Chapter 11: Managing Capacity and Demand
Chapter 11
Managing Capacity and Demand
Service Management: Operations, Strategy, Information Technology, 7th edition, by
Fitzsimmons & Fitzsimmons (McGraw-Hill Irwin 2011)
TEACHING NOTE
Many of the approaches to matching capacity and demand discussed in the chapter should be familiar to students
and prompt lively discussion. The controversial practice of "overbooking" is treated in economic terms, but the
behavioral implications of this policy for both service personnel and consumers should be discussed. This chapter
presents a number of techniques to assist management in matching supply and demand, but leaves to the
imaginative student the opportunity for other creative approaches. The chapter concludes with a discussion of a
technique that was pioneered by American Airlines called yield management. Yield management is a computerbased approach for maximizing revenues. Instructions for playing the yield management game are given in the
textbook and students should be encouraged to familiarize themselves with the material before “game day.” The
tally sheets and other instructional materials are presented in this manual.
SUPPLEMENTARY MATERIALS
Case: American Airlines, Inc. Revenue Management (HBS 9-190-029)
Following deregulation, American Airlines embarked on an ambitious program of revenue management by
building on its SABRE reservation system. The case presents the reader with pricing decisions on two routes and
describes the development of yield management.
Case: University Health Services: Walk-In Clinic (HBS 9-681-061)
In response to complaints of excessive waiting, a triage system was introduced and one year later the director
reviews the results to decide whether or not the patient waiting time is acceptable.
LECTURE OUTLINE
I.
Generic Strategies of Level Capacity and Chase Demand (Table 11.1)
II.
Strategies for Managing Demand (Figure 11.1)
Customer-induced Variability (Table 11.2)
Segmenting Demand (Figure 11.2)
Offering Price Incentives (Tables 11.4 and 11.5)
Promoting Off-peak Demand
Developing Complementary Services
Reservation Systems and Overbooking (Tables 11.9 and 11.7)
11-1
III.
Strategies for Managing Capacity
Defining Service Capacity
Daily Workshift Scheduling (Figures 11.3, 11.4, 11.5)
Weekly Workshift Scheduling with Days-off Constraint (Table 11.8)
Increasing Consumer Participation
Creating Adjustable Capacity
Sharing Capacity
Cross-training Employees
Using Part-time Employees
Scheduling Part-time Employees at a Drive-in-Bank (Figures 11.6 and 11.7)
IV.
Yield Management (Figures 11.8, 11.9, 11.10, 11.11, 11.12)
Yield Management Applications
TOPICS FOR DISCUSSION
1. What organizational problems can arise from the use of part-time employees?
The use of part-time employees can be very helpful to businesses that have peak demand periods such as
restaurants, supermarkets, and banks. Part-time employees are usually paid lower wages and they enjoy fewer, if
any, of the company benefits that are provided for full-time employees. Also, it is not generally feasible for a
company to offer career development incentives to part-time employees and it is also more difficult to fit them
into the organizational structure. In addition, part-time employees generally have lower experience levels than do
full-time workers. As a result, part-time employees may have poorer attitudes and less loyalty and commitment,
which could affect reliability, performance, and the quality of work. This situation can have a direct impact on
customers and the business. In view of these conditions, part-time employees may require greater supervision and
control than would be necessary for full-time employees. Also, there is usually a greater turnover in part-time
employees, so more time must be spent in training new employees. Finally, the business must hire more part-time
employees than full-time people to staff positions, which creates more administrative work for work scheduling,
personnel records, and payroll.
2. How can computer-based reservation systems increase service capacity utilization?
The main function of the reservation system is to pre-sell the service. A reservation system allows the customer
to reserve a service long before it is actually utilized. Allowing the customer to make reservations has numerous
advantages.
The reservation system can be used to deflect demand to other times or locations where service capacity is
available. For example, if a passenger wants a flight that is full, a reservation clerk can suggest immediately
alternative flights that are available. Thus, the demand for service capacity has been effectively rerouted to underutilized capacity. Reservation systems also allow the service to overbook its capacity when it reasonably expects
to have no-shows.
11-2
Finally, when computer-based systems are standardized among different service organizations in the same
industry, each participant may make use of a larger information base. For example, a passenger can call a Delta
Air Lines reservation agent and book a flight on United Airlines if a Delta flight is unavailable. Such a system
benefits all the participants and the customers.
3. Illustrate how a particular service has implemented successfully strategies for managing both demand and
capacity.
Because a service is consumed and produced simultaneously, a failure to provide enough capacity to serve results
in idle servers and facilities. Public schools have been experiencing this variability in demand recently owing to
fluctuations in the numbers of school age children, a move of families from central cities to the suburbs, and
increasing enrollments in private schools. The agrarian pattern of closing school during summer months has
contributed to idle servers and facilities during that period.
As a public service, education has been somewhat slow at adapting to its fluctuating demand, but the recent trend
of more competition for scarce resources in public services and the public outcry for educational accountability
has spurred some strategies for managing the demand and supply. Some examples follow:
Managing demand:

Expand educational services by offering adult education, early childhood education programs, and before and
after school childcare programs.

Offer social services such as community education programs, job retraining programs, programs for senior
citizens, and summer recreation programs.

Offer incentives to attend particular schools by making them magnet schools, open-area schools, or openenrollment schools, which any child in the community may attend.

Promote off-peak demand in some areas by offering summer school classes.

Partition demand by using educational voucher systems.
Managing capacity:

Use school buildings where classes are no longer held because of declining enrollment for warehousing
school supplies or for administrative offices or lease those to other organizations until school demographics
change.

Arrange flexible schedules on a daily basis and a school-year basis for crowded schools. For example, on a
daily basis in an elementary school, schedule hours for kindergarten through third grade students from 8 a.m.
to 2 p.m. and hours for fourth through sixth grade students from 10 a.m. to 4 p.m. Another alternative would
be to stagger the students throughout the calendar year instead of offering classes only during the typical ninemonth school year, i.e., some students would attend school December to September, others would go from
June to March, and still others would attend the typical September to June period.

Merge two or more grades in under-utilized schools or, in a crowded grade school, shift sixth graders to a
junior high/middle school.

Hire teachers who are trained in more than one subject in order to cope with a fluctuating demand for courses.

Hire part-time teachers in small schools for support areas such as art, music and physical education.

