# Kelson Sporting Equipment_ Inc

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```					                           ADDITIONAL PROBLEMS
Attempt as many as you can
There may be problems that were not covered in your class this semester – ignore those

INVENTORY MANAGEMENT:

1. The following table contains figures on the annual volume and unit costs for a random
sample of 16 items for a list of 2000 inventory items at a health care facility.

Item        Unit Cost Usage
1           75     3
2           30   274
3           20   397
4           13   139
5           24    41
6           20   123
7           24   379
8           19   372
9           22   114
10           55  3270
11          105   594
12           55   482
13           35   198
14           40    37
15           12   215
16           20   116

Develop an ABC classification for these items.

2. A large bakery buys flour in 25-pound bags. The bakery uses an average of 4860 bags a
year. Preparing an order and receiving a shipment of flour involves a cost of \$4/order.
Annual carrying costs are \$30 per bag.
a. Determine the economic order quantity.
b. What is the average number of bags on hand?
c. How many orders per year will there be?
d. Compute the total cost of ordering and carrying flour.
e. If ordering costs were to increase by \$1 per order, how much would that affect
the total annual cost?
Solution: 36 bags, 18 bags, 135, \$1080, increase by \$127.48

3. A hardware store sells approximately 27,000 cans of a white paint a month. Because of
storage limitations, a lot size of 4000 cans has been used. Monthly holding cost is 18
cents per can, and reordering cost is \$60 per order. The company operates an average
of 20 days a month.
a. What penalty is the company incurring by its present order size on annual costs?
b. The manager would prefer ordering 10 times each month but would have to
justify any change in order size. One possibility is to simplify order processing to
reduce the ordering cost (say using web orders). What ordering cost would
enable the manager to justify ordering every other day?
Solution: \$16, \$52.06

4. (If quantity discounts have been covered in class) A mail order house uses 18000
boxes a year. Carrying costs are 20 cents per box a year, and the ordering costs are
\$32. The following price schedule applies.

Number of Boxes        Price per box
1000 to 1999           \$1.25
2000 to 4999           \$1.20
5000 to 9999           \$1.18
10000 or more          \$1.15

Determine the optimal order quantity.
Solution: 10,000 boxes

5. Suppose the expected demand during lead time for a particular item is 300 units, with a
standard deviation of daily demand of 30 . Suppose the lead time is 4 days.
a. What is the reorder point that provides a 1% risk of stock out during lead time?
b. The safety stock needed to provide a 1% risk of stock out during lead time?
c. The safety stock needed to provide a 7% risk of stock out during lead time?
Solution: 439.8, 139.8, 88.55

6. In the above problem, if a reorder point of 275 is used, what is the probability of stock
out?
Solution: 66.15%

SIMULATION:

1. A car rental agency has collected data on the demand for luxury-class automobiles over
the past 25 days. The data are shown below.

Daily Demand                 Number of days
7                           2
8                           5
9                           8
10                           7
11                           3
Total            25

Because customers drop cars at another location, the agency only has 9 cars available
currently. Assume single day rentals.

a. Use the following five random numbers to generate 5 days of demand for
the rental agency: 15 48 71 56 90.
b. What is the average number of cars rented for the 5 days?
c. How many rentals will be lost over the 5 days?
d. What is the average daily demand for the 5 days?
Solution: b. 8.8 cars, c. 1 lost on day 3, 2 lost on day 5, d. 9.4 cars

2. A service technician for a major photocopier company is trained to service two models of
copier: the X100 and the X200. Approximately 60% of the technician's service calls are
for the X100, and 40% are for the X200. The service time distributions for the two
models are as follows:

X100 Copier                           X200 Copier
Time (mins)    Relative Freq          Time (mins)    Relative Freq
25              0.50                  20              0.40
30              0.25                  25              0.40
35              0.15                  30              0.10
40              0.10                  35              0.10

a. Show the random number intervals that can be used to simulate the type of
machine to be serviced and the length of the service time for each model.
b. Simulate 20 service calls. What is the total service time spent on the 20 calls?
See Excel solution on website

3. Three discount pharmacies (Super Z, Devco, and Floorgreen) compete for business in a
suburban area. Customers often make a purchase at one of the stores and then make
their, next purchase at another store. The matrix below shows the probability that a
customer will switch stores from one purchase to the next.

