Automated optimal design methodology of elevator systems

Article

Automated optimal designmethodology of elevator systemsusing rules and graphical methods(the HARint plane)

Lutfi Al-Sharif, Ahmad M Abu Alqumsan andOsama F Abdel Aal

Abstract

The design of an elevator system heavily relies on the calculation of the round-trip time under up-peak

(incoming) traffic conditions. The round-trip time can either be calculated analytically or by the use of

Monte Carlo simulation. However, the calculation of the round-trip time is only part of the design

methodology. This paper does not discuss the round-trip time calculation methodology as this has

been addressed in detail elsewhere. This paper presents a step-by-step automated design methodology

which gives the optimum number of elevators in very specific, constrained arrival situations. A range of

situations can be considered and a judgement can be made as to what is the best cost–performance

tradeoff. It uses the round trip value calculated by the use of other tools to automatically arrive at an

optimal elevator design for a building. It employs rules and graphical methods. The methodology starts

from the user requirements in the form of three parameters: the target interval; the expected passenger

arrival rate (AR%) which is the passenger arrival in the busiest 5 min expressed as a percentage of the

building population; and the total building population. Using these requirements, the expected number of

passengers boarding an elevator car is calculated. Then, the round-trip time is calculated (using other

tools) and the optimum number of elevators is calculated. Further iterations are carried out to refine the

actual number of passengers boarding the elevator and the actual achieved target. The optimal car capacity

is then calculated based on the final expected passengers boarding the car. The HARint plane is presented

as a graphical tool that allows the designer to visualise the solution. Three different rated speeds are

suggested and used in order to explore the possibility of reducing the number of elevator cars. Moreover,

the average passenger travel time is used to indicate the need for zoning of buildings.

Practical application: This paper has an important application in allowing the designer to arrive at the

optimum design for the elevator system using a clearly defined methodology. This ensures that the

number of elevators, their speed and their capacity are optimised, thus ensuring that the cost of the

elevator system and the space it occupies within the building are minimised. The method also employs a

Mechatronics Engineering Department, University of Jordan,

Amman, Jordan

Corresponding author:

Lutfi Al-Sharif, Mechatronics Engineering Department,

University of Jordan, Amman 11942, Jordan.

Email: [email protected]

Building Serv. Eng. Res. Technol.

34(3) 275–293

! The Chartered Institution of Building

Services Engineers 2012

DOI: 10.1177/0143624412441615

bse.sagepub.com

at PENNSYLVANIA STATE UNIV on May 11, 2016bse.sagepub.comDownloaded from

http://bse.sagepub.com/

graphical method (the HARint) in order to allow the designer to visualise the optimality and the feasibility

of the different design options.

Keywords

Elevator, lift, round-trip time, interval, up-peak traffic, rule base, Monte Carlo simulation, average travel

time, HARint plane

Introduction

The round-trip time is the time needed by theelevator to complete a full journey in the build-ing, taking passengers from the main entrance/entrances and delivering them to their destin-ations and then expressing back to the mainentrance, under up-peak (incoming) trafficconditions.

However, having calculated the round-triptime, no non-proprietary automated optimalcalculation methodology has been publishedthat guides the designer to find a suitable opti-mum design. Most methods rely on trial anderror, and might not lead to an optimal design.The car capacities are usually oversized; thehandling capacity might be in excess of the arri-val rate.

This paper presents a step-by-step automatedmethod for elevator design under specific arrivalconditions, assuming that a method exists forcalculating the round-trip time. It uses a com-bination of rules and graphical methods toarrive at an optimal solution. It allows thedesigner to find the optimum number of cars,the optimum elevator speed and the minimalcar capacity (CC). It also presents the designerwith an objective criterion as to when to split thebuilding into zones based on the average traveltime of the passengers. A graphical tool, calledthe HARint plane, is presented as a means tovisualise the solution. Having obtained ananswer for the specific arrival condition, thedesigner can then examine other arrival condi-tions and carry out a cost–performance tradeoff.

The fact that the methodology is fully auto-mated makes it very attractive for implementa-tion as a software package. It has also beenused for teaching elevator traffic analysis tofinal-year undergraduate mechatronics engineer-ing students.

Commentary on the methods used for calcu-lating the round-trip time is presented in the sec-tion Calculation of the round-trip time. Thedrawbacks of the existing elevator systemdesign methods are discussed in section Theproblem with conventional design methods,where it shows that the results from these meth-ods are not optimal and can be wasteful. Thesection Analysis and development of the formulaeis the core of this paper, where it presents the fullmathematical analysis for the method. The sec-tion Graphical representation: the HARint planeshows how graphical tools can visually conveythe optimum solution. The section Examplesprovides two examples to illustrate the method-ology. The section Notes on optimality and con-vergence presents two important points relatingto the optimality of the method and the assur-ance of convergence to a solution. The sectionTriggers for zoning provides another example toillustrate the use of the passenger average travel-ling time as a trigger for zoning. The last sectiondraws conclusions from the work.

