# A fuzzy approach to critical path analysis

**Matthew J. Liberatore, Ph.D., John F. Connelly Chair in Management, Professor, Decision and Information Technologies,Villanova University,Villanova, PA**

Measuring and coping with project risk and uncertainty continues to be of interest to both academics (Tavares, Ferreira, and Coelho 1998; Fernandez, Armacost, and Pet-Edwards 1998; Van Dorp and Duffey 1999) and project management professionals (Ruskin 2000; Jones 2000; Hulett 2000; Royer 2000). Program Evaluation and Review Technique (PERT) and Monte Carlo simulation have been the traditional approaches use to address project schedule uncertainty. Both of these methods are based on critical path analysis, and apply probability theory to quantify the uncertainty related to the time required to complete project activities. Concerning the use of these methods, a recent survey of project management professionals found that 70 percent used critical path analysis, and 17 percent used probabilistic analysis and/or simulation. In addition, the responding individuals also expressed interest in research concerning project scheduling (40 percent) and uncertainty/risk analysis (10 percent) (Liberatore and Pollack-Johnson 2002). About 20 percent of popular project management software packages have Monte Carlo simulation capability (Project Management Institute 1999).

However, there is an alternative approach for measuring schedule uncertainty based on fuzzy logic. Fuzzy logic is an approach for measuring imprecision or vagueness in estimation, and may be preferred to probability theory in capturing activity duration uncertainty in some situations. Several authors have proposed or investigated the fuzzy logic approach as applied to project schedule uncertainty (for example, Dubois and Prade 1979, 1980; Prade 1980; Chanas and Kamburowski 1981; Chanas 1982; Kamburowski 1983; Chanas 1987; McCahon and Lee 1988; Lootsma 1989; Buckley 1989; McCahon 1993; Rommelfanger 1994; Hapke et al 1994; Chang et al 1995; Shipley et al 1997; Chanas and Zielinksi (2001); see Chanas and Kuchta 1998 for a summary). The dominant approach presented in the literature is the fuzzy extension of the standard critical path method (CPM) algorithm, where forward and backward passes are taken in the network to compute earliest and latest start times and slack. However, the backward pass has been reported to yield incorrect results when the minimum and subtraction operations are replaced by their fuzzy counterparts (Kamburowski 1983, Rommelfanger, 1994, Chanas and Kuchta 1998). The direct use of the fuzzy backward pass was commonly used in some of the early work on fuzzy critical path analysis, and is still being employed by some authors (e.g., Mares 1994; Hapke and Slowinski 1996). Rommelfanger (1994) offers an alternate approach for computing the fuzzy backward pass that has some limitations. A recent article by Chanas and Zielinski (2001) presents a method for analyzing the degree of criticality of the fuzzy paths in the project network, but does not present a method for determining the fuzzy criticality of project activities.

The fuzzy logic approach for project schedule uncertainty analysis is not widely understood, and as is evident from the above discussion, generally accepted computational approaches are not available. The purpose of this paper is to describe the differences between the probabilistic and fuzzy approaches for critical path analysis and to highlight the advantages of the latter, to investigate the incorporation of fuzzy activity durations within critical path analysis, and to present a straightforward computational approach for fuzzy critical path analysis.

**Background on Fuzzy Logic**

Zadeh (1965) introduced fuzzy sets to represent knowledge that is vague or imprecise; that is, “fuzzy.” In classical set theory, an element either is or is not a member of a set. In contrast to the sharp or “crisp” boundaries of classical sets, fuzzy sets allow degrees of membership in a set, as expressed by a number between 0 and 1.

For example, suppose we wish to define membership in the set of warm air temperatures. In classical set theory we might define the set of warm air temperatures as those that are 85 degrees Fahrenheit or warmer. But what if the temperature is 84 ? degrees? This temperature might naturally be described as somewhat warm, as would a temperature of 84 degrees. However, neither temperature would be a member of our set of warm temperatures. This example demonstrates that “warm temperatures” might be better represented as a vague or imprecise concept, without sharp borders separating the warm from the not warm.

