Conditional uncertainty distribution of multiple experts’ data

Abstract

The uncertainty theory is a branch of mathematics for studying subjective uncertainty phenomenon, and its role in subjective uncertain problems helps people make better decisions. But in real life, there is not a standard method to deal with multiple experts’ data problem. A simple method is to average all experts’ data to get a result. The other is to use the Delphi method to collect data many times and then get a normal result. This paper gives two new methods to handle this problem through conditional distributions. Compared to traditional method, they do not require all experts’ data from the beginning and the result obtained by these methods can be updated easily when new expert’s data is given.

Keywords

Uncertain statistics multiple experts model conditional uncertainty distribution Delphi

1 Introduction

In real life policymakers often face uncertain situations. People divide uncertainty into two categories, one that can be repeated and the other that only happens once. For repeated situations, probability theory is a very good tool for dealing with uncertainty. But when we measure events that happen only once, probability theory is not always the best choice. In such situation, people usually use some vague vocabulary to describe the event. These ambiguous words always are not the frequency of the event [13], but express how much people believe that the events will happen or not.

In order to deal with the uncertainty that only happens once, many methods for uncertainty measure have been proposed, such as fuzzy set [17], DS evidence theory or Z-number, and there has been a lot of research about fuzzy set lately [2 , 7]. Liu founded the uncertainty theory in 2007 and refined as a branch of axiomatic mathematic for modeling human uncertainty in 2010.

For modeling human uncertainty, gathering and analysing data about how people think the uncertainty is important.

Based on uncertainty theory, uncertain statistics was introduced by Liu [15] in 2010. Liu gave a way to collect expert’s experimental data by questionnaire survey, and introduced empirical uncertainty distribution and principle of least squares to estimate the uncertainty distribution of an uncertain variable. Chen-Ralexcu [1] successfully acquired the expert’s experimental data of the travel distance between Beijing and Tianjin by questionnaire survey. Wang [23] introduced uncertain hypothesis testing for two experts’ empirical data, and proposed the method of moments, to determine the distribution like empirical uncertainty distribution. In order to rank uncertain variables, Liu [9] proposed the concept of expected value. Liu and Ha [16] derived a useful formula for calculating the expected values of strictly monotone functions of independent uncertain variables. In order to find the fitted regression model, Lio-Liu [8], Yao-Liu [24], Guo-Wang-Gao [4], Song [20] did a lot of work.

In real life, the situation faced by decision makers may be more complicated. First of all, decision makers cannot be content to make decisions simply by asking an expert. They may ask many experts before making a decision. Then, experts may not understand the theory of uncertainty such as probability theory or uncertainty theory. So, experts do not give an accurate distribution of an uncertain event. More common thing is that they can only give empirical data. Therefore, the problem that decision makers always encounter is that they need to make a decision based on many group experimental data.

When multiple experts are available, Liu [9] suggests a new uncertainty distribution aggregated by uncertainty distributions. Wang, Gao and Guo [21] recast Delphi method as a process to determine uncertainty distributions. But in some situations, these methods are not flexible or convenient to use. For example, the government investigates how much entrepreneurs understand the current policy, or school surveys students’ perceptions of the difficulty of the course. If experts only give the experts’ experimental data, the result of Liu’s method is the empirical uncertainty distribution with average of all experts’ experimental data. This method needs all experts’ data at the beginning, and Delphi method even needs to ask experts many times. It sounds impossible that the decision maker ask entrepreneurs or students many times for their views. And all experts’ data is difficult to get at the beginning.

This paper will propose two methods, condition iteration method and expected value iteration method, that only need to ask experts once. And the result obtained by these methods can be updated easily when new expert’s data is given. So, it is not necessary to require all experts’ data at the beginning.

The condition iteration method through condition distribution combines the opinions of multiple experts. And the expected value iteration method combines the opinions of various experts by averaging the new distribution and the total distribution. Both methods implement the idea of iteration. No matter how many experts give their views (even if there is only one), these methods always get a more objective result.

In this paper, two new methods will be introduced to integrate multiple experts’ data by conditionaldistribution. In the next section, we will review some concepts in uncertainty theory, such as uncertain measure, uncertainty space, expected value of uncertain variable, and conditional uncertainty distribution. In Section 3, we will introduce some basic concepts of uncertain statistics like experimental data and empirical uncertainty distribution. And then, in Section 4, we will propose two methods and give the relevant properties. In Section 5, we will give two examples to verify the validity of the methods. Finally, a conclusion is drawn in Section 6.

2 Preliminary

In convenience, we give some useful concepts of uncertainty theory at first. Let Γ be a nonempty set, and ℒ be a σ-algebra over Γ. Each element Λ∈ ℒ is called an event.

Liu [9] defined a set function ℳ on the σ-algebra ℒ as an uncertain measure if it satisfies the following three axioms: (Normality) ℳ {Γ} =1 for the universal set Γ; (Self-Duality) ℳ {Λ} + ℳ {Λ^c} =1 for any event Λ; (Countable Subadditivity) For every countable sequence of events Λ₁, Λ₂, ⋯ , we have $ℳ {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} ℳ {Λ_{i}} .$ Then the uncertainty space is defined by uncertain measure. Let Γ be a nonempty set, ℒ be the σ-algebra over Γ, and ℳ be an uncertain measure. Then the triplet (Γ, ℒ , ℳ) is said to be an uncertainty space.

Based on the uncertainty space, to deal with the operational law, Liu defined a product uncertain measure by the fourth axiom of uncertainty theory: (Product Axiom) Let (Γ_k, ℒ _k, _ℳk) be uncertainty spaces for k = 1, 2, ⋯, and Γ_k are nonempty sets, _ℳk defined on ℒ_k. The product measure ℳ is an uncertain measure on the product σ-algebra ℒ =ℒ ₁ × ℒ ₂ × ⋯ satisfying $ℳ {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} ℳ_{k} {Λ_{k}} .$ where Λ_k are arbitrarily chosen events from ℒ_k for k = 1, 2, ⋯, respectively. Gao [15] and Peng and Iwamura [19] studied some properties of an uncertain measure.

An uncertain variable is a function ξ from an uncertainty space (Γ, ℒ , ℳ) to the set of real numbers such that ξ ∈ B is an event for any Borel set B of real numbers. Then we use the uncertainty distribution to describe uncertain variable. The uncertainty distribution $Φ : R \to [0, 1]$ of an uncertain variable ξ is defined by Φ (x) = ℳ {ξ ≤ x}.

