A general approach to fuzzy TOPSIS based on the concept of fuzzy multicriteria acceptability analysis

Abstract

Within Multi-Criteria Decision Analysis (MCDA), the TOPSIS method and its fuzzy extensions, fuzzy TOPSIS (FTOPSIS) models, are widespread ones for solving multi-criteria decision problems. At the same time, FTOPSIS models, as a rule, are implemented based on approximate computations with the use of triangular and trapezoidal fuzzy numbers. This paper introduces a novel approach to fuzzy extension of TOPSIS with the use of fuzzy criteria values and fuzzy weight coefficients of the general type and implementing functions of fuzzy numbers based on standard fuzzy arithmetic and transformation methods. Within FTOPSIS, for ranking of fuzzy numbers/alternatives the concept of Fuzzy Multi-criteria Acceptability Analysis (FMAA) is implemented. The use of FMAA within Fuzzy MCDA (FMCDA) represents a systematical implementation of the concept of fuzzy decision analysis that “the decision taken in the fuzzy environment must be inherently fuzzy”. FTOPSIS-FMAA model not only allows ranking the set of alternatives, but also provides the confidence measure for the rank obtained by this model. This approach also considers the overestimation problem, which arises within FMCDA and FTOPSIS-FMAA implementation. A case study on a multi-criteria housing development decision problem is introduced and explored by several FTOPSIS-FMAA models. Finally, a comparison of different FTOPSIS-FMAA models is implemented with the use of Monte Carlo simulation.

Keywords

Fuzzy number fuzzy preference relation ranking of fuzzy numbers MCDA TOPSIS fuzzy TOPSIS FMAA overestimation

1 Introduction

The use of Multi-Criteria Decision Analysis (MCDA) comes for the necessity of avoiding oversimplification of decision problems by a single criterion that leads mostly to unrealistic decisions. MCDA is a relevant branch of operations research field devoted to implement and support the decision process in complex real world decision problems that involve multiple dimensions, goals, criteria and objective of conflicting nature [1].

MCDA includes two different decision making analyses: Multi-Attribute Decision Making (MADM or discrete MCDA, when a finite number of explicitly given alternatives is analyzed with respect to several criteria) and Multi-Objective Decision Making (MODM or continuous MCDA, when infinite or a big number of implicitly given alternatives are analyzed with the use of several criteria) [2 –6]. Among the well-known MADM methods, there are SAW (Simple Additive Weighting), WP (Weighted Product Method), AHP (Analytic Hierarchy Process), ANP (Analytic Network Process), TOPSIS (Technique for Order Preference by Similarity to Ideal Solution), MAVT (Multi-Attribute Value Theory) and MAUT (Multi-Attribute Utility Theory), ELECTRE (ELimination and Choice Expressing the REality), PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluation), VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje or Multicriteria Optimization and Compromise Solution), MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique), and DEMATEL (Decision Making Trial and Evaluation Laboratory). There are also recently developed MADM techniques in the literature: EDAS (Evaluation Based on Distance from Average Solution), MULTIMOORA (Multiobjective Optimization by Ratio Analysis plus Full Multiplicative Form), WASPAS (Weighted Aggregated Sum Product Assessment), COPRAS (Complex Proportional Assessment), SIMUS (Sequential Interactive Model for Urban Systems), and others [5 –8].

TOPSIS approach was introduced in [9] by Hwang and Yoon.

Behzadian et al. [10] reviewed 266 major research papers published in 103 journals, which use TOPSIS as the solution technique for multi-criteria analysis of various topical problems. They categorized these papers into nine classes according to the problems they deal with: (1) Supply Chain Management and Logistics, (2) Design, Engineering and Manufacturing Systems, (3) Business and Marketing Management, (4) Health, Safety and Environment Management, (5) Human Resources Management, (6) Energy Management, (7) Chemical Engineering, (8) Water Resources Management, and (9) Other topics.

In [11] Chen and Hwang extended the TOPSIS method to the Fuzzy TOPSIS which has been widely used. A search for fuzzy TOPSIS in largest abstract and citation database of peer-reviewed literature, SCOPUS (all fields), in May 2018 gives about 14,002 published papers. The pioneer research in this area started at 1983, but the usage of fuzzy TOPSIS approaches dramatically increased since 2006. Some of these works have also extended the ordinary fuzzy TOPSIS to other fuzzy extensions of TOPSIS such as hesitant fuzzy TOPSIS, intuitionistic fuzzy TOPSIS, Pythagorean fuzzy TOPSIS, Type-2 fuzzy TOPSIS, and Neutrosophic TOPSIS [12 –29].

In this contribution, a general approach to Fuzzy TOPSIS (FTOPSIS) method is presented. This approach consists in the possibility to use weight coefficients and criteria values as fuzzy numbers of a general type, and no defuzzification method is used at all in the stages of FTOPSIS implementation. Novelty of this method is also based on the FMAA (Fuzzy Multicriteria Acceptability Analysis) concept [30, 31]. FMAA is in complete agreement with the conception in decision analysis that “the decision taken in the fuzzy environment must be inherently fuzzy” [32]. FMAA provides ranking of alternatives along with the degree of confidence (fuzzy measure) for each alternative to have the appropriate rank.

FMAA is based on Fuzzy Rank Acceptability Analysis (FRAA) method [31, 33] that implements ranking of FNs and provides a degree of confidence to the ranking obtained. FMAA, as a methodological approach and an analytical method, extends the acceptability ideas of SMAA (Stochastic MAA, which is based on Monte Carlo simulation) [34] and ProMAA (Probabilistic MAA) [35] to a fuzzy context.

This paper is structured as follows. The survey of TOPSIS and FTOPSIS methods is presented in Section 2. Section 3 briefly revises concepts of FNs, fuzzy preference relations, ranking of FNs, and FRAA based approach to ranking of FNs. Section 4 introduces FTOPSIS-FMAA method. Implementation of FTOPSIS-FMAA within a case study on housing development is presented in Section 5 along with comparison of output results with those by several FTOPSIS-FMAA models and TOPSIS methods. In Section 6, the proposals provided across the paper are discussed, and Section 7 concludes this paper.

2 Survey of TOPSIS and FTOPSIS methods

TOPSIS method needs a performance table, {C_ij}, where C_ij is the criterion value of alternative i with respect to criterion j, and is based on the concept of similarity (or closeness) of an alternative to the ideal and anti-ideal alternatives. The similarity is based on distance measurements in many-dimensional space. To assess distances in multicriteria problem, where criteria have different dimensions (risk, kilometers, euros, tons, etc.), the various approaches to normalization of criteria values have been suggested with subsequent assessing distances using one of the possible metrics. Different approaches to criteria normalization, assessment of the distances to ideal and anti-ideal solutions/alternatives, and evaluation of the similarity index (or generalized criterion) for each alternative constitute different variants of TOPSIS/FTOPSIS methods.

2.1 Classical TOPSIS

TOPSIS has been often preferred as a decision making method because of its well defined and easily understandable steps. Even some slight differences in its steps may occur in the literature, the classical TOPSIS method is composed of the following steps.

Step 1. Determine the alternatives A_i, i = 1, 2, . . . , n, and the evaluation criteria C_j, j = 1, 2, . . . , m, and construct the performance table (decision matrix) D = {C_ij} as in Equation (1): $D = [\begin{matrix} C_{11} \dots C_{1 m} \\ ⋮ ⋱ ⋮ \\ C_{n 1 \dots} C_{nm} \end{matrix}]$ (1) where C_ij is the value of alternative i with respect to criterion j.

Step 2. Normalize the criteria values. There are several approaches to normalization of criteria values, which make the elements of normalized decision matrix unit-free. We consider here two of the most popular ones. The first approach is based on Equation (2) both for benefit and cost criteria (below, i = 1, ⋯ , n ; j = 1, . . . , m). $x_{ij} = \frac{C_{ij}}{(\sum_{i = 1}^{n} C_{ij}^{p})^{1 / p}}$ (2)

Here and below p is a natural number, and in most cases p = 2 is considered.

