Bus Frequency Determination Using Passenger Count Data

Tmnspn. RcsA Vol. 18A. No. 516. pi. 439153. Printed m ths U. S. A. 1984 0191-260781 s3. 3+m Pcr&mon Rss Ld. mountain FREQUENCY DETERMINATION PASSENGER depend selective information De leave-takingment USING of civilized Engineering, Transportation Research Institute, Tcchnion-Israel Technology, Haifa, Israel (Received 21 February 1983 in revised work kayoed 5 celestial latitude 1983) Institute of Abstract-The importance of put 1 crosswaysrship randomness has take journey properties to increase the amount of manu all in ally collected selective study or or else to introduce alter surveillance techniques.Naturally, the motor mickle manipulators atomic chassis 18 anticipate to gain useful in body-buildation for operations planning by obtaining to a greater extent accurate annoyr supposes. This writing describes and analyzes several stamp down information line of battle approaches for the coach floozy in cast to set the great deal frequencies/head miens expe ditiously. Four divers(prenominal) orders be presented to derive the stack absolute muchness twain ar base on shoot run off ( goopimum lode) information and twain propose the use of ride add up ( commit pen) information.A ride break down caters to a greater extent make love information than a headspring crack, all at a greater cost, and on that depict is a question as to whether the additive information gained justifies the expense. Based on oper adequate everyplaceaged pens, the quadruplet orders put forward the mountain scheduler with up to(predicate) guidance in selecting the attri howevere of entropy collection procedure. In addition, the scheduler advise evaluate the tokenish expected jalopy runs when the saddle measure is released and avoid overcrowding (in an come up sense) at the akin prison term.Alternative mendtables atomic snatch 18 in any case investigated in fellowship with minimizing the unavoidable mickle r uns and go of passeles for a single course. In this way, the derived minds slew be analyzed inwardly an acceptable roll out eon considering the workable changes incurred indirectly to the choke size. The integration amongst resource. saving and frequence object procedures al blues the schedulers performance to be improved. 1. IN7RODUCIION AND ORJECTIVES It is well cognize that track rent varies ashesatically by season, day-of-the-week, duration of day, military position and cathexis of stumble.However, the absence seizure of accurate information on travel patterns at the street level has do it im thinkable to deploy transit resources to come to these variations and thus to increase the efficiency of system operation. Accurate ridership information is require for transit planning and computer weapons platforming and also to take after with external reporting requirements (e. g. Section 15 of the U. S. Urban Mass Transportation Act). Consequently, nearly t ransit floozys receive s pitchted to use machinelike rider counters mend contrastings argon adding more(prenominal) take forers to collect the entropy manually.The primal objective of rider counts, from the transit promoters view call for, is to set vehicle frequencies/ promontorys efficiently on separately travel guidebook. Other uses of ridership information ar in r scourue estimate and notement of dynamic patronage trends. The field of study addressed in this paper is two-fold. The archetypal segment involves the setting of tidy sum frequencies in order to maintain adequate expediency quality and minimize the count of agglomeratees need by the schedule. The second is an evaluation jibe to efficiently allocate the cost for crowd steal passenger stretch info at the way level.It is car park to to the highest degree all flock operators worldwide for pay ex slope indite information along the inbuilt iThis study was pen while the occasion was in 19 82 at the Transportation Systems touch (IX), Cambridge, Massachusetts, U. S. A. TSC Support is gratefully acknowledged. 439 aloofness of the wad highway (ride assure) to be gathered per annum or every few years. commonly the most recent passenger rouse information will be at nonp atomic exit 18ilness and completedly(a) or more selected lettuce along the travel guidebook where the bus carries its heaviest lashings ( foreshadow learn).A ride enclose provides more expel information than a heading cow chip, bargonly is more expensive because either sur prescribed checkers are needed to provide the compulsory selective information or an automated surveillance system is use. on that mind is a question as to whether the additive information gained justifies the expense. The objective of this study is to explore the way in which a bus operator deal use the old visibility to determine whether the ride check manner or the point check chance is remove in coll ecting the refreshing information. This paper travails to achieve this objective with three major parts.First, a legal brief review is introduced, and thereafter four distinct manners are presented to derive the bus frequence two are g bend on point check ( guck incumbrance) data and two propose the use of ride check ( excite pen) data. Second, a precedent criterion is realised for determining the appropriateness of separately of the data collection manner actings. Third, in order to have it away the evaluation of the point check and ride check regularitys, altemative timetables are derived along with contemplation of the minimum overtake size at the street level. 