Calculation of the round-trip time

The design of elevator systems is a multi-objective optimisation problem. It aims tominimise a number of parameters: the number,

276 Journal of Building Services Engineering Research & Technology 34(3)



capacity and speed of the elevators used; theaverage waiting time and the average travellingtime of the passengers using the elevators; theenergy consumption of the elevators and thecore space used. This paper concentrates onminimising the number, speed and capacity ofthe elevators, while meeting the quality of ser-vice and quantity of service requirement as sti-pulated by the client.

The calculation of the round-trip time is thebasis for the classical design method of elevatordesign. This paper does not address the issue ofcalculating the round trip or the passenger aver-age travelling time. It assumes that these vari-ables have been calculated by other means. Thecalculation of the round-trip time can either becarried out by the use of analytical equationssuch as those outlined in References [1]–[8] orby the use of the Monte Carlo simulationmethod as outlined in Reference [9].

The problem with conventionaldesign methods

The following is an overview of the traditionalelevator design method. The round-trip time isusually calculated analytically using equation (1)below assuming a single entrance and up-peaktraffic conditions

�¼2 �H � tvð Þþ Sþ1ð Þ � tf� tvþ tdoþ tdcþ tsd� tao� �

þP tpiþ tpo� �

ð1Þ

� is the round-trip time in s.H is the highest reversal floor (where floors are

numbered 0, 1, 2,. . ., N.S is the probable number of stops.df is the typical height of one floor in m.tf is the time taken to complete a one floor jour-

ney in s.tv is the time taken to transverse one floor jour-

ney in s when travelling at rated speed.P is the number of passengers in the car when it

leaves the ground (does not need to be aninteger).

tdo is the door opening time in s.

tdc is the door closing time in s.

tsd is the motor start delay in s.

tao is the door advance opening time in s (wherethe door starts opening before the car comesto a complete standstill).

tpi is the passenger boarding time in s.

tpo is the passenger alighting time in s.

Other methods can be used, such as theMonte Carlo simulation method, to calculatethe round-trip time.9 The Monte Carlo simula-tion offers the advantage over equation (1) inthat it can provide an accurate value for theround trip even where any combination of thefollowing special conditions (or all of them)exist:

a. Unequal floor populations.

b. Unequal floor heights.

c. Multiple entrance floors.

d. Top speed not attained in one floor journey.

It will be assumed that the designer startswith the knowledge of the following parametersthat are given either by the architect or thebuilding owner or that can be inferred fromthe type of occupancy (e.g. office, residential,etc.). These represent the user requirements.

a. The total building population, U. If this isnot given directly, it can be calculated fromeither net floor area or the gross floor area.

b. The expected arrival rate, AR%. This is thepercentage of the building population arriv-ing in the building during the busiest 5min.This value depends on the type of buildingoccupancy.

c. The target interval.

A sufficient design meets the following twoconditions

HC% � AR% ð2Þ

intact � inttar ð3Þ

Al-Sharif et al. 277



where HC% is the handling capacity expressedas a percentage of the building population in5min.

A design that meets equations (2) and (3) isan acceptable design, but might not be an opti-mum design (i.e. it could be a wastefuldesign). The optimum design is one that meetsthe two equations shown below, equations (4)and (5).

HC% � AR% ð4Þ

intact ¼ inttar ð5Þ

In practice however, it is nearly impossible tofind a design that meets both of equations (4)and (5) above. This is due the fact that thenumber of cars in the group, L, cannot be afraction (it has to be a whole number). Hence,in practice, an optimum solution will satisfy thetwo equations shown below

HC% ¼ AR% ð6Þ

intact 5 inttar ð7Þ

This section illustrates the main problem withthe conventional design method. It relies on theuser picking a suitable speed, v, and a suitablecar capacity, CC. The user then assumes that thecars will fill up to 80% of the CC.

The round-trip time is then calculated basedon the selected speed and the selected CC usingequation (1) or any other suitable means. Thisprovides a value for the round-trip time, �.

Dividing the round-trip time by the targetinterval and rounding up the answer providesthe required number of elevators.

The user has now two values that representthe qualitative and quantitative performance ofthe systems: The handling capacity and theactual value of the interval, respectively.

Comparing these values to the desired valuesresults in four possible cases discussed in detailbelow.

Quantitative

design

criterion

Qualitative design criterion

intact 4 inttar intact 5 inttar

HC%< AR% Case I Case II

Unacceptable

design. Cannot be

addressed by redu-

cing the car loading.

The designer will

have to increase

the number of

elevators and

repeat the analysis.

Unacceptable

design. Might be

addressed by

increasing the car

loading and using a

larger car capacity if

needed.

HC%> AR% Case III Case IV

Unacceptable

design. But, it might

be addressed by

reducing the car

loading.

Acceptable design

but might not be

an optimum one.

There may be fur-

ther scope in redu-

cing the number of

elevators, reducing

the rated speed or

both.

Specifically, there are two problems with thismethod:

1. In the three cases where the design isunacceptable, the designer does not have aclear set of rules of how to move to an accept-able design (as defined in Case IV). It is amixture of judgement, experience and trialand error.

