Charter vs. Scheduled Airlines
by
Gautam Gupta
B.Tech. (Indian Institute of Technology, Bombay) 2003
M.S. (University of California, Berkeley) 2004
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Engineering-Civil and Environmental Engineering
in the
Graduate Division
of the
University of California, Berkeley
Committee in charge:
Professor Mark Hansen, Chair
Professor Alper Atamturk
Professor Carlos Daganzo
Professor Pravin Varaiya
Fall 2008
i
ii
Contents ........................................................................................................................ ii
List of Tables .............................................................................................................. vii
List of Figures ................................................................................................................x
Acknowledgements ..................................................................................................... xii
Abstract ..........................................................................................................................1
1. Introduction ..............................................................................................................3
1.1 Brief History of Commercial Aviation in USA ...................................................4
1.1.1 Early History: The Regulation Era ........................................................4
1.1.2 Deregulation and Beyond .....................................................................8
1.2 Scheduled Airline Characteristics ..................................................................... 10
1.3 Alternative Service Concepts and Group Demand ............................................ 11
2. Mathematical Models of Competition over a Single Link ..................................... 13
2.1 Assumptions and Profit Functions .................................................................... 15
2.1.1 Market Setting .................................................................................... 15
2.1.2 Cost functions .................................................................................... 16
2.1.3 Scheduled Service Frequency ............................................................. 17
2.1.4 Individual and Group Demand ............................................................ 17
iii
2.1.5 Group Desired Departure Time and Service Choice ............................ 19
2.1.6 Profit Functions .................................................................................. 20
2.2 Models of Competition and Possible Scenarios ................................................ 22
2.2.1 Simultaneous Game and Leader-Follower Game ................................ 23
2.2.2 Scenarios of Competition ................................................................... 25
2.2.3 Interior Equilibrium and Stability ....................................................... 26
2.3 Analysis of Different Scenarios ........................................................................ 27
2.3.1 Scenario 1: SS Serves Individuals and Groups, CS Absent ................. 27
2.3.2 Scenario 2: SS Monopolistic Behavior, Optimal CS Prices ................. 30
2.3.3 Scenario 3(a): Single SS Price for Individuals and Groups, Unchanged
Frequency; Optimal CS Prices ............................................................ 30
2.3.4 Scenario 3(b): Distinct SS Individual and Group Prices, Unchanged
Frequency; Optimal CS Prices ............................................................ 34
2.3.5 Scenario 4: Optimal SS Frequency and Prices; Optimal CS Prices ...... 37
3. Numerical Illustration of Competition Models over a Single Link ....................... 41
3.1 Setting and Parameter Values ........................................................................... 42
3.1.1 Individual Demand Parameters ........................................................... 42
3.1.2 Group Demand Parameters ................................................................. 44
3.1.3 Scheduled Service Operating Costs .................................................... 45
3.1.4 Charter Operating Cost ....................................................................... 48
3.2 Analysis ........................................................................................................... 49
3.2.1 Preliminary Analysis and Notation ..................................................... 50
3.2.2 Results Assuming no Additional Charter Benefit (θ = 0) .................... 52
iv
3.2.3 Results Assuming Charter Benefit (θ > 0) .......................................... 63
3.3 Conclusions ..................................................................................................... 71
4. Charter Strategic Planning over a Large Network ................................................ 72
4.1 Introduction ..................................................................................................... 72
4.1.1 Scheduled Airline Planning and Airline Fleet Assignment .................. 72
4.1.2 Problem Statement ............................................................................. 73
4.2 Model Inputs .................................................................................................... 74
4.2 Basic Mathematical Formulation ...................................................................... 77
4.4 Optional Constraints and Considerations .......................................................... 83
4.5 Computational Experiments and Improvements ............................................... 84
4.5.1 Computational Experiments on Basic Formulation ............................. 84
4.5.2 Valid Inequalities ............................................................................... 86
4.5.3 Computational Experiments with Valid Inequalities ........................... 90
5. Case Study for Charter Strategic Planning ........................................................... 95
5.1 Introduction: Charter Service for Student Athlete Travel .................................. 95
5.2 Big Sky Conference, Charter Costs and Scheduled Costs ................................. 98
5.2.1 Event Data, Resulting Demand and Demand Variation ..................... 100
5.2.2 Team Travel Needs .......................................................................... 103
5.2.3 Scheduled Service and Charter Service Assumptions ........................ 104
5.2.4 Value of Time and Charter Fleet Size ............................................... 106
5.3 Analysis and Results ...................................................................................... 107
5.3.1 Charter Market Penetration and Time Savings .................................. 107
5.3.2 Best Fleet Size .................................................................................. 108
v
5.3.3 Cost Components and Comparison with All-Scheduled Case............ 110
5.3.4 Sensitivity to Base of Operations ...................................................... 112
5.4 Variation and Distribution of Charter Benefits ............................................... 114
5.4.1 Weekly Variation in Time Savings and Demand Fraction Served by
Charter ............................................................................................. 115
5.4.2 Variation across Schools .................................................................. 117
5.4.3 Variation across Sports ..................................................................... 119
5.5 Conclusions ................................................................................................... 121
6. Conclusions............................................................................................................ 123
Bibliography .............................................................................................................. 126
Appendix 1: Properties of Charter and Scheduled Profit Functions ................ 134
A1.1 Charter Profit Function ............................................................................. 134
A1.2 Scheduled Profit Function ........................................................................ 135
Appendix 2: Mathematical Derivations of Equilibrium in Chapter 2 ............... 138
A2.1 Scenario 1: SS Serves Individuals and Groups, CS absent ........................ 138
A2.2 Scenario 2: Optimal CS Prices, no SS response ........................................ 140
A2.3 Scenario 3(a): Single SS Price for Individuals and Groups, Unchanged
Frequency; Optimal CS Prices................................................................. 143
A2.3.1 Simultaneous Game ......................................................................... 143
A2.3.2 Leader-Follower Game .................................................................... 145
A2.4 Scenario 3(b): Distinct SS Individual and Group Prices, Unchanged
Frequency; Optimal CS Prices................................................................. 147
A2.4.1 Simultaneous Game ......................................................................... 147
vi
A2.4.2 Leader-Follower Game .................................................................... 150
A2.4.3 Comparing SS Profit in Simultaneous and Leader-Follower Game .. 152
A2.5 Scenario 4: Optimal SS Frequency and Prices; Optimal CS Prices ............ 154
A2.5.1 Simultaneous Game ......................................................................... 154
A2.5.2 Leader-Follower Game .................................................................... 154
Appendix 3: Individual Value of Time in Demand Function ............................. 156
vii
Table 3.1: Individual demand parameters ....................................................................... 43
Table 3.2: Group demand parameters ............................................................................. 45
Table 3.3: Parameters used in determining SS operating costs ....................................... 47
Table 3.4: Linear SS operating cost values for different stage lengths ............................ 48
Table 3.5: CS operating costs......................................................................................... 49
Table 3.6: Maximum SS frequency for non-zero CS market share ................................. 51
Table 3.7: Results for short-haul, low-density corridor without charter benefit ............... 57
Table 3.8: Results for short-haul, high-density corridor without charter benefit .............. 58
Table 3.9: Results for medium-haul, low-density corridor without charter benefit .......... 59
Table 3.10: Results for medium-haul, high-density corridor without charter benefit ....... 60
Table 3.11: Results for long-haul, low-density corridor without charter benefit .............. 61
Table 3.12: Results for long-haul, high-density corridor without charter benefit ............ 62
Table 3.13: Results for short-haul, low-density corridor including additional charter
benefit .................................................................................................................. 65
Table 3.14: Results for short-haul, high-density corridor including additional charter
benefit .................................................................................................................. 66
viii
Table 3.15: Results for medium-haul, low-density corridor including additional charter
benefit .................................................................................................................. 67
Table 3.16: Results for medium-haul, high-density corridor including additional charter
benefit .................................................................................................................. 68
Table 3.17: Results for long-haul, low-density corridor including additional charter
benefit .................................................................................................................. 69
Table 3.18: Results for long-haul, high-density corridor including additional charter
benefit .................................................................................................................. 70
Table 4.1: Computational experiments on the basic formulation .................................... 85
Table 4.2: Comparison of LP Relaxation with and without valid inequalities ................. 92
Table 4.3: Comparison of solution times with and without valid inequalities ................. 93
Table 5.1: Universities in the NCAA Big Sky Conference ............................................. 99
Table 5.2: Sports in the NCAA Big Sky Conference ...................................................... 99
Table 5.3: Flights demanded across origin and destination ........................................... 103
Table 5.4: Cost and time assumptions for the scheduled service ................................... 105
Table 5.5: Cost and time assumptions for the charter service ....................................... 106
Table 5.6: Time savings and percentage demand served by charter for different
operational configurations .................................................................................. 108
Table 5.7(a): Expenditure and components for the two optimal charter fleet
configurations and comparison with existing scheduled service (values in 1000$)
........................................................................................................................... 111
Table 5.7(b): Reduction in cost from using optimal charter configurations as compared to
existing scheduled service options (values in 1000$) .......................................... 111
ix
Table 5.8: Cost comparison for the best and worst location for operational base .......... 114
Table 5.7: Total movements split over flying team and venue ...................................... 117
Table 5.8: Movements served by charter with 1 aircraft and $3/pr-hr split over team and
venue .................................................................................................................. 118
Table 5.9: Movements served by charter with 2 aircraft and $30/pr-hr split over team and
venue .................................................................................................................. 118
Table 5.10: Total movements served by charter for different schools in eight
configurations ..................................................................................................... 119
x
Figure 3.1: Operating cost for 300 mile stage length using Cobb-Douglas form and its
linear approximation ............................................................................................. 48
Figure 3.2: Charter market share for low-density markets for different scenarios in the
simultaneous game, assuming no additional charter benefit .................................. 53
Figure 3.3: Frequency change in scenario 4 for different groups in short-haul, low density
corridor ................................................................................................................ 54
Figure 4.1: Departure time and associated penalties ....................................................... 76
Figure 5.1: Location of the nine Big Sky schools ......................................................... 100
Figure 5.2: Variation in team movements over different weeks .................................... 102
Figure 5.3: Total expenditure (time and money) for the eight configurations ................ 109
Figure 5.4: Total expenditure for different operational bases and charter fleet with value
of time $3/pr-hr .................................................................................................. 113
Figure 5.5: Total expenditure for different operational bases and charter fleet with value
of time $30/pr-hr ................................................................................................ 113
Figure 5.6: Weekly variation in demand served and time savings with 1 charter aircraft
and VOT as $3/pr-hr ........................................................................................... 115
xi
Figure 5.7: Weekly variation in demand served and time savings with 2 charter aircraft
and VOT as $30/pr-hr ......................................................................................... 116
Figure 5.8: Time savings per person per game across sport. ......................................... 120
Figure 5.9: Total cost of travel per game and ratio of charter to scheduled spending across
sport for one aircraft and value of time $3/hour. ................................................. 121
xii
Ph. D. has been an immense learning and growing experience for me. Besides new
concepts, I have learnt a lot about myself, and have found many changes in me (hopefully
for the best).
Professor Mark Hansen, the committee chair, supported this throughout. I must say
I was a brat when I joined the program. With hook, crook and example, he has taught me
to think. He has led me with amazing patience through all phases of sloth and immaturity.
Besides being the thesis advisor, he has been a friend, philosopher and guide throughout
my stay at Berkeley. Perhaps the most interesting aspect has been his sense of humor,
part of which I think has rubbed on to me. He can count on me to keep bugging him for
advice for a long time.
The faculty at Berkeley is fantastic. Special thanks goes to all the ITS faculty. The
dissertation committee has been very supportive and encouraging. Prof Atamturk’s
course on computational optimization has been a big help in this work, and in other
things that I have done and am doing after Ph. D. Prof Daganzo’s emphasis on simplicity
of solution while maintaining mathematical elegance was useful in many places. Prof
Varaiya’s comments on grounding mathematical results in the real world during one of
xiii
the discussions are directly responsible for one chapter of this work. Besides the
committee, special thanks go to Prof Madanat. It was a pleasure being his GSI in CE-252.
I learnt a lot about effective presentation observing him teach, and will be using it for
many years to come.
I have made many friends while my stay at Berkeley. Shankar Bhamidi, Nikolas
Geroliminis and Vishnu Narayanan deserve a special mention, for all the fun times,
discussions and mischief. Avijit Mukherjee deserves a mention for getting me interested
in a doctorate, and for all the laughs. A lot of the work here started with my interaction
with Anne Goodchild, and I am grateful for all her help. A whole set of ITS and
NEXTOR students gave valuable feedback, were appreciative spectators, or were just fun
being around.
My family deserves a special mention. My two brothers have always been co-
conspirators in all the mischief. My guru, Gurbax sir, has been a pillar of light, guiding
me throughout. Last and foremost, my parents have been laying a foundation for all this
for many years. I have nothing but pride and gratitude for it.
1
Abstract
Charter vs. Scheduled Airlines
by
Gautam Gupta
Doctor of Philosophy in Engineering – Civil and Environmental Engineering
University of California, Berkeley
Professor Mark Hansen, Chair
Scheduled airlines cause delays to passengers due to compromises with the schedule.
Poor service to the pertinent region coupled with high passenger value of time may
warrant the use of a non-scheduled airline, particularly the use of a charter airline by a
group of passengers. Recent advances in communication technology coupled with recent
introduction of very light jet aircraft could further fuel the use of charter service.
The entry of such a non-scheduled airline would result in a price and/or frequency
competition with existing service. Further, the competition would be at the network level.
This research analyzes the competition between the charter service and scheduled service,
and develops a model for charter strategic planning in light of such competition.
To analyze competition, we consider a market composed of individual and group
passengers. The charter service competes by setting prices for various groups, and
scheduled service sets the price and frequency. Using simple forms of demand and cost
models, we analyze different scenarios of change in price and frequency for a
simultaneous equilibrium and leader-follower equilibrium. The analysis is over a
2
generalized setting of various groups with varying sizes and value of time. The analytical
expressions are followed by a numerical treatment for a variety of cases, and the results
show that charter is successful when the group sizes are large, and the entry of charter
service benefits both the individual and group passengers.
We develop a mixed integer linear program for charter strategic planning over a
network. In light of competition with the scheduled service, the model selects the most
profitable or beneficial group-movements over a network, and assigns a limited charter
fleet to them. The scheduling of charter group-movements is done over continuous and
penalized time windows. The computational aspects of the model are tested and the
solution time is reasonable for real-world problems. The model is used to explore the
benefits of a charter service to student athlete travel. The time savings are substantial
with little change in dollar cost of travel.
______________________________________
Mark Hansen, Chair
3
Scheduled airlines are public transport systems whose business model is based on
common carriage, published schedules, and published (if dynamic) fares. By their very
definition, these airlines cause delays to passengers. Besides the delays associated with
deviation from schedule, very often the desired departure time for the passenger does not
coincide with the departure time of scheduled airlines, and the passenger is forced to
compromise with the existing schedule. Such delays become even more prominent when
the frequency of service between the desired origin destination pair is low. If the value of
delay is sufficiently high for the passengers, it may warrant the introduction of un-
scheduled service. There are instances where individual passenger’s value of delay is
sufficiently high (executive travelers), and passengers flying in groups would definitely
have a higher value of delay due to accumulation. With advances in communication
technology, regional jets (RJ’s) and the introduction of the very light jets (VLJ’s), ―on-
demand‖ air travel could be a profitable enterprise. Recently there have been studies on
certain models of on-demand air travel, for example dial-a-flight service (Espinoza et al
2008a, 2008b).
4
The entry of such on-demand service in any market would result in price and/or
frequency competition with scheduled service. Further, the competition would occur at a
network level. The goal of this research is to analyze the competition between a charter
service and scheduled service, and to analyze the subsequent effect on charter planning.
The difference between charter service and other on-demand services is that a single
passenger group uses the aircraft for a flight, and there is no grouping of passengers
based on origin and destination. A review of existing literature reveals that there has been
a lot of work in analyzing scheduled airline competition, scheduled and charter airline
operations, but there is no evidence of work (in the open domain) on analyzing charter
and scheduled airline competition.
To motivate the problem further, the next section briefly describes the growth of
commercial aviation in US, and the role played by un-scheduled airlines. Next, we review
existing scheduled airline service, indicating the opportunities for charter operation. We
then review existing literature on alternative service concepts, particularly for charter and
group travel. At the end, we briefly describe the contents and structure of the rest of
document.
Regularly scheduled air services were offered in the US for the first time in 1918. The
Post Office, through the development of airmail, was directly responsible for the
beginning of commercial air transportation. The first important act with regard to airlines,
the Airmail Act (also known as the Kelly Mail Act) was passed in 1925. It required that
5
award of airmail contract be made by competitive bidding, and provided subsidies for
commercial airmail. The aim was to encourage commercial aviation and to transfer
airmail operations to private carriers. As a result, the mail service mileage increased
rapidly. However, carriers concentrated on airmail service, and were not particularly
interested in providing passenger services.
The Air Commerce Act, designed to encourage passenger service, was passed in
1926. It initiated the development of civil airways, navigational aids and provided for the
regulation of safety by federal government. The underlying basis was the idea of having a
stable and viable airline industry.
The Airmail Act (McNary-Watres Act) was passed in 1930 in order to unify the
industry. It gave the Post Master General (PMG) wide ranging powers, like granting
contracts without competitive bidding. The mail payment rates were changed from
weight-based to volume-based rates. As a result, the airlines started acquiring larger
planes, giving a boost to passenger transportations. Walter Brown, the then PMG,
initiated this act, and his four year tenure was marked by mergers and consolidation of
airlines. Most of the airlines of today emerged during this period. The Air Mail Act
(Black-McKeller Act) of 1934 reversed the contracting procedure to competitive bidding.
It also required the separation of airline carriers and aircraft manufacturers.
The airline industry was hit hard by the Great Depression. With the resumption of
competitive bidding, some airlines submitted very low bids. The incomes of several
airlines fell dramatically. It was clear that airmail contract were not enough for survival,
so most of the airlines started developing their passenger traffic. By December 1936, the
income from passengers overtook the income from airmail services (Sinha, 2001).
6
After a long legislative history, the Civil Aeronautics Act was passed in 1938. This
act, along with some administrative changes in 1940, established the Civil Aeronautics
Board (CAB) was as an independent agency, giving it the responsibility of conducting the
following activities (Richmond, 1961):
Control of entry of new carriers into the industry, and entry of existing carriers
into new or existing routes
Control of exit by requiring approval of the Board before a carrier’s abandonment
of service to a point or on the route
Regulation of fares on every route
Fix and award subsidies, control mergers and eliminate rate discrimination and
unfair competition, or unfair and deceptive practices in air transportation
The Civil Aeronautics Act had far-reaching effects on the commercial aviation in the
USA. In most cases, only one or two airlines were allowed to serve a particular route.
Prices tended to be high and to increase over time, since CAB permitted increased costs
to be passed along in higher fares. This, coupled with the relative absence of competition,
acted as disincentives for airlines to seek out ways to reduce costs. Under regulation,
efforts were made to ensure that no airline ever went out of business. The ability to pass
on costs via high fares allowed inefficient work rules.
Initially, the non-scheduled airlines were exempted from having to obtain
certificates in 1938. At that time, the airlines were small with limited resources and the
opportunities for acquiring load-carrying transport aircraft were small. However, this
changed after World War II. The cessation of hostilities released many surplus aircrafts
and thousands of aircrew, and since no operating certificate was needed for contract
7
services, many new companies sprang up. As a result, in 1946 CAB modified the
exemption regulation of 1938, and required the non-scheduled carriers to obtain a Letter
of Registration from the Board. Further, in 1948, the Board stopped issuing more Letters
of Registration beyond the 142 listed at that time. It also ruled that non-scheduled
operations would be limited to eight to twelve flights per month between the same two
points. In 1949, regulations were further tightened by removing blanket exemption in
favor of individual exemption (Davies (1982)). The non-scheduled airlines made
significant contributions during times of national crisis. During the Berlin airlift between
July 1948 and August 1949, the Large Irregular Carriers moved 25 percent of the
passengers and 57 percent of the cargo tonnage. During the Korean War in the early
1950’s, they flew 50 percent of the total commercial airlift. In 1955, the Board concluded
that the non-scheduled air carriers were performing a useful service to the public, and that
henceforth they would be known as Supplemental Air Carriers. They were allowed
unlimited charter business, and regularly scheduled individual services up to a maximum
of ten one-way flights per month between any two points in the US (later on extended to
foreign points). In 1962, Congress enacted Public Law 87-528 that terminated the
participation of Supplementals in individually ticketed service. Further, the Department
of Defense announced that in 1964 that for the fiscal year 1966, contracts would be
granted only to carriers deriving 30 percent of their revenue from commercial sources.
Even though the growth of Supplemental Carriers was phenomenal post-1962 (revenues
doubled from 1962 in 1966 helped by the Vietnam airlift in 1965). Further, in the late
1960’s and early 1970’s, a loophole in the Supplemental regulations allowed affinity
groups to charter commercial aircraft to Europe. This led to formation of many ―phony‖
8
affinity groups to take advantage of the loophole and get rates below those charged by
scheduled airlines. The CAB crackdown on such affinity groups was one of the many
precipitating factors leading to airline deregulation.
In October 1974, CAB prepared a study of the domestic route system, and found the
overall level of service was excessive in relation to demand. Various other studies
brought out the inefficiencies of regulation (Sinha, 2001). Further, the Ford
administration established a National Commission on Regulatory Reform in 1974.
Various such factors led to the passing of the Airline Deregulation Act (ADA) on 24
October 1978, which phased out CAB controls on routes and pricing, and eventually the
CAB itself.
During the first 10 years of deregulation, the major airlines shifted from point-to-
point system to hub-spoke-system. Hubbing led to increased service for those living in
hub cities, and also gave the people living at spokes access to large number of
destinations at the expense of a little point-to-point service. But growing congestion at the
major hub airports created opportunities for alternatives, and led to the return of point-to-
point model in the form of low-fare, no-frills airlines. Southwest airlines is probably the
biggest success story of this model, and carved out a thriving niche for the point-to-point
service model. Aggressive pricing also expanded the market.
In 1997, a new type of small jet airliner, called the ―regional jet‖ (RJ) began
entering into service. These were 30 to 70 seat aircraft jet aircraft (Poole and Butler,
1999), and became highly popular because of 2 reasons:
9
It was preferred to the small turboprop aircraft by the air travelers
They had a low seat-mile cost for medium length routes, enabling them to support
a modest number of passengers.
The introduction of RJ’s was advantageous for both the hub-and-spoke system as well as
the point-to-point system. A RJ could serve as a feeder to a hub, enabling more-frequent
service to existing spokes, and the addition to new spokes. RJ’s also enabled new markets
for the point-to-point service. Consequently, the aircraft manufacturers estimated a large
market for such RJ’s. Fairchild Dornier estimated an additional US market for over 400
30-seat RJ’s (Poole and Butler, 1999). But the carnage and uncertainty in the airline
industry post September 11 had a tremendous effect on the ―RJ boom‖ as well (for
example, Fairchild Dornier went bankrupt in April 2002).
With the recent introduction of the very light jet aircraft (VLJ), on-demand air taxi
service is becoming a reality. VLJ’s are small jet aircraft approved for single-pilot
operation, with a maximum take-off weight of under 10,000 lb. VLJs are intended to
have lower operating costs than conventional jets. A significant example of use of VLJ
for a ―non-scheduled‖ airline is DayJet (DayJet, 2008), which planned to offer on-
demand air travel service using the Eclipse 500 jet, but has recently suspended operations
citing the current economic downturn.
Post deregulation, activity in the non-scheduled sector has been dormant as
compared to the pre-deregulation era of affinity groups and ―nonskeds‖. However, the
introduction of new aircraft is definitely a factor that could lead to the revival of such
models. Yet another factor is the increased online social networking. The primary driving
force behind nonskeds in the pre-deregulation period was the formation of affinity groups
10
seeking to minimize travel cost by avoiding mandated scheduled airline prices. In the
post-deregulation period, formation of affinity groups looking for reduced travel time
could be seeded by online networking. Further, characteristics of existing scheduled
service, as described in the next section, bodes well for different service models in certain
regions.
Scheduled airlines are public transport systems whose business model is based on
common carriage, published schedules, and published (if dynamic) fares. Airline
networks feature link economies of scale and stage length (Hansen,1990; Wei and
Hansen, 2001) and firm level economies of scope that have led the predominant carriers
to establish national or international-scale networks featuring multiple hubs (Bania et al.,
1998; Gillen and Morrison, 2005). These networks afford excellent service for regions
served by hub airports but force passengers traveling in thinner markets to take
connecting service. Such service was the only alternative available to 70 percent of O&D
markets in 2005 (Government Accounting Office, 2006). The same forces have led to
reduced service to smaller communities who, despite government subsidy, saw a 17
percent reduction in flights between 2000 and 2005 (General Accounting Office, 2006).
