Computational method for monotonic hesitant fuzzy linguistic terms by fitting Weibull distribution function concerns explicit and implicit appraisal information

Abstract

Natural monotonic linguistic language is widely used to express experts’ uncertain subjective appraisal opinion, such as “More than”, “At least”, “Less than” and “At most”, which reveals explicit information about performance range and implicit information about his hiding preference on a linguistic scale. A novel computational method for monotonic hesitant fuzzy linguistic terms is developed to transfer experts’ uncertain appraisal information to decision-making data, which can systematically consider and mine expert’s obvious explicit and hidden implicit appraisal information. Specially, the comprehensive meanings of monotone decreasing and increasing hesitant fuzzy linguistic terms are investigated, in which both explicit and implicit appraisal information are explored to reveal its actual meaning. Additionally, Weibull distribution functions with three parameters are fitted considering the comprehensive meaning of monotone increasing appraisals, which is determined by a multi-objective programming model following ABC classification method. Symmetry principle is employed to confirm the expression of monotone decreasing appraisals, which are transferring from monotone increasing appraisals with same length of domain field. Moreover, feasibility analysis is explored to show the influence of parameters on decision-making precision. Finally, a numerical study is conducted to show the feasibility and advantage of the new method, which can effectively improve the precision of computational transfer by comparing to previous method.

Keywords

Monotonic hesitant fuzzy linguistic term set implicit appraisal information Weibull distribution ABC analysis

1 Introduction

Natural language, which can present the experts’ professional knowledge and previous experience, provides subjective information support in decision-making and artificial intelligence fields. Zadeh [1 –3] introduced the linguistic variable with a context-free grammar approach which can generate the linguistic terms as sentences which give experts more alternatives to express their preference, especially in complex and uncertain decision making environment. Because of complicated environment, limited knowledge and bounded rationality, monotonic uncertain linguistic terms are commonly used to express expert’s uncertain appraisal information, which cover multiple linguistic scales. For example, if an expert cannot give an exact precise appraise on a supplier’s quality assurance activity, he may compare the supplier’s performance to others’ and give subjective opinions as “better than average”, “at least good”, “weaker than poor level” or “at most as excellent”. These types of linguistic terms were described by the comparative linguistic term expression develop by Rodríguez et al. [4, 5]. They also defined a function to transform a comparative linguistic term expression to hesitant fuzzy linguistic term set (HFLTS) [4, 5]. In recent years, extended research on HFLTS was made in many areas with additional information, such as the priority with the computation of envelope [6, 7], probabilistic distribution [8, 9] and membership degree in discrete fuzzy numbers [10, 11] of possible terms. This transformation function reflects experts’ explicit appraisal information which is denoted by a linguistic scale interval. In this paper, implicit appraisal information as well as explicit information is considered in the process of transformation of monotonic uncertain linguistic terms.

According to monotonic uncertain linguistic terms, an expert presents his explicit appraisal information as linguistic scale interval. What is more important, he also implies his implicit appraisal information about his hidden fuzzy preference on one linguistic grade, which should be explored and utilized by transforming linguistic information to data. For example, a seven-point Likert scale S = {s₀ = Extremely Weak ; s₁ = Very Weak ; s₂ = Weak; s₃ = Medium ; s₄ = Good ; s₅ = Very Good ; s₆ = Excellent} is employed to collect expert’s opinion. An expert gives his appraisal information on supplier’s performance as “at least good”. His explicit information is that supplier’s performance covers grade interval from “good” to “excellent” as [s₄, s₆]. Because “at least good” covers “at least very good”, [s₄, s₆] ⊃ [s₅, s₆]. The expert chooses “at least good ” as [s₄, s₆] other than “at least very good” as [s₅, s₆], which implies that he believes that the supplier’s performance is located in the grade interval [s₄, s₅] with higher probability than other adjacent intervals. Such implicit appraisal information comes from expert’s intrinsic psychological activity and logic thinking process to give explicit appraisal result.

However, the existing related research, which mainly focuses on expert’s explicit appraisal information about monotonic uncertain linguistic terms, always utilized a normal distribution function to estimate expert’s quantitative opinion as membership grade [12 –17]. For example, monotonous linguistic expression “at least good” was transferred as a normal distribution function in domain field [s₄, s₆]. Mean value is in the middle of domain field [4, 6] as $E (x) = \int_{4}^{6} xf (x) dx = 5$ and standard deviation is σ = 1/3, both of which are determined by the maximum and minimum of domain field. This is unreasonable when experts’ implicit appraisal information is extracted from monotonic hesitant fuzzy linguistic terms to express their comprehensive preference. In this case, the normal distribution function obviously is not enough to realize the quantitative transformation of monotonic hesitant fuzzy terms.

Weibull distribution is a continuous probability distribution with three parameters, which is widely used in many domains, such as reliability engineering, information retrieval and weather forecasting. The advantage of Weibull distribution is that different combinations of shape, scale and location parameters can easily simulate various curves, such as exponential distribution, uniform distribution and normal distribution. Consequently, one can employ Weibull distribution to reflect the comprehensive meaning of monotonic hesitant fuzzy by determining the potential selection of shape, scale and location parameters concerning expert’s implicit hidden appraisal attitude. Not only normal distribution curve, but also exponential distribution, uniform distribution, S sharp curves can be displayed according to logic rules.

In recent years, many researchers described uncertain linguistic terms as hesitant fuzzy linguistic information and explored their quantitative meanings in decision-making problems. Wang et al. [18] summarized the current methodologies for hesitant fuzzy linguistic decision-making. In the domain of computing strategy with possible terms dealing with HFLTS, the main contributions were focused in the following areas as fusion operator, entropy and cross-entropy as well as distance calculation and its applications.

Fusion operator of HFLTS.

In order to aggregate scattered information into alternative’s overall performance, fusion operators were designed and developed in various decision-making situations as multi-criteria, multi-stage or group negotiation, which concludes hesitant fuzzy linguistic integration operator [19 –24], ordered weighted average (OWG) operator [25 –27], and the hesitant fuzzy linguistic prioritized weighted geometric (HFLPWG) operator [28 –30]. Lee et al. [7] studied the comparative relation of hesitant fuzzy linguistic terms possibility in group decision making and analyzed the influence of various kind of hesitant fuzzy linguistic operators on the fuzzy group decision making. Krishankumar et al. [31] proposed hierarchy hesitant fuzzy hybrid aggregation operator for sensibly aggregating of experts’ preference information.

