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Transportation Research Record (^818 )
Setting Frequencies on Bus Routes: Theory and Practice
PETER G. FURTH AND NIGEL H.M. WILSON
Since most transit systems have relatively stable route structures and politically determined levels of subsidy, one of the main recurrent decisions the transit planner must make is the service frequencies to be provided on each route in the system. Current practical and theoretical approaches to this problem are re- viewed and, in light of their seeming inadequacies, a new model for setting fre- quencies is developed. The model allocates the available buses between time periods and between routes so as to maximize net social benefit subject to con- straints on total subsidy, fleet size, and levels of vehicle loading. An algorithm is developed to solve this nonlinear program that can be applied by using a small computer program or, simplified in some generally acceptable way, by using a pocket calculator. In a case study the model is shown to produce results quite different from the existing allocation, which suggests changes that are insensi- tive to the specific set of parameters and objectives. It is shown that the model can readily be applied to evaluate the impacts of an alternative vehicle capacity and to investigate the value of changing service policies.
The North American public transit industry has, in the past decade, emerged from a long period of stag- nation and decline to become a major focus in stra- tegic planning to deal with the energy problem. In- creasing attention is being given to the problem of using the ever-mounting public resources being de- voted to transit more efficiently. This attention has revealed an apparent enigma: Although there is a wealth of academic research on how transit planning should be done, methods in use in the tran- sit industry are generally crude and dominated by the planners' experience and judgment, sometimes codified into simple rules of thumb. In this paper, one important part of the short- range transit planning process is selected and used to investigate whether significant differences exist between current practice and reasonable theory (]J. The topic is setting frequencies on bus routes, a problem that must be addressed, either explicitly or implicitly, several times each year by all transit operators. After a discussion of existing industry practice in setting frequencies, prior research is briefly reviewed. In light of the weaknesses iden- tified in this prior work, a new model is proposed that accurately reflects the objectives and con- straints with which the transit industry must deal. Finally, a case study of part of the Massachusetts Bay Transportation Authority (MBTA) system shows the differences between the actual allocation of buses and that suggested by the theory.
CURRENT PRACTICE
Methods used by schedulers to set frequencies on routes are generally poorly documented and seem to vary among operators. Typically, however, only a small number of rules of thumb have been used that can be overridden by the judgment and experience of the scheduler. The best way of assessing industry practice is to refer to the service standards that have been widely adopted by many operators in the past five years. Service standards cover a broad range of planning, operations, and management and (of interest here) usually include specific guide- lines on service frequencies. These service stan- dards are a result of both codification of existing rules of thumb and a statement of policy. As such they do not always accurately reflect decisions made by schedulers (and others) but are likely to include factors traditionally used in decision making. Based on a survey of existing service standards (2), the most frequently used methods for setting f-;equencies are policy headways, peak-load factor,
revenue/cost ratio, and vehicle productivity. of these is described briefly below.
Policy Headways
Each
Policy headways are used by virtually all operators and serve as a lower bound on the frequency. Routes are categorized by factors such as orientation (radial or crosstown), function (line-haul or feeder), and location (urban or suburban); and each category is assigned a set of policy headways for each period of the day. Policy headways are most effective in systems that operate principally as a low-demand social service. However, in large cities, during peak hours, and whenever demand is high, policy headways lose their relevance and other methods must be used to assign headways.
Peak-lDad Factor
The ratio of the number of passengers on board at the peak-load point to the seating capacity of the vehicle is widely used under heavier demand. A lower bound on frequency is based on maximum peak- load factors established by route category and time period. These factors are based on the physical capacity of the vehicle and on comfort and opera- tional considerations.
Revenue/Cost Ratio
The revenue/cost ratio is often used to define an upper bound on the amount of service to be provided on a route. This ratio is a rough measure of ef- ficiency and equity in the distribution of service and has the important advantage of being readily understood by both elected officials and the general public.
