Associated Probabilities in Insufficient Expert Data Analysis

Abstract

Problems of modeling uncertainty and imprecision for the analysis of insufficient expert data (IED) are considered in the environment of interactive multi-group decision-making (MGDM). Based on the Choquet finite integral, a moments’ method for the IED is developed for the evaluation of the associated probabilities class (APC) of Choquet’s second-order capacity based on the informational entropy maximum principle. Based on the IED new approach of the lower and upper Choquet’s second-order capacities, identification is developed. The second pole of insufficient expert data, the data imprecision indicator, is presented in the form of a fuzzy subset and image on the alternatives set. In the environment of the Dempster–Shafer belief structure, connections between an associated possibilities class (APosC), with the APC, and an associated focal probabilities class (AFPC) are constructed. In the approach of A. Kaufman’s theory of expertons, based on the APosC and the AFPC unique fuzzy subset, the IED image on the alternatives set is constructed. Based on Sugeno’s finite integral most typical value (MTV), as a prediction on possible alternatives set, the IED is constructed. In the example, a sensitive and comparative analysis is provided for the evaluation of the new approach’s stability and reliability.

1. Introduction

It is known that intelligent decision-making tools play an important role in the problem of improving almost all aspects of human activity. Among them, probabilistic-statistical or deterministic modeling is mostly used in decision-making problems [1]. However, when working on complex and difficult decision-making problems, the assumption of fuzziness has a special place today [2,3,4,5,6,7,8,9]. As always, this assumption is related to the difficulty of studying the problem and/or the lack of objective statistical data. In this case, the use of expert data is necessary to obtain reliable estimates. However, with the admission of fuzziness, new difficulties arise. It is clear that our ability to make reliable decisions with the increasing complexity of information decreases to a certain level [2,5,7,10,11,12]. Therefore, the two-pole characteristics of the incompleteness of expert data, fuzzy uncertainty and imprecision, fall into a certain conflicting resistance [10,13,14,15]. Using fuzzy modeling approaches, the authors in their research [4,11,12,13,14,16,17,18,19,20,21,22,23,24,25,26] try to reduce the impact of the uncertainty index on decision-making, but instead, the imprecision characteristics—the levels of fuzzy assignment of the fuzzy set increase. Our research is concerned with the information analysis of this complex uncertainty of IED and its use in modeling more precise decisions with minimal decision risks from the point of view of systems research [2,3,4,10,11,12,16,27,28,29]. Here, we imply the following: if a decision-making specified mechanism is involved in the decision-making model, then we are dealing with minimal risk of decision-making within the model environment, of course.

Today, the complicated problems of analyzing and synthesizing complex phenomena often require the creation of new, deeply detailed decision-making models. There are two classical approaches to objective data analysis. If experimental data are “sufficiently” exact, then, for their processing and estimation of general characteristics, probabilistic-statistical methods can be used. If data are presented with “imprecision” by intervals, then, for their study, the methods of theory of errors will be used. However, there are cases when both the methods of statistics and the theory of errors do not give satisfactory results. The reason consists of the nature of the data and the means of measurement, description, scaling, etc. When data are presented by intervals, and their description is “vague,” characterized by “overlapping,” and in the receipt of data the expert is intervened, it is clear that the nature of data is combined; parallel to the probabilistic-statistical uncertainty, there exists the fuzzy uncertainty which is received by the activity of expert. This fact gives rise to the necessity of the use of fuzzy methods. Only the probabilistic-fuzzy analysis guarantees more or less adequate results. When we assume the expert’s evaluation of some decision-making approaches, the problems of modeling uncertainty and imprecision must be solved under the data information of the MGDM or multi-attribute group decision-making (MAGDM) [3,11,12,16,17,22,23,29,30,31]. One way to solve this problem is presented in this article.

The tasks of MGDMs and MAGDM are especially complicated when interactions between possible alternatives and attributes are observed in the decision-making process [28,32,33]. Often, in such cases, statistical evaluations of the possible alternatives (ratings, benefits, etc.) from objective experiments are almost non-existent, and expert evaluations and knowledge are the only sources of information for certain evaluations of alternatives [7,8]. However, expert evaluations of alternatives are incomplete due to their statistical insufficiency [4,11,12] and uncertainty, and imprecision appears in their evaluations [4,10,15,24,25]. Problems of modeling uncertainty and imprecision for the analysis of the IED are considered in the environment of MGDM or MAGDM.

Let us now return to the phenomenon of decision alternative interactions in MGDM, decision attribute interactions in MAGDM models, and related fuzzy measure identification problems [28,32,33,34,35]. Modern MAGDM modeling is concerned with the analysis of complex problems and processes. MAGDM modeling, which takes into account the interaction or dependence of possible alternatives, becomes particularly relevant. At the same time, such studies are more and more in demand, which consider expert evaluations of a bipolar nature, in the form of fuzzy uncertainty and imprecision [11,12,22,23]. In such MAGDM modeling of the experts’ data aggregations, the authors largely use non-additive aggregation tools [10,36,37]. In many cases, in the construction of the relevant aggregation integral operators, monotone (fuzzy) measures are used as the uncertainty index [24,25], and the imprecision descriptor is the integrative function identified by expert evaluations, also known as the compatibility function [5,6,7,10]. Sugeno [15] was the first to define a monotone (fuzzy) measure. As further studies have shown, the monotone measure theory has been found to be the most effective tool for describing the uncertainty index in interactive MGDM modeling [16,28,30,32,33,34,38] when the additive and probabilistic measure in the aggregation integral operators should be replaced by monotone, fuzzy measures are used for building more believable and plausible models. Within the framework of interactive MAGDM modeling, the authors of [32,34] discussed the behavior analysis of invited experts and the related uncertainty index, or fuzzy measure identification problems. There are several approaches to solving this problem: linear polynomial approaches [30,32,33,39], optimization approaches for fuzzy measure identification in the environment of non-additive finite integrals of Sugeno, Choquet [16,28,34,38], and others. The k-order additive fuzzy measure identification problem is given in [25,30,33,34]. There is a review article [40] that discusses a number of problems related to non-additive measure identification. As already mentioned, the Choquet finite integral aggregation is the most common and popular tool in interactive MAGDM modeling. It should be noted that, in order to better represent the intellectual activity of experts in modern MAGDM or MGDM problems, the semantic forms of expert evaluations take rather complex forms in the form of fuzzy sets [7,8,9,10]. That is why the construction of extensions of the Choquet-type aggregation operators for different fuzzy environments continues [3,14,17,18,19]. Many authors dedicated their studies to the Choquet integral type extended aggregation fuzzy operators and applying them to decision-making models with interactive alternatives [11,13,14,17,30,32,33,34,35,40]. It should be said that, as practice has shown, when the interaction between alternatives in interactive MAGDM models is revealed to a low degree, the use of the Choquet-type aggregation operators loses reliability. This is related to the fact that aggregation with the Choquet integral reflects the interaction of only certain types of combinations of alternatives, such as the focal elements of consonant structure [10,41]. Of course, a general solution to this problem is not possible. However, the authors of this article tried to solve this problem for different fuzzy environments in the presence of certain conditions [19,20,21,35,42,43,44]. The same problem was set in the case of this research.

