- Email: [email protected]
Mathematical programming models for airport site selection
Tronspn Res-B Vol. 166, No. 6. pp 43547. Printed m Great Britain.
1982
0191-!615/82/oho435-13$03.0010 0 1982 Pergamon Prw Ltd.
MATHEMATICAL PROGRAMMING MODELS FOR AIRPORT SITE SELECTION OMER SAATCICCLUt, Middle East Technical University, (Received 13 February
Ankara, Turkey
1978; in revised form 11 May 1981)
Abstract-Three mathematical programming models are developed in this paper, using different criteria, for the airport site selection problem in developing countries. Estimation of parameters, based on Turkish data, and the sensitivity analyses are presented. The first is a typical set covering model used to determine the minimum number of airports required for a given population of passengers. The other two are locationallocation type models used to find the optimal airport location patterns, the number of passengers to be bussed to cities having airports, and the optimal frequencies of transportation modes between the city pairs.
1. INTRODUCTION
Airport site selection problem is one of the macro-level planning problems of the transportation system and has to be treated within the entire system domain. However, because of the complexity of the transportation system and the lack of certain socio-economic indicators and parameters, this particular problem is treated in isolation. Most of the time it is handled within the scope of Airport Master Plan recommended by either the International Civil Aviation Organization or the Federal Aviation Administration of the USA (see Ashford; Wright, 1979). Furthermore, the number of alternatives available for the airport site selection problem is, in general, too large, and it is very d&cult to generate all ‘of them. Therefore, researchers and planners are inclined to consider a few intuitively generated alternatives for evaluation. In turn, the evaluation process, directed to the identification of the decision makers and of the best criterion or a set of criteria, appears to be a critical issue. There are various costs and benefits associated with the airport site selection problem. These costs and benefits have to be traded off to find the optimal location pattern of airports and their capacities. However, each nation generally has its own concept of the role of government, of what constitutes public benefits, and of the role of private sector agencies, and evaluation procedures. DeNeufville (1976) presented a country’s concept of the public interest, who participates in the decisions, what kind of evaluation will occur and where power will lie in relation to airport site selection problem. This problem is all the more important in developing countries where resources are scarce, and data necessary to make rational decisions are either insufficient or inaccurate. In the literature, economic and decision analysis approaches have been used in selecting the optimal airport site and the most effective strategy for developing the airport facilities (see, e.g., Howard, 1974; Keeney, 1973). In this study, three different mathematical programming models are developed for the airport site selection problem in developing countries. Air transportation system is represented by a network with airports and routes being the nodes and links, respectively. Planned airport constructions in appropriate cities or regions are treated as additions of nodes and links to the transportation network. The models are tested for the land-locked eastern Turkey where road transportation is difficult. Figure 1 shows a map of the region and the cities selected as potential airport sites. Socio-economic interactions between the cities of the region were taken into consideration. At the time of this investigation, three airports were reported to be operational. Air travel demand in the region was met by these three airports located in Erzurum, Diyarbakir and Van. Previous studies, Alaybeyoglu et al. (1974), and Aviation Planning Services (1973) concluded that additional airports were necessary for the region.
Kurrently
Visiting Associate Professor
at the University of Wisconsin-Milwaukee. 435
OMER SAATCIOGIL
436
‘\
0 Gumushane
:\ .’ ‘..a
0
0 Agri
Erzurum
.-.-
\
i’
‘L
Erzincan
0
:\
0
Bingo1
0 Van
MUS
l
0
Elazig
Biilis
0 Siitl
0
Hakkari ,.;-.-._.i
Diyarbakir
0 Urfa --4
..(._.-.-
./ -. H’
.’
A’
.
Other
o Cities
0
C, .,. i
P.. <_.I
cities selected
potentialawpclrt
BS CIUOS
airport5 Approximatescale:
@Cities
5 i
with
1 cm=4OKm
Fig. 1. Eastern Turkey with the cities selected as potential airport sites.
2. MODEL
I
The first model used in this study to find the optimal locations for the airports, is a typical set covering model. A common formulation of a set covering problem was given by Francis and White (1974), Revelle and Swain (1970) and Toregas et al. (1971), and involved the determination of the minimum number of facilities required to cover a set of customers. The same problem was presented by Schilling (1975) and Cohon (1978) as a multiobjective facility location problem. The model has a simple structure and can be applied to airport site selection problem as shown below:
CPjYj
Minimize
j=l
Subject to
TdiiYj?lfori=l,...,l
Yj=OorYj=l
where Pj is the per-capita construction cost of the airport in city j, dij is the covering coefficient. (dij = 1 if air transportation is feasible between cities i and j and d, = 0 otherwise), Yj is the O-l decision variable (Y, = 1 if an airport is constructed in jth city, Yj = 0 otherwise). The objective function gives the total per-capita airport construction cost in the cities of the entire region. The underlying assumption is that the airport construction cost per person can be used as a criterion to evaluate the alternatives. One way of estimating the cost parameters, Pj, of the objective function is to divide the airport construction costs by the total population to be served by an airport. An alternative approach would be to divide the construction costs by the air travel potential of each city and its vicinity. The constraint of the model generates alternative airport location patterns based on a specific criterion. This criterion could be distance, travel time or travel cost between cities. Aggregate average bus fare was chosen as the criterion to identify the potential city pairs for air transportation. The reasoning behind this choice was that if the bus fare between two cities
Mathematical
programming
431
mpdelsfor airport site selection
exceeded a certain amount, then the air travel could become preferable. In other words, it was assumed that a passenger would prefer air travel over bus service beyond a certain distance and corresponding bus fare. The covering matrix is formed by dii’s in the constraint on the basis of bus fares between cities. If the bus fare between a city pair is equal to or greater than a specified level, then the covering coefficient is equal to “1”. If this condition is not satisfied, the covering coefficient becomes “0”. Multiplication of the covering matrix by the vector Yi’s identifies the cities, which can be reached by air transportation from a particular city i. The model is applied to the region mentioned above for testing. Three types of information were required for the model: bus fares between cities, construction costs of airports, and populations of the cities as given in Tables 1 and 2. Two covering matrices were used for testing. Both matrices were obtained by using the trade off points between bus and air transportation. These trade off points were taken as 70 Turkish Liras (TL) and 50TL for the first and the second matrix, respectively. Table 3 illustrates the first matrix. Evidently, these values are subject to change in time. MPSX package was used to solve the problem, because of its availability at the time. MPSX solves the linear analog of an integer programming problem as part of the branch-and-bound procedure. For large scale problems it may be advisable to use more efficient set covering algorithms as presented by Garfinkel and Nemhauser (1972).
Table I. Population and cost estimates Airport ClXIStlXCtiOIl Costs (TL)
Cities
City Population(l977)
Cost Per Person (Pj)
ErZUKUII 5.960
Kars
lS3.000
32.5
Agri
3.880.100
89.000
43.6
:1us
4.620.000
54.000
85.5
Bingal
4.620.000
28.000
165.0
3.849.300
62.000
62.1
Siirt
7.199.900
126.000
57.1
Grdin
6.020.000
135.000
44.6
Hakkari
a.022.000
12.000
668.5
Diyarbakir Bitlis Van
Table 2. Bus fares between cities EIYZ. Kars E~7.Ui?Ull
0
Agri
Plus Bin.
Diy.
Bitlis Van
Siirt Mad.
Hakkr.
