Entropy and probability based Fuzzy Induced Ordered Weighted Averaging operator

Abstract

The traditional Ordered Weighting Average (OWA) operator is suitable for aggregating numerical attributes. However, this method fails when the attribute values are given in a linguistic form. In this paper, a novel aggregating method named Entropy and Probability based Fuzzy Induced Ordered Weighted Averaging (EPFIOWA) is proposed for Gaussian-fuzzy-number-based linguistic attributes. A method is first designed to obtain a reasonable weighting vector based on probability distribution and maximal entropy. Such optimal weighting vectors can be obtained under any given level of optimism, and the symmetric properties of the proposed model are proven. The linguistic attributes of EPFIOWA are represented by Gaussian fuzzy numbers because of their concise form and good operational properties. In particular, the arithmetic operations and distance measures of Gaussian fuzzy numbers required by EPFIOWA are given systematically. A novel method to obtain the order-inducing variables of linguistic attribute values is proposed in the EPFIOWA operators by calculating the distances between any Gaussian fuzzy number and a set of ordered grades. Finally, two numerical examples are used to illustrate the proposed approach, with evaluation results consistent with the observed situation.

Keywords

Gaussian fuzzy numbers induced ordered weighted averaging operators order-inducing variables probability distribution maximal entropy

1 Introduction

Multiple attribute comprehensive evaluation refers to an evaluation process of ranking a number of alternatives (objects) based on particular attribute values. As a solution for multiple attribute evaluation, Ordered Weighted Averaging (OWA) operators were first proposed by Yager in 1988 [1], with the output of an OWA operator being defined as the dot product of a weighting vector and an attribute value vector wherein any attribute values are rearranged in non-decreasing order. OWA rapidly attracted significant attention among researchers, and has since been applied across multiple fields, including risk assessment [2], project planning [3], industry scheduling [4], book recommendation technique [5], safety evaluation [6]. Several surveys on the field, as noted in [7, 8] and [9], have been undertaken to summarize the development of OWA. And across the relevant literature, the generalization of OWA operators and the optimization of the OWA weighting vector have emerged as two equally important issues.

With respect to the first issue, several generalization methods have been proposed since the advent of OWA, such as the Generalized WOA (GOWA) [10], Ordered Weighted Geometric Averaging (OWGA) [11], Probabilistic OWA (POWA) [12], Weighted OWA (WOWA) [13], and Olympic type OWA [14]. These improvements provide a parameterized family of comprehensive evaluation methods, which includes the most commonly used evaluation operators such as the maximum, the minimum and the average evaluation.

Yager and Filev generalized OWA operator theory further by proposing a method known as the Induced OWA (IOWA) operator [15]. They introduced another type of variables to the IOWA operator named the associated order-inducing variables, such that the ordered position of the attributes is no longer determined by the attribute values, but by these order-inducing variables. IOWA has grown in popularity in recent years [16 –20], because this method permits the use of linguistic values that may have no inherent order as attribute values, as long as these linguistic values are associated with appropriate order-inducing variables. However, this method cannot be used when the order-induced variables are unknown.

Linguistic values are now commonly used across various evaluation fields [21, 22], as many researchers find it more reasonable to adopt unspecified linguistic values that are more consistent with human cognition than numerical data. Due to the ground-breaking work of Zadeh et al. [23], fuzzy logic based linguistic values have become efficient tools for modelling uncertain information about the performance of attributes, and, in recent years, several significant developments have occurred in the modelling of uncertainty, incomplete information and subjective views in OWA and IOWA formats [24 –27]. However, the operational laws and distance measures of fuzzy linguistic variables still require further analysis for the effective use of IOWA, especially when order-inducing variables are not provided initially.

With respect to the second issue, there are numerous papers dealing with the optimization of OWA weighting vectors. The survey given in [28] summarizes many of them. One direct way to generate OWA weights is to utilize the normal probability distribution [29], which assumes that attributes of unduly high and low preference should be assigned very low weights, while non-preferential attributes should be assigned high weights. This basic and direct method can also be generalized to cover other types of probability distribution and multiple OWA weight forms, as in [30, 31] and [32]. However, the obtained weighting vectors may be susceptible to subjective judgments with respect to evaluators.

Another way to generate OWA weights is to use the well-known concept “orness” introduced by Yager in [1]. As a linear combination of weighting vectors, orness can reflect the optimism level of an evaluator. Evaluators often need to rank alternatives under a predefined level of optimism. Unsurprisingly, many papers investigate the optimization of weighting vectors under a given orness level. Some orness-based methods obtain optimal weighting vectors by maximizing the entropy of the relevant OWA weights. O’Hagan was probably the first to suggest a maximal entropy model subject to a given level of orness [33], and this model had been proven to have an analytical solution [34]. In [35], the authors maximized Bayesian entropy under a given orness level. By using the Lagrange multiplier method, this model had also been proven to have an analytic solution [36]. A model based on Rényi entropy was proposed in [37]. Other researches on entropy-based models can be found in [38, 39] and so on. Another kind of orness-based models obtain optimal weighting vectors by minimizing the differences among weights. Such models include the least square deviation model [40], the minimax disparity model [41, 42], the equidifferent OWA operator [43], the χ² method [40] and so on. Other kinds of orness-based methods can be found in a recent survey [28]. It is also worth mentioning that there exist papers where the optimal weighting vector is derived to obtain the desired evaluation result of the evaluator [44].

In order to design reasonable OWA weights which integrate subjective and objective factors, and to use linguistic attribute variables in IOWA when their associated order-inducing variables are not provided, this paper focuses on developing a novel Entropy and Probability based Fuzzy Induced Ordered Weighted Averaging (EPFIOWA) model based on the methods discussed above. We mainly carry out the following work in this study.

A new method that combines subjective and objective criteria to determine the weights of OWA is proposed. An initial probability distribution and an optimism level are given subjectively by the evaluator, while the optimal weights are objectively determined by using a maximal entropy programming model. The symmetric properties of the optimal weights are proven.

Arithmetic operation properties and distance measurements for Gaussian fuzzy numbers are proposed. Based on these results, the similarity of two Gaussian-fuzzy-number-based linguistic values can be calculated.

A novel method to obtain the order-inducing variables is proposed for IOWA operators. When Gaussian fuzzy numbers are used as linguistic attribute values, their order-inducing variables can be obtained by calculating the similarities between any Gaussian fuzzy number and a set of ordered grades as provided by the evaluator.

Finally, an EPFIOWA method is proposed for multiple attribute comprehensive evaluation using linguistic values, with evaluation results consistent with the observed situation.

