Planning and Design of Flex-Route Transit Services

Transcription

1 Transortation Research Record Paer No Planning and Design of Flex-Route Transit Services Liing Fu A theoretical investigation is resented of various issues involved in the lanning and design of flex-route transit services. An analytical model is roosed for an idealized oerating environment with the objective of determining the otimal slack time that should be allocated to a flexroute segment. The otimization objective is defined to minimize total oerator and user cost, which enables a systematic examination of comlex interactions among the system arameters. An equation is derived for the relationshi between the number of feasible deviations and various system arameters such as slack time, zone size, and dwell time. Subsequent analysis shows that the analytical model is elaborate enough to rovide substantial insights into various issues that may arise in designing a flex-route service. A simulation analysis is conducted to validate some of the conclusions drawn from the analytical model and to further analyze the imlications of stochastic variation in assenger demand. Flex-route transit is a hybrid of conventional fixed-route transit and demand-resonsive aratransit service. It assimilates conventional transit in that its main route covers a service corridor with a set of fixed stos and schedules. Flex-route transit is also demand resonsive as its service vehicles are allowed to deviate from the main route to rovide door-to-door or checkoint-to-checkoint service to users who either have tri ends located out of the service coverage of the main route or require accessible services such as aratransit. Thus, flexroute service is targeted to two grous of riders: one includes mainly the general ublic transit users who use the fixed stos and the other includes mostly the aratransit users who use the deviation service. By integrating the regularity of conventional fixed-schedule transit and the flexibility of demand-resonsive variable-route aratransit, flex-route transit has the otential to become a vital transit otion, esecially for low-density areas where demand for general ublic transit is too low to be efficiently serviced by conventional transit. The major economic otential of flex-route service is that it may reduce systemwide costs by covering a roortion of aratransit riders who would otherwise have to be accommodated by a dedicated but more costly aratransit service (1, 2). Although concetually simle and attractive, flex-route transit has roven to be much more difficult to lan, design, and oerate than its counterart modes regular transit and aratransit. The reason is twofold. First, a flex-route service has to maintain a balance between the needs of the two grous of riders the general ublic transit users and the aratransit users who usually have very different service requirements, exectations, and attitudes toward transit service. Second, oerating a flex-route service requires a dedicated real-time decision suort system resonsible for registration, tri booking, scheduling, and disatching. The real-time scheduling and Deartment of Civil Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. disatching tasks can become esecially challenging when the system has to deal with a large volume of requests for deviation and highly varied demand at fixed stos. This research deals with the roblem of lanning and designing flex-route services, with the secific focus on issues such as where to set fixed stos, what zone size to use, and how much slack time to allocate. The flex-route design roblem has been addressed in several ast studies involving emirical analysis and field exeriments. Rosenbloom (1) rovided a recent account of current ractices on the flexroute service concet and acknowledged that the imlementation of flex-route services is still limited to rural and small urban areas. The main research on flex-route transit was recently initiated by Durvasula et al. (3), who rovided an extensive discussion on critical issues and challenges involved in flex-route oerations and conducted an emirical study using data from North Virginia s Potomac and Raahannock Transortation Commission. Welch et al. (4) described a methodology for determining whether or not art of a conventional service route should be modified for out-of-direction travel. To the author s knowledge, no research has been attemted to formulate the flex-route service-design roblem in a systematic manner. The objective is to develo an analytical otimization model that can be used to identify critical factors and relationshis that need to be considered in flex-route design. First discussed is how a simlified analytical