Charter vs. Scheduled Airlines - NEXTORnextor.org/pubs/GuptaDissertation2008.pdfCharter vs....

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

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


vi


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



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)


(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)


(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


(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


(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


0%

20%

40%

60%

80%

100%

2 3(a) 3(b) 4

CS

mar

ket

shar

e

n=2, k=25


0%

20%

40%

60%

80%

100%

2 3(a) 3(b) 4

CS

mar

ket

shar

e

n=3, k=25


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



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



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



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



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



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



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



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



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



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



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



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:

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(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,

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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).

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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%

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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


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


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


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


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


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

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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.

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