2. Even where the user manages to get to anacceptable design by arriving at Case IV,he/she cannot be sure that he/she has an opti-mum solution, despite the fact that the designmeets both qualitative and quantitative cri-teria. The designer will have to do furthertrial and error iterations to check thatthe design is optimum (e.g. further reducethe number of elevators, L, and thenrepeat the calculation of the round-triptime). The main reason for this is that thedesigner starts from an arbitrary car sizeand assumes it fills up to 80% of its capacity




rather than calculating the actual passengerarrival expected.

The next section attempts to address thedrawback with this traditional methodology.

Analysis and development of theformulae

The design methodology developed in this sec-tion allows the designer to arrive directly at adesign that is optimum and in fixed number ofsteps without the need for trial and errorsearches or iterations. This section develops themethod and the associated formulae.

Developing a clearly defined methodology fordesign with concrete steps offers the followingadvantages:

1. It allows designers to carry out the designregardless of their level of expertise, throughclearly defined sets of rules.

2. It offers the opportunity to automate thedesign process in software.

The methodology presented here uses the fol-lowing rule:

The following parameters should be minimised inan optimal design in the following order of import-ance (that reflect the cost of the wholeinstallation):

a. Number of elevators.b. Elevator speed.c. Elevator capacity.

So where two solutions have different number ofelevators, the one with fewer elevators is selected;for solutions with the same number of elevators,the one with lower speed is selected; for solutionswith the same number of elevators and the samespeed, the one with the smaller CC is selected.

Nevertheless, it is accepted that there are situ-ations where the order of priority above is not

correct (e.g. the restricted headroom in thebuilding might restrict the rated elevator speedand force the designer to use a larger number ofelevators in order to force a lower rated speed).In such conditions, the designer can alter therule for the priorities and select the answersaccordingly.

Optimising the number of elevators

The design process starts by finding the actualnumber of passengers that will board the eleva-tor in any round trip journey. In effect, this is thenumber of passengers that will board the eleva-tor from the main entrance (in the case of asingle-entrance arrangement) or the number ofpassengers in the car when it leaves the highestarrival floor heading upwards to the destinationfloors (in case of multiple contiguous entrances).This depends on three parameters that are allknown at the start of the design process andare usually provided by the client, the developeror the architect. These are the target interval,inttar, the arrival rate, AR%, and the total build-ing population, U. This is shown in equation (9)below.

The number of passengers arriving in thepeak 5min can be found by multiplying the arri-val rate by the total population, as shown below(the 5-min period has traditionally been used asthe design basis in elevator systems)

P5 min ¼ AR% �U ð9Þ

The arrival rate can then be expressed in unitsof persons per second by dividing by 300 s/min.

l ¼AR% �U

300

� �ð10Þ

Thus, an initial estimate of the actual numberof passengers that will arrive in a single intervalcan be found by multiplying the target intervalby the arrival rate of passengers.

Pact i ¼ inttar � lð Þ ð11Þ




The subscript i denotes the fact that it is aninitial estimate. It is worth noting that equation(11) is used in Reference [3] as a tool to assessthe actual interval at partial car loading but notas a sizing tool.

There is no need at this stage to consider theCC. This can be done later when the finalnumber of the passengers in the car has beendetermined.

Having arrived at an estimate for the actualnumber of passengers in the car, the next step isto find the corresponding round-trip time. Usinga classical method of calculating the round-triptime or using Monte Carlo simulation, the valueof the round-trip time can be found, where oneor more of the four special conditions exist, thena variation of equation (1) can be used or onecan resort to the Monte Carlo simulationmethod. The round-trip time is in effect a func-tion of the actual number of passengers if allother parameters are kept constant (such asthe kinematics, number of floors, total buildingpopulation, door timings, floor heights).This provides an initial value for the round-trip time.

�i ¼ f Pactð Þ ð12Þ

From the calculated value of the round-triptime, the required number of elevators can becalculated

L ¼ ROUNDUP�i

inttar

� �ð13Þ

It is worth noting that this act of rounding upis unavoidable as a whole number of elevatorscan only be selected. The resultant number ofelevators, L, is the nearest to the optimum, aspractically as possible.

Due to the process of rounding up to find thewhole, the actual value of the interval will beslightly lower than the target interval, and theactual value of the handling capacity,HC%, willbe slightly higher than the arrival rate, AR%.

The actual value of the interval will befound by dividing the round-trip time by the

number of elevators, as shown below in equa-tion (14)

intact i ¼�iL

ð14Þ

The actual handling capacity can also befound by using equation (15) below

HC%i ¼300 � Pact i

U � intact ið15Þ

This provides an acceptable solution thatsatisfies Case IV discussed in the last section.This is the optimum number of elevatorsrequired to meet the design criterion. However,it is not an optimum design regarding therequired CC. The next subsection examines themethodology to find the optimum CC.

Optimising the car capacity

The solution developed so far optimises thenumber of elevators, but does not provide theoptimum CC. Thus, such a solution is not com-plete, as it will not exist in practice due to thefact that the actual arrival rate, AR%, is lessthan the handling capacity, HC%i, and hence,fewer passengers will fill up car than Pact i. Ineffect, the actual number of passengers will befewer than Pact i. This results in a value of theinterval that is lower than the actual interval,intact i.

