When a bill is mailed from a region to a processing center

PROBLEM 1

CitiSavings Bill-processing Operations

CitiSavings
operates three bill-processing centers as a part of its credit card business.
These centers are located in the Los Angeles, Chicago, and New York areas, and
they can process the following numbers of bills each day:

Los Angeles

Chicago

New York

Daily bill-processing
capacity

60,000

105,000

100,000

Customers from
around the country mail payments on their credit card bills to the three
centers for processing. The numbers of bills to be processed daily from each
region are as follows:

Number to be Processed

West

70,000

Midwest

50,000

East

80,000

South

40,000

When a bill is
mailed from a region to a processing center, it spends time in the U.S. Postal
Service (USPS) delivery system. The table below shows the average number of
days a bill spends in transit between each region and processing center:

Los Angeles

Chicago

New York

West

2

6

8

Midwest

6

2

5

East

8

5

2

South

8

5

5

Each day that a
bill spends in transit is a day’s worth of interest CitiSavings has lost on the
payment received. At a 5% annual rate, the interest lost on an average payment
is approximately 10 cents per day.

CitiSavings has
formulated a linear program (LP) to help it determine to which processing
center customers from the various regions should mail their payments.
CitiSavings would like to minimize the interest income lost due to transit
times of the payments. The model they have formulated is as follows:

Xij
= number of bills to be mailed from region i to processing center j each day
Min 2
XWL + 6 Xwc + 8 XWN + 6 XML + 2 XMC
+ 5 XMN +
8 XEL + 5 XEC + 2
XEN + 8 XSL + 5 Xsc + 5 XSN
s.t.

XWL + XML
+ XEL + XSL ≤ 60,000 (Los
Angeles capacity)
Xwc
+ XMC + XEC + Xsc ≤ 105,000 (Chicago
capacity)
XWN
+ XMN + XEN + XSN ≤ 100,000 (New York capacity)
XWL
+ Xwc + XWN ≥ 70,000 (West
requirements)
XML
+ XMC + XMN
≥ 50,000 (Midwest
requirements)
XEL
+ XEC + XEN ≥ 80,000
(East requirements)
XSL
+ Xsc + XSN ≥ 40,000 (South
requirements)
Xij
≥ 0
for all i and j (non negativity)

The spreadsheet
results and sensitivity report for the CitiSavings LP are shown below:

Answer
questions (a) through (g) below. Each question is independent of the others. If
an answer cannot be determined from the information in the spreadsheet results
or the sensitivity report, write “Cannot be Determined” and explain
why it cannot be determined.

a) Under the
optimal plan, what is the total $ interest lost each day?

b) The USPS is
changing its delivery network, and CitiSavings has discovered that the average
number of days that a bill spends in transit from the West to Chicago will increase
from 6 to 7 days.

How will this change the optimal
solution values?
How much will the total interest
lost each day change?

c) By how much
would the bill-processing capacity in New York have to decrease before the
optimal solution values would change? Explain why?

d) Suppose the
company could expand the production capacity in Los Angeles by 15,000 units at
a cost of $3,000 per day. Should this be done? Explain.

CitiSavings
intends to merge with BankZero, which has its own set of processing centers for
credit card bills. BankZero’s processing centers are in Toledo (Ohio) and
Atlanta. The processing capacities of the full set of processing centers are as
follows:

Los Angeles

Chicago

New York

Toledo

Atlanta

Daily
bill-processing capacity

60,000

105,000

100,000

135,000

155,000

By region, the
number of bills to be processed each day for the two companies’ combined
customer bases and the transit times are as follows:

Number to be Processed

Los Angeles

Chicago

New York

Toledo

Atlanta

West

100,000

2

6

8

6

8

Midwest

150,000

6

2

5

2

5

East

100,000

8

5

2

5

5

South

100,000

8

5

5

5

2

The new optimal
solution would be found by solving the following LP:

