Last Name ______________________ First Name _________________________ ID ___________________________
Operations Management I 73-331 Winter 2002
Odette School of Business
University of Windsor
Midterm Exam II Solution
Wednesday, March 27, 10:00 – 11:20 am
Instructor: Mohammed Fazle Baki
Aids Permitted: Calculator, straightedge, and a one-sided formula sheet.
Time available: 1 hour 20 min
Instructions:
 This exam has 17 pages including this cover page and 8 pages of tables.
 Please be sure to put your name and student ID number on each page.
 Show your work.
Grading:
Question
Marks:
1
/10
2
/10
3
/10
4
/10
5
/10
6
/15
Total:
/65
Name:_________________________________________________
ID:_________________________
Question 1: (10 points)
1.1 The annual holding cost equals _____________________ near EOQ
a. the annual ordering cost
b. the annual stock-out cost
1.2 The total cost curve is flat near
a. EOQ
b. EPQ
c. Both
d. None
1.3 In a rotation cycle policy the products are produced
a. once in each production cycle
b. in the same sequence in each production cycle
c. both
d. none
1.4 In a single-period model, the items unsold at the end of the period is __________ over to the next
period.
a. carried
b. not carried
1.5 When the demand is uncertain, the reorder point, R includes
a. the expected demand during the lead time
b. safety stock
c. both
d. none
1.6 The standardized loss function is used to compute
a. the probability of stocking out during the lead time
b. the proportion of demands that are met from the stock
c. both
d. none
1.7 Storage cost is a part of
a. holding cost
b. ordering cost
c. setup cost
d. stock-out cost
e. none of the above
1.8 In a multi-period inventory model it is assumed that the ending inventory
a. is salvaged
b. is salvaged and transferred to the next period
c. of one period is the beginning inventory of the next period
2
Name:_________________________________________________
ID:_________________________
1.9 The fact that the EOQ cost curve is flat near the optimal order quantity implies that
a. if there are some managerial reasons to order Q units such that Q  EOQ, but Q is near
EOQ, then one may order Q units without causing a large increase in inventory cost
b. inventory cost is not sensitive to the cost of buying items
1.10 If the shortages are back-ordered, then the annual number of units purchased equals the
annual demand
a. True
b. False
3
Name:_________________________________________________
ID:_________________________
Question 2: (10 points)
Montgomery Associates produces switches for scientific equipment and has gathered information
about the production of its 6-13 switches:
Annual production rate
3,000 units
Annual demand rate 1,800 units
Cost per switch
$12
Annual holding cost
Setup cost
$120
25%
a. (4 points) Find the optimal size of each production run.
EPQ=
2 K

h'
2 K
2120 3,000 

 600 units
 
 1,800 
h1  
120.251 

 P
 3,000 
b. (2 points) Find the optimal cycle time.
Cycle time 
Q 600

 0.3333 years
 1,800
c. (2 points) Compute the uptime and downtime in each cycle.
Uptime 
Q
600

 0.20 years
P 3,000
Downtime = Cycle time – up time = 0.3333 – 0.2000 = 0.1333 years
Or, downtime 
Q    600  1,800 
1 
  0.1333 years
1   
  P  1,800  3,000 
d. (2 points) What is the maximum dollar amount invested in the inventory?
 1,800 
 
Maximum inventory  Q1    6001 
  240 units
 P
 3,000 
Cost of maximum inventory = 24012 = $2,880
4
Name:_________________________________________________
ID:_________________________
Question 3: (10 points)
Harold Gwynne is considering starting a sandwich-making business from his dormitory room to earn
some extra income. However, he has only a limited budget of $1600 to make his initial purchase.
Harold divides his needs into three areas: bread, meats and cheeses, and condiments. He estimates
that he will be able to use all of the products he purchases before they spoil, so perishability is not an
issue. The demand and cost parameters are given below:
Breads
Meats and Cheeses
Condiments
Weekly demand
30 packages
25 packages
10 pounds
Cost per unit
$1.5
$5
$3
Fixed order cost
$30
$20
$25
The choice of these fixed costs is based on the fact that these items are purchased at different
locations in town. They include the cost of Harold’s time in making the purchase. Assume that
holding costs are based on an annual interest rate of 30 percent. Find the optimal quantities that
Harold should purchase of each type of product so that he does not exceed his budget.
EOQB=
2K
23030  52

 456.07 units
h
1.5  0.30
EOQM=
2 K
22025  52

 186.19 units
h
5  0.30
EOQC=
2 K
22510  52

 169.97 units
h
3  0.30
Fund required = 1.5  456.07 + 5  186.19 + 3  169.97 = $2,124.96
Fund available = $1,600 < $2,124.96 = Fund required
Hence, order quantities are obtained by reducing the EOQ values proportionately
Compute m 
1,600
 0.753
2,124.96
QB  0.753456.07  343.42 units
QB  0.753186.19  140.20 units
QB  0.753169.97  127.99 units
5
Name:_________________________________________________
ID:_________________________
Question 4: (10 points)
Irwin sells a particular model of fan, with most of the sales being made in the summer months. Irwin
makes a one-time purchase of the fans prior to each summer season at a cost of $50 each and sells
each fan for $100. Any fans unsold at the end of summer season are marked down to $20 and sold
in a special fall sale.
a. (2 points) What is the underage cost per unit?
cu  Selling price – purchase price = 100-50 = $50/unit
b. (2 points) What is the overage cost per unit?
co  Purchase price – salvage value = 50-20 = $30/unit
c. (3 points) If the demand is uniformly distributed between 300 and 900 units, find the optimal order
quantity.
For the optimal order quantity Q , Probability(demand  Q ), p 
cu
50

