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8/6/2019 The Airline Capacity_ Planning Problem for Network Dominated by Low Traffic Sectors
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The Airline Capacity,Planning Problem
for Network Dominated by
Low Traffic Sectors
BALRAMV~ITATJXJRAssociate Professor, IIM Calcutta, Diamond Harbour Road
Joka P.O., Kolkata700 104,[email protected]
BANIK. SINHADirector; Management Education Centre; Heritage Institute of Technology;
Chowbaga Road, Anandapur; Kolkata 700 107, [email protected]
ABSTRACT
Owing to severe congestions at major airports and growing passenger discontent with non-directflights, the traditional hub and spoke network is expected to increasingly give way to direct routesnetwork. Iftheformer is characterised by lesser number of high trafic sectors, the latteris characterisedby higher number of low-traffic sectors. The airline capacity planning problem for a networkdominated by low-traffic sectors is comparatively more complex. Bigger aircraft have higher
operational cost economies in comparison to small aircraft. However, owing to low trafi c, demand-capacity matching is more complex when an airlinefleet comprises only of big aircraft. Usage ofbig aircraft in low-traffic sectors also implies infrequent schedules, which again is a practice thatgenerates discontent among passengers. In this paper, our aim is to formulate the airline capacityplanning problem for an airlinefirm whose network is dominated by low-trafic sectors with theobjective of minimising total operational cost. An attempt is also made to apply this model to areal-life situation in India. It is shown that optimal total cost per day for afleet comprising of onesmall and one big aircraft types is lesser than that forafleet consisting ofone aircraft type only, asis the practice adopted by many domestic airline firms.
Keywords: Airlines, Mixed-Integer Linear Programming, Capacity Planning
1. INTRODUCTION
Increasing competition in the airline industry to increasingly give way to direct routes
has resulted in airline firms considering network. If the former is characterised by
passenger preferences more seriously than lesser number of predominantly high traffic
ever before. Owing to severe congestions at sectors, the latter is characterised by higher
major airports and growing passenger number of predominantly low-traffic sectors.
discontent with non-direct flights, the The airline capacity planning problem for
traditional hub and spoke network is expected network dominated by low-traffic sectors is
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comparatively more complex. Bigger aircraft
have high operational cost economies in
comparison to small aircraft. However, owing
to low traffic, demand-capacity matching is
more complex when an airline fleet comprisesonly of big aircraft. Usage of big aircraft in
low-traffic sectors also implies infrequent
schedules, which again is a practice that
generates discontent among passengers. Thisis expected to make airline firms increasingly
switch from defining air schedules as 'number
of flights per week' to 'number of flights per
day'.
The motivation for this paper arose in the
context of studying the Indian domestic
airline industry. It was the monopoly of thestate run Indian Airlines company till the
introduction of the 'Open Skies' policy in
1994. This saw the entry of several private
domestic airline firms. The entry of private
airline firms in the industry hardly succeeded
in shifting long distance passenger traffic that
is presently done by road or rail to air travel.
The biggest hurdle to this has been the inabi-
lity of the airline firms to match the expecta-
tions of the road or rail passengers. No airline
firm, through their aircraft capacity decisions,
really considered tapping passenger segments
other than that of businessmen and
executives, which even today is the largest
segment. The growth of the other passenger
segments of the domestic airline industry that
are highly price sensitive depends heavily on
the efficient running of the airline firms.
Normally an airline firm has to deal with high
cost of equipment (aircraft ) and h igh
operational cost, both of which contribute
significantly to the cost structure. Most of the
new airline firms started with fleet of sameaircraft type, with capacities of 120 to 150
seats, to cater to high traffic sectors. Their
subsequent entry into low-traffic sectors was
also through these aircrafts. This resulted in
low capacity utilisation or infrequent
schedules in these traffic sectors. This added
to inefficient running of the firms. By 1998
many of these firms went out of business.
Furthermore, till recently, firms had been
following an approach of maintaining or
increasing profitability by regular hikes in
fares. However, this approach is successful
only in a monopolistic situation and does notcontribute to the growth of the market. A
policy of maintaining profitability by cost
reduction enables cheaper air travel and
results in the growth of the market. The cost
reduction approach has significantly
contributed to the growth of domestic air
travel in developed countries (Heskett 1994,
Sull 1999). In this context, the capacity
planning problem assumes considerable
significance, especially in countries like India
that are characterised by airline networkswith high proportion of low-traffic sectors,
and where the industry competition has
increased significantly after opening up to
private sector participation.
Minoux (1986) presents a generalised
assignment problem for the assignment of
aircraft types to various routes. In the classical
airline fleet-mix problem, which is an integer
linear programme model, the objective is to
determine how many of each type of aircraft
should be in a fleet to satisfy demands and
minimise total operational cost. Marsten andMuller (1980) extended the airline fleet-mix
problem to design an air cargo carrier's route
and plane assignment for spider-shaped
networks. This network, also referred to as
hub and spoke, has received sufficient re-
search attention. Lederer and Nambimadom
(1998) review various research articles on hub
and spoke networks. They also postulate the
ideal network choice for various environ-
ments. From their findings, it can be con-
cluded that the hub and spoke network isideally suited for an airline firm operating a
network dominated by low-traffic sectors
when the firm's objective is to minimise
operational cost. However, the choice of hub
and spoke network by the airline firm may
not ensure passenger convenience.According to Heskett (1994), flight
frequency and travel time are two important
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aspects of passenger convenience. He traces
the policy of operating frequent direct flights
as an important reason for the rapid growth
and success of Southwest Airlines in the USA.