Theorem 2.1 (Peng-Iwamura [18]) A function $Φ : R \to [0, 1]$ is an uncertainty distribution if and only if it is an increasing function except Φ (x) ≡0 and Φ (x) ≡1.

In addition to the distribution, we can get more information about the uncertain variable through expected value.

Definition 2.1 (Liu [9]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by

$\begin{matrix} E [ξ] & = & \int_{0}^{+ \infty} ℳ {ξ \geq r} r \\ - \int_{- \infty}^{0} ℳ {ξ \leq r} d r \end{matrix}$ (1) provided that at least one of the two integrals is finite.

Let ξ be an uncertain variable with regular uncertainty distribution Φ (x). That is, Φ (x) is continuous and strictly increasing function with respect to x at which 0 < φ (x) <1, and $lim_{x \to - \infty} Φ (x) = 0$ and $lim_{x \to + \infty} Φ (x) = 1$ . Then the inverse function Φ^-1 (α) is called the inverse uncertainty distribution of ξ.

Theorem 2.2 (Liu [9]) Let ξ be an uncertain variable with regular uncertainty distribution Φ (x). Then we have $E [ξ] = \int_{- \infty}^{+ \infty} x d Φ (x) = \int_{0}^{1} Φ^{- 1} (α) d α .$ (2)

The uncertain variable ξ₁, ξ₂, ⋯ , ξ_n are said to be independent if $ℳ {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋀_{i = 1}^{n} ℳ {ξ_{i} \in B_{i}}$ for any Borel sets B₁, B₂, ⋯ , B_n of real numbers. Let ξ and η be independent uncertain variables with finite expected values. Then for any numbers a and b, we have E [aξ + bη] = aE [ξ] + bE [η].

Next, we will give some concepts of conditional uncertainty distribution.

Definition 2.2 (Liu [15]) The conditional uncertainty distribution Φ of an uncertain variable ξ given B is defined by $Φ (x ∣ B) = ℳ {ξ \leq x ∣ B}$ provided that ℳ {B} >0.

Theorem 2.3 (Liu [9]) Let ξ be an uncertain variable with uncertainty distribution Φ (x), and θ be a real number with Φ (θ) <1. Then the conditional uncertainty distribution of ξ given ξ > θ is

$\begin{matrix} Φ (x ∣ (θ, + \infty)) \\ = {\begin{matrix} 0 & if Φ (x) \leq Φ (θ), \\ \frac{Φ (x)}{1 - Φ (θ)} \land 0.5 & if Φ (θ) < Φ (x) \\ and Φ (x) < \frac{1 + Φ (θ)}{2}, \\ \frac{Φ (x) - Φ (θ)}{1 - Φ (θ)} & if \frac{1 + Φ (θ)}{2} \leq Φ (x) . \end{matrix} \end{matrix}$ (3) Similarly, when the given condition is ξ ≤ θ, and θ is a real number with Φ (θ) >0,we have

$\begin{matrix} Φ (x ∣ (- \infty, θ]) \\ = {\begin{matrix} \frac{Φ (x)}{Φ (θ)} & if Φ (x) \leq \frac{Φ (θ)}{2}, \\ \frac{Φ (x) + Φ (θ) - 1}{Φ (θ)} \lor 0.5 & if \frac{Φ (θ)}{2} < Φ (x) \\ and Φ (x) < Φ (θ) \\ 1 & if Φ (θ) \leq Φ (x) . \end{matrix} \end{matrix}$ (4)

3 Expert’s experimental data

Because not all experts understand uncertainty theory, it is difficult to get the accurate uncertainty distribution of an uncertain variable for experts. Generally, the decision maker can collect expert’s experimental data by a questionnaire survey proposed by Liu [15].

First, we ask an expert to choose a possible value x that the uncertain variable ξ may take, and then quiz him “How likely is ξ less than or equal to x?”. Denote the expert’s belief degree by α. So an expert’s experimental data (x, α) is obtained. Repeating this process, the following expert’s experimental data are obtained by the questionnaire, $(x_{1}, α_{1}), (x_{2}, α_{2}), \dots, (x_{n}, α_{n}) .$

And these data should meet the following consistence conditions:

$x_{1} < x_{2} < \dots < x_{n}; 0 \leq α_{1} \leq α_{2} \leq \dots \leq α_{n} \leq 1 .$

In an actual problem, what decision makers really need is the uncertainty distribution of an uncertain variable given by experts. In order to obtain a distribution, the simplest and most effective method is using empirical distribution method. Based on the expert’s experiment data (x_i, α_i) (1 ≤ i ≤ n), we can get the empirical uncertainty distribution:

$\begin{matrix} Φ (x) \\ = {\begin{matrix} 0 & if x < x_{1}, \\ α_{i} + \frac{(α_{i + 1} - α_{i}) (x - x_{i})}{x_{i + 1} - x_{i}} & if x_{i} \leq x \leq x_{i + 1}, \\ 1 \leq i < n, \\ 1 & if x > x_{n}, \end{matrix} \end{matrix}$ (5) and the empirical uncertainty distribution Φ determined by (5) has an expected value

$\begin{matrix} \begin{matrix} E (ξ) = & \frac{α_{1} + α_{2}}{2} x_{1} + \sum_{i = 2}^{n - 1} \frac{α_{i + 1} - α_{i - 1}}{2} x_{i} \\ + (1 - \frac{α_{n - 1} + α_{n}}{2}) x_{n} . \end{matrix} \end{matrix}$

Next, we can give the inverse uncertainty distribution of the empirical uncertainty distribution based on (x_i, α_i) (1 ≤ i ≤ n):

$\begin{matrix} \begin{matrix} Φ^{- 1} (α) \\ = & {\begin{matrix} x_{1} & if α < α1 \\ x_{i} + \frac{(x_{i + 1} - x_{i}) (α - α_{i})}{α_{i + 1} - α_{i}} & if αi ≤ x≤ αi+1, \\ 1 ≤ i < n \\ x_{n} & if α > αn \end{matrix} \end{matrix} \end{matrix}$

4 Two new methods of multiple experts’ data

In the practical problems, the decision maker often need to make decisions based on the different opinions of experts. Suppose there are m experts giving m uncertainty distributions Φ₁ (x) , Φ₂ (x) , ⋯ , Φ_m (x). How to integrate the various opinions of these experts? Liu [15] gives a method that m uncertainty distributions could be aggregated to an uncertainty distribution. We have $Φ (x) = ω_{1} Φ_{1} (x) + ω_{2} Φ_{2} (x) + \dots + ω_{m} Φ_{m} (x)$ where ω₁, ω₂, ⋯ , ω_m are convex combination coefficients. Φ (x) is also an uncertainty distribution by Peng-Iwamura theorem [18]. Liu suggests set $ω_{i} = \frac{1}{m}$ , i = 1, ⋯ , m.