Current                       Next Purchase                       Total
Purchase           Super Z         DevCo     Floorgreen
Super Z            0.70            0.10        0.20                1.00
Devco             0.30            0.55        0.15                1.00
Floorgreen          0.10            0.10        0.80                1.00

a. Show the probability distribution and the intervals of random numbers that could
be used to generate the next purchase for a customer who last made a purchase
at Super Z.
b. Repeat part (a) for a customer who last made a purchase at Devco.
c. Repeat part (a) for a customer who last made a purchase at Floorgreen.
d. Gary Hatcher made his last purchase at Super Z. Use the following four random
numbers to simulate the store at which he makes his next four purchases: 42, 81,
16, 57.
Solution: See Excel solution on website

4. The New York City corner newsstand orders 250 copies of The New York Times daily.
Primarily due to weather conditions, the demand for newspapers varies from day to day.
The probability distribution of the demand for newspapers is as follows:

# of newspapers         150          175         200        225        250
Probability          0.10         0.30        0.30       0.20       0.10
The newsstand makes a 15-cent profit on every paper sold, but it loses 10 cents on
every paper unsold by the end of the day. Use 10 days of simulated results to determine
whether the newsstand should order 200, 225, or 250 papers per day. What is the
average daily profit the newsstand can anticipate based on your recommendation?

Revenue Management in Supply Chains: (if covered in class)

1. Felgas, a manufacturer of felt gaskets, has production capacity of 1000 units per day.
Currently, the firm sells production capacity for \$5 per unit. At this price, all the
production capacity is booked about one week in advance. A group of customers have
said that they would be willing to pay \$15 per unit if only Felgas accepted their orders on
the last day. The demand from this high paying segment is normally distributed with a
mean of 250 and a standard deviation of 100. How much capacity should Felgas reserve
for the last day?

Solution: 293 units of capacity
See Excel solution on website

Bonus: Suppose the high paying demand distribution was uniformly distributed between 175 and
325, how much capacity should be reserved?
Solution: 275 units of capacity (Hint: use the same methodology as used for Normal, except that
now you cannot use a Normal table. For uniform distribution you really don’t need a table).

2. A manufacturer sources several components from China and has monthly transportation
needs that are normally distributed with a mean of 10 million units and a standard
deviation of 4 million. The manufacturer must decide on the portfolio of transportation
contracts to carry. A long term bulk contract costs \$10,000 per month for a million units.
Transportation capacity is also available in the spot market at an average price of
\$12,500 per million units. How much transportation capacity should the manufacturer
sign a long-term bulk contract for?

Solution: 6.63 million units
See Excel solution on website

WAITING LINES

(Note: ta=1/arrival rate and ts=1/service rate. If no ca or cs is given, assume them to be 1
(Poisson))

1. The reference desk of a university library receives requests for assistance. Assume that
a Poisson probability distribution with a mean rate of 10 requests per hour can be used
to describe the arrival pattern and that service times follow the exponential probability
distribution with a mean service rate of 12 requests per hour.

a. What is the average number of requests that will be waiting for service?
b. What is the average waiting time in minutes before service be-ins?
c. What is the average time at the reference desk in minutes (waiting time plus
service time)?
Solution: 4.17, 0.42hrs, 0.5hrs
2. Agan Interior Design provides home and office decorating assistance to its customers. In
normal operation, an average of 2.5 customers arrive each hour. One design consultant
is available to answer customer questions and make product re-commendations. The
consultant averages 10 minutes with each customer.

a. Compute the length of the queue and the waiting times, assuming Poisson
arrivals and exponential service times.
b. Service goals dictate that an arriving customer should not wait for service more
than an average of 5 minutes. Is this goal being met? What action do you
recommend?
c. If the consultant can reduce the average time spent per customer to 8 minutes,
what is the mean service rate? Will the service goal be met?
Solution: a. 0.297, 7.14mins
b. No. Increase service rate or hire a second consultant
c. 7.5 customers/hr, Lq=0.1667, Wq=4 mins. Yes.