Vehicle Productivity
Either in the form of passengers per vehicle mile or per vehicle hour, vehicle productivity is also oc- casionally used to set upper bounds on the fre- quency. As in the case of the revenue/cost ratio, vehicle productivity is used to approximate the benefit/cost ratio of a specific service and to guard against inefficient allocation of resources. Al though service standards are an advance in the state of the art of transit planning, this brief re- view shows that they fall far short of ensuring that transit resources are allocated most efficiently (l). Specifically, the standards focus on upper and lower bounds for setting frequencies but say nothing about setting frequencies to maximize efficiency within these constraints. To better understand how frequencies are actually established, a set of MBTA routes was analyzed and a set of empirical relationships tested by linear re- gression. The 17 routes analyzed all belong to the Arborway Garage of the MBTA and include a wide variety: radial and crosstown, high- and low-fre- quency, that serve affluent and poor neighborhoods. Results show that the midday frequencies are heavily constrained by the policy headways; only four routes have a higher frequency. The following three empirical relationships for setting frequencies were tested:
2
F.qual load factor (2-h peak):
Q= b(PLP)
Q= a+ b(PLP)
Equal load factor (30-min peak):
Q= b(PLP)
Q= a+ b(PLP)
Square-root rule:
InQ =a+ bln(r/T)
where
a,b coefficients, Q scheduled frequency (round trips/h), PLP peak-load-point count (riders/h that pass peak-load point),
(1)
(2)
(3)
r = ridership per hour (total boardings in both directions) , and T =round-trip run time (min).
The most important results for the morning peak period are shown below (all estimates of coeff i- cients are significant at the 99 percent level):
Empirical Coeff:i.cients
Re 1:ationshi2 ~ £ R'
Equal load factor 0.024^ o.^85 (2-h peak) 1.54 0.020 0. F.qual load factor 0.018 0. (30-min peak) 1. 32 0.016 0. Square-root rule 0.43 o. 72 0.
It appears that existing frequencies are very well explained by setting the peak-load factor equal on all routes, particularly during the peak half-hour. The average peak load on these routes during the peak half-hour was 1.2, which is about 13 percent below the policy peak-load factor of 1.4. This case study suggests that schedulers do fol- low a clear decision-making process, which revolves around the rule of an equal peak-load factor. These results are strikingly similar to those found by Morlok in Chicago (_!) i the important point is that in both cases the equal peak loads were signifi- cantly below actual bus capacity. As demonstrated in the next section, this fact makes the rule inef- ficient with respect to the optimization of passen- ger service.
PREVIOUS THEORY
The best-known theory for setting frequencies on bus routes is the square-root rule, which is based on the minimization of the sum of total passenger wait- time costs and total operator cost. In the general case when routes of different lengths exist, the rule states that the service frequency provided on a route should be proportional to the square root of the ridership per unit distance (or time) for that route (_~). Major weaknesses of the square-root rule, which explains its lack of acceptance by the industry, are that it does not consider bus capacity constraints and that it assumes that ridership is fixed and in- dependent of the service frequency. Ignoring the capacity constraint means that on some heavily used routes not enough capacity will be provided (i.e., the solution is infeasible). The assumption of fixed demand means that the user benefits are limited to minimization of wait time, which is probably only a minor part of the public benefit of transit service. A second, almost trivial, theory is that if the
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objective is simply to minimize operator cost, the frequencies should be set so that the capacity will equal the peak load on each route. If the system is at capacity, each route will have the same peak-load factors, but if the system is operating below capa- city, efficiency arguments do not lead to equal peak-load factors. Guinn used a linear-programming approximation to allocate buses to maximize revenue subject to a fleet-size constraint. Although his objective and constraint set are too approximate for direct appli- cation, the model presented later in this paper uses the same general optimization framework. Scheele Ill proposed a more complex mathematical programming approach to determine optimal service frequencies in the long-run case in which the distribution of trips (but not total trip generation and production) is allowed to vary in response to the service provided. Several models have been developed for the simul- taneous choice of routes and frequencies (8-10). The frequency components of these models typically minimize passenger wait time subject to capacity constraints under an assumption of fixed demand. Recent work at the Volvo Bus Corporation (11) has resulted in a package for choosing routes and fre- quencies that has been successfully applied in numerous cities in Europe and elsewhere. Most of these models and theori.es are designed for one-time application when the entire transit network is redesigned--by definition a major and in- frequent undertaking. Furthermore, only one of these models has been applied frequently, and none has been accepted for use by transit operators. This is both because of their orientatidn to large- scale system change and because they are either complex and hard to use or crude and hard to be- lieve. There is need for a model that accurately reflects the frequency-choice decision, that is simple enough in its data and application require- ments to be used frequently by operators, and that focuses on small changes so it can be applied re- peatedly over the years. Such a model is developed in the remainder of this paper.