The second approach to normalization is based on Equation (2) for benefit criteria and Equation (3) for cost criteria: $x_{ij} = \frac{1 / C_{ij}}{(\sum_{i = 1}^{n} 1 / Cijp)^{1 / p}}$ (3)

Step 3. Assign the weight coefficients {w_j, j = 1, . . . , m} for the criteria. There are several approaches to setting weight coefficients for TOPSIS based on so-called subjective and objective approaches [9 , 37].

Step 4. Identify the ideal (or positive ideal), I ⁺, and anti-ideal (negative ideal), I ^-, solutions/alternatives. The following (local) approaches to setting I ⁺ and I ^- points in m-dimensional space are considered.

If normalization of both benefit and cost criteria is based on Equation (2), ideal and anti-ideal alternatives are defined as follows: $I^{+} = (x_{1}^{+}, x_{2}^{+}, \dots, x_{m}^{+}), x_{j}^{+} = x_{opt, j}^{+}$ (5) where $x_{opt, j}^{+} = max_{i} x_{ij}$ for benefit and $x_{opt, j}^{+} = min_{i} x_{ij}$ for cost criteria {j}; correspondingly, $I^{-} = (x_{1}^{-}, x_{2}^{-}, \dots, x_{m}^{-}), x_{j}^{-} = x_{nopt, j}^{-}$ (6) where $x_{nopt, j}^{+} = min_{i} x_{ij}$ for benefit and $x_{nopt, j}^{+} = max_{i} x_{ij}$ for cost criteria {j}.

If normalization of benefit and cost criteria is based on Equations (2) and (3) correspondingly, then ideal and anti-ideal alternatives are defined as follows: $I^{+} = (x_{1}^{+}, x_{2}^{+}, \dots, x_{m}^{+}), x_{j}^{+} = max_{i} x_{ij}, j = 1, \dots, m,$ (7) $I^{-} = (x_{1}^{-}, x_{2}^{-}, \dots, x_{m}^{-}), x_{j}^{-} = min_{i} x_{ij}, j = 1, \dots, m,$ (8)

The following transformation (scaling normalized criteria values) is often used within the description of TOPSIS steps: $v_{ij} = w_{j} x_{ij}, i = 1, \dots, n; j = 1, \dots, m$ (9) with corresponding use of v_ij for forming $v_{j}^{+}$ and $v_{j}^{-}$ in formulas (5)–(8) and forming ideal and anti-ideal solutions I ⁺ (v) and I ^- (v).

Step 5. Calculate the distance of each alternative i from the ideal solution, $D_{j}^{+}$ , and anti-ideal solution, $D_{j}^{-}$ : $D_{i}^{+} = d (x_{i}, I^{+}) = {(\sum w_{k}^{p} (x_{ik} - x_{k}^{+})^{p})}^{1 / p}$ (10) $D_{i}^{-} = d (x_{i}, I^{-}) = {(\sum w_{k}^{p} (x_{ik} - x_{k}^{-})^{p})}^{1 / p}$ (11)

In the case of using transformation (9) at the Step 4, distances $D_{i}^{+}$ and $D_{i}^{-}$ are determined as $D_{i}^{+} = {(\sum (v_{ik} - v_{k}^{+})^{p})}^{1 / p}$ (12) $D_{i}^{-} = {(\sum (v_{ik} - v_{k}^{-})^{p})}^{1 / p}$ (13)

Step 6. Assess the relative coefficient of closeness (a generalized criterion for TOPSIS) of each alterative to the ideal solution: ${CC}_{i} = \frac{D_{i}^{-}}{D_{i}^{+} + D_{i}^{-}}$ (14)

Step 7. Ranking of the alternatives with respect to CC_i values so that the largest value means the highest ranked alternative.

2.2 Fuzzy TOPSIS

Classical TOPSIS method has been extended to its fuzzy versions in many publications, see, e.g., survey [2] and [11, 29]. Table 1 presents the most recent fuzzy extensions of TOPSIS method.

Table 1
Most recent fuzzy extensions of TOPSIS method

Authors Type of Fuzzy Sets Type of Fuzzy Numbers Weighting Normalization

Dymova et al. [20] interval type-2 fuzzy sets Triangular type-2 fuzzy numbers Readily given weights Linear normalization of Triangular type-2 fuzzy numbers

Wang and Chen [21] intuitionistic fuzzy sets Interval-valued Through linear programming –

Walczak and Rutkowska [22] Ordinary fuzzy sets Triangular fuzzy numbers (TrFNs) Readily given weights Fuzzy division of TrFNs

Baykasoglu and Gölcük, [23] type-2 fuzzy sets Interval-valued Through hierarchical pairwise comparison normalized direct-relation matrix

Rudnik, and Kacprzak, [24] Ordinary fuzzy sets Triangular fuzzy numbers (TrFNs) Arithmetic mean of experts weights Fuzzy division of TrFNs

Hatami-Marbini and Kangi [25] Ordinary fuzzy sets Trapezoidal fuzzy numbers (TpFNs) Geometric mean of readily given criteria weights Fuzzy division of TpFNs

Xu and Zhang [26] Hesitant fuzzy sets Interval-valued Through a constrained optimization model Crisp normalization of weights

Liang and Xu [27] hesitant Pythagorean fuzzy sets Single valued Readily given weights A special normalization algorithm

Joshi and Kumar [28] intuitionistic hesitant fuzzy sets Interval-valued Through fuzzy measures of criteria Normalized Hamming distance and Euclidean distance

Nadaban and Dzitac [29] Neutrosophic sets Single-valued Readily given weights A special normalization algorithm

Authors	Type of Fuzzy Sets	Type of Fuzzy Numbers	Weighting	Normalization
Dymova et al. [20]	interval type-2 fuzzy sets	Triangular type-2 fuzzy numbers	Readily given weights	Linear normalization of Triangular type-2 fuzzy numbers
Wang and Chen [21]	intuitionistic fuzzy sets	Interval-valued	Through linear programming	–
Walczak and Rutkowska [22]	Ordinary fuzzy sets	Triangular fuzzy numbers (TrFNs)	Readily given weights	Fuzzy division of TrFNs
Baykasoglu and Gölcük, [23]	type-2 fuzzy sets	Interval-valued	Through hierarchical pairwise comparison	normalized direct-relation matrix
Rudnik, and Kacprzak, [24]	Ordinary fuzzy sets	Triangular fuzzy numbers (TrFNs)	Arithmetic mean of experts weights	Fuzzy division of TrFNs
Hatami-Marbini and Kangi [25]	Ordinary fuzzy sets	Trapezoidal fuzzy numbers (TpFNs)	Geometric mean of readily given criteria weights	Fuzzy division of TpFNs
Xu and Zhang [26]	Hesitant fuzzy sets	Interval-valued	Through a constrained optimization model	Crisp normalization of weights
Liang and Xu [27]	hesitant Pythagorean fuzzy sets	Single valued	Readily given weights	A special normalization algorithm
Joshi and Kumar [28]	intuitionistic hesitant fuzzy sets	Interval-valued	Through fuzzy measures of criteria	Normalized Hamming distance and Euclidean distance
Nadaban and Dzitac [29]	Neutrosophic sets	Single-valued	Readily given weights	A special normalization algorithm

The steps for implementing ordinary Fuzzy TOPSIS (FTOPSIS) method constitute a modification of those described in subsection 2.1 and are presented below.

Step 1. Determine the alternatives A_i, i = 1, 2, . . . , n, and the evaluation criteria C_j, j = 1, 2, . . . , m, and construct the performance table (decision matrix) D = {C_ij} as in Equation (1), where C_ij is a fuzzy or crisp value of alternative i with respect to criterion j. As a rule, triangular or/and trapezoidal FNs are used as input values to present uncertain criteria values in fuzzy environment.