3. Fore and Aft Marina is a newly planned marina that will be located on the Ohio River
near Madison, Indiana. Assume that Fore and Aft has decided to build a docking facility
where one boat at a time can stop for gas and servicing. Assume that arrivals follow a
Poisson probability distribution, with a mean of 5 boats per hour, and that service times
follow exponential probability distribution, with a mean of 10 boats per hour. Consider
the following questions:

a. What is the average number of boats that will be waiting for service?
b. What is the average time a boat will spend waiting for service?
c. What is the average time a boat will spend at the dock?
d. If you were the management of Fore and Aft Marina, would you be satisfied
with service level your system will be providing?
Solution: 0.5 boats, 6 mins, 12 mins

4. The management of the Fore and Aft Marina in the above problem wants to investigate
the possibility of enlarging the docking facility so that two boats can stop for gas and
servicing simultaneously. Assume that the mean arrival rate is 5 boats per hour and that
the mean service rate for each of the channels is 10 boats per hour.

a. What is the average number of boats that will be waiting for service?
b. What is the average time a boat will spend waiting for service?
c. What is the average time a boat will spend at the dock?
d. If you were the manager of Fore and Aft Marina, would you be satisfied with
the - service level your system will be providing?
Solution: 0.03 boats, 0.4 mins, 6.4 mins

5. Consider a two-channel waiting line with Poisson arrivals and exponential service times.
The mean arrival rate is 14 units per hour, and the mean service rate is 10 units per hour
for each channel.

a. What is the average number of units in the system?
b. What is the average time a unit waits for service?
c. What is the average time a unit is in the system?
Solution: 1.35 customers in queue, 2.75 in system; 5.76 mins, 11.76 mins
6. Melvin's Market has an "express" checkout for customers with twelve items or less. The
inter-arrival time for customers at the express checkout has an exponential probability
distribution with a mean time between arrivals of 90 seconds. The checkout time for a
customer at the express checkout has an exponential probability distribution with a mean
checkout time that depends on whether a cashier has the help of a bagger. With a
bagger's help, the average time a cashier needs to check out a customer is 50 seconds;
without a bagger's help, the average time is 72 seconds. Consider the situations in which
a cashier has and does not have a bagger's help, and construct a table that compares
these situations with respect to the following operating characteristics:
a. The average number of customers in queue at the express checkout.
b. The average total time a customer spends at the express checkout.
Solution:      With bagger: =40, =72, Lq=0.69, Wtot=1.88 mins
Without bagger:  =40, =50, Lq=3.2, Wtot=6.0 mins

7. To promote its reputation for fast service, Earl's While-U-Wait Automotive Tune-up Shop
promises to reduce a customer's bill by \$0.20 for every minute the customer must wait
until his or her car's tune-up is finished. The inter-arrival time for Earl's customers has an
exponential probability distribution with a mean arrival rate of 5 customers per hour. The
time required by a mechanic to perform a tune-up has an exponential probability
distribution with a mean tune-up rate of 2 cars per hour. Earl is considering maintaining a
staff of 3, 4, or 5 mechanics. A mechanic's salary is \$20 per hour. Define Earl's total
hourly cost as the sum of two components: (1) the cost per hour of the mechanics and
(2) the profit lost per hour because of reductions of customers' bills. Estimate the total
hourly cost if Earl employs 3, 4, or 5 mechanics. Which number of mechanics results in
the lowest total?
Solution: \$132.13, \$116.40, \$131.56. Choose 4 mechanics
See Excel solution on website

Material Requirements Planning: (if covered in class)

1. A table is assembled using three components, 2X of Wood sections, 3X of Braces and
4X of Legs. The company that makes the table wants to ship 100 units at the beginning
of day 4, 150 units at the beginning of day 5, and 200 units at the beginning of day 7.
Receipts of 100 wood sections are scheduled at the beginning of day 2. There are 120
legs on hand. An additional 10% of the order size of legs is added for safety stock. There
are 60 braces on hand and no safety stock is required for it. Lead time in days for all
items is as follows: If quantity ordered is 1-200, lead time is 1 day, for 201-550 it is 2
days, and for 551-999 it is 3 days. Prepare an MRP plan using lot-for-lot ordering.