In this research, a new two-pole (uncertainty and imprecision) aggregation approach to the solution of the mentioned problem is presented, which can be briefly described as follows. In Section 2, based on the Choquet finite integral, the moments’ method for the IED is developed for the evaluation of the insufficient associated probabilities class (IAPC) of Choquet’s second-order capacity by use of the principle of maximum informational entropy. Based on the IED, a new approach to the problem of identification of the lower and upper Choquet’s second-order capacities is developed. In Section 3, the second pole of insufficient expert data, the data imprecision indicator, is presented in the form of a fuzzy subset image on an alternative set. In the environment of Dempster–Shafer, they believe structure connections between the APosC with the IAPC and the AFPC are constructed. In the approach of A. Kaufman’s theory of expertons, based on the APosC and the AFPC, unique fuzzy subset, as the IED image, on the alternatives set is constructed in Section 3. In Section 4, based on Sugeno’s finite integral, both poles of IED are aggregated into the most typical value (MTV) as a prediction of optimal alternatives is constructed. In Section 5, a new approach for the MAGDM general model is developed. In Section 6, an illustrative example of the presentation of the new approach is constructed. In Section 7, the sensitivity and comparative analysis for the evaluation of a new approach’s reliability is developed. Based on the IED, the predictive alternatives are selected from all possible alternatives. The basic conclusions and possible future perspective studies on the theme presented in this work are considered in Section 8. Appendix A discusses the exact scheme for identifying the class of associated probabilities based on the complete statistics of monotone moments.

2. Fuzzy Measure Identification Problem on the Insufficient Experts Data

In this section, we consider a new approach for fuzzy measure identification, an index of uncertainty generated by insufficient expert data on possible alternatives in the general MGDM model. Using the method of generalized moments based on the aggregation of the Choquet integral, an identification scheme will be formed, where the estimate of the class of fuzzy measure associated probabilities will be reduced to the solution of the system of transcendental equations built on the principle of maximum entropy.

Let it be given a finite set, $X=\{x_1,x_2,...,x_n\}$, which consists of elements of all possible alternatives in some MGDM models.

Definition 1 

[24]. Let $X=\{x_1,x_2,...,x_n\}$ be a finite set. Let $g$ be a set function $g:2^X\to \left[\mathrm{0,1}\right]$. $g$ is a fuzzy measure on $X$ if

$$\left(i\right)\hspace{1em}g\left(\varnothing \right)=0;g\left(X\right)=1; \left(ii\right)\hspace{1em}\forall G,H\subseteq X,if G\subseteq H,theng\left(G\right)\le g\left(H\right)\hspace{0.33em}.$$

Let $\xi ,\xi :X\to R_0^{+},{ \xi }_i\equiv \xi \left(x_i\right)\ge 0$ represent the results of some variable measurements of the finite set $X$ by some non-negative certain scale. The numerical characteristics of the variable $\xi$ are identical to data usually given by experimenters and can be easily evaluated by statistical methods. However, sometimes, in MGDM experiment data is wholly absent, and an expert’s knowledge is a unique source for receiving information on the variable $\xi$. We assume the expert evaluations for the variable $\xi$ (ratings, utilities, gains, and others) on the set $X$. Therefore, the problems of modeling fuzzy uncertainty and imprecision must be solved under the data information of the MGDM. We also assume that the uncertainty index is presented by a fuzzy measure. It is known that fuzzy measures on a finite set $X$ can be represented by the associated probabilities class $\{P_{\sigma },\hspace{0.33em}\sigma \in S_n\}$ [13,39,45], where $S_n$ is the group of all permutations of the set $X=\{x_1,x_2,\dots ,x_n\}$. Then, for $\forall \sigma =\left\{\sigma \right(1),\sigma (2),\dots ,\sigma (n\left)\right\}\in S_n$, the probability distribution (associated probability [45]),

$$P_{\sigma }=\left\{P_{\sigma }\right(x_{\sigma \left(1\right)}),{ P}_{\sigma }(x_{\sigma \left(2\right)}),\dots ,P_{\sigma }(x_{\sigma \left(n\right)}\left)\right\},$$

(1)

is connected with the fuzzy measure $g$ in the following form:

$$\begin{gathered} \hspace{1em}{P}_{\sigma }\left(x_{\sigma \left(1\right)}\right)=g\left(\left\{x_{\sigma \left(1\right)}\right\}\right), \\ \hspace{1em}\dots \\ \hspace{1em}{P}_{\sigma }\left(x_{\sigma \left(i\right)}\right)=g\left(\left\{x_{\sigma \left(1\right)},\dots ,x_{\sigma \left(i\right)}\right\}\right)-g\left(\left\{x_{\sigma \left(1\right)},\dots ,x_{\sigma \left(i-1\right)}\right\}\right), \\ \dots \\ \hspace{1em}{P}_{\sigma }\left(x_{\sigma \left(n\right)}\right)=1-g\left(\right\{x_{\sigma \left(1\right)},\dots ,x_{\sigma \left(n-1\right)}\left\}\right). \end{gathered}$$

(2)

From reference [45], it is known that for $\forall A\subseteq X$ and $\exists \tau \in S_n$, such that if $A=\{x_{i_1},x_{i_2},\dots ,x_{i_r}\}$, then, $\tau \left(1\right)=i_1,\dots ,\tau \left(r\right)=i_r$ and

$$g(A)=g\left(\{x_{\tau \left(1\right)},\dots ,x_{\tau \left(r\right)}\}\right)=\sum _{j=1}^r{P}_{\tau }\left(x_{\tau \left(j\right)}\right)=\sum _{j=1}^r{P}_{\tau }\left(x_{i_j}\right).$$

(3)

To any associated probability, $P_{\sigma }$ on $X$ corresponds with the sufficient class of a statistical moment of order $t$ of the random variable $\xi \left(t=0, 1, \dots , n-1; \overline{{\xi }_{\sigma }^0}\equiv 1\right)$:

$$\sum _{i=1}^n{P}_{\sigma }\left(x_{\sigma \left(i\right)}\right)({\xi }_{\sigma \left(i\right)}{)}^t=\overline{{\xi }_{\sigma }^t} .$$

(4)

On each permutation, the probability distribution is defined by corresponding moments, that is, the possibility of $n!$ representations. From Equation (4), one can find the associated probabilities:

$$P_{\sigma }\left(x_{\sigma \left(i\right)}\right)=\sum _{l=0}^{n-1}{a}_l^{\sigma \left(i\right)}\overline{{\xi }_{\sigma }^l}.$$

(5)

where $i=1,\dots ,n$ and the coefficients $a_l^{\sigma \left(i\right)}$ are received from data in Appendix A.

Definition 2.

(a) The sets of moments,

$$\{\overline{{\xi }_{\sigma }^t}{\}}_{\sigma \in S_n},t=1,\dots ,n-1,$$

 is called an associated moments class (AMC); (b) the set of probabilities $\{P_{\sigma }{\}}_{\sigma \in S_n}$is called an associated probabilities class (APC).

