Complexity

Volume 2018, Article ID 3927951, 11 pages

https://doi.org/10.1155/2018/3927951

## A Hybrid Approach for Modular Neural Network Design Using Intercriteria Analysis and Intuitionistic Fuzzy Logic

^{1}Intelligent Systems Laboratory, “Prof. Dr. Asen Zlatarov” University, Burgas, Bulgaria^{2}Bioinformatics and Mathematical Modelling Department, IBPhBME-Bulgarian Academy of Sciences, Sofia, Bulgaria^{3}Tijuana Institute of Technology, Tijuana, BC, Mexico

Correspondence should be addressed to Oscar Castillo; xm.anaujitcet@ollitsaco

Received 20 September 2017; Revised 30 December 2017; Accepted 8 March 2018; Published 19 April 2018

Academic Editor: Enzo Pasquale Scilingo

Copyright © 2018 Sotir Sotirov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Intercriteria analysis (ICA) is a new method, which is based on the concepts of index matrices and intuitionistic fuzzy sets, aiming at detection of possible correlations between pairs of criteria, expressed as coefficients of the positive and negative consonance between each pair of criteria. Here, the proposed method is applied to study the behavior of one type of neural networks, the modular neural networks (MNN), that combine several simple neural models for simplifying a solution to a complex problem. They are a tool that can be used for object recognition and identification. Usually the inputs of the MNN can be fed with independent data. However, there are certain limits when we may use MNN, and the number of the neurons is one of the major parameters during the implementation of the MNN. On the other hand, a high number of neurons can slow down the learning process, which is not desired. In this paper, we propose a method for removing part of the inputs and, hence, the neurons, which in addition leads to a decrease of the error between the desired goal value and the real value obtained on the output of the MNN. In the research work reported here the authors have applied the ICA method to the data from real datasets with measurements of crude oil probes, glass, and iris plant. The method can also be used to assess the independence of data with good results.

#### 1. Introduction

One of the open and important questions in biology is the ability of biological systems to adapt to new environments, a concept termed evolvability [1]. A typical feature of evolvability is the fact that many biological systems have modularity; especially many biological processes and structures can be modeled as networks, such as metabolic pathways, gene regulation, protein interactions, and brains [1–5]. This feature has motivated important concepts in intelligent systems, such as modular neural network and evolutionary computation.

Neural networks are considered modular if they are comprised of highly connected clusters of nodes that are connected to nodes in other clusters [4, 6, 7]. Despite importance and continuous research in this area, there is no agreement on why modular biological systems can evolve [4, 8, 9]. There is evidence that modular systems look more adaptable in nature [10] than the monolithic networks [11, 12]. Consequently, there are many papers dedicated to this problem, for example, the work in [12].

In this paper, we introduce a hybrid combination between the intercriteria analysis (ICA, see [13–18]) method and modular neural network models. The ICA employs the apparatus of the intuitionistic fuzzy sets (IFS) for detecting possible correlations between pairs of criteria. Introduced in [19], IFSs are one of the extensions of Zadeh’s fuzzy sets [20]. In contrast to fuzzy sets, IFS [21–24] have two degrees: of membership (validity, etc., ) and of nonmembership (nonvalidity, etc., ), so that for each element of the universe, over which an IFS is defined, the following inequality is valid: . In this case, a pair , where , is called an intuitionistic fuzzy pair (IF pair). The ICA method produces the so-called positive and negative consonance coefficients between the different criteria used for evaluation of different objects.

The main contribution of the paper is the proposed hybrid approach combining intuitionistic fuzzy logic (through the ICA method) with modular neural networks for designing a powerful neural model for classification. The neural network model is tested with benchmark problems and a real world case to show the advantages of the proposed approach. As regards existing works that could be considered similar to this one, we can mention that intuitionistic fuzzy logic has not been considered in conjunction with modular neural networks previously, so it can be considered an original contribution to the area of computational intelligence that combines the advantages of the two methods. For the purpose of testing the proposed method for preprocessing the information going into MNNs, we use data from the LUKOIL Neftochim Burgas AD from the measurements of a set of crude oil probes (objects, in terms of ICA) against a set of technological properties (criteria, in terms of ICA), which precedes and conditions the process of production of petrochemical products from the crude oil [25], dataset for iris plant [26], and glass types [27].

The remainder of the paper is organized as follows. In Section 2 some short remarks about the intercriteria analysis method are given, which is based on intuitionistic fuzzy logic. Section 3 describes basic concepts about modular neural networks. Section 4 describes the simulations and a discussion of the results. Finally, Section 5 offers the conclusions and outlines future work in this area.

#### 2. Short Remarks on the Index Matrices and Intercriteria Analysis Method

As we mentioned above, the ICA method [13, 14] is based on two main concepts: intuitionistic fuzzy sets and index matrices. A brief description is offered below for completeness. Index matrices allow summarizing the criteria relevant to a particular decision making problem.

Let be a fixed set of indices and let be the set of the real numbers. An index matrix (IM) with sets of indices and is defined by (see [13])where , , for , and .

For any two IMs, a series of relations, operations, and operators have been defined. The theory behind the IMs is described in a more detailed fashion in [13].

Here, following the description of the ICA approach, given by [14], we will start with the IM called with index sets with rows and columns , where for every , , is an evaluated object, is an evaluation criterion, and is the evaluation of the th object against the th criterion, defined as a real number that is comparable according to relation with all the remaining elements of the IM .

From the requirement for comparability above, it follows that for each , , the relation (, ) holds. The relation has a dual relation , which is true in the cases when the relation is false, and vice versa. For instance, if is “greater,” the dual relation is “less.”

For the requirements of the proposed method, pairwise comparisons between every two different criteria are made along all evaluated objects. During the comparison, a counter is maintained for the number of times when the relation holds, as well as another counter for the dual relation.

Let be the number of cases in which the relations () and () are simultaneously satisfied. Let also be the number of cases in which the relations (, ) and the dual ) are simultaneously satisfied. As the total number of pairwise comparisons between the objects is given by , it can be verified that the following inequalities hold:

For every , , such that and for two numbers are defined:

The pair constructed from these two numbers plays the role of the intuitionistic fuzzy evaluation of the relations that can be established between any two criteria and . In this way, the IM that relates evaluated objects with evaluating criteria can be transformed to another IM that gives the relations detected among the criteria, where stronger correlation exists where the first component is higher while the second component is lower.

From practical considerations, it has been more flexible to work with two IMs and , rather than with the IM of IF pairs. IM contains as elements the first components of the IFPs of , while contains the second components of the IFPs of . Once the intercriteria pairs have been calculated, for example, using the software described in [28], the question arises about defining the thresholds against which the membership and the nonmembership parts are evaluated [29, 30].

As has been discussed in some publications on ICA, for example, in [31, 32], the ICA results are very close to those obtained with the correlation analyses of Spearman, Pearson, and Kendall. It is worth noting the so far empirically observed fact that when in the data there are mistakes (e.g., shift of the decimal separator) these three correlation analyses give a larger deviation of the value than ICA; that is, ICA is less sensitive, so the use of them together can be used as a way of detecting errors in the input data.

#### 3. Modular Neural Networks

Modular neural networks [33, 34] are one of the models that can be used for object recognition, classification, and identification (see Figure 1). A modular neural network can be viewed as a set of monolithic neural networks [35–37] that deal with a part of a problem, and then their individual outputs are combined by an integration unit to form a global solution to the complete problem. The main idea is that a complex problem can be divided into simpler subproblems that can be solved by simpler neural networks and then the total solution will be a combination of the outputs of the simple monolithic neural networks.