**Exhibit 1. Fuzzy Set of Warm Temperatures**

In fuzzy set theory we can accommodate this imprecision by associating a degree of membership in the set of warm temperatures for varying temperature values. We might associate a value of 0.95 to represent the degree to which 84 ^{½} degrees is a member of the fuzzy set of warm temperatures, and a lower value, say 0.90, for the degree to which 84 degrees is a member. For a temperature of 85 degrees we would assign a membership value of 1.00, implying certain membership in the set. For the temperature of 75 degrees we might assign a 0, implying certain exclusion from the set. Because the meaning of the term “warm temperature” can vary by situation, a fuzzy set must defined in a context.

Formally, we could define the membership function m_{T}(84 ^{1}?2) = 0.95 to represent the degree to which a temperature of 84 ^{½} is a member of the set of warm temperatures (T). Using “x” to represent temperature, we could specify that m_{T}(x) = 1 if x>=85, and m_{T}(x) = 0 if x<=75. For those values in-between, we could assess specific values for each temperature, or use a formula such as m_{T}(x) = [(x-75)/10], if 75<x<85. For x = 84 ^{½}, applying the formula yields 0.95. This membership function is shown graphically as Exhibit 1.

Fuzziness does not assume randomness, but is concerned with event ambiguity. It measures the degree to which an event occurs, not whether it occurs. Using fuzzy logic, we might say that an activity takes “about 4 weeks” to complete. The membership of 4 within this set is 1.0, since 4 is certainly a member. Other possible times such as 5 are assigned membership values, also called beliefs, based on our meaning of the vague notion of “about 4 weeks.” The beliefs are not required to sum to 1.0. After the event occurs fuzziness remains, since the actual time is one of the possible times in our fuzzy set and so is still “about 4 weeks.”

**Fuzzy Logic versus Probability**

As mentioned earlier probability theory is often used to model the uncertain duration of a project activity. We might say that the probability that a given activity will take 7 days to complete is 0.8 and the probability that it will take 8 days is 0.2. Since these two probabilities sum to 1.0, no other activity durations are possible in this example. The interpretation of these probability statements is that after repeating this activity many times, in 80 percent of the cases the activity time was 7 days, and in 20 percent of the cases the activity time was 8 days. The *next* time the activity is completed the resulting duration is random—there is an 80 percent chance it will take 7 days, and a 20 percent chance it will require 8 days. This example shows that randomness describes the uncertainty *of event occurrence* in the future: the event “activity time takes 7 days” occurs or not, and the event “activity time takes 8 days” occurs or not. After the activity is completed, randomness dissipates since the activity is known to have taken a specific amount of time; that is, one of these events occurred.

Unlike probability theory, fuzziness does *not *assume randomness. Fuzziness is concerned with *event ambiguity*. It measures the *degree* to which an event occurs, not whether it occurs. Using fuzzy logic, we might say that the given activity takes “about 7 days” to complete. This statement reflects the imprecision or vagueness of our time estimate. Here we define “about 7 days” as a particular type of fuzzy set called a* fuzzy quantity*.The membership of 7 within this set is 1.0, since 7 is certainly a member. Other possible times such as 8 are assigned membership values, also called *beliefs*, based on our meaning of the vague notion of “about seven days.” Note that the beliefs are not required to sum to 1.0. After the event occurs fuzziness remains, since the actual time is one of the possible times in our fuzzy set and so is still “about 7 days.”

From the previous discussion it is evident that, depending upon the situation, both the probability and fuzzy approaches can be applied to model uncertain activity duration. The probabilistic approach seems best suited for those situations where the same or a very similar activity has been completed several times in the past and some historical information on the activity’s duration is available. The fuzzy logic approach may be suitable in those situations where past data is either unavailable or not relevant, the definition of the activity itself is somewhat unclear, or the notion of the activity’s completion is vague. Of course, there are situations that lie between these cases, where perhaps both of these approaches either individually or in tandem can be applied.

For a concise introduction to fuzzy models and the relationship of fuzziness to probability see Bezdek’s editorial (1993) and comments on it by Woodall and Davis (1994) and a rejoinder by Bezdek (1994). For a fuller discussion see Kosko (1990) and Yen and Langari (1999).

**Background on Critical Path Analysis**

We begin with a summary of the essentials of critical path analysis. A project network is defined as a set of nodes representing activities that are connected by directed arcs to represent precedence relationships. A precedence relationship between two activities means that the predecessor activity must be completed before its immediate successor can begin. We assume that the project has a unique start activity (no predecessor activities) and a unique finish activity (no successor activities). If the project network does not have unique start and finish activities, we add dummy activities (which have zero duration) for this purpose. A *path* is a finite sequence of activities that connects the start activity to the finish activity. The length of the longest path is the minimum project completion time and is called the *critical path*, and the activities along it are called *critical path activities*.