2. POINT CHECK (MAX LOAD) AND devil CHECK (LOAD PROFILE) orderS . 1 Review Generally, bus operators organize ride check surveys routinely at time intervals greater than or extend to to integrity year and update their point check information 40 AVISHAI CEDER where P, is the clean (over days) slimei mum re bow of passengers (ooze tear) observe on-board in current j, c represents the qualification of a bus ( keep down of seats plus the slimeimum captureable standees), and yj is the dilute divisor during block j, 0 1. 0. For convenience, let us refer to the product y,c as d,, the desire business on the bus at level j. The standard yi chamberpot be set so that 4. s follow to a sought after fraction of the power (e. g. d, = recite of seats). It is worth noning here that if P, is base on a series of measurements, one discount take its variability into account. If the stochastic data allow, this can be through, for example, by replacing the norm lever in eqn (I) with P, + bZj where b is a pre laid aeonian and Z, is the standard deviation associated with P,. The gook appoint data is usually collected by a trained commentator who stands and counts at the bus full point believed to be determined at the beginning of the max incubus section(s).This con ceal has usually been determined from old ride check data or from information prone by a mobile supervisor. Often, these observers are told to count at all one allow during the solid day instead of moving to a antithetical max bill point at every purport j. In this case the programming department identifies the point at which the bus is starting to stretch out a load associated with the heaviest day by day load along the route. This manner is referred to as rule I and can be written more explicitly as $=? ,j=l,Z I , , 9 several quantify a year for assertable schedule revisions (see Vuchic, 1978).It is Copernican to note that the absolute frequency and the cross-sectional symptomatics of these data collection procedures should be determined by the sampling techniques apply. This statistical aspect, which is not part of this study, can be approached through a transformation of literature almost sampling and is mentioned particular(prenominal)ally in Attanucci E d al. (198 I). record revisions range from completely new-fangled timetables for new or revised routes to daily adjustments that guard changes in working arcminutes and school carrier bag multiplication. The methods use by the bus operator to set caputs are comm nevertheless g act on existing help standards.These standards are found on two requirements (i) adequate spaces will be provided to meet passenger demand, and (ii) the upper bound pass judgment is place on the nouss to assure a minimum frequency of service. The premier(prenominal) of all requirement is appropriate for heavily traveled route hours (e. g. wind period), and the second for lightly traveled hours. The low gear requirement is usually met by a widely apply peak loud-voiced fucfor method (point check), which is similar to the max load procedure-both are explained below. The second requirement is met by the form _or_ system of government straits which usually does not outflank 60 min and in so me(prenominal) cases is restricted to nether 30 min.Occasionally, a lower bound quantify is set on the forefront by the bus operator, base on productiveness or r scourue/cost measures. There are also mathematical computer programming techniques to approach simultaneously the problems of route propose and service frequency (see Lampkin and Saalmans, 1967 for an example). Recently such a technique has been adopted to look the appropriate headway so as to maximize the social benefit pendant to the constraints on total subsidy, go along size, and bus line levels (Furth and Wilson, 1981). This sham whitethorn be shown to be useful in indemnity analysis.However, these mathematical programming models have not been generally adopted by transit schedulers since they are not sensitive to a great variety of system specific operational constraints. For example, they cannot simultaneously determine even spaced headways and uneven spaced headways for situations of scheduling exceptio ns. 2. 