Share capacity with other public service agencies to utilize space better.
4. What possible dangers are associated with developing complementary services?
One of the possible dangers associated with developing complementary services is the possibility of increasing
the firm's liability because of those added services. For instance, adding cold and hot sandwiches at convenience
grocery stores incur the possibility that a customer might suffer food poisoning and take legal action against the
11-3
store. Another example is seen with the addition of self-service gas pumps, which involve a risk of fire, damage
and injury.
Another problem associated with offering complementary services may occur when the complementary service
attracts customers who may hurt business. Consider a shopping mall that installs a video arcade. The arcade may
become a haven for noisy and rowdy teenagers who will drive away those customers who want to shop in peace.
Finally, a complementary service should enhance the firm's image; as noted above, a poorly selected
complementary service could serious compromise business.
5. Will the widespread use of yield management eventually erode the concept of fixed prices for any
service?
Most services exhibit, to some extent, the capacity-constrained dilemma faced by airlines and hotels that are
unable to inventory their products (seats on a flight or rooms for a night). To avoid losing the revenue from this
perishable capacity, capacity- constrained services are motivated to pre-sell the inventory when possible by using
reservations and giving discounts to avoid lost sales. For example, travelers have found that the published room
rates (rack rates) for underutilized hotels are quickly abandoned if the guest requests a discount (e.g., government
or AARP). The exceptions to businesses that practice this strategy are the budget motels that fill all their rooms
each night. Yield management has alerted customers to the perishable nature of capacity-constrained services and
this knowledge will destroy the myth of fixed prices for many such services and lead to price negotiation for all
services.
6. Go to http://en.wikipedia.org/wiki/Yield_management/ and discuss the ethical issues associated with yield
management.
Yield management is a form of price discrimination, and thus could be seen as unfair. For example, why should
two customers receive the identical service (coach seat on an airline flight) and pay different prices. However,
demand-responsive pricing in which last minute purchases are more expensive than purchases far in advance is
considered by most customers as reasonable.
INTERACTIVE CLASS EXERCISE
Watch the PowerPoint presentation concerning the overbooking experience at the Doubletree Hotel in Houston,
Texas. How could this situation been handled differently?
Clearly, the night desk clerk was not properly prepared to handle such a late arrival for a reserved room. The desk
clerk should have apologized for the problem and immediately secured alternative accommodations and
transportation, if necessary. We note that the guests contributed to the situation by not notifying the hotel of their
late arrival. Figure 6.13 the service recovery framework could be displayed and used as a focus of the discussion.
11-4
EXERCISES
11.1 (a)
Day
Mon.
Tue.
Wed.
Th.
Fri.
Capacity
75
75
75
75
75
Walk-ins
50
30
40
35
40
Appointments
25
45
35
40
35
(b) Appointments should not be scheduled at their maximum level because the walk-ins are only expected values.
There will be some variation both in time of arrival and in the actual demand. We learned from queuing theory
that capacity to serve must always exceed demand. Thus, some slack time should be scheduled into the system in
order to avoid excessive patient waiting.
(c) In order to avoid physician waiting, a few appointments should be made at the very beginning of the day. The
majority of appointments should be made in the late afternoon to avoid excessive waits for walk-in patients.
11.2
Now C u  $100 and C o remains $100
P(d  x) 
Cu
100

.5
C u  C o 100  100
From Table 13.4 in the textbook, a strategy of overbooking three rooms guarantees that 48 percent of the
no-shows will be covered. This represents a change from the previous policy of overbooking two rooms.
11.3
No Shows
0
1
2
3
4
Frequency
6
5
4
3
2
P(d  x)
Probability
.30
.25
.20
.15
.10
0
.30
.55
.75
.90
If overbook by 1, then P(d  x ) must be at least .30 and less than .55.
P(d  x ) 
Cu
20

 0.30
Cu  Co 20  Co
Solving for Co : Co 
20  0.30(20)  $46.67 (maximum overbooking opportunity loss)
0.30
11-5
11.4
Daily
Demand
Frequency
Probability
P(d  x)
10
11
12
13
14
15
16
17
18
19
20
1
1
2
2
2
3
3
2
2
1
1
.05
.05
.10
.10
.10
.15
.15
.10
.10
.05
.05
0
.05
.10
.20
.30
.40
.55
.70
.80
.90
.95
P(d  x ) 
Cu
(30  10)
20


 .67
Cu  Co (30  10)  10 30
Crazy Joe should lease 16 canoes.
11.5
Using a payoff matrix approach:
Passengers Overbooked
No
Shows
P(NS)
0
.30
1
.25
2
.20
3
.15
4
.10
Expected Profit:
0
180*
100
20
– 60
–140
$ 60
1
130
180
100
20
– 60
$101
2
80
130
180
100
20
$109.50
3
30
80
130
180
100
$ 92
4
– 20
30
80
130
180
$ 55
* Cell value = 6  $80  $300  $180
Thus, overbook 2 passengers (i.e., take 8 reservations) and expect a profit of $109.50 per flight.
Using the critical fractile model:
P( d  x ) 
Cu
80

 . 61
Cu  Co 80  50
Where:
Cu = Cost of underestimating no-shows (i.e., there are more no-shows than expected and the airline
must fly empty seats)
11-6
Co
=
Cost of overestimating no-shows (i.e., there are fewer no-shows than expected and the
passengers who cannot be accommodated get free lift tickets)
11.6
Emergency Room Nurse Staffing
Objective Function:
Minimize x1  x2  x3  x4  x5  x6  x7
Constraints:
Sun.
Mon.
Tue.
Wed.
Th.
Fri.
Sat.
x1
x2
x3
x4
x5
x6
x7
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
0
0
0
1
1
1
1
1
0







RHS
3
6
5
6
6
6
5
For: i  1 to 7, xi  0 and integer
Where:
xi = Number of nurses assigned to a shift that has two consecutive days off beginning with day i (i =
1 for Sunday, 2 for Monday, etc.)
Solution Summary for Emergency Room
Objective
Variable
Solution
Coefficient
Variable
Solution
x1
1
1
x5
2
x2
1
1
x6
0
x3
1
1
x7
3
x4
0
1
Minimized OBJ = 8 Iteration = 7 Elapsed CPU seconds = .109375
11-7
Objective
Coefficient
1
1
1
Weekly Staff Schedule for Emergency Room
x2 x3 x4 x5
x6
x7 Req. Staff
1
1
0
2
0
3
4
1
0
2
0
3
6
6
0
2
0
3
5
6
1
2
0
3
6
7
1
1
0
3
6
6
1
1
0
3
6
6
1
1
0
2
5
5
x1
Day
Sun.
Mon.
Tue.
Wed.
Th.
Fri.
Sat.
1
1
1
1
1
Excess
1
0
1
1
0
0
0
11.7
Sheriff Patrol Problem:
Objective Function:
Minimize x1  x2  x3  x4  x5  x6  x7
Constraints:
Sun.
Mon.
Tue.
Wed.
Th.
Fri.
Sat.
x1
x2
x3
x4
x5
x6
x7
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
0
0
0
1
1
1
1
1
0







