Hybrid fuzzy MCDM model for Z-numbers using intuitive vectorial centroid

Abstract

This paper presents a hybrid fuzzy multi criteria decision making model for z-numbers using intuitive vectorial centroid. There are two novelty aspects discussed: 1) development of intuitive vectorial centroid defuzzification and 2) development of hybrid fuzzy multi criteria decision making model based on consistent fuzzy preference relations and fuzzy technique for order performance by similarity to ideal solution for z-numbers. The latter are taken into consideration due to their capability of capturing human knowledge and extensive use in decision making problems under linguistic uncertainty. Fuzziness is not sufficient enough when dealing with real information and a degree of reliability of the information is very critical. The proposed combination of z-numbers with multi criteria decision making techniques facilitates the use of fuzzy linguistic aspects of human intuition in decision making. The proposed model is applied to a staff recruitment case study.

Keywords

Multi criteria decision making consistent fuzzy preference relations fuzzy TOPSIS z-numbers intuitive vectorial centroid human intuition

1 Introduction

Multiple criteria decision making (MCDM) approach has become a discipline of operational research that has been widely explored by experts and practitioners [1]. It is the process of making decisions in the presence of multiple criteria or objectives. Nowadays, uncertainty is present to a large extent in the world whereby much of the information on which decisions are based is uncertain [2]. In particular, selecting the best alternatives in decision making is subject to uncertainty due to imprecision and subjectivity in the decision makers judgement. However, this uncertainty can be captured by fuzzy knowledge based on linguistic variables due to their ability to represent uncertain information. Also, choices in decision making which are determined by numerous factors are usually dependent on human ability that is difficult to model [3]. In this case, the use of fuzzy logic in MCDM knowledge can address this problem.

In the literature of fuzzy sets, Zadeh [4] introduced fuzzy set theory in representing vagueness or imprecision in a mathematical approach. In order to do so, the main motivation of using fuzzy sets shows its ability in appropriately dealing with imprecise numerical quantities and subjective preferences of decision makers [5]. Zadeh [6] proposed a notion of z-numbers, which is an order pair of fuzzy numbers $(\tilde{A}, \tilde{B})$ . The $\tilde{A}$ component plays the role of a fuzzy restriction and represents the information about an uncertain variable, while the $\tilde{B}$ component is a reliability of $\tilde{A}$ component and enable to represent an idea of certainty or probability [7, 8]. The idea of z-numbers is to provide a basis for computation with numbers which are not completely reliable and more intelligent to describe the knowledge of human beings and capable to cater for uncertain information.

In dealing with fuzzy systems, defuzzification plays a significant role in the performance of fuzzy system modelling [9]. The defuzzification process is based on the output fuzzy subset such whereby a single crisp value is selected as the system output. The centroid defuzzification methods of fuzzy numbers have been explored for the last decade. In most centroid methods fuzzy numbers are normally extracted from geometric aspects whereby various order relationships are constructed from the perspective of membership function to some extent. In this context, fuzzy set theory has been used in every stage of the formal analysis when dealing with vagueness and imprecision in human decision making. In this paper, the intuitive vectorial centroid defuzzification is introduced as an improvement of the classical vectorial centroid [10]. In this sense, the intuitive vectorial centroid defuzzification is relevant in the context of human intuition that is capable of considering representing all possible fuzzy numbers properly. This proposed method is incorporated into the development of an integrated fuzzy MCDM model. The computational process of the intuitive vectorial centroid is illustrated in Section 3.

The latest trend with respect to MCDM modelling is to combine or integrate two or more techniques to make up or handle shortcomings appropriately in any single particular technique [11]. In much of the literature, usually two techniques are combined or integrated in order to tackle the evaluation of criteria and the evaluation of alternatives respectively [12 –15]. The evaluation process of criteria and alternatives play important role in MCDM techniques requirements. To identify the best decision to be made among the various alternatives with several criteria, the methodology has to study the preferences among the criteria to make sure the weights of criteria are reliable enough to be implemented in the selection of alternatives. In this paper, the combination of consistent fuzzy preference relations and fuzzy technique for order of preference by similarity to ideal solution (TOPSIS) using new centroid defuzzification is proposed in dealing with imprecise judgements.

The consistent fuzzy preference relations was proposed by [16] for constructing the decision matrices of pairwise comparisons based on additive transitivity property. In reality, the decision maker is generally unsure of his/her preferences in the partnership selection process because information about the future partners and their performance is incomplete and uncertain. Consistency is crucial for achieving correct solutions in decision process. Due to each positive reciprocal matrix is described by fuzzy numbers in fuzzy linguistic terms, it is very difficult to satisfy the consistency [17]. Besides this, establishing a fuzzy positive reciprocal matrix requires $\frac{n \times (n - 1)}{2}$ judgments to be made for a level with n criteria. Hence, the number of comparisons increase with the number of criteria and inconsistent conditions are likely to occur. To resolve the inconsistency problem, the consistent fuzzy preference relations technique is adopted in order to construct fuzzy decision matrix instead of fuzzy positive reciprocal matrix. The utilisation of consistent fuzzy preference relations in this phase yields decision matrices for making pairwise comparison matrices using additive transitivity. There are only n - 1 comparison judgements required to ensure consistency on a level that contains n criteria.

According to [18], TOPSIS provides a unique way to approach problems due to its intuitiveness and easiness of understanding. In addition, it also represents the rationale of individual choice of a scalar value that records both the best and worst alternatives concurrently using a straightforward computational algorithm. Fuzzy TOPSIS is an extended version classical TOPSIS with considered fuzzy component as an added value in order to deal with human perceptions. The concept of fuzzy TOPSIS is that the most preferred alternative should have the shortest distance from the fuzzy positive ideal solution (FPIS) and longest distance from the fuzzy negative ideal solution (FNIS) [19]. Fuzzy TOPSIS at present offers a solution for decision makers when dealing with real world data that are usually multi criteria and involves a complex decision making process. Regarding the level of interaction of with decision makers to imprecise data collection, fuzzy TOPSIS provides good agility in the decision process.

The MCDM techniques always deal with unbalanced and changeable data inputs. Therefore, sensitivity analysis after problem solving can effectively contribute to making accurate decisions by assuming that a set of weights for criteria or alternatives is obtained in a new round whereby the efficiency of alternatives is equal or their order has changed [20]. Sensitivity analysis is a valuable tool for identifying important model parameters, testing the model conceptualization and improving the model structure [21]. It clearly indicates that the sensitivity analysis calculates the changing in the final score of alternatives when a change is occurred in the weights of alternatives. Sensitivity analysis can be beneficial for a wide range of purposes including [22]: testing the robustness of the output from a model or system under uncertainty; improving the understanding of the relationship between input and output variables in a model or system; reducing uncertainty; easing the calibration stage. In this paper, sensitivity analysis is applied to validate the proposedmodel.