In the AMC, there are $(n-1)n!$ elements. With the aid of Equation (5), the AMC uniquely defines the APC, and, consequently, one may receive the possibility of the fuzzy measure identification on the set $X$:

$$g\leftrightarrow \{P_{\sigma }{\}}_{\sigma \in S_n}\leftrightarrow \{\overline{{\xi }_{\sigma }^t}{\}}_{t=1;\sigma \in S_n}^{n-1}.$$

(6)

In general, associated moments $\overline{{\xi }_{\sigma }^t}$ will be numerically evaluated from objective statistical measurements of the variable $\xi$. However, sometimes, for decision-making problems (for example, emergency logistics management decision-making problems, strategic management decision-making problems, and others), objective statistical measurements practically do not exist, and expert (network dispatchers, managers, and others) knowledge and experience are the unique information source for the reflection of some evaluations on the model’s input variable $\xi$. We consider such cases. Let the expert’s group be given $E\equiv \{e_1,e_2,\dots ,e_s\}$, which participates in the evaluation process of the variable $\xi$. Let us denote the $i$-th expert’s $e_i$ evaluations of the variable $\xi$ by the value ${\xi }^i$ and ${\xi }_{ji}\equiv {\xi }^i\left(x_j\right), \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} j=1,\dots ,n, \mathrm{and} i=1,2,\dots ,s$.

We will match the ${\xi }_{ji},j=1,\dots ,n$, of the expert’s data to such a permutation, ${\sigma }^{\left(i\right)}\in S_n$, for which these non-negative values will be arranged in a non-increasing order:

$${\xi }_{{\sigma }^{\left(i\right)}\left(1\right)i}\ge {\xi }_{{\sigma }^{\left(i\right)}\left(2\right)i}\ge \dots \ge {\xi }_{{\sigma }^{\left(i\right)}\left(n\right)i}.$$

(7)

Definition 3.

The monotone $t$-order moment ($M{M}_t$) of the value ${\xi }^i$, with respect to the fuzzy measure $g,$is called the value of the Choquet finite integral [36] of the $t$-th power of the same value with respect to the fuzzy measure $g,$

$$\begin{gathered} M{M}_t({\xi }^i)=Ch\int ({\xi }^i{)}^t\circ g(\cdot )={\int }_0^{+\infty }g\left(\right\{x/x\in X,\left({\xi }^i\right(x){)}^t\ge \alpha \})d\alpha = \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\sum _{j=1}^n{P}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)({\xi }_{{\sigma }^{\left(i\right)}\left(j\right)i}{)}^t, \end{gathered}$$

(8)

i.e., $M{M}_t$ of the value ${\xi }^i$ coincides with some $t$-order-associated moment of the same value:

$$M{M}_t\left({\xi }^i\right)=\overline{{\xi }_{{\sigma }^{\left(i\right)}}^t},i=1,\dots ,s;t\dots \dots ,n-1;{\sigma }^{\left(i\right)}\in S_n.$$

(9)

Definition 4.

The data in Equation (9) is called total monotone moments expert data (TMMED) associated with the experts $E$.

For the expert $e_i$ and respective permutation ${\sigma }^{\left(i\right)}\in S_n$, let us denote the following TMMED as follows:

$$E_{{\sigma }^{\left(i\right)}}^i\equiv \{\overline{{\xi }_{{\sigma }^{\left(i\right)}}^1},\overline{{\xi }_{{\sigma }^{\left(i\right)}}^2},\dots ,\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{n-1}}\}.$$

(10)

Obviously, depending on the specific problem, some elements from the TMMED will have some practical content value for experts. For example, first and second-order moments, and others. Moreover, as is often the case, from certain considerations or even from prehistoric data, experts may have even estimated these important elements. Then, let us suppose the following. We assume that in the TMMED from Equation (10) that, for the expert $e_i$, only some monotone moment’s numerical values are known by his/her intellectual activity. Let us denote these data as follows:

$${\widehat{E}}_{{\sigma }^{\left(i\right)}}^i\equiv \left\{\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i1}}},\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i2}}},\dots ,\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i{k}_i}}}\right\},1\le q_{il}\le n-1;1\le k_i\le n-1,i=1,\dots ,s.$$

(11)

Definition 5.

The data of Equation (11), from Equation (10), is called the insufficient monotone moment’s expert data (IMMED) associated with the expert $e_i$ from $E$.

Based on the IMMED ${\widehat{E}}_{{\sigma }^{\left(i\right)}}^i$ for the permutation ${\sigma }^{\left(i\right)}\in S_n$, the problem of evaluation of associated probability $P_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right),j=1,\dots ,n$, of a fuzzy measure $g$ is reduced to the following mathematical programming problem with the use of Shannon’s entropy maximum principle [38,46]:

$$\begin{gathered} \mathrm{m}\mathrm{a}\mathrm{x}\left\{-\sum _{j=1}^n{P}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)\log P_{\sigma }\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)\right\}, \\ \sum _{j=1}^n{P}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)=1, P_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)\in \left[\mathrm{0,1}\right], j=1,..,n, \end{gathered}$$

(12)

$$\sum _{j=1}^n{P}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right){\left({\xi }_{{\sigma }^{\left(i\right)}\left(j\right)i}\right)}^{q_{il}}=\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{il}}}; 1\le l\le k_i.$$

The corresponding Lagrangian has the following form:

$$\begin{gathered} L=-\sum _{j=1}^n{P}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)\log P_{\sigma }\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)+ \\ \sum _{l=1}^{k_i}{\lambda }_i^l\left(\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{il}}}-\sum _{j=1}^n{P}_{\sigma }\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)({\xi }_{{\sigma }^{\left(i\right)}\left(j\right)i}{)}^{q_{il}}\right)+ \\ {\lambda }_i^0\left(1-\sum _{j=1}^n{P}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)\right). \end{gathered}$$

(13)

Consider the system of equations:

$$\frac{\partial L}{\partial P_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(t\right)}\right)}=0, t=1,\dots ,n,$$

(14)

or

$$-\log P_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(t\right)}\right)-{\lambda }_i^0-\sum _{l=1}^{k_i}{\lambda }_i^l({\xi }_{{\sigma }^{\left(i\right)}\left(t\right)i}{)}^{q_{il}}=0,$$

(15)

where, $t=1, 2, \dots , n;\hspace{1em}{\lambda }_i^0\equiv 1+{\lambda }_i^0$. Then,

$$P_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(t\right)}\right)=\exp \left(-{\lambda }_i^0-\sum _{l=1}^{k_i}{\lambda }_i^l{\left({\xi }_{{\sigma }^{\left(i\right)}\left(t\right)i}\right)}^{q_{\mathit{il}}}\right),$$

(16)

where, $t=1 ,2, \dots ,n$; ${\sigma }^{\left(i\right)}\in S_n$.