RHS
6
4
4
4
5
5
6
For: i  1 to 7, xi  0 and integer
Where:
xi = Number of deputies assigned to a shift that has two consecutive days off beginning with day i (i = 1 for
Sunday, 2 for Monday, etc.)
11-8
Summarized Report for Sheriff Patrol
Opportunity
Objective
Number
Variable
Solution
Cost
Coefficient
1
x1
+1.0000000
0 +1.0000000
2
x2
+2.0000000
0 +1.0000000
3
x3
+1.0000000
0 +1.0000000
4
x4
+2.0000000
0 +1.0000000
5
x5
0
0 +1.0000000
6
x6
+1.0000000
0 +1.0000000
7
x7
0
0 +1.0000000
Minimized OBJ = 8 Iteration = 7 Elapsed CPU seconds = .109375
Weekly Staff Schedule for Summer Months
Day
x2 x3 x4 x5 x6
x1
Sun.
–
2
1
2
0
1
Mon.
–
–
1
2
0
1
Tue.
1
–
–
2
0
1
Wed.
1
2
–
–
0
1
Th.
1
2
1
–
–
1
Fri.
1
2
1
2
–
–
Sat.
1
2
1
2
0
–
x7
–
0
0
0
0
0
–
Req.
6
4
4
4
5
5
6
Minimum
Obj. Coeff.
0
+ .66666663
+ 1.0000000
+ .66666663
+ 1.0000000
0
+ 1.0000000
Staff
6
4
4
4
5
6
6
Maximum
Obj. Coeff.
+1.5000000
+1.0000000
+1.5000000
+1.0000000
+1.5000000
+1.0000000
+ Infinity
Excess
0
0
0
0
0
1
0
11.8
F = $79
D = $59
p = 0.8
 = 75
 = 15
P(d  x ) 
( F  D) (79  59)

 0.316
p( F )
0.8(79)
Full-fare seats reserved =  +z = 75 + (-.48)(15) = 67
11.9 (a)
Model
Car
Compact
Midsize
Rental
Rate (F)
$30
$45
Discount
Rate (D)
$20
$30
% Discount
Seekers (p)
80
60
Daily
Demand ()
50
30
11-9
Standard
Deviation ()
15
10
Fleet
Size
60
30
P (d compact  xcompact ) 
P (d midsize  xmidsize ) 
30  20
 . 417  xcompact  50  (. 21)(15)  46
30(.8)
45  30
 .555  xmidsize  30  (.14)(10)  31
45(. 6)
(b)
Because the current fleet of midsize cars is smaller than the optimal number of midsize cars that should be
reserved for full-price paying customers, an expansion of the midsize fleet should be considered.
CASE: RIVER CITY NATIONAL BANK
Assignment
As Ms. Chen's top aide, you are assigned the task of providing an analysis of the situation and recommending
a solution. This is your opportunity to serve your company and community as well as to make yourself "look
good" and earn points toward your raise and promotion.
Gary Miller is faced with the problem of fluctuating demand at the recently constructed drive-in facility. The
customer complaints of excessive waiting occur for the most part on Fridays. Little congestion is experienced
during the remainder of the week. This represents the classical service problem of managing capacity. The
problem can be addressed from two perspectives: managing both demand and supply.
Altering demand:
The bank customers could be encouraged to use the facility on off-peak days, thus avoiding delays in service. The
customers can be informed of the peak times with notices in their monthly statements or signs outside the remote
facility. A further inducement to try the facility on an off-peak day could be accomplished by giving those
customers some small gift such as a pocket calendar or pen. A more ambitious incentive would be an offer of free
or reduced monthly service charges for customers who bank regularly on off-peak days. Cashing payroll checks
represents a major transaction on Fridays. Customers should be informed of the convenient direct deposit option
where employers can deposit paychecks directly into employee bank accounts. The bank may wish to approach
large employers in the community to promote the direct deposit idea. Finally, customers may need
encouragement to be better prepared to conduct their transactions as soon as they reach the tellers' windows. A
preparation stand containing deposit slips and pens could be placed just before the entry to the drive-in station so
customers could do their paperwork while waiting for the customer currently being served.
Controlling capacity:
With the use of part-time tellers and proper shift scheduling, the bank could better match service capacity with the
fluctuations in daily demand. Because there are no problems on Saturdays, we will only consider weekdays. Our
analysis begins with an assessment of teller availability. Tellers available for the remote drive-in consists of 2
full-time tellers who work the morning shift only (7 a.m. to 2 p.m.), Monday through Friday and 4 part-time
tellers who work various combinations of morning shifts (10 a.m. to 2 p.m.) or afternoon shifts (2 p.m. to 7 p.m.).
Each part-time teller works 20 hours per week, excluding Saturdays. This yields a total of 80 part-time teller
hours per week to supplement the 70 full-time hours for the two full-time tellers. The problem involves allocating
these 150 teller hours to meet the daily demand levels for the week.
Table 1 is developed from the data provided in Table 11.8 of the case in the textbook. This table summarizes the
daily transaction demand by calculating the mean and standard deviation for each day by morning and afternoon
shift. The last column of the table gives the 95 percent demand level assuming normal distribution. This demand
level divided by teller capacity represents the percent of utilization. From a queuing standpoint, this percent
11-10
utilization must be less than 100 percent and probably not more than 80 percent to avoid excessive customerwaiting times.
Table 1
Statistical Analysis of Transactions by Day of Week and Shift
Mean
Standard
95 Percent
Number
Deviation
Demand Level
Monday:
Morning
168.75
14.20
192.18
Afternoon
133.75
26.42
177.34
Tuesday:
Morning
131.75
20.73
165.95
Afternoon
97.00
27.13
141.76
Wednesday:
Morning
147.80
43.87
220.19
Afternoon
138.80
47.84
217.74
Thursday:
Morning
140.80
25.74
183.27
Afternoon
125.40
30.07
175.02
Friday:
Morning
227.80
20.63
261.84
Afternoon
232.00
39.11
296.53
In order to calculate teller capacity it is assumed that each teller could service two lanes and that all eight lanes
could be used when needed. The number of transactions per lane that are possible is based on a typical
transaction, which takes approximately five minutes per customer when we account for all the delays such as
moving the car into position, filling out forms, counting money and pulling away. Thus, each open lane can
accommodate 12 transactions per hour. For the morning shift (7 a.m. – 2 p.m.), one teller can accommodate 168
transactions, a teller working during the midday period (l0 a.m. – 2 p.m.) can process 96 transactions and one
afternoon (2 p.m. – 7 p.m.) teller can handle 120 transactions. Because of the prevailing policy, the two full-time
tellers are scheduled to cover the morning shift each day. The remaining 80 hours of part-time teller capacity is
allocated to the midday and afternoon to create a schedule that has a reasonably consistent percent utilization
across the weekdays. Table 2 shows the proposed teller schedule. Note that 81 part-time teller hours have been
scheduled. In Table 3 the percent utilization for-each period is calculated. It is interesting to note that now
customers may perceive Monday and Thursday afternoons as being busier than Friday.
11-11
Table 2
Monday:
Teller Schedule
Morning
Afternoon
two full-time 7-2
two part-time 2-7
one part-time 10-2
Tuesday:
two full-time
7-2
two part-time 2-7
Wednesday:
two full-time 7-2
one part-time 10-2
three part-time 2-7
Thursday:
two full-time
7-2
two part-time 2-7
Friday:
two full-time 7-2
two part-time 10-2
four part-time 2-7
Total teller hours: 70 full-time and 81 part-time per week
Monday:
Table 3
Transaction Capabilities of Tellers as Scheduled in Table 2
Lanes
Total
Percent
1
2
Open
Capacity
Utilization3
Morning
5.14*
432
44
Afternoon
4.00
240
74
Tuesday:
Morning
Afternoon
4.00
4.00
336
240
49
59
Wednesday:
Morning
Afternoon
5.14*
6.00
432
360
51
60
Thursday:
Morning
Afternoon
4.00
4.00
336
240
54
73
528
480
50
62
Friday:
Morning
6.29*
Afternoon
8.00
1 Lanes Open assumes each teller serves two lanes.
2 Total Capacity is calculated as follows:
(lanes open)  (hours/shift)  (12 transactions/hour/lane)
3 Percent Utilization is calculated as follows:
95 percent demand level/total capacity
*Calculated using a weighted average, because the two full-time tellers work from 7 to 2 and the parttime teller(s) work(s) from 10 to 2.
11-12
CASE: GATEWAY INTERNATIONAL AIRPORT
Questions
1. Assume that you are the assistant to the manager for operations at the FAA. Use the techniques of workshift
scheduling to develop an analysis of the total workforce requirements and days-off schedule.
The most restrictive problem assumes the following constraints:

Each controller must be allowed two consecutive days off.

All controllers must work five 8-hour days.

All controllers on a shift begin work at the same time (i.e., no starting times may overlap.

The shifts run from 00:01 to 08:00, 08:01 to 16:00, and 16:01 to 24:00.

The ratio of take-offs and landings to the number of controllers on duty cannot exceed 16.
Shown below is the number of operations and controller requirements for each hour of each day.
Controller Requirements
Hour
0-1
1-2
2-3
3-4
4-5
5-6
6-7
7-8
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
22-23
23-24
S
6.4
4.
3.2
1.6
2.4
7.2
9.6
16.
20.
26.4
32.8
35.2
27.2
30.4
32.
32.8
33.6
40.8
44.
50.4
28.
19.2
13.6
12.
Hourly Demand Adjusted for Daily Variation
M
T
W
T
F
8.64
8.32
8.
8.32
8.96
5.4
5.2
5.
5.2
5.6
4.32
4.16
4.
4.16
4.48
2.16
2.08
2.
2.08
2.24
3.24
3.12
3.
3.12
3.36
9.72
9.36
9.
9.36
10.08
12.96
12.48
12.
12.48
13.44
21.6
20.8
20.
20.8
22.4
27.
26.
25.
26.
28.
35.64
34.32
33.
34.32
36.96
44.28
42.64
41.
42.64
45.92
47.52
45.76
44.
45.76
49.28
36.72
35.36
34.
35.36
38.08
41.04
39.52
38.
39.52
42.56
43.2
41.6
40.
41.6
44.8
44.28
42.64
41.
42.64
45.92
45.36
43.68
42.
43.68
47.04
55.08
53.04
51.
53.04
57.12
59.4
57.2
55.
57.2
61.6
68.04
65.52
63.
65.52
70.56
37.8
36.4
35.
36.4
39.2
25.92
24.96
24.
24.96
26.88
18.36
17.68
17.
17.68
19.04
16.2
15.6
15.
15.6
16.8
Controllers Required (Integer Value of Demand  16)
S
M
T
W
T
F
S
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
2
3
3
3
3
3
3
3
3
3
3
3
3
4
3
2
3
3
3
3
3
2
2
3
3
3
3
3
3
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
3
3
4
4
4
4
4
4
4
5
5
4
5
5
4
2
3
3
3
3
3
3
2
2
2
2
2
2
2
1
2
2
2
2
2
1
1
2
1
1
1
2
1
S
7.36
4.6
3.68
1.84
2.76
8.28
11.04
18.4
23.
30.36
37.72
40.48
31.28
34.96
36.8
37.72
38.64
46.92
50.6
57.96
32.2
22.08
15.64
13.8
The minimum number of controllers needed for the basic problem is shown below:
Shift
00-08 (night)
08-16 (day)
16-24 (evening)
S
1
3
4
M
2
3
5
T
2
3
5
W
2
3
4
T
2
3
5
F
2
4
5
S
2
3
4
The size of the workforce that is needed to meet the minimum requirements considering the constraints of this
problem is determined using the following integer linear programming (ILP) model:
11-13
Let x j = Number of ATCs assigned to work schedule j
Minimize: x1  x2  x3  x4  x5  x6  x7
Shift 1
8 a.m. - 4 p.m.
Subject
to:
Sun.
Mon.
Tue.
Wed.
Th.
Fri.
Sat.
x 2  x 3  x 4  x5  x 6
x 3  x 4  x5  x 6  x 7
x1 
x 4  x5  x 6  x 7
x1  x2 
x5  x 6  x 7
x1  x2  x3 
x6  x7
x1  x2  x3  x4 
x7
x1  x2  x3  x4  x5
3
3
3
3
3
4
3
Right Hand Sides
Shift 2
Shift 3
4 p.m. - midnight midnight - 8 a.m.
4
5
5
4
5
5
4
Summarized Results for GIA (8am–4pm)
Variables
Obj. Fnctn.
Variables
Obj. Fnctn.
No. Names
Solution
Coefficient
No. Names
Solution
Coefficient
1
x1
+1.0000000 +1.0000000 5
x5
0 +1.0000000
2
x2
+1.0000000 +1.0000000 6
x6
+1.0000000 +1.0000000
3
x3
0 +1.0000000 7
x7
0 +1.0000000
4
x4
+2.0000000 +1.0000000
Minimized OBJ = 5 Iteration = 11 Elapsed CPU seconds = 10.38281
Summarized Results for GIA (4pm–MN)
Variables
Obj. Fnctn.
Variables
Obj. Fnctn.
No. Names
Solution
Coefficient
No. Names
Solution
Coefficient
1
x1
+2.0000000 +1.0000000 5
x5
0 +1.0000000
2
x2
0 +1.0000000 6
x6
+2.0000000 +1.0000000
3
x3
+1.0000000 +1.0000000 7
x7
0 +1.0000000
4
x4
+2.0000000 +1.0000000
Minimized OBJ = 7 Iteration = 9 Elapsed CPU seconds = 8.953125
Summarized Results for GIA (MN–8am)
Variables
Obj. Fnctn.
Variables
Obj. Fnctn.
No. Names
Solution
Coefficient
No. Names
Solution
Coefficient
1
x1
+1.0000000 +1.0000000 5
x5
0 +1.0000000
2
x2
0 +1.0000000 6
x6
+1.0000000 +1.0000000
3
x3
0 +1.0000000 7
x7
0 +1.0000000
4
x4
+1.0000000 +1.0000000
Minimized OBJ = 3 Iteration = 3 Elapsed CPU seconds = 3.234375
11-14
1
2
2
2
2
2
2
Work Schedules (W = workday, X = day off)
8 a.m.-4 p.m. Shift:
S
M
T
W
T
F
S
Jean
X
X
W
W
W
W
W
Laura
W
X
X
W
W
W
W
Patty
W
W
W
X
X
W
W
David M.
W
W
W
X
X
W
W
Chon
W
W
W
W
W
X
X
Staffing
Requirements
Excess
4
3
1
3
3
0
4
3
1
3
3
0
3
3
0
4
4
0
4
3
1
4 p.m.-midnight Shift:
Mario
Yao (Joe)
Shan Ling
Kathy
Jeff
Scott
John
X
X
W
W
W
W
W
X
X
W
W
W
W
W
W
W
X
W
W
W
W
W
W
X
X
X
W
W
W
W
W
X
X
W
W
W
W
W
W
W
X
X
W
W
W
W
W
X
X
Staffing
Requirements
Excess
5
4
1
5
5
0
6
5
1
4
4
0
5
5
0
5
5
0
5
4
1
Midnight-8 a.m. Shift:
Lynda
Claudia
Ron
X
W
W
X
W
W
W
W
W
W
X
W
W
X
W
W
W
X
W
W
X
Staffing
Requirements
Excess
2
1
1
2
2
0
3
2
1
2
2
0
2
2
0
2
2
0
2
2
0
Total Controllers Required: 15
If each shift is started one hour earlier, the workload will be smoothed enough to eliminate one controller's
position. The new shifts begin at 7 a.m. (day), 3 p.m. (evening), and 11 p.m. (night) and yield the following ILP
model:
Let x j
= Number of ATCs assigned to work schedule j
Minimize: x1  x2  x3  x4  x5  x6  x7
11-15
Shift 1
7 a.m. - 3 p.m.
Subject to:
Sun.
Mon.
Tue.
Wed.
Th.
Fri.
Sat.
x 2  x 3  x 4  x5  x 6
x 3  x 4  x5  x 6  x 7
x1 
x 4  x5  x 6  x 7
x1  x2 
