The market size of a city-pair route at an airport

The market size of a city-pair route at an airport

Chaug-ing Hsu *, Yai-hui Wu **

Department of Transportation Engineering and Management, National Chiao-Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 30050, Republic of China (Fax: 0 35-72 08 44) Received: August 1995 / Accepted in revised form: February 1997

* Associate Professor, corresponding author. ** Graduate Research Assistant.

Abstract A model is developed for estimating the size of the market for a

city-pair route at an airport from both the demand and supply sides of air transportation. The average airport access cost, average passenger delay cost, and average airline operating cost all either increase or decrease with an increase in the market size of a city-pair route at an airport, so the opti-mum market size can be determined from trade-offs among these costs. A nonlinear mathematical programming problem is formulated to determine the optimal number of passengers, the local service area of a city-pair mar-ket and to perform sensitivity analyses. The results show that long-haul vices ought to be concentrated in one large airport, while short-haul ser-vices might be dispersed among many small airports. Improvements in the technology of the airport access mode or increases in the average income of the cities served can expand the market size and service area, but at a declining expansion rate. In metropolitan areas with high population den-sity, airlines can operate more efficiently and distribute air services among more airports. City-pair markets with stable passenger demand, or markets served by airlines with efficient scheduling technology are shown to exhibit high cost efficiencies.

1. Introduction

Airlines frequently need to estimate passenger demand for city-pair routes at a particular airport in order to plan marketing, scheduling, and routing strategies. Although much research has been done on the development of city-pair air passenger demand models, there are two issues that need to be considered further:

1. Most city-pair demand models or airport demand models have been in-vestigated by aggregate gravity-type or econometric regression approaches (e.g., Verleger 1972; Morrison and Winston 1985; Kaemmerle 1991; Rus-son and Riley 1993) or disaggregate utility maximizing approaches (e.g., Harvey 1987; Hansen and Kanafani 1988, 1990) from the perspective of either passengers or airlines. The discussion has focused on models in which the supply characteristics are implicitly assumed to be exogenous. However, in long-term equilibrium, the market for city-pair routes at an air-port should be an endogenous variable resulting from interactions between decisions made by air passengers, airlines, and airport authorities.

2. O’Kelly (1986) combined a facility location problem with a model of spatial interaction and described a normative framework which finds the best location for hubs. However, a normative framework which simulta-neously finds the optimal passenger number and local market area for the airport market of a city-pair air route has not been investigated. Estimates of the number of passengers in the city-pair market of an airport are useful for many purposes. Estimating the optimal local service area for the city-pair market of an airport could be the key to determining the number of airports needed to serve a multi-airport region or metropolitan area. More-over, simultaneously determining both the optimal market size and the opti-mal service area may unveil trade-offs among airline operation-related costs (e.g., air fares, schedule delays) and airport ground access costs.

This paper attempts to develop a unified model to determine both the number of passengers of the optimal market and the optimal local service area for the city-pair route served by an airport. Economies of density exist if unit costs decline as airlines add flights or use larger aircraft with no changes in load factor and stage length within a given city-pair route (Caves et al. 1984). Economies of density are reached within any city-pair market as a response to expanding market size. If more flights are added, passenger delay costs are reduced, and if larger aircraft are used, the unit operating cost declines. Reductions in the above costs result in better quali-ty of service and lower air fares, and thus attract more passengers; conse-quently, the size of the market for the city-pair route can be expected to ex-pand continuously. However, on the other hand, the airport ground access cost for the city-pair increases with market size, since the increased market size leads to an expanded market area and extended access distance. Conse-quently, the market size will not expand continuously, because the rising airport ground access cost will reduce passenger demand. Also, an airport congestion cost might arise if passenger flows in an airport approach the airport capacity. We assume in this study that the airport has adequate ca-pacity so that increases in the size of the single city-pair market will not cause airport congestion. The average airline operating cost, passenger de-lay cost, and airport access cost either decrease or increase with an increase in the size of a city-pair market at an airport, so the optimum market size as well as optimal service area for a city-pair route can be determined from trade-offs among these costs.