25
25
50
55
63
70
50
83
72
80
Kars
25
0
30
75
80
115
95
65
100
109
95
Agri
25
30
0
50
70
88
50
30
65
97
60
j\lus
50
75
50
0
15
40
15
30
30
44
60
Bingcl
55
80
70
15
0
35
20
40
35
39
70
Diyarbakir 63
115
88
40
35
0
30
40
30
9
80
95
50
15
20
30
0
20
15
30
50
Bitlis
70
Van
50
65
30
30
40
40
20
0
35
49
30
Siirt
83
100
65
30
35
30
15
35
0
39
65
Mardin
72
109
97
44
39
9
30
49
39
0
79
Hakkari
80
95
60
60
70
80
50
30
65
79
0
IX-BVol. 16. No.6-B
438
OUER SAATCIO(;LU
Table 3. Airport covering matrix Er2. EKZU?XlU
Kars
0
0
Agri
Nus
0
Bin.
0
0
Diy.
Bitlis Van
0
1
Siirt Plard. Hakkr.
0
1
1
1
KZXlTS
0
0
0
1
1
1
1
0
1
1
1
Agri
0
0
0
0
1
1
0
0
0
1
0
El"S
0
1
0
0
0
0
0
0
0
0
0
Bingtil
0
1
1
0
0
0
0
0
0
0
1
Diyarbakir 0
1
1
0
0
0
0
0
0
0
1
Bitlis
1
1
0
0
0
0
0
0
0
0
0
"a"
0
0
0
0
0
0
0
0
0
0
0
Siirt
1
1
0
0
0
0
0
0
0
0
0
Mardin
1
1
1
0
0
0
0
0
0
0
1
Hakkari
1
1
0
0
1
1
0
0
0
1
0
Table 4. Cost ranges of the sensitive variables in Model I Variables
Activity Level
Input COSt
Lower Cost -m
0
UPPer cost 0
YOl (Erzurum)
1.
YO2 (Kars)
1.
32.5
Y09 (Siirt)
0
57.1
44.6
62.1
YlO (Mardin)
1.
44.6
-m
57.1
0
43.6
When 70TL trade off point was used to solve the model, the results indicated an airport requirement for Kars, in addition to the existing airports of Erzurum, Diyarbakir and Van. However, no additional airport was recommended when SOTL was used as a trade off point. This clearly indicates that the model is sensitive to the covering matrix. The postoptimality analysis was conducted on coefficients of the objective function (i.e. on the construction costs of airports per person). The ranges of the cost coefficients were determined and a parametric analysis was carried out. When the range analysis was conducted, it was observed that only four of the eleven cities considered were sensitive to the lower and upper cost limits. Table 4 summarizes the ranges, activity levels, and input costs of the variables associated with these four cities. All the variables remain at their present activity levels as long as the cost coefficients remain between their lower and upper limits. In addition, the range analysis indicated the status of these variables if the coefficients go above and below their limits. For example, if the unit construction cost of the airport in Mardin increases above 57.1, Y 10 goes to zero, indicating an airport should not be considered for Mardin. Similarly, if the cost of variable Y09 is reduced to 44.6 or below, the variable takes a value of 1. This causes YlO to enter into the solution at its upper bound of 1, proposing airports for Siirt and Mardin, simultaneously. In the parametric analysis, cost coefficients were decreased by 30 TLlperson but the original solution did not change. This indicates that the optimal airport location pattern is not sensitive to population growth as long as the rate of increase remains the same. In this analysis it was assumed that the rate of population increase was the same in each city. 3. MODEL
II
The second model is developed on the basis of total infrastructural cost and cost-of-travel criteria. In this model, it is assumed that the airline management would provide a bus service for the air passengers of towns without airports to cities with airports. The model is a discrete
Mathematical
modelsfor airport site selection
programming
439
plant-location model. It determines the locations and capacities of the airports as well as the number of passengers transported from cities without airports. The formulation used is given below: Minimize
Subject to:
g.ZijcKj,Yj,forj=l
,...,
iZij=ti
nandl=l,...,
L
for i=l,...,m
j=l
x
Yj, 5 1 for j = 1,. . . , Tl
2 5 Aj,Yj, 5
B
I=I
j=l
Z,, Yj, 2 0 and Yj, = 0 or 1 where Zij = number of passengers to be carried between ith and jth cities, Yjl = O-l variable ( Yj, = 1 if an Ith type airport will be constructed in jth city, Yj, = 0 otherwise), Uij = unit transportation cost between ith and jth cities, Ajl = annual cost of an Ith type of airport in jth city, Kjl = annual capacity (in terms of the number of passengers) of an Ith type airport constructed in jth city, Ti = trip generation in ith city, and B = total budget. The first part of the objective function defines the total transportation cost of the passengers transported to cities with airports. Uij’s can be estimated either by considering a unit bus fare between ith and jth cities or by using the following formula: Uij
=
0, +
UjO
where 0, = bus fare between ith and jth cities and Vi, = air fare between jth city and cities with major airports. Here it is assumed that the air fare between airports in the region and the major connecting airport would affect the choice of routes between ith and jth cities. The second part of the objective function represents a total airport construction cost. A, is a function of the airport capacity and city characteristics. The first constraint of the model is a capacity constraint, restricting the number of pas: sengers. Because of the airport capacity limitations, there is a constraint imposed on the number of bus passengers. Airport capacity of the jth city is determined by a O-l variable, Yjl. The second constraint of the model is a demand constraint. By this constraint, the number of bus passengers carried from a given city is restricted by forecasted demand in that city. The third constraint limits the number of airports to one in each city. The last constraint is a budget constraint. Model II was tested in the same region used for testing Model I. Some modifications were deemed necessary in the model due to lack of data. As a result of these modifications the budget constraint was eliminated, and only one type of airport was considered. The data required were obtained from Alaybeyoglu et al. (1974), and 11 Merkezleri Arasi Yolcu ve Esya Tasima Ucretleri (1974). The parameter, Uij was estimated in two different ways, as previously explained. One way of estimating Vij was to sum the bus fares between city pairs, O,, and the air fare up to the major connecting airport, Uj,. The values of Uj, were found by multiplying the distance by average unit fare. The average unit fare was found to be l.O6TL/KM. From the air fare data given in Table 5. Values of the parameter, Uij are given in Table 6. Data required for other parameters, demand forecast, cost of airports and the capacity of airports are given in Table 7.
440
OMER SAATCIOGLL
Table 5. Flight distances and air fares for existing routes Fares (TL)
Fare per Km (TL/Km)
300
0.29
597
300
0.50
498
395
0.79
378
828
570
0.68
696
485
0.69
72
165
2.29
171
260
1.52
108
195
1.80
1014
560
0.55
Table 6. Sum of bus and air fares between cities Erz. Kars
Agri
Mus
Bin.
Diy.
Bitlis Van
E?ZZU?XXll
-
-
_
-
-
-
KarS
25
-
155
140
-
-
150
-
Siirt Mard.
Hakk.
185
-
-
Agri
25
167
-
160
200
-
150
-
-
MUS
50
-
-
-
40
165
-
155
-
-
BingGl
55
-
-
-
35
-
-
-
Bitlis
50
-
-
125
30
-
-
-
_
-
Van
50
200
-
-
Diyarbakir
155
140
40
167
-
155
Siirt
-
140
30
-
-
-
Mardin
-
152
-
160
-
-
80
200
65
190
Hakkari
-
Table 7. Demand forecast, airport costs and airport capacities
Cities
Annual Airport Cost (TL Millions)
Erzurum" Kars
5.96
Agri
3.88
MUS
4.62
Bingal
4.62
Diyarbakir* Bitlis
3.85
vatI* Siirt
7.20
Mardin
6.02
Hakkari
8.02
*Cities with airports
Demand FOreCaStS 0977)
Airport Capacity (number of Passengers)
30168
71731
22000
II
3808
I,
1650
(1
450
,I
36972
,I
1909
II
14200
(1
8090
3,
10000
11
624
,1
Mathematical
programming
441
models for airport site selection
Fig. 2. Network representation of the computer output of Model II, when U,, = 0,, t U,,,.
