Several important parts were mistakenly omitted from Mr. Crandall’s article which appeared in the June, 1973 PMQ. We are reprinting the complete article in this issue.
The precedence networking convention using lead and lag notation provides a meaningful tool in the early stages of project planning and estimating. These concepts allow the user to formulate a viable scheduling network in much the same fashion as would be done to develop a standard bar chart.
Although lead/lag concepts have been utilized for some time the resulting network calculations most often ignore interruption to activities that can result when they are used. A formulation accounting for this interruption is presented and its use is demonstrated by example. The formulation is purposely described in a manner for easy conversion to computer usage but it is also usable in manual processes.
Computer generated output is shown in support of the basic conclusion that the more accurate results obtained are also more easily understood and are useful in all aspects of the scheduling process.
The introduction of network scheduling techniques in the late fifties greatly aided the timely completion of complex construction projects. For the scheduling of large projects, these methods have generally replaced the older bar charting (gantt chart) techniques. One area where the bar chart is still extensively utilized is in project planning and estimating. Reasons for this usage of bar charts include the vagueness of detail when projects are in the planning phase and familiarity of bar charting methods by the senior personnel associated with planning and estimating.
The Precedence Networking Convention (1, 2, 3, 4, 5) provides the flexibility to combine the logic processes used in developing bar charts with the more rigid accounting for time associated with networking techniques. The Precedence Networking Convention requires that every precedence constraint between the various activities in a network be shown with a precedence arc or arrow. Figure 1 illustrates this point by comparing the arrow and precedence conventions. In Figure la only one additional arc (arrow) is required to show that activity A precedes activity D; while all constraints in Figure 1b are shown by arcs.
Since each constraint is defined by an arc additional information can be placed on these arcs that would not be possible under the arrow convention. This information is described in detail once lead and lag are defined but it provides for network description similar to bar charting concepts. This additional information is currently included in many commercial computer programs, but the methods of evaluation do not provide the detailed results proposed by this paper. The significance of this additional detail is presented in the discussion of the example problem.
A program developed by the author and currently used by a Washington State firm does include the formulation suggested in this paper. The firm has found the techniques very useful, although no studies have been prepared to reflect benefits derived from their use. The author introduced the concepts to a colleague who found their use correctly predicted a time savings exceeding one and a half months on a hospital addition. This time savings reflected the difference between the results presented from a computer program with the old lead lag formulation and the hand calculation utilizing the formulation in this paper.
The terms lead and lag are often used to express the time relationship between an activity and its logical follower or predecessor (1, 4, 5). The term “lead” is utilized to indicate the amount of time the referenced activity preceeds its follower. While “lag” indicates the amount of time the referenced activity follows its predecessor. The following definitions are based on the “lead” concept and develop factors used in this paper. These factors will represent actual days of progress for the purpose of this paper, other references allow additional concepts such as percent complete (1,4). These additional concepts are straight forward and inclusion was considered unnecessary.
Start to start lead is the number of completed units of time required on the predecessor activity prior to the start of its follower. The symbol Sij will represent this factor, and indicates the number of units of time expended on activity (i) prior to the start of activity (j).
Finish to finish lead is the number of completed units of time required on the follower activity after the completion of its predecessor. The symbol Fij will represent this factor, and indicates the number of time units from the completion of activity (i) to the completion of activity (j).
Finish to start lead is the required number of time units from the completion of the precedessor to the start of its follower. The symbol Cij will represent this factor and indicates the number of time units from the completion of activity (i) to the start of activity (j).
One possible additional factor is a “start to finish lead”. Since the factor has not proven of value to the author to date, it is not included in this paper. The formulation of the necessary equations for the factor is straight forward and if desired the factor would be easy to add. The remainder of the paper refers to the general use of these concepts under the term Lead/Lag.
Figure 2 graphically demonstrates these factors as they are applied to the precedence arcs and then in bar chart time scale. (There is currently a great need to adopt a standard convention to graphically depict these factors; the method shown in this paper merely uses the previously defined symbols.) Note that the start to start and finish to finish factors were used in conjunction on the arc between activities 1 and 2 (A and B). This joint usage is not required but was shown to indicate that it is allowable. The finish to start must, on the other hand, be used independently. All precedence constraints that require no lead/lag relations are shown as arcs without the S, F, and C factors, this will be demonstrated in the example problem.