Xij
= number of bills to be mailed from region i to processing center j each day

Min 2
XWL + 6 Xwc + 8 XWN + 6 XWT + 8 XWA
+
6 XML + 2 XMC + 5
XMN + 2 XMT + 5 XMA
+
8 XEL + 5 XEC + 2
XEN + 5 XET + 5 XEA+
8 XSL + 5 Xsc + 5 XSN
+ 5 XST + 2 XSA
s.t.
XWL + XML + XEL + XSL ≤ 60,000 (Los Angeles capacity)
Xwc + XMC + XEC + Xsc ≤ 105,000 (Chicago
capacity)
XWN + XMN + XEN + XSN ≤ 100,000 (New York capacity)
XWT + XMT + XET + XST ≤ 135,000 (Toledo capacity)
XWA + XMA + XEA + XSA ≤ 155,000 (Atlanta capacity)
XWL
+ Xwc + XWN+ XWT+ XWA ≥ 70,000
(West requirements)
XML
+ XMC + XMN + XMT+ XMA ≥ 50,000 (Midwest requirements)
XEL
+ XEC + XEN+ XET+ XEA ≥ 80,000
(East requirements)
XSL
+ Xsc + XSN+ XST+ XSA ≥ 40,000 (South requirements)
Xij
≥ 0 for all i and j (non negativity)

e) Management
would like to ensure that Toledo handles at least 2/3 of all the bills
processed by Toledo and Chicago combined. Extend the formulation to assure that
this requirement is met.

Citisavings
recognizes mat the combined operations would have significant excess capacity
and would like to decide which processing facilities to keep open and which to
close. Ignore the cost of closing a facility, and also ignore the Toledo –
Chicago requirement described in part (e). The fixed daily cost of operating
the five facilities are as follows:

Los Angeles

Chicago

New York

Toledo

Atlanta

fixed daily
operating cost

12,000

21,000

20,000

40,000

30,000

f) Revise the
linear program to help CitiSavings decide which processing centers to close. Be
sure to define any new decision variables. Make sure that part of the objective
function has consistent units. Write below only the modifications you make to
the formulation on page 5. Do not use IF statements in the constraints.

g) CitiSavings
management would like to consider the expansion of capacity at the Los Angeles
processing center as another option within the overall consolidation plan. That
is, in addition to either closing the center or leaving it at the current
capacity, they want to consider adding 15,000 bills-per-day of additional
capacity. The additional cost would be $3,000 per day. Modify/extend the
formulation of (f) to include this additional option. Be sure to define any new
decision variables. Do not use IF statements in the constraints.

PROBLEM 2

The CIAA-TREF Pension Fund

Naomi Nakakura
manages a pension fund for CIAA-TREF, a large financial services company that
offers retirement annuities and mutual funds. She is considering a number of
fixed income securities (treasury bonds) to finance a series of pension
liabilities (cash outflows) over the next five years. The cash requirement a
year from now is projected to be $45 million and is expected to grow by $10
million a year, as summarized in Table 1 below. (All figures are in $million):

Table 1: Annual
Cash Requirements (in $mm)

Years form
now (t)

1

2

3

4

5

Expected cash
liability

45

55

65

75

85

Naomi is
considering the following seven types of bonds for inclusion in her portfolio
to finance the above liabilities. Each bond has a face value of $100. Table 2
below provides information on the current price, annual coupon amount, and
maturity of the bonds:

Table 2: Bond
Characteristics

Bond 1

Bond 2

Bond 3

Bond 4

Bond 5

Bond 6

Bond 7

Current Price

100

97

98

99

103

93

98

Annual Coupon

5

6

5.8

7

4.7

5

5.9

Maturity (Yr)

