8 TRANSPORTATION RESEARCH RECORD 1221 Optimal Design of Transit Short-Turn Trips A VISHAI CEDER A set of procedures is presented for efficiently designing transit lines (crosstown routes, downtown-oriented routes, feeder timetables with trips that are initiated beyond the route departure routes, etc.). point, or terminated before the route arrival point, or both ("short­ The major objectives set forth herein are as follows: turn trips"). In practice, transit frequency is determined at the route segment with heaviest load whereas at othcl' segment · the • To identify feasible short-turn points based on passenger operation may be inefficient because of partial load (empty seats). load profile data; Transit schedulers attempt to overcome this problem by manually • To derive U1e minimum fleet size required to carry on a constructing short-turn trips to 1·cduc the number of vehicles given timetable (including the consideration of deadheading, re<auired to carry out U1e transit timetable. The study presented i.e., nonrevenue trip ); herein was meant to improve and automate thi task by identi­ • To adjust the number of departures at each short-turn l"ying feasible short-turn points de.riving lbe minimum fleet ize required by a given chedule, and adjusting the number of depar­ point to that required by the load data, provided that the tures at each short-turn point to that required by lhe load data maximum headway to be obtained is minimized (this objective (provided 'lltat the maximum headwa associated with passenger results in the maximum possible short-turn trips and the min­ wait lime i minimized). Other object.ives included minimizing imum required fleet size); lhe number of short-tum trip · while en uring I hat the minimum • T minimize the number f short-turn trips, provided Oect size i preserved and crcafotg vehicle schedules (blocks). that the minimum fleet ize i maintained (for a given time­ simple example is used throughout to illustrate the procedures table chis bjective r ults in increasing the level f ervice developed. seen by the passengers); and • To create vehicle blocks for the final derived timetable The first phase of this research, which has been completed (a block is a sequence of revenue and nonrevenue activities and documented (1, 2), provides procedures for using pas­ for an individual vehicle). senger load data to derive alternative timetables along an entire transit route, without short-turn trips. A short-turn trip To satisfy the. e objectives, several methods were developed. begins beyond the route departure terminal or is terminated These methods are ba ed on procedures and algorithms that before the route arrival terminal, or both. The possibility of use data commonly inventoried or collected by most transit generating short lines permits further saving of vehicles while properties. Furth (4) uses origin-destination (0-D) data to ensuring that the passenger load in each route segment will asse hort-tum ·1rategies for route .16 in Los Angele. (SCRTD) not exceed the desired occupancy (load factor). between West Hollywood and downtown. Although use of Schedulers at most transit properties usually include the 0-D data ca n improve the cheduling of hort-tmn trips, this short-turn operating strategy in their efforts to reduce the cost informati n i n t c mmonly available al transit agencie . The of service. The procedures commonly used are based only on current work is not based on 0-D data, but its methods can visual observation of the load profile, that is, the distribution be extended to include such data whenever they are available. of the loads along the entire route. A potential turn point is determined at the adequate time point (major stop) nearest to the stop at which a sharp decrease or increase in the pas­ senger load is observed. Although this procedure is intuitively APPROACH AND BACKGROUND correct, the schedulers do not know if all the short-turn trips are actually needed to reduce the fleet size. Unfortunately, Framework each short-turn trip limits service and hence tends to reduce the passenger level of service. The initial information required for constructing the short­ Furth et al. (3) presented an overview of operating strat­ turn trips includes egies on major downtown-oriented bus routes. Among the strategies discussed were short-turn trips, in which the service • A complete timetable for each route timepoint; trip begins farther along the route, but the arrival point of all • Passenger loads for each time period across all time­ the trips is the same. The present work designates all the points; possible categories of short-turn trips for any type of transit • Minimum frequency or policy headway; and • A set of candidate short-turn points. Department of Civil ngineering, Transponmion Research Institute, This information is given for both route directions (each direc­ T chnion-1 rael Institute of Technology, Haifa. Israel 32000. tion require its own data). The complete timetable can be Ceder 9 Initialization : • Timetable (set of departure times at all timepoints) • Passenger Loads (for each time period across all time points) • Minimum Frequency (policy headway) • Set of candidate short-turn po ints • Creation of R = set of feasible short-tum points, for each time period .,, ~ construction of deficit functions at the end points (departure and arrival termi nals ) Result: Deficit A procedure to insert deadheading trips I+ i1in. Fleet Function size ~·1i thou t tneory and to minimize the route fleet size short turns, algorithms ... '-i_!..,J..1;·.:,:n..___ -' Determination of the required number of 11~ departures (max. load/desired occ.) for each timepoint ER Vj Network­ based A procedure to exclude departure times from each timepoint minimax H £R Vj ~xceot the max. load points. It is carried out to theory and algorithm ]"-+ minimize the maximllll obtained headway while reducing the '--- - --"' number of departures to that required. Result : A ~ew timetable with the maximum possible short-turn I trips Theory and I Construction of deficit functions at each timepoint £R Vj algori thms to maxi mi z1 t the ~A procedure to insert deadheading trips to minimize the sum of extensions l all the maximal deficit function values of short tum! I tC===::::oi11 _____ _ ___ ..... 0 Procedures Apply the procedure to extend ~< II n Min . fleet size with short turns, N . to construct " the short turns toward their ? mi n. vehicle ?rigina~ tri ~s without schedules 1ncreas1ng mi n. 0 t --~--~~-t Resul t: final optimal Tiiiieta'b le with short turns I Result: a timetable Crea ti on of vehicle blocks I 't without any short .__ _______ _t~~l-----"-----------------4..._tu_r_n_.s_·--- --........J FIGURE I Flow chart describing the design of transit timetables and vehicle schedules with short turns. provided by the scheduler or may be derived from the pas­ number of vehicles required to carry out all the trips in the senger load information (1, 2). Candidate short-turn points complete two-direction timetable, Nmin· The required number are usually all the major route stops (timepoints) at which a of departures is determined at each feasible short-turn point, public timetable is posted. In some cases, the scheduler may and then the so-called minimum H algorithm is applied. The limit the candidate points to only those timepoints at which basis of the algorithm is the elimination of some departures the vehicles can actually turn back. from the complete timetable to obtain the number of depar­ The overall program to accomplish the objectives of this tures required. In that procedure, the algorithm minimizes work is presented in flowchart form in Figure 1. It starts with the maximum difference between two adjacent departure times a procedure to determine the set of feasible short-turn points, (headway). At this stage, as shown in Figure 1, the deficit Ri, among the candidate points. Then rile deficit function function me!!!od derives the minimum required fleet size with theory, as explained below, is used to derive the minimum short turns, Nmin· If this minimum is less than the size required 10 TRANSPORTATION RESEARCH RECORD 1221 without short turns, then another procedure is applied. This by Bartlett (5), Gertsbach and Gurevich (6), and Salzborn second procedure in. e1 ls the m11ximum po· !Ille departures (7, 8). Mathematically, for a given fixed schedule: back among those p~viously eliminated, pr vided that the N = L D(k) = L max d(k, t) (1) minimum fleet size, Nm'"' is maintained. The final step of kEE kEE t the overall program is to create vehicle blocks to cover all the trips that appear in the last version of the two-direction where N is the minimum number of vehicles required to ser­ timetable. vice the set E. When deadheading (DH) llips a1e alluweu, the fleet size may be reduced below the level described in Equation 1. Deficit Function: Background Ceder and Stern (9) describe this procedure in the construc­ tion of a unit reduction deadheading chain (URDHC). Such The deficit function approach for assigning the minimum num­ a chain is a set of nonoverlapping DH trips that, when inserted ber of vehicles to carry out a given timetable can be described into the schedule, reduces the fleet size by one. The procedure as follows. A deficit function is simply a step function that continues to insert URDHCs until no more can be inserted increases by one at the time of each trip departure and decreases or until a lower bound on the minimum fleet size is reached. by one at the time of each trip arrival. Such a func1ion may [determination of the lower bound is detailed in the work of be constructed for each terminal in a multi-terminal transit Stern and Ceder (10)].