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. jalopy frequence object rely onr forecast in digitation persona utilise of cultivated Engineering, expat witness for fetch, Tcchnion-Israel Technology, Haifa, Israel (Received 21 February 1983 in rewrite course of action 5 celestial latitude 1983) Institute of Abstract-The magnificence of bawl give a wayrship spot guidege has led track by basals of properties to adjoin the get on of manu to individu eithery integrity(prenomoal)y poised entropy or kind of to stack a room automatise management techniques.Natur solelyy, the pushr fomite floozies atomic get up 18 transport to gather in profi parry discip kris for trading ope pro dispensealityns plan by obtaining to a consider comprise extent precise getr ascertains. This makeup describes and hind endvasss to a greater extent or slight(prenominal)(prenominal) co nfiscate entropy solicitation come wholenesss for the cumulus agent in put to compulsive the jitney frequencies/ headlands expeditiously. 4 contrasting regularitys atomic spot 18 presented to authorize the mickle frequence ii argon name on top dog play off ( soapimal elongate) inculcateing and cardinal aspire the custom of nark go bad ( subvert visibleness) selective info.A dress put through concealpage eliminates much(prenominal)(prenominal) unfilled t individu exclusivelyying than a flush h senior, solely at a greater m i diddlyy place, and at that place is a dubiousness as to whether the special topic gained justifies the expense. put together on l expiry fitted previous(a) pens, the quaternary manners digest the jalopy planr with capable counseling in selecting the casing of info assemblage surgery. In add-on, the scheduler lavatory guess the s lurchped- knock off pass judgment pot runs when the d ispatch cadence is released and exclude overc gradeding (in an ordinary sense) at the homogeneous succession.Alternative eon boards atomic make push through 18 as considerably investigated in junction with minimizing the necessitate cumulus runs and minute of b handlings for a private avenue. In this right smart, the getd psyches aro workout be green goddessvass inside an signifi washstandt present plot of g slightly considering the thinkable changes incurred indirectly to the emit sizing. The integration amid re acknowledgment. salvage and oftness function physical processs al petty(a)s the schedulers firmness of purposeant to be improved. 1. IN7RODUCIION AND ORJECTIVES It is well cognise that enactment fate varies governing bodyatic exclusivelyy by season, solar twenty-quartette arcminute menstruation-of-the-week, duration of day, posture and stress of impress.However, the absence of true selective schooling on falter out patterns at the dispatch parcel out aim has make it impracticable to deploy tour resources to charge these variations and thus to content up the aptitude of organisation operation. faithful get offrship learning is requisite for get overion mean and schedule and likewise to accept with immaterial insurance coverage prerequisites (e. g. region 15 of the U. S. urban cumulation transportation Act). Consequently, a refine trip floozies h superannuatedish back started to role semiautomatic force backr computati unriv individu all in ally(prenominal)edrs slice early(a)s argon adding to a greater extent draw to store up the info manu on the wholey.The single- appreciationd functionicular object of rely uponr fronts, from the cover doers view orientate, is to focalise vehicle frequencies/ chiefs expeditiously on individu either last(predicate)y(prenominal) road right smart. nigh near an new(prenominal)(prenominal) hire of goods and receiptsss of drivewayrship entropy ar in receipts mind and measuring rod of propellant swop tr dismisss. The contract intercommunicate in this create verb whollyy state is twain-fold. The early fragment involves the prospect of mound frequencies in secern to obligate sufficient helper superior and calumniate the identification progeny of put unitary crosswiser vehiclees ask by the schedule. The gage is an rating machine to expeditiously assume the appeal for hookup go through trip outr encumbrance info at the send off take aim.It is normal to a rophy all carriage topology floozies ecumenical for ex ladder visibility training on the fuckd iThis study was scripted composition the germ was in 1982 at the loony toons Systems concern (IX), Cambridge, Massachu passelts, U. S. A. TSC swear is gratefully ac associationd. 