visual interactive fitting of beta distributionsmanagement system is based on a network model [for...

VISUAL INTERACTIVE FITTING OF BETA DISTRIBUTIONS

By Simaan M. AbouRizk,1 Associate Member, ASCE, Daniel W. Halpin,2 Member, ASCE,

and James R. Wilson3

ABSTRACT: This paper describes a visual interactive procedure for fitting beta distributions to activity times in a simulation model when sample data are not available for statistical analysis of the model's input processes. Using subjective information about a given activity time the modeler specifies the activity's minimum and maximum times together with two of the following characteristics: mode, mean, variance, or selected percentiles. The fitting procedure includes efficient methods for computing the shape parameters of the beta distribution that most nearly matches the specified characteristics. The user can manipulate the fitted distribution by either revising the specified characteristics or directly altering a visual display of the density. This fitting procedure was implemented in a public-domain microcomputer-based software system called VIBES (visual interactive beta estimation system). An example from construction engineering illustrates the operation of VIBES.

INTRODUCTION

Planning and analyzing the operations comprising a large-scale engineering project generally require accurate estimates of selected numerical characteristics of the input processes for those operations. Whether the project management system is based on a network model [for example, the critical path method (CPM), program evaluation and review technique (PERT), or precedence diagramming], velocity diagrams, line of balance diagrams, or simulation models, the validity of the system's performance measures (outputs) depends directly on the quality of the estimates of the input characteristics. With respect to the availability of sample information describing the input processes, three situations commonly occur: (1) Sample observations are readily available and can be reduced to appropriate deterministic or probabilistic models of the input processes; (2) sample observations are not available, and the properties of the input processes must be based on subjective information elicited from individuals who are experts on those processes; and (3) sample observations are available in relatively small quantities so that the sample information must be combined effectively with subjective information to obtain the required input models. In this paper we are concerned primarily with situation 2, which frequently arises when one is planning a new operation rather than analyzing the performance of an existing operation.

In the planning phase of an engineering project a first-cut simulation model of the project typically involves uniform or triangular distributions

'Asst. Prof., Dept. of Civ. Engrg., Univ. of Alberta, Edmonton, Alberta, T6G 2G7, Canada.

2Prof. and Head, Div. of Constr. Engrg. and Mgmt., Purdue Univ., West Lafayette, IN 47907.

3Assoc. Prof., School of Industrial Engrg., Purdue Univ., West Lafayette, IN. Note. Discussion open until May 1, 1992. To extend the closing date one month,

a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on July 3, 1990. This paper is part of the Journal of Construction Engineering and Management, Vol. 117, No. 4, December, 1991. ©ASCE, ISSN 0733-9364/91/0004-0589/$1.00 + $.15 per page. Paper No. 26406.

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for activity times to reflect the modeler's uncertainty about the duration of those activities as well as inherent variability in the time required to perform repetitions of those activities. To facilitate the development of more flexible statistical models for such activity times DcBrota et al. (1989) developed a public-domain computer program for visual interactive fitting of bounded Johnson distributions (usually called SB distributions). The chief drawback of the SB family of distributions is that it is neither as well known nor as widely used in project management systems as the generalized beta family. Moreover, the generalized beta family has almost the same degree of flexibility as the SB family. For a comprehensive analysis of the relative advantages of using beta, gamma, lognormal, SB, triangular, uniform, or Weibull distributions to model the activity times in large-scale discrete simulation experiments see Klein and Baris (1991). In the spirit of the approach of DeBrota et al. (1989) to simulation input modeling, we discuss in this paper a public-domain microcomputer-based software system that was specifically designed for subjective estimation of generalized beta distributions—the Visual Interactive Beta Estimation System (VIBES).

When using VIBES, the simulation modeler initially specifies the end points of the target distribution together with two of the following numerical characteristics: the mode, the mean, the variance, or selected percentiles of the distribution. The following steps are then performed: (1) VIBES automatically solves for the shape parameters of the generalized beta distribution that most nearly matches the specified characteristics; (2) the user iteratively revises the specified characteristics until a satisfactory fit is obtained; (3) VIBES generates a visual display of the resulting probability density function (PDF); and (4) the user interactively modifies the shape of the fitted density via the four arrow keys until the fit is finalized. VIBES is menu-driven, and it exploits efficient numerical methods and programming tools to enhance responsiveness and accuracy.