So, the car loading has to be graduallyreduced until the handling capacity, HC%i, isexactly equal to the arrival rate, AR%. Thisgives the final value of the interval, intact f, thatis less than the target interval, inttar, and a newvalue for the number of passengers that we willdenote as Pact f. This is carried out by applyingthe four equations (16)–(19).

Pact f ¼ l � intact i ð16Þ

This new value of Pact f is then used to evalu-ate the new value of the round-trip time usingeither equation (1), Monte Carlo simulation, or




any other suitable method. This is simply shownbelow as the evaluation of the round-trip timebased on the new value of Pact f.

�f ¼ f Pact f

� �ð17Þ

This resultant value of the round-trip time isthen used to re-evaluate the final value of theinterval, as shown below.

intact f ¼�fL

ð18Þ

This value of the interval is then set to theoriginal value of the interval in order to carryout another iteration of equations (16)–(18)

intact i ¼ intact f ð19Þ

The four equations (16)–(19) are thenrepeated in a loop as many times as necessary,until an acceptable convergence has beenachieved. For example, an acceptable conver-gence criterion might be set such that the iter-ation process is stopped if the change in P islower than a certain predetermined limit, asshown in equation (20) below. This effectivelyprovides a tool for specifying the resolution ofthe final solution.

Pact i � Pact f 5�Pmin ð20Þ

From the value of Pact f, the required CC canbe calculated by dividing Pact f by the assumedmaximum car loading factor and then roundingup to the nearest standard CC size. The max-imum car loading factor has been assumedhere to be 80%, but any other value can beused depending on the application.

CC ¼ ROUNDUPPact f

0:8

� �ð21Þ

It is important to note that the rounding upfunction shown in equation (21) above does notmerely round up to the nearest integer; it alsorounds up to the nearest CC preferred size (in

units of passengers). Examples of standard carsizes are 8, 10, 13, 16, 21 and 26 persons.

The actual car loading can then be found bydividing the number of passengers by the CC

CL% ¼Pact f

CCð22Þ

By definition, the final value of the handlingcapacity is equal to the arrival rate, AR%(depending the convergence criterion shown inequation (20))

HC%f ¼ AR% ð23Þ

And, the final value of the interval, intact f,is less than the initial estimate of the interval,intact i, which in turn is less than the target inter-val, inttar

intact f 5 intact i 5 inttar ð24Þ

This final optimum solution optimises the CCas well as the number of elevators. It is repre-sented by the three parameters

1. The number of elevators, L.2. The car capacity, CC.3. The car loading, CL%.

Its performance is characterised by the twoequations that set the quantitative and qualita-tive performance, respectively, reproducedbelow

HC%f ¼ AR% ð25Þ

intact f 5 inttar ð26Þ

Optimising the rated speed of the elevators

The development so far has assumed that thevalue of the rated speed is set. In many situ-ations this is not the case, and the designer hasthe freedom to find an optimum value for therated elevator speed.




A general accepted industry standard is tofind the rated speed by dividing the total traveldistance by 20. This is an approximation thatallows the elevator to travel the distancebetween terminal floors in 20 s. In practice, theelevator will take longer to do this due to thetime required for accelerating and decelerating,and placing a limit on the value of the jerk.

It is thus suggested that three initial values forthe speed are found by dividing by 20, 25 and 30and rounding up the answer to the neareststandard speed (e.g. 1.6m/s, 2.0m/s, 2.5m/s,3.15m/s, 4.0m/s, etc.).

v1 ¼dtotal20

ð22Þ

v2 ¼dtotal25

ð23Þ

v3 ¼dtotal30

ð24Þ

Using each one of the three suggested speedsfrom equations (22) to (24), the full-design pro-cess defined by equations (9)–(21) is repeated

giving three possible solutions. It is worthnoting that equation (12) that should be modi-fied in order to clearly show the round-trip timeis now dependent on both the actual number ofpassenger in the car as well as the rated speed.This is shown below in equation (25).

�i ¼ f Pact, vnð Þ

n ¼ 1, 2, 3ð25Þ

The three design solutions are then com-pared. The solution that provides the fewestnumber of elevators is chosen. Where two solu-tions require the same number of elevators, thesolution with the lower speed is selected. Thisrationale is based on the assumption that redu-cing the number of elevators results in a costsaving more than the reduction in speed offers.

A general block diagram of the software isshown in Figure 1. The user requirements areshown inside a ‘cloud’ and they specify the per-formance requirements that must be achieved.The arrival rate is extracted from these param-eters, as well as the expected number of passen-gers to arrive in the interval. A round-trip timecalculator (not described in this paper) is used to

Total travel in m

(only used for the first estimate)Target interval (inttar) Target interval (inttar)

Target interval (inttar)

Arrival rate (AR%)

User Requirements

Total building population (U)

Suggestedrated speedscalculator (3suggestions)

Building andelevator data(e.g.,numberof floors, door

times...)

Building and elevatordata

Round trip timecalculator (Analytical or

Monte Carlo)

arrival ratecalculator(pass/sec)

Arrivalrate

(pass/s)

Actualno.ofpass.)