The model developed in the rest of the paper addresses the considera-tions raised above. It explores demand-supply interaction and combines a spatial market area problem with a city-pair model of air travel demand. A nonlinear mathematical programming problem is formulated to determine the optimal number of passengers and the optimal service area of a city-pair market by minimizing the sum of various unit costs, subject to the de-mand-supply equality. The objective function of the model is formulated on the assumption that both passenger travel costs and airline operating costs should be minimized when planning the city-pair market of an airport.

Section 2 describes the city-pair market of an airport, defines its market area and passenger demand, and analyzes demand-supply equilibrium. A variety of unit costs, including airport access cost, passenger delay cost, and airline operating cost, are described in Sect. 3. In Sect. 4, we formulate a nonlinear mathematical program to determine the optimal passenger num-ber and market area, and analyze how the optimal passenger numnum-ber and market area are affected by changes in the parameters representing stage length, population density, average income, demand variation, and airline technology. Finally, Sect. 5 summarizes the study and presents our conclu-sions.

2. Passenger demand, market area, and demand-supply equilibrium

The market size of a city-pair route at an airport is defined in this paper as including both the total passenger demand and geographical market area served by the city-pair service. In studying the market area, this paper de-parts from earlier research, which focused only on passenger demand. The market area of the city-pair route of an airport not only defines the number and spatial distribution of airports providing city-pair service in a region, but also influences the average airport access cost, which affects passenger demand in the market in reverse.

Airports are usually located in outlying areas accessible via direct ac-cess freeways or transit lines rather than at the center of metropolitan areas restricted by local grid networks, so their market area can be measured by the Euclidean distance. Also, airports and airlines usually exhibit econo-mies of scale, and are suitable for monopoly or oligopoly operation; there-fore, this paper assumes that airports have circular market areas (Morrill 1974). For a circular market area with the airport at its center, as shown in Fig. 1, the local passenger demand for a city-pair route i, J_i, can be ob-tained by integral (1):

Jiˆ

ZRi 0

wherea_i…r† is a passenger demand density function within the market area of route i that is a decreasing function of the airport access distance, r, (trips/person); p_o is the average population density of the origin city (per-sons/mile2); andR_i is the radius of the market area (miles).

Factors affecting demand for any city-pair route are assumed to include not only the socioeconomic characteristics of the origin and destination ci-ties, for instance, their total population and average personal income, but also the generalized travel cost of the route, including average fare, in-flight time cost, access cost, and passenger delay cost associated with inconvenience of schedules. We assume the following gravity type function fora_i…r†:

ai…r† ˆ d0Pdo1Pdi2God3Gdi4exp‰ÿe…ACi…r† ‡ DTi‡ TTi‡ CCi†Š …2† where P_o and P_i are the total population of the origin and destination ci-ties; G_o; G_i are the average personal income of the origin and destination cities; AC_i…r† is access cost for a passenger departing from an origin that lies access distance r from the airport for city-pair route i; DT_i and TT_i are, respectively, average passenger delay cost and average in-flight travel time cost for city-pair route i; CC_i is the average fare of city-pair routei, which is assumed to be the average operating cost per passenger; and

d0; d1; d2; d3; d4, ande are parameters.

Fig. 1. The geographical market area of the airport.r: Airport access distance, Ri: radius of market area

In normal situations, higher total population and average personal in-come entail greater passenger demand, so the values of parametersd₀; d₁;

d2; d3; d4 would be positive.ACi…r†; DTi; TTi; and CCi are various travel cost components that constitute the generalized cost of an air trip; these components will be defined in detail in Sect. 3. The higher generalized travel cost will result in less passenger demand, so the parameter e would normally be positive. Though Eq. (2) does not explicitly take into account alternative airports or other modes of transportation, the various cost com-ponents of the generalized cost indicate that passenger demand increases as the service level of the city-pair route i at the airport increases and as the passenger’s access distance decreases. The AC_i…r† in Eq. (2) implies that passengers departing from an origin that lies closer to the airport are more likely to undertake their intercity trips via the airport while those departing from an origin that lies farther from the airport are more likely to undertake their trips via alternative airports or other modes. However, major factors affecting passenger demand are the service level of city-pair route i at the airport (e.g., DT_i; TT_i, and CC_i in Eq. (2)). Finally, the parameter d₀ ac-counts for other qualitative factors that also affect passenger demand, such as safety and cultural characteristics.

