Fish vendor: Optimist

Maximum –maximum rule

 Optimistic rule (Maximax) Maximum Sales 10 15 20 25 30 Payoff Alternatives (Purchase Qty. )  kg 10 150 150 150 150 150 150 15 125 225 225 225 225 225 20 100 200 300 300 300 300 25 75 175 275 375 375 375 30 50 150 250 350 450 450

Maximum payoff =450 and thus, 30kg should be bought by fish vendor.

3.Fish vendor: Pessimistic

Maximum –minimum rule

 Pessimistic rule(Maximin) Minimum Sales 10 15 20 25 30 Payoff Alternatives (Purchase Qty. )  kg 10 150 150 150 150 150 150 15 125 225 225 225 225 125 20 100 200 300 300 300 100 25 75 175 275 375 375 75 30 50 150 250 350 450 50

Maximum payoff =150 and thus, 10kg should be bought by fish vendor.

4. Fish vendor is using Laplace criterion.

Average rule

 Laplace rule (Average) Average Sales 10 15 20 25 30 Payoff Alternatives (Purchase Qty. )  kg 10 150 150 150 150 150 150 15 125 225 225 225 225 205 20 100 200 300 300 300 240 25 75 175 275 375 375 255 30 50 150 250 350 450 250

Maximum payoff =255 and thus, 25kg should be bought by fish vendor.

5. Fish vendor is using criterion of regret.

Regret rule

 Regret matrix Maximum Sales 10 15 20 25 30 Regret Alternatives (Purchase Qty. )  kg 10 0 75 150 225 300 300 15 25 0 75 150 225 225 20 50 25 0 75 150 150 25 75 50 25 0 75 75 30 100 75 50 25 0 100

Minimum regret =75 and thus, 25kg should be bought by fish vendor.

6. Fish vendor is using criterion of regret.

Expected monetary value rule

 Maximizing expected value Prob 0.1 0.2 0.4 0.2 0.1 Expected Sales 10 15 20 25 30 Value Alternatives (Purchase Qty. )  kg 10 150 150 150 150 150 150 15 125 225 225 225 225 215 20 100 200 300 300 300 260 25 75 175 275 375 375 265 30 50 150 250 350 450 250

Maximum value =75 and thus, 25kg should be bought by fish vendor.

7. The optimal sea food quantity that fish vendor needs to buy each week so to maximize the profit (Hillier, 2006).

Cost of underage = \$30-\$15 = \$15

Cost of overage =\$15-\$10 =\$5

Hence,

Critical factor = 15 / (15 +5) = 0.75

Z value for the critical factor = NORMSINV (0.75) = 0.675

Therefore, optimal seafood quantity = (20) + (5 *0.675) = 23.37 kg

2. Prior probability of success = 0.3

a.Probability of success p = 0.3

Probability of failure q = 1- p = 0.7

Expected value of profit = (0.3 * 1,000,000) – (0.7*600,000) = -120,000

b. Expected value of perfect information regarding success and failure of product (Hastie, Tibshirani and Friedman, 2006).

Expected value of perfect of success =0.3 * 1,000,000 = 300,000

Expected value of perfect of failure = 0

c. Prior probabilities

P (favourable | success) = 0.7

P (unfavourable | success) 1-0.7 = 0.3

## Question 2: Value of Information

Similarly,

Similarly,

P (unfavourable | failure) = 0.8

P (favourable | failure) =1-0.8 = 0.2

Hence,

P (favourable) = (0.3*0.7) + (0.7*0.2) = .35

P (unfavourable) = (0.3*0.3) + (0.7*0.8) = .65

d.Posterior probability of success for a given favourable

P (success | favourable) = (0.7 *0.3)/0.35 = 0.60

P (failure | unfavourable) = (0.8*0.7)/0.65 = 0.8615

e. Maximum the firm should pay for market survey

Company would take the project when the survey predictions are favourable (Harmon, 2011).

• Expected value of profit with information

= (0.35*0.60*1,000,000) = 210,000

• Expected value of profit with information

EVPI = 210,000 -0 =210,000

3. a. Simulation model for 1 month operation.

b. Normal and formula views of simulation model (Lieberman, et. al., 2013)

c. The minimum average cost as per overbooked rooms are shown below:

 Sensitivity Analysis Overbooked Rooms Average Cost 0 113.43 1 94.96 2 37.4 3 17 4 0 5 0

d. To: The Hotel Manager

From: STUDENT NAME

Date : May 8, 2018

Dear Sir

I have carried out the simulation as advised in order to ascertain the respective costs associated with the overbooked rooms. The results in this regards are highlighted in part C. From that   analysis, it clearly emerges that there needs to be a change in the hotel’s current overbooking policy as the average costs associated with overbooking by 3 rooms is about \$ 17. However,   the optimum choice in this regards is to overbook by 4 rooms as it would reduce the cost further based on the given trend of no-shows.