 0.625 (1 point)
cu  co 50  30
Hence, Q *  a  pb  a   300  0.625900  300  300  375  675 units (2 points)
d. (3 points) If the demand is normally distributed with a mean of 600 and a standard deviation of
120, find the optimal order quantity.
For the optimal order quantity Q , Probability(demand  Q ), p 
cu
50

 0.625
cu  co 50  30
Find the standard normal z -value for which cumulative area on the left, F z   0.625 .
Since Table A-1 gives area between z  0 and positive z -values, find z -value for which Table A1 area is 0.625-0.50 = 0.125. Hence, z  0.32 (1 point)
Q *    z  600  0.32  120  638.4 units (2 points)
6
Name:_________________________________________________
ID:_________________________
Question 5: (10 points) Comptek Computers wants to reduce a large stock of personal computers it
is discontinuing. It has offered the University Bookstore a quantity discount pricing schedule if the
store will purchase the personal computers in volume, as follows:
Quantity
Price
1-9
$1800
10-49
1500
50+
1200
The annual inventory holding cost is 40%, the ordering cost is $200, and annual demand for this
particular model is estimated to be 216 units. Compute the optimal order size.

First, consider the cheapest price level of c3  $1,200 per unit. (1 point)
h3  Ic3  0.40  1,200  $480 /unit/year
EOQ3 
2 K

h3
2200216
 13.42 units (1 point)
0.401,200
Since the price level of c3  $1,200 is not available for an order quantity Q  EOQ3 = 13.42 units,
EOQ3 is infeasible and a candidate for optimal order quantity is Q3  50 , because 50 is the
minimum order quantity for the price level of c3  $1,200. (1 point)

Now, consider the next price level, c2  $1,500 per unit. (1 point)
h2  Ic2  0.40 1,500  $600 /unit/year
EOQ2 
2 K

h2
2200216
 12 (1 point)
600
Since the price level of c2  $920 is available for an order quantity Q  EOQ2 = 12 units, EOQ2 is
feasible and a candidate for optimal order quantity is Q2  12 . (1 point)

It’s not necessary to consider the other price level.

Now, compute total cost for each candidate for optimal order quantity:
j
3
2

(1 point)
Holding cost
Ordering cost
Cost of item
Total cost
Candidate
h jQ j
c j
Qj
2
K
Qj
= Holding cost +
Ordering cost +
Cost of item
48  50
 1,200
2
200  216
 864
50
1,200  216  259,200
$272,064
600  12
 3,600
2
200  216
 3,600
12
1,500  216  324,000
Q3  50
Q2  12
(1 point)
$331,200
(1 point)
Conclusion: The total cost is minimum, $272,064 for Q3  50 . Therefore, an optimal order
quantity is Q3  50 . (1 point)
7
Name:_________________________________________________
ID:_________________________
Question 6: (15 points)
The home appliance department of a large department store is using a lot size-reorder point system
to control the replenishment of a particular model of FM table radio. The store sells an average of
1,200 radios each year. The annual demand follows a normal distribution with a standard deviation
of 100. The store pays $40 for each radio, which it sells for $80. The holding cost is 30 percent per
year. Fixed costs of replenishment amount to $98. If a customer demands the radio when it is out of
stock, the customer will generally go elsewhere. Loss-of-goodwill costs are estimated to be about
$15 per radio. Replenishment lead time is one month. Currently, the store is using Q  150 and
R  140 . Compute
a. (2 points) the mean and standard deviation of the lead time demand

1
1
 0.0833 years,     1200  100 units, (1 point)
12
12
     100
1
 28.87 units (1 point)
12
b. (1 point) the annual holding cost per unit
h  Ic  0.3  40  $12 per unit per year
c. (1 point) the stock-out cost per unit
p = loss of profit + good will = (80-40) +15 =$55 per unit
d. (1 point) the safety stock
R    140  100  40 units
e. (1 point) the expected number of units stock-out per cycle
z
R   140  100

 1.3855

28.87
L z  
0.0383  0.0375
 0.0379 (since Lz   0.0383 for z  1.38 and Lz   0.0375 for z  1.39 )
2
n  Lz   28.870.0379  1.09 units per cycle
(Continued…)
f. (2 points) the annual holding cost
8
Name:_________________________________________________
ID:_________________________
hQ
12  150
 hR    
 12140  100  900  480  $1,380 (1 point for each part)
2
2
g. (2 points) the annual ordering cost
K 98  1,200

 $784
Q
150
h. (2 points) the annual stock-out cost
np 1.09  55  1,200

 $479.6
Q
150
i.
(1 point) the total annual holding, ordering and stock-out cost
1,380  784  479.6  $2,643.6
j.
(1 point) the probability of not stocking out during the lead time
z
R   140  100

 1.3855

28.86
Table A-4: The probability of not stocking out during the lead time = F z  1.3855

0.9162  0.9177
 0.91695 (since F z   0.9162 for z  1.38 and F z   0.9177 for z  1.39 )
2
Table A-1: The probability of not stocking out during the lead time
= the area on the left of z  1.3855
= P   z  1.3855  P   z  0  P0  z  1.3855
= 0.50  P0  z  1.3855  0.5  0.41695  0.91695
k. (1 point) the fill rate, up to four decimal places
  1
n
1.09
 1
 0.9927  99.27%
Q
150
9