x5  x 6  x 7
x1  x2  x3 
x6  x7
x1  x2  x3  x4 
x7
x1  x2  x3  x4  x5
3
3
3
3
3
4
3
Right Hand Sides
Shift 2
Shift 3
3 p.m. - 11 p.m. 11 p.m. - 7 a.m.
4
5
5
4
5
5
4
Summarized Results for GIA (7am–3pm)
Variables
Obj. Fnctn.
Variables
Obj. Fnctn.
No. Names
Solution
Coefficient
No. Names
Solution
Coefficient
1
x1
+1.0000000 +1.0000000 5
x5
0 +1.0000000
2
x2
+1.0000000 +1.0000000 6
x6
+1.0000000 +1.0000000
3
x3
0 +1.0000000 7
x7
0 +1.0000000
4
x4
+2.0000000 +1.0000000
Minimized OBJ = 5 Iteration = 11 Elapsed CPU seconds = 10.32813
Summarized Results for GIA (3pm–11pm)
Variables
Obj. Fnctn.
Variables
Obj. Fnctn.
No. Names
Solution
Coefficient
No. Names
Solution
Coefficient
1
x1
+2.0000000 +1.0000000 5
x5
0 +1.0000000
2
x2
0 +1.0000000 6
x6
+2.0000000 +1.0000000
3
x3
+1.0000000 +1.0000000 7
x7
0 +1.0000000
4
x4
+2.0000000 +1.0000000
Minimized OBJ = 7 Iteration = 9 Elapsed CPU seconds = 9.007813
Summarized Results for GIA (11pm–7am)
Variables
Obj. Fnctn.
Variables
Obj. Fnctn.
No. Names
Solution
Coefficient
No. Names
Solution
Coefficient
1
x1
+1.0000000 +1.0000000 5
x5
0 +1.0000000
2
x2
0 +1.0000000 6
x6
+1.0000000 +1.0000000
3
x3
0 +1.0000000 7
x7
0 +1.0000000
4
x4
+1.0000000 +1.0000000
Minimized OBJ = 2 Iteration = 1 Elapsed CPU seconds = 1.320313
11-16
1
2
1
1
1
2
1
Work Schedules (W = workday, X = day off)
7 a.m.-3 p.m. Shift:
Jean
Laura
Patty
David M.
Chon
S
X
W
W
W
W
M
X
X
W
W
W
T
W
X
W
W
W
W
W
W
X
X
W
T
W
W
X
X
W
F
W
W
W
W
X
S
W
W
W
W
X
Staffing
Requirements
Excess
4
3
1
3
3
0
4
3
1
3
3
0
3
3
0
4
4
0
4
3
1
3 p.m.-11 p.m. Shift:
Mario
Yao (Joe)
Shan Ling
Kathy
Jeff
Scott
John
X
X
W
W
W
W
W
X
X
W
W
W
W
W
W
W
X
W
W
W
W
W
W
X
X
X
W
W
W
W
W
X
X
W
W
W
W
W
W
W
X
X
W
W
W
W
W
X
X
Staffing
Requirements
Excess
5
4
1
5
5
0
6
5
1
4
4
0
5
5
0
5
5
0
5
4
1
11 p.m.-7 a.m. Shift:
Lynda
Ron
W
X
W
W
W
W
X
W
X
W
W
W
W
X
Staffing
Requirements
Excess
1
1
0
2
2
0
2
1
1
1
1
0
1
1
0
2
2
0
1
1
0
Total Controllers Required:
14
2. On the basis of your primary analysis, discuss the potential implications for workforce requirements and daysoff scheduling if assumptions (a) and (b) above are relaxed so that analysis can be based on the hourly demand
without the constraints of a preset number of shifts and no overlapping of shifts. In other words, discuss the
effects of analyzing hourly demand requirements on the basis of each ATC position essentially having its own
shift, which can overlap with any other ATC shift to meet that demand.
Relaxing the shift time constraint:
As shown above, beginning each shift one hour earlier smoothed the workload enough to eliminate one ATC
position. Flexibility in shift times allows a better match between service requirements and employee scheduling.
The most efficient shift times will minimize the number of controllers needed, subject to the
operations/controllers ratio.
11-17
Relaxing the shift demand with no overlap of shifts constraint:
Relaxing the shift profile of demand requirements and allowing shifts to overlap permit better utilization of the
workforce, because the service requirements and workforce capacity can be matched better.
The
operations/controller ratio could be met with a smaller workforce. However, a tradeoff exists between efficient
workforce utilization and scheduling difficulties. In this case, it will become easier to accommodate the general
requirements of the problem, but the effort involved in scheduling the workshifts will increase.
Relaxing other constraints:
Relaxing the days-off constraint would equalize the workload and facilitate the scheduling task. Using part-time
employees or a split-shift schedule can reduce workforce requirements and allow the work schedule to better
match demand. However, implementing these two strategies might be inadvisable because they might have an
adverse effect on employee morale.
3. Do you feel that this would result in a larger or smaller degree of difficulty in meeting the four general
constraints? Why?
Relaxing the shift time constraint seems an attractive option. It improves the match between service requirements
and employee schedules and reduces the number of employees by one. Except for the sensitive issue of
eliminating jobs, this option does not have major drawbacks.
As noted in question 2, relaxing the shift demand and no-shift overlap constraint offers some benefits but the
implementation of this policy requires considerable expertise on the part of the scheduler.
Using part-time employees and split-shift scheduling, on the other hand, could make scheduling easier, but such a
method might have adverse effects on employee morale. Moreover, these alternatives might also have
implications for maintaining a high level of ATC skills.
4. What additional suggestions could you make to the manager of operations to minimize the workforce
requirements level and days-off scheduling difficulty?