In addition the low profit potential of small community services, stronger safety
regulation for commuter aircraft and increased post-9/11 security requirements at smaller
airports have also contributed to this trend (General Accounting Office, 2007). Such
trends point to the possibility of charter success in serving regional market groups, such
11
as college athletes (colleges and universities, for a variety of economic and historical
reasons, are often located in rural or semi-rural locations).
The recent emergence of low cost carriers, whose networks feature point-to-point
services rather than hubs (Gillen and Morrison, 2005), has done little to improve service
to such regions. Low cost carriers often serve secondary airports, but ones that are
proximate to large urban areas. The pressure on yields resulting from low cost airlines
may have caused legacy airlines to cross-subsidize markets in which they face low cost
competition by raising fares in others. An analysis by the General Accountability Office
found that markets in the lowest of five passenger traffic volume categories were the only
ones that in which fares increased between 1998 and 2005. O&D markets in this lowest
traffic category represent 85 percent of the total in which there is measurable traffic,
although they account for just 1/5 of the total passenger traffic volume (General
Accountability Office, 2006).
While the deficiencies of scheduled airline service to non-urban areas have been studied
for some time, the potential of alternative service concepts to compete for this niche has
only recently received attention. Espinoza et al (2008a, 2008b), analyze per-seat, on-
demand services using very light jets. This service is clearly targeted at individuals and
infeasible for group travel. Other studies do consider the special issues associated large
group air travel, but focus on pricing (Goel and Haghani, 2000) and yield management
(Svrek, 1991) decisions of scheduled airlines.
12
Charter airline services have also been considered in previous work, but in applications
very different from the one considered here. Kim and Barnhardt (2005) consider use of a
charter to serve a market featuring day-to-day variation in O-D travel demand consisting
of passengers with different levels of price sensitivity. Other studies examine pricing of
(Bishop and Thompson, 1992) and demand for (Karlaftis and Papastrvrou, 1998)
international charters, which traditionally target price-conscious leisure travelers
traveling in group tours or individually.
The rise of new business models in aviation, introduction of new aircraft and
characteristics of existing scheduled service plead the case for analyzing the competition
between an un-scheduled or charter operator and a scheduled operator. As stated before,
this competition could be in price and/or frequency. But any change in frequency would
have a cascading effect on operations over the network (in terms of fleet assignment,
routing, crew scheduling etc). This translates into increasing complexity with increasing
network size. Thus, based on network size, this research is divided into two parts: single
link models in chapter 2 and 3, and large network models in chapter 4 and 5.
In chapter 2, we develop models of competition between a charter and scheduled
airline. These models are generic and different scenarios are considered, and chapter 3
includes a numerical treatment of these models over a variety of settings. In chapter 4, we
develop a mixed integer program for charter strategic planning over a network, and
briefly discuss the computational aspects of this mixed integer program. We then
illustrate the model with a case study based on real data in chapter 5.
13
Our study addresses the competition between scheduled and charter airline service, and
this form of competition that has not been considered previously. There is, however,
considerable literature on other forms of competition involving airlines. Most models
allocate demand between competing services using a random utility framework, in which
utility is systematically related to fare, service frequency, flight time, and service
directness. Hansen (1990) develops a model for airline hub competition between hub
carriers with different hub airports as well as point-to-point carriers. Dobson and Lederer
(1993) address the problem of competitive choice of flight schedules and route prices by
airlines operating in a hub and spoke configuration, employing random utility models that
take into account travelers desired departure times. Adler (2001, 2005) models
competition involving hub-and-spoke networks and its implication for optimal network
design. Pels et al (1999) investigate airport and airline competition in a metropolitan area
14
with multiple departure airports, Wei and Hansen (2006) analyze an airline’s choices of
aircraft size and frequency in duopoly markets between city pairs, while Inzerilli and
Jara-Diaz (1993) introduce a microeconomic approach to model and analyze the price-
capacity combinations that maximize welfare in the operation of an airline facing modal
competition (train, car etc) in the presence of a random total demand.
Other researchers employ models that allocate demand deterministically, based on
similar factors as the random utility models but building from an assumed distribution of
desired departure times and a resulting ―schedule delay‖ (Douglass and Miller, 1974)
arising from the difference between desired time and when a flight is available. Schipper
et al (2006) use this approach to model airline competition as a two-stage game in
frequency and prices. Alderighi et al (2005) analyze network competition on a simplified
4-node network, identifying cases when a hub-and-spoke system is preferred and over a
point-to-point system and vice versa. Borenstein and Netz (1999) empirically assess the
role of schedule delay in airline competitive behavior by comparing flight schedule
differentiation using US airline departure times from 1976 (when the fares were
regulated) and 1986 (when fares were not regulated). Hendricks et al. (1999) consider
monopoly and duopoly markets, and show that if carriers compete aggressively, e.g.
Bertrand-like behavior, one carrier operating a single hub-spoke network is an
equilibrium outcome. Competing hub-spoke networks are not an equilibrium outcome,
although duopoly equilibrium in non-hub networks can exist. If carriers do not compete
aggressively, an equilibrium with competing hub-spoke networks exists as long as the
number of cities is not too small. They provide conditions under which all equilibrium
consists of hub-spoke networks.
15
Although airline competition has been extensively studied for the scheduled service
case, there is a lack of literature on charter and scheduled service competition. To this
end, we develop mathematical models for the competition between a charter service (CS)
and a scheduled service (SS). In the first section, we detail the various demand and cost
assumptions and the appropriate mathematical functions. We also describe the
operational setup, including arrival of passengers and flight departure, and define the
expected profit function for both CS and SS. These profit functions can be used to
analyze a variety of possible scenarios under different models of competition, and in
section 2.2 we describe models of competition and the possible scenarios that are
analyzed, and comment on the complexity of analysis of each scenario. Section 2.3 gives
a mathematical description of the equilibrium for each scenario, and describes the
properties of the equilibrium in each case. The detailed derivations of the mathematical
expressions for each scenario are provided in Appendix 2.
We consider a market composed of individual passengers and groups, where SS has a
monopoly on individual travel, and there is a duopoly for group travel with both SS and
CS offering flights to groups. Even though the competition is for group travel, any
changes in frequency by SS to accommodate the group travel will have an effect on its
service for individual passengers. Further, the effect would depend on the relative size of
the individual passenger market and group passenger market. Thus, we include the
individual passengers in the analysis, and treat them as consumers in a monopoly market.
16
The SS flights depart at scheduled times (determined as a variable in competition),
whereas the CS flights depart at the group’s discretion. We assume that only one group of
passengers occupies each charter flight, and there is no ―assimilation‖ of multiple groups
over a single flight, even when the desired departure times might be the same. The
variable that CS can alter (strategy space) is price alone (dependent on group size and
value of time), whereas for the SS it is both price and frequency. The CS price is modeled
as a group price, whereas the SS price is modeled as an individual ticket price. Of course,
the SS ticket price could be different for individuals and groups (and different for groups
depending on group size and value of time), and scenarios with a single ticket price over
the entire market are examined by constraining the prices to be the same. In the models
presented here, we do not assume any capacity constraint on CS (in terms of number of
available aircraft) or on SS (in terms of seats available on a single SS flight). However,
the cost models incorporate the additional cost of operating a larger aircraft, and are
discussed in detail later.
For both the CS and SS, we assume a linear cost function, comprising of a fixed cost of
operating a flight, and a variable cost for each flight based on the number of passengers
in the flight. These cost functions can be used to approximate the cost of an ―elastic‖
aircraft based on the number of passengers served, or be used to model the use of a single
aircraft size only, with the assumption that this aircraft is sufficiently large to serve any
potential increase in passengers, and the variable cost reflect the cost of handling
passengers. The cost parameters used are as follows:
17
is the variable cost of serving each additional passenger for SS
is the fixed cost of operating a scheduled flight
is the variable cost of serving each additional passenger for CS
is the fixed cost of operating a charter flight
2.1.3 Scheduled Service Frequency
The SS frequency in the market is a variable for competition. If be the time period of
study and be the number of SS flights during this time interval, then we assume that all
the scheduled service flights are equally spaced. Thus, the first scheduled flight in the
interval is at time , and subsequently there is a scheduled flight after every
time. We call the SS service frequency. Individual passenger and the groups chose the
SS flight which is closest to their desired departure time.
As stated before, individual demand is served by SS only. We assume a linear variation
of individual demand with price, and an inverse relationship with service frequency
where an increase in service frequency would result in a higher demand. The functional
form for individual demand is given below in equation (2.1)
(2.1)
where
is the ticket price for each individual
is the service frequency as defined before
18
are non-negative parameters
The group demand is modeled as a constant number of groups (price-inelastic) willing to
use either CS or SS as long as the price is below a pre-defined limit. If the price is greater
than this limit, the group demand is zero. The motivation behind using a ―price-inelastic‖
group demand is to keep the problem tractable and to obtain analytical expressions of
equilibrium. Such analytical expressions are useful in understanding how competitive
equilibria depend on different demand and cost factors. Even with price inelasticity, there
are some scenarios where explicit equilibrium expressions are not feasible, as
demonstrated in the following sections. The price limit for group demand is introduced to
avoid scenarios where an operator might charge an infinite price for profit maximization.
We denote this price limit for group as , and assume that , where is the
individual price for zero demand at infinite frequency. It seems reasonable that group
demand would be zero at the same price the individual demand is zero at theoretically-
infinite frequency.
Thus, if
J is the set of all types of groups in the market,
is the number of groups of type j,
is the number of passengers in a group of type j
is the dollar value of time of group of type j
is the price of flying CS for a group of type j (per group)
is the price of flying SS for a group of type j (per person)
is the schedule delay associated with using SS
is the additional disutility of using SS service by group
19
then the group would chose CS over SS if
(2.2)
The parameter quantifies the additional difference in utility of using either the CS or
SS. Besides reduction in schedule delay, use of CS might have certain other advantages
over SS. These could be better access to the aircraft (less time spent in the terminal area),
quicker baggage check and baggage claim, or other factors that make CS more
―attractive‖ than SS. We assume that the is always greater or equal to zero, i.e. for the
same price and no schedule delay case, the utility of using SS is less than the utility of
using CS.
The schedule delay the group experiences from using SS is dependent on the desired
departure time of the group. The desired departure time needs to be modeled as a
probability distribution in order to analyze the equilibrium prices and frequency for the
competition. We assume that the desired departure time for the group is uniformly
distributed over the entire analysis period . The assumption of uniformly distributed
desired departure time leads to a simple piecewise linear function for the probability of a
group choosing CS over SS, as demonstrated below.
Since a group’s desired departure time is unlikely to exactly match the departure
time for the SS flight, we assume that the group chooses the SS flight closest to its
desired departure, and this flight could be either before or after. Given the uniformly
distributed desired departure, group prices being less than , the equally spaced SS
20
flights and the expression for schedule delay values when CS is chosen over SS, the
probability of group choosing to fly CS can be expressed as:
(2.3)
where, to simplify the notation, we have defined . It is clear that the interval
is where the choice probability (and thus the group market
share) is fractional, and neither CS nor SS has a monopoly on the group. Let us call this
interval as the competitive interval.
Based on the probability of choosing CS by group defined in equation (2.3), the
expected profit functions for CS and SS are given in equations (2.4) and (2.5).
(2.4)
(2.5)
In equation (2.4), the profit from each group is multiplied by the probability of the group
selecting CS over SS. This is summated over all the groups to get the expected CS profit.
In the expected SS profit in equation (2.5), the first term represents the profit from
individuals, and is the product of profit from one individual and the demand. The second
21
term is the profit from all groups, and includes the probability of selecting SS by a group
. The last term is the fixed cost of operating the flights by SS.
The variables or strategy space for CS is the price it charges for each group ,
whereas the SS strategy space includes price for individuals , price for each group
and the frequency of flights . Analysis of the behavior of the profit functions
over the respective strategy spaces yields the following theorem:
Proposition 2.1: For any set of charter and scheduled variables, the following statements
are true for the competitive interval (i.e. if ).
(a) The charter profit function is concave in charter prices for given scheduled
prices and frequency.
(b) The scheduled profit function is concave in scheduled prices (both
individual and group) for fixed frequency and charter prices, and is also concave
in scheduled frequency for fixed scheduled and charter prices.
Proof: The proof of the above stated proposition is given in Appendix 1.
As a consequence of proposition 2.1, for the competitive interval there is a unique,
optimal CS price for every SS price and frequency, and there is a unique, optimal SS
price for every SS frequency and CS price. Of course, the competitive interval is itself
defined as a function of prices and frequency, and coupled with the cases where CS
market share might be zero or one, this does not yield a well-defined zone for uniqueness
and optimality of prices. In the following section, we define various scenarios of
competition, and address the issue of the optimal prices on a case-by-case basis.
22
In order to keep the profit functions non-negative, we define some restrictions on
the prices and the frequency. These constraints are simply the fact that price has to be
greater than operating cost, and that frequency has to be non-negative:
(2.6)
(2.7)
(2.8)
The profit functions defined in equations (2.4) and (2.5) consist of a set of CS and
SS variables, and depending on the subset of these variables that the competitor changes,
different scenarios of market competition can be constructed and analyzed. However,
before we define these scenarios, it is important to define the model of competition itself.
We consider two different models of competition: the simultaneous game and the leader-
follower game. The following section gives details of the two models and the possible
scenarios considered for both the models.
Utilizing the profit functions (developed in the previous section) for analyzing different
conditions requires some assumptions on the behavior of the competitors. In this section
we detail some of these assumptions. For all the models and scenarios, we assume
conditions of complete information, i.e., each competitor’s profit function is completely
known to the rival (Gibbons, 1992). Further, we assume that the competitors act
rationally, and pursue a course which maximizes their profit given the rivals’ actions.
With these assumptions, two different models or games can be formulated based on the
23
sequence of decision making by the players. Additionally, for each such model different
scenarios can be characterized based on which subset of the variables the players alter
(price and/or frequency). In the following sub-sections, we describe the two models and
the scenarios. This is followed by a description of equilibrium conditions for each case in
the next section.
The aim behind analyzing competition between CS and SS is to understand the long term
behavior of the competitors over different markets. Since scheduled airlines might have a
strong prior presence as a regional or even a low-cost carrier, a key component in the
analysis is the sequence of decision making. The outcomes are different when the
competitors make decisions on price and/or frequency simultaneously, as compared to
when one player has advantage over the other by being the first to decide. To address
this, we use two different games or models of competition: simultaneous game and
leader-follower game.
In the simultaneous game, the competitors make simultaneous decisions about their
variables to maximize their profit. Each player assumes that the competitor will also
make a choice to maximize profit, and that choice will not change. This model is similar
to the classical Cournot duopoly (Gibbons 1992) in some ways. We find the Nash
equilibrium (NE) where, given the competitors strategy (price and/or frequency
combination), the player does not gain from changing their own strategy. Mathematically,
this model is solved by first finding each player’s best response to any of the rival’s
actions, and then finding the ―intersection‖ of these best response functions.
24
In the leader-follower game, the players do not make decisions simultaneously. One
of the players, called the market leader, has the advantage of acting first and the other
player (follower) responds to those actions. The reasons for the market leader’s advantage
could be historical dominance, or even prior presence in the market. In our case, we
assume that SS is the market leader and CS is the follower. This seems reasonable in the
light of the fact that SS is also serving individuals in the market. The leader-follower
model is similar to the classical Stackelberg duopoly (Gibbons 1992). This model is a
two-step sequential model, where in the first step the leader acts in a way to maximize its
profit, and in the next step the follower acts. Fundamental to this model is the assumption
of perfect information, which means that the players have up-to-date information of all
the prior steps in the game. This is in addition to the previous assumption of complete
information. Since this is a two-step model, we solve for the sub-game perfect Nash
equilibrium (SPNE) in this case, where each stage is a NE itself. SPNE is a stronger
equilibrium concept for multiple stage models, and excludes the possibilities of ―non-
credible threats‖ (Gibbons 1992). Mathematically, this model is solved backwards, by
first evaluating the follower’s best response to any action by the leader. This best
response is then included in the leader’s profit function before maximization. The leader,
thus, anticipates the actions of the follower and acts accordingly.
In the following sub-section, we describe the various market scenarios. It should be
noted that the above two models are applicable to each of these scenarios, and in section
2.3 we give the mathematical expressions of equilibrium for both the models for all the
scenarios.
25
In a given demand and cost situation, the competitors can alter their ―variables of
service‖ as actions in response to different conditions. The only variables for CS are the
different group prices, since CS charges a single group price based on group
characteristics. SS, however, has individual ticket price, group ticket prices and
frequency as variables. SS can change either all or a subset of these variables. Further, it
can choose to have a single ticket price for the entire market. Based on which variables a
competitor changes, different scenarios can be defined as follows:
1) SS serves both individuals and groups, CS absent
2) CS enters the market and competes, SS does not respond with any changes
3) SS changes prices in response to CS, does not alter frequency; CS alters price
reflecting SS changes
a) SS sets a single, optimized ticket price for everyone (individuals and
groups), CS sets optimal prices based on group characteristics (size and
value of time)
b) SS sets separate, optimal prices for individuals and different groups, CS
sets optimal prices based on group characteristics
4) SS changes group and individual price as well as frequency, CS sets optimal
prices based on group characteristics
Besides the above mentioned, more scenarios can be constructed. One possibility is that
SS changes frequency but does not alter the prices and CS changes its prices accordingly.
This, however, is not a very realistic case since SS would find it easier to alter prices than
frequency. A scheduled airline’s fleet is spread over the entire network, and any changes
26
in frequency would require accounting for all the cascading effects over the network.
Compared to this, changes in price can be restricted to a particular market. Thus, it seems
unreasonable that a competitor will try to optimize by making relatively complex changes
while fixing prices that are easier to change.
In the four scenarios listed above, as we move from scenario one to four the
complexity of the problem increases since maximization is done over more variables.
This also represents an increasing degree of competition between CS and SS. SS will
consider responding to CS only if it CS poses a substantial challenge. Further, since
changing frequency is a much more involved decision as compared to changing prices,
SS will only do so if it sees that CS has captured a substantial portion of the market share.
Based on the same logic, scenario one to four represent a timeline of changes, with
scenario one occurring earlier and scenario four later. Again, since changing price is
―easier‖ than frequency the first response SS will have is to change the price, and later
the frequency, if at all.
In the following section, we give analytical expressions for equilibrium prices and
frequency for the scenarios mentioned above. The expressions are for a generalized
setting of numerous groups with different value of time and group size. It should be
noted, however, that these expressions are for the competitive interval defined in equation
(2.3). Let us call this interior equilibrium. When the prices and frequency are outside the
competitive interval, either CS or SS has the complete market share for the particular
group. Given the interior equilibrium expressions, conditions when either competitor has
27
complete market share are derived using the expressions in equation (2.3). Thus,
identifying the interior equilibrium is the essential step in determining market share in
different scenarios.
The expressions for interior equilibrium are for the four different scenarios listed
above for both models of competition (simultaneous game and leader-follower game).
Along with presenting the interior equilibrium expressions for different scenarios, we
discuss their stability and uniqueness. We also describe the cases where the resulting
equilibrium might not be an interior equilibrium, and discuss the stability and uniqueness
in such cases.
In this scenario, CS is absent from the market and SS serves both individuals and groups.
Thus, there is no competition and the two models of competition defined earlier are not
applicable here. However, SS sets a combination of profit maximizing price and
frequency, and we determine this combination.
As stated before, group demand is inelastic to price as long as the price is below the
value . Also, frequency has no effect on group market share, since SS is the
only operator and groups cannot select a service based on their schedule delay, which is
linked to frequency. In such a case, SS would make the most profit by charging each
group their maximum price , and setting an optimal individual price and market
frequency by maximizing the first part of the profit function in equation (2.5), i.e.
28
(2.9)
However, this means that groups are being charged more than individual passengers,
since the resulting is less than , whereas the group price is . It seems
improbable that such a situation would persist, since the group passengers can ―pretend‖
to be individuals and buy individual tickets rather than approaching the SS as a group. If
this is the case, the SS profit would be sub-optimal. SS would fare better by charging a
single ticket price to all the passengers, individuals or group, and to base this single ticket
price as well as frequency on the entire market demand. Further, since SS cannot
decide to forgo the individual market completely and set a ticket price greater than ,
capturing groups (either all or some) and charging them a high price. Thus, SS would
either pick a price less than and an optimal frequency, or pick the price and set
the frequency as one. If be the SS ticket price set for both individuals and groups, then
this can be represented mathematically as:
(2.10)
In the second case in equation (2.10), the optimal frequency would be one under the
assumption that there has to be at least one SS flight in the time horizon. Analysis of the
Hessian matrix of the above expression for the case reveals that for generic
demand and cost parameters, the profit function is not uniformly concave, and hence
29
there is no unique set that maximizes the profit, even when restricted to the
realistic case of and . Further, taking the first derivative of this
expression with respect to and and solving for equilibrium frequency yields the
following cubic equation in equilibrium frequency when :
(2.11)
Even though analytical solution methods exist for cubic equations, the expressions
obtained from such methods are very involved to yield any insights into the sign of the
solution itself. However, we can identify the equilibrium price in terms of the
equilibrium frequency:
(2.12)
It should be noted that for any frequency, there is a unique, optimal price as given by
equation (2.12).
The optimal profit, and the frequency and price that yield that profit, can be found
by maximizing the function in equation (2.10). However, the expression in equation
(2.10) is non-differentiable at , and first-order conditions result in a cubic
equation as shown in (2.11). This hinders the search for a closed form expression for
optimal SS profit, and optimal strategy as well (whether to set price below or not).
However in numerical cases, the above expressions give sufficient information to
generate candidate price and frequency combinations and compare these with boundary
values. We demonstrate this in the next chapter with examples.
30
This scenario can be regarded as the entry of CS in the market, with CS competing with
SS but SS not responding. The profit maximizing CS price for any group can be obtained
from the profit function in equation (2.4), and the expression for optimal price is
(2.13)
The first case is when CS does not fly that group at all, since at any price above the
operating cost, the SS price for the group is sufficiently small to overcome the
disadvantage from schedule delay. The second case is where the market splits, with some
groups using CS and some using SS, and market share can be evaluated from the
probability function in equation (2.3) as . The last case is where all the
groups are served by CS, and the profit maximizing CS price in this case is the total cost
of flying SS for the group minus the schedule delay. This is the case where the SS prices
are sufficiently high for the group to choose CS for any desired departure time.
In this case, SS does not alter its frequency and neither does it set special group
prices (special group prices could be given in the form of discounts, as stated before).
However, as a response to the entry of CS, SS may adjust the single ticket price charged
31
to everyone. In this scenario, both SS and CS are competing for groups, and the two
models of competition discussed before are applicable. It should be noted that SS
frequency is a parameter and not a variable, and represents the existing SS frequency. As
described in section 2.2.3, we look for an interior equilibrium in this case. Below are the
equilibrium SS and CS prices for the both the models of competition, where
is the equilibrium SS ticket price
is the equilibrium CS group price for group
Simultaneous Game:
(2.14)
(2.15)
Leader-Follower Game:
(2.16)
(2.17)
In both the simultaneous game and leader-follower game, the interior-equilibrium
defined above is unique and stable when it actually is an interior equilibrium. This
32
follows from the structure of the strategy space and concavity of the profit functions as
shown in proposition 1 (Nikaido and Isoda, 1955) and can be explained as follows: Since
there is no change in SS frequency, the CS and SS profit functions are concave, or there
is a unique set of profit maximizing prices for any set of competitor’s prices. Thus, any
deviation from the equilibrium prices would lead to a reduction in profit, and hence the
interior equilibrium is stable.
However, stability needs to be addressed for the case when the parameters do not
result in an interior equilibrium for a particular group. In such a case, either CS or SS has
the complete market share for the group. If SS has the complete market share, it will have
no reason to change the optimal price since it already has the complete market share at
the profit maximizing price. CS’s profit maximizing price is resulting in zero market
share, and since CS actions cannot result in ―negative‖ profit, this equilibrium is stable.