Step 2. Normalization of the decision matrix. To avoid the use of complicated normalization according to Equations (2) and/or (3), different simplified expressions have been suggested for normalization decision matrix $D = {{\tilde{C}}_{ij}}$ within FTOPSIS. One of them is based on the transformation of (positive) source values ${\tilde{C}}_{ij}$ to normalized values x_ij with the use of expression (15) [15]: $x_{ij} = {\begin{matrix} {\tilde{C}}_{ij} / {\tilde{C}}_{j}^{+}, for benefit attributes j \\ {\tilde{C}}_{j}^{-} / {\tilde{C}}_{ij}, for cost attributes j \end{matrix}$ (15) where ${\tilde{C}}_{ij}$ are considered as triangular or trapezoidal FNs, ${\tilde{C}}_{ij} = (a_{ij}, b_{ij}, c_{ij}, d_{ij}, e_{ij})$ ; ${\tilde{C}}_{j}^{+} = (a_{j}^{+}, b_{j}^{+}, c_{j}^{+}, d_{j}^{+}, e_{j}^{+})$ and ${\tilde{C}}_{j}^{-} = (a_{j}^{-}, b_{j}^{-}, c_{j}^{-}, d_{j}^{-}, e_{j}^{-})$ are dfined based on ${\tilde{C}}_{ij}, i = 1, \dots n; j = 1, \dots, m$ , with the use of approaches close to those in (5) and (6). For determination of values x_ij (15), standard simplification approach with approximation for dividing two FN is implemented: $x_{ij} = {\begin{matrix} {\tilde{C}}_{ij} / {\tilde{C}}_{j}^{+} = (\frac{a_{ij}}{e_{j}^{+}}, \frac{b_{ij}}{d_{j}^{+}}, \frac{d_{ij}}{b_{j}^{+}}, \frac{e_{ij}}{a_{j}^{+}}), for benefit attributes j \\ {\tilde{C}}_{j}^{-} / {\tilde{C}}_{ij} = (\frac{a_{i}^{-}}{e_{ij}}, \frac{b_{i}^{-}}{d_{ij}}, \frac{d_{i}^{-}}{b_{ij}}, \frac{e_{i}^{-}}{a_{ij}}), for cos t attributes j \end{matrix}$ (16)

Another approach to normalization of decision matrix $D = {{\tilde{C}}_{ij}}$ , which is based on a linear transformation of values ${\tilde{C}}_{ij}$ without approximation, is considered in this contribution, subsection 4.2.

Step 3. Assigning the weight coefficients {w_j, j = 1, ⋯ , m} for the criteria. There are several approaches to setting crisp or fuzzy weight coefficients in FTOPSIS. They include setting uncertainties of weight coefficients for classical TOPSIS, determination of objective entropy based weight coefficients, and the use of linguistic approaches [38 –42].

Step 4. The choice of ideal and anti-ideal alternatives. Several approaches are used here to define the ideal, I⁺, and anti-ideal, I^-, solutions. For normalized trapezoidal fuzzy criteria values ${\tilde{x}}_{ij} = (a_{ij}, b_{ij}, c_{ij}, d_{ij})$ , the ideal, and and anti-ideal solutions

$I^{+} = ({\tilde{x}}_{1}^{+}, \dots, {\tilde{x}}_{m}^{+}) = ((a_{1}^{+}, b_{1}^{+}, c_{1}^{+}, d_{1}^{+}), \dots, (a_{m}^{+}, b_{m}^{+}, c_{m}^{+}, d_{m}^{+})),$

$I^{-} = ({\tilde{x}}_{1}^{-}, \dots, {\tilde{x}}_{m}^{-}) = ((a_{1}^{-}, b_{1}^{-}, c_{1}^{-}, d_{1}^{-}), \dots, (a_{m}^{-}, b_{m}^{-}, c_{m}^{-}, d_{m}^{-})),$ are formed in m-dimensional fuzzy space as “best” and “worst” alternatives with the use of approaches similar to those in (5) and (6). At this step, transformation (9) is also often used with subsequent forming ideal and anti-ideal solutions I⁺ (v) and I^- (v).

For the cases, when after normalization all the criteria are benefit ones in a new x-scale and for all fuzzy numbers $x_{ij}, 0 \leq {\tilde{x}}_{ij} \leq 1$ , the following global ideal and anti-ideal crisp solutions may be considered [12]: $I^{+} = (1, \dots, 1), I^{-} = (0, \dots, 0)$ (17)

The approach (17) for setting ideal and anti-ideal solutions is used in this contribution, subsection 4.2.

Step 5. Determination of the distances for each alternative i from the ideal, $D_{j}^{+}$ , and anti-ideal, $D_{j}^{-}$ , solutions. This is the most complicated step within FTOPSIS. Instead of direct implementation of (10), (11) or (12), (13) as functions of fuzzy variables based on extension principle, various approaches for simplification of the expressions for determining distances $D_{i}^{+}$ and $D_{i}^{-}$ have been suggested to overcome significant computational efforts. These approaches constitute different modifications of FTOPSIS and are based on defuzzification methods [43 –46] or simplification of basic expressions (10)–(13) along with approximation of calculations [47 –50].

Implementation of transformation (9) with subsequent determination of distances $D_{i}^{+}$ and $D_{i}^{-}$ based on formulas (12), (13) is suffered from overestimation, which is discussed in subsection 4.1.

Step 6. Determination of the relative coefficient of closeness CC_i for each alterative based on Equation (14). In most cases, coefficients CC_i are considered as crisp numbers. If distances to ideal and anti-ideal solutions, $D_{i}^{+}$ and $D_{i}^{-}$ , are computed as FNs, coefficients CC_i can also be considered as FNs.

Step 7. Ranking of the alternatives with respect to CC_i values. If values CC_i are FNs, one of the methods for ranking of FNs is implemented [51 –53].

2.3 Application of FTOPSIS

A SCOPUS based review of FTOPSIS from 1983 till the end of 2018 gives 18,678 papers. Among them, 2,922 papers mention fuzzy TOPSIS in “keywords, abstract, or article title,” and 1011 papers in their titles. The distribution of the papers with respect to publication years is given in Fig. 1.

Fig. 1

Number of fuzzy TOPSIS publications with respect to years.

In Table 2, application areas, associated percentages and detailed information about the most recent studies in that area are given. Supplier selection and performance/effectiveness measurement using FTOPSIS are the most used problem areas in the literature. Triangular fuzzy numbers (TrFNs) are the most preferred numbers in the applications of FTOPSIS.

Table 2

Application Papers using Fuzzy TOPSIS

Application area	Percentage of the area	Representative recent studies	Type of fuzzy set	Other integrated methods
Supplier selection	18%	Gupta and Barua [54]	Triangular fuzzy sets	Best Worst Method, AHP
Performance/effectiveness	18%	Sangaiah et al. [55]	Triangular fuzzy sets	DEMATEL, ELECTRE
Technology selection	11%	Onu et al. [56]	Triangular fuzzy sets	ρ
Problem/factor diagnosis	9%	Liu et al. [57]	Intuitionistic fuzzy sets	RPN method, Fuzzy TOPSIS
System selection/design	7%	Aikhuele and Turan [58]	Interval-valued intuitionistic fuzzy sets	Delphi
Operations management	6%	Rudnik and Kacprzak [26]	Triangular fuzzy sets, Trapezoidal fuzzy sets	Ordered Fuzzy Numbers
Service/quality evaluation	5%	Kang et al. [25]	Triangular fuzzy sets	E-S-QUAL
Location selection	5%	Mohammed et al. [60]	Triangular fuzzy sets	Fuzzy AHP-TOPSIS

3 Ranking of fuzzy numbers and FRAA concept

In this section are reviewed necessary concepts for the development and understanding of our proposal such as fuzzy numbers (FNs), fuzzy preference relations [61–63 , 72], and fuzzy ranking, including ranking of FNs based on FRAA (Fuzzy Rank Acceptability Analysis) concept.

3.1 Fuzzy numbers and their ranking

Definition 1. A FN Z is a bounded, normal, and convex fuzzy set in $ℝ$ with either a continuous or a piecewise upper continuous membership function μ_z (x).

It implies that exist real numbers c₁ and c₂, c₁ < c₂, such that:

$\begin{matrix} Z = {(x, μ_{z} (x)) : μ_{z} (x) > 0, x \in (c_{1}, c_{2}), \\ μ_{z} (x) = 0, x \notin [c_{1}, c_{2}] \end{matrix}$ (18)

Crisp numbers are also considered as a special type of FNs, fuzzy singletons Z = c, with the membership function μ_z (x) = {0 for x < c ; 1 for x = c ; 0 for x > c}.

Let $F$ be the set of FNs according to Definition 1, Equation (18), united with the set of fuzzy singletons.