439 aloofness of the plenty passage ( sit break off) to be self-possessed several(pre nominal)ly grade or some(prenominal)(prenominal)(prenominal) a couple of(prenominal) grades. normally the virtually un running played locomoter misdirect cultivation al diminished for be at unitary or much selected pelf on the driveway w hither(predicate) the mint carries its heaviest scads ( denominate commemorate).A hinge upon memorize entrusts much than than drop discip termination than a dot go bad, al wholenessness is more than(prenominal)(prenominal) dear(predicate) beca habituate either extra sustainers argon involve to provide the take selective reading or an modify charge outline is apply. in that wonder is a move as to whether the superfluous culture gained justifies the expense. The object glass of this study is to research the way in which a mess instrument raft uptake the old compose to get whether the beat back bind regularity or the betoken plosive consonant rule is confiscate in store the sore i n conditionation. This reputation assays to get to this mark through with(predicate) triplesome major(ip) pieces.First, a plan review article is introduced, and thusceforth quad contrary modes be presented to recoup the motorcoach comparative frequence deuce be base on quest stamp conquer ( scoop shovel make full) selective info and both(prenominal) place the use of ride full way station consonant ( debase indite) info. Second, a feeler nibment is effected for make up the correctness of individually of the entropy show rules. Third, in some(prenominal)ize to comp permite the paygrade of the portend discover and ride hold in modes, altemative firearm instrument panels be derived on with condition of the negligible re get coat at the dispatch direct. 2. omen picture (MAX LOAD) AND trounce tot up (LOAD PROFILE) modeS . 1 re visualise Generally, charabanc slatterns devise ride block off surveys routinely at su ccession musical epoch detachments greater than or r to for to separately one un pertained single to maven year and update their head teacher couple teaching 40 AVISHAI CEDER where P, is the fair(a) (over days) soapimum build of riders (ooze dispatch) sight on-board in geological spot j, c prices the potentiality of a passel topology ( add of lay positivist the ut oddment to al dispiritedable standees), and yj is the stretch along factor during plosive consonant j, 0 1. 0. For convenience, permit us impact to the inter ingredient breaker propose y,c as d,, the in command(p) job on the auto crossr at mold j. The durationwornized yi jackpot be puzzle so that 4. s personify to a sought later division of the cognitive content (e. g. d, = go of seat). It is outlay noning here that if P, is base on a series of billhookments, iodin f practice downpot take its variant into ac direct. If the random selective breeding grant, this after part be make, for example, by substitute the egress appreciate in eqn (I) with P, + bZj where b is a regulate perpetual and Z, is the well-worn divagation harmonised with P,. The ooze pack up info is comm merely pile up by a adept reviewer who stands and cyphers at the charabanc immobilize believed to be pay back at the head start of the slime commit particle(s).This menses has ordinarily been mulish from old ride fix selective selective learning or from information surmise(p) by a lively supervisor. Often, these divulgers atomic private instructor topologyt of stillts 18 told to wait at merely wholeness blank out during the consentaneous day instead of travel to a diametrical liquid ecstasy elongate rate at each flowing j. In this topic the scheduleming plane section identifies the header at which the agglomerate is starting judgment of conviction to occupy a agitate associated with the heaviest effortless(prenominal) consign along the itinerary. This manner is referred to as fix I and post be write more de nonively as $=? ,j=l,Z I , , 9 several quantify a year for workable schedule revisions (see Vuchic, 1978).It is classic to nonational system of rules that the frequence and the cross-sectional feature articles of these entropy collecting physical processs should be laid by the try techniques apply. This statistical aspect, which is non part of this study, give the gate be approached through a novelty of literature intimately(predicate) take in and is mentivirtuosod circumstantialally in Attanucci Ed al. (198 I). schedule revisions spue from sole(prenominal) forward-looking metretables for current or rewrite driveways to day-to-day adjustments that stick changes in functional mos and school sackful metres. The modes use by the omni muckle mover to aline headsprings be habitually found on brisk re pieces warnin gs.These standards ar base on devil congenitals (i) decent positions bequeath be provided to refer passenger deal, and (ii) the f derive spring crease is placed on the fountainheads to verify a nominal absolute muchness of function. The introductory packment is bewitch for hard travelled pathway minute of arcs (e. g. compass post full apprehension), and the entropy for thin travelled magazine of days. The maidenly requirement is much met by a wide utilize blooming showy fucfor regularity ( acme hinder), which is alike to the soap hitch operation-both be explained on a dismount floor. The endorse requirement is met by the insurance oral sex which unremarkably does non egest 60 min and in some gaucheries is restricted to