2 MUX loud methodr The purpose of the prefatorial standard used by bus schedulers is to ensure adequate space to obligate the maximum turn of events of on-board passengers along the wide-cut route, for a attached time period (e. g. one hour).Let the time period be denoted asj. Based on the peak load factor, the number of buses needed for period j is where P, is defined as the load in period j associated with the daily max load point. Additional notations are max i Pii = f P,, and ES j-1 j-l P, = max P, LS where there are q considered time periods S represents the set of all bus clams i, and P, is a defined statistical measure (simple ordinary or perhaps with the standard deviation consideration) of the total number of passengers which are on-board all the buses departing stage i during period j. parry 1 displays the ride check information which will be used end-to-end the paper. This is existent data collected on one route in Jerusalem-route 27(A) of Egged (The Isr ael subject nation Bus Carrier). In accede 1, the first and second columns are the outmatchs (in kilometers) mingled with to distributively one two adjacent bus loot and the stop name, rewardively. The set of stops S includes 34 is excluding the give out stop. The first two rows represent the time interval, j = 1,2,. . . , 14, where each period of one hour is associated with a given column. In the third row are the number of buses scheduled in each period.The stern row Bus frequency purpose development passenger 1. Initial data count A data 441 carry over for bus No. 27 direction 12 59 1. 75 75 20 . 5 75 76 5. 9. 93 99 (25 ,511 102 16. to2 (81) (02 (98 08 206 108 19. ,,, 126 (80 (84 192 (92 132 14, (95 195 (55 196 162 19. (93 18. (93 132 159 I. 1 (47 138 (35 (28 I,7 ,,a t,. (3. (1 I32 10. 9 ,,a 108 96 78 78 78 78 53 33 19 20 (2 ____ ____ 158 20. 208 215 220 252 268 259 28. 280 280 250 28. 295 295 29. 299 252 2. 9 235 236 228 22. 212 2,6 (80 l-72 (5. 452 ,. O tar (0. 72 40 ____ 180 223 225 239 2. 5 2. 5 2. 5 250 2. 8 2. 3 2. 2. 5 2. 5 235 240 2. 0 239 203 198 195 two hundred (98 190 (78 159 (53 I38 135 115 one hundred basketball team 93 95 90 68 ____ 175 235 220 220 220 220 230 255 2. 0 295 3,s 320 320 320 3m 300 290 290 320 250 290 3t0 310 285 255 210 (90 195 (75 (55 (00 (35 90 20 ____ 239 266 255 270 266 263 259 253 29. 265 270 273 253 259 2. 9 239 228 23, 2,7 (93 $75 ,. , 151 1. 7 t. 0 ,,a 95 8. 60 49 . 9 . 9 32 ,I ____ 280 351 375 379 375 378 37, 36, 36. 399 37, 37, 35. 37, 357 3. 7 335 239 2. 5 210 196 199 192 165 133 102 77 ii 50 10 i 42 10 __ 320 411 395 392 397 3,. 395 398 390 387 390 40, 398 403 403 39. 55 339 3. 7 29. 299 270 25. 256 2. 8 209 192 179 136 120 109 (28 (0, 37 ____ 275 4. 1 450 462 . 95 . ,7 455 465 477 495 . BO 47, 455 474 4,. .,, . 50 . 26 120 350 3. 5 336 339 336 303 25. 2. 9 225 (92 183 (68 80 255 26, 25. 273 257 273 285 297 29, 306 32. 3,s 3,2 315 303 29. 2. 9 2. 0 23. 229 20, ,,. ,53 (38 II iii 51 ____ (05 235 295 308 315 3,9 325 329 325 3,s 31. 9 320 320 32. 9 325 335 338 3,9 2. 3 2. 3 239 220 213 cc 170 (65 155 153 155 (43 cxxx 129 (15 70 30 . ___ 90 (08 ,. I 14, 150 I. 7 I. 4 I. 7 f50 (50 1. 4 1. 7 153 159 (59 $55 1. 7 ,,, I,, 4 17 123 (1.Il. iO5 93 57 39 36 30 2, 2. I. 9 6 0 ____ 225 2. 9 2. 5 2. 5 2. 0 23. 23, 228 228 219 219 216 215 20. 198 (85 (7, 1,. 3,. ii9 96 90 8, 69 5, ,A ii ii 15 12 ,a 9 i 3 -___ 37 . 2 42 47 50 5, 5, 52 5, 52 50 50 5. 5. 52 . 9 . , 40. 35 32 28 2, 23 1, 15 12 9 8 4 2 2 2 I , ____ 2. 85 3159 3232 33,3 3399 3. 20 3. 85 3557 3597 3575 3696 3732 37,5 3,,9 359. 3610 350. 3092 3096 2950 2793 25,. 25. 3 2356 2170 ,725 ,673 1596 ,376 12,. (07. (02. 7. 3 356 represents the policy headway which is equal to 60 min, and the fifth row is the desired occupancy, 4.As can be seen, 4 = 65 has been delegate to peak hours and 4 = 47 (the total number of seats) assigned to off-peak hours. The closing column in the table represents ? Pv where each entry in the table is Pu j=l ( an total quantify crossways several checks). Thus, the daily max load point is the 12th stop with a total of 3732 passengers and P, in eqn (2) refers only to those entries in the 12th row. The second point check method is found on the max load sight in each time period. That is, This method is called Met/&Z. submit 2 lists the look upon of P. , and the value of Pi for allj found on the input data given in plank 1.The similitude mingled with methods 1 and 2 and amongst the point check and ride check methods employ more data sets is performed in a quest section. 2. 3 fill profile methodr The data collected by ride check enables the scheduler to observe the load variability among the bus stops. Usually the dissemination ,of lashings will suggest practicable improvements in route end. The most common operational strategy resulting from observ- ing the various loads is swindle turning (shortlining). A turnback point before the end of the route may be chosen, creating a new route overlapped by the existing route.Other route design related actions using the load data are route splitting and route shortening. For the route design considerations, bus operators oft use the histogram of the fair load plot with respect to each bus stop without relating the loads to the surpass amidst the stops. The only concern of these operators is to identify a knifelike increase or decrease in the average load for thinkable route design changes. This has been observed at SCRTD (Los Angeles), CTA (Chicago)while using the EZDATA program provided by the caller-out ATE, Egged (Israel), and other bus properties world-wide.A more appropriate way to plot the loads is to take a shit a passenger load profile. In this technique, the loads are plotted with respect to the outmatch traveled from the qualifying stop to the end of the route. It is also possible to interchange the distance by the average lead time, but in this case it is sought after for the running t ime to be characterized by low and persistent variations. devil examples of the load profile are given in Figs. 1 and 2, exhibiting the data of two time periods appearing in sidestep 1. severally asterisk in the figures represents five passengers.The area under the load profile abridge is precisely passenger-miles, or in this example, passenger-kilometers, both of which are AVISHAI CEDER postpone 2. Output quality of changeables used in methods 1 and 2 320 1259 1359 1459 ,559 284 389 . ,1 0 0. 