In real world decision making problems, linguistic variables tend to be very complex to handle but they still make more sense than classical fuzzy numbers [8]. Rather than using classical fuzzy numbers, the linguistic scales of the proposed integrated consistent fuzzy preference relations and TOPSIS are expressed in more detailed and flexible way byz-numbers. The membership functions of type-1 and type-2 fuzzy sets do not contain information regarding knowledge of human beings. This issue is the motivation for the proposed hybrid MCDM model that has the capability of handling knowledge of human beings and uncertain information properly using z-numbers. The proposed model is constructed without losing the generality of the consistent fuzzy preference relations and fuzzy TOPSIS for z-numbers (Z-CFPR-TOPSIS). The rest of this paper is organised as follows: Section 2 introduces the concepts of z-numbers and intuitive vectorial centroid defuzzification. Section 3 describes the methodology of intuitive vectorial centroid method for z-numbers and the integration of consistent fuzzy preference relations and fuzzy TOPSIS that incorporates the intuitive vectorial centroid method. Section 4 discusses a case study in MESSRS SAPRUDIN, IDRIS & CO Company to demonstrate the feasibility of the hybrid model. Section 5 summarises the conclusions.

2 Preliminaries

In this section, we briefly review some basic concepts and definitions that are illustrated as follows.

2.1 A. Z-numbers

A z-number is an ordered pair of fuzzy numbers

$\begin{matrix} μ_{\tilde{A}} (x) & = & (a_{1}, a_{2}, a_{3}, a_{4}) \\ = & {\begin{matrix} \begin{matrix} \frac{(x - a_{1})}{(a_{2} - a_{1})} & \begin{matrix} if & a_{1} \leq x \leq a_{2} \end{matrix} \\ 1, & a_{2} \leq x \leq a_{3} \end{matrix} \\ \begin{matrix} \frac{(a_{4} - x)}{(a_{4} - a_{3})} & \begin{matrix} if & a_{3} \leq x \leq a_{4} \end{matrix} \\ 0 & otherwise \end{matrix} \end{matrix} \\ μ_{\tilde{B}} (x) & = & (b_{1}, b_{2}, b_{3}, b_{4}) \\ = & {\begin{matrix} \begin{matrix} \frac{(x - b_{1})}{(b_{2} - b_{1})} & \begin{matrix} if & b_{1} \leq x \leq b_{2} \end{matrix} \\ 1, & b_{2} \leq x \leq b_{3} \end{matrix} \\ \begin{matrix} \frac{(b_{4} - x)}{(b_{4} - b_{3})} & \begin{matrix} if & b_{3} \leq x \leq b_{4} \end{matrix} \\ 0 & otherwise \end{matrix} \end{matrix} \end{matrix}$ (1) denoted as $Z = (\tilde{A}, \tilde{B})$ . First component, $\tilde{A}$ is known as restriction component whereby it is a real-valued uncertain on X while second component, $\tilde{B}$ is a measure of reliability for $\tilde{A}$ [6]. The illustration for z-number is depicted in Fig. 1 [7].

Fig.1

Z-number, $Z = (\tilde{A}, \tilde{B})$ .

3 Proposed methodology

As noted in the introduction, z-numbers are widely applied in many research areas to deal with uncertain information in data analysis which consistent with human intuition. Most of researchers attempt to eliminate the need of human intuition in data analysis processes. Human intuition is strictly can’t be eliminated because it can lead us towards uncertain problems. This section focuses on the development of hybrid MCDM model that incorporated with intuitive vectorial centroid. The proposed methodology consist of two stages as illustrated below.

3.1 A. Stage one

The development of intuitive vectorial centroid defuzzification methodology for z-numbers.

The intuitive vectorial centroid is an extension of the classical vectorial centroid methods for fuzzy numbers that proposed by [10]. The concept is similar like other centroid methods, to find the best centre point of fuzzy number that represent in crisp value or single value. Compare to other centroid methods, the way to get the value is more intelligent manner, easy to compute, more balance, and consider all feasible cases of fuzzy number.

Let $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4}; h_{A})$ as the generalised trapezoidal fuzzy number. The method process for intuitive vectorial centroid is showed as follows:

Step 1. Find the centroids of the three parts of α, β and γ in trapezoid plane representation as shown in Fig. 2. $α (x_{\tilde{A}}, y_{\tilde{A}}) = (a_{1} + [\frac{2}{3} (a_{2} - a_{1})], \frac{h_{\tilde{A}}}{3})$ (2) $β (x_{\tilde{A}}, y_{\tilde{A}}) = (\frac{a_{2} + a_{3}}{2}, \frac{h_{\tilde{A}}}{2})$ (3) $γ (x_{\tilde{A}}, y_{\tilde{A}}) = (a_{4} + [\frac{2}{3} (a_{3} - a_{4})], \frac{h_{\tilde{A}}}{3})$ (4)

Fig.2

Intuitive vectorial centroid representation.

Step 2. Connect all vertices centroids points of α, β and γ each other, where it will create another triangular plane inside of trapezoid plane.

Step 3. The centroid index of intuitive vectorial centroid of $(\tilde{x}, \tilde{y})$ with vertices α, β and γ can be calculated as:

$\begin{matrix} IVC ({\tilde{x}}_{\tilde{A}}, {\tilde{y}}_{\tilde{A}}) \\ = (β ({\tilde{x}}_{\tilde{A}}) + [\frac{2}{3} (\frac{α ({\tilde{x}}_{\tilde{A}}) + γ ({\tilde{x}}_{\tilde{A}})}{2} - β ({\tilde{x}}_{\tilde{A}}))], \\ \dots β ({\tilde{y}}_{\tilde{A}}) + [\frac{2}{3} (\frac{α ({\tilde{y}}_{\tilde{A}}) + γ ({\tilde{y}}_{\tilde{A}})}{2} - β ({\tilde{y}}_{\tilde{A}}))]) \end{matrix}$ (5)

Intuitive vectorial centroid can be summarised as: $IVC ({\tilde{x}}_{\tilde{A}}, {\tilde{y}}_{\tilde{A}}) = (\frac{2 (a_{1} + a_{4}) + 7 (a_{2} + a_{3})}{18}, \frac{7 h_{\tilde{A}}}{18})$ (6) where

α: the centroid coordinate of first triangle plane

β: the centroid coordinate of rectangle plane

γ: the centroid coordinate of second triangle plane

$\tilde{x}$ : the centroid point on the horizontal x-axis

$\tilde{y}$ : the centroid point on the vertical y-axis

$(\tilde{x}, \tilde{y})$ : the centroid coordinate of fuzzy number $\tilde{A}$

Centroid index of intuitive vectorial centroid can be generated using Euclidean Distance by [23] as: $R (\tilde{A}) = \sqrt{{\tilde{x}}^{2} + {\tilde{y}}^{2}}$ (7)

Assume a z-number, $Z = (\tilde{A}, \tilde{B})$ , which is describe in Fig. 1. Let $\tilde{A} = {〈 x, u_{\tilde{A}} (x) 〉 | x \in [0, 1]}$ and $\tilde{B} = {〈 x, u_{\tilde{B}} (x) 〉 | x \in [0, 1]}$ , $u_{\tilde{A}} (x)$ and $u_{\tilde{B}} (x)$ are trapezoidal membership function.

