## Fuzzy inference

Each inference stage, or inference engine, of the SAFE model has its own set of rules, or rule base, and combines certain input indicators into a composite output indicator.

The inference engines of SAFE uses product-sum algebra to compute the membership grades of the output indicator to the corresponding fuzzy sets. Products and sums correspond to the logical operations of conjunction (“and”) and disjunction (“or”). The operation “and” is involved in the rules and the operation “or” corresponds to an operation that aggregates all rules. Product-sum inference is described below by means of an example.

Each rule is assigned a *firing strength* which measures the degree
to which the rule matches the inputs. Suppose, for example, that ECOS is A
(Average) with membership grade 0.4 and G (Good) with grade 0.6, and HUMS
is A with membership grade 0.9 and G with grade 0.1. Consider four rules of
the rule base for OSUS:

a. `R` 1

if ECOS is AandHUMS is A then OSUS is I (Intermediate).

b. `R` 2

if ECOS is AandHUMS is G then OSUS is FH (Fairly High).

c. `R` 3

if ECOS is GandHUMS is A then OSUS is FH (Fairly High).

d. `R` 4

if ECOS is GandHUMS is G then OSUS is H (High)

The firing strength of a rule is given by the product of the input membership grades, and this value is passed to the membership grade of the output to the corresponding fuzzy set. Thus,

- firing strength of
*R*_{1}= 0.4 × 0.9 = 0.36 = membership grade of OSUS to the fuzzy set I - firing strength of
*R*_{2}= 0.4 × 0.1 = 0.04 = membership grade of OSUS to the fuzzy set FH - firing strength of
*R*_{3}= 0.6 × 0.9 = 0.54 = membership grade of OSUS to the fuzzy set FH - firing strength of
*R*_{4}= 0.6 × 0.1 = 0.06 = membership grade of OSUS to the fuzzy set H.

If several rules assign the same fuzzy set to the output variable (here
we have a disjunction or union of rules), then the overall membership grade
of the output is the sum of the individual firing strengths. In the above
example, both rules *R*_{2} and *R*_{3} assign
the fuzzy FH to OSUS. Thus, the output of the inference engine is:

*µ*_{I}(OSUS) = 0.36, *µ*_{FH}(OSUS)
= 0.04 + 0.54 = 0.58, *µ*_{H}(OSUS) = 0.06.