PROPOGED MODEL
The fleet-allocation problem can be formulated as an optimization in which an objective function is maxi- mized (minimized) subject to a set of constraints. Before the formulation is presented and discussed in detail, however, it is useful to illustrate the style of solution by using a simple example. Suppose that a bus company operates three routes, charges a flat fare per passenger, and has allocated a fixed amount to cover the deficit that will result from providing the service. The single objective of the company is to maximize ridership by means of al- locating buses, given that fares, routes, and operating speeds are fixed. The problem can be viewed as a resource-alloca- tion problem: How can the limited resources (sub- sidy) be allocated to maximize the benefit (rider- ship)? As shown in Figure 1, for each route a curve that relates net cost (deficit) to benefit can be obtained by varying the frequency of service on the route. At an optimal allocation, the ratio of mar- ginal benefit to marginal cost should be the same for each route. Denoting the benefit of route i by Bi, its net cost by Ci, and its frequency by Qi, this rule can be written as follows:
(dB;/dQi)/(dC;/dQi) = (dB;dC;) = (B/C) 0 (4)
As suggested in Figure 1, at the optimum some routes may be operating at a profit and others at a loss; however, the total benefit cannot be increased by
4
Headway:
h;;" X;;, j =I, - - - • p i= 1, ... , N;
where
number of time periods, number of routes operated during time period j, duration of period j, value of wait time, headway on route i during period j, surplus marginal ridership benefit on route i during period j (marginal rider- ship benefit minus fare), ridership on route i during period j (a function of hijl, operating cost per run on route i during period j, c fare on route i during period j, n subsidy available, run time (round trip) on route i during period j, fleet size during period j, and maximum headway for route i during period j.
(9)
The objective function (Equation 6) can easily be shown to be equivalent to maximizing the wait-time savings plus the social-ridership benefit minus operating costi this is the net social benefit. Equation 7 simply states that the operating cost minus the revenue must be equal to the known sub- sidy. Equation 8 is the fleet-size constraint, and Equation 9 constrains the headway to be less than the policy headway and the headway at which the loading constraint is binding. This;; general formulation could be simplified or made more complex (for example, by defining classes of riders, each of which has a separate marginal benefit) in specific applications, but all important facets of the problem are included. Before the method developed to solve this mathematical program is presented, it is necessary to recognize and dis- cuss perhaps the most important limitation of the model--the assumption of the independence of all routes in the system. It is because both costs and benefits due to a headway on a specific route have been assumed inde- pendent of headways on other routes that the problem formulation is so straightforward, but this assump- tion is not always true, at least on the benefit side. In this model, ridership on a route depends on the headway of only that route, whereas, in general, ridership will also depend on the headways on competing and complementary routes. When passengers have a choice among several routes, an improvement in service on one of those routes will divert riders from the other routes. Such route competition is less common in North America than in other parts of the world, in which an approach that directly considers route competi- tion is called for <1JJ. An improvement in service on one route can also raise the demand on another route when there is a large transfer volume between the two routes. Care must be taken, therefore, both in applying the model and in interpreting its re- sults in situations in which strong route competi- tion or complementarity exists.
THE ALGORITHM
Optimality (Kuhn-Tucker) conditions can be derived as a set of equations that relate headways to the
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other variables in the modeli these equations then become the optimal decision rules for the operator. These optimality conditions are applied in the fol- lowing step-by-step algorithm to determine the opti- mal set of headways by route and by period:
Step 1: Relax the fleet-size and maximum-headway constraints on all routes and for all time periods not yet constrained and solve the following set of equations for the headways, hij'
(10)
where X is determined to exhaust the available subsidy. If no routes violate their maximum-headway constraint, go to step 3. Step 2: For routes and periods for which the maximum-headway constraint (Equation 9) is violated, set hij = Xij• Compute the deficit incurred on those routes and reduce the availabl e subsidy by that amount. Go to step 1. Step 3: Identify time periods in which fleet-size constraint (Equation 8) is binding. each of these time periods solve the following of equations:
the For set
(11)
where wj is the shadow price of run time during period j and is determined to use all available buses. Step 4: If no routes violate their maximum-head- way constraint (Equation 9), go to 15tep S. Other- wise, for every route that violates its maximum- headway constra int , set hil' = Xij• Compute the number of buses requ ired by a 1 such rou tes in each period j and reduce the number of available buses in period j by this amount. Go to step 3. Step 5: Compute the deficit incurred by the fleet-constrained time periods and reduce the avail- able subsidy by this amount. Let Xe = x. Step 6: Repeat steps 1 and 2 for the uncon- 15trained time periods to find a new value of X, which is Xu. Step 7: If Xu:: Xe' stopi otherwise set X = Xu and return to step 3.
