9.4 Defining Precedence Relationships Among Activities
Once work activities have been defined, the relationships among the activities can be specified. Precedence relations between activities signify that the activities must take place in a particular sequence. Numerous natural sequences exist for construction activities due to requirements for structural integrity, regulations, and other technical requirements. For example, design drawings cannot be checked before they are drawn. Diagramatically, precedence relationships can be illustrated by a network or graph in which the activities are represented by arrows as in Figure 9-0. The arrows in Figure 9-3 are called branches or links in the activity network, while the circles marking the beginning or end of each arrow are called nodes or events. In this figure, links represent particular activities, while the nodes represent milestone events.
Figure 9-3 Illustrative Set of Four Activities with Precedences
More complicated precedence relationships can also be specified. For example, one activity might not be able to start for several days after the completion of another activity. As a common example, concrete might have to cure (or set) for several days before formwork is removed. This restriction on the removal of forms activity is called a lag between the completion of one activity (i.e., pouring concrete in this case) and the start of another activity (i.e., removing formwork in this case). Many computer based scheduling programs permit the use of a variety of precedence relationships.
Three mistakes should be avoided in specifying predecessor relationships for construction plans. First, a circle of activity precedences will result in an impossible plan. For example, if activity A precedes activity B, activity B precedes activity C, and activity C precedes activity A, then the project can never be started or completed! Figure 9-4 illustrates the resulting activity network. Fortunately, formal scheduling methods and good computer scheduling programs will find any such errors in the logic of the construction plan.
Figure 9-4 Example of an Impossible Work Plan
Forgetting a necessary precedence relationship can be more insidious. For example, suppose that installation of dry wall should be done prior to floor finishing. Ignoring this precedence relationship may result in both activities being scheduled at the same time. Corrections on the spot may result in increased costs or problems of quality in the completed project. Unfortunately, there are few ways in which precedence omissions can be found other than with checks by knowledgeable managers or by comparison to comparable projects. One other possible but little used mechanism for checking precedences is to conduct a physical or computer based simulation of the construction process and observe any problems.
Finally, it is important to realize that different types of precedence relationships can be defined and that each has different implications for the schedule of activities:
In revising schedules as work proceeds, it is important to realize that different types of precedence relationships have quite different implications for the flexibility and cost of changing the construction plan. Unfortunately, many formal scheduling systems do not possess the capability of indicating this type of flexibility. As a result, the burden is placed upon the manager of making such decisions and insuring realistic and effective schedules. With all the other responsibilities of a project manager, it is no surprise that preparing or revising the formal, computer based construction plan is a low priority to a manager in such cases. Nevertheless, formal construction plans may be essential for good management of complicated projects.
Example 9-4: Precedence Definition for Site Preparation and Foundation Work
Suppose that a site preparation and concrete slab foundation construction project consists of nine different activities:
A. Site clearing (of brush and minor debris),
B. Removal of trees,
C. General excavation,
D. Grading general area,
E. Excavation for utility trenches,
F. Placing formwork and reinforcement for concrete,
G. Installing sewer lines,
H. Installing other utilities,
I. Pouring concrete.
Activities A (site clearing) and B (tree removal) do not have preceding activities since they depend on none of the other activities. We assume that activities C (general excavation) and D (general grading) are preceded by activity A (site clearing). It might also be the case that the planner wished to delay any excavation until trees were removed, so that B (tree removal) would be a precedent activity to C (general excavation) and D (general grading). Activities E (trench excavation) and F (concrete preparation) cannot begin until the completion of general excavation and grading, since they involve subsequent excavation and trench preparation. Activities G (install lines) and H (install utilities) represent installation in the utility trenches and cannot be attempted until the trenches are prepared, so that activity E (trench excavation) is a preceding activity. We also assume that the utilities should not be installed until grading is completed to avoid equipment conflicts, so activity D (general grading) is also preceding activities G (install sewers) and H (install utilities). Finally, activity I (pour concrete) cannot begin until the sewer line is installed and formwork and reinforcement are ready, so activities F and G are preceding. Other utilities may be routed over the slab foundation, so activity H (install utilities) is not necessarily a preceding activity for activity I (pour concrete). The result of our planning are the immediate precedences shown in Table 9-1.
