Maximizing net benefits for conventional and flexible bus services

Transportation Research Part A 80 (2015) 116–133

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Transportation Research Part A journal homepage: www.elsevier.com/locate/tra

Maximizing net benefits for conventional and flexible bus services Myungseob (Edward) Kim a,1, Paul Schonfeld b,⇑ a b

Parsons Corporation, 100 M Street, South East, Washington, DC 20003, United States Department of Civil & Environmental Engineering, University of Maryland, College Park, 1173 Martin Hall, College Park, MD 20742, United States

a r t i c l e

i n f o

Article history: Received 15 July 2014 Received in revised form 24 July 2015 Accepted 28 July 2015 Available online 26 August 2015 Keywords: Public transportation Conventional bus Flexible bus Maximum social welfare Genetic algorithm Constrained optimization

a b s t r a c t Transit ridership is usually sensitive to fares, travel times, waiting times, and access times, among other factors. Therefore, the elasticities of demand with respect to such factors should be considered in modeling bus transit services and must be considered when maximizing net benefits (i.e. ‘‘system welfare’’ = consumer surplus + producer surplus) rather just minimizing costs. In this paper welfare is maximized with elastic demand relations for both conventional (fixed route) and flexible-route services in systems with multiple dissimilar regions and periods. As maximum welfare formulations are usually too complex for exact solutions, they have only been used in a few studies focused on conventional transit services. This limitation is overcome here for both conventional and flexible transit services by using a Real Coded Genetic Algorithm to solve such mixed integer nonlinear welfare maximization problems with constraints on capacities and subsidies. The optimized variables include service type, zone sizes, headways and fares. We also determine the maximum welfare threshold between optimized conventional and flexible services) and explore the effects of subsidies. The proposed planning models should be useful in selecting the service type and optimizing other service characteristics based on local geographic characteristics and financial constraints. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction In public transportation systems, feeder services may serve an essential purpose for transit users, who start and end their journeys with them. Thus trips may be sufficiently concentrated into sufficient densities for economical use of mass transportation. Feeder services include two main types: conventional services, which are also known as fixed-route and flexible-route bus services. Fixed (i.e. pre-determined) headways are assumed in this paper for both service types. Bus systems with both conventional and flexible services have been studied extensively. Kocur and Hendrickson (1982) optimized conventional services connecting a terminal with a local region. Since then, various studies have analyzed conventional services (Chang, 1990; Chang and Schonfeld, 1991a, 1991b, 1991c, 1993; Lee et al., 1995; Kim and Schonfeld, 2012, 2013; Diana et al., 2009). Chang and Schonfeld (1991a) compared conventional and flexible bus services for serving a local region. They used an analytic approach to obtain closed-form solutions. Chang and Schonfeld (1993) maximized the social system welfare for conventional services. Their solutions were found through analytic optimization (with approximations), but such

⇑ Corresponding author. Tel.: +1 301 405 1954; fax: +1 301 405 2585. 1

E-mail addresses: [email protected] (Myungseob (Edward) Kim), [email protected] (P. Schonfeld). Tel.: +1 202 469 6568.

http://dx.doi.org/10.1016/j.tra.2015.07.016 0965-8564/Ó 2015 Elsevier Ltd. All rights reserved.

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methods seem difficult to extend to more complex problems, such as those with multiple interrelated regions. Lee et al. (1995) considered mixed bus fleets for conventional services and found them beneficial for single-fleet conventional services when the demand fluctuates over regions and periods. Flexible feeder services have also been actively explored since the 1970s. Stein (1978) estimated the optimal tour distance for flexible services. Daganzo (1984) compared demand responsive services, considering both rectilinear and Euclidean distances. Various aspects of flexible services have been studied (Chang and Schonfeld, 1991a, 1993; Chandra and Quadrifoglio, 2013a, 2013b; Chandra et al., 2011; Diana et al., 2007; Li and Quadrifoglio, 2009, 2011; Quadrifoglio and Dessouky, 2007; Quadrifoglio et al., 2007, 2008; Shen and Quadrifoglio, 2012; Horn, 2002; Luo and Schonfeld, 2011a,b; Zhou et al., 2008). Luo and Schonfeld (2011a) proposed an online rejected-reinsertion heuristic for a dynamic dial-a-ride problem. Luo and Schonfeld (2011b) also developed metamodels for dial-a-ride services. Chandra and Quadrifoglio (2013a) developed an analytic queuing model for estimating the tour length for demand-responsive services. Conventional services are generally preferable (with large bus sizes) at high demand densities. Conversely, flexible services are usually preferable, with smaller buses, when demand densities are low (Chang and Schonfeld, 1991a; Kim and Schonfeld, 2012, 2013). Thus, if conventional and flexible services are jointly provided, it may be possible to provide more efficient feeder services than solely with either of these types. To address such bus transit integration problems, Chang and Schonfeld (1991c) developed an optimization method for integrating conventional and flexible services by switching between them as demand varies over time. Kim and Schonfeld (2012) proposed a variable-type bus service, which operates conventional services in higher demand periods and changes to flexible services in low demand periods. In that study the model re-optimizes the flexible service decision variables, which are headways, fleet size, and service area, with given the vehicle size. Kim and Schonfeld (2013) considered conventional and flexible services as well as a mix of bus sizes. Their results confirmed that the integration of different types of services (e.g., conventional and flexible services) with mixed fleets can reduce total costs when demand varies over time and over regions. Aldaihani et al. (2004) suggested an integrated transit network with fixed route and demand-responsive services. The role of fixed route services was to transport passengers between zones while demand responsive services pick-up and drop-off passengers within zones. For demand-responsive services, the model assumed that only one passenger can be carried, thus precluding ridesharing. Diana et al. (2009) developed a model for comparing the distance traveled in fixed route and demand responsive bus services for the same set of customers. As expected, they found that when demand is moderate, demand-responsive services outperform traditional fixed-route services in minimizing the traveled distances. They also found that demand-responsive services perform better in a ring-radial network rather than a grid network. Quadrifoglio and Li (2009) analyzed the optimal number of zones in designing feeder bus services using fixed-route and demand-responsive services. They provided a closed-form solution for conventional services and an approximation formula for flexible services. The results were verified with simulation. As an extension to Quadrifoglio and Li (2009), Li and Quadrifoglio (2010) explored the demand threshold between fixed-route and demand-responsive services with one vehicle. A limitation of their models was their assumption that the entire region was covered by one vehicle for either fixed-route and demand-responsive services. Transit ridership may be sensitive to the elasticity of fares and other factors such as in-vehicle times, waiting times and access times. However, most studies on feeder transit services mentioned above did not consider demand elasticity. A few studies did explore demand elasticity in public transportation services, mostly for conventional bus transit systems (Kocur and Hendrickson, 1982; Imam, 1998; Chang and Schonfeld, 1993; Zhou et al., 2008; Chien and Spasovic, 2002). When considering the demand elasticity, formulations typically become maximization problems, presumably because it makes little sense to minimize costs if demand is elastic (and may be driven to zero by minimal service or high fares). Kocur and Hendrickson (1982) optimized transit decision variables, namely route spacing, headway, and fare, while taking into account demand elasticities. They assumed a linear transit utility function rather than more complex forms. Their justifications for the linear utility approximation were that it is analytically tractable, it is easily differentiated and manipulated, and it is convex within its upper and lower bounds. They considered wait time, walk time, in-vehicle time, fare, and auto time and cost in the demand model. They provided analytic closed-form solutions, but their study was limited to a conventional bus service for one local region. Imam (1998) extended Kocur and Hendrickson (1982)’s study by relaxing the linear demand function and applying a log-additive demand function. Chang and Schonfeld (1993) considered time-dependent supply and demand characteristics for a transit welfare maximization problem. They used a linear demand function as in Kocur and Hendrickson (1982). Their decision variables were route spacing, headways, and fare. Since their study considered multiple time periods, they optimized headways for each time period. Their objective was to maximize the sum of consumer surplus and producer surplus. They solved this welfare maximization problem with alternative financial constraints, namely without any constraint, with a break-even constraint, or a with subsidy constraint. Their problem size extended to one local region and multiple periods. Solutions were obtained analytically with some approximations. For the formulations with constraints, a Lagrange multipliers method was applied. The vehicle size was considered an input rather than a decision variable. Zhou et al. (2008) formulated the welfare for conventional bus and flexible bus services, but only for a system connecting a terminal to one local region in one period. They found solutions analytically since the formulation of a system connecting a terminal to one local region in one period was analytically tractable. However, analyses of system