The entropy and cross-entropy of HFLTS.

In some complicated multi-criteria decision-making (MCDM) problems, the weights of criteria, stage or expert can be calculated concerning hesitant fuzzy linguistic entropy and cross entropy [32 –37], which concerns independent effect of a hesitant fuzzy linguistic element (HFLE) and interactive effect between HFLEs with respect to each criterion. Gou et al. [35] developed the definitions of hesitant fuzzy linguistic entropy and cross-entropy and proposed an alternative queuing method to deal with the MCDM problems. Yang et al. [36] established an optimization model for obtaining criteria weights concerning the cross entropy of hesitation fuzzy intuitionistic information.

Distance between HFLTS and its applications

Distance between hesitant fuzzy linguistic terms is widely used in decision-making problems for checking alternatives’ priority sequence. Various distance calculation formula of hesitant fuzzy linguistic term sets were designed to solve this problem, such as the Hamming distance, the Euclidean distance, the Hausdorff distance and their generalized [38 –41] or weighted forms [42 –45]. Jin and Zheng [38] designed the Hamming distance formula, the Euclidean distance formula and the Hausdorff distance formula, which can be used to achieve hierarchy equality between arbitrary adjacent hesitant fuzzy linguistic terms. Hesamian and Shams [44] studied the distance and similarity measures for hesitant fuzzy linguistic term sets (HFLTSs) to characterize the preference priority between different HFLTSs. Additionally, HFLTSs and its distance calculation methods are widely utilized in many areas, such as cluster [46, 47], emergency decision [48] and hazard assessment [49].

According to literature review, existing research about hesitant fuzzy linguistic information has already achieved many contributions in theoretical and practical levels, especially on linguistic information management method, criteria weight determination and distance calculation measurement. But the study on the characters of monotonic linguistic information and its computational method need to be expended. What is more important, except for explicit linguistic information, expert’s implicit linguistic information, such as presentation preference hidden under his obvious expression, should be widely concerned and explored to obtain a suitable computational result. In other words, not only “What is expert’s saying” (explicit information), but also “What is expert’s unwilling saying” (implicit information) should be comprehensively considered and utilized to appropriately estimate expert’s subjective appraisal opinions. Consequently, there are some questions need to be answered according to the monotonous uncertain linguistic terms. First, except for the explicit expression about linguistic interval grade, is there any implicit meaning under the expression surface about expert’s preference to a certain linguistic grade? Second, how to fit continuous functions with several parameters systematically reflect the expert’s explicit and implicit appraisal? Third, how to comprehensively use the numbers within linguistic continuity interval to estimate the expert’s appraisal result, other than some discrete grade remarks?

Consequently, this paper contributes to design the computational method especially for monotonic linguistic information terms comprehensively using expert’s explicit and implicit appraisal information. In section 2, some related definitions are introduced. In section 3, the characters of monotonic linguistic information are revealed, which are expressed in the form of HFLTs. In section 4, according to monotone decreasing linguistic terms, a multi-objective programming model is constructed considering ABC analysis and Weibull function with 3 parameters can be determined. The transferring method for monotone increasing linguistic terms is developed with symmetry principle. Finally, a numerical study is conducted in section 5 in which comparison result which shows the effectiveness and advantages of the new method mentioned in this paper.

2 Preliminaries

In order to facilitate the following presentation, some important definitions are introduced in this section, which include linguistic term set, transformed function and Weibull distribution.

Definition 1 [50]. Let S = {s_i|i = 0, 1, ⋯ , g} , g > 0 be a linguistic term set and s_i represents linguistic term given by expert. The total linguistic term set S satisfies the following characteristics:

s_i > s_j, only if i > j;

The negation operator is defined as: Neg (s_i) = s_j, j = g - i.

Definition 2 [4]. An HFLTS, H_S, is an ordered finite subset of consecutive linguistic terms of S, where S = {s_i|i = 0, 1, ⋯ , g} , g > 0 is a linguistic term set.

Liao et al. [41] proposed another more mathematical definition of HFLTS based on Rodríguez et al.’s. In this paper, the following definition is utilized as the base of expression concerning HFLTS.

Definition 3 [51]. Leta_i ∈ A,i = 1, 2, ⋯ , N, and S = {s_i|i = 0, 1, ⋯ , g} , g > 0, be a linguistic term set. The ordered finite subset of consecutive hesitant fuzzy linguistic term set on A,H_S, is in mathematical terms of $H_{S} = {〈 a_{i}, h_{S} (a_{i}) 〉 | a_{i} \in A}$ (1)

In Equation 1, h_S (a_i) : A → S is the set of values in the linguistic term S and can be expressed as h_S (a_i) = {s_{φ
_l} (a_i) |s_{φ
_l} (a_i) ∈ S, l = 1, ⋯ , L (a_i)}, in which L (a_i) is the number of linguistic terms in h_S (a_i) denoting the possible degrees of the linguistic variable a_i to the linguistic term set S. For convenience, h_S (a_i) is called the hesitant fuzzy linguistic element (HFLE) and H_S is the set of all HFLEs.

Rodríguez et al. [4, 5] proposed the comparative linguistic expression with context-free grammar. For differentiating between monotone increasing and monotone decreasing hesitant fuzzy linguistic terms, we use the following similar denotation.

Definition 4 [4, 5]. Let E : s_j → H_S be a function that transforms the linguistic term expressions concerning s_j into HFLTSs. The linguistic expressions are transformed into monotonous hesitant fuzzy linguistic terms through the following transformations: $E (not less than s_{j}) = {s_{i} | s_{i} \geq s_{j}, i \in [j, g], s_{i} \in S};$ $E (better than s_{j}) = {s_{i} | s_{i} > s_{j}, i \in [j + 1, g], s_{i} \in S};$ $E (not more than s_{j}) = {s_{i} | s_{i} \leq s_{j}, i \in [0, j], s_{i} \in S};$ $E (worse than s_{j}) = {s_{i} | s_{i} < s_{j}, i \in [0, j - 1], s_{i} \in S} .$

MHFLTs can be classified into monotone increasing hesitant fuzzy linguistic terms (MIHFLTs) and monotone decreasing hesitant fuzzy linguistic terms (MDHFLTs). MIHFLTs contain “not more than s_j” and “worse than s_j”. MDHFLTs contain “not less than s_j” and “better than s_j”.