Roundtrip

time (s)

No. ofelevatorscalculator No. of

elevatorsN.B Finalanswer;

not affectedby iterations

Actualinterval

calculator(intoct)

Actualintervalintoct

Actual intervalintoct

Passenger calculator

(pass)

Iterations repeated until convergence achieved

Rated speeds (V1,V2,V3)

v

l tPactL

Figure 1. Block diagram showing the automated optimal design methodology.




calculate the round time based on the buildingparameters and the number of passengers. Thisvalue of round trip is divided by the target inter-val in order to find the required number ofelevators.

An iteration loop is then followed by usingthe actual value of interval to find a new valuefor the number of passengers expected to arriveduring the interval. This is then used again tofind a new value for the round-trip time, untilfinal convergence is achieved. From this finalvalue of the round-trip time, the final value ofthe interval and the final value of the passengersarriving during the interval are calculated.

Graphical representation: TheHARint plane

The methodology described in the last sectioncan be represented in a graphical format. Theaim of the graphical representation in this case

is to allow the designer to understand the effectof changes on the resulting solution and be ableto assess how far it is from the optimumsolution.

In order to develop the graphical representa-tion, a plane is presented. This plane has twoaxes; the x-axis represents the interval in secondsand the y-axis represents the handing capacity.Each point on the plane represents a possiblesolution (not necessarily an acceptable or correctone). The point representing the optimum solu-tion can be located by the intersection of thevertical line representing inttar and the horizon-tal line representing AR%, as shown in Figure 2.The plane is referred to as the HARint plane, asit contains the HC% and the AR% on the y-axisand the int on the x-axis (HCARint abbreviatedto HARint).

Plotting lines of equal L (number of eleva-tors) values produces the HARint plane shownin Figure 3. Plotting lines of equal P (number of

Direction of the arrow showsincreasingly wasteful design

The handlingcapcity,HC%

AR%

The HCINTPlane

OptimumHypothetical

Solution

Case IV: Rangeof acceptable

solutions

Case III: Design fails tomeet qualityrequirment

Case II: Design fails tomeet quantity

requirment

The interval, int (s)

Case I: Designfails to meet both

requirment

HC%=AR%

int=inttar

inttar

Figure 2. The HARint plane.




passengers) produces the HARint plane shownin Figure 4. The HARint plane can be veryuseful in visualising a specific solution andappreciating the optimality or otherwise of sug-gested solutions.

Figure 5 shows the position of the hypothet-ical optimum solution on the HARint planewhich is the intersection point of the AR% hori-zontal line and the inttar vertical line. It is hypo-thetical because it is not achievable in practice asit requires a fractional number of elevators, L.Applying the rounding up equation (13) andthen applying the iterations from equations(16) to (19) moves the solution to the practicaloptimum solution (that lies on the AR% lineand uses a whole number of elevators, L. Thisis shown in more detail in Figure 6 where theintermediate point is also shown (after applyingequation (13) and before applying theiterations).

The use of the HARint plane shown inFigures 2–6 has been useful for visualising thesolution but has not been used to actually find asolution by the use of graphical methods. Itmight be possible to find a graphical methodof finding a solution for a problem by theusing the HARint plane as a solution chart(e.g. as the Smith chart is used in radio engin-eering and the Nichols chart is used in controlsystems). However, before this can be achieved,a method of normalisation needs to be intro-duced in order to make the HARint plane a uni-versal tool.

Examples

A number of numerical examples will be pre-sented in this section, in order to illustrate themethodology developed in the section Analysisand development of the formulae.

The handlingcapacity,

HC%

AR%

inttar

L=6L=5

L=4

L=3

L=2

L=1

L=3,4

The HCINTPlane

The interval, int (s)

Figure 3. The lines on the HARint plane that represent different values for L.




Example 1

A vertical transportation design is required foran office building with 10 floors above themain entrance, no basements and a total popu-lation of 550 persons. A target interval of 30 s isset by the client. An arrival rate of 12% is alsogiven.

Other parameters are given as shown below:

a. All floor heights are equal, at 4.5m perfloor.

b. Rated speed is 1.6m/s.c. Rated acceleration is 1m/s2.d. Rated jerk is given as 1m/s3.e. Door opening time is given as 2 s.f. Door closing time is given as 3 s.g. Start delay is given as 1 s.h. Advance door opening is given as 0.5 s.i. Passenger boarding time is given as 1.2 sj. Passenger alighting time is given as 1.2 s.

Applying equation (9) provides a value forthe number of passengers arriving in the busiest5min

P5 min ¼ AR% �U ¼ 12% � 550 ¼ 66 pass=5 min

ð26Þ

Applying equation (10) gives the value of thepassenger arrival rate in units of passenger persecond

l ¼P5 min

300

� �¼

66

300¼ 0:22 pass=s ð27Þ

Using equation (11) provides an initial valuefor the number of passengers boarding each car

Pact i ¼ inttar � lð Þ ¼ 30 � 0:22 ¼ 6:6 pass ð28Þ

0.35

0.3

0.25

0.2

HC

0.15

0.1

0.05

010 20 30 40

INT50 60 70 80

Figure 4. HARint plane with curves of equal L and equal P added and the optimum solution marked with a small

circle.