1

1

2

3

3

4

5

The coupon is
the annual interest payment to the bond holder. For example, Bond 4 sells for
$99 today, pays $7 in Year 1, $7 in Year 2, and $107 in Year 3. In other words,
if Naomi were to buy 5 million units of Bond 4, this will require $495 million
today and return $35 million in Year 1, $35 million in Year 2, and $535 million
in Year 3. All seven bonds are available in essentially unlimited amounts.
Naomi would
like to determine the mix of bonds she should purchase now to meet the pension
fund’s cash liabilities over each of the next: five years with the least initial
investment. Assume that a cash surplus in any year can be reinvested at an
annual interest rater of 4% and will be available to meet the next year’s
liability.
Formulate the
problem algebraically as a linear program.

a) Decision
Variables (be sure to mention the units)

b) Objective
Function (also mention units)
The
objective is to minimize the initial investment which is given by:

c) Constraints
(label each constraint clearly):

PROBLEM 3

The H.S.
Daugherty Company has been manufactured industrial vacuum cleaning systems for
a number of years. Recently a member of the company’s. new-product research
team submitted a report suggesting that the company consider manufacturing a
cordless vacuum cleaner. The vacuum cleaner, referred to as a Porta-Vac, could
contribute to Daugherty’s expansion into the household market. Management hopes
that the new product can be manufactured at a reasonable cost and that its
portability and no-cord convenience will make it extremely attractive.

Given below is
information about the activities that must be carried out in order to realize
this project. Times are in weeks and cost in thousands of dollars.

Activity

Description

Immediate
Predecessor

Expected
Time

Expected
Cost

A

Prepare
R&D product design

6

90

B

Plan market
research

2

16

C

Prepare
routing (manufacturing engineering)

A

3

3

D

Build
prototype model

A

5

100

E

Prepare
marketing brochure

A

3

6

F

Prepare cost
estimates (industrial engineering)

C

2

2

G

Do
preliminary product testing

D

3

60

H

Complete
market survey

B, E

4

20

I

Prepare
pricing and forecast report

H

2

4

J

Prepare final
report

F, G, I

2

2

a) Construct a
network diagram for this project.

b) Fill in the
table bellow

Activity

Earliest Start

Latest Start

Slack

A

B

C

D

E

F

G

H

I

J

c) Give the critical path for the project and the
minimum completion time.

d) At the end
of the 10th week, this is the situation of the project (*):

A,
B, E : Completed
C:
1 week completed
D:
4 weeks completed

Will the
project be completed in time? If not what is the duration of the project?
(*) If an
activity is not listed below, assume that it has not been started.

e) Prepare a Pert/Cost analysis for each of the 2
points in time. For each case, show the percent overrun or under-run for the
project to date, and indicate any correction action that should be undertaken. NOTE:
If an activity is not listed below, assume that it has not been started.

At the end of
the 10th week:

Activity

Actual Cost (th$)

Percent Complete

A

85

100%

B

16

100%

C

1

33%

D

100

80%

E

4

100%

H

10

25%

At the end of
the 15th week:

Activity

Actual Cost (th$)

Percent Complete

A

85

100%

B

16

100%

C

3

100%

D

105

100%

E

4

100%

F

3

100%

G

55

100%

H

25

100%

I

4

100%

PROBLEM 4

A major
operation in an outpatient medical office is answering the telephones. This is
especially true in primary care, such as pediatrics. Patients mostly use the
telephone to communicate with the physician’s office. In pediatrics, such
interactions include calling for appointments, refills, medical advice,
referrals, and forms (for example: school forms, camp forms.) Because of the
frequent use of the telephone in outpatient pediatrics, it is an important
focus for assessing productivity and efficiency.

A pediatric
practice consists of nine physicians and two nurse practitioners. The practice
has two offices. The patient population is approximately ten thousand children,
with nearly fifty thousand visits per year. The phone system consists of
sixteen telephone lines, most of them at the main office.

As the practice
has grown, there have been increasing complaints from patients about wait time
on the phone lines. All incoming calls are routed to the main office. When a
patient dials the practice’s office telephone number, a voicemail system directs
the caller to press a number according to the purpose of the call (for example,
“Press ‘one’ for appointments.”) The system also distributes the
phone calls according to whether the person calling is a patient, physician,
laboratory, or hospital.
During the
winter months, when the volume of sick patients is highest, a patient’s wait
can sometimes be as long as ten to fifteen minutes on the appointment line
before speaking to a person. Since most customer service guidelines recommend
telephone hold times no longer than one minute, this is an area that greatly
needs improvement.