Definition 5 [52]. The probability density function f (x) and cumulative distribution function F (x) of Weibull distribution are given as bellows. $f (x) = \frac{β}{α} {(\frac{x - θ}{α})}^{β - 1} e^{- {(\frac{x - θ}{α})}^{β}}, (0 < θ < x)$ (2) $F (x) = 1 - e^{- {(\frac{x - θ}{α})}^{β}}, (0 < θ < x)$ (3)

θ is the location parameter and x = θ is the asymptotic line of f (x) α and β are the scale parameter and shape parameter of f (x), respectively. If β > 1, probability density function f (x) monotonically increases. If β < 1, probability density function f (x) monotonically decreases. If β = 1, Weibull distribution is an exponential distribution.

3 Monotonic hesitant fuzzy linguistic terms concerning implicit preference

In this section, the definition of monotonic hesitant fuzzy linguistic terms (MHFLTs) is introduced. Additionally, the expert’s hidden implicit attitude is studied and the comprehensive meanings of MHFLTs are explored.

3.1 Definition of monotonic hesitant fuzzy linguistic terms

Definition 6. Monotonic hesitant fuzzy linguistic terms (MHFLTs) denote expert’s increasing or decreasing subjective appraisal information in discourse domain, such as “not less than”, “better than”, “not more than” and “worse than”, which determine the lower control limit or upper control limit.

While giving the monotonous linguistic expressions s_j, expert revealed his implicit information hidden under the explicit information, which reflects expert’s internal consideration and fuzzy preferences. Consequently, the computational method for monotonous linguistic terms should be comprehensively explored and utilized both explicit and implicit information.

3.2 Comprehensive meaning of monotone decree-sing hesitant fuzzy linguistic terms

The MDHFLTs mainly contain two forms of monotonous natural linguistic expressions as “not less than s_j” and “better than s_j”. The comprehensive meaning of MDHFLTs is given as Table 1.

Table 1
Comprehensive meaning of MDHFLTs

Linguistic term s_j HFLTS H_s(Explicit Information) Implicit Information

Not less than s_j 0 < j ≤ g - 1 {s_j, s_j+1, ⋯ , s_g} h_S (s_i|s_i ∈ [s_j, s_j+1])> h_S (s_i|s_i ∈ [s_j+1, s_g])

j > g - 1 {s_g} H_S (not less than s_j) = {g, 1}

Better than s_j 0 ≤ j < g - 1 {s_j+1, s_j+2, ⋯ , s_g} h_S (s_i|s_i ∈ [s_j+1, s_j+2])> h_S (s_i|s_i ∈ [s_j+1, s_g])

j ≥ g - 1 {s_g} H_S (better than s_j) = {g, 1}

Linguistic term s_j	HFLTS H_s(Explicit Information)	Implicit Information
Not less than s_j	0 < j ≤ g - 1	{s_j, s_j+1, ⋯ , s_g}	h_S (s_i\|s_i ∈ [s_j, s_j+1])> h_S (s_i\|s_i ∈ [s_j+1, s_g])
	j > g - 1	{s_g}	H_S (not less than s_j) = {g, 1}
Better than s_j	0 ≤ j < g - 1	{s_j+1, s_j+2, ⋯ , s_g}	h_S (s_i\|s_i ∈ [s_j+1, s_j+2])> h_S (s_i\|s_i ∈ [s_j+1, s_g])
	j ≥ g - 1	{s_g}	H_S (better than s_j) = {g, 1}

Comprehensive meaning of “not less than s_j”

Assume that an expert gives his uncertain appraisal information as “not less than s_j”, E (not less than s_j) = {s_i|s_i ≥ s_j, i ∈ [j, g] , s_i ∈S}, his explicit information is that appraisal result should obviously cover linguistic grade interval [s_j, s_g]. If 0 < j ≤ g - 1, because of [j + 1, g] ∈ [j, g], the expert gives judgment “not less than s_j” rather than “not less than s_j+1”, which implies that he believes that the appraisal object’s performance is located in the grade interval [j, j + 1] with much higher probability than other intervals. From Definition 2, h_S (s_i|s_i ∈ [s_j, s_j+1]) >> h_S (s_i| s_i ∈ [s_j+1, s_g]). If j > g - 1, E (not less than s_j) = {s_i|s_i = g, i ∈ [j, g]} , H_S (not less than s_j) = {g, 1}, which means expert’s linguistic expression “not less than s_g-1” shows his appraisal opinion is s_g with 100% probability.

Comprehensive meaning of “better than s_j”

According to uncertain appraisal information as “better than s_j”, E (better than s_j) = {s_i|s_i > s_j, i ∈ [j + 1, g]}, expert’s explicit information is that appraisal result should obviously cover linguistic grade interval [s_j+1, s_g]. If 0 ≤ j < g - 1, the expert gives judgment “better than s_j” rather than “better than s_j+1”, which implies that the experts believes that appraisal object’s performance is located in the grade interval [j + 1, j + 2] with higher probability, h_S (s_i|s_i∈ [s_j+1, s_j+2]) >> h_S (s_i|s_i ∈ [s_j+1, s_g]). If j ≥ g - 1, H_S (better than s_j) = {g, 1}.

3.3 Comprehensive meaning of monotone increasing hesitant fuzzy linguistic terms

The MIHFLTs mainly contain two forms of monotonous natural linguistic expressions as “not more than s_j” and “worse than s_j”. The comprehensive meaning of MIHFLTs is given as Table 2.

