*Ed. note: Critical path practitioners have for some time discounted the value of three time estimates and variance calculations as being unrealistic and impractical to use. In this article, the authors take a new look at the uses of variance which may be of practical use in certain types of projects.*

**Introduction**

Project management frequently uses network diagrams to plan the project, evaluate alternatives and control progress toward completion. The two most common networking techniques, CPM (Critical Path Method) and PERT (Program Evaluation and Review Technique), while having much in common, were independently derived and are based on different concepts. Both techniques define the duration of a project and the relationships among the project’s component activities. CPM uses a single deterministic time estimate to emphasize minimum project costs while downgrading consideration of time restraints. PERT, on the other hand, uses three time estimates to define a probabilistic distribution of activity times which emphasizes minimum project duration while downgrading consideration of cost restraints. It is therefore not surprising that CPM is often the choice of cost-conscious private industry, while PERT tends to be used more frequently in critically-timed government-related projects.

While these two techniques are based on different assumptions, they cannot be altogether independent of one another because of the obvious relationship between time and cost. Perhaps the “ideal” network technique would combine the concepts of CPM’s crashing strategy with PERT’s probability distribution of activity times to derive the optimum project duration and cost. While this goal has yet to be achieved for application to practical projects, much insight can be obtained into the “real world” effects of crashing strategies by using PERT’s probability distribution to allow the activity times to vary from the estimate, as they obviously do in actual projects.

When PERT is used on a project, the three time estimates (optimistic, most likely, and pessimistic) are combined to determine the expected duration and the variance for each activity. The expected times determine the critical path, and the variances for the activities on this path are summed to obtain the duration variance for the project. A probability distribution for the project completion time can also be constructed from this information (a procedure that is all too frequently ignored in practice). However, the variances of activities which do not lie on the critical path are not considered when developing this project variance, and this fact can lead to serious errors in the estimate of project duration.

A similar problem exists when the CPM technique is used to develop a crashing strategy where two or more paths through the network have nearly the same length. If the usual assumption of deterministic activity times is dropped and the activity duration is allowed to vary, a decrease in the length of the critical path may not result in an equivalent decrease in the project duration because of the variances inherent in the parallel or alternate paths. These variations of activity times can even allow the alternate path to become critical in specific instances, as indicated in figure 1. Thus, simply allowing the activity times to vary slightly from their estimates, as they do in every actual project, can cause serious errors in a CPM crashing strategy and lead to wasted resources and cost overuns.

In this paper, probabilistic activity times are derived for a simplified project network, and the project’s duration is determined at successive crash levels using simulation techniques. At each crash level this simulated project duration is compared to the traditional CPM results. The purpose is to analyze the effect of the probabilistic activity times on the estimates of project duration and cost as both the activity variances and the number of parallel paths are allowed to increase. The ultimate goal of this research is to develop a series of “rules-of-thumb” which the practitioner can use to develop a more accurate and cost-effective crashing strategy for his project.

**Previous Development**

As it was originally developed, CPM was principally concerned with establishing activity prerequisits (technological ordering) and determining basic network solutions (7,5). It was useful primarily as a vehicle for keeping track of activities in the project and for identifying and analyzing activity conflicts or sequencing flexibilities which could effect project completion. Later development led to the use of cost slope analysis as a means of calculating the shortest project time within the constraints of overhead costs and subject to the assumption of deterministic time estimates (10). Basically, this technique solves for the effective savings in time when it is possible to “crash” or shorten the individual activity times. The time actually saved (effective time) is divided into the cost increase resulting from compressing the activity to determine the net cost slope (see exhibit #5). These slopes are then compared to the overhead costs to determine if reducing the project duration will result in an overall cost savings. Finally, linear programming techniques were developed to solve the entire problem (11) resulting in a method for scheduling a project’s activities at the most cost effective time. Note, however, that this entire development is based on the basic CPM assumption of deterministic activity time estimates, as assumption which has already been described as leading to potentially serious errors in practical applications.

PERT, on the other hand, was specifically designed to overcome the problems inherent in assuming deterministic activity time estimates (9). Three time estimates for each activity (optimistic, most likely, and pessimistic) are required, but these allow the user to develop a probability distribution for the length of each activity. With this accomplished, the technological ordering and network solutions are calculated in a manner practically identical to that of CPM. Since the initial development of PERT, several studies have analyzed the activity time errors that can arise from using the basic PERT quantitative assumptions (8,12), but research has also shown that use of the PERT estimates leads to results which are much more accurate than those obtained from CPM (13).

