A Counterexample to the Forward Recursion in Fuzzy Critical Path Analysis Under Discrete Fuzzy Sets

TL;DR

This paper demonstrates that the commonly used fuzzy forward recursion method in fuzzy critical path analysis with discrete fuzzy sets can produce inconsistent results, highlighting the need for alternative approaches.

Contribution

The paper provides a counterexample showing the failure of fuzzy forward recursion with discrete fuzzy sets and discusses methods that align with the extension principle.

Findings

Fuzzy forward recursion can produce inconsistent results with discrete fuzzy sets.

The application of fuzzy maximum causes the inconsistency.

Alternative methods are identified that satisfy the extension principle.

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.