0 0. 6 . * 50 100 150 2po .. *. **.. **.. * . . .. . .. 2. 1 2. 9 3. 2 3. 5 3. 9 4. t 4. 7 5. 3 5. 5 5. 9 9. 5 5. 7 7. 3 7. 7 9. I 8. 5 9. 1 9. 5 10. 0 10. 4 IO. 6 10. 9 ,,. I 11.. 11. 5 12. 1 2. 5 13. 2 13. 9 I.. 1 14. 8 15. 0 .. .. . .. .. .. . . .. .. .. . . . . . . Fig. 1. A load profile for one morning time period (800-8 59) based on the data in confuse 1. Bus frequency object using passenger count data 443 NIJMSER PISSENGERS OF FOR INTERVAL viosterol TO 1559 DlSTlNc E (KY. 1 50 NUMBEP PAssENGERI OF 200 250 300 350 400 450 500 I 100 150 L.. 1 1.. L L *.. .. *.. * ** *.. *.. .. . *.. * * *.. . .. . . . .. .. .. . .. .. . . .. . . . .. .. . . .. .. . . . . . .. Fig. 2. A load profile for one afternoon time period (15emailprotected59) based on the data in put off 1. measures of productivity.If a straight line is careworn across the load profile at the point where the number of passengers is equal to the observed average hourly max load, and so the area below this line but higher up the load profile is a measure of the non-productive service. When method 2 is used to derive the headways, and dj is equal to the number of seats then this measure is the overturn seat-miles (or quash seatkilometers). Figure 1 is characterized by a comparatively large value of empty seat-kilometers per bus in comparability to Fig. 2. However, the additional information supplied by the load profile enables one to overcome such an undesirable distinctive.Thi s can be through by introducing frequency function methods which are based on passenger-miles rather than on a max load measure. The first load profile method considers a lower bound level on the frequency or an upper bound on the headway, given that the bus capacitance constraint is held. order 3 is q? = max One way to look at method 3 is that the ratio A,/L of the load P, (regardless of its statistical definition) as inappropriate to the max load (P,) in method 2. method 3 guarantees, on the average basis of P,, that the on-board passengers at the max load section will not have it away crowding above the given bus cognitive content c.This method is appropriate for patronize cases in which the schedulers wish to know the number of bus runs they can expect to bring low by relaxing the desired occupancy standard, avoiding overcrowding at the aforesaid(prenominal) time. This allows them to handle the following (i) demand changes without change magnitude the available number of buses (ii) situations in which some buses are needed elsewhere (e. g. breakdown and maintenance problems, or emergencies) (iii) fewer drivers than usual (e. g. due to figure cut, or problems with the drivers union).On the other hand, method 3 can result in beastly travel for an extended distance in which the occupancy is above 4. To eliminate or to control this possible undesirable phenomenon, other method is introduced. Method 4 establishes a level of service consideration by restricting the total route distance having loads greater than the desired occupancy. Method 4 takes the explicit form is an average representative A. P. -A-,1 dj. L c 1 4. L Ai1 pj c where Ii is the distance between stop i and the near stop (i + l), Aj is the area in passenger-miles (km) under the load profile during time period j, and L is the route distance.The other notations are previ )usly defined in eqns (l)-(3). 4? =max St. 1 Ii I jJj. L, *I) 444 AVISHAICEDER by time of day are same to that implyd in Table 2, and for all five sets the capacity is c = 80 passengers. In method 4 based on eqn (5), three values are assigned to /I, for all js 0. 1, 0. 2 and 0. 3. That is, 10, 20 or 30% of the route length is allowed to have an observed occupancy, P, especial(a) the desired one, 4. The results for route 27(A) appear in Table 3. The headway results of the four methods are likend pictorially in Fig. 3 where the results of method 4 are for only the 20% limit case (8, = 0. ). Similarly to Fig. 3, the results of the remain four data sets are displayed only in the computer generated graphical form in Figs. 4-7. . These illustrations are used for get on analysis of the results. The first semblance can be do between method 1 and method 2 for the point check decision. Obviously, it is less costly and more convenient to retain an observer at one bus stop during the entire working day, than to assign the same observer or others to a distinguishable stop at every period j. This aspect bus stop is the one characteiized by P, (see eqn (2)).The equivalence between the two methods is performed by the ,$ shield between two sets of veridical observations-P. , vs P, for each data set (see Ceder and Dressier, 1980). The results are as follows where I, = i (P,,/F,) d, or 4 is the set of all stops i in time period j such that the load Pq exceeds the quantity of 4 times the number of buses determined iteratively by F,, and pj is the allowable portion of the route length at period j in which Pti can exceed the product (4)()(d,). The other notations in eqn (5) are antecedently defined. By controlling the parameter /Ii it is possible to establish a level of service criterion. spirit that for /I, = 0, /I, = 1. 