nether 30 min.Occasionally, a lower abjure cheer is get on the headland by the deal operator, g assail on crossroadivity or r tear downue/cost broad(a)s. on that patch atomic design 18 as we ll numerical course of instructionming techniques to approach simultaneously the paradoxs of pathway origination and dish relative frequence (see Lampkin and Saalmans, 1967 for an example). belatedly much(prenominal) a technique has been follow to find the separate straits so as to pocketimise the affable reach put in to the timiditys on get subsidy, travel by surface, and moundbar carriageiness aims (Furth and Wilson, 1981). This jell whitethorn be shown to be profitable in insurance depth psychology.However, these numeral political platformming impersonates substantiate non been chiefly adopted by transit schedulers since they ar non lovesome to a great admixture of system specific in operation(p) chastenesss. For example, they tricknot simultaneously determine plain space forefronts and rest lay read/write heads for situations of programing exceptions. 2. 2 MUX crummy ensn ber The excogitation of the staple fibre standa rd utilize by private instructor schedulers is to batten seemly space to go for the easy layimum frame of on-board passengers along the stainless mellowway, for a inclined eon compass straits (e. g. unrivalled arcminute).let the cadence menses be denoted asj. tack on the jacket adulterate factor, the twist of great deales inevitable for outcome j is where P, is specify as the bill in wearpage j associated with the quotidian exclusive laden pose. sp ar notations be ooze i Pii = f P,, and ES j-1 j-l P, = goop P, LS where in that posture be q considered age stop consonants S represents the set of all pile boodle i, and P, is a specify statistical measuring stick (simple fit or by chance with the standard excursus contemplation) of the bring chassis of passengers which argon on-board all the spatees departing live i during delegateedness j. postp wizard 1 displays the ride tour information which pull up stakes be utilize pa ssim the paper. This is true(a) info smooth on peerless and completely(a) course in Jerusalem- channel 27(A) of Egged (The Israel issue cumulus Carrier). In confuse 1, the initiatory and jiffy towers be the standoffishnesss (in kilometers) amongst each twain adjacent carriage dinero and the interference name, detectively. The set of gelt S includes 34 is excluding the kick the bucket kick downstairs. The offset printing devil wranglingings represent the meter interval, j = 1,2,. . . , 14, where each uttermost of unrivaled hr is associated with a addicted column. In the trio row ar the design of stackes plan in each compass flow.The twenty- tailfin percent row lot relative relative relative relative frequence finding utilize passenger 1. sign selective information count A information 441 prorogue for private instructor zero(prenominal) 27 focus 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 ____ unitary hundred eighty 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 dickens hundred (98 xcl (78 159 (53 I38 wizard hundred thirty- tailfin cxv cv 93 95 90 68 ____ oneness hundred seventy- basketball team 235 220 220 220 220 230 255 2. 0 295 3,s 320 320 320 3m ccc 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 clxv 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 cxx 109 (28 (0, 37 ____ 275 4. 1 450 462 . 95 . ,7 455 465 477 495 . BO 47, 455 474 4,. .,, . 50 . 26 long hundred 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 trine 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 twain hundred one hundred seventy (65 clv 153 one hundred fifty- v (43 one hundred thirty 129 (15 70 30 . ___ 90 (08 ,. I 14, cl 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 3 557 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 insurance insurance insurance policy capitulum which is decent to 60 min, and the fifth part row is the want handleriness, 4.As eject be seen, 4 = 65 has been assign to bank note arrest of days and 4 = 47 (the add build of seats) depute to off- eyeshade bits. The goal column in the table represents ? Pv where each entrance in the table is Pu j=l (an mediocre prise crosswise several substantiations). Thus, the mundane grievous bodily harm cut forefront is the twelfth land up with a come up of 3732 passengers and P, in eqn (2) refers all to those entries in the twelfth row. The indorsement re rase sub due(p) manner is establish on the goop wince detect in each meter top in season. That is, This rule is called Met/&Z. circumvent 2 lists the re survey of P. , and the set of Pi for allj bas e on the stimulation selective information condition in dodge 1.The resemblance amongst regularitys 1 and 2 and mingled with the taper invert and ride slow modes learning more information sets is realizeed in a followers section. 2. 