Using equation (1) and this initial value forPact i of 6.6 passengers provide an initial valuefor the round-trip time

�i ¼ f Pactð Þ ¼ f 6:6ð Þ ¼ 115:9 s ð29Þ

Using equation (30) provides the optimumnumber of elevators

L ¼ ROUNDUP�i

inttar

� �

¼ ROUNDUP115:9

30

� �¼ 4 elevators

ð30Þ

As mentioned previously, this act of roundingup (from 3.86 to 4) is inevitable, as a wholenumber of elevators must be used. It is thereason why the two equations (4) and (5) thatrepresent the ultimate optimum solution cannotbe simultaneously met, and why the designer hasto resort to simultaneously meeting the othertwo equations (6) and (7).

Once the optimum number of elevators hasbeen determined, the next step is to determinethe optimum CC by iteratively finding the actual

The handlingcapacity, HC%

The HARint Plane

inttarThe interval, int (s)

Practical optimumsolution

hypotheticaloptimum solution

AR%

P=6

P=7

L=6L=5

L=4

L=3

L=1L=2

L=3,4

Figure 5. Hypothetical optimum solution and practical optimum solution on the HARint plane.

IterationsEqu. (16)-(19)

Rounding upEqu. (13)

Figure 6. Details of movement from hypothetical

optimum to practical optimum.




number of passenger boarding the car and theactual achieve interval.

The first step is to find the initial estimate forthe interval that will be achieved in practice,based on the optimum number of elevatorsand the initial estimate of the round-trip time.This is done using equation (14)

intact i ¼�iL¼

115:9

4¼ 28:98 s ð31Þ

The interval has been given the subscript i toemphasise that it is an initial estimate. As thisvalue is lower than the target interval, inttar, andconsidering that the arrival rate in passengersper second is still constant (l), the actualnumber of passengers that will in fact boardeach car will be lower, and an estimate for itcan be found using equation (16)

Pact f ¼ l � intact i ¼ 0:22 � 28:98 ¼ 6:375 s ð32Þ

This new value of Pact f is then used to evalu-ate the new value of the round-trip time usingeither equation (1), Monte Carlo simulation, orany other suitable method. This provides a newvalue for the round-trip time

�f ¼ f Pact f

� �¼ f 6:375ð Þ ¼ 114:2 s ð33Þ

Using equation (18), a new estimate of theactual interval (given a subscript f to denotefinal) can found

intact f ¼�fL¼

114:2

4¼ 28:55 s ð34Þ

This value of the interval is then set to theoriginal value of the interval in order to carryout another iteration of equations (32)–(34)

intact i ¼ intact f ¼ 28:55 s ð35Þ

Following this first iteration, a number ofiterations can now be carried out using equa-tions (32)–(35), until convergence is deemed tohave taken place in the value of Pact. Applyingfour more iterations, the results are shown inTable 1.

It can be seen that change in Pact in the lastiteration was around 0.01 which is deemedsufficient for convergence.

Using equation (21), the optimum car carry-ing capacity (taking into consideration the pre-ferred value) can be calculated as follows

CC ¼ ROUNDUPPact f

0:8

� �

¼ ROUNDUP6:21

0:8

� �¼ 8 persons

ð36Þ

Where the car carrying capacity of eight ispreferred value and corresponds to a preferred

Table 1. Iteration results for Example 1.

intact i using

equation (35)

Pact f using

equation (32)

�i using

equation (1) or

Monte Carlo

simulation

intact f using

equation (34)

Second iteration 28.55 6.281 113.4 28.35

Third iteration 28.35 6.237 113.1 28.275

Fourth iteration 28.275 6.22 112.95 28.225

Fifth iteration 28.225 6.21 112.87 28.218




load of 630 kg. The actual car loading can thenbe found using equation (22)

CL% ¼Pact f

CC¼

6:21

8¼ 77:6% ð37Þ

This result provides an optimum solution forthe selected speed of 1.6m/s, as shown in Table 2(described by the four parameters: number ofcars, car carrying capacity, car loading and topspeed). The table also shows the resultantround-trip time and the resultant interval

The fact that the actual number of passengersis not a whole number is not a problem, as itrepresents an average value over the consecutiveround trip journeys.

As expected, the final handling capacity isnearly equal to the arrival rate (AR%), as ver-ified by equation (38)

HC%i ¼300 � Pact i

U � intact i¼

300 � 6:21

550 � 28:218

¼ 12:004% � 12% ð38Þ

The solution based on a speed of 1.6m/s isshown graphically on the HARint plane inFigure 7 below, where the final optimum solu-tion is shown by the small circle.

However, the solution shown in Table 2 isonly applicable to a speed of 1.6m/s. It is pos-sible that using a speed of 2m/s or a speed of2.5m/s could result in a more economical solu-tion by reducing the number of elevators. Thecomplete design process is then carried out byapplying equations (9)–(22) to the two caseswith speeds of 2m/s and 2.5m/s.