Definition 2. An α-cut of FN $Z \in F, α \in (0, 1]$ is defined as: $Z_{a} = [A_{a}, B_{a}] = {x \in ℝ : μ_{z} (x) \geq α} .$ (19)

For α = 0, denote segment [c₁, c₂] (in accordance with (18)) as [A₀, B₀]. Then, FN Z is identified with the set of segments $Z = {[A_{α}, B_{α}]} (α \in [0, 1])$ (20)

For FNs $Z_{i} = {[A_{α}^{i}, B_{α}^{i}]}$ and $Z_{j} = {[A_{α}^{j}, B_{α}^{j}]} \in F$ , according to standard fuzzy arithmetic [61, 62]:

$\begin{matrix} Z_{ij} = Z_{i} - Z_{j} \\ = {[A_{α}^{ij}, B_{α}^{ij}]} = {[A_{α}^{i} - B_{α}^{i}, B_{α}^{i} - A_{α}^{i}]} \end{matrix}$ (21)

Definition 3. Fuzzy preference relation R = ((Z_i, Z_j) , μ_R (Z_i, Z_j)), is a fuzzy relation on $F \times F$ , where grade of membership μ_R (Z_i, Z_j) ∈ [0, 1] designates the degree of preference Z_i over Z_j.

Fuzzy preference relation R is reciprocal [62, 63] if the following property has the place: $μ_{R} (Z_{i}, Z_{j}) = 1 - μ_{R} (Z_{j}, Z_{i})$ (22)

Note that from (22): if Z_i = Z = Z_j, then μ_R (Z_i, Z_j) = 0.5.

Definition 4. Ranking of FNs, $Z_{i}, Z_{j} \in F$ , on the basis of preference relation R is defined as follows:

$\begin{matrix} Z_{i} ≽ Z_{j} if μ_{R} (Z_{i}, Z_{j}) \\ \geq 0.5, Z_{i} ≻ Z_{j} if μ_{R} (Z_{i}, Z_{j}) > 0.5, \\ and Z_{i} \sim Z_{j} if μ_{R} (Z_{i}, Z_{j}) = 0.5 \end{matrix}$ (23)

Reciprocal fuzzy preference relation R is consistent within ranking of FNs: from μ_R (Z_i, Z_j) >0.5 follows μ_R (Z_j, Z_i) <0.5.

The following denotations are also used for fuzzy preference relation R: $μ_{ij} = μ_{R} (Z_{i}, Z_{j}) = μ_{R} (Z_{i} \geq Z_{j}) = μ_{R} (Z_{j} \leq Z_{i})$ (24)

It should be stressed here that symbols ≽, ≼ in (23) imply ranking of FNs and differ from the symbols ≥, ≤ (24), which are used for notational purposes.

Below, a brief review of the main classes of ranking methods is provided. A comprehensive survey of ranking methods has been presented in [64 –66].

Ranking methods based on defuzzification. These methods substitute FNs by respective real numbers with their subsequent ranking [67 –70].

Etalon set based ranking methods. FNs are ranked according to the distances to an etalon set (the closest FN has the higher rank). Different approaches are used for forming the etalon set modeling the distances [62, 64].

Ranking methods based on pairwise comparison. Various approaches for pairwise comparisons and subsequent ranking of FNs have been suggested, and these are the most extensively studied methods [62 , 71].

The basic requirements (axioms) to ranking methods, which represent their reasonable properties and are in accordance with our intuition, have been presented in [64].

This proposal implements the FRAA (Fuzzy Rank Acceptability Analysis) method for ranking FNs, which is a pairwise comparison method that can use different (including intransitive) fuzzy preference relations [33]. In our proposal, the FRAA is implemented with the Yuan’s preference relation, which is revised below.

3.2 Yuan’s fuzzy preference relation

Let FNs Z_i and $Z_{j} \in F, Z_{ij} = Z_{i} - Z_{j}$ , and Z_ij = {[A_α, B_α]}. Within Yuan’s preference relation [71], R_Y = ((Z_i, Z_j) , μ_Y (Z_i, Z_j)), the value $S_{Y}^{+} (Z_{ij})$ is assessed by the expression [33] $S_{Y}^{+} (Z_{ij}) = \int_{0}^{1} (B_{α} θ (B_{α}) + A_{α} θ (A_{α})) d α,$ (25) where θ(x) is the Heaviside function: θ (x) = {1, x ≥ 0 ;0, x < 0} $S_{Y}^{+} (Z_{ij})$ is interpreted here as an area between positive parts of A_α and B_α and the axis OY. The modified “total area” under membership function of Z_ij is evaluated as $S_{Y} (Z_{ij}) = S_{Y}^{+} (Z_{ij}) + S_{Y}^{+} (Z_{ji})$ (26)

For Yuan’s fuzzy preference relation, μ_Y (Z_i ≥ Z_j) is defined as

$\begin{matrix} μ_{ij} = μ_{Y} (Z_{i} \geq Z_{j}) \\ = S_{Y}^{+} (Z_{ij}) / S_{Y} (Z_{ij}), S_{Y} (Z_{ij}) > 0 \end{matrix}$ (27) where $S_{Y}^{+}$ and S_Y are assessed according to (25), (26); for singletons Z_i = c_i, Z_j = c_j, c_ij = c_i - c_j:

$\begin{matrix} μ_{ij} = μ_{Y} (Z_{i} \geq Z_{j}) = \\ {(1 if c_{ij} > 0), (0 if c_{ij} < 0), (0.5 if C_{ij} = 0)} \end{matrix}$ (28)

Yuan’s fuzzy preference relation is reciprocal, transitive, and satisfies the key axioms for ranking methods [65, 71].

3.3 FRAA ranking

Fuzzy Rank Acceptability Analysis (FRAA) is a methodology that consecutively propagates the concept of fuzziness within fuzzy decision analysis [32]. FRAA is based on Fuzzy Rank Acceptability Indices (FRAIs) [33], which are directly used within ranking of FNs.

FRAIs are evaluated as a fuzzy measure of fuzzy logical expressions or Fuzzy Rank Statements, FRSs [31, 33]. For a finite set of FNs Z = {Z_i, i = 1, . . . , n}, FRS, F_ik, is a statement of the type: $F_{ik} = (Z_{i}, k) = {Z_{i} has a rank k}, i, k = 1, \dots n .$ (29)

Formalization of the FRSs with the use of a chosen fuzzy preference relation R (Z_i, Z_j) ≡ (Z_i ≥ Z_j) ≡ (Z_j ≤ Z_i)) is based on the following fuzzy logical expressions: $F_{i 1} = {\land_{j \neq i}^{n} (Z_{i} \geq Z_{j})}$ (30) $F_{i 2} = {\lor_{l \neq i}^{n} (Z_{i} \leq Z_{l}) \land_{j \neq i, j \neq l}^{n} (Z_{i} \geq Z_{j}))$ (31) $\begin{matrix} F_{ik} = {\underset{(l_{1} < l_{2} < \dots < l_{k - 1}) l_{s} \neq i, s = 1, \dots, k - 1}{\lor} ((\land_{s = 1}^{k - 1} (Z_{i} \leq Z_{l_{s}}) \land_{j \neq i, j \neq l_{s}, s = 1, . . ., k - 1}^{n} (Z_{i} \geq Z_{j}))} \end{matrix}$ (32) $F_{in} = {\land_{j \neq i}^{n} (Z_{i} \leq Z_{j})}$ (33)

FRAIs, μ (i, k) , i, k = 1, …, n, represent a fuzzy measure (index or degree of confidence) of the logical expressions (30)–(33), μ (i, k) = μ (Z_i, k) = μ_R (F_ik), and are interpreted, for given i and k, as an acceptability of FN Z_i with a rank k. For formalization of FRAIs the following model is used: $μ (i, 1) = μ (F_{i 1}) = \land_{j \neq i}^{n} μ_{ij},$ (34) $μ (i, 2) = μ (F_{i 2}) = \lor_{l \neq i}^{n} (μ_{li} \land_{j \neq i, j \neq l}^{n} μ_{ij})$ (35) $μ (i, k) = μ (F_{i k}) = \underset{\begin{array}{l} (l_{1} < l_{2} < ... < l_{k - 1}) \\ l_{s} \neq i, s = 1, .., k - 1 \end{array}}{\lor} ((\land_{s = 1}^{k - 1} μ_{l_{s} i}) (\land_{\begin{array}{l} j \neq i, j \neq l_{s}, \\ s = 1, .., k - 1 \end{array}}^{n} μ_{i j}))$ (36) $μ (i, n) = μ (F_{i n}) = \land_{j \neq i,}^{n} μ_{i j} .$ (37)

The analysis of FRSs and FRAIs in details is presented in [33].

