Background

Probably dating back to the first use of the critical path method (CPM) for project scheduling, project managers have realized the shortcomings of CPM for dealing with uncertainties that are inherent in any plan. As pointed out by Hulett (1995, 2000) the project duration calculated by CPM is accurate only if everything goes according to plan. This is rare in real projects. Furthermore, given uncertain activity duration times, the completion date provided by CPM is often not even the most likely project completion date, and the path using traditional CPM techniques may not be the one that will be most likely to delay the project and need management attention. The reasons for this phenomenon are well known and include the facts that CPM uses only point estimates of duration times (thereby ignoring duration variances) and only the longest path in the project network (using these point estimates) is used to compute the project length.

To circumvent the above difficulties with CPM, several methods have been investigated to obtain more realistic estimates of project completion time. Bendell et al. (1995) give a nice review of some of these methods. Probably the earliest attempt is what has become known as the program evaluation and review technique (PERT) method, whereby three duration estimates (optimistic, most likely, and pessimistic) are used for each activity. The critical path is then found using the most-likely duration values. The probability distribution of project completion time is taken to be the distribution provided by the distribution of the sum of random variables describing the durations of activities on the critical path. Difficulties with this approach include the assumption that activity durations are independent variables, and that durations of non-critical activities (even if their durations have a large variance) are ignored. In spite of these difficulties, the PERT method seems to be widely adopted and is a feature included in most project management software systems.

Another approach, often referred to as the analytical approach, involves the computation of the cumulative distribution function (CDF) of project completion time as a multiple integral distribution (Ringer 1969). Due to the complexity of the computations, this approach is feasible only if the network is small and the probability density functions of the activities are in analytical form. When exact methods are impractical, Ringer has proposed computer-based numerical integration methods to approximate the completion time CDF.

Another approximation method, called the moments method, is proposed by Sculli (1983) and it depends on being able to compute the first four central moments of the distribution of the sum and maximum of activity times. In this method, the project network is progressively reduced to a single arc, by collapsing serial and parallel arcs. Davis and Stephens (1983) have developed computer software to support this approach. Bendell et al. (1995) developed the moments method in the special case where activity times are Erlang distributed. Gong and Hugsted (1993) proposed a method they call backward-forward uncertainty estimation as a means to include non-critical activity time uncertainties in the risk analysis of a project network.

Other methods have also been proposed in the literature for managing risk. These include managerial approaches such as the Planned Contingency Allowance (PCA) technique proposed by Eichhorn (1997), using "unders" to offset "overs" as proposed by Ruskin (2000), or the application of the Theory of Constraints (TOC) to project management proposed by Goldratt (1997); as well as more analytical approaches such as those proposed by Gong and Rowlings (1997) and Gong (1997).

Computer simulation is another approach that is becoming even more popular as desktop computing power increases and special simulation software becomes available at an affordable price. Early simulation approaches include those of Van Slyke (1963), Gray and Reiman (1969), and Burt and Garman (1971a, 1971b). In this approach, each iteration of the activity time distribution of each activity is sampled, and the resulting values are used to compute the longest path in the network. This exercise is repeated a large number of times and a distribution of project completion times is then developed. In addition, other useful information such as the fraction of times a given activity appears in the critical path is gathered. Clearly, for the technological reasons mentioned above, this approach to estimating the distribution of project completion time will continue to become more popular (for example, see Levine 1996; Gump 1997). For a further discussion of simulation in project risk management see Grey (1995).

Simister (1994), whose paper provides a nice overview of various project risk analysis and management techniques, reported on the results of a mail survey of various methods that expert practitioners use to manage project risk. Although the number of respondents to his survey was quite modest, he concluded that computer applications, e.g., packages such as @RISK (@Risk, Inc.), are used by the majority of practitioners. He also concluded that one of the simplest of all possible techniques (checklists) was the most favored of all techniques suggested in his survey instrument. We remark that the results of that survey may be quite different now (in the year 2000) than in the year of his survey (1994).

Indeed, recent advances in such software have made them even more relevant and user-friendly. As an example, the @RISK add-on for Microsoft Project 98 now allows "conditional branching" in the simulation analysis of project schedules. By conditional branching we mean that specific branches of the decision tree are "sampled" only if certain conditions are met. However, the user must specify the branching rules prior to the simulation run.

Specifying branching rules is a not easy. Yet it is a fundamental problem that must be faced. In this chapter we investigate the use of decision theory (Jones 2000; Johnson and Schou 1990) to make such decisions. Specifically, the problem with which we are concerned involves contracting opportunities to reduce task times on various project activities. These contracts involve financial commitments and in some cases lead-times to accept contract terms. Also, task times are uncertain but have known probability distributions. Thus, we are concerned with the decision process of crashing activities with the overall objective of minimizing expected project cost. A part of the planning process is to determine if, and when, to elect the crashing option for various tasks. Thus, the setting is ripe for the use of decision theory.

In the next three sections we consider a relatively simple (core) project with the characteristics mentioned previously. We show that when crash-decision lead-times are zero, the problem is quite easily solved. However, when lead-times are positive, the problem becomes much more challenging. We use the core project to demonstrate the relationship of (optimal) expected project cost to various problem parameters. Our analysis allows us to gain insight into such problems, but clearly only scratches the surface of possible research efforts. In the section "Research Issues and Practical Considerations in Utilizing Decision Analysis", we discuss research issues as well as practical issues in utilizing this approach. In the last section of the chapter, we give some concluding observations and discuss other risk containment strategies that are possible to use in this problem environment.