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welfare for larger problems (i.e., multiple regions and multiple periods) for both conventional and flexible services are desirable. Chien and Spasovic (2002) studied a grid bus transit system with an elastic demand pattern. They optimized route spacings, station spacings, headways, and fare with the objectives of maximum total operator profit and social welfare. The elastic demand was formulated linearly with functions of potential demand, time components and fare. The optimal solutions were found analytically. This work was applicable to conventional bus services. Tsai et al. (2013) found headway and fare solutions for a Taiwan High Speed Rail (THSR) line, with a maximum welfare objective. They considered demand elasticity, and obtained solutions with a genetic algorithm (GA). They compared GA solutions with those from an SSM (Successive Substitution Method), which is an optimization algorithm proposed by Chien and Schonfeld (1997) and used by Tsai et al. (2008). Tsai et al. (2013) found that the solutions from GA and SSM were fairly close. Most of the literature on welfare maximization in bus transit systems covers conventional services. Most existing problems in transit welfare maximization are solved with analytic optimization methods. Analytic optimization can find solutions quickly and possibly in closed forms, but becomes intractable for more complex problems (e.g., for multiple region analysis). Thus, in the literature, the problem of maximizing welfare for conventional services is limited to one local region with multiple periods. For flexible services, the solved problem size encompasses one local region and one period. With numerical solutions it seems possible and desirable to consider problems with multiple regions as well as multiple periods for both conventional and flexible services, as undertaken in our present study. In this paper different service levels and parameters that affect the demand for and competitiveness of alternative service types are considered in conventional and flexible service formulations. An elastic demand function is applied for both conventional and flexible services. Analyzing and comparing transportation systems with elasticity in their demand functions, especially when the optimized characteristics of such systems are affected differently by the demand, seems very important. The maximization of welfare (i.e. the sum of net benefits to both suppliers and consumers) is a more general and comprehensive objective than the minimization of costs since benefits are also considered in the optimization. Using elastic demand functions, we optimize here various decision variables, which include fares on conventional and flexible services, bus sizes, headways and fleet sizes for both service types, route spacings for conventional services, and zones for flexible services. Several contributions are made in this paper that overcome important weaknesses in previous studies, namely in (1) their assumed fixed demand (i.e. with zero elasticity), (2) their total cost optimization objective (rather than the more general maximum welfare objective), and (3) their overly simple system configuration. In addition, (4) we determine the demand density threshold between conventional and flexible services after optimizing each type for maximum welfare, subject to capacity and subsidy constraints. These contributions should be helpful to public transportation planners and managers in determining the most appropriate service for various regions and periods.

2. System specifications and assumptions This section addresses assumptions for analyzing a general system (shown in Fig. 1) with multiple local regions as well as multiple periods. A region is assumed to be a rectangular region and subdivided into zones for fixed routes or flexible routes, as shown in Fig. 1. The zone dimensions and route headways are optimized by our proposed model. The flexible routes (as well as conventional services) operate with fixed schedules (with headways optimized separately for each service area and period) and the same optimized vehicle size in all zones and periods. Assumptions adopted from Kim and Schonfeld (2013) are still applicable, and additional assumptions (for the welfare analysis) are introduced in the following sections when required. Henceforth, superscripts k and i correspond to region and time period, respectively, while subscripts c and f represent conventional and flexible services, respectively. The definitions, units and default values of variables used in this paper are presented in Table 1. 2.1. Common assumptions for conventional and flexible services All service regions, 1 . . . k, are rectangular, with lengths Lk and widths Wk. These regions may have different line haul distances Jk (miles, in region k) connecting a terminal and each region’s nearest corner. (a) The demand is uniformly distributed over space within each region and over time within each specified period. (b) The optimized bus sizes (Sc for conventional, Sf for flexible) are uniform throughout regions and time periods. (c) The average waiting time of passengers is approximated as a constant fraction alpha of the headway (hc for conventional, hf for flexible). Alpha is assumed to be 0.5. (d) Bus layover time is negligible. (e) Within each local region k, the average speed (V ic for conventional bus, V if for flexible bus) includes stopping times. (f) External costs are assumed to be negligible.

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Fig. 1. Local regions and bus operations.

2.2. Assumptions for conventional bus only (a) The region k is divided into N kc parallel zones with a width rk = W/Nkc for conventional bus, as shown in Fig. 1. Local routes branch from the line haul route segment to run along the middle of each zone, at a route spacing rk = Wk/N kc . (b) Qki trips/mile2/h, entirely channeled to (or through) the single terminal, are uniformly distributed over the service area. (c) In each round trip, as shown in Fig. 1, buses travel from the terminal a line haul distance Jk at non-stop speed yV ic to a corner of the local regions, then travel an average of Wk/2 miles at local non-stop speed zV ic from the corner to the assigned zone, then run a local route of length Lk at local speed V ic along the central axis of the zone while stopping for passengers every d miles, and then reverse the above process in returning to the terminal. 2.2.1. Assumptions for flexible bus only (a) To simplify the flexible bus formulation, region k is divided into N kf equal zones, each having an optimizable zone area Ak = LkWk/N kf . The zones should be ‘‘fairly compact and fairly convex’’ (Stein, 1978). (b) Buses travel from the terminal line haul distance Jk at non-stop speed yV if and an average distance (Lk + Wk)/2 miles at local non-stop speed zV if to the center of each zone. They collect (or distribute) passengers at their door steps through i ki an efficiently routed tour of n stops and length Dki c at local speed V f . Dc is approximated according to Stein (1978), in pffiffiffiffiffiffiffiffi nAk ; and ;  1:15 for the rectilinear space assumed here (Daganzo, 1984). The values of n and Dki which Dki c  ; c are

endogenously determined. To return to their starting point the buses retrace an average of (Lk + Wk)/2 miles at zV if miles per hour and Jk miles at yV if miles per hour. (c) Buses operate on schedules with preset headways and with flexible routing designed to minimize each tour distance Dki c . (d) Tour departure headways are equal for all zones of each region and uniform within each period.