Consider the generalized statistical sum:

$$Z_{{\sigma }^{\left(i\right)}}\left(\frac{{\lambda }_i^l}{l}=1,\dots ,k_i\right)=\sum _{t=1}^n\exp \left(-\sum _{l=1}^{k_i}{\lambda }_i^l({\xi }_{{\sigma }^{\left(i\right)}\left(t\right)i}{)}^{q_{il}}\right).$$

(17)

It is known that

$$\frac{\partial Z_{{\sigma }^{\left(i\right)}}({\lambda }_i^l/l=1,\dots ,k_i)}{\partial {\lambda }_i^l}=\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{il}}},\ln (Z_{{\sigma }^{\left(i\right)}}({\lambda }_i^l/l=1,\dots ,k_i))={\lambda }_i^0,$$

(18)

are identities. Expert data $E_{{\sigma }^{\left(i\right)}}^i\equiv \left\{\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i1}}},\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i2}}},\dots ,\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i{k}_i}}}\right\}$ is known for every expert $e_i\in E$. With respect to ${\lambda }_i^0,{ \lambda }_i^l, l=1,\dots ,k_i$, based on the necessary condition of extreme points existing for a function from Equation (18), we obtain the following equations system:

$$\sum _{t=1}^n\left((({\xi }_{{\sigma }^{\left(i\right)}\left(t\right)i}{)}^{q_{il}}-\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{il}}})\exp \left(-\sum _{k=1}^{k_i}{\lambda }_i^k({\xi }_{{\sigma }^{\left(i\right)}\left(t\right)i}{)}^{q_{ik}}\right)\right)=0,l=1,\dots ,k_i,$$

(19)

$${\lambda }_i^0=\ln \left(\sum _{t=1}^n\exp \left(-\sum _{l=1}^{k_i}{\lambda }_i^l({\xi }_{{\sigma }^{\left(i\right)}\left(t\right)i}{)}^{q_{il}}\right)\right).$$

If ${\widehat{\lambda }}_i^0,{\widehat{\lambda }}_i^l, l=1,\dots ,k_i$, are some numerical solutions of Equation (19), then by Equation (16) we evaluate the probability distribution $\left\{{\widehat{P}}_{{\sigma }^{\left(i\right)}}\right(x_{{\sigma }^{\left(i\right)}\left(j\right)}){\}}_{j=1}^n$ for the expert $e_i\in E$. If we solve Equation (19) for every expert, then the insufficient associated probabilities class (IAPC) $\{{\widehat{P}}_{{\sigma }^{\left(i\right)}}{\}}_{i=1}^s$, instead of the total APC $\{P_{\sigma }{\}}_{\sigma \in S_n}$, will be received. Thus, now we can construct the lower and upper dual Choquet’s second-order capacities [36] based on the IAPC $\{{\widehat{P}}_{{\sigma }^{\left(i\right)}}{\}}_{i=1}^s$: $\forall A\subseteq X$:

$$\widehat{g}\left(A\right)=\underset{e_i\in E}{\mathrm{m}\mathrm{i}\mathrm{n}}{\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(A\right),\hspace{1em}{\widehat{g}}^{\ast }\left(A\right)=\underset{e_i\in E}{\mathrm{m}\mathrm{a}\mathrm{x}}{\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(A\right).$$

(20)

We have come to an important definition:

Definition 6.

Choquet’s dual second-order capacities $(\widehat{g},{\widehat{g}}^{\ast })$ by Equation (20) are called associated fuzzy measures (AFMs) induced by the experts IMMEDs $\{{\widehat{E}}_{{\sigma }^{\left(i\right)}}^i{\}}_{i=1}^s$on the set $X$ with respect to variable $\xi$.

3. Construction of Fuzzy Subset (Image) on the Insufficient Experts Data

Every expert $e_i\in E,$ based on his/her intellectual activities, possesses insufficient data, or IMMED,

$${\widehat{E}}_{{\sigma }^{\left(i\right)}}^i\equiv \left\{\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i1}}},\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i2}}},\dots ,\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i{k}_i}}}\right\}, 1\le q_{il}\le n-1; 1\le k_i\le n-1, i=1,\dots ,s,$$

and induces the corresponding associated probability, $\left\{{\widehat{P}}_{{\sigma }^{\left(i\right)}}\right(x_1),...,{\widehat{P}}_{{\sigma }^{\left(i\right)}}(x_n\left)\right\}$. Therefore, all experts IMMEDs $\{{\widehat{E}}_{{\sigma }^{\left(i\right)}}^i{\}}_{i=1}^s$ induce a corresponding IAPC $\{{\widehat{P}}_{{\sigma }^{\left(i\right)}}{\}}_{i=1}^s$. In turn, based on this class, AFMs (20), as the expert data uncertainty index, are constructed in Section 2.

In this section, we consider the scheme of constructing a certain fuzzy subset of the set of possible alternatives, a data image and a descriptor of the second pole of insufficient expert data in the MGDM environment.

Constructed dual fuzzy measure AFMs contain information about a body of evidence on $X$ [37,41]. It is obvious that for each expert $e_i\in E$, and its corresponding permutation ${\sigma }^{\left(i\right)}\in S_n$, this body of evidence is consonant [10]:

$$K_1^{{\sigma }^{\left(i\right)}}=\left\{x_{{\sigma }^{\left(i\right)}\left(1\right)}\right\}\subseteq K_2^{{\sigma }^{\left(i\right)}}=\left\{x_{{\sigma }^{\left(i\right)}\left(1\right)},x_{{\sigma }^{\left(i\right)}\left(2\right)}\right\}\subseteq \dots \subseteq K_n^{{\sigma }^{\left(i\right)}}=\left\{x_{{\sigma }^{\left(i\right)}\left(1\right)},\dots ,x_{{\sigma }^{\left(i\right)}\left(n\right)}\right\}.$$

(21)

On this body of evidence one can define the possibility distribution $\left\{{\pi }_{{\sigma }^{\left(i\right)}}\right(x_1),\dots ,{\pi }_{{\sigma }^{\left(i\right)}}(x_n\left)\right\}$ [10] on the set $X$ which is connected with focal subsets and corresponding focal probability values [41]:

$$m^{{\sigma }^{\left(i\right)}}\left(\varnothing \right)=0,\hspace{1em}\hspace{1em}\sum _{j=1}^n{m}^{{\sigma }^{\left(i\right)}}\left(K_j^{{\sigma }^{\left(i\right)}}\right)=1.$$

