Fuzzy weighted averages revisited

The fuzzy weighted average operations are common operations in risk and decision analysis. Based on Zadeh's extension principle, Dong and Wang (Fuzzy Sets and Systems 21 (1987) 183) proposed a FWA method: a method based on the alpha-cut representation of fuzzy sets and interval analysis. The FWA method provides a discrete approximation of the alpha-cuts of fuzzy weighted average. The FWA method has an exponential complexity. Subsequently, Liou and Wang (Fuzzy Sets and Systems 49 (1992) 307) proved that indeed the FWA method can be simplified to be a nonconstrainted linear fractional programming problem with lower and upper bounds for each of the variables. They proposed an improved algorithm, called IFWA, which is of O(N-2) Complexity, where N is the number of decision variables in IFWA. Guh et al. (Comput, Math. Appl. 32 (8) (1996) 115) proposed a "max-min paired elimination method" (PFWA). They claimed their PFWA was of order N in complexity. Yet, the PFWA is a heuristic method lacking any proof of convergence. In 1997, Lee and Park proposed an algorithm (EFWA) which is of O(N log N) complexity, Most recently, Kao and Liu (Fuzzy Sets and Systems 120 (2001) 435) solved the FWA by transforming the fractional program into a linear program. In this note, we shall relate the IFWA framework with the nonconstrainted 0-1 integer linear fractional programming problem. In the mathematical programming literature, several O(N)-time algorithms have been existing, for instance, Robillard (Naval Res. Lot-list. Quart. 18 (1971) 47) and Hansen et al. (Math. Programming 52 (1991) 255). The algorithm of Hansen et al. will be presented for easy reference. (C) 2002 Elsevier Science B.V. All rights reserved.