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Table 1 Notation. Variable

Definition

Baseline value

a Ak b D Dki c Dkf

Hourly fixed cost coefficient for operating bus ($/bus h) Service area (i.e., zone) (miles2) = LkWk/Nk Hourly variable cost coefficient for bus operation ($/seat h) Bus stop spacing (miles) Distance of one flexible bus tour in local region k and period i (miles) Equivalent line haul distance for flexible bus on region k (=(Lk + Wk)/z + 2Jk/y), (miles) Equivalent average bus round trip distance for conventional bus on region k (=2Jk/y + Wk/z + 2 Lk), (miles) Directional demand split factor

30.0 – 0.2 0.2 – –

Dk ki

ds

F ki ki

hc ; h c

ki

hf ; hf k, i Jk lc ; lf Lk, Wk Mk N N kc ; N kf Q ki qki rk Rki c Sc ; Sf tki u ki U ki c ; Uf V ic V if Vx

vv ; vw; vx y Ø ⁄

ki Y ki c ; Yf ki P ki c ; Pf

Rki c ;

Rki f

ki C ki c ; Cf

f c; f f ki Gki c ; Gf ev ; ew ; ex ; ep Yc; Yf FSki

Fleet size for region k and period i (buses) Subscript corresponds to (c = conventional, f = flexible) Headway for conventional bus; for region k and period i (h/bus) Headway for flexible bus; for region k period i (h/bus) Index (k: region, i: period) Line haul distance for region k (miles) Load factor for conventional and flexible bus (passengers/seat) Length and width of local region k (miles) equivalent average trip distance for region k (=(Jk/yc + Wk/2zc + Lk/2)) Number of passengers in one flexible bus tour Number of zones in local region for conventional and flexible bus Actual demand density (trips/h) Potential demand density (trips/mile2/h) Route spacing for conventional bus at region k (miles) Round trip time of conventional bus for region k and period i (h) Round trip time of flexible bus for region k and period i (h) Sizes for conventional and flexible bus (seats/bus) Time duration for region k and period i Average number of passengers per stop for flexible bus The total user benefit in region k and period i Subscript: c = conventional, f = flexible Local service speed for conventional bus in period i (miles/h) Local service speed for flexible bus in period i (miles/h) Average passenger access speed (miles/h) Value of in-vehicle time, wait time and access time ($/passenger h) Express speed/local speed ratio for conventional bus Constant in the flexible bus tour equation (Daganzo, 1984) Superscript indicating optimal value; subscript: c = conventional, f = flexible Total social welfare in region k and period i Subscript: c = conventional, f = flexible Producer surplus (revenue–cost) in region k and period i Subscript: c = conventional, f = flexible Revenue in region k and period i Subscript: c = conventional, f = flexible Operating cost in region k and period i Subscript: c = conventional, f = flexible Fares on the system; subscript: c = conventional, f = flexible Consumer surplus in region k and period i Subscript: c = conventional, f = flexible Elasticity factors The total welfare of system Subscript: c = conventional, f = flexible The amount of subsidy for region k and period i

– 1.0 – – – – – 1.0 – – – – – – – – – – – 1.2 – 30 25 2.5 5, 12, 12 Conventional bus = 1.8 flexible bus = 2.0 1.15 – – – – – –

0.35, 0.7, 0.7, 0.07

3. Elastic demand functions and operating costs 3.1. Conventional bus services In this section, the linear elastic demand function and the operating cost for conventional services are formulated. Chang and Schonfeld (1993) consider elastic demand for conventional bus services for one region and multiple periods. Their elastic demand function for conventional services is modified here to accommodate multiple regions as well as multiple periods. 3.1.1. Elastic demand function for conventional bus services The demand density may be sensitive to in-vehicle time, waiting time, access time, and the fare of the system. A linear demand function, Q ki c , in region k and period i is formulated as follows:

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( ki

k k ki Q ki 1  ew z1 hc  ex z2 c ¼ L W q

ðr k þ dÞ Mk  ev ic  ep f c Vx Vc

) ð1Þ

where z1 = 0.5 for uniform passenger arrivals, uniform bus arrivals and sufficient bus capacity; z2 = 0.25 for rectilinear network. The elastic demand function in Eq. (1) can be rewritten as:

  rk ki k k ki Q ki K kc  ew z1 hc  ex z2  ep f c c ¼ L W q Vx

ð2Þ

k

c where K kc ¼ 1  ex z2 Vdx  ev M . Vi c

3.1.2. Conventional bus operating cost The conventional bus operating cost in region k and time period i is formulated below. Unit operating cost, Bc, is assumed to be a function of vehicle size (i.e., Bc ¼ a þ bSc ): k C ki c ¼ Bc N c

Dk tki

ð3Þ

ki

V ic hc

3.2. Flexible bus services 3.2.1. Elastic demand function for flexible bus services The demand density of flexible bus services is affected by in-vehicle time, waiting time and fare. The access time is not considered for flexible services. Zhou et al. (2008) considered a flexible service with elastic demand for only one time period and one region. Their solutions were obtainable with basic calculus since the problem was relatively simple. Here, the elastic demand function for flexible services is modified for multiple regions as well as multiple periods. The actual demand in region k and period i is formulated as: ki

k k ki ki Q ki f ¼ L W q f1  ew z1 hf  ev M f  ep f f g

ð4Þ

where z1 = 0.5 uniform passenger arrivals, uniform bus arrivals and sufficient bus capacity. Eq. (4) can be rewritten as

 qffiffiffiffiffiffiffiffiki  8 9 qki hf > > > > ;Ak < = u ki k k ki ki Q ki ¼ L W q K  e z h  e  e f w 1 v p f f f f i > > 2V f > > : ; where K ki f ¼ 1  ev



Lk þW k 2V if yf

k

þ VJi y

ð5Þ

 .

f f

3.2.2. Flexible bus operating cost Flexible bus operating cost, C ki f , is formulated by multiplying unit bus operating cost, the number of zones in region k, and round travel time: k C ki f ¼ Bf N f

ki ðDkf þ Dki c Þt

ð6Þ

ki

V if hf

pffiffiffiffiffiffiffiffi ki nAk , and ; = 1.15 for the Dki c is the approximated flexible bus tour distance according to Stein (1978), in which Dc ¼ ; k

k

rectilinear space assumed here (Daganzo, 1984). The service area, Ak , of flexible bus in region k is equal to L NWk . Thus, by subf

stituting average tour distance Dki c into Eq. (6), the flexible bus operating cost in region k and time period i is estimated as:

pffiffiffiffiffiffiffiffi k nAk Þtki k ðDf þ ;

C ki f  Bf N f

ki

V if hf

¼

Bf N kf Dkf t ki ki

V if hf

þ

;Bf Lk W k tki

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ki qki hf =u ki

V if hf

ð7Þ

4. Welfare maximization without financial constraints For public transit services and in general, the social welfare is the sum of the consumer surplus and the producer surplus. In this section, social welfare functions are formulated for both conventional and flexible bus services.