For any focal subset we introduce the uniform conditional probability distribution $P_{{\sigma }^{\left(i\right)}}(\cdot /K_j^{{\sigma }^{\left(i\right)}}), j=1,\dots ,n$. Then, the possibility distribution $\left\{{\pi }_{{\sigma }^{\left(i\right)}}\right(x_1),\dots ,{\pi }_{{\sigma }^{\left(i\right)}}(x_n\left)\right\}$ will be approximated by the associated probability ${\widehat{P}}_{{\sigma }^{\left(i\right)}}(\cdot )$. Based on the total probability formula, for $\forall x_{{\sigma }^{\left(i\right)}\left(j\right)},j=1,\dots ,n$, we have the following:

$$\begin{gathered} {\widehat{P}}_{{\sigma }^{\left(i\right)}}(x_{{\sigma }^{\left(i\right)}\left(j\right)})=\sum _{j=1}^n{P}_{{\sigma }^{\left(i\right)}}\left(\frac{x_{{\sigma }^{\left(i\right)}\left(j\right)}}{K_j^{{\sigma }^{\left(i\right)}}}\right)m^{{\sigma }^{\left(i\right)}}\left(K_j^{{\sigma }^{\left(i\right)}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{j=1}^n\frac{m^{{\sigma }^{\left(i\right)}}\left(K_j^{{\sigma }^{\left(i\right)}}\right)}{\left|K_j^{{\sigma }^{\left(i\right)}}\right|}{I}_{K_j^{{\sigma }^{\left(i\right)}}}(x_{{\sigma }^{\left(i\right)}\left(j\right)}), \end{gathered}$$

(22)

where, $\left|K_j^{{\sigma }^{\left(i\right)}}\right|$ is the cardinality of the focal element $K_j^{{\sigma }^{\left(i\right)}}$ and $I_{K_j^{{\sigma }^{\left(i\right)}}}$ is the indicator of $K_j^{{\sigma }^{\left(i\right)}}$. It is evident that associated probability ${\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(\cdot \right)$ for $\forall A\subseteq X$ will be within the limits of the necessity and possibility measures $N^{{\sigma }^{\left(i\right)}}\left(A\right) \mathrm{and}{ \Pi }^{{\sigma }^{\left(i\right)}}\left(A\right)$, respectively [10,41], as follows:

$$N^{{\sigma }^{\left(i\right)}}\left(A\right)\le {\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(A\right)\le {\Pi }^{{\sigma }^{\left(i\right)}}\left(A\right).$$

(23)

It is easy to connect the associated probability ${\widehat{P}}_{{\sigma }^{\left(i\right)}}(\cdot )$ on the permutation ${\sigma }^{\left(i\right)}$ with the respective possibility distribution as follows:

$${\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{\sigma \left(l\right)}\right)=\sum _{j=l}^n\frac{1}{j}\left({\pi }_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)-{\pi }_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j+1\right)}\right)\right),$$

(24)

$$l=1,\dots ,n; {\pi }_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(n+1\right)}\right)\equiv 0,$$

where,

$$1={\pi }^{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(1\right)}\right)\ge {\pi }^{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(2\right)}\right)\ge \dots \ge {\pi }^{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(n+1\right)}\right)\equiv 0.$$

(25)

From (24) one can obtain for the following:

$${\pi }_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(l\right)}\right)=\sum _{j=1}^n\min \{{\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(l\right)}\right),{\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)\}, l=1,\dots ,n,$$

(26)

Based on the work in [10], we assume that the possibility distribution $\left\{{\pi }_{{\sigma }^{\left(i\right)}}\right(x_1),\dots ,{\pi }_{{\sigma }^{\left(i\right)}}(x_n\left)\right\}$ on $X$ induces the fuzzy subset ${\tilde{\xi }}^{{\sigma }^{\left(i\right)}}$ with following membership function:

$${\mu }_{{\sigma }^{\left(i\right)}\left(j\right)}^i\equiv {\mu }_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)={\pi }_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right), j=1,\dots ,n.$$

(27)

Definition 7.

Fuzzy subset ${\tilde{\xi }}^{{\sigma }^{\left(i\right)}},{ \sigma }^{\left(i\right)}\in S_n$ is called an associate fuzzy subset (AFS) induced by the activity of the expert $e_i\in E$.

(22) is the equations system for focal probabilities $m^{{\sigma }^{\left(i\right)}}\left(K_j^{{\sigma }^{\left(i\right)}}\right), j=1,\dots ,n$. We have:

$${\Pi }^{{\sigma }^{(i)}}(A)=\underset{x_{{\sigma }^{\left(i\right)}\left(j\right)}\in A}{\mathrm{m}\mathrm{a}\mathrm{x}}{\pi }_{{\sigma }^{\left(i\right)}}(x_{{\sigma }^{\left(i\right)}\left(j\right)})=\underset{x_{{\sigma }^{\left(i\right)}\left(j\right)}\in A}{\mathrm{m}\mathrm{a}\mathrm{x}}\sum _{l=1}^n\min \{{\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(l\right)}\right),{\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)\},$$

(28)

and

$$N^{{\sigma }^{\left(i\right)}}\left(A\right)=1-{\Pi }^{{\sigma }^{\left(i\right)}}\left(\overline{A}\right), \mathrm{where} \overline{A}=X\backslash A.$$

(29)

From (22) we obtain the following:

$$\begin{gathered} {\widehat{P}}_{{\sigma }^{\left(i\right)}}(x_{{\sigma }^{\left(i\right)}\left(1\right)})=\frac{m^{{\sigma }^{\left(i\right)}}\left(K_1^{{\sigma }^{\left(i\right)}}\right)}{1}+\frac{m^{{\sigma }^{\left(i\right)}}\left(K_2^{{\sigma }^{\left(i\right)}}\right)}{2}+\dots +\frac{m^{{\sigma }^{\left(i\right)}}\left(K_n^{{\sigma }^{\left(i\right)}}\right)}{n}, \\ \cdots \\ {\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)=\frac{m^{{\sigma }^{\left(i\right)}}\left(K_j^{{\sigma }^{\left(i\right)}}\right)}{j}+\dots +\frac{m^{{\sigma }^{\left(i\right)}}\left(K_n^{{\sigma }^{\left(i\right)}}\right)}{n}, \\ \cdots \\ \hspace{1em}\hspace{1em}\hspace{1em}{\widehat{P}}_{{\sigma }^{\left(i\right)}}(x_{{\sigma }^{\left(i\right)}\left(n\right)})=\frac{m^{{\sigma }^{\left(i\right)}}\left(K_n^{{\sigma }^{\left(i\right)}}\right)}{n}. \end{gathered}$$

(30)

For the focal probabilities we obtain the following:

$$\begin{gathered} m^{{\sigma }^{\left(i\right)}}\left(K_n^{{\sigma }^{\left(i\right)}}\right)=n{\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(n\right)}\right), \\ \cdots \\ {m}^{{\sigma }^{\left(i\right)}}\left(K_j^{\sigma }\right)=j\left({\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)-{\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j+1\right)}\right)\right), \\ \cdots \\ {m}^{{\sigma }^{\left(i\right)}}\left(K_1^{\sigma }\right)=\left({\widehat{P}}_{{\sigma }^{\left(i\right)}}\right(x_{{\sigma }^{\left(i\right)}\left(1\right)})-{\widehat{P}}_{{\sigma }^{\left(i\right)}}(x_{{\sigma }^{\left(i\right)}\left(2\right)}\left)\right). \end{gathered}$$

(31)

Definition 8. 

The set ${\left\{m^{{\sigma }^{\left(i\right)}}\right\}}_{e_i\in E}$ of focal probabilities is called an associated focal probabilities class (AFPC) induced by the IMMEDs $\{{\widehat{E}}_{{\sigma }^{\left(i\right)}}^i{\}}_{i=1}^s$.