The standard computational approach used for critical path analysis requires computing a forward and backward pass in the project network. The forward pass includes computing the earliest start (ES) and the earliest finish (EF) for each activity, while the backward pass requires computing the latest start (LS) and latest finish (LF). Once both passes are completed the activity slack (S) for can be computed. The formulas used in standard CPM are defined as follows. For the forward pass the earliest start of activity *i*,or *ES(i),* is defined as the maximum of the earliest finishes of activity *i*’s immediate predecessor activities *j*.We define *b _{i}* as the set of immediate predecessors of activity

*i*.

The earliest finish of activity *i* is the sum of its earliest finish plus its duration* t _{i}*:

The backward pass requires first computing the latest finish of activity *i, LF(i)*:

In the previous equation, stating that the minimization occurs over all activities *j* such that *i* is an immediate predecessor of* j *is just a way of stating that these *j *are immediate *successors* of activity *i*.The latest start of activity *i* is its latest finish minus its duration:

Finally, critical path activities are those that have zero slack, which is defined as

**Fuzzy Critical Path Analysis**

We begin by assuming that the activity times,* t _{i},* are uncertain and can be represented using fuzzy logic. As an example of a fuzzy activity duration, the vague statement that the duration of activity

*i*is “about nine weeks” might be represented by a fuzzy set having the following membership function: 8/0.3, 9/1.0, 10/0.5. That is, 9 weeks is certainly a member of this set since it has a belief value of 1.0. However, 8 and 10 weeks are also possible activity times, and have been assigned belief values of 0.3 and 0.5, respectively, to reflect the degree to which these times are thought to be possible. A variety of methods can be used to determine membership functions. These include direct estimation, and the use of statistical and other experimental procedures with one or a group of individuals (Li and Yen 1995; Klir and Yuan 1995).

A straightforward approach to determining the membership function for the fuzzy set of critical path times and activity criticality is to use equations (1)–(5) above and replace the mathematical operators addition, subtraction, maximum, and minimum by their fuzzy counterparts. However, application of the forward and backward pass in the fuzzy case lead to different definitions of the fuzzy critical path which give different estimations of the degree of criticality for the same path in the network (Chanas and Zielinski 2001). As a result, there is no generally accepted approach to compute fuzzy activity criticality. In what follows we provide an approach to determine the fuzzy critical path and fuzzy activity criticality for a project network.

Let the set of project activities be *A _{1},A_{2},… ,A_{N}* , and their activity times

*t*be fuzzy quantities with membership functions

_{1},t_{2},… ,t_{N}*m*. Also let

_{i}(t_{I})*m*be the membership function for the fuzzy critical path length

_{CP}(L)*L*.We define

*f(t*to be the function that determines the critical path length

_{1},t_{2}, ...,t_{N}|S)*given*a set of specific activity times

*t*and a specific network structure

_{1},t_{2},…,t_{N}*S*(nodes and precedence relationships). Using the extension principle initially developed by Zadeh (1975), we can generalize the non-fuzzy or “crisp” concept of critical path length to the situation where fuzzy logic is used to represent the activity times as follows:

**Exhibit 2. Network for Fuzzy Critical Path Example**

**Exhibit 3. Immediate Predecessors and Activity Duration Membership Functions by Activity for Fuzzy Critical Path Example**

The above equation is Zadeh’s extension principle where the function *f* has been specified as above to represent a mapping of a project network with specific activity times *(t _{1}, t_{2}, . . ., t_{N})* and given project network with structure

*S*to a critical path length

*L*.

This equation can be applied as follows. For a given project network, we identify all possible combinations of activity times *(t _{1}, t_{2}, . . ., t_{N})* that lead to a specific critical path length. We need consider only those activity times that have a positive belief values from the associated membership functions. The belief associated with each combination of activity times leading to this critical path length is the

*minimum*of the beliefs of all the individual activity times in that combination. Finally, the belief of this critical path length is the

*maximum*of the belief values over all combinations yielding

*L*.This process would then be repeated for all possible critical path lengths. In a similar fashion, we can apply the extension principle to obtain membership functions for all path lengths, the belief that each path is critical, and the belief that each activity lies on the critical path.

