Test 1 with solution

Kulliyyah of Economics and Management Sciences Test 1 MGT 3050 Maximum Marks : 30

Date : 30/05/09 Time : 2pm – 3.30pm

Answer all questions.

Question 1 A machine shop owner is attempting to decide whether to purchase a new drill press, a lathe or a grinder. The return from each will be determined by whether the company succeeds in getting a government military contract. The profit or loss from each purchase and the probabilities associated with each contract outcome are shown in the following payoff table:

Purchase

Contract 0.40

No Contract 0.60

Drill press Lathe Grinder

$ 40,000 20,000 12,000

$ − 8,000 4,000 10,000

(a) Which machine should be purchased? The machine shop owner is considering hiring a military consultant to ascertain whether the shop will get the government contract. The consultant is a former military officer who uses various personal contacts to find out such information. By talking to other shop owners who have hired the consultant, the owner has estimated a 0.70 probability that the consultant would present a favorable report, given that the contract is awarded to the shop (P(f | c)), and a 0.80 probability that the consultant would present an unfavorable report, given that the contract is not awarded (P(u | n)). Using decision tree analysis, (b) Determine (i) the decision strategy the owner should follow, (ii) the expected value of this strategy, and (iii) the maximum fee the owner should pay the consultant. [2+17+1+2=22 marks]

1

2. The Northwoods Outdoor Company specializes in outdoor recreational clothing. Demand for its items is very seasonal peaking during October-December and AprilJune. It has accumulated the following data for orders per quarter during the last two years. Orders (1000s) 2006

2007

January – March

18

19

April – June

23

25

July – September

20

22

October – December

41

45

(a) Compute the indices for the four quarters. (b) Develop the trend line and forecast for the four quarters in 2008. [4+9 = 13 marks] Note. No need to adjust the seasonal irregular components if their sum is larger or equal to 3.99.

  b0 = Y − b1 X , 

b1 =

∑ x y − (∑ x ∑ y ) ∑ x − (∑ x ) n i

i

i

i

2

2

i

2

i

n  

Question 1 (a)

(b) P(c) = probability of contract = 0.40; P(n) = probability of no contract = 0.60; P(f | c) = 0.70; P(u | c) = 0.30 P(u | n) = 0.80; P(f | n) = 0.20 Computation of posterior probabilities: If f - Favorable State of Nature

P(sj)

P(f | sj)

P(f∩sj)

P(sj | f)

c

P(c) = 0.40

P(f | c) = 0.70

P(fc) = 0.28

P(c | f) = 0.70

n

P(n) = 0.60

P(f | n) = 0.20 P(fn) = 0.12 P(f) = 0.40

P(n | f) = 0.30

If u - Unfavorable State of Nature

(i)

P(sj)

P(u | sj)

P(u∩sj)

P(sj | u)

c

P(c) = 0.40

P(u | c) = 0.30 P(uc) = 0.12

P(c | u) = 0.20

n

P(n) = 0.60

P(u | n) = 0.80 P(un) = 0.48 P(u) = 0.60

P(n | u) = 0.80

Decision strategy: If report is favorable, purchase a Drill press; If report is unfavorable, purchase a Grinder. 3

(ii)

EV (strategy) = $ 16,480.

(iii)

EVSI = EVwSI – EvwoSI = $16,480 – $11,200 = $ 5,280

4

2. Computation of seasonal indices: Year

Quarter

Sales

4-Qtr. Moving

Centered

(1000s)

Average

Moving Avg.

Seasonal Irregular Component

1

18

2

23 102/4 = 25.5

1

3

20

25.625

0.7805

26

1.5769

26.5

0.7170

27.25

0.9194

103/4 = 25.75 4

41 105/4 = 26.25

1

19 107/4 = 26.75

2

25

2

111/4 = 27.75 3

22

4

45

Sum of the indices is 3.9938 ≈ 4.00. Seasonal irregular components are considered as indices of the corresponding quarters.

5

Deseasonalization of the time series data

Year

Quarter

Yt

St

Yt/St

1

18 23

0.7170

25.10

0.9194

25.01

20 41

0.7805

25.62

1.5769

26

19 25

0.7170

26.50

0.9194

27.19

22 45

0.7805

28.19

1.5769

28.54

2

1

3 4 1 2

2

3 4

Trend projection calculation:

Total

t

Yt

t Yt

t2

1

25.10

25.10

1

2

25.01

50.02

4

3

25.62

76.86

9

4

26

104

16

5

26.50

132.5

25

6

27.19

163.14

36

7

28.19

197.33

49

8

28.54

228.32

64

212.15

977.27

204

36

6

We have

36 212 .15 = 4.5, Y = = 26 .52 8 8 977 .27 − (36 × 212 .15 ) / 8 b1 = = 0.5381 . 204 − (36 × 36 ) / 8 b0 = 26 .52 − 0.5381 × 4.5 = 24 .10

t=

Tt = 24 .10 + 0.5381 t.

Forecasting without seasonal effect:

T9 = 24.10 + 0.5381 × 9 = 28.94 T10 = 24.10 + 0.5381 × 10 = 29.48 T11 = 24.10 + 0.5381 × 11 = 30.02 T12 = 24.10 + 0.5381 × 12 = 30.56

Forecasting with seasonal adjustment:

t

Trend Forecast

Seasonal Index

Quarterly forecast

13

28.94

0.7170

20.75

14

29.48

0.9194

27.10

15

30.02

0.7805

23.43

16

30.56

1.5769

48.19

7

150 61

8