AIRLINE ASSIGNMENTAND

ROUTING PROBLEM:A NETWORK PROGRAMMING MODEL

byMargarita D. Garcia

OREM 4390

SENIOR DESIGN INENGINEERING MANAGEMENT

8 May 1982

Dr. Richard S. Barr, Professor

TABLE OF CONTENTS

PageIntroduction 1

Background and Overview 2

Model Description 4

Solution Phase 9

Analysis 10

Conclusion 12

FIGURES AND TABLES

Figure 1 14

Figure 2 15

Figure 3 16

Figure 4 17

Figure 5 18

Figure 6 19

Figure 7 20

Figure 8 21

Table A 22

INTRODUCTION

The airline assignment and routing problem was originally

developed and solved as a standard linear programming model by

Jorge Casaus, a senior engineering student at SMU, in 1981.

Professor Richard Barr, however, has suggested that a network pro-

gramming approach be taken in finding an optimal solution to the

problem. The following report describes and discusses how a net-

work programming model can be used in the selection of aircraft

assigned to cover various flight schedules. This model utilizes a

computer program available at SMU's computer center, called OPTNET,

for generating the best possible solution to the problem.

BACKGROUND & OVERVIEW

The problem addressed by this network programming model is

described as follows:

An airline company owns different types of aircraft, and for each

type, a maximum number is available to cover various flight sche-

dules. The flight schedule for a given day identifies, for each

flight number, the origination city and departure time, and the

termination city and arrival time. Based on forecasts of passenger

demand and operating costs, the profitability of operating each

aircraft type on every flight has been estimated (for example, the

profit contribution on a particular flight may be $1000 if a 7271d~?

is used, -$500 with a 707, and $400 with a DC-10).

In addition, due to ground servicing requirements, there is

a minimum ground time between the arrival and subsequent departure

of an aircraft. This ground time may vary by city but is the same

for all aircraft types at a given city. In an airline routing (a

one-day sequence of flights), each departure follows the preceding

arrival by at least the ninimum ground time specific to the city

i n question. In general, there may be more than one aircraft of a

particular type on the ground that is available for use at the next

departure flight needing that type of plane. If such instances occur

the airline routing should follow the first-in-first-out (FIFO)

rule: the aircraft type required which has been on the ground the

l ongest should be assigned to the next departure.

Having collected all this information, the airline company

wishes to determine the optimal assignment of aircrafts to flights

2

on a single day so as to maximize profit and -minimize the number

of planes i.n the system. Data used in solving the model was generated

in spring 1981 and has not been updated.

3

( 3) the

MODEL DESCRIPTION

Before any analysis of the problem can be done, the flight

schedule data has to be constructed into a circularized network

flow diagram, consisting of a series of nodes connected by a set

of lines (see figure 1a). In an airline routing problem, the nodes

represent the cities (or airports), and the lines connecting them

represent the different flight routes (see figure 1b). Note that a

line connecting two nodes with the same city name represents any

number of planes on the ground waiting for the next departure

flights.

For simplification, this airline routing problem is formulated

and solved with the assumption that only one type of plane exists.

In addition, other assumptions have been incorporated into the

model, and they include the following:

( 1) a flight schedule is said to be "balanced" if it ends

the day with the same number of planes(at each city)

that-it started out with. A lack of balance may repre-

sent the existence of flights which operate only on

certain days of the week. No imbalance penalty is charged

to any city that has a balanced schedule;

( 2) the cost of covering a flight (= -$100) and the minimum

ground time required (= 45 minutes) are assumed to be

the same for all cities;

cost of planes waiting on the ground for the next

departure flights is assumed to be zero; and

( 4) each airport begins the day with a starting inventory

equal to the sum of all flights coming in and going out

4

of the city on a single day. In addition, based on

assumption #1, the city's ending inventory should equal

i ts starting inventory.

There are 25 cities and 61 flights in this routing problem,

and the complete airline network is drawn out in figure 2. Notice

that some nodes have arrows coming in that are not coming from any

of the other nodes (for example, SAN715, SAN915, PHX1445, PHX1609).

