 Using Program Evaluation and Review Technique (PERT) to Determine Project Duration

A Report Submitted to the Department of Civil Engineering at Salahaddin University-Erbil in Partial Fulfillment of the Requirement for the Degree of Master of Science in Civil Engineering

Prepared by:
Shamal Ali Othman
(MSc Student)

Supervised by:
Dr. Saman
(Assistant Professor, Ph.D.)

Erbil-Kurdistan
June 2018

Table of Content
Table of Content 1
List of Tables 1
List of Figures 1
Acknowledgment 2
Abstract 3
1. Introduction 4
1.1. Objective 5
1.2. Literature Review 5
2. Materials and Methods 8
2.1. Program Evaluation and Review Technique (PERT) scheduling steps: 8
2.2. Probability of Project Completion 10
3. Results and Discussions 11
3.1. Program Evaluation and Review Technique (PERT) scheduling: 11
3.2. Probability of Project Completion Calculation: 15
4. Conclusions 20
References 21
Appendix A 22

List of Tables
Table 1 Activity List for Construction House 11
Table 2 Activity List with Predecessors 12
Table 3 Pert data calculation 14
Table 4 Variance Calculation 17
Table 5 standard deviation Calculation 18

List of Figures
Figure 1 AOA Diagram for Construction House 13
Figure 2 AOA Diagram for Construction House With Total Calculated Expected Duration 15
Figure 3 normal distribution With Mean Equal to 89 16

Acknowledgment

I would like to thank my advisor Asst. Prof. Dr. Khalil Saman for his direction and support throughout this research. Also, I wish to offer my great acknowledgment to the academic staff of Department of Civil Engineering at Salahaddin University-Erbil for their facilities and assistance throughout this work.
Finally, I would like to thank my wife, for her understanding, love, and assistance in making this thesis.

Abstract
The traditional PERT (Program Evaluation and Review Technique) model uses beta distribution as the distribution of activity duration and estimates the mean and the variance of activity duration. It has three-time estimates, which are optimistic, pessimistic and most likely, and all the time estimates mentioned follows the beta distribution. Besides, the probability in completing the project within certain duration is calculated by using the standard normal distribution. For illustration, the data used for the construction of a 125² house was studied.
The results show that the minimum completion time for the project is 89 days and with probability 90% completion time equal to 93 days. In conclusion, CPM and PERT are practical tool in the project scheduling.

Key words: Critical Path Method, Program Evaluation Review Technique, Probabilistic Completion Time.

Introduction

The three dimensional goal for any project is well known, i.e. to finish the project on time, within budget and with satisfactory performance or quality (Ganame and Chaudhari, 2015). Any construction project is expected to be completed within certain period of time. And if the project gets delayed it results in increase in cost of the project and contractor may have to face penalty for causing delay. Hence it is very important for both owner and contractor to follow project schedule.
Scheduling is an important part of the construction project management. Planning and scheduling of construction activities helps engineers to complete the project in time and within the budget. The early endeavors of project management and scheduling date back to the development of the Gantt chart by Henry Gantt (1861–1919). This charting system for production scheduling formed the basis for two scheduling techniques, which were developed to assist in planning, managing and controlling complex organizations: The Critical path Method (CPM) and Program Evaluation and Review Technique (PERT).
The Project Evaluation and Review Technique (PERT), in conjunction with the Critical Path Method (CPM), were developed in the 1958’s to address the uncertainty in project duration for complex projects (Kerzner, 2017). During the 1960s and 1970s, various authors proposed a revision of the hypotheses put forward by the creators of the PERT method and several authors have modified the original three point PERT estimators to improve the accuracy of the estimates(Pleguezuelo et al., 2003). Hahn (2008) argues that the tail-area decay of an activity time distribution is a key factor which has been insufficiently considered previously and provides a distribution which permits varying amounts of dispersion and greater likelihoods of more extreme tail-area events that is straightforward to implement with expert judgments.
The expected value or mean value for each activity of the project network was calculated by applying the beta distribution and three estimates for the duration of the activity. The total project duration was determined by adding all the duration values of the activities on the critical path (Visser, 2016).

Objective
The main objective of this study was to determine the project duration by using Program Evaluation and Review Technique (PERT) in conjunction with the Critical Path Method. A residential house took as a case study, objective to calculate total duration and percentage probability of finishing with estimated duration.

Literature Review
Chin et al. (2017) In this paper, the probabilistic completion time of a project scheduling was discussed and the researchers used two methods to find the minimum completion time for a project scheduling. These methods are Critical Path Method (CPM) and Program Evaluation Review Technique (PERT). In CPM, a network diagram, which is Activity on Node (AON), is drawn and the slack time for every activity is calculated such that the project’s critical path could be found. The researchers explained the difference between these methods which is CPM has only one determined time estimate, while PERT has three-time estimates, which shows the uncertainty in the duration of an activity in a project. In this paper, the case study used is construction of a three-room house. The results show that the minimum completion time for the project is 44 days with a success probability 0.91.

Kim et al. (2014) This paper quantifies the accuracy of a wide range of PERT mean-variance estimation formulas. In addition, they develop a new PERT variant using common percentiles. The proposed method uses three points for estimation, just like the classical PERT. However, it provides options for the selection of the three points. It provides different set of probability weights by the selection of the three points and what parameter to estimate, i.e., mean or variance, which minimizes the estimation error. The researchers compare the accuracy of them approach with existing methods using the Pearson distribution system. The use of the Pearson system allows us to systematically compare different PERT methods over a wider range of distribution shapes than has previously been considered. This analysis shows that, despite its simple structure, their new method outperforms existing methods when estimating means and variances of most bell-shaped and J-shaped beta distributions. They also demonstrate how practitioners could use our new methods in actual project settings.

Chrysafis and Papadopoulos (2014) taking into account the intrinsic drawbacks of both classic and fuzzy PERT, the authors present a new approach in an attempt to deal with them. The tools for this approach are fuzzy statistics and fuzzy probabilities. A necessary condition for this approach is the existence of statistical data for the project activity duration. The final findings are the fuzzy activity times (earliest/latest start – earliest/latest finish), the fuzzy float, the Criticality Degree both for each activity and for each path, a fuzzy probability for the activity duration to be in a certain time interval and a fuzzy probability for the project completion time to be in certain time interval. A numerical example is provided for thorough comprehension.

Shankar et al. (2010) In this paper, on the basis of the study of the PERT assumptions, they present an improvement of these estimates. At the end of the paper, an example is presented to compare with those obtained using the proposed method as well as other method. The comparisons reveal that the method proposed in this paper is more effective in determining the activity criticalities and finding the critical path.

Pleguezuelo et al. (2003) In this paper the expression for the mean in the PERT method is considered. This mean involves a parameter k, that sometimes has seen set to 4. Insisting on the similarity between the beta and the normal distributions, certain hypotheses are proposed that lead to k necessarily being exactly 4. More speci6cally, by using the moments of the second and fourth orders, it is shown that the usual beta distribution in the PERT method is mesocratic (? = 3) and of constant variance (?² = 1/36).

Shipley et al. (1997) This paper presents two variations of a fuzzy probability based model for project management. The Belief in Fuzzy Probability Estimations of Time (BIFPET) model uses human judgment instead of stochastic assumptions to determine project completion times. Following a literature review of PERT critiques, background information is provided for BIFPET. Next, a foam block production machine project is described and solved based on three estimates of time for each activity. A variation of BIFPET that uses ranges on these time estimates is presented and the case is solved for fuzzy expected completion times. The results are compared to those derived by using PERT and benefits of the BIFPET approach are detailed. The paper concludes with a description of our ongoing research initiative in the area of fuzzy probability applications to project management.
Materials and Methods

Program Evaluation and Review Technique (PERT) scheduling steps:
Firstly, it is necessary to discuss the methodology for preparing PERT schedules. PERT scheduling is a six-step process. The steps listed below (Kerzner, 2017):
Step one
Define the Project and all of its significant activities or tasks. The Project (made up of several tasks) should have only a single start activity and a single finish activity.

Step Two
Develop the relationships among the activities. Decide which activities must precede and which must follow others.
Step three
Draw the “Network” connecting all the activities. Each Activity should have unique event numbers. Dummy arrows are used where required to avoid giving the same numbering to two activities.
The project network needs to convey all this information. Two alternative types of project networks are available for doing this.
One type is the activity-on-arc (AOA) project network, where each activity is represented by an arc. A node is used to separate an activity (an outgoing arc) from each of its immediate predecessors (an incoming arc). The sequencing of the arcs thereby shows the precedence relationships between the activities (Hillier and Lieberman, 2001).
The second type is the activity-on-node (AON) project network, where each activity is represented by a node. The arcs then are used just to show the precedence relationships between the activities. In particular, the node for each activity with immediate predecessors has an arc coming in from each of these predecessors (Hillier and Lieberman, 2001).
In this paper, activity-on-arc had been used for draw the network.

step four
In this step, the functional manager converts the arrow diagram to a PERT chart by identifying the time duration for each activity. It should be noted here that the time estimates that the line managers provide are based on the assumption of unlimited resources because the calendar dates have not yet been defined. The standard calculated values for PERT calculations are as follows (Mubarak, 2010) :
The weighted average activity time T:
T = (O+4M+P)/6 ………….. Equation 1
where:
O = optimistic activity time (1 chance in 100 of completing the activity earlier under normal conditions)
M = most likely activity time (without any learning curve effects)
P = pessimistic activity time (1 chance in 100 of completing the activity later
under normal conditions)
Step five
Compute the longest time path through the network. This is called the critical path. It is here that the project manager looks at the critical calendar dates in the definition of the project’s requirements.
Step six
Step six is often the most overlooked step. Here the project manager places calendar dates on each event in the PERT chart, thus converting from planning under unlimited resources to planning with limited resources. Even though the line manager has given you a time estimate, there is no guarantee that the correct resources will be available when needed. That is why this step is crucial.
Probability of Project Completion
The PERT method for computing project completion probability, however, assumes that the one path identified as having the longest expected completion time will always be the critical path. Thus, probability computations for the entire network are based solely on the mean and variance of this one path. The general approach first requires computing the weighted average activity time (T) and standard deviation, the mean and variance of the critical path. If (To ) represents a desired completion time (such as a project deadline or target date) then the Z-transform (Zigli):

Z = (To-T)/?te ………………….2
Where:
Z: value is used to compute the probability of completion in a standard
normal distribution table. This probability is regarded as the probability of
project completion by time To.
To: represents a desired completion time.
T: The weighted average activity time
?te: is the standard deviation of the expected time this can be found
from the expression (Kerzner, 2017):
?te=(P-O)/6 ……………….. 3
Where:
O = optimistic activity time (1 chance in 100 of completing the activity
P = pessimistic activity time (1 chance in 100 of completing the activity
later under normal conditions)

Therefore, an approximate formula for variance = (?te)²
Variance =( (P-O)/6 )² ………….. 4

Results and Discussions
Program Evaluation and Review Technique (PERT) scheduling:
As a case study, we will create a project schedule to construct a residential house 125 m² which consist two story. Schedule process consists of a number of sequential steps:

First step:
According to house plan, the researcher could define the Project and all of its significant activities or tasks which is consist the following activity as showed in Table 1:
Table 1 Activity List for Construction House
Activity Activity Description
A Site Preparation
B Excavation
C Concrete Foundation
D Foundation Filling
E Ground Floor Hollow Block Wall
F Ground Floor Slab
G First Floor Hollow Block Wall
H First Floor Slab
I Rough Electric Work
J Rough Mechanical Work
K Wall Ceramic Tile
L Granit Façade Work
M Flooring Tiles
N Cement Plastering
O Gypsum Plastering
P Secondary Ceiling Work
Q Door
R Window
S Electric Fittings
T Mechanical Fittings

Second step:
Develop the relationships among the activities. Decide which activities must precede and which must follow others. Firstly, for a built schedule to be successful, the critical path method (CPM) network should be developed at a level of detail that shows the important project structure. Secondly, enter each activity name and their predecessors with relationship between each activity.

Table 2 Activity List with Predecessors
Activity Activity Description Predecessors
A Site Preparation —
B Excavation A
C Concrete Foundation B
D Foundation Filling C
E Ground Floor Hollow Block Wall D
F Ground Floor Slab E
G First Floor Hollow Block Wall F
H First Floor Slab G
I Rough Electric Work H
J Rough Mechanical Work H
K Wall Ceramic Tile I, J
L Granit Façade Work I, J
M Flooring Tiles K
N Cement Plastering L
O Gypsum Plastering N
P Secondary Ceiling Work M, O
Q Door O
R Window O
S Electric Fittings P, Q, R
T Mechanical Fittings P, Q, R

Third Step
Based on the list of activity with its predecessors in (Table 2) the researcher draws the activity-on-arc (AOA) Network connecting all the activities as shown in Figure 1.

Fourth Step
In this step, the researcher converts the arrow diagram to a PERT chart by identifying the time duration for each activity. The data to be collected consist of 3-point estimates of duration (optimistic, most likely and pessimistic scenarios).
For determine the activity duration the researcher has made an interview with expert engineers for estimate the three-point duration for each task according them experience. After collecting data, the researcher calculates each task duration by using equation 1 (T = (O+4M+P)/6 ) and use this formula for each activity, to find the mean time for each activity, and then computed data use to perform the calculations on the network, the estimated three-point duration and determined each expected mean duration by using Equation 1 illustrated in Table:

Table 3 Pert data calculation
Activity Activity Description O M P Calculation T
A Site Preparation 1.0 2.0 3.0 (1+4*2+3)/6 2.0
B Excavation 2.0 3.0 4.0 (2+4*3+4)/6 3.0
C Concrete Foundation 4.0 5.0 6.0 (4+4*5+6)/6 5.0
D Foundation Filling 1.00 2.00 3.00 (1+4*2+3)/6 2.0
E Ground Floor Hollow Block Wall 4.0 6.0 8.0 (4+4*6+8)/6 6.0
F Ground Floor Slab 10.0 12.0 14.0 (10+4*12+14)/6 12.0
G First Floor Hollow Block Wall 4.0 6.0 8.0 (4+4*6+8)/6 6.0
H First Floor Slab 10.0 12.0 14.0 (10+4*12+14)/6 12.0
I Rough Electric Work 3.0 5.0 7.0 (3+4*5+7)/6 5.0
J Rough Mechanical Work 2.0 4.0 6.0 (2+4*4+6)/6 4.0
K Wall Ceramic Tile 12.0 18.0 24.0 (12+4*18+24)/6 18.0
L Granit Façade Work 7.0 9.0 11.0 (7+4*9+11)/6 9.0
M Flooring Tiles 4.0 6.0 8.0 (4+4*6+8)/6 6.0
N Cement Plastering 4.0 6.0 8.0 (4+4*6+8)/6 6.0
O Gypsum Plastering 4.0 6.0 8.0 (4+4*6+8)/6 6.0
P Secondary Ceiling Work 5.0 8.0 11.0 (5+4*8+11)/6 8.0
Q Door 2.0 4.0 6.0 (2+4*4+6)/6 4.0
R Window 2.0 4.0 6.0 (2+4*4+6)/6 4.0
S Electric Fittings 2.0 3.0 4.0 (2+4*3+4)/6 3.0
T Mechanical Fittings 2.0 4.0 6.0 (2+4*4+6)/6 4.0

Fifth Step
Based on the computed data in forth step the expected time for each activity is copied onto the network and used to calculate the Early start, Early finish, Late Finish, Late Start, Critical Path and total expected duration for project as shown in Figure 2.

Based on the AOA diagram the longest path is (A, B, C, D, E, F, G, H, I, K, M, P, T) and the estimated project duration equal to 89 days.

Probability of Project Completion Calculation:
Now the researcher tries to use PERT method for computing project completion probability
the expected activity times are simply the means of Beta distributions, so they are a set of random variables. Also, the project completion time is simply the sum of the expected activity times for the critical path activities (Vanhoucke, 2012). Therefore, the project completion time should be the mean of a normal distribution, and we can use the Normal Distribution Table at the back of your text to find out how likely it is that any particular deadline will be met. Below is a Figure 3 of the normal distribution for the project, with the mean of 89.

Figure 3 normal distribution With Mean Equal to 89

Now, a normal distribution needs two pieces of information or parameters. The first is the mean, and we have that, and the second is the variance or standard deviation. To calculate that we go back to our data and calculate the variance of each activity. The formula we use is: ( (P-O)/6 )² . Therefore, we get Table 4. We only need the variance for the critical path activities, so researcher has again marked them with asterisks. Adding up the fourteen critical path variances, we get a total variance of 8.56, and taking the square root of that number, we get a standard deviation of 2.94 as shown in Table5.

Table 4 Variance Calculation
Activity Activity Description O M P T calculations variance Critical
A Site Preparation 1 2 3 2 ( (3-1)/6 )² 0.11 *
B Excavation 2 3 4 3 ( (4-2)/6 )² 0.11 *
C Concrete Foundation 4 5 6 5 ( (6-4)/6 )² 0.11 *
D Foundation Filling 1 2 3 2 ( (3-1)/6 )² 0.11 *
E Ground Floor Hollow Block Wall 4 6 8 6 ( (8-4)/6 )² 0.44 *
F Ground Floor Slab 10 12 14 12 ( (14-10)/6 )² 0.44 *
G First Floor Hollow Block Wall 4 6 8 6 ( (8-4)/6 )² 0.44 *
H First Floor Slab 10 12 14 12 ( (14-10)/6 )² 0.44 *
I Rough Electric Work 3 5 7 5 ( (7-3)/6 )² 0.44 *
J Rough Mechanical Work 2 4 6 4 ( (6-2)/6 )² 0.44
K Wall Ceramic Tile 12 18 24 18 ( (24-12)/6 )² 4.00 *
L Granit Façade Work 7 9 11 9 ( (11-7)/6 )² 0.44
M Flooring Tiles 4 6 8 6 ( (8-4)/6 )² 0.44 *
N Cement Plastering 4 6 8 6 ( (8-4)/6 )² 0.44
O Gypsum Plastering 4 6 8 6 ( (8-4)/6 )² 0.44
P Secondary Ceiling Work 5 8 11 8 ( (11-5)/6 )² 1.00 *
Q Door 2 4 6 4 ( (6-2)/6 )² 0.44
R Window 2 4 6 4 ( (6-2)/6 )² 0.44
S Electric Fittings 2 3 4 3 ( (4-2)/6 )² 0.11
T Mechanical Fittings 2 4 6 4 ( (6-2)/6 )² 0.44 *

Table 5 standard deviation Calculation
Activity Critical Activity Description variance
A Site Preparation 0.11
B Excavation 0.11
C Concrete Foundation 0.11
D Foundation Filling 0.11
E Ground Floor Hollow Block Wall 0.44
F Ground Floor Slab 0.44
G First Floor Hollow Block Wall 0.44
H First Floor Slab 0.44
I Rough Electric Work 0.44
K Wall Ceramic Tile 4.00
M Flooring Tiles 0.44
P Secondary Ceiling Work 1.00
T Mechanical Fittings 0.44
total variance 8.56
standard deviation 2.92

Based on the calculated data we can finding the probability of meeting a particular completion time by determining Z value the formula we use is,
Z = (To-T)/?te .
“Z” is the number you can look up in the border of the Normal Distribution Table in Appendix A. Then read the probability out of the body of the table. We try to determine the probability of project done in 88-day, 90 day and completion time corresponding to 90% probability.
Now have all the data needed to calculate Z. The desired completion time(To) is 88, the mean is 89, and the standard deviation is 2.92. Putting these values into the Equation 2 so get:

Z = (To-T)/?te
Z = (88-89)/2.92
Z= -0.34

Referring to normal curve distribution table, we get
P (project will be done in 88 days) = 0.3669 = 36.69%. a Low probability

When the desired completion time(To) equal to 90 days:

Z = (To-T)/?te
Z = (90-89)/2.92
z=0.34
Referring to normal curve distribution, we get
P (project will be done in 90 days) = 0.6331 = 63.31%. a Medium probability.

Completion time corresponding to 90% probability:
From normal distribution table for probability 0.9 Z equal to 1.29

Z = (To-T)/?te

1.29 = (To-89)/2.92
To= 4.1+89
To= 93.1 Day To = 93 Day

the desired completion equal to 93 Day corresponding to 90% probability.

Conclusions
In this paper, we have scheduled construction house with the Project Evaluation and Review Technique (PERT), in conjunction with the Critical Path Method (CPM). Firstly, the methodology for preparing PERT schedules has been discussed and the procedure of determining the probability of finishing project in determined time was explained. Secondly, as a case study the researcher had taken a house 125 m² for determine the project duration by using Project Evaluation and Review Technique (PERT). Total expected duration for house equal to 89 day and the probability project will be done in 90 days equal to= 63.31%. Farther more, the desired completion equal to 93 Day corresponding to 90% probability. Uncertainty in the project can be minimized by applying CPM and PERT. The project manager will be able to cope with these methods as they know the use of these two approaches. CPM is more suitable for construction process as the project has fairly accurate in the time estimation. However, if it is dealing with large and high capital construction project, PERT would be a better choice.

References
CHIN, M. K., KEK, S. L., SIM, S. Y. ; SEOW, T. W. 2017. Probabilistic Completion Time in Project Scheduling.
CHRYSAFIS, K. A. ; PAPADOPOULOS, B. K. 2014. Approaching activity duration in PERT by means of fuzzy sets theory and statistics. Journal of Intelligent ; Fuzzy Systems, 26, 577-587.
GANAME, P. ; CHAUDHARI, P. 2015. Construction building schedule risk analysis using Monte-Carlo simulation. International Research Journal of Engineering and Technologi, 2, 1402-1406.
HAHN, E. D. 2008. Mixture densities for project management activity times: A robust approach to PERT. European Journal of Operational Research, 188, 450-459.
HILLIER, F. ; LIEBERMAN, G. 2001. Project management with pert/cpm. Introduction to Operations Research.
KERZNER, H. 2017. Project management: a systems approach to planning, scheduling, and controlling, John Wiley ; Sons.
KIM, S. D., HAMMOND, R. K. ; BICKEL, J. E. 2014. Improved mean and variance estimating formulas for pert analyses. IEEE Transactions on Engineering Management, 61, 362-369.
MUBARAK, S. 2010. Construction project scheduling and control. Wiley.
PLEGUEZUELO, R. H. A., PÉREZ, J. G. A. ; RAMBAUD, S. C. 2003. A note on the reasonableness of PERT hypotheses. Operations Research Letters, 31, 60-62.
SHANKAR, N. R., RAO, K. S. N. ; SIREESHA, V. Estimating the Mean and Variance of. International Mathematical Forum, 2010. 861-868.
SHIPLEY, M. F., DE KORVIN, A. ; OMER, K. 1997. BIFPET methodology versus PERT in project management: fuzzy probability instead of the beta distribution. Journal of Engineering and Technology management, 14, 49-65.
VANHOUCKE, M. 2012. Project management with dynamic scheduling, Springer.
VISSER, J. K. Suitability of different probability distributions for performing schedule risk simulations in project management. Management of Engineering and Technology (PICMET), 2016 Portland International Conference on, 2016. IEEE, 2031-2039.
ZIGLI, R. M. Pert probability.

Appendix A