model can be established over an idealized oerating environment. The analytical model is subsequently used to analyze various design issues in flex-route service. Finally, simulation exeriments are conducted to test some of the conclusions drawn from the analytical model and analyze the effect of random variation in assenger demand on the quality of a flex-route service. ANALYTICAL MODEL This section introduces an analytical model for designing and analyzing flex-route service systems. The model is derived on the basis of an idealized oerating environment with the following settings (see Figure 1 and Table 1): Both flex-route and aratransit services are rovided by the same oerator. All aratransit tris need to be accommodated either by the flex-route service or by a secialized aratransit service. The flex-route service is to be oerated within a rectangular service area of length l and width 2w. The service area is covered by a uniformly distributed grid road network with a link travel seed of v. The main route of the flex-route service is located at the middle of the zone and includes one route segment (or one service zone) between two fixed stos: Stos A and B. The service vehicle dearts from Stos A and B at rescheduled dearture times. The difference between the dearture times at Stos B and A is the scheduled running time, denoted as T, which is also defined as the analysis eriod

2 6 Paer No Transortation Research Record 1791 l Fixed sto Served aratransit sto 2 w Rejected aratransit sto Bus route FIGURE 1 Flex-route service in an idealized network. in this study. To accommodate ossible deviations, the scheduled running time (T ) must be greater than the direct running time between the fixed stos (T ). The difference between them is called slack time, denoted by (T = T + ), which reresents the main decision variable in designing flex-route service. There are N aratransit stos that are exected during the analysis eriod (T, the scheduled time for a flex-route vehicle tri from Stos A to B). These deviated stos are to be serviced during the service headway. The stos are uniformly distributed over the service zone. There are N t general ublic transit riders traveling from A to B for each flex-route tri. Both transit and aratransit demands are erfectly inelastic, that is, they are not affected by service quality. With this assumed oerating environment, the service-design roblem is defined as to determine the otimal amount of slack time that should be allocated to this service route. The otimization objec- tive is assumed to minimize the total net cost that would be incurred to the oerator as well as to the assengers, as discussed in the following section. It is imortant to oint out that the analysis erformed in this study is not concerned with the feasibility of the flex-route service; that is, the decision to run the flex-route service is assumed to be made before this analysis. The question to be addressed is simly how to best oerate the service. Oerator Cost Because of route deviation, flex-route service incurs additional costs to the service oerator. The total marginal oerating cost deends on increases in mileage and service hours due to the route deviation service, and can be assumed to be roortional to the additional slack time ( ) built into the schedule, that is, TABLE 1 Symbol Variable Definitions Definition Samle Value l Length of the service zone (km) 1. w One-side width of the service zone reresenting zone size (km) 1.2 v Average vehicle travel seed on road (km/h) 2. T Direct running time between the fixed stos (h), T = l/v.5 T Scheduled running time between the fixed stos (h), T = T +. / N Total number of deviated stos requested er analysis eriod T 4 n Total number of feasible deviations er analysis eriod 2 T Exected total route length to visit n stos (h) / N t Number of transit riders from sto A to sto B er flex-route tri (or 1 analysis eriod) c f marginal hourly oerating cost of flex-route service ($/hour) 12 c the cost that would result from not servicing a deviated sto, or 8 marginal benefit of servicing a deviated sto, or marginal oerating cost of servicing a deviated sto using aratransit ($/sto) c t Cost coefficient reresenting assengers' disutility toward increase 1 in ride time due to deviation C o Oerator cost due to deviation from the main route for analysis / eriod T ($) B o Oerator benefit due to covering deviated stos for each analysis / eriod T ($) C u User cost associated with transit riders due to increase in ride time / for each analysis eriod T ($) M Vehicle caacity (seats) 2 Slack time - decision variable (hour) / NOTE: / = for corresonding variables, default values are either not available or not necessary.