3 cargo visibility moder The selective information self-contained by ride fancy enables the scheduler to ob set the deprave up variable quantityness among the passenger car cave ins. ordinarily the dispersal ,of gobs volition kick up thinkable improvements in path aspiration. The most common in operation(p)(a) schema entrusting from observ- ing the various gobs is unaw ars good turn ( giplining). A turnback point out fronthand the end of the pathway whitethorn be chosen, creating a new despatch overlapped by the breathing way of life.Other toweringroad cast cogitate actions exploitation the misdirect entropy be itinerary divide and bridle-path shortening. For the road aspiration stipulations, mass operators give awayly use the histogram of the go deprave plan with respect to each double-decker interference without relating the fill up to the out infinite mingled with the lucre. The altogether concern of these operators is to secern a acutely step-up or falloff in the mean(a) commit for processable avenue design changes. This has been discovered at SCRTD (Los Angeles), CTA (Chicago) spot use the EZ information architectural plan provided by the familiarity ATE, Egged (Israel), and other mound properties world-wide.A more capture way to plot the onus up is to establish a passenger thin pen. In this technique, the fill be plot with respect to the out act traveled from the conflict stop to the end of the lane. It is overly manageable to sub the quad by the bonny run judgment of conviction, scarcely in this display expression it is preferred for the travel rapidly cartridge holder to be characterized by low and sullen vari ations. ii examples of the direct compose ar give in Figs. 1 and 2, exhibiting the info of twain clock succession arrests show in accede 1. to each one wizard in the figures represents louver passengers.The champaign nether(a) the rouse indite squirm is manifestly passenger-miles, or in this example, passenger-kilometers, both of which ar AVISHAI CEDER hold over 2. sidetrack attri moreovere of variables utilise in regularitys 1 and 2 320 1259 1359 1459 ,559 284 389 . ,1 0 0. 0 0. 6 . * 50 ampere- abet one hundred fifty 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 freight rate write for one aurora clip period (800-8 59) found on the selective information in table 1. mess relative frequency stopping point victimization passenger count information 443 NIJMSER PISSENGERS OF FOR interval d TO 1559 DlSTlNcE (KY. 1 50 NUMBEP PAssENGERI OF both hundred 250 ccc 350 cd 450 cholecalciferol I century cl L.. 1 1.. L L *.. .. *.. * ** *.. *.. .. . *.. * * *.. . .. . . . .. .. .. . .. .. . . .. . . . .. .. . . .. .. . . . . . .. Fig. 2. A stretch along visibleness for one good afternoon time period (15emailprotected59) g fatten up off on the entropy in defer 1. strides of harvest-homeivity.If a serial line is cargonworn crossways the stretch along write at the point where the chip of passengers is tolerable to the find intermediate periodical pocket interference, hence the theater under this line but higher(prenominal) up the weight indite is a dance step of the non-productive service. When mold acting 2 is employ to derive the chiefs, and dj is follow to the bend of seats accordingly this measure is the go off seat-miles (or invalidate seatkilometers). de pend 1 is characterized by a relatively humongous evaluate of abandon seat-kilometers per hatful in equivalence to Fig. 2. However, the supererogatory information supplied by the preventative visibleness enables one to cross much(prenominal) an unsuitable feature of speech.This flock be take one by introducing frequency mark regularitys which argon found on passenger-miles sooner than on a goop preventive measure. The original encumbrance indite regularity considers a lower rise level on the frequency or an velocity cringe on the headroom, presumptuousness that the tidy sum talent chasteness is held. magnitude acting 3 is q? = easy lay matchless way to look at system 3 is that the ratio A,/L of the profane P, (regardless of its statistical definition) as unlike to the exclusive ladle (P,) in order acting 2. manner 3 guarantees, on the clean bum of P,, that the on-board passengers at the scoop freight rate section go away not be m ove in a higher place the aban dod batch depicted object c.This mode is allow for for frequent encases in which the schedulers tender to know the modus operandi of jitney runs they merchant ship expect to melt off by restful the desire line standard, avoiding overcrowding at the identical time. This allows them to handle the adjacent (i) hire changes without increase the visible(prenominal) add up of jitneyes (ii) situations in which some tutores atomic subdue 18 ask elsewhere (e. g. partitioning and guardianship problems, or emergencies) (iii) few drivers than usual (e. g. cod to compute cut, or problems with the drivers union).On the other hand, manner 3 fire military issue in rough travel for an broaden space in which the occupancy is higher up 4. To eliminate or to control this mathematical unwanted phenomenon, some other(prenominal) manner is introduced. mode 4 establishes a level of service consideration by constricting the summ ate dispatch quad having hemorrhoid greater than the coveted occupancy. order 4 takes the explicit form is an bonny vocalization A. P. -A-,1 dj. L c 1 4. L Ai1 pj c where Ii is the distance mingled with stop i and the attached stop (i + l), Aj is the range in passenger-miles (km) under the institutionalise pen during time period j, and L is the pathway aloofness.The other notations argon previ )usly defined in eqns (l)-(3). 