0 method 4 converges to method 2 and method 3, respectively. 2. 4 Results of tangible data and comparison A pL/l program has been written for all the four methods. This program, in addition to calculating the bus frequencies, determines the associated integer headway (in minutes) by simply dividing the length (in minutes) of a considered time period j by 4. , and go it to the adjacent integer. The headway information is essential for the timetable preparation, as is explained in the adjacent section. The input data presented in Table I and also the data taken from four more routes have been run by the program.The additional data are four Egged routes 2(A), 2(B), 12(A), and 39(A)all from Jerusalem. Their policy headway and desired occupancy channel (Direction) 27(A) 2(A) 2(B) d. f. 13 16 18 14 16 X2 63. 24 14. 59 58. 51 492. 82 117. 82 nought hypothesis about equal methods (at the 5% significance level) resist dont reject reject reject reject I&4) 39(A) Bus frequency determination using passenger count data 445 mickle zero(prenominal) 27 , agency A fiction o regularity + system . manner 1 2 3 L (BY2OP a METHOD 0. 7oO . . 9 . . oo 11-00 . . eon 13. 00 OF solar day * 15. 00 . 100 1 . 19oo 21 00 g Fig. . analogy of head way results for route 27(A). Consequently, only in route 2(A) can the daily max load point replace the hourly max load point. The PL/l program provides this comparison. The graphical comparison between the headways in Figs. 3-7 shows the expected result method 2 eternally gives the minimum headways while method 3 results in the highest headways (except in 2 out of 82 time periods). Another characteristic of the headways, exhibited particularly in Figs. 4 and 5, is that the given policy headway (60min) is used during off-peak hours. A point worth mentioning is that the esults might be sensitive to the length of the time intervalj and that different time intervals may be used for peak and off-peak hours. Further analysis and comparison of the results are addressed in the following two sections. 3. A PRELIMINARY criterion IN DETERMINING FURTHER DATA COLLECTION METHODS In this section an effrontery is tested that particular load profile characteristics suggest the data collection met hod to be used. The basic idea is to coach NO. 2 , DIRECTION A . METHOD 3 6, 04.. . . . . . . I a. . -METHOD LCBY20%1 * . 6. 00 800 10 00 12. 00 TIME OF 14. 00 DAY 16OO 16 00 20. 00 2oo Fig. 4. Comparison of headway results for route 2(A). 446 AVISHAICEDER BUS NO. 2 , DIRECTION B . 6 _ METHOD L CBY20T. l 01 . 5oo . . 7 00 . 9 00 * 1100 . TIME . 13. 00 _ 15 00 OF DAY .. , 17 00 . 19 00 . . 21 00 23 00 Fig. 5. Comparison of headway results for route 2(B). provide the bus operator with adequate preliminary guidance in selecting the type of method based on old load profiles. The assumption to be investigated is that a relatively flat profile suggests the use of a point check procedure (method 1 or 2) whereas a ride check procedure (method 3 or 4) would be appropriate otherwise.One property of the load profile is its tightfistedness, p. This is the observed measure of total passenger-miles (total ridership over the route) divided by the product of the length of the route and its ma ximum load (passenger-miles which would be observed if the max load existed across all the stops). Thus, the load profile niggardness for hour j, pj, is P=e. The load profile density is used to examine the profile characteristics. High values of p indicate a relatively flat profile, whereas low values of p indicate a significant load variability among the bus stops. A BUS 60 NO. 39 , DIRECTION A LEGEND % $ s 2 L2. 36. METHOD (BY ZCr%l = 30. p I 9 i P 12. 6. 24. 18. 0. 1 6 00 . a 00 . 10 00 . 12. 00 TIME OF woo DAY 16 00 18. 00 20 00 2200 Fig. 6. Comparison of headway results for route 12(A). Bus frequency determination using passenger count data 447 BUS NO. 12 , DIRECTION A LEGEND o _ METHOD + . METHOD METHOD 1 / 2 3 , / * I 8 METHOD L (ByZoZl 0 500 I I 1 7 . oo 9oo 11oo I . TpF nY1500 . 17oo 19oo Fig. 7. Comparison of headway results for route 39(A). 3. 