Next, we will give two new methods based on the above method. In these methods, distributions are not only just added by a fixed weight, but also can be updated constantly.

4.1 Condition iteration method

When decision makers need to refer to the opinions of multiple experts, they always choose one expert firstly, then another. Finally, decision makers aggregate the opinions of all experts into one. For decision makers, after each expert gives advice, they will get two types of information. One is given by former expert or from the prior information, the other is given by current expert. In uncertainty theory, expected value is the average value of an uncertain variable in the sense of uncertain measure.

So when some experts give uncertainty distributions about an uncertain variable, decision makers can know the general opinion of experts on this issue through the expected values. Then they can use this information to make a decision based on conditional uncertainty distribution.

Set θ₁ as the expected value of the expert’s uncertainty distribution Φ₁ (x), and Φ₂ (x) is the uncertainty distribution given by the next expert. So the uncertainty distribution Φ^∗ (x) based on the condition that the expert’s uncertainty distribution Φ₁ (x) has known is

$\begin{matrix} Φ^{*} (x) = & (1 - Φ_{1} (θ_{1})) Φ_{2} (x ∣ (θ_{1}, + \infty)) \\ + Φ_{1} (θ_{1}) Φ_{2} (x ∣ (- \infty, θ_{1}]) . \end{matrix}$ (6) Since Φ₂ (x ∣ (θ₁, + ∞)) and Φ₂ (x ∣ (- ∞ , θ₁)) are all uncertainty distribution, (1 - Φ₁ (θ₁)) and Φ₁ (θ₁) are constants and greater than 0. So Φ^∗ (x) is an increasing function except Φ (x) ≡0 and Φ (x) ≡1. According to Theorem 2.1, Φ^∗ (x) is also an uncertainty distribution.

The new uncertainty distribution Φ^∗ (x) integrates the opinions of new and old experts. So when another expert gives his opinion, decision makers can set Φ₁ (x) = Φ^∗ (x) and calculate the new distribution by Eq.(6). As long as there is new distribution given by experts, they can always get the new distribution Φ^∗ (x) that contains all the information they know.

Furthermore, we may calculate the expected value of the uncertain variable with the distribution Φ^∗ (x).

Theorem 4.1 Let ξ be the uncertain variable with uncertainty distribution Φ^∗ (x) defined by Eq.(6) and Φ₁ (x) , Φ₂ (x) are respectively two experts’ distributions. θ₁ is the expected value of Φ₁ (x). The expected value of ξ with Φ^∗ (x) is

$\begin{matrix} E [ξ] = \\ {\begin{matrix} (1 - Φ_{1} (θ_{1})) \int_{- \infty}^{+ \infty} x d Φ_{2} (x), if Φ_{2} (θ_{1}) = 0, \\ \frac{1 - Φ_{1} (θ_{1})}{1 - Φ_{2} (θ_{1})} (\int_{θ_{1}}^{α_{1}} x d Φ_{2} (x) + \int_{α_{2}}^{+ \infty} x d Φ_{2} (x)) \\ + \frac{Φ_{1} (θ_{1})}{Φ_{2} (θ_{1})} \int_{- \infty}^{β_{1}} x d Φ_{2} (x), if 0 < Φ_{2} (θ_{1}) < \frac{1}{3}, \\ \frac{1 - Φ_{1} (θ_{1})}{1 - Φ_{2} (θ_{1})} \int_{α_{2}}^{+ \infty} x d Φ_{2} (x) \\ + \frac{Φ_{1} (θ_{1})}{Φ_{2} (θ_{1})} \int_{- \infty}^{β_{1}} x d Φ_{2} (x), if \frac{1}{3} \leq Φ_{2} (θ_{1}) \leq \frac{2}{3}, \\ \frac{Φ_{1} (θ_{1})}{Φ_{2} (θ_{1})} (\int_{- \infty}^{β_{1}} x d Φ_{2} (x) + \int_{β_{2}}^{θ_{1}} x d Φ_{2} (x)) \\ + \frac{1 - Φ_{1} (θ_{1})}{1 - Φ_{2} (θ_{1})} \int_{α_{2}}^{+ \infty} x d Φ_{2} (x), \\ if \frac{2}{3} < Φ_{2} (θ_{1}) < 1, \\ Φ_{1} (θ_{1}) \int_{- \infty}^{+ \infty} x d Φ_{2} (x), if Φ_{2} (θ_{1}) = 1, \end{matrix} \end{matrix}$ (7) where $\begin{matrix} α_{1} & = & max {α ∣ Φ_{2} (α) = \frac{1 - Φ_{2} (θ_{1})}{2}}, \\ α_{2} & = & min {α ∣ Φ_{2} (α) = \frac{1 + Φ_{2} (θ_{1})}{2}}, \\ β_{1} & = & max {β ∣ Φ_{2} (β) = \frac{Φ_{2} (θ_{1})}{2}}, \\ β_{2} & = & min {β ∣ Φ_{2} (β) = 1 - \frac{Φ_{2} (θ_{1})}{2}} . \end{matrix}$

Proof. If Φ₂ (θ₁) =0, applying the Eqs. (3) and (4), we have $\begin{matrix} Φ_{2} (x ∣ (θ_{1}, + \infty)) & = & Φ_{2} (x), \\ Φ_{2} (x ∣ (- \infty, θ_{1}]) & = & 1 . \end{matrix}$ So the new uncertainty distribution Φ^∗ is $Φ^{*} (x) = (1 - Φ_{1} (θ_{1})) Φ_{2} (x) + Φ_{1} (θ_{1}),$ and the expected value of ξ with Φ^∗ (x) is

$E [ξ] = (1 - Φ_{1} (θ_{1})) \int_{- \infty}^{+ \infty} x d Φ_{2} (x) .$ (8) If Φ₂ (θ₁) =1, we have $\begin{matrix} Φ_{2} (x ∣ (θ_{1}, + \infty)) & = & 0, \\ Φ_{2} (x ∣ (- \infty, θ_{1}]) & = & Φ_{2} (x) . \end{matrix}$ So Φ^∗ is $Φ^{*} (x) = Φ_{1} (θ_{1}) Φ_{2} (x),$ and the expected value of ξ with Φ^∗ (x) is