FRAA_R ranking of FNs {Z_i, i = 1, . . . , n} , with the use of fuzzy preference relation R is based on analysis of the matrix M = {μ (i, k)} of FRAIs, i, k = 1, … n.

Definition 7. [33] Let $Z = {Z_{i}, i = 1, . . ., n} \subset F$ be a set of FNs. FN Z_m has a FRAA_R rank k, r (Z_m) = k, if it has the maximal FRAI for that k: $μ_{R} (m, k) = max_{i = 1, . . ., n} μ_{R} (i, k), 1 \leq k, m \leq n$ (38)

Thus, it can be pointed out that FN Z_m has the rank k with the degree of confidence μ_R (m, k) = μ_R (F_mk).

It has been proved [33], FRAA ranking with the use of Yuan’s fuzzy preference relations, FRAA_Y, satisfies the key axioms for ranking methods, and FRAA_Y ranking coincides with Yuan’s ranking. Comparison of ranking FNs by different ranking methods (including FRAA_Y and FRAA_I ranking with intransitive Integral fuzzy preference relation), and advantages of FRAA ranking is analyzed in details in [33].

4 Fuzzy multicriteria acceptability analysis and FTOPSIS-FMAA method

The use of FRAA ranking within MCDA in fuzzy environment forms FMAA (Fuzzy Multicriteria Acceptability Analysis) conception. Thus, FMAA provides not only ranking of alternatives but also a measure of confidence that a given alternative has the corresponding rank.

In this section the integration of FMAA with an FTOPSIS model, FTOPSIS-FMAA, is introduced and implemented. Below, the use of FMAA within an FMCDA is considered along with overestimation problem [72], which arises at some steps of FMCDA models implementation.

4.1 General FMCDA model and the overestimation problem

A discrete MCDA method is based on assessing a generalized criterion G and can be presented by a model $V (a_{i}) = G (a_{i}, w, a, p); (a, w, p) \in U,$ (39) here a = ( a ₁, . . . , a _n) is a set of alternatives, a _i = (C_i1, …, C_im) , C_ik is the criterion value for the alternative a _i according to criterion k, i = 1, …, n, k = 1, …, m; w = (w₁, … w_m) and w_k is a weight coefficients of criterion k, p is a set of parameters, and U is a set of coupling and restrictions. Within FMCDA, criteria values and weight coefficients are considered as FNs of a given type (as a rule, triangular or trapezoidal FNs).

Integration of FMAA with the FMCDA model (39) consists of the use of FRAA ranking for the set of FNs Z = {Z_i = V ( a _i) , i = 1, … n}. Differences Z_ij = Z_i - Z_j are assessed by the expression $Z_{ij} = G (a_{i}, w, a, p) - G (a_{j}, w, a, p)$ (40)

Taking into account (40), the following feature of FMAA and FMCDA as a whole should be pointed out.

Weight coefficients w_k, k = 1, … m, and, in a general case, criteria values C_ik occurs in the right part of the expression (40) at least two times. It means, formula (40) contains dependent variables/FNs, therefore, an overestimation of the output values should be taken into account [72].

In fuzzy modeling, the overestimation problem can be briefly presented by the following expressions: $Z_{0} = (a + b) / a$ (41) $Z_{T} = 1 + b / a$ (42) here a and b are positive FNs. Implementation of α-cuts for assessing (41) and (42) demonstrates that, Supp (Z_T) ⊂ Supp (Z₀) (O and T means, correspondingly, overestimation and transformation). Thus, being equivalent in class of real numbers, expressions (41) and (42) lead to different outcomes if standard fuzzy arithmetic with independent FNs a in nominator and denominator in (41) is implemented (it means in fact that FNs Z₁ = a and Z₂ = a are used in nominator and denominator correspondingly). It should be stressed, in contrast to Z₀ (41), FN Z_T (42) presents the proper value of (41) assessed in accordance with the fuzzy extension principle [61, 62].

For some FMCDA models (e.g., FMAVT-FMAA, [31]), the overestimation within implementation of FMAA can be overcome by using a transformation of the source expression (e.g., similar to above example with the use of (42) instead of (41)). However, in the case of the FTOPSIS, see below Equations (47) and (48), overestimation has the place both for source and transformed models.

To overcome overestimation problem, the following variants of the Transformation Method (TMs) [72] can be used depending on the complexity and peculiarity of the mathematical expression: Reduced Transformation Method (RTM), General Transformation Method (GTM), and Extended Transformation Method (ETM).

Comment 1. In a brief form, the variants of TMs can be presented as follows.

Let $f : ℝ^{n} \to ℝ$ be a real function, and propagation of fuzziness by a function f (x₁, ⋯ , x_n) is implemented, i.e., instead of real numbers x_i, FNs Z_i are used, and fuzzy extension function, f (Z₁, ⋯ , Z_n), is considered with application of the extension principle [62, 72]. The goal of TMs is numerical determination of FN Z = f (Z₁, ⋯ , Z_n) using an appropriate number M of α-cuts $Z_{α}^{i}$ for each FN $Z_{i} = {[A_{α}^{i}, B_{α}^{i}]}$ .

If function f (x₁, ⋯ , x_n) is monotonic for each x_i in segment $U_{i} = \bar{supp (Z_{i})} = [A_{0}^{i}, B_{0}^{i}], i = 1, \dots, n$ , that is (for differentiable functions), $\frac{\partial f}{\partial x_{i}}$ does not change its sign in the given segment U_i (for fixed values of all other variables in corresponding segments), RTM is implemented: for each α -cut, values Y = f (X₁, …, X_n) are determined for all combinations (X₁, …, X_n), where X_i is one of the marginal points of $Z_{a}^{i} : X_{i} \in {A_{α}^{i}, B_{α}^{i}}$ , with subsequent assessing minimal and maximal value of Y for forming α-cut Z_a = [A_α, B_α] of FN Z = f (Z₁, ⋯ , Z_n).

If function f (x₁, ⋯ , x_n) is not monotonic for each x_i in segment U_i, GTM is used: for each α-cut, values Y = f (X₁, …, X_n) are determined for all combinations (X₁, …, X_n), where X_i is one of the N_α points in the segment $[A_{α}^{i}, B_{α}^{i}]$ ; for this, segment $[A_{α}^{i}, B_{α}^{i}]$ is subdivided for (N_α - 1) intervals, in accordance with a specific algorithm, by the points C₁, …, C_{N_α-2}, i.e., $X_{i} \in {A_{α}^{i}, C_{1}, \dots, C_{N_{α} - 2}, B_{α}^{i}}$ ; then minimal and maximal value of Y for forming α-cut Z_a = [A_α, B_α] is found. Let’s stress that RTM may also be used (as an approximation) in this case too. However, in both cases 1 and 2, the special algorithm for organizing cycles for α-cuts from α = 1 to α = 0 is implemented.

In the general case, function f (x₁, ⋯ , x_n) can be monotonic for some variables x_i in segments U_i, i = 1, …, n₁, and non-monotonic for other variables in their segments. In this case, to diminish number/time of computations, instead of GTM, an ETM can be used: for “monotonic variables” x_i, RTM approach is used, for remaining variables, GTM is implemented.

The following notions are used below concerning accepting or avoiding the overestimation problem:

Model-O, when standard fuzzy arithmetic (SFA) is implemented despite existing dependent variables in corresponding expressions;

Model-T/TM, when transformations of the model formula(s) along with subsequent SFA (as for the cases (41), (42)) are used, or when one of the TMs is implemented; in the last case, the following denotations can also be used in accordance with Comment 1: Model-R (RTM is implemented). Model-G (GTM), and Model-E (ETM).