But if CS has the complete market share, SS would alter its universal ticket price to
incorporate the fact that some groups will always use CS. Thus, it becomes important to
identify groups for which CS has the complete market share. For this, say is the SS
ticket price in either the simultaneous or leader follower game. Then the optimal CS price
is given by equation (2.17). The condition for CS having complete market share can be
written as:
(2.18)
In Appendix 2 section A2.3, we show that this condition is always satisfied when:
(2.19)
It should be noted that the above condition is a special case of (2.18). There might be
other group types for which (2.18) is true but (2.19) is not. Thus, finding true equilibrium
33
would involve an iterative process, where optimal price including the ―suspect‖ group is
calculated and the output price is evaluated for (2.18). If true, SS has no market share for
this group and the optimal price needs to be re-evaluated. To this end, let us call the
subset of that satisfies (2.18) . Thus, the actual equilibrium prices can be re-written as:
Simultaneous Game:
(2.20)
(2.21)
(2.22)
Leader-Follower Game:
(2.23)
(2.24)
(2.25)
The equilibrium defined by the above equations is stable in all the three cases:
when CS has complete market share for a group, when CS and SS have partial market
shares or when SS has the complete market share. When CS has the complete market
share, it sets the maximum price that it can charge beyond which SS might get some
market share (equations (2.22) and (2.25)). SS discounts these groups when setting its
34
single ticket price. When SS has the complete market share, CS does not compete. When
both SS and CS have partial market shares, they are charging their unique, profit
maximizing prices, any deviation from which would not benefit either competitor. Hence,
the equilibrium is stable.
The above optimal prices are subject to the condition that the group ticket price
should be less than (or for the charter price), and we constrained
earlier. These constraints are not explicitly stated in the above equations. Based on the
demand parameters and the operational costs of the two competitors, we evaluate the case
when the optimal values are always less than (or ). These conditions are as
follows:
(2.26)
(2.27)
In equation (2.26), the third and fourth term on the left side represent the total cost of
serving the group by charter, and the first two terms represent the maximum benefit that
can be derived from using charter. Similarly, equation (2.27) describes the operating cost
for serving each passenger in the group as well as the ―additional cost‖ of using SS. Thus,
the conditions that the optimal prices by CS and SS are always less than have a
practical interpretation, and we assume they are always true.
As in scenario 3a, SS does not change the frequency in response to competition, but
charges separate prices for individuals and groups, where, like CS optimal prices, each
35
groups’ SS price is dependent on the group characteristics. Again, we look at both the
models of competition and develop expressions for interior equilibrium. These
expressions are given below, where
is the equilibrium SS ticket price for individuals
is the equilibrium SS ticket price for group
Simultaneous Game:
(2.28)
(2.29)
(2.30)
Leader-Follower Game:
(2.31)
(2.32)
(2.33)
The above equilibrium prices are obtained by maximizing the profit functions
defined in section 2.1.6. They do not explicitly consider the case when one of the
competitors has the complete market share for a particular group. In order to arrive at
more explicit equilibrium expressions including the above mentioned case, we use the
36
above expression along with the choice probability expression defined in equation (2.3)
to arrive at conditions when either CS or SS has the complete market share. We also
evaluate the maximum group price that can be charged by CS or SS when either has the
complete market share. Each set of prices below is divided into three expressions as the
choice probability in equation (2.3): when CS has the complete market share, when CS
and SS have fractional market share, and when SS has the complete market share.
Simultaneous Game:
(2.34)
(2.35)
(2.36)
where
Leader-Follower Game:
(2.37)
(2.38)
37
(2.39)
where
The above equations show that the ―zone for interior equilibrium‖ is larger in the
leader-follower case as compared to the simultaneous case
. At first glance, this seems counter-intuitive, since SS being the market
leader could leverage more market share in the leader-follower case. However, the goal is
profit maximization, and in appendix 2 (section A2.4.3) we show that even though the CS
market share might be higher in the leader-follower case, SS profit is always higher in the
leader follower case.
In both the simultaneous and leader-follower game, the equilibria defined above are
stable. If neither CS nor SS has the complete market share for a particular group, then
they charge their profit maximizing prices described above, and given the concavity of
the profit function described earlier, any deviation from such prices would result in bring
the prices back to the equilibrium. Is either CS or SS has the complete market share, then
they charge the maximum price, beyond which the competitor enters the market and the
market share and profit drops. Charging a price lower than this would have no effect on
market share, but would yield less profit.
In this scenario, SS alters both the frequency and its prices because of the competition
with CS, and CS respond to these by altering its prices. As before in scenario 3, SS can
38
either charge a single ticket price for the entire group as in scenario 3(a), or set different
prices for individuals and groups as in scenario 3(b). However, it should be noted that a
single ticket price for the entire market is a special, more constrained solution to the
scenario with different prices (the additional constraint being ), and will
therefore yield less profit for any given CS prices. As described before, a change in
frequency by SS is in response to perceived greater threat from CS, and is a more
involved decision as compared to price change alone. Thus, it seems unlikely that change
in frequency would be accompanied by a constrained pricing scheme with a single ticket
price. Therefore, we assume that SS charges different prices to individuals and different
categories of groups.
Using the first order conditions obtained by differentiating the profit functions
defined in section 2.1.6, we can get relations between optimal prices and frequency.
Utilizing all these relations, the search for equilibrium results in a cubic equation in
equilibrium frequency ( ) in both the simultaneous and leader follower game. These
equations are as follows:
Simultaneous Game:
(2.40)
where
39
Leader-Follower Game:
(2.41)
where
The above cubic equations are of the form , and the
properties of such equations as well as their solutions have been studied extensively, and
a brief description of these follows:
Let be the discriminant of the equation, the expression for is
(2.42)
Based on the value of , it can be determined whether some of the roots of the equation
are complex or not. If , the equation has three distinct real roots. If , the
equation has one real root and a pair of complex conjugate roots. If , then at least
two roots coincide, i.e., the equation may have a double real root and another distinct
single real root, or it may have a triple real root. There exist involved, closed form
expressions for the different roots of the cubic equation, but given the size and
complexity of the coefficients, closed form solutions for the equilibrium frequency that
satisfy the constraint are not apparent. Further, the cubic equation raises the
possibility of multiple equilibrium frequencies, and thus, multiple equilibria.
Additionally, frequency here is synonymous with the number of flights in the analysis
time period. Integral values of might be more desirable or practical than fractional
values. Lastly, it should be noted that the above expressions are from competitive interval
price. In case either CS or SS has complete market share, the equilibrium prices would be
40
different than the ones used in deriving the above cubic equation (refer scenario 3(b) in
section 2.3.4). But since the competitive interval itself is based on values, this would
involve an iterative process of testing each suspect group. Given all these factors,
obtaining analytic generic expressions of equilibrium frequency is not feasible. In the
next chapter, we apply the above cubic equations for certain numeric cases and discuss
the various issues.
However, it should be noted that given an equilibrium frequency, there is a unique
set of SS and CS prices. Effectively, given an equilibrium frequency the problem of
finding the equilibrium prices reduces to scenario 3(b). Further, any such equilibrium is
stable in price change only. Thus, even though the cubic equations in frequency could
result in multiple equilibria, each such equilibrium is stable in price change alone.
41
In this chapter, we illustrate some of the theoretical results from the previous chapter in a
numerical setting. Of particular interest here is scenario 4 as mentioned before (section
2.3.5), where solution to a cubic equation in frequency yields the equilibrium. The goal in
this chapter is to identify conditions where the charter would be ―successful‖. To this end,
we use a range of values for the demand and cost parameters. All the scenarios and the
two models of competition in the previous chapter are analyzed, and besides the
equilibrium market share, prices and frequency, we evaluate the benefit of charter entry
to individual and group passengers. To understand the sensitivity of results to group
characteristics like size and number, we use only a single group type in this analysis.
The rest of this chapter is organized as follows: We first detail the range of values
used for charter (CS) and scheduled (SS) operating costs, as well as for individual and
group demand. This is followed by approximate results from simple calculations using
expressions from the previous chapter. We then give detailed results for all scenarios and
42
all parameter ranges in a tabular form. These detailed results are from implementation of
the mathematical expressions in the previous chapter, and we use the same notations and
symbols as used in the previous chapter, re-stating them where needed.
For this analysis, we use a study period of 24 hours, corresponding to daily operations.
Further, we assume three different markets based on distance: short-haul (300 miles),
medium-haul (600 miles) and long-haul (1200 miles). This definition of long-haul differs
from that used in existing literature Wei and Hansen (2007), where long-haul is typically
used for distances of 2400 miles. However, charter aircrafts would be smaller than typical
commercial jet aircraft, and their range would be limited. A case in point is the aircraft
used in the case study in chapter 5 (Dornier D328), which has an approximate range of
1300 miles. This analysis is limited to direct flights, and serving distances more than
1200 miles would involve additional time in re-fueling for the charter aircraft. Thus, we
restrict our study to above mentioned distances, and use the names short, medium and
long haul in this context. In the rest of this section, we first give the individual and group
demand parameter ranges, and then detail the SS operating costs and CS operating costs.
The individual demand function is defined in equation (2.1), and the relevant parameters
are , and . The following observations can be made about the demand function
parameters:
represents the demand at zero price and infinite frequency
43
represents the price for zero demand at infinite frequency. It is also the
maximum price that can be charged to any group
The individual value of time can be evaluated as , as explained in
appendix 3.
Calculating unit elasticity of demand at infinite frequency gives the unit
elastic price as , and unit elastic demand at this frequency as .
For the individual value of travel time we use $30/hour, on the value adopted by the U. S.
Department of Transportation (DOT 1997). Based on the above, we identify two types of
markets: a high density corridor with unit elastic demand of 5000 (approximately 50
flights of 100 passengers), and a low density corridor with unit elastic demand of 250
(approximately five flights of 50 passengers each). Further, we use a unit elastic price of
$150 for a stage length of 300 miles, $250 for 600 miles and $350 for 1200 miles. For a
group size of 10, these values result in a maximum price of $3000, $5000 and $7000
for distances of 300, 600 and 1200 miles respectively. The resulting values of , and
are summarized below in table 3.1.
Table 3.1: Individual demand parameters
Stage Length (miles)
Short-haul (300 miles)
Med-haul (600 miles)
Long-haul (1200 miles)
High Density
10000.0 10000.0 10000.0
33.3 20.0 14.3
6000.0 3600.0 2571.4
Low Density
500 500 500
1.7 1.0 0.7
300.0 180.0 128.6
44
The generic model has been devised for analyzing multiple groups with different
characteristics, but for the sake of this analysis, we use only one group type, and vary its
characteristics to isolate the effect of such groups on overall scheduled service. We use
two different groups, 10 and 25 passengers, and use the same value of travel time as for
individuals ($30/hour).
To define the number of groups, we first select the total number of group
passengers as a percentage of unit elastic individual demand . For a high density
corridor, we use the values of 1%, 5% and 10% to show varying levels of group demand.
A 10% group demand implies 500 group passengers, or 20 groups of 25 people per day.
For a low density corridor, we use 10%, 20% and 30%, with 30% demand implying 3
groups of 25 people. The percentages are used as an approximation to result in an integral
number of groups. Higher percentages of group demand can be used, of course, but there
is no evidence of a corridor where the group demand is so high as compared to individual
demand, and hence we use conservative numbers.
In the previous chapter, we use to define the additional disutility of using SS for
the group . At one end of the spectrum, we assume there is no difference between the SS
and CS besides the schedule delay as identified in equation (2.2), and set The
other extreme is the case in which the charter flying time is used productively by the
group, for purposes like on-board meetings or work, so that charter flight time is
effectively zero. In this case, the value of time spent in flying via SS is included in .
Further, we assume an additional time savings of one hour from charter flying in this
case, due to expedited terminal services such as baggage check-in, security check and
45
baggage claim. To determine the average gate-to-gate travel time for the three distances,
we use the formula from a recent study Smirti and Hansen (2009), and get the values 1.3
hours for 300 miles, 1.9 hours for 600 miles and 3.1 hours for 1200 miles. Thus, the value
of is the dollar value of the group size time 2.3 hours, 2.9 hours or 4.1 hours. In table
3.2, we summarize the range of values for the group demand parameters of group size
, group number and .
Table 3.2: Group demand parameters
Corridor Type High Density Low Density
Group Size 10 25 10 25
Group Number 5 25 50 2 10 20 2 5 8 1 2 3
in $
(Distance in miles)
0 0
690 (300)
870 (600)
1230 (1200)
1725 (300)
2175 (600)
3075 (1200)
690 (300)
870 (600)
1230 (1200)
1725 (300)
2175 (600)
3075 (1200)
Airline operating costs have been approximated by a fixed cost and variable cost.
As stated earlier in section 2.1.2, this model can be used to approximate the cost of an
―elastic‖ aircraft based on the number of passengers it serves, or can be used to model the
use of a single aircraft size only, with the variable cost reflecting the cost of handling
passengers. For SS we use the elastic aircraft model, since the number of passengers per
flight would vary over the different scenarios. Using a large size aircraft that
accommodates all possible passenger loads would be too simplistic and in many cases
overestimate SS costs, and using a smaller aircraft might give passenger loads in excess
of capacity.
46
Airline operation costs depend on a variety of factors, like fuel cost, stage length,
number of passengers, aircraft type etc. Wei (2000) built a comprehensive model of
airline costs based on 10 years of cost data. He divides the costs into direct operating
cost, indirect operating cost and passenger service cost, and uses a Cobb-Douglas
functional form for each of these. The total operating cost function used is:
(3.1)
where
is the average seat capacity (available seat-mile per plane-mile) for a
particular aircraft type
is the average stage length
is the unit fuel price per gallon
is the unit pilot cost per block hour
is the total number of departures
is the unit handling labor cost per handling personnel
is the average number of passengers per flight
are airline specific parameters.
The three components of the total cost above represent direct cost, indirect cost and
passenger service cost respectively. For the purpose of this analysis, we use a linear
approximation of the above model for total cost vs. number of seats. We use three
different stage lengths: 300, 600 and 1200 miles. Due to the substantial fluctuations in jet
fuel prices in 2008, as evident from statistics provided by the Bureau of Transportation
47
Statistics (DOT 2008), we use the average 2007 value of $2.09/gallon. American Airlines
Negotiations website (AAN, 2008) provides the average pilot cost per block hour as
$255, and we use $30/hour for the unit handling labor cost. We use an average of five
departures (changes in intercept and slope of linear approximation are less than 1% with
10 departures) and a load factor of 80% (changes in slope and intercept are less than
2.5% between load factor of 70% and 90%). Wei (2000) uses data from 10 different
airlines and estimate for each airline; we use an average of these values.
The input data for the model is summarized in table 3.3 below.
Table 3.3: Parameters used in determining SS operating costs
Value DC coeff. IC coeff. PC coeff.
Stage Length (miles) (variable) 0.826 0.758 0
Fuel Cost ($/gallon) 2.09 0.312 0 0
Pilot Cost ($/block hour) 255 0.491 0 0
Load Factor 0.8 0 0 0
Avg number of departures 5 0 -0.604 0
Handling cost ($/hour) 30 0 0.215 0.094
Number of Seats (variable) 0.77 0.684 0
Number of passengers (variable) 0 0 -0.926
DC airline specific parameter -3.6634
IC airline specific parameter -4.7743
PC airline specific parameter 6.6331
In figure 3.1, we plot the Cobb-Douglas form given in equation (3.1) and the linear
approximation with respect to number of seats for a stage length of 300 miles. As evident
from the plot, linearly increasing cost with increasing number of seats is a good
approximation of airline cost for a given stage length. The plots for 600 miles and 1200
miles are very similar to that for 300 miles. Table 3.4 summarizes the slope and intercept
for the linear approximation for the three different stage lengths.
48
Figure 3.1: Operating cost for 300 mile stage length using Cobb-Douglas form and its linear
approximation
Table 3.4: Linear SS operating cost values for different stage lengths
Stage length (miles)
Intercept (fixed cost in $)
Slope (variable cost) ($/passenger)
300 2,022 12.86
600 2,525 22.33
1200 3,417 39.11
Similar to the SS operating cost model, charter operating cost has been modeled as
a fixed cost and variable cost. Again, this can be used as an elastic aircraft or an aircraft
y = 12.861x + 2022.1R² = 0.9916
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 100 200 300 400 500 600
Op
era
tin
g C
ost
($
)
Number of Seats
Total Operating Cost from Cobb-Douglas function and Linear Approximation for stage length 300 miles
Operating Cost ($)
Linear Approximation
49
of single size. For CS, we use the single aircraft size model rather than an elastic aircraft.
Presumably CS would be a smaller operator catering to a limited number of groups, and
would use a very limited fleet based on average group sizes. Further, we assume that the
group demand is known to CS, and hence CS would use an aircraft suited to those
conditions, basing the aircraft size on the largest group size it intend to serve. To this end,
using a load factor of 80% as before, we assume a 13 seat aircraft for group size 10 and
31 seat aircraft for group size 25. The actual operating costs are calculated directly from
the Cobb-Douglas model specified in equation (3.1) and table 3.3. Table 3.5 shows these
values, where cost represents the total operating cost .
Table 3.5: CS operating costs
Stage Length (miles)
Cost for Group Size 10 ($)
Cost for Group Size 25 ($)
300 1,640 2,099
600 1,946 2,696
1200 2,487 3,753
In this section, we present the equilibrium conditions for the various conditions outlined
previously. These conditions are based on the mathematical expressions presented in
chapter 2. We first conduct a preliminary analysis of the cost and demand parameters to
identify maximum SS frequency for non-zero CS market share. We then present the
detailed results for the two cases: when there is no additional charter benefit and
when additional charter benefit is included as shown in table 3.2 .
50
In section 2.3.4 of chapter 2, we show that CS has non-zero market share in the
simultaneous game when , and a non-zero market share in the leader follower
game when . Based on this, for each group size and stage length, the
maximum SS frequency for ―charter success‖ can be calculated for both the cases of
excluding and including charter benefit. CS would have a non-zero market share in
scenario 3(b) if the SS frequency is less than this maximum frequency. Similarly, if the
optimal SS frequency in scenario 4 is less than this value, then CS would have a non-zero
market share. Further, a non-zero CS market share in scenario 3(b) guarantees a non-zero
market share in scenario 3(a) and 2, since SS pricing is much more constrained in these
cases. Thus, the maximum SS frequency can serve as an approximate measure of charter
market penetration. In table 3.6, we calculate this maximum SS frequency excluding and
including charter benefit, for both the simultaneous and leader-follower models. These
values indicate that for a group size of 10 charter service would not be viable in high
frequency markets with 15 or more SS flights. Even in low frequency markets of around
5 SS flights, charter profitability is questionable when excluding the additional benefit.
However, charter would be highly competitive for a group size of 25 when the high-end
value for is assumed (∞ indicates the case where CS market share is non-zero for any
SS frequency).
In the following sections, we present the complete equilibrium results as stated
before. The results are presented for all combinations of corridor type (short, medium or
long haul), with either high or low density of individual travel. Results from all the
scenarios as well as the two models of competition (simultaneous and leader-follower)
51
are presented. For scenario 4, we assume that if CS market share is zero in scenario 3(b),
then there is no change in SS frequency in scenario 4. In other words, if SS gets the
complete market share by altering prices only, it does not alter the frequency.
Table 3.6: Maximum SS frequency for non-zero CS market share
Group Size
Stage Length
Simultaneous ( )
Leader-Follower ( )
Simultaneous ( )
Leader-Follower ( )
10
300 4.76 7.15 8.77 13.15
600 4.18 6.27 8.45 12.67
1200 3.43 5.15 8.31 12.47
25
300 10.13 15.19 341.65 512.48
600 8.42 12.63 ∞ ∞
1200 6.49 9.73 ∞ ∞
The notation used is the same as in chapter 2, and is re-stated below. CS prices are
in grey cells when they are equal to the CS operating cost, signifying cases with zero
charter market share. Besides equilibrium prices, frequency and market share, we also
quantify the change in consumer surplus from scenario 1 for the individual passengers, as
well as the decrease in total cost of travel (time and money) for all groups from scenario
1. The formulae used for these calculations are described below along with the notation:
represents the number of groups
represents the group size (persons in the group)
represents the CS market share of group travel in percentage
represents the CS price in $
52
represents the SS group price in $
represents the SS individual price in $
represents the equilibrium SS frequency for the scenario
ind represents the change in consumer surplus (in $/day) for the individual
passengers from scenario 1, calculated using the ―rule of one-half‖ as
, where and are the demand at scenario 1 and new
demand, and and are the respective prices. Thus, higher value implies
more benefit.
grp represents the decrease in total cost of travel (in $/day) for all groups from
scenario 1, calculated as
, with and being the individual price and SS
frequency in scenario 1 and and being the relevant quantities
in new scenario. Thus, higher value implies more benefit.
θ = 0
Table 3.7 to 3.12 give the complete results for the six corridor cases identified earlier
(short, medium or long haul, with high or low density of individual travel).
As expected, charter market share decreases as the SS competitive response moves
scenario 2 (no response) to 4 (changing price and freq). This is because moving from
scenario 2 to 4 represents a more effective response by SS to CS competition. In fact, CS
is not profitable in scenario 3(b) and 4 in any of the high-density corridors. For the low-
53
density corridor, we present the CS market share for certain group numbers and sizes in
figure 3.2. These results are for the simultaneous game, and are a graphical representation
of some results in table 3.7 to 3.12.
(a) (b)
(c)
(d)
Figure 3.2: Charter market share for low-density markets for different scenarios in the
simultaneous game, assuming no additional charter benefit
Figure 3.2 shows that market share decreases with increasing corridor length in
scenario 3(b) and 4, but increases in scenario 3(a). This points to the fact when SS starts
offering ―special‖ group prices, charter is more profitable in shorter corridors with
relatively low density of individual demand.
0%
20%
40%
60%
80%
100%
2 3(a) 3(b) 4
CS
mar
ket
shar
e
n=5, k=10
Short-haulMedium-haulLong-haul
0%
20%
40%
60%
80%
100%
2 3(a) 3(b) 4
CS
mar
ket
shar
e
n=8, k=10
Short-haulMedium-haulLong-haul
0%
20%
40%
60%
80%
100%
2 3(a) 3(b) 4
CS
mar
ket
shar
e
n=2, k=25
Short-haulMedium-haulLong-haul
0%
20%
40%
60%
80%
100%
2 3(a) 3(b) 4
CS
mar
ket
shar
e
n=3, k=25
Short-haulMedium-haulLong-haul
54
In all the cases, CS market share decreases with increasing group demand
percentage. Increasing group demand percent also translates to increasing group numbers,
and as evident from the results, charter market share decreases with increasing number of
groups. Figure 3.2(a) and (c) represent the same percent group demand, and similarly
figure 3.2(b) and (d) represent the same percent group demand. For the same percent
group demand, thus, CS market share is higher for larger groups. This matches the
expectation that larger groups would benefit more from the charter since their total value
of time is higher.
Figure 3.3: Frequency change in scenario 4 for different groups in short-haul, low density corridor
In the high density corridor, there is no change in SS frequency from scenario 1 to 4
(scenario 4 is the case where SS changes its frequency in response to CS). This is
consistent with the fact that CS has no market share in scenario 3(b) in all the cases,
4
4.25
4.5
4.75
5
5.25
5.5
5.75
6
k=10n=2
k=10n=5
k=10n=8
k=25n=1
k=25n=2
k=25n=3
SS fr
eq
ue
ncy
Scenario 1-3(b)Scenario 4: SimultaneousScenario 4: Leader-Folloer
55
giving SS no stimulus to alter the frequency. For the low-density corridor, we plot the
frequency change from scenario 3(b) to 4 for both the simultaneous and leader-follower
game for the short haul case. The figure shows that, frequency change decreases with a
larger group size. The change in SS frequency increases with increasing percentage of
group travelers, but for the same % group demand, the change is higher for a smaller
group size. Trends in frequency change are the same for the medium and long-haul,
although the magnitude of the change is smaller. In all the cases the frequency response is
very small, to the point that if frequency were integer valued, there would most likely be
no response at all.
Increasing competition between CS and SS (in other words, moving from scenario
2 to 4) is definitely beneficial for the group passengers, as evident from increasing grp
values. For the individual passengers, however, scenario 3(a) is consistently better in all
the cases. This is expected, because it’s the case where SS alters its single price in the
market in response to CS entry. The price in this scenario is the lowest individual price
over all the scenarios, since SS is responding to CS by altering its single ticket price,
thereby lowering it as a response to competition. Ultimately, CS entry benefits the
individuals too over all the scenarios, although the magnitude of the benefit decreases
with increasing competition between the two operators.
Between the two models of competition, CS market share is consistently higher in
the leader-follower game than in the simultaneous game. As described before in chapter
2, this seems counterintuitive since SS has a distinct advantage in the leader-follower
case, which it could leverage to obtain a higher market share. The CS objective, however,
is not market share, but profit. Although not shown here, SS total profit is always higher
56
in the leader-follower case than in the simultaneous case, even with the lower market
share. SS is able to capture a larger share of consumer benefit (both individual and
group), and thus is more profitable in the leader-follower game. This can be seen from
the consistently lower ind and grp values for the leader-follower game as compared
to the simultaneous game. In the end, the consumer is the biggest loser in the leader-
follower case.