Comprehensive meaning of “not more than s_j”

If an expert gives uncertain appraisal information as “not more than s_j”, E (not more than s_j) = {s_i|s_i ≤ s_j, i ∈ [0, j]}, his explicit information is that appraisal result should cover linguistic grade interval [s₀, s_j]. If 1 ≤ j < g, because of [j + 1, g] ∈ [j, g], the expert gives judgment “not more than s_j” rather than “not more than s_j-1”, which implies that the performance is located in the grade interval of [j - 1, j] with higher probability than other intervals, h_S (s_i|s_i ∈ [s_j-1, s_j]) >> h_S (s_i|s_i ∈ [s₀, s_j-1]). If j < 1, E (not more than s_j) = {s_i |s_i = 0, i ∈ [0, j]}, H_S (not more than s_j) = {0, 1}, which means expert’s linguistic expression “not more than s₀” shows his appraisal opinion is s₀ with 100% probability.

Comprehensive meaning of “worse than s_j”

Referring to uncertain appraisal information “worse than s_j”, E (worse than s_j) = {s_i|s_i < s_j, i ∈ [0, j - 1]}, his explicit information is that appraisal result should obviously cover linguistic grade interval [s₀, s_j-1]. If 1 < j ≤ g, the expert gives judgment “worse than s_j” rather than “worse than s_j-1”, which implies that the appraisal object’s performance is located in the grade interval from “j - 2” and “j - 1” as[j - 2, j - 1] with higher probability than other intervals, h_S (s_i|s_i ∈ [s_j-2, s_j-1]) >> h_S (s_i|s_i ∈ [s₀, s_j-2]). If j ≤ 1, H_S (worse than s_j) = {0, 1}.

Table 2
Comprehensive meaning of MIHFLTs

Linguistic term s_j HFLTS H_s (Explicit Information) Implicit Information

Not more than s_j 1 ≤ j < g {s₀, s₁, ⋯ , s_j} h_S (s_i|s_i ∈ [s_j-1, s_j])> h_S (s_i|s_i ∈ [s₀, s_j-1])

j < 1 {s₀} H_S (not more than s_j) = {0, 1}

Worse than s_j 1 < j ≤ g {s₀, s₁, ⋯ , s_j-1} h_S (s_i|s_i ∈ [s_j-1, s_j])> h_S (s_i|s_i ∈ [s₀, s_j-1])

j ≤ 1 {s₀} H_S (worse than s_j) = {0, 1}

Linguistic term s_j	HFLTS H_s (Explicit Information)	Implicit Information
Not more than s_j	1 ≤ j < g	{s₀, s₁, ⋯ , s_j}	h_S (s_i\|s_i ∈ [s_j-1, s_j])> h_S (s_i\|s_i ∈ [s₀, s_j-1])
	j < 1	{s₀}	H_S (not more than s_j) = {0, 1}
Worse than s_j	1 < j ≤ g	{s₀, s₁, ⋯ , s_j-1}	h_S (s_i\|s_i ∈ [s_j-1, s_j])> h_S (s_i\|s_i ∈ [s₀, s_j-1])
	j ≤ 1	{s₀}	H_S (worse than s_j) = {0, 1}

4 Computational method of monotone hesitant fuzzy linguistic terms

Concerning the comprehensive meaning of MHFLTs, the computational method is developed to determine Weibull distribution function with three parameters following ABC classification and symmetry principle. Besides, the sensitivity analysis of three parameters is analyzed to reveal some important rules.

4.1 Weibull distribution function fit for monotone decreasing hesitant fuzzy linguistic terms based on ABC classification analysis

The Weibull distribution function is widely used in reliability engineering, forecasting, and industrial engineering, et al. The advantage of Weibull distribution function is that various combinations of three parameters α, β and θ which can formed to constitute various curves shown in Fig. 1.

Fig. 1

Various curves of Weibull distribution functions.

Consequently, Weibull distribution function can be fitted by expressing expert’s explicit and implicit linguistic information.

In order to express the functional curve of MDHFLTs in linguistic term set S = {s_i|i = 0, 1, ⋯ , g} , g > 0, Fig. 2 is designed in which x-coordinate means the subscript of s_i, i = 0, 1, ⋯ , g, and ordinate denotes the probability density function f (x) of MDHFLTs. If j ∈ [1, ⋯ , g], MDHFLTs “Not less than s_j ” and “better than s_j-1” cover the same linguistic scales {j, j + 1, ⋯ , g} and have the same meaning according to Table 1. Consequently, “not less than s_j” is selected to show its quantized process related to Weibull function fitting based on ABC analysis.

Fig. 2

The probability density function of MDHFLTs.

Step 1: According to the domain of “not less than s_j”, [j, g ], let x_λ = j + λ * (g - j), x_μ = j + μ * (g - j), 0 < λ < 0.5 < 1 - μ < 1. As shown in Fig. 2, one can find that [j, x_λ ] occupies a small proportion and [x_μ, g] takes a large scale of definition domain [j, g ]. Generally, λ ∈ [0.15, 0.5] , μ ∈ [0.2, 0.4]. According to ABC analysis [54 –56], λ = 15% , μ = 40 %.

Step 2: Let m and 1 - n be the m percentile and 1 - n percentile of Weibull distribution f (x), $0 < n < \frac{1}{2} < m < 1$ . As shown in Fig. 2, Area 1 (Orange part) denotes the larger probability,m, of expert’s appraisal opinion located in the smaller interval [j, x_λ ] and Area 3 (Blue part) means the smaller probability, n, of expert’s appraisal opinion located in the larger interval [x_μ, g]. Generally, m ∈ [0.7, 0.8] , n ∈ [0.05, 0.1]. According to ABC analysis methods [47 –49], m = 75% , n = 5 %.