Previous research regarding passenger demand or airline supply strate-gies usually assumed that either demand or supply was exogenous. In this paper, we assume there are interactions among the market size of city-pair routes, the operating strategies of airlines, and the demand behavior of pas-sengers. Load factor is the ratio of demand to supply, representing the effi-ciency of resource use and the level of service. It is defined by

LFi ˆ Ji=Si …3†

whereLF_i is the load factor of route i and S_i is the total supply of route i. If airlines take the load factor as a measurement of the service level and set the ideal load factor to be a constant, then an equilibrium supply, the prod-uct of aircraft size and flight frequency, can be determined from a given de-mand by the following equation:

Siˆ Ji=LFiˆ BiFi …4†

whereLF_i is the ideal load factor;B_i is the aircraft size for route i; andF_i is the flight frequency of route i. The ideal load factors that airlines deter-mine in the short run would change if the reservation process improved in the future. A poor communication and reservation system prevents airlines from making full use of seats. Hence the ideal load factor cannot be very high. In the long run, if computer reservation systems or reconfirmation re-quirements for passengers holding reservations are made more effective, the ideal load factor could be raised.

In deregulated markets, passenger airlines have the freedom to choose flight schedules, aircraft sizes, and fare prices for city-pair routes. The

larger aircraft and fewer flights or one of using smaller aircraft and more flights so as to maintain the fixed supply, S_i. The former strategy would lower the airline’s unit operating cost and average fare but raise the passen-ger schedule delay cost; the latter would raise the fare but reduce the delay. Both schedule delay and fare prices affect the airline’s ability to attract pas-sengers, as shown by (2). So, there is a trade-off between these two strate-gies. Moreover, changes in the number of passengers attracted make air-lines adjust flight schedules and aircraft sizes, and these adjusted strategies may affect the number of passengers attracted, and so on. So, there are de-mand-supply interactions; and the equilibrium number of passengers and market area could be determined from these interactions and trade-offs among different travel costs, as discussed below.

3. Supply function and passengers’ travel costs

The supply function of air transportation for city-pair route i at the airport is represented by the perceived passenger average cost function. This defi-nition is analogous to the way that Kanafani (1983) defined it. He argues that the supply function in classical microeconomic theory that gives the quantity of a good that a supplier is willing to offer in a market at a given price is not appropriate in transportation, where there are nonmonetary as-pects of supply that are as important as the price charged by the supplier. Travel time and delay time associated with schedule inconvenience are very important non-monetary attributes of supply and much of what deter-mines the nature of transportation supply is a result of both traveler and air-line behavior. The discussion in this paper is limited to supply analysis only as it relates to supply-demand equilibrium.

The components of an air passenger’s perceived cost consist of airport access cost, passenger delay cost, in-flight travel time cost, and the fare. To simplify the analysis, for a specific city-pair route, we assume there is no transfer or connection, thus the cost of the route duration, i.e., the in-flight time cost, is constant if en route flying speed and unit time value are con-stant. The city-pair market is assumed to be competitive in which the fare is set according to the lowest level offered by any airline in the market and is not at the discretion of any single airline, in principle (Kanafani 1983). We also assume the average operating cost of airlines serving any route re-presents approximately the average fare for that route. The equilibrium out-put of the model is assumed to be the total supply of airlines serving the city-pair market without further considering the number of airlines in the market. Therefore, only passenger delay costs, average airline operating cost, and airport access cost will be discussed here.