Solution: See Excel solution on website

2. The BOM for an item is as follows. Product A consists of B (1X) and D(2X). Every B in
turn consists of C(2X), and every D consists of B(1X).

If the Master Production Schedule of A has a requirement of 500 units in week 6, and
the lead time for assembly of A is 1 week, develop the MRP plan for B,C, and D for the
next 6 weeks given the following information.
Item
Data category             B                  C                D
Lot sizing rule          FP(3)          MOQ=1500                L4L
Safety Stock              50                100                  0
Lead time               1 week            1 week              2 weeks
Sch Receipts             None          2000 (week 1)           None
Beg. Inventory            50                200                  0

Solution: See Excel solution on website

LINEAR PROGRAMMING

1.    Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular
model and a catcher's model. The firm has 900 hours of production time available in its
cutting and sewing department, 300 hours available in its finishing department, and 100
hours available in its packaging and shipping department. The production time
requirements and the profit contribution per glove are given in the table at the top of the
next page.

Production Time (hours)
Model            Cutting and      Finishing    Pack and Ship       Profit/Glove
Sewing
Regular glove            1               1/2              1/8              \$5
Catcher’s glove          2/3              1/3              1/4              \$8

Assuming that the company is interested in maximizing the total profit contribution, answer
the following:
a. What is the linear programming model for this problem?
b. Find the optimal solution. How many gloves of each model should Kelson
manufacture?
c. What is the total profit contribution Kelson can earn with the above production
quantities?
d. How many hours of production time will be scheduled in each department?

Solution: b. 500, 150
c. \$3700
d. 600, 300, 100
See Excel solution on website

2.     Yard Care, Inc., manufactures a variety of lawn care products, including two well-
known lawn fertilizers. Each fertilizer product is a blend of two raw materials known as
K40 and K50. During the current production period, 900 pounds of K40 and 400 pounds
of K50 are available. Each pound of the product known as "Green Lawn" uses 3/5 pound
of K40 and 2/5, pound of K50. Each pound of the product known as "Lawn Care" uses
3/4 pound of K40 and 1/4 pound of K50. In addition, a current limit on the availability of
packaging materials restricts the production of Lawn Care to a maximum of 500 pounds.
Assume that the profit contribution for both products is \$3 per pound.
a. What is the linear programming model for this problem?
b. Find the optimal solution. How many pounds of each product should be
manufactured?
Solution: b. 687.5, 500.
3562.5
See Excel solution on website

3.     Greentree Kennels, Inc., provides overnight lodging for a variety of pets. A particular
feature at Greentree is the quality of care the pets receive, including excellent food. The
kennel's dog food is made by mixing two brand-name dog food products to obtain what
the kennel calls the "well-balanced dog diet." The data for the two dog foods are as
follows:
Dog food               Cost/ounce            Protein%            Fat%
Bark bits                 \$0.06                  30                15
Canine Chow                 \$0.05                  20                30

If Greentree wants to be sure that the dogs receive at least 5 ounces of protein and at
least 3 ounces of fat per day, what is the minimum cost mix of the two dog food
products?

Solution: 15oz, 2.5 oz
\$1.025
See Excel solution on website

4.      Photo Chemicals produces two types of photographic developing fluids. Both products
cost Photo Chemicals \$1 per gallon to produce. Based on an analysis of current
inventory levels and outstanding orders for the next month, Photo Chemicals'
management has specified that at least 30 gallons of product 1 and at least 20 gallons of
product 2 must be produced during the next 2 weeks. Management has also stated that
an existing inventory of highly perishable raw material required in the production of both
fluids must be used within the next 2 weeks. The current inventory of the perishable raw
material is 80 pounds. While more of this raw material can be ordered if necessary, any
of the current inventory that is not used within the next 2 weeks will spoil - hence, the
management requirement that at least 80 pounds be used in the next 2 weeks.
Furthermore, it is known that product 1 requires 1 pound of this perishable raw material
per gallon and product 2 requires 2 pounds of the raw material per gallon. Since Photo
Chemicals' objective is to keep its production costs at the minimum possible level, the
firm's management is looking for a minimum-cost production plan that uses all the 80
pounds of perishable raw material and provides at least 30 gallons of product 1 and at
least 20 gallons of product 2. What is the minimum-cost solution?
Solution: 30, 25
\$55
See Excel solution on website