The rest of this paper is organized as follows. Section 2 briefly reviews OWA theory and the relevant IOWA operators. In Section 3, a novel optimization model to obtain the OWA weights based on probability, entropy and orness level is proposed, and symmetric properties of this model are proven. The arithmetic system and distance measures used for Gaussian fuzzy numbers are also introduced in this section. In Section 4, an EPFIOWA method based on Gaussian fuzzy numbers is proposed. In Section 5, two examples are given to illustrate how to use the proposed EPFIOWA operators.

2 Ordered weighted average operators

This section offers the necessary preliminaries regarding the OWA theory.

2.1 OWA theory

An OWA operator is an aggregation operator for multi-attribute decision-making.

Definition 1 ([1]). The OWA operator of dimension n s a mapping OWA_W : Rⁿ → R that has an associated n-dimensional vector W = (w₁, w₂, ⋯ , w_n), with w_i ∈ [0, 1], and $\sum_{i = 1}^{n} w_{i} = 1$ , such that ${OWA}_{W} (a_{1}, a_{2}, \dots, a_{n}) = \sum_{i = 1}^{n} w_{i} b_{i},$ where b_i is the ith largest element of the attribute values (a₁, a₂, ⋯ , a_n).

OWA_W is a useful decision-making method due to the fact that it takes the optimism level of the evaluator into account [21]. An optimistic evaluator would select a weighting vector W_max = (1, 0, ⋯ , 0) to retrieve the largest attribute,

${OWA}_{W_{\max}} (a_{1}, a_{2}, \dots, a_{n}) = b_{1}$ (1) as the final evaluation result; while a pessimistic evaluator would select the weighting vector W_min = (0, ⋯ , 0, 1) to retrieve the smallest attribute

${OWA}_{W_{\min}} (a_{1}, a_{2}, \dots, a_{n}) = b_{n}$ (2) as the final evaluation result. The OWA method thus provides a parameterized family of comprehensive evaluations between the minimum and the maximum.

2.2 Induced OWA (IOWA) operators

Yager and Filev proposed a more general type of OWA operator known as the IOWA operator [15]. IOWA is an extension of OWA, with the difference being that the ordering step of the IOWA operator is not sorted by the attribute values, but rather by a set of order-inducing variables associated with the attribute values.

Definition 2 ([15]). Let 〈u₁, a₁〉, 〈u₂, a₂〉, ⋯ , 〈u_n, a_n〉 be n two-tuples formed for the relevant attributes, where a_i is the attribute value, and u_i is the associated order-inducing variable of a_i. An IOWA operator associated with a weighting vector W = (w₁, w₂, ⋯ , w_n), with w_i ∈ [0, 1], and $\sum_{i = 1}^{n} w_{i} = 1$ is a mapping Rⁿ × Rⁿ → R, such that:

${IOWA}_{W} (〈 u_{1}, a_{1} 〉, 〈 u_{2}, a_{2} 〉, \dots, 〈 u_{n}, a_{n} 〉) = \sum_{i = 1}^{n} w_{i} b_{i}$

where b_i is the attribute value of the two-tuple with the ith largest value of order-inducing variable (u₁, u₂, ⋯ , u_n).

Example 1. Consider the following collection of attribute values with their respective order-inducing variables: 〈u₁, a₁〉 = 〈Grade1, 25〉, 〈u₂, a₂〉 = 〈Grade2, 60〉, 〈u₃, a₃〉 = 〈Grade4, 90〉, 〈u₄, a₄〉 = 〈Grade3, 80〉, where Grade4 represents the best and Grade1 represents the worst. If the weighting vector is W = (0.2, 0.2, 0.3, 0.3), then the aggregation formula becomes $\begin{matrix} {IOWA}_{W} (〈 u_{1}, a_{1} 〉, \dots, 〈 u_{4}, a_{4} 〉) \\ = 0.2 \times 90 + 0.2 \times 80 + 0.3 \times 60 + 0.3 \times 25 \\ = 59.5 \end{matrix}$ where a₁, a₂, a₃, a₄ are sorted according to the order-inducing variables u₁, u₂, u₃, u₄, which are expressed as four ordered grades: $Grade 1 < Grade 2 < Grade 3 < Grade 4 .$

The IOWA is especially useful when the attribute values are expressed as linguistic values which have no explicit order relationship. In such case, these linguistic values can be sorted by their corresponding order-inducing variables; however, it remains difficult to sort the linguistic attribute values if the corresponding associated order-inducing variables are missing.

2.3 Orness levels

As a linear combination of the OWA weights, the orness level associated with the OWA operator [45] is defined as follows.

Definition 3 ([1]). The orness level of a weighting vector W = (w₁, w₂, ⋯ , w_n) is defined as $α (W) = Orness (w_{1}, w_{2}, \dots, w_{n}) = \sum_{i = 1}^{n} \frac{n - i}{n - 1} w_{i} .$

The orness level α (W) ∈ [0, 1] is a parameter that can be used to determine the optimism level of the evaluator. If α (W) = 1, we have $\begin{matrix} 1 = α (W) \\ = \sum_{i = 1}^{n} \frac{n - i}{n - 1} w_{i} \\ = \sum_{i = 1}^{n} w_{i} - \sum_{i = 1}^{n} \frac{i - 1}{n - 1} w_{i} \\ = 1 - \sum_{i = 1}^{n} \frac{i - 1}{n - 1} w_{i}, \end{matrix}$ which implies that w₁ = 1, w₂ = w₃ = ⋯ = w_n = 0, thus that OWA_Wis the maximization operator (1). Based on the same reasoning, if α (W) = 0, then OWA_W is the minimum operator (2). Thus, the orness level can reflect the optimism level of an evaluator, allowing alternatives to be evaluated under a predefined level of optimism.

3 Entropy and probability based weights and gaussian fuzzy numbers

This section introduces a novel programming model to determine a weighting vector based on entropy and probability density function. Some symmetric properties of this model are then proven. And then the arithmetic operations and distance measure of Gaussian fuzzy numbers are proposed.