3.1 Passenger delay cost

The passenger delay cost associated with schedule inconvenience includes schedule delay cost and stochastic delay cost. In reality, in any city-pair

market, there may be passenger demand at any time; though for profit or cost reasons, airlines do not schedule departures whenever there is ger demand. The cost of departing at a time that differs from the passen-ger’s preferred departure time is called schedule delay cost. We assume pas-sengers will choose the flight scheduled nearest to their preferred departure time. Schedule delay time is related to the headway of scheduled flights and the degree to which an airline’s scheduling matches passengers’ de-mand (Teodorovic 1988). We can formulate the average schedule delay cost for city-pair route i,SC_i, as follows:

SCi ˆ 0:5s…7T=Fi†rdt …5†

where T is the daily operating time of the airport (hrs), F_i is weekly flight frequency on the city-pair route,…7T=F_i† is the average headway between flights serving route i,r_dt is the value of time for delay ($/hr), and parame-ter s represents the degree of agreement between the airline’s scheduling and passengers’ demand. In other words, the value of s represents how ef-fectively an airline schedules its flights: when an airline’s flight schedules comply more closely with the demand distribution, the value of s is smal-ler, and the passenger’s schedule delay cost is lower, if the other terms in (5) do not change.

The stochastic delay cost appears when the passenger intends to reserve a seat on the flight of his choice, but is refused because there is no room on that flight. The cost of the time difference between the departure time of the flight on which the passenger finds a seat and the departure time of the flight that the passenger first chose is called the stochastic delay cost. Fac-tors affecting the stochastic delay cost include flight frequency, load factor, variability in air traffic demand, airline reservation technology, and the val-ue of time for delay. We follow the stochastic delay function proposed by Swan (1979) and extend the meaning of the parametera to represent airline reservation technology so as to obtain the average stochastic delay cost for city-pair route i,ST_i,

STi ˆ a…7T=Fi†…LFi†brdt …6†

whereLF_i is the ideal load factor; and two parameters,a and b, represent, respectively, airline reservation technology and variability in air traffic de-mand. A smallera value represents more effective airline reservation tech-nology, i.e., more efficient utilization of aircraft seats. As a decreases, the average stochastic delay cost determined by (6) decreases, if flight headway and ideal load factor are kept constant. Furthermore, suppose the aircraft size, B_i, and weekly flight frequency, F_i, do not change; then an increase in passenger demand, J_i, would cause an increase in the ideal load factor,

LF

i, so as to maintain demand-supply equilibrium shown in (4), and, con-sequently, result in an increased average stochastic delay cost. However, airlines could alleviate the impact of this increased stochastic delay cost by enhancing their reservation technology, i.e., reducing the value of a. The

parameterb stands for variability in air traffic demand. The higher the vari-ation in air demand distribution is, i.e., the smaller the value of b is, the less likely it will be for a passenger to find a seat on the flight of his choice; that is, the average stochastic delay cost will increase providing the other variables in (6) do not change.

Assume the daily operating time of the airport, T, is equal to 18 hours across cities due to so few departure flights scheduled from midnight to 6 am. Then combine (5) and (6), rearrange, and the result yields, the average passenger delay cost for city-pair route i,DT_i,

DTiˆ l=Fi …7†

where

l ˆ 126rdt…0:5s ‡ a…LFi†b† …8†

3.2 Average operating cost of airlines

The operating costs of airlines serving any city-pair market consist of direct operating costs, including flight operating costs, landing fees, fuel, depre-ciation, and maintenance costs, and indirect operating costs, including ad-ministration, customer service, and marketing costs. Direct operating costs are usually correlated with the number of seats provided, while indirect op-erating costs are related to the size of the airline and are measured in terms of passengers served.

Unit costs of airlines serving a city-pair market decline markedly as density of service and stage length increase (Caves et al. 1984); therefore, the average total operating cost of providing city-pair service is assumed to be a nonlinear function, which can be expressed as follows:

Ciˆ w0…Bi; Fl†
Fi‡ w1…Bi; Li†
LiBiFi …9† whereC_i is the total weekly operating cost for serving the city-pair route i,

w₀…Bi; Fl† is the landing fee ($/flight), a function of aircraft size, Bi, and unit landing fee, Fl ($/1000 lb); and w₁…B_i; L_i† is the unit operating cost ($/seat-mile) excluding the landing fee.w₀…B_i; Fl† is an increasing function of aircraft size,B_i, as defined by

w₀…Bi; Fl† ˆ h0‡ h1BiFl …10†

where h₀ and h₁ are parameters and h₁> 0. w₁…B_i; L_i† is a decreasing function of aircraft size, B_i, and stage length, L_i, which can be written as follows:

where h₂; h₃; h₄, and h₅ are parameters, h₅< 0 and dw₁…B_i; L_i†=dB_i< 0. The average operating cost per passenger for city-pair route i,CC_i, is