Travel time is lesser for a direct flight, whichmakes direct flight network much more
popular than hub and spoke network.
Besides, it is felt that passengers would
always prefer travel flexibility, which is
ensured by higher frequency of direct flights.
In a competitive market it is obvious that
passenger convenience is an important
success factor.
A quick look at airline industry research
literature from 1980 reveals that while not
much attention has been given to the fleet-mix problem, there are many modelling
papers in the area of airline scheduling or crew
scheduling. Some of these papers have been
reviewed by Subramanian et al. (1994).
Avittathur and Sinha (2000) developed amathematical programming model for the
fleet-mix problem where flights between two
cities are either direct or one-hop (flight with
a halt at an intermediate city). The objective
is to minimise the total of operational and
opportunity costs, for an airline firm. Besides,
they applied this model to a real-life situationin India. However, this model is analytically
large and complex.
Operating frequent direct flights by
employing a fleet of just one aircraft type is
one of the key strategies adopted by some of
the successful airline firms in recent years
(Heskett 1994, Sull 1999). However, this
strategy may not be successful when the
proportion of low-traffic sectors in the airline
network is high, which is the case with Indian
domestic airline firms. If the fleet werecomposed only of small aircraft type, the
firms would be able to operate frequent direct
flights. However, they can be operated
economically in the low-traffic sectors only.
If the fleet were composed only of big aircraft
type, objective of operating economically
would force firms either to operate less
frequently in the low-traffic sectors or opt
for a hub and spoke network that would weed
out many of the low-traffic sectors. As
described above, the passenger perceives
both options as inconvenient.
The disadvantages of having a fleet ofsingle aircraft type that is either small or big
aircraft thus leads one to evaluate the com-
bination option - fleet comprising of aircraft
of different seating capacities. In the context
of enhancing passenger convenience by
operating a network of frequent direct flights,
it is important to understand the benefits that
could accrue to the airline firm by using a
combination of small and big aircraft types.
Our literature search did not reveal any study
that has jointly evaluated the issues of
passenger convenience and cost benefits to
an airline firm by employing an aircraft fleet
of different seating capacities instead of a
fleet of just one aircraft type. An attempt is
made in this paper to develop a capacity
planning model for determining the com-
bination of small and big aircraft so as to
minimise the total of operational costs, for
an airline firm that operates a network of
direct flights. The model is used in a real-life
situation - an Indian domestic airline firm.
The capacity planning problem ispresented in 2. The solution methodologyis described in 3 . This model is applied to areal-life example in 4.The paper concludeswith a discussion in 55.
2. THE CAPACITY PLANNING
PROBLEM
Consider a network of cities to be covered
by an airline firm. We use the term sector to
indicate a pair of cities having demand forpassenger air traffic between them. The airline
firm serves different sectors in the network
by providing appropriate number of flights
of different aircraft types. The model
determines the optimal number of flights per
period of different aircraft types for each
sector. Discussions with airline managers
revealed that airline firms would consider
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fleets of at most two aircraft types in order
to avoid complexities in aircraft and crew
scheduling, maintenance and spare parts
management. Hence, it is assumed that the
airline firm would like to serve all the sectorswith at most two different types of aircraft -
small aircraft and/or big aircraft types. Theobjective is to minimise the total operationalcost per period for serving all the sectors by
the airline firm.
It is obvious that the number of flights
per period is a decision variable that takes
integer values. This has serious bearing on
the definition of period. Optimal solution will
ensure that there is at least one flight per day
when period is defined as a day as demandfor each day has to be fully satisfied. For
instance, let daily demand be 35 in either
direction of a sector. Whether a 70-seater or
a 150-seater aircraft is employed, one flight
per day (or seven flights per week) has to be
operated in either direction when period is
defined as a day. If period is defined as a
week, demand to be satisfied becomes 245 in
either direction. This would mean that
number of flights to be offered per week in
either direction is four when a 70-seater
aircraft is employed. The number of flights
to be offered per week in either direction is
two when a 150-seater aircraft is employed.
Thus, increase in length of the period from a
day to a week results in schedules with lesser
frequency, particularly for aircraft with higher
seating capacities. Also, the optimal solution
will not ensure that there is at least one flight
per day. The passenger will construe this as
causing more inconvenience.In this paper, weincorporate the frequency aspect of passenger
convenience objective through the definitionof period. It is assumed that smaller the
period higher is the passenger convenience
on the frequency aspect.