These arrows indicate the points where each of the 61 flights be-

gin. They also connect the city nodes to a special node called

SOURCE, as. seen in figure 3. Similarly, the last node of each city

is

connected to another special node called SINK (see figure 4).

The purpose of adding these special nodes is to ensure that the

network model becomes circularized. In addition, each arc (or line)

in the network must haveJspecific upper and lower bounds. This en-

sures that the number of planes used in the system will not be more

than what the company can assign. Consequently, lines representing

flight paths have upper and lower bounds equal to 1, ensuring that

all flights are covered and that each flight is flown only by one

plane. Also, lines representing planes waiting on the ground have

no upper limits but must have lower bounds equal to zero (there

i s no such thing as a negative number of planes). Combining all

these characteristics of a network into one, figure 6 illustrates

how a segment of the network looks like once its upper/lower bounds

and costs have been defined.

For this airline routing problem, there exists a cost called

i mbalance penalty (=$1000) that comes into effect if the number of

of planes udd by each city exceeds its starting inventory. Inven-

5

tory is calculated by adding up the flights departing and ar-

riving in a city, so LAS has a starting inventory of 8 planes,

BOS has 18 planes, and LAX has 12. Figure 5 shows more clearly how

the imbalance penalty is modelled into the network. Thus, if SAN's

supply of planes for the day is less than or equal to its start-

i ng inventory, no penalty is incurred; otherwise, $1000 is added

to thelmodel's total cost.for

Since the OPTNET/program only solves the optimal minimum

cost value, the objective function of this problem, that calls

for maximizing profit, has to be changed to minimizing cost. Thus,

i f P j is the profit from operating flight j, and xj

stands for

flight j, then the objective function of this problem is:

61

max

P.xj=1

J j

This is equivalent to

61min 1; (-C)x

J=1

J

J

i f it is assumed that changing the signs of the profit values

would turn them into cost values. There are only two other con-

straints that have not been discussed in this report, and they

include:

(1) flow balance equations which ensure that at no airport

can a plane be created or destroyed. That is,

/ units of

flow into the

node

/

/ units

+

of supply

at the node/

units of

units

flow out of + of dema

the node

nd

at the node /

and

( 2) nonnegativity constraints which ensure that the flow of

the network will travel in only one direction.

One special constraint that has been mentioned at the

beginning of the report is called the minimum ground time re-

striction. To find out just how much this constraint will

affect the optimal solution, Two models are set up: one with the

constraint, and one without. Model 1 requires that all flights are

covered and that the starting inventory, costs, and bounds for

each node and arc are specified. Model 2 carries all these re-

quirements, too, but'it has the minimum ground time constraint

added in. To better understand how this constraint works, consider

figure 7. The figure shows a segment of the BOS network, and the

numbers inside the nodes indicate thejdeparture/arrival times.

When calculating the lower bounds of theinodes affected by the

constraint, only arrival nodes are considered. So the nodes

affected by the constraint from the BOS segment are BOS-1209,

1214, 1301, and 1346. To each of these times, the minimum ground

time of 45 minutes is added; this gives the next departure times

that the planes (used in the 4 flights) can cover. Once departure

time is estimated, it is then necessary to change the lower

bounds of the inventory arcs affected. This will force the planes,

that had just arrived, to stay on the ground until the time re-

striction has expired. Thus, if a plane arrives in BUS at 1209

( and can only leave at 1254), the arcs between 1209-1214 and

1214-1255 will have a lower bound greater than or equal to 1.

Similarly, the next plane arriving at 1214 (and leaves at 1259)

7

will cause the arcs between 1214-1255 and 1255-1301 to have a

l ower bound greater than or equal to 1. Notice that there is an

overlap of lower bounds for the arc between 1214-1255. Because two

planes have just arrived and are affected by the time constraint,

the lower bound for this arc ensures that'those planes will not leave

the city until they have fulfilled their 45-minute restriction.

Notice also that the arc between 1435-1520 has a lower bound greater

than or equal to 0. This indicates that any plane can be used to

cover the flights departing at these times.

8

SOLUTION PHASE

Once formulated, both models are put into input files to be

used in the OPTNET program (see computer listings #1 and #2).