57
Table 3.7: Results for short-haul, low-density corridor without charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 2 k = 10
Wj 0% 37% 30% 5% 1% 27% 29% 29%
Pc,j 0 1,941 1,888 1,685 1,649 1,865 1,879 1,876
k.Ps,j 1,419 1,419 1,314 907 889 1,268 1,296 1,294
Ps,0 142 142 131 136 137 127 136 136
m 4.37 4.37 4.37 4.37 4.68 4.37 4.37 4.40
Δ ind - - 2,130 1,200 935 3,122 1,200 1,179
Δ grp - 724 776 1,104 1,184 803 786 793
n = 5 k = 10
Wj 0% 42% 26% 3% 0% 20% 27% 26%
Pc,j 0 1,975 1,848 1,665 1,640 1,800 1,857 1,844
k.Ps,j 1,516 1,516 1,262 897 884 1,167 1,281 1,272
Ps,0 152 152 126 137 138 117 137 137
m 4.54 4.54 4.54 4.54 4.76 4.54 4.54 4.64
Δ ind -- - 5,132 2,906 2,733 7,339 2,906 2,825
Δ grp -- 2,162 2,445 3,209 3,345 2,604 2,417 2,483
n = 8
k = 10
Wj 0% 48% 22% 1% 0% 14% 26% 23%
Pc,j 0 2,010 1,813 1,648 1,640 1,748 1,838 1,807
k.Ps,j 1,612 1,612 1,218 888 884 1,088 1,268 1,248
Ps,0 161 161 122 137 138 109 137 138
m 4.69 4.69 4.69 4.69 4.76 4.69 4.69 4.95
Δ ind - - 7,901 4,495 4,446 11,060 4,495 4,327
Δ grp - 4,088 4,710 5,851 5,919 5,091 4,586 4,819
n = 1 k = 25
Wj 0% 86% 76% 38% 37% 73% 53% 54%
Pc,j 0 3,865 3,663 2,870 2,846 3,595 3,188 3,247
k.Ps,j 3,587 3,587 3,181 1,596 1,584 3,046 2,233 2,272
Ps,0 143 143 127 136 136 122 136 135
m 4.40 4.40 4.40 4.40 4.48 4.40 4.40 4.24
Δ ind - - 3,346 1,492 1,423 4,564 1,492 1,642
Δ grp - 3,016 2,920 2,932 2,958 2,897 2,853 2,791
n = 2 k = 25
Wj 0% 93% 73% 37% 36% 67% 53% 54%
Pc,j 0 3,936 3,546 2,829 2,782 3,432 3,143 3,255
k.Ps,j 3,790 3,790 3,008 1,575 1,552 2,780 2,202 2,277
Ps,0 152 152 120 137 137 111 137 135
m 4.54 4.54 4.54 4.54 4.71 4.54 4.54 4.22
Δ ind - - 6,477 2,906 2,776 8,678 2,906 3,175
Δ grp - 6,598 6,188 6,236 6,337 6,126 6,087 5,847
n = 3 k = 25
Wj 0% 99% 70% 36% 34% 62% 52% 54%
Pc,j 0 4,010 3,443 2,792 2,722 3,297 3,101 3,264
k.Ps,j 3,991 3,991 2,859 1,557 1,522 2,566 2,175 2,283
Ps,0 160 160 114 137 138 103 137 135
m 4.67 4.67 4.67 4.67 4.94 4.67 4.67 4.19
Δ ind - - 9,390 4,239 4,058 12,376 4,239 4,603
Δ grp - 10,802 9,814 9,910 10,132 9,720 9,700 9,179
58
Table 3.8: Results for short-haul, high-density corridor without charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 5 k = 10
Wj 0% 18% 2% 0% 0% 0% 0% 0%
Pc,j 0 1,672 1,644 1,640 1,640 1,640 1,640 1,640
k.Ps,j 1,528 1,528 1,471 1,463 1,463 1,439 1,463 1,463
Ps,0 153 153 147 152 152 144 152 152
m 20.37 20.37 20.37 20.37 20.37 20.37 20.37 20.37
Δ ind - - 26,891 3,469 3,469 42,080 3,469 3,469
Δ grp - 659 358 322 322 442 322 322
n = 25 k = 10
Wj 0% 27% 0% 0% 0% 0% 0% 0%
Pc,j 0 1,686 1,640 1,640 1,640 1,640 1,640 1,640
k.Ps,j 1,558 1,558 1,319 1,465 1,465 1,210 1,465 1,465
Ps,0 156 156 132 152 152 121 152 152
m 20.59 20.59 20.59 20.59 20.59 20.59 20.59 20.59
Δ ind -- - 117,718 17,166 17,166 177,398 17,166 17,166
Δ grp -- 4,900 5,989 2,326 2,326 8,704 2,326 2,326
n = 50
k = 10
Wj 0% 37% 0% 0% 0% 0% 0% 0%
Pc,j 0 1,704 1,640 1,640 1,640 1,640 1,640 1,640
k.Ps,j 1,596 1,596 1,193 1,467 1,467 1,039 1,467 1,467
Ps,0 160 160 119 152 152 104 152 152
m 20.87 20.87 20.87 20.87 20.87 20.87 20.87 20.87
Δ ind - - 204,123 33,877 33,877 296,194 33,877 33,877
Δ grp - 14,083 20,155 6,441 6,441 27,839 6,441 6,441
n = 2 k = 25
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 3,819 3,171 2,099 2,099 3,171 2,099 2,099
k.Ps,j 3,819 3,819 3,800 1,657 1,657 3,800 1,657 1,657
Ps,0 153 153 152 152 152 152 152 152
m 20.37 20.37 20.37 20.37 20.37 20.37 20.37 20.37
Δ ind - - 3,469 3,469 3,469 3,469 3,469 3,469
Δ grp - 4,334 5,630 4,323 4,323 5,630 4,323 4,323
n = 10 k = 25
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 3,895 3,169 2,099 2,099 3,169 2,099 2,099
k.Ps,j 3,895 3,895 3,801 1,662 1,662 3,801 1,662 1,662
Ps,0 156 156 152 152 152 152 152 152
m 20.59 20.59 20.59 20.59 20.59 20.59 20.59 20.59
Δ ind - - 17,166 17,166 17,166 17,166 17,166 17,166
Δ grp - 21,620 28,884 22,331 22,331 28,884 22,331 22,331
n = 20 k = 25
Wj 0% 100% 100% 0% 0% 81% 0% 0%
Pc,j 0 3,990 3,167 2,099 2,099 2,448 2,099 2,099
k.Ps,j 3,990 3,990 3,803 1,668 1,668 2,366 1,668 1,668
Ps,0 160 160 152 152 152 95 152 152
m 20.87 20.87 20.87 20.87 20.87 20.87 20.87 20.87
Δ ind - - 33,877 33,877 33,877 355,765 33,877 33,877
Δ grp - 43,127 59,601 46,451 46,451 66,055 46,451 46,451
59
Table 3.9: Results for medium-haul, low-density corridor without charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 2 k = 10
Wj 0% 80% 64% 3% 0% 58% 27% 27%
Pc,j 0 2,665 2,522 1,969 1,946 2,467 2,187 2,181
k.Ps,j 2,488 2,488 2,202 1,096 1,110 2,092 1,532 1,529
Ps,0 249 249 220 239 240 209 239 239
m 4.02 4.02 4.02 4.02 4.31 4.02 4.02 4.05
Δ ind - - 6,294 2,114 1,801 8,954 2,114 2,074
Δ grp - 2,550 2,432 2,830 2,876 2,410 2,509 2,520
n = 5 k = 10
Wj 0% 90% 53% 0.5% 0% 41% 25% 24%
Pc,j 0 2,729 2,406 1,949 1,946 2,304 2,165 2,145
k.Ps,j 2,645 2,645 2,000 1,086 1,106 1,796 1,518 1,504
Ps,0 264 264 200 239 240 180 239 240
m 4.15 4.15 4.15 4.15 4.30 4.15 4.15 4.29
Δ ind -- - 14,483 5,117 4,990 19,909 5,117 4,987
Δ grp -- 7,468 6,760 7,812 7,843 6,786 7,015 7,106
n = 8
k = 10
Wj 0% 100% 44% 0% 0% 29% 24% 20%
Pc,j 0 2,802 2,318 1,946 1,946 2,191 2,145 2,103
k.Ps,j 2,802 2,802 1,850 1,106 1,106 1,597 1,504 1,477
Ps,0 280 280 185 240 240 160 240 242
m 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.59
Δ ind - - 21,459 7,914 7,914 28,682 7,914 7,666
Δ grp - 13,679 12,017 13,568 13,568 12,248 12,409 12,698
n = 1 k = 25
Wj 0% 100% 100% 35% 34% 100% 51% 52%
Pc,j 0 6,285 5,448 3,468 3,438 5,448 3,832 3,886
k.Ps,j 6,285 6,285 5,972 2,013 1,998 5,972 2,741 2,777
Ps,0 251 251 239 239 239 239 239 238
m 4.04 4.04 4.04 4.04 4.13 4.04 4.04 3.91
Δ ind - - 2,629 2,629 2,535 2,629 2,629 2,775
Δ grp - 4,403 5,240 5,294 5,325 5,240 5,233 5,178
n = 2 k = 25
Wj 0% 100% 98% 34% 32% 87% 50% 52%
Pc,j 0 6,612 4,822 3,427 3,367 4,577 3,786 3,890
k.Ps,j 6,612 6,612 4,781 1,993 1,963 4,293 2,710 2,779
Ps,0 264 264 191 239 240 172 239 238
m 4.15 4.15 4.15 4.15 4.34 4.15 4.15 3.91
Δ ind - - 16,762 5,117 4,942 22,138 5,117 5,381
Δ grp - 8,683 12,104 11,203 11,323 11,687 11,092 10,882
n = 3 k = 25
Wj 0% 100% 90% 33% 31% 76% 50% 52%
Pc,j 0 6,939 4,598 3,390 3,300 4,307 3,744 3,894
k.Ps,j 6,939 6,939 4,391 1,974 1,929 3,809 2,682 2,782
Ps,0 278 278 176 240 241 152 240 238
m 4.26 4.26 4.26 4.26 4.56 4.26 4.26 3.90
Δ ind - - 23,564 7,462 7,217 30,405 7,462 7,819
Δ grp - 12,856 18,674 17,727 17,986 18,067 17,575 17,121
60
Table 3.10: Results for medium-haul, high-density corridor without charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 5 k = 10
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 2,575 2,352 1,946 1,946 2,352 1,946 1,946
k.Ps,j 2,575 2,575 2,562 1,749 1,749 2,562 1,749 1,749
Ps,0 257 257 256 256 256 256 256 256
m 18.31 18.31 18.31 18.31 18.31 18.31 18.31 18.31
Δ ind - - 5,832 5,832 5,832 5,832 5,832 5,832
Δ grp - 5,333 6,446 4,130 4,130 6,446 4,130 4,130
n = 25 k = 10
Wj 0% 100% 67% 0% 0% 10% 0% 0%
Pc,j 0 2,625 2,075 1,946 1,946 1,966 1,946 1,946
k.Ps,j 2,625 2,625 2,010 1,751 1,751 1,792 1,751 1,751
Ps,0 263 263 201 256 256 179 256 256
m 18.50 18.50 18.50 18.50 18.50 18.50 18.50 18.50
Δ ind -- - 318,243 28,856 28,856 449,363 28,856 28,856
Δ grp -- 26,614 32,018 21,863 21,863 23,169 21,863 21,863
n = 50
k = 10
Wj 0% 100% 0% 0% 0% 0% 0% 0%
Pc,j 0 2,689 1,946 1,946 1,946 1,946 1,946 1,946
k.Ps,j 2,689 2,689 1,723 1,753 1,753 1,448 1,753 1,753
Ps,0 269 269 172 256 256 145 256 256
m 18.75 18.75 18.75 18.75 18.75 18.75 18.75 18.75
Δ ind - - 521,274 56,946 56,946 703,901 56,946 56,946
Δ grp - 53,102 48,296 46,757 46,757 62,053 46,757 46,757
n = 2 k = 25
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 6,437 4,797 2,696 2,696 4,797 2,696 2,696
k.Ps,j 6,437 6,437 6,406 2,204 2,204 6,406 2,204 2,204
Ps,0 257 257 256 256 256 256 256 256
m 18.31 18.31 18.31 18.31 18.31 18.31 18.31 18.31
Δ ind - - 5,832 5,832 5,832 5,832 5,832 5,832
Δ grp - 5,333 8,615 8,467 8,467 8,615 8,467 8,467
n = 10 k = 25
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 6,564 4,795 2,696 2,696 4,795 2,696 2,696
k.Ps,j 6,564 6,564 6,407 2,209 2,209 6,407 2,209 2,209
Ps,0 263 263 256 256 256 256 256 256
m 18.50 18.50 18.50 18.50 18.50 18.50 18.50 18.50
Δ ind - - 28,856 28,856 28,856 28,856 28,856 28,856
Δ grp - 26,614 44,303 43,544 43,544 44,303 43,544 43,544
n = 20 k = 25
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 6,722 4,792 2,696 2,696 4,792 2,696 2,696
k.Ps,j 6,722 6,722 6,409 2,216 2,216 6,409 2,216 2,216
Ps,0 269 269 256 256 256 256 256 256
m 18.75 18.75 18.75 18.75 18.75 18.75 18.75 18.75
Δ ind - - 56,946 56,946 56,946 56,946 56,946 56,946
Δ grp - 53,102 91,685 90,119 90,119 91,685 90,119 90,119
61
Table 3.11: Results for long-haul, low-density corridor without charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 2 k = 10
Wj 0% 100% 79% 0% 0% 71% 25% 24%
Pc,j 0 3,576 3,314 2,487 2,487 3,221 2,743 2,733
k.Ps,j 3,576 3,576 3,100 1,447 1,447 2,915 1,959 1,952
Ps,0 358 358 310 344 344 291 344 344
m 3.46 3.46 3.46 3.46 3.46 3.46 3.46 3.51
Δ ind - - 10,680 2,974 2,974 15,271 2,974 2,904
Δ grp - 4,540 4,219 4,257 4,257 4,092 3,964 3,980
n = 5 k = 10
Wj 0% 100% 63% 0% 0% 47% 23% 21%
Pc,j 0 3,794 3,123 2,487 2,487 2,961 2,718 2,685
k.Ps,j 3,794 3,794 2,752 1,481 1,481 2,429 1,942 1,920
Ps,0 379 379 275 344 344 243 344 345
m 3.58 3.58 3.58 3.58 3.58 3.58 3.58 3.74
Δ ind -- - 24,001 7,195 7,195 32,995 7,195 6,983
Δ grp -- 11,181 11,101 11,565 11,565 10,839 10,932 11,054
n = 8
k = 10
Wj 0% 100% 51% 0% 0% 31% 21% 16%
Pc,j 0 4,012 2,986 2,487 2,487 2,792 2,695 2,630
k.Ps,j 4,012 4,012 2,509 1,512 1,512 2,122 1,927 1,884
Ps,0 401 401 251 345 345 212 345 347
m 3.69 3.69 3.69 3.69 3.69 3.69 3.69 4.05
Δ ind - - 34,896 11,122 11,122 46,514 11,122 10,735
Δ grp - 17,644 19,090 19,999 19,999 18,959 19,125 19,479
n = 1 k = 25
Wj 0% 100% 100% 31% 30% 100% 48% 49%
Pc,j 0 9,030 7,465 4,552 4,506 7,465 4,998 5,053
k.Ps,j 9,030 9,030 8,592 2,765 2,742 8,592 3,658 3,695
Ps,0 361 361 344 344 344 344 344 343
m 3.48 3.48 3.48 3.48 3.57 3.48 3.48 3.39
Δ ind - - 3,698 3,698 3,559 3,698 3,698 3,848
Δ grp - 5,661 7,225 7,462 7,501 7,225 7,453 7,402
n = 2 k = 25
Wj 0% 100% 100% 30% 28% 91% 47% 49%
Pc,j 0 9,485 7,439 4,505 4,416 6,045 4,946 5,050
k.Ps,j 9,485 9,485 8,610 2,741 2,697 5,821 3,623 3,693
Ps,0 379 379 344 344 346 233 344 343
m 3.58 3.58 3.58 3.58 3.78 3.58 3.58 3.39
Δ ind - - 7,195 7,195 6,936 35,970 7,195 7,463
Δ grp - 11,181 15,272 15,775 15,924 17,107 15,771 15,579
n = 3 k = 25
Wj 0% 100% 96% 29% 26% 77% 47% 49%
Pc,j 0 9,939 6,100 4,462 4,330 5,649 4,897 5,047
k.Ps,j 9,939 9,939 5,995 2,720 2,654 5,094 3,591 3,691
Ps,0 398 398 240 345 347 204 345 343
m 3.67 3.67 3.67 3.67 3.99 3.67 3.67 3.40
Δ ind - - 37,441 10,488 10,129 48,496 10,488 10,848
Δ grp - 16,578 27,402 24,940 25,253 26,070 24,954 24,544
62
Table 3.12: Results for long-haul, high-density corridor without charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 5 k = 10
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 3,656 3,178 2,487 2,487 3,178 2,487 2,487
k.Ps,j 3,656 3,656 3,638 2,258 2,258 3,638 2,258 2,258
Ps,0 366 366 364 364 364 364 364 364
m 15.67 15.67 15.67 15.67 15.67 15.67 15.67 15.67
Δ ind - - 8,096 8,096 8,096 8,096 8,096 8,096
Δ grp - 7,299 9,689 6,990 6,990 9,689 6,990 6,990
n = 25 k = 10
Wj 0% 100% 100% 0% 0% 29% 0% 0%
Pc,j 0 3,726 3,177 2,487 2,487 2,554 2,487 2,487
k.Ps,j 3,726 3,726 3,639 2,260 2,260 2,393 2,260 2,260
Ps,0 373 373 364 364 364 239 364 364
m 15.84 15.84 15.84 15.84 15.84 15.84 15.84 15.84
Δ ind -- - 40,049 40,049 40,049 728,529 40,049 40,049
Δ grp -- 36,432 50,172 36,656 36,656 42,842 36,656 36,656
n = 50
k = 10
Wj 0% 100% 9% 0% 0% 0% 0% 0%
Pc,j 0 3,814 2,508 2,487 2,487 2,487 2,487 2,487
k.Ps,j 3,814 3,814 2,305 2,263 2,263 1,894 2,263 2,263
Ps,0 381 381 230 364 364 189 364 364
m 16.05 16.05 16.05 16.05 16.05 16.05 16.05 16.05
Δ ind - - 825,520 79,022 79,022 1,106,491 79,022 79,022
Δ grp - 72,716 81,335 77,577 77,577 96,013 77,577 77,577
n = 2 k = 25
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 9,139 6,711 3,753 3,753 6,711 3,753 3,753
k.Ps,j 9,139 9,139 9,095 3,179 3,179 9,095 3,179 3,179
Ps,0 366 366 364 364 364 364 364 364
m 15.67 15.67 15.67 15.67 15.67 15.67 15.67 15.67
Δ ind - - 8,096 8,096 8,096 8,096 8,096 8,096
Δ grp - 7,299 12,154 11,921 11,921 12,154 11,921 11,921
n = 10 k = 25
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 9,316 6,709 3,753 3,753 6,709 3,753 3,753
k.Ps,j 9,316 9,316 9,097 3,185 3,185 9,097 3,185 3,185
Ps,0 373 373 364 364 364 364 364 364
m 15.84 15.84 15.84 15.84 15.84 15.84 15.84 15.84
Δ ind - - 40,049 40,049 40,049 40,049 40,049 40,049
Δ grp - 36,432 62,497 61,307 61,307 62,497 61,307 61,307
n = 20 k = 25
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 9,536 6,706 3,753 3,753 6,706 3,753 3,753
k.Ps,j 9,536 9,536 9,099 3,192 3,192 9,099 3,192 3,192
Ps,0 381 381 364 364 364 364 364 364
m 16.05 16.05 16.05 16.05 16.05 16.05 16.05 16.05
Δ ind - - 79,022 79,022 79,022 79,022 79,022 79,022
Δ grp - 72,716 129,315 126,879 126,879 129,315 126,879 126,879
63
θ > 0
Table 3.13 to 3.18 give the results for the six corridor cases as before, but this time
assuming that group passengers have much lower effective travel time when they fly on
charter. As mentioned previously, CS prices are in grey cells when they are equal to the
CS operating cost, and SS group prices are shown strikethrough when the price is equal
to .
As in the case with , charter market share consistently falls from scenario 2 to
4. However, CS market share is consistently higher than before, with CS being profitable
even in the long-haul high density corridor for the larger group size . A larger
group size is more suitable for CS, since total time savings for a larger group are more
resulting in a higher value of . In short, including actual travel time as savings gives CS
a distinct advantage, making it viable even in corridors where it could not compete in the
previous case.
The SS frequency change from scenario 3(b) to 4 is noticeably smaller here, even in
the short-haul low density case. Interestingly, SS frequency in scenario 4 is actually
lower than in 3(b) for in all cases. This can be explained as follows: In the
absence of CS, SS set a slightly higher frequency in scenario 1 based on the expectation
that groups would fly SS. However, with the entry of CS with the advantage of , a large
number of the groups prefer CS, and SS decreases the frequency by a small amount to
reflect the groups’ preference for CS.
The trend in Δ grp values is the same as before, with Δ grp increasing from scenario
2 to 4, signifying more benefit to group passengers with increasing competition.
However, it is not the case with Δ ind values. The Δ ind values for scenario 3(a) are not
64
very different from that of scenario 3(b) and 4, except in the case for short and
medium-haul, low density corridor. This reflects that SS sets a more individual demand
based price in scenario 3(a) than before, recognizing the distinct advantage that CS has
from .