$E [ξ] = Φ_{1} (θ_{1}) \int_{- \infty}^{+ \infty} x d Φ_{2} (x) .$ (9)

Applying the Eqs.(3) and (4), we have $\begin{matrix} \int_{- \infty}^{θ_{1}} x d Φ_{2} (x ∣ (θ_{1}, + \infty)) = 0, \\ \int_{θ_{1}}^{+ \infty} x d Φ_{2} (x ∣ (- \infty, θ_{1}]) = 0 . \end{matrix}$ Hence the expected value of ξ with Φ^∗ (x) is $\begin{matrix} E [ξ] & = & (1 - Φ_{1} (θ_{1})) \int_{θ_{1}}^{+ \infty} x d Φ_{2} (x) \\ + Φ_{1} (θ_{1}) \int_{- \infty}^{θ_{1}} x d Φ_{2} (x) . \end{matrix}$ Set $\begin{matrix} \begin{matrix} α_{1} & = max {α ∣ Φ_{2} (α) = \frac{1 - Φ_{2} (θ_{1})}{2}}, \\ α_{2} & = min {α ∣ Φ_{2} (α) = \frac{1 + Φ_{2} (θ_{1})}{2}}, \\ β_{1} & = max {β ∣ Φ_{2} (β) = \frac{Φ_{2} (θ_{1})}{2}}, \\ β_{2} & = min {β ∣ Φ_{2} (β) = 1 - \frac{Φ_{2} (θ_{1})}{2}} . \end{matrix} \end{matrix}$

(i) If $0 < Φ_{2} (θ_{1}) < \frac{1}{3}$ , we have $Φ_{2} (θ_{1}) < \frac{1 - Φ_{2} (θ_{1})}{2} < \frac{1 + Φ_{2} (θ_{1})}{2} .$ So when θ₁ < x < α₁ we have $Φ_{2} (x ∣ (θ_{1}, + \infty)) = \frac{Φ_{2} (x)}{1 - Φ_{2} (θ_{1})} < 0.5 .$ So $\begin{matrix} \int_{θ_{1}}^{α_{1}} x d Φ_{2} (x ∣ (θ_{1}, + \infty)) \\ = & \int_{θ_{1}}^{α_{1}} \frac{x}{1 - Φ_{2} (θ_{1})} d Φ_{2} (x) \\ = & \frac{1}{1 - Φ_{2} (θ_{1})} \int_{θ_{1}}^{α_{1}} x d Φ_{2} (x) \end{matrix}$ When α₁ < x < α₂ we have $Φ_{2} (x ∣ (θ_{1}, + \infty)) = 0.5 < \frac{Φ_{2} (x)}{1 - Φ_{2} (θ_{1})} .$ So $\int_{α_{1}}^{α_{2}} x d Φ_{2} (x ∣ (θ_{1}, + \infty)) = 0 .$ When x > α₂ we have $\begin{matrix} \begin{matrix} \int_{α_{2}}^{+ \infty} x d & Φ_{2} (x ∣ (θ_{1}, + \infty)) \\ = & \frac{1}{1 - Φ_{2} (θ_{1})} \int_{α_{2}}^{+ \infty} x d (Φ_{2} (x) - Φ_{2} (θ_{1})) \\ = & \frac{1}{1 - Φ_{2} (θ_{1})} \int_{α_{2}}^{+ \infty} x d Φ_{2} (x) . \end{matrix} \end{matrix}$ Hence we have $\begin{matrix} \begin{matrix} \int_{θ_{1}}^{+ \infty} x d Φ_{2} (x) \\ = \frac{1}{1 - Φ_{2} (θ_{1})} (\int_{θ_{1}}^{α_{1}} x d Φ_{2} (x) + \int_{α_{2}}^{+ \infty} x d Φ_{2} (x)) . \end{matrix} \end{matrix}$ Similarly, when x ≤ β₁ we have $\begin{matrix} \begin{matrix} \int_{- \infty}^{β_{1}} x d & Φ_{2} (x ∣ (- \infty, θ_{1}]) \\ = & \frac{1}{Φ_{2} (θ_{1})} \int_{- \infty}^{β_{1}} x d Φ_{2} (x) . \end{matrix} \end{matrix}$ Because 0 < Φ₂ (θ₁) <1/3, we have Φ₂ (θ₁)/2 < Φ₂ (θ₁) ≤1 - Φ₂ (θ₁)/2. So when β₁ < x < θ₁ we have $Φ_{2} (x ∣ (- \infty, θ_{1}]) = 0.5 > \frac{Φ_{2} (x) + Φ_{2} (θ_{1}) - 1}{Φ_{2} (θ_{1})} .$ So $\int_{β_{1}}^{θ_{1}} x d Φ_{2} (x ∣ (- \infty, θ_{1}]) = 0 .$ Hence we have $\begin{matrix} \begin{matrix} \int_{- \infty}^{θ_{1}} x d & Φ_{2} (x ∣ (- \infty, θ_{1}]) \\ = & \frac{1}{Φ_{2} (θ_{1})} \int_{- \infty}^{β_{1}} x d Φ_{2} (x) . \end{matrix} \end{matrix}$ Therefore, the expected value of ξ with Φ^∗ (x) is $\begin{matrix} E [ξ] \\ = \frac{1 - Φ_{1} (θ_{1})}{1 - Φ_{2} (θ_{1})} (\int_{θ_{1}}^{α_{1}} x d Φ_{2} (x) \\ + \int_{α_{2}}^{+ \infty} x d Φ_{2} (x)) + \frac{Φ_{1} (θ_{1})}{Φ_{2} (θ_{1})} \int_{- \infty}^{β_{1}} x d Φ_{2} (x) . \end{matrix}$

(ii) If 1/3 ≤ Φ₂ (θ₁) ≤2/3, we have $(1 - Φ_{2} (θ_{1})) / 2 \leq Φ_{2} (θ_{1}) \leq (1 + Φ_{2} (θ_{1})) / 2$ and Φ₂ (θ₁)/2 < Φ₂ (θ₁) ≤1 - Φ₂ (θ₁)/2. When α₁ < θ₁ < x < α₂, we have $Φ_{2} (x ∣ (θ_{1}, + \infty)) = 0.5 < \frac{Φ_{2} (x)}{1 - Φ_{2} (θ_{1})} .$ So $\int_{θ_{1}}^{α_{2}} x d Φ_{2} (x ∣ (θ_{1}, + \infty)) = 0 .$ When x > α₂ we have $\begin{matrix} \begin{matrix} \int_{α_{2}}^{+ \infty} x d & Φ_{2} (x ∣ (θ_{1}, + \infty)) \\ = & \frac{1}{1 - Φ_{2} (θ_{1})} \int_{α_{2}}^{+ \infty} x d Φ_{2} (x) . \end{matrix} \end{matrix}$