The different costs considered are those
that are relevant to operating aircraft. They
are: (i) depreciation, labour and maintenance
cost; (ii) fuel cost for flying and (iii) additional
fuel cost for take-off and landing. The
depreciation cost is equivalent to either the
investment cost amortised over the expected
life of aircraft or the leasing cost. Labour cost
refers to the wages paid to relevant labour
namely pilots, crew and aircraft maintenancestaff.
The indices used in the model are as follows:
Sector i, where i =1, 2, 3, . . ., I
Small aircraft type j = 1, 2, 3, . . ., JBig aircraft type k=1, 2, 3, . . ., K
The parameters used in the model are asfollows:
c,: maximum passenger capacity ofaircraft type j.
C,: maximum passenger capacity ofaircraft type k. It may be noted that
C, is greater than c,, for all values of jand k.
d,: a random stochastic variable with aknown probability density function.
This indicates the demand per period
in either directions of sector i.
f ; : depreciation, labour and maintenancecost per hour of operating time for
small aircraft type j, wheref;= af+ bp,,af and bf are non-negative constants.
F,: depreciation, labour and maintenancecost per hour of operating time for
big aircraft type k, where F, = af+ bp,.g,: fuel cost per hour of flying time for
small aircraft type j, where g, = a, +b,c,, a, and b, are non-negativeconstants.G,: fuel cost per hour of flying time forbig aircraft type k, where G, = a, +b,Ck.
h,: additional fuel cost per flight in take-off and landing only for small aircraft
type j, where h,= ah+ bhc,, ahand bharenon-negative constants.
H,: additional fuel cost per flight in
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take-off and landing only for big
aircraft type k, where Hk= a, + bhCk.M: a very large constant.
a : minimum probability that capacity
provided in a sector is greater than
the corresponding demand in that
sector in any period (demand service
level), where 0 Ia I1. This is takento be same for all sectors.
ti: aircraft flying time in hours in eitherdirection of sector i. We assume that
this does not vary from one aircraft
type to another because aircrafts of
similar technology (e.g. Boeing 737,Airbus 320, etc.) only are considered
in this study, although these aircraftmay have different seating capacities.
T,: total operation time (sum of flight
preparation time at origination city
plus flying time) in hours in either
direction in sector i.
The variables used in the model are as
follows:
1, if aircraft typej is chosen as the
small aircraft for the airline fleet
(0, otherwise1, if aircraft type kis chosen as the
big aircraft for the airline fleet
0, otherwiseu,,: Number of flights to be provided per
period in each direction in sector i by
aircraft type j.U,,: Number of flights to be provided per
period in each direction in sector i by
aircraft type k.
It may be observed from the definition
of each of the cost parameters, that there is a
fixed component and a variable component
with respect to the capacity. This assumption
has been motivated by the operational data
available on aircrafts of similar technology. A
similar assumption has been made by Lederer
and Nambimadom (1998). This assumption
implies that the total operational cost per seat
reduces as aircraft capacity increases.
It may be noted that the term 'operating
time' that appears in the definition of the cost
parameters comprises of time spen t in
activities on ground prior to a particular flight
(flight preparation time) and the time spent
in that flight. The term 'flying time' refers to
the time the aircraft is in air.
The stochastic model for determining the
number of flights of different aircraft types
to be provided to all the sectors is as shownin (1)-(8).
Min TC= Ci2 {Xj.iV;Ti+jti+ h,)uij+ Ck(FkTi+ Gkti + Hk)Uik}
Subject to
zjyj=1, (2)Myj- u,,> 0, Vi, j (3)CkYk= I, (4)MY, - U, 2 0, Vi, k (5)P(CjcjuV+ CkCkUikd i ) 2 a, Vi (6)Yjf Yk E (0, 1) (7)Uiif uik=o,, 2,. . . @)The objective is to minimise the total of
per period operational cost, TC, which
comprises of depreciation, labour and
maintenance cost, fuel cost for flying and
additional fuel cost for take-off and landing,
which are as shown below.
(a) depreciation, labour and maintenance
cost = Ci2Ti{Cf;uq+ &FkUik];(b) fuel cost for flying = Xi2ti[C,gjui,+
CkGkUik];and(c) additional fuel cost for take-off and
landing = Ci2{CjhjuV+ CH,Uik].It may be noted that each of the terms
(a), (b) and (c) includes a factor 2, which
indicates the costs incurred in both directions
for all sectors.
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The constraints involved in the problem
are represented by (2) through (8). The
constraints (2) and (4) respectively indicate
that at most only one type each of small and
big aircraft can be considered in the airline
fleet. The constraints (3) and (5) respectively
indicate the arbitrary upper bounds of the
number of flights per period in different
sectors of the various aircraft types. The
constraints (3) and (5) ensure that the upper
bounds are zero if a particular type of aircraft
is not considered for the airline fleet.
Constraint (6) indicates the probability that
total capacity provided in sector i is greater
than the demand in that sector is at least a
(demand service level) in any period. Theprobability constraint as indicated by (6)
results in a stochastic programming
formulation of the problem. The conditions
(7) and (8) indicate binary and integerconstraints of the respective variables. As u,and Ulkare integer variables describingnumber of flights to be provided per period,
the optimal solution for a given network, asmentioned earlier, would also be dependent
on the definition of period. For instance in a
particular sector i, if u, and Ulkvalues whenperiod is a day are respectively 1 and 0, itcould respectively be 0 and 3 when period isa week, in order to satisfy the same demand.