65
Table 3.13: Results for short-haul, low-density corridor including additional charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 2 k = 10
Wj 0% 78% 71% 33% 33% 68% 50% 51%
Pc,j 0 2,286 2,222 1,915 1,904 2,200 2,052 2,069
k.Ps,j 1,419 1,419 1,291 677 671 1,246 951 962
Ps,0 142 142 129 136 136 125 136 135
m 4.37 4.37 4.37 4.37 4.46 4.37 4.37 4.26
Δ ind - - 2,615 1,200 1,120 3,609 1,200 1,309
Δ grp - 1,014 1,078 1,667 1,700 1,106 1,348 1,303
n = 5 k = 10
Wj 0% 86% 66% 32% 30% 61% 49% 51%
Pc,j 0 2,320 2,167 1,895 1,869 2,122 2,030 2,069
k.Ps,j 1,516 1,516 1,211 667 654 1,120 936 962
Ps,0 152 152 121 137 138 112 137 135
m 4.54 4.54 4.54 4.54 4.78 4.54 4.54 4.25
Δ ind -- - 6,289 2,906 2,724 8,491 2,906 3,143
Δ grp -- 2,912 3,276 4,656 4,859 3,443 3,857 3,593
n = 8
k = 10
Wj 0% 93% 63% 31% 28% 55% 48% 51%
Pc,j 0 2,355 2,121 1,878 1,836 2,060 2,010 2,070
k.Ps,j 1,612 1,612 1,144 658 637 1,022 923 963
Ps,0 161 161 114 137 139 102 137 135
m 4.69 4.69 4.69 4.69 5.11 4.69 4.69 4.25
Δ ind - - 9,658 4,495 4,231 12,787 4,495 4,827
Δ grp - 5,329 6,156 8,224 8,740 6,561 6,945 6,303
n = 1 k = 25
Wj 0% 100% 100% 66% 66% 100% 74% 74%
Pc,j 0 5,312 4,634 3,445 3,468 4,634 3,619 3,697
k.Ps,j 3,587 3,587 3,400 1,021 1,032 3,400 1,370 1,422
Ps,0 143 143 136 136 136 136 136 135
m 4.40 4.40 4.40 4.40 4.33 4.40 4.40 4.19
Δ ind - - 1,492 1,492 1,559 1,492 1,492 1,691
Δ grp - 2,045 2,723 3,452 3,421 2,723 3,347 3,250
n = 2 k = 25
Wj 0% 100% 100% 66% 66% 100% 74% 74%
Pc,j 0 5,515 4,611 3,404 3,447 4,611 3,574 3,726
k.Ps,j 3,790 3,790 3,415 1,000 1,022 3,415 1,340 1,441
Ps,0 152 152 137 137 136 137 137 135
m 4.54 4.54 4.54 4.54 4.39 4.54 4.54 4.12
Δ ind - - 2,906 2,906 3,023 2,906 2,906 3,270
Δ grp - 3,968 5,775 7,296 7,181 5,775 7,092 6,713
n = 3 k = 25
Wj 0% 100% 100% 66% 66% 100% 74% 74%
Pc,j 0 5,716 4,591 3,367 3,428 4,591 3,533 3,757
k.Ps,j 3,991 3,991 3,429 982 1,012 3,429 1,312 1,462
Ps,0 160 160 137 137 136 137 137 134
m 4.67 4.67 4.67 4.67 4.46 4.67 4.67 4.04
Δ ind - - 4,239 4,239 4,392 4,239 4,239 4,738
Δ grp - 5,786 9,162 11,529 11,287 9,162 11,232 10,392
66
Table 3.14: Results for short-haul, high-density corridor including additional charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 5 k = 10
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 2,218 2,013 1,640 1,640 2,013 1,640 1,640
k.Ps,j 1,528 1,528 1,520 773 773 1,520 773 773
Ps,0 153 153 152 152 152 152 152 152
m 20.37 20.37 20.37 20.37 20.37 20.37 20.37 20.37
Δ ind - - 3,469 3,469 3,469 3,469 3,469 3,469
Δ grp - 884 1,905 3,772 3,772 1,905 3,772 3,772
n = 25 k = 10
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 2,248 2,013 1,640 1,640 2,013 1,640 1,640
k.Ps,j 1,558 1,558 1,521 775 775 1,521 775 775
Ps,0 156 156 152 152 152 152 152 152
m 20.59 20.59 20.59 20.59 20.59 20.59 20.59 20.59
Δ ind -- - 17,166 17,166 17,166 17,166 17,166 17,166
Δ grp -- 4,370 10,257 19,576 19,576 10,257 19,576 19,576
n = 50
k = 10
Wj 0% 100% 94% 0% 0% 53% 0% 0%
Pc,j 0 2,286 1,802 1,640 1,640 1,731 1,640 1,640
k.Ps,j 1,596 1,596 1,102 777 777 959 777 777
Ps,0 160 160 110 152 152 96 152 152
m 20.87 20.87 20.87 20.87 20.87 20.87 20.87 20.87
Δ ind - - 257,563 33,877 33,877 347,390 33,877 33,877
Δ grp - 8,627 32,344 40,941 40,941 34,245 40,941 40,941
n = 2 k = 25
Wj 0% 100% 100% 63% 63% 100% 72% 72%
Pc,j 0 5,544 4,033 2,376 2,376 4,033 2,417 2,419
k.Ps,j 3,819 3,819 3,800 486 486 3,800 569 570
Ps,0 153 153 152 152 152 152 152 152
m 20.37 20.37 20.37 20.37 20.35 20.37 20.37 20.29
Δ ind - - 3,469 3,469 3,494 3,469 3,469 3,553
Δ grp - 884 3,905 7,013 7,012 3,905 6,959 6,956
n = 10 k = 25
Wj 0% 100% 100% 63% 63% 100% 72% 72%
Pc,j 0 5,620 4,031 2,373 2,375 4,031 2,414 2,421
k.Ps,j 3,895 3,895 3,801 485 486 3,801 566 571
Ps,0 156 156 152 152 152 152 152 152
m 20.59 20.59 20.59 20.59 20.47 20.59 20.59 20.18
Δ ind - - 17,166 17,166 17,283 17,166 17,166 17,570
Δ grp - 4,370 20,259 35,820 35,797 20,259 35,554 35,469
n = 20 k = 25
Wj 0% 100% 100% 63% 63% 100% 72% 72%
Pc,j 0 5,715 4,029 2,369 2,373 4,029 2,410 2,423
k.Ps,j 3,990 3,990 3,803 483 485 3,803 563 572
Ps,0 160 160 152 152 152 152 152 152
m 20.87 20.87 20.87 20.87 20.63 20.87 20.87 20.04
Δ ind - - 33,877 33,877 34,098 33,877 33,877 34,656
Δ grp - 8,627 42,351 73,532 73,442 42,351 73,004 72,670
67
Table 3.15: Results for medium-haul, low-density corridor including additional charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 2 k = 10
Wj 0% 100% 100% 35% 34% 100% 51% 52%
Pc,j 0 3,358 3,050 2,259 2,249 3,050 2,405 2,422
k.Ps,j 2,488 2,488 2,388 806 801 2,388 1,097 1,109
Ps,0 249 249 239 239 239 239 239 238
m 4.02 4.02 4.02 4.02 4.08 4.02 4.02 3.92
Δ ind - - 2,114 2,114 2,039 2,114 2,114 2,234
Δ grp - 1,792 2,408 3,582 3,611 2,408 3,251 3,204
n = 5 k = 10
Wj 0% 100% 98% 34% 32% 87% 50% 52%
Pc,j 0 3,515 2,797 2,239 2,215 2,699 2,382 2,424
k.Ps,j 2,645 2,645 1,912 796 784 1,717 1,083 1,110
Ps,0 264 264 191 239 240 172 239 238
m 4.15 4.15 4.15 4.15 4.33 4.15 4.15 3.90
Δ ind -- - 16,770 5,117 4,943 22,146 5,117 5,382
Δ grp -- 4,333 7,849 9,741 9,922 7,921 8,913 8,638
n = 8
k = 10
Wj 0% 100% 89% 33% 30% 75% 50% 52%
Pc,j 0 3,672 2,692 2,221 2,183 2,573 2,362 2,426
k.Ps,j 2,802 2,802 1,729 787 768 1,491 1,069 1,112
Ps,0 280 280 173 240 242 149 240 238
m 4.29 4.29 4.29 4.29 4.60 4.29 4.29 3.89
Δ ind - - 24,832 7,914 7,659 31,904 7,914 8,288
Δ grp - 6,719 13,892 16,840 17,302 14,238 15,514 14,838
n = 1 k = 25
Wj 0% 100% 100% 67% 67% 100% 75% 75%
Pc,j 0 8,460 6,535 4,193 4,218 6,535 4,376 4,453
k.Ps,j 6,285 6,285 5,972 1,288 1,301 5,972 1,653 1,705
Ps,0 251 251 239 239 239 239 239 238
m 4.04 4.04 4.04 4.04 3.97 4.04 4.04 3.86
Δ ind - - 2,629 2,629 2,705 2,629 2,629 2,838
Δ grp - 2,228 4,152 6,003 5,970 4,152 5,899 5,802
n = 2 k = 25
Wj 0% 100% 100% 67% 67% 100% 75% 75%
Pc,j 0 8,787 6,512 4,152 4,199 6,512 4,330 4,481
k.Ps,j 6,612 6,612 5,987 1,268 1,291 5,987 1,623 1,724
Ps,0 264 264 239 239 239 239 239 237
m 4.15 4.15 4.15 4.15 4.02 4.15 4.15 3.80
Δ ind - - 5,117 5,117 5,251 5,117 5,117 5,503
Δ grp - 4,333 8,883 12,648 12,524 8,883 12,445 12,067
n = 3 k = 25
Wj 0% 100% 100% 67% 67% 100% 75% 75%
Pc,j 0 9,114 6,491 4,115 4,181 6,491 4,288 4,511
k.Ps,j 6,939 6,939 6,002 1,249 1,282 6,002 1,595 1,743
Ps,0 278 278 240 240 239 240 240 237
m 4.26 4.26 4.26 4.26 4.07 4.26 4.26 3.74
Δ ind - - 7,462 7,462 7,640 7,462 7,462 7,994
Δ grp - 6,331 14,199 19,934 19,669 14,199 19,637 18,799
68
Table 3.16: Results for medium-haul, high-density corridor including additional charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 5 k = 10
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 3,445 2,787 1,946 1,946 2,787 1,946 1,946
k.Ps,j 2,575 2,575 2,562 879 879 2,562 879 879
Ps,0 257 257 256 256 256 256 256 256
m 18.31 18.31 18.31 18.31 18.31 18.31 18.31 18.31
Δ ind - - 5,832 5,832 5,832 5,832 5,832 5,832
Δ grp - 983 4,271 8,480 8,480 4,271 8,480 8,480
n = 25 k = 10
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 3,495 2,787 1,946 1,946 2,787 1,946 1,946
k.Ps,j 2,625 2,625 2,563 881 881 2,563 881 881
Ps,0 263 263 256 256 256 256 256 256
m 18.50 18.50 18.50 18.50 18.50 18.50 18.50 18.50
Δ ind -- - 28,856 28,856 28,856 28,856 28,856 28,856
Δ grp -- 4,864 22,588 43,613 43,613 22,588 43,613 43,613
n = 50
k = 10
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 3,559 2,786 1,946 1,946 2,786 1,946 1,946
k.Ps,j 2,689 2,689 2,564 883 883 2,564 883 883
Ps,0 269 269 256 256 256 256 256 256
m 18.75 18.75 18.75 18.75 18.75 18.75 18.75 18.75
Δ ind - - 56,946 56,946 56,946 56,946 56,946 56,946
Δ grp - 9,602 48,253 90,257 90,257 48,253 90,257 90,257
n = 2 k = 25
Wj 0% 100% 100% 69% 69% 100% 77% 77%
Pc,j 0 8,612 5,884 3,036 3,036 5,884 3,074 3,075
k.Ps,j 6,437 6,437 6,406 710 710 6,406 785 786
Ps,0 257 257 256 256 256 256 256 256
m 18.31 18.31 18.31 18.31 18.28 18.31 18.31 18.23
Δ ind - - 5,832 5,832 5,871 5,832 5,832 5,925
Δ grp - 983 6,440 11,927 11,925 6,440 11,886 11,882
n = 10 k = 25
Wj 0% 100% 100% 69% 69% 100% 77% 77%
Pc,j 0 8,739 5,882 3,032 3,035 5,882 3,070 3,077
k.Ps,j 6,564 6,564 6,407 708 709 6,407 783 788
Ps,0 263 263 256 256 256 256 256 256
m 18.50 18.50 18.50 18.50 18.35 18.50 18.50 18.14
Δ ind - - 28,856 28,856 29,039 28,856 28,856 29,305
Δ grp - 4,864 33,428 60,891 60,855 33,428 60,689 60,597
n = 20 k = 25
Wj 0% 100% 100% 69% 69% 100% 77% 77%
Pc,j 0 8,897 5,880 3,028 3,034 5,880 3,065 3,080
k.Ps,j 6,722 6,722 6,409 706 708 6,409 780 789
Ps,0 269 269 256 256 256 256 256 256
m 18.75 18.75 18.75 18.75 18.44 18.75 18.75 18.01
Δ ind - - 56,946 56,946 57,293 56,946 56,946 57,812
Δ grp - 9,602 69,935 124,924 124,784 69,935 124,527 124,160
69
Table 3.17: Results for long-haul, low-density corridor including additional charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 2 k = 10
Wj 0% 100% 100% 39% 39% 100% 54% 55%
Pc,j 0 4,806 4,096 2,892 2,884 4,096 3,051 3,073
k.Ps,j 3,576 3,576 3,436 1,026 1,023 3,436 1,344 1,359
Ps,0 358 358 344 344 344 344 344 343
m 3.46 3.46 3.46 3.46 3.50 3.46 3.46 3.37
Δ ind - - 2,974 2,974 2,913 2,974 2,974 3,126
Δ grp - 2,080 3,498 5,413 5,436 3,498 5,073 5,016
n = 5 k = 10
Wj 0% 100% 100% 38% 37% 100% 53% 55%
Pc,j 0 5,024 4,084 2,869 2,849 4,084 3,025 3,078
k.Ps,j 3,794 3,794 3,444 1,015 1,005 3,444 1,327 1,362
Ps,0 379 379 344 344 345 344 344 343
m 3.58 3.58 3.58 3.58 3.69 3.58 3.58 3.34
Δ ind -- - 7,195 7,195 7,050 7,195 7,195 7,532
Δ grp -- 5,031 9,732 14,619 14,765 9,732 13,773 13,428
n = 8
k = 10
Wj 0% 100% 100% 37% 36% 86% 53% 55%
Pc,j 0 5,242 4,072 2,849 2,817 3,325 3,002 3,084
k.Ps,j 4,012 4,012 3,452 1,005 989 1,957 1,312 1,366
Ps,0 401 401 345 345 346 196 345 342
m 3.69 3.69 3.69 3.69 3.88 3.69 3.69 3.32
Δ ind - - 11,122 11,122 10,909 51,764 11,122 11,604
Δ grp - 7,804 17,160 25,126 25,501 22,191 23,774 22,922
n = 1 k = 25
Wj 0% 100% 100% 71% 70% 100% 78% 78%
Pc,j 0 12,105 9,003 5,577 5,609 9,003 5,767 5,855
k.Ps,j 9,030 9,030 8,592 1,740 1,756 8,592 2,121 2,179
Ps,0 361 361 344 344 343 344 344 343
m 3.48 3.48 3.48 3.48 3.42 3.48 3.48 3.33
Δ ind - - 3,698 3,698 3,799 3,698 3,698 3,938
Δ grp - 2,586 5,688 8,577 8,533 5,688 8,478 8,368
n = 2 k = 25
Wj 0% 100% 100% 71% 70% 100% 78% 78%
Pc,j 0 12,560 8,977 5,530 5,592 8,977 5,714 5,887
k.Ps,j 9,485 9,485 8,610 1,716 1,747 8,610 2,086 2,200
Ps,0 379 379 344 344 343 344 344 342
m 3.58 3.58 3.58 3.58 3.45 3.58 3.58 3.28
Δ ind - - 7,195 7,195 7,373 7,195 7,195 7,637
Δ grp - 5,031 12,197 18,048 17,883 12,197 17,858 17,427
n = 3 k = 25
Wj 0% 100% 100% 71% 71% 100% 78% 78%
Pc,j 0 13,014 8,953 5,487 5,575 8,953 5,666 5,920
k.Ps,j 9,939 9,939 8,626 1,695 1,739 8,626 2,053 2,223
Ps,0 398 398 345 345 344 345 345 342
m 3.67 3.67 3.67 3.67 3.49 3.67 3.67 3.23
Δ ind - - 10,488 10,488 10,726 10,488 10,488 11,098
Δ grp - 7,353 19,536 28,411 28,060 19,536 28,136 27,181
70
Table 3.18: Results for long-haul, high-density corridor including additional charter benefit
Simultaneous Game Leader Follower Game
Scenario 1 2 3a 3b 4 3a 3b 4
n = 5 k = 10
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 4,886 3,793 2,487 2,487 3,793 2,487 2,487
k.Ps,j 3,656 3,656 3,638 1,028 1,028 3,638 1,028 1,028
Ps,0 366 366 364 364 364 364 364 364
m 15.67 15.67 15.67 15.67 15.67 15.67 15.67 15.67
Δ ind - - 8,096 8,096 8,096 8,096 8,096 8,096
Δ grp - 1,149 6,614 13,140 13,140 6,614 13,140 13,140
n = 25 k = 10
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 4,956 3,792 2,487 2,487 3,792 2,487 2,487
k.Ps,j 3,726 3,726 3,639 1,030 1,030 3,639 1,030 1,030
Ps,0 373 373 364 364 364 364 364 364
m 15.84 15.84 15.84 15.84 15.84 15.84 15.84 15.84
Δ ind -- - 40,049 40,049 40,049 40,049 40,049 40,049
Δ grp -- 5,682 34,797 67,406 67,406 34,797 67,406 67,406
n = 50
k = 10
Wj 0% 100% 100% 0% 0% 100% 0% 0%
Pc,j 0 5,044 3,791 2,487 2,487 3,791 2,487 2,487
k.Ps,j 3,814 3,814 3,639 1,033 1,033 3,639 1,033 1,033
Ps,0 381 381 364 364 364 364 364 364
m 16.05 16.05 16.05 16.05 16.05 16.05 16.05 16.05
Δ ind - - 79,022 79,022 79,022 79,022 79,022 79,022
Δ grp - 11,216 73,913 139,077 139,077 73,913 139,077 139,077
n = 2 k = 25
Wj 0% 100% 100% 84% 84% 100% 88% 88%
Pc,j 0 12,214 8,249 4,236 4,237 8,249 4,259 4,260
k.Ps,j 9,139 9,139 9,095 1,069 1,070 9,095 1,115 1,116
Ps,0 366 366 364 364 364 364 364 364
m 15.67 15.67 15.67 15.67 15.63 15.67 15.67 15.61
Δ ind - - 8,096 8,096 8,160 8,096 8,096 8,194
Δ grp - 1,149 9,079 16,951 16,949 9,079 16,939 16,934
n = 10 k = 25
Wj 0% 100% 100% 84% 84% 100% 88% 88%
Pc,j 0 12,391 8,246 4,232 4,236 8,246 4,254 4,262
k.Ps,j 9,316 9,316 9,097 1,067 1,069 9,097 1,112 1,117
Ps,0 373 373 364 364 364 364 364 364
m 15.84 15.84 15.84 15.84 15.65 15.84 15.84 15.55
Δ ind - - 40,049 40,049 40,359 40,049 40,049 40,525
Δ grp - 5,682 47,122 86,517 86,453 47,122 86,455 86,354
n = 20 k = 25
Wj 0% 100% 100% 84% 84% 100% 88% 88%
Pc,j 0 12,611 8,244 4,227 4,236 8,244 4,248 4,264
k.Ps,j 9,536 9,536 9,099 1,065 1,069 9,099 1,108 1,119
Ps,0 381 381 364 364 364 364 364 364
m 16.05 16.05 16.05 16.05 15.67 16.05 16.05 15.47
Δ ind - - 79,022 79,022 79,613 79,022 79,022 79,939
Δ grp - 11,216 98,565 177,433 177,186 98,565 177,315 176,917
71
The results indicate that CS performs better in shorter and lower density corridors when
SS has special group prices, although including additional time savings makes it
profitable in almost all cases. Further, CS is more competitive when the group size if
larger, while its market share (of group travelers) decreases as the number of groups
increases. The entry of CS is beneficial to both individual travelers as well as group
travelers, although the benefit is less when SS is the market leader as opposed to being an
equal competitor.
In all the cases, the cubic equations in section 2.3.5 for scenario 4 result in a
unique, positive SS frequency for scenario 4. Further, the variation in frequency from
scenario 3(b) to 4 is fairly small. Only in low density corridors, with smaller group sizes,
a larger number of groups, and no inherent service advantage of CS does the frequency
change exceed one flight per day.
72
The focus in this chapter is to develop a charter strategic planning model. To this end, we
first describe the schedule planning process for scheduled airlines, particularly the fleet
assignment problem which bears some similarity to our model here. We then describe the
problem and the differences from the schedule airline planning and airline fleet
assignment problem.
Scheduled airlines use demand estimates and desired travel time profiles for the various
links of a network, to plan flight schedules and assign their fleet to scheduled flights in
order to maximize profit. Schedule planning involves determining when and where to
offer flights so that profits are maximized (e.g. Rushmeier et al, 1995), while fleet
assignment involves assigning aircraft types to flight legs to minimize operating cost (e.g.
73
Barnhart and Lohatepanont, 2004). Schedule planning and fleet assignment are not
independent of each other, since scheduling a certain flight requires the availability of an
aircraft with sufficient capacity to adequately serve the demand for the flight. However,
given the large size of the problem, historically the decision making step has been broken
down into steps that are solved sequentially. Typically, fleet assignment is modeled as a
multi-commodity network flow problem. The nodes in the underlying network flow
problem correspond to time intervals in which interconnection activity occurs. The
equipment types are the commodities, and the decision variables correspond to flow
along the network arcs (Barnhart et al, 2002). A significant weakness of this approach is
the requirement of fixed departure times. Rexing et al (2000) address this by assigning
time windows to each flight, and discretizing this window to allow for flexibility in
schedules. This approach is computationally expensive, and they present two algorithmic
approaches for solving this model.
In many ways, charter planning parallels the scheduled planning process as described
above. However, the basic difference between the services arises in the way the schedule
of flights is determined. A scheduled airline (SS) develops a schedule based on market
analysis and historical service, also considering fleet and crew constraints. This schedule
is then fixed for all the passengers. A charter service (CS) does not have a ―schedule‖ per
se. A charter service flies according to the groups’ desires. Thus, any ―schedule‖ of
flights or group-movements that the charter decides to serve is planned in response to the
desires of the group, with consideration to availability of resources (aircraft, crew) and
74
ferrying cost of aircraft. Once the charter ―schedule‖ is determined, it might seem that the
steps of fleet assignment, routing and crew scheduling can be performed as in scheduled
service planning. However, we assume a movement can be served by CS only if an
aircraft is available. In other words, treating schedule design and fleet assignment as two
sequential problems is not a practical approach for charter planning.
Given a set of required group-movements (teams traveling to pre-scheduled athletic
events), service characteristics of the best SS option for each movement, cost factors for
travel time, overnight stays, etc, and a charter fleet size and operating characteristics, we
find the subset of movements to be served by the CS, and assigns aircraft to the service,
so as to maximize total cost savings from operating the charter. The cost savings take the
form of charter profit of reductions in the total travel cost incurred by the school and its
athletes. Thus the cost minimization objective is equivalent to a profit maximization
objective assuming that the CS charge for each movement is the precise amount that
would make the school indifferent between the CS and SS alternative if it took into
account the full costs of each, including both money and time. . The movements not
served by charter use SS.
CS competitiveness depends on efficient fleet utilization. To increase efficiency, it is
desirable to introduce some flexibility into the scheduling of group-movements, while at
the same time recognizing that this market is very schedule-sensitive. Consider a
particular group-movement . Each group-movement consists of a set of inputs like
origin, destination, the number of people in the group, desired departure time, and the
75
relevant details for the best-scheduled service option. We assume each group-movement
has a desired departure time interval, rather than a fixed departure time. The size of such
an interval depends on the group and its event, and is an input to the model. The interval
size can be set to zero, if the time of departure is completely inflexible.
We thus assume that for each group-movement there is a departure interval from
to , and the movement is not served by CS unless it can depart within this time
interval. Further, we consider three subdivisions of this interval based on times and
. The interval from to is the ideal service interval
for group-movement , with the group being indifferent to the actual departure time
when it lies in this interval. However, if the departure time lies between the intervals
or , the group experiences a disutility, where cost increases linearly
with the difference from the ideal interval, or schedule delay. This penalty increases the
cost of CS travel. In the profit-maximizing formulation, this cost is absorbed by the CS in
the form of reduced fares. This scheme of linear penalties and piecewise linear time-
window is shown in figure 4.1, where and represent the slope of the linearly
penalties. It should be noted that the charter departure times are determined in advance so
that the teams can plan their activity schedules around them, reducing the cost of
deviating from the ideal service interval.
76
Figure 4.1: Departure time and associated penalties
The base of operations for the charter aircraft is an input to the model. This is the
location where the charter aircraft must be at the beginning of the planning horizon, and
to which they must return at the end of the planning horizon. We assume a single base,
but this could be relaxed with a simple modification in the formulation. The planning
horizon is the time period for which the model solves the assignment and routing
problem. This planning horizon could be based on some cyclical patterns in the
demanded group-movements, or maintenance constraints for the charter fleet as
elaborated below.
The set of group-movements to be served by either CS or SS, are an input to the
model. In our formulation, the group-movements must form an ordered sequence. This
pre-determined sequence establishes the order in which charter flights are served. If the
charter departure times were fixed, then the group-movements could be ordered by
departure time. But with the use of time intervals as shown in figure 4.1, the sorting order
Slope =
Slope =
Departure time for flight
Pen
alt
y f
or
Ch
arte
r
Dep
artu
re D
ela
y
77
is not so clear. With sufficiently long time intervals, it is possible two group-movements
(say and ) could be served by the same charter aircraft in either order. In our
application, the sequence is based on . In cases where more than one sequence is
possible, the model can be run multiple times for the various sequences to determine the
optimal solution. While the model could be altered to determine the sequence
endogenously, the problem size and subsequent solution times would increase
tremendously while the benefit from re-ordering would likely be small.
Based on the above concept of time windows and the pre-determined sequence, we
detail the mathematical formulation used to determine the subset of group-movements to
be served by charter aircraft. The mathematical formulation presented here determines
this subset, decides which group-movement is served by which aircraft, the time of
departure for the flight and the associated schedule delay penalty, and the routing of the
aircraft. The objective is, equivalently, cost minimization or CS profit maximization.
Cost components include air transport costs, airport access cost, time cost including
schedule delay, and accommodation cost. In the case of SS, air transport cost consists of
airfare, while in the case of CS, it includes aircraft operating and ownership cost. Other
cost components have the same definition irrespective of the service type. In the cost
minimization formulation, we seek to minimize the sum of all these costs. In the profit
maximization formulation, we define CS profit as the CS fare revenue minus CS air
transport cost, and set the fares for each flight at the sum of SS fare, costs of additional
travel time, airport access costs, and accommodation costs.
78
We first give the notation for parameters and variables, followed by the
mathematical formulation and its explanation. It should be noted that the price of
scheduled option includes the entire cost of using the scheduled option, including
the value of time spent in travel.