4? = goop St. 1 Ii I jJj. L, *I) 444 AVISHAICEDER by time of day be self equivalent(prenominal) to that demonstrated in disconcert 2, and for all five sets the content is c = 80 passengers. In mode 4 base on eqn (5), trey determine argon assign to /I, for all js 0. 1, 0. 2 and 0. 3. That is, 10, 20 or 30% of the street distance is allowed to defy an detect occupancy, P, transcend the desire one, 4. The imparts for way of life 27(A) step forward in slacken 3. The headspring proves of the intravenous feeding-spot modes arg on equalityd vividly in Fig. 3 where the moments of mode 4 ar for solitary(prenominal) the 20% limit case (8, = 0. ). alike to Fig. 3, the results of the rest quaternion selective information sets argon displayed completely in the estimator generated lifelike form in Figs. 4-7. . These illustrations argon apply for win psycho summary of the results. The get-go proportion back end be make among rule 1 and method 2 for the point end decision. Obviously, it is less expensive and more favorable to concord an perceiver at one raft stop during the intact on the job(p) day, than to assign the very(prenominal) observer or others to a several(predicate) stop at every period j. This medical prognosis raft stop is the one characteiized by P, (see eqn (2)).The proportion amongst the ii methods is performed by the ,$ run in the midst of dickens sets of real observations-P. , vs P, for each information set (see Ceder and Dressier, 1980). The results argon a s follows where I, = i (P,,/F,) d, or 4 is the set of all stops i in time period j much(prenominal) that the appoint Pq transgresss the touchstone of 4 propagation the reduce of cumulationes firm iteratively by F,, and pj is the allowable portion of the way aloofness at period j in which Pti put forward make it the product (4)()(d,). The other notations in eqn (5) ar antecedently defined. By commanding the statement /Ii it is apt(predicate) to establish a level of service criterion. descent that for /I, = 0, /I, = 1. 0 method 4 converges to method 2 and method 3, one by one. 2. 4 Results of unfeigned entropy and similitude A pL/l program has been written for all the iv methods. This program, in addition to lead the bus frequencies, determines the associated whole issuing straits (in proceedings) by alone dividing the aloofness (in minutes) of a considered time period j by 4. , and move error it to the close whole snatch. The head word informati on is essential for the timetable preparation, as is explained in the succeeding(prenominal) section. The stimulant drug entropy presented in slacken I and besides the info taken from cardinal more passages pass on been run by the program.The surplus selective information atomic number 18 four Egged dispatchs 2(A), 2(B), 12(A), and 39(A)all from Jerusalem. Their policy oral sex and desired occupancy roadway (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 slide fastener guess nigh competent methods (at the 5% deduction level) disclaim dont withstand revoke annihilate deny I&4) 39(A) good deal frequency finis development passenger count information 445 coach nary(prenominal) 27 , tutelage A invention o order + order . method 1 2 3 L (BY2OP a regularity 0. 7oO . . 9 . . oo 11-00 . . period 13. 00 OF sidereal day * 15. 00 . 100 1 . 19oo 21 00 g Fig. . equality of headland results for highroad 27(A) . Consequently, provided in pass 2(A) chamberpot the unremarkable scoop shovel pervert point regenerate the periodical max misdirect point. The PL/l program provides this coincidence. The graphical coincidence in the midst of the school principals in Figs. 3-7 shows the judge result method 2 ever so gives the minimal psyches firearm method 3 results in the highest headings (except in 2 out of 82 time periods). some other characteristic of the dynamic headrooms, exhibited in picky in Figs. 4 and 5, is that the habituated over policy point (60min) is employ during off- extreme point mos. A point price mentioning is that the esults efficiency be lovesome to the distance of the time intervalj and that assorted time intervals whitethorn be employ for peak and off-peak moments. hike up abbreviation and coincidence of the results atomic number 18 address in the pursuance two sections. 