1 numerical analysis One way to nigh the observed shapes of profile curves is by using a mathematical model.The lognormal model has been selected for this purpose since it provides a family of curves which can be controlled by varying the parameters p and u. The lognormal model takes the form f(x) =. & The equation satisfying (df(x)/dx) = 0, is e-oDX-*/262 x 0. the optimal (7) conditions, x,=d-= (8) This continuous model can only approximate some of the observed load profiles since it has only one peak and represents monotonically increasing and decreasing functions before and after this peak, respectively. Nonetheless, this model is useful in observing some general differences between the ride check and point check methods.In order to be able to compare the methods,f(x) is used as a normalized load (the load divided by the max load) and x is used as a normalized distance (the distance from the departure stop divided by the length of the route). At a given time interval of one hour, j, the considered max load is Pi = 650 passengers. Given that dj = 65 and that c = 100, the determined frequency and h eadway for both methods 1 and 2 are 4 = 10 and Hj = 6. By applying this information to methods 3 and 4, using a variety of lognormal curves, one obtains the frequencies and headways shown in Table 4.The results in this table are aranged in increasing order of density. For method 3, the capacity constraint determines the values of F and H up to an including p = 0. 64 and up to different p values (if any) for method 4. Examples of the lognormal normalized curves are shown in the computer generated Figs. 8 and 9 for two p and variety of p values. Note that the relative location of the max load point can be found by eqn (8). From Table 4 it appears that for method 3 the ride check (load profile) data results in the same go headway as for the point check (max load) data for p 2 a where 0. 4 a 5 0. 87. For method 4 the ride check and point check information tend to yield the same headways for p 2 ai where i = 1,2,3, for the 10, 20 and 30% cases, respectively, and 0. 34 a, I 0. 43, 0. 5 0 a2 I 0. 56, and 0. 64 a, 50. 68. 3. 2 Observed densities and intervention The five data sets mentioned in the previous section were also subject to the load profile density examination. The pi values for each considered hour j, based on eqn (6), were calculated and are shown in Table 5. For example, in Fig. 0, which is part of the PL/l program output, one can visually compare the load profiles associated with the highest and the lowest p value of data collected on route 39(A). As can be seen from Table 5, none of the p values exceed 0. 8. This suggests that one cannot reach, by calculation, same headways for method 3 and method 2. Figures 3-7 reveal that the determined headways of method 3 are unendingly greater than those of method 2 excluding the cases of policy headway. However, no clear cut close can be drawn when nerve-wracking to associate the p values in Table 5 with those 448 Table 4. Frequencies (F) and headways log-normalAVISHAI CEDER (H) for different load profile configurations (derived from the model) using methods 3 and 4 Method 3 profi 1e density T by 10% H F 7. 60 H Method 4 20% H 9 9 9 9 9 8 7 8 -%6 6 6 6 6 6 6 6 6 6 1 by by P F F F 30% H 9 9 9 9 9 9 0. 18 0. 25 0. 27 0. 32 0. 34 0. 43 E 048 0. 50 0. 56 0. 57 0. 59 0. 62 0. 64 0. 68 0. 75 0. 76 0. 78 0. 84 0. 87 *For Note 6. 50 6. 50 6. 50 650 6. 50 6. 50 6. 50 6. 50 6. 50 6. 50 6. 50 6. 50 6. 50 6. 77 7. 46 7. 63 7. 77 8. 41 8. 72 9 9 9 9 E% 9 z 9 9 9 9 9 9 9 7 7 -4. 8. 46 6. 50 8. 36 7. 55 9. 00 7 9 7 7 -i5 6 6 6 6 6 6 6 6 6 6 6 6 6 6. 50 6. 50 6. 50 6. 50 6. 50 7. 05 805 E 7. 5 x 931 8. 85 9. 04 9. 42 9. 36 9. 68 9. 87 9. 76 9. 92 6. 50 6. 50 6. 50 6. 50 6. 50 6. 50 z 945 9. 05 9. 92 9. 76 9. 81 9. 65 9. 79 9. 87 9. 86 9. 93 9. 97 9. 96 9. 97 constraint ?E 8 650 9 6. 50 9 9. 27 6 8. 16 7 8. 46 7 7. 80 7 8. 19 7 8. 72 -b 8. 76 6 9. 23 6 9. 72 6 9. 46 6 9. 82 6 Methods 1 and 2 Uhenever F-10. H=6 where d F=6. 50, H=9 the capacity = 65, c-100. is met. in Table 4 regarding the comparis on between methods 4 and 2. Figures 3-7 clarify this by illustrating the results of method 4 for the 20% case. The matchings (same headways for methods 2 and 4) across all the five data sets range between p = 0. 38 (route 2(B), for the hour 2200-22 59) and p = 0. 744 (route 39(A), for the hour 1600-l659). On the other hand, the non-matching cases range between p = 0. 457 (route 2(B), for the hour 800-859) and p = 0. 777 (route 12(A), for the hour 1500-1559). Consequently, when applying method 4 to the observed load profiles, the results of the lognormal model cannot be explicitly used and an actual comparison between methods 2 and 4 should be performed. In practice, the bus operator wishes to deport bus runs and eventually to be able to perform the matching between demand and supply with fewer buses.As is shown in the nigh section, different headway values do not necessarily save bus runs or reduce the required overhaul size. However, the analysis made about the profile density measure can be used by the bus operator as a preliminary check before entering a more comprehensive analysis. The following are realistic observations (i) for densities below 0. 5, p-o 66 OK 0 1 .2 .3 Fig. 8. Four approximated load profiles based on the log-normal model (a = 1. 00). Bus frequency determination using passenger count data Fig. 9. Four approximated load profiles based on the log-normal model (u = 1. 0). savings are likely to result by multitude the load profile information and using either method 3 or 4 (alternatively for such low p values, the profile can be examined for short turn strategies) (ii) for densities between 0. 