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PROBABILITY DISTRIBUTION ANALYSIS OF ACTIVITY DURATION IN

INDIAN HIGHWAY PROJECTS

C.Arun1and Bh. Nagabhushana Rao2

ABSTRACT : Traditionally, project scheduling is carried out using CPM or PERT model on a network.

CPM assumes that activity durations are deterministic while PERT assumes that they are probabilistic andfollow Beta distribution. In most practical cases, a number of random factors influence the activity

durations. Whether this randomness would lead to a Beta probability distribution for the activity duration is

debatable. The local regional conditions are expected to have an influence on this. Considering this, aprobability analysis has been performed for activity durations in several highway projects in India. Details

regarding the various activity durations have been collected from a large number of highway projects from

all over India. The data obtained from these sources was analyzed using standard statistical tools and theresults of the analysis discussed. It is observed that none of the activity durations followed the Beta

distribution as assumed in the PERT model. Rather most of the activities followed the log-log distribution.

This analysis is helpful for scheduling highway projects whose activity durations are highly random. The

probability distribution curves for activity durations, derived from this study, can be used for formulating a

simulation model whose simulated runs can realistically predict the completion time of the project andassociated confidence limits.

1. INTRODUCTION

A high level of uncertainty is associated with the activities in the highway construction projects in India.

The uncertainties in the project duration are mainly attributed due to the highly random characteristics likeunexpected variations in ground water table, delay in obtaining statutory clearances, political uncertainties,

etc. These uncertainties are observed when there is variation in the planned duration and the observed

duration for the activities. Hence the project personnel have to reschedule the activities in accordance with

the site parameters.

Due to the randomness associated with activity durations, a probability analysis is essential for the activity

duration. Studies conducted on process duration (S.M.Abourizk, D.W.Halpin 1992) indicated the

requirement of a flexible distribution function to model diversified construction activity durations. Thequality of probability distribution associated with the activity is essential for any simulation model and the

quality is essentially dependent on the sample size of the activity duration. The model put forward by

Christino Mio, et.al. (2000) provided method to fit probability distribution to activity durations recorded

from Truck payload monitoring system.

The scheduling of highway projects in India is highly dependent on the site characteristics. Hence, the

activity durations require rigorous stochastic analysis to determine the activity duration probability

function, which will stand as the backbone for any simulation model for construction projects.

1Teaching/Research Associate,Structural Engineering Division,College of Engineering, Guindy,Anna University, Chennai-25 IndiaEmail: arun_c77@hotmail.com

2Chairman Faculty of Civil EngineeringAnna University, Chennai-25IndiaEmail: nags@annauniv.edu

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2. STOCHASTIC MODEL BUILDING

The probability model developed for the activity durations of highway projects is based on the actual

duration of the activities on the project site. The data regarding the activity durations has been collected

from one hundred and twenty highway construction sites located geographically all over India. The datacollected from all over India represents wide variations in geographic, topographic and climatic conditions,

which are likely to create hindrance to construction project. This also provides the various scenarios at the

construction site viz., the unexpected variations in the soil conditions, ground water table, labor unrest, etcthat are likely to influence the activity durations. From the data, collected eleven major activities

associated with highway construction have been identified for detailed study. These cleaning and grubbing,

laying of sub grade, laying of capping layers, laying of sub base, laying of road base, laying of base course,laying of wearing course, construction of foundation for cross drainage, construction of substructure for

cross drainage, construction of super structure for cross drainage and construction of drainage works. The

duration of these activities in a project varies with the length of the highway to be constructed at any

instant. In order to have the uniform comparison amongst various projects, the activity durations are

reduced to duration of unit length. This is based on the assumption that the site factors that affect theactivity duration will influence the activity throughout the length.

The adequacy of the sample size is tested using Students T test given by.

where M is the median of the sample

2

25.1

3.0

=

t

Mn

n is the samplesizet is the Students t value for 90% confidence limit and corresponding degree of freedom

and is the standard deviation of the sample.

The sample size required at 90 percent confidence level has been determined. It was found from theanalysis that the size of the data obtained for each activity at this confidence level is far higher than the

required sample size. The minimum sample size required and the sizes of the data collected for each

activity are given in Table1.

The data collected is analyzed for fitting the probability distribution using Best Fit software from PallisadeInc. This software fits the data for each activity with sixteen standard probability distribution functions and

tests for Goodness of Fit (GOF) using Kolmogrov-Smirnov (K-S), Anderson Darling (A-D), Chi-square(2) criteria. This software will fit the probability distribution function and arrives at the parameters for the

distribution obtained by Moment matching method.

The sixteen standard distribution functions then formulated for each of the activity under each GOF criteria

are ranked based on the root mean square error (RMSE) between the probability obtained from the fitted

distribution function and the actual probability value.

= 2)(1

... ii yfxn

ESMR

Where f(xi) is the probability of xi from the probability function

yiis the probability of xifrom the observed data

n is the number of samples

3. STOCHASTIC ANALYSIS

All the major activities involved in the highway construction project are analyzed using Best Fit by varyingthe sample size. The sample size is varied randomly from thirty to the maximum available with increments

of ten samples taken uniformly throughout the population. The study was carried out for A-D test and K-S

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test and the variation in the probability distribution function was studied. From this analysis it was found

that the samples showed variation in the distribution function when the sample size was less than forty in

the K-S test and all the activities except one followed log-log distribution when the sample size was more

than forty. When the same activities were analyzed, using A-D criteria five of them converged to Log

logistic distribution while two converged to Inverse Gauss two converged to Exponential and one followed

Extreme value distribution. 2 test is not taken into account due to the inherent disadvantage of this test that

the distribution function is likely to vary with the size and number of bins used in the analysis. Table 2shows the variation and the convergence of the probability distribution function for various activities with

varying sample size employing both A-D and K-S tests.

It is observed from Table 2 that none of the activity durations is following the Beta distribution irrespective

4. DISCUSSION AND CONCLUSION

he stochastic analysis of the data collected from one hundred and twenty different highway construction

he probability distribution for most of the activities for the highway construction project is following the

he actual probability distribution for the activities can be incorporated in simulation model of project

. REFERENCES

hua D.K.H., Li G.M. (2002) RISim: Resource interacted simulation modeling in construction, ASCE

Der

a probability distribution function for

Jona d,

Lero abilistic cost and schedule

Pho ecission

Sim duration data, ASCE

of the sample size is varied. Two examples of the final distribution curves and the frequency histograms for

various activities obtained with A-D and K-S tests are given in Figures 1 and 2. It can be seen from these

figures that the probability distribution in all the cases is skewed towards the left, the skewness and kurtosis

of the final distribution for each activity is given in Table 3.

T

sites shows that the duration of most of the major activities associated with highway construction followsLog-logistic distribution. Further this analysis shows that Beta distribution is not followed in any case evenwhen the sample size is increased five fold than the minimum required. This probability can be attributed to

the highly random nature of reasons for time and cost overruns in developing countries. Thus in such cases,

network analysis with Beta distribution of PERT is debatable. It is essential that network analysis should

follow the actual distribution function for activity duration.

T

log logistic distribution model. The skewness and kurtosis for this distribution is tending towards positive

infinity that clearly shows that the distributions are biased to the left. This implies that for the activity the

most likely duration is very close to the optimistic duration as compared to pessimistic duration.

T

network. This will result in more realistic estimate of the project duration and the associated confidencelevel. This will also enable the project personnel to provide an intermediate schedule for the project

depending on the ambient and temporal site conditions and provide necessary changes in the whole project

schedule.

5

C

journal of Construction Engineering and Management (pp195-202)

k T Beeston (1983) Statistical methods for building price data

Javier Fente, Cliff Schexnayder, Kraig Knutson (2000) Definingconstruction simulation, ASCE journal of Construction Engineering and Management (pp 234- 241)

than Jingsheng Shi (1999) Activity based construction (ABC) modelling and simulation metho

ASCE journal of Construction Engineering and Management (pp 354-360)

y. J Isidore, W.Edward Back (2002) Multiple simulation analysis for probintegration, ASCE journal of Construction Engineering and Management (pp 211-219)

tios G Ioannou, Julio C Matinez (1998) Project scheduling using state based probabilistic d

network, Proceedings of the 1998 Winter simulation conference(pp 1287-1295)aan M Aburizk, Daniel W Halpin(1992) Statistical properties of construction

journal of Construction Engineering and Management (pp 525-544)

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Table 1 Sample size required and available

ActivitySample size required

(90% confidence level)

Available

sample

size

Clearing 8 94Subgrade 4 94

Capping layer 4 74

Sub base 12 80

Road base 12 94

Base Course 12 94

Wearing Course 40 94

C.D. Foundation 4 74

C.D.Substructure 4 74

C.D.Super structure 3 74

Drainage 4 90

Table 2 Convergence of Distribution Function for Activities

A.D.Test K.S.TestActivity

< 40 40 50 > 50 < 40 40 50 > 50

Clearing Lognormal Lognormal Exponential Lognormal Loglog Exponential

Sub grade Lognormal Loglog Loglog Inv. gauss Loglog Loglog

Capping layer Inv. gauss Lognormal Lognormal Inv. gauss Lognormal Lognormal

Sub base course. Inv. gauss Inv. gauss Inv. Gauss Loglog Loglog Loglog

Road Base Exp. Loglog Loglog Exp. Loglog Loglog

Base course Pearson Loglog Loglog Pearson Loglog Loglog

Wearing Course Exp. Pearson Inv. Gauss Loglog Loglog Loglog

C.D. Foundation Loglog Loglog Loglog Exp. Loglog Loglog

C.D.Substructure Loglog Loglog Lognormal Inv. gauss Pearson Loglog

C.D.Super structure Ext. value Ext.value Ext.value Pearson Pearson Loglog

Drainage Exp. Loglog Loglog Exp. Loglog Loglog

Table 3. Probability distribution for activities with skewness and kurtosis

Activity Probabaility Distribution Skewness Kurtosis

Clearing y=Exp(8.7740) Shift= +0.90391 1.1395 5.4

Subgrade y= Loglog(-0.25766,14.356,1.7980) +Infinity +Infinity

Capping Layer y=Lognormal2(1.6585,11.46)shift=+0.73546 9.389 328.8852

Sub base y=loglog(3.878,7.3434,1.7212) +Infinity +Infinity

Road Base y=Loglog(0.037730,13.425,1.6813) +Infinity +Infinity

Base Course y=Loglog(1.1471,11.283,1.6399) +Infinity +Infinity

Wearing Course y=loglog(0.42242,10.653,1.5038) +Infinity +Infinity

C.D. Foundation y=Loglog(0.30439,11.616,1.8924) +Infinity +InfinityC.D.Substructure y=Loglog(0.30439,11.616,1.8924) +Infinity +Infinity

C.D.Super structure y=loglog(-1.0579,12.290,2.6968) +Infinity +Infinity

Drainage y=Loglog(0.31680,10.658,2.0930) +Infinity +Infinity

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Figure 1. Probability Distribution for the Activity Clearing

Figure 2. Probability Distribution for the Activity Sub Grade

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Figure 3. Probability Distribution for the Activity Capping Layer

Figure 4. Probability Distribution for the Activity Sub Base Course

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Figure 5. Probability Distribution for the Activity Road Base

Figure 6. Probability Distribution for the Activity Base Course

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Figure 7. Probability Distribution for the Activity Wearing Course

Figure 8. Probability Distribution for the Activity -CD-Foundation

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Figure 9. Probability Distribution for the Activity CD-Substructure

Figure 10. Probability Distribution for the Activity CD-Superstructure

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Figure 11. Probability Distribution for the Activity Drainage

APPENDIX

Distributions Probability Distribution Function

Erlang( )

x

ex

mmf

=1

!1

1),(

m>0 (integer shape parameter)

>0(continuous scale parameter)

Exp f(x)=

x

e

=mean

Ext. Value f(x)=

)(

1

zezeb

+

b

axz

=

a=Continuous location parameterb=Continuous shape parameter

Gamma

x

ex

f

=1

)(.

1),(

>0 (scale parameter)

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>0 (shape parameter)

Inv. Gauss

=x

x

e

x

xf22

2)(

32

)( ,>0

Loglog

2)1(

1)(

t

xtxf

+

=

=

xt

= Continuous shape parameter= Continuous scale parameter= Continuous location parameter

LogNormal

2

)ln(

2

1

2

1

)(

=

x

exxf

,are continuous parameters>0

Pearson1

.

)(.

1)(

+

=

x

xe

x

xf

>0 (continuous shape parameter)

>0 (continuous scale parameter)

Weibull

=

1.

),(

x

ex

f

>0 (scale parameter)>0 (shape parameter)