Step 1. Converting the reliability component on x-coordinate into crisp number or weight using intuitive vectorial centroid method from Equation (6). ${IVC}_{\tilde{B}} (\tilde{x}) = \frac{2 (b_{1} + b_{4}) + 7 (b_{2} + b_{3})}{18} = α$

Step 2. Add the weight of reliability component to the restriction component. The weighted z-number can be denoted as ${\tilde{Z}}^{α} = {〈 x, μ_{{\tilde{B}}^{α}} (x) 〉 | μ_{{\tilde{B}}^{α}} (x) = α μ_{{\tilde{B}}^{α}} (x), x \in [0, 1]}$

Theorem 1. $E_{{\tilde{A}}^{α}} (x) = α E_{\tilde{A}} (x), x \in X$ (8) Subject to: $μ_{{\tilde{A}}^{α}} (x) = α μ_{{\tilde{A}}^{α}} (x), x \in X$ (9)

Proof.

$\begin{matrix} E_{{\tilde{A}}^{α}} (x) & = & [a_{1}, a_{2}, a_{3}, a_{4}; \frac{2 (b_{1} + b_{4}) + 7 (b_{2} + b_{3})}{18}] \\ = & [a_{1}, a_{2}, a_{3}, a_{4}; α] = α E_{\tilde{A}} (x) \end{matrix}$ (10) which can be denoted by the Fig. 3 [7].

Fig.3

Z-number after multiplying the reliability.

Step 3. Convert the irregular fuzzy number (weighted restriction) to regular fuzzy number that denoted as ${\tilde{Z}}^{'} = {〈 x, μ_{{\tilde{Z}}^{'}} (x) 〉 | μ_{{\tilde{Z}}^{'}} (x) = μ_{\tilde{A}} (\sqrt{α} x), x \in [0, 1]}$ . In accordance with the Theorem 3, the conclusion can be made that ${\tilde{Z}}^{'}$ has the same fuzzy expectation with ${\tilde{Z}}^{α}$ where both are equal with fuzzy expectation.

Theorem 2. $E_{{\tilde{Z}}^{'}} (x) = α E_{\tilde{A}} (x), x \in \sqrt{α} X$ (11) Subject to: $μ_{{\tilde{Z}}^{'}} (x) = μ_{\tilde{A}} (\sqrt{α} x), x \in \sqrt{α} X$ (12)

Proof.

$\begin{matrix} E_{{\tilde{Z}}^{'}} (x) \\ = [a_{1}, a_{2}, a_{3}, a_{4}; \sqrt{\frac{2 (b_{1} + b_{4}) + 7 (b_{2} + b_{3})}{18}}] \\ = [a_{1}, a_{2}, a_{3}, a_{4}; \sqrt{α}] = \sqrt{α} E_{\tilde{A}} (x) \end{matrix}$ (13) which can be denoted by the Fig. 4 below [7]:

Fig.4

The regular fuzzy number transformed from z-number.

Theorem 3. $E_{{\tilde{Z}}^{'}} (x) = E_{{\tilde{A}}^{α}} (x)$ (14)

Proof. $E_{{\tilde{A}}^{α}} (x) = α E_{\tilde{A}} (x)$ (15) $E_{{\tilde{Z}}^{'}} (x) = α E_{\tilde{A}} (x)$ (16) $E_{{\tilde{Z}}^{'}} (x) = E_{{\tilde{A}}^{α}} (x)$ (17)

3.2 B. Stage two

The development of hybrid fuzzy consistent fuzzy preference relations – fuzzy TOPSIS using the intuitive vectorial centroid.

The methodology of proposed hybrid fuzzy MCDM model consist of four phases as illustrated below in Fig. 5.

Fig.5

Hybrid consistent fuzzy preference relations – fuzzy TOPSIS framework.

Phase 1: Linguistic evaluation

The fuzzy linguistic terms are used to present the important of criteria preferences based on z-numbers. These preferences are computed using consistent fuzzy preferences. For fuzzy TOPSIS evaluation, another fuzzy linguistic terms are used to represent the evaluating values of the alternatives with respect to difference criteria with degree of confidence respectively.

Phase 2: Fuzzy weights evaluation of criteria using consistent fuzzy preference relations

Step 1. Construct a hierarchy structure.

The construction of hierarchy model needs judgement matrix filled by decision makers about the evaluation of all criteria.

Step 2. Construct a pair-wise comparisonmatrices

Consistent fuzzy preference relations is adopted to evaluate the weights of difference criteria for the performance of alternatives. The pairwise comparison matrices are constructed among all criteria in the dimension of the hierarchy systems based on the decision makers’ preferences in phase 1 as following matrix A:

$\begin{matrix} A & = & [\begin{matrix} 1 & {\tilde{a}}_{12} & \dots & {\tilde{a}}_{1 n} \\ {\tilde{a}}_{21} & 1 & \dots & {\tilde{a}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{a}}_{n 1} & {\tilde{a}}_{n 2} & \dots & 1 \end{matrix}] \\ = & [\begin{matrix} 1 & {\tilde{a}}_{12} & \dots & {\tilde{a}}_{1 n} \\ 1 / {\tilde{a}}_{12} & 1 & \dots & {\tilde{a}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 1 / {\tilde{a}}_{1 n} & 1 / {\tilde{a}}_{2 n} & \dots & 1 \end{matrix}] \end{matrix}$ (18)

Step 3. Convert decision makers’ preferences from z-numbers into regular fuzzy numbers.

Conversion process is computed by using Equation (13). $\begin{matrix} E_{\tilde{Z}} (x) & = & [a_{1}, a_{2}, a_{3}, a_{4}; \sqrt{\frac{2 (b_{1} + b_{4}) + 7 (b_{2} + b_{3})}{18}}] \\ = & [a_{1}, a_{2}, a_{3}, a_{4}; \sqrt{α}] = \sqrt{α} E_{\tilde{A}} (x) \end{matrix}$

Step 4. Aggregate the decision makers’ preferences.

The pairwise comparison matrices of decision makers’ preferences are aggregated using equation below: ${\tilde{a}}_{ij} = ({\tilde{a}}_{ij}^{1} \times {\tilde{a}}_{ij}^{2} \times \dots \times {\tilde{a}}_{ij}^{n})^{1 / k}$ (19) where k is the number of decision makers and i = 1, 2, … m; j = 1, 2, … n.

Step 5. Defuzzify the fuzzy numbers of aggregation’s result of decision makers’ preferences.

Intuitive vectorial centroid for z-numbers is used for conversion process using equation below:

$\begin{matrix} IVC ({\tilde{x}}_{\tilde{A}, \tilde{R}}, {\tilde{y}}_{\tilde{A}, \tilde{R}}) \\ = (\frac{2 (\sqrt{φ} a_{1} + \sqrt{φ} a_{4}) + 7 (\sqrt{φ} a_{2} + \sqrt{φ} a_{3})}{18}, \frac{7 h_{\tilde{A}, \tilde{R}}}{18}) \end{matrix}$ (20)

Step 6. Compute the criteria values as weightage for alternatives’ evaluation using consistent fuzzy preference relations.

In order to avoid misleading solutions in expressing the decision makers’ opinions, the study of consistency by means of preference relations becomes a very important aspect. In decision making problems based on fuzzy preference relations, the study of consistency is associated with the study of transitivity properties. In this study, the new characterisation of consistency property defined by the additive transitivity property of fuzzy preference relation is developed.