Parameters
is the set of all aircraft, and represents any aircraft in this set
is the set of all potential group-movements, and represents any group-
movement in this set
is the set of group-movement pairs , which cannot be
served by the same aircraft because of time conflict
is total cost of using SS for group-movement
is the total non-air transport cost of using CS for group-movement
is the operating cost for serving group-movement using aircraft
are the time-window parameters for each group-movement , as described in
figure 4.1
is the unit penalty for serving group-movement before or after , as
described in figure 4.1
is the relocation cost of aircraft between group-movements and
is the relocation cost for aircraft from base to origin of group-movement
79
is the relocation cost for aircraft from destination of group-movement to
base
is flying time for group-movement
is the time to relocate the aircraft from the destination of group-movement
to the origin of group-movement ,
is the time to relocate the aircraft from the base to the origin of group-
movement
is the time to relocate the aircraft from the destination of group-movement
to the base
Variables
is a binary variable, = 1 if group-movement is served by plane ; 0
otherwise
is a binary variable, = 1 if plane serves group-movement and then serves
group-movement , and does not serve any group-movement in between
; 0 otherwise
is a binary variable, = 1 if is the first group-movement that plane serves
during the planning horizon; 0 otherwise
is a binary variable, = 1 if is the last group-movement that plane serves
during the planning horizon; 0 otherwise
80
is a non-negative real number, and is the time of departure for group-
movement if served by plane . If group-movement is not served by
plane , then
is a non-negative real number, and is the schedule delay for departure times
before when group-movement is served by plane . If group-
movement is not served by plane , then
is a non-negative real number, and is the schedule delay for departure times
after when group-movement is served by plane . If group-movement
is not served by plane , then
Mathematical Formulation
(4.1)
such that
(4.2)
(4.3)
(4.4)
81
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
Equation (4.1) gives the objective function, which is the cost savings from serving
the specific set of group-movements with the specific aircraft as determined by . This
includes the savings in cost incurred from using SS, including fare, travel time,
accommodation, and airport access cost, minus the cost incurred from using CS,
including the operating cost of charter planes, the cost of relocating the charter aircraft
between flights, the cost of relocating the charter aircraft at the beginning and end of the
planning horizon, additional cost due to schedule delay, and travel time and airport access
costs incurred from using the CS. The ownership costs for the charter aircraft are not
explicitly included in the objective function, since they are assumed independent of
which group-movements are actually served. They are, however, introduced for the final
benefits assessment.
82
Constraint (4.2) ensures that each group-movement is served by no more than 1
charter aircraft. Constraint (4.3) ensures that the relocation variable is 1 if the
charter plane serves group-movement and then serves group-movement , and does
not serve any group-movement in between. If group-movement and are not served or
if there is another group-movement between the two, then could be 0 or 1. Since we
are maximizing the objective and the quantities pertinent to subtract from the
objective function, optimization forces to be 0 in this case. Equations (4.4) and (4.5)
are constraints on the departure time of the charter aircraft for any group-movement, and
ensure that if the group-movement is served by charter plane , then the time of
departure must lie between and . Equation (4.6) is the constraint on aircraft
availability, and ensures that if two group-movements and are served by the same
charter aircraft , then there should be sufficient time-difference between the respective
departure times to serve the first group-movement, and relocate to the origin of the
second one. Equations (4.7), (4.8) and (4.9) are constraints that evaluate the schedule
delay for charter as described in figure 4.1. Together, these equations ensure that is
the delay before and is the delay beyond only when group-movement
is served by charter aircraft . Equations (4.10) and (4.11) are relocation constraints at
the beginning and end of the planning horizon. Equation (4.10) states that the relocation
variable from the base is 1 if plane serves group-movement , and does not
serve any flight before this. Otherwise, could be 0 or 1, and since it subtracts a
positive quantity from the objective function, optimization forces to be 0.
83
Constraint (4.11) ensures the same constraint for relocation to the base at the end of the
planning horizon.
The charter fleet could be composed of multiple aircraft with different seat capacities.
With movements for different groups of varying sizes, the charter fleet needs to be
assigned the movements based on the compatibility of the aircraft seats and group size. In
the basic mathematical formulation presented in equations (4.1) to (4.11), the group size
and number of seats in the aircraft are not matched explicitly. The assignment of charter
aircraft can include group size with an additional inequality. Let be the seat capacity of
the aircraft and be the number of people in the group for movement
. The additional inequality given in equation (4.12) states that if plane serves
the movement , then number of seats in should be greater than or equal to the number
of people in the group.
(4.12)
An additional constraint could also be put on the total hours of operation of a charter
aircraft in the planning horizon. If be the limit on the total number of flight hours of
plane during the planning horizon, then the constraint can be formulated as:
(4.13)
84
In this section, we examine the computational aspects of the MIP model defined in
equations (4.1) to (4.11). We conduct experiments on the basic formulation by applying it
to realistic instances of varying sizes. This is followed by a description of certain
measures to improve the computational efficiency, and the quantification of benefits from
these improvements for the same instances.
For computational experiments we use 27 instances, and solve them using CPLEX 11.0
(ILOG, 2008) using AMPL as the interface on a single 3GHz CPU computer with 1 GB
RAM and 2 GB swap space. These instances are based on the case study described in the
chapter 5, where CS serves student athlete travel for the Big Sky conference. This
involves movement of groups over a network of nine nodes with a planning horizon of a
week. In all the instances, four planes of the same type are used as the charter fleet, and
the aircraft type is assumed to have sufficient capacity to serve all the groups. The 27
instances have different number of group movements, resulting in different problem
sizes. The case study analyzes the sports schedule for the entire year, and the number of
movements in each instance as well as the associated chronological week number is
given in table 4.1, along with a summary of computational times and results. The solution
times are reported for optimality gap of 0.1% (if IS denotes the best integer solution and
LP denotes the best linear relaxation bound, then optimality gap is ). For the
sake of convenience, the table has been sorted in the number of movements, since the
number of movements is related to the problem size.
85
Table 4.1: Computational experiments on the basic formulation
Instance #
Week #
Number of Movements
Solution Time (CPU second)
MIP Simplex Iterations
Branch and Bound Nodes
1 23 5 1.7 141 7
2 21 7 0.2 183 -
3 10 8 0.9 2,759 255
4 24 8 0.4 135 9
5 20 10 2.2 3,423 167
6 2 12 6.0 30,203 1,557
7 9 12 2.4 5,947 443
8 8 13 5.2 16,121 938
9 1 14 10.7 61,278 3,147
10 3 14 1.3 2,114 42
11 6 16 66.2 493,119 32,494
12 22 16 62.0 534,806 23,031
13 25 19 9.8 41,544 1,453
14 17 20 1,421.7 7,555,295 474,625
15 26 21 325.8 1,916,774 91,997
16 13 24 2,875.9 8,922,365 466,567
17 5 25 3,035.0 14,336,147 341,159
18 11 25 21,822.1 92,726,432 2,357,978
19 27 25 115,017.0 - -
20 4 26 60,519.8 - -
21 19 26 103,872.0 527,754,168 15,786,968
22 12 28 3,528.0 15,614,000 314,410
23 15 31 21,150.8 43,248,184 2,381,559
24 18 33 44,257.9 - -
25 14 34 62,471.5 108,845,191 3,171,149
26 16 36 133,742.0 - -
27 7 45 72,103.6 - -
(Shaded boxes are instances where CPLEX ran out of memory)
The results in table 4.1 show that solution time is greater than 20,000 CPU seconds
in 9 out of 11 instances where the number of movements was greater than 25. Further, in
5 out of these 11, CPLEX had to quit before the optimality gap was achieved due to
inadequate memory. In the rest of the high movement instances, the number of branch
and bound nodes is substantial. In summary, the basic formulation is computationally
86
very expensive, and methods to reduce the computational time are needed. In the next
section, we define some additional inequalities to reduce the computational time, and the
test the benefits from these inequalities.
An inequality is called a valid inequality for a polyhedron if it is satisfied by all the points
in the polyhedron Nemhauser and Wolsey (1988). A similar notion is true for discreet
sets (which is the case in our formulation), where an inequality is a valid inequality for a
discreet set if it is true for all points within the set. Such inequalities are not part of the
basic formulation, but can be used to reduce the computational times of the problem. One
method to utilize valid inequalities is to use ―branch-and-cut‖ instead of branch-and-
bound (used in section 4.5.1). Branch-and-cut is a variant of the branch-and-bound
mechanism, where after solving for the LP relaxation and failure in pruning the node
based on the LP solution, we try to find a ―cut‖ or valid inequality based on the integral
nature of the solution. If such violated cuts are found, they are added to the formulation
and the LP is solved again. If such cuts are not found, we branch on the node.
In our case, however, we utilize valid inequalities by including them within the
basic formulation before branching. Branch-and-bound is used on this augmented
formulation, and results in ―tighter‖ LP relaxations at each node. The valid inequalities
are formulated based on the structure of the problem, with emphasis on redundancy in
certain set of variables when a subset is fixed. In the following paragraphs, we explain
the logic behind each valid inequality and give a mathematical representation.
87
While describing the basic formulation, we defined the set , comprised of
movement pairs that cannot be served by the same aircraft. This is because even at the
limit of the time windows there is not enough time to serve the first and ferry the aircraft
to the origin of the second movement. This aspect of the solution is implicit in equation
(4.6), but can be made explicit as given below in equation (4.14).
(4.14)
The large problem size in the basic formulation is primarily a result of the variables,
which track the sequence in which movements are served by an aircraft and the resulting
ferrying of empty aircraft. is defined for all possible movement pairs for each aircraft,
and for a 50 movement 5 aircraft problem this results in 6,625 binary variables. Since
these variables are defined for each movement pair to determine the sequence, this
introduces a lot of redundancy. In other words, fixing certain or variables as either
zero or one would pre-determine some other variables for a feasible solution. In the
next set of equations, we consider such aspects of the feasible solution to determine valid
inequalities.
It is obvious that if a particular aircraft serves a movement , that aircraft would
either serve a movement before and fly empty to the origin of , or could be the first
movement the aircraft serves in that planning horizon. A similar argument can be made
for the sequence of events after movement , which would include either flying empty to
the origin of the next movement, or to the base for the end of the planning horizon. The
above conditions are represented mathematically in equations (4.15) and (4.16). In
equation (4.15), if flight is served by plane (in other words, ), then either one
of the movement pair variables is one (meaning that the aircraft relocated from the
88
destination of a previous movement , or the origin of the subsequent movement
coincides with the destination of the previous movement) or the variable for start of
planning period is one (meaning the aircraft relocates from the base to the origin of
movement ). If is not served by this aircraft, then all these variables are necessarily
zero. A similar case is made for the aircraft relocation after flight in equation (4.16).
(4.15)
(4.16)
The equations (4.15) and (4.16) relate the variable to the selection of the movements
and . Besides this, the definition of the variable states that it is zero if aircraft
serves any other movement between and . Thus, if any other flight is served in
between, would be zero. This is stated explicitly in equation (4.17).
(4.17)
Similar to this is the case for the relocation from the base at the beginning and to the base
at the end of the planning period. If is the first movement for plane in the planning
horizon, then plane cannot serve any movement that occurs before . A similar
argument holds for the last movement of the aircraft and the relocation to the base, and
these are represented as valid inequalities in equations (4.18) and (4.19).
(4.18)
(4.19)
89
Further, it is clear that only one movement can be the plane’s first or last movement for
the planning period, and this is stated in equations (4.20) and (4.21). These constraints are
inequalities because aircraft may not be used to serve any movement in a given planning
period; if it is used then the constraints can be written as equalities.
(4.20)
(4.21)
In addition to constraints on the binary variables, constrains can be formulated for the
positive, real valued variables too. and are variables defined to measure the
deviation of charter service from the ideal window, as shown in figure 4.1. In case
movement is not served by plane , both and will be zero. Although this is
implicit in the feasible solution from equations (4.7), (4.8) and (4.9), it can stated
explicitly as shown in equations (4.22) and (4.23), which constrain and to be
zero when , and otherwise yield an upper bound.
(4.22)
(4.23)
Next consider the departure time of an aircraft serving its first movement of the planning
period. All else being equal, the earlier this time the better, because this will make the
plane available sooner for subsequent movements. Thus the departure time for the first
movement should always be at or before the time when the schedule delay penalty for
earliness takes effect. This implies:
90
(4.24)
If be the first movement served by plane , then the departure time for this flight would
lie between and . Equation (4.4) will set the lower bound and above equation
(4.24) will set the upper bound since . If is not the first movement
for plane , then , and the upper limit for if . A similar argument can
be made for the last movement served by plane where the departure time would be
greater than or equal to , and inequality for this case is given in equation (4.25).
(4.25)
It should be noted that the inequalities in equation (4.24) and (4.25) are valid only if
there are no constraints on when an aircraft departs from or arrives at the operational
base. If there are such constraints, these inequalities may no longer be valid, but can be
modified to reflect the required departure time or arrival time at the base.
The valid inequalities described in the previous section are based on the attributes of the
optimal solution, and these inequalities are included within the basic formulation when
being input into CPLEX. These inequalities could, thus, lead to better linear-
programming relaxations (LP relaxation), where the integrality of the discrete variables is
relaxed and the resulting linear program is solved to optimality. For the 27 problem
instances described before in table 4.1, we compared the LP relaxation from the basic
formulation and the ―augmented‖ formulation realized by including the extra constraints,
91
and the results are given in table 4.2. There is a consistent improvement in the LP
objective of the augmented formulation, with a minimum improvement of 12% and an
average improvement of 21%. There is no trend of increasing improvement with problem
size, which is expected since the bounds resulting from inclusion of valid inequalities are
based on the instance data (primarily the time-window size for each movement).
Table 4.3 compares the solution times, number of simplex iterations and branch-
and-bound modes for the basic formulation and the augmented one for an optimality gap
of 0.1%. The contrast between the basic and augmented formulation is much more
pronounced here as compared to table 4.2. In all but one instance, the solution time is less
than 1,000 CPU seconds. None of the instances terminated prematurely due to memory
allocation issues. The number of iterations is also considerably reduced with the inclusion
of valid inequalities. For small instances of the problem as well as for some large sized
instances, branching did not occur at all. The LP relaxation and the in-built heuristics in
CPLEX for generating a feasible solution resulted in a solution within the optimality gap.
Where branching did occur, the number of branch-and-bound nodes was considerably
smaller than the basic formulation. Overall, there is only one instance where the solution
time was considerably high (the third instance with 25 movements).
92
Table 4.2: Comparison of LP Relaxation with and without valid inequalities
Instance # Week # Number of Movements
LP Relaxation Objective % Improvement with Valid
Inequalities Basic
Formulation With Valid
Inequalities
1 23 5 26,392.29 33,588.81 27.3%
2 21 7 31,121.50 34,863.33 12.0%
3 10 8 35,131.63 43,855.47 24.8%
4 24 8 33,483.23 40,314.54 20.4%
5 20 10 39,284.05 49,816.22 26.8%
6 2 12 54,843.32 65,991.15 20.3%
7 9 12 48,223.36 58,917.81 22.2%
8 8 13 44,768.60 57,976.95 29.5%
9 1 14 61,643.29 74,231.12 20.4%
10 3 14 62,942.10 76,023.68 20.8%
11 6 16 77,711.56 92,052.43 18.5%
12 22 16 76,979.20 92,421.34 20.1%
13 25 19 67,896.32 77,687.94 14.4%
14 17 20 81,308.99 94,292.21 16.0%
15 26 21 54,427.54 61,788.44 13.5%
16 13 24 103,298.18 126,477.31 22.4%
17 5 25 102,358.51 122,261.95 19.4%
18 11 25 110,480.26 134,153.31 21.4%
19 27 25 124,847.11 152,978.66 22.5%
20 4 26 108,487.58 132,216.28 21.9%
21 19 26 111,227.96 136,609.95 22.8%
22 12 28 115,101.42 140,183.70 21.8%
23 15 31 121,444.61 152,561.95 25.6%
24 18 33 143,514.72 174,195.79 21.4%
25 14 34 146,428.54 175,008.04 19.5%
26 16 36 144,598.04 169,673.55 17.3%
27 7 45 196,386.12 245,956.71 25.2%
93
Table 4.3: Comparison of solution times with and without valid inequalities
Number of Movements in Instance
Solution Time (CPU second)
MIP Simplex Iterations Branch and Bound
Nodes
Basic With Valid
Inequalities Basic
With Valid Inequalities
Basic With Valid
Inequalities
5 1.7 0.0 141 50 7 -
7 0.2 0.0 183 66 - -
8 0.9 0.0 2,759 56 255 -
8 0.4 0.1 135 112 9 -
10 2.2 0.1 3,423 171 167 -
12 6.0 0.1 30,203 169 1,557 -
12 2.4 0.1 5,947 177 443 -
13 5.2 0.1 16,121 177 938 -
14 10.7 0.2 61,278 306 3,147 -
14 1.3 0.2 2,114 203 42 -
16 66.2 0.4 493,119 445 32,494 -
16 62.0 3.3 534,806 495 23,031 6
19 9.8 0.6 41,544 157 1,453 -
20 1,421.7 4.6 7,555,295 2,548 474,625 16
21 325.8 1.4 1,916,774 1,659 91,997 -
24 2,875.9 25.7 8,922,365 13,364 466,567 112
25 3,035.0 29.3 14,336,147 10,224 341,159 112
25 21,822.1 213.2 92,726,432 90,350 2,357,978 929
25 115,017.0 146,519.0 - 73,789,424 - 337,061
26 60,519.8 61.7 - 21,122 - 214
26 103,872.0 13.6 527,754,168 6,873 15,786,968 1
28 3,528.0 51.5 15,614,000 13,200 314,410 110
31 21,150.8 72.1 43,248,184 13,912 2,381,559 14
33 44,257.9 8.5 - 8,399 - -
34 62,471.5 838.2 108,845,191 106,588 3,171,149 537
36 133,742.0 17.2 - 6,898 - -
45 72,103.6 310.0 - 30,518 - 59
(Shaded boxes are instances where CPLEX ran out of memory)
We examined the computationally expensive instance to identify the reasons for
inefficiency and the methods to mitigate them. The instance includes track and field
competition involving nine teams at a single location. Such simultaneous events
94
introduce redundancy in the basic formulation even when the valid inequalities are
included. Consider the case where the schedule comprises of individual events over
various nodes in the network. The inclusion of valid inequalities here links the selection
of movements closely, identifying various movements that cannot be served with the
same aircraft when a certain movement is served. This is because serving multiple
movements involves relocation of the aircraft between the nodes, and can occur only if
the relocation time falls within the time windows. However, in a conference event at a
single location, the time windows almost coincide since the groups leave only when the
event is over. Further, the relocation time is almost negligible, since all aircraft converge
to a single node around the same time when the groups arrive, and depart from that node
itself. Thus, even with the inclusion of valid inequalities, selection of a particular
―arrival‖ movement for an aircraft still leaves the possibility of serving any ―departure‖
movement, or turning around and serving another arrival movement. In such a case, the
formulation leads to many branches and is thus computationally expensive.
If the time windows are sufficiently small such that the possibility of serving
multiple arrivals by the same aircraft is very limited, a heuristic can be designed to
circumvent the large computational times for the conference-event case. The heuristic
centers on the expectation that each aircraft will serve only one arrival and one departure.
If this is true, both arrivals and departures can be sorted in terms of their addition to the
objective function, and the most profitable arrivals and departures could be selected. This
would yield a feasible solution potentially close to the optimal solution, which could be
used as an input to branching to reduce computational time.
95
As an application of the strategic planning model described in the previous chapter, we
apply the model to plan charter service for student athlete travel for the Big Sky
Conference for the 2006-2007 season. Besides demonstrating the model, we investigate
the potential of charter air service to reduce the burden of intercollegiate athletic travel on
student athletes, and develop a model intended to optimize the realization of that
potential. For some time now there has been attention to the issue of whether or not
student athletes bear an unreasonable burden for participation in college athletics (Knorr,
2003). Athletic events provide a significant amount of publicity, and at times revenue, for
Universities, but students bear the burden of carrying a full course load as well as an
athletic career.
The impact on student welfare is not clear (Pascarella et al, 1995). A substantial
amount of evidence suggests that athletic participation may be negatively linked with
such outcomes as involvement and satisfaction with the overall college experience, career
96
maturity, and clarity in educational and occupational plans [(Blann, 1985, Kennedy and
Dimmick, 1987, Sowa and Gressard, 1983, Stone and Strange, 1989), but there is also a
body of evidence that indicates various objective indexes of career success are not
correlated with collegiate athletic participation (DuBois, 1978, Howard, 1986).
Graduation rates and GPAs for student athletes can even exceed University averages
(Wright State University, 2004). It is also clear these effects may vary by sport and
gender (Pascarella et al, 1995). It is often more difficult for student athletes to be science
and engineering majors because of the increased class time and conflict with lab course
meeting times (Pascarella et al, 1995).
A significant portion of an athlete’s time is spent traveling. This reduces the
amount of time available to spend on academic pursuits, which may impact their
Academic Performance Rate. The Academic Performance Rate is a key metric defined
by the National Collegiate Athletic Association (NCAA) to track student performance. It
includes the number of student athletes that remain academically eligible, remain full-
time students, and graduate. Institutions that do not graduate at least 50% of their student
athletes will be at risk of losing funding from the NCAA (Knorr, 2003).
Most coaches currently choose to have their team travel to an event the day before
the game, in part to guard against unreliable travel and ensure the team reaches the
destination on time, further burdening the student by increasing the time spent away from
their home campus. Although the NCAA limits required team-related activities to 20
hours a week, travel to and from events is not included in this 20 hours (NCAA, 2006).
Less time spent on athletic travel means less time spent away from the student’s home
97
campus and the academic resources—from labs and libraries to professors and tutors—
available there.
Currently most college sports teams travel between events using commercial air
service. Many Universities are located in places where commercial air service is
infrequent, not conveniently scheduled, and often indirect. Thus, using a charter service
sports teams would benefit from reduced travel times in
Accessing the origin and destination airport
Check-in and security screening at the airport
Actual flying time (including time spent at connecting airports)
Baggage claim at the destination airport
Schedule delay (defined as the deviation from desired departure/arrival time for
the scheduled service)
The primary disadvantage of a dedicated charter could be the cost of operation. These
include the actual cost of flying between the intended airports, the cost of ferrying the
empty aircraft and ownership cost of the aircraft.
The objective of this study was to identify the ―total benefit pool‖ for the student
athletes and schools from using the charter service. These benefits would include savings
in time, and savings in money (which might be negative, signifying an increased cost).
To identify the benefits, a benchmark is needed against which the comparison can be
done. We use the existing scheduled service as the baseline. In practice this total benefit
pool would be shared among the participating Universities, in the form of time and
money cost savings, the charter carrier operator, in the form of profit. We do not consider
impacts of traffic diversion on scheduled airlines profits or the remaining passengers.
98
Obviously, the comparison between the scheduled service (SS) and the charter
service (CS) cannot be done at an individual event or flight basis, since ferrying cost for
the charter aircraft is substantial. Thus, we consider the entire set of team movements for
comparison. Each team movement could be served either by the charter or the scheduled
airline service. Given the costs involved (including time converted to costs using an
appropriate value of time), the problem reduces to routing a limited fleet of aircraft to
serve a set of team movements required by a pre-determined event schedule in a cost
effective manner.
In the following sections, we describe the data and assumptions for the Big Sky
conference as well as the charter and scheduled service. This is followed by a description
of the results from applying the charter strategic planning model for different operational
assumptions. The implementation of MIP was done in CPLEX 11.0, similar to discussion
on computational aspects in the previous chapter.
The Big Sky Conference was used as a case study for charter planning. Big Sky is a
Division I NCAA conference made up of universities in Arizona, California, Colorado,
Idaho, Oregon, Montana, Utah, and Washington (figure 5.1 shows the geographical
locations of the nine universities). Table 5.1 identifies the universities, their location and
the short code used as a reference in the following analysis. As can be seen from the
table, most of the universities are located at quite a distance from well-served commercial
airports. We assume that the charter aircraft use an airport closer to the campus wherever
possible, reducing the time and money spent for airport access as well as the waiting time
99
for the scheduled flight when the service is not frequent. Table 5.2 shows the sports in the
NCAA Big Sky Conference.