Allow workers to choose whether they want consecutive days or non-consecutive days off.

Allow overlapping shift times. This will equalize workloads of all of the controllers, allow smooth shift
changes, and create a better match between workload capacity and requirements.

Decrease the length of workshifts or allow controllers to work split-shifts. For example, if a controller were
to work from 5 to 8 p.m. on Wednesday, the airport would save more than one-half of a day's ATC pay and
still meet FAA requirements.

Consider allowing controllers to work more than five days per week (within the bounds of safety) or fewer
than five days per week (within the bounds of maintaining skills).

Attempt to alter demand. GIA could offer airlines incentives, such as lower take-off/landing fees during offpeak times, to alter timing of their operations.
11-18
CASE: THE YIELD MANAGEMENT ANALYST
[This material is reprinted with permission of Kevin Baker and Robert Freund.]
YIELD MANAGEMENT GAME TALLY SHEET
Booking ID
A (example)
# Passengers
100
# Passengers Carried
Given in Phase III
Revenue
Phase III
Passengers
Gross Revenue
SUBTOTALS
Oversale Adjustment (if over 100)
Spoilage Adjustment (if under 100)
NET REVENUE (Gross Revenue – Adjustment)
Airplane Capacity: 100 seats
Historical Market Information:
Average no-show, misconnect, and cancellation rate: 20 percent
Average revenue per passenger: $400
Spoilage Penalty:
Oversale Penalty:
1-5 seats
6-9
10-15
16+
$200 per empty seat
$200 per passenger
$500 per passenger
$800 per passenger
$1,000 per passenger
Example Oversale Penalty (108 passengers): (5)($200)+(3)($500)=$2500
11-19
After reading the game instructions presented in the textbook and explaining the tally sheet, students should be
ready to begin the game. The game begins with revealing each of the demand opportunities in Phase I in turn, one
at a time, beginning with opportunity (A) and proceeding to the next opportunity on the list only after everyone
has made his and her decision. This demand revelation process is accomplished using the PowerPoint
presentation available on the CD-ROM. Students should make their decisions in ink, because once a passenger
group has been accepted, it is final. Also, students cannot return to an earlier passenger group that has been
previously rejected. Thus, as each group is revealed, the students must make a non-reversible YES or NO
decision to accept or reject that opportunity and be ready to move on without knowledge of the future
opportunities. Students record the group ID and number of passengers in the first two columns of the tally sheet,
keeping in mind a running total of seats sold in order to hold some in reserve for Phase II, but also realizing that
not all passengers will show up.
Phase I: Passenger Demand Received Outside of 13 days Prior to Departure
(A)
(B)
(C)
(D)
(E)
(F)
(G)
(H)
(I)
(J)
(K)
(L)
(M)
(N)
(O)
(P)
100
60
25
5
45
35
70
20
40
80
25
15
10
40
45
20
Sales Representatives - $275 per passenger
Professors - $200 per passenger
MBAs - $150 per passenger
Person Family - $300 per passenger
Conventioneers - $180 per passenger
Conventioneers - $300 per passenger
Sportswriters - $350 per passenger
Person High School Basketball Team - $300 per passenger
High School Band Members - $275 per passenger
Government Officials - $390 per passenger
Marketing Managers - $450 per passenger
Golfers - $325 per passenger
Nurses for Convention - $275 per passenger
Republican Congressmen - $450 per passenger
Club Med Vacationers - $375 per passenger
Shouldice Hernia Patients - $300 per passenger
Phase II: Passenger Demand Between 13 days
Prior to Departure and Day of Departure
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
5
1
4
2
10
8
11
8
2
Business People - $875 per passenger
CEO - $1000 per passenger
IBM Executives - $850 per passenger
Last-minute Persons - $1500 per passenger
Passengers from another airline’s oversold flight - $1000 per passenger
Travel Agents - $875 per passenger
Sales Mangers - $900 per passenger
Passengers from another airline’s oversold flight - $1300 per passenger
CEO’s running up to the plane at departure - $1500 per passenger
11-20
The final stage of the game requires each student to complete the last two columns of the tally sheet, given the
actual passengers who show up for each of the demand groups they accepted (i.e., booking ID in column one). In
Phase III the number of passengers who show up and the revenue contribution are reported, again using the
PowerPoint presentation.
Phase III: Actual Passengers Who Show Up for Flight and Resulting Revenue
(A)
70 Sales Representatives Show Up – $19,250
(B)
50 Professors Show Up – $10,000
(C)
25 Reliable MBAs Show Up – $3,750
(D)
5 Person Family Show Up – $1,500
(E)
35 Conventioneers Show Up – $6,300
