IEEE Systems, Man, and Cybernetics Magazine - January 2018 - 25

responses and to optimize resources by reducing the
restrictions of time and location.
Fuzzy logic attempts to mimic human thinking; it plays
an important role in modeling and representing imprecise
linguistic human concepts, such as "close" and "far," performing well despite the uncertainty, noise, and imprecision in real-world settings [46]. Fuzzy logic extends
classical set theory to calculate intermediate values
between true and false, providing a smooth transition
when encountering uncertainties. In addition, it uses
linguistic IF-THEN rules to model behavior in a humanreadable form. Fuzzy logic approaches for learning and
tracking users' behavior have been employed in different
scenarios [47], [48].
Type-2 (T2) fuzzy sets were conceptually introduced
by Zadeh [49]. While type-1 (T1) fuzzy sets do well managing short-term uncertainties (e.g., imprecision associated
with sensors and actuators, or slight user-behavior changes), T2 fuzzy sets are designed to model and handle higher
levels of uncertainty, such as those present in real-world
environments. This is because T1 fuzzy sets use precise
and crisp membership functions, whereas T2 fuzzy sets
use fuzzy membership functions (i.e., the membership
value for each element is a fuzzy set) instead of a crisp

number-as in [0,1]. Thus, T2 fuzzy sets provide two
degrees of freedom via the footprint of uncertainty and
the 3-D nature of these sets [50].
Figure 3(c) shows the structure of a T2 fuzzy logic system [52], in which crisp inputs are first fuzzified, converting them to input T2 fuzzy sets. Then, the inference engine
identifies the rules fired from a previously defined rule
base, combining them to produce output T2 fuzzy sets.
Then, the T2 fuzzy output sets are reduced, mapping them
to T1 fuzzy sets (also known as type-reduced sets) by combining them and performing a centroid calculation. Finally, the type-reduced sets are defuzzified (by taking the
average of the type-reduced set) to obtain crisp outputs
[52]. In our study case, we used an interval T2 fuzzy logic
system to simplify the computation (as opposed to a general T2 fuzzy logic system) by setting all of the third-dimension values equal to one [51]-[53]. This makes it more
suitable for real-time applications. An interval T2 fuzzy set
is illustrated in Figure 3(d). Next, we present two specific
implementations using immersive technologies and fuzzy
logic toward the creation of intelligent immersive systems
for service operations.
In [44], we introduced a novel system using immersive
technology and fuzzy logic for equipment monitoring and

Type 1 Fuzzy Logic

Fuzzifier

T1 Input
Fuzzy Sets

Crisp Output

Rules

Inference

Defuzzifier

T1 Output
Fuzzy Sets

Low
Degree of Membership

Crisp Input

y = f (x )
(a)

Medium

High

1

0

a

b

x

c
(b)

Type 2 Fuzzy Logic

Fuzzifier

Crisp Output

Rules

Defuzzifier
T1 Output
Reduced
Fuzzy Sets
Type Reduction

T2 Input
Fuzzy Sets

Inference

T2 Output
Fuzzy Sets

y = f (x )
(c)

Low

Medium
Upper
Upper
Membership Membership

1
Degree of Membership

Crisp Input

High

Lower
Membership
Footprint
of Uncertainty

0

a

b

c

x

(d)

Figure 3. (a) The structure of a T1 fuzzy logic system [51], (b) a T1 fuzzy set, (c) the structure of a T2 fuzzy logic

system [52], and (d) an interval T2 fuzzy set.

Ja nua r y 2 01 8

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25



Table of Contents for the Digital Edition of IEEE Systems, Man, and Cybernetics Magazine - January 2018

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