With these three factors a planner can describe a precedence network in much the same fashion as he would develop a bar chart. The principle advantage of using these factors, and the following formulation, is that the solution is that the solution obtained is the same as one which results from a semi-detailed breakdown of activities in the arrow convention. This advantage will be demonstrated in the example problem but basically the planner is allowed to consider each activity or component in its total, as with bar charts, rather than separating it into parts to provide the desired staging of time.
LEAD/LAG SOLUTION FORMULA
Before demonstrating the usefulness of these factors it is necessary to develop the procedure for evaluating the normal time values of early and late starts and finishes. Since most of these networking problems are solved on computers the following formulation will be structured for adoption to computer solution if desired. (Existing precedence programs can readily be modified to accept the proposed formulation. Basically, only the portions of the program that perform the forward and backward iterations need change. The structure of the formulation allowed the author to revise an existing program in one day.)
The following formulation considers that the use of lead/lag factors will cause certain activities to be interrupted, and the valuation will provide correct results considering these interruptions.
The lead/lag factors are indicated by pairs of subscripts, with the first representing the predecessor activity number and the second representing the follower activity number. For clarity in the following description a subscript convention is hereby established and shown in Figure 3. The activity being evaluated will always have the subscript (j). During the forward pass when early starts and finishes are found, the predecessors to the activity under evaluation are used and they will have the subscript (i). The backward iteration determines the late start and finish times and require the followers of the activity under evaluation and these will have the subscript (k).
Since actual durations will be applied to the activities during evaluation two special cases can develop when using both Sij and Fij factors jointly as shown in Fig. 2. These special cases are shown graphically to time scale in Figure 4 and indicate the aforementioned activity interruptions.
First when considering the early time relations, Figure 4a shows that activity (j) is interrupted when one considers its early start and its early finish as defined by the Sij factors. This type of interruption is correctly evaluated but less apparent when using arrow techniques and activity breakdown. In order to aid in the evaluation process aj is shown on the figure and is defined to be equal to the duration of activity (j) minus the Fij factor.
The second special case relates to the late times and is shown in Figure 4b. In this case a factor Bj is shown and is equal to the duration of activity (j) minus the Sjk factor. Both aj and Bj are utilized in the evaluation process to insure that sufficient lead time has occurred when subsequent Sjk and Fjk factors are incurred. These special cases will be demonstrated in the example problem.
It should be noted that although the interruptions are shown as two separate segments that the user retains the ability to schedule actual work within the allowable floats. The advantage of the segments is that they more clearly depict those times when the user can exercise his scheduling prerogative.
The evaluation process starts by determining the node of all activities. (Ranking provides for a one pass computer solution during both forward and backward iterations and is not required for a manual solution.) Those activities with no predecessors are assigned the rank of zero. All other activities will be assigned a rank equal to the maximum of arcs (i-j precedence relations) between them and a beginning activity. Activities are then evaluated in this rank order to assure that values assigned previously are correct. This ranking process is utilized in the example problem in order to demonstrate its application.
The forward iteration evaluates the early start and finish of all activities.
For computer solution; the early start of all activities is set at zero, the early finish and aj are set equal to the activity’s duration. For manual solution the early start of beginning activities is set to zero, their early finish is set equal to their duration; the a- values are evaluated later as required under the following formulation.
If the largest rank obtained from the network ranking is defined as “n”, select the “activity under evaluation” from the ranked order starting with those activities with a rank of (1) and ending with those of rank (n).
For each activity under evaluation calculate early start (ESj) and early finish (EFj) using the following expressions depending on the lead/lag values shown on the valid i-j arcs. (For each activity under evaluation all predecessors indicated by the i-j arcs must be considered).
Since the evaluation process follows the ranking order, the required values of early start, early finish and aj for predecessors of activities under evaluation will be correct.
The backward iteration evaluates the late start and finish for all activities.
