Proceedings of the 1999 Winter Simulation ConferenceP. A. Farrington, H. B. Nembhard, D. T. Sturrock, and G. W. Evans, eds.

DEFINING A BETA DISTRIBUTION FUNCTION FOR CONSTRUCTION SIMULATION

Javier FenteKraig Knutson

Cliff Schexnayder

Del E. Webb School of ConstructionArizona State University

P.O. Box 870204Tempe, AZ 85287-0204, U.S.A.

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ABSTRACT

In most applications of simulation to construction, thunderlying Probability Distribution Function (PDF) isgenerally unknown. Consequently, an expert will haveselect a PDF hoping that the one that is chosen matchesshape of the underlying distribution. This researattempts to quantify, through a sensitivity analysis, teffect of subjective information in choosing parameters a Beta distribution to be used in earthmoving simulatimodels.

1 INTRODUCTION

Although simulation is widely applied as a practical tofor planning and analysis in many industries, in the caseconstruction engineering it has not yet emerged from research stage to practice. Modeling construction activtime elements is crucial to developing a usable simulatprocedure because the time elements affect procproduction rates, the completion of a job, and resouutilization. The lack of confidence in the subjectivselection of a Probability Distribution Function (PDF) foan activity time element is due to the unpredictable natof construction processes. This concern has limited the of simulation as a practical tool for constructors.

It is often recommended that, in order to model tduration parameter of a construction activity in an efficieand accurate way, a flexible family of PDFs capable attaining a wide variety of shapes should be useAbouRizk and Halpin (1992) demonstrated through modified β1-β2 plane analysis that most earthmovinconstruction operations can be described by the Beta PIn addition, according to McCrimmon et al. (1964), thPDF used for construction simulations should bcontinuous and limited between two positive timintercepts, and have a unique mode in its defined ranThe Beta PDF satisfies all the conditions stated.

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1.1 Study Data

The data used in this research was acquired from Atkinson-Washington-Zachry (AWZ) joint venture on thEastside Reservoir Project in California. AWZ operatesfleet of Caterpillar (CAT) trucks equipped with a VitaInformation Management System (VIMS) and a TrucPayload Monitoring System (TPMS) on the project. Thetwo systems automatically record truck performance d(e.g., payload weight, load time, haul distance, travel timand dump time). TPMS data from this projecrepresenting 54,000 truck cycles, is the original data for of the subsequent statistical research presented in paper.

2 THE BETA PDF AND ITS PARAMETERS

The Beta PDF is a continuous distribution defined overrange. Additionally, both of its end points are fixed exact locations and it belongs to the flexible family odistributions. Because of its extreme flexibility, thdistribution appears ideally suited for the description subjective time estimates of activity duration. The shaof the Beta PDF depends on the choice of its twparameters “a” and “b.” The parameters are any rnumber greater than negative one, and depending on tvalues the Beta PDF generated will have the “U”, the “Jthe triangle or the general bell shape of the unimodfunction. Estimating these parameters is controlled by davailability.

In the absence of data, an experienced constructor usubjective information to estimate the parameters of tBeta PDF expected to describe the construction operatThe subjective information needed to determine the twBeta parameters that will describe a unique Beta cuinclude four characteristics: the minimum and maximutimes, as well as two of the following subjective statisticmode, mean, variance or selected percentiles. In

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context of fitting a Beta PDF for construction simulationthe percentiles represent the percent chance that analyzed activity duration will not exceed a time “t.” Thnatural limitations on the input data range for construction simulation PDF were defined by McCrimmoet al. (1964) as: having a unique mode in its defined ranand the parameters “a” and “b” of the Beta PDF must positive and greater than one. Consequently a restricsubset of Beta density curves have often been appliedconstruction modeling.

Weiler (1965) concluded that many errors in thoutput of a simulation are those of assigning wrong valuto the parameters of a distribution. If, on the other hanthe form of the PDF is unknown, a further error may bintroduced by assuming a Beta distribution when in fasome other distribution would have been appropriaHowever, this type of error is likely to be small incomparison to errors in the parameters.

Maio (1998) studied the effect of PDF input in a haucycle simulation. The distributions compared were a Bedistribution whose parameters were estimated using same data, and a PDF ranked, by a goodness of fit testhe closest fit to the actual data set in the case of eaoperation. The simulation results using these twdistributions were then compared to a simulation usingtrace of the actual cycle data as input. The reseaconcluded that the choice of a distribution function, derived from the same original data, did not influence toutput results of his SLAM II model.

3 SUBJECTIVE ASSESSMENT OF PDFS

Estimating the productivity of an earthmoving operatioinvolves the investigation of events for which significanstatistical records usually do not exist. This lack of dacreates problems concerning the selection of a PDF. that point the judgement of an experienced construcbecomes very important. In such a case, an expert whave to assume a PDF and hope that its shape matchesof the underlying distribution. For this reason the flexiblBeta distribution capable of attaining a variety of shapesoften used in construction simulation applications.

