Wei [57] proposed a new aggregation operator called fuzzy induced ordered weighted

harmonic mean (FIOWHM) for fuzzy multi criteria group decision making.

For [alpha] = [q.sub.p], the p-means [A.sub.p], [G.sub.p] and [H.sub.p] are reduced to ordinary arithmetic mean, geometric mean, and

harmonic mean, respectively.

In these situations, the

harmonic mean price may be expressed as follows:

The power exponential mean and Contra

Harmonic mean are deduced directly from weighted Arithmetic and Geometric Mean, as follows.

The total

harmonic mean thickness of the air-hemolymph barrier ([[tau].sub.h] ([micro]m)) and those of the cuticle and hypodermis layers were estimated by a stereological method in a square lattice grid as suggested by Weibel (1970/71) and Maina (2002):

The case p = 1 yields the arithmetic mean and the case p = -1 yields the

harmonic mean. As p approaches 0, the limit of [M.sub.p] is the geometric mean of the given numbers.

Equation 3 shows that the SCOP should be estimated by the weighted

harmonic mean of COPs at each rating point.

The

harmonic mean (mZ in our notations) of the quantities [2.sup.Max] is then likely to be of the order of n/m.

This relies on the

harmonic mean of the ratio of realized to expected survivorship rates across cohorts.

(9) Since the

harmonic mean of positive numbers is less than or equal to their arithmetic mean, V([X.sub.W]) [less than or equal to] V([X.sub.A]).

For example, the cases [alpha] = 0 and [alpha] > 80 correspond respectively to arithmetic and

harmonic mean models, which are the upper and lower thermal conductivity limits of the layered media.