The Airline Capacity_ Planning Problem for Network Dominated by Low Traffic Sectors

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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15/16

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.

REFERENCES

Avittathur,B.

andB.K. Sinha, 'The Airline Fleet-

Mix Cum Aircraft Allocation Problem',

Proceedings of O M Conference 4, IIT Madras,

India, 2000, pp. 87-93.Heskett, J.L., 'Southwest Airlines:1993 (Abridged

Update)', Harvard Business School Case,

1994.Lederer, P.J. and R.S. Nambimadom, 'Airline

Network Design', Operations Research, vol.46,no. 6, pp. 785-804.

Marianov, V., D. Serra and C.ReVelle, 'Location ofHubs in a Competitive Environment',

European Journal of Operational Research,

vol. 114, 1999, pp. 363-71.

Marsten, R.E. and M.R. Muller, 'A Mixed-IntegerProgramming Approach to Air Cargo Fleet

Planning', Management Science, vol. 26, 1980,

pp. 1096-

1107.Minoux, M., Mathematical Programming:

Theory and Algorithms, John Wiley & Sons,1986.

Subramanian, R., R.P. Scheff, J.D. Quillinan, D.S.

Wiper and R.E. Marsten, 'Coldstart: FleetAssignment at Delta Air Lines', Interfaces,vol. 24, no. 1, 1994, pp. 104-20.

Sull, D., 'easyJets $500Million Gamble', European Management Journal, vol. 17, no. 1, 1999,

pp. 20-38.Taha, H.A., Operations Research: A n Introduc tion,

Prentice-Hall of India, 1997.

The Airline Capacity_ Planning Problem for Network Dominated by Low Traffic Sectors

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