3 Fu Paer No C o f = c () 1 f With this routing strategy, the exected total time to visit n random stos is where C o f is the total marginal cost for each analysis eriod T and c f is the marginal hourly oerating cost of the flex-route service. It should be emhasized that this marginal hourly oerating cost should include only the increment in cost due to route deviation as comared with a service without route deviation (regular transit). It is different from the hourly oerating cost of the flex-route service, which would include all costs such as vehicles, crews, and management. Therefore, the main contributor to the marginal cost should be the cost of gasoline. Service Benefit The benefit of oerating a deviation service is that it will cover a certain number of tris that would otherwise not be served or have to be served by a more costly otion, such as driving, regular transit (which means oening of new transit routes), or secialized aratransit. The benefit should therefore be equivalent to the costs that would result if the flex-route service was not rovided. However, quantifying such costs is extremely difficult, if not imossible, esecially in those cases in which the deviation service is oen to the general ublic. For the general ublic, the tangible benefits of flex-route service could be increased mobility and reduced traffic congestion. But the question is how to measure the magnitude of those benefits. In this study, the analysis was limited to cases in which the deviation service was only for aratransit users and there was a dedicated aratransit service available to cover aratransit requests if not serviced by the flex-route service. With this assumtion, we can assume the oerator benefit, denoted by B o, is a function of the number of aratransit tris that can be covered by the route deviation service and the marginal oerating cost of aratransit service, that is, o B = n c ( 2) where c is the marginal oerating cost of the aratransit service ($/sto) and n is the number of feasible deviation that can be covered by route deviation during a flex-route tri. Note that the marginal oerating cost of aratransit service can be readily obtained from aratransit service roviders. The number of deviated stos that can be covered (n ) is limited by the amount of slack time allocated to the route segment. For oerational efficiency, the amount of slack time allocated should be aroximately equal to the difference between the exected route time (with n deviated stos) and the direct running time, that is, = T T 3 where T is the direct running time, T = l/v; T is the exected time required to cover n deviated starting from Sto A and ending at Sto B. T deends on how the stos are routed or sequenced, which in turn deends on how the deviated stos are distributed over the zone. Under the idealized conditions assumed in this study, the route length can be aroximated with the following simlified routing strategy: deviated stos are first ordered based on their horizontal distances from the starting sto (A) and then visited sequentially; once a sto is visited, the vehicle will return to the main route (center line) and then to the next sto via the shortest ath (see Figure 1). ( ) T where δ (= w / v +τ) = exected time er deviation or the additional time needed to visit one additional deviated sto, τ=average dwell time at a deviated sto, and v = average vehicle travel seed. Relace T in Equation 3 with Equation 4; the exected number of stos that can be serviced within a given amount of slack time can be obtained as follows: n Equation 5 indicates that the number of feasible deviation is a linear function of the slack time. It should be noted that this relationshi is aroximate resulting from the continuous aroximation over an idealized network. Further discussion on this result is included in the following simulation study. With Equation 5, the exected oerator benefit shown in Equation 2 can therefore be exressed as a function of slack time as follows: B o In addition, the number of feasible deviations is also limited by the total number of available deviations requested (n N ), and vehicle caacity (n + N t M ). With Equation 5, we can obtain the following constraints: N δ ( 7) l + n w = + n τ v = lv+ ( wv+ τ) n = T + δ n T T = = wv+ τ = c δ ( 6) ( M N t ) δ () 8 The number of aratransit riders that the flex-route system can accet is also limited by the vehicle seating caacity. To avoid this issue, we assume the vehicle size is not a limiting factor for roviding the service; that is, the flex-route vehicles are large enough to handle all ossible tris. User Cost 4 δ ( ) ( 5) Route deviation will however cause inconvenience to the transit riders because of increased ride time. The larger the deviation or slack time the higher the inconvenience will become. Such inconvenience could even lead to loss of transit riders when it exceeds a certain amount. It is therefore necessary to consider this consequence in designing a flex-route service. The inconvenience resulting from route deviation is modeled as a user cost which is assumed to be a function of the increase in transit rider travel time ( ) as follows:

4 62 Paer No Transortation Research Record 1791 u r C = N c ( 9) where N t = average number of transit riders for each flex-route tri from Sto A to Sto B, c t = cost coefficient that can be calibrated on the basis of assenger attitude toward increased ride time, and r = model arameter reresenting transit rider sensitivity to deviation time. In this study, we assume r = 1, and consequently the corresonding cost coefficient c t can be considered as the value of time of the transit riders. A threshold is used to consider the maximum allowable deviation as follows: β T where β is the maximum allowable deviation ratio. For the aratransit riders, quality of service rovided by flex-route service and aratransit service are assumed similar and no user cost is therefore considered in this analysis. Problem Formulation and Solution The roblem of identifying otimal slack time can now be formulated as a linear rogramming roblem: min Z( ) = Oerator cost + User cost Service benefit o u o = Cf + C B c cf = δ Nc t t δ subject to t t or Tβ ( 1) { t } ( 11) min N δ, Tβ, ( M N ) δ ( 12) where Z is total marginal cost. This roblem can be solved analytically, yielding the otimal slack time ( *): c cf δ δ Nc t t * = min { Nδ, Tβ, ( M Nt) δ} otherwise where c f δ=unit oerating cost of the flex-route service, or, the oerating cost for the flex-route to service one additional deviated sto; as a result, (c c f δ)/δ =unit net oerating benefit er deviation hour, and N t c t = the total cost of the transit riders er deviation hour. Figure 2 shows an examle relationshi between total marginal cost and slack time when l = 5 km, w = 1 km, v = 2 km/h, c f = $12/hr, c = $8/sto (or $16/tri), c t = $6/hr (that is, transit riders value of time is $6/h, or they would feel a loss of $1 for an increase of 1 min for deviation), N t = 5, N = 2, τ =1 min, β =4%, and M = 9 seats. The otimal slack time in this examle is 6 min. A more detailed analysis is rovided in the following section to show the alication of this analytical result. ANALYSIS WITH ANALYTICAL MODEL The analytical model established reviously, although simlistic, can rovide some meaningful insights into various design issues involved in flex-route services. For examle, in lanning a flex-route service, it is often useful to be able to analyze how various service arameters such as service zone size and aratransit dwell time influence the number of deviations that can be accommodated by a flex-route service. The analytical model facilitates these tyes of investigations as described in the following section. Otimal Slack Time ( ) ( 13) The otimal amount of slack time that should be rovided to a route segment between two consecutive fixed stos is a function of many factors, including transit and aratransit demands, marginal oerating costs of flex-route service and aratransit service, zone size, and aratransit dwell time. Indeed, on the basis of Equation 13, the deviation service would make sense only when the unit net oerating 12 Total marginal cost, Z, ($) Otimal SlackTime Infeasible according to Equ. 1 Infeasible according to Equ. 7 Infeasible according to Equ Slack time (min) FIGURE 2 Otimal slack time.

5 Fu Paer No benefit of the flex-route transit is greater than the collective value of time of the transit riders, that is, ( c cf δ) > Nc t t δ ( 14) An interesting observation of Equation 13 is that the economic viability of flex-route transit does not deend on aratransit demand, but the transit demand only. However, when deviation is in the beneficial region, the slack time should be as large as ossible, when this is feasible [limited by maximum accetable deviation for transit riders, T β, and caacity, (M N t )δ] and meaningful (limited by demand, N δ). Note that the otimal slack time does not deend on the cost rates (c f, c, and c t ). Figure 3 shows the relationshi between the ercentage of otimal slack time ( * / T ) and aratransit demand under various ratios of unit deviation time to direct running time. The slack time should be increased as aratransit demand, dwell time at deviated stos, and the length of the route segment increase, and decreased as transit rider deviation tolerance decreases. When transit rider tolerance is high and aratransit demand is low, the otimal slack time mainly deends on aratransit demand and unit time er deviation. accommodated by flex-route vehicles under a given amount of slack time. The larger a service zone is, the larger the average distance from deviated stos to the main route, and the more time is needed to visit a deviated sto, therefore, the fewer number of stos that can be accommodated. This conclusion has also been shown emirically by Durvasula et al. (3). It should, however, be ointed out that this effect of zone size is mainly due to the assumed service olicy of the first-come-first-served olicy. This oerating olicy is necessary in order to guarantee equitable access to the service, esecially when the deviation service is also made available to the general ublic. However, it may not be necessary when the deviation service is available only to aratransit users such as the elderly and disabled. Rejecting these tris would not cause any judicial roblems because they will be covered by a secialized aratransit