5.     A millionaire wants to invest \$150 million by purchasing some or all of the following
properties: a shopping center that costs \$90 million, a professional basketball franchise
that costs \$50 million, and a 20-story office building that costs \$100 million. The annual
return on the shopping center is \$10 million, from the basketball franchise is \$4 million,
and from the office building is \$ 12 million. The investor has hired a manager who works
50 hours per week. The time required to oversee operations of the shopping center is 30
hours, for the basketball franchise is 10 hrs, and the office building is 20 hours. Because
of potential problems due to traffic conditions at the shopping center and fan reaction to
the basketball team, the investor wishes to invest in either the shopping center or the
basketball franchise, not both. What would be his optimal investment in the alternatives?
Solution: \$16 million; Basketball franchise and Office building
See Excel solution on website

6.    A beer company has breweries in two cities and has distributors in six states. The
monthly capacities in the breweries, the monthly demand per state, and shipping costs
per barrel are shown in the table below.
Shipping cost per barrel
Tampa    St. Louis Demand
Tennessee                   2.5       1.25  1600
Georgia                    1.75       3.25  1800
North Carolina                3          2  1500
South Carolina             2.25       2.75   950
Kentucky                      4          1  2000
Virginia                   3.75       3.25  1400
Capacity                  3500       5000

How should the firm distribute its product at minimum total cost?

Solution: Total cost \$14,825; Ship 1600 1800 1500 950 2000 650
See Excel solution on website

Project Management:

1. Consider the following project

Immediate                    Activity Times             Cost to
Activity    Predecessor      Optimistic       Most      Pessimistic crash/day
Likely
A             -               2              6           8         500
B             -               3              3           6         600
C             -               5              7          12        1000
D            A,B              3              6           6         250
E             B               3              6           8         100
F            D,C              5              6           9         350
G            E,C              4              8           9         700
H            F,G              8             12          13         450

Draw the network, specify the critical path, figure out the ES and LS for each activity and
their slacks. Round numbers as necessary.
Solution: See Excel solution on website

2. What is the probability that the above project will be complete in 30 days or less? In 27
days or less?
Solution: See Excel solution on website
3. What activities would you crash to reduce the duration of the above project by 2 days.
Solution: See Excel solution on website

Quality Management:

1. For the following process, find the control limits.

Observations
Sample             1            2          3         4         5             6
1           5.02          5.02       4.94     4.99       4.96          4.90
2           5.01          5.03       5.07     4.95       4.96          4.96
3           4.99          5.00       4.93     4.92       4.99          4.99
4           5.03          4.91       5.01     4.98       4.89          4.93
5           4.95          4.92       5.03     5.05       5.01          4.95
6           4.97          5.06       5.06     4.96       5.03          4.87
7           5.05          5.01       5.10     4.96       4.99          5.00
8           5.09          5.10       5.00     4.99       5.08          5.02
9           5.14          5.10       4.99     5.08       5.09          4.94
10           5.01          4.98       5.08     5.07       4.99          4.99

Is the process in control with respect to both mean and range in all periods?

Solution: See Excel solution on website

2. In controlling the number of defectives, you take samples of size 100, and get the
following number of defectives

Sample          #Defectives
1                  5
2                  3
3                  6
4                  7
5                  4
6                  6
7                  8
8                  4
9                  5
10                 8
11                 3
12                 4
13                 5
14                 6
15                 6
16                 7
17                 5
18                 3
19                 5
20                 6
Draw 3 control limits for this process. Is the process in control?

Solution: See Excel solution on website

3. In controlling defects in a particular process, the count of defects were as shown below.

Sample           #Defects
1                12
2                 8
3                16
4                14
5                10
6                11
7                 9
8                14
9                13
10               15
11               12
12               10
13               14
14               17
15               15

Is the process in control? Draw 3 control limits for this process

Solution: See Excel solution on website

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