```UNIVERSITY OF THE COMMONWEALTH CARIBBEAN
FALL SEMESTER 2021 (Online)
FINAL SEMESTER PRACTICE PAPERQUESTIONS &amp; ANSWER KEY
QUESTION 1
Given the following data on hotel check-ins for a 6-month period:
Month
July
August
September
October
November
December
# of Rooms
70 rooms
105 rooms
90 rooms
120 rooms
130 rooms
115 rooms
A). What is the 3-month moving average forecast for January? (1 mark)
B). With alpha = 0.3, what is the simple exponential smoothing forecast for October? Assume
the forecast for July was 75 rooms. (3 marks)
Demand
Forecast
July 70
75
Aug 105
73.5
Sept 90
82.95
Oct 120
85.065
Nov 110
95.546
Dec 115
105.882
October forecast is 85.065 rooms
C) Using a weighted moving average with weights of 0.6, 0.3, and 0.1 (from most recent), how
many check-ins can be forecasted for January? (1 mark)
D) The last four weekly values of sales were 80, 100, 105, and 90 units, respectively. The last
four forecasts (for the same four weeks) were 60, 80, 95, and 75 units, respectively. Calculate the
MAD, MSE, and MAPE for these four weeks. 91 +2 +2 marks)
Sales
80
100
105
90
Forecast
60
80
95
75
Error
Error squared
Pct. error
Answer: MAD = 65/4 = 16.25; MSE = 1125/4 = 281.25; MAPE = 0.712/4 = .178 or 17.8%
QUESTION 2
A network consists of the following list. Times are given in weeks.
Activity
Preceding
Optimistic
Probable
Pessimistic
A
--
5
11
14
B
--
3
3
9
C
--
6
10
14
D
A, B
3
5
7
E
B
4
6
11
F
C
6
8
13
G
D, E
2
4
6
H
F
3
3
9
(a)
Draw the network diagram. (3 marks)
(b)
Calculate the expected duration and variance of each activity. (3+2 marks)
(c)
Calculate the expected duration and variance of the critical path. (2 marks)
(a)
(b, c)
Expected
time
Variance Std. dev.
A
10.5
2.25
1.5
3
B
4
1
1
8
C
10
1.778
1.333
0
D
5
0.444
0.667
3
E
6.5
1.361
1.167
8
F
8.5
1.361
1.167
0
G
4
0.444
0.667
3
H
4
1
1
0
1
Project
22.5
Project
4.139
Std. dev.
2.034
(d) z = (28 - 22.5)/2.03 = 2.71, (P ≤ 28) = .997
Slack
Variance
1.778
1.361
QUESTION 3
The sales manager of a large apartment rental complex feels the demand for apartments may be
related to the number of newspaper ads placed during the previous month. She has collected the
data shown in the accompanying table.
Apartments leased, (Y)
15
6
9
4
40
16
20
6
25
13
25
9
15
10
35
16
a) Find the least squares regression equation. (5 marks)
b) Estimate sale if the number of advertisements purchased is 30. (1 mark)
c). Find the coefficient of determination (4 marks)
soln:
(X – X )2
(X – X )(Y – Y )
6
15
64
32
4
9
196
84
16
40
289
102
6
20
9
12
13
25
4
6
9
25
4
–2
10
15
64
0
16
35
144
72
Leases, Y
Y = 80
X = 184
(X – X )2 = 774
(X – X )(Y – Y ) = 306
from :
so
b1 
 XY  nXY
 X  n( X )
2
2
and
b 0 = Y-b1 X
Y 
80
184
 10; X 
 23
8
8
b1 = 306/774 = 0.395
b0 = 10 – 0.395(23) = 0.915
The estimated regression equation isŶ = 0.915 + 0.395X
or
Apartments leased = 0.915 + 0.395 ads placed
b).If the number of ads is 30, we can estimate the number of apartments leased with the regression
equation
0.915 + 0.395(30) = 12.76 or 13 apartments
The coefficient of determination.