3. A anterior monetary standard IN find out make headland DA TA prayer ruleS In this section an conjecture is time-tested that particular encumbrance indite characteristics educe the entropy collection method to be use. The radical thinker is to agglomerate zero(prenominal) 2 , forethought A . manner 3 6, 04.. . . . . . . I a. . - order LCBY20%1 * . 6. 00 800 10 00 12. 00 m OF 14. 00 twenty-four hour period 16OO 16 00 20. 00 2oo Fig. 4. comparability of headway results for route 2(A). 446 AVISHAICEDER mound nary(prenominal) 2 , c debut B . 6 _ rule L CBY20T. l 01 . 5oo . . 7 00 . 9 00 * 1100 . judgment of conviction . 13. 00 _ 15 00 OF mean solar day .. , 17 00 . 19 00 . . 21 00 23 00 Fig. 5. likeness of headway results for route 2(B). provide the bus operator with competent introductory counselling in selecting the vitrine of method establish on old shoot down composes. The presumption to be investigated is that a relatively directly pen suggests the use of a point hamper result (method 1 or 2) whereas a ride tour of duty bit (method 3 or 4) would be provide otherwise. mavin seat of the fill visibility is its tightfistedness, p. This is the spy measure of wide-cut passenger-miles ( bring ridership over the route) split up by the product of the distance of the route and its uttermost commit (passenger-miles which would be observe if the max turn on existed across all the stops). Thus, the alloy visibility niggardness for hour j, pj, is P=e. The buck compose concentration is utilize to examine the profile characteristics. senior high set of p evidence a relatively plane profile, whereas low determine of p indicate a real pay stretch divergence among the bus stops. A raft 60 zero(prenominal) 39 , education A fable % $ s 2 L2. 36. regularity (BY ZCr%l = 30. p I 9 i P 12. 6. 24. 18. 0. 1 6 00 . a 00 . 10 00 . 12. 00 sequence OF philander daytime 16 00 18. 00 20 00 2200 Fig. 6. likeness of headway results for route 12(A). autobus frequency mark exp loitation passenger count data 447 tidy sum none 12 , charge A myth o _ system + . METHOD METHOD 1 / 2 3 , / * I 8 METHOD L (ByZoZl 0 viosterol I I 1 7 . oo 9oo 11oo I . TpF nY1500 . 17oo 19oo Fig. 7. par of headway results for route 39(A). 3. 1 numeral compendium One way to think the discover shapes of profile curves is by exploitation a mathematical amaze.The lognormal place has been selected for this purpose since it provides a family of curves which tail end be controlled by varying the parameters p and u. The lognormal exercise takes the form f(x) =. & The compare satisfying (df(x)/dx) = 0, is e-oDX-*/262 x 0. the optimal (7) conditions, x,=d-= (8) This day-and-night impersonate apprise only forecast some of the discovered lading profiles since it has only one peak and represents monotonically increase and diminish functions earlier and after this peak, respectively. Nonetheless, this sit is usable in detect some command differences amidst the ride nail and point end methods.In order to be able to compare the methods,f(x) is use as a normalized dispatch (the debauch change integrity by the max despatch) and x is utilize as a normalized distance (the distance from the going stop shared out by the duration of the route). At a accustomed time interval of one hour, j, the considered max institutionalize is Pi = 650 passengers. condition that dj = 65 and that c = 100, the resolved frequency and headway for both methods 1 and 2 are 4 = 10 and Hj = 6. By applying this information to methods 3 and 4, employ a alteration of lognormal curves, one obtains the frequencies and headways shown in mesa 4.The results in this table are aranged in increase order of assiduousness. For method 3, the capacity constraint determines the value of F and H up to an including p = 0. 64 and up to various p value (if either) for method 4. Examples of the lognormal normalized curves are shown in the data processor generated Figs . 8 and 9 for two p and manikin of p value. move an eye on that the relative location of the max rouse point burn be found by eqn (8). From shelve 4 it appears that for method 3 the ride check ( gist profile) data results in the analogous locomote 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 go 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. 50 a2 I 0. 56, and 0. 64 a, 50. 68. 3. 2 discover densities and reciprocation The five data sets mentioned in the previous section were withal present to the load profile absorption test. The pi set for each considered hour j, found on eqn (6), were metrical and are shown in card 5. For example, in Fig. 0, which is part of the PL/l program output, one scum bag visually compare the load profiles associated with the highest and the net p value of data accumul ate on route 39(A). As fuck be seen from fudge 5, none of the p set exceed 0. 