Definition 9. 

The set of fuzzy subsets $\{{\tilde{\xi }}^{{\sigma }^{\left(i\right)}}{\}}_{e_i\in E}$ is called an associated fuzzy subsets class (AFSC) induced by the experts IMMEDs $\{{\widehat{E}}_{{\sigma }^{\left(i\right)}}^i{\}}_{i=1}^s$.

Now we consider IMMEDs  $\{{\widehat{E}}_{{\sigma }^{\left(i\right)}}^i{\}}_{i=1}^s$ in the Kaufman’s theory of expertons [3,16]. In the discrete case of the experts’ table of membership, function values are represented by the following quantities: $\{{\mu }_{{\sigma }^{\left(i\right)}}^j\equiv {\mu }_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right) \mathrm{and} j=1,\dots ,n;{\sigma }^{\left(i\right)}\in S_n\}$, where, ${\mu }_{{\sigma }^{\left(i\right)}}^j,j=1,\dots ,n$ is the value of membership level of ${\tilde{\xi }}^{{\sigma }^{\left(i\right)}}$. Following the scheme of Kaufman’s theory of expertons, if $a_0=0\le a_1\le a_2\le \dots \le a_L=1$ is an increasing sequence of membership levels, the corresponding expert is presented as follows [16]:

$$E_{{\sigma }^{\left(i\right)}\left(j\right)}^i=\frac{1}{L}\sum _{l=1}^L{I}_{\left[a_l;1\right]}\left({\mu }_{{\sigma }^{\left(i\right)}\left(j\right)}^i\right),$$

(32)

where, $i=1,\dots ,s,\hspace{0.33em}j=1,\dots ,n,$ $I_{\left[a_l;1\right]}\left(\cdot \right)$ is an indicator function, and

$${\mu }_j=\sum _{i=1}^s\left(w_i{E}_{{\sigma }^{\left(i\right)}\left(j\right)}^i\right),\hspace{1em}j=1,\dots ,n,$$

(33)

we call the weighed averaging expert of the membership of $x_j\in X$.

Definition 10.

The fuzzy subset

$$\tilde{\xi }=⟨\begin{array}{ccccccc}{x}_1& x_2& \cdots & x_i& \cdots & x_n& X\\ {\mu }_1& {\mu }_2& \cdots & {\mu }_i& \cdots & {\mu }_n& \mu \end{array}⟩$$

(34)

is called a fuzzy subset (an image) induced by the IMMEDs $\{{\widehat{E}}_{{\sigma }^{\left(i\right)}}^i{\}}_{i=1}^s$.

Therefore, we construct a unique image, a fuzzy subset $\tilde{\xi }$ on insufficient expert data, in which the second pole of insufficient data is represented in the form of imprecision.

4. Fuzzy Expected Value (FEV) and Most Typical Value (MTV) on the Insufficient Expert Data

It was shown [8,9,10] that in the MGDM, the nature of expert data is the source of the fuzziness and in our approach its uncertainty index is represented by the Choquet’s dual capacities $(\widehat{g},{\widehat{g}}^{\ast })$ (Section 2). In Section 3 we demonstrated that the IED on the finite set $X$ in the expertons approach induced the unique fuzzy set, the image $\tilde{\xi }$.

In this section, we consider the problem of constructing the most typical alternatives based on IEDs in the MGDM environment, which actually represents the problem of aggregating both poles from the IEDs on the same alternatives. The Sugeno’s finite integral [24] is taken as an aggregation tool, the calculated value of which in the MGDM is the fuzzy expected value (FEV) of the image [4,11,12]; its preface represents the most typical values of the MGDM [13,27,39], and, in our cases, alternatives. From a practical point of view, this means that we can consider this preface as recommended or optimal predictive alternatives. In short, all this can be formally described as follows.

From the point of view of decision making, there are all possible conditions for choosing the MTV corresponding to $\tilde{\xi }$. In our case the following occurs:

$$MTV={\mu }^{-1}\left(FEV\left(\mu \right)\right),$$

(35)

where, $FEV\left(\mu \right)$ is the fuzzy expected value or the Sugeno finite integral’s value [24]. Indeed, consider the ordered sequence of $\mu$’s values:

$${\mu }_{\tau \left(1\right)}\le {\mu }_{\tau \left(2\right)}\le ...\le {\mu }_{\tau \left(n\right)}.$$

(36)

Let us introduce the notations:

$$K_i=\{{\mu }_{\tau \left(i\right)}\dots {\mu }_{\tau \left(n\right)}\},\hspace{1em}{g}_i=\widehat{g}\left(K_i\right),\hspace{1em}{g}_i^{\ast }={\widehat{g}}^{\ast }\left(K_i\right),\hspace{1em}i=1\dots ...,n.$$

On the finite set $X$, the $FEV$ can be calculated by the Sugeno integral’s formula [24]:

$$FE{V}_{\widehat{g}}\left({\mu }_{\tilde{\xi }}\right)={\int }_X{\mu }_{\tilde{\xi }}\left(x\right)\circ \widehat{g}\left(\cdot \right)=\underset{i}{max}\left\{{\mu }_{\tau \left(i\right)}\wedge g_i\right\}.$$

(37)

In our case, one receives thr lower and upper FEV’s, respectively, as follows:

$$F\equiv FE{V}_{\widehat{g}}\left({\mu }_{\tilde{\xi }}\right),\hspace{1em}{F}^{\ast }\equiv FE{V}_{{\widehat{g}}^{\ast }}\left({\mu }_{\tilde{\xi }}\right)=\underset{i}{max}\left\{{\mu }_{\tau \left(i\right)}\wedge g_i^{\ast }\right\}.$$

(38)

It is evident that $F\le F^{\ast }$ (because $\widehat{g}\le {\widehat{g}}^{\ast }$) and on [0; 1] the pair $F,F^{\ast }$ determines the interval $[F,F^{\ast }]$. Naturally Equation (35) must be changed by

$$MTV=\left\{x\in X| F\le {\mu }_{\tilde{\xi }}\left(x\right)\le F^{\ast }\right\}.$$

(39)

In the usual decision-making models, MTVs are predictive or recommendatory elements [13,27,39]. Based on the IED of some complex experiments, the constructed model, to a certain extent, represents the multi-group decision-making model, which predicts MTVs from the alternatives set $X$ with respect to the fuzzy variable $\tilde{\xi }$.