Fig. 3. Network representation of the computer output of Model II, when Uii = Oij
The model was solved several times using MPSX for different transportation costs, U,. When Uij’s were taken as the sum of bus and air fares, the solution of the model proposed an additional airport in Bitlis. Furthermore, the solution provided the number of passengers carried by bus. The air transportation network representing the solution of the model is given in Fig. 2. The model, with the same values of cost parameters, was also solved assuming that the airports in Erzurum, Diyarbakir and Van did not exist. This time the solution proposed airports only for Erzurum and Mus. When the model was solved for Uij = Oi,, the solution did not recommend any airport other than the existing ones. The network representation of the solution for this case is given in Fig. 3. The solutions of the model in their network representations clearly show the regions that are affected by the proposed or existing airports. This information can be used for the management of the transportation system in establishing the required bus routes. Range and parametric analysis were performed on cost and right-hand side parameters of the model. The results indicated that ranges on the bus fares between cities did not lead to a meaningful analysis. This can be explained by negative lower cost limits of the bus fares. Most of the airport construction cost ranges had infinity as upper bounds. Although the lower cost limits had positive values, they did not lead to a change of status in the variables. The analysis of the right-hand side ranges for the travel demand produced reasonably Table 8. Demand ranges of Model II Cities
Activity
Demand
Lower Limit
Upper Limit
30167.97
30167.97
0
49922.95
21999.98
21999.98
0
37754.96
3807.99
3807.99
0
19562.97
1649.99
1649.99
0
41291.98
449.99
449.99
0
24758.99
36971.99
36971.99
0
61280.98
1908.99
1908.99
0
41550.98
8
14199.99
14199.99
0
53841.99
3
8089.99
3039.99
0
47731.98
10
9999.99
9999.99
0
34308.98
11
6239.99
6239.99
0
458S1.95
6
442
OMER SAATCIOGLU
valuable information with respect to solution status. Table 8 summarizes demand ranges, original demand figures, and activity levels for each constraint. The solution remains optimal with current basis as long as the demands range between zero and their upper limits. In this analysis, the output also indicates state of airport capacity if demand figures reaches their lower and upper limits. Accordingly, if the demands for Erzurum, Kars and Agri reach their upper limits, the airport capacity of Erzurum reaches its maximum. Capacity of Diyarbakir airport also goes to its upper limit when the travel demands of Bingol, Diyarbakir and Mardin reach their upper limits. Airport capacity of Van is fully utilized if the travel demands of Bitlis, Van, Siirt and Hakkari raise to their upper limits. When the parametric analysis was conducted on the demand vector, up to a maximum increase of 30,000, the same three cities were proposed for airport construction. This indicates that proposed airports are sufficient at least up to 30,000 additional travel demand in each city. 4. MODEL
III
Model III is an extension of the aircraft assignment model developed by Miller (1967). In this model, the air and road line costs between the cities were used as criteria in addition to the annualized airport construction costs. The model is a mixed integer programming model which minimizes the following objective function.
2i 2 i=l
Subject to
j=]
CiikXijk
+i
AjYj
j=l
k=l
2 X,, - 2 Xjik = 0 for j = 1, . . , n and k=l,...,r
2 2 bijkXijk5 S, for k = 1, . . . , r [=I j=l r
~giikXij,>Tijfori=l,...,mand k=l
j=l,...,n
Yi, Xijk 2 0 and Yj is O-l variable. where C,, = line cost of kth type of bus and/or aircraft between ith and jth cities, Ai = annual cost of an airport to be located in jth city S, = available operation time of kth mode of transportation per year in minutes, Tij = annual travel demand between ith and jth cities gijk = the number of passengers to be transported by kth mode of transportation between ith and jth cities, bijk = travel time (in minutes) of kth mode of transportation between ith and jth cities, Kj = annual capacity of an airport to be located in jth city in terms of number of passengers, and Xii, = frequency of kth mode of transportation from ith to jth city. The first part of the objective function is composed of the total annual line costs of transportation modes and total annual airport cost. The line costs for aircrafts and buses are defined separately. The aircraft line cost is defined by the following formula for all types of aircraft: c,,
=
vii c, + tVfk ’
where Vii = the distance between ith and jth cities, Ck = cost of kth type of aircraft per km, and Mk = fixed take off and landing cost of kth type of aircraft. Bus line cost is assumed to he composed of infrastructure cost, operation cost and travel time
Mathematical
programming
cost. The following formula is proposed Cij(bus)
=
models for airport site selection
443
for this purpose:
Iij + T[bij(bus) - bijcaircrart)l
where Iii = cost of infrastructure and operation between ith and jth cities and T = the value of time (TL/hr). The second part of the objective function which expresses total airport cost is common to all models. The constraints of the model essentially provide for the flow of transportation modes in the transportation network. The first constraint is called conservation of flow equation. There is one such equation for each node in the network. The second constraint is a supply constraint, restricting the frequency of an aircraft to its annual flight time. The third constraint is formulated to meet the travel demand of the city pairs, and the last is a capacity constraint for the frequency of transportation modes. The maximum capacity referred to here is the capacity of the airport to be constructed. The model was tested in the same region but on a specified network shown in Fig. 4. The parameters of the model were estimated by using Turkish Transportation Coordination Agency data covered in Alaybeyoglu (1973), Alaybeyoglu et al. (1974), and Hava Yolu Tasimasi (1974). The line costs of aircraft were estimated by use the formula given previously. For this purpose, flight times between the city pairs were multiplied by the cost per minute of each type of aircraft. In order to estimate the flight times, first, distance-minutes per kilometer were obtained as shown in Fig. 5 from the information given in Table 9. This relationship was used to
Fig. 4. Network used for the test application of Model III.
Min./Km.
100
I
90 60 70 60 50 40 30 20 10 I I
100
200
300
400
500
600
700
600 Km
Fig. 5. Distance-min/km
relationship.
444
OMERSAATCIOGLU Table 9. Distance-time
and min/km relationship of some lines
Distance (Km)
Flight Time (Min.)
in operation Min./Km
378
45
0.12
595
50
0.085
498
55
0.11
825
75
0.09
696
70
0.10
72
25
0.34
171
40
0.23
108
30
0.27
1014
95
0.09
Table 10. Flight times and the line costs of type B and C aircrafts between the city pairs
Distance
(W
Flight Time (Xin)
Line Cc 1 (TL) TYPO B
TYPO C
171
34
3921.9
1391.2
1, - Agri
156
33
3806.5
1350.3
!I - Mus
135
32
3691.2
1309.4
II - Bitlis
180
35
4037.2
1432.2
II - Van
225
40
4614.0
1636.8
Dlyarbakir - Mus
141
32
3691.2
1309.4
II - Bitlis
183
35
4037.2
1432.2
II - Van
282
42
4844.7
1718.6
II - siirt
150
33
3806.5
1350.3
determine the corresponding flight times between the city pairs. Two types of aircraft were selected for testing the model: types B and C, with respective costs per minute of 115.35 and 40.92 TL. The line costs for each type of aircraft were estimated as shown in Table 10, assuming that the flight times of both types were the same. Line costs of buses were estimated by using the formula previously proposed. To obtain the operating and infrastructure costs, distances between city pairs were multiplied by 3.45 TL/Km which was estimated by Alaybeyoglu (1973). The total value of time was obtained by multiplying travel time differences between bus and aircraft by the value of time (see Table 11). Since the airport costs defined in the objective function were identical to that of the previous model, the same cost estimates were also used in this model. The annual flight time of aircrafts were obtained from the existing schedules of Turkish Airlines. When the annual flight times of all the aircraft were analyzed, it was observed that they were available for flight 83% of the time. Using this information, the annual flight times of both types of aircraft were estimated. Annual flight times of the two types of aircraft and the number of passengers carried are given in Table 12. Demand forecasts between the city pairs for 1977 were obtained from Alaybeyoglu (1973). This data are given in Table 12. Demand forecasts for the return flights were assumed to be the same as the forecasts given in Table 13. MPSX package was used to solve the model. According to the solution, airports are proposed for the cities of Erzurum and Bitlis. In addition, the information regarding C-type aircraft schedule between the proposed airports and frequency of buses between the other cities are given.