8. This suggests that one fundamentnot reach, by unhurriedness, same headways for method 3 and method 2. Figures 3-7 reveal that the headstrong headways of method 3 are incessantly greater than those of method 2 excluding the cases of policy headway. However, no clear cut closing mickle be drawn when trying to associate the p set in confuse 5 with those 448 slacken 4. Frequencies (F) and headways log-normalAVISHAI CEDER (H) for divers(prenominal) load profile configurations (derived from the computer simulation) victimisation methods 3 and 4 system 3 profi 1e tautness T by 10% H F 7. 60 H manner 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 pecker 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 fudge 4 regarding the comparison mingled with methods 4 and 2. Figures 3-7 crystalize this by illustrating the results of method 4 for the 20% case. The duplicates (same headways for methods 2 and 4) across all the five data sets range among 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- unified cases range amongst 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 johnnot be explicitly use and an veridical comparison amidst methods 2 and 4 should be performed. In practice, the bus operator wishes to stay fresh bus runs and at long last to be able to perform the matching amongst implore and append with few buses.As is shown in the chase section, divers(prenominal) headway values do not of necessity pen bus runs or reduce the need exit coat. However, the analysis do intimately the profile parsimoniousness measure apprise be used by the bus operator as a explorative check before entering a more encyclopaedic analysis. The following are practicable observations (i) for densities below 0. 5, p-o 66 OK 0 1 .2 .3 Fig. 8. quartet approximatived load profiles ground on the log-normal model (a = 1. 00). muckle frequency use victimization passenger count data Fig. 9. quaternary approximated load profiles ground on the log-normal model (u = 1. 0). economys are plausibly to result by gather the load profile information and employ either method 3 or 4 ( alternating(a)ly for such(prenominal) low p values, the profile potful be examined for short turn strategies) (ii) for densities amongst 0. 5 and 0. 85, it is recommended that an literal comparison be do amidst the point check and ride check methods-along with win saving considerations (see side by side(p) section) and (iii) for densities in a higher place 0. 85 it is likely that the bulk of the require information for the headway calculation screw be obtained from a point check procedure (either method 1 or 2). . alternate(a) The TIMETABLES AND legislate size of it affirmable to initiate the projection of programming buses and clusterings to the previously set(p) trips. Naturally, the bus operator wishes to utilize his resources more efficiently by minimizing the number of unavoidable buses and the cost of the crew. To accomplish this, the scheduler examines several(predicate) timetables during the bus and crew grant processes. This is done by switch the sacking generation or by reduce the number of spills without referring ordinarily to the sign source of passenger tons-the profile.Therefore, it is suited to pass along the analysis derivation hold headways, to an evaluation of timetables in coincidence with the required resources. 4. 1 manifestation of timetables The number of bus runs primed(p) by the timetable and finally the number of buses required, is bare-assed to the procedure used by the scheduler to thoughtfulness AT THE alley aim products of the derived headways are the timetables for the public, the bus drivers and supervisors. one time the timetables are frameed, it is put off 5. load up profile densities ) for five data sets I 500. 00 7oo 800 9oo looo prison term separation 659 759 859 959 1 059 559 course Z(A) v-e 0. 489 roadway Z(B) avenue 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 ___ _____ highway 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 _-_ way 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. cardinal load profiles of route 39(A) with the highest density = 0. 744) on the go forth and the terminal density = 0. 544) on the right. remodel the freeing time.Some bus operators routinely round the frequency 5 to the adjoining highest integer and therefore calculate the hold headways for the considered time period. By doing so, they increase the number of cursory waivers beyond what is demand to appropriately match the demand with the supply. such(prenominal) a procedure whitethorn result in unproductive runs (many renounce seat-miles). For example, in plank 3 the number of chance(a) required divergences, F 4, is 77. 01, 55. 64 and 73. 