3.1 An OWA weight generation model using entropy and probability distribution

Xu [29] proposed a Gaussian probability density function to generate OWA weights. The Gaussian distribution is one of the most important two-parameter probability distributions in statistics, as many data sets drawn from nature display similar bell-shaped curves. For a n-attribute alternative to be aggregated, weights based on a Gaussian density function can thus be computed as

$p_{i}^{N} = \frac{1}{σ_{n} \sqrt{2 π}} exp (- \frac{{(i - μ)}^{2}}{2 σ_{n}^{2}})$ (3) where μ_n and σ_n are the mean and modified standard deviation of 1, 2, ⋯ , n, respectively. Based on this, μ_n and σ_n can be computed as

$μ_{n} = \frac{1 + n}{2}$ (4) and

$σ_{n} = K \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(i - μ_{n})}^{2}}$ (5) respectively, where K ∈ (0, + ∞) is a user-specified parameter reflecting the degree of weight dispersion. When K = 1, the σ_n defined in (5) is equal to the standard deviation of 1, 2, ⋯ , n. According to the sum-one constraint, weights obtained by (3) must be normalized to become an OWA weighting vector W = (w₁, w₂, ⋯ , w_n), that is

$w_{i}^{N} = \frac{p_{i}^{N}}{\sum_{i = 1}^{n} p_{i}^{N}}, i = 1, 2, \dots, n$ (6)

As shown in Fig. 1, in order to analyze the effect of the parameter K, different OWA weights are generated using (3) and (6) under different values of K∈ { 1/2, 2/3, 1, 3/2, 2 }, where the OWA dimension n = 7. Obviously, the larger the K value, the more similar these weights are. Therefore, K can reflect the attitude of the evaluator towards the maximum and minimum evaluation values. If K approaches 0, the unduly high and low evaluation values are assigned very low weights.

Fig. 1

OWA weights generated using a Gaussian distribution.

Apparently, OWA weights based on Gaussian function are symmetrical about the center position. Weighting vectors generated from such probability functions are very common, because the weights of extreme evaluation values are relatively small compared with those of moderate evaluation values, and such systems belong to the broader class of OWA operators that are averaging around the center. However, in such method, weights on the left of the center are mirrors of weights on the right of the center, causing such systems to be unable to measure the level of optimism of a given evaluator.

Unlike Xu’s method, which gives the weighting vector directly from the Gaussian distribution, this paper uses the a more comprehensive model to determine the OWA weights based on considering probability distributions of any given type and the weight entropy under a predefined level of optimism.

$\begin{matrix} Model (α, W^{Pro}) : \\ min_{W} Q = β_{1} \sum_{i = 1}^{n} {(w_{i} - w_{i}^{Pro})}^{2} \\ + β_{2} \sum_{i = 1}^{n} w_{i} ln w_{i} \\ s . t . \sum_{i = 1}^{n} \frac{n - i}{n - 1} w_{i} = α \\ \sum_{i = 1}^{n} w_{i} = 1 \\ w_{i} ⩾ 0, for i = 1, 2, \dots, n \end{matrix}$ (7) where $w_{i}^{Pro}$ is an OWA weight generated by a given type of probability distribution, i = 1, 2, ⋯ , n. For example, a Gaussian distribution based weighting vector W^N can be obtained if the $w_{i}^{Pro}$ is determined by (3) and (6). While a symmetric binomial distribution based OWA weighting vector can be written by using the following formula

$\begin{matrix} w_{i}^{B} = (\begin{matrix} n - 1 \\ i \end{matrix}) p^{i} {(1 - p)}^{n - 1 - i} \\ = \frac{C_{n - 1}^{i - 1}}{2^{n - 1}}, i = 1, 2, \dots, n, \end{matrix}$ (8) where n - 1 and p are the parameters of the binomial probability density function. If p = 1/2, the obtained weighting vector $W^{B} = {w_{i}^{B}}$ is symmetric. If p is set to other values, however, an asymmetric weighting vector emerges.

The first term of the objective function in (7) makes w_i equal to $w_{i}^{Pro}$ insofar as possible, and the second term $\sum_{i = 1}^{n} w_{i} ln w_{i}$ in (7) represents the entropy of the weights, making w₁, w₂, ⋯ , w_n as equal as possible. β₁ and β₂ are then the weight coefficients for adjusting these two terms in the objective function.

It can be shown that if the given probability distribution is symmetric, such that $w_{i}^{Pro} = w_{n + 1 - i}^{Pro}$ , then the optimal solution for Model (α, W^Pro) has the following properties.

Theorem 1. Given a vector W^Pro generated from a symmetric probability distribution, if $W^{*} = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*})$ is the optimal solution for Model (α, W^Pro), then ${\tilde{W}}^{*} = (w_{n}^{*}, w_{n - 1}^{*}, \dots, w_{1}^{*})$ is the optimal solution for Model (1 - α, W^Pro).

Proof: Let ${\tilde{w}}_{i}^{*} = w_{n - i + 1}^{*}$ . As the objective function Q (W) of Model (α, W^Pro) reaches its minimal value at $W^{*} = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*})$ , the objective function Q (W) of Model (1 - α, W^Pro) obtains the same minimal value where ${\tilde{W}}^{*} = ({\tilde{w}}_{1}^{*}, {\tilde{w}}_{2}^{*}, \dots, {\tilde{w}}_{n}^{*}) = (w_{n}^{*}, w_{n - 1}^{*}, \dots, w_{1}^{*})$ because $\begin{matrix} \sum_{i = 1}^{n} w_{i}^{*} ln w_{i}^{*} = \sum_{i = 1}^{n} w_{n + 1 - i}^{*} ln w_{n + 1 - i}^{*} \\ = \sum_{i = 1}^{n} {\tilde{w}}_{i}^{*} ln {\tilde{w}}_{i}^{*} . \end{matrix} .$

And $\begin{matrix} \sum_{i = 1}^{n} {(w_{i}^{*} - w_{i}^{Pro})}^{2} & = \sum_{i = 1}^{n} {(w_{n + 1 - i}^{*} - w_{n + 1 - i}^{Pro})}^{2} \\ = \sum_{i = 1}^{n} {({\tilde{w}}_{i}^{*} - w_{i}^{Pro})}^{2} \end{matrix}$

is correct since $w_{i}^{Pro} = w_{n + 1 - i}^{Pro}$ . Therefore, we have $β_{1} \sum_{i = 1}^{n} {(w_{i}^{*} - w_{i}^{Pro})}^{2} + β_{2} \sum_{i = 1}^{n} w_{i}^{*} ln w_{i}^{*}$ $= β_{1} \sum_{i = 1}^{n} {({\tilde{w}}_{i}^{*} - w_{i}^{Pro})}^{2} + β_{2} \sum_{i = 1}^{n} {\tilde{w}}_{i}^{*} ln {\tilde{w}}_{i}^{*} .$

Now, we show that ${\tilde{W}}^{*} = ({\tilde{w}}_{1}^{*}, {\tilde{w}}_{2}^{*}, \dots, {\tilde{w}}_{n}^{*}) = (w_{n}^{*}, w_{n - 1}^{*}, \dots, w_{1}^{*})$ is a feasible solution for Model (1 - α, W^Pro). For one thing, because $W^{*} = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*})$ is a feasible solution of Model (α, W^Pro), we have $\sum_{i = 1}^{n} w_{i}^{*} = 1$ , which means

$\sum_{i = 1}^{n} {\tilde{w}}_{i}^{*} = \sum_{i = 1}^{n} {\tilde{w}}_{n + 1 - i}^{*} = 1 .$ (9)

For the other thing, because

$\sum_{i = 1}^{n} \frac{n - i}{n - 1} w_{i}^{*} = α,$ (10) we have

$\begin{matrix} \sum_{i = 1}^{n} \frac{n - i}{n - 1} {\tilde{w}}_{i}^{*} = \sum_{i = 1}^{n} (1 - \frac{i - 1}{n - 1}) w_{n + 1 - i}^{*} \\ = 1 - \sum_{i = 1}^{n} \frac{i - 1}{n - 1} w_{n + 1 - i}^{*} \\ = 1 - \sum_{i = 1}^{n} \frac{n - i}{n - 1} w_{i}^{*} \\ = 1 - α . \end{matrix}$ (11)

Thus, ${\tilde{W}}^{*}$ is the optimal solution for Model (1 - α, W^Pro). This completes the proof of Theorem 1.□

Corollary 1. If the W^Pro is generated from a symmetric probability distribution, then the optimal solution $W^{*} = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*})$ for Model (1/2, W^Pro) is also symmetric, such that $w_{i}^{*} = w_{n - i + 1}^{*}$ .

Proof: since α = 1/2 = 1 - α, the optimal solution for Model (1/2, W^Pro) can be written as $\begin{matrix} {\tilde{W}}^{*} = (w_{n}^{*}, w_{n - 1}^{*}, \dots, w_{1}^{*}) \\ = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*}) \\ = W^{*}, \end{matrix}$ according to Theorem 1. Therefore, we have $w_{i}^{*} = w_{n - i + 1}^{*}$ .□

3.2 Arithmetic operations and distance measurements with Gaussian fuzzy numbers

In real-life applications, attribute values are sometimes given using linguistic values. Thus, just as Gaussian distributions are important in statistics and are often used in the natural and social sciences, Gaussian fuzzy numbers are used as a very common and remarkably useful way of expressing linguistic attribute values, based on their ability to represent membership degrees in a natural way.

Such linguistic values may be estimated by using (historical) numerical evaluation scores, or Gaussian fuzzy numbers that are given by some experts in the field, who do not necessarily include the evaluator. However, it may be difficult for the evaluator to sort such linguistic values. To resolve this problem, this section introduces the arithmetic operations and distance measures used with Gaussian fuzzy numbers. By applying distance measurements, an evaluator can determine the ranking of linguistic values based on a comparison of these values with a set of ordered grades given by the evaluator.

3.2.1 Arithmetic operations of Gaussian fuzzy numbers

Gaussian fuzzy numbers are basic concepts in fuzzy theory, being useful tools for modeling the uncertainty of attribute information. A Gaussian fuzzy number in R is defined as a special type of fuzzy set whose membership function f (x) is

$f (x; μ, σ) = exp (- \frac{{(x - μ)}^{2}}{2 σ^{2}})$ (12) where μ represents the average value of an attribute, and the standard deviation σ determines the degree of uncertainty around that attribute value. A given linguistic attribute value may be given directly by an expert, or be estimated based on a number of (historical) numerical evaluation values. To facilitate analysis, the resulting Gaussian fuzzy number, whose membership is f (x ; μ, σ), is denoted as G (μ, σ) in this paper.

The following example describes how to use evaluation data to construct a Gaussian-fuzzy-number-based linguistic value and the comparison of two Gaussian fuzzy numbers.

Example 2. This example illustrates how to construct linguistic evaluation values for two students based on their historical scores. Assume that Student A’s five evaluation scores are 3, 4, 5, 6 and 7; and those of Student B are 4.2, 4.4, 4.6, 4.8 and 5. Assume also that these two groups of values have Gaussian distributions N (μ₁, σ₁) and N (μ₂, σ₂), respectively. Then, the fact that $μ_{1} = 5, σ_{1} = \sqrt{2};$ $μ_{2} = \frac{23}{5}, σ_{2} = \frac{\sqrt{2}}{5}$ can be inferred by the classical moment estimation method. As shown in Fig. 2, these two groups of numerical values can thus be represented by two Gaussian fuzzy numbers $G (5, \sqrt{2})$ and $G (23 / 5, \sqrt{2} / 5)$ . From the perspective of average value μ, it can be considered that $G (5, \sqrt{2})$ is better than $G (23 / 5, \sqrt{2} / 5)$ since 5 > 23/5; however, from the perspective of standard deviation σ, it can also be considered that $G (5, \sqrt{2})$ is worse than $G (23 / 5, \sqrt{2} / 5)$ since $\sqrt{2} ⪢ \sqrt{2} / 5$ , and sometimes the student with stable performance is more preferred.

Fig. 2

Membership function of a Gaussian fuzzy number.

One advantage of using Gaussian fuzzy numbers to represent linguistic values is that: the arithmetic operations and distance measures of Gaussian fuzzy numbers have simple forms that are similar to those of real numbers. Thus, for n Gaussian fuzzy numbers G (μ_i, σ_i) , i = 1, 2, ⋯ , n, and n real numbers w₁, w₂, ⋯ , w_n ∈ R, according to the extension principle [46], we have

$\sum_{i = 1}^{n} w_{i} G (μ_{i}, σ_{i}) = G (\sum_{i = 1}^{n} w_{i} μ_{i}, \sum_{i = 1}^{n} w_{i} σ_{i})$ (13) which mean that the linear combination of two Gaussian fuzzy numbers is also a Gaussian fuzzy number. This property is useful in the construction of the IOWA operator based on Gaussian fuzzy numbers, and it is guaranteed by the following theorem.

Theorem 2. For n Gaussian fuzzy numbers G (μ_i, σ_i) , i = 1, 2, ⋯ , n, and a weighting vector W = (w₁, w₂, ⋯ , w_n) ∈ Rⁿ, we have $\sum_{i = 1}^{n} w_{i} G (μ_{i}, σ_{i}) = G (\sum_{i = 1}^{n} w_{i} μ_{i}, \sum_{i = 1}^{n} w_{i} σ_{i})$

Proof: The r-level (or r-cut) set [46] of G (μ_i, σ_i) consists of those points whose membership function values are greater than r, i.e.

$\begin{matrix} R_{r} (G (μ_{i}, σ_{i})) = {x | f_{i} (x) ⩾ r} \\ = [μ_{i} - \sqrt{2 ln (\frac{1}{r})} σ_{i}, μ_{i} + \sqrt{2 ln (\frac{1}{r})} σ_{i}] \end{matrix}$ (14) where f_i (x) is defined as in (12). Consequently, the r-level set of $\sum_{i = 1}^{n} w_{i} G (μ_{i}, σ_{i})$ can be written as $\begin{matrix} \sum_{i = 1}^{n} w_{i} [G_{i, r}^{-}, G_{i, r}^{+}] \\ = \sum_{i = 1}^{n} w_{i} [μ_{i} - \sqrt{2 ln (\frac{1}{r})} σ_{i}, μ_{i} + \sqrt{2 ln (\frac{1}{r})} σ_{i}] \\ = \sum_{i = 1}^{n} [w_{i} μ_{i} - \sqrt{2 ln (\frac{1}{r})} w_{i} σ_{i}, w_{i} μ_{i} \\ + \sqrt{2 ln (\frac{1}{r})} w_{i} σ_{i}] \\ = [\sum_{i = 1}^{n} w_{i} μ_{i} - \sqrt{2 ln (\frac{1}{r})} (\sum_{i = 1}^{n} w_{i} σ_{i}), \sum_{i = 1}^{n} w_{i} μ_{i} \\ + \sqrt{2 ln (\frac{1}{r})} (\sum_{i = 1}^{n} w_{i} σ_{i})] \end{matrix}$ which is simply the r-level set of $G (\sum_{i = 1}^{n} w_{i} μ_{i}, \sum_{i = 1}^{n} w_{i} σ_{i})$ . This thus offers a proof of Theorem 2. □

3.2.2 Distance measurement

An evaluator may have several ordered evaluation grades ${{\hat{G}}_{i} | i = 1, 2, \dots, m}$ that are represented by linguistic values, where ${\hat{G}}_{1}$ represents the worst and ${\hat{G}}_{m}$ represents the best. To evaluate an alternative, a set of linguistic values {G_j|j = 1, 2, ⋯ , n} is collected from experts with respective to all n attributes of the alternative, whose associated order-inducing variables are not provided. Note that G_j is not necessarily an element of ${{\hat{G}}_{i} | i = 1, 2, \dots, m}$ . In order to use these {G_j} in the IOWA operator, the order of these linguistic values must be determined.

It is possible to obtain the order-inducing variable of {G_j} by calculating any similarity between a linguistic value G_j and an evaluation grade ${\hat{G}}_{i}$ . For this purpose, it is necessary to determine the distance between two Gaussian fuzzy numbers based on the Hausdorff metric. The L₁-type Hausdorff distance for two fuzzy numbers G₁ = G (μ₁, σ₁) and G₂ = G (μ₂, σ₂) can be calculated by using Theorem 2 offered below [47, 48].

Theorem 3. ([49]) The L₁-type Hausdorff distance D (G₁, G₂) between G₁ = G (μ₁, σ₁) and G₂ = G (μ₂, σ₂) can be written as

$\begin{matrix} D (G_{1}, G_{2}) = abs (μ_{1} - μ_{2}) \\ + \frac{\sqrt{2 π} (abs (σ_{1} - σ_{2}))}{2} \end{matrix}$ (15)

Proof: The L_p-type Hausdorff distance [47, 48] of G₁and G₂ is defined as

$\begin{matrix} D_{p} (G_{1}, G_{2}) = \\ {(\int_{0}^{1} max {| G_{1 r}^{-} - G_{2 r}^{-} |, | G_{1 r}^{+} - G_{2 r}^{+} |}^{p} d r)}^{\frac{1}{p}}, \end{matrix}$ (16) with p > 0, where $R_{r} (G_{1} (μ_{1}, σ_{1})) = [G_{1 r}^{-}, G_{1 r}^{+}],$ and $R_{r} (G_{2} (μ_{2}, σ_{2})) = [G_{2 r}^{-}, G_{2 r}^{+}]$ are the r-level sets of G₁ = G (μ₁, σ₁) and G₂ = G (μ₂, σ₂), respectively. More specifically, for Gaussian fuzzy numbers G₁ and G₂, the lower and upper bounds of the r-level set of G_i (μ_i, σ_i) take the following forms based on (12):

$\begin{matrix} G_{ir}^{-} = μ_{i} - \sqrt{2 ln (\frac{1}{r})} σ_{i}, \\ G_{ir}^{+} = μ_{i} + \sqrt{2 ln (\frac{1}{r})} σ_{i}, i = 1, 2 . \end{matrix}$ (17)

Substituting (17) into (16) gives the L₁-type Hausdorff distance of G₁and G₂ $D (G_{1}, G_{2}) = abs (μ_{1} - μ_{2}) + \frac{\sqrt{2 π} (abs (σ_{1} - σ_{2}))}{2} .$

□

4 Entropy and probability based Fuzzy Induced Ordered Weighted Averaging Operators

The Fuzzy Induced OWA (FIOWA) operator is an extension of the IOWA operator, with the difference that the attribute values of FIOWA are not numerical numbers, but instead more general linguistic values represented by fuzzy sets.

Gaussian fuzzy numbers are suitable for representing fuzzy linguistic values. For one thing, Gaussian fuzzy numbers are easy to understand, and are commonly used as in social sciences; for another thing, they have excellent operation properties as shown in Theorem 2 and Theorem 3. Based on these considerations, Gaussian fuzzy numbers are used as linguistic values in the following Gaussian fuzzy number based FIOWA operator.

Definition 4. (Gaussian fuzzy number based FIOWA) Let 〈u₁, G₁〉, 〈u₂, G₂〉, ⋯ , 〈u_n, G_n〉 be n two-tuples for the FIOWA attributes, where G_i is a linguistic attribute value represented by a Gaussian fuzzy number, and u_i is the associating order-inducing variable of G_i. A Gaussian fuzzy number based FIOWA operator with a weighting vector W = (w₁, w₂, ⋯ , w_n) with w_i ∈ [0, 1], and $\sum_{i = 1}^{n} w_{i} = 1$ is thus defined according to the following expression:

$\begin{matrix} {FIOWA}_{W} (〈 u_{1}, G_{1} 〉, 〈 u_{2}, G_{2} 〉, \dots, 〈 u_{n}, G_{n} 〉) \\ = \sum_{i = 1}^{n} w_{i} {\hat{G}}_{i} \end{matrix}$ (18) where $({\hat{G}}_{1}, {\hat{G}}_{2}, \dots, {\hat{G}}_{n})$ is (G₁, G₂, ⋯ , G_n) as reordered in decreasing order of the order-inducing values of u_i, i = 1, 2, ⋯ , n, and the weighted sum of $〈 {\hat{G}}_{1}, {\hat{G}}_{2}, \dots, {\hat{G}}_{n} 〉$ is calculated by (13).

Synthesizing all the methods mentioned above, this work develops an EPFIOWA approach based on the Gaussian-fuzzy-number based FIOWA, the maximal entropy OWA model, and probabilistic OWA weighting vectors for multi-attribute evaluation problems. This EPFIOWA can be accomplished by implementing the following steps:

Step 1. Determining the OWA weights.

Given the required optimism level (orness level) α, and a probability density-based weighting vector W^Pro, the programming model Model (α, W^Pro) in (7) is utilized to determine the necessary IOWA weighting vector W.

Step 2. Collecting attribute information from experts and evaluators.

Linguistic attribute values {G_i|i = 1, 2, ⋯ , n} are collected from experts or constructed from numerical evaluation data with respective to n attributes, where every G_i is represented by a Gaussian fuzzy number.

The evaluator defines m ordered grades within a dimensionless domain (such as [0, 1]), which are expressed by m two-tuples $〈 {\tilde{u}}_{k}, {\tilde{G}}_{k} 〉, k = 1, 2, \dots, m$ , such that ${\tilde{G}}_{k}$ is a linguistic value represented by a Gaussian fuzzy number, the order-inducing variable ${\tilde{u}}_{k} \in {1, 2, \dots, m}$ represents the rank of ${\tilde{G}}_{k}$ with ${\tilde{u}}_{1} < {\tilde{u}}_{2} < \dots < {\tilde{u}}_{m}$ , and ${\tilde{u}}_{k}$ can be interpreted as a specific evaluation grade. In general, m is an odd number. As an example, an evaluator may choose to implement seven order-inducing variables as follows

$\begin{matrix} {\tilde{u}}_{1} = Grade 1 : VeryPoor; \\ {\tilde{u}}_{2} = Grade 2 : FairlyPoor; \\ {\tilde{u}}_{3} = Grade 3 : SlightlyPoor; \\ {\tilde{u}}_{4} = Grade 4 : Moderate; \\ {\tilde{u}}_{5} = Grade 5 : SlightlyGood; \\ {\tilde{u}}_{6} = Grade 6 : FairlyGood; \\ {\tilde{u}}_{7} = Grade 7 : VeryGood . \end{matrix}$ (19)

As shown in Fig. 3, the corresponding Gaussian fuzzy numbers ${\tilde{G}}_{i} (μ_{i}, σ_{i})$ may be selected, where $μ_{i} = {\begin{matrix} \frac{1}{2} - \frac{1}{2} \cdot \frac{a^{l - i} - 1}{a^{l - 1} - 1}, i = 1, 2, \dots, l \\ \frac{1}{2} + \frac{1}{2} \cdot \frac{a^{i - l} - 1}{a^{l - 1} - 1}, i = l + 1, \dots, n \end{matrix}$

Fig. 3

Gaussian fuzzy numbers.

$σ_{i} = 0.04 + 0.01 \cdot abs (i - \frac{n + 1}{2}),$ where $a = 1.2, l = \frac{n + 1}{2}$ , i = 1, 2, ⋯ , n.

Step 3. Calculation of associated order-inducing variables.

In this step, the order-inducing variables {u_i|i = 1, 2, ⋯ , n} that are associated with the linguistic values {G_i|i = 1, 2, ⋯ , n} are determined by calculating the distance between every element in {G_i|i = 1, 2, ⋯ , n} as given by the historical data or experts, and every element in ${{\tilde{G}}_{1}, {\tilde{G}}_{2}, \dots, {\tilde{G}}_{m}}$ as given by the evaluator in Step 2. Elements in {G_i|i = 1, 2, ⋯ , n} are first normalized into the same dimensionless domain as those in ${{\tilde{G}}_{1}, {\tilde{G}}_{2}, \dots, {\tilde{G}}_{m}}$ , and the order-inducing variable u_i is defined as a weighted sum of the associated order-inducing variables ${{\tilde{u}}_{1}, {\tilde{u}}_{2}, \dots, {\tilde{u}}_{m}} = {1, 2, \dots, m}$ :

$\begin{matrix} u_{i} = \sum_{k = 1}^{m} {weight}_{i, k} \cdot {\tilde{u}}_{k} \\ \underline{\underline{def}} \sum_{k = 1}^{m} (\frac{\frac{1}{D^{q} (G_{i}, {\tilde{G}}_{k})}}{\sum_{k = 1}^{m} \frac{1}{D^{q} (G_{i}, {\tilde{G}}_{k})}}) \cdot k \end{matrix}$ (20) where D represents the L₁-type Hausdorff distance defined in (15). Parameter q is generally set to 2. When q→ + ∞, the method of determining u_i via (20) becomes the nearest neighbor method, i.e.

$u_{i} = {\tilde{u}}_{j} = j = \underset{k \in {1, 2, \dots, m}}{argmin} D (G_{i}, {\tilde{G}}_{k})$ (21)

Thus, the linguistic attribute value for each attribute is obtained by the evaluator.

Step 4. Calculation of the evaluation result

Aggregate the linguistic attribute values <u_i, G_i>, i = 1, 2, ⋯ , n, obtained in Step 3 according to the Gaussian fuzzy number based FIOWA operator (18), then denote the aggregation result as Z.

Step 5. Matching the aggregation result

Based on Theorem 2, the aggregation result Z obtained in Step 4 is also a Gaussian fuzzy number. However, in general, $Z \notin {{\tilde{G}}_{1}, {\tilde{G}}_{2}, \dots, {\tilde{G}}_{m}}$ is not among the fuzzy linguistic values defined by the evaluator. The nearest neighbor method must thus be utilized again to select a Gaussian fuzzy number from among ${{\tilde{G}}_{1}, {\tilde{G}}_{2}, \dots, {\tilde{G}}_{m}}$ as the final evaluation result $\tilde{Z}$ , such that

$\tilde{Z} = {\tilde{G}}_{j}, j = \underset{k \in {1, 2, \dots, m}}{argmin} D (Z, {\tilde{G}}_{k})$ (22) where D represents the L₁-type Hausdorff distance defined in (15).

5 Numerical examples

This section offers two experiments to illustrate the use of the proposed EPFIOWA approach in dealing with linguistic multi-attribute evaluation problems. Section 5.1 uses a simple example to calculate the evaluation results at different optimism levels to analyze the parameters α (Orness level) in the EPFIOWA method. Section 5.2 discusses a problem related to the battery recycling process, two classical OWA methods are used as comparative methods to verify the effectiveness of the proposed method.

5.1 An illustrative example

Assume that an object to be evaluated has n = 4 attributes. The attribute values given by some experts for these four attributes are $G_{1} = G (0.5, 0.02), G_{2} = G (0.9, 0.04),$ $G_{3} = G (0.2, 0.05), G_{4} = G (0.4, 0.10),$ respectively, and the Gaussian fuzzy numbers given by an evaluator are as shown in Fig. 3. The level of optimism of the evaluator is α = 0.6. The predefined weighting vector W^N is based on a Gaussian distribution given in (3) and (6) with K = 1. The evaluation result is obtained by the following steps.

Step 1. Given the level of optimism α = 0.6, set β₁ = β₂ = 1 with the predefined weighting vector $W^{N} = [0.1550, 0.3450, 0.3450, 0.1550] .$

According to (3) and (6), the obtained OWA weighting vector W via Model (α, W^Pro) is thus $W = [0.2450, 0.3750, 0.3150, 0.0650]$

Step 2. The linguistic attribute values given by experts are G_i, i = 1, ⋯ , 4. And the evaluation two-tuples $〈 {\tilde{u}}_{i}, {\tilde{G}}_{i} 〉, i = 1, 2, \dots, 7$ , as given by evaluators have values as shown in (19) and Fig. 3.

Step 3. Calculate the distances between each G_i and every element in ${{\tilde{G}}_{1}, {\tilde{G}}_{2}, \dots, {\tilde{G}}_{7}}$ by applying (21), where i = 1, 2, 3, 4. The four linguistic attribute values thus become four two-tuples in the form: $G_{1} \to 〈 u_{1}, G_{1} 〉, G_{2} \to 〈 u_{2}, G_{2} 〉,$ $G_{3} \to 〈 u_{3}, G_{3} 〉, G_{4} \to 〈 u_{4}, G_{4} 〉,$ where u₁ = 4.00, u₂ = 6.03, u₃ = 2.01, u₄ = 3.31 are calculated by (20) and their order is u₃ < u₄ < u₁ < u₂.

Step 4. Calculate the evaluation result by applying the Gaussian fuzzy number based FIOWA operator. $Z = G (0.5470, 0.0520)$

Step 5. Calculate the distance between Z and ${\tilde{G}}_{k}, k = 1, 2, \dots, 7$ , as in (21): $D (Z, {\tilde{G}}_{1}) = 0.5695$ $D (Z, {\tilde{G}}_{2}) = 0.3592$ $D (Z, {\tilde{G}}_{3}) = 0.1869$ $D (Z, {\tilde{G}}_{4}) = 0.0621$ $D (Z, {\tilde{G}}_{5}) = 0.0929$ $D (Z, {\tilde{G}}_{6}) = 0.2652$ $D (Z, {\tilde{G}}_{7}) = 0.4755$

As $D (Z, {\tilde{G}}_{4}) = 0.0621$ is the smallest among all $D (Z, {\tilde{G}}_{k}), k = 1, 2, \dots, 7$ , the final linguistic evaluation is ${\tilde{u}}_{4} = Moderate$ .

Repeating this experiment under different levels of optimism α_i = 0, 0.1, 0.2, ⋯ , 1, gives the evaluation results Z_i = G (μ_i, σ_i) as shown in Fig. 4. The mean value of the evaluation result increases gradually with the increase in α. While, the variance shows no obvious change. The reason for this is that a large difference emerges in the mean value of the linguistic attribute values {G_i|i = 1, 2, 3, 4}, while the difference in variance remains relatively small. As α increases, the evaluation result thus gives higher weights to those terms with better attribute values. Consequently, if the alternative has some attributes with good evaluation grades (near ${\tilde{G}}_{m}$ ), the overall evaluation result will be significantly improved when the level of optimism increases.

Fig. 4

Evaluation Results under different value of orness level α.

5.2 Selecting a third-party reverse logistics provider

This section discusses a problem related to the battery recycling process. The aim is to select an appropriate third-party reverse logistics supplier to carry out reverse logistics activities, with the example adapted from [50] and [25]. In order to verify the effectiveness of the proposed method, two classic evaluation methods mentioned above, namely, OWA and IOWA, are used as comparison methods and applied to the same problem.

In this example, the evaluator needs to evaluate five potential suppliers by considering seven factors (attributes). These attributes are: (1) F₁ : quality; (2) F₂ : delivery; (3) F₃ : cost; (4) F₄ : rejection rate; (5) F₅ : technological capability; (6) F₆ : development potential; and (7) F₇ : Service attitude. Five suppliers X_i, i = 1, 2, ⋯ , 5, are offered as potential suppliers, and several experts are invited to evaluate the competencies of these five suppliers in these areas. The numerical attribute values from these experts are then converted into Gaussian fuzzy number-based linguistic values in the interval [0,1], the resulting means and standard deviations of these Gaussian fuzzy numbers are listed in Table 1. Based on these statistical results, the linguistic value G_ij of candidate X_i for any attribute F_j can thus be obtained. For example, the linguistic value of X₁ in F₂ is expressed by the Gaussian fuzzy number G₁₂ = G (0.5, 0.02). Note that G₄₂ and G₄₅ have extremely large variances, which indicates that there are considerable differences in the experts’ evaluation.

Table 1
Linguistic values collected from experts

G _ij F ₁ F ₂ F ₃ F ₄ F ₅ F ₆ F ₇

X ₁ G (0.5, 0.05) G (0.5, 0.02) G (0.4, 0.04) G (0.4, 0.04) G (0.6, 0.03) G (0.5, 0.05) G (0.8, 0.01)

X ₂ G (0.6, 0.03) G (0.1, 0.03) G (0.6, 0.02) G (0.3, 0.02) G (0.5, 0.04) G (0.6, 0.02) G (0.4, 0.04)

X ₃ G (0.3, 0.03) G (0.7, 0.05) G (0.3, 0.03) G (0.7, 0.07) G (0.8, 0.02) G (0.6, 0.03) G (0.7, 0.07)

X ₄ G (0.6, 0.02) G (0.7, 0.2) G (0.4, 0.02) G (0.5, 0.02) G (0.7, 0.4) G (0.8, 0.01) G (0.7, 0.01)

X ₅ G (0.5, 0.03) G (0.6, 0.02) G (0.5, 0.01) G (0.6, 0.04) G (0.5, 0.05) G (0.5, 0.03) G (0.6, 0.02)

G _ij	F ₁	F ₂	F ₃	F ₄	F ₅	F ₆	F ₇
X ₁	G (0.5, 0.05)	G (0.5, 0.02)	G (0.4, 0.04)	G (0.4, 0.04)	G (0.6, 0.03)	G (0.5, 0.05)	G (0.8, 0.01)
X ₂	G (0.6, 0.03)	G (0.1, 0.03)	G (0.6, 0.02)	G (0.3, 0.02)	G (0.5, 0.04)	G (0.6, 0.02)	G (0.4, 0.04)
X ₃	G (0.3, 0.03)	G (0.7, 0.05)	G (0.3, 0.03)	G (0.7, 0.07)	G (0.8, 0.02)	G (0.6, 0.03)	G (0.7, 0.07)
X ₄	G (0.6, 0.02)	G (0.7, 0.2)	G (0.4, 0.02)	G (0.5, 0.02)	G (0.7, 0.4)	G (0.8, 0.01)	G (0.7, 0.01)
X ₅	G (0.5, 0.03)	G (0.6, 0.02)	G (0.5, 0.01)	G (0.6, 0.04)	G (0.5, 0.05)	G (0.5, 0.03)	G (0.6, 0.02)

*Data adapted from [25]. Values in bold indicate that the linguistic values have large variances.

Assuming there are seven evaluation grades within the cognitive scope of the evaluator, as listed in (19), with corresponding Gaussian fuzzy numbers as shown in Fig. 3. The optimistic evaluator selects an orness level of α = 0.6. The initial weighting vector $W^{N} = (0.1344, 0.1739, 0.2121,$ $0.2161, 0.1693, 0.0882, 0.0059)$ is thus obtained based on (3) and (6), with K = 1.

The final linguistic evaluation results for all five candidates are listed in Table 2. This highlights that X₃ and X₄ are the two best suppliers based on the proposed model, with linguistic evaluation values

Table 2

Linguistic evaluation results for all candidates

Suppliers	Evaluation Results Z_i	Order-inducing variables u_i	Rank	Matching Grades
X ₁	G (0.5483, 0.0338)	4.1624	4	${\tilde{G}}_{4}$ Moderate
X ₂	G (0.5151, 0.0291)	4.0105	5	${\tilde{G}}_{4}$ Moderate
X ₃	G (0.6589, 0.0484)	4.9838	1	${\tilde{G}}_{5}$ Slightly Good
X ₄	G (0.6771, 0.1372)	4.9350	2	${\tilde{G}}_{5}$ Slightly Good
X ₅	G (0.5521, 0.0265)	4.2070	3	${\tilde{G}}_{4}$ Moderate

$Z_{3} = G (μ_{3}, σ_{3}) = G (0.6589, 0.0484),$ $Z_{4} = G (μ_{4}, σ_{4}) = G (0.6771, 0.1372) .$

From the perspective of mean value alone, Z₄ appears better than Z₃ as μ₄ > μ₃. However, in terms of standard deviation, σ₃ ⪡ σ₄, which means the uncertainty of result Z₄ is very large. The order-inducing variable calculated by (20) are given in Table 2, from which one can see that Z₃ is slightly better than Z₄, consistent with the fact that suppliers with stable evaluation results may be preferred over highly variable suppliers.

The membership functions of Z₃ and Z₄ are shown in Fig. 5.

Fig. 5

Comparison of evaluation results for X₃ and X₄.

In order to verify the effectiveness of the proposed method, two classic OWA and IOWA methods are applied to the same evaluation problem. In the OWA method, the weighting vector is the same as that used in the EPFIOWA, the mean value information in Table 1 is used as numerical attribute values, and the standard deviation information in Table 1 is ignored. In the IOWA method, the weighting vector is set to (1/7, 1/7, ⋯ , 1/7), and the other calculation steps are the same as those in EPFIOWA. The evaluation results of the three methods are shown in Table 3. As the standard deviation information is not fully considered, the OWA method regards X₄ as the best supplier, which has the largest difference in attribute values in F₂ and F₅. The IOWA method also regards X₄ as the best supplier due to the fact that the optimism level of the evaluator is not fully considered. In this example, therefore, the EPFIOWA method proposed in this paper can get more reasonable results than the other methods.

Table 3

Evaluation results for different methods

Suppliers	OWA	IOWA	EPFIOWA
X ₁	4	4	4
X ₂	5	5	5
X ₃	2	2	1
X ₄	1	1	2
X ₅	3	3	3

6 Conclusions

The traditional induced OWA operator is suitable for aggregating the information in the form of numerical values. However, it cannot handle problems in which the attribute values are given in a linguistic format, especially when these linguistic attribute values are difficult to sort. This paper thus introduces a frequently used representation type for linguistic values, i.e. the Gaussian fuzzy numbers, to allow the arithmetic operations and distance measures required for IOWA to be deduced. Based on this, a Gaussian-fuzzy-number-based IOWA operator is defined that takes argument pairs as its inputs, of which one component is the Gaussian-fuzzy-number-based attribute value, and the other is an order-inducing variable associated with the Gaussian fuzzy number.

A critical step in developing an OWA operator is the determination of OWA operator weights. In order to obtain a reasonable weighting vector for the proposed FIOWA operator, a novel programming method is thus proposed to determine the OWA weights according to the optimism level of the evaluator and an initial probability distribution. Some of the desired properties of the proposed method, such as its symmetry property, are examined in this study.

The research results show that in the proposed model, if an alternative has some good attributes, the evaluation results of the proposed EPFIOWA method will be significantly improved with the improvement of the optimistic level. The proposed EPFIOWA operator can obtain reasonable evaluation results that are consistent with the observed situation because the proposed method fully takes into account the uncertainty of attribute values and the optimistic level of the evaluator.

The extension and application of the developed EPFIOWA to more fields is a problem worthy of further study.

Footnotes

Acknowledgment

This research is supported by the Science and Technology Programs of Quanzhou (2021CT0010, 2022N002S), the Fujian Natural Science Foundation Project (2021J01001), and Fujian Key Laboratory of Financial Information Processing (Putian University, JXC202205).

The authors would like to thank the editors and the anonymous reviewers for their insightful and constructive comments and suggestions, which improved the quality of this article. The authors would like to acknowledge the support by Fujian provincial key laboratory of data-intensive computing, Fujian university laboratory of intelligent computing and information processing, Fujian provincial big data research institute of intelligent manufacturing.

Conflicts of interest

The authors declare no conflicts of interest.

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