It can be seen that the above formulation
has (J + K) binary variables and I x (J + K)
integer variables. There are I x (J + K + 1)
inequality constraints in this formulation. Ina realistic situation, both J and K are small
(about 3 to 4 aircraft types each). Accordingly,
the number of sectors, I (for a country like
India it would be about 300), in the networkis the more important factor that determinesthe size of the problem. Hence, for a country
like India, the above problem is such that I =
300, J = 4 and K = 4, which has 8 binary
variables, 2,400 integer variables and 2,700
inequality constraints. This is a large integer
programming problem and may be difficult
to solve using available software. Anotheraspect of the formulation to be noted is the
size of the parameter M involved in (3) and
(5). The larger the value ofM the longer will
be the time to search the optimal solution.
Consequently, an efficient algorithm to solve
the problem is presented in 3.This algorithmexploits the separabi lity property of the
problem.
Furthermore, the stochastic nature of the
problem because of (6) can be handled easily
if we consider its deterministic equivalent.
We denote the deterministic equivalent of the
problem by P', which is presented below.
If dl is an i.i.d, random variable that isnormally distributed with mean E{d,]andvariance var{d,],the equivalent deterministiclinear constraint of (6) is
Z,c,u, + z,C,U, >E{d,}+ z,Jvar(d,/ b'i(6')
where, z is standard normal value corres-ponding to a. In this case the problem
becomes a deterministic mathematical
programming problem, which will be referredto as P'.
3. SOLUTION METHODOLOGY FOR P'The objective function and constraints in
deterministic formulation P' are separable(refer Separable Programming [Taha 19971).
Accordingly, the problem P'can be separatedinto I x J x K sub-problems. The sub-problem
when sector is i, small aircarft type deployed
is j and big aircraft type deployed is k is
denoted by Pik and is as shown below by(91, (10) and (11).
Sub-problem &jk :Min tcij,= 2 {Cf;Ti+ gjfi+ hi)uij
+ (FkTi+ Gkti+ Hk)Uik (9)Subject to
c j u q + C kU i k > E { d i } + z a ~ v a r ( d i ) ( lo)Uij/ Uik= 0, I, 2, . . . (11)
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tqjkvalues that is also less thanis the
lowest total cost per period. Asmentioned above, the optimal
* I *solution of Pjk is {u,, U ik, tcijh.
Step 3: Determining least costs for
combination whose small aircraft type
is j and big aircraft type is k.
The least total cost per period when airline
fleet is of small aircraft typej and big aircraft
type k, which will be indicated as TCjk', isexpressed as T C ~= zit.;.Phase 2: Choosing the optimal combi-nation
of one small and one big aircraft types. TC,,'for the J x K combinations of one small andone big aircraft types are compared. The small
aircraft typej' and big aircraft type k'for whichTC,,' is minimum among the J x K combina-tions is the optimal combination of one small
and one big aircraft types. In other words,
y,., Y,, =1.It may be noted that for optimal
combination where small aircraft type is j'and big aircraft type is k' if uir is equal tozero for all i, then optimal combination
comprises only of one aircraft type - aircrafttype k'. Similarly, if u,;,is equal to zero forall i, then optimal combination comprises only
of one aircraft type - aircraft type j'.A solver was developed to run in
Microsoft3Excel spreadsheet software. It wasdeveloped by coding the three steps
described in Phase 1 as a series of macros in
Visual Basic. The solver is run for each J x K
combination of one small and one big aircraft
types. The outputs of each run are the optimal
number of flights per period of small aircraft
type j in each sector, the optimal number of
flights per period of big aircraft type kin each
sector and TC,,'. It took about a second for aproblem with 15 sectors when run on a
Pentium I11PC (833 MHz, 128 MB RAM). Fora problem with 15 sectors and 2 aircraft types
each of small and big aircraft (four J x Kcombinations), it took less than one minute
to determine the optimal combination of one
small and one big aircraft types. This time
also included the time spent on changing input
data.
4. A REAL-LIFE EXAMPLE
This section has been included for comparingperformance of different network and fleet
types on total cost per period and passenger
convenience aspects for a real-life situation.The real-life example is based on a domestic
airline firm of India with a national presence.
For convenience of illustration only a portion
of the actual network, comprising of six cities
of one region, is considered. For the sake of
confidentiality, the airline firm name and thecities covered are concealed. The cities and
network types are as shown in Figure 1.
City 1, which is almost at the centre of the
6-cities network, is a major airport with many
domestic and international flights. It is also
one of the five big cities in the country. The
remaining cities in the network are small in
comparison to City 1. None of the sectors in
this network are trunk sectors, which is the
terminology used by Indian airline firms for
sectors that pair any of the top five cities
(Bombay, Delhi, Bangalore, Madras and
Calcutta).
