Complexity

Volume 2018, Article ID 2602376, 10 pages

https://doi.org/10.1155/2018/2602376

## MOORA under Pythagorean Fuzzy Set for Multiple Criteria Decision Making

^{1}Department of Industrial and Manufacturing Engineering, Universidad Autónoma de Ciudad Juárez, Ave. Del Charro 450 Norte, 32315 Ciudad Juárez, CHIH, Mexico^{2}Business School, Sichuan University, Chengdu, Sichuan 610064, China

Correspondence should be addressed to Luis Pérez-Domínguez; xm.jcau@zeugnimod.siul

Received 30 October 2017; Revised 12 February 2018; Accepted 12 March 2018; Published 23 April 2018

Academic Editor: Rosario Domingo

Copyright © 2018 Luis Pérez-Domínguez 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

The multiobjective optimization on the basis of ratio analysis (MOORA) method captures diverse features such as the criteria and alternatives of appraising a multiple criteria decision-making (MCDM) problem. At the same time, the multiple criteria problem includes a set of decision makers with diverse expertise and preferences. In fact, the literature lists numerous approaches to aid in this problematic task of choosing the best alternative. Nevertheless, in the MCDM field, there is a challenge regarding intangible information which is commonly involved in multiple criteria decision-making problem; hence, it is substantial in order to advance beyond the research related to this field. Thus, the objective of this paper is to present a fused method between multiobjective optimization on the basis of ratio analysis and Pythagorean fuzzy sets for the choice of an alternative. Besides, multiobjective optimization on the basis of ratio analysis is utilized to choose the best alternatives. Finally, two decision-making problems are applied to illustrate the feasibility and practicality of the proposed method.

#### 1. Introduction

Recently, the area of multiple criteria decision making (MCDM) had suffered a rapid development. MCDM aims to provide methods of ranking alternatives or select the optimal alternatives among a set of possible alternatives regarding several criteria [1]. Due to the commonness of the MCDM problems in modern life, its theories have been widely applied in various domains like military affair, industrial engineering, macroeconomic domain, and management [2]. Likewise, there are numerous multicriteria methodologies to deliver aid in the problematic task of making this decision [3]. In this sense, the most commonly reported methodologies in the literature are elimination and choice translation reality (ELECTRE, 1968) [4], decision support system (DSS, 1971) [5], data envelopment analysis (DEA, 1978) [6], analytic hierarchy process (AHP, 1980) [7], technique for order of preference by similarity to ideal solution (TOPSIS, 1981) [8], dimensional analysis (DA, 1993) [9], multicriteria optimization and compromise solution (vlsekriterijumska optimizacija i kompromisno resenje, VIKOR, 1998) [10], analytic network process (ANP, 1996) [11], multiobjective optimization on the basis of ratio analysis (MOORA, 2006) [12], and preference selection index (PSI, 2010) [13]. In addition, there are conventional MCDM problems that only consider nonfuzzy (crisp) type for appraising the alternatives with respect to each criterion and preferences of the criteria. In this logic, the conventional MOORA method is proficient for establishing the evaluations and rankings of the alternatives without any complexity. Nonetheless, in real-world, there are MCDM problems, where the opinions (feeling, preferences) of the DMs for appraising the alternatives and criteria weights are commonly expressed by means of linguistic terms embracing ambiguity and hesitation [14, 15]. In this manner, the classical MOORA method presents drawback for manipulating the nonfuzzy (crisp) and fuzzy (qualitative) information involved in a problem of MCDM [16, 17]. Then, there exists the panorama to continue developing investigation in decision making to approaches to deal with incomplete and imprecise information involved in MCDM problems.

Moreover, there are frequently reported hybrid methods with fuzzy sets and equally fuzzy set theory has been generalized in order to manipulate vagueness [18]. Nevertheless, these methods by themselves in addition to the hybrids with fuzzy sets still have some drawbacks and there is an imperative demand to present new MCDM methods [3, 19–21].

Additionally, current investigations assert that multicriteria methods are being combined with intuitionistic fuzzy sets (IFS). Principally, the IFS, introduced by Atanassov [22], become of a generality of the conventional fuzzy sets stated by Zadeh [23]. According to literature over the last decade, the academics have paid great attention to the use of IFS in MCDM [3, 24, 25]. The IFS are proficient at imprecise treatment and inexact data [26–28].