4.2 FTOPSIS-FMAA method

In this subsection, an approach to implementation of the FTOPSIS-FMAA is introduced. Within this method, weight coefficients and criteria values can be considered as FNs of a general type.

The following denotations are used below: n and m are number of alternatives and number of criteria correspondingly; C_ij –value of alternative i for criterion j, $C_{ij} = {[A_{α}^{ij}, B_{α}^{ij}]} w_{j}$ ; –weight coefficient for criterion j, $w_{j} = {[A_{α}^{j}, B_{α}^{j}]}, j = 1, \dots, m, i = 1, \dots, n$ .

Normalization of criteria values . Several approaches to normalization of criteria values within TOPSIS/FTOPSIS were discussed in Introduction section. In this paper, the suggested approach is effective for computations within FTOPSIS.

For criteria values, $C_{ij} = {[A_{α}^{ij}, B_{α}^{ij}]}$ , according to denotations (19) and (20), the following boundary values for each criterion j are introduced: $B_{0}^{0 j} = max_{i = 1, \dots, n} B_{0}^{ij}, A_{0}^{0 j} = min_{i = 1, \dots, n} A_{0}^{ij}$ (43)

Normalization presents a mapping of criteria values to a dimensionless quantity: C_ij → x_ij. The following mapping is considered:

for benefit criteria: $x_{ij} = (C_{ij} - A_{0}^{0 j}) / (B_{0}^{0 j} - A_{0}^{0 j})$ ;

for cost criteria: $x_{ij} = (B_{0}^{0 j} - C_{ij}) / (B_{0}^{0 j} - A_{0}^{0 j})$ .

Thus, for FNs x_ij, $\bar{supp (x_{ij})} \subseteq [0, 1]$ , and in new dimensionless x - scale all the criteria are benefit ones.

Choice of ideal and anti-ideal solutions . The following ideal, I ⁺, and anti-ideal, I ^-, solutions (as m-dimensional crisp points) are considered following [12]: $I^{+} = (1, \dots, 1), I^{-} = (0, \dots, 0)$ (44)

Such a choice significantly simplifies computations of functions of fuzzy variables for the FTOPSIS method.

Setting weight coefficients . Several methods are used for assigning weight coefficients in TOPSIS [36 –42]. For classical approaches, in the case of m criteria, degree of freedom for set of weight coefficients is m-1 (due to normalization of weight coefficients with the sum equal 1). Taking into account this property, the following approach to assigning weights as coefficients of relative importance of criteria can be recommended (this is a variant of swing methods used in MAVT/FMAVT [3 , 31]):

for the most important criterion, say C₁, the weight coefficient w₁ = 1 is assigned;

for criterion C_k, k = 2, …, m, weight coefficient as a fraction from the most important criterion is assigned by experts, taking into account the relative importance of this criterion in comparison with $C_{1} : w_{j} = {[A_{α}^{j}, B_{α}^{j}]}$ , with $0 < A_{0}^{j} \leq B_{0}^{j} \leq 1$ ; the forms of membership functions for weight coefficients w_j are assigned by experts.

Determination of distances . In the normalized criterion space, the weighted distances from an alternative x _i = (x_i1, . . . . , x_im) (with fuzzy values x_ij) to the ideal, I ⁺, and to anti-ideal, I ^-, alternatives, taking into account (44), are determined based on the expressions (10) and (11) with implementation of the fuzzy extension principle [62]:

$\begin{matrix} D_{i}^{+} = d (x_{i}, I^{+}) = ({\sum w_{k}^{p} (x_{ik} - I_{k}^{+})}^{p})^{1 / p} \\ = (\sum w_{k}^{p} (1 - x_{ik}) p)^{1 / p} \end{matrix}$ (45)

$\begin{matrix} D_{i}^{-} = d (x_{i}, I^{-}) = ({\sum w_{k}^{p} (x_{ik} - I_{k}^{-})}^{p})^{1 / p} \\ = {(\sum w_{k}^{p} x_{ik}^{p})}^{1 / p} \end{matrix}$ (46) where $D_{i}^{+}$ and $D_{i}^{-}$ are FNs. In this paper, classical case p = 2 is considered.

Determination of generalized criterion

With denotations introduced above, the generalized criterion (coefficient of closeness) Z_i for alternative A_i is assessed with the use of Equation (14), taking into account (45) and (46), in an adjusted form: $Z_{i} = 1 / (1 + \frac{D_{i}^{+}}{D_{i}^{-}}) = 1 / (1 + \frac{(\sum w_{k}^{p} (1 - x_{ik})^{p})^{\frac{1}{p}}}{(\sum w_{k}^{p} x_{ik}^{p}) \frac{1}{p}}$ (47)

FNs w_k and x_ik occur both in numerator and denominator, that leads to the overestimation when using SFA. There is no transformation of the expression (47) to avoid overestimation (as for the case (42)). Evidently, Z_i = Z_i (x_ik) as a real function of real value x_ik is monotonic in interval (0, 1). According to additional analysis, $\frac{\partial Z_{i} (w_{k})}{\partial w_{k}}$ does not change its sign for real values w_k in interval (0, 1) if other variables are fixed. Thus, for determination of generalized criterion Z_i (47) without overestimation, RTM [72] may be used.

Ranking of the alternatives . In FTOPSIS-FMAA (with the use of Yuan’s ranking method) ranking of alternatives/FNs {Z_i, i = 1, 2, . . . , n, } is based on assessing differences Z_ij = Z_i - Z_j:

$\begin{matrix} Z_{ij} = Z_{i} - Z_{j} = 1 / (1 + \frac{(\sum w_{k}^{p} (1 - x_{ik})^{p})^{\frac{1}{p}}}{(\sum w_{k}^{p} x_{ik}^{p}) \frac{1}{p}} \\ - 1 / (1 + \frac{(\sum w_{k}^{p} (1 - x_{jk})^{p})^{\frac{1}{p}}}{(\sum w_{k}^{p} x_{jk}^{p}) \frac{1}{p}} \end{matrix}$ (48) with subsequent implementation of FRAA ranking based on the chosen fuzzy preference relation. As above, Z_ij = Z_ij (x) is monotonic as a function of real value x = x_ik (and x = x_ik). However, in a general case (for real values w_k), $\frac{\partial Z_{i} (w_{k})}{\partial w_{k}}$ can change its sign in interval (0, 1). Thus, for proper determination of Z_ij (48), ETM should be implemented: RTM may be used for fuzzy variables x_ik and x_jk, and GTM is implemented for fuzzy weight coefficients w_k, k = 1, …, m ; i, j = 1, …, n.

5. Application of FTOPSIS within a housing development problem

Here, FTOPSIS-FMAA is applied to a case study on housing development (the selection of a site in a region for building cottages) [73].

The group of stakeholders and experts considered 11 criteria for analysis of potential sites by using GIS functions to conform the appropriate set of alternatives.

The experts and stakeholders imposed constraints and restrictions concerning 6 criteria, which were elaborated with the use of GIS at the first stage of the problem analysis to specify and diminish the number of possible alternatives (suitable sites). In the second stage, 5 alternatives (A₁ - A₅) and the following 5 criteria are considered within the multi-criteria problem: C₁ –proximity to animal breeding farms (maximize), C₂ –proximity to ecologically adverse objects (maximize), C₃ –level of radioactive contamination (minimize), C₄ –local landscape quality (maximize), C₅ –total expected cost (minimize). The indicated 5 criteria, C₁ –C₅, are used for analysis of 5 alternatives, A₁ –A₅, which were formed through GIS-based conjunctive screening process.

In the original work [73], averaged criteria values and weight coefficients were considered by experts with implementation of MAVT method. In this contribution, uncertainties/imprecisions of all values are taken into account with the use of FNs. Corresponding values, which are based on the range of previous estimates and experts’ judgments, are considered as symmetrical trapezoidal (TpFNs) and triangular (TrFNs) FNs regarding source mean values. The performance table with criteria values and basic weight coefficients for housing development problem are represented in Figs. 2 and 3, respectively. For criteria values, TpFNs with ±10% for [A₁, B₁] and ±35% for [A₀, B₀] (according to denotation (20)) with respect to mean values, indicated in Fig. 2, are used for criteria C₁, C₂, and C₃; for criteria C₄ and C₅, TrFNs with ±20% and ±1 score with respect to mean values are used correspondingly for respective [A₀, B₀].