The models are specified as a series of arcs entered into their

respective files as follows:

line-

from-

to-

unit-

upper-

l ower

number

node

node

cost

bound

bound

wich one arc per linenuinber. All numeric values must be integers

with no decimal points, and to simplify data input, the following

defaults are used: (a) if upper and lower bounds are not specified,

-chey are assumed to be 9999999 and zero, respectively, and (b) if

the unit cost is not speciifed, it is assumed to be zero.

When

OPTNET is executed, the problem is read in and solved, and the

program provides a summary report of the network in its optimized

state. The optimal solution for model 1 resulted in a minimum cost

of -$6100 with 33 planes being used in the system (see computer

listing #3). After the model was modified to include the minimum

ground time constraint, the optimal solution for model 2 generated

a minimum cost of -$6100 with 50 planes covering the various flights

(see computer listing #4).

9

ANALYSIS

Now that optimal solutions for both models have been gene-

rated, analysis of the output can begin. For this airline routing

problem, output is analyzed by taking a particular plane used in

the system and following its flight schedule through the network.

OPTNET generated the same minimum cost for models 1 and 2, and this

i ndicates that covering all 61 flights with this type of plane is

profitable. But model 2 is different from model 1 since the

minimum ground time restriction had been added. This constraint

may appear redundant (since it did not change the optimal cost

value for model 2), but it did affect the number of planes pre-

sent in the system: 33 in the first model against 50 in the se-

cond. Figure 8 shows how such a constraint determines the flights

a plane will cover and the city at which it will stop at the end

of the day.

The assumption that only one type of plane exists now seem to

be unrealistic since an airline company would surely have at least

two types of planes available for use in its flights. There is

usually a limit to the load of flights that one type of planes can

take on a single day.

The balance of schedule and starting inventory assumptions

prevented any imbalance penalty to be incurred by any city;

consequently, the cost accumulated from this part of the network

model was zero and had no effect on the optimal solution. In

addition, these assumptions seem to be valid since it would be

l ogical to require that the number of planes be the same at the

beginning and end of a day. If such was not the case, the airline

10

company would have empty planes flown around (to keep the in-

ventory levels at each city balanced) and incur more costs.

Furthermore, the cost of covering a flight may not be the

same at all times because there are some places where it would

cost more to fly to than others. Similarly, the cost of planes

waiting for the next departure flights may not be zero at all

times, too. Nevertheless, the fact that these costs were zero for

this type of plane affected the optimal solution to the problem

in a positive way: it kept the minimum cost from getting worse.

11

CONCLUSION

To summarize, there were several steps taken to find the

best solution to the problem, and they included the following:

(1) construction of a circularized network diagram based

on the flight schedule information provided;

( 2) defintion of costs, upper and lower bounds, and

starting inventory levels forithe different nodes and

arcs of the network;

( 3) generation of the optimal flight routes for models 1

and 2 using the OPTNET computer program; and

( 4) analysis of the different solutions generated. Flight

routes for the models were mapped into the network

diagram.

Based on the solution generated by OPTNET, the optimal mini-

mum cost for both models was the same. The only effect of adding

the minimum ground time constraint (in model 2) was to increase

the number of planes in the starting inventory for some airports.

Because the problem had only considered using one type of

plane, several limitations that would have been used were omitted.

For future projects concerning the airline routing problem, it

would be interesting to consider the following:

(1) airports are now charging landing fees

according to the type of plane used. This cost would have

to be included in the objective function of the problem.

(it becomes a fixed cost in the model);

( 2) realistically, the minimum ground time restriction will

vary for the different cities

12

that vary

(3) in response to various marketing considerations, it may

be necessary to specify the type of plane assigned to

certain flights, so an initial assignment of an aircraft

type to each flight may be included in the model; and

( 4) a multiperiod routing problem should be considered since

there are fluctuations in passenger demand at certain

months of the year (for example, more people fly to

Colorado during -che skiing season).

In conclusion, the network programming model provided an

efficient alternative means of integrating data for the analysis

of an airline routing problem. It was easy to use and to under-

stand it made visualizing the system (before and after a solu-

tion was found) easier. As mentioned at the beginning of the,

report, this problem was solved with the assumption that only one

type of aircraft existed. This model, however, can be extended

to solve a multi-aircraft type routing problem.