CCiˆ ‰w0…Bi; Fl†
Fi‡ w1…Bi; Li†
LiBiFiŠ=Ji …12† By substituting (4) into this expression, we obtain an expression that is also a function of the ideal load factor,LF_i:

CCiˆ w0…Bi; Fl†=…Bi
LFi† ‡ w1…Bi; Li†
Li=LFi …13† The equation above shows that when demand-supply equilibrium is main-tained, an increase in passenger demand J_i, would allow airlines to utilize larger aircraft,B_i, without changing the flight frequency, F_i, and the ideal load factor, LF_i, thereby yielding a lower operating cost per passenger,

CCi. If, on the other hand, aircraft size,Bi, and flight frequency,Fi, do not change, the increased passenger demand could raise the ideal load factor,

LF

i, and reduce the average operating cost per passenger. The economies deriving from the increase in the ideal load factor are comparatively strong in comparison with those deriving from the use of larger aircraft.

3.2 Average airport access cost

The access cost for a passenger traveling from his origin to the airport is assumed to include the travel time cost and the travel monetary cost. The access cost for a passenger departing from an origin that lies access dis-tance r from the airport for city-pair route i,AC_i…r†, is specified by

ACi…r† ˆ …r=m†
rac‡ f
r …14†

where m is the average travel speed of access modes (miles/hr), f is the average monetary cost of access mode per unit distance ($/mile), andr_ac is the time value of access time. The average access cost per passenger, AC_i, is obtained by averaging the total access costs for all passengers originating within the market area. From (1), (2), and (14), AC_i can be formulated as follows: ACiˆ ZRi 0 2prpo
ai…r†
ACi…r†dr  Jiˆ ‰…rac=m† ‡ f Šf…2=E† ÿ ‰R2

i exp…ÿRiE†Š=‰1=E ÿ …1=E ‡ Ri† exp…ÿRiE†Š …15† whereE ˆ e…r_ac=m ‡ f † and R_i is the radius of the market area.

4. Formulation of the optimization problem and sensitivity analysis

In this paper, we assume that both the economic efficiencies of airlines and the travel costs of passengers are considered when planners decide the mar-ket size of an airport for a city-pair marmar-ket. The average airport access cost, average passenger delay cost, and average airline operating cost in-creases or dein-creases with the size of the city-pair market of an airport. Therefore the optimum equilibrium market size can be determined from trade-offs among these costs. A nonlinear mathematical programming prob-lem is formulated here to determine the optimal market size by minimizing the sum of the various costs, subject to demand-supply equilibrium. The nonlinear optimization problem is formulated as follows:

min JiRiBiFiTCiˆ …ACi‡ DTi‡ CCi†=Li …16† s.t. Bi
Fiˆ Ji=LFi …17† and Ji; Bi; Fi; Ri> 0

The formulations for the three different costs in (16), DT_i; CC_i, and AC_i, are expressed, respectively, by (7), (13), and (15); J_i in (17) is expressed by (1); and (17) is an equilibrium condition for demand and supply, as de-rived by (4) in Sect. 2. This nonlinear optimization model may be solved by means of a variety of algorithms. We solve the optimization problem by using GINO, a computer-modeling program developed by Liebman et al. (1986) based on a generalized reduced gradient algorithm. The outputs for this model include the minimum average unit total cost ($/passenger-mile), the optimal market size (passengers/week), and the optimal market area (radius of market area, in miles) for the city-pair market and the optimal aircraft size and weekly flight frequency for airlines serving this market. The inputs for the model include the total population, the average personal income of the origin and destination cities, the population density of the origin city, a parameter value representing the scheduling technology of air-lines, the average travel speed and momentary cost of access modes, and the time values of different costs, etc.