Similarly, when x ≤ β₁ we have $\begin{matrix} \begin{matrix} \int_{- \infty}^{β_{1}} x d & Φ_{2} (x ∣ (- \infty, θ_{1}]) \\ = & \frac{1}{Φ_{2} (θ_{1})} \int_{- \infty}^{β_{1}} x d Φ_{2} (x) . \end{matrix} \end{matrix}$ When β₁ < x < θ₁ ≤ β₂ we have $\begin{matrix} Φ_{2} (x ∣ (- \infty, θ_{1}]) = 0.5 > \frac{Φ_{2} (x) + Φ_{2} (θ_{1}) - 1}{Φ_{2} (θ_{1})} . \end{matrix}$ So $\int_{β_{1}}^{θ_{1}} x d Φ_{2} (x ∣ (- \infty, θ_{1}]) = 0 .$ Hence we have $\begin{matrix} \begin{matrix} \int_{- \infty}^{θ_{1}} x d & Φ_{2} (x ∣ (- \infty, θ_{1}]) \\ = & \frac{1}{Φ_{2} (θ_{1})} \int_{- \infty}^{β_{1}} x d Φ_{2} (x) . \end{matrix} \end{matrix}$ Therefore, the expected value of ξ with Φ^∗ (x) is $\begin{matrix} \begin{matrix} E [ξ] = & \frac{1 - Φ_{1} (θ_{1})}{1 - Φ_{2} (θ_{1})} \int_{α_{2}}^{+ \infty} x d Φ_{2} (x) \\ + \frac{Φ_{1} (θ_{1})}{Φ_{2} (θ_{1})} \int_{- \infty}^{β_{1}} x d Φ_{2} (x) . \end{matrix} \end{matrix}$

(iii) If 2/3 < Φ₂ (θ₁) <1, we have $\frac{1 - Φ_{2} (θ_{1})}{2} < Φ_{2} (θ_{1}) < \frac{1 + Φ_{2} (θ_{1})}{2}$ and $\frac{Φ_{2} (θ_{1})}{2} < 1 - \frac{Φ_{2} (θ_{1})}{2} < Φ_{2} (θ_{1}) .$ When α₁ < θ₁ < x < α₂, we have $Φ_{2} (x ∣ (θ_{1}, + \infty)) = 0.5 < \frac{Φ_{2} (x)}{1 - Φ_{2} (θ_{1})} .$ So $\int_{θ_{1}}^{α_{2}} x d Φ_{2} (x ∣ (θ_{1}, + \infty)) = 0 .$ When x > α₂ we have $\begin{matrix} \begin{matrix} \int_{α_{2}}^{+ \infty} x d & Φ_{2} (x ∣ (θ_{1}, + \infty)) \\ = & \frac{1}{1 - Φ_{2} (θ_{1})} \int_{α_{2}}^{+ \infty} x d Φ_{2} (x) . \end{matrix} \end{matrix}$ When x ≤ β₁ we have $\begin{matrix} \begin{matrix} \int_{- \infty}^{β_{1}} x d & Φ_{2} (x ∣ (- \infty, θ_{1}]) \\ = & \frac{1}{Φ_{2} (θ_{1})} \int_{- \infty}^{β_{1}} x d Φ_{2} (x) . \end{matrix} \end{matrix}$ When β₁ < x < β₂ we have $Φ_{2} (x ∣ (- \infty, θ_{1}]) = 0.5 > \frac{Φ_{2} (x) + Φ_{2} (θ_{1}) - 1}{Φ_{2} (θ_{1})} .$ So $\int_{β_{1}}^{β_{2}} x d Φ_{2} (x ∣ (- \infty, θ_{1}]) = 0 .$ When β₂ < x < θ₁ we have $Φ_{2} (x ∣ (- \infty, θ_{1}]) = \frac{Φ_{2} (x) + Φ_{2} (θ_{1}) - 1}{Φ_{2} (θ_{1})} > 0.5 .$ So $\int_{β_{2}}^{θ_{1}} x d Φ_{2} (x ∣ (- \infty, θ_{1}]) = \frac{1}{Φ_{2} (θ_{1})} \int_{β_{2}}^{θ_{1}} x d Φ_{2} (x) .$ Therefore, the expected value of ξ with Φ^∗ (x) is $\begin{matrix} \begin{matrix} E [ξ] = & \frac{1 - Φ_{1} (θ_{1})}{1 - Φ_{2} (θ_{1})} \int_{α_{2}}^{+ \infty} x d Φ_{2} (x) \\ + \frac{Φ_{1} (θ_{1})}{Φ_{2} (θ_{1})} (\int_{- \infty}^{β_{1}} x d Φ_{2} (x) + \int_{β_{2}}^{θ_{1}} x d Φ_{2} (x)) . \end{matrix} \end{matrix}$ The proof of the theorem is completed.

The method by using Eq.(6) to get the distribution integrating previous information may be called the condition iteration method (C-I Method). Whether Φ₂ (θ₁) is too big or too small indicates that there is disagreement among experts. When Φ₂ (θ₁) =0 or Φ₂ (θ₁) =1, it shows that there is a huge difference between the two experts. The specific steps of this method are as follows.

Algorithm 1. Condition iteration method (C-I Method)

There are m distributions ${\tilde{Φ}}_{1} (x), \dots, {\tilde{Φ}}_{m} (x)$ given by m experts. Let $Φ_{1}^{*} (x) = {\tilde{Φ}}_{1} (x)$ , k = 1.

If k = m go to Step 7. Otherwise, let $Φ_{2} (x) = {\tilde{Φ}}_{k + 1} (x)$ .

Calculate the expected value θ_k of $Φ_{k}^{*} (x)$ by Eq. (7).

Calculate the conditional distributions Φ₂ (x ∣ (θ_k, + ∞)) and Φ₂ (x ∣ (- ∞ , θ_k]) by Eqs.(3) and (4), respectively.

