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Uncertainty in Task Duration and Cost Estimates: Fusion of Probabilistic Forecasts and Deterministic Scheduling

Homayoun Khamooshi, Ph.D.1; and Denis F. Cioffi, Ph.D.2

Abstract: Amodel for project budgeting and scheduling with uncertainty is developed. The traditional critical-path method (CPM) misleads because there are few, if any, real-life deterministic situations for which CPM is a great match; program evaluation and review technique (PERT) has been seen to have its problems, too (e.g., merge bias, unavailability of data, difficulty of implementation by practitioners). A dual focus on the distributions of the possible errors in the time and cost estimates and on the reliability of the estimates used as planned values suggests an approach for developing reliable schedules and budgets with buffers for time and cost. This method for budgeting and scheduling is executed through either simulation or a simple analytical approximation. The dynamic buffers provide much-needed flexibility, accounting for the errors in cost and duration estimates associated with planning any real project, thus providing a realistic, practical, and dynamic approach to planning and scheduling. DOI: 10.1061/(ASCE)CO.1943-7862.0000616. © 2013 American Society of Civil Engineers.

CE Database subject headings: Estimation; Scheduling; Construction costs; Budgets; Probability; Project management; Simulation; Uncertainty principles; Forecasting.

Author keywords: Estimate; Reliability; Scheduling; Cost; Budget; Probability; Contingency; Project management; Simulation.

Introduction and Background

Inaccurate estimation has long been identified as one of the major causes of project failure (Flyvbjerg et al. 2009; Chan and Kumaraswamy 1997; Pinto and Mantel 1990), and Standish Group reports (1998, 2009) show more projects failing and fewer success- ful projects. Not easily achieved are good measures of worker pro- ductivity and the total amount of the work, which when combined determine task durations. For a specific activity, underestimation is generally caused by oversight or lack of familiarity or understand- ing of the job at hand, but it may even be driven by organizational culture or political causes.

Estimating errors on work packages or activities may delay achieving a milestone and disrupt the remaining project schedule. The delay and disruption caused by bad estimation may lead to project failure (Lee et al. 2009) or at best to project management failure, that is, not delivering the project on time, within budget, and per specifications. Abundant literature provides statistics on project management failure that link the failure to an absence of good planning and scheduling, the causes of which are either the estimates or the process used for planning and scheduling (e.g., Williams 1995; Herroelen et al. 1998; Ritch et al. 2002; De Meyer et al. 2002; Herroelen and Leus 2004, 2005).

Williams (2005) provides overwhelming literature on project management failure and project overruns. He questions the

underlying assumptions of project management theory, specifically the critical-path method (CPM) model and its suitability for man- aging specific types of projects. Although the focus of Williams’ paper is on a specific category of projects, i.e., those that are large and complex (with high levels of uncertainty), inaccurate estimates and a rigid scheduling approach are generic problems for almost all projects, which by definition are about achieving something new. Touran (2010) also claims that “the use of probabilistic risk assessment in major infrastructure projects is increasing to cope with major cost overruns and schedule delays.”

As argued by Howick (2003), the delay and disruption caused by the uncertainty associated with estimates (as well as their presumed certainty) can drastically affect the cost, quality, and du- ration of the project. Although the objective should be developing the most accurate estimates, the inaccuracy inherent in estima- tion needs to be accounted for through some flexible mechanism (Khamooshi 1999), or the impact may cause more damage than would be tolerated by the stakeholders, leading to project failure. To solve the problem of unwanted scheduling iterations in design and development, Ballard (1999, 2000) and the Lean Construction Institute (LCI) at Berkeley introduced the concept of so-called phase scheduling, in which much emphasis is put on team work and on the reliability of the estimate. Ballard goes further and sug- gests, “The key point is to deliberately and publicly generate, quan- tify, and allocate schedule contingency.” The following assertions are derived from the literature: • The time needed to complete individual tasks or work packages

is almost always greater than or at best equal to the original estimates used for establishing the baseline schedule. In other words, Parkinson’s law (Parkinson 1957) holds: work expands to fill available time. Work can be further delayed because of the student syndrome, i.e., waiting until the end to finish a task.

• Statistics on classical project management success, that is, de- livering the project on time, within budget, and per specifica- tion, show that most projects finish late (Eden et al. 2005) and overrun their budgeted costs (Hughes 1986; Standish Group 2009).

1Assistant Professor, Dept. of Decision Sciences, School of Business, George Washington Univ., Funger Hall, Suite 415, 2201 G St., NW, Washington, DC 20052 (corresponding author). E-mail: hkh@gwu.edu

2Associate Professor, Dept. of Decision Sciences, School of Business, George Washington Univ., Funger Hall, Suite 415, 2201 G St., NW, Washington, DC 20052. E-mail: dcioffi@gwu.edu

Note. This manuscript was submitted on October 25, 2010; approved on May 29, 2012; published online on July 24, 2012. Discussion period open until October 1, 2013; separate discussions must be submitted for indivi- dual papers. This paper is part of the Journal of Construction Engineering and Management, Vol. 139, No. 5, May 1, 2013. © ASCE, ISSN 0733- 9364/2013/5-488-497/$25.00.

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http://dx.doi.org/10.1061/(ASCE)CO.1943-7862.0000616
http://dx.doi.org/10.1061/(ASCE)CO.1943-7862.0000616
http://dx.doi.org/10.1061/(ASCE)CO.1943-7862.0000616
http://dx.doi.org/10.1061/(ASCE)CO.1943-7862.0000616
• In organizations with less emphasis on planning and control, inaccuracy of the baseline estimates due to padding (Burt and Kemp 1994) or even intentional underestimation could be accepted as norm, leading to deviation from reliable, accu- rate, and ethical estimates. González et al. (2010) developed an approach to improve planning reliability at an operational level. They argue that reliability and commitment are key factors for improving project performance. The lack of reliability in turn can lead to delayed and overrun projects (Flyvbjerg et al. 2002).

• The cost of underestimation, which leads to rescheduling or scheduling on the go (e.g., critical chain project management, agile scheduling, and other similar approaches), is normally ignored. Although dynamism and flexibility are a must for the uncertain and volatile environments in which projects operate, the objective should be to develop as stable a schedule as possible (Khamooshi 1999; Emhejellen et al. 2003). The perceived agility costs money, especially in large and complex project environments. Williams (2005, 1995), Lichtenberg (1983), and many others

have questioned the efficacy of deterministic approaches, and more than two decades ago, the British Petroleum company, for example, committed itself to probabilistic approaches for time and cost analysis for its major projects. These probabilistic-based ap- proaches, however, provide for contingency; they do not focus on scheduling tasks. More realistic, reliable, and stable schedules are still needed.

Certainty, Uncertainty, and Errors in Scheduling Estimates

Program Evaluation and Review Technique and Monte Carlo Focus on Project Duration

The accuracy and the resulting reliability of estimates should be taken into account more seriously in developing project schedules. After the early stages of selection and approval, planning for a project continues with defining the scope of the project in greater detail, developing a work breakdown structure, and establishing estimates for the tasks and activities inside the work packages in preparation for schedule creation. The estimation process is used to specify the effort: the product of the duration needed to deliver the work-package products and the types and quantities of resour- ces needed to achieve these objectives. The figures developed in this process are only estimates that are subject to uncertainty and, hence, inaccuracy. The lower the certainty, the higher the chance of exceeding the planned duration.

Two divergent approaches are used for planning project dura- tions: (1) deterministic (CPM) and (2) probabilistic, e.g., program evaluation and review technique (PERT) or Monte Carlo. In all these methods, single-valued task durations are used to develop a baseline schedule, i.e., the schedule against which the real work must be accomplished.

With PERT, by taking advantage of the central-limit theorem and using mean activity durations from the assumed distributions of duration times for each task on the critical path, one can estimate the probabilities that correspond to completion times less or more than the planned project duration that is based on those mean times (which has a 50% probability). AMonte Carlo simulation, although a more-detailed calculation that eliminates PERT’s merge-bias problem and its ignorance of near-critical paths, uses what is fun- damentally a similar plan of attack. Again, a distribution is assumed for each task on the critical path, and the probabilities of various project durations are found.

More than 50 years ago, PERT was developed to deal with uncertainties associated with estimated activity durations (and through them the project as a whole), but the model is not used widely as a real project planning and scheduling tool because few project managers are willing to schedule projects using PERT’s mean value for the tasks’ durations. There has been much criticism of PERT (MacCrimmon and Ryanec 1964; Klingel 1966; Britney 1976; Schonberger 1981; Baker 1986). Wayne and Cottrell (1998) introduced a simplified version to overcome some of its difficulties.

The PERT method is based on and supported by the central- limit theorem, which implies using the expected value duration, Te ¼ ðaþ 4mþ bÞ=6, of each task as its planned duration (the mode or other values could be used as well). The project duration, which is the sum of the activities on the critical path, will be the average or expected project completion time, i.e., with only a 50% chance of being accomplished within that duration.

The analysis assumes that the actual duration of each activity could be any value from the range given by the assumed distribu- tion. Realistically, however, the planned duration of each task (the duration estimate used to develop the baseline schedule) is at best the time needed to finish the job; it is often exceeded. In other words, the probability of each task taking less time than the planned value is low (much less than the theoretical value of 50% if the average value is used as the planned value). This underestimation is not a theoretical problem or issue with statistical foundation of PERT analysis. Rather, it is a behavioral or perceptional one explained by Parkinson’s law (Parkinson 1957; Gutierrez and Kouvelis 1991) and the student syndrome law that work expands to fill the time available or that the job is delayed.

At the task level, the studies by Buehler et al. (1994) and Roy et al. (2005) discuss the rationale and support the reality that indi- viduals often underestimate the time needed to do a job. Because the actual time needed to do a task is normally more than what one plans for, the actual duration typically lands on the right-hand side of the time axis of the planned duration.

At the system and project levels there is also ample evidence of optimism bias in time and cost estimation (Flyvbjerg 2008; Wook and Rojas 2011). These observations suggest that the planned duration at best is almost always perceived as, and ends up being, the minimum actual time needed to do the job even if the worker may know, for example, that the planned duration is the most likely value (the mode), per historical data. Thus, the possible values to the left of planned durations become very unlikely as soon as the planned duration is confirmed. This problem does not exist when the system (project) is simulated mechanically, where all values above and below the planned value have their probabilities of occurrence as per given assumed distribution. In rare cases of a “routine” project (Cioffi 2006) and management environment, the tasks may be completed at their planned durations but hardly in less time. Both PERT and the traditional Monte Carlo simulation do not account for the reality of these reduced probabilities; thus, they overestimate greatly the chances of project durations less than the scheduled duration.

To develop a real working schedule, one must assume single- point values for the individual task durations. In this paper, by con- sidering the reliability of the particular single-point values used in developing the actual schedule and by also examining the size and distribution of the error associated with those values, it is shown how one can develop a robust schedule and determine the proper size of its buffer; this method of then extended to incorporate cost budgeting. The overestimates of shorter project durations by PERT and Monte Carlo suggest using a new distribution in the statistical calculations; therefore, more-realistic duration distributions are first examined.

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Which Distributions Best Describe Task Durations?

As planning and scheduling specialists start developing an estimate for the duration of an activity, they look for previous experiences and data. If the data for a particular task have been recorded and are available, they can be used to develop the duration distribution of the task. A distribution can be presented with values of the dura- tions shown on the x-axis and the occurrence likelihood (continu- ous) or count (discrete) depicted on the y-axis. Figs. 1(a and b) show examples of continuous and discrete probability distributions.

Fente et al. (2000) developed a model for defining a beta dis- tribution for simulating construction project tasks, but Back et al. (2000) suggested using a triangular probability distribution func- tion for costs in construction projects.

What types of estimates best describe project tasks and provide an accurate indication of their durations? • Optimistic estimates (minimum durations) that define the lower

limits of beta or triangular distributions have, by definition, a probability of order 1%. Thus, if n is the number of activities on the critical path, the minimum project duration derived when using these distributions will occur at a probability of approximately 0.01 to the nth power. This probability will be vanishingly small.

• Using the expected value of the durations as per central-limit theorem analysis in PERT or the modes as a rationale choice (because these are the most frequently occurring value of each estimate) instead of using optimistic values will result in longer project duration with a better chance of being achieved. But, in reality, how much better? Assuming that the tasks will not finish in less time than planned for (Buehler et al. 1994; Roy et al. 2005), the probability of finishing the project in a time less than or equal to this planned duration is approximately 0.5 to the nth power, where n is the number of activities on the critical path.

For a critical path containing only 10 tasks, this probability is less than one-tenth of 1%.

• As explained previously with regard to Parkinson’s law and the student syndrome, as soon as any value is fixed as the planned value, the actual task duration has little or no chance of being realized in a time much less than its planned duration but has some probability of lasting longer than planned, depending on the given reliability for the duration. Thus, some tasks on the critical path will exceed their planned durations, and the project will overrun.

• Pessimistic estimates could be used as the planned values. This choice increases the chances of success, with minimum delay, but it would most likely waste resources because the opportu- nity for shorter delivery has been lost. The environment and characteristics in which tasks are executed

and projects are developed are unique. Any historical data gathered should not necessarily be interpreted as a replication of the same experiment, and so reproducing the duration distributions in a sim- ulation will not result in a true prediction of the new project under consideration unless valid, detailed data are available. Research by Kirytopoulos et al. (2008) suggests that when good historical data are applied properly, these data can guide the selection of the spe- cific duration distribution to be used for scheduling, and then PERT and Monte Carlo simulations produce essentially the same duration results.

Before scheduling, the continuous probability density or the discrete distribution function for the task duration developed from historical data normally contains a mode or most likely value that is greater than or equal to the minimum duration. After scheduling, Parkinson’s law takes effect, and the duration distribution should be truncated on the left-hand side of the scheduled duration. Choosing values less than the mode could produce underestimates in the task durations, which causes overruns and frequent need of reschedul- ing. Thus, in most cases, the scheduled duration should be equal to or greater than the mode, and this new duration (i.e., the minimum of the now-truncated distribution) is the one to be used for the most realistic probabilistic forecast of the behavior of the project.

Hence, for durations less than the scheduled duration, zero prob- ability is assumed. A truncated version of a beta distribution or a right-angle triangle distribution may now show the new probability density function for the duration. The distribution to the right of the planned duration shows the probability distribution of the error in the estimate. This probability distribution of the error is expected to typically decrease (if a continuous distribution) with increasing duration.

The size and shape of the distribution characterize the error and its uncertainty. A small error will be reflected in a small range to the right of the estimate. There are many possible ways to characterize the error distribution. A decreasing function [Fig. 2(a)] gives greater reliability because it indicates a greater chance of the real- ized duration being close to the minimum; a constant indicates that, within some fixed limits, all possible extensions to the duration or cost are equally likely. Lastly, there could be situations in which the probability of the error increases with the size of the error [Fig. 2(b)].

Because high reliability of the mode was established more than four decades ago (Peterson and Miller 1964; Clark 1961), the trun- cation idea can be extended to a discrete, binomial distribution, where the two values of the distribution represent the minimum (the mode) and maximum durations. Now the error is represented by only a single duration: the difference between the maximum and the minimum durations is the error. This method, with either con- tinuous or discrete distributions, shifts the estimating focus for any

Fig. 1. (a) Continuous and (b) discrete probability distributions for duration of task

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given task to establishing a single reliable task duration, the error, and its associated probability.

In contrast to critical chain, which takes the estimator’s number and arbitrary assigns a fraction of it, say 50%, as planned duration, hoping that some resources will be saved, the best estimate of the planned duration is obtained by allowing the estimator to take full responsibility, and then this estimate is used in the schedule. This decision shows great trust in the estimator’s judgment.

These arguments are now tested with a numerical example, in which project durations obtained with this method are compared with those obtained from traditional and modified Monte Carlo simulations.

Numerical Example Using Three Different Distributions

A house-refurbishing project from the Pertmaster sample files is used to illustrate the effects of using the different distributions pre- viously discussed; Table 1 shows the project data. With Pertmaster, four separate simulations of the project’s schedule were conducted. For each run, one of the following distributions was used when modeling the activity durations. For each task, ta is the minimum duration, tm is the mode, and tb is the maximum duration. The distributions used are as follows: 1. Truncated beta distribution: ta ¼ tm and tb specified; 2. Right-angle triangular distribution: ta ¼ tm and tb specified; 3. Binomial distribution: In contrast to the previous distributions,

a discrete distribution with only two possible outcomes, ta ¼ tm and tb, where δt ¼ tb − ta is the error of the duration estimate. Table 1 shows the data for this example; and

4. Binomial distribution with half the original maximum error; the new maximum ðta þ tbÞ=2 yields an error δt ¼ ðtb − taÞ=2.

All the distributions need two input values, ta and tb. In the simulations, only the fraction of possibilities equal to the error percentage (1 minus the reliability percentage) sits to the right of the minimum; e.g., with a reliability of x percent, x percent of the simulation selections will equal the minimum duration, with

the remainder ð1 − xÞ at the maximum for the two binomials and somewhere between the minimum and the maximum for the two continuous distributions. Hence, the simulations differ from the traditional Monte Carlo because each activity is modeled with a branched duration. One branch yields the fixed minimum duration at a frequency equal to the reliability percentage (given by the es- timator), and the other branch uses the given distribution to extend the duration beyond the minimum. Changing the reliability of the estimation does not necessarily change the shape of the distribution or the size of the estimation error; it does change the branching probability, i.e., the probability of the minimum duration being realized.

The comparison between the discrete and continuous distribu- tions will differ depending on the numbers used in the discrete dis- tribution. For example, a binomial distribution where the longer duration is set to the maximum error (number 3 in the preceding list) is much more conservative than continuous distributions because when the error occurs, the estimated duration falls at the duration that corresponds to the maximum error. Thus, a less conservative approach was also used by using half the maximum error, ðtb − taÞ=2, i.e., the average error, as the error duration in the binomial distribution; these behaviors are illustrated in the follow- ing. The practical advantages of the binomial outweigh minor con- cerns about the exact shape of the truncated duration distribution.

Now the results of numerical simulations that used the preced- ing distributions are examined. The following tables show probabi- listic duration results from simulations of 1,000 and 10,000 iterations. Table 2 shows the results of six different simulations, each using a different distribution for all project tasks: (1) a tradi- tional beta distribution; (2) a truncated beta distribution that necessitates branched simulation; (3) a binomial distribution that uses ta as the default duration and tb as the extended duration, i.e., the default plus the total error; and (4) a binomial distribution that uses ta as the default duration but uses the average error, ðta þ tbÞ=2, for the extended duration instead of the maximum er- ror; (5) a triangular distribution; and (6) a right-angle triangular distribution. The truncated beta, right-angle triangular, and the bi- nomial distributions were calculated with a 90% reliable planned duration (i.e., 10% probability of error).