Telephone calls
form a single waiting line and are served on a first-come, first-served basis.
Arrival rates can be described by Poisson distribution, and service times can
be described by negative exponential distribution. With these characteristics,
a multiple-channel model for queuing analysis is most appropriate.

The queuing
analysis of the practice’s phone system can be divided into three parts of the
workday, which lasts from 8:00 A.M. to 5:00 P.M. For the first hour of the day
(8:00 A.M. to 9:00 A.M.) there are usually three receptionists working to
answer telephone calls only. For the last hour of the day (4:00 P.M. to 5:00
P.M.), there are usually five receptionists answering phones as well as
checking patients in and out. For the bulk of the day, there are usually six
receptionists working. The use of fewer servers during the first and last hours
is primarily because fewer patients are being seen during those hours, so fewer
servers are needed for checking patients in and out.

To determine
the customer arrival rate (or phone calls/hour), incoming monthly phone call
data for the previous year were obtained from the telephone company (Table 1.)

TABLE 1

Month Phone Calls
January 6,640
February
6,756
March
6,860
April
6,226
May
6,671
June
7,168
July
6,802
August
6,971
September
7,205
October
6,944
November
6,623
December
6,875
Total 81,741

From examining
previous studies of the office’s phone call volume distribution, it is
estimated that 30% of the phone calls occur between 8 A.M. and 9 A.M.; 40%
between 9 A.M. and 4 P.M., and the remaining 30% arriving from 4 P.M. to 5 P.M.
(Table 2).

TABLE 2

Customer
Arrival Rates (λ)

8:00
A.M. to 9:00 A.M. 31 phone
calls/hr
9:00
A.M. to 4:00 P.M. 42 phone
calls/hr
4:00
P.M. to 5:00 P.M. 31
phone calls/hr

To estimate the service rate (or phone
calls/hour/receptionist), several sample studies were performed by an office
administrator. It is important to note that the receptionists perform functions
other than answering phones, such as checking patients in and out. Therefore,
the number of phone calls that a server can answer per hour depends on the
other responsibilities that the person has that day. In order to arrive at a
service rate, the assumption was made that the average maximum of phone calls per
hour for the sample days would represent the servers operating at the maximum
phone-call-answering capacity when having other responsibilities. While this
assumption may underestimate actual server rate, for purposes of this study,
the conservative estimate is acceptable in the absence of further data.

There is one exception to this
assumption. During the first hour of the day, from
8:00 A.M. to
9:00 A.M., patients are not yet being seen in the office. Therefore, during
that hour the servers have a faster telephone service rate, since they have no
other primary duties (Table 3) From samples studied, we have determined that
the maximum service capability when only answering phones is approximately four
minutes per phone call, or fifteen calls per hour per server. This number was
used for the service rate for the first hour.

TABLE
3
Service
Rate μ.

8:00
A.M. to 9:00 A.M. 15 phone calls/hr
9:00
A.M to 4:00 P.M. 8
phone calls/hr
4:00
P.M to 5:00 P.M. 8 phone
calls/hr

Cost studies
were then performed based on financial data from the previous year. Capacity
costs were calculated based on salary and benefits per server and a percentage
of the equipment maintenance, phone line costs, rent, and other capital
expenditures (Table 4). With a total of fifty employees and a total of thirty
full-time equivalents (FTEs), the portion of capital expenditures was
determined as 1/30 of costs. Phone line charges were determine by a per line
charge, since one server would utilize one line each day.

