Question 1: Decision Analysis

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

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.

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.