Table 5.1: Universities in the NCAA Big Sky Conference
University Code Location of University
Airport used with charter aircraft
(approx. distance from University)*
Airport used with scheduled aircraft
(approx. distance from University)*
Eastern Washington University
EAW Cheney, Washington
GEG (Spokane) (13 miles)
GEG (Spokane) (13 miles)
Idaho State University
IDS Pocatello, Idaho
PIH (Pocatello) (12 miles)
SLC (Salt Lake City) (166 miles)
University of Montana
UMO Missoula, Montana
MSO (Missoula Int.) (7 miles)
MSO (Missoula Int.) (7 miles)
Montana State University
MOS Bozeman, Montana
BZN (Gallatin Field) (10 miles)
BZN (Gallatin Field) (10 miles)
Northern Arizona University
NAU Flagstaff, Arizona
FLG (Flagstaff Pulliam) (5 miles)
PHX (Phoenix) (148 miles)
University of Northern Colorado
UNCO Greeley, Colorado
GXY (Greeley-Weld County) (5 miles)
DEN (Denver) (56 miles)
Portland State University
PSU Portland, Oregon
PDX (Portland) (13 miles)
PDX (Portland) (13 miles)
Weber State University
WSU Ogden, Utah SLC (Salt Lake City) (35 miles)
SLC (Salt Lake City) (35 miles)
California State University at Sacramento
CSU Sacramento, California
SMF(Sacramento) (17 miles)
SMF(Sacramento) (17 miles)
* Source: Google Maps (maps.google.com)
Table 5.2: Sports in the NCAA Big Sky Conference
Men’s Sports Women’s Sports
Basketball Basketball
Cross Country Cross Country
Football Soccer
Tennis Golf
Track and Field Tennis
Track and Field
Volleyball
100
Figure 5.1: Location of the nine Big Sky schools
Outside of football, all sports can be handled with 30 seat aircraft. This provides
significant flexibility in terms of terminal access and uniformity of fleet, and hence we
excluded football from the schedule of events considered. For these 9 universities and 11
sports teams, a table of events (including game times) for the 2006-2007 season was
developed from published schedules on the university websites. In case the start time for
the game was not available, we assume a start time based on other similar events. The
game duration is assumed to be 3 hours for all games. In the case that a team has two
events on consecutive dates away from their home university, we assume the team would
travel directly from one event to the next, and not return to the home university between
events.
101
A demanded flight is defined as an origin-destination team movement required for
participation in a Big Sky Conference event. In reality, this movement could also be
served by a bus, but we assume that all such movements are served either by a scheduled
airline or a charter airline. Our aim here is to demonstrate the planning model developed
earlier and to highlight the contrast between SS and CS in the case of group travel. All of
these flights will be served by either chartered flights, or scheduled service. The schedule
of events spans 27 weeks and includes 557 team movements. All of these movements
occur on either Wednesday, Thursday, Friday, Saturday, or Sunday, leaving at least 24
hours between Monday and Tuesday for scheduled maintenance to the aircraft. We
assume this occurs at a home base, which is located at one of the airports used to access
the charter service listed in Table 5.1. As discussed later in section 5.3.4 we carried out
the optimization for six of the nine base locations that were judged to be most promising
and found that PIH, the charter airport used by Idaho State, was the least cost base,
although the differences among the alternatives were quite small. Further, with the
cyclical nature of the demand over the week, we use a one week planning horizon.
Parameters defined above include the set of events to be served, which changes from
week to week, so that for a given set of parameter values the model must be run 27 times.
While analyzing the data for each week, we observed that the number of
movements varies significantly over different weeks. In figure 5.2 we show the number
of movements for each week. As can be seen from this figure, the number of movements
varies from 5 to 45 over the 27 weeks. Such variation in demand across weeks implies
that there are diminishing returns from increasing the size of the charter fleet. As the
102
number of charter planes increases, so do the number of weeks when a given aircraft is
under-utilized or idle.
Figure 5.2: Variation in team movements over different weeks
We analyzed the spatial pattern of the movements by developing an
origin/destination table, shown in table 5.3. As can be seen, there is variation among total
flights for each origin or destination (the right-most column and the bottom row), as well
as over individual origin-destination pairs. Further, as expected given the balanced nature
of athletic schedules, there is no single station that clearly predominates. Thus there is no
clear-cut choice for the base of operations. We evaluated the best base of operations by
performing multiple runs of the model, as discussed in section 5.3.4.
0
5
10
15
20
25
30
35
40
45
50
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Team
Mo
vem
en
ts d
uri
ng
We
ek
Week Number
103
Table 5.3: Flights demanded across origin and destination
Destination
CSU EAW IDS MOS NAU PSU UNCO WSU
Ori
gin
CSU . 7 12 7 11 8 5 14 64
EAW 9 . 3 4 4 14 6 5 45
IDS 12 4 . 10 10 6 8 14 64
MOS 9 4 5 . 4 3 7 6 38
NAU 10 7 10 6 . 4 17 7 61
PSU 6 13 7 3 6 . 4 5 44
UNCO 10 6 9 3 20 5 . 3 56
WSU 11 5 16 5 6 5 8 . 56
67 46 62 38 61 45 55 54
The team size was estimated using an average of the number of players on the roster at
each of the Big Sky Universities. To this we added 5 to account for coaches and trainers.
For travel to the venue (inbound flight), we assume that the team arrives at the
venue on the day before the game. This results in accommodation cost for a day, assumed
at $100 per person. Further, we assume that the teams would be willing to leave after
4pm from home campus, and would want to reach the destination before 9pm in order to
rest and prepare before the day of the game. Thus, scheduled service flights that met this
criterion had no schedule delay, and if the best flight went beyond this window, the extra
time was treated as schedule delay. Similarly, if charter served these flights between 4pm
and 9pm, there was no charter delay. For charter service, we assume that the teams would
not be willing to fly before 4pm, and would want to reach the venue by 11pm, or at least
by the time the scheduled service flight would do so. Thus, if the charter flight results in
arrival at venue after 9pm, the time beyond 9pm is penalized as delay with slope equal to
104
the value of time, and the limit of this penalized window is the maximum of 11pm or the
arrival using scheduled service.
For movements after the event (outbound), the earliest departure time is the game
end time plus an hour. The first scheduled service flight after this time was identified, and
in cases in which there was no flight on the day of the game itself, extra overnight
lodging expenses are included at $100 per person. For charter, we also assumed that the
team cannot leave before the end of game plus 1 hour. If the end of game enables the
team to be ready to fly before 12 midnight, the no-penalty time window is 1 hour,
followed by a 2 hour window with penalty. If the end of game forces the team to leave
after 12, the no-penalty time window is half hour, followed by a 2 hour window with
penalty. This is based on the observation that in almost all cases, the best scheduled
service option for departure right after the game (when available) fly before 12:30 am.
For movements between successive games, travel from one venue to another was
treated as an outbound flight with the exception that no extra overnight expenses were
factored in, since these will apply to both service options and therefore do not affect the
choice between them.
For both SS and CS, we detail the parameters for cost and time spent in service. In table
5.4, we give the assumptions for the scheduled service. We assume the reference week as
representative of the entire year’s airline schedule and availability, and determine the best
scheduled flight for the same day of the week as the demand. We assume that teams
arrive at the venue of the game the day before the game, and depart on the day of the
105
game itself when a scheduled flight is available within that time. Schedule delay is
calculated as the absolute value of the difference between the preferred departure time
and the scheduled departure time.
Table 5.4: Cost and time assumptions for the scheduled service
Scheduled Service Data and Assumptions
Source for scheduled service data (flight duration, departure time, refundable ticket price)
Expedia (www.expedia.com)
Reference week for schedule September 5th to 10
th, 2007
Time for arrival at airport 1 hour prior to boarding
Time for leaving the airport 1 hour after flight arrival
Average speed for airport access 50 miles per hour
Driving cost for airport access 40 cents per mile (one way)
For the charter service, we used the Fairchild Dornier Envoy aircraft for operational data.
This aircraft can seat approximately 30 people, and thus is suited for all the sports under
consideration. Table 5.5 gives the various assumptions about the charter aircraft.
The ownership cost of the aircraft is included in the analysis. Our discussion with
aircraft lease holders yielded an estimated annual aircraft lease cost of a million dollars
for the Do328-310. Rather than include the lease cost for the entire year, we included the
lease cost for 27 weeks only. On one hand this assumption may be an under-estimate of
the lease cost, since leasing the aircraft for a period less than a year may not be possible.
Further, the 27 week period is not continuous, making the possibility of lease for that
period alone even more remote. On the other hand, attributing the entire 27-week least
cost to the charter service could be an over-estimate because of the fact that the aircraft
may not be needed for athletic travel for several days or even longer. In these cases, it is
possible that the aircraft could be used for other purposes and any revenue from this
106
would offset the ownership cost. Given these two divergent considerations, we use the
lease cost for 27 weeks only, and in the later section comment on the additional costs
from leasing for the entire year.
Table 5.5: Cost and time assumptions for the charter service
Charter Service Data and Assumptions
Time buffer for arrival at airport 15 minutes prior to boarding
Time buffer for leaving the airport 30 minutes after flight arrival
Average aircraft speed 460 statute miles per hour †
Time for landing, take-off and taxiing 30 minutes
Hourly operating cost $1600 (including crew costs) †
Driving cost to for airport access 40 cents per mile (one way)
† Source: Technical specifications of Fairchild Dornier Envoy (Do328-310)
The majority of the benefits from using charter are in time savings, and to monetize these
we need an estimate of the value of travel time for Big Sky intercollegiate athletes. The
appropriate value of time (VOT) to be used in this setting is difficult to know. Additional
research beyond the scope of this study would be required establish an appropriate value.
For this reason, we will attempt to bracket the potential benefits by using low and high
VOTs: $3/pr-hr (per person hour), which is less than the federal minimum wage, and
$30/pr-hr, which is close to the Federal Aviation Administration’s recommended value of
travel time for use in investment analyses. Sources of variability within this range include
whether to take the perspective of the university or the athlete, how to incorporate
students’ present income, anticipated future income, and family income in assessing
willingness to pay for travel time savings, and the actual opportunity costs of extra time
spent away from campus. While recognizing these as important issues, our aim here is to
107
demonstrate the application of the charter planning model and roughly quantify the
benefits to students. As we show in the later sections, these benefits are substantial even
when the low $3/pr-hr value is used.
Besides the assumption of the value of time, another input to the model is charter
fleet. As stated before, we assume that the charter operates the Fairchild Dornier Envoy
Do328-310 only. Further, we assumed four different fleet sizes, including one, two, three
or four charter aircraft. With a fleet size of five, it was observed that the total cost (money
and time) for serving all the demand exceeded the base scenario (where every movement
is served using existing scheduled options) for both values of time. Also, we demonstrate
later that charter with four aircraft serves almost 80% of the movements. Thus, we
limited the fleet size to four. With the two values of time and four different fleet sizes, we
analyze eight different operational scenarios.
We now summarize results for the optimization for the case in which PIH (the
charter airport used for IDS) is used as the home base, and begin with market penetration
and time savings
In table 5.6, we show the market penetration total time savings from using charter with
the eight different scenarios mentioned in the previous section. As expected, market
penetration and time savings increase with increasing fleet size. We also observe
diminishing returns in both of these quantities with increasing fleet size. The time savings
108
are greater when the assumed value of time is higher, which is expected since a higher
value of time gives more ―weight‖ to time savings in the overall costs. However, even
though time savings are greater when the assumed value of time is higher, the difference
is not large. With just one charter aircraft when the assumed value of time is increased
tenfold from $3/pr-hr to $30/pr-hr, the increase in time savings is only 20%.
Table 5.6: Time savings and percentage demand served by charter for different operational
configurations
Charter Fleet size
Value of time $3/pr-hr Value of time $30/pr-hr
Time Savings in pr-hr **
Avg. trip length (hr)
†
Movements served by Charter
Time Savings in pr-hr **
Avg. trip length (hr)
†
Movements served by Charter
1 22,863 (31%) 5.68 45% 27,605 (38%) 5.14 50%
2 32,856 (45%) 4.55 65% 38,162 (52%) 3.95 73%
3 38,329 (52%) 3.93 73% 42,612 (58%) 3.45 82%
4 39,926 (55%) 3.75 76% 43,764 (60%) 3.32 87%
** Values in brackets are percentage out of total travel time when flying scheduled only † Average trip time in all scheduled case is 8.26 hours. This includes schedule delay, airport access,
terminal services and gate-to-gate travel time.
The time savings from using charter are significant. With just one charter aircraft
operated under the assumption that VOT is $3/pr-hr, the average travel time per
movement decreases from 8.26 hours to 5.68 hours, and almost half the movements are
served by charter.
The optimal fleet size is one that minimizes the total cost, including charter flying and
ownership, scheduled flying, accommodation and student time for the entire 27 week
season. As stated before in section 5.2.3, the charter aircraft ownership cost is included
109
only for 27 weeks. We compare the expenditures separately for the value of time $3/pr-hr
and $30/pr-hr for the four charter fleet sizes. The results of the comparison are given in
figure 5.3.
For $3/pr-hr, the optimal fleet size is one charter aircraft, although the difference
between total cost for the one-aircraft and two-aircraft fleets is not substantial. For
$30/pr-hr, the optimal fleet size is two charter aircraft. Since the majority of benefits of
charter come from time savings, a higher value of time results in such benefits being
given more weight when compared to the extra dollar expenditure of operating a larger
fleet. Thus, the size of optimal fleet will be larger when the assumed value of time for
planning is higher.
Figure 5.3: Total expenditure (time and money) for the eight configurations
3,500
4,000
4,500
5,000
5,500
6,000
Value of time $3/pr-hr Value of time $30/pr-hr
Tota
l Co
st (
mo
ne
y an
d t
ime
) x 1
00
0$
1 Charter Aircraft
2 Charter Aircraft
3 Charter Aircraft
4 Charter Aircraft
110
In the previous section, we compared the four fleet sizes for the charter service, and
determined the optimal fleet size for the two values of time used (for $3/pr-hr, one charter
aircraft result in lowest cost, and for $30/pr-hr, two charter aircraft give the lowest cost).
In table 5.7(a), we compare these two scenarios with the case where all the movements
are served using scheduled service only. This includes the corresponding value of travel
time, including the schedule delay. For both $3/pr-hr and $30/pr-hr, the optimal charter
scenario leads to a significant reduction in total cost (time and money), and in table
5.7(b), we present the reduction in total cost, and its split into the various components.
We consider the two cases where the charter aircraft ownership cost is included for the 27
weeks only, and when it is included for the entire year.
Usage of charter aircraft reduces the overall travel cost substantially. Also, the
reduction in travel time is accompanied with reduction in dollar expenditure. Even when
the aircraft ownership cost is altered to reflect a lease period of one year (represented by
the values in brackets in table 5.7(a) and (b)), the total expenditure is still substantially
smaller than under the all-scheduled case. The reduction in total expenditure in this case
is 12% for $3/pr-hr and 15% for $30/pr-hr.
111
Table 5.7(a): Expenditure and components for the two optimal charter fleet configurations and
comparison with existing scheduled service (values in 1000$)
Cost Component
VOT $3/pr-hr VOT $30/pr-hr
1 Charter Aircraft
(in 1000$)
All Scheduled (in 1000$)
2 Charter Aircraft
(in 1000$)
All Scheduled (in 1000$)
Value of time spent in charter service 35 - 576 -
Value of time spent in scheduled service
93 165 421 1,659
Value of incurred schedule delay 25 55 91 553
Accommodation costs 538 589 520 589
Charter flying (aircraft operation and airport access)
931 - 1,525 -
Charter aircraft ownership 545
(1,051) -
1,091 (2,102)
-
Scheduled flying (tickets and airport access)
1,675 4,151 634 4,151
Total cost (money and time) 3,845
(4,350) 4,962
4,861 (5,872)
6,953
(Values in bracket are values assuming charter ownership for 1 year instead of 27 weeks)
Table 5.7(b): Reduction in cost from using optimal charter configurations as compared to existing scheduled service options (values in 1000$)
Cost Component
1 Charter Aircraft with VOT $3/pr-hr
2 Charter Aircraft with VOT $30/pr-hr
Value (in 1000$)
Percentage Value
(in 1000$) Percentage
Time savings ($ value) 66 30.2% 1,122 50.8%
Accommodation cost savings 51 8.7% 68 11.7%
Flying cost savings (excluding ownership)
1,545 37.2% 1,992 48%
Flying cost savings (including ownership)
999 (493)
24.1% (11.9%)
900 (-110)
21.7% (-2.7%)
Total savings 1,117 (611)
22.5% (12.3%)
2,092 (1,081)
30.1% (15.5%)
(Values in bracket are values assuming charter ownership for 1 year instead of 27 weeks)
112
The above results assume that PIH - the charter airport used by Idaho State (IDS) - is the
operational base. Here we present cost results for several other bases. While the relevant
input costs of fuel, aircraft storage, etc., may vary by location, we assume that these are
equivalent to focus on comparing these bases strictly from a geographical point of view.
In addition to finding the best base, we sought to gauge the importance of base location in
affecting costs. Six alternatives were considered. Idaho State and Washington State were
selected because of their central location. In addition, we analyzed the charter airports for
CSU, NAU, PSU and UNCO as possible bases, due to the larger volume of movements
demanded at these locations (refer table 5.3, which shows the demand over various origin
destination pairs). Figure 5.4 shows the variability in total cost for these 6 bases of
operation for the four charter fleet sizes and value of time $3/pr-hr, and figure 5.5 shows
the same variability with value of time $30/pr-hr. The cost is generally higher for the
locations on the edge of the operating region, even if they are stronger demand
generators, and lower for those locations closer to the center. The cost sensitivity is not
great, however, making it possible that factor price differences could easily tilt the
balance toward a location that is less efficient geographically.
113
Figure 5.4: Total expenditure for different operational bases and charter fleet with value of time
$3/pr-hr
Figure 5.5: Total expenditure for different operational bases and charter fleet with value of time $30/pr-hr
3.6
3.8
4
4.2
4.4
4.6
4.8
5
1 2 3 4
Tota
l exp
end
itu
re in
mill
ion
do
llars
Charter aircraft fleet size
CSU IDS NAU PSU UNCO WSU
4.6
4.8
5
5.2
5.4
5.6
5.8
6
1 2 3 4
Tota
l exp
end
itu
re in
mill
ion
do
llars
Charter aircraft fleet size
CSU IDS NAU PSU UNCO WSU
114
Table 5.8 shows the least expensive and most expensive base of operations for each
fleet size and value of time. The percentage difference is small, never exceeding 4%. The
centrally located bases (IDS and WSU) are consistently the optimal ones, while the
locations farthest from the geographical center (PSU, NAU, CSU and UNCO) are the
worst. The difference between the least and most expensive location grows with fleet
size. This is expected, since the number of ferrying operations to and from the base will
increase with the fleet. Thus, with very large fleets, the total costs would be more
sensitive to the choice of base of operations.
Table 5.8: Cost comparison for the best and worst location for operational base
Value of time
Charter fleet size
Least expensive base Most expensive base Difference
Location Total Cost ($) Location Total Cost ($)
$3/hour
1 IDS 3,845,110 NAU 3,894,145 1.3%
2 IDS 3,854,033 PSU 3,931,494 2.0%
3 IDS 4,223,347 PSU 4,326,963 2.5%
4 IDS 4,708,899 CSU 4,822,908 2.4%
$30/hour
1 IDS 5,141,538 NAU 5,190,326 0.9%
2 IDS 4,861,601 PSU 4,926,251 1.3%
3 IDS 5,094,796 UNCO 5,234,072 2.7%
4 WSU 5,519,782 NAU 5,686,321 3.0%
In this section, we consider the distribution of benefits, first amongst weeks in the 27
week schedule, second amongst participating conference universities, and third across
conference sports.
115
In figures 5.6 and 5.7, we show the weekly variation in demand, the number of flights
served by charter and the weekly time savings for the two optimal operational
configurations defined above with values of time $3/pr-hr and $30/pr-hr and IDS as the
operational base. Charter-served flights and resulting time savings increase significantly
from figure 5.6 to 5.7, due to increased value of time and larger charter fleet size.
Figure 5.6: Weekly variation in demand served and time savings with 1 charter aircraft and VOT as $3/pr-hr
0
5
10
15
20
25
30
35
0
5
10
15
20
25
30
35
40
45
50
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Time
savings in
hu
nd
red
pe
rson
-ho
urs
Team
Mo
vem
en
ts d
uri
ng
We
ek
Week Number
Flights served with 1 aircraft Flights demanded Time savings using 1 aircraft(on secondary axis)
116
Figure 5.7: Weekly variation in demand served and time savings with 2 charter aircraft and VOT as $30/pr-hr
As evident from these two figures, there is wide week-to-week variation in the total
flights demanded, charter flights, and time savings resulting from charter services.
However, there is limited correlation between these variables. In weeks, such as 12 and
14, when the numbers of flights served by charter are almost the same, the time savings
realized can be quite different. Nonetheless, the weeks in which the time savings are
greatest (weeks 5, 12, 16 and 18) overlap considerably with those featuring the largest
number of charter flights (12, 14, 16, and 18) and total flight demand (7, 14, 16, and 18).
The week-to-week variability shown in Figures 5.6 and 5.7 has two important
implications. First, there could be considerable benefit from outsourcing the service to a
larger charter company that so that the number of charter aircraft available for Big Sky
athletic travel can vary from week to week. Second, if a dedicated charter fleet is used, it
0
5
10
15
20
25
30
35
0
5
10
15
20
25
30
35
40
45
50
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Time
savings in
hu
nd
red
pe
rson
ho
urs
Team
Mo
vem
en
ts d
uri
ng
We
ek
Week Number
Flights served with 2 aircraft Flights demanded Time savings using 2 aircraft(on secondary axis)
117
would be beneficial to modify the event schedule so as to reduce the week-to-week
variability in charter service needs.
In table 5.7, we disaggregate the movements by team traveling for the event and the event
venue, which gives a representation of the share of each school in the total travel. The
total number of movements to various venues is almost uniformly distributed over all the
schools. The number of movements associated with a particular venue, however, show
considerable variation, with the CSU and IDS having more than double the movements
with PSU as venue.
Table 5.7: Total movements split over flying team and venue
Venue
CSU EAW IDS MOS NAU PSU UNCO WSU
Fly
ing
Team
CSU . 8 12 4 10 6 4 11 55
EAW 10 . 8 6 8 6 9 8 55
IDS 7 6 . 8 12 4 8 10 55
MOS 14 8 8 . 4 3 8 6 51
NAU 12 6 10 8 . 6 6 3 51
PSU 9 6 10 3 7 . 6 7 48
UNCO 11 6 12 6 14 5 . 4 58
WSU 10 5 13 8 10 4 12 . 62
73 45 73 43 65 34 53 49
0 - 3 4 - 7 8 - 11 12 - 15
In table 5.8, we show the number of movements served by charter assuming value
of time as $3/pr-hr and using one charter aircraft with IDS as the operational base.
Considerable variation is seen in number of charter movements for a particular school.
While NAU has only 11 out of 51 (~21%) movements served by charter, WSU has 35 out
of 62 (~56%) movements served by charter. The variation is noticeably reduced when
118
using value of time $30/pr-hr and two charter aircraft, as shown in table 5.9. This
indicates that the spread of charter movements over all the schools is more uniform for
either larger fleet sizes or higher values of time or a combination of both.
Table 5.8: Movements served by charter with 1 aircraft and $3/pr-hr split over team and venue
Venue
CSU EAW IDS MOS NAU PSU UNCO WSU
Fly
ing
Team
CSU . 0 7 0 2 0 1 9 19
EAW 0 . 5 3 2 0 2 6 18
IDS 3 2 . 8 1 2 5 5 26
MOS 8 6 6 . 1 0 4 6 31
NAU 1 2 1 5 . 0 1 1 11
PSU 2 1 5 1 1 . 0 7 17
UNCO 1 4 5 6 4 1 . 3 24
WSU 5 4 6 7 5 3 5 . 35
20 19 35 30 16 6 18 37
0 - 3 4 - 7 8 - 11 12 - 15
Table 5.9: Movements served by charter with 2 aircraft and $30/pr-hr split over team and venue
Venue
CSU EAW IDS MOS NAU PSU UNCO WSU
Fly
ing
Team
CSU 0 1 10 3 6 1 2 9 32
EAW 2 0 7 6 6 2 5 7 35
IDS 4 6 0 8 8 4 7 6 43
MOS 13 7 8 0 2 0 6 6 42
NAU 5 5 6 8 0 3 3 3 33
PSU 2 3 6 1 4 0 2 7 25
UNCO 6 5 10 6 11 1 0 3 42
WSU 5 4 7 8 8 4 11 0 47
37 31 54 40 45 15 36 41
0 - 3 4 - 7 8 - 11 12 - 15
We now consider whether the difference seen in tables 8 and 9 is a result of the
difference in value of time or size of fleet. In table 5.10, total charter movements for each
school are given for the eight operational configurations (four charter fleet size and two
119
different values of time). A larger value of time leads to more movements being served
by charter (refer table 5.6), but does not result in a uniform distribution of charter service
across the conference participants. A larger charter fleet does result in even spread of
charter flights over the various schools.