For a given set of input values, the optimal values for the decision vari-ables can be found by numerically solving the model. Since we are inter-ested in exploring how the market size of the city-pair service at an airport is affected by changes in the stage length of the city-pair route, the average travel speed of access modes, the population density and income of the ori-gin city, and the operating technologies of airlines, we shall present the results of numerical experiments conducted by a set of base values for the model parameters. The base parameter values, as shown in Table 1, are

assumed or found in previous studies (Caves et al. 1984; Teodorovic 1988; Kane 1990; Kling et al. 1991).

The values of parameters h₂; h₃; h₄, and h₅ in Table 1 are initially as-sumed on the basis of work by Kling et al. (1991). They collected operat-ing cost data of 305 routes operated by U. S. major airlines and calibrated these parameter values using multiple regression methods. Then, these pa-rameter values were validated using the statistical data of airline and air-craft operating costs listed in Kane (1990). However, all of the airair-craft size data shown in Kling et al. (1991) and Kane (1990) is no larger than 412 seats. Thus, the use of these parameter values is noted to be limited to air-craft size, B_i, less than 416, since dw₁…B_i; L_i†=dB_i < 0 cannot be valid when B_i > 416. The base parameter values listed in Table 1 are only for demonstration purposes. Estimates based on actual data should be used in future applications of the model for specific airports. The numerical experi-ments are illustrated here to observe the behavior and results of the model. Each numerical experiment was conducted by varying the value of one or a few parameters while holding the others constant.

Changes in the stage length of the city-pair route

Analysis of changes in the stage length of the city-pair route should allow us to elaborate the differences between short-haul and long-haul city-pair

Table 1. Base parameter values for numerical experiments

Parameter Value Unit

T 18 Hours/day Li 1 000 Miles LF i 0.65 po 10 000 Persons/mile2 Po; Pi 106 Persons Go; Gi 9 000 $/year d0 exp (–23.6) d1; d2 0.5 d3; d4 0.14 e 0.018 m 40 Miles/hr f 0.25 $/mile s 0.7 a 2.5 b 9 rac 2.5 $/hr rdt 1.25 $/hr Fl 50 Cents/1000 lb h0 61.91 h1 1.97 h2 12.204 h3 –0.07 h4 0.000168 h5 –0.0009

markets. The results of varying the stage length of the city-pair route are shown in Table 2 and Fig. 2. As the stage length of the route increases, the number of passengers of the optimal market declines while the radius of the optimal market area expands. Also, as shown by the results in Table 2, as the stage length increases, airlines should utilize larger aircraft and schedule fewer flights so as to achieve economic efficiencies. These results imply that since the average demand level per person declines as the stage

Table 2. Model results for city-pair routes with different stage length

Stage length Objective value Market size Radius of Aircraft size Flights per (miles) ($/pax-mile) (pax/week) market area (seats) week

(miles) 500 0.09086 4088 20.74 205.76 30.56 1000 0.08503 3173 26.70 205.86 23.71 1500 0.08346 2463 34.54 206.15 18.38 2000 0.08305 2084 46.98 206.17 15.55 2500 0.08317 1482 59.08 206.21 11.06 3000 0.08373 1141 78.34 205.59 8.54 3500 0.08469 857 104.58 205.52 6.42 4000 0.08615 650 144.38 205.29 4.87 4500 0.08838 472 204.98 204.57 3.55 5000 0.09201 341 322.04 203.67 2.57

length of the route increases, the market area will tend to expand so as to accumulate more passengers and achieve density economies. Additionally, the ratio of access and delay costs to the total travel cost decreases as the stage length of the route increases, so passengers would be willing to travel longer distances and accept less frequent flights. Scheduling flights less fre-quently enables airlines to use larger aircraft in long-haul markets. On the other hand, the results also indicate that shorter haul routes with very high passenger demand should be operated by airlines in more airports, who should use slightly smaller aircraft but much more frequent flights. The ob-jective value, average total cost per passenger-mile, for short-haul routes is also higher than that for long-haul routes. For extremely long-haul routes, because of significantly lower passenger demand, the flight frequency should be markedly reduced. Note that the optimal average aircraft size is not significantly different for short-haul and long-haul routes. This might be because the effect of high passenger flows compensates for that of short stage length in determining the aircraft size used for short-haul service.