TABLE
4

Total Hourly Cost for
Busy Server Summary

Salary $13.00
Benefit $
3.75
Telephone Charges $ 4.73
Capital Expenses $ 4.83
Total Hourly Cost for Busy Server $26.31

Capacity cost
or busy server cost would be equivalent to idle server cost. Regardless of
whether or not the receptionist is answering the phone, she is paid the same salary
and benefits and is using the same space and utilities. In addition, the
practice must pay the phone line and equipment maintenance charges, regardless
of usage.

For calculation
purposes, a value was assigned to the cost to the customer of waiting. A value
of $50/hour was assigned to customer waiting costs. In reality, though,
customer waiting costs are likely to vary with the length of time waited, with
a steep exponential increase in cost to the patient for longer times waited.

The cost of
being balked would represent a lost patient if a patient’s call was not
answered. In pediatrics, the patients generally prefer continuity of care
throughout their child’s life. Therefore, a truly balked customer might
represent a child’s lifetime worth of visits. However, one might also define a
balked customer as one who will not come for a visit that day because the phone
call was not answered promptly. This person would be likely to return to the
practice on another day if he or she established a doctor-patient relationship
with the practice. Therefore, for the purposes of this model, it is assumed
that the cost of being balked is the lost revenue from an office visit, which
is approximately $80. (See Table 5)

TABLE
EX 14.5.5

Cost
Summary

Busy
server cost/hr $
26.31
Idle
server cost/hr $
26.31
Customer
waiting cost/hr $ 50.00
Cost
of customer being balked $
80.00

Using EXCEL,
perform a queuing analysis for the pediatric practice’s telephone system to
determine the optimal server capacity for the volume of phone calls that they
receive. Are there enough servers/receptionists and enough phone lines?

PROBLEM 5

Alabama Airlines
opened its doors in June 2002 as a commuter service with its headquarters and
only hub located in Birmingham. A product of airline deregulation. Alabama Air
joined the growing number of successful short-haul, point-to-point airlines,
including Lone Star, Comair, Atlantic Southeast, Skywest, and Business Express.

Alabama Air was
started and managed by two former pilots, David Douglas (who had been with the
defunct Eastern Airlines) and Michael Hanna (formerly with Pan’ Am). It
acquired a fleet of 12 used prop-jet planes and the airport gates vacated by
Delta Airlines’ 1996 downsizing.

With business growing
quickly, Douglas turned his attention to Alabama Air’s “SOO”
reservations system. Between midnight and 6:00 A.M., only one telephone
reservations agent had been on duty. The time between incoming calls during
this period is distributed as shown in Table1. Douglas carefully observed and
timed the agent and estimated that the time taken to process passenger
inquiries is distributed as shown in Table 2.

TABLE 1: Incoming Call Distribution

Time
Between Calls

Probability

1

0.11

2

0.21

3

0.22

4

0.20

5

0.16

6

0.10

TABLE 2: Service Time Distribution

Time
Between Calls

Probability

1

0.20

2

0.19

3

0.18

4

0.17

5

0.13

6

0.10

7

0.03

All customers
calling Alabama Air go “on hold” and are served in the order of the
calls unless the reservations agent is available for immediate service. Douglas
is deciding whether a second agent should be on duty to cope with customer
demand. To maintain customer satisfaction, Alabama Air does not want a customer
“on hold” for more than 3 to 4 minutes and also wants to maintain a
“high” operator utilization.

Further, the
airline is planning a new TV advertising campaign. As a result, it expects an
increase in “8oo”-line phone inquiries. Based on similar campaigns in
the past, the incoming call distribution from midnight to 6 A.M. is expected to
be as shown in Table 2. (The same service time distribution will apply.)

TABLE 3: Incoming Call Distribution

Time
Between Calls

Probability

1

0.22

2

0.25

3

0.19

4

0.15

5

0.12

6

0.07

Discussion Questions

1. What would
you advise Alabama Air to do for the current reservation system based on the
original call distribution? Create a simulation model to investigate the
scenario. Describe the model carefully and justify the duration of the
simulation, assumptions, and measures of performance.

2. What are your
recommendations regarding operator utilization and customer satisfaction if the
airline proceeds with the advertising campaign?

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