1. Department of Civil and Environmental Engineering CEE 491 – Decision and Risk Analysis
University of Illinois at Urbana-Champaign Instructor: Paolo Gardoni
1
Forecasting Construction Productivity:
A Pilot Study
M.G. Hmaidi
Hmaidi2, 662308970
University of Illinois at Urbana-Champaign, US
ABSTRACT
In an effort to create a fully automated 4D scheduler that can effectively predict and model activity progression in
BIM, it is necessary that construction productivity is forecasted. The final model would be designed to automatically
gather data on-site and self-correct so that the productivity forecast is tailored to each unique construction site. The
objective of this paper is to construct the base probability density function of construction productivity. Herein, two
factors that affect labour productivity are considered. The first considers the productivity gains made as a crew
becomes accustomed to a repetitive task and the second accounts for fatigue resulting from temperature variations.
More factors will be considered in further studies.
Keywords: Determining the PDF of construction productivity. Temperature Effect on Fatigue. Learning-Curve.
1. INTRODUCTION
Current industry practice relies heavily on the experience of construction managers along with generalized
productivity and duration estimates to determine order of activity precedence and project finish times. As
a result, a large uncertainty is created when calculating funds and manpower required to complete a project
which at best has to be added into the bid as an extra cost to protect the firm from financial loss. At worst,
the firm goes significantly over budget; creating a forecasting model of construction productivity would
significantly mitigate this risk. Furthermore, such a model would allow the construction manager to clearly
see where future problems may arise and work to prevent this outcome. Productivity is defined as:
ker
Output CY
X
Input Wor hours
 

(1.1)
One of the most significant drivers of productivity is the learning curve a crew experiences as it
completes a repetitive task. The learning curve portrays productivity increase as crews gain experience
and familiarity with a task (Thomas et al. 1986). Repetitive tasks (e.g. pouring concrete, formwork
installation, rebar fixing, column erection, etc.) form the vast majority of construction work; for this
reason, they will be used as the main proponent of the model and all other factors will be measured
against them.
The other exogenous variable used in the model is fatigue due to temperature fluctuations. Physical work
in construction is very taxing and unfavorable temperatures amplify exhaustion. Even with breaks,
workers’ ability to perform labor will invariably drop (Thomas et al. 1987).

2. Department of Civil and Environmental Engineering CEE 491 – Decision and Risk Analysis
University of Illinois at Urbana-Champaign Instructor: Paolo Gardoni
2
2. PREVIOUS WORKS AND PDF DISTRIBUTIONS OF FACTORS
2.1. Learning curve
2.1.1. Relation to Productivity
Crews learn and enhance their productivity the more they conduct a repetitive task. This implies that
productivity can be related to total worker-hours spent on an activity. The research presented in (Kim et al.
(2015)) discussed this relationship extensively and hypothesized productivity to be related to time through
a natural logarithmic relationship as shown in figure 1. Through multiple time series analyses, the lognormal
relationship was demonstrated to be appropriate as shown in figure 2. This result was also confirmed in
(Kisi et al. (2016)). Therefore, the learning curve is modelled to follow a general natural logarithmic
function as shown below:
1 2lnWX a a  (2.1)
where X represents productivity, a1, a2 are constants, and W represents worker-hours:
2.1.2. Probability distribution function of random variable T
The uncertainty of the random variable W can be effectively modeled by a beta-PERT distribution which
for simplicity’s sake can be approximated as a triangular distribution with only a marginal accuracy loss
(Davis 2005). The mean and PDF of the triangular distribution are represented below:
min max mod
3
e
W
w w w

 
 (2.2)
min
min
min mod
max mod mod min
max
mod max
max min max mod
max
0
2(w )
(w )(w )
(w)
2(w )
(w )(w )
0
e
e e
W
e
e
w w
w
w w w
w w
f
w
w w w
w w
w w
 
   
  
  
  
  
 
 
(2.3)
where w represents the amount of worker hours taken to complete an activity. w will be calculated using
the following formula:
Figure 1: Learning Curve Based on Kim (2015) Figure 2: Regression Model based on Kim (2015)

3. Department of Civil and Environmental Engineering CEE 491 – Decision and Risk Analysis
University of Illinois at Urbana-Champaign Instructor: Paolo Gardoni
3
mod
R
e
RSmeans
Quantity
w
X
 (2.4)
where QuantityR is a known constant of the amount of work that must be completed in cubic yards and
XRSmeans is the productivity for that activity provided in the R.S means book. This function states that the
most likely amount of worker-hours to complete an activity is based on average productivity from the R.S
means book. wmax and wmin are going to be related to wmode through the following equations:
min 1 mod
1 2
max 2 mode
1
ew d w
d d
w d w
 
  
 
(2.5)
where d1 and d2 are constants. Now using the result in Eqn. 2.5, we can write the new PDF of W in Eqn.
2.3 as follows:
1 mode
1 mod
1 mod mod
2 1 mod 1 mod
2 mod
mod 2 mod
2 1 mod 2 mod
2 mod
0
2(w )
[(d d )w ][(1 )w ]
(w)
2(d )
[(d d )w )[(d 1)w ]
0
e
e e
e e
W
e
e e
e e
e
w d w
d w
d w w w
d
f
w w
w w d w
w d w
 
   
  
  
  
  
 
 
(2.6)
2.2 Temperature
2.2.1. Relation to efficiency
Temperatures both hot and cold have an adverse effect on labour productivity as highlighted in figure 3
(Thomas et al. 1987). This figure was generated through countless regression models; it was found that
fluctuations of temperature away from approximately 55 degrees Fahrenheit reduced worker efficiency.
Any days where the relative humidity exceeded 65% were considered anomalies and discounted. (Thomas
et al. (1987)) found that productivity efficiency varied with temperature in accordance with the following
equation:
1 2 3 ln( )b b t b t    (2.7)
where ε is the efficiency factor, b1, b2, and b3 are constants, and t is the temperature in degrees Fahrenheit.

4. Department of Civil and Environmental Engineering CEE 491 – Decision and Risk Analysis
University of Illinois at Urbana-Champaign Instructor: Paolo Gardoni
4
2.2.2 PDF of temperature
The spread of daily high and low temperatures for a given State on any day this year can be found on (IEM).
Figure 4 shows an example distribution of the spread for August 1st
in Illinois. Depending on the activity
start date, the distribution would be selected accordingly. A linear combination of the low and high
temperature distributions is completed to find a new average distribution of temperatures throughout the
work day. This is represented in the following equation:
1 2
1 2
2 1
{L H
c c Day Shift
T cT c T
c c Night Shift

 

(2.8)
where T represents the normal distribution of daily temperatures, TL and TH represent the normal
distributions of daily low and high temperatures respectively, and c1 and c2 are constants. The constants c1
and c2 represent the proportion of worker-hours spent during times of day within the respective high or low
temperature distributions. For example, laborers working a typical morning shift are going to spend a
greater proportion of their day operating within the TH zone. This would imply that c2 > c1. From Eqn. 2.8,
we can derive the mean, standard deviation, and PDF of the random variable T:
1 2[T] [T ] [T ]L HE c E c E  (2.9)
2 2 2 2
1 2 1 2 ,T2L H L H L HT T T T T Tc c c c        (2.10)
21 1
(t) exp[ ( ) ] {
22
T
T
TT
t
f t



      (2.11)
Figure 3: Regression Curve of Temperature Effect based on Thomas (1987)

5. Department of Civil and Environmental Engineering CEE 491 – Decision and Risk Analysis
University of Illinois at Urbana-Champaign Instructor: Paolo Gardoni
5
Figure 4: Daily Temperature Spread Model by IEM (2016)
3. PDF OF PRODUCTIVITY
Efficiency due to temperature Eqn. 2.7 can be multiplied by the relation of productivity to the learning
curve Eqn. 2.1 to create the modified productivity curve shown below:
1 2( log )X a W a 
1 2 3 1 2(b lnT)( lnW )X b T b a a    (3.1)
where X is a random variable representing productivity, W is a random variable representing worker-
hours, T is a random variable denoting temperature in degrees Fahrenheit, ε is the efficiency factor, and
a1, a2, b1, b2, c1, and c2 are constants. This equation can be written in the generalized form as:
( ,T)X g W (3.2)
and the inverse, g-1
is given by
1 2 3 1 2(b ln )( ln )X b T b T a W a   
1 2
1 2 3
2
1 2 3 1
2 1 2 2 2 3
1 1 1 2 1 3
lnW
lnT
1
lnW ( )
lnT
lnT
exp( )
lnT
X
a a
b b T b
X
a
b b T b a
X a b a b T a b
W
a b a b T a b
 
 
 
 
  

 
1 2 1 2 2 2 3
1 1 1 2 1 3
lnT
( , ) exp( )
lnT
X a b a b T a b
g X T
a b a b T a b
   

 
(3.3)
Since W and T are both basic continuous random variables, the CDF and PDF of X would be:

6. Department of Civil and Environmental Engineering CEE 491 – Decision and Risk Analysis
University of Illinois at Urbana-Champaign Instructor: Paolo Gardoni
6
1
W,T
1
1
W,T
( ) (w,t)
( ) (g ,t)
g
X
X
X
F x f dwdt
g
F x f dwdt
x


 
 

 




 
 
1
1
W,T( ) (g ,t)X
g
f x f dt
x
 




 (3.4)
The partial derivative with respect to x is given below:
1
2 1 2 2 2 3
1 1 1 2 1 3 1 1 1 2 1 3
lnT1
exp( )
lnT lnT
X a b a b T a bg
x a b a b T a b a b a b T a b

  

    
(3.5)
Since the W and T are statistically independent, we can write the joint probability as follows:
W,T ( ,t) ( ) ( )W Tf w f w f t  (3.6)
1 mod
21 mod
1 mod mod
2 1 mod 1 mod
W,T
22 mod
mod 2 mod
2 1 mod 2 mod
0 ,
2(w ) 1 1
( )[ exp( ( ) )] ,
[(d d )w ][(1 )w ] 22
( ,t)
2(d ) 1 1
( )[ exp( ( ) )] ,
[(d d )w )[(d 1)w ] 22
e
e T
e e
e e TT
e T
e e
e e TT
w d w t
d w t
d w w w t
d
f w
w w t
w w d w t




    
 
       
 

 
     
 
2 mode0 ,w d w t
 
 
 
  
 
 
 
 
      
(3.7)
The PDF is obtained by substituting Eqns. 3.5 and 3.7 into Eqn. 3.4 and integrating with respect to t
yields the probability distribution function of productivity. This integral would then be completed on
Mathematica. The expected value and standard deviation would them be computed again through
Mathematic using the following two equations,
[ ] ( )X XE X xf x dx


   (3.8)
2 2
( )X X xx f x dx 


  (3.9)
For the sake of this paper, these integral will not be computed and will be left in the form shown in Eqns.
3.4, 3.8 and 3.9. This is because more factors will be added in future works and the distribution of X in its
current form is not of interest.
4. CONCLUSIONS AND FUTURE WORKS
It is possible to create a theoretical framework to estimate the probability distribution of productivity.
However, when creating the above distribution a simplifying assumption was made that only two factors
affect productivity rate. Future work will be completed to incorporate more factors to get a better
representation of the PDF of productivity. These factors include but are not limited to the effects of

7. Department of Civil and Environmental Engineering CEE 491 – Decision and Risk Analysis
University of Illinois at Urbana-Champaign Instructor: Paolo Gardoni
7
humidity, labour unavailability, material unavailability and change orders on efficiency. Furthermore, a
mechanism will be introduced to automatically gather data on site and update the PDFs of the factors to
make each PDF unique to a project site.
AKCNOWLEDGEMENT
I would like to stretch a hand of gratitude to Professor Paolo Gardoni, Paul Gharzouzi and Robert Koulakes for their
helpful discussions and suggestions on the topic.
REFERENCES
Davis, R. (2006). Stochastic Project Duration Analysis using PERT beta Distributions. Marrketing and Decision
Sciences.
Ibbs, W. (2005). Impact of Change’s Timing on Labor Productivity. Journal of Construction Engineering and
Management. 131:11
Kim, H. Lee, H. Park, M. Ahn, C. Hwang, S. (2015). Productivity Forecasting of Newly Added Workers Based on
Time-Series Analysis and Site Learning. Journal of Construction Engineering and Management. 141:9.
Thomas, H. Yiakoumis, I. (1987). Factor Model of Construction Productivity. Journal of Construction Engineering
and Management. 113:4f
Nguyen, L. and Nguyen, H. (2012). Construction Research Congress 2012.
akrherz@iastate.edu, daryl. "IEM :: Automated Data Plotter". Mesonet.agron.iastate.edu. N.p., 2016. Web. 14 Dec.
2016.