With these higher speeds, it is no longer pos-sible to use equation (1) as the top speed will notbe attained in a one-floor journey. Monte Carlo

simulation will be used in this case to produce asolution.

Examination of Table 3 reveals that the speedof 1.6 results in the most economical solution, asthe solutions using the other two speeds havenot resulted in reduction in the number of ele-vators. Thus, the optimum solution is the onethat uses four elevators running at a speed of1.6m/s and rated at a car carrying capacity ofeight persons (630 kg). The car loading in thiscase will be 77.6%.

Example 2

Example 1 will be repeated with a larger popu-lation of 650 persons. The three speeds used arethe same as the total travel distance has notchanged. The results for the three speeds areshown in Table 4.

Examination of Table 4 reveals that the use ofa slightly higher speed of 2m/s instead of 1.6m/shas resulted in the saving of one elevator. So theoptimum solution, in this case, employs four ele-vators running at a speed of 2m/s with a ratedcar carrying capacity of 10 persons. The carloading is 70.4%.

The solution utilising 2m/s speed is shown ina graphical format on the HARint plane inFigure 8 where the small circle represents thefinal solution.

Notes on optimality andconvergence

Equations (11) and (13) are fundamental to thesuccess of the method. Two points are in order,relating to optimality and convergence.

The first point relates to optimality. The useof equation (11) ensures that the number of ele-vators, L, calculated in equation (13) is optimal.

Table 2. Optimum design solution for Example 1 at a speed of 1.6 m/s.

v (m/s1) L CC (persons) CL% Pact (persons) � (s) intact (s)

1.6 4 8 77.6% 6.21 112.87 28.218




It will not change no matter how many iter-ations are carried out and no matter whatvalue for Pact is used.

The second point relates to convergence ofthe solution. The rounding up that is appliedto the number of elevators (L) in equation (13)is essential for convergence and finding the cor-rect solution. It ensures that the final value ofthe actual interval converges to its correct valuefollowing a sufficient number of iterations.In the first iteration, the actual number of pas-sengers reduces, and this leads to a reduction inthe value of the round-trip time that, in turn,leads to a reduction in the value of the actualinterval. This in turn leads to a reduction in theactual number of passengers and so on.Eventually, the iterations converge and no fur-ther changes take place.

On the other hand, if rounding down hadbeen applied, then the value of the round-triptime and the actual interval would diverge to

0.6

0.5

0.4

HC

0.3

0.2

0.1

010 20 30 40

INT

50 7060 80

Figure 7. HARint plane showing solution of Example 1 (with a speed of 1.6 m/s).

Table 3. Optimum design solution for Example 1 for all

three speeds.

v (m/s1) L

CC

(persons) CL%

Pact

(persons) � (s)

intact

(s)

1.6 4 8 77.6% 6.21 112.87 28.218

2.0 4 8 68.1% 5.45 99.8 24.7

2.5 4 8 61.0% 4.862 88.3 22.1

Table 4. Optimum design solution for Example 2 for all

three speeds.

v (m/s1) L

CC

(persons) CL%

Pact

(persons) � (s)

intact

(s)

1.6 5 8 70.7% 5.655 108.84 21.75

2.0 4 10 70.4% 7.439 114.57 28.61

2.5 4 10 67.8% 6.779 104.21 26.07




infinity. As iterations are applied, the value ofthe round-trip time will increase, leading to anincrease in the interval, which leads in turn to anincrease in the actual number of passenger,which then leads to an increase in the value ofthe round-trip time and so on, until the value ofthe round-trip time reaches a very large valueand the CC reaches a very large unrealisticvalue.

This can also be clearly seen on the HARintplane, where rounding down leads to an L linethat is outside the solution zone and no matterhow many passengers board the car, the line cannever pass through the solution zone. Roundingup, however, gives an L line that passes throughthe solution zone.

In general, the iterations will converge if thefollowing condition applies

L4�i

inttarð39Þ

and, will diverge if the following conditionapplies

L5�i

inttarð40Þ

where �i is the initial result for the round-triptime found from equation (12).

Triggers for zoning

It is well known that buildings exceeding 20floors high are usually zoned. However, thereis no clear objective criterion that can be usedas a trigger, although some guidelines are pro-vided in Reference [10] based on simulation. Inthis paper, the following rule is presented as atrigger for zoning a building. The rule states thata building must be zoned if any of the followingconditions exists:

Number of required elevators exceeds 8

OR

The average travelling time exceeds 90 s

0.6

0.5

0.4

0.3

0.2

0.1

10 20 30 40 50 60 70 80INT

0

HC

Figure 8. HARint plane graphical representation of the solution of Example 2.




The average travelling time can either be cal-culated using analytical methods (such asReference [11]) or Monte Carlo simulationmethods.12 Example 3 illustrates this rule.

Example 3

The elevator system design needs to be carriedout for the building, the parameters of which areshown below.

User requirements:a. Passenger arrival rate (AR%) is 12%.b. Total building population of 1300 persons.c. Target interval of 30 s.

Building details:d. Number of floors above ground is 40 floors.e. Floor height is 4.5m (finished floor level to

finished floor level).f. A single main entrance and no basements.g. Equal floor populations are assumed.