The theory behind this algorithm will not be pre- sented in detail here (1) • However, the computa- tional burden of the alg;rithm is very small, since it consists basically of a sequence of one-dimen- sional searches that are performed very rapidly. Equations 10 and 11 can be solved very efficiently by using the Newton method (provided the demand function has continuous second derivatives), and values of A and Wj can be found by making suc- cessive linear app roximations.
CASE STUDY
The Arborway Garage of MBTA, which serves 21 bus routes, was chosen to illustrate the capabilities of the model. Fifteen of these routes were included in the analysis i the others were excluded for one of the following reasons: incomplete ridership data, highly irregularly scheduled runs, and interdepen- dence of routes. The most important data and assumptions made in the study are summarized here:
- Two time periods were examined--the morning peak (7-9 a.m.) and midday (10 a.m.-2 p.m.). Scheduled headways were approximately constant dur- ing each of these periods.
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- Scheduled round-trip running times were used with a layover time of 25 percent of the run time. This is above MBTA's policy of 10-20 percent of run time for layovers but slightly below the average ob- served figure of 28 percent.
- Costs per run were based on MBTA's figures of cost per vehicle mile and per driver hour.
- The systemwide average revenue per bus ride of 18¢ was used.
- The current deficit incurred on this part of the system was used as the available subsidy, and the current number of vehicles used in the morning peak was used as the fleet-size constraint' in each period.
- Policy headways of 30 min in the peak period and 60 min in the off-peak period were used based on current MBTA service policies.
- MBTA uses two peak-period load-factor stan- dards: For the 2-h peak period, the peak-load fac- tor: should be no greater: than 1.2: for: the peak half-hour: it should be no greater: than 1. 4. Actual pea k loads were estimated for: both periods for: each route based on peak-point counts taken over a three- year: period. Only one of these load-factor constraints will be binding for: each route. The off-peak policy load factor of 1. 0 adopted by MBTA was also used.
- Bus seating capacity of 45 was used for the load-factor constraints.
- Current route ridership was taken from a 1978 on-board survey.
More-detailed discussion is warranted about the demand model and the operator's objectives. A binary legit demand model was used that has assumed coefficients for wait time taken from another study. Estimates of the base transit market share
Table 1. Frequency on case-study routes: (^) Actual actual and recommended. Peak Frequency \l,.h^ Load
5
were also made based on mode-split characteristics of the Boston area. These assumptions implied wait- time elasticities of demand of -0.2 in the peak period and -0. 5 in the off-peak period, which ar:e within the range observed in other U.S. cities (14). It is hoped that advances in the state of the art of demand forecasting at the route level will soon obviate the need for: such assumptions. In this case study, sensitivity analyses demonstrated that the results were very robust with respect to these parameters. The model allows an objective function that con- sists of a weighted sum of total passengers and total passenger wait-time savings. The absolute coefficients of these terms do not have to be exo- genously specified, but their ratio does. The ini- tial ratio chosen implied a tr:ade-of f of one passen- ger for 12 passenger-min of wait time. Table 1 shows the resource allocation between routes and between periods as suggested by the model compare d with the current MBTA allocation. The re- sults in terms of deficit, number: of buses, and changes in wait-time and ridership benefits are given in Table 2. The most striking result is that only 59 of the 70 available buses ar:e used in the peak period, and the peak period's share of the deficit declines accordingly. Only 44 percent of the total subsidy is allocated to the peak period by the model compared with 58 percent in the current system. The peak period is heavily constrained by capacity: nine routes operate at the maximum load during the peak half-hour. In general, the shorter routes have the smallest loads and so do not neces- sarily have the highest revenue/cost ratio. As ex- pected, midday loads are much lower: than those dur- ing the peak period. Several factors contribute to this large shift in
Recommended
Peak Revenue/Cost Frequency 'h-h Load Revenue/Cost Route (buses/h) (passengers) Ratio (buses/h) (passengers) Ratio
Morning Peak
21 5.0 63' 0. 55 5.0 63' 0. 24 4.0^42 0.50^ 4.0^42 0. 25 5.0 35 0.37 3.7 44 0. 28 3.0 54 0.50 3.6 48 0. 29 13.3 57 0.60 12.0 63'^ 0. 31 4.0 39 0.3 0 2.9 49 0. 32 15.0 60 0 .52 14.2 63' o. 35 5.0 53 0.32 4.0 63 3 0. 36 10.0 51 0.49 8.0 63' 0. 37 5.0 54 0.42 4.2 63'^ 0. 38 5.5 34 0.39 2.6 59 0. 41 6.0^56 0.67^ 5.3 63'^ 0. 46 2.0 29 0.63 3.1 21 0. 50 3.3 59 0.39 3.0 63'^ 0. 51 4.0 SS 0.26 3.3 63' 0.