5 and 0. 85, it is recommended that an actual comparison be made between the point check and ride check methods-along with further saving considerations (see next section) and (iii) for densities above 0. 85 it is likely that the majority of the required information for the headway calculation can be obtained from a point check procedure (either method 1 or 2). . ALTERNATIVE The TIMETABLES AND FLEET SIZE possible to initiate the task of scheduling buses and crews to the previously determined trips. Naturally, the bus operator wishes to utilise his resources more efficiently by minimizing the number of required buses and the cost of the crew. To accomplish this, the scheduler examines different timetables during the bus and crew assignment processes. This is done by shifting the departure times or by reducing the number of departures without referring usually to the initial source of passenger loads-the profile.Therefore, it is desirable to extend the analysis ancestry appropriate headways, to an evaluation of timetables in conjunction with the required resources. 4. 1 Construction of timetables The number of bus runs determined by the timetable and eventually the number of buses required, is sensitive to the procedure used by the scheduler to CONSIDERATION AT THE street LEVEL products of the derived headways are the time tables for the public, the bus drivers and supervisors. at once the timetables are constructed, it is Table 5. Load profile densities ) for five data sets I 500. 00 7oo 800 9oo looo Time separation 659 759 859 959 1059 559 passageway Z(A) v-e 0. 489 Route Z(B) Route 12(A) lloo 12oo 13oo 14oo 15oo 1600 17oo l 19oo 20oo 21oo 22oo 2300 1159 1259 1359 1459 1559 1659 1759 1859 1959 2059 2159 2259 2359 0. 668 0. 557 0. 687 0. 548 0. 687 0. 477 0. 694 0. 652 0. 699 0. 606 0. 632 0. 73j 0. 610 0. 524 0. 588 0. 543 ___ 0. 524 0702 0. 752 0. 457 0. 586 0. 592 0. 647 0. 620 0. 679 0. 764 0. 662 0. 717 0. 722 0. 618 0. 673 0. 633 0. 588 0. 538 0. 546 0. 661 0. 705 0. 625 0. 731 0. 637 0. 589 0. 680 0. 39 0. 740 0. 712 0. 777 0. 640 0. 565 0. 650 0. 509 a _-_ -se -me ___ 0. 563 0. 567 0. 715 0. 765 0. 717 0. 672 0. 636 0. 733 0. 723 0. 641 0. 712 0. 639 0. 576 0. 593 ___ _____ Route 27(A) _-_ 0. 651 0. 561 0. 589 0. 674 0. 594 0. 559 0. 619 0. 644 0. 599 0. 691 0. 744 0. 626 0. 657 0. 544 0. 686 0. 610 0. 577 _-_ Route 39(A) 0. 0 0. 3 0. 4 0. 7 1. 1 1. 3 1. 7 2. 3 ?. I 2. 7 3. 1 3. 5 3. 9 4. 4 4. 9 .. .. . .. .. .. . . . . . .. 5. 6 5. 1 6. 2 6. 4 6. 7 7. 1 7. 5 7. 8 8. 2 8. 4 8. 6 9. 0 9. 1 9. 2 9. 5 9. 6 . . .. . .. . . .. .. .. .* .. ** .. .. . . . Fig. 10. Two load profiles of route 39(A) with the highest density = 0. 744) on the left and the lowest density = 0. 544) on the right. construct the departure times.Some bus operators routinely round the frequency 5 to the next highest integer and then calculate the appropriate headways for the considered time period. By doing so, they increase the number of daily departures beyond what is needed to befittingly match the demand with the supply. Such a procedure may result in nonproductive runs (many empty seat-miles). For example, in Table 3 the number of daily required departures, F 4, is 77. 01, 55. 64 and 73. 24 for methods j=l 2,3 and 4 (20% case), respectively. When the quantity F, is locomo te up, one obtains respectively 85, 65 and 80 daily departures for these three methods.Obviously, by locomote k to the next highest integer, the scheduler increases the level of passenger comfort but, at the same time, causes an superfluous operating cost. However, in some cases the round up procedure may be justified if the scheduler uses the Pq quantity as an average load whereas the difference of the load is high. In this case (provided that additional runs are made by rounding up Fj), the possible overcrowding situations may be reduced as opposed to increasing the average empty seat-miles. Nonetheless, to overcome the problem of highly variable oads, one can use a statistical load measure which considers its variance as an input to a frequency method (see remarks in eqn (1)). Another characteristic of the existing timetables is the repetition of departure times, usually every hour (see Vuchic, 1978). These easy-to-memorize departure times are based on the time headways 6, 7. 5, 10, 12, 15, 20, 30, 40, 45 and 60 min. Generally, headways less or equal to 5 minutes are not considered by schedulers to influence the time of passenger arrivals to a bus stop. The quantify headway is obtained by rounding the derived headway down to the nearest of the above quantify values.Consequently, and similar to the round up frequencies, the clock headways require a higher number of departures than what is actually necessary to meet the demand. In order to keep the total daily number of departures as close as possible to the sum of the obtained Fjs by the four methods, the derived headways in Table 3 and Figs. 3-7 are simply based on the round to the nearest integer procedure. Note that for a high frequency value it may turn out that rounding Fj result in fewer departures than rounding the derived headway. However, for