Math

MGMT 2012- Introduction to Quantitative Methods-

Graded Assignment Two S1- 2015-16- Due November 21st – 10 percent Final Grade

Students should note that this assignment should be submitted in a Word document.

Question 1A

A CEO of a multihospital system is planning to expand operations into various states. It will take several years to get certificate of need (CON) approvals so that the new facilities can be constructed. The eventual cost (in millions of dollars) of building a facility will differ among states, depending upon finances, labor, and the economic and political climate. An outside consulting firm estimated the costs for the new facilities as based on declining, similar, or improving economies, and the associated probabilities as shown in the Table below. Decision alternatives

Declining

Same Improving

Kentucky 22.00 19.00 15.00

Maryland 19.00 18.50 18.00

North Carolina 19.50 17.00 15.50

Tennesee 23.00 17.00 14.00

Virginia 25.00 21.00 13.00

(a) Which alternative should the firm choose under the maximax criterion? (1 mark)

(b) Which option should the firm choose under the maximin criterion? (1 mark)

(c) Which option should the firm choose under the LaPlace criterion? (2 marks)

(d) Which option should the firm choose with the Hurwicz criterion with α = 0.4? (2 marks)

(e) Using a minimax regret approach, what alternative should the firm choose? (4 marks)

(f) Economists have assigned probabilities of 0.25, 0.40, and 0.35 to the possible economic climate of declining, same and improving respectively. Using expected monetary values, what option should be

chosen and what is that optimal expected value? (4 marks)

(g) What is the most that the firm should be willing to pay for additional information? Use Expected Regret

(3 marks)

(h) Use the alternative method to verify EVPI (3 marks)

Question 1B

Warren Buffy is an enormously wealthy investor who has built his fortune through his legendary busimen acumen. He currently has been offered three major investments and would like to choose one. The first one is a conservative investment that would perform well in an improving economy and only suffer a small loss in a worsening economy. The second is a speculative investment that would perform extremely well in an improving economy but would do very badly in a worsening economy. The third is a countercyclical investment that would lose some money in an improving economy but would perform well in a worsening economy. The respective payoffs (millions) and the associated probabilities as shown in the Table below.

Decision alternatives

Improving

Stable Worsening

Conservative 30.00 5.00 -8.00

Speculative 40.00 10.50 -30.00

Countercyclical -5.0 .00 15.0

probabilities .10 .50 .40

(i) Which alternative should Warren choose under the maximax criterion? (1 mark)

(j) Which option should Warren choose under the maximin criterion? (1 mark)

(k) Which option should Warren choose under the LaPlace criterion? (2 marks)

(l) Which option should Warren choose with the Hurwicz criterion with α = 0.4? (2 marks)

(m) Using a minimax regret approach, what alternative should the firm choose? (4 marks)

(n) Economists have assigned probabilities of 0.10, 0.50, and 0.40 to the possible economic climates of improving, stable and worsening respectively. Using expected monetary values, what option should be

chosen and what is that optimal expected value? (4 marks)

(o) What is the most that Warren should be willing to pay for additional information? Use Expected Regret (3

marks)

(p) Use the alternative method to verify EVPI (3 marks)

Question 1C

Technology Inc needs to decide how many of the new Ipads to purchase for it new store. The company has assumed that demand will be 30, 55, 90, or 110 Ipads next month and needs to decide whether to order 30,55,90 or 110 Ipads for this period. Each Ipad costs Technology inc $2500 and can be sold for $4000, Technology Inc can sell any unsold Ipads back to their supplier for $500.00.