Step 3: According to the comprehensive meaning of “not less than s_j”, h_S (s_i|s_i ∈ [s_j, s_j+1])>> h_S (s_i|s_i ∈ [s_j+1, s_g]), which means that f′ (x) <0, f″ (x) >0. Consequently, ABC classification analysis can be utilized to satisfy it. Specifically, the cumulative probability of HFLTs can reach the larger area as Area 1 in Fig. 2 with m percentage in smaller length interval [j, x_λ]. $\int_{j}^{x_{λ}} f (x) dx \geq m$ (4)

The cumulative probability of HFLTs can reach a smaller area as Area 3 in Fig. 2 with n percentage in larger length interval [x_μ, g]. $\int_{x_{μ}}^{g} f (x) dx \leq n$ (5) Consequently, the percentage of Area 2 can be $\int_{x_{λ}}^{x_{μ}} f (x) dx \leq 1 - n - m$ (6)

Step 4: Let F (x) be the Weibull distribution function at x, and a multi-objective programming model can be constructed as below, in which Equation (4) and Equation (6) are designed as objective 1 and objective 2 to be satisfied. $\begin{matrix} min Z = P_{1} d_{1}^{-} + P_{2} d_{2}^{+} \\ s . t {\begin{matrix} \int_{j}^{x_{λ}} f (x) dx - m + d_{1}^{-} - d_{1}^{+} = 0 \\ \int_{x_{λ}}^{x_{μ}} f (x) dx - (1 - m - n) + d_{2}^{-} - d_{2}^{+} = 0 \\ d_{i}^{-}, d_{i}^{+} \geq 0, i = 1, 2 \end{matrix} \end{matrix}$ (7) where P₁ and P₂ indicate the priority of the objectives, P₁ ≻ P₂. $d_{i}^{-}, d_{i}^{+}$ indicate the positive and negative deviation values of thei-th objective.

The optimal value marked α^* and β^* can be obtained from Equation (7) through MATLAB. Obviously, θ indicates the asymptote. If an expert gives his uncertain linguistic term as “not less than s_j”, θ = j.

4.2 Distribution function fit for monotone increasing hesitant fuzzy linguistic terms based on symmetry principle

According to the comprehensive meanings of Monotonic hesitant fuzzy linguistic terms, MLHFLTs and MDHFLTs have absolutely opposite meanings if they cover same interval length but from different starting nodes. Specially, “Not more than s_j” and “Worse than s_j +1” covers the same multiple linguistic scales {0, 1, ⋯ , j} and have the same comprehensive meaning. “Not more than s_j” is selected to show its quantized process related to symmetry principle.

As shown in Fig. 3, According to linguistic term set S = {s_i|i = 0, 1, ⋯ , g} , g > 0, “Not more than s_j” covers linguistic interval [0, j] and “Not less than s_g-j” covers linguistic interval [g - j, g] with the same interval length as j. The expert’s preference of “Not more than s_j” in [0, j] and “Not less than s_g-j” in [g - j, g] have absolutely opposite variation tendency from asymptote x = j negatively and x = g - j positively. Consequently, the distribution function q (x) of “Not more than s_j” can be obtained using symmetry principle which is oppositely transferred from the Weibull distribution function f (x) of “Not less than s_g-j” from Programming Model (7).

Fig. 3

Different curves of “not less than s_g-j”and “not more than s_j”.

Step 1: Let the MDHFLT “not more than s_j” and MIHELT “not less than s_g-j” be h_{S
₁} (s_i|s_i ∈ [0, s_j]) and h_{S
₂} (s_i|s_i ∈ [s_g-j, g]), which have same linguistic interval length and their asymptotes are x = j and x = g - j, respectively.

Step 2: Programming Equation (7) can be used to obtain Weibull distribution function f (x) of “Not less than s_g-j”.

Step 3: Because $x = \frac{g}{2}$ is the symmetric line of the distribution function q (x) of “Not more than s_j” and Weibull distribution function f (x) of “Not less than s_g-j”. $q (x) = f (\frac{g}{2} - x)$ .

4.3 Sensitivity analysis of parameters

Because distribution function q (x) of “Not more than s_j” and Weibull distribution function f (x) of “Not less than s_g-j” are symmetric. Weibull distribution function f (x) of “Not less than s_g-j”is selected for analyze the sensitivity of parameters.

Sensitivity analysis of parameter θ

If an expert gives his uncertain linguistic information as “not less than s_j”, the definition domain of Weibull distribution function is [j, g]. x = θ = j is the left asymptote of f (x). As shown in Fig. 4, if θ increases, the expert’s appraisal opinion is focused on smaller domain length g - j.

Fig. 4

Weibull function in different θ values.

Theorem 1: According to the Weibull distribution function f (x) of “not less than s_j”, if θ increases, expert’s appraisal information is more accurate.

Proof: If an expert gives information “not less than s_j”, its Weibull function f (x) is monotone decreasing, β < 1.□

(a) The mean E (x, θ) and variance D (x, θ) of Weibull function f (x) can be expressed as

Mean is $E (x, θ) = \int_{θ}^{g} xf (x) dx$ ,

Variance is $D (x, θ) = E (x^{2}, θ) - [E (x, θ)]^{2} = \int_{θ}^{g} x^{2} f (x) dx - [\int_{θ_{1}}^{g} xf (x) dx]^{2}$ .

(b) Suppose that 0 ≤ θ₁ < θ₂ < g. As f (x) is a monotone decreasing function, there exists ɛ, ɛ ∈ (θ₁, θ₂) , $\begin{matrix} E (x, θ_{2}) - E (x, θ_{1}) = E^{'} (ɛ) (θ_{2} - θ_{1}) = - ɛ f^{'} (ɛ) \\ (θ_{2} - θ_{1}) > 0, E (x, θ_{2}) > E (x, θ_{1}) \end{matrix}$ $\begin{matrix} D (x, θ_{2}) - D (x, θ_{1}) = [E (x, θ_{1})]^{2} - [E (x, θ_{2})]^{2} \\ - \int_{θ_{1}}^{θ_{2}} x^{2} f (x) dx < 0, D (x, θ_{2}) < D (x, θ_{1}) \end{matrix}$

Consequently, with the increasing value of θ, expert’s appraisal information “not less than s_j” is more accurate, which can be described as Fig. 4.

(2) Sensitivity analysis of parameter α

α is the scale parameter of Weibull distribution function, which determines the convexity of the curve. According to “not less than s_j”, Weibull function is a monotone decreasing function, 0 < β < 1, 0 < θ < x < g. Therefore, the second derivative $f^{″} (x) = (\begin{matrix} \frac{β (β - 1) (β - 2)}{α^{β}} (x - θ)^{β - 3} \\ - \frac{3 β^{2} (β - 1)}{α^{2 β}} (x - θ)^{2 β - 3} \\ + \frac{β^{3}}{α^{3 β}} (x - θ)^{3 β - 2} \end{matrix}) e^{- \frac{(x - θ)^{β}}{α^{β}}} > 0$ (8)

As shown in Fig. 5, f (x) is convex to (j, 0). With the increasing value of α, the curve of f (x) is more convex. Let β = 0.789, θ = 2, probability density function curves of Weibull distribution f₁ (x, α₁ = 0.594), f₂ (x, α₂ = 2.97), f₃ (x, α₃ = 5.94) and f₄ (x, α₄ = 11.88) is given as Fig. 5.