As described in 52, the definition ofperiod influences the frequency aspect of
Direct Flight Network Hub and Spoke Network(15 sectors) (15 spokes)
FIGURE1 Six Cities Network Diagram
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passenger convenience objective. It is shown
there that increase in period duration results
in schedules with lesser frequency. On the
basis of interactions with the managers on
the frequency aspect of passenger conve-
nience, the period is defined as a day for this
example. Doing so would ensure that there
is at least one flight per day in either direction
of each sector of the direct flight network and
each spoke of the hub and spoke network.
The managers feel that at least one flight per
day should be operated in either direction if
the airline timetable were to be perceived as
convenient to passenger.
In the example, two each of small and big
aircraft types are considered. Small aircrafttypes are 50-seater and 70-seater aircraft. Big
aircraft types are 120-seater and 150-seater
aircraft. The cruising speeds are similar for
all aircraft types. Hence, it is assumed that
flying time, for a given distance, is same for
all aircraft types. The relevant cost parameters
for these four aircraft are as shown in
Table1.These cost parameters were obtained
from the discussions with the airline
managers.
Currently, the firm employs a hub andspoke network that is serviced by 120-seater
or 150-seater aircraft. City1is the hub airport
with direct flights to the remaining five cities.
However, there are no direct flights between
the other cities. Thus, firm is flying only in
five of the 15 sectors possible for a 6-cities
network. The firm provided daily demand
data for the existing five sectors. Caution was
taken to ensure that this demand included
only passengers who were travelling to/from
City1or flying to/ from international destina-
tions or destinations belonging to other re-
gions of the country. Based on market studies,
the firm provided demand estimates for the
ten sectors that were not covered presently
by direct flights. According to the firm, only
a small percentage of this estimated demand
travelled by the present hub and spoke
network with changeover at City1.This was
attributed to factors like poor flight connec-
tion and long waiting time for flight change-
over at City 1, and good direct connectivity
by rail.
For this example, the performance of the
different network and fleet types on cost,
airline operational and passenger convenienceaspects are as shown in Table 2. For each
sector, it is assumed that fares are same for
all network and fleet combinations. For either
direction of each sector, it is also assumed
that the time unit of demand is day and entire
demand can be met at any point(s) in the dayas long as there is at least one flight per day.
In other words, there is no demand loss if at
least one flight per day is offered in either
direction of each sector. This aspect is taken
care by the definition of period as a day. Inreality this may not be true for hub and spoke
network, particularly if competitors offer
direct flights or there are other travel alterna-
tives like rail or road. The first four columns
of Table 2 describe respectively the sectors,
the cities they connect, t, and E{d,] values. Aflight p reparation time of 0.75 hours
(45 minutes) is considered for all sectors;
hence, T, is equal to t ,+ 0.75 hours for all
sectors. Based on demand data collected, a
TABLE 1: Aircraft Specific Data
Small Aircraft Big Aircraft-7:50- 120- 150-seater seater seater seater-Cost ItemDepreciation, labour and maintenance costper hour of operating time ('000 Rupees)Fuel cost per hour of flying time ('000 Rupees)Fuel cost per flight in take-off and landing
('000 Rupees)
Fixed
Cost
3.30
1.50
5.00
Variable
Cost(per seat)
0.18
0.09
0.30
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Hours of operation per day: Total time spent in flight preparation on ground and flying for all flights in both directions of all sectors.Capacity utilisation: Ratio of average traffic to seats provided for all sectors together, expressed as a percentage.Flight frequency per day: Demand-weighted average flights per day in either direction of each sector. It i s calculated similarly for all networks (based on demand of 15sectors). Hence, for hub and spoke network figures are lower than if i t had been calculated on basis of the six spokes only.