Mathematical programming models for airport site selection
445
Table 11. Bus line costs Distance (Km)
city Pair
227
Erzurum - Kars 1,
- Agri
1,
- van
1,
189 419 259
- Mus
272
Diyarbakir - Mus 11
Infrastructure and Operating costs (TL)
231
- Bitlis
1,
- van
406
,I
- Siirt
215
Total Value of Time
Line Cost (TL)
784.60
4.000
4784.60
652.25
3,
4652.25
1448.23
(1
5448.23
895.20
(1
4895.20
950.51
,I
4950.51
798.42
11
4798.42
1403.29
11
5403.29
743.12
11
4743.12
Table 12. Aircraft information Rate of Utilization
Number of Passengers Carried
Annual Flight Times (Hrs.)
Aircraft TYPO
Seat Capacity
B
48
70
34
400
C
19
70
14
400
Table 13. Demand forecasts (one way) between the city pairs (1977) Demand Forecasts (T,,)
City Pairs
Fxzurum
-
Kars
-
Agri
D,
1,
52.000 9.000
- Mus
2.500
11
- Bitlis
2.500
!I
- Van
0
- Siirt
Diyarbakir-Kars
19.500 52.000
11
- Agri
9.000
1,
- Mus
2.500
#1
- Bitlis
2.500
1,
- Van
19.500
1,
- Siirt
19.500
Postoptimality analyses were conducted on the costs and the right-hand side parameters. When range analysis was performed, it was observed that almost 50% of the variables had negative values for their upper and lower cost limits. The other 50% had meaningful ranges identifying lower and upper activity levels. Table 14 gives cost ranges for the sample variables. The results indicate that if the line cost of type-B aircraft between City 1 (Erzurum) and City 2 (Kars) is reduced to 2744.99 TL, the corresponding frequency variable X,,, reaches its upper activity level of 352.94. As an extreme case, if the bus line cost between Kars and Erzurum is increased to 5306.19 TL, the variable X,,, takes its lower activity limit of 999.9, dropping from its present activity level of 1299.99.
OMER SAATCIOGLU
446
Table 14. Cost ranges of Model III Frequency (Xijk) From 1
To 2
14
Mode
Input COSt
Activity
LOW% Activity
UPPer Activity
Lower cost
UPPer Cost
1
0
3921.90
0
352.94
2744.99
1
0
3691.20
0
73.52
2519.43
73.52
2519.43 11500.39
5
4
1
0
3691.20
0
7
5
1
0
4844.69
0
0 178.5
m 5163.66
3372.23 12653.89
5
4
2
0
3691.20
0
-1171.76
3841.15
2
1
3
1299.99
3921.59
999.9
m
-3921.90
5306.19
4
1
3
62.49
3691.20
0
62.49
-3460.49
5069.74
Table 15. Right hand side ranges of Model III Row sup 1
Activity 0
LOWfY Activity
UPP‘= Activity
0
103999.83 252570.85
sup 2
0
0
Dem 1
51999.98
51999.98
Dan 2
8999.99
8999.99
Dem 3
2499.99
2499.99
0
0
0
19499.99
0
19499.99
DenI4 Dan 15
m 2499.99
When the right-hand side ranges of the constraints were analyzed, it was observed that only ranges for demand and supply constraints had practical meanings. Table 15 shows the right-hand side ranges for sample demand and supply constraints. As it can be seen in Table 15, travel demand of Erzurum and Kars can increase to infinity without changing the present optimal solution. In the parametric analysis, only the right-hand side values of demand constraints were increased by 30,000 as the maximum. The resulting output was very similar to the original output. The same variables came up with positive but different values in the solution, implying that the optimal airport allocation pattern remained the same. 5. SUMMARY
AND CONCLUSIONS
In this paper, it is shown that airport site selection problem in developing countries can be approached through mathematical programming models using different evaluation criteria. In the first model, airport cost per person and the bus fares between the cities were used to determine the optimal airport sites. The second model included both annual airport costs and the cost of transporting passengers from other cities as criteria. The model found the optimal airport sites as well as the number of passengers to be transported from the cities without airport. In the third model the criteria used were the airport costs and the line costs of aircraft and buses between the cities. The model selected the optimal airport sites, and, at the same time, found the optimal frequencies of transportation modes between the city pairs. Results of the first and second models were in general agreement with the existing situation. However, when the trade off point used in the first model was 70TL, and when Uij’s were estimated by considering both bus and air fares in the second model, the solutions proposed an additional airport. The output of the third model was completely different from the others. This shows that the structure of the models, evaluation criteria used, parameter estimation procedures and the information requirements of the model used can yield different solutions to the same problem. Therefore, selection of a model would depend on the formulation of the problem as well as the availability of data.
Mathematical
programming
models for airport site selection
447
The range and parametric analyses were conducted on each model with respect to their costs and right-hand side parameters. In Model I it was observed that four cities were sensitive to the cost parameters but none of them were sensitive to population growth as long as the rate of increase remains the same. The postoptimality analysis in the other models either did not lead to meaningful results or the variables were not sensitive to the changes in data. Acknowledgement-The
author would like to express his sincere thanks for the suggestions made by the referees.
REFERENCES Alaybeyoglu Y. (1973) Katma Deger ve Karayolu Tasimasi. Unpublished Research Project of Turkish Coordination Agency. Ministry of Transportation, Ankara, Turkey. Alaybeyoght, Y. et al. (1974) Ttirkiye’ de Besleyici Havayolu Sistemi Yapilabihrlik Etudti. Ministry of Transportation Publications, 106436. Ministry of Transportation, Ankara, Turkey. Ashford N. and Wright P. H. (1979) New York, Airport Engineering J. Wiley, Inc. Aviation Planning Services. (1973) The feasibility of an expanded domestic airline route structure in Turkey for the international civil aviation organization. Final Rep. Ministry of Transportation, Ankara, Turkey. Cohon J. L. (1978) Multiobjective Programming and Planning. Academic Press New York. DeNeufville R. (1976) Airport Systems Planning. The MIT Press, Mass. Francis R. and White J. A. (1974) Facilities Layout and Location: An Analytical Approach. Prentice-Hall, New Jersey. Garfinkel R. S. and Nemhauser, G. L. (1972) Integer Programming Wiley, Inc. New York. Havayolu Tasimasi. (THY) (1974) Tasima sistemlerinde isletme maliyeti arastirma projesi. Ministry of Transportation Publication, 073 UK1 4-043. Ministry of Transportation, Ankara, Turkey. Howard, G. P. (1974). Airport Economic Planning. The MIT Press, Mass. II Merkezleri Arasi Yolcu ve Esya Tasima Ucretleri. (1974) General Directory of Road Transportation Publication. Ministry of Transportation, Ankara, Turkey. Keeney R. L. (1973) A decision analysis with multiple objectives: The Mexico City Airport. Bell1 Econ. Management Sci. 4, 101-117. Miller R. E. (1%7) An optimization model for transportation planning. Trans. Rex 1, 271-284. Revelle C. and Swain, R: (1970) Central facilities location. Geographical Anal. 2, 30-42. Schilling D. (1976). Multiobjectioe and Temporal Considerations in Public Facility Location. Ph.D. Thesis, Department of Geography and Environmental Engineering, Johns Hopkins University, Baltimore, Maryland. Toregas C. et al. (1971). The location of emergency service facilities. Gper. Res. 19, 1363-1373.