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4.1. Welfare formulation for conventional services ki The welfare of conventional bus services, Y ki c , in region k and period i is the sum of the producer surplus, P c and the con-

sumer surplus, Gki c : ki ki Y ki c ¼ P c þ Gc

ð8Þ

ki ki The producer surplus Pki c is the total revenue Rc minus the operating cost C c of the conventional bus service:

ki ki Pki c ¼ Rc  C c

ð9Þ

The total revenue Rki c in region k and period i is the fare multiplied by the total demand density in region k and time period i:

  rk ki ki ki k k ki ki k Rki ¼ f Q t ¼ f L W q t K  e z h  e z  e f w 1 x 2 p c c c c c c c Vx

ð10Þ

k

c where K kc ¼ 1  ex z2 Vdx  ev M . Vi c

The producer surplus in Eq. (8) can be now rewritten as:

  rk Dk tki ki k k ki ki k  Bc Nkc i ki Pki ¼ f L W q t K  e z h  e z  e f w 1 x 2 p c c c c c Vx V h

ð11Þ

c c

The consumer surplus Gki c is formulated for region k and time period i. The consumer surplus is the total user benefit minus the prices that transit users actually pay. The total user benefit function can be obtained by using the willingness to pay function in Eq. (2). The fare in Eq. (2) is formulated as a function of the demand density, Q ki c :

fc ¼

  1 rk Q ki ki c  K kc  ew z1 hc  ex z2 ep Vx ep Lk W k qki

ð12Þ

The total user benefit is then obtained by integrating Eq. (12) over the demand density, Q ki c , which is expressed as:

Z

ki

f c dQ c ¼

2   1 rk ðQ ki ki c Þ Q ki K kc  ew z1 hc  ex z2  c ep Vx 2ep Lk W k qki

ð13Þ

Eq. (13) is rearranged by substituting the potential demand density qki from Eq. (2). The total user benefit U ki c is formulated as follows:

U ki c ¼

  ki Q ki rk ki c t K kc  ew z1 hc  ex z2 þ ep f c 2ep Vx

ð14Þ

The consumer surplus is formulated as the total user benefit minus the fares that users actually pay to the conventional bus providers:

Gki c ¼

 2 Lk W k qki t ki rk ki K kc  ew z1 hc  ex z2  ep f c 2ep Vx

ð15Þ

The total welfare in Eq. (8) that sums the producer surplus and consumer surplus in region k and period i is then expressed as:

   2 rk Dk t ki Lk W k qki tki rk ki ki k k ki ki Y ki K kc  ew z1 hc  ex z2  ep f c  Bc Nkc i ki þ K kc  ew z1 hc  ex z2  ep f c c ¼ f cL W q t Vx 2ep Vx V h

ð16Þ

c c

The total welfare for the entire system is formulated as follows:

Yc ¼

K X I K X I X X ki Y ki ðPki c ¼ c þ Gc Þ k¼1 i¼1

k¼1 i¼1

( ) 2 k ki K X I X rk Lk W k qki t ki k rk ki ki k k ki ki k k D t Yc ¼ f c L W q t ðK c  ew z1 hc  ex z2  ep f c Þ  Bc Nc i ki þ ðK c  ew z1 hc  ex z2  ep f c Þ Vx 2ep Vx V h k¼1 i¼1 c c

ð17Þ

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Eq. (17) can be written as:

Yc ¼

!   X K X I  K X I X rk Dk t ki ki  f c Lk W k qki t ki K kc  ew z1 hc  ex z2  ep f c Bc Nkc i ki Vx V c hc k¼1 i¼1 k¼1 i¼1 ( )   2 k k K I ki ki k XX L W q t r ki K kc  ew z1 hc  ex z2  ep f c þ 2e V p x k¼1 i¼1

ð18Þ

The social welfare in Eq. (18) is maximized by optimizing the decision variables of vehicle size, fares, headways, fleet sizes, and route spacings (or numbers of zones). 4.2. Welfare formulation for flexible services ki The welfare of flexible bus services in region k and period i, Y ki f , is formulated as the sum of producer surplus P f and

consumer surplus Gki f : ki ki Y ki f ¼ P f þ Gf

ð19Þ

The producer surplus Pki f is computed by subtracting the flexible bus operating cost from the revenue of the flexible bus services: ki ki Pki f ¼ Rf  C f

ð20Þ

The total revenue of the flexible bus services in region k and period i, Rki f , is the flexible bus service fare multiplied by total demand density: ki

ki ki k k ki ki ki Rki f ¼ f f Q f t ¼ f f L W qf t ð1  ew z1 hf  ev M f  ep f f Þ

ð21Þ

Then, the producer surplus in region k and period i P ki f is:

0 Pki f

k

¼ ffL

ki W k qki f t ð1

ki ew z1 hf





ev Mki f

 ep f f Þ  @

Bf Nkf Dkf t ki ki

V if hf

þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ki qki f hf =u A ki V if hf

;Bf Lk W k tki

ð22Þ

The consumer surplus Gki f in region k and period i is formulated as the total user benefit of the flexible bus services minus the price that flexible bus users actually pay. The total user benefit of the flexible bus service U ki f can be found by integrating the willingness to pay function:

U ki f

¼

U ki f ¼

Z

ki f f dQ f

¼

Z (

1 Dki ki K kf  ew z1 hf  ev c i ep 2V f

Q ki Dki ki f K kf  ew z1 hf  ev c i ep 2V f

! 

ðQ ki f Þ

! 

Q ki f ep Lk W k qki

) ki

dQ f

2

2ep Lk W k qki

ð23Þ

By substituting the potential demand density qki from Eq. (4), the total user benefit of the flexible buses in region k and period i U ki f becomes:

U ki f

ki Q ki Dki ki f t ¼ K kf  ew z1 hf  ev c i þ ep f f 2ep 2V f

! ð24Þ

The consumer surplus of the flexible bus services is formulated as the total user benefit minus the price that users actually pay to the flexible bus providers:

Gki f ¼

Lk W k qki t ki Dki ki K kf  ew z1 hf  ev c i  ep f f 2ep 2V f

!2

The total welfare of the flexible bus service in region k and period i is expressed as: ki ki Y ki f ¼ Gf þ P f

ð25Þ

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Y ki f

Lk W k qki t ki Dki ki ¼ K kf  ew z1 hf  ev c i  ep f f 2ep 2V f qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 ki k k ki qki hf =u Bf Nkf Dkf tki ;Bf L W t A @ þ ki ki V if hf V if hf

!2

n o ki þ f f Lk W k qki t ki 1  ew z1 hf  ev M ki f  ep f f

ð26Þ

The social welfare for all flexible bus services is formulated as follows:

Yf ¼

K X I K X I   X X ki Y ki Gki f þ Pf f ¼ k¼1 i¼1

Yf ¼

k¼1 i¼1

8 K X I < k X L W k qki tki k¼1 i¼1

0 @

:

2ep

Bf Nkf Dkf t ki ki

V if hf

þ

ki

K kf  ew z1 hf  ev

Dki c 2V if

!2  ep f f

n o ki þ f f Lk W k qki tki 1  ew z1 hf  ev M ki f  ep f f

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi19 ki qki hf =u = A ki ; V if hf

;Bf Lk W k tki

ð27Þ

Eq. (27) can be written as:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8 ki k k ki k k ki K X I n K X I < o X X qki hf =u= ;B L W t B N D t f f f f ki Yf ¼ f f Lk W k qki t ki ð1  ew z1 hf  ev Mki  e f Þ þ  p f f ki : V i hki ; V if hf k¼1 i¼1 k¼1 i¼1 f f 8 9 !2 K X I < k = X L W k qki tki Dki ki k c K f  ew z1 hf  ev  ep f f þ i : 2ep ; 2V k¼1 i¼1

