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The first design constraint is that the fuzzy sets must conform to Eq. 1. Equation 1 states that for all points x in the state space, the degrees of membership mi(x) in all the fuzzy sets sum to exactly one. This also implies that there is full coverage of the state space.
.mi (x)= 1 "x (1)
i
The other requirement is that the fuzzy logic connectives, used to perform logical operations on fuzzy sets, must be based on multiplication (for fuzzy AND) and addition (for fuzzy OR). These are in contrast to the more widely used min function (for fuzzy AND) and max function (for fuzzy OR). The former was first proposed by Bellman and Zadeh12 as the “soft connectives” in addition to the “hard connectives” of min and max. The soft connectives have the advantage of being mathematically consistent with Bayesian decision theory. This is important because it is the basis for the next section on designing multi-dimensional fuzzy sets.
C. Hypertrapezoidal Fuzzy Sets
To overcome the shortcomings of the one-dimensional state-of-the-art in fuzzy set theory, we explored the options for multidimensional fuzzy sets. Those options included fine-grained rule-base composition of multidimensional relationships, conditional fuzzy membership functions, and multi-dimensional Gaussian functions. For various reasons all these options are still inadequate for the engineering problem of flight segment identification.13
An important consideration in the development of N-dimensional membership functions is that they be specified with only a few parameters. The standard method for defining one-dimensional trapezoidal membership functions is with four points – a, b, c, and d, as shown in Fig. 12. This method, however, is impractical for defining membership functions on multiple dimensions. The extension of the trapezoidal membership function into a two-dimensional space would require at least eight points, as shown in Fig. 13.

ab cd
x
m(x1, x2)

x1
Another important consideration is that the multidimensional fuzzy sets should enforce the requirement of Eq. 1 and use the alternate fuzzy logic connectives that are isomorphic with Bayesian probabilistic reasoning mentioned in the previous section. Membership functions defined in such a manner are referred to as a fuzzy partitioning. Fuzzy membership functions based on Gaussian probability density functions can easily be extended to N dimensions. However, they do not exhibit the desirable property of Eq. 1. Trapezoidal membership functions, on the other hand, can be defined with the design constraint of Eq. 1.
Based on the requirements outlined above, we developed a new mechanism for specifying and calculating multidimensional fuzzy membership functions.13-15 Termed hypertrapezoidal fuzzy membership functions (HFMF), this new development is a major advancement in the practical application of fuzzy logic to engineering problems.
As an alternative to trying to define all the corners of N-dimensional fuzzy sets, consider the use of a single point in the state space as the defining parameter of an N-dimensional fuzzy set. Each fuzzy set in a fuzzy partitioning would then have an associated N-dimensional vector which is a typical value for that set. We chose to call such an N-dimensional vector the prototype point. The prototype point, li, for a fuzzy set, Si, with a membership function,
mi(x), satisfies the following equations.
mi ()li = 1
(2)
ml= 0 ji
() .
ji
Figure 14a shows a simple example of a fuzzy partitioning in two dimensions using three prototype points to define three fuzzy sets leaving some area of overlap between the sets.

(a) (b) (c)
A measured value, x, which is an N-dimensional point in the state space of a fuzzy partitioning, has a degree of membership in a fuzzy set based on its Euclidean distance from the prototype point for that set. For example, if x= l1, then m1(x) = 1, m2(x) = 0, and m3(x) = 0. As another example, if xis equidistant from all three prototype points,
　
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