Number of Ipads demanded

Amount ordered 30 55 90 110

30 45,000

55 -5000

90 135,000

110 -27,500

(a) Complete the Payoff table( 3 marks) (b) Using an optimistic approach (maximax), which option would you choose? 1 mark (c) Using a pessimistic approach (maximin), which option would you choose?1 mark (d) If you are a LaPlace decision maker, which option would you choose? 2 marks (e) If you are a Hurwicz decision maker, which option would you choose with α = 0.60?1 mark (f) Using a minimax regret approach, which option would you choose? 4 marks (g) Using the probabilities of 0.20, 0.35, .25, and 0.20 for possible level of demand respectively, . Which

decision alternative will maximise the expected profit? 4 marks (h) What is the most that should be paid for additional information? Use Expected Regret (2 marks)

(i) Use the alternative method to verify EVPI (2 marks

Question 2

CWD Winter Inc is considering entering the snowplowing business for the coming winter season. CWD can purchase either a snowplow blade attachment for the station's pick-up truck or a new heavy-duty snowplow truck. CWD has analyzed the situation and believes that either alternative would be a profitable investment if the snowfall is heavy. Smaller profits would result if the snowfall is moderate, and losses would result if the snowfall is light. The

following profits have been determined. CWD is willing to pay $50 for additional information on the weather which can be normal or unreasonable.

Decision Alternatives Heavy, s1

Moderate, s2 Light, s3

Blade Attachment, d1 3500 1000 -1500

New Snowplow, d2 7000 2000 -9000

Probability 0.4 0.3 0.3

Conditional Probability For A Given State of Nature

Test Results Heavy Moderate Light

Normal 0.7 0.8 0.9

Unreasonable 0.3 0.2 0.1

That is P (N/H) =0.7; P (N/M) =0.8, P(N/L) =0.9, P (U/H) =0.3; P (U/M) =0.2, P(U/L) =0.1

After you have computed the revised probabilities round to two decimal places

a) Construct the appropriate decision tree to help CWD make the appropriate decisions. This tree must be constructed in logical order with labels and net payoffs ( 5 marks ) It also includes the revised probabilities (5 marks) (10 marks)

b) Fold back the decision tree (5 marks) to determine the best strategy for CWD; you must state this strategy (1 mark). What is the final expected profit?( 1 mark)(7 marks)

c) What is the expected value of sample information(EVSI)- the most that should be paid to seismic testing firm for the test?( 1 mark)

d) Calculate the expected value of perfect information (EVPI)- the most that should be paid to an expert for perfect prediction of the uncertain outcomes.(1 mark)

e) What is the efficiency of sample information?(1 mark)

Question 3

Quarterly Cash donations (in thousands of dollars) at the SDA Mission for 2004-2006 were as shown in the following table.

Period Value 1 242 2 282

3 254

4 345 5 253 6 290 7 262 8 352 9 270 10 286 11 271 12 378

a) Using a weighted moving average with three periods, determine the donations for period 13. Use 3, 2, and 1 for the weights of the most recent, second most recent, and third most recent periods, respectively b) Use exponential smoothing with a smoothing constant of 0.20 to forecast the donations (Assume that last period’s forecast for period 1 is equal to actual to begin the procedure(Ft1=Yt1). Which method do you think is best? Is this an improvement over the weighted average- use MAD only ( Six marks) c) Use regression analysis to estimate the line of best fit -use the manual method with the formulas shown below. (Eight marks)

Question four

A new project is to be completed. The following activities need to be completed in the order shown, where times are in weeks.

A new project is to be completed. The following activities need to be completed in the order shown, where times are in weeks. Activity Immediate

predecessors Optimistic time

Most likely time

Most pessimistic time

Variance

A - 2 3 4 0.11111

B - 2 4 6 0.44444

C A,B 1 2 3 0.11111

D B 1 3 5 0.44444

E A 2 3 4 0.11111

F C 1 4 7 1

G E,F 2 2 2 0

H D,F 2 5 8 1

I G,H 1 3 5 0.44444

J I 2 3 4 0.11111

You must use two decimal places at all times for expected times when they are not integer (you are not permitted to round off to integers). (a) Draw the activity network for this problem (6 marks) (b) Determine the Expected Times. (1 mark) (c) Determine the activity schedule (ES, EF, LS, and LF) as well as slack. (7 marks) (d) Determine and state the critical path for this project. (1 mark) (e) What are the expected time and the variance of the project? (2 marks) (f) What is the probability that the project is completed in 22 or fewer weeks(1 mark) (g) What date should be set such that there is a 94% chance of completion within that time? (2

marks)