Consistent fuzzy preference relations was proposed by [16] for constructing the decision matrices of pairwise comparisons based on additive transitivity property. Referring to [24], a fuzzy preference relation R on the set of the criteria or alternatives A is a fuzzy set stated on the Cartesian product set A × A with the membership function μ_R : A × A → [0, 1]. The preference relation is denoted by n × n matrix R = (r_ij) where r_ij = μ_R (a_i, a_j) ∀i, j ∈ {1, …, n}. The preference ratio, r_ij of the alternative a_i to a_j is determined by: $r_{ij} = {\begin{matrix} 0.5 & a_{i} is different to a_{j} \\ (0.5, 1) & a_{i} is preferred than a_{j} \\ 1 & a_{i} is absolutely preferred than a_{j} \end{matrix}$

The preference matrix R is presumed to be additive reciprocal, p_ij + p_ji = 1, ∀ i, j ∈ {1, …, n}. Several propositions are associated to the consistent additive preference relations as follows:

Proposition 1. [25] Consider a set of criteria or alternatives, X = {x₁, …, x_n}, and associated with a reciprocal multiplicative preference relation A = (a_ij) for $a_{ij} \in [\frac{1}{9}, 9]$ . Then, the corresponding reciprocal fuzzy preference relation, P = (p_ij) with p_ij ∈ [0, 1] associated with A is given by the following formulation: $p_{ij} = g (a_{ij}) = \frac{1}{2} (1 + {log}_{9} a_{ij})$ (21)

Generally, if $a_{ij} \in [\frac{1}{n}, n]$ , then log _na_ij is used, in particular, when $a_{ij} \in [\frac{1}{9}, 9]$ ; log ₉a_ij is considered as in the above proposition because a_ij is between $\frac{1}{9}$ and 9. If a_ij is between $\frac{1}{7}$ and 7, then log ₇a_ij is used.

Proposition 2. [25] For a reciprocal fuzzy preference relation P = (p_ij), the following statements are equivalent $(1) p_{ij} + p_{jk} + p_{ki} = \frac{3}{2}, \forall i, j, k$ (22) $(2) p_{ij} + p_{jk} + p_{ki} = \frac{3}{2}, \forall i < j < k$ (23)

Proposition 3. [25] For a reciprocal fuzzy preference relation P = (p_ij), the following statements are equivalent:

$\begin{matrix} (1) p_{ij} + p_{jk} + p_{ki} = \frac{3}{2}, \forall i < j < k \\ (2) p_{i (i + 1)} + p_{(i + 1) (i + 2)} + \dots + p_{(j - 1) j} + p_{ji} \\ = \frac{j - i + 1}{2}, \forall i < j \end{matrix}$ (24)

Proposition 3 is crucial because it can be used to construct a consistent fuzzy preference relations form the set of n - 1 values {p₁₂, p₂₃, …, p_n-1}. A decision matrix with entries that are not in the interval, [0, 1], but in an interval [- c, 1 + c], c > 0, can be obtained by transforming the obtained values using a transformation function that preserves reciprocity and additive consistency with the function: $f : [- c, 1 + c] \to [0, 1], f (x) = \frac{(x + c)}{(1 + 2 c)}$ (25)

Phase 3: Fuzzy TOPSIS evaluation for alternatives selection

Step 1. Determine the weights of evaluation criteria.

The weighting of evaluation criteria are employed from consistent fuzzy preference relations process before.

Step 2. Construct the fuzzy decision matrix for alternatives’ evaluation using fuzzy TOPSIS.

The proposed methodology for fuzzy TOPSIS is illustrated differ from others in terms of the usage of defuzzification method, normalization process and ranking process.

The fuzzy decision matrix is constructed and the linguistic terms from Table 2 is used to evaluate the alternatives with respect to criteria. Then, aggregate DMs’ preferences:

Table 1

Trapezoidal fuzzy numbers preference scale [30]

Linguistic variables	Scale of relative important	Trapezoidal fuzzy	Reciprocal trapezoidal
	of crisp numbers	numbers	fuzzy number
Equally important (EI)	1	(1, 1, 1, 1; 1)	(1, 1, 1, 1; 1)
Intermediate value (IV)	2	(1, 3/2, 5/2, 3; 1)	(1/3, 2/5, 2/3, 1; 1)
Moderately more important (MMI)	3	(2, 5/2, 7/2, 4; 1)	(1/4, 2/9, 2/5, 1/2; 1)
Intermediate value (IV)	4	(3, 7/2, 9/2, 5; 1)	(1/5, 2/9, 2/7, 1/3; 1)
Strongly more important (SMI)	5	(4, 9/2, 11/2, 6; 1)	(1/6, 2/11, 2/9, 1/4; 1)
Intermediate value (IV)	6	(5, 11/2, 13/2, 7; 1)	(1/7, 2/13, 2/11, 1/5; 1)
Very strong more important (VSMI)	7	(6, 13/2, 15/2, 8; 1)	(1/8, 2/15, 2/13, 1/6; 1)
Intermediate value (IV)	8	(7, 15/2, 17/2, 9; 1)	(1/9, 2/17, 2/15, 1/7; 1)
Extremely more important (EMI)	9	(8, 17/2, 9, 9; 1)	(1/9, 1/9, 2/17, 1/8; 1)

Table 2

Linguistic terms and their corresponding generalised fuzzy numbers [30]

Linguistic terms	Generalised fuzzy numbers
Absolutely-low (AL)	(0.0, 0.0, 0.0, 0.0; 1)
Very-low (VL)	(0.0,0.0, 0.02, 0.07;1)
Low (L)	(0.04, 0.10, 0.18, 0.23; 1)
Fairly-low (FL)	(0.17, 0.22, 0.36, 0.42; 1)
Medium (M)	(0.32, 0.41, 0.58, 0.6; 1)
Fairly-high (FH)	(0.58, 0.63, 0.80, 0.86; 1)
High (H)	(0.72, 0.78, 0.92, 0.97; 1)
Very-high (VH)	(0.93, 0.98, 1.0, 1.0; 1)
Absolutely-high (AH)	(1.0, 1.0, 1.0, 1.0; 1)

$\bar{D M} = \overset{}{\begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \end{matrix} \overset{\begin{matrix} C_{1} & C_{2} & \dots & C_{n} \end{matrix}}{[\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} & \dots & {\tilde{x}}_{1 n} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} & \dots & {\tilde{a}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{x}}_{m 1} & {\tilde{x}}_{m 1} & \dots & {\tilde{x}}_{m n} \end{matrix}]}}$ (26)i = 1, 2, … , m ; j = 1, 2, …, n and ${\tilde{x}}_{ij} = \frac{1}{K} ({\tilde{x}}_{ij}^{1} \oplus \dots \oplus {\tilde{x}}_{ij}^{k} \oplus \dots {\tilde{x}}_{ij}^{K})$ .

where ${\tilde{x}}_{ij}$ is the performance rating of alternatives, A_i with respect to criterion C_j evaluated by kth experts and ${\tilde{x}}_{ij} = (a_{1}^{k}, a_{2}^{k}, a_{3}^{k}, a_{4}^{k}; h^{k})$ .