Experiments in psychology summarized by Petersand Miller (1964) and Peterson and Beach (1967) haindicated that when attempting to estimate the mean oskewed distribution there is a bias to estimate the metoward the median. Alpert and Raiffa in their 198progress report on the training of probability assessorecommended direct fractile assessments as the mreliable method and less prone to bias. In this method, planners express their uncertainties about an unknoquantity in terms of percentiles of a distributionrepresenting their state of knowledge. This led Mosleh aApostolakis (1982) to recommend that “assessors upercentiles to quantify their beliefs and to avoid the dire

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assessment of other measures like mean value or standeviation.” This procedure has been used in applicatiofrom nuclear energy to experimental psychology anmathematics.

The Beta distribution is a continuous PDF boundeover a range; that is, the end points, or minimum amaximum durations, are defined at exact locationAccording to Wilson et al. (1982), an expert can estimathe end points of an activity duration distributionsomewhat easily and accurately due to the experfamiliarity with the technological constraints on the targactivity. For construction applications, the mode or beguess, of a construction activity duration can be eliciteaccurately from an expert using deterministic methodThe end points and a value for the mode define a numof different Beta PDFs. To fit a unique beta distributioone more characteristic of the PDF must be specified.

AbouRizk and Halpin (1992) developed amicrocomputer based software system specificadesigned for the subjective estimation of Beta PDFs. Tsoftware is the Visual Interactive Beta Estimation Syste(VIBES). VIBES uses a combination of four activity-timecharacteristics, two of which must be the end points, determine the parameters of the unique Beta PDF. Tpossible combinations include the end points and 1) meand standard deviation; 2) mean and a selected percen3) mode and a selected percentile; and 4) two selecpercentiles.

For the purpose of this study, VIBES was used as ttool to determine the Beta shape parameters “a” and from given subjective information. The input combinatiowas the mode, the 75th percentile and the two end points asuggested by the investigations of Wilson et al. (1982), aLichtenstein et al. (1982). The input combinations usinthe mean and/or the standard deviations were rejectedthe basis of the study presented by Peterson and Mi(1964) which explains the difficulty of assessing the meand standard deviation of skewed distributions.

The following example illustrates an application oVIBES when a combination of the mode, the 75th

percentile, and the end points is used to find the parameof the underlying Beta PDF.

3.1 Application of VIBES

The following example from a hypothetical earthmovinoperation illustrates the use of VIBES to fit Betdistributions to subjective information developed fromdeterministic calculations of truck travel time. The modand the 75th percentile, along with the minimum andmaximum durations are specified as inputs.

The Beta PDF of the travel time for an empty CA785 off-highway truck moving in a range of 2.4-2.5 miles

Defining a Beta Distribution Function for Construction Simulation

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can be estimated using VIBES by inputting the followinsubjective information:

• The minimum possible time in which a truckcan complete a cycle is 7.5 minutes,considering the physical characteristics of theproject site and manufacturer’s truck data.

• The most likely duration for the return timewill be 9.3 minutes, allowing for accelerationand deceleration, and applying an efficiencyfactor.

• The maximum possible time will beestimated to be 18.7 minutes, two times themode. It is assumed that management wouldnotice such a laggard truck and take action.

• There is at least a 75% chance that theactivity duration will not exceed 10.6minutes, which is 1.13 times greater than themode.

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Using this information, VIBES yields the distributionand parameters shown in Figure 1.

The shape parameters estimated by VIBES wecompared to the parameters from an actual data set of travel durations for an empty CAT 785 in a 2.4-2.5 mirange. These parameters were calculated using Maximum Likelihood Estimates (MLE), Method of MomenMatching (MM) and the Method of Ordinary Least SquaMinimization (OLS) minimization procedures implementein the Betafit software developed at Purdue UniversiBetafit can fit a Beta distribution to a data set based on user’s choice of fitting procedure. The software computhe statistics of the sample and of the fitted PDF. Tsample and fitted data distributions graphs, Figures 2, 3 4, show how the methods of MLE and OLS yield a better to the given data histogram than the MM technique.

The parameters subjectively estimated by VIBES athose calculated by Betafit with the actual project data ashown in Table 1.

50th percentile: 9.704 Shape Parameter a: 3.15695th percentile: 11.967 Shape Parameter b: 11.782Mode: 9.350 Standard Deviation: 1.147Mean: 9.851 Variance: 1.316

Figure 1: Output from VIBES

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Beta distribution for return travel. (2.4-2.5 miles)

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Figure 2: Maximum Likelihood Estimates (MLE)

Figure 3: Method of Moment Matching (MM)

Figure 4: Method of Ordinary Least Square Minimization (OLS)

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Defining a Beta Distribution Function for Construction Simulation

Table 1: Comparison of Calculated vs. Estimated Shape Parameters

VIBES MLE MM OLS

Parameter a 3.156 3.894 0.159 2.943

Parameter b 11.782 19.396 4.196 15.781

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Figure 5 illustrates the minor differences among tPDF yielded by VIBES and those given by the MLE aOLS methods from Betafit. For this case, the differencethe output of a simulation model derived from using eithof the three mentioned Beta PDFs, would be relativinsignificant. The Beta PDF rendered by the MM methvaries considerably from the data histogram in Figure 5.