ð28Þ

f

Eq. (28) must be maximized by optimizing the decision variables of flexible bus size, fares, headways, fleet sizes, and zones (or service areas). 4.3. Solution method The welfare formulations for both conventional and flexible services are nonlinear and contain both integer and continuous variables. When multiple dissimilar regions and time-dependent demand are included in the formulation, exact solutions become unreachable. Among the numerical approaches considered for solving this problem we chose a Real Coded Genetic Algorithm (RCGA), which is especially suitable for efficiently solving mixed integer problems (Deb, 2000, 2001; Deep et al., 2009). Such an RCGA algorithm, including a new crossover and a new mutation operators as well as a special truncation procedure for integer variables, is presented in considerable detail elsewhere (Deep and Thakur, 2007a, 2007b; Deep et al., 2009; Kim and Schonfeld, 2013). 4.4. Numerical example In this section, a numerical case study is designed to check formulations without financial constraints. For this case study, the maximum allowable headway constraints are enforced. The vehicle size (seats/bus) is one of the input values, and its sensitivity to the system welfare is also analyzed. 4.4.1. Input values The models are formulated for serving dissimilar regions and time-dependent demand patterns. In order to keep the presentation simple and the results easy to follow, only three local regions and four time periods are shown in this numerical example. However, the demonstrated method is quite capable of analyzing more complex systems. For instance, regions can be further subdivided in order to reduce the variability of demand density, traffic speeds and other characteristics within regions. The baseline input values are shown in Table 1. The input values for the various service types, regions and periods were generated by the authors based on only two considerations: reasonableness (i.e. consistency with typical public transportation operations in the U.S.) and sufficient variability among regions and periods to provide an interesting case for demonstrating model capabilities. The potential demand densities, sizes of regions, and time periods are shown in Table 2. The minimum and maximum headways are assumed to be 3 and 60 min, respectively. The lower and upper fleet sizes are obtained from the minimum and maximum headways. For vehicle sizes, 7, 10, 16, 20, 25, 35, and 45 seats are the acceptable input values. Regions A, B, and C have the same demand densities but other regional characteristics differ. Region A is 4 mile2, region B is 12.25 mile2, and region C is 25 mile2. Therefore, despite the equal demand densities, the total demand in region C exceeds

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Myungseob (Edward) Kim, P. Schonfeld / Transportation Research Part A 80 (2015) 116–133 Table 2 Potential demand, service time, line-haul distance, and sizes of regions. Period

Region A

B

C

Demand (trips/mile2/h) 1 2 3 4

90 40 20 10

90 40 20 10

90 40 20 10

Time (h) 1 2 3 4

4 6 8 6

4 6 8 6

4 6 8 6

Line-haul distance (miles) Length of region (miles) Width of region (miles) Regional area (mile2)

6 2 2 4

6 3.5 3.5 12.25

6 5 5 25

demands in regions A or B. The same demand density inputs are assumed initially for all regions, in order to identify the effects of region size.

4.4.2. Discussion of results We constrained the minimum headway at 3 min. Table 3 shows that fares optimized by RCGA for conventional services with 7 and 10 seat vehicles are positive. However, with these smaller vehicles, resulting headways are below 3 min, thereby violating the low boundary of the headway. Thus, it is confirmed that the optimization model cannot find any feasible solutions for conventional services with 7-seat or 10-seat buses because the input demand is too high or, alternatively because the minimum headways of 3 min are too high. It is possible that if demand inputs or minimum headways were lower, conventional services with 7 or 10 seats buses could have solutions with optimized fares of zero. The flexible services also cannot find any feasible solutions for vehicle sizes of 7, 10, and 16 seats. In this study the genetic algorithm is run up to 2000 generations, although the solution converges in about 550 generations. The total computation time for the conventional service using 25 seat vehicles with IntelÒ Core™ i7-3610QM CPU @ 2.30 GHz is 465 s, which is about 0.84 s per generation. Fig. 2 shows that the welfare is maximized (at 120,219$/day) when the conventional bus size is 25 seats. For the flexible services, welfare is maximized at 113,999$/day with 20 seat vehicles. Table 4 shows the actual trip density results for conventional and flexible services. The resulting actual trip densities of conventional services in regions A, B and C are very similar (less than one trip/mile2/h). The areas of regions A, B, and C are 4, 12.25, and 25 miles2, respectively, as shown in Table 2. These results confirm that the size of regions does not significantly affect the density of actual trips. However, it is found that the actual trip densities for flexible services exceed those for conventional services. The main reason why flexible services attract more actual trips is that they provide door-to-door services, and hence have zero access times for passengers. However, conventional services include access times in their elastic demand function. Thus, as shown in Table 4, flexible services attract more actual trips than conventional services, but the difference in trip density decreases as service regions get larger. Table 10 shows that flexible services yield higher consumer surplus than conventional services because they attract more actual trips than conventional services (shown in Table 4). However, flexible services have higher operating costs than conventional services (Table 9). Thus, conventional services still yield higher system welfare than flexible services. Table 5 shows the optimized number of zones for conventional and flexible services. The numbers of zones for conventional services are two, four, and six for regions A, B, and C, respectively. The route spacings (which are Width of region/Number of zones) are then 1.0, 0.875, and 0.833 miles for regions A, B, and C, respectively. As expected, the number of zones increases with the width of a region. The numbers of zones for flexible services are one, three, and five for regions A, B, and C, respectively. Hence, optimized zones for flexible services are 4.0, 4.08, and 5.0 mile2 for regions A, B, and C, respectively. Thus, we conclude that, with the given inputs, the optimized zone sizes for flexible services range between 4 and 5 square miles. The optimized headways for both conventional and flexible services are provided in Table 6. For conventional services, period 1, which has higher demand densities than other periods, has optimized headways of about four to six minutes. Optimized service frequencies decrease as demand densities decrease. It is also found that headways of flexible services are generally lower than headways of conventional services if they are compared for the same period and region. This indirectly helps explain why flexible services yield more actual trips than conventional services. For period 1, flexible service headways are slightly above 3 min, which is the specified minimum headway.

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Table 3 Optimized fares with vehicle size inputs. Vehicle size (seats) Fare for conventional services ($/trip) Fare for flexible services ($/trip)

7 2.58 1.67

10 1.14 1.20

16 0.00 0.74

125000

Welfare ($/day)

120000

116949

115000

109338

110000 105000 100000 95000

20 0.00 0.00

25 0.00 0.00

30 0.00 0.00

40 0.00 0.00

119127 120219 120052 119010

112157

113999 113038 111751

109187

101745 95733 93837

90000

7

10

16

20

25

30

40

Vehicle Size (seats) Welfare for Convenonal Services

Welfare for Flexible Services

Fig. 2. Welfare vs. vehicle sizes.