For weight coefficients, Fig. 3, symmetrical TrFNs with indication of the ranges [A₀, B₀] (according to denotation (20)) are specified. The number of α-cuts for assessing functions of fuzzy variables for all FTOPSIS-FMAA models within the case study under consideration is 20.

Fig. 2

Performance Table: housing development case study.

Fig. 3

Fuzzy criterion weights: housing development case study.

In this contribution, FMAA is based on FRAA_Y. For research purposes, the following models of FTOPSIS-FMAA implementation are considered:

O (overestimation): Standard Fuzzy Arithmetic (SFA) is used for determination of Z_i, i = 1, . . . , n, based on (45)–(47), and then Z_ij is assessed as Z_ij = Z_i - Z_j;

RO: RTM is implemented for determination of Z_i (47), and then Z_ij is assessed as Z_ij = Z_i - Z_j;

R: RTM is implemented for determination of Z_ij according to expression (48);

E: ETM is implemented for determination of Z_ij (48): RTM is used for variables x_ik and x_jk, and GTM is implemented for weight coefficients w_k.

For the case study under consideration with basic weight scenario (scenario WS1) according to Fig. 3, ranks of alternatives by FTOPSIS-FMAA for models O, RO, R, E, and classical TOPSIS (with mean criteria values and weight coefficients according to Figs. 2 and 3) are presented in Table 3. For FTOPSIS-FMAA, ranks of alternatives are provided also with corresponding confidence degree according to FRAA_Y algorithm (i..e., fuzzy measure of the statement that alternative A_i has FRAA_Y rank k.)

According to Table 3, alternatives A₄ and A₅ both have ranks 1 and 2 for Model-O, alternatives A₄ and A₅ have ranks 1 and 2 correspondingly for TOPSIS and all other models of FTOPSIS-FMAA. Comparison of the differences Z₄₅ = Z₄ - Z₅ for Model-O and Model-R, Fig. 4, demonstrates a perceptible influence of overestimation in the case of Model-O. The degree of overestimation [72] for this case, ω₄₅, is assessed as

Table 3

Weight scenario WS1: Ranks of alternatives by FTOPSIS-FMAA (with degree of confidence) and TOPSIS methods

Methods/Alternatives	A ₁	A ₂	A ₃	A ₄	A ₅
FTOPSIS-FMAA/O	5 (0.541)	4 (0.525)	3 (0.525)	1-2 (0.5)	1-2(0.5)
FTOPSIS-FMAA/RO	5(0.6434)	4(0.5094)	3(0.5094)	1(0.5613)	2(0.5613)
FTOPSIS-FMAA/R	5(0.6449)	4(0.5106)	3(0.5106)	1(0.5562)	2(0.5562)
FTOPSIS-FMAA/E	5(0.6449)	4 (0.5106)	3(0.5106)	1 (0.5562)	2(0.5562)
TOPSIS	5	3	4	1	2

Fig. 4

Scenario WS1: Z₄₅ for Model-O (overestimation) and for model-R (RTM).

$\begin{matrix} ω_{45} ≃ ((0.76 + 0.84) - (0.3 + 0.27)) / \\ (0.3 + 0.27) ≃ 1.81 (181 %), \end{matrix}$ that may be considered as significant overestimation of results by Model-O in comparison with proper assessments based on model Model-R.

The weight of criterion C₂ (distance from ecologically adverse objects), Fig. 3, was originally considered [73] as about 0.3 regarding the most important criterion. According to subsequent discussion of this case study, experts stressed that contamination of the environment by chemical substances increases, and relative importance of criterion C₂ (distance from ecologically adverse objects) should be reconsidered and significantly raised. The up-dated weight coefficient w₂ for criterion C₂ is considered as TrFN (0.85, 0.9, 0.95), and along with other weights according to Fig. 3 the weight scenario WS2 is formed. Ranking of alternatives for scenario WS2 is presented in Table 4.

Table 4

Weight scenario WS2: Ranking alternatives by FTOPSIS-FMAA (with degree of confidence) and TOPSIS methods

Method/Alternative	A ₁	A ₂	A ₃	A ₄	A ₅
FTOPSIS-FMAA/O	5(0.5010)	4(0.5010)	3(0.5354)	2(0.5883)	1(0.6044)
FTOPSIS-FMAA/RO	5(0.5725)	4(0.5301)	3(0.5301)	2(0.6150)	1(0.6150)
FTOPSIS-FMAA/R	5(0.5722)	4(0.5314)	3(0.5314)	2(0.6240)	1(0.6240)
FTOPSIS-FMAA/E	5(0.5722)	4(0.5314)	3(0.5314)	2(0.6240)	1(0.6240)
TOPSIS	5	3	4	1	2

For this scenario, ranks of alternatives for all FTOPSIS-FMAA models (O, RO, R, and E) coincide and differ from TOPSIS ranks. To clarify, why ranking of alternatives by TOPSIS remains the same for weight scenarios WS1 and WS2, consider weight sensitivity analysis changing weight coefficient w₂ within TOPSIS, Fig. 5. Varying weight w₂ from scenario WS1 to WS2 changes normalized weight coefficient (which is implemented within classical TOPSIS) from 0.12 to 0.29; and A₅ exceeds A₄ when w₂ ≥ 0.308 (with remaining proportions of all other normalized weight coefficients).

Fig. 5

Weight sensitivity analysis (changing w₂) for TOPSIS.

Further varying weight coefficient w₂ (e.g., if weight coefficients w₂ is set as TrFN (0.39, 0.49, 0.59) and other weights, Fig. 3, remain the same) demonstrates difference of the ranks for alternatives A₄ and A₅ for models O and RO. However, for the model RO, the degree of confidence that alternative A₄ has FRAA_Y rank 1 and A₅ has rank 2 is 0.5030, that may be considered as a negligible difference (and these alternatives may be considered as having the same ranks).

To understand, why for scenarios WS1 and WS2, Tables 3 and 4, the measures/degrees of confidence that alternatives A₄ and A₅ have ranks 1 and 2 can coincide (for models RO, R, and E), consider Table 4 with the degree of preference of alternative A_i over A_j (correspondingly), degree of preference of FN Z_i over Z_j, μ_ij = μ_Y (Z_i ≥ Z_j), for scenario WS1 and model-R as an example.

Taking into account the degree of preferences μ_ij, Table 5, and FRAIs μ(4,1) and μ(5,2) in accordance with (34) and (35), the following equalities have the place: μ (4, 1) = μ (5, 2) = μ₄₅ = μ_Y (Z₄ ≥ Z₅) =0.5562.

Table 5

FTOPSIS-FMAA, Scenario WS1, Model-R: Degree of preference of A_i over A_j

	A ₁	A ₂	A ₃	A ₄	A ₅
A ₁	0.5000	0.3551	0.3244	0.0589	0.0868
A ₂	0.6449	0.5000	0.4894	0.0833	0.1353
A ₃	0.6756	0.5106	0.5000	0.1429	0.1864
A ₄	0.9411	0.9167	0.8571	0.5000	0.5562
A ₅	0.9132	0.8647	0.8136	0.4438	0.5000

As the results of multi-criteria analysis of HD case study, experts have recommended for decision makers the alternative A₄ as the most justified for the scenario WS1 and A₅ for the scenario WS2.

6 Discussion

The suggested FTOPSIS-FMAA model shows adequate results (Tables 3 and 4) for different scenarios and may be considered as a validated FMCDA method.

As distinct from other previously published works on FTOPSIS, the suggested approach to implementation of FTOPSIS-FMAA presumes utilization of input values (criteria values and weight coefficients) as fuzzy numbers of a general type (according to Definition 1). Though, in most cases of FTOPSIS implementation, the basic types of FNs (triangular, trapezoidal, and crisp ones) are used, the suggested general approach can be claimed when modeled input data are used (e.g., model assessments of risk values, results of non-linear functions of fuzzy variables).