1982
0191-!615/82/oho435-13$03.0010 0 1982 Pergamon Prw Ltd.
MATHEMATICAL PROGRAMMING MODELS FOR AIRPORT SITE SELECTION OMER SAATCICCLUt, Middle East Technical University, (Received 13 February
Ankara, Turkey
1978; in revised form 11 May 1981)
Abstract-Three mathematical programming models are developed in this paper, using different criteria, for the airport site selection problem in developing countries. Estimation of parameters, based on Turkish data, and the sensitivity analyses are presented. The first is a typical set covering model used to determine the minimum number of airports required for a given population of passengers. The other two are locationallocation type models used to find the optimal airport location patterns, the number of passengers to be bussed to cities having airports, and the optimal frequencies of transportation modes between the city pairs.
1. INTRODUCTION
Airport site selection problem is one of the macro-level planning problems of the transportation system and has to be treated within the entire system domain. However, because of the complexity of the transportation system and the lack of certain socio-economic indicators and parameters, this particular problem is treated in isolation. Most of the time it is handled within the scope of Airport Master Plan recommended by either the International Civil Aviation Organization or the Federal Aviation Administration of the USA (see Ashford; Wright, 1979). Furthermore, the number of alternatives available for the airport site selection problem is, in general, too large, and it is very d&cult to generate all ‘of them. Therefore, researchers and planners are inclined to consider a few intuitively generated alternatives for evaluation. In turn, the evaluation process, directed to the identification of the decision makers and of the best criterion or a set of criteria, appears to be a critical issue. There are various costs and benefits associated with the airport site selection problem. These costs and benefits have to be traded off to find the optimal location pattern of airports and their capacities. However, each nation generally has its own concept of the role of government, of what constitutes public benefits, and of the role of private sector agencies, and evaluation procedures. DeNeufville (1976) presented a country’s concept of the public interest, who participates in the decisions, what kind of evaluation will occur and where power will lie in relation to airport site selection problem. This problem is all the more important in developing countries where resources are scarce, and data necessary to make rational decisions are either insufficient or inaccurate. In the literature, economic and decision analysis approaches have been used in selecting the optimal airport site and the most effective strategy for developing the airport facilities (see, e.g., Howard, 1974; Keeney, 1973). In this study, three different mathematical programming models are developed for the airport site selection problem in developing countries. Air transportation system is represented by a network with airports and routes being the nodes and links, respectively. Planned airport constructions in appropriate cities or regions are treated as additions of nodes and links to the transportation network. The models are tested for the land-locked eastern Turkey where road transportation is difficult. Figure 1 shows a map of the region and the cities selected as potential airport sites. Socio-economic interactions between the cities of the region were taken into consideration. At the time of this investigation, three airports were reported to be operational. Air travel demand in the region was met by these three airports located in Erzurum, Diyarbakir and Van. Previous studies, Alaybeyoglu et al. (1974), and Aviation Planning Services (1973) concluded that additional airports were necessary for the region.
Kurrently
Visiting Associate Professor
at the University of Wisconsin-Milwaukee. 435
OMER SAATCIOGIL
436
‘\
0 Gumushane
:\ .’ ‘..a
0
0 Agri
Erzurum
.-.-
\
i’
‘L
Erzincan
0
:\
0
Bingo1
0 Van
MUS
l
0
Elazig
Biilis
0 Siitl
0
Hakkari ,.;-.-._.i
Diyarbakir
0 Urfa --4
..(._.-.-
./ -. H’
.’
A’
.
Other
o Cities
0
C, .,. i
P.. <_.I
cities selected
potentialawpclrt
BS CIUOS
airport5 Approximatescale:
@Cities
5 i
with
1 cm=4OKm
Fig. 1. Eastern Turkey with the cities selected as potential airport sites.
2. MODEL
I
The first model used in this study to find the optimal locations for the airports, is a typical set covering model. A common formulation of a set covering problem was given by Francis and White (1974), Revelle and Swain (1970) and Toregas et al. (1971), and involved the determination of the minimum number of facilities required to cover a set of customers. The same problem was presented by Schilling (1975) and Cohon (1978) as a multiobjective facility location problem. The model has a simple structure and can be applied to airport site selection problem as shown below:
CPjYj
Minimize
j=l
Subject to
TdiiYj?lfori=l,...,l
Yj=OorYj=l
where Pj is the per-capita construction cost of the airport in city j, dij is the covering coefficient. (dij = 1 if air transportation is feasible between cities i and j and d, = 0 otherwise), Yj is the O-l decision variable (Y, = 1 if an airport is constructed in jth city, Yj = 0 otherwise). The objective function gives the total per-capita airport construction cost in the cities of the entire region. The underlying assumption is that the airport construction cost per person can be used as a criterion to evaluate the alternatives. One way of estimating the cost parameters, Pj, of the objective function is to divide the airport construction costs by the total population to be served by an airport. An alternative approach would be to divide the construction costs by the air travel potential of each city and its vicinity. The constraint of the model generates alternative airport location patterns based on a specific criterion. This criterion could be distance, travel time or travel cost between cities. Aggregate average bus fare was chosen as the criterion to identify the potential city pairs for air transportation. The reasoning behind this choice was that if the bus fare between two cities
Mathematical
programming
431
mpdelsfor airport site selection
exceeded a certain amount, then the air travel could become preferable. In other words, it was assumed that a passenger would prefer air travel over bus service beyond a certain distance and corresponding bus fare. The covering matrix is formed by dii’s in the constraint on the basis of bus fares between cities. If the bus fare between a city pair is equal to or greater than a specified level, then the covering coefficient is equal to “1”. If this condition is not satisfied, the covering coefficient becomes “0”. Multiplication of the covering matrix by the vector Yi’s identifies the cities, which can be reached by air transportation from a particular city i. The model is applied to the region mentioned above for testing. Three types of information were required for the model: bus fares between cities, construction costs of airports, and populations of the cities as given in Tables 1 and 2. Two covering matrices were used for testing. Both matrices were obtained by using the trade off points between bus and air transportation. These trade off points were taken as 70 Turkish Liras (TL) and 50TL for the first and the second matrix, respectively. Table 3 illustrates the first matrix. Evidently, these values are subject to change in time. MPSX package was used to solve the problem, because of its availability at the time. MPSX solves the linear analog of an integer programming problem as part of the branch-and-bound procedure. For large scale problems it may be advisable to use more efficient set covering algorithms as presented by Garfinkel and Nemhauser (1972).
Table I. Population and cost estimates Airport ClXIStlXCtiOIl Costs (TL)
Cities
City Population(l977)
Cost Per Person (Pj)
ErZUKUII 5.960
Kars
lS3.000
32.5
Agri
3.880.100
89.000
43.6
:1us
4.620.000
54.000
85.5
Bingal
4.620.000
28.000
165.0
3.849.300
62.000
62.1
Siirt
7.199.900
126.000
57.1
Grdin
6.020.000
135.000
44.6
Hakkari
a.022.000
12.000
668.5
Diyarbakir Bitlis Van
Table 2. Bus fares between cities EIYZ. Kars E~7.Ui?Ull
0
Agri
Plus Bin.
Diy.
Bitlis Van
Siirt Mad.
Hakkr.
25
25
50
55
63
70
50
83
72
80
Kars
25
0
30
75
80
115
95
65
100
109
95
Agri
25
30
0
50
70
88
50
30
65
97
60
j\lus
50
75
50
0
15
40
15
30
30
44
60
Bingcl
55
80
70
15
0
35
20
40
35
39
70
Diyarbakir 63
115
88
40
35
0
30
40
30
9
80
95
50
15
20
30
0
20
15
30
50
Bitlis
70
Van
50
65
30
30
40
40
20
0
35
49
30
Siirt
83
100
65
30
35
30
15
35
0
39
65
Mardin
72
109
97
44
39
9
30
49
39
0
79
Hakkari
80
95
60
60
70
80
50
30
65
79
0
IX-BVol. 16. No.6-B
438
OUER SAATCIO(;LU
Table 3. Airport covering matrix Er2. EKZU?XlU
Kars
0
0
Agri
Nus
0
Bin.