service anyway. Therefore, if only aratransit tris are to be served and they are known in advance to the flex-route oerator, there will be no advantage to restrict service to a given buffer area. By designating a larger service area, the oerator will have more choices of deviations and therefore have a higher likelihood of (a) maximizing the number of deviations that can be accommodated and (b) minimizing the ossible leftover slack time or idle time at the fixed stos. Otimal Slack Time Distribution Perhas the most likely use of the analytical formula is in determining the otimal ratios by which to distribute a given amount of route slack time to individual route segments. In situations in which aratransit demand is relatively low, the otimal slack time should be dominated by the first item in Equation 13; therefore, the total slack time should be distributed in roortion to the roduct of aratransit demand (N ) and zone size (δ). In the case that all route segments have similar zone size, the distribution ratio should be roortional to aratransit demand. This finding is consistent with the emirical results of Durvasula et al. (3). However, when both transit and aratransit demand are high, the otimal slack time should be roortional to the direct running time; that is, the higher the direct running time between a segment is, the larger slack time should be allocated. Effect of Zone Size on Feasible Deviations As shown in Equation 5, the size of the service zone (w) is an imortant factor influencing the number of feasible deviations that can be Relative otimal slack time, / T (%) 8% 7% 6% 5% 4% 3% 2% 1% % / T = β = 5% Deviation intolerable to transit riders δ / T = Paratransit demand, N (deviation stos/t) FIGURE 3 Otimal slack time as a function of aratransit demand and unit deviation travel time. Effect of Dwell Time on Feasible Deviations Equation 5 also reveals the effect of dwell times at deviated stos on the exected number of feasible stos that can be made within a given amount of slack time. As would be exected, the number of feasible deviations is inversely roortional to the average dwell time at each deviated sto. This relationshi is consistent with the emirical analysis of Durvasula et al. (3). A further examination of the equation indicates that the relative effect deends on the associated service zone size (w) and the average vehicle running seed (v). This attern can be shown using a simle numerical examle in which a fixed vehicle running seed of 2 km/h is assumed. If the service zone size is 12-m wide on each side, a dwell time of 6 s would yield n = 1 / [(1.2 * 6 / 2) + 1.] = 2.17 feasible deviations. If the dwell time was increased to 66 s (a 1% increase), the average number of feasible deviations would be reduced to 1 / [(1.2 * 6 / 2) + 1.1] = 2.13, which is a 3% reduction. However, if the zone size were instead 4 m, the same amount of increase in dwell time would induce a 9% of reduction in total number of feasible deviations. SIMULATION ANALYSIS The analytical model discussed in the revious section was established on the basis of several imortant assumtions including idealized network, continuous aroximation of route length, and deterministic demand. The model has the following limitations: 1. It may overestimate the number of feasible deviations that can be made within a given amount of slack time. 2. It cannot redict the idle time at the end sto or intermediate fixed stos, which is clearly imortant to the transit riders who have to wait for the bus to deart during the idle time. 3. It identifies otimal slack time for a single route segment. However, it is unclear how this result can be alied to determine the otimal slack time for a flex-route system with multile segments. The objective of this section is to address these limitations through a simulation analysis. The simulation is erformed using a simulation model called SimParatransit, which was originally develoed as a tool

6 64 Paer No Transortation Research Record 1791 for evaluating aratransit systems under a variety of oerating conditions and service concets (5). The system was extended with the functionality to model flex-route service. The simulation exeriments were erformed on a hyothetical network as shown in Figure 4. The following secifications were used: 1. The service area covers a corridor of two subzones labeled Zone 1 (5.6 km 2.4 km) and Zone 2 (1.6 km 1.2 km). Each zone is covered by a uniform grid road network with all neighboring nodes (intersections) connected by two links, one in each direction. Each link has a length of 4 m and a seed 2 km/h. 2. All service vehicles are identical and each has a seating caacity of 3 assengers. (Assuming this large caacity is to eliminate the ossibility of rejecting tris because of the caacity constraint.) Vehicles run sequentially from Stos A to B and then to Sto C with a fixed headway of 3 min. Dearture time at each fixed sto was created on the basis of the direct running times between the fixed stos and a resecified slack time. For examle, if the slack time is 14-min total, of which 1 min are allocated to Zone 1 and 4 min to Zone 2, and the scheduled dearture time at Sto A is 8: a.m., the dearture times at Stos B and C would be 8:27 a.m. and 8:36 a.m., resectively. 