(F)
15 Conventioneers Show Up – $4,500
(G)
70 Sportswriters Show Up – $24,500
(H)
20 Person High School Basketball Team Shows Up – $6,000
(I)
25 High School Band Members Show Up – $6,875
(J)
68 Government Officials Show Up – $26,520
(K)
25 Marketing Managers Show Up – $11,250
(L)
10 Golfers Show Up – $3,250
(M
8 Nurses Show Up – $2,200
(N)
30 Republican Congressmen Show Up – $13,500
(O)
30 Club Med Vacationers Show Up – $11,250
(P)
20 Shouldice Hernia Patients Show Up – $6,000
(1)
2 Business People Show Up – $1,750
(2)
1 CEO Shows Up – $1,000
(3)
3 IBM Executives Show Up – $2,550
(4)
2 Last-minute People Show Up – $3,000
(5)
5 Oversold Passengers Show Up – $5,000
(6)
8 Travel Agents Show Up – $7,000
(7)
11 Sales Managers Show Up – $9,900
(8)
4 Oversold Passengers Show Up – $5,200
(9)
2 CEO’s Show Up – $3,000
After Phase III is revealed, the total number of passengers and gross revenue is known. Net revenue is
determined by adjusting for the costs of oversale or spoilage. Upon completion of the game, the instructor
records the total revenue for the flight for each student on the black board. We find a range of $20,000 to $60,000
is not uncommon.
CASE: SEQUOIA AIRLINES
1. For the forecast period (i.e., July - December), determine the number of new trainees who must be hired at the
beginning of each month so that total personnel costs for the flight attendant staff and training program are
minimized. Formulate the problem as a linear programming model and solve.
Sequoia Airlines faces the problem of planning the optimal use of its resources during the next six months. It
must hire new employees and develop them through three stages of training, first as trainees, then as junior flight
attendants, and finally as full flight attendants. Moreover, Sequoia must meet staff requirements for all flights
during the training period. This problem is analyzed using total personnel cost as the selection criterion.
Other resources in addition to the available personnel create constraints that must be considered when hiring and
training new employees. We will solve the problem by using linear programming and then follow up with an
11-21
analysis of the objective function and the constraints and assumptions on which they are based. We will also
offer an interpretation of the results.
Because cost is the only criterion we will use to solve this problem, the objective function is straightforward. One
assumption is noteworthy: we assume that once an experienced attendant becomes an instructor, he or she
continues to receive the instructor's salary even when there is a surplus of instructors who act as attendants. The
objective function, therefore, becomes the sum of the number of employees in each class during each period
multiplied by the appropriate wage.
Minimize:
Z  750T1  T2  T3  T4  T5  T6 
 1050 J 1  J 2  J 3  J 4  J 5  J 6 
 1400F1  F2  F3  F4  F5  F6 
 1500I 1  I 2  I 3  I 4  I 5  I 6 
We must also deal with several resource constraints. First, we must meet the requirement for hours of in-flight
attendant service. We assume that a surplus instructor works the same number of hours as an experienced
attendant does.
140 J1  125F1  125S1  14000
140 J2  125F2  125S2  16000
140 J3  125F3  125S3  13000
140 J4  125F4  125S4  12000
140 J5  125F5  125S5  18000
140 J6  125F6  125S6  20000
11-22
Next, we must satisfy the constraint that requires the total number of hours for junior attendants to be less than or
equal to 25 percent of the total attendant hours in each month.
140 J i  0.25140 J i  125Fi  125S i 
fori  1,2,3,4,5,6
Also, there must be one instructor for every five trainees.
I i  0.2Ti
fori  1,2,3,4,5,6
The remaining constraints deal with the relationships between the status of employees during the current and prior
periods. In this case we make assumptions that pertain to the initial staffing levels for the positions of junior
attendant, full attendant, and instructor.
Ji  0.8Ti 1
Fi  0.95 Ji 1  0.92 Fi 1
Ii  0.95Ii 1  0.005Fi 1
Si  Ii  0.2Ti
For i  1,2,3,4,5,6
Except J1  8, F1  119, I1  6
We see a total of 42 constraints in the following computer model.
11-23
Min 1050J1+ 1050J2+ 1050J3+ 1050J4+ 1050J5+ 1050J6+ 1400F1+ 1400F2+ …
Subject to
(1) 140J1+ 125F1+ 125S1 >= 14000
(2) 140J2+ 125F2+ 125S2 >= 16000
(3) 140J3+ 125F3+ 125S3 >= 13000
(4) 140J4+ 125F4+ 125S4 >= 12000
(5) 140J5+ 125F5+ 125S5 >= 18000
(6) 140J6+ 125F6+ 125S6 >= 20000
(7) 105Jl-31.25Fl-31.25S1 <= 0
(8) 105J2-31.25F2-31.25S2 <= 0
(9) 105J3-31.25F3-31.25S3 <= 0
(10) 105J4-31.25F4-31.25S4 <= 0
(11) 105J5-31.25F5-31.25S5 <= 0
(12) 105J6-31.25F6-31.25S6 <= 0
(13) 1I1-.2T1 >= 0
(14) 112-.2T2 >= 0
(15) 113-.2T3 >= 0
(16) 114-.2T4 >= 0
(17) 115-.2T5 >= 0
(18) 116-.2T6 >= 0
(19) 1J1 = 8
(20) 1F1 = 119
(21) 111 = 6
(22) -111+ 1Sl+ .2T1 = 0