SST = 150; SSE = 29.02; SSR = 120.76
Y
X
(Y– Ῡ)2
Yˆ = 0.915 +
(Y – Yˆ )2
( Yˆ –Ῡ)2
0.395X
6.00
15.00
16
6.84
0.706
9.986
4.00
9.00
36
4.47
0.221
30.581
16.00
40.00
36
16.715
0.511
45.091
6.00
20.00
16
8.815
7.924
1.404
13.00
25.00
9
10.79
4.884
0.624
9.00
25.00
1
10.79
3.204
0.624
10.00
15.00
0
6.84
9.986
9.986
16.00
35.00
36
14.74
1.588
22.468
80.00
184.00
SST=150.00
80.00
SSE=29.02
SSR=120.76
From this the coefficient of determination is
r2 = SSR/SST = 120.76/150 = 0.81
QUESTION 4
A) Perform an ABC analysis on the following set of products.( 3maeks)
Annual
Item
Demand
Unit Cost
A211
800
\$9
B390
100
\$90
C003
450
\$6
D100
400
\$100
E707
85
\$2,000
F660
250
\$320
G473
500
\$75
H921
100
\$75
Answer: The table below details the contribution of each of the eight products. Item E707 is
clearly an A item, and items B390, H921, A211, and C003 are all C items. Other classifications
are somewhat subjective, but one choice is to label E707 and F660 as A items, and D100 and
G473 as B items.
Annual Unit
Cumulative Cumulative
Item Demand Cost
Volume
volume
percent
E707
85 \$2,000 \$170,000
\$170,000
48.04%
F660
250
\$320 \$80,000
\$250,000
70.64%
D100
400
\$100 \$40,000
\$290,000
81.94%
G473
500
\$75 \$37,500
\$327,500
92.54%
B390
100
\$90
\$9,000
\$336,500
95.08%
H921
100
\$75
\$7,500
\$344,000
97.20%
A211
800
\$9
\$7,200
\$351,200
99.24%
C003
450
\$6
\$2,700
\$353,900
100.00%
\$353,900
B) A local artisan uses supplies purchased from an overseas supplier. The owner believes the
assumptions of the EOQ model are met reasonably well. Minimization of inventory costs is her
objective. Relevant data, from the files of the craft firm, are annual demand (D) =150 units,
ordering cost (S) = \$42 per order, and holding cost (H) = \$4 per unit per year
(a)
How many should she order at one time? (2 marks)
(b)
How many times per year will she replenish her inventory of this material?(1 mark)
(c)
What will be the total annual inventory (holding and setup) costs associated with this
material (rounded to the nearest dollar)? (2 marks)
(d)
If she discovered that the carrying cost had been overstated, and was in reality only \$1
per unit per year, what is the corrected value of EOQ?( 2 marks)
(a)
Q* =
(b)
(c)
N=
= 2.67 She should place about 2.67 orders per year.
Setup costs = 2.67(\$42) = \$112. Holding costs = (56.12/2)(\$4) = \$112.
Total costs = \$112 + \$112 = \$224.
At the lower value for H, the EOQ will be doubled to 112.25.
(d)
= 56.12. She should order 56 units at a time.
QUESTION 5
A) A crew of mechanics at the Highway Department Garage repair vehicles that break down at
an average of λ = 5 vehicles per day (approximately Poisson in nature). The mechanic crew can
service an average of μ = 8 vehicles per day with a repair time distribution that approximates a
negative exponential distribution.
(a)
What is the probability that the system is empty? (1 mark)
(b)
What is the probability that there is precisely one vehicle in the system?(1 mark)
(c)
What is the probability that there are more than two vehicles in the system?(2 marks)
(d)
What is the probability of 5 or more vehicles in the system?(2 marks)
Answer: (a) P0 = 1 - 5/8 = 0.375
(b) Pn&gt; 1 = (5/8)2 = 0.391; the probability of exactly one is 1 - .391 - .375 = .234
(c) Pn&gt; 2 = (5/8)2+1 = 0.244
(d) the probability of five or more is Pn &gt; 4 = (5/8)5 = 0.0954.
B) What combination of a and b will yield the optimum for this problem?(2 marks)
Maximize \$6a + \$15b, subject to (1) 4a + 2b ≤ 16 and (2) 5a + 2b ≤ 20 and (3) a, b ≥ 0.
Answer a = 0, b = 8
C) Suppose that the feasible region of a maximization LP problem has corners of (0,0), (10,0),
(5,5), and (0,7). If profit is given to be \$X + \$3Y what is the maximum profit the company can
earn?(1 marks)