3. Two grous of aratransit tris are modeled. The first grou of tris originates in Zone 1 and arrives at the same location Sto B, whereas the second grou of tris is from Zone 2 to the same location Sto C. Tri origins of both grous are uniformly distributed over their corresonding originating zone and reresent the deviation demand for the flex-route service. Each tri is assumed to have a icku dwell time of 2 min and a dro-off dwell time of zero. The simulated demand rates are 2.5 tris/h for both grous, which were used to generate deviation requests with desired icku times based on a stationary Poisson distribution. For each simulation exeriment setting, the service system was simulated continuously for 2 h, including a total of 4 bus tris from Sto A to Sto C, which should rovide statistically reliable estimates of various erformance measures. The following section resents key findings obtained from the simulation exeriments. Effects of Slack Time on Feasible Deviations The first set of exeriments focuses on the relationshi among the number of deviated stos accommodated by the flex-route vehicles, the idle time at a fixed sto, and the assigned slack time. Only the route segment in Zone 1 (A-B) was allocated with slack time with values varying from to 15 min. The simulated and theoretical results are shown in Figure 5 for the relationshi between the ercentage of feasible deviations and allocated slack time (Zone 1 only). It can be observed that the theoretical aroximation (Equation 5 with δ = w / v +τ=(6 * 1.2 / 2) + 2 = 5.6 min) has catured the general trend of the relationshi between the number of feasible deviations and the amount of slack time. However, the theoretical formula has consistently overestimated the number of feasible deviations. The overestimation is mainly caused by the variation in aratransit demand. Under a small amount of slack time (although on average the demand is higher than what can be accommodated) the variation in demand will cause some unsaturated eriods, which would then lead to leftover slack time. When the slack time is increased to a certain oint, the deviation demand becomes too low to use u all of the assigned slack time that is why the overestimation tends to increase as the slack time increases. Effect of Slack Time on Idle Time Distribution Based on the analytical model, the idle time at a fixed sto should be equal to min{, N δ}, which would then redict zero idle time when the average deviation demand is high and the allocated slack time is small, or, <N δ. However, because of the satial and temoral variation in deviation demand, there may not be a sufficient number of requests in some eriods to comletely fill in the allocated slack time, even when the average demand is much higher than what the flex-route system can accommodate with the given slack time. This means a flex-route vehicle may have to wait (idle) at a fixed sto for its scheduled dearture. The resulting idle time has a significant design imlication as transit users have to sit and wait for the bus to deart and are likely to develo negative ercetions and attitudes toward the rovided service. Figure 6 shows results from simulation and an analytical model under a given level of demand (note that in the analytical model, idle time = min{, N δ}, = min{, 2 * 5.6}). As exected, the analytical model significantly underestimates the idle time. It can also be observed that the mean and standard deviation of the idle time at Sto B were aroximately in roortion to the assigned slack time. This suggests that although use of larger slack time will have the benefit of being able to cover larger number of deviations, it will have a negative consequence of larger idle time at fixed stos. This result is not revealed in a deterministic model but is imortant in designing a 5.6 km 1.6 km Zone 1 2 X 1.2 km A B Zone 2 C 2 X.6 km Fixed sto Paratransit sto Bus route FIGURE 4 Flex-route service in an idealized network.

7 Fu Paer No Feasible deviations (%) 1% 8% 6% 4% 2% % FIGURE 5 By Model (Equ. 5) By Simulation Slack time (min) Feasible number of deviations versus slack time. Feasible deviations (%) 1% 9% 8% 7% 6% 5% Slack-time allocation ratio (Zone 1 : Zone 2) FIGURE 7 Feasible deviations versus slack-time distribution ratio. flex-route system; this will be further examined when the issue of slack time allocation is discussed in the next section. Slack Time Allocation for Maximal Number of Feasible Deviations Figure 7 gives the relationshi between the roortion of aratransit stos made by the flex-route service and the ratio of slack times allocated to Zone 1 and Zone 2. It is evident that there is an otimal allocation ratio at which the total number of aratransit tris covered by the flex-route service is maximized. In this simulated case, the otimal slack-time-allocation ratio is aroximately 1.4. Note that this ratio is quite different from the distance-based allocation logic (2.) and demand-based allocation method (1.). Interestingly, it is close to what we would obtain based on the roduct of N and δ, or, δ zone1 / δ zone2 = [(6 * 1.2 / 2) + 2] / [(6 *.6 / 2) + 2] = This evidence further suorts the need to consider both the aratransit demand and demand area even when the regular transit demand is uniformly distributed over the route segments. Figure 8 shows the mean and standard deviation of the total idle time at the fixed stos as a function of the slack-time-allocation ratio. Three imortant observations can be made. First, similar to the single zone case, the variation in idle time is fairly high no matter what ratio was used in allocating the total slack time. The coefficients of variation (which equal standard deviation and mean) range from 8% to 9%. Secondly, both mean and standard deviation of the idle time are fairly uniform across the slack-time-allocation ratio, suggesting that it may not be effective to reduce the idle time by selecting an aroriate slack-time-allocation ratio. With the results from the single zone simulation, it can be concluded that allocating a smaller amount of slack time is the only way to reduce idle time and idle time variation. As discussed reviously, when a bus is idling with riders from revious segments on board, it will cause a negative view of the service and thus have a negative effect on future demand; this negative effect can be reduced by minimizing the number of fixed stos. An ideal system would be one with a single fixed sto (feeder service) or two fixed stos (flexible route shuttle). CONCLUSIONS Flex-route service reresents an innovative integration of conventional fixed-route, fixed-schedule transit, and dial-a-ride demandresonsive service. The necessity to meet a fixed schedule and at the same time rovide deviation service to other stos oses a significant challenge for the lanning, design, and management of such services. This research has mainly focused on various issues associated with flex-route transit and is the first to aroach the roblem through a combination of theoretical and simulation analyses. Our roosed analytical model is simle but elaborate enough to reveal the fundamental relationshis between system erformance and design arameters. The otimal slack time should be determined with a consideration of the trade-off between the savings that can be achieved from serving aratransit riders and the inconvenience that may result to the transit users. The critical factors that should be considered include level of aratransit demand, zone size, and aratransit dwell time. The 8 Idle time at fixed sto (min) Mean by Simulation By Model Stdev by Simulation Idle time (min) Mean Stdev Slack time (min) Slack-time allocation ratio (Zone 1 : Zone 2) FIGURE 6 Idle time at fixed sto versus slack time (Stdev standard deviation). FIGURE 8 Idle time versus slack-time distribution ratio (Stdev standard deviation).

8 66 Paer No Transortation Research Record 1791 roosed analytical model rovides a framework for a systematic trade-off analysis in determining the otimal slack time as well as other design arameters such as zone size and length. If the roblem is just to distribute a given amount of slack time to individual route segments, the distribution scheme should consider aratransit demand and zone size. If the distribution objective is to maximize the total number of feasible deviations, the otimal distribution method should be based on the roduct of the exected aratransit demand and the average additional time that is required to visit a deviated sto. Idle time at a fixed sto in the middle of a flex-route system has a negative effect on those riders who are already on the bus and are heading to a destination beyond the fixed sto. Unfortunately, idle time cannot be eliminated comletely and in fact will increase as the allocated slack time increases. This finding suggests that the flexroute concet might be viable only with a minimal number of fixed stos, such as in feeder routes (one fixed sto) and shuttle routes (two fixed stos). ACKNOWLEDGMENTS This research was suorted by the Natural Sciences and Engineering Research Council of Canada. We also thank the reviewers for various constructive suggestions that have imroved the resentation of our aer. REFERENCES 1. Rosenbloom, S. Service Routes, Route Deviation, and General Public Paratransit in Urban, Suburban, and Rural Transit Systems. AZ U.S. Deartment of Transortation, FTA, Washington, D.C., Farwell, R. G., and E. Marx. Planning, Imlementation, and Evaluation of OmniRide Demand-Driven Transit Oerations: Feeder and Flex-Route Services. In Transortation Research Record 1557, TRB, National Research Council, Washington, DC, 1996, Durvasula, K. P., B. L. Smith, S. C. Brich, and M. J. Demetsky. An Investigation of Route Deviation Transit Service Design. Presented at the 78th Annual Meeting of the Transortation Research Board, Washington, D.C, Welch, W., R. Chisholm, D. Schumacher, and S. R. Mundle. Methodology for Evaluating Out-of-Direction Bus Route Segments. Transortation Research Record 138, TRB, National Research Council, Washington, D.C., 1991, Fu, L. Simulation Model for Evaluating Intelligent Paratransit Systems. In Transortation Research Record: Journal of the Transortation Research Board, No. 176, TRB, National Research Council, Washington, D.C, 21, Publication of this aer sonsored by Committee on Paratransit.