Fig. 5

Curves in different α values.

Theorem 2: According to the Weibull distribution function f (x) of “not less than s_j”, if α increases, expert’s appraisal information is with weaker preference to a particularly linguistic grade.

Proof: Let $x_{λ} = j + λ * (g - j), 0 < λ < \frac{1}{2}$ .□

The cumulative distribution function for the Weibull distribution in smaller interval [j, x_λ] is $F (x, α) = \int_{j}^{x_{λ}} f (x, α) dx = F (x_{λ}, α) - F (j, α)$

Suppose that α₁ < α₂, $F (x, α_{2}) - F (x, α_{1}) = \int_{j}^{x_{λ}} f (x, α_{2}) dx - \int_{j}^{x_{λ}} f (x, α_{1}) dx$ $= F (j, α_{2}) - F (x_{λ}, α_{2}) - [F (j, α_{1}) - F (x_{λ}, α_{1})]$ $\begin{matrix} = F (x_{λ}, α_{2}) - F (j, α_{2}) - F (x_{λ}, α_{1}) + F (j, α_{1}) \\ = F (x_{λ}, α_{2}) - F (x_{λ}, α_{1}) \end{matrix}$ $= e^{- (\frac{x_{λ} - θ}{α_{1}})^{β}} - e^{- (\frac{x_{λ} - θ}{α_{2}})^{β}} < 0$ $F (x, α_{2}) < F (x, α_{1}) .$

Consequently, if α increases, the expert’s appraisal opinion “not less than s_j”is less centralized in smaller interval [j, x_λ], starting from the beginning of discourse domain [j, g]. In other words, as the curve of f (x) become more flat, the expert’s appraisal opinion “not less than s_j” is with weaker preference to a particularly linguistic grade.

(3) Sensitivity analysis of parameter β

β is the shape parameter of Weibull distribution, which determines the monotonicity of the curve. According to “not less than s_j”, Weibull function is a monotone decreasing function and satisfies 0 < β < 1, 0 < θ < x < g.

As shown in Fig. 6, f (x) is convex to point (j, 0). With the decreasing value of β, the curve of f (x) is more convex. Let α = 0.594, θ = 2, probability density function curves of Weibull distribution f₁ (x, β₁ = 0.95), f₂ (x, β₂ = 0.786), f₃ (x, β₃ = 0.55) and f₄ (x, β₄ = 0.25) are given as in Fig. 6.

Fig. 6

Curves with different β values.

Theorem 3: According to the Weibull distribution function f (x) of “not less than s_j”, if β decreases, expert’s appraisal information is with stronger preference to a particularly linguistic grade.

Proof: Let x_λ = j + λ * (g - j), $0 < λ < \frac{1}{2}$ , the cumulative distribution function for the Weibull distribution in smaller interval [j, x_λ] is□ $F (x, β) = \int_{j}^{x_{λ}} f (x, β) dx = F (x_{λ}, β) - F (j, β)$

Suppose that β₁ > β₂ $F (x, β_{2}) - F (x, β_{1}) = \int_{j}^{x_{λ}} f (x, β_{2}) dx - \int_{j}^{x_{λ}} f (x, β_{1}) dx$ $\begin{matrix} = F (x_{λ}, β_{2}) - F (j, β_{2}) - F (x_{λ}, β_{1}) + F (j, β_{1}) \\ = F (x_{λ}, β_{2}) - F (x_{λ}, β_{1}) \end{matrix}$ $= e^{- (\frac{x_{λ} - θ}{α})^{β_{1}}} - e^{- (\frac{x_{λ} - θ}{α})^{β_{2}}} > 0$ $F (x, β_{2}) > F (x, β_{1}) .$

Consequently, if βdecreases, the expert’s appraisal opinion “not less than s_j” is more centralized in smaller interval [j, x_λ] starting from the beginning of discourse domain [j, g]. In other words, as the curve of f (x) become more steep, the expert’s appraisal opinion “not less than s_j” is with stronger preference to a particularly linguistic grade.

5 Numerical study and comparative analysis

In this section, a numerical case about Likert scale with 7 linguistic labels is studied, which can show the feasibility and effectiveness of the above analysis. Additionally, the advantage of the method in this paper is given by comparing to others’ papers.

5.1 Function fitting and quantitative result

In Reference [12], experts gave linguistic appraisal information with the criteria as SS = {s₀ = Extremely Poor ; s₁ = Very Poor ; s₂ = Poor ;s₃ = Slightlypoor ; s₄ = Fair ; s₅ = Slightly Good ; s₆ = Good ; s₇ = Very Good ; s₈ = Extremely} = {s₀ = ExtremelyPoor ; s₁ = VeryPoor ; s₂ = Poor ; s₃ = Slightly poor ; s₄ = Fair ; s₅ = SlightlyGood ; s₆ = Good ; s₇ = Very Good ; s₈ = Extremely}. Consequently, g = 8 and {s_i|i = 0, 1, ⋯ , 8}.

According to ABC analysis methods [53 –55], suppose that λ = 15%, μ = 40%, m = 75%, n = 5%, which means that the cumulative probability of HFLTs can reach a smaller area as Area 3 with 5% percentage in larger length interval [j + (8 - j) *40 % , 8], the cumulative probability of HFLTs can reach a larger area as Area 1 with 75% percentage in smaller length interval [j, j + (8 - j) *15 %], as displayed by Fig. 7.

Fig. 7

The probability density function of MDHFLTs in optimal condition.

If an expect gives monotonous decreasing appraisal information “not less than s_j”, j = 0, 1, ⋯ , 8, an multi-objective programming model can be established according to Model (7), in which the parameter value of Weibull distribution function, mean and variance in definition domain [j, g] in optimal condition can be obtained as shown in Table 3. Please note that s₈ is the maximum value and “not less than s₈ ”only covers appraisal grade {s₈}, which has certain information.