TABLE 9-1 Precedence Relations for a Nine-Activity Project Example
With this information, the next problem is to represent the activities in a network diagram and to determine all the precedence relationships among the activities. One network representation of these nine activities is shown in Figure 9-5, in which the activities appear as branches or links between nodes. The nodes represent milestones of possible beginning and starting times. This representation is called an activity-on-branch diagram. Note that an initial event beginning activity is defined (Node 0 in Figure 9-5), while node 5 represents the completion of all activities.
Figure 9-5 Activity-on-Branch Representation of a Nine Activity Project
Alternatively, the nine activities could be represented by nodes and predecessor relationships by branches or links, as in Figure 9-6. The result is an activity-on-node diagram. In Figure 9-6, new activity nodes representing the beginning and the end of construction have been added to mark these important milestones.
These network representations of activities can be very helpful in visualizing the various activities and their relationships for a project. Whether activities are represented as branches (as in Figure 9-5) or as nodes (as in Figure 9-5) is largely a matter of organizational or personal choice. Some considerations in choosing one form or another are discussed in Chapter 10.
Figure 9-6 Activity-on-Node Representation of a Nine Activity Project
It is also notable that Table 9-1 lists only the immediate predecessor relationships. Clearly, there are other precedence relationships which involve more than one activity. For example, "installing sewer lines" (activity G) cannot be undertaken before "site clearing" (Activity A) is complete since the activity "grading general area" (Activity D) must precede activity G and must follow activity A. Table 9-1 is an implicit precedence list since only immediate predecessors are recorded. An explicit predecessor list would include all of the preceding activities for activity G. Table 9-2 shows all such predecessor relationships implied by the project plan. This table can be produced by tracing all paths through the network back from a particular activity and can be performed algorithmically. For example, inspecting Figure 9-6 reveals that each activity except for activity B depends upon the completion of activity A.
TABLE 9-2 All Activity Precedence Relationships for a Nine-Activity Project
9.5 Estimating Activity Durations
In most scheduling procedures, each work activity has an associated time duration. These durations are used extensively in preparing a schedule. For example, suppose that the durations shown in Table 9-3 were estimated for the project diagrammed in Figure 9-0. The entire set of activities would then require at least 3 days, since the activities follow one another directly and require a total of 1.0 + 0.5 + 0.5 + 1.0 = 3 days. If another activity proceeded in parallel with this sequence, the 3 day minimum duration of these four activities is unaffected. More than 3 days would be required for the sequence if there was a delay or a lag between the completion of one activity and the start of another.
TABLE 9-3 Durations and Predecessors for a Four Activity Project Illustration
All formal scheduling procedures rely upon estimates of the durations of the various project activities as well as the definitions of the predecessor relationships among tasks. The variability of an activity's duration may also be considered. Formally, the probability distribution of an activity's duration as well as the expected or most likely duration may be used in scheduling. A probability distribution indicates the chance that a particular activity duration will occur. In advance of actually doing a particular task, we cannot be certain exactly how long the task will require.
A straightforward approach to the estimation of activity durations is to keep historical records of particular activities and rely on the average durations from this experience in making new duration estimates. Since the scope of activities are unlikely to be identical between different projects, unit productivity rates are typically employed for this purpose. For example, the duration of an activity Dij such as concrete formwork assembly might be estimated as:
where Aij is the required formwork area to assemble (in square yards), Pij is the average productivity of a standard crew in this task (measured in square yards per hour), and Nij is the number of crews assigned to the task. In some organizations, unit production time, Tij, is defined as the time required to complete a unit of work by a standard crew (measured in hours per square yards) is used as a productivity measure such that Tij is a reciprocal of Pij.