(23) -112+ 1S2+ .2T2 = 0
(24) -113+ 1S3+ .2T3 = 0
(25) -114+ 1S4+ .2T4 = 0
(26) -115+ 1S5+ .2T5 = 0
(27) -116+ 1S6+ .iT6 = 0
(28) 1J2-.8T1 = 0
(29) 1J3-.8T2 = 0
(30) 1J4-.8T3 = 0
(31) 1J5-.8T4 = 0
(32) 1J6-.8T5 = 0
(33) -.95J1-.92F1+ 1F2 = 0
(34) -.95J2-.92F2+ 1F3 = 0
(35) -.95J3-.92F3+ 1F4 = 0
(36) -.95J4-.92F4+ 1F5 = 0
(37) -.95J5-.92F5+ 1F6 = 0
(38) -.005F1-.95I1+ 112 = 0
(39) -.005F2-.9512+ 113 = 0
(40) -.005F3-.9513+ 114 = 0
(41) -.005F4-.9514+ 115 = 0
(42) -.005F5-.9515+ 116 = 0
2. How would you deal with the noninteger results?
The real number value solution for this problem is shown in Table 1. However, it should be noted that Sequoia
could only hire whole people, not fractions of people. Ordinarily we would attempt to apply integer programming
to this problem, but, unfortunately, an integer solution is not possible owing to the large number of constraints
that have decimal parameters. Nevertheless, we can still obtain an integer solution by assigning ranges to some of
the decimal parameters. An example integer solution is presented in Table 2.
11-24
TABLE 1 Sequoia Airlines Real Number Solution
Variables
No. Names
1
J1
2
J2
3
J3
4
J4
5
J5
6
J6
7
F1
8
F2
9
F3
10
F4
11
F5
12
F6
13
I1
14
I2
15
I3
Minimized OBJ
Final Solution for SEQUOIA AIRLINES
Page : 1
Opportunity
Variables
Solution
Cost
No. Names
Solution
+8.0000000
0 16
I4
+6.7955503
+4.1294594
0 17
I5
+6.9693007
0
0 18
I6
+7.1805620
+18.374796
0 19
S1
+4.9676352
+27.182201
0 20
S2
+6.2950001
+21.434549
0 21
S3
+1.9719510
+119.00001
0 22
S4
0
+117.08000
0 23
S5
+1.6106634
+111.63660
0 24
S6
+7.1805620
+102.70567
0 25
T1
+5.1618242
+111.94527
0 26
T2
0
+128.81274
0 27
T3
+22.968494
+6.0000000
0 28
T4
+33.977749
+6.2950001
0 29
T5
+26.793188
+6.5656500
0 30
0
T6
= 1177115 Iteration = 35 Elapsed CPU seconds = 16.85938
Opportunity
Cost
0
0
0
0
0
0
0
0
0
0
+1686.3875
0
0
0
+1221.3268
Variables
No. Names
31
S1
32
A1
33
S2
34
A2
35
S3
36
A3
37
S4
38
A4
39
S5
40
A5
41
S6
42
A6
43
S7
44
S8
45
S9
Minimized OBJ
Final Solution for SEQUOIA AIRLINES
Page : 2
Opportunity
Variables
Solution
Cost
No. Names
Solution
+2615.9539
0 46
S10
+1280.1985
0
0 47
S11
+694.49170
0
+20.099571 48
S12
+1999.1627
0
–20.099571 49
S13
+4.9676352
+1201.0670
0 50
S14
+6.2950001
0
0 51
S15
+1.9719510
+3410.6787
0 52
S16
0
0
0 53
S17
+1.6106634
0
+20.861754 54
S18
+7.1805620
0
–20.861754 55
T19
+5.1618242
0
+18.853069 56
T2O
0
0
–18.853069 57
T21
0
+3033.9885
0 58
T22
0
+3421.8755
0 59
T23
0
+3550.2668
0 60
0
T24
= 1177115 Iteration = 35 Elapsed CPU seconds = 16.85938
Opportunity
Cost
0
0
0
0
0
0
+7367.7900
0
0
–725.91003
–1053.4916
+4704.0610
0
+2512.4463
0
11-25
Final Solution for SEQUOIA AIRLINES
Page : 3
Variables
Opportunity
Variables
No. Names
Solution
Cost
No. Names
Solution
61
A25
0
0 70
A34
0
62
A26
0
+2607.7190 71
A35
0
63
A27
0
+2356.6335 72
A36
0
64
A28
0
+937.50000 73
A37
0
65
A29
0
–542.37274 74
A38
0
66
A30
0
+937.50000 75
A39
0
67
A31
0
+2779.4475 76
A40
0
68
A32
0
+1589.4298 77
A41
0
69
A33
0
+341.14731 78
0
A42
Minimized OBJ = 1177115 Iteration = 35 Elapsed CPU seconds = 16.85938
J1
J2
J3
J4
J5
J6
= 8
= 0
= 0
= 0
= 32
= 13
F1
F2
F3
F4
F5
F6
Opportunity
Cost
–869.93689
+534.34448
+2092.1052
+956.63367
+6530.5908
+5808.5732
+7693.2349
+1921.5211
+856.63373
TABLE 2 Sequoia Airlines Integer Solution
= 119
I1 = 6
T1 = 0
S1 = 6
= 122
I2 = 6
T2 = 0
S2 = 6
= 104
I3 = 6
T3 = 0
S3 = 6
= 102
I4 = 8
T4 = 36
S4 = 0
= 102
I5 = 10
T5 = 15
S5 = 7
= 134
I6 = 12
T6 = 0
S6 = 12
Note that the integer solution is radically different from the continuous solution. This difference is explained by
the large ranges given to the constraint parameters. Which solution should be used? Perhaps rounding up the
continuous solution and then adjusting for constraint requirements would provide the best solution. In fact, this
solution is preferable because the integer solution implicitly assumes that management can control the number of
employees who quit each month which is clearly not the case. In order to use the integer solution, we might first
use forecasting methods to estimate the number of employees who will quit and then use integer programming to
plan for hiring new employees.
3. Discuss how you would use the LP model to make your hiring decision for the next six months.
Following the hiring decisions for the current month (July in this case), the program can be run again at a later
date by dropping the July requirements and adding the January requirements. This "leap frog" approach may be
used to fine tune proposed hiring levels by extending the planning horizon for another six months. The decision
on future hiring (July through December in this case) is postponed until a commitment must be made. At that
time the decision can be based on a six-month projection of requirements.
11-26