0
0
Diy.
Bitlis Van
0
1
Siirt Plard. Hakkr.
0
1
1
1
KZXlTS
0
0
0
1
1
1
1
0
1
1
1
Agri
0
0
0
0
1
1
0
0
0
1
0
El"S
0
1
0
0
0
0
0
0
0
0
0
Bingtil
0
1
1
0
0
0
0
0
0
0
1
Diyarbakir 0
1
1
0
0
0
0
0
0
0
1
Bitlis
1
1
0
0
0
0
0
0
0
0
0
"a"
0
0
0
0
0
0
0
0
0
0
0
Siirt
1
1
0
0
0
0
0
0
0
0
0
Mardin
1
1
1
0
0
0
0
0
0
0
1
Hakkari
1
1
0
0
1
1
0
0
0
1
0
Table 4. Cost ranges of the sensitive variables in Model I Variables
Activity Level
Input COSt
Lower Cost -m
0
UPPer cost 0
YOl (Erzurum)
1.
YO2 (Kars)
1.
32.5
Y09 (Siirt)
0
57.1
44.6
62.1
YlO (Mardin)
1.
44.6
-m
57.1
0
43.6
When 70TL trade off point was used to solve the model, the results indicated an airport requirement for Kars, in addition to the existing airports of Erzurum, Diyarbakir and Van. However, no additional airport was recommended when SOTL was used as a trade off point. This clearly indicates that the model is sensitive to the covering matrix. The postoptimality analysis was conducted on coefficients of the objective function (i.e. on the construction costs of airports per person). The ranges of the cost coefficients were determined and a parametric analysis was carried out. When the range analysis was conducted, it was observed that only four of the eleven cities considered were sensitive to the lower and upper cost limits. Table 4 summarizes the ranges, activity levels, and input costs of the variables associated with these four cities. All the variables remain at their present activity levels as long as the cost coefficients remain between their lower and upper limits. In addition, the range analysis indicated the status of these variables if the coefficients go above and below their limits. For example, if the unit construction cost of the airport in Mardin increases above 57.1, Y 10 goes to zero, indicating an airport should not be considered for Mardin. Similarly, if the cost of variable Y09 is reduced to 44.6 or below, the variable takes a value of 1. This causes YlO to enter into the solution at its upper bound of 1, proposing airports for Siirt and Mardin, simultaneously. In the parametric analysis, cost coefficients were decreased by 30 TLlperson but the original solution did not change. This indicates that the optimal airport location pattern is not sensitive to population growth as long as the rate of increase remains the same. In this analysis it was assumed that the rate of population increase was the same in each city. 3. MODEL
II
The second model is developed on the basis of total infrastructural cost and cost-of-travel criteria. In this model, it is assumed that the airline management would provide a bus service for the air passengers of towns without airports to cities with airports. The model is a discrete
Mathematical
modelsfor airport site selection
programming
439
plant-location model. It determines the locations and capacities of the airports as well as the number of passengers transported from cities without airports. The formulation used is given below: Minimize
Subject to:
g.ZijcKj,Yj,forj=l
,...,
iZij=ti
nandl=l,...,
L
for i=l,...,m
j=l
x
Yj, 5 1 for j = 1,. . . , Tl
2 5 Aj,Yj, 5
B
I=I
j=l
Z,, Yj, 2 0 and Yj, = 0 or 1 where Zij = number of passengers to be carried between ith and jth cities, Yjl = O-l variable ( Yj, = 1 if an Ith type airport will be constructed in jth city, Yj, = 0 otherwise), Uij = unit transportation cost between ith and jth cities, Ajl = annual cost of an Ith type of airport in jth city, Kjl = annual capacity (in terms of the number of passengers) of an Ith type airport constructed in jth city, Ti = trip generation in ith city, and B = total budget. The first part of the objective function defines the total transportation cost of the passengers transported to cities with airports. Uij’s can be estimated either by considering a unit bus fare between ith and jth cities or by using the following formula: Uij
=
0, +
UjO
where 0, = bus fare between ith and jth cities and Vi, = air fare between jth city and cities with major airports. Here it is assumed that the air fare between airports in the region and the major connecting airport would affect the choice of routes between ith and jth cities. The second part of the objective function represents a total airport construction cost. A, is a function of the airport capacity and city characteristics. The first constraint of the model is a capacity constraint, restricting the number of pas: sengers. Because of the airport capacity limitations, there is a constraint imposed on the number of bus passengers. Airport capacity of the jth city is determined by a O-l variable, Yjl. The second constraint of the model is a demand constraint. By this constraint, the number of bus passengers carried from a given city is restricted by forecasted demand in that city. The third constraint limits the number of airports to one in each city. The last constraint is a budget constraint. Model II was tested in the same region used for testing Model I. Some modifications were deemed necessary in the model due to lack of data. As a result of these modifications the budget constraint was eliminated, and only one type of airport was considered. The data required were obtained from Alaybeyoglu et al. (1974), and 11 Merkezleri Arasi Yolcu ve Esya Tasima Ucretleri (1974). The parameter, Uij was estimated in two different ways, as previously explained. One way of estimating Vij was to sum the bus fares between city pairs, O,, and the air fare up to the major connecting airport, Uj,. The values of Uj, were found by multiplying the distance by average unit fare. The average unit fare was found to be l.O6TL/KM. From the air fare data given in Table 5. Values of the parameter, Uij are given in Table 6. Data required for other parameters, demand forecast, cost of airports and the capacity of airports are given in Table 7.
440
OMER SAATCIOGLL
Table 5. Flight distances and air fares for existing routes Fares (TL)
Fare per Km (TL/Km)
300
0.29
597
300
0.50
498
395
0.79
378
828
570
0.68
696
485
0.69
72
165
2.29
171
260
1.52
108
195
1.80
1014
560
0.55
Table 6. Sum of bus and air fares between cities Erz. Kars
Agri
Mus
Bin.
Diy.
Bitlis Van
E?ZZU?XXll
-
-
_
-
-
-
KarS
25
-
155
140
-
-
150
-
Siirt Mard.
Hakk.
185
-
-
Agri
25
167
-
160
200
-
150
-
-
MUS
50
-
-
-
40
165
-
155
-
-
BingGl
55
-
-
-
35
-
-
-
Bitlis
50
-
-
125
30
-
-
-
_
-
Van
50
200
-
-
Diyarbakir
155
140
40
167
-
155
Siirt
-
140
30
-
-
-
Mardin
-
152
-
160
-
-
80
200
65
190
Hakkari
-
Table 7. Demand forecast, airport costs and airport capacities
Cities
Annual Airport Cost (TL Millions)
Erzurum" Kars
5.96
Agri
3.88
MUS
4.62
Bingal
4.62
Diyarbakir* Bitlis
3.85
vatI* Siirt
7.20
Mardin
6.02
Hakkari
8.02
*Cities with airports
Demand FOreCaStS 0977)
Airport Capacity (number of Passengers)
30168
71731
22000
II
3808
I,
1650
(1
450
,I
36972
,I
1909
II
14200
(1
8090
3,
10000
11
624
,1
Mathematical
programming
441
models for airport site selection
Fig. 2. Network representation of the computer output of Model II, when U,, = 0,, t U,,,.