TABLE 2: Performance Comparison of Different Network and Fleet Types
Sector
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Cities
Connec-ted
l a n d 2l a n d 3
l a n d 5l a n d 6 0 . 9 02and 3
2and42and52 and 6
3and43 and 5
3and64and54and65and6
Hours of operation per day
Optimal Total Cost(Thousand Rupees per day)
Best solution in each category
Capacity Utilisation (%)
Flight frequency per day
# of departures from City 1
(hub in Hub and S~ okeNetwork)
f(hours)
0.90
0.70
l a n d 4 0 . 7 0 1 2 0 1 5 01.20
0.60
1.00
1.70
1.40
0.90
1.50
1.20
0.90
1.00
1.20
93.5 73.3 57.0 54.1
2548.1 2583.5 31 37.2 3613.2
465.2 59.9 45.8 39.1
2.12 1.67 1.14 1.00
13 11 6 5
E{d)(perday)
80
80
96
48
48
32
32
48
32
32
40
56
40
100.7 71.4 48.9 35.5
2803.8 2578.8 2772.3 2460.7
483.2 83.7 71.6 78.1
3.35 2.39 1.61 1.21
31 22 15 11
Network: Direct FlightsFleet: One Aircraft Type Only
n, 50 70 120 150(per Sea- Sea- Sea- Sea-
day) ter ter ter ter
100 2 2 1 1
100 2 2 1 1
3 3 2
120 3 2 1 1
9 6 1 2 0 3 2 1
60 2 1 1 1
60 2 1 1 1
40 1 1 1 1
40 1 1 1 1
60 2 1 \ 1 140 1 1 1 1
40 1 1 1 1
50 1 1 1 1
70 2 1 1 1
50 1 1 1 1
71.9 7.2
2361.7
68.2
1.68
9
Optimal Number of Flights per Day
Network: Hub and SpokeFleet: One Aircraft Type Only
, 50 70 120 150
(per Sea- Sea- Sea- Sea-
day) ter ter ter ter
270 6 4 3 2
270 6 4 3 2
1 3 5 0 7 5 3 3
270 6 4 3 2
1 2 9 0 6 5 3 2
63.2 10.1
2404.6
65.2
1.41
7
in each Direction of Sector iNetwork: Direct Flights
n,(perday)
100
100
150
120
120
40.7 16.3
2332.0
464.2
1.14
6
37.8 16.3
2424.8
60.7
1.OO
5
Fleet: Combination
Cmbtn1
Small Big50 120
2 02 03 00 10 1
of a Big and
Cmbtn2
Small Big50 150
2 02 00 10 10 1
6 0 2 0 2 0 1 0 1 0
6 0 2 0 2 0 1 0 1 0
4 0 1 0 1 0 1 0 1 0
4 0 1 0 1 0 1 0 1 0
6 0 2 0 2 0 1 0 1 0
4 0 1 0 1 0 1 0 1 0
4 0 1 0 1 0 1 0 1 0
5 0 1 0 1 0 1 0 1 0
7 0 2 0 2 0 1 0 1 0
5 0 1 0 1 0 1 0 1 0
a Small
Cmbtn3
Small Big70 120
0 10 11 1
0 10 1
Aircraft Types
Cmbtn4
Small Big70 150
0 10 10 10 10 1
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coefficient of variation of 0.125 is considered
for daily one-direction demand for all the 15
sectors. Hence, for z, value of 2, the minimumtarget capacity values, n,, for the 15 sectorsare 1.25 times the E{d,}values.
The remaining columns of Table 2 display
the optimal number of flights per day in each
direction for the 15 sectors, hours of operation
per day, optimal total cost, overall capacity
utilisation, flight frequency per day and
number of departures per day from City 1
(hub airport in hub and spoke network) for
different fleet types in each of three categories
titled: (i)direct flight network with fleet of
one aircraft type only, (ii) hub and spoke
network with fleet of one aircraft type only
and (iii) direct flight network with fleet of
one small aircraft and one big aircraft type.
The fleet types in first two categories are
50-seater, 70-seater, 120-seater and 150-seater
aircraft types. The fleet types in third category
are 50-seater and 120-seater combination, 50-seater and 150-seater combination, 70-seater
and 120-seater combination, and 70-seater
and 150-seater combination. Deriving the
optimal results for the first two categories is
straightforward. The results for the directflight network with fleet of one big aircraft
and one small aircraft type are generated
using the solver developed for this problem,
which is described in 3.The hub and spoke network in this
example is developed using hub location
competitive model (Marianov et al. 1999). The
number of hubs is specified as one. City 1
turned out to be the hub as per this model.
Thus, network generated by this model is
same as that followed presently by the airlinefirm. It may be noted that this is not a pure
hub and spoke network as there is demand
originating or terminating at City 1. As
mentioned earlier, it is assumed that there is
no demand loss in the hub and spoke
network. However, this assumption is not
true in reality for this airline firm. There are
five spokes in this hub and spoke network.
The demand for these spokes is determined
by aggregating the concerned sectors. It is
assumed that there is zero correlation
between sector demands. Hence, the
coefficient of variation of daily one-directiondemand in the hub and spoke network is
lesser owing to demand pooling. Then,valuesfor the five spokes are shown in Table 2. It
may be noted that for the satisfying the same
demand per day, the passengers carried in
hub and spoke network is greater than that
of the direct flight network as passengers
flying between non-hub cities in the former
network are counted twice.
For the direct flight network with fleet
of one aircraft type only, best solution is
Rupees 2,548.1 thousand per day for fleet of
50-seater aircraft. For the hub and spokenetwork with fleet of one aircraft type only,
best solution is Rupees 2,460.7 thousand per
day for fleet of 150-seater aircraft. For the
direct flight network with fleet of one small
aircraft and one big aircraft type, best solution
is Rupees 2,332 thousand per day for fleet
combination of 70-seater and 120-seater
aircraft. This solution is confirmed by using
General Algebraic Modelling System (GAMS
2.25.089 DOS Extended/C) software. How-ever, it may be noted that the problem is small
enough for GAMS to be used (GAMS may
not be efficient if the problem is large). The
GAMS software provides solution of the
primal problem and also gives the shadow
prices. The interpretation of the shadow
prices for this particular example will be
discussed briefly later.