In addition, Van Slyke has used a Monte Carlo technique to simulate a PERT network in an effort to decrease the effect of network errors (14), errors which are common to both PERT and CPM. This present paper continues this analysis of network errors, and also incorporates a consideration of activity time variance into the development of a more realistic CPM crashing strategy.

**Experiment:**

BASIC NETWORK. To analyze the effects of probabilistic activity times and parallel paths on the crashing strategy of a network it was necessary to design an investigation technique that would be sensitive to slight variations in input data. This would permit changes, such as an increase in variance or an increase in the number of paths, to directly effect the network solution. A very basic network was constructed consisting of two paths with three activities on each path (exhibit #1). allowing for both ease of calculation and flexibility. The supporting data for this network and the time activity diagram are presented in exhibits #1 and #4 respectively. This data includes estimates of the optimistic time (a), the most likely time (m) and the pessimistic time (b) for each activity at the normal and three crash levels. These estimates allow the variance to be calculated as per the normal PERT procedure. A symmetrical distribution of activity times was used to avoid contaminating the results with any possible effects of skewness. In the experiment the estimates were used to determine an expected time and variance for each activity and to provide an input to the program which simulated the actual activity completion times.

Three crash levels were included to insure enough compressions so that the initial critical path would be shortened to the same length as the alternate path. As can be seen from exhibit #4 Part A, the difference between the two paths using PERT expected times is six days. The three crash levels allow each path to be shortened a total of nine days, one day at a time; therefore, at some level in the crashing sequence the length of the large path will approach the length of the shorter path. Exhibit #1 also includes the costs associated with the normal and crash times for each activity. The cost slopes were computed by dividing the increase in costs by the expected time saved. For example, if activity A is crashed from normal to 1st crash (N-S^{1}) the cost slope would be 5/1. Since the costs of an activity usually increase in actual projects when the time available is shortened, all cost slopes were assumed to increase as the crash level increased (see exhibit #5). Care was taken to exclude any data which would lead to a unique solution not representative of the effect of changes in variances of activities or in network structure.

ANALYSIS OF THE BASIC TWO PATH NETWORK BY TRADITIONAL CPM METHODS. First, the basic PERT approach was used to determine the expected times for each activity. These times were then used as if they were deterministic to solve for the critical path using the usual CPM techniques. ABC was found to be the critical path with a length of 55 days. The parallel path DEF had a length of 49 days. At this point a crashing strategy, following the CPM approach of cost slope evaluation, was developed. The results of this strategy are presented in exhibit #6. On the 7th compression the two paths had the same length and it was necessary to crash both paths simultaneously (parallel crash) to further reduce the project length.

ANALYSIS OF THE BASIC NETWORK BY SIMULATION. The activity times were simulated by using the a, m, and b values to determine a distribution for each activity as per the usual PERT technique. Then a random number system was used to define an assumed actual activity duration for each activity at each trial. The critical path and project duration were determined at each trial by summing the durations of the activities on each path. Each compression of each trial required a separate simulation. The results of these simulations are presented and compared with those derived from the traditional CPM technique in exhibit #6.

The complete experiment was conducted using the following variations in the data and/or the network:

- The simulation was conducted with the basic two path network. All activity distributions were symmetrical with a range of 3 and a variance of .25.
- The variance was increased to .694 and the range to 5. The data for this trial is presented in exhibit #2.
- A third path was added to the network. This path was given an initial length equal to the shorter of the two original paths. The basic data for this network is presented in exhibit #3.

The information generated from these experiments is summarized in exhibit #6. The simulation was ended after 500 iterations. The mean results were calculated and are plotted against the traditional CPM results in exhibits #7 and #8.

**Analysis Of Results**

Exhibits #7 and #8 indicate that when the simulated activity times are used the sub-critical path begins to effect the project duration earlier in the compression sequence than when the deterministic estimates are used. Specifically, in exhibit #7 the third compression produces an effective time saved of only .994 days while the traditional CPM would yield an entire day. This effect was caused by the variance in the activity times. The effective time saved decreases even further as the parallel path lengths converge at successive compressions. This decrease in the effective time saved caused the net cost slopes to increase, indicating that the variance could have a direct effect on the crashing strategy. The data in exhibit #6 shows that in all cases the net cost slope in the 6th compression was greater than the corresponding value in the 7th compression.