Fig. 3. Network representation of the computer output of Model II, when Uii = Oij
The model was solved several times using MPSX for different transportation costs, U,. When Uij’s were taken as the sum of bus and air fares, the solution of the model proposed an additional airport in Bitlis. Furthermore, the solution provided the number of passengers carried by bus. The air transportation network representing the solution of the model is given in Fig. 2. The model, with the same values of cost parameters, was also solved assuming that the airports in Erzurum, Diyarbakir and Van did not exist. This time the solution proposed airports only for Erzurum and Mus. When the model was solved for Uij = Oi,, the solution did not recommend any airport other than the existing ones. The network representation of the solution for this case is given in Fig. 3. The solutions of the model in their network representations clearly show the regions that are affected by the proposed or existing airports. This information can be used for the management of the transportation system in establishing the required bus routes. Range and parametric analysis were performed on cost and right-hand side parameters of the model. The results indicated that ranges on the bus fares between cities did not lead to a meaningful analysis. This can be explained by negative lower cost limits of the bus fares. Most of the airport construction cost ranges had infinity as upper bounds. Although the lower cost limits had positive values, they did not lead to a change of status in the variables. The analysis of the right-hand side ranges for the travel demand produced reasonably Table 8. Demand ranges of Model II Cities
Activity
Demand
Lower Limit
Upper Limit
30167.97
30167.97
0
49922.95
21999.98
21999.98
0
37754.96
3807.99
3807.99
0
19562.97
1649.99
1649.99
0
41291.98
449.99
449.99
0
24758.99
36971.99
36971.99
0
61280.98
1908.99
1908.99
0
41550.98
8
14199.99
14199.99
0
53841.99
3
8089.99
3039.99
0
47731.98
10
9999.99
9999.99
0
34308.98
11
6239.99
6239.99
0
458S1.95
6
442
OMER SAATCIOGLU
valuable information with respect to solution status. Table 8 summarizes demand ranges, original demand figures, and activity levels for each constraint. The solution remains optimal with current basis as long as the demands range between zero and their upper limits. In this analysis, the output also indicates state of airport capacity if demand figures reaches their lower and upper limits. Accordingly, if the demands for Erzurum, Kars and Agri reach their upper limits, the airport capacity of Erzurum reaches its maximum. Capacity of Diyarbakir airport also goes to its upper limit when the travel demands of Bingol, Diyarbakir and Mardin reach their upper limits. Airport capacity of Van is fully utilized if the travel demands of Bitlis, Van, Siirt and Hakkari raise to their upper limits. When the parametric analysis was conducted on the demand vector, up to a maximum increase of 30,000, the same three cities were proposed for airport construction. This indicates that proposed airports are sufficient at least up to 30,000 additional travel demand in each city. 4. MODEL
III
Model III is an extension of the aircraft assignment model developed by Miller (1967). In this model, the air and road line costs between the cities were used as criteria in addition to the annualized airport construction costs. The model is a mixed integer programming model which minimizes the following objective function.
2i 2 i=l
Subject to
j=]
CiikXijk
+i
AjYj
j=l
k=l
2 X,, - 2 Xjik = 0 for j = 1, . . , n and k=l,...,r
2 2 bijkXijk5 S, for k = 1, . . . , r [=I j=l r
~giikXij,>Tijfori=l,...,mand k=l
j=l,...,n
Yi, Xijk 2 0 and Yj is O-l variable. where C,, = line cost of kth type of bus and/or aircraft between ith and jth cities, Ai = annual cost of an airport to be located in jth city S, = available operation time of kth mode of transportation per year in minutes, Tij = annual travel demand between ith and jth cities gijk = the number of passengers to be transported by kth mode of transportation between ith and jth cities, bijk = travel time (in minutes) of kth mode of transportation between ith and jth cities, Kj = annual capacity of an airport to be located in jth city in terms of number of passengers, and Xii, = frequency of kth mode of transportation from ith to jth city. The first part of the objective function is composed of the total annual line costs of transportation modes and total annual airport cost. The line costs for aircrafts and buses are defined separately. The aircraft line cost is defined by the following formula for all types of aircraft: c,,
=
vii c, + tVfk ’
where Vii = the distance between ith and jth cities, Ck = cost of kth type of aircraft per km, and Mk = fixed take off and landing cost of kth type of aircraft. Bus line cost is assumed to he composed of infrastructure cost, operation cost and travel time
Mathematical
programming
cost. The following formula is proposed Cij(bus)
=
models for airport site selection
443
for this purpose:
Iij + T[bij(bus) - bijcaircrart)l
where Iii = cost of infrastructure and operation between ith and jth cities and T = the value of time (TL/hr). The second part of the objective function which expresses total airport cost is common to all models. The constraints of the model essentially provide for the flow of transportation modes in the transportation network. The first constraint is called conservation of flow equation. There is one such equation for each node in the network. The second constraint is a supply constraint, restricting the frequency of an aircraft to its annual flight time. The third constraint is formulated to meet the travel demand of the city pairs, and the last is a capacity constraint for the frequency of transportation modes. The maximum capacity referred to here is the capacity of the airport to be constructed. The model was tested in the same region but on a specified network shown in Fig. 4. The parameters of the model were estimated by using Turkish Transportation Coordination Agency data covered in Alaybeyoglu (1973), Alaybeyoglu et al. (1974), and Hava Yolu Tasimasi (1974). The line costs of aircraft were estimated by use the formula given previously. For this purpose, flight times between the city pairs were multiplied by the cost per minute of each type of aircraft. In order to estimate the flight times, first, distance-minutes per kilometer were obtained as shown in Fig. 5 from the information given in Table 9. This relationship was used to
Fig. 4. Network used for the test application of Model III.
Min./Km.
100
I
90 60 70 60 50 40 30 20 10 I I
100
200
300
400
500
600
700
600 Km
Fig. 5. Distance-min/km
relationship.
444
OMERSAATCIOGLU Table 9. Distance-time
and min/km relationship of some lines
Distance (Km)
Flight Time (Min.)
in operation Min./Km
378
45
0.12
595
50
0.085
498
55
0.11
825
75
0.09
696
70
0.10
72
25
0.34
171
40
0.23
108
30
0.27
1014
95
0.09
Table 10. Flight times and the line costs of type B and C aircrafts between the city pairs
Distance
(W
Flight Time (Xin)
Line Cc 1 (TL) TYPO B
TYPO C
171
34
3921.9
1391.2
1, - Agri
156
33
3806.5
1350.3
!I - Mus
135
32
3691.2
1309.4
II - Bitlis
180
35
4037.2
1432.2
II - Van
225
40
4614.0
1636.8
Dlyarbakir - Mus
141
32
3691.2
1309.4
II - Bitlis
183
35
4037.2
1432.2
II - Van
282
42
4844.7
1718.6
II - siirt
150
33
3806.5
1350.3
determine the corresponding flight times between the city pairs. Two types of aircraft were selected for testing the model: types B and C, with respective costs per minute of 115.35 and 40.92 TL. The line costs for each type of aircraft were estimated as shown in Table 10, assuming that the flight times of both types were the same. Line costs of buses were estimated by using the formula previously proposed. To obtain the operating and infrastructure costs, distances between city pairs were multiplied by 3.45 TL/Km which was estimated by Alaybeyoglu (1973). The total value of time was obtained by multiplying travel time differences between bus and aircraft by the value of time (see Table 11). Since the airport costs defined in the objective function were identical to that of the previous model, the same cost estimates were also used in this model. The annual flight time of aircrafts were obtained from the existing schedules of Turkish Airlines. When the annual flight times of all the aircraft were analyzed, it was observed that they were available for flight 83% of the time. Using this information, the annual flight times of both types of aircraft were estimated. Annual flight times of the two types of aircraft and the number of passengers carried are given in Table 12. Demand forecasts between the city pairs for 1977 were obtained from Alaybeyoglu (1973). This data are given in Table 12. Demand forecasts for the return flights were assumed to be the same as the forecasts given in Table 13. MPSX package was used to solve the model. According to the solution, airports are proposed for the cities of Erzurum and Bitlis. In addition, the information regarding C-type aircraft schedule between the proposed airports and frequency of buses between the other cities are given.