Table 4 Demand with elasticity. Period

Demand (trips/mile2/h) Conventional services Region

1 2 3 4

Flexible services

A

B

C

A

B

C

74.21 32.53 15.81 7.22

73.59 31.74 15.26 6.72

73.35 31.47 15.36 7.30

77.01 33.68 16.05 7.85

75.86 32.83 16.32 7.55

74.20 32.30 15.98 8.10

Table 5 Optimized number of zones. Period

Conventional services Region

Number of zones Route spacings (mile) Zones (mile2)

Flexible services

A

B

C

A

B

C

2 1.0 –

4 0.875 –

6 0.833 –

1 – 4.0

3 – 4.08

5 – 5.0

Table 6 Optimized headways in minutes. Period

Headways (min) Conventional services Region

1 2 3 4

Flexible services

A

B

C

A

B

C

5.89 7.85 11.78 23.56

6.24 10.41 15.61 31.22

4.86 9.72 12.96 19.44

3.53 6.77 13.54 18.43

3.59 7.68 10.52 21.03

3.17 6.60 9.84 10.83

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Myungseob (Edward) Kim, P. Schonfeld / Transportation Research Part A 80 (2015) 116–133 Table 7 Load factors. Period

Conventional services Region

Flexible services

A

B

C

A

B

C

1 2 3 4

0.58 0.34 0.25 0.23

0.94 0.67 0.49 0.43

0.99 0.85 0.55 0.39

0.91 0.76 0.72 0.48

0.93 0.86 0.58 0.54

0.98 0.89 0.66 0.37

Average

0.56

0.72

Table 8 Optimized fleet sizes. Period

Fleet sizes (buses) Conventional services Region

Flexible services

A

B

C

A

B

C

1 2 3 4

4 3 2 1

5 3 2 1

8 4 3 2

12 6 3 2

13 6 4 2

17 8 5 4

Total fleet size (buses)

166

268

For conventional services, the longest headway, which is about 31 min, is used for period 4 in region B. For flexible services, period 4 in region B has headways of about 21 min. With optimized results from elastic demand, zone sizes vehicle sizes and headways, the load factors for given inputs are shown in Table 7. The average load factor for conventional services is 0.56 while the average load factor for flexible services is 0.72. The maximum load factor is set at 1.0, implying that seats are available for every passenger. The vehicle’s capacity for standees can represent the margin for accommodating probabilistic demand variability in each period and region. Table 8 shows optimized fleet sizes for conventional and flexible services. As expected, period 1 requires larger fleets than other periods. It is also noted that flexible services require much larger fleets than conventional services, and thus have higher operating costs. When optimized, conventional services require a total of 166 vehicles with 25 seats, while flexible services require 268 vehicles with 20 seats. Table 9 provides costs and profits for conventional and flexible services. The analytically optimized fares using Eqs. (18) and (28) are zero. Numerically optimized fares for both conventional and flexible services are also zero. Thus, without subsidy (i.e., in financially unconstrained solutions), revenues of conventional and flexible services in any periods are zero, and thus total revenues are also zero. Therefore, profits are simply the negative values of the supplier costs in each period and region. Costs and profits are shown per hour because periods have different durations. As shown in Table 9, the total cost of flexible services exceeds that of conventional services by about 52%. Table 10 shows the consumer surplus results for conventional and flexible services. Period 1 in region A has the highest consumer surplus, which is 42,705$/period for conventional services and 43,692$/period for flexible services. It is found that the consumer surplus of flexible services exceeds that of conventional services. The main reason is that, with elasticity to access time, door-to-door flexible services attract more actual trips than conventional services. The total consumer surplus of flexible services is $162,143/day for flexible services and $151,859/day for conventional services. Flexible services also have higher operating costs than conventional services, which explain why they also have a larger negative profit (i.e., loss) than conventional services. Table 11 provides welfare results of conventional and flexible services for each period and region for financially unconstrained cases. With given demand and other parameters, it is noted that the total welfare of conventional services exceeds that of flexible services. For region A, the welfare of flexible services exceeds that of conventional services. For regions B and C, conventional services produce greater welfare than flexible services. Here, the total welfare difference between conventional and flexible services is about 5.45% (120,219 vs. 113,999). 4.5. Welfare threshold between conventional and flexible services As discussed in the introduction, a welfare threshold should exist between conventional and flexible services. Determining the appropriate type of transit services (i.e., fixed-route or flexible service) under various circumstances is an important contribution of this study, as it can help in planning efficient transit systems that match the demand variability over space and time. For instance, Table 11 shows that either conventional or flexible services may be preferred, depending

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Table 9 Costs and profits for conventional and flexible services. Period

Conventional services Region

Flexible services

A

B

C

A

B

C

Cost ($/h) 1 2 3 4

280.0 210.0 140.0 70.0

700.0 420.0 280.0 140.0

1680.0 840.0 630.0 420.0

408.0 204.0 102.0 68.0

1326.0 612.0 408.0 204.0

2890.0 1360.0 850.0 680.0

Total cost ($/day)

31,640

Profit ($/h) 1 2 3 4

280.0 210.0 140.0 70.0

1326.0 612.0 408.0 204.0

2890.0 1360.0 850.0 680.0

Total profit ($/day)

31,640

48,144 700.0 420.0 280.0 140.0

1680.0 840.0 630.0 420.0

408.0 204.0 102.0 68.0 48,144

Table 10 Consumer surplus. Period

Consumer Surplus ($/period) Conventional services Region

Flexible services

A

B

C

A

B

C

1 2 3 4

6994.1 4534.1 2854.9 892.5

21060.7 13218.9 8151.2 2370.6

42705.5 26523.8 16842.9 5709.4

7531.7 4862.4 2944.8 1056.3

22381.2 14148.1 9321.6 2989.8

43692.5 27936.9 18240.9 7037.4

Total ($/day)

151,859

162,143

Table 11 Social welfare. Period

Welfare ($/period) Conventional services Region

Flexible services

A

B

C

A

B

C

1 2 3 4

5874.1 3274.1 1734.9 472.5

18260.7 10698.9 5911.2 1530.6

35985.5 21483.8 11802.9 3189.4

5899.7 3638.4 2128.8 648.3

17077.2 10476.1 6057.6 1765.8

32132.5 19776.9 11440.9 2957.4

Total ($/day)

120,219

113,999

on the given parameters. In period 4 for region A, flexible services produce greater system welfare than conventional services. However, in period 1 for region C, conventional services generate greater system welfare than flexible services. Thus, in this section, we find the welfare threshold numerically since formulations are too complex to yield analytic closed form solutions. In order to find the threshold between the two service types (conventional and flexible), we consider an additional case study, which has one local region connected to the terminal, a line haul distance of five miles, a region length of 3 miles, and a region width of 2 miles. We analyze conventional and flexible services for one hour and for various the potential demand density inputs (trips/mile2/h). Other inputs are identical with those in Table 1. System welfare results for financially unconstrained conventional and flexible services are shown in Table 12. Optimized values of decision variables, which are the vehicle size, fleet, the number of zones, and headway, are omitted for brevity in Table 12. Table 12 confirms that flexible services are preferable to conventional services when potential demand density is low. For higher demand density, in the other hand, conventional services produce greater system welfare than flexible services. With given input demand density and input parameter values, the welfare threshold lies between 65 and 70 potential trips/mile2h. Such results confirm that integrated services may be preferable to purely conventional or purely flexible services.