Step 3. Fuzzy decision matrix is weighted and normalised. Then, defuzzify the standardised generalised fuzzy numbers into coordinate form $(\tilde{x}, \tilde{y})$ . The weighted fuzzy normalised decision matrix is denoted by $\tilde{V}$ as depicted below: $\tilde{V} = {[{\tilde{v}}_{ij}]}_{m \otimes n}; i = 1, 2, \dots, m; j = 1, 2, \dots, n$ (27) where ${\tilde{v}}_{ij} = {\tilde{x}}_{ij} \times {\tilde{w}}_{j}$ (28)

Normalised each generalised trapezoidal fuzzy numbers into standardised generalised fuzzy numbers using [26]:

$\begin{array}{l} a_{' 1}^{'} = \frac{a_{1} - \min (a_{1}, b_{1})}{\max (a_{4}, b_{4}) - \min (a_{1}, b_{1})}, \\ a_{' 2}^{'} = \frac{a_{2} - \min (a_{1}, b_{1})}{\max (a_{4}, b_{4}) - \min (a_{1}, b_{1})}, \\ a_{' 3}^{'} = \frac{a_{3} - \min (a_{1}, b_{1})}{\max (a_{4}, b_{4}) - \min (a_{1}, b_{1})}, \\ a_{' 4}^{'} = \frac{a_{4} - \min (a_{1}, b_{1})}{\max (a_{4}, b_{4}) - \min (a_{1}, b_{1})}, \end{array}$ (29)

The weights from consistent fuzzy preference relations are adopted here. Defuzzify the standardised generalised fuzzy numbers using intuitive vectorial centroid, then translate them into the index point proposed by [27] as shown as follows: $x_{{\tilde{A}}_{i}^{*}} = x_{{\tilde{A}}_{i}^{*}}$ (30) $y_{{\tilde{A}}_{i}^{S}} = 0.5 \times h_{{\tilde{A}}_{i}} - y_{{\tilde{A}}_{i}^{*}} \times {STD}_{{\tilde{A}}_{i}}$ (31) $where spread, {STD}_{{\tilde{A}}_{i}} = \sqrt{\frac{\sum_{j = 1}^{4} (a_{ij}^{*} - x_{{\tilde{A}}_{i}^{*}})}{n - 1}}$ (32)

Use the new point of $y_{{\tilde{A}}_{i}^{*}}$ to compute fuzzy positive-ideal solution and fuzzy negative-ideal solution.

Step 4. Determine the fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS).

Referring to normalise trapezoidal fuzzy weights, the FPIS, A⁺ represents the compromise solution while FNIS, A^- represents the worst possible solution. The range belong to the closed interval [0,1]. The FPIS A⁺ (aspiration levels) and FNIS A^- (worst levels) as follows. $A^{+} = [1, 1, 1, 1; 1] A^{-} = [- 1, - 1, - 1, - 1; 1]$

The FPIS, A⁺ and FNIS, A^- can be obtained by centroid method for (x_{A
⁺}, y_{A
⁺}) and (x_{A
^-}, y_{A
^-}).

Step 5. Calculate the distance of each alternative from FPIS and FNIS.

The distance ${\tilde{d}}_{i}^{+}$ and ${\tilde{d}}_{i}^{-}$ of each alternative from formulation A⁺ and A^- can be calculated by the area of compensation method: ${\bar{d}}_{i}^{+} ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{+}) = \sqrt{(x_{{\tilde{A}}_{i}^{*}} - x_{A^{+}})^{2} + (y_{{\tilde{A}}_{i}^{*}} - y_{A^{+}})^{2}}$ (33) ${\bar{d}}_{i}^{-} ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{-}) = \sqrt{(x_{{\tilde{A}}_{i}^{*}} - x_{A^{-}})^{2} + (y_{{\tilde{A}}_{i}^{*}} - y_{A^{-}})^{2}}$ (34)

Step 6. Find the closeness coefficient, CC_i and improve alternatives for achieving aspiration levels in each criteria. The decision rules for five classes are depicted in Table 1. Notice that the highest CC_i value is used to determine the rank. ${\bar{CC}}_{i} = \frac{{\bar{d}}_{i}^{-}}{{\bar{d}}_{i}^{+} + {\bar{d}}_{i}^{-}} = 1 - \frac{{\bar{d}}_{i}^{+}}{{\bar{d}}_{i}^{+} + {\bar{d}}_{i}^{-}}$ (35) where, $\frac{{\bar{d}}_{i}^{-}}{{\bar{d}}_{i}^{+} + {\bar{d}}_{i}^{-}}$ is satisfaction degree in ith alternative and $\frac{{\bar{d}}_{i}^{+}}{{\bar{d}}_{i}^{+} + {\bar{d}}_{i}^{-}}$ is fuzzy gaps degree in ith alternative.

Fuzzy gap should be improvised for reaching aspiration levels and get the best mutually beneficial strategy from among a fuzzy set of feasible alternatives.

Phase 4: Validation process using sensitivity analysis

Sensitivity analysis can effectively contributes in making accurate decisions by assuming that a set of weights for criteria or alternatives then obtained a new round of weights for them, so that the efficiency of alternatives has become equal or their order has changed. The results of MCDM models are importantly needed to validate using sensitivity analysis method to analyse the quality and how robustness of MCDM model to reach a right decision under different conditions. In this paper, sensitivity analysis technique by [20] is utilised.

4 Case study

A legal company in Malaysia, MESSRS SAPRUDIN, IDRIS & CO plans to recruit new staff from the several applicants/ candidate in some aspects with the lowest of him/ her to resign. There are three decision makers (DMs) DM1, DM2, and DM3 of a firm to evaluate the candidates and four candidates or alternatives x1, x2, x3 and x4. Several criteria are considered to evaluate the candidates which are: Emotional steadiness (ES), Oration (O), Personality (P), Past experience (PE) and, Self-confidence (S-C). These criteria are used based on [28]. This study simplify the concept of attributes to $μ_{\tilde{A}} \in [0, 1]$ for fuzzy events. The values of attributes correspond to z-numbers. The proposed model (Z-CFPR-TOPSIS) is compared with Z-AHP [8] and Z-TOPSIS [29] from the literature for comparative study.

Phase 1: Linguistic evaluation

The decision makers use the linguistic terms as depicted in Table 1 to present the weights using consistent fuzzy preference relations evaluation. The linguistic terms in Table 1 present the important of criteria preferences. In Table 2, the decision makers (DMs) use the linguistic terms for fuzzy TOPSIS evaluation to represent the evaluating values of the alternatives with respect to difference criteria with degree of confidence (reliability) respectively as shown in Table 3.

Table 3
Reliability linguistic terms and their corresponding z-numbers [31]

Linguistic terms Generalised fuzzy numbers

Very-low (VL) (0, 0, 0, 0.25; 1)

Low (L) (0, 25, 0.25, 0.5; 1)

Medium (M) (0.25, 0.5, 0.5, 0.75; 1)

High (H) (0.5, 0.75, 0.75, 1; 1)

Very-high (VH) (0.75, 1, 1, 1; 1)

Linguistic terms	Generalised fuzzy numbers
Very-low (VL)	(0, 0, 0, 0.25; 1)
Low (L)	(0, 25, 0.25, 0.5; 1)
Medium (M)	(0.25, 0.5, 0.5, 0.75; 1)
High (H)	(0.5, 0.75, 0.75, 1; 1)
Very-high (VH)	(0.75, 1, 1, 1; 1)

Phase 2: Fuzzy weights evaluation using consistent fuzzy preference relations

Step 1. Construct a hierarchy structure.