On the other hand, the Pythagorean fuzzy set (PFS) [29–32] has arisen as an operational instrument for handling the vagueness of MCDM problems. The PFS is categorized by means of the affiliation degree and the nonaffiliation degree, whose sum of squares is less than or equal to . In the circumstance, the PFS can explain the difficulties that the IFS cannot; for example, if a DM gives the membership degree and the nonmembership degree as 0.8 and 0.3, respectively, then it is just operative for the PFS. In this sense, all the IFS degrees are a part of the PFS degrees, which specifies that the PFS is more proficient in handling problems of vagueness. Motivated by the advantages of the MOORA method and PF, this paper proposes two algorithms of MCDM by extending the MOORA to PF environments. Additionally, dealing with the last two challenges mentioned in the paragraph above arises. In this sense, the originality and contribution of this paper can be summarized as follows. First, we propose MOORA under PF environments to overcome the limitation of MOORA for dealing with any other type of arguments rather than crisp data and extend its potential applications to more extensive areas. Second, our approach can simultaneously handle quantitative (tangible) and qualitative (intangible) information, commonly presented in an MCDM problem. Hereafter, the intention of this paper is to extend the MOORA method under the PFS environment for the MCDM field.

The remainder of this paper is organized as follows. Section 2 briefly presents the concepts related to PFS. Section 3 presents the explanation of MOORA. Section 4 pronounces the method proposed in this work. In the Section 5 two numerical cases are presented to describe the proposed methodology and the conclusions are presented in Section 6.

#### 2. Pythagorean Fuzzy Set

Pythagorean fuzzy set (PFS) presented by [29, 33]is explained as follows.

*Definition 1 (see [33]). *Let be an arbitrary nonempty set. A PFS is a mathematical object of the form

Thus, a Pythagorean fuzzy set in is given by .

Here and depict the affiliation function and nonaffiliation function of the fuzzy set ; depict the affiliation of in . At the same time, a PFS in is defined as , and . With the condition , , the numbers and depict the degree of affiliation and degree of nonaffiliation of element with respect .

The number is named the Pythagorean index degree of the hesitancy of in and can be stated aswhere for each .

Hence, a PFS in is fully defined with the form . Here ; and . Thus, diverse operations are presented over the PFSs [29]; some of them are revealed in (3), (4), and (5).

*Definition 2 (see [32, 34]). *Assuming ,, and are three PFNs, then,

In fact, to rank the PFNs the next definition is presented.

*Definition 3. *Let describe a PFN; then the total function of is presented asThe large depict the best PFN.

*Definition 4. *Let represent a PFN; at that time the precision function of is introduced as

Obviously, Thus, . The superior rate of describes the higher precision of the PFN .

Thus, per (2) and (8), can be determined. The inferior hesitant degree makes higher accuracy of the PFN .

Hence, with the total function and the precision function of PFNs, the ranking method for any two PFNs can be defined as follows.

*Definition 5. *Let and depict two PFNs. Here, and describe the rate and the precision of and . Then, (i)if , then ;(ii)if , then,(1) if , then ;(2) if , then ;(3) if , then

#### 3. MOORA

The MOORA method was introduced by [12], which analyzes the complete throughput for each alternative as the variance between the sums of cost criteria and benefit criteria. Therefore, the MOORA method is definite through steps.

*Step 1. *Establish the decision-making matrix called In this manner collects rows that denote the alternatives in assessment and columns that characterize criteria in appraisal ( quantitative criteria and qualitative criteria). In this mode, per (9), the decision-making matrix can be obtained as follows:where denote the alternatives, aimed at , and reflect the inputs of the alternative with reference to criterion .

*Step 2. *Proceed with the normalization of matrix. Here the Euclidean norm to the criterion is obtained by using

Therefore, the normalization of each entry in the is calculated by using

*Step 3. *Create the balanced normalized decision-making matrix called . Following [12] the different preferences of criteria, the evaluations are computed by using

*Step 4. *Analyze the global assessments of cost and benefit criteria for each .

In this mode, the global evaluations of benefit criteria are estimated as the sum of weights normalized per where is related to .

Likewise, the global assessments of cost criteria are calculated by mean of where is related to .

*Step 5. *Establish the contribution value. is obtained via (15) originated by [12].where represents the contribution of each alternative , are the maximum criteria, and are the lowest criteria.

#### 4. MOORA under Pythagorean Fuzzy Environment (PF-MOORA)

Let represent a set of alternatives and depict a collection of criteria to be appraised. The PF-MOORA method is described in the following steps.

*Step 1. *Establish a team of DMs and capture the preferences of each one. Here denote a group of decision makers (DMs). The preferences of each DM are evaluated via a linguistic term mapping by PFN. The scale and their corresponding PFN used are shown in Table 1.