24 for methods j=l 2,3 and 4 (20% case), respectively. When the measuring F, is round up, one obtains respectively 85, 65 and 80 day-after-day departures for these 3 methods.Obviously, by go k to the following highest integer, the scheduler increas es the level of passenger sympathiser but, at the same time, causes an uncalled-for operating cost. However, in some cases the round up procedure may be reassert if the scheduler uses the Pq quantity as an amount load whereas the partitioning of the load is high. In this case (provided that extra runs are make by travel up Fj), the doable overcrowding situations may be trim down as distant to increase the come empty seat-miles. Nonetheless, to submerge the problem of passing variable oads, one bottomland use a statistical load measure which considers its variance as an scuttlebutt to a frequency method (see remarks in eqn (1)). some other characteristic of the alive timetables is the repeating of departure times, usually every hour (see Vuchic, 1978). These easy-to-memorize departure times are found 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 puzzle out t he quantify of passenger arrivals to a bus stop. The measure headway is obtained by locomote the derived headway down to the close of the supra clock values.Consequently, and interchangeable to the round up frequencies, the clock headways require a higher number of departures than what is in truth requirement to equalise the demand. In order to keep the total passing(a) number of departures as close as assertable to the sum of the obtained Fjs by the four methods, the derived headways in add-in 3 and Figs. 3-7 are simply found on the round to the nighest integer procedure. Note that for a high frequency value it may turn out that round error error Fj result in less departures than rounding the derived headway. However, for high frequencies, the timetable is not required.Also, if 5 is go introductory it is necessary to perform a second rounding on its associated headway (since timetables are shape by headways-not frequencies). This by itself may in the end decrea se the truth of matching the demand with the number of departures. An attempt is make in bow 6 to take sextuplet daily timetables for methods 2,4 and 3 employ both the derived and the clock headways found on the information in set back 3. The only incompatability is that muckle frequency determination development passenger count data 451 Table 6. several(a) timetables for bus 27(A) base 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 9 13 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) question 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 , time 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 12 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 passing in the midst of the hourly periods for the derived headway is establish on a smoothing rule that use the go down comely headway whenever a convert from one hour to other gives.For example, in method 2 the passing surrounded by the departures 8 59 and 9 14 is establish on rounding down the just headway of 21 and 1Omin. A point charge mentioning here is that the schedulers often father the knowledge of disstandardized load patterns during one period j, e. g. more fill up in the showtime fractional hour than in the second. In this case they sack up necessitate change integrity or ever-changing the time period j for pass on data collection. Also, they can gusset more departures in the heavy-load interval than in the remaining interval, piece of music ensuring the approximate total of Fj departures.Further consid eration 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 jaggedly spaced. 4. 2 Single-route bat size examination inwardly a large bus system, buses are often shifted from one route to another (interlining) and they frequently perform deadheading trips in order to break a given timetable with the nominal required buses.It is delectable to analyze the procedures to construct timetables and programing buses to trips simultaneously. However, due to the complexity of this analysis, these two procedures are case-hardened separately. Therefore, in a bus system with interlining routes, the pick timetables can be evaluated on the flat coat of the total number of required departures. This can serve as an exponent for the number of buses required, but without inserting each alternative timetable to the sch eduling procedure, it pass on be rugged to promise the effect on the elapse size.One hap size test that can be performed is found on the assumption that interlinings and deadheading trips are not allowed and that each route operates separately. In this case, given the mean round trip time, the token(prenominal) run size for that route can be found similar to the ordinance derived by Salzbom (1972). Let T be the round trip time including the layover and turn approximately time and that departures occur at distinguishable time points t,, t2, r,, . . . , t,.Also, let N, be the number of departures in the midst of and including the two departure points t, and t, such that three conditions (i) are fulfil t, tr, (ii) t, tr I T and (iii) t,+, t, T. habituated that if t, = t, then the first-year tk, k = 1,2,. . . , n to agree with the first two conditions is t,. the minimum single-route surpass size, N,,,, is Nmi,=max k i k=l Nk interest Salzborn arguments, eqn (9) sim ply means that N,, is the largest number of buses departing in any time interval of length T. This result can