Myungseob (Edward) Kim, P. Schonfeld / Transportation Research Part A 80 (2015) 116–133

129

Table 12 Welfare threshold with numerical analysis. Potential demand (trips/sq.mile/h)

Conv. service welfare ($/h)

Flex. service welfare ($/h)

Difference (conv-flex welfare, $/h)

60 65 70 75 80

1,488.42 1637.36 1779.09 1,928.03 2,076.96

1,497.15 1639.26 1775.49 1,923.76 2,070.45

8.73 1.90 3.60 4.27 6.52

5. Welfare maximization with financial constraints In addition to vehicle capacity constraints, financial (i.e., subsidy) constraints are considered in this section. With various subsidy inputs, the resulting variations of fares, headways and fleet sizes are explored. To consider additional financial constraints, formulations for conventional and flexible services are modified. 5.1. Formulations 5.1.1. Conventional service formulations The total welfare Yc is the sum of the welfare for all time and all regions, shown in Eq. (29). The financial constraint is expressed in Eq. (30). The amount of subsidy is an input value. If zero subsidy is provided, the financial constraint simply becomes that the profit should be non-negative. The maximum allowable headway (service capacity) constraints are also applied in Eq. (31):

Maximize Y c ¼

K X I X fY ki c g

ð29Þ

k¼1 i¼1

subject to K X I K X I X X fPki fFSki g P 0 c gþ k¼1 i¼1

ki

ð30Þ

k¼1 i¼1

Sc lc

ki

hc 6 hc;max ¼

ki

rk W k ds Q ki c

ð31Þ

5.1.2. Flexible service formulations Flexible service formulations that consider financial constraints are provided in Eqs. (32)–(34). The maximum allowable headway constraints for flexible services in Eq. (34) differ from those for conventional services in Eq. (31):

Maximize Y f ¼

K X I X fY ki f g

ð32Þ

k¼1 i¼1

subject to K X I K X I X X fPki fFSki g P 0 f gþ k¼1 i¼1

ki

ð33Þ

k¼1 i¼1

ki

hf 6 hf ;max ¼

Sf l f Ak Q ki f

ð34Þ

5.2. Numerical examples In the following numerical examples, financial (subsidy) constraints are added to maximum allowable headway constraints. Different input values for subsidy are considered through sensitivity analysis. As explained below, the sum of the total revenue and the total subsidy should be larger or equal to the total cost. If the total subsidy is zero, the total revenue minus the total cost (i.e. the profit) should be non-negative. When financial constraints are imposed, the objective functions for both conventional and flexible services become more complex because those constraints are moved to the objective function using the Lagrange multiplier method. Among the numerical approaches we considered for solving this problem we chose a Real Coded Genetic Algorithm. More details regarding the RCGA are presented in Section 4.3. The total subsidy is an input value; unit subsidy ($/potential trip) is used to calculate the total amount of subsidy in this numerical analysis.

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It is possible to jointly optimize vehicle sizes, numbers of zones, headways, and fleet sizes with a financial constraint. However, computation times are much longer and optimized vehicle sizes and numbers of zones are not significantly different from the ones in the financially unconstrained case. Thus, by using the optimized vehicle sizes and numbers of zones from financially unconstrained results, the complexity of financially constrained welfare formulations is reduced and converged solutions are found relatively quickly. It is also reasonable to assume that route spacings of conventional services, zones of flexible services, and vehicle sizes can be determined in an earlier planning stage. The service providers (operators) may then wish to re-optimize service frequencies and fares based on the available subsidy. Thus, headways, fleet sizes, and fares are optimized here with the various subsidy inputs. The value of route spacings for conventional services, zone sizes for flexible services, and vehicle sizes are adapted from the solution of the financially unconstrained optimization model. The main focus of this numerical analysis is on exploring how optimized fares are changed along with other decision variables for different financial constraints (i.e., subsidies). 5.2.1. Results for conventional services Assumed subsidies are applied from zero to 1.2$/potential trip with increments of 0.2$/potential trip. Table 13 summarizes all results for conventional services with various subsidy levels. One further finding is that actual trips increase as the subsidy increases. For instance, the zero subsidy case serves about 68.7% of the total potential demand, but the fully subsidized case serves about 79.2% of the total potential demand. For conventional services, seven sensitivity cases are considered, as shown in Table 13. The amount of subsidy increases linearly. Thus, when the unit subsidy is 1.0$/potential trip, the total subsidy is $33,825/day. For the zero subsidy case, the conventional service fare is optimized at 1.3$/actual trip. As the subsidy increases, as presented in Table 13, the optimized fare decreases fairly linearly. When the unit subsidy is about 1.0$/potential trip, the fare becomes zero, which means the total revenue is zero, and all the costs of bus operations are covered by the subsidy. We also analyzed the profit under various subsidy levels. With the subsidy provision, the revenue decreases since the optimized fare decreases. As expected, the profit also decreases because the revenue decreases. In the formulation, as shown in Eq. (30), the sum of the profit and subsidy can be either zero or positive. For the zero subsidy case, the profit is positive, which means the optimized fare could have been slightly reduced if a break-even solution had been sought. The cost of the zero subsidy case is lower than other subsidy cases. From the 0.2 $/potential trip to 1.2$/potential trip, total costs are identical, which means, their resulting fleet sizes do not change over different subsidy inputs. It explains why fleet sizes and headways do not change significantly for conventional services with financial constraints while the fare decreases as the subsidy increases. The total consumer surplus of conventional services in the zero subsidy case is $114,699/day. Results of other subsidy cases show that the consumer surplus increases until the unit subsidy is 1.0$/potential trip. Beyond that, the consumer surplus does not change significantly. When the unit subsidy is 1.2$/potential trip, the consumer surplus for the conventional services is 151,859$/day. The consumer surplus difference between the subsidies 1.0 and 1.2$/potential trip is 39$/day, which is tiny if divided by the total actual trips served per day. Thus, it can be confirmed that the consumer surplus does not increase when the unit subsidy exceeds 1.0$/potential trip. The welfare of conventional services of the zero subsidy case is 118,321$/day, while the maximum system welfare is found as 120,219$/day from unit subsidies of 1.0$/potential trip or more. As expected, when the total cost is fully covered by the subsidy, the system welfare becomes identical to the one without financial constraints. Numerical results confirm that an RCGA used here finds good and consistent solutions, although it does not guarantee global optimality. 5.2.2. Results for flexible services Table 14 shows detailed results for flexible services with financially constrained cases (i.e., sensitivity analyses of subsidy with respect to welfare). Table 14 also includes a row for ‘‘Profit + Subsidy’’. When the unit subsidy is 1.8$/potential trip, the sum of the profit and the subsidy is positive, which means that some unused budget is still available. In the formulation, the sum of profit and subsidy is no smaller than the cost. Therefore, the system does not have to use the entire sum of profit and subsidy. With this leftover amount of subsidy, the system welfare stops decreasing after reaching its maximum. Therefore, results confirm that unit subsidies beyond about 1.2$/potential trip yield no additional social benefits in this case. In the zero subsidy case, 67.7% of the total potential demand yields actual trips. However, when the operating cost is fully subsidized, about 81.9% of the potential demand is served. In the zero subsidy case, the optimized fare for flexible services is 1.91$/actual trip, which exceeds the optimized fare (1.30$/actual trip in Table 13) of conventional services with the zero subsidy. The higher operating cost for flexible service increases the flexible service fare. The optimized fares decrease as the subsidy increases. When the unit subsidy is 1.4$/potential trip, the optimized fare for flexible services is close to zero (at three cents per actual trip). When the unit subsidy is 1.4$/potential trip, the total subsidy is 47,355$/day. For conventional services (shown in Table 13), the optimized fare becomes zero when the subsidy reaches 1.0$/potential trip. Thus, it is found that flexible services require larger subsidies than conventional services to fully cover the operating cost. For the zero subsidy case, flexible services have zero profit, as expected; this means the operating cost is exactly equal to the revenue. The profit decreases as the subsidy increases because the optimized fare decreases. The cost of flexible service increases with the subsidy since the subsidy enables higher service frequencies. As shown in Table 14, the total operating cost increases with the larger subsidy. As the subsidy increases, the optimized fare and the