This paper is organized as follows. In the second section we summarize conventional approaches for modeling activity durations in project management systems, we discuss relevant properties of the generalized beta family of distributions, and we briefly review methods for fitting beta distributions, to activity times. In the third section we describe the operation of VIBES, we discuss a specific application of VIBES in the field of construction engineering, and we explain the program's hardware and software requirements. Our conclusions and recommendations are summarized in the fourth section. Appendix I details the numerical methods used to solve for the shape parameters of the fitted beta distribution given user-specified characteristics of a target population. In Appendix II we discuss the scheme for controlling interactive modification of the displayed beta density. Appendix IV summarizes the notation used throughout the paper.

MODELING RANDOM ACTIVITY TIMES

Planning a large-scale project requires estimation of the time required to complete each of the activities comprising the project. To represent uncertainty or random variation in a given activity time X, the analyst usually postulates an appropriate probability density function/(x) for that quantity. MacCrimmon and Ryavec (1964) proposed the following properties for the density of an activity time.

• The function f(x) should be continuous on an interval (L, U), where L = minimum activity time; and U = maximum activity time.

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• The function f(x) should have a unique mode m so that f(x) < f(m) when x 4= m.

• The end points L and U should be nonnegative, distinct, and finite so that the activity duration is a bounded, nondegenerate random variable.

These criteria are assumed throughout the rest of this paper.

Generalized Beta Family of Distributions In project management systems, the most widely used probabilistic model

for the duration of activities is the generalized beta density

tt w m r(a + *>) (*- L)-l(U - x)b~l .„ r

fM,L,U) = ^ - ^ ( t / - L y + »-i *Lxx*U. (la)

f(x;a,b,L,U) = 0 otherwise (lb)

where T( ) = gamma function

r ( 2 ) EEE J tz-le-'dt for all z > 0. (2)

The three requirements for an activity-time density are satisfied by (1), provided that the modeler chooses the shape parameters a and b and the end points L and U so that

a > 1 (3a)

b > 1 (3b)

0 < L < U <o3 (3C)

Fig. 1 depicts the wide variety of shapes that can be obtained with beta densities satisfying condition (3). Although the generalized beta family of distributions does not possess the same degree of flexibility as the Johnson SB family or the Pearson family (Johnson and Kotz 1970), an extensive study by AbouRizk (1990) demonstrated that beta densities satisfying (3) are capable of faithfully representing most of the distributional shapes encountered in modeling the activities of construction engineering projects.

Another advantage of input modeling with the beta distribution [(1)] is that many familiar numerical characteristics of this distribution are mathematically tractable functions of the basic parameters a, b, L, and U. The mean and variance, respectively, of the beta distribution are given by

all + bL ,, s

» = ̂ rw- (4fl)

and

^ = < ? - L > 2 f „ (4b) (a + b)2(a + b + 1) y '

Provided (3) holds, the unique mode of the beta distribution is

W = < a - ^ : <* - I ) L (5)

a + b - 2 v ' Let F(x;a,b,L,U) = f*xf(t;a,b,L,U)dt denote the cumulative distribution

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3.00 q

0.00 0.20 0.40 0.60 0.80 1.00 X

FIG. 1. Generalized Beta Densities Satisfying Condition (3)

function (CDF) corresponding to the density [(1)]. For 0 < q < 1, the 100g percentile xq of the beta distribution is defined in terms of the inverse CDF

JC, = F-\q;a,b,L,U) (6)

(Note that.*,, is sometimes called the fractile or quantile of order q.) Efficient numerical approximations to the inverse beta CDF (6) are available (Griffiths and Hill 1985).

Fitting Beta Distributions to Activity Times When we have sample data on the duration of a given activity we can

apply a variety of statistical-estimation methods to fit a beta distribution to that activity. If the end points L and U are known, then the conventional methods of moment matching or maximum likelihood easily yield estimates of the shape parameters a and b. On the other hand if L and U are unknown, then these estimation methods can fail, and other techniques such as the method of least squares may be preferred; see AbouRizk et al. (1990).

When sample activity times are not available we can fit a beta distribution to a given activity by eliciting from an expert subjective estimates of certain numerical characteristics of the overall population of activity times. The mode m of the population can be readily interpreted as the "best guess" of what we are most likely to see on a single realization of the target activity, subjective estimates of the mode are not only easier to elicit but also more reliable than subjective estimates of other characteristics such as the mean and variance (Peterson and Miller 1964). Although the standard PERT procedure for modeling an activity time also requires elicitation of the end points L and U, subjective estimation of these quantities is much more

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difficult and unreliable (Grubbs 1962; Alpert and Raiffa 1982). In this paper we assume that subjective estimates of the end points are sufficiently accurate because of the expert's familiarity with the technological constraints on the target activity (Wilson et al. 1982).

In addition to the end points and the mode at least one other characteristic of the population of activity times must be specified to fit a unique beta distribution to the activity. There is substantial evidence that subjective estimates of certain percentiles of a population can be reasonably accurate— in particular, subjective estimates are commonly elicited for the 25th, 50th, or 75th percentiles of a target population (Lichtenstein et al. 1982). By contrast, subjective estimates of the mean are often biased, and the magnitude of the bias increases with the variance or skewness of the population (Beach and Swenson 1966; Spencer 1963). Moreover, subjective estimates of the variance decrease as the mean of the population increases (Lathrop 1967). If the mean or variance of an activity time must be estimated to complete the specification of a beta distribution as the model of a given activity, then we strongly recommend that the user collect some sample activity times so that the corresponding sample mean and variance can be used at least to corroborate if not to replace subjective estimates of the population mean and variance.

VIBES—VISUAL INTERACTIVE BETA ESTIMATION SYSTEM

The design and operation of VIBES are based on three main principles: (1) For ease of use, the arrow keys should be the primary means of interaction with the user; (2) for portability and responsiveness, the distribution-fitting and display-updating procedures should execute quickly on a broad range of IBM-compatible microcomputers; and (3) for greater flexibility, VIBES should be easy to reconfigure so that advanced users can alter the performance of the procedures for distribution fitting, display updating, and report generation. The operation of VIBES proceeds as follows.

1. The user chooses the desired combination of activity-time characteristics to be specified. Currently the following combinations are available.

a. Mean and variance. b. Mean and mode. c. Mode and variance. d. Mode and an arbitrary percentile. e. Two arbitrary percentiles.

2. The user specifies the minimum and maximum activity times together with the combination of characteristics selected in step 1.

3. VIBES computes the shape parameters of the beta distribution that most closely matches the user-specified characteristics of the activity time.

4. VIBES displays the requested values of the activity-time characteristics along with actual values of those characteristics for the fitted beta distribution. If the user is satisfied with the fit, VIBES proceeds to step 5; otherwise VIBES returns to step 1.

5. VIBES plots the fitted beta density on the display to allow direct interactive modification of the displayed density via the arrow keys as follows.

a. Pressing the right-arrow key, -» , moves the mode toward the upper end point.

b. Pressing the left-arrow key, <—, moves the mode toward the lower end point.

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c. Pressing the up-arrow key, t . increases the variance of the displayed distribution.

d. Pressing the down-arrow key, | , decreases the variance of the displayed distribution.

With each keystroke, VIBES recomputes the shape parameters of the fitted beta density and displays the updated density. Repeatedly pressing the same arrow key accelerates the desired change so that even drastic changes in the shape of the displayed density can be obtained quickly and with few keystrokes. Alternately pressing the arrow keys for opposing directions enables the user to home in on the desired density with progressively greater precision.

6. When the user is satisfied'with the shape of the displayed beta density, a hard copy of the display can be generated by simultaneously pressing the "Ctrl" and "P" keys.

Some general remarks on the design and operation of VIBES should be made at this point. The user's text-oriented interaction with VIBES is limited to the initial system configuration and the initial elicitation of the desired activity-time characteristics. By selecting the "Configuration" option in the main menu, the user may change the printer type, certain properties of the display, or the parameters controlling the speed and accuracy of the distribution-fitting algorithms. Appendix IV contains a detailed description of the distribution-fitting algorithms developed for VIBES. The scheme for interactivity updating the displayed density is based on the adaptive seeking strategy of DeBrota et al. (1989) and is summarized in Appendix I. In the next subsection we illustrate the operation of VIBES with an application from the field of construction engineering.

Application of VIBES In a hypothetical construction project, the contract requires resurfacing

the only runway in a small airport. This project involves a number of activities to be completed between the times of the last scheduled flight on one day and the first scheduled flight on the next day. Normally these two times are 8:30 p.m. and 6:30 a.m., respectively. The required construction activities included bringing in all necessary equipment, setting up light towers, delivering asphalt from a nearby asphalt batch tower, paving and compacting, and, finally, clearing the area for the first morning flight. To keep the discussion concise, we focus on the paving and compaction of one section of the runway. This activity was thought to be subject to random variation because of a large number of uncontrollable factors (such as bad weather, equipment breakdowns, employee illness, asphalt delivery delays, etc.) that could affect its duration. The following information about the activity-time distribution was available to the project engineer:

• Under the best circumstances the minimum feasible time to complete this activity is 1 hr.

• This activity might be completed within 8 hr so that succeeding activities can be completed before the first morning flight.

• In the past this activity has most frequently lasted about 3 hr. • There is at least a 75% chance that the activity duration will not exceed

6hr .

Thus the project engineer initially specifies the minimum L = 1; the max-

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imum U = 8; the mode m = 3; and the 75th percentile x0 75 = 6 for the population of possible activity times.

To model this activity with a beta distribution, the project engineer ran VIBES; and in the main menu, shown in Fig. 2, he moved the highlighted selection bar to the option labeled "Specify Percentile and Mode." The dialog box depicted in Fig. 3 shows how he entered his initial estimates of the selected characteristics. [To avoid awkward and confusing language in the rest of this paper, we use the words "he" and "his" in a generic sense to mean, respectively, "he or she" and "his or her"; see pages 75-76 of Fowler and Fowler (1931) and pages 4-5 of van Leunen (1986).]

The VIBES output presented in Fig. 4 reports the shape parameters, the mode, and the 75th percentile of the fitted beta distribution along with the values of the mode and 75th percentile that were originally requested by the project engineer. In this example, VIBES matches the user-specified activity-time characteristics to three significant figures. (The default parameter values for the distribution-fitting algorithms in VIBES are "tuned" to yield this degree of accuracy in a broad class of applications.) To handle situations in which the user-specified characteristics are not matched with sufficient accuracy, the menu shown in the lower right-hand corner of Fig. 4 provides the user with the following options: (1) Reconfigure the parameters of the distribution-fitting algorithms, respecify the activity-time characteristics, and redo the fit; (2) proceed to the next stage of graphical interactive density modification based on the current fit; or (3) exit the program immediately. For his application, the project engineer choses to proceed with option 2.

FIG. 2. Main Menu for VIBES

Enter the Lower End Point: Enter the Upper End Point:

Enter Percentage corresponding to the quantile [0-100] Enter Quantile point:

Enter Mode

1 8

75 6

3

FIG. 3. Initial Entry of Activity-Time Characteristics

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(a)

(b)

75th perc. B

Mode 0

USER SPECIFIED PARAMETERS FITTED PARAMETERS

6.80 6.80

3.00 3.00

Shape Paraneters of the fitted Beta

a = 1,068 b = 1.149

Respecify paraneters

Exit Progran

FIG. 4. Beta Distribution Fitted to User's Initial Specifications

UIBES: Visual Interactive Beta Estimation System Uer. 1.1

B.1SE+8B .

1.0000E+08 B.0000E+00 X-AXIS

5th xtile B X = 95th xtile 0 X = Mode S X -Mean B X =

1.363 7.589 3.000 4.358

Shape Parameter (a) = Shape Parameter (b) = Standard Deviation -Uariance =

1.060 1.149 1.952 3.812

FIG. 5. Beta Density Fitted to User's Initial Specifications

In practice, a VIBES user generally does not develop an intuitive "feel" for the target distribution until he sees the graphical display of the fitted density function. Fig. 5 shows the fitted density for the application at hand. In this case visual inspection of the density immediately reveals a grossly different shape than the project engineer expected. The user was especially concerned about the size of the tail above the cutoff value of 7 hr; and although he was "reasonably sure" the activity would last at most 7 hr in the "great majority" of cases, he was unwilling to quantify this belief more precisely without further experimentation on the current fit.

To obtain a more realistic fitted density while simultaneously trying to refine his ideas about the shape of the extreme upper tail of the activity-time distribution, the project engineer presses the "Esc" key to return to the main menu (Fig. 2), he reconfigures the display to show both the 75th and 99th percentiles of the fitted distribution, and he reruns the original fit.

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(For brevity and simplicity, the menu and dialog box used to reconfigure the display are not presented.) Upon reentering the visual interactive density modification step, the project engineer presses the down-arrow key seven times in succession to compress the extreme upper tail of the activity-time distribution while keeping the mode at 3. After examining the modified density as shown in Fig. 6, the project engineer decides that he has overshot the target density and must now partially uncompress the upper tail. Fig. 7 depicts the fitted density after the user presses the up-arrow key twice in succession. At this point the display reveals a tail weight of at most 1% above the cutoff value of 6.919 hr, and the project engineer decides that the fit is acceptable. Note that the final estimates of the 75th percentile and the shape parameters (x0.75 = 4.605; a = 1.819; b = 3.048) differ substantially from the corresponding values for the initial fit as shown in Fig. 4.

UIBES: Uisual Interactive Beta Estimation System Uer. 1.1

0.29E+00 .

0.00E+0B 4 r 1.8BBBE+BB 8.80B8E+B0

X-AXIS

75th y.tile 0 X = 99th xtile 0 X = Mode 0 X = Mean 0 X =

4.39B 6.619 3.000 3.S1Z

Shape Parameter (a) = Shape Parameter (b) -Standard Deviation = Uariance =

2.104 3.760 1.ZB2 1.642

FIG. 6. Intermediate Fit during Interactive Curve Modification

UIBES: Uisual Interactive Beta Estimation System Uer. 1.1

B.26E+8B .

» ;/ B.00E+B0 4-

1.0000E+00

\

\

8.0000E+00

75th xtile 0 X = 99th xtile 0 X = Mode 0 X -Mean 0 X =

4.605 6.919 3.000 3.616

Shape Parameter (a) = Shape Parameter (b) = Standard Deviation = Uariance =

1.819 3.B4B 1.39B 1.955

FIG. 7. Final Beta PDF Fitted by Interactive Curve Modification

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Hardware and Software Requirements of VIBES Although VIBES was completely written in the QuickBASIC program

ming language (Microsoft 1989), the program invokes many Intel assembly-language routines from the QuickPack professional library (QuickPack 1988) to increase its speed, to reduce its memory requirements, and to enhance its user interface. VIBES runs on IBM-compatible microcomputers under DOS 3.0 or higher, and it requires an EGA- or VGA-compatible display. Currently, VIBES supports printers that are compatible with Hewlett-Packard LaserJet printers or Epson dot-matrix printers. VIBES will use a numeric coprocessor if one is present; and since the distribution-fitting algorithms in VIBES require extensive floating-point computations, a numeric coprocessor will significantly improve the responsiveness of the program.

CONCLUSIONS AND RECOMMENDATIONS

In this paper we presented VIBES, a visual interactive software system for subjective estimation of generalized beta distributions. Although the discussion was oriented toward improved probabilistic modeling of activity times in construction engineering simulations, the potential applications of this software extend beyond the fields of construction engineering and stochastic simulation. The generalized beta family of distributions plays a prominent role in a broad diversity of stochastic modeling and analysis techniques, and we believe that VIBES can be used successfully with these techniques in situations where sample data are scarce but reliable expert advice is readily available.

Follow-up research is required on a number of issues that arose in the course of this research. Subjective estimation of probability distributions is still fraught with substantial risks that are incompletely understood. More research is required to yield definitive formulations of: (1) The characteristics of a target distribution for which subjective estimation is both feasible and sufficiently reliable in practice; and (2) the proper procedures for eliciting subjective estimates of those characteristics. We believe that software tools like VIBES will play a key role in these lines of research.

In another direction, we foresee the need to develop effective techniques for subjective estimation of the stochastic dependencies among multiple input processes. Within the context of modeling the activity times in an engineering project using beta distributions this means that we should explicitly account for correlations among those activities by fitting a multivariate beta distribution to the vector of activity times. We hope that the developments presented in this paper will stimulate future research on all of these issues.

ACKNOWLEDGMENTS

We thank Dr. David J. DeBrota, Mr. Robert W. Klein, and Prof. Stephen D. Roberts for many enlightening discussions on simulation input modeling. This work was supported in part by the National Science Foundation under Grant No. DMS-8717799.

APPENDIX I. NUMERICAL METHODS USED IN VIBES

Throughout this appendix, the symbols that denote subjective estimates of user-specified characteristics of \ generalized beta distribution are overscored with a hat ("); thus L and U denote, respectively, the estimated

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minimum and maximum of the fitted beta distribution. Given L and U, we require subjective estimates of two additional activity-time characteristics to uniquely determine the fitted beta distribution. In general this setup yields a nonlinear system of two equations in the two unknown shape parameters a and b; and in some cases it is possible to determine closed-form solutions to this equation system. In other cases the equation system or its solution (or both) cannot be expressed in closed form, and then we must resort to mathematical approximations or iterative numerical solution procedures (or both). In this appendix we summarize the distribution-fitting algorithms that are implemented in VIBES for each combination of activity-time characteristics that can be specified by the user.

MEAN AND VARIANCE SPECIFIED

Given subjective estimates |i and <r2, respectively, for the mean and variance of the target activity time, we see that the resulting equation system has the form

A all + bL P- = 7-7— (7a)

a + b

cr (a + bf {a + b + 1) y '

In terms of the "standardized" estimates

^IrA ^ * = ( 7 T ^ ^ the analytic solution of (7) yields the corresponding estimates of the shape parameters

B = ML-i£ + ? - 1 ( 9 f l)

& = rh- (%) MEAN AND MODE SPECIFIED

Given subjective estimates |i and m, respectively, for the mean and mode of the target activity time, we see that the resulting equation system has the form

A all + bL

* = ^rvv (10fl)

,» = ( « - i ) & + ( » - - ^ { m )

a + b - 2 v '


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f = I1 ~ L (11a) e 0 - L

v = irrz {nb)

the analytical solution of (10) yields the corresponding estimates of the shape parameters

6 = ( 1 - *><2 ' - *> (12a) v — t,

d = rh {12b)

MODE AND VARIANCE SPECIFIED

Given subjective estimates m and a2, respectively, for the mode and variance of the target activity time, we see that the resulting equation system has the form

m = , — (13a) a + b - 2 v '

ff = (« + &)*(« + 6 + 1 ) " ( 1 }


*f = W^T? (14fl)

-W^T ™ T = ( T ^ (14c)

the desired solution of equation system (13) corresponds to the largest positive real root of the following cubic equation in the shape parameter b:

c3b3 + c2b

2 + db + c0 = 0 (15)

whose coefficients are given by

c0 = - 12TV3 + 20TV2 - IITV + 2T (16)

q = 16TV2 + (2 - 18T)V + 5T - 1 , (17)

c2 = - (7T + l)v + 4T (18a)

c3 = T (186)

In VIBES the desired root B of (15) is computed by the Muller method (Conte and de Boor 1980). The corresponding estimate of the other shape parameter of the fitted beta distribution is then given by

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a = <* - 2>v+ ' (19)

This option in VIBES allows the user to fit a so-called Beta-PERT distribution to subjective estimates L, U, and m by taking & = (U - L)/6. Thus the (mathematically) exact procedure described earlier in (13)-(19) extends the approximate technique of McBride and McClelland (1967) for fitting a Beta-PERT distribution to subjective estimates of the end points and the mode; moreover, the numerical implementation of the VIBES procedure is not appreciably slower than the McBride-McClelland approximation on microcomputers equipped with a numeric coprocessor.

Two ARBITRARY PERCENTILE POINTS SPECIFIED

Given subjective estimates xq and xq , respectively, for the percentiles of order ql and q2, respectively, we see that the resulting system of equations

T(a + b) [*-. (t - L)"-l((l - t)"-1 J

cannot be expressed algebraically in terms of the unknown shape parameters a and b. Thus it is not surprising that a closed-form solution to (20) is not known.

To approximate the solution to (20) numerically, we formulated the following nonlinear programming problem in terms of the inverse beta distribution function F~l(q\a,b,L,U) defined for all q in the interval (0, 1):

1 2

Minimize -j j - Y, {F-\qn;a,b,L,U) - x ]2 (21) a, b (U ~ L) „ = l

subject to

a > 1 (22a)

b > 1 (22b)

To evaluate the objective function in (21), we approximated the inverse beta distribution function by the method of Majumder and Bhattacharjee (Griffiths and Hill 1985). In VIBES, the minimization problem defined by (21) and (22) is solved by the Nelder-Mead simplex search procedure (Olsson 1974). The simplex search is started by initially estimating the mean and standard deviation of the target activity-time distribution as follows:

P, = £ + 4 i * . i

+o

4 * * + & (23)

and

^ M then the solution (9) of equation system (7) provides the starting values for the shape parameters a and b. By selecting the configuration option in the main menu of VIBES, the user can reset the various parameters controlling the speed and accuracy of the simplex search procedure.

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MODE AND AN ARBITRARY PERCENTILE SPECIFIED

Given subjective estimates m and x„, respectively, for the mode and lOOtfth percentile of the target activity time, we compute the corresponding estimates of the shape parameters by setting up a nonlinear programming problem analogous to (21) and (22)

Minimize a, b

[F-\q;a,b£,tJ) - xq}2 +.

(a - 1)0 + (b - 1)L — m

a + b - 2 (25)

(U - LY

subject to

a > 1 (26a)

b > 1 (266)

As in the case that the user specifies two arbitrary percentiles, the minimization problem defined by (25) and (26) is solved by the simplex search procedure. The simplex search is started by initially estimating the mean and standard deviation of the target activity-time^ distribution using the standard PERT approximations |i = (L + Am + t))/6 and d = (U - L)l 6; then the solution (9) of equation system (7) provides the starting values for the shape parameters a and b.

APPENDIX II. INTERACTIVELY UPDATING DISPLAYED DENSITY

After the user has completed the initial text-oriented specification of the desired activity-time characteristics, VIBES computes and plots the fitted beta density as shown in Fig. 5. In the subsequent stage of graphical interactive distribution fitting, the user controls the density-modification process by pressing the arrow keys. Suppose that after the (k — l)st keystroke in this stage of operation, the current estimates of the mode and variance are fhk_l and d-jt-i, respectively, where m0 and 6-g similarly denote the mode and variance of the initially specified beta distribution. Since the user's estimates L and U of the end points remain constant during the process of interactive density modification, no subscripts are attached to these symbols. After the kth keystroke on the arrow keys, VIBES executes the following algorithm to modify the shape of the displayed density.

1. Determine the type of arrow key pressed. a. If the up- or down-arrow key was pressed, then the variance of the

displayed distribution is to be changed. Let 8k denote the direction of the requested change in the variance

8* = 1 if T was pressed (27a)

8̂ = — 1 if J, was pressed (27b)

The magnitude ak of the change in variance is determined by the adaptive seeking strategy of DeBrota et al. (1989) as a function of the previous value &l_1 of the variance and the last two keystrokes. The

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mode and variance for the fitted density at this iteration are updated as follows:

mk = mk^x (28a)

H = &2-i + »*«* (28ft)

Go to step 2. b. If the left- or right-arrow key was pressed, then the mode of the dis

played distribution is to be changed. Let 8,. denote the direction of the requested change in the mode

SA = 1 if —> was pressed (29a)

Sjt = - 1 if <— was pressed (295)

The magnitude ak of the change in mode is determined by the adaptive seeking strategy of DeBrota et al. (1989) as a function of the previous value mk_1 of the mode and the last two keystrokes. The mode and variance for the fitted density at this iteration are updated as follows:

Wt = tfik-i + §kak (30a)

n = frj-i (so/?) Go to step 2.

2. Check for feasibility of the updated estimates of the mode and variance

L< mk< U (31a)

0 < H < (U ~2L)2 (31ft)

If (31) is satisfied, go to step 3; otherwise set

mk = w,t_i (32a)

n = fri-i (32ft)

and go to step 3. 3. Solve the equation system (13) as described in equations (14)-(19) to

obtain the updated shape parameter values ak and Bk. 4. Check for the feasibility of the new parameters:

ak > 1 (33a)

Bk> 1 (33ft)

If (33) is satisfied, go to step 5; otherwise set

ak = ak_ i (34a)

K = K-x (34ft)

and go to step 5. 5. Plot the updated density f(x;ak,6k,L,U) for L < x < U. 6. Increment the iteration counter

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k <r- k + 1 (35)

and wait for the user to enter another keystroke.

APPENDIX III. REFERENCES

AbouRizk, S. M. (1989). VIBES user's guide. Division of Construction Engineering and Management, Purdue University, West Lafayette, Ind.

AbouRizk, S. M. (1990). "Input modeling for construction simulation," thesis presented to Purdue University, at West Lafayette, Ind., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

AbouRizk, S. M., Halpin, D. W., and Wilson, J. R. (1990). "Modeling input data with the beta distribution—a description of the BetaFit software." Division of Construction Engineering and Management, Purdue University, West Lafayette, Ind.

Alpert, M., and Raiffa, H. (1982). "A progress report on the training of probability assessors." Judgment under uncertainty: Heuristics and biases, D. Kahneman, P. Slovic, and A. Tversky, eds., Cambridge University Press, Cambridge, England.

Beach, L. R., and Swenson, R. G. (1966). "Intuitive estimation of means." Psy-chonomic Sci., 5(4), 161-162.

Conte, S. D., and de Boor, C. (1980). Elementary numerical analysis: An algorithmic approach. McGraw-Hill Book Co., Inc., New York, N.Y.

Debrota, D. J., Dittus, R. S., Roberts, S. D., and Wilson, J. R. (1989). "Visual interactive fitting of bounded Johnson distributions." Simulation, 52(5), 199-205.

Fowler, H. W., and Fowler, F. G. (1931). The king's English. 3rd Ed., Oxford University Press, Oxford, England.

Griffiths, P., and Hill, I. D. (1985). Applied statistics algorithms. Royal Statistical Society, London, England.

Grubbs, F. E. (1962). "Attempts to validate certain PERT statistics or 'picking on PERT,'" Operations Res., 10(2), 912-915.

Johnson, N. L., and Kotz, S. (1970). Distributions in statistics: Continuous univariate distributions—1. John Wiley and Sons, Inc., New York, N.Y.

Klein, R. W., and Baris, P. M. (1991). "Selecting and generating variates for modeling service times." Computers and Industrial Engineering, 20(1), 27-33.

Lathrop, R. G. (1967). "Perceived variability."/. Experimental Psychology, 73(4), 498-502.

Lichtenstein, S., Fischhoff, B., and Phillips, L. D. (1982). "Calibration of probabilities: The state of the art to 1980." Judgment under uncertainty: Heuristics and biases, D. Kahneman, P. Slovic, and A. Tversky, eds., Cambridge University Press, Cambridge, England.

MacCrimmon, K. R., and Ryavec, C. A. (1964). "An analytical study of the PERT assumptions." Operations Res., 12(1), 16-37.

McBride, W. J., and McClelland, C. W. (1967). "PERT and the beta distribution." IEEE Trans, on Engrg. Mgmt., EM-14(4), 166-169.

Microsoft QuickBASIC programmer's guide. (1989). Microsoft Corporation, Redmond, Wash.

Olsson, D. M. (1974). "A sequential simplex program for solving minimization problems." J. Quality Tech., 6(1), 53-57.

Peterson, C , and Miller, A. (1964). "Mode, median, and mean as optimal strategies." /. Experimental Psychology, 68(4), 363-367.

QuickPak professional—advanced programming library for BASIC compilers, version 3.0. (1989). Crescent Software, Stamford, Conn.

Spencer, J. (1963). "A further study of estimating averages." Ergonomics, 6, 255-265.

van Leunen, M.-C. (1986). A handbook for scholars. Alfred A. Knopf, Inc., New York, N.Y.

Wilson, J. R., Vaughan, D. K., Naylor, E., and Voss, R. G. (1982). "Analysis of Space Shuttle ground operations." Simulation, 38(6), 187-203.

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APPENDIX IV. NOTATION

The following symbols are used in this paper:

a = first shape parameter of beta distribution representing the duration of given activity;

b = second shape parameter of beta distribution representing the duration of given activity;

f(x;a,b,L,U) = beta probability density function evaluated at the cutoff value x as defined in (1);

F(x;a,b,L,U) = cumulative beta distribution function evaluated at cutoff value x;

F~1{q;a,b,L,U) = inverse beta distribution function evaluated at fraction q between 0 and 1;

L = lower end point of beta distribution—that is, minimum time to complete the corresponding activity;

m = mode of a beta distribution—that is, most likely time to complete corresponding activity;

q = fraction between 0 and 1 specifying percentage of given population of activity times;

X = duration of random activity time that is to be represented by beta distribution F(x;a,b,L,U);

xq = lOOgth percentile of distribution of random variable X as defined in (6);

U = upper end point of beta distribution—that is, maximum time to complete corresponding activity;

u, = mean of given population of activity times; and cr2 = variance of given population of activity times.

Subscripts k = accumulated number of keystrokes on arrow keys dur

ing process of visual interactive density modifications; and

n = index counting number of user-specified percentiles.

Embellishments = indicator for subjective estimate (like L) of property

(like L) of given activity-time distribution.

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visual interactive fitting of beta distributionsmanagement system is based on a network model [for...

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