Improvements in average travel speed of access modes

The optimal market size and area increase with increasing average travel speed, but the rate of increase diminishes gradually, as shown in Fig. 3. In

other words, improvements in the average travel speed of access modes will reduce the access cost, which will stimulate more passengers to travel from farther areas. This in turn will permit airlines at the airport to achieve economic efficiencies, and passengers will share in the surplus created by those efficiencies. This situation can be accounted for by the objective val-ues in Table 3, which indicates that average total cost per passenger mile decreases as the average travel speed of access modes increases. However,

Table 3. Optimal objective values, market sizes and areas for different average access mode

speeds

Average speed of access modes Objective value Market size Radius of market area (miles/hr) ($/pax-mile) (pax/week) (miles)

10 0.08822 2254 23.17 20 0.08615 2773 25.21 30 0.08542 2919 25.70 40 0.08504 3091 26.35 50 0.08481 3243 26.94 60 0.08466 3243 26.90 70 0.08455 3244 26.88 80 0.08446 3389 27.45 90 0.08440 3389 27.43 100 0.08434 3389 27.42

the diminishing effect of improvement in airport access modes suggests that other approaches are needed in order to expand the market size for a city-pair route at an airport, especially when the average travel speed of ac-cess modes is already high.

Changes in the average population density of the origin city

Figure 4 shows that as the average population density of the origin city in-creases, the number of passengers in the optimal market increases and the radius of the optimal market area decreases. The index of population den-sity roughly represents the extent of urbanization and the scale of a metro-politan area. A higher population density for the origin city means that it is a highly urbanized city with a strong demand for air travel, which creates more density efficiencies for airlines serving city-pair markets that include this city and allows airlines to operate in more airports of this city to share in these markets.

Changes in the average income of the origin city

Figure 5 shows that with an increase in the average personal income of the origin city, the optimal market size as well as the optimal area increases, but at a declining rate of increase. The number of passengers in a city-pair

market increases with the average income through an income effect, since air transportation is a normal good, and through a substitution effect, be-cause the average income determines the opportunity cost of time. It should be noted that the opportunity cost of time is assumed to be a function of in-come in the given set of input values. An increasing number of passengers in the optimal market may reduce schedule delay cost and the fare of pas-sengers through economies of schedule and economies of density. How-ever, continued increase in market size would result in an expanded market area and thus raise the average access cost at the airport which in turn would cause the rate of increase in market size to decline. These results im-ply that, in high income areas, city-pair air services should concentrate on one or a small number of large airports.

Variation in the distribution of passenger demand

The parameterb in (6) represents variability in air traffic demand. The sto-chastic delay cost resulting from passengers who can not reserve a seat for the flight of their choice increases with increasing variation in air traffic de-mand if teh flight schedule is fixed. Table 4 and Fig. 6 show that as the variability of air traffic demand decreases, i.e., as the value of b increases, even when the optimal market size and area are smaller, the objective value of the average total travel cost is lower. In other words, economic efficien-cies can still be obtained in small markets with stable passenger demand.

Changes in the scheduling technology of airlines

Parameters in (5) stands for the scheduling technology that airlines use to match flight schedules with passenger demand. Table 5 indicates that the average total cost per passenger-mile is low for high scheduling technol-ogy, symbolized by a lows value, but high for low scheduling technology, even when the number of passengers in the optimal market is low for the

Table 4. Optimal objective values, market sizes and areas, and flight frequencies for different

variations in air travel demand,b

b Objective value Market size Radius of market area Flights per week ($/pax-mile) (pax/week) (miles)