5. Application in the MAGDM

Now we consider the realization of the constructed approach for the general model of the MAGDM. The formation of the expert’s input data for construction of attributes is an important task of the alternatives ranking-prediction problem. To decide on the selection of an optimal alternative (decision), it is assumed that a set of candidate alternatives (CAs) already exists. This set is denoted by $CA=\{x_1,x_2,\dots ,x_n\}$, where we can describe attributes set and $S=\{s_1,s_2,\dots ,s_m\}$ is the set of all attributes (minimized attributes are transformed into the form of maximized attributes) which define the CA’s selection. Let $W=\{w_1,w_2,\dots ,w_s\}$ be the weights of experts. For each expert $e_k$, from the invited group of experts $E=\{e_1,e_2,\dots ,e_s\}$, let ${\xi }_l^k(x_j,s_i),l=1,\dots ,t_k$ be the $t_k$ number of parameters (rating various generalized weighted values of alternatives for all attributes or generalized weighted values of value functions) of his/her evaluations in non-negative real numbers for each candidate alternative $x_j (j=1,\dots ,n)$, with respect to each attribute $s_i,(i=1,\dots ,m)$. For the expert $e_k$ and ${\xi }_l^k, l=1,\dots ,r_k$ parameters, we construct $r_k$ number of binary relations $A_l^k=\{{\xi }_{l,ij}^k={\xi }_l^k(x_j,s_i), i=1,\dots ,m;j=1,\dots ,n\}$ as a decision-making matrix. Therefore, instead of Equation (4) in our scheme we will have the following:

$$\sum _{j=1}^n{P}_{{\sigma }_{li}^{\left(k\right)}}\left(s_{{\sigma }_{li}^{\left(k\right)}\left(j\right)}\right)\left({\xi }_{l,i{\sigma }_l^{\left(k\right)}\left(j\right)}^k\right)=\overline{{\xi }_{l,i}^k}, l=1,\dots ,r_k; i=1,\dots ,m; k=1,\dots ,s,$$

(40)

where, ${\sigma }_{li}^{\left(k\right)}\in S_n$ is a permutation for which the values ${\xi }_{l,ij}^k$ are ordered non-incrementally—${\xi }_{l,i{\sigma }_l^{\left(k\right)}\left(1\right)}^k\ge {\xi }_{l,i{\sigma }_l^{\left(k\right)}\left(2\right)}^k\ge \dots \ge {\xi }_{l,i{\sigma }_l^{\left(k\right)}\left(n\right)}^k$, and the value $\overline{{\xi }_{l,i}^k}$ is the $k$-th expert’s monotone expectation estimate of the value ${\xi }_l^k$ with respect to the attribute $s_i$. Data in Equation (13) ${\widehat{E}}_{{\sigma }^{\left(i\right)}}^i\equiv \left\{\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i1}}},\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i2}}},\dots ,\overline{{\xi }_{{\sigma }^{\left(i\right)}}^{q_{i{k}_i}}}\right\}, 1\le q_{il}\le n-1;1\le k_i\le n-1, i=1,\dots ,$ s, is transformed in the following insufficient monotone moments expert data (IMMED) associated with the experts $E$. For each $e_k, k=1,\dots ,s,$ expert,

$${\widehat{E}}^k=\{\overline{{\xi }_{l,i}^k}, l=1,\dots ,r_k; i=1,\dots ,m;{ \sigma }_{li}^{\left(k\right)}\in S_n\}.$$

(41)

Furhter, for each $e_k, k=1,\dots ,s,$ expert, the mathematical programming problem is transformed into the following form:

$$\underset{{\sigma }_{li}^{\left(k\right)}\in S_n}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{\underset{\begin{array}{c}i=1,\dots ,m;\\ l\dots \dots ,r_k\end{array}}{\mathrm{m}\mathrm{i}\mathrm{n}}\left(-\sum _{j=1}^n{P}_{{\sigma }_{li}^{\left(k\right)}}\left(x_{{\sigma }_{li}^{\left(k\right)}\left(j\right)}\right)\log \left(P_{{\sigma }_{li}^{\left(k\right)}}\left(x_{{\sigma }_{li}^{\left(k\right)}\left(j\right)}\right)\right)\right)\right\},$$

$$\sum _{j=1}^n{P}_{{\sigma }_{li}^{\left(k\right)}}\left(x_{{\sigma }_{li}^{\left(k\right)}\left(j\right)}\right)=1,P_{{\sigma }_{li}^{\left(k\right)}}\left(x_{{\sigma }_{li}^{\left(k\right)}\left(j\right)}\right)\in \left[\mathrm{0,1}\right],j=1,..,n;l=1,\dots ,r_k,i=1,\dots ,m$$

(42)

$$\sum _{j=1}^n{P}_{{\sigma }_{li}^{\left(k\right)}}\left(x_{{\sigma }_{li}^{\left(k\right)}\left(j\right)}\right)\left({\xi }_{l,i{\sigma }_l^{\left(k\right)}\left(j\right)}^k\right)=\overline{{\xi }_{l,i}^k},\dots =1,\dots ,\dots i=1,\dots ,m.$$

Therefore, for each expert $e_k$, by solving problem (37) we obtain the $m\times r_k$ associated probabilities, $\{P_{{\sigma }_{li}^{\left(k\right)}}(\cdot ){\}}_{\begin{array}{c}l=1,\dots ,r_k\\ i=1,\dots ,m\end{array}}$. In total, taking into account all the expert data, $m\times r_k\times s$ associated probabilities is $\{P_{{\sigma }_{li}^{\left(k\right)}}(\cdot ){\}}_{\begin{array}{c}l=1,\dots ,r_k\\ i=1,\dots ,m\\ k=1,\dots s\end{array}}$. For each expert’s data, the scheme of Equations (13)–(19) will be applied and the associated probability estimates $\{{\widehat{P}}_{{\sigma }_{li}^{\left(k\right)}}(\cdot ){\}}_{\begin{array}{c}l=1,\dots ,r_k\\ i=1,\dots ,m\\ k=1,\dots ,s\end{array}}$ will be obtained. Similarly, to Equation (20), the dual Choquet capacities of the second order will be also constructed as follows:

$$\widehat{g}\left(A\right)=\underset{\begin{array}{c}{e}_k\in E\\ l=1,\dots ,r_k\\ i=1,\dots ,m\end{array}}{\mathrm{m}\mathrm{i}\mathrm{n}}{P}_{{\sigma }_{li}^{\left(k\right)}}\left(A\right),\hspace{1em}{\widehat{g}}^{\ast }\left(A\right)=\underset{\begin{array}{c}{e}_k\in E\\ l=1,\dots ,r_k\\ i=1,\dots ,m\end{array}}{\mathrm{m}\mathrm{a}\mathrm{x}}{P}_{{\sigma }_{li}^{\left(k\right)}}\left(A\right), A\subseteq X.$$

(43)

The calculation of the distribution of $m\times r_k\times s$ number of possibilities for the alternatives will be performed with the following transformations:

$${\pi }_{{\sigma }_{li}^{\left(k\right)}}\left(x_{{\sigma }_{li}^{\left(k\right)}\left(h\right)}\right)=\sum _{j=1}^n\min \{{\widehat{P}}_{{\sigma }_{li}^{\left(k\right)}}\left(x_{{\sigma }_{li}^{\left(k\right)}\left(h\right)}\right),\left({\widehat{P}}_{{\sigma }_{li}^{\left(k\right)}}\left(x_{{\sigma }_{li}^{\left(k\right)}\left(j\right)}\right)\right\},h=1,\dots ,n;$$