Mathematical programming models for airport site selection
445
Table 11. Bus line costs Distance (Km)
city Pair
227
Erzurum - Kars 1,
- Agri
1,
- van
1,
189 419 259
- Mus
272
Diyarbakir - Mus 11
Infrastructure and Operating costs (TL)
231
- Bitlis
1,
- van
406
,I
- Siirt
215
Total Value of Time
Line Cost (TL)
784.60
4.000
4784.60
652.25
3,
4652.25
1448.23
(1
5448.23
895.20
(1
4895.20
950.51
,I
4950.51
798.42
11
4798.42
1403.29
11
5403.29
743.12
11
4743.12
Table 12. Aircraft information Rate of Utilization
Number of Passengers Carried
Annual Flight Times (Hrs.)
Aircraft TYPO
Seat Capacity
B
48
70
34
400
C
19
70
14
400
Table 13. Demand forecasts (one way) between the city pairs (1977) Demand Forecasts (T,,)
City Pairs
Fxzurum
-
Kars
-
Agri
D,
1,
52.000 9.000
- Mus
2.500
11
- Bitlis
2.500
!I
- Van
0
- Siirt
Diyarbakir-Kars
19.500 52.000
11
- Agri
9.000
1,
- Mus
2.500
#1
- Bitlis
2.500
1,
- Van
19.500
1,
- Siirt
19.500
Postoptimality analyses were conducted on the costs and the right-hand side parameters. When range analysis was performed, it was observed that almost 50% of the variables had negative values for their upper and lower cost limits. The other 50% had meaningful ranges identifying lower and upper activity levels. Table 14 gives cost ranges for the sample variables. The results indicate that if the line cost of type-B aircraft between City 1 (Erzurum) and City 2 (Kars) is reduced to 2744.99 TL, the corresponding frequency variable X,,, reaches its upper activity level of 352.94. As an extreme case, if the bus line cost between Kars and Erzurum is increased to 5306.19 TL, the variable X,,, takes its lower activity limit of 999.9, dropping from its present activity level of 1299.99.
OMER SAATCIOGLU
446
Table 14. Cost ranges of Model III Frequency (Xijk) From 1
To 2
14
Mode
Input COSt
Activity
LOW% Activity
UPPer Activity
Lower cost
UPPer Cost
1
0
3921.90
0
352.94
2744.99
1
0
3691.20
0
73.52
2519.43
73.52
2519.43 11500.39
5
4
1
0
3691.20
0
7
5
1
0
4844.69
0
0 178.5
m 5163.66
3372.23 12653.89
5
4
2
0
3691.20
0
-1171.76
3841.15
2
1
3
1299.99
3921.59
999.9
m
-3921.90
5306.19
4
1
3
62.49
3691.20
0
62.49
-3460.49
5069.74
Table 15. Right hand side ranges of Model III Row sup 1
Activity 0
LOWfY Activity
UPP‘= Activity
0
103999.83 252570.85
sup 2
0
0
Dem 1
51999.98
51999.98
Dan 2
8999.99
8999.99
Dem 3
2499.99
2499.99
0
0
0
19499.99
0
19499.99
DenI4 Dan 15
m 2499.99
When the right-hand side ranges of the constraints were analyzed, it was observed that only ranges for demand and supply constraints had practical meanings. Table 15 shows the right-hand side ranges for sample demand and supply constraints. As it can be seen in Table 15, travel demand of Erzurum and Kars can increase to infinity without changing the present optimal solution. In the parametric analysis, only the right-hand side values of demand constraints were increased by 30,000 as the maximum. The resulting output was very similar to the original output. The same variables came up with positive but different values in the solution, implying that the optimal airport allocation pattern remained the same. 5. SUMMARY
AND CONCLUSIONS
In this paper, it is shown that airport site selection problem in developing countries can be approached through mathematical programming models using different evaluation criteria. In the first model, airport cost per person and the bus fares between the cities were used to determine the optimal airport sites. The second model included both annual airport costs and the cost of transporting passengers from other cities as criteria. The model found the optimal airport sites as well as the number of passengers to be transported from the cities without airport. In the third model the criteria used were the airport costs and the line costs of aircraft and buses between the cities. The model selected the optimal airport sites, and, at the same time, found the optimal frequencies of transportation modes between the city pairs. Results of the first and second models were in general agreement with the existing situation. However, when the trade off point used in the first model was 70TL, and when Uij’s were estimated by considering both bus and air fares in the second model, the solutions proposed an additional airport. The output of the third model was completely different from the others. This shows that the structure of the models, evaluation criteria used, parameter estimation procedures and the information requirements of the model used can yield different solutions to the same problem. Therefore, selection of a model would depend on the formulation of the problem as well as the availability of data.
Mathematical
programming
models for airport site selection
447
The range and parametric analyses were conducted on each model with respect to their costs and right-hand side parameters. In Model I it was observed that four cities were sensitive to the cost parameters but none of them were sensitive to population growth as long as the rate of increase remains the same. The postoptimality analysis in the other models either did not lead to meaningful results or the variables were not sensitive to the changes in data. Acknowledgement-The
author would like to express his sincere thanks for the suggestions made by the referees.
REFERENCES Alaybeyoglu Y. (1973) Katma Deger ve Karayolu Tasimasi. Unpublished Research Project of Turkish Coordination Agency. Ministry of Transportation, Ankara, Turkey. Alaybeyoght, Y. et al. (1974) Ttirkiye’ de Besleyici Havayolu Sistemi Yapilabihrlik Etudti. Ministry of Transportation Publications, 106436. Ministry of Transportation, Ankara, Turkey. Ashford N. and Wright P. H. (1979) New York, Airport Engineering J. Wiley, Inc. Aviation Planning Services. (1973) The feasibility of an expanded domestic airline route structure in Turkey for the international civil aviation organization. Final Rep. Ministry of Transportation, Ankara, Turkey. Cohon J. L. (1978) Multiobjective Programming and Planning. Academic Press New York. DeNeufville R. (1976) Airport Systems Planning. The MIT Press, Mass. Francis R. and White J. A. (1974) Facilities Layout and Location: An Analytical Approach. Prentice-Hall, New Jersey. Garfinkel R. S. and Nemhauser, G. L. (1972) Integer Programming Wiley, Inc. New York. Havayolu Tasimasi. (THY) (1974) Tasima sistemlerinde isletme maliyeti arastirma projesi. Ministry of Transportation Publication, 073 UK1 4-043. Ministry of Transportation, Ankara, Turkey. Howard, G. P. (1974). Airport Economic Planning. The MIT Press, Mass. II Merkezleri Arasi Yolcu ve Esya Tasima Ucretleri. (1974) General Directory of Road Transportation Publication. Ministry of Transportation, Ankara, Turkey. Keeney R. L. (1973) A decision analysis with multiple objectives: The Mexico City Airport. Bell1 Econ. Management Sci. 4, 101-117. Miller R. E. (1%7) An optimization model for transportation planning. Trans. Rex 1, 271-284. Revelle C. and Swain, R: (1970) Central facilities location. Geographical Anal. 2, 30-42. Schilling D. (1976). Multiobjectioe and Temporal Considerations in Public Facility Location. Ph.D. Thesis, Department of Geography and Environmental Engineering, Johns Hopkins University, Baltimore, Maryland. Toregas C. et al. (1971). The location of emergency service facilities. Gper. Res. 19, 1363-1373.