A formula such as Eq. (9.1) can be used for nearly all construction activities. Typically, the required quantity of work, Aij is determined from detailed examination of the final facility design. This quantity-take-off to obtain the required amounts of materials, volumes, and areas is a very common process in bid preparation by contractors. In some countries, specialized quantity surveyors provide the information on required quantities for all potential contractors and the owner. The number of crews working, Nij, is decided by the planner. In many cases, the number or amount of resources applied to particular activities may be modified in light of the resulting project plan and schedule. Finally, some estimate of the expected work productivity, Pij must be provided to apply Equation (9.1). As with cost factors, commercial services can provide average productivity figures for many standard activities of this sort. Historical records in a firm can also provide data for estimation of productivities.
The calculation of a duration as in Equation (9.1) is only an approximation to the actual activity duration for a number of reasons. First, it is usually the case that peculiarities of the project make the accomplishment of a particular activity more or less difficult. For example, access to the forms in a particular location may be difficult; as a result, the productivity of assembling forms may be lower than the average value for a particular project. Often, adjustments based on engineering judgment are made to the calculated durations from Equation (9.1) for this reason.
In addition, productivity rates may vary in both systematic and random fashions from the average. An example of systematic variation is the effect of learning on productivity. As a crew becomes familiar with an activity and the work habits of the crew, their productivity will typically improve. Figure 9-7 illustrates the type of productivity increase that might occur with experience; this curve is called a learning curve. The result is that productivity Pij is a function of the duration of an activity or project. A common construction example is that the assembly of floors in a building might go faster at higher levels due to improved productivity even though the transportation time up to the active construction area is longer. Again, historical records or subjective adjustments might be made to represent learning curve variations in average productivity.
Figure 9-7 Illustration of Productivity Changes Due to Learning
Random factors will also influence productivity rates and make estimation of activity durations uncertain. For example, a scheduler will typically not know at the time of making the initial schedule how skillful the crew and manager will be that are assigned to a particular project. The productivity of a skilled designer may be many times that of an unskilled engineer. In the absence of specific knowledge, the estimator can only use average values of productivity.
Weather effects are often very important and thus deserve particular attention in estimating durations. Weather has both systematic and random influences on activity durations. Whether or not a rainstorm will come on a particular day is certainly a random effect that will influence the productivity of many activities. However, the likelihood of a rainstorm is likely to vary systematically from one month or one site to the next. Adjustment factors for inclement weather as well as meteorological records can be used to incorporate the effects of weather on durations. As a simple example, an activity might require ten days in perfect weather, but the activity could not proceed in the rain. Furthermore, suppose that rain is expected ten percent of the days in a particular month. In this case, the expected activity duration is eleven days including one expected rain day.
Finally, the use of average productivity factors themselves cause problems in the calculation presented in Equation (9.1). The expected value of the multiplicative reciprocal of a variable is not exactly equal to the reciprocal of the variable's expected value. For example, if productivity on an activity is either six in good weather (ie., P=6) or two in bad weather (ie., P=2) and good or bad weather is equally likely, then the expected productivity is P = (6)(0.5) + (2)(0.5) = 4, and the reciprocal of expected productivity is 1/4. However, the expected reciprocal of productivity is E[1/P] = (0.5)/6 + (0.5)/2 = 1/3. The reciprocal of expected productivity is 25% less than the expected value of the reciprocal in this case! By representing only two possible productivity values, this example represents an extreme case, but it is always true that the use of average productivity factors in Equation (9.1) will result in optimistic estimates of activity durations. The use of actual averages for the reciprocals of productivity or small adjustment factors may be used to correct for this non-linearity problem.