The construction of hierarchy model needs judgement matrix filled by DMs about the evaluation of all criteria (Fig. 6).

Step 2. Construct a pair-wise comparison matrices.

Fig.6

The hierarchy of staff recruitment problem.

Consistent fuzzy preference relations is adopted to evaluate the weights of difference criteria for the performance of alternatives. The pair-wise comparison matrices are constructed among all criteria in the dimension of the hierarchy systems based on the DMs’ preferences in phase 1 using Equation (18) that are depicted in Figs. 7–9.

Fig.7

Pairwise comparison matrix of criteria with reliability component from DM1.

Fig.8

Pairwise comparison matrix of criteria with reliability component from DM2.

Fig.9

Pairwise comparison matrix of criteria with reliability component from DM3.

Step 3. Convert decision makers’ preferences from z-numbers into regular fuzzy numbers.

Convert the z-numbers into regular numbers using Equation (13). Defuzzify the reliability component using intuitive vectorial centroid, Equation (6) for x-axis. For the evaluation of criteria by DMs for this stage, the degree of confidence of the DMs’ opinions are agreed as highest degree which is 1. The fuzzy decision matrices of DMs’ preferences are aggregated as shown in example as follows:

Decision maker 1 (DM1)/ES×O: $\begin{matrix} \tilde{A} & = & (1 / 7, 2 / 13, 2 / 11, 1 / 5; 1) \\ \tilde{B} & = & (0.75, 1, 1, 1; 1) \end{matrix}$

The DM’s knowledge can be expressed to z-number as: $Z = [(1 / 7, 2 / 13, 2 / 11, 1 / 5; 1) (0.75, 1, 1, 1; 1)]$

At first, the reliability component should be converted into crisp using Equation (7) for x-axis. $\begin{matrix} α & = & \frac{2 (0.75 + 1) + 7 (1 + 1)}{18} = 0.9722 \\ Z^{α} & = & [(1 / 7, 2 / 13, 2 / 11, 1 / 5; 0.9722)] \end{matrix}$

Add the weight of the reliability to the constraint. Convert the weighted z-number to regular fuzzy number. $\begin{matrix} {\tilde{Z}}^{'} & = & (\sqrt{0.9722} \times 1 / 7, \sqrt{0.9722} \times 2 / 13.5, \\ \sqrt{0.9722} \times 2 / 11, \sqrt{0.9722} \times 1 / 5; 1) \\ {\tilde{Z}}^{'} & = & (0.1409, 0.1517, 0.1793, 0.1972; 1) \end{matrix}$

Step 4. Aggregate the decision makers’ preferences.

The aggregated weighted pairwise comparison matrix for each criterion is calculated using Equation (19) as follows: $\begin{matrix} {\tilde{a}}_{ES \times O} & = & ((0.1409 \times 0.1643 \times 0.1643)^{1 / 3}, \\ (0.1517 \times 0.1793 \times 0.1793)^{1 / 3}, \\ \dots (0.1793 \times 0.2191 \times 0.2191)^{1 / 3}, \\ (0.1972 \times 0.2465 \times 0.2465)^{1 / 3}; 1) \\ {\tilde{a}}_{ES \times O} & = & (0.1561, 0.1696, 0.2049, 0.2288; 1) \end{matrix}$

Step 5. Defuzzify the fuzzy numbers of aggregation’s result of decision makers’ preferences.

The fuzzy decision matrices of DM1s’ preferences for all preference. ES × O are defuzzified using Equation (6) for x-axis only as follows:

Decision maker 1 (DM1)/ES×O: $\begin{matrix} {Defuzzify}_{DM 1 / ES \times O} \\ = (0.1561, 0.1696, 0.2049, 0.2288; 1) \\ {Defuzziy}_{DM 1 / ES \times O} \\ = \frac{2 (0.1561 + 0.2288) + 7 (0.1696 + 0.2049)}{18} \\ {Defuzzify}_{DM 1 / ES \times O} = 0.1884 \end{matrix}$

Do the same computation for DMs’ judgement for all criteria. Figure 10 shows the defuzzification results of aggregated matrix comparison.

Fig.10

Defuzzification results of aggregated matrix comparison.

Step 6. Compute the criteria values as weights for alternatives’ evaluation using consistent fuzzy preference relations.

The aggregated matrix comparison of each criterion is calculated in Fig. 11.

Fig.11

The consistent fuzzy preference relations matrix for criteria.

By having five criteria, n = 5 so only (n - 1) =5 - 1 =4 entry values (p₁₂, p₂₃, p₃₄ and p₄₅) are required in constructing the consistent fuzzy preference relations matrix from Fig. 10 where: $\begin{matrix} p_{12} & = & \frac{1}{2} (1 + {log}_{9} 0.1884) = 0.1202 \\ p_{23} & = & \frac{1}{2} (1 + {log}_{9} 4.8622) = 0.8599 \\ p_{34} & = & \frac{1}{2} (1 + {log}_{9} 0.1770) = 0.1059 \\ p_{45} & = & \frac{1}{2} (1 + {log}_{9} 1.7033) = 0.6212 \end{matrix}$

The remains of the entries can be determined by utilizing Proposition 2 and 3 by presented as follows: $\begin{matrix} p_{21} & = & 1 - p_{12} = 1 - 0.1202 = 0.8798 \\ p_{31} & = & \frac{3}{2} - p_{12} - p_{23} \\ = & \frac{3}{2} - 0.1202 - 0.8599 = 0.5199 \\ p_{51} & = & \frac{5 - 1 + 1}{2} - p_{12} - p_{23} - p_{34} - p_{45} \\ = & \frac{5}{2} - 0.1202 - 0.8599 - 0.1059 - 0.6212 \\ = & 0.7928 \end{matrix}$

Notes: Some of remains entries are not shown for calculation.

Then, the average and weights of each criterion are illustrated in Fig. 12. Referring to Fig. 12, ‘past experience’ criterion has highest weights value with 0.2771 (27.71%). Followed by ‘oration’ 0.2634 (26.34%), ‘self-confidence’ 0.2286 (22.86%), ‘personality’ 0.1195 (11.95%) and ‘emotional steadiness’ with 0.1115 (11.15%). Which mean, based on decision makers evaluations, ‘past experience, ‘oration’ and ‘self-confidence’ criteria play important aspects in recruiting new staff.

Fig.12

The average and weights for criteria.

Referring to Table 4, the comparison weights of criteria of established and proposed models (Z-CFPR-TOPSIS) are presented. Z-AHP [8] and Z-TOPSIS [29] give same ranking results for criteria with O>PE>S-C>ES>P, but different with proposed model which the ranking results of criteria is PE>O>S-C>P>ES. Both Z-AHP [8] and Z-TOPSIS [29] evaluate criteria simply by getting the aggregation results from several decision matrices The authors prefer to utilise consistent fuzzy preference relations technique order to avoid misleading solution in expressing the decision makers’ opinions by means of preference relations.

Table 4

Comparison of weights of criteria

Z-numbers MCDM Model	Criteria weight values					Ranking Results
	(ES)	(O)	(P)	(PE)	(S-C)
Z-AHP [8]	0.0887	0.3661	0.045	0.3405	0.1602	O>PE>S-C>ES>P
Z-TOPSIS [29]	0.0631	0.4044	0.009	0.3736	0.1499	O>PE>S-C>ES>P
Z-CFPR-TOPSIS (Proposed)	0.1115	0.2634	0.1195	0.2771	0.2286	PE>O>S-C>P>ES

Phase 3: Ranking evaluation of alternatives using fuzzy TOPSIS

The steps of fuzzy TOPSIS are illustrated as follows [12]:

Step 1. Determine the weights of evaluation criteria.

The weights of evaluation criteria are employed from consistent fuzzy preference relations process before.

Step 2. Construct the fuzzy decision matrix, covert z-numbers into regular numbers and aggregate them.

The fuzzy decision matrix is constructed and the linguistic terms from Tables 2 and 3 (reliability) are used to evaluate the alternatives with respect to criteria. The alternatives’ evaluations are presented in Table 5. Then, convert z-numbers into regular numbers and aggregate them.

Table 5

Evaluating linguistic terms of the alternatives with reliability components given by the decision makers with respect to different criteria

Criteria	Alternatives/Candidates	Decision Marker
		DM1	DM2	DM3
Emotional Steadiness	x1	FH (VH)	H (VH)	VH (H)
	x2	H (H)	H (VH)	FH (VH)
	x3	VH (VH)	H (VH)	VH (VH)
	x4	M (VH)	FH (VH)	M (VH)
Oral	x1	VH (VH)	H (VH)	VH (VH)
	x2	H (VH)	H (H)	VH (VH)
	x3	VH (VH)	VH (VH)	H (VH)
	x4	FH (H)	M (VH)	FH (VH)
Personality	x1	VH (VH)	VH (VH)	VH (VH)
	x2	H (VH)	H (VH)	VH (VH)
	x3	VH (VH)	VH (VH)	VH (VH)
	x4	H (VH)	H (VH)	H (VH)
Past Experience	x1	FL (VH)	L (VH)	FL (VH)
	x2	M (VH)	M (H)	M (VH)
	x3	H (H)	M (VH)	H (VH)
	x4	FL (VH)	FL (VH)	FL (VH)
Self-Confidence	x1	H (VH)	FH (VH)	FH (VH)
	x2	VH (H)	H (VH)	H (H)
	x3	VH (VH)	VH (VH)	VH (VH)
	x4	M (VH)	FH (VH)	FH (VH)

Step 3. Fuzzy decision matrix is weighted and normalised. Then, defuzzify the standardised generalised fuzzy numbers into coordinate form, $(\tilde{x}, \tilde{y})$ .

The weights from consistent fuzzy preference relations are adopted here. The weighted fuzzy decision matrix is denoted by $\tilde{V}$ as depicted in Fig. 13 as computed below: $\begin{matrix} {\tilde{v}}_{Aggregated, x 1, ES} \\ = (0.6889, 0.7401, 0.8526, 0.8889; 1) \times 0.1115 \\ {\tilde{v}}_{Aggregated, x 1, ES} \\ = (0.0768, 0.0825, 0.095, 0.0099; 1) \end{matrix}$

Fig.13

The weighted fuzzy pairwise comparison matrix for alternatives evaluation.

Convert each generalised trapezoidal fuzzy numbers into standardised generalised fuzzy numbers using normalisation process by [26] from Equation (29) as presented in Fig. 14. Defuzzify the standardised generalised trapezoidal fuzzy numbers using intuitive vectorial centroid method, then translate them into the index point proposed by [27], as depicted in Figs. 15 and 16. Use the new point of $y_{{\tilde{A}}_{i}^{*}}$ to compute fuzzy positive-ideal solution and fuzzy negative ideal solution. Get the average for all of alternatives evaluation with respect to criteria.

Fig.14

The normalised fuzzy pairwise comparison matrix for alternatives evaluation.

Fig.15

The defuzzified pairwise comparison matrix for alternatives evaluation.

Fig.16

The average translated defuzzified pairwise comparison matrix for alternatives evaluation.

Step 4. Determine the fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS).

The FPIS A⁺ (aspiration levels) and FNIS A^- (worst levels) as following below. $A^{+} = [1, 1, 1, 1; 1] A^{-} = [- 1, - 1, - 1, - 1; 1]$

The FPIS, A⁺ and FNIS, A^- can be obtained by centroid method for (x_{A
⁺}, y_{A
⁺}) and (x_{A
^-}, y_{A
^-}).

Step 5. Calculate the distance of each alternative from FPIS and FNIS.

The distance ${\tilde{d}}_{i}^{+}$ and ${\tilde{d}}_{i}^{-}$ of each alternative from formulation A⁺ and A^- can be calculated by the area of compensation method from Equations (33) and (34). The numerical calculation is shown as follows: $\begin{matrix} {\bar{d}}_{candidate 1}^{+} \\ = \sqrt{(0.4703 - 1)^{2} + (0.4695 - 0.5)^{2}} \\ {\bar{d}}_{candidate 1}^{+} = 0.5306 \\ {\bar{d}}_{candidate 1}^{-} \\ = \sqrt{(0.4603 - (- 1))^{2} + (0.4695 - 0.5)^{2}} \\ {\bar{d}}_{candidae 1}^{-} = 1.4706 \end{matrix}$

Step 6. Find the closeness coefficient, CC_i and improve alternatives for achieving aspiration levels in each criteria.

The closeness coefficient formulation can be obtained from Equation (35). Notice that the highest CCi value is used to determine the rank. The numerical calculation is shown as: ${\bar{CC}}_{candidate 1} = \frac{1.4706}{1.4706 + 0.5306}$ (36) ${\bar{CC}}_{candidate 1} = 0.7348$ (37)

Table 6 presents the results of closeness coefficient, CCi values present that the candidate 3 achieves the highest rank with 0.8173 followed by candidate 2 with 0.7538, candidate 1 with 0.7348 and candidate 4 with 0.6785 for the last ranked. The results reveal that the candidate 3 is most suitable for this recruitment because has highest CCi value.

Table 6

Closeness coefficients computation

Alternative	Closeness Coefficient, CC_i
Candidate 1	0.7348 (Rank 3)
Candidate 2	0.7538 (Rank 2)
Candidate 3	0.8173 (Rank 1)
Candidate 4	0.6785 (Rank 4)

Table 7 depicts the ranking results of all the established and proposed models for alternatives. All models present same ranking for alternatives/ candidates with Alt3>Alt2>Alt1>Alt4. This is showed that the proposed model is consistent with established models for z-numbers in terms of ranking results.

Table 7

Ranking results of alternatives for hybrid fuzzy MCDM models

Z-numbers MCDM Model	Alternative ranking values				Ranking Results
	(Alt1)	(Alt2)	(Alt3)	(Alt4)
Z-AHP [8]	0.2321	0.2584	0.3016	0.2079	Alt3>Alt2>Alt1>Alt4
Z-TOPSIS [29]	0.5503	0.5562	0.5678	0.5404	Alt3>Alt2>Alt1>Alt4
Z-CFPR-TOPSIS (Proposed)	0.7348	0.7538	0.8173	0.6785	Alt3>Alt2>Alt1>Alt4

Phase 3: Ranking evaluation of alternatives using fuzzy TOPSIS

In sensitivity analysis evaluation, the focus is to test the effect of the criteria weights on the ranking of the results. The tests are process by increasing each original criteria weight by 50%, 100% and 150%. While one criterion is increased, the values of the remaining criteria are decreased by certain amount, such that the total amount of criteria are equal to one. Referring to Table 8, the proposed Z-CFPR-TOPSIS is quite robust and stable, since changes in the criteria weights significantly affect for several cases in the final ranking order of the alternatives candidates. As related before, the consistency of correct ranking order based on original rank presents 86.67% level of consistency. Even the ranking values are changed, but the ranking order are significantly consistent with the original ranking. However, when criterion ‘Oration’ are increased by 100% and 150%, the ranking order are changed to Alt3>Alt1>Alt2>Alt4 both of them. In the context of sensitivity analysis evaluation, it presents that the proposed hybrid fuzzy MCDM model for z-numbers is consistent even the weights of criteria are changed.

Table 8

Sensitivity analysis results of proposed hybrid fuzzy MCDM model for z-numbers

Changes of	Alt1	Alt2	Alt3	Alt4	Ranking results	Consistency based on
criteria (%)						original result
ES’ (50%)	0.7580	0.7759	0.8442	0.6945	Alt3>Alt2>Alt1>Alt4	Consistent
ES’ (100%)	0.7845	0.8010	0.8750	0.7127	Alt3>Alt2>Alt1>Alt4	Consistent
ES’ (150%)	0.7352	0.7465	0.8064	0.6746	Alt3>Alt2>Alt1>Alt4	Consistent
O’ (50%)	0.6673	0.6732	0.7101	0.6242	Alt3>Alt2>Alt1>Alt4	Consistent
O’ (100%)	0.6362	0.6359	0.6606	0.5991	Alt3>Alt1>Alt2>Alt4	Inconsistent
O’ (150%)	0.6182	0.6145	0.6321	0.5847	Alt3>Alt1>Alt2>Alt4	Inconsistent
P’ (50%)	0.7657	0.7817	0.8481	0.7050	Alt3>Alt2>Alt1>Alt4	Consistent
P’ (100%)	0.7823	0.7941	0.8595	0.7208	Alt3>Alt2>Alt1>Alt4	Consistent
P’ (150%)	0.7301	0.7359	0.7853	0.6823	Alt3>Alt2>Alt1>Alt4	Consistent
PE’ (50%)	0.6405	0.6678	0.7200	0.6097	Alt3>Alt2>Alt1>Alt4	Consistent
PE’ (100%)	0.5893	0.6179	0.6578	0.5761	Alt3>Alt2>Alt1>Alt4	Consistent
PE’ (150%)	0.5608	0.5901	0.6233	0.5574	Alt3>Alt2>Alt1>Alt4	Consistent
S-C’ (50%)	0.6810	0.6957	0.7428	0.6402	Alt3>Alt2>Alt1>Alt4	Consistent
S-C’ (100%)	0.6391	0.6503	0.6853	0.6096	Alt3>Alt2>Alt1>Alt4	Consistent
S-C’ (150%)	0.6151	0.6243	0.6524	0.5920	Alt3>Alt2>Alt1>Alt4	Consistent
Level of consistency	86.67%

Table 9 summarises the sensitivity analysis results for all three comparative studies in this research work. Representing both established models (Z-AHP and Z-TOPSIS) achieve 66.67% of level of consistency while the proposed Z-CFPR-TOPSIS achieves 86.67%. This is depicted that the proposed Z-CFPR-TOPSIS model is more robust and reliable than Z-AHP [8] and Z-TOPSIS [29] to deal with uncertain environment in studying knowledge of human being. From the consistency results, the proposed Z-CFPR-TOPSIS model is recommended to deal with bigger case study in real world phenomena in order to solve human based decision making problems under fuzzy environment.

Table 9

Ranking results of alternatives

Z-numbers MCDM Model	Level of Consistency
Z-AHP [8]	66.67%
Z-TOPSIS [29]	66.67%
Z-CFPR-TOPSIS (Proposed)	86.67%

5 Conclusion

This study introduces the concept of a hybrid fuzzy MCDM model. The latter consists of consistent preference relations and fuzzy TOPSIS (Z-CFPR-TOPSIS) using an intuitive vectorial centroid defuzzification method to deal with z-numbers. In dealing with the uncertainty and complexity in the information, the reliability of information is taken into consideration efficiently. Z-numbers have better capability in describing uncertain and complex knowledge. For this reason, z-numbers are used in this work.

The development of an extension of an intuitive vectorial centroid provides an efficient computational defuzzification procedure for uncertain environment. It is presented by a simple formula that is based on the perspective of analytic geometric principles. In developing an intuitionistic defuzzification, a novel manner of computing the intuitive vectorial centroid method makes it capable of dealing with all possible cases of fuzzy numbers. The novel Z-CFPR-TOPSIS model is developed by improvising several steps in computing the consistent fuzzy preference relations and fuzzy TOPSIS to make sure both techniques are perfectly integrated. The proposed model is capable of interacting or cooperating with unlimited criteria in dealing with real world decision making problems.

The proposed Z-CFPR-TOPSIS model provides better selection of alternatives in human based decision making by capturing uncertainty in human judgement. Due to access information and availability of the incomplete and uncertain data, it is hard to make a right decision. In this sense, it is important to modify some classical techniques and models by adding intuitive reasoning and human subjectivity. As a consequence, the proposed model offers a robust and reliable methodology that provides best alternatives with regard to available resources. Therefore, this methodology can be further extended in order by considering complicated case studies drawn from a wide range of human based decision making problems.

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Hybrid fuzzy MCDM model for Z-numbers using intuitive vectorial centroid

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 A. Z-numbers

3.1 A. Stage one

Table 3 Reliability linguistic terms and their corresponding z-numbers [31] Linguistic terms Generalised fuzzy numbers Very-low (VL) (0, 0, 0, 0.25; 1) Low (L) (0, 25, 0.25, 0.5; 1) Medium (M) (0.25, 0.5, 0.5, 0.75; 1) High (H) (0.5, 0.75, 0.75, 1; 1) Very-high (VH) (0.75, 1, 1, 1; 1)

References

Table 3
Reliability linguistic terms and their corresponding z-numbers [31]

Linguistic terms Generalised fuzzy numbers

Very-low (VL) (0, 0, 0, 0.25; 1)

Low (L) (0, 25, 0.25, 0.5; 1)

Medium (M) (0.25, 0.5, 0.5, 0.75; 1)

High (H) (0.5, 0.75, 0.75, 1; 1)

Very-high (VH) (0.75, 1, 1, 1; 1)