Get the new distribution by $\begin{matrix} \begin{matrix} Φ_{k + 1}^{*} (x) = & (1 - Φ_{k}^{*} (θ_{k})) Φ_{2} (x ∣ (θ_{k}, + \infty)) \\ + Φ_{k}^{*} (θ_{k}) Φ_{2} (x ∣ (- \infty, θ_{k}]) \end{matrix} \end{matrix}$

Let k = k + 1 and go to Step 2.

The final distribution is $Φ^{*} (x) = Φ_{m}^{*} (x)$ .

4.2 Expected value iteration method

On the other hand, we can use another way to get the new uncertainty distribution Φ^∗ (x) by giving condition that Φ₁ (x) and Φ₂ (x) have known.

Let Φ₁ (x) and Φ₂ (x) be the uncertainty distribution given by two experts about the uncertain variable and θ₁, θ₂ are respectively the expected value of Φ₁ (x), Φ₂ (x). Suppose there exist four constants a₁, a₂, b₁, b₂ such that Φ₁ (a₁) =0, Φ₁ (b₁) =1, Φ₂ (a₂) =0, Φ₂ (b₂) =1. Set $L_{1} = min {b_{1} - a_{1} ∣ Φ_{1} (a_{1}) = 0, Φ_{1} (b_{1}) = 1},$ (10) $L_{2} = min {b_{2} - a_{2} ∣ Φ_{2} (a_{2}) = 0, Φ_{2} (b_{2}) = 1} .$ (11) Denote L = max {L₁, L₂}. So the new uncertainty distribution Φ^∗ (x) is given by

$\begin{matrix} Φ^{*} (x) & = & (1 - \frac{| θ_{2} - θ_{1} |}{L}) Φ_{1} (x) \\ + \frac{| θ_{2} - θ_{1} |}{L} Φ_{2} (x) \end{matrix}$ (12)

When another expert gives his opinion, we can set his distribution as Φ₂ (x) and Φ^∗ (x) as Φ₁ (x). Then we will get a new uncertainty distribution Φ^∗ (x) included all experts’ opinion.

Theorem 4.2 Let ξ be the uncertain variable with uncertainty distribution Φ^∗ (x) defined by Eq.(12). Then the expected value of ξ is

$E [ξ] = θ_{1} - \frac{| θ_{2} - θ_{1} | (θ_{1} - θ_{2})}{L}$ (13)

Proof. Because $Φ^{*} (x) = (1 - \frac{| θ_{2} - θ_{1} |}{L}) Φ_{1} (x) + \frac{| θ_{2} - θ_{1} |}{L} Φ_{2} (x)$ . Applying Theorem 2.2, we have

$\begin{matrix} E [ξ] & = & \int_{- \infty}^{+ \infty} x d Φ^{*} (x) \\ = & (1 - \frac{| θ_{2} - θ_{1} |}{L}) \int_{- \infty}^{+ \infty} x d Φ_{1} (x) \\ + \frac{| θ_{2} - θ_{1} |}{L} \int_{- \infty}^{+ \infty} x d Φ_{2} (x) \\ = & (1 - \frac{| θ_{2} - θ_{1} |}{L}) θ_{1} + \frac{| θ_{2} - θ_{1} |}{L} θ_{2} \\ = & θ_{1} - \frac{| θ_{2} - θ_{1} | (θ_{1} - θ_{2})}{L} \end{matrix}$ (14) So the expected value of Φ^∗ (x) is $E [ξ] = θ_{1} - \frac{| θ_{2} - θ_{1} | (θ_{1} - θ_{2})}{L}$ . The proof is completed.

The method by using Eq.(12) to get the distribution integrating previous information may be called the expected value iteration method (E-V-I Method). The specific steps of this method are as follows.

Algorithm 2. Expected value iteration method (E-V-I Method)

There are m distributions ${\tilde{Φ}}_{1}, \dots, {\tilde{Φ}}_{m}$ given by m experts. Let $Φ_{1}^{*} (x) = {\tilde{Φ}}_{1} (x)$ , k = 1. Compute the expected value of $Φ_{1}^{*} (x)$ by Eq. (2).

If k = m go to Step 1. Otherwise, let $Φ_{1} (x) = Φ_{k}^{*} (x)$ and $Φ_{2} (x) = {\tilde{Φ}}_{k + 1} (x)$ .

Calculate the expected value θ₁ of $Φ_{1} (x) = Φ_{k}^{*} (x)$ by Eq. (13) if k > 1, and the expected value θ₂ of Φ₂ (x) by Eq. (2). Set L₁ and L₂ by Eqs. (10) and (11), respectively. Denote L = max {L₁, L₂}. Compute the new distribution by $\begin{matrix} \begin{matrix} Φ_{k + 1}^{*} (x) = (1 - \frac{| θ_{2} - θ_{1} |}{L}) Φ_{1} (x) \\ + \frac{| θ_{2} - θ_{1} |}{L} Φ_{2} (x) \end{matrix} \end{matrix}$

Let k = k + 1 and go to Step 2.

The final distribution is $Φ^{*} (x) = Φ_{m}^{*} (x)$ .

Thus, we see, when θ₂ = θ₁, the expected value of new distribution is equal to θ₁ and θ₂; when θ₂ > θ₁ (θ₁ < θ₂) the expected value is in the middle between θ₁ and θ₂, θ₁ < E [ξ] < θ₂ (θ₂ < E (ξ) < θ₁). Because Φ₁ (x) may contain all former experts’ opinions, and Φ₂ (x) comes from new or next expert, so the expected value θ^∗ of Φ^∗ (x) is the synthesis of all expected value.

4.3 Prior distribution

When there is only one expert, the final distribution from the expert is often too subjective to appropriate. So we give a prior distribution to balance the opinions of experts. Decision makers can set the linear uncertainty distribution ℒ (a, b) as the prior distribution, and a, b are respectively the upper and lower limits of the expert’s empirical data. Then they can calculate the distribution by the above methods, even if there is only one expert. There is another benefit to this approach, which is to assume that they don’t know anything about the problem and that all possibilities are equal.

5 Examples

To illustrate that the new distributions by two methods are all appropriate and compare with the existing methods, two examples will be given. Through these examples, the rationality and practicability of the two algorithms proposed in this paper are well reflected.