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Myungseob (Edward) Kim, P. Schonfeld / Transportation Research Part A 80 (2015) 116–133 Table 13 Results of conventional services with financial constraints. Unit subsidy ($/trip) Fare Revenue Cost Profit Subsidy Profit + subsidy Consumer surplus Welfare Total actual trips Total actual trips/total potential trips

0.0 1.30 30221.6 26600.0 3621.6 0.0 3621.6 114699.0 118321.0 23247 68.7%

0.2 1.02 24875.0 31640.0 6765.0 6765.0 0.0 125751.3 118986.3 24381 72.1%

0.4 0.72 18110.4 31640.0 13529.6 13530.0 0.4 133131.3 119601.7 25087 74.2%

0.6 0.44 11345.0 31640.0 20295.0 20295.0 0.0 140283.9 119988.9 25754 76.1%

0.8 0.17 4580.0 31640.0 27060.0 27060.0 0.0 147242.9 120183.0 26386 78.0%

1.0 0.00 38.8 31640.0 31601.2 33825.0 2223.8 151819.8 120218.6 26793 79.2%

1.2 0.00 0.0 31640.0 31640.0 40590.0 8950.0 151858.6 120218.6 26797 79.2%

Table 14 Results of flexible services with financial constraints. Unit subsidy ($/trip) Fare Revenue Cost Profit Subsidy Profit + subsidy Consumer surplus Welfare Total actual trips Total actual trips/total potential trips

0.0 1.91 43452.0 43452.0 0.0 0.0 0.0 109692.6 109692.6 22774.8 67.3%

0.2 1.59 37419.1 43792.0 6372.9 6765.0 392.1 117674.1 111301.2 23597.9 69.8%

0.4 1.28 31146.0 44676.0 13530.0 13530.0 0.0 125490.7 111960.7 24359.6 72.0%

0.6 1.00 25197.0 45492.0 20295.0 20295.0 0.0 133186.4 112891.4 25102.6 74.2%

0.8 0.73 18842.8 45900.0 27057.2 27060.0 2.8 140684.4 113627.2 25803.4 76.3%

1.0 0.48 12755.2 46580.0 33824.8 33825.0 0.2 147646.5 113821.7 26434.1 78.1%

1.2 0.29 7842.3 47600.0 39757.7 40590.0 832.3 153750.4 113992.6 26977.1 79.8%

1.4 0.03 789.0 48144.0 47355.0 47355.0 0.0 161353.5 113998.5 27635.3 81.7%

1.6 0.00 0.0 48144.0 48144.0 54120.0 5976.0 162143.5 113999.5 27702.9 81.9%

1.8 0.00 0.2 48144.0 48143.8 60885.0 12741.2 162143.3 113999.5 27702.9 81.9%

revenue decreases. When the operating cost is fully subsidized, the optimized fare should be zero so that the revenue is also zero. Since the profit is the revenue minus the cost, the absolute value of the profit equals the absolute value of the cost if the revenue is zero. The absolute value of the minimum profit and the absolute value of the maximum cost in Table 14 are identical (i.e., unit subsidy of 1.8$/actual trip). This explains why the optimized fare and revenue are zero for the fully subsidized case. The consumer surplus with the zero subsidy case is 109,693$/day. The higher subsidy results in lower fare and more actual trips. As expected, the consumer surplus increases as the subsidy increases. The maximum consumer surplus is 162,143$/day when the unit subsidy exceeds 1.6$/potential trip. The welfare in the zero subsidy case is 109,693$/day, which is identical to the consumer surplus because the profit for the zero subsidy case is zero. For flexible services, the zero subsidy case is the break-even case. The system welfare of flexible services converges to 113,999$/day without exhausting the available subsidy. 6. Conclusions This paper contributes several improvements in optimization models for the analysis and comparison of conventional and flexible services. First, it considers the elasticity of demand to in-vehicle times, wait times, access times and fares and the resulting effects on the comparisons of service types. Secondly, the models comprehensively maximize net benefits (i.e. ‘‘welfare’’) instead of just minimizing costs. More specifically, the welfare maximization problem for conventional services had been solved in the literature for a terminal connecting multiple regions, but for only for one time period. For flexible services, welfare had only been maximized for small problems, e.g. for one local region with one time period. A third contribution of this study is that it solves the maximum welfare problem for both conventional and flexible services considering dissimilar multiple regions as well as time-dependent demand patterns. Fourthly, this study numerically finds the system demand density threshold between fixed-route conventional and flexible transit services when both are optimized for maximum welfare. With inputs from the empirical transit data or planning scenarios, this model should be useful in planning efficient bus transit services. The formulations (i.e., net benefit functions for both conventional and flexible services) presented here are highly nonlinear and have mixed integer decision variables. Such nonlinear mixed integer formulations are known to be NP-hard, and no proven methods exist for exactly finding their global optima unless the problems are very small. Deep et al. (2009) showed that a Real Coded Genetic Algorithm (RCGA) was capable of efficiently finding optimal or nearly optimal solutions for highly complex problems, including mixed integer problems. Deep’s research team proposed a specialized crossover and a mutation operator as well as a truncation procedure for integer restrictions (2001, 2007A, 2007B, and 2009). Thus, we selected that RCGA for solving our formulations because these formulations are highly nonlinear, mixed integer and constrained

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optimization problems. The reliability of RCGA is discussed in previous studies, where it is shown to efficiently find near-optimal solutions (Kim and Schonfeld, 2013; Kim, 2013). Two constrained optimization models are analyzed. Their objectives are: (1) maximum system welfare with service capacity (maximum headway) constraints, and (2) maximum system welfare with service capacity and financial constraints, for both conventional and flexible services. In numerical examples, the fares, route spacings for conventional services, zones for flexible services, headways and fleet sizes are optimized. The numerical examples show that, with the given input parameters, conventional services generate higher system welfare than flexible services when the operating costs are fully subsidized. Numerical examples also explore the sensitivity of vehicle sizes and the sensitivity of the subsidy with respect to the social welfare for conventional and flexible services. For both conventional and flexible services, the actual trips increase as the subsidy increases. In reality governments may not subsidize all the supplier costs of transit services. When the available subsidies and potential demand levels are given, the models proposed here can help transit planners compare optimized service types and choose the most appropriate one for each region and period. In choosing between conventional and flexible services, transit planners and managers should treat the demand density as the most important factor. In general, conventional services are increasingly preferable to flexible-route services as demand densities and route lengths increase. For instance, our numerical results in Table 11 show that Region C, which has longer routes and higher demand density than Region A, yields a higher system welfare with conventional services than with flexible services. However, managerial decisions regarding service type should also consider other factors such as unit costs, expected service levels and available resources. The models presented in this paper take such other factors into account and allow practitioners to estimate their effects, as shown in the sensitivity analyses presented above. To overcome the limitations of the present study, various extensions may be of interest, including the following: 1. In real bus transit operations, the elasticity factors of in-vehicle time, waiting time, access time, and fares may be a nonlinearly related to the actual transit ridership. Thus, the assumption of a linear demand function may be revised in further studies. 2. It would be interesting to compare the results obtained with Stein’s approximation for efficient tours with those from microscopic dial-a-ride algorithms. 3. Other solution algorithms and additional customized operators for genetic algorithms could be tested. 4. Methods may be developed for determining the boundaries of regions for analyzing heterogeneous metropolitan areas. 5. If actual ridership data are available, the optimization models presented here may be able to provide more realistic guidance to the transit service planners and managers in what situation conventional services are preferable to flexible services or vice versa.

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