Computational Intelligence - August 2013 - 31

where M is the total number of rules.
The objectives considered in
Start
the design of the FC include the
improvement of the controller
Population Intialization
RACACCO-Based Solution Generation
performance and the minimization of the number of fuzzy rules.
Objective Function Evaluation
N New Solutions
The former may entail several
objectives such as the optimization
Nondominated Sorting
Phase Three: Gaussian Sampling
of the wall-robot distance and the
robot speed in the wall-following
Crowding Distance Sorting
Phase Two: Rule-Based Mutation
control task. This paper applies the
MO-RACACO algorithm to
solve the multi-objective FC optiReserve the Top N Best
Phase One: Ant Path Selection
Performing Solutions
mization problem. All of the free
parameters in the antecedent and
consequent parts of the FC are
Iteration IC
IC + 1
optimized through the MOYes
RACACO algorithm. It is supIC < Imax
posed that the maximum number
No
of rules of interest is denoted as
+
End
M. The MO-RACACO optimizes the FCs with the number of
rules allowed to vary between 3
Figure 2 Flow chart of the MO-RACACO algorithm.
+
and M. For an FC consisting of M
rules, the number of fuzzy sets A ij in each input variable x j is
also equal to M. That is, neither the number of rules nor the
Crowding
Non-Dominated Distance
fuzzy set partition number has to be assigned in advance. The
Sorting
Iteration Ic Sorting
MO-RACACO finds the Pareto-optimal solutions with differF1
ent numbers of rules.

The MO-RACACO algorithm uses the following methods:
tag-oriented rule coding of the solutions and rule-based mutation operations to find Pareto-optimal solutions with different
numbers of rules, the non-dominated sorting technique in the
NSGA II to find the non-dominated solutions, and the rulecoded modified CACO approach to generate new solutions in
a population to improve the optimization performance. The
following provides a detailed introduction of the MORACACO-based FC optimization algorithm.

MO-RACACO

Discarded

Next Iteration
Figure 3 Learning configuration in the MO-RACACO algorithm.

A. Rule-Coded Individuals (Solutions)

Each individual (solution) S j in the MO-RACACO represents an FC. As shown in Fig. 1, each individual in the MO+
RACACO consists of M possible rules, with each one being
assigned with a tag. A rule in an individual is either active or
inactive when the tag value is "1" or "0", respectively. An individual consisting of M active rules corresponds to an FC consisting of M rules. The motivation for using the tag is to enable
the MO-RACACO to optimize FCs with a different number
of rules in the Pareto-optimal front. Each solution vector S j is
described as
1
2
M
S j = 8 s j , s j , f, s j B ! R (2n +c) M ,
+

Iteration
Ic + 1

III. Multi-Objective, Rule-Coded, Advanced,
Continuous-Ant-Colony-Optimized Fuzzy Controller

Performance
Ranking

(5)

where vector s j represents all of the free parameters in the
i
rule R i. If the rule R i is active, then s j is described as
s j = 6s ij1, s ij2, f, s iD
j @
i1
i1
= [m j , v j , f, m inj , v inj , a ij1, f, a icj ] ! R D, D = 2n + c. (6)
i

On the contrary, if rule R i is inactive, then s j ! R D is a null
vector. This solution representation method is different from
previous CACO-designed FSs in which the numbers of free
parameters in all the individuals are identical and no optimization of rules is considered [12], [13], [27]. For the multi-objective FC design problem considered in this paper, the number of

August 2013 | IEEE ComputAtIonAl IntEllIgEnCE mAgAzInE

Table of Contents for the Digital Edition of Computational Intelligence - August 2013