Note that the rules presented for processing beliefs, called max-min composition, are different from the way that probabilities are processed. In a probability analysis of a project network, the probability associated with each combination of activity times leading to a specific value for the critical path length is obtained by multiplying the probabilities of all the individual activity times in that combination. The probability of a specific critical path length is the sum of the probabilities over all combinations yielding that length*.* More importantly, as explained earlier, the meaning and interpretation of the results of the probability and fuzzy approaches are quite different.

**A Computational Approach**

In what follows we describe and illustrate a straightforward procedure for fuzzy critical path analysis using the ideas from the previous section. A Visual Basic program has been written to automate the enumeration procedure. This enumeration algorithm consists of the following steps:

1.* Enumerate all possible combinations of activity durations*. Each combination is referred to as a case. The number of cases is found by multiplying the number of possible positive belief values across all project activities.

2.* For each case compute all path lengths and the case belief value, and identify the critical path*.

The belief value for each case is the minimum of the activity beliefs comprising this case. The longest path is the critical path for that case. The path lengths and the critical path for this case all are assigned the same case belief value.

3.* Construct membership functions for the length of all paths including the critical path*.

For a given path, all cases are reviewed and those having a particular path length are identified. The belief for this particular length is the maximum of the beliefs of all cases having this length. This process is repeated for all possible path lengths.

**Exhibit 4. Case Analysis for Fuzzy Critical Path Example**

**Exhibit 5. Membership Functions for Path and Critical Path Lengths, and Activity and Path Criticalities**

4.* Determine the fuzzy criticality for each activity*.

For a given activity, all cases are reviewed and those having this activity on the critical path are identified. The belief that this particular activity is critical is the maximum of the beliefs for all cases having this activity on its critical path.

5.* Compute the centroid or center of area (COA) for all paths including the critical path to provide “crisp” values for their lengths*.

The COA, or center of area, is a special weighted average whose numerator is obtained by multiplying each path length by its belief and summing over all possible path lengths, while its denominator is the sum of the beliefs of all possible path lengths (remember that the beliefs will not sum to 1). The COA is our best estimate of each path length.

**Example**

A simple project is chosen to illustrate the application of the enumeration method, and is explained with the help of Exhibits 2–5. To avoid possible confusion with activity duration, we label the activities using letters without subscripts. Using the precedence information given in Exhibit 3, the project network shown as Exhibit 2 can be drawn. This network is displayed using activity-on-node (AON) notation. Exhibit 3 also provides data on the membership functions for each activity. The data used are discrete, as contrasted with the continuous approach described in our height example and shown in Exhibit 1. Note that the durations for activities B, D, E, and F can be expressed as “about 2,” “about 3,” “about 7,” and “about 2,” respectively. In this example we are using three possible values along with their belief values to define the uncertainty associated with each of these four activities. Three point estimates such as these are relatively easy to make. The information on the membership function for B can be interpreted as having possible values of 1, 2, and 3 with belief values of 0.3, 1.0, and 0.5, respectively. In contrast, there is only one possible time with a belief of 1.0 for activities A, C, and G. That is, it is assumed that these three time estimates are certain. The application of each of the five steps follows.

*Steps*

1. There are 1x3x1x3x3x3x1 = 81 possible cases that must be evaluated.

2. An examination of Exhibit 1 shows that there are three possible paths in the network. These paths are as follows:

P_{1}: A–C–E–G

P_{2}: A–B–D–F–G

P_{3}: A–C–D–F–G

For each of the 81 cases, the lengths of each path are computed, and the critical path(s) are identified. For example, the belief for the case A = 3, B = 3, C = 4, D = 1, E = 7, F = 1, and G = 2 is

*Min*(1.0,0.5,1.0,0.1,1.0,0.5,1.0,)= 0.1.

The computations of the path lengths and beliefs, and the identification of the critical path(s) are given in Exhibit 4.

**Exhibit 6. Membership Function for Critical Path Length**

3. Using the information given in Exhibit 4, we can form the necessary membership functions. For example, there are 18 cases where the critical path is 14. The beliefs over these cases are either 0.1 or 0.3, leading to a belief of 0.3 for a length of 14 since the maximum value is selected. The complete results are given in Exhibit 5 and shown graphically for the critical path in Exhibit 6. In a similar fashion the path criticalities can be determined.