For computer solution the late finish for all activities is set equal to the project duration, the late start is set equal to (LFj-dj) and Bj is set equal to the duration. For manual solutions evaluate only the late start and finish for the end activities.
In the backward iteration the progression is backward through the rank order for the activities, starting with those activities of rank (n-1) to those with a rank of (0).
For each activity under evaluation calculate the late finish (LFj) and late start (LSj) using the following expressions depending on the lead/lag values shown on the j-k arcs.
Once again the reverse order of ranked activities assures the required values to obtain a one pass computer solution.
The formulation has three possible areas that can cause problems during use. First, since the criteria allow activities to be interrupted depending on lead lag values, it may be desirable to flag certain activities as non-interrupt-able if indeed they cannot be split. Second, the sole use of start factors on all arcs to and from an activity will determine its start relationships, but the completion will not be defined and will be equal to the project completion date. If the completion is actually related to other activities in the network this relationship must be shown. Finally, the sole use of finish to finish factors will likewise leave the start of an activity undefined and the project start time will be utilized.
An example problem2 is presented to demonstrate the precedence lead/lag evaluation and show the corresponding detail required to evaluate the problem under the arrow convention. The problem is described below in much the same logic as that required to develop a bar chart. The problem is intended to convey the material presented previously and is not considered to represent a valid description of level of detail or durations for actual networking practice. The basic problem is to pour three consecutive floor slabs in a multi-floor building.
EXAMPLE PROBLEM REQUIREMENTS:
- Complete the pour of floor slabs for three floors with the following constraints:
- Complete the floors in order so that the crew for each activity is shown to move from one floor to the next with the exception of pouring which is described below.
- Start by placing bottom reinforcing steel. Once adequately under way with bottom steel start mechanical and electrical rough ins. Place the top reinforcing mesh once the rough ins are sufficiently advanced. When complete allow one day-for inspection and pour preparation. Finally the pour completes the work on a given floor.
- Lead/Lag constraints:
- Mechanical rough in can start once the bottom reinforcing is two days under way and completes no sooner than one day after the bottom reinforcing.
- Electrical rough in can start three days after bottom reinforcing yet cannot complete until two days after bottom reinforcing.
- Top reinforcing mesh can start once mechanical and electrical rough ins are two days complete. Top mesh must complete two days after the completion of mechanical rough in and cannot complete until the same day as electrical rough in.
- The pours can be over-lapped as long as one day of pour has been completed on the previous floor.
- Estimated durations for each floor:
- Bottom reinforcing 5 days
- Mechanical rough in 10 days
- Electrical rough in 3 days
- Top mesh 3 days
- Delay for inspection* 1 day
- Pour 2 days
*Shown in network as a “C” Lead/Lag factor.
These requirements are translated into networks using the arrow and precedence conventions. For the sake of legibility, the arrow network was separated by floor level into three networks shown on Figures 5 through 7. The precedence network is shown on Figure 8, note that the required lead/lag notation is provided on the precedence arcs. In order to show the use of Cij lead/lag factors the delay for inspection on the precedence network is shown as a Cij factor.
Solutions for both the arrow and precedence networks are compared in Table 1. The activities from the arrow network were regrouped into the separate operation by floor for direct comparison with the precedence activities. (This was necessary since the arrow network required seventy-one activities to describe the same logic as shown on the fifteen activity precedence network). The lead/lag notation generates solutions that may appear invalid to an analysist familiar with arrow networking techniques. Note the variation between (he (LS-ES) and (LF-EF) shown on bottom reinforcing first floor. (These differences are numerically the same for the separate parts of an activity in the detailed activity breakdown shown in the arrow convention and are called total float.) What is noted here is that the start of the composite activity is critical but its finish has float. Referring back to the arrow network, Figure 5, it can be seen that the first two days of the bottom reinforcing activity is critical, since the mechanical rough in is critical, while the remainder is not. The same information is available from the precedence network but not yet as readily apparent. (Comparison of computer bar chart output later will demonstrate that computer output clearly shows this delineation of the critical portion of an activity.) The precedence evaluation establishes the proper project duration and activity interrelations.