In the two path network with the variance increased to .694, the net time saved decreased even further. This caused the subcritical path to influence the expected project duration at an earlier compression and with greater magnitude. This in turn increased the net cost slope which would further modify the crashing strategy.

When the third path was added to the network and the variance held at .25 the additional path increased the parallelism effect. This increase was considerably pronounced in the 5th and 6th compressions, and provides further evidence that the variance will have a direct effect on the crashing strategy.

The experiment has clearly shown that an increase in variance or an increase in the number of alternate paths through the network will cause estimates of project durations to increase. Since the PERT-type estimates have been shown to be more accurate than the CPM deterministic estimates (13) this indicates that the CPM estimating techniques will tend to underestimate project durations. In addition, traditional methods will continue to crash activities along the critical path after the variance of a shorter path is effecting the completion time. This could cause the project manager to underestimate both his individual activity and his total project costs. There are two reasons. The cost of crashing the project is actually greater than expected because, if needed time is actually to be saved, the manager must parallel crash his independent paths earlier than traditional techniques would indicate. Second, if he follows the traditional crashing strategy, the expected time saved cannot be realized and hence the overhead costs are not reduced as expected. In summary, the net cost slope calculations have shown that in all cases parallel crashing should be considered at a earlier time in the crashing sequence than the traditional approach would suggest.

**Summary And Conclusions**

In this paper, simulation techniques were used to determine project duration at successive crash levels while considering activity time variance for each activity, and the project duration at each level was compared to the traditional CPM results. The experiment clearly indicates that traditional CPM methods used by project management to estimate project duration and costs may be overly optimistic. The optimism arises from the traditional failure to consider the effects of variability in path completion times on the crashing strategy.

In situations where accuracy is important this optimism can be reduced by recognizing the following situations and adjusting the estimating technique accordingly:

- The expected activity times derived from a three estimate, PERT-type calculation provides a more accurate estimate and also allows the activity time variance to be calculated and included in the estimates of project duration.
- As the variances of the activities increase, the estimate of project duration calculated by traditional methods becomes less accurate. Hence, the greater the activity variance the greater the optimism demonstrated by traditional CPM methods.
- As the number of parallel paths through the network increases the estimates of project duration calculated by traditional methods become less accurate. Hence, the greater the number of paths through the network, the greater the optimism demonstrated by traditional CPM methods.

The use of activity variances and simulation techniques to derive the project duration leads to more accurate calculations of effective time saved and net cost slopes, which in turn yield a crashing strategy incorporating parallel crashing at an earlier time in the compression sequence. In this way, much of the optimism can be removed from estimates of project completion and project costs.

In some special instances it may also be possible to decrease the optimism by employing a special technique:

- If several activities exist which have large variances relative to the rest of the project, it may be possible to decrease the effect of the variance in any associated path by crashing these activities. This would effectively decrease the influence of that particular path on the completion time.
- In a case where the cost slopes along the critical path are relatively high compared to the project overhead costs, it would not be economical to crash these activities. However, if cost slopes on alternate paths were low relative to overhead costs it might be possible to reduce any further effect of the alternate paths by crashing at the lower cost.
- When activity time distributions are skewed in either a positive or a negative direction, estimates and actual time occurances should be expected to fall in the direction of the skewness. Awareness of this possibility could be a considerable benefit in the development of a crashing strategy.

It should also be noted, however, that this experiment was based on a network which presupposed independence of the alternate paths. If cases occur where there is some degree of dependence between the paths, caused perhaps by common or interconnecting activities, the effect of the variance in individual paths should be expected to decrease slightly from the results shown by this experiment.

As he derives his estimates, each project manager must face the possibility of this optimism effecting his crashing strategy. Although it may not always be possible to anticipate problem areas in a large and complicated project, it may be possible to reduce their impact by being aware of the effect that activity time variance can have on project estimates and crashing strategies.

To this point the research has defined the optimism which exists in project estimates, and the effect it can have on the development of a crashing strategy has been clarified. This continuing research effort is rapidly developing “rules of thumb” which may in the future prove useful in providing more accurate and cost effective crashing strategies for realistic projects.