In Tables 2 and 3, ranks of alternatives by FTOPSIS-FMAA and (classical) TOPSIS have been represented. Comparison of different MCDA models, including comparison of classical ones and their fuzzy “analogues”, is a methodological problem. FTOPSIS-FMAA and TOPSIS are different MCDA models, and possible distinctions in ranks of alternatives for the “same” scenarios may be considered as inevitable. Several problems of comparing TOPSIS and FTOPSIS models may be pointed out (in [31], the problems of comparing FMAVT-FMAA and MAVT ranking have been discussed):

setting fuzzy values for criteria and weight coefficients for FTOPSIS based on real (crisp) values of TOPSIS (and vice versa), presents a model approach and can be implemented with different ways;

methods for normalization of criteria values in (classical) TOPSIS and FTOPSIS-FMAA are methodologically different: in TOPSIS, normalized criteria values x_ij are on the sphere with radius 1: $\sum_{i = 1}^{n} x_{ij}^{2} = 1$ ; in FTOPSIS-FMAA linear functions for normalization are used;

the approaches to setting ideal and anti-ideal points in TOPSIS and FTOPSIS-FMAA are also different;

the use of non-symmetric FNs for setting criteria values and weight coefficients in FTOPSIS-FMAA can lead to significant differences in output results in comparison with TOPSIS;

the choose of method for ranking FNs/alternatives can also influence on the results of ranking.

Though, FRAA_R ranking, and first of all, determination of FRAIs (34)–(37) and realization of TMs, seems complicated, however, the use of appropriate computer system (extension of DecernsMCDA [31] to corresponding fuzzy system) allows users/experts to implement FTOPSIS-FMAA method effectively even they do not go into details of corresponding mathematical methods.

In this contribution, instead of standard practice of using simplified methods for determination of functions of fuzzy variables, TMs are implemented in FTOPSIS-FMAA providing methodologically consistent and mathematically strong approach to implementation of TOPSIS model in the fuzzy environment. In addition, to the best of our knowledge, the suggested FTOPSIS-FMAA model is the only in FMCDA, where overestimation problem [72] is taken into account.

In Section 5, four models for determination of generalized FTOPSIS criterion (47) and criterion differences (48) are considered depending on the use of SFA (Standard Fuzzy Arithmetic) based on standard α-cuts procedures for independent arguments or/and TMs (Transformation Methods): models O, R, RO, and E. Model-O is based on the use of SFA at all the stages of FTOPSIS-FMAA implementation and leads to overestimation of the output results, Fig. 4. The absence of distinctions in ranking alternatives between model-R and model-E of FTOPSIS-FMAA implementation within the case-study under consideration, Section 5, Tables 3 and 4, can be explained by the properties of functions Z_ij (w_k) for the model (48) as well as by the features of RTM algorithm (model-R), which can catch the properties of some non-monotonic function and leads to the proper assessment or insignificant underestimation of the output fuzzy values in comparison with ETM algorithm (model-E).

According to Tables 3 and 4 (despite the significant level of overestimation for model-O, Fig. 4), the ranks of alternatives for the considered case-study coincide for models with different complexity (O, RO, R, and E). Thus, the question about the necessity of using complicated FTOPSIS-FMAA models (i.e., implementation of models E, R, or RO instead of model O) is justified and needs a discussion.

For further research, to analyze the ranking of alternatives depending on the different models of FTOPSIS-FMAA under consideration, Monte Carlo simulation of input data (criteria values and weight coefficients) has been implemented with subsequent computations and ranking according to models O, RO, and R. Within Monte Carlo simulation (based on uniform probability distribution for generated points in corresponding segments), the following scenario, SM1, is implemented:

symmetrical TrFNs of a special form (range ±20% with respect to mean value is used for respective support (A₀, B₀) of this TrFN) for criteria values in the segment [0, 10] are randomly generated with subsequent normalization according to subsection 4.2;

symmetrical TrFNs for weight coefficients are randomly generated according to weighting process, described in section 4.2;

FTOPSIS-FMAA models are considered with 5 criteria and 5 alternatives;

models O, RO, and R are implemented for each Monte Carlo iteration and corresponding distinctions in rank 1 comparing with model R are counted;

the following numbers of Monte Carlo iterations N and the numbers of α -cuts n_α have been implemented: N = 1000, and n_α = 20.

The number of differences in choice problem (i.e., choice of the “best” alternative), based on the models O and RO in comparison with model R are presented in Table 6.

Table 6
FTOPSIS-FMAA: Monte Carlo simulation, scenario SM1, number of differences for rank 1 (the “best” alternative) for models RO and O in comparison with model R; number of iteration N = 1000

R RO O

Rank 1 7 29

R	RO	O
Rank 1	7	29

Thus, difference in ranks 1 according to ‘proper’ assessments by model R and models with overestimation O and RO, for the scenario under consideration, has the probability (statistically assessed relative frequency) about 0.007 for model RO, and about 0.03 for model O. These distinctions may be considered as insignificant.

It can be stressed here that the scenario for Monte Carlo simulation considered above should be extended for more general and entire exploring. Scenarios with the use of symmetric TrFNs without additional restrictions as well as linguistic variables for criteria values and weight coefficients can result in values, which are different from those in Table 6.

Authors believe, comprehensive exploring FMCDA models with different complexity and evaluating differences of their outputs is a subject for the further specific research.

7 Conclusions

The importance of TOPSIS method and its extension to fuzzy context to deal with uncertainty within MCDA is clear in the related literature, but the inherent difficulty of dealing with fuzzy values for operating and ranking has led to the use of the Fuzzy Multicriteria Acceptability Analysis (FMAA) to develop the FTOPSIS-FMAA method to obtain not only ranking of alternatives but also the confidence about such a rank for each alternative.

In this paper, FTOPSIS-FMAA model has been implemented for analysis of a multi-criteria problem on housing development. Within this analysis, several models for fuzzy extension of TOPSIS with the use of FMAA concept for ranking alternatives are considered. These models are based on SFA (Standard Fuzzy Arithmetic), which results in overestimation of the generalized criterion, and TM (Transformation Method) for proper assessing functions of fuzzy variables in accordance with fuzzy extension principle.

Taking into account that TMs are resource and time consuming, a comparison of the FTOPSIS-FMAA models with different complexity with the use of Monte Carlo simulation has been implemented and discussed. The comparison of the model-O (which is based on implementation of SFA for each function within FTOPSIS), model-R (implementing Reduced TM for determining the differences (48)), and model RO (determining generalized criterion (47) based on RTM, and the use of SFA for assessing differences of generalized criterion (48)) has been implemented and analyzed. Within the scenario considered for comparing ranks of alternatives according to models O, RO, and R, the evaluated distinctions can be considered as insignificant (however, in general case, such a statement can require additional research and discussions). Comprehensive exploring FTOPSIS-FMAA as well as other FMCDA models with different complexity and evaluating differences of their outputs is a subject for the specific research.

Authors believe linking FMAA concept with different FMCDA methods and adapting FMAA to other extensions of fuzzy sets may be considered as fruitful for treatment and analysis of uncertainty in the fuzzy environment. Besides, new extensions of ordinary fuzzy sets such as intuitionistic fuzzy sets [74], hesitant fuzzy sets [75], Pythagorean fuzzy sets [76], and spherical fuzzy sets [77] can be employed for the Fuzzy Multicriteria Acceptability Analysis in TOPSIS method.

Footnotes

Acknowledgments

This work is partially supported by the Spanish National research project TIN2015-66524-P, PGC2018-099402-B-I00 and ERDF, and the Russian National research project RFBR-19-07-01039.

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A general approach to fuzzy TOPSIS based on the concept of fuzzy multicriteria acceptability analysis

Abstract

Keywords

1 Introduction

2 Survey of TOPSIS and FTOPSIS methods

2.1 Classical TOPSIS

3.1 Fuzzy numbers and their ranking

4.1 General FMCDA model and the overestimation problem

Table 6 FTOPSIS-FMAA: Monte Carlo simulation, scenario SM1, number of differences for rank 1 (the “best” alternative) for models RO and O in comparison with model R; number of iteration N = 1000 R RO O Rank 1 7 29

Footnotes

Acknowledgments

References

Table 6
FTOPSIS-FMAA: Monte Carlo simulation, scenario SM1, number of differences for rank 1 (the “best” alternative) for models RO and O in comparison with model R; number of iteration N = 1000

R RO O

Rank 1 7 29