Yours Sincerely

STUDENT NAME

4. a. The requisite regression model with price as the dependent variable and mileage as the independent variable is given below.

 SUMMARY OUTPUT Regression Statistics Multiple R 0.8498 R Square 0.7221 Adjusted R Square 0.6873 Standard Error 1532.3929 Observations 10 ANOVA df SS MS F Significance F Regression 1 48810175.9145 48810175.9145 20.7860 0.0019 Residual 8 18785824.0855 2348228.0107 Total 9 67596000.0000 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 17227.294 1188.417 14.496 0.000 14486.799 19967.788 14486.799 19967.788 Mileage -0.096 0.021 -4.559 0.002 -0.144 -0.047 -0.144 -0.047

Comment

In the above model, the R2 value is 0.7221 which implies that Mileage is able to explain 72.21% variation in the price of the car. Further, the slope coefficient of mileage is also statistically significant since the p value is 0.002 and hence significance at 1% can also be established (Hastie, Tibshirani and  Friedman, 2006).

The requisite regression model with price as the dependent variable and age of the car as the independent variable is highlighted below (Harmon, 2011).

 SUMMARY OUTPUT Regression Statistics Multiple R 0.8551 R Square 0.7311 Adjusted R Square 0.6975 Standard Error 1507.2223 Observations 10 ANOVA df SS MS F Significance F Regression 1 49422248.2168 49422248.2168 21.7554 0.0016 Residual 8 18173751.7832 2271718.9729 Total 9 67596000.0000 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 16226.391 971.102 16.709 0.000 13987.026 18465.756 13987.026 18465.756 Age (years) -839.658 180.019 -4.664 0.002 -1254.782 -424.533 -1254.782 -424.533

Comment

In the above model, the R2 value is 0.7311 which implies that age is able to explain 73.11% variation in the price of the car. Further, the slope coefficient of mileage is also statistically significant since the p value is 0.0016 and hence significance at 1% can also be established.

The requisite regression model with price as the dependent variable and age & mileage of the car as the independent variables is highlighted below.

 SUMMARY OUTPUT Regression Statistics Multiple R 0.8595 R Square 0.7388 Adjusted R Square 0.6642 Standard Error 1588.2080 Observations 10 ANOVA df SS MS F Significance F Regression 2 49939168.3072 24969584.1536 9.8991 0.0091 Residual 7 17656831.6928 2522404.5275 Total 9 67596000.0000 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 16699.519 1462.676 11.417 0.000 13240.839 20158.199 13240.839 20158.199 Mileage -0.039 0.087 -0.453 0.664 -0.245 0.166 -0.245 0.166 Age (years) -507.303 758.280 -0.669 0.525 -2300.351 1285.744 -2300.351 1285.744

Comment

In the above model, the R2 value is 0.7388 which implies that age & mileage are jointly able to explain 73.88% variation in the price of the car. Further, neither the slope coefficient of mileage nor age is statistically significant since the p value for both the slopes is greater than 0.50 (Hastie, Tibshirani and  Friedman, 2006).

Best Model

The best model to use would be simple regression model with age as the independent variable. This is because for this model the R2 is higher than the corresponding R2 for the simple regression model. The increase in R2 in the multiple regression model is on account of increase in predictor variables but since both of these end of being insignificant, the adjusted R2 value is the lowest for the multiple regression model (Lieberman, et. al., 2013).

a. Owing to the higher R2 value produced by the model with age as the independent variable, this simple regression model would be preferred. Further, the negative coefficient for age and mileage are acceptable considering the fact that both with age and mileage, there is wear and tear in the car leading to depreciation in the values. Hence, the inverse relationship between age &mileage with the price of the car(Lind, Marchal and Waten, 2012).

b. No, Barry should not use the multiple regression models. This is because for this model, both the slopes are not significant as reflected from the respective p values. One of the major reasons for this is the high amount of correlation between age and mileage which violates a key assumption of multiple regression and leads to multi-collinearity. Also, the adjusted R2 value for this model is lower than the two simple regression models. Hence, this model has poor predictive power and hence the usage of this model is not recommended (Hastie, Tibshirani and Friedman, 2006).

5. a. Let the number of unit is x.

At break even, profit =0 (Hair, et. al., 2015)

Model

Therefore, the number of units at breakeven point 300

b. Model

Let the number of unit is x.

At profit = \$1600

Therefore, the number of units for profit of \$1600 is 300.

c. One more product B is also produced by the company.

Total profit = \$20,000

Ratio A to B = 2: 1

Let the number of units produced for B is x and hence, A is 2x.

Therefore, the number of units produced of B is 1000 and of A is 2000.