Kinematics:h. Rated acceleration, a, is 1.5m/s2.i. Rated jerk, j, is 1.5m/s3.

Passenger data:j. Passenger transfer time out of the car is 1.2 s.k. Passenger transfer time into the car is 1.2 s.

Elevator data:l. Door opening time is 2 s.m. Door closing time is 3 s.n. Start delay is 1 s.o. Advanced door opening is 0.5 s.

Preferred car speeds are: 1, 1.25, 1.6, 2.0, 2.5,3.1 and 4.0m/s.

Preferred car capacities are: 630, 800, 1000,1250, 1600 and 2000 kg.

Solution

Using the automated design methodology givesthe following solution:

Round-trip time: 213.7 s

Interval: 26.7 sNumber of elevators: 8 elevatorsAverage travelling time: 97.6 sHandling capacity: 12%Optimum speed: 8m/sActual number of passengers: 13.9 passengersOptimum CC: 21 personsCar loading: 66.2%

However, although the number of requiredelevators is less than nine elevators, the averagetravelling time is 97.6 s which is more than theimposed limit of 90 s. This is excessive andshould trigger zoning of the building.

When zoning the building, a general rule of

thumb is to split the building population in the

ratio of 60% for the lower zone and 40% for the

upper zone to make up for the extra travel

requirement for the upper zone and attempt to

equalise the interval and the number of elevator

for both zones. This rule of thumb is used for

deciding on the zoning cut-off point.

Applying this rule would split the buildinginto two zones: 23 floors for the lower zoneand 17 floors for the upper zone (ratio of57%:43%). This gives the following solutionfor the two zones:

Lower zoneRound-trip time: 147 sInterval: 29.4 sNumber of elevators: 5 elevatorsAverage travelling time: 64 sHandling capacity: 12%Optimum speed: 4m/sActual number of passengers: 9 passengersOptimum CC: 13 personsCar loading: 69.2%Upper zoneRound-trip time: 138.5 sInterval: 27.7 sNumber of elevators: 5 elevatorsAverage travelling time: 66 sHandling capacity: 12%Optimum speed: 6.3m/sActual number of passengers: 6.63 passengers




Optimum CC: 10 personsCar loading: 66.3%It can be clearly seen that the interval is

approximately equal between the two zones;and the number of elevators is equal. It is pref-erable to have an equal number of elevatorsbetween the two zones for architectural andstructural reasons.

Advantage over conventionalmethods

It is worth highlighting at this point, the maindifference between this automated method andthe conventional methods. Most of the conven-tional methods rely on the designer selecting aCC. The designer then finds the number of pas-sengers by multiplying the CC by 80% (or anyother figures that he/she deems appropriate).This provides the value of P and he/she thenproceeds to find the value of the round-triptime using a round-trip time calculator tool.The result is then divided by the target intervalto find required number of elevators. Althoughthis guarantees that the quality of service is met,it does not guarantee that the quantity of serviceis met. The user then finds the handling capacityand compares it to the arrival rate. If the hand-ling capacity is lower than the arrival rate, he/she has to either increase the number of eleva-tors, the CC or both, but there is currently noclear guidance as to which one to adjust and byhow much. If the handling capacity is more thanthe arrival rate, the user has no way of knowingwhether a better solution can be found that isless wasteful.

The proposed method in this paper over-comes this problem by offering a clear step-by-step method to find the most suitable number ofelevators in one calculation.

Conclusions

A new methodology has been introduced thatprovides a set of rules and graphical methods

that can be used to design elevator systems inbuildings. The method optimises the number ofelevators in the group of elevators for a buildingbased on the user requirements of arrival rate(AR%), target interval (inttar) and the totalbuilding population (U). The methodologythen optimises the speed of the elevators andthen the elevator CC. The method allows theuser to work backwards from the actual arrivalrate in the building in order to find the optimumnumber of elevators, instead of the trial anderror method.

The methodology then presents a rule thatcan be used for the decision on zoning of thebuilding where the number of required elevatorsexceeds eight elevators or where the passengeraverage travelling time exceeds 90 s.

The methodology assumes that a method existsfor accurately calculating the round-trip time andthe passenger average travelling time. Both ana-lytical and Monte Carlo simulation methods canbe used to calculate these parameters.

Due to the automated and rule-based natureof the methodology, it is very attractive forimplementation in a software tool for thedesign of elevator systems.

The method has also been successfully used inteaching the principles of elevator traffic analysisto final-year undergraduate mechatronic engin-eering students at the University of Jordan. Oneof the main reasons for its success is that stu-dents do not possess any past experience in ele-vator traffic design, and hence, rely on the rulebase and graphical methods in reaching an opti-mal and convergent design.

It is worth noting that the analysis in thismethodology has assumed a constant uniformpas-senger arrival process. Further work is currentlybeing done in understanding the effect of a randomarrival process on the results and the final answersand will be the subject of a future paper.

Funding

This research received no specific grant from any

funding agency in the public, commercial, or not-for-profit sectors.




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Automated optimal design methodology of elevator systems