A Comparison of Discrete Algorithms for Fuzzy Weighted Average

Fuzzy weighted average (FWA), which can be applied to various fields such as engineering design, decision analysis, etc., and as function of fuzzy numbers, is suitable for the problem of multiple occurrences of fuzzy parameters. Additional fuzziness may be introduced in the
$alpha$
-cut arithmetic. This paper reviews and compares discrete algorithms for the FWAs in both theoretical comparison and numerical comparisons, as opposed to the linear programming algorithms that may be efficient but require the help of linear programming software. An alternative efficient algorithm is also proposed. The algorithm introduces an all-candidate (criteria ratings) weights-replaced benchmark adjusting procedure other than a binary (dichotomy) search in the existing methods. The theoretical worst-case comparison shows that the algorithms of Guu and Guh and our alternative algorithm require the same order of calculation complexity
$O(n)$
, where
$n$
is the number of FWA terms. Yet, the Guh algorithm always requires
$2(n-1)$
calculations to be performed. The algorithm of Guu requires the elemental comparison complexity
$O(n)$
, and is superior to the other algorithms, in the worst case. On the other hand, our alternative algorithm and Lee and Park algorithm have the same comparison complexity
$O(nlog n)$
. Yet, our alternative algorithm requires
$O(n)$
calculation complexity which is better than
$O(nlog n)$
of the Lee and Park algorithm. The numerical experiments show that the algorithm of Guu and our alternative algorithm generally perform and converge faster by requiring fewer calculations than that of Lee and Park's algorithm. The alternative algorithm requires a slightly smaller number of calculations than that of the Guu algorithm, due to the use of the convergent benchmark adjustment procedure rather than a somewhat fixed binary search. Conversely, in the number of element comparisons, Lee and Park's algorithm is shown numerically generally slightly better than the alternative algorithm due to the simple binary search scheme used. The alternative algorithm also requires fewer comparisons than that of the Guu algorithm. Furthermore, in comparison of the average CPU time requirements, the ranks of the results of these algorithms are consistent with those of the algorithms for the average numbers of calculations. In general, the alternative algorithm may provide an efficient alternative algorithm to the FWAs. In the worse-case situation, the Guu algorithm still may be considered as an efficient alternative.