The simple duration calculation shown in Equation (9.1) also assumes an inverse linear relationship between the number of crews assigned to an activity and the total duration of work. While this is a reasonable assumption in situations for which crews can work independently and require no special coordination, it need not always be true. For example, design tasks may be divided among numerous architects and engineers, but delays to insure proper coordination and communication increase as the number of workers increase. As another example, insuring a smooth flow of material to all crews on a site may be increasingly difficult as the number of crews increase. In these latter cases, the relationship between activity duration and the number of crews is unlikely to be inversely proportional as shown in Equation (9.1). As a result, adjustments to the estimated productivity from Equation (9.1) must be made. Alternatively, more complicated functional relationships might be estimated between duration and resources used in the same way that nonlinear preliminary or conceptual cost estimate models are prepared.
One mechanism to formalize the estimation of activity durations is to employ a hierarchical estimation framework. This approach decomposes the estimation problem into component parts in which the higher levels in the hierarchy represent attributes which depend upon the details of lower level adjustments and calculations. For example, Figure 9-8 represents various levels in the estimation of the duration of masonry construction. At the lowest level, the maximum productivity for the activity is estimated based upon general work conditions. Table 9-4 illustrates some possible maximum productivity values that might be employed in this estimation. At the next higher level, adjustments to these maximum productivities are made to account for special site conditions and crew compositions; table 9-5 illustrates some possible adjustment rules. At the highest level, adjustments for overall effects such as weather are introduced. Also shown in Figure 9-8 are nodes to estimate down or unproductive time associated with the masonry construction activity. The formalization of the estimation process illustrated in Figure 9-8 permits the development of computer aids for the estimation process or can serve as a conceptual framework for a human estimator.
TABLE 9-4 Maximum Productivity Estimates for Masonry Work
TABLE 9-5 Possible Adjustments to Maximum Productivities for Masonry Construction/caption
Figure 9-8 A Hierarchical Estimation Framework for Masonry Construction
In addition to the problem of estimating the expected duration of an activity, some scheduling procedures explicitly consider the uncertainty in activity duration estimates by using the probabilistic distribution of activity durations. That is, the duration of a particular activity is assu med to be a random variable that is distributed in a particular fashion. For example, an activity duration might be assumed to be distributed as a normal or a beta distributed random variable as illustrated in Figure 9-9. This figure shows the probability or chance of experiencing a particular activity duration based on a probabilistic distribution. The beta distribution is often used to characterize activity durations, since it can have an absolute minimum and an absolute maximum of possible duration times. The normal distribution is a good approximation to the beta distribution in the center of the distribution and is easy to work with, so it is often used as an approximation.
Figure 9-9 Beta and Normally Distributed Activity Durations
If a standard random variable is used to characterize the distribution of activity durations, then only a few parameters are required to calculate the probability of any particular duration. Still, the estimation problem is increased considerably since more than one parameter is required to characterize most of the probabilistic distribution used to represent activity durations. For the beta distribution, three or four parameters are required depending on its generality, whereas the normal distribution requires two parameters.
As an example, the normal distribution is characterized by two parameters, and representing the average duration and the standard deviation of the duration, respectively. Alternatively, the variance of the distribution could be used to describe or characterize the variability of duration times; the variance is the value of the standard deviation multiplied by itself. From historical data, these two parameters can be estimated as:
where we assume that n different observations xk of the random variable x are available. This estimation process might be applied to activity durations directly (so that xk would be a record of an activity duration Dij on a past project) or to the estimation of the distribution of productivities (so that xk would be a record of the productivity in an activity Pi) on a past project) which, in turn, is used to estimate durations using Equation (9.4). If more accuracy is desired, the estimation equations for mean and standard deviation, Equations (9.2) and (9.3) would be used to estimate the mean and standard deviation of the reciprocal of productivity to avoid non-linear effects. Using estimates of productivities, the standard deviation of activity duration would be calculated as:
where is the estimated standard deviation of the reciprocal of productivity that is calculated from Equation (9.3) by substituting 1/P for x.