For each sector, as it is assumed that fares
and passenger demand are same for all
network and fleet combination categories, itcan be concluded that revenue generated is
also same for all combination categories. For
an equal revenue assumption, the direct flight
network with fleet of one small aircraft and
one big aircraft type is more profitable thanthe other two categories.
The travel time comparison of direct flight
and hub and spoke networks is shown in
Table 3. Travel time here refers to time
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TABLE 3: Travel Time Comparison of Direct Flight and Hub and spoke Networks
1 and 2
1 and 31 and 4
1 and 5
1 and 6
2 and 3
2 and 4
2 and 5
2 and 6
3 and 4
3 and 5
3 and 6
4 and 5
4 and 6
5 and 6
Sector
i
Yub and Spoke NetworkAverage Waiting at Hub (hrs)50 70 120 150
Seater Seater Seater Seater
0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.000.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00
1.20 2.00 3.00 6.00
1 I0 1.75 3.00 4.501.20 2.00 3.00 6.00
1.20 1.75 3.00 6.00
1 I0 1.75 3.00 4.501.20 2.00 3.00 6.00
1.20 1.75 3.00 6.00
1.10 1.75 3.00 4.50
1.10 1.50 3.00 4.50
1.20 1.75 3.00 6.00
Cities
Connec-
ted
Average Travel Time (hours)
Travel Time lhrs)
0.98
50 70 120 150
Seater Seater Seater Seater
f(hrs)
Travel time: It is sum of flying times, average waiting time at hub and transit time at hub. Journey is completed in one flight in direct flight network, hence,travel time is equal to flying time. Transit time at hub is assumed to be 0.25 hours (15minutes) in above example.
Average waiting at hub: It is calculated on the basis of number of deparatures in a 12-hour time frame. For example, there are 6 flights per day fromcity 1 to city 2 and7 flights per day from city 1 to 4 in the hub and spoke network that employs a 50-seater aircraft fleet. Assuming a gap of 12hours betweenfirst and last flight, and an uniform spacing of departures of intermediate flights in the 12-hour horizon, there is a flight every 2.4 hours and 2 hours,respectively, in these two sectors. Assuming a likelihood of flight arrival at hub that is same for any instant in the 12-hour horizon, the average waiting timeat hub (city 1 airport) is 1.2 hours for passengers travelling from city 2 to city 4 and 1 hour for passengers travelling from city 4 to city 2. As traffic is equalin both directions, the average waiting time at hub for passengers travelling between cities 2 and 4 is 1.1 hours.
n,(per
day)
Average travel time: Demand-weighted average travel time from one city to another for the 6 cities network example.
Direct Flight
Network
TravelTime (hrs)
Number of Flights per Day Time iri o i FlightsSeater Seater Seater Seater (hrs)
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between boarding aircraft at originating city
and disembarking aircraft at destination city.
Travel time is equal to aircraft flying time (ti)in all sectors of direct flight network and
those sectors that have direct flights (sectors
involving hub city) in hub and spoke network.
For sectors where travel is routed through
hub, travel time is sum of flying times, average
waiting time at hub and transit time at hub.
The average waiting time at hub is deter-
mined based on the frequency of flights from
hub. Hence, it differs from one fleet type to
another. The demand-weighted averagetravel time is used to compare the network
and fleet types. For all fleet types in direct
network, the average travel time is 0.98hours. For hub and spoke network, it is 1.92
hours, 2.22 hours, 2.78 hours and 3.84 hours
respectively for 50-seater only, 70-seater only,
120-seater only and 150-seater only fleettypes. In the hub and spoke network,
frequency reduces as aircraft capacity
increases. Hence, average travel time is least
for 50-seater fleet and highest for 150-seaterfleet. In a competitive environment, where
competition could be from other airline firms
or from other modes of transport like railand road, a travel time with very substantial
waiting time at hub has strong potential for
demand loss.
The capacity utilisation definition in
Table 2 is the definition followed by the
airline firm. Looking at the hours of operation
per day, optimal total cost per day and
capacity utilisation for 70-seater fleet for the
first two categories, it is seen that hours of
operation per day and optimal total cost show
values that are close to each other (73.3 hours
and 71.4 hours, respectively and Rupees
2,583.5 thousand and Rupees 2,578.8 thou-
sand, respectively) while capacity utilisation
values differ significantly (59.9 per cent and
83.7 per cent, respectively). The capacityutilisation is higher for hub and spoke
network, though, hours of operation per day
and demand satisfied are similar. The hub and
spoke network displays higher figure as
passengers travelling between non-hub citiesare 'counted twice. The average demand per
day for all sectors together is 1,760 passen-gers. Though, the demand satisfied by hub
and spoke network is also 1,760 passengers,
the number of passengers it flies daily
becomes 2,576 passengers a day if passengers
are counted separately for each spoke. The
difference, 816 passengers a day in this case,is the number of passengers who change flight
at the hub everyday. In other words, the
higher utilisation for hub and spoke network
is as a result of this double counting. As fares
are same for a sector whether travel is by
direct flight or through a hub, revenue
generated is same for both networks. Hence,for this definition of capacity utilisation, it is
obvious that capacity utilisation required to
breakeven is lesser for direct flight network
than that for hub and spoke network.