4 CONCLUSION

Simulation applications normally use the flexible familieof probability distributions due to their capability oattaining a wide variety of shapes. Among such familis the Beta distribution. A modified β1-β2 plane analysisrevealed that most of the construction data sets analyby AbouRizk and Halpin (1992) lay in the Beta regioThus, it was concluded that the Beta distribution suitable for modeling durations of construction activities

Due to the natural limitations of the input data ranand mode in construction activities, the parameters of Beta distribution must be positive and greater than oConsequently, a unique and restricted family of Be

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curves is specified to the model durations of constructioactivities. The restricted subset of Beta curvecorresponds to the unimodal polynomial graphs, (i.ethose with parameters “a” and “b” greater than one).

When sample observations are not available, a Bedistribution can be defined by using subjective information provided by experts about a given activity durationBased on the research in experimental psychology, it believed that a construction expert using deterministmethods can accurately estimate the minimum anmaximum activity durations, as well as the mode and th75th percentiles to use in defining a Beta distribution.

The lack of accuracy in the subjective estimation oprobability distributions is still a subject of research in thsimulation field across various engineering disciplines. It one of the most important factors for its limited use as planning tool in the construction industry. This researcattempted to quantify, through a sensitivity analysis, theffect of subjective information in developing earthmovingsimulation models. The simulation model was developeby Maio (1998) in SLAM II (Pritsker, 1995).

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Figure 5: Comparison of Beta PDF given by VIBES and Betafit

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REFERENCES

AbouRizk, S.M., and Halpin, D. W. 1992. “Statisticalproperties of construction duration data.” J. of Constr.Engrg. and Mgmt., ASCE, 118(3), 525-543

Alpert, M., and Raiffa, H. 1982. “A progress report on thetraining of probability assessors.” Judgement underuncertainty: Heuristics and biases, D. Kahneman, P.Slovic, and A. Tversky, eds, Cambridge UniversityPress, Cambridge, England. 294-305

Lichtenstein, S., Fischhoff, B., and Phillips, L.D. 1982“Calibration of probabilities: The state of the art to1980. Judgement under uncertainty: Heuristics andbiases, D. Kahneman, P. Slovic, and A. Tversky, edsCambridge University Press, Cambridge, England306-334

MacCrimmon, K.R., and Rayvec, C.A. 1964. “Ananalytical study of the PERT assumptions.Operations Research, 12(1), 16-37

Maio, C. 1998. Investigation of probability distributionfunctions for haul cycle simulation. MS Thesis,Arizona State University, Tempe, AZ.

Mosleh, A., and Apostolakis, G. 1982. “Some propertieof distributions useful in the study of rare events.IEEE Transactions on Reliability, R-31(1), 87-94

Peterson, C., and Miller, A. 1964. “Mode, median, anmean as optimal strategies.” J. of Exp. Psych. 68(4),363-367

Peterson, C., and Beach, L.R. 1967. “Man as an intuitivstatistician.” Psych. Bull., 68, 29-46

Pritsker, A.A.B. 1995. Introduction to simulation andSLAM II, John Wiley, New York.

Weiler, H. 1965. “The use of the incomplete betfunctions for prior distributions in binomial sampling.”Technometrics, 7(3), 335-347

Wilson, J.R., Vaughan, D.K., Naylor, E., and Voss, R.G1982. “Analysis of space shuttle ground operationsSimulation. 38(6), 187-203

AUTHOR BIOGRAPHIES

JAVIER FENTE is a graduate student in the Del E. WebbSchool of Construction at Arizona State University (ASUin Tempe, Arizona. He holds a M.S. in Civil Engineeringand a M.S. in Construction from Arizona State Universitas well as a B.S.C.E. from Lafayette College, EastoPennsylvania. Mr. Fente is a native of Madrid, Spain.

KRAIG KNUTSON , CPC, is an Assistant Professor in theDel E. Webb School of Construction at Arizona StatUniversity (ASU) in Tempe, Arizona. He holds a Ph.D. inIndustrial Engineering from Arizona State University awell as bachelor's and master's degrees in constructiHis research interests are related to the design, simulatand optimization of manufacturing systems and con

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struction processes and to the decommissioning semiconductor facilities. He is a member of AACE, ASSEIIE, INFORMS, and SCS.

CLIFF SCHEXNAYDER is the Eminent Scholar in theDel E. Webb School of Construction at Arizona StaUniversity (ASU) in Tempe, Arizona. He holds a Ph.D. iCivil Engineering from Purdue University as well as M.S.C.E. and B.C.E. from Georgia Institute oTechnology. Prior to joining ASU, Dr. Schexnayder wathe Chief Engineer for the Nello L. Teer Company oDurham, North Carolina. His research interests incluconstruction engineering and heavy constructioequipment and estimating. He is an ASCE Fellow aChair of the Transportation Research Board’s ConstructiSection. Dr. Schexnayder is co-author of the McGraw-Htextbook Construction Planning, Equipment and Methods.