13

FIGURESAND

TABLES

(Q)

(b)

FIGURE 1.(Q) CIRCULARIZED NETWORK 'IAGg J

(b) SEGM ' OF THE AHZUNE NETWORk

I

FrGvRE 3.

1 6

LINK €ETWEEN ZOU(ZCE

AND GT1y NODES

1 7

r--rGvRE 4. LINK b"EEN SINk

AND CITY NODES

~rGvRE ~. NETWo J4 WM IMBALANCE

PENALTY C05TS

18

wM¢ac C1 < C2

SUPPLYt4

CITY i

( nix )

FIGVRE C.

1 9

CITY 2

( CLE)

t.E6 E N D

C

(

' •)

5EG11AENT of AT2uN E N "O(ZK

141Tµ COSTS `1 SOUNDS TEPT14ED

SurPLY+2

Bos:

2 0

;~s

(PCPART. 134C)

(DEPWs 1431)

PiCVRer 1. SECMFI4T of CITY WETWo(ZK WITH

MINI MUM GROUND TIME GoNgT(ZAINT

:~i

MODEL i

I IaURE ~. COMPAJSON of FLIGHT Rouiss

$ oR MOVE .S i A 2

2 1

(2

D

LAX-SINK

MODEL 2

TABLE A

LIGHT SCHEDULE DATA

22

INDEX FLT. NO . ORIG DEST DEPART ARRIVE

1 420 lax bos 105 1214

2 321 bos lax 1335 1959

3 410 lax las 700 759

4 486 las bos 945 1833

5 153 bos dfw 1918 2327

6 084 dfw ind 640 929

7 234 ind lga 1037 1347

8 243 lga san 1430 1943

9 284 san bos 715 1650

10 383 bos cvg 1815 2146

11 071 cvg ord 755 759

12 226 ord lga 900 1150

13 285 lga bos 1230 1346

14 309 bos cvg 1435 1923

15 047 cvg sfo 2020 2332

16 220 sfo bos 710 1543

17 217 bos ont 1650 1952

18 408 ont lga 840 1756

19 259 lga ord 1900 2033

20 46o ord bdl 655 946

21 599 bdl iah 1035 1506

22 490 iah msp 1625 2037

23 302 msp lga 755 1247

2 3

INDEX FLT. NO . ORIG DEST DEPART ARRIVE

24 491 lga lax 1345 1805

25 304 lax dfw 2100 10137

26 244 dfw lga 640 1033

27 573 lga dtw 1230 1407

28 236 dtw lga 1500 1619

29 391 lga sat 1700 2140

30 367 sat lax 855 1218

31 092 lax sat 1400 2048

32 328 sat lga 710 1418

33 041 lga lax 1530 1920

34 580 tul bos 715 1301

35 545 bos tus 1520 2022

36 108 tus lga 845 1753

37 223 lga buf 1845 1953

38 566 buf lga 700 803

39 565 lga las 905 1450

40 522 las lax 740 1415

41 193 lax las 1635 1935

42 156 las sat 1010 1613

43 578 sat dfw 1705 1802

44 278 dfw pit 1250 1616

45 283 pit dfw 1705 2021

46 113 dfw phx 1115 1324

47 274 phx cle 1445 2123

48 347 cle phx 955 1328

Cost of waiting for the next departure -- $0.00

Cost of covering a flight-- -$100 (equal to profit)

Imbalance penalty -- $1000

24

INDEX FLT. NO . ORIG DEST DEPART ARRIVE

49 286 phx okc 1609 2036

50 504 okc bos 705 1209

51 409 bos las 1255 1624

52 239 las lax 1025 1119

53 072 lax roc 121.0 2037

54 501 roc iah 715 1116

55 680 iah dca 1225 2000

56 245 dca bos 725 1024

57 351 bos sfo 1110 2029

58 140 sfo bos 925 2214

59 481 bos san 1015 1507

60 028 san iga 915 2001

61 231 lga tul 825 1318