A hub and spoke network has another
disadvantage in the form of number of flights
operating from hub city. Looking at Table 2,
it is seen that number of departures from
City1,hub in hub and spoke network, is more
than two times the number of departures in
the direct flight network. Thus, from aviewpoint of avoiding the congestion at hub
airports, airline firms should opt for direct
flight network.
Another interesting observation is that
the highest optimal total cost per day for the
direct flight network with fleet of one small
aircraft and one big aircraft type, which is
equal to Rupees 2,424.8 thousand for fleet
combination of 70-seater and 150-seater
aircraft, is lesser than the best solution total
cost per day for direct flight as well as hub
and spoke networks with fleet of one aircraft
type only. This again substantiates the
superiority of the direct network with fleet
of one small aircraft and one big aircraft type
over the other network categories.
In a d y n ~m i cmarket where demandexhibits seasonality, growth, etc., the average
demand per day will change from time to time
indicating a non-stationary demand situation.
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robustness of the optimal solution with
respect to change in demand. It is, however,
incorrect to come to this conclusion on the
basis of evaluating a few cases. However, as
the number of cities in the network increases,
which implies increase in number of sectors,the optimal combination of small and big
aircraft types is expected to remain same as
demand changes within certain bounds.
The shadow prices of the demand
constraint as given by the GAMS solution are
as shown in Table 5. Shadow prices for sectors
4, 5 and 14 are positive and zero for theremaining sectors. This indicates that capacity
provided in these three sectors are exactly
equal to the corresponding demands.Accordingly, these shadow prices indicate the
increase in operating cost if the demand in
each of these sectors where to increase by
one unit from the current level. Table 5 also
shows the marginal costs of round-trip flights
per day in each of the 15 sectors for the four
aircraft types. These marginal costs are the
same as the operating costs for each round-trip that appear as the coefficients of the
decision variables (number of round-trip
flights) in the objective function.
5. DISCUSSION
From the real-life example, it is observed that
optimal total cost per day for direct network
with fleet of one small and one big aircraft
types is lesser than that for a fleet of one
aircraft type only. However, in practice, most
domestic airline firms in India operate non-trunk sectors with fleets consisting of aircraft
of similar capacities, in the 120-150 seats
range. As mentioned earlier, the cost terms'
definition implies that, for a particular capacity
utilisation, the total operating cost per pas-
senger seat decreases as the aircraft capacity
increases. This is a major reason why airline
firms prefer to employ big aircraft fleets.
According to Heskett (1994), SouthwestAirlines, one of the most successful domestic
airline firms in the United States, operates a
direct flight network with a fleet of Boeing
737 aircraft (seating capacity in the range of
120 to 150 seats). As a policy, it enters only
sectors where demand guarantees at least one
flight a day in either direction. Thus, it can
be concluded that this airline firm operates a
direct flight network dominated by medium
or high traffic sectors. When operating a
direct flight network dominated by high
traffic sectors, difference in optimal total cost
per day between network with fleet of one
small and one big aircraft types and network
with fleet of big aircraft type will not be too
significant to warrant a fleet of two aircraft
types. This might be an important reason why
Southwest Airlines uses only a fleet of Boeing737 aircraft.
In India, where most of the non-trunk
sectors are still low-traffic ones, it is difficult
to operate direct frequent flights economi-
cally with fleet of big aircraft type only. The
largest domestic airline in India covers
roughly 60 cities, which implies 1,800 sector
combinations roughly. It employs aircraft
with seating capacity in the range 120 seats
to 200 seats. It is felt that there is potential to
operate economically at least one flight a dayin at least 300 sectors if airline firm employed
a fleet of one small and one big aircraft types.
Presently, the firm covers only about 160
sectors within India. A good proportion of
these sectors have schedules of just two to
four flights a week in either direction. The
situation is quite similar for the other existing
airline firms. Much of the traffic not met by
these airline firms is satisfied by other
transportation modes like rail or road. Thus,
it is felt that there is a good potential for mar-ket growth in the country. The predominant
use of hub and spoke network with fleet of
big aircraft types has been inhibiting the
market growth.
Small aircraft are at a disadvantage with
respect to economies of operation (on a per
seat basis). However, when they are com-
bined with big aircraft, it is easier to balancedemand and capacity. The importance of
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having a combination of big and small aircraft
cannot be ignored in a competitive envirofl-ment. Based on the modelling and analysisin this paper, it is concluded that this wouldbe a good strategy, particularly for low-trafficand high growth sectors. It will also be agood strategy for entering new sectors.Though, cost minimisation has been cited as
justification for fleet with same type aircraft,the formulation in this paper and thesubsequent example disproves this perception.
For each sector, the paper assumes zerodemand loss and same fare for any networkand fleet combination. This implies samerevenue for each combination. In such a
situation, the comparisons in 54 are valid.However, network and fleet type could haveimplications on fare, demand and, hence, onrevenue. Another issue that has to be studied
in detail is the sensitivity of optimal solution
to change in demand for large regional
networks. These issues are identified astopics for future research.
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