A Counterexample to the Forward Recursion in Fuzzy Critical Path Analysis Under Discrete Fuzzy Sets
📝 Abstract
Fuzzy logic is an alternate approach for quantifying uncertainty relating to activity duration. The fuzzy version of the backward recursion has been shown to produce results that incorrectly amplify the level of uncertainty. However, the fuzzy version of the forward recursion has been widely proposed as an approach for determining the fuzzy set of critical path lengths. In this paper, the direct application of the extension principle leads to a proposition that must be satisfied in fuzzy critical path analysis. Using a counterexample it is demonstrated that the fuzzy forward recursion when discrete fuzzy sets are used to represent activity durations produces results that are not consistent with the theory presented. The problem is shown to be the application of the fuzzy maximum. Several methods presented in the literature are described and shown to provide results that are consistent with the extension principle.
💡 Analysis
Fuzzy logic is an alternate approach for quantifying uncertainty relating to activity duration. The fuzzy version of the backward recursion has been shown to produce results that incorrectly amplify the level of uncertainty. However, the fuzzy version of the forward recursion has been widely proposed as an approach for determining the fuzzy set of critical path lengths. In this paper, the direct application of the extension principle leads to a proposition that must be satisfied in fuzzy critical path analysis. Using a counterexample it is demonstrated that the fuzzy forward recursion when discrete fuzzy sets are used to represent activity durations produces results that are not consistent with the theory presented. The problem is shown to be the application of the fuzzy maximum. Several methods presented in the literature are described and shown to provide results that are consistent with the extension principle.
📄 Content
International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
DOI : 10.5121/ijfls.2016.6204 53
A COUNTEREXAMPLE TO THE FORWARD RECURSION IN FUZZY CRITICAL PATH ANALYSIS UNDER DISCRETE FUZZY SETS
Matthew J. Liberatore1
1Department of Management and Operations, Villanova School of Business, Villanova University, Villanova, PA 19085
ABSTRACT
Fuzzy logic is an alternate approach for quantifying uncertainty relating to activity duration. The fuzzy version of the backward recursion has been shown to produce results that incorrectly amplify the level of uncertainty. However, the fuzzy version of the forward recursion has been widely proposed as an approach for determining the fuzzy set of critical path lengths. In this paper, the direct application of the extension principle leads to a proposition that must be satisfied in fuzzy critical path analysis. Using a counterexample it is demonstrated that the fuzzy forward recursion when discrete fuzzy sets are used to represent activity durations produces results that are not consistent with the theory presented. The problem is shown to be the application of the fuzzy maximum. Several methods presented in the literature are described and shown to provide results that are consistent with the extension principle.
KEYWORDS
critical path analysis, discrete fuzzy sets, forward recursion, counterexample, project scheduling
- INTRODUCTION
CPM or the critical path method [11] has been successfully applied to plan and control projects that are organized as a set of inter-related activities. PERT or Program Evaluation and Review Technique [16] and Monte Carlo simulation apply probability analysis to address situations where there is uncertainty related to activity duration. PERT models uncertainty by collecting optimistic, most likely and pessimistic duration estimates of all activities and makes certain assumptions about the underlying probability distributions. Since the basic version of PERT tends to underestimate the expected minimum project duration [15]. Monte Carlo simulation is often preferred in practice when activity durations are uncertain.
However, the information required to estimate probabilities related to activity duration may not always be known. Fuzzy logic is an alternative approach for measuring uncertainty related to activity duration. Fuzzy logic measures imprecision or vagueness in estimation, and may be preferred to probability theory in those situations where past data concerning activity duration is either unavailable or not relevant, the definition of the activity itself is somewhat unclear, or the International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
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notion of the activity’s completion is vague. Many authors including Chanas and colleagues have investigated the situation when activity duration can be described by fuzzy numbers [1], [2], [3], 5], [6], [7].
The dominant approach presented in the fuzzy critical path analysis literature is the fuzzy extension of the forward and backward recursions taken in the project network. This approach computes the earliest and latest start and finish times and slack, where the maximum, minimum, addition, and subtraction operators are replaced by their fuzzy counterparts. The application of the forward recursion with fuzzy activity times was first demonstrated in [3]. In a review of the fuzzy critical path analysis literature two approaches are described for applying the forward recursion [4]. They also indicate that the application of the backward recursion would cause a considerable increase in the range of uncertainty in the start and finish times that are calculated. These authors present a modification of the backward recursion that has been proposed to eliminate this disadvantage [12]. Some authors directly apply the backward recursion, while other authors have proposed different approaches for modifying the backward pass when the activity times are fuzzy [18], [19], [20], [22]. The backward recursion was found not to compute the sets of the possible values of the latest starting times and floats of activities [22]. In a stream of research that uses the joint possibility distribution of activity durations, several authors [8], [9] have conducted preliminary work for computing fuzzy latest starting times and fuzzy floats, especially for series-parallel graphs. Polynomial algorithms for determining the intervals of the latest starting times in the general project network are presented in [22].
Unlike the fuzzy backward recursion, the use of the fuzzy forward recursion is generally accepted in the literature as providing correct results. As mentioned by several authors [3], [10], [18], [19], the forward recursion
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