4. Using the information given in Exhibit 4, we can determine the activity criticalities. For example, D is on paths P_{2 }and P_{3}, and an inspection of Exhibit 4 shows that P_{2 }is never critical and that P_{3 }is critical in 23 cases. The maximum of the beliefs of these 23 cases is 0.4, and so this value is the fuzzy criticality of D. This same result can be obtained by noticing that the beliefs that P_{2 }and P_{3} are critical are 0 and 0.4, respectively. Since D lies on both of these paths, we take the maximum (0.0.4) = 0.4 to obtain its criticality. The same result holds for F. The activities along P_{1} (A, C, E, G) are all certainly critical since P_{1 }is certainly critical. On the other hand B is certainly not critical since it is only on path P_{2 }and this path is certainly not critical. The complete results are given in Exhibit 5.

5. The COAs for each path are also given in Exhibit 5.

To further interpret the results, consider the case where all activities assume the length associated with a belief of 1.0 (A = 3, B = 2, C = 4, D = 3, E =7, F = 2, and G = 2). The critical path for this case is P_{1 }with a length of 16 and a belief of 1.0 as shown in Exhibit 4. The COA of 16.17 for the critical path is close to the value for this case because the network is small and the uncertainty is not that great. However, there is uncertainty in completing this project, since P_{3} is moderately critical, and as a result D and F are also moderately critical. There is also some uncertainty associated with the critical path length, which has possible values ranging from 14 through 19 (Exhibit 6).

**Comparison with the Forward and Backward Passes**

The forward pass defined by equations (1) and (2) was applied to solve this problem using the fuzzy operators for addition and maximum. Using the extension principle, we obtain:

*Fuzzy Addition*: Let *t _{1}* and

*t*have membership functions

_{2}*m*and

_{1}(t_{1})*m*,respectively. Then the membership function for

_{2}(t_{2})*t*+

_{1}*t*is defined as:

_{2}For example, if the membership functions for *t _{1}* and

*t*are 6/0.2, 8/1.0, 9/0.3, and 5/0.4, 7/1.0, 8/0.5, respectively, the membership function for their sum is 11/0.2, 13/0.4, 14/0.3, 15/1.0, 16/0.5, 17/0.3. A similar approach is used for the fuzzy maximum.

_{2}*Fuzzy Maximum*: Let *t _{1}* and

*t*have membership functions

_{2}*m*and

_{1}(t_{1})*m*,respectively. Then the membership function for the maximum of

_{2}(t_{2})*t*and

_{1}*t*is defined as:

_{2}For example, if the membership functions for the fuzzy max of* t _{1}* and

*t*as defined above, then the membership function for their maximum is 6/0.2, 7/0.2, 8/1.0, 9/0.3.

_{2}A Visual Basic program was written to automate computation of the forward pass using equations (1) and (2) with addition and maximum replaced by their fuzzy counterparts using equations (6) and (7). Using the example given in Exhibit 3, we obtain the same fuzzy membership function for the critical path length as found using the enumeration procedure as given in Exhibit 5. However, taking the fuzzy backward pass using equations (3) and (4) with fuzzy subtraction and minimum defined similarly to equations (6) and (7) above yields a result that is inconsistent with the forward pass.Hence,the forward pass can be used only if the fuzzy critical path length membership function is desired. If fuzzy activity criticality is needed, then the enumeration procedure should be used.

**Summary and Conclusions**

In this paper we have offered fuzzy logic as an alternative approach for modeling uncertainty in project schedule analysis. The fuzzy logic approach has been compared and contrasted with probability theory. Fuzzy logic may be suitable in those situations where past data is either unavailable or not relevant, the definition of the activity itself is somewhat unclear, or the notion of the activity’s completion is vague. An enumeration procedure for analyzing a project network whose uncertainty is represented by fuzzy logic has been presented and demonstrated. It was also shown that the fuzzy forward pass yields results that are consistent with the enumeration procedure, while the fuzzy backward pass does not. Future work is needed to better identify those circumstances where project managers prefer fuzzy logic to probability theory or where either approach could be used. In addition, improved computational procedures are needed so that fuzzy logic can be applied to projects of arbitrary size. These efforts should lead to increased practitioner use of the fuzzy logic approach for measuring project schedule uncertainty.

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Proceedings of PMI Research Conference 2002

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