(44)

$$k=1,\dots ,s; i=1,\dots ,m; l=1,\dots ,r_k.$$

Analogous to Equation (27), these possibility distributions for each expert $e_k$ also induce the same number of Zadeh fuzzy sets over all combinations of alternatives:

$$\left\{{\mu }_{{\sigma }_{li}^{\left(k\right)}}\left(x_1\right)={\pi }_{{\sigma }_{li}^{\left(k\right)}}\left(x_1\right),\dots ,{\mu }_{{\sigma }_{li}^{\left(k\right)}}\left(x_n\right)={\pi }_{{\sigma }_{li}^{\left(k\right)}}\left(x_n\right)\right\}, l=1,\dots ,r_k; i=1,\dots ,m.$$

(45)

In the environment of experton theory, we construct only one fuzzy set over a group of alternatives. The scheme repeats itself. First, we construct expertons as follows:

$$E_{j,kli}=\frac{1}{L}\sum _{c=1}^L{I}_{\left[a_c;1\right]}\left({\pi }_{{\sigma }_{li}^{\left(k\right)}}(x_j)\right),j=1,\dots n,$$

(46)

where, $i=1,\dots ,s,\hspace{0.33em}j=1,\dots ,n$, $I_{\left[a_l;1\right]}\left(\cdot \right)$ is an indicator function, and

$${\mu }_j=\sum _{k=1}^s\frac{1}{r_{k}m}\left(\sum _{l=1}^{r_k}\sum _{i=1}^m\left(w_k{E}_{j,kli}\right)\right),\hspace{1em}j=1,\dots ,n.$$

(47)

Then the resulting fuzzy set has the following face:

$$\tilde{\xi }=⟨\begin{array}{ccccccc}{s}_1& s_2& \cdots & s_i& \cdots & s_n& S\\ {\mu }_1& {\mu }_2& \cdots & {\mu }_i& \cdots & {\mu }_n& \mu \end{array}⟩.$$

(48)

The recommended optimal alternative is selected according to the last paragraph of Section 4.

6. An Example

To illustrate the results obtained in this research, let us consider the following simple MGDM model. Suppose $X=\{x_1,x_2,x_3,x_4\}$ is a set of alternatives for making a complex decision. Let us assume that a predictive alternative (alternatives) should be chosen by a group of experts, $E=\{e_1,e_2,e_3\}$, from the estimations of any non-negative variable $\xi$ (rating, usefulness, etc.) of the alternatives. In our case we assume that $W=\{w_1,w_2,w_3\}=\{1/\mathrm{3,1}/\mathrm{3,1}/3\}$. Within the framework of the approach built in the article, let us assume that the experts presented the following assessments (

Table 1. Experts’ evaluations on the variable $\xi$ by the variables ${\xi }^i$ and IMMEDs (${\xi }_{ji}\equiv {\xi }^i\left(x_j\right),j=1,2,3,4; i=1,2,3$ and IMMEDs $\{{\widehat{E}}_{{\sigma }^{\left(i\right)}}^i{\}}_{i=1}^3$).

	${\mathit{e}}_1\leftrightarrow {\mathit{\xi }}^1$	${\mathit{e}}_2\leftrightarrow {\mathit{\xi }}^2$	${\mathit{e}}_3\leftrightarrow {\mathit{\xi }}^3$
$x_1$	0.65	0.75	0.45
$x_2$	0.75	0.65	0.55
$x_3$	0.85	0.73	0.35
$x_4$	0.45	0.85	0.65
${\sigma }^{(i)}=({\sigma }^{(i)}(1),{\sigma }^{(i)}(2),{\sigma }^{(i)}(3),{\sigma }^{(i)}(4))$	${\sigma }^{(1)}=(3,2,1,4)$	${\sigma }^{(2)}=(4,1,3,2)$	${\sigma }^{(3)}=(4,2,1,3)$
${\overline{\xi }}_{{\sigma }^{(i)}}^1$	0.70	0.70	0.50
${\overline{\xi }}_{{\sigma }^{(i)}}^2$	0.45	0.50	0.35

).

Based on the analysis of insufficient expert data developed in the article, the solution to the prediction problem can be described using the following scheme.

Step 1. The numerical estimation of Lagrangian parameters ${\widehat{\lambda }}_i^0,{\widehat{\lambda }}_i^l, l=1,\dots ,k_i$.

For all experts we have the following: $q_i=2,{ q}_{i1}=1,{ q}_{i2}=1, \mathrm{and} i=1, 2, 3;$ or ${\mathrm{q}}_{11}={\mathrm{q}}_{21}={\mathrm{q}}_{31}=1\mathrm{and}{\mathrm{q}}_{12}={\mathrm{q}}_{22}={\mathrm{q}}_{32}=2.$

We composed the system of Equation (16) for each expert. Namely, for expert $e_1$ we obtain the following system of nonlinear equations:

$$\left\{\begin{array}{c}(0.85-0.7)\exp (-(0.85{\lambda }_1^1+0.85^2{\lambda }_1^2))+\\ (0.75-0.7)\exp (-(0.75{\lambda }_1^1+0.75^2{\lambda }_1^2))+\\ (0.65-0.7)\exp (-(0.65{\lambda }_1^1+0.65^2{\lambda }_1^2))+\\ (0.45-0.7)\exp (-(0.45{\lambda }_1^1+0.45^2{\lambda }_1^2))=0,\\ \\ (0.85^2-0.45)\exp (-(0.85{\lambda }_1^1+0.85^2{\lambda }_1^2))+\\ (0.75^2-0.45)\exp (-(0.75{\lambda }_1^1+0.75^2{\lambda }_1^2))+\\ (0.65^2-0.45)\exp (-(0.65{\lambda }_1^1+0.65^2{\lambda }_1^2))+\\ (0.45^2-0.45)\exp (-(0.45{\lambda }_1^1+0.45^2{\lambda }_1^2))=0.\end{array}\right.$$

After solving this system, ${\lambda }_1^0$ is calculated by the following expression:

$$\begin{gathered} {\lambda }_1^0=\ln (\exp (-(0.85{\lambda }_1^1+0.85^2{\lambda }_1^2\left)\right)+\exp (-(0.75{\lambda }_1^1+0.75^2{\lambda }_1^2))+ \\ \exp (-(0.65{\lambda }_1^1+0.65^2{\lambda }_1^2))+\exp (-(0.45{\lambda }_1^1+0.45^2{\lambda }_1^2\left)\right)). \end{gathered}$$

We conducted similar calculations for the second and third experts (the results are not given here).

The above-mentioned systems of equations (cases of all three experts) have complex roots. Consequently, in each case we replaced the solution of these systems by finding in [0, 1] × [0, 1] the local minimum of a functional, which is a norm of transcendental functions, involved in the corresponding system of equations.

For the numerical experiment we generated the random real numbers uniformly distributed in [0, 1] × [0, 1]. The experiments show that calculations are quite stable with respect to numerical errors.

Step 2. Computation of associated probabilities was performed according to the following formula:

$$\begin{gathