project management for construction chapter 11

First, the uncertainty in activity durations may simply be ignored and scheduling done using the expected or most likely time duration for each activity.. Using expected activity duratio

11 Advanced Scheduling Techniques

11.1 Use of Advanced Scheduling Techniques

Construction project scheduling is a topic that has received extensive research over a number of

decades The previous chapter described the fundamental scheduling techniques widely used and supported by numerous commercial scheduling systems A variety of special techniques have also been developed to address specific circumstances or problems With the availability of more powerful computers and software, the use of advanced scheduling techniques is becoming easier and of greater relevance to practice In this chapter, we survey some of the techniques that can be employed in this regard These techniques address some important practical problems, such as:

• scheduling in the face of uncertain estimates on activity durations,

• integrated planning of scheduling and resource allocation,

• scheduling in unstructured or poorly formulated circumstances

A final section in the chapter describes some possible improvements in the project scheduling process

In Chapter 14, we consider issues of computer based implementation of scheduling procedures,

particularly in the context of integrating scheduling with other project management procedures

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11.2 Scheduling with Uncertain Durations

Section 10.3 described the application of critical path scheduling for the situation in which activity

durations are fixed and known Unfortunately, activity durations are estimates of the actual time

required, and there is liable to be a significant amount of uncertainty associated with the actual

durations During the preliminary planning stages for a project, the uncertainty in activity durations is particularly large since the scope and obstacles to the project are still undefined Activities that are outside of the control of the owner are likely to be more uncertain For example, the time required to gain regulatory approval for projects may vary tremendously Other external events such as adverse weather, trench collapses, or labor strikes make duration estimates particularly uncertain

Two simple approaches to dealing with the uncertainty in activity durations warrant some discussion before introducing more formal scheduling procedures to deal with uncertainty First, the uncertainty

in activity durations may simply be ignored and scheduling done using the expected or most likely time duration for each activity Since only one duration estimate needs to be made for each activity, this approach reduces the required work in setting up the original schedule Formal methods of

introducing uncertainty into the scheduling process require more work and assumptions While this simple approach might be defended, it has two drawbacks First, the use of expected activity durations typically results in overly optimistic schedules for completion; a numerical example of this optimism appears below Second, the use of single activity durations often produces a rigid, inflexible mindset

on the part of schedulers As field managers appreciate, activity durations vary considerable and can

be influenced by good leadership and close attention As a result, field managers may loose confidence

in the realism of a schedule based upon fixed activity durations Clearly, the use of fixed activity durations in setting up a schedule makes a continual process of monitoring and updating the schedule

in light of actual experience imperative Otherwise, the project schedule is rapidly outdated

A second simple approach to incorporation uncertainty also deserves mention Many managers

recognize that the use of expected durations may result in overly optimistic schedules, so they include

a contingency allowance in their estimate of activity durations For example, an activity with an

expected duration of two days might be scheduled for a period of 2.2 days, including a ten percent contingency Systematic application of this contingency would result in a ten percent increase in the

expected time to complete the project While the use of this rule-of-thumb or heuristic contingency

factor can result in more accurate schedules, it is likely that formal scheduling methods that

incorporate uncertainty more formally are useful as a means of obtaining greater accuracy or in

understanding the effects of activity delays

The most common formal approach to incorporate uncertainty in the scheduling process is to apply the critical path scheduling process (as described in Section 10.3) and then analyze the results from a probabilistic perspective This process is usually referred to as the PERT scheduling or evaluation method [1] As noted earlier, the duration of the critical path represents the minimum time required to complete the project Using expected activity durations and critical path scheduling, a critical path of activities can be identified This critical path is then used to analyze the duration of the project

incorporating the uncertainty of the activity durations along the critical path The expected project duration is equal to the sum of the expected durations of the activities along the critical path

Assuming that activity durations are independent random variables, the variance or variation in the duration of this critical path is calculated as the sum of the variances along the critical path With the mean and variance of the identified critical path known, the distribution of activity durations can also

be computed

The mean and variance for each activity duration are typically computed from estimates of

"optimistic" (ai,j), "most likely" (mi,j), and "pessimistic" (bi,j) activity durations using the formulas:

probabilistic beta distribution under a restrictive condition [2] The probability density function of a beta distributions for a random varable x is given by:

If + = 4, then Eq (11.4) will result in Eq (11.1) Thus, the use of Eqs (11.1) and (11.2) impose

an additional condition on the beta distribution In particular, the restriction that = (b - a)/6 is imposed

Figure 11-1 Illustration of Several Beta Distributions

Since absolute limits on the optimistic and pessimistic activity durations are extremely difficult to estimate from historical data, a common practice is to use the ninety-fifth percentile of activity

durations for these points Thus, the optimistic time would be such that there is only a one in twenty (five percent) chance that the actual duration would be less than the estimated optimistic time

Similarly, the pessimistic time is chosen so that there is only a five percent chance of exceeding this duration Thus, there is a ninety percent chance of having the actual duration of an activity fall

between the optimistic and pessimistic duration time estimates With the use of ninety-fifth percentile values for the optimistic and pessimistic activity duration, the calculation of the expected duration according to Eq (11.1) is unchanged but the formula for calculating the activity variance becomes:

(11.5)

The difference between Eqs (11.2) and (11.5) comes only in the value of the divisor, with 36 used for absolute limits and 10 used for ninety-five percentile limits This difference might be expected since

the difference between bi,j and ai,j would be larger for absolute limits than for the ninety-fifth

percentile limits

While the PERT method has been made widely available, it suffers from three major problems First, the procedure focuses upon a single critical path, when many paths might become critical due to random fluctuations For example, suppose that the critical path with longest expected time happened

to be completed early Unfortunately, this does not necessarily mean that the project is completed early since another path or sequence of activities might take longer Similarly, a longer than expected

duration for an activity not on the critical path might result in that activity suddenly becoming critical

As a result of the focus on only a single path, the PERT method typically underestimates the actual

project duration

As a second problem with the PERT procedure, it is incorrect to assume that most construction activity

durations are independent random variables In practice, durations are correlated with one another For

example, if problems are encountered in the delivery of concrete for a project, this problem is likely to influence the expected duration of numerous activities involving concrete pours on a project Positive

correlations of this type between activity durations imply that the PERT method underestimates the

variance of the critical path and thereby produces over-optimistic expectations of the probability of meeting a particular project completion deadline

Finally, the PERT method requires three duration estimates for each activity rather than the single estimate developed for critical path scheduling Thus, the difficulty and labor of estimating activity characteristics is multiplied threefold

As an alternative to the PERT procedure, a straightforward method of obtaining information about the

distribution of project completion times (as well as other schedule information) is through the use of

Monte Carlo simulation This technique calculates sets of artificial (but realistic) activity duration times and then applies a deterministic scheduling procedure to each set of durations Numerous

calculations are required in this process since simulated activity durations must be calculated and the scheduling procedure applied many times For realistic project networks, 40 to 1,000 separate sets of activity durations might be used in a single scheduling simulation The calculations associated with Monte Carlo simulation are described in the following section

A number of different indicators of the project schedule can be estimated from the results of a Monte Carlo simulation:

• Estimates of the expected time and variance of the project completion

• An estimate of the distribution of completion times, so that the probability of meeting a

particular completion date can be estimated

• The probability that a particular activity will lie on the critical path This is of interest since the longest or critical path through the network may change as activity durations change

The disadvantage of Monte Carlo simulation results from the additional information about activity durations that is required and the computational effort involved in numerous scheduling applications for each set of simulated durations For each activity, the distribution of possible durations as well as the parameters of this distribution must be specified For example, durations might be assumed or

estimated to be uniformly distributed between a lower and upper value In addition, correlations

between activity durations should be specified For example, if two activities involve assembling forms in different locations and at different times for a project, then the time required for each activity

is likely to be closely related If the forms pose some problems, then assembling them on both

occasions might take longer than expected This is an example of a positive correlation in activity times In application, such correlations are commonly ignored, leading to errors in results As a final problem and discouragement, easy to use software systems for Monte Carlo simulation of project schedules are not generally available This is particularly the case when correlations between activity durations are desired

Another approach to the simulation of different activity durations is to develop specific scenarios of events and determine the effect on the overall project schedule This is a type of "what-if" problem solving in which a manager simulates events that might occur and sees the result For example, the effects of different weather patterns on activity durations could be estimated and the resulting

schedules for the different weather patterns compared One method of obtaining information about the range of possible schedules is to apply the scheduling procedure using all optimistic, all most likely, and then all pessimistic activity durations The result is three project schedules representing a range of possible outcomes This process of "what-if" analysis is similar to that undertaken during the process

of construction planning or during analysis of project crashing

Example 11-1: Scheduling activities with uncertain time durations

Suppose that the nine activity example project shown in Table 10-2 and Figure 10-4 of Chapter 10 was thought to have very uncertain activity time durations As a result, project scheduling considering this uncertainty is desired All three methods (PERT, Monte Carlo simulation, and "What-if" simulation) will be applied

Table 11-1 shows the estimated optimistic, most likely and pessimistic durations for the nine activities From these estimates, the mean, variance and standard deviation are calculated In this calculation, ninety-fifth percentile estimates of optimistic and pessimistic duration times are assumed, so that Equation (11.5) is applied The critical path for this project ignoring uncertainty in activity durations consists of activities A, C, F and I as found in Table 10-3 (Section 10.3) Applying the PERT analysis procedure suggests that the duration of the project would be approximately normally distributed The sum of the means for the critical activities is 4.0 + 8.0 + 12.0 + 6.0 = 30.0 days, and the sum of the variances is 0.4 + 1.6 + 1.6 + 1.6 = 5.2 leading to a standard deviation of 2.3 days

With a normally distributed project duration, the probability of meeting a project deadline is equal to the probability that the standard normal distribution is less than or equal to (PD - D)| D where PD

is the project deadline, D is the expected duration and D is the standard deviation of project

duration For example, the probability of project completion within 35 days is:

where z is the standard normal distribution tabulated value of the cumulative standard distribution appears in Table B.1 of Appendix B

Monte Carlo simulation results provide slightly different estimates of the project duration

characteristics Assuming that activity durations are independent and approximately normally

distributed random variables with the mean and variances shown in Table 11-1, a simulation can be performed by obtaining simulated duration realization for each of the nine activities and applying critical path scheduling to the resulting network Applying this procedure 500 times, the average project duration is found to be 30.9 days with a standard deviation of 2.5 days The PERT result is less than this estimate by 0.9 days or three percent Also, the critical path considered in the PERT

procedure (consisting of activities A, C, F and I) is found to be the critical path in the simulated

networks less than half the time

TABLE 11-1 Activity Duration Estimates for a Nine Activity Project

Activity Optimistic Duration Most Likely Duration Pessimistic Duration Mean Variance

0.4 0.9 1.6 0.9 6.4 1.6 0.4 1.6 1.6

If there are correlations among the activity durations, then significantly different results can be

obtained For example, suppose that activities C, E, G and H are all positively correlated random variables with a correlation of 0.5 for each pair of variables Applying Monte Carlo simulation using

500 activity network simulations results in an average project duration of 36.5 days and a standard deviation of 4.9 days This estimated average duration is 6.5 days or 20 percent longer than the PERT estimate or the estimate obtained ignoring uncertainty in durations If correlations like this exist, these methods can seriously underestimate the actual project duration

Finally, the project durations obtained by assuming all optimistic and all pessimistic activity durations are 23 and 41 days respectively Other "what-if" simulations might be conducted for cases in which

peculiar soil characteristics might make excavation difficult; these soil peculiarities might be

responsible for the correlations of excavation activity durations described above

Results from the different methods are summarized in Table 11-2 Note that positive correlations among some activity durations results in relatively large increases in the expected project duration and variability

TABLE 11-2 Project Duration Results from Various Techniques and Assumptions for an Example

Procedure and Assumptions Project Duration (days)

30.9 36.5 23.0 30.0 41.0

NA 2.3

2.5 4.9

NA

NA

NA

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11.3 Calculations for Monte Carlo Schedule Simulation

In this section, we outline the procedures required to perform Monte Carlo simulation for the purpose

of schedule analysis These procedures presume that the various steps involved in forming a network plan and estimating the characteristics of the probability distributions for the various activities have been completed Given a plan and the activity duration distributions, the heart of the Monte Carlo

simulation procedure is the derivation of a realization or synthetic outcome of the relevant activity

durations Once these realizations are generated, standard scheduling techniques can be applied We shall present the formulas associated with the generation of normally distributed activity durations, and then comment on the requirements for other distributions in an example

To generate normally distributed realizations of activity durations, we can use a two step procedure First, we generate uniformly distributed random variables, ui in the interval from zero to one

Numerous techniques can be used for this purpose For example, a general formula for random number generation can be of the form:

(11.6)

where = 3.14159265 and ui-1 was the previously generated random number or a pre-selected

beginning or seed number For example, a seed of u0 = 0.215 in Eq (11.6) results in u1 = 0.0820, and

by applying this value of u1, the result is u2 = 0.1029 This formula is a special case of the mixed congruential method of random number generation While Equation (11.6) will result in a series of numbers that have the appearance and the necessary statistical properties of true random numbers, we should note that these are actually "pseudo" random numbers since the sequence of numbers will repeat given a long enough time

With a method of generating uniformly distributed random numbers, we can generate normally

distributed random numbers using two uniformly distributed realizations with the equations: [3]

(11.7)

with

where xk is the normal realization, x is the mean of x, x is the standard deviation of x, and u1 and

u2 are the two uniformly distributed random variable realizations For the case in which the mean of an activity is 2.5 days and the standard deviation of the duration is 1.5 days, a corresponding realization

of the duration is s = 2.2365, t = 0.6465 and xk = 2.525 days, using the two uniform random numbers generated from a seed of 0.215 above

Correlated random number realizations may be generated making use of conditional distributions For example, suppose that the duration of an activity d is normally distributed and correlated with a second normally distributed random variable x which may be another activity duration or a separate factor such as a weather effect Given a realization xk of x, the conditional distribution of d is still normal, but it is a function of the value xk In particular, the conditional mean ( 'd|x = xk) and standard

deviation ( 'd|x = xk) of a normally distributed variable given a realization of the second variable is:

(11.8)

where dx is the correlation coefficient between d and x Once xk is known, the conditional mean and standard deviation can be calculated from Eq (11.8) and then a realization of d obtained by applying Equation (11.7)

Correlation coefficients indicate the extent to which two random variables will tend to vary together Positive correlation coefficients indicate one random variable will tend to exceed its mean when the other random variable does the same From a set of n historical observations of two random variables,

x and y, the correlation coefficient can be estimated as:

Example 11-2: A Three-Activity Project Example

Suppose that we wish to apply a Monte Carlo simulation procedure to a simple project involving three activities in series As a result, the critical path for the project includes all three activities We assume that the durations of the activities are normally distributed with the following parameters:

Activity Mean (Days) Standard Deviation (Days)

A

B

C

2.5 5.6 2.4

1.5 2.4 2.0

To simulate the schedule effects, we generate the duration realizations shown in Table 11-3 and

calculate the project duration for each set of three activity duration realizations

For the twelve sets of realizations shown in the table, the mean and standard deviation of the project duration can be estimated to be 10.49 days and 4.06 days respectively In this simple case, we can also obtain an analytic solution for this duration, since it is only the sum of three independent normally distributed variables The actual project duration has a mean of 10.5 days, and a standard deviation of

days With only a limited number of simulations, the mean obtained from simulations is close to the actual mean, while the estimated standard deviation from the

nature of the set of realizations used in the simulations; using a larger number of simulated durations would result in a more accurate estimate of the standard deviation

TABLE 11-3 Duration Realizations for a Monte Carlo Schedule Simulation

Simulation Number Activity A Activity B Activity C Project Duration

6.94 4.83 6.86 7.65 5.82 8.71 2.05 10.57 3.68 0.86 9.47 6.66

1.04 2.17 5.56 2.17 1.74 4.03 1.10 3.24 2.47 1.37 0.13 1.70

9.51 9.66 15.78 10.22 10.06 15.51 6.96 17.53 7.22 3.40 11.27 8.72 Estimated Mean Project Duration = 10.49Estimated Standard Deviation of Project Duration = 4.06Note: All durations in days

Example 11-3: Generation of Realizations from Triangular Distributions

To simplify calculations for Monte Carlo simulation of schedules, the use of a triangular distribution is advantageous compared to the normal or the beta distributions Triangular distributions also have the advantage relative to the normal distribution that negative durations cannot be estimated As illustrated

in Figure 11-2, the triangular distribution can be skewed to the right or left and has finite limits like the beta distribution If a is the lower limit, b the upper limit and m the most likely value, then the mean and standard deviation of a triangular distribution are:

(11.10)

(11.11)

The cumulative probability function for the triangular distribution is:

(11.12)

where F(x) is the probability that the random variable is less than or equal to the value of x

Figure 11-2 Illustration of Two Triangular Activity Duration Distributions

Generating a random variable from this distribution can be accomplished with a single uniform

random variable realization using the inversion method In this method, a realization of the cumulative probability function, F(x) is generated and the corresponding value of x is calculated Since the

cumulative probability function varies from zero to one, the density function realization can be

obtained from the uniform value random number generator, Equation (11.6) The calculation of the corresponding value of x is obtained from inverting Equation (11.12):

(11.13)

For example, if a = 3.2, m = 4.5 and b = 6.0, then x = 4.8 and x = 2.7 With a uniform realization

of u = 0.215, then for (m-a)/(b-a) 0.215, x will lie between a and m and is found to have a value of 4.1 from Equation (11.13)

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11.4 Crashing and Time/Cost Tradeoffs

The previous sections discussed the duration of activities as either fixed or random numbers with known characteristics However, activity durations can often vary depending upon the type and

amount of resources that are applied Assigning more workers to a particular activity will normally result in a shorter duration [6] Greater speed may result in higher costs and lower quality, however In this section, we shall consider the impacts of time, cost and quality tradeoffs in activity durations In

this process, we shall discuss the procedure of project crashing as described below

A simple representation of the possible relationship between the duration of an activity and its direct costs appears in Figure 11-3 Considering only this activity in isolation and without reference to the project completion deadline, a manager would undoubtedly choose a duration which implies minimum direct cost, represented by Dij and Cij in the figure Unfortunately, if each activity was scheduled for the duration that resulted in the minimum direct cost in this way, the time to complete the entire

project might be too long and substantial penalties associated with the late project start-up might be

incurred This is a small example of sub-optimization, in which a small component of a project is

optimized or improved to the detriment of the entire project performance Avoiding this problem of sub-optimization is a fundamental concern of project managers