5 0.08647 4249 31.30 31.96 6 0.08591 3799 29.45 28.54 7 0.08551 3490 28.13 26.19 8 0.08523 3277 27.18 24.57 9 0.08503 3133 26.53 23.48 10 0.08491 3133 26.50 23.48 11 0.08482 2979 25.82 22.31 12 0.08476 2979 25.81 22.31 13 0.08473 2979 25.80 22.31 14 0.08470 2979 25.80 22.31

former case. This implies that, even in a small market, cost efficiencies achieved by airlines with high scheduling technology are comparatively strong in comparison to those achieved by airlines with low scheduling technology. However, if there is a single airline in the market, there is no incentive to reduce the value of s since a larger market (always consistent with the model equilibrium condition) compensates for low-scheduling technology, but the average total cost per passenger-mile for high-schedul-ing technology is still lower than that for low-schedulhigh-schedul-ing technology.

Fig. 6. Optimal market size and area vs. variation in air demand

Table 5. Optimal objective values and market sizes for different scheduling technologies,s

s Objective value Market size

($/pax-mile) (pax/week) 0.1 0.08196 1272 0.2 0.08271 1620 0.3 0.08330 1977 0.4 0.08382 2308 0.5 0.08426 2555 0.6 0.08467 2869 0.7 0.08503 3137 0.8 0.08537 3389 0.9 0.08570 3399 1.0 0.08599 3776 1.1 0.08627 4089 1.2 0.08654 4087

5. Conclusion

A model for determining both the number of passengers and local market area for city-pair routes served by an airport has been formulated and em-ployed to analyze how changes in significant parameters affect the opti-mum market size and service area. A variety of unit costs change in differ-ent ways as the market size of a given city-pair route changes. The unit air-line operating cost is a decreasing function of aircraft size and load factor: hence, when the market for a given city-pair route expands, airlines can either utilize larger aircraft or raise the load factor to reduce the average op-erating cost per passenger. Therefore, the average airline opop-erating cost per passenger is a decreasing function of the market size. The schedule delay cost and the stochastic delay cost are primarily related to flight frequency. They are also decreasing functions of the market size, since flight fre-quency can be raised to serve a large number of passengers if aircraft size and the ideal load factor are fixed. On the other hand, the average access cost of passengers increases with the market size of the airport serving the city-pair route, because a larger market size leads to a larger market area and longer access distance (providing the average air travel demand level of the city-pair market and other variables do not change).

This paper has formulated a nonlinear programming problem to deter-mine the optimal market size and service area for the city-pair route served by an airport by finding an optimal trade-off between the above costs sub-ject to demand-supply equilibrium. Such a trade-off must succeed in both minimizing average total travel cost for passengers and achieving economic efficiencies for airlines. Coupled with numerical experiments, the model provides a basis for examining the effects of variations in stage length, pop-ulation density and average income of the origin city, average travel speed of access modes, and scheduling technology of airlines. In regard to stage length, the service area for long-haul markets is larger than that for short-haul markets, indicating that long-short-haul services might be concentrated in one or a small number of large airports, while short-haul services might be dispersed among many small airports. Improving the technology of the air-port access mode can expand the market size and service area for a city-pair service at one airport by reducing access cost directly and airline oper-ating cost and delay cost indirectly. In areas with higher population density, which results in higher demand density and a denser market, airlines serv-ing a city-pair market can operate more efficiently through economies of density and distribute air services among more airports. When the cities served have a high personal income, the market size and the market area both increase, but at a declining rate of increase. Finally, city-pair markets with relatively stable passenger demand or markets that are served by air-lines with more effective scheduling technology exhibit higher cost efficien-cies.

The findings of this research are exploratory, because the input data used in the numerical experiments are only hypothetical. Estimates based on actual data should be used in future applications of the proposed model.

Only local markets have been considered here, so issues regarding transit passengers should also be addressed in future research. Furthermore, future studies should explicitly address issues such as including competition from alternative airports or other modes of transportation in the demand function and should explicitly consider the impact of different types of market struc-tures, pricing, and the regulatory environment on the evolution of the sup-ply function.

Acknowledgement. This work was supported by the National Science Council, R. O. C.,

through grant NSC82-0410-E-009-150. The constructive comments of referee and editor are highly appreciated.

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