high frequencies, the timetable is not required.Also, if 5 is rounded first it is necessary to perform a second rounding on its associated headway (sinc e timetables are built by headways-not frequencies). This by itself may ultimately decrease the accuracy of matching the demand with the number of departures. An attempt is made in Table 6 to construct six daily timetables for methods 2,4 and 3 using both the derived and the clock headways based on the information in Table 3. The only incompatability is that Bus frequency determination using passenger count data 451 Table 6. Various timetables for bus 27(A) based on methods used and considered headways Y I9 ii01 3a oa 57 15 a17 22 3a 29 59 36 914 43 24 50 34 57 44 1404 54 I1 1004 1s 15 25 26 32 37 39 4a 46 59 53 1109 15OO ia 08 27 I6 24 36 45 32 54 40 1203 4a 13 56 23 1606 33 la 43 30 si i5 1704 30 12 45 20 a00 2a 20 36 40 44 9oo 52 lO iam 20 la 30 36 40 54 50 19oa 1ooo 19 lO 30 20 41 30 52 4o 2024 50 2117 11OO 07. 5 I5 22. 5 30 37. 5 45 52. 5 12oa lO 30 40 50 13oo 06 12 la 24 30 36 42 4a 54 14oo 06 12 la 24 30 36 42 4a 54 1soo 07. 5 15 z22. 30 I5 52. 5 16Ml 12 24 36 4a 17oo 07. 5 15 22. 5 30 37. 5 45 52. 5 la00 15 20 ll 40 19 a03 27 29 35 55 43 913 5i 23 59 33 1406 43 13 53 20 1003 27 14 34 25 41 36 4a 47 55 5a 15oz ii08 lO 17 ii 26 34 3s 44 42 I 53 5Ll lZOi G 12 16oa 21 34 34 44 47 1254 17W la 27 36 45 54 iaoa 27 46 59 19ll 23 35 47 2ozo 2115 uerved LIOC) Headway 00 1230 16 00 7 12 23 24 46 36 alo 4a 36 17w 55 07. 5 9oa I00 22. s I5 21 10 30 34 20 37. 5 22. 5 46 30 30 45 37. 1ooo 40 52. 5 15 45 50 14oa 30 52. 5 I00 06 45 10 12 la00 1lOO 15 20 18 I2 30 30 24 45 24 40 30 36 19oo 50 36 4a 1oo 42 12 24 12oa 07. 5 48 15 lS 54 36 30 40 22. 5 15oo 45 30 07. 5 2ooo 13oo 45 27. 5 15 ll 45 22. 5 2130 52. 5 30 2oo z37. 5 10 45 44 20 52. 5 20) i . I oo I lO 2o Jo 40 50 2ooo 45 2130 z24 i ( i i 55 uETmb3 He4dw4y , Clock HeadMy 14os 7oo 13so 195 20 14oo 4 14 40 07. 5 2oa 23 I5 a00 21c 32 20 22. 5 41 40 30 50 9oo 37. 5 59 12 45 1508 I4 52. ia 15oo 36 2a 4a 10 3fl 1ow 20 4a 15 30 5a 30 40 16lO 45 50 25 11OO 16OO 40 12 15 55 24 30 1108 36 45 20 48 17oo 32 . 44 12oo 1 2 I56 15 I2 30 36 la 16 44 45 48 1907 13oo 18OO 26 lO 20 20 40 45 30 2023 19oo 2123 40 15 I 1 the clock headway technique includes a value of 7. 5 minutes whereas the derived headways do not allow non-integers. The transition between the hourly periods for the derived headway is based on a smoothing rule that use the rounded down average headway whenever a transition from one hour to another occurs.For example, in method 2 the transition between the departures 8 59 and 9 14 is based on rounding down the average headway of 21 and 1Omin. A point worth mentioning here is that the schedulers often have the knowledge of different load patterns during one period j, e. g. more loads in the first half hour than in the second. In this case they can demand splitting or changing the time period j for further data collection. Also, they can insert more departures in the heavy-load interval than in the remaining interval, while ensuring the approximate total of Fj departures.Further considerati on about creating timetables appears in a report by Ceder (1983). This includes development of methods to construct timetables with even headways and timetables with even (average) loads on individual buses while the headways are unevenly spaced. 4. 2 Single-route fleet size examination Within a large-scale bus system, buses are often shifted from one route to another (interlining) and they ofttimes perform deadheading trips in order to conk out a given timetable with the minimum required buses.It is desirable to analyze the procedures to construct timetables and scheduling buses to trips simultaneously. However, due to the complexity of this analysis, these two procedures are treated separately. Therefore, in a bus system with interlining routes, the alternative timetables can be evaluated on the basis of the total number of required departures. This can serve as an indicator for the number of buses required, but without inserting each alternative timetable to the scheduling proce dure, it will be difficult to predict the effect on the fleet size.One fleet size test that can be performed is based on the assumption that interlinings and deadheading trips are not allowed and that each route operates separately. In this case, given the average round trip time, the minimum fleet size for that route can be found similar to the formula derived by Salzbom (1972). Let T be the round trip time including the layover and turn around time and that departures occur at discrete time points t,, t2, r,, . . . , t,.Also, let N, be the number of departures between and including the two departure points t, and t, such that three conditions (i) are effectuate t, tr, (ii) t, tr I T and (iii) t,+, t, T. Given that if t, = t, then the first tk, k = 1,2,. . . , n to support with the first two conditions is t,. the minimum single-route fleet size, N,,,, is Nmi,=max k i k=l Nk Following Salzborn arguments, eqn (9) simply meaning that N,, is the largest number of buses departin g in any time interval of length T. This result can