Table 5.10: Total movements served by charter for different schools in eight configurations
Value of time $3/pr-hr Value of time $30/pr-hr
Charter Fleet Size 1 2 3 4 1 2 3 4
Fly
ing
Team
CSU 19 26 34 35 23 32 38 41
EAW 18 30 37 38 23 35 42 46
IDS 26 38 42 44 29 43 47 51
MOS 31 40 43 43 34 42 44 48
NAU 11 24 34 37 15 33 41 45
PSU 17 22 25 31 16 25 29 36
UNCO 24 36 42 44 26 42 47 52
WSU 35 45 46 47 41 47 48 52
Implementation of the charter service by the entire conference would lead to issues
of distribution of costs of charter among the various schools, and a larger charter fleet
would mitigate the concerns of ―equality‖ to some extent since most schools have similar
numbers of movement served by charter. In case a smaller fleet is used, the costs could be
allocated across schools on the basis of the number of movements served by charter.
Here we consider how the benefits of using a charter service vary across sports
participating in the Big Sky Conference. The average hours per person saved per event is
a fairly consistent value of approximately 3.5 hours, with the largest value for women’s
120
golf. This is shown in Figure 5.8. This is also the sport most heavily utilizing charter (as a
percentage of total trips).
Figure 5.8: Time savings per person per game across sport.
Figure 5.9 shows the total cost of travel, per game, for each sport, and the ratio of
expenditure on charter flights versus the expenditure on scheduled flights. We can
observe that the sport with highest expenditure per event (women’s tennis) is also the
sport that spends least on charter flights. Similarly, the sport with the smallest total travel
expenditure per event (women’s basketball), is the sport with the highest charter to
scheduled expenditure ratio.
Hours per person per game average savings (except schedule delay)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Wom
en's Bas
ketb
all
Wom
en's Cro
ss C
ountry
Wom
en's G
olf
Wom
en's Soc
cer
Wom
en's T&
F
Wom
en's Tenn
is
Wom
en's Volleyb
all
Men
's B
aske
tball
Men
's C
ross
Cou
ntry
Men
's T
&F
Men
's T
ennis
ho
urs
per
pers
on
avera
ge s
avin
gs
121
Figure 5.9: Total cost of travel per game and ratio of charter to scheduled spending across sport
for one aircraft and value of time $3/hour.
The case study shows little differentiation in the utilization of charter versus
scheduled service with respect to sport or University location and therefore underscores
the necessity of a model to identify fleet routing and assignment. Further, since the
benefits of charter are diffused among campuses and sports, a conference wide initiative
would be better than a campus or sport-specific service.
The model also shows significant time and cost savings, particularly with a small
fleet of aircraft, and distinct benefits for student athletes. This is true for a range of values
of time, and in the face of highly variable and temporally concentrated demand. Not only
$-
$2,000.00
$4,000.00
$6,000.00
$8,000.00
$10,000.00
$12,000.00
$14,000.00
Wom
en's Tenn
is
Men
's T
&F
Wom
en's G
olf
Wom
en's Volleyb
all
Men
's C
ross
Cou
ntry
Men
's B
aske
tball
Men
's T
ennis
Wom
en's Cro
ss C
ountry
Wom
en's Soc
cer
Wom
en's T&
F
Wom
en's Bas
ketb
all
Sport
To
tal
Exp
en
dit
ure
per
gam
e o
n t
ravel
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
Total
Expenditure
per event
Charter to
scheduled
expenditure
122
does the demand vary by day of the week, but also by season. In practice, additional
markets would need to fill the summer months when intercollegiate athletics do not
compete.
The benefits of the charter would be further increased if Universities started
traveling on the day of the game itself, rather than fly in the previous day. But through
discussions with coaches we have found this practice to be pervasive, so the study
presents a conservative but realistic comparison of the charter service to current practice.
Also, recall that the majority of the benefit from charter service is derived from the real
cost of travel.
123
We have developed two sets of models: one for charter and scheduled service
competition, and the other for charter planning over a network. The models of
competition give simple, closed-form expressions for equilibrium solutions prices in
many scenarios. A numerical treatment of these solutions over a variety of situations
indicates that charter would be successful over shorter routes with low density of travel.
But incorporating additional time savings from charter (due to reduced opportunity cost
of total travel time) makes it profitable in almost all cases. Additionally, the entry of such
a charter service is beneficial to the consumer. In all the cases, the resulting change in
scheduled service frequency was fairly small, to the point that if the frequency was
integral, there would probably be no change. The changes in prices, however, were
substantial and varied over different cases. Further, when the reduced opportunity cost is
not considered in the high density market, the success of the charter would depend on the
scheduled service response. With changes in scheduled prices, charter would almost
124
always lose complete market share. For small group sizes, this is true even in the case
when the reduced opportunity costs are considered.
The charter planning model is developed as a mixed integer program bearing some
similarities to the airline fleet assignment problem. The model was operationalized to
estimate the benefits to student athletes in the Big Sky conference using actual game
schedules, airline flying times and fares, and charter operating costs. Although our
assumptions on charter operations were conservative (for example, teams always arrived
a day earlier), there are substantial benefits to students in terms of time savings.
Moreover, the distribution of benefits was fairly uniform over all conference members
and sports. There were substantial time savings were realized with little change in the
dollar cost of travel.
The above analysis seems to make a compelling case for charter success. However,
there are many issues that need to be addressed and researched to operate such a service
in the real world. An underlining assumption in the above models is the equivalence of
time and money. Further, the value of time spent in various activities is included. For
student travel, besides the value of schedule delay and flight time, the value of time spent
away from home is used. Estimating this value is an issue, since student athletes could
potentially use the saved time for academic purposes. It is possible that with savings in
time, student athletes undertake more demanding coursework, or even a different major.
The opportunity cost of total travel time has also been included, and in a charter aircraft
flying the time could be used productively for meetings and presentations, which would
make travel time an asset rather than a liability. A prospective charter operator would
need to research the above issues for implementation.
125
Historically, charter or un-scheduled airline service had a strong presence in the
pre-deregulation period. This was due to the formation of affinity groups driven by the
desire for lower fares than the regulated prices. With de-regulation, this driving force was
no longer present. However, recent advances in communication technology, specially
social networking (Facebook) and micro-blogging (Twitter) could very well be the new
medium for passengers to combine into groups for better travel times, and perhaps better
prices too in regions with poor scheduled service. Our research indicates that charter
could perform well in pre-scheduled group travel. The case study uses student athlete
travel as an example, and other examples could be musicians and other traveling
performers. Such charter service could potentially attract more groups through the use of
above mentioned mediums, further increasing the charter fleet utilization and thereby
increasing profit. The possibility of the formation of such groups and the potential role
played of the different mediums is a direction for future research.
126
AAN. (2008). "Pilot Cost Per Block Hour." from http://www.aanegotiations.com/,
retrieved November 20, 2008.
Adler, N. (2001). "Competition in a Deregulated Air Transportation Market." European
Journal of Operational Research 129(2): 337-345.
Adler, N. (2005). "Hub-Spoke Network Choice under Competition with an Application to
Western Europe." Transportation Science 39: 58-72.
Alderighi, M., A. Cento, et al. (2005). "Network Competition-the Coexistence of Hub-
and-Spoke and Point-to-Point Systems." Journal of Air Transport Management 11(5):
328-334.
Bania, N., P. W. Bauer, et al. (1998). "U.S. Air Passenger Service: A Taxonomy of Route
Networks, Hub Locations, and Competition." Transportation Research Part E:
Logistics and Transportation Review 34(1): 53-74.
127
Barla, P. and C. Constantatos (2000). "Airline Network Structure under Demand
Uncertainty." Transportation Research Part E: Logistics and Transportation Review
36(3): 173-180.
Barnhart, C., P. Belobaba, et al. (2003). "Applications of Operations Research in the Air
Transport Industry." Transportation Science 37(4): 368-391.
Barnhart, C., T. S. Kniker, et al. (2002). "Itinerary-Based Airline Fleet Assignment."
Transportation Science 36: 199-217.
Bishop, M. and D. Thompson (1992). "Peak-Load Pricing in Aviation : The Case of
Charter Air Fares." Journal of Transport Economics and Policy. 26(1).
Blann, F. W. (1985). "Intercollegiate Athletic Competition and Students’ Educational and
Career Plans." Journal of College Student Personnel 26(2): 4.
Borenstein, S. and J. Netz (1999). "Why Do All the Flights Leave at 8 Am?: Competition
and Departure-Time Differentiation in Airline Markets." International Journal of
Industrial Organization 17(5): 611-640.
CNET News (2005). "On-demand Private Jets Nearing Take-off", March 23, 2005.
Davies, R. (1982). Airlines of the United States since 1914, Smithsonian Institution Press.
DayJet (2008). "DayJet", from http://www.dayjet.com/, retrieved November 20, 2008.
Department of Intercollegiate Athletics, W. S. U. (2003-2004). Assessment Report,
Wright State University: 3.
128
Dobson, G. and P. J. Lederer (1993). "Airline Scheduling and Routing in a Hub-and-
Spoke System." Transportation Science 27(3): 281.
DOT (1997). Memorandum: Departmental Guidance for the Valuation of Travel Time in
Economic Analysis. D. o. Transportation.
DOT. (2008). "Airline Fuel Cost and Consumption." from
http://www.transtats.bts.gov/fuel.asp, retrieved November 20, 2008.
Dubois, P. E. (1978). "Participation in Sports and Occupational Attainment: A
Comparative Study." Research quarterly 49(1): 28-37.
Ernest T. Pascarella , L. B., Amaury Nora , Patrick T. Terenzini (1999). "Intercollegiate
Althletic Participation and Freshman-Year Cognitive Outcomes." Journal of Higher
Education 70(1): 27.
Espinoza, D., R. Garcia, et al. (2008). "Per-Seat, on-Demand Air Transportation Part I:
Problem Description and an Integer Multicommodity Flow Model." Transportation
Science 42(3): 263-278.
Espinoza, D., R. Garcia, et al. (2008). "Per-Seat, on-Demand Air Transportation Part II:
Parallel Local Search." Transportation Science 42(3): 279-291.
Forbes (2004). "Sky King", August 16, 2004. Retrieved from
http://www.forbes.com/forbes/2004/0816/076.html on November 20, 2008.
129
Force, N. P. T. (2006). Ncaa Presidential Task Force on the Future of Division I
Intercollegiate Athletics Student Athlete Well-Being Subcommittee, Twenty-Hour
Rule and Lack of Enforcement, NCAA Presidential Task Force.
Franke, M. (2004). "Competition between Network Carriers and Low-Cost Carriers--
Retreat Battle or Breakthrough to a New Level of Efficiency?" Journal of Air
Transport Management 10(1): 15-21.
Gibbons, R. (1992). Game Theory for Applied Economists, Princeton University Press
Princeton, NJ.
Gillen, D. and W. G. Morrison (2005). "Regulation, Competition and Network Evolution
in Aviation." Journal of Air Transport Management 11(3): 161-174.
Goel, P. and A. Haghani (2000). "Model for Determining Airline Fares for Meeting or
Convention Demand." Journal of Transportation Engineering 126(2): 107.
Government Accountability Office (2006). "AIRLINE DEREGULATION: Reregulating
the Airline Industry Would Likely Reverse Consumer Benefits and Not Save Airline
Pensions". Report to the Congressional Committees, GAO-06-630, June 2006.
Government Accountability Office (2007). "COMMERCIAL AVIATION: Programs and
Options for Providing Air Service to Small Communities." Testimony Before the
Subcommittee on Aviation, Committee on Transportation and Infrastructure, U.S.
House of Representatives, GAO-07-793T, April 2007.
130
Hansen, M. (1990). "Airline Competition in a Hub-Dominated Environment: An
Application of Noncooperative Game Theory." Transportation Research Part B:
Methodological 24(1): 27-43.
Hendricks, K., M. Piccione, et al. (1999). "Equilibria in Networks." Econometrica 67(6):
1407-1434.
Howard, A. (1986). "College Experiences and Managerial Performance." Journal of
Applied Psychology 71(3): 12.
Hsu, C.-I. and Y.-H. Wen (2003). "Determining Flight Frequencies on an Airline
Network with Demand-Supply Interactions." Transportation Research Part E:
Logistics and Transportation Review 39(6): 417-441.
ILOG (2008). "ILOG CPLEX: High-Performance Software for Mathematical
Programming and Optimization." from http://www.ilog.com/products/cplex/. retrieved
November 20, 2008.
Inzerilli, F. and S. Jara-Díaz (1994). "Uncertain Demand, Modal Competition and
Optimal Price-Capacity Adjustments in Air Transportation." Transportation 21(1): 91-
101.
Karlaftis, M. G. and J. D. Papastavrou (1998). "Demand Characteristics for a Charter Air-
Travel." Rivista internazionale di economia dei trasporti. 25(1): 19.
131
Kennedy, S. R. D., Kenneth M. (1987). "Career Maturity and Professional Sports
Expectations of College Football and Basketball Players." Journal of College Student
Personnel 28(4): 4.
Kim, D. and C. Barnhart (2007). "Flight Schedule Design for a Charter Airline."
Computers and Operations Research 34(6 SPEC ISS): 1516-1531.
Knorr, J. (2003). "Athletics on Campus: Refocusing on Academic Outcomes."
Perspectives in Business at St. Edwards University 2(1): 4.
Lohatepanont, M. and C. Barnhart (2004). "Airline Schedule Planning: Integrated Models
and Algorithms for Schedule Design and Fleet Assignment." Transportation Science
38: 19-32.
NCAA Presidential Task Force on the Future of Division I Intercollegiate Athletics
Student Athlete Well-Being Subcommittee, Twenty-Hour Rule and Lack of
Enforcement, January 19. 2006
Nemhauser, G. L. and L. A. Wolsey (1988). Integer and Combinatorial Optimization.
New York, Wiley.
Nikaido, H. and K. Isoda (1955). "Note on Noncooperative Convex Games." Pacific
Journal of Mathematics 5(1): 807-815.
Pascarella, Ernest T., Bohr, Louise, Nora, Amaury, Terensini, Patrick T. ―Intercollegiate
Althletic Participation and Freshman-Year Cognitive Outcomes‖, The Journal of
Higher Education, July 1995; 66, 4
132
Pels, E., P. Nijkamp, et al. (2000). "Airport and Airline Competition for Passengers
Departing from a Large Metropolitan Area." Journal of Urban Economics 48(1): 29-
45.
Poole, R. and V. Butler (1999). "Airline Deregulation: The Unfinished Revolution."
Regulation 22: 44-51.
Rexing, B., C. Barnhart, et al. (2000). "Airline Fleet Assignment with Time Windows."
Transportation Science 34: 1-20.
Rushmeier, R. A., K. L. Hoffman and M. Padberg (1995). "Recent advances in exact
optimization of airline scheduling problems." Technical Report, George Mason
University, 1995.
Rushmeier, R. A. and S. A. Kontogiorgis (1997). "Advances in the Optimization of
Airline Fleet Assignment." Transportation Science 31(2): 159.
Schipper, Y., P. Nijkamp, et al. (2007). "Deregulation and Welfare in Airline Markets:
An Analysis of Frequency Equilibria." European Journal of Operational Research
178(1): 194-206.
Sinha, D. (2001). Deregulation and Liberalisation of the Airline Industry, Ashgate
Burlington, VT.
Smirti, M. and M. Hansen (2009). The Effect of Fuel Prices on Comparative Aircraft
Costs. Transportation Research Board's 88th Annual Meeting, January 11-15, 2009.
Washington, DC.
133
Sowa, C. J. G., Charles F. (2000). "Career Development in College Varsity Athletes."
Journal of College Student Personnel 24(3): 4.
Stone, J. A. S., C. Carney (1989). "Quality of Student Experiences of Freshman
Intercollegiate Athletes." Journal of College Student Development 30(2): 6.
Svrcek, T. (1991). Modeling Airline Group Passenger Demand for Revenue
Optimization. Cambridge, Mass., Massachusetts Institute of Technology, Flight
Transportation Laboratory.
Wei, W. (2000). Airlines' Choice of Aircraft Size in a Competitive Environment. Civil
and Environmental Engineering. Berkeley, University of California. PhD.
Wei, W. and M. Hansen (2005). "Impact of Aircraft Size and Seat Availability on
Airlines' Demand and Market Share in Duopoly Markets." Transportation Research
Part E: Logistics and Transportation Review 41(4): 315-327.
Wei, W. and M. Hansen (2007). "Airlines' Competition in Aircraft Size and Service
Frequency in Duopoly Markets." Transportation Research Part E: Logistics and
Transportation Review 43(4): 409-424.
134
The CS profit function is:
To evaluate the Hessian matrix of the above function for constant SS frequency and
price, we calculate the partial second derivatives as follows:
Thus the Hessian matrix can be written as:
135
Since the above Hessian matrix is diagonal, the eigenvalues of the matrix are the diagonal
entries themselves. Thus, all the eigenvalues of the above Hessian matrix are negative,
which means that the matrix is negative-definite and the profit function is concave in
prices.
The SS profit function can be written as:
To evaluate the Hessian matrix of the above function for constant SS frequency and
price, we calculate the partial second derivatives as follows:
136
Thus, the Hessian matrix can be written as:
The sign quantities in the first row and first column (where is a variable) in the above
matrix cannot be determined for a generic case, and thus the nature of the function is not
certain. However, for the case when frequency is fixed (or is fixed), the resulting
Hessian is the same as above matrix without the first row and column. This sub-matrix is
diagonal, and the eigenvalues are negative. Thus, for the price change only case, the
Hessian is negative definite and the profit function is concave.
137
For the case when SS charges a single ticket price and the frequency is not altered, the SS
profit function can be evaluated as:
Second derivative with respect to yields:
which is less than zero always. Thus, for this case too, the profit function is concave.
138
Equation (2.10) gives the SS profit as
(1)
As stated before, the optimal frequency would be one for the case when under
the assumption that there has to be at least one SS flight. First order conditions for the
case give
139
(2)
and
(3)
Solving (2) for gives the expression for equilibrium price in terms of frequency, as
described previously in equation (2.12):
(4)
Inserting (4) in equation (3), we get a cubic equation in equilibrium frequency when
, as given before in equation (2.11):
(5)
140
The expression for CS profit, as given in equation (2.4) is:
(6)
where
(7)
Given a SS price for each group (it could be a single price over the entire market, , or
group based prices, ) as well as the SS service frequency , first order conditions can
be applied to (6) to yield the optimal CS price for each group . However, since or
yields a linear CS profit function, evaluating the boundary conditions yields the
CS response. The boundary conditions that need to be evaluated are the conditions when
or , and the optimal CS price being greater than or equal to the operating
cost.
For the interior equilibrium (as defined in section 2.2.3), applying first order
conditions we get:
141
(8)
Using the condition for interior equilibrium
(9)
Also, checking the condition for , we get ,
which is satisfied above. Thus, the condition for interior equilibrium can be expressed in
terms of as shown in equation (9), and the CS price is given in equation (8).
Now, consider the case when . Here,
where is defined before in equation (8). We identify the condition for
, resulting in .
The case when is when the optimal CS price results in complete market share for
CS, or . Evaluating this condition, we get
. Thus, for this condition CS would charge the maximum price at which ,
which is . Also, under this condition, . Thus, the optimal
CS price would be
142
(10)
143
We first find the optimal CS and SS prices as a function of the competitor’s price
and SS frequency for the interior equilibrium. We then evaluate the various conditions of
market share and the lower bounds set by operating cost. This is done separately for the
simultaneous and leader-follower setup.
Using the first order conditions on the CS profit function, we get the optimal CS price for
a fixed SS price as follows:
(11)
The SS profit function can be written in terms of a single SS price as follows:
Using first order conditions for the interior equilibrium case, we get:
144
The above yields the ticket price as a function of the CS group prices. Using the optimal
CS price for a given SS price from equation (11), we get
(12)
The above expression includes all the groups, irrespective of SS market share. But if
there are groups for which SS market share is always zero, SS would exclude them from
setting the ticket price. For this, consider the condition for for a group :
Using the expression for from equation (11), we get
(13)
If the above expression is satisfied for a group , SS market share for that group would be
zero. Since , the expression is always true when . Thus any group
that satisfies will be excluded from SS optimal price calculation. Thus, the
final expressions can be written as:
145
(14)
(15)
(16)
where represents the groups satisfying .
The CS profit function is the same as before, resulting in the same optimal CS price for a
given SS price as shown in equation (11). However, since SS is the market leader, it
accounts for CS response while optimizing price. Thus, the optimal CS price in equation
(11) is included in the SS profit function before applying first order conditions. The
modified SS profit function is:
From first order conditions, we obtain:
146
(17)
The analysis for the case when a group always uses CS is identical to the simultaneous
game. Thus, the final expressions for optimal prices in this case are as follows:
(18)
(19)
(20)
where represents the groups satisfying .
147
As before, we first find the optimal CS and SS prices as a function of the competitor’s
price and SS frequency for the interior equilibrium. We then evaluate the various
conditions of market share and the lower bounds set by operating cost. This is done
separately for the simultaneous and leader-follower setup. In the end, we evaluate the CS
market share in both the cases and the SS profit and draw comparisons.
Using first order conditions on the profit functions, we get the optimal CS group price
and optimal SS individual and group price as follows:
(21)
(22)
148
(23)
Solving equations (21) and (23) simultaneously, we get the optimal CS and SS group
prices for interior equilibrium ( and ) as:
(24)
(25)
We next evaluate conditions when these prices are less than the operating cost.
Using the expression from equation (24), we get
Similarly
Evaluating the resulting from and , we get
149
(26)
Thus, for , , and and are greater than the respective
operating costs.
Now consider the case when . The optimal CS price is
lower than the operating cost. Thus, the actual CS price would be equal to the operating
cost. Thus, for , . The resulting SS price from equation (23)
is . Calculating , we get
Thus, for , . Hence, the SS prices would be given by
. In other words
(27)
A similar argument can be made for the case , resulting in
(28)
150
Combining all the above expressions, we get
(29)
(30)
(31)
In this case, the first order conditions are applied to the SS profit function after including
the optimal CS response. The optimal CS response for a fixed SS price and frequency is
given in equation (21). Putting this in the expression for , we get
(32)
Using first order conditions, we get
(33)
151
(34)
Putting equation (34) in equation (21), we get the optimal CS and SS group prices for
interior equilibrium ( and ) as:
(35)
(36)
We next evaluate conditions when these prices are less than the operating cost.
Using the expression from equation (35), we get
Similarly
Evaluating the resulting from and , we get
152
(37)
Thus, for , , and and are greater than the respective
operating costs. The analysis for the rest of the values of is similar to the analysis in
the simultaneous game. All the expressions can then be combined to give
(38)
(39)
(40)
Here we compare SS profit in both the simultaneous and leader-follower games for the
same frequency . Since the individual prices are the same, the individual profit would
be the same. Further, we compare the profit for a generic group , with the results being
true for any group type. The comparison is done for the case .
Based on the expressions in equations (26) and (31), SS profit from group in the
simultaneous game can be evaluated as:
153
(41)
Similarly, SS profit in the leader-follower game is as follows:
(42)
Clearly, .
For the case when , the expression for SS profit in the leader follower
case remains the same as in equation (42), but the expression for the simultaneous game
changes since here. Thus
(43)
To compare the profit from the leader-follower game and simultaneous game, we
evaluate the difference between the two. Thus
Thus, SS profit is greater in the leader-follower case than the simultaneous case when
.
154
First order conditions on CS profit as function of price for particular SS price and
frequency yields the same optimal price as in equation (21). Similarly, optimal SS prices
for given frequency are given in equation (22) and (23). First order conditions on SS
frequency keeping CS and SS prices constant result in the following:
For the interior equilibrium, we solve the above expression simultaneously with the
prices from equations (21), (22) and (23), which yields:
(44)
As before, SS being the market leader incorporates the CS optimal prices in its profit
function. Thus, first order conditions are applied to the modified SS profit function in
equation (32), resulting in the optimal individual price in equation (33) and group price in
equation (34). First order conditions on profit in equation (32) as a function of frequency
155
yields:
For the interior equilibrium, we solve the above expression simultaneously with the
prices from equations (33) and (34), which yields:
(45)
156
The individual demand function is given by :
Consider the case when and . We evaluate the value of average
time savings of one hour by evaluating the change in price with change in frequency
keeping demand constant. To this end, consider the case when the price increases by
and frequency changes from to . Thus
(46)
Now, the average delay for an individual passenger when there are flights over a
period is . Since we are assuming one hour average delay saving from changing
frequency from to we get:
157
Using this in equation (46), we get:
where represents the value of average time savings of one hour.