Example 5.1

By using the questionnaire survey, we collected multiple experts’ experimental data of the distance between place A and place B as following: $\begin{matrix} \begin{matrix} A_{1} : & (0.8, 0.1), (1, 0.5), (1.2, 0.95), (1.4, 0.995), \\ (1.6, 0.999), (1.8, 1), (2, 1), (2.2, 1), (2.4, 1) \\ A_{2} : & (0.8, 0.05), (1, 0.1), (1.2, 0.5), (1.4, 0.7), \\ (1.6, 1), (1.8, 1), (2, 1), (2.2, 1), (2.4, 1) \\ A_{3} : & (0.8, 0), (1, 0), (1.2, 0.2), (1.4, 0.4), (1.6, 0.6), \\ (1.8, 0.65), (2, 0.7), (2.2, 0.75), (2.4, 0.75) \\ A_{4} : & (0.8, 0), (1, 0.333), (1.2, 0.8), (1.4, 0.933), \\ (1.6, 1), (1.8, 1), (2, 1), (2.2, 1), (2.4, 1) \\ A_{5} : & (0.8, 0.2), (1, 0.5), (1.2, 0.7), (1.4, 0.8), \\ (1.6, 0.9), (1.8, 1), (2, 1), (2.2, 1), (2.4, 1) \\ A_{6} : & (0.8, 0), (1, 0), (1.2, 0), (1.4, 0), (1.6, 0.3), \\ (1.8, 0.4), (2, 0.6), (2.2, 0.9), (2.4, 0.95) \end{matrix} \end{matrix}$ where A_i represents ith domain expert, i = 1, 2, ⋯ , 6. We can see that each expert gives his/her own opinion, but does not give a certain distribution Φ (x). We use the empirical uncertainty distribution as the uncertainty distribution, and get m distribution functions Φ₁, ⋯ , Φ_m. The total possible values provided by the six experts are 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2, 2.2, 2.4 (km). Then calculate the corresponding values of Φ_i (x) for x = 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2, 2.2, 2.4. These data are shown in Table 1.

Table 1
Data for distance between place A and B

x (km) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Φ₁ (x) 0.1 0.5 0.95 0.995 0.999 1 1 1 1

Φ₂ (x) 0.05 0.1 0.5 0.7 1 1 1 1 1

Φ₃ (x) 0 0 0.2 0.4 0.6 0.65 0.7 0.75 0.75

Φ₄ (x) 0 0.333 0.8 0.933 1 1 1 1 1

Φ₅ (x) 0.2 0.5 0.7 0.8 0.9 1 1 1 1

Φ₆ (x) 0 0 0 0 0.3 0.4 0.6 0.9 0.95

First, we use Algorithm 1 to get the final distribution. We set first expert’s empirical uncertainty distribution as Φ₁ (x), and the linear uncertainty distribution ℒ (a, b) as Φ₂ (x) to assume that we don’t know anything about this problem, a, b are respectively the upper and lower limits of x in empirical data. Then calculate the new distribution Φ^∗ (x) by Eq.(6).

Next, we set Φ₂ (x) = Φ^∗ (x), and let next expert’s empirical uncertainty distribution as Φ₁ (x). For condition distribution, we also need to calculate θ₁, the expected value of Φ₁ (x). Then we update the new distribution contain all experts’ opinion who have already appeared. Repeat this step untill all experts have been used. Finally we obtain the distribution Φ^∗ (x) as following:

$\begin{matrix} Φ^{*} (x) = {\begin{matrix} 0 & if x < 0.8, \\ - 0.7548 + 0.7548 x & if 0.8 \leq x \leq 1, \\ - 0.3854 + 0.5363 x & if 1 \leq x \leq 1.2, \\ - 1.5319 + 1.79 x & if 1.2 \leq x \leq 1.4, \\ 0.61625 & if 1.4 \leq x \leq 2, \\ - 1.3025 + 0.9594 x & if 2 \leq x \leq 2.4, \\ 1 & if x > 2.4 . \end{matrix} \end{matrix}$

On the other hand, we use Algorithm 2 to compute the final uncertainty distribution Φ^∗∗ (x). The process is basically the same as the condition iteration method. The difference between two methods is that we need to calculate two expected value of Φ₁ (x), Φ₂ (x), and calculate Φ^∗ (x) by Eq. (12). Finally we obtain the distribution Φ^∗∗ (x) as following:

$\begin{matrix} Φ^{* *} (x) = {\begin{matrix} 0 & if x < 0.8, \\ - 0.6755 + 0.8829 x & if 0.8 \leq x \leq 1, \\ - 1.1431 + 1.3505 x & if 1 \leq x \leq 1.2, \\ - 0.135 + 0.5105 x & if 1.2 \leq x \leq 1.4, \\ - 0.4429 + 0.7303 x & if 1.4 \leq x \leq 1.6, \\ 0.363 + 0.2267 x & if 1.6 \leq x \leq 1.8, \\ 0.209 + 0.3122 x & if 1.8 \leq x \leq 2, \\ 0.006 + 0.4137 x & if 2 \leq x \leq 2.2, \\ 0.679 + 0.1078 x & if 2.2 \leq x \leq 2.4, \\ 1 & if x > 2.4 . \end{matrix} \end{matrix}$

Figure 1 shows the difference of the final distributions by two methods and expert’s empirical distribution. From Figure 1, we can find that the distribution obtained by the expected value iteration method is smoother than the other one. Because expected value iteration method is distinguished by the relative size of the experts’ expected values, and condition iteration method is reflected by the conditional uncertainty distribution. Compared with the expected value iteration method, the condition iteration method can reflect the conservativeness of uncertainty theory. And the expected values of two final distribution by above methods are, respectively, ${\hat{θ}}^{*} = 1.318915$ , ${\hat{θ}}^{* *} = 1.397226$ . Not only are we interested in the final result, we are also interested in this process. Figure 2 shows the expected value of every iteration and each expert. We can see that the new expected value of every iteration is always between the expected values of new expert and the former expert.

Fig.1

The distributions of experts and two methods of Example 5.1.

Fig.2

Expected values with iteration of Example 5.1.