Table 3

The optimal parameters of the linguistic terms “not less than s_j” and “better than s_j - 1”

Linguistic term s_j	HFLTSsH_s	α ^*	β ^*	θ	E (x)	D (x)
Not less than s₀	{s₀, s₁, s₂, s₃, s₄, s₅, s₆, s₇, s₈}	0.792	0.786	0	0.888	1.195
Not less than s₁ (better than s₀)	{s₁, s₂, s₃, s₄, s₅, s₆, s₇, s₈}	0.693	0.786	1	1.775	0.92
Not less than s₂ (better than s₁)	{s₂, s₃, s₄, s₅, s₆, s₇, s₈}	0.594	0.786	2	2.662	0.687
Not less than s₃ (better than s₂)	{s₃, s₄, s₅, s₆, s₇, s₈}	0.495	0.786	3	3.549	0.494
Not less than s₄ (better than s₃)	{s₄, s₅, s₆, s₇, s₈}	0.396	0.786	4	4.436	0.338
Not less than s₅ (better than s₄)	{s₅, s₆, s₇, s₈}	0.297	0.786	5	5.323	0.224
Not less than s₆ (better than s₅)	{s₆, s₇, s₈}	0.198	0.786	6	6.209	0.161
Not less than s₇ (better than s₆)	{s₇, s₈}	0.099	0.786	7	7.096	0.129

Because distribution function curve “not more than s_j” and “not less than s_g-j” are symmetry with $x = \frac{g}{2} = 4$ . Suppose that Weibull distribution function of “not less than s_g-j” is f (x′), Weibull distribution function of “not more than s_j” is f (x′) = f (g - x). Consequently, the mean and variance of in “not more than s_j” definition domain [0, j] can be calculated in Table 4. Please note that s₀ is the minimum value and “not more than s₀ ” only covers appraisal grade {s₀} as certain information.

Table 4

The optimal parameters under the “not more than s_j” and “worse than s_j+1”

Linguistic term s_j	HFLTSs H_s	θ	E (x)	D (x)
Not more than s₀ (worse than s₁)	{s₀}	0	0	0
Not more than s₁ (worse than s₂)	{s₀, s₁}	1	0.904	0.129
Not more than s₂ (worse than s₃)	{s₀, s₁, s₂}	2	1.791	0.161
Not more than s₃ (worse than s₄)	{s₀, s₁, s₂, s₃}	3	2.677	0.224
Not more than s₄ (worse than s₅)	{s₀, s₁, s₂, s₃, s₄}	4	3.564	0.338
Not more than s₅ (worse than s₆)	{s₀, s₁, s₂, s₃, s₄, s₅}	5	4.451	0.494
Not more than s₆ (worse than s₇)	{s₀, s₁, s₂, s₃, s₄, s₅, s₆}	6	5.338	0.687
Not more than s₇ (worse than s₈)	{s₀, s₁, s₂, s₃, s₄, s₅, s₆, s₇}	7	6.225	0.92

5.2 Comparison analysis

In Reference [12], Chen gave HFLTSs based on the assumption that all linguistic terms in HFLTSs following a normal distribution. An assessing attitude-driven approach was proposed based on probability density functions to generate HFLTSs possibility distributions. The probability distribution of monotone linguistic information is designed as probability distribution. If “not less than s_j (0 ≤ j ≤ 8)” are given, mean E (x), variance D (x) and deviation coefficient $C = \sqrt{D (x)} / E (x)$ following the methods in this paper and Reference [12] can be compared in Table 5. Additionally, comparison result of “not more than s_j” following the methods in this paper and Reference [12] can be seen in Table 6.

Table 5
Comparison on the means and variances of MDHFLTs

Linguistic Term s_j Literature [12] This paper Precision increase rate

E₁(x) D₁(x) C ₁ E₂(x) D₂(x) C ₂ $1 - \sqrt{D_{2} (x) / D_{1} (x)}$

Not less than s₁ (better than s₀) 4.5 3.522 0.417 1.775 0.920 0.540 48.89%

Not less than s₂ (better than s₁) 5 2.69 0.328 2.662 0.687 0.311 49.46%

Not less than s₃ (better than s₂) 5.5 2.043 0.260 3.549 0.494 0.198 50.83%

Not less than s₄ (better than s₃) 6 1.368 0.195 4.436 0.338 0.131 50.29%

Not less than s₅ (better than s₄) 6.5 0.870 0.143 5.323 0.224 0.090 49.26%

Not less than s₆ (better than s₅) 7 0.346 0.084 6.209 0.161 0.065 31.79%

Not less than s₇ (better than s₆) 7.5 0.250 0.067 7.096 0.129 0.051 28.17%

Linguistic Term s_j		Literature [12]		This paper			Precision increase rate
Not less than s₁ (better than s₀)	4.5	3.522	0.417	1.775	0.920	0.540	48.89%
Not less than s₂ (better than s₁)	5	2.69	0.328	2.662	0.687	0.311	49.46%
Not less than s₃ (better than s₂)	5.5	2.043	0.260	3.549	0.494	0.198	50.83%
Not less than s₄ (better than s₃)	6	1.368	0.195	4.436	0.338	0.131	50.29%
Not less than s₅ (better than s₄)	6.5	0.870	0.143	5.323	0.224	0.090	49.26%
Not less than s₆ (better than s₅)	7	0.346	0.084	6.209	0.161	0.065	31.79%
Not less than s₇ (better than s₆)	7.5	0.250	0.067	7.096	0.129	0.051	28.17%

Table 6

Comparison on the means and variances of MIHFLTs

Linguistic Term s_j		Literature [12]			This paper		Precision increase rate
	E₁(x)	D₁(x)	C ₁	E₂(x)	D₂(x)	C ₂	$1 - \sqrt{D_{2} (x) / D_{1} (x)}$
Not more than s₁ (worse than s₂)	0.7	0.214	0.661	0.904	0.129	0.397	22.36%
Not more than s₂ (worse than s₃)	1	0.346	0.588	1.791	0.161	0.224	31.79%
Not more than s₃ (worse than s₄)	1.5	0.87	0.622	2.677	0.224	0.177	49.26%
Not more than s₄ (worse than s₅)	2	1.368	0.585	3.564	0.338	0.163	50.29%
Not more than s₅ (worse than s₆)	2.5	5.870	0.969	4.451	0.494	0.160	70.99%
Not more than s₆ (worse than s₇)	3	2.69	0.547	5.338	0.687	0.155	49.46%
Not more than s₇ (worse than s₈)	3.5	3.526	0.537	6.225	0.92	0.154	48.92%

According to the comparison, some advantages of this paper can be found.

Both expert’s explicit and implicit appraisal information are comprehensively utilized to achieve more reasonable quantitative result. If an expert gives uncertain linguistic opinion as “not less than s_j ”, he not only expresses his explicit information about domain fields as [j, g], but also indicates his implicit preference that his evaluation had a large probability within [j, j + 1]. That is why he didn’t choose the linguistic expression as [j + 1, g]. In Reference [12], the author only use expert’s explicit appraisal information to determine the domain fields as [j, g], but the expert’s implicit fuzzy preference information was not explored which was hidden under the explicit appraisal information. Similarly, “not more than s_j” analysis has the same phenomenon.

The mean of monotonous linguistic expression is determined by all linguistic grades in the domain field, other than the minimum and maximum only. If an expert gives uncertain linguistic opinion as “not less than s_j ”, continuous Weibull distribution function can be obtained in the domain fields as [j, g]. The mean value of Weibull distribution function is determined by all the number in domain fields [j, g]. In Reference [12], uncertain linguistic expressions, “not less than s_j ”, are assumed following normal unbiased distribution in domain fields [j, g] and the mean value is located as the middle of domain field [j, g], E (x) = (i + g)/2, which is only determined by the maximum and minimum of the domain field. The analysis is similarly as “not more than s_j”.

The computational precision is highly increasing comparing to the method in Reference [12]. According to the deviations obtained in Table 5 and Table 6, one can find the new method is better to control the deviation of fitting results and the precision increases greatly. According to “not less than s_j ” and “not more than s_j”, 0 ≤ j ≤ 8, the average precision increase 44.10% and 46.15% comparing to the method in Reference [12], which shows improvements on computational effect.

5.3 Discussion

In this section, a discussion about the differences between present paper and References is made, in which Reference [56, 57] treat HFLTS in discrete forms and Reference [30, 58] regards HFLTS in continuous forms

In Reference [30 , 56–58], only the explicit information of HFLTs is reflected as the linguistic labels. The implicit information of HFLTs, which can be regarded as expert’s preference on a specific linguistic label, is ignored. In the present paper, both expert’s explicit and implicit appraisal information are comprehensively utilized to achieve more reasonable quantitative result.

Reference [56, 57] mainly focused on the computational methods of monotonic HFLTS in discrete forms. The trend of unbalanced linguistic term set between adjacent linguistic labels is linear and is not continuous in definition domain. But the present paper employs continuous Weibull distribution function which covers all the numbers within definition domain with various curves, other than several independent linguistic labels.

In Reference [30, 58], uncertain linguistic items are described in the form as “Between s_L and s_U” with expert’s proportion attitude on a linguistic label. In order to calculate the semantic, three kinds of transformation functions are designed according to different standard deviation changes. But the expectation of the uncertain linguistic item is the average number of minimum and maximum numbers g (s_L) and g (s_U) obtained from the above transformation functions, other than all the numbers in the definition domain following a certain distribution. Consequently, some information is artificially lost in computational process. In current paper, the computational result is obtained from the expectation result of continuous Weibull distribution with all numbers in the definition domain, which can avoid information loss.

In compared References, there are many important parameters which are difficult to be determined. In Reference [56, 57], threshold ψ should be set up beforehand for transforming monotonic HFLTS into scores, which is not easy to be scientifically confirmed. In Reference [30, 58], ρ, λ and γ should be determined based on the practical cases, which are complicated to be obtained with a strong theoretical support. But present paper follows ABC classification idea to obtain the parameter combination of Weibull distribution function, which is easier for implementation.

6 Conclusion and future works

A novel computational method for quantitatively transferring monotonic hesitant fuzzy linguistic terms is developed to comprehensively reflect the expert’s explicit and implicit appraisal opinion, which is estimated by fitting Weibull distribution functions with parameters. Monotonic uncertain linguistic terms are defined and their comprehensive meanings are revealed, which include explicit information about linguistic interval and expert’s potential preference about his hidden fuzzy preference on linguistic grade. According to ABC classification method, a multi-objective programming model is proposed related to the comprehensive meanings of monotonic increasing linguistic terms, in which continuous Weibull distribution functions with three parameters are explored. Additionally, symmetry principle is employed to fit the expression of monotone decreasing appraisals, which are transferring from monotone increasing appraisals with same length of domain field.

Moreover, the mean value and variance are estimated related to continuous Weibull distribution functions as well as the sensitivity analysis is explored to show the influence of parameters on decision-making precision. With the increasing value of scale parameter α, expert’s appraisal information is weaker preference to a particularly linguistic grade. If shape parameter β increase, expert’s appraisal information is stronger preference to a particularly linguistic grade. With the increasing value of location parameter θ, expert’s appraisal information is more accurate. From the quantitative results, this paper calculated the mean of MHFLTs by all linguistic grades in the domain field, other than the minimum and maximum only, and its contributes to the highly computational precision. Comparison to existing method in paper [1], the method mentioned in this paper is with more precision and less deviation.

However, as the exploration on the computational method of MHFLTs concerning expert’s implicit attitude, there are still some limitations, which may be some valuable research aspects in the future.

The value of λ, m and n are from the empirical data from ABC cluster analysis. But expert’s preference on a linguistic label may be fluctuated because of uncertain appraisal environment. λ, m and n should be an uncertain number, such as fuzzy numbers, interval and grey numbers.

Expert’s implicit attitude is not consistently stable and may be explored from psychological conditions to check whether his appraisal is from his true thinking, which can be reflected by brain wave, eye movement and pulse change.

In some complicated group decision-making problems, experts’ implicit attitudes may be different, which can influence on the final result of appraisal result. How to access the influence of subjective attitude on computational effect is another important issue in the further.

Footnotes

Acknowledgment

The research work is supported by National Natural Science Foundation of China (No. 71603242), Humanity and Social Science foundation of Ministry of Education (No. 19YJA630047), and Natural Science Foundation of Zhejiang Province (No. LY17G010002).

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