Midday
21 1.3 13 0.22 1.8 13 0. 24 1. 5 15 0.37^ 2.5 13 0. 25 3.0 11 0.35 3.0 II 0. 28b (^) 0.0 0. 29 5.0 38 0.85 5.6 35 0. 31 1.5 13 0.58 3.4 9 0. 32 4.6 28 0. 60 4.7 28 0. 35 2.0 33 0.48 3.2 26 0. 36 2. 0 40 0.57 3.5 30 0. 37 2. 0 24 0.44 3.0 20 0. 38 2.7 15 0. 20 2.0 17 0. 41 3.5 24 0.66 4.2 21 0. 46 2.0 7 0.1 8 1.5 7 0. 50 2. 0 20 0.28^ 2.3 19 0. 51 2.0 21 0.23 2.0 21 0. (^8) Capacity constra in ed. bRoute 28 is nol operated in the orf-p eak period.
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Transport: Initial Concepts and Model. In Traffic F.quilibrium Methods (M.A. Florian, ed.), Proc., International Symposium on Traffic F.quilibrium Methods, Montreal, 1974, Springer Verlag, Berlin, 1976, pp. 322-367.
- H. Mohring. Optimization and Scale Economies in Urban Bus Transportation. American Economic Review, Sept. 1972.
- C.R. Guinn. Transit Optimization--Test 1 Analysis; Transit Optimization--Test 2 Analy- sis. New York State Department of Transporta- tion, Albany, Preliminary Res. Repts. 2 and 12,
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Planning the Route System for Urban Buses. Computers and Operations Research, Vol. 1, 1974, pp. 201-211.
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Publication of this paper sponsored by Committee on Traveler Behavior and Values.
Strategies for Improving Reliability of
Bus Transit Service
MARK A. TURNQUIST
Four major classes of strategies for improving reliability of bus transit service are analyzed: vehicle-holding strategies, reduction of the number of stops made by each bus, signal preemption, and provision of exclusive right-of-way. The principal findings are that (a) strategies to improve service reliability can have very substantial impacts on overall service quality, including improvements in average wait and in-vehicle time as well, and (b) the best strategy to use in a particular situation depends on several factors, but service frequency is the most important. For low-frequency services (less than 10 buses per hour), schedule-based holding strategies or zone scheduling is likely to work best. For midfrequency services (10-30 buses per hour) zone scheduling or signal pre- emption is likely to be most effective, although headway-based holding can also work well if an appropriate control point can be found. In high-frequency situations (more than 30 buses per hour), an exclusive lane combined with signal preemption should be considered.
The concept of service reliability has come into in- creasing prominence in recent years as an important characteristic of the quality of service provided by transportation systems. A basic definition of re- liability, as the term is used here, is the var ia- bili ty of a system performance measure over time. The focus is on stochastic variation in performance rather than on more-traditional engineering concepts of probability of component or system failure. The level-of-service measure most clearly subject to variation is travel time, and this variability is often described in terms of nonadherence to schedule. Service reliability is important to both the transit user and the transit operator. To the user, nonadherence to schedule results in increased wait time, makes transferring more difficult, and causes uncertain arrival time at the destination. The im- portance of some measure of reliability to trip- making behavior has been emphasized in several atti-
tudinal studies. For example, Paine and others !1)
found that potential users ranked "arriving when planned" as the single most important service char- acteristic of a transit system. This finding has been substantiated in further studies by Golob and others (~) and by Wallin and Wright (l). In addition to its importance to transit users, unreliability in operations is a source of reduced productivity and increased costs for transit opera- tors. This is due to the need to build substantial slack time into timetables in order to absorb devia- tions from the schedule. This leads to reduced use of both equipment and personnel. The recent report by Abkowitz and others (!) provides an excellent summary of the major issues in transit-service reliability from the perspectives of both the user and the operator. In light of the current need for more cost- effective public transportation in urban areas, it is important to understand the sources of unrelia- bility and to investigate the potential of several alternative control strategies to improve both the quality of service provided and the productivity of the equipment and the personnel in the system. The research on which this paper is based has had four major objectives:
- Investigation of the sources of service- reliability problems in bus transit networks,
- Identification of potential strategies for improving reliability of service,
- Development of models to allow these strate- gies to be analyzed and evaluated, and
- General evaluation of the relative effective- ness of these strategies.