The actual evaluation of the precedence network is summarized in Tables 2 and 3, using the steps outlined in the paper. Table 2 provides the ranking of each activity and Table 3 shows the controlling condition of the basic formulation for each separate calculation, the actual variables of each calculation and results obtained. Of special interest is the evaluation of early start for the first floor top mesh and late finish for the third floor bottom reinforcing.
The only occurrence of Sij>ai in the forward pass occurs during the evaluation of activity (4), the first floor top mesh. During this calculation S3,4=2 and a3=1; therefore expression 3a was used to evaluate the early start as shown in Table 3. The other special case occurs during the backward pass evaluation of activity (11), when Fjk>Bk. These details are also shown in Table 3.
Figures 9 and 10 show bar chart outputs from computer programs for these two formulations. (This output is from the program developed by the author using the proposed formulation.) Note that the precedence bar chart shows exactly the same time staging as the arrow bar chart but is much easier to understand since each activity is a single line entry. As indicated in the introduction; this display of actual time staging under the composite precedence activity description aids the planner and ultimate user of the network analysis. Referring to the two figures note how much easier it is for a user to observe the interruption to the top mesh and electrical rough ins when using the precedence output. Manual resource assignments would be greatly aided by this type of presentation.
The use of lead/lag factors with the precedence networking convention allows a user to define the network constraints in much the same logic as was formally used to create the bar chart. The resulting network contains fewer activities than a corresponding arrow network of the same logic yet the proposed evaluation yields the identical solution. In addition an arrow network with its requirement for activity subdivision or breakdown has inherent in it a much greater potential for a logical networking error.
It should be noted that the proposed techniques do not resolve all problems of activity subdivision; indeed if subdivision is desired beyond the three segments shown in the arrow network, the corresponding precedence activities will require some subdivision. The major benefit is that the early networks developed in the planning stage where subdivision is minimal are directly adaptable to the proposed techniques. Any revisions to these early networks requiring additional subdivision can be made just as easily if not easier in the precedence formulation with continued use of the lead lag notation.
The results obtained from the precedence notation can be output from the computer in a bar chart form showing the same staging of time as the more complicated arrow network and yet is easier to understand since each activity is still a single item.
The methodology has been applied to networks requiring five to eight hundred activities in the precedence lead lag notation with very significant success. Although the program developed by the author will process networks of up to five thousand activities there is some danger in using the techniques for very large networks. This stems from the inadequacy of present networking symbols to depict adequately the effect of the lead lag offsets. This is especially true when many activities are found to be inter-ruptable during evaluation. This concern will diminish as the analyst becomes familar with results obtained from these techniques. Once the concepts are mastered the analyst will be able to define actual network logic constraints in a form more consistent with the mental process used to envision project work interaction.
The user can utilize this information to modify the actual progress by scheduling actual work accomplishment within the constraints generated. There are several ways to achieve this revised scheduling including a modification in the crew size with a corresponding change in duration, to work intermittently, or to postpone early segments and do the work all at once. Clearly the user must understand the constraints in order to effectively modify the actual progress, and the information generated provides this in the most straight forward manner.
This approach should be far more acceptable to the user during the early planning and project estimating stage. It also can provide an excellent basis for a more detailed network to control the project during actual construction if such a detailed network is desired.
1. Baker, Wilson C., SPREAD AND LEVEL CPM, Technical Report No. 56, Department of Civil Engineering, Stanford University, 1967.
2. Fondahl, John W„ A NON-COMPUTER APPROACH TO THE CRITICAL PATH METHOD FOR THE CONSTRUCTION INDUSTRY, Technical Report No. 9, Department of Civil Engineering, Stanford University, 1962.
3. Fondahl, John W., METHODS FOR EXTENDING THE RANGE OF NON-COMPUTER CRITICAL PATH APPLICATIONS, Technical Report No. 47, Department of Civil Engineering, Stanford University, 1964.
4. IBM, Project Management System, Application Description Manual (H20-0210), IBM, 1968.
5. Moder, Joseph J. and Phillips, Cecil R., PROJECT MANAGEMENT WITH CPM AND PERT (2ND ED.), Van Norstrand Reinhold, New York, 1970.