Example 5.2

Consider 6 experts’ experimental data from Wang [21] as follows:

$\begin{matrix} \begin{matrix} T_{1} : & (70, 0.15), (75, 0.75), (80, 1) \\ T_{2} : & (60, 0.2), (70, 0.4), (75, 0.5), (80, 1) \\ T_{3} : & (50, 0.5), (60, 0.6), (70, 0.7), (80, 0.9) \\ T_{4} : & (60, 0.05), (70, 0.15), (80, 0.51), (85, 0.85), \\ (90, 0.95) \\ T_{5} : & (75, 0.5), (80, 0.8), (90, 0.95) \\ T_{6} : & (60, 0.1), (70, 0.2), (75, 0.5), (80, 0.7), \\ (90, 0.9) \end{matrix} \end{matrix}$ where T_i represent the i-th expert, i = 1, 2, ⋯ , 6. We can see that each expert gives his or her own opinion, but does not give a certain distribution Φ (x). We use the empirical uncertainty distribution as the uncertainty distribution, and get m distribution functions Φ₁ (x) , ⋯ , Φ₆ (x). The total possible values provided by the six experts are 50, 60, 70, 75, 80, 85, 90. Then calculate the corresponding values of Φ_i (x) when x = 50, 60, 70, 75, 80, 85, 90. These data are shown inTable 2.

Table 2

Data for average score by six teachers

x	50	60	70	75	80	85	90
Φ₁ (x)	0	0	0.15	0.75	1	1	1
Φ₂ (x)	0	0.2	0.4	0.5	1	1	1
Φ₃ (x)	0.5	0.6	0.7	0.8	0.9	0.95	1
Φ₄ (x)	0	0.05	0.15	0.33	0.51	0.85	0.95
Φ₅ (x)	0	0	0	0.5	0.8	0.875	0.95
Φ₆ (x)	0	0.1	0.2	0.5	0.7	0.9	1

Next, we need a new distribution that contains all experts’ opinions. Referring to the steps in the first example, we can obtain two final uncertaintydistribution Φ^∗ (x), Φ^∗∗ (x) by two methods respectively as follows, $\begin{matrix} Φ^{*} (x) = {\begin{matrix} 0 & if x < 50, \\ - 0.8357 + 0.0167 x & if 50 \leq x \leq 60, \\ - 0.2401 + 0.0068 x & if 60 \leq x \leq 70, \\ - 0.465 + 0.1 x & if 70 \leq x \leq 75, \\ 0.735 & if 75 \leq x \leq 80, \\ 0.4334 + 0.0038 x & if 80 \leq x \leq 85, \\ - 3.4307 + 0.0492 x & if 85 \leq x \leq 90, \\ 1 & if x > 90, \end{matrix} \end{matrix}$ and $\begin{matrix} Φ^{* *} (x) = {\begin{matrix} 0 & if x < 50, \\ - 0.6001 + 0.0134 x & if 50 \leq x \leq 60, \\ - 0.6835 + 0.0148 x & if 60 \leq x \leq 70, \\ - 2.667 + 0.0431 x & if 70 \leq x \leq 75, \\ - 1.9599 + 0.0337 x & if 75 \leq x \leq 80, \\ - 1.6718 + 0.0301 x & if 80 \leq x \leq 85, \\ - 0.7526 + 0.0193 x & if 85 \leq x \leq 90, \\ 1 & if x > 90 . \end{matrix} \end{matrix}$

On the other hand, according to Wang [21], the distribution obtained by the Delphi method after asking the experts twice is as follows $\begin{matrix} Φ_{Delphi} (x) = {\begin{matrix} 0 & if x < 60, \\ - 0.792 + 0.015 x & if 60 \leq x \leq 65, \\ - 1.988 + 0.0334 x & if 65 \leq x \leq 70, \\ - 2.744 + 0.0442 x & if 70 \leq x \leq 75, \\ - 3.614 + 0.0558 x & if 75 \leq x \leq 80, \\ - 0.414 + 0.0158 x & if 80 \leq x \leq 85, \\ 0.011 + 0.0108 x & if 85 \leq x \leq 90, \\ 1 & if x > 90 . \end{matrix} \end{matrix}$

And the expected values respectively are ${\hat{θ}}^{*} = 72.94691255$ , ${\hat{θ}}^{* *} = 71.56367141$ , ${\hat{θ}}_{Delphi}$ =72.8575. Figure 3 can show the difference between our two new methods and Delphi method clearly.

Fig.3

The distributions of experts and three methods of Example 5.2.

As in the first example, the following figure (see Figure 4) show that the different of expected value of each iteration by two new methods and each expert. We can see that the new expected value of each iteration is always between the expected values of new expert and the former expert too.

Fig.4

Expected values with iteration of Example 5.2.

From the Wang’s paper, we know that the expectation in practice is 72.9558. And the estimated expected value by our two new methods are all close to it.

6 Conclusion

How to deal with multi-expert data is an important issue. It not only requires us to maintain the objectivity of the result, but also allows us to better balance the opinions of various experts. Delphi method may be a good way, but not the best way in some situations. Because we need to ask the experts’ opinions many times, until the final result meets our requirements. Based on the concept of conditional distribution and expected value, this paper provided two methods to deal with this problem. The condition iteration method through conditional distribution combines the opinions of multiple experts. And the expected value iteration method combines the opinions of various experts by averaging the new distribution and the total distributions. Both methods implement the idea of iteration. When decison maker deal with multi-expert data, no matter how many experts they have (even if there is only one), they can always get a more objective result. The two methods have their own advantages and disadvantages, expected value iteration method is distinguished by the relative size of the experts’ expected values, and condition iteration method is reflected by the conditional uncertainty distribution. So the two methods are suitable for different situations. If the decision maker only pays attention to the expected value of the uncertain variable, the expected value iteration method would be a better choice. If the uncertainty distribution of an uncertain variable is more important, then the condition iteration method would be a better choice. Because the two methods discussed in this paper are designed to solve the problem of integrating multiple expert opinions, there may be some new research in practical application decisions, especially in game theory, or making business decisions in uncertain finance.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No.61673011) and Science Foundation of Jiangsu Province (China) for Young Scientists (No.BK20170916) and Postgraduate Research and Practice Innovation Program of Jiangsu Province(China) (No.KYCX19_0249).

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Liu , Uncertain regression analysis: An approach for imprecise observations, Soft Computing 22 (2018), 5579–5582.

x (km)	0.8	1	1.2	1.4	1.6	1.8	2	2.2	2.4
Φ₁ (x)	0.1	0.5	0.95	0.995	0.999	1	1	1	1
Φ₂ (x)	0.05	0.1	0.5	0.7	1	1	1	1	1
Φ₃ (x)	0	0	0.2	0.4	0.6	0.65	0.7	0.75	0.75
Φ₄ (x)	0	0.333	0.8	0.933	1	1	1	1	1
Φ₅ (x)	0.2	0.5	0.7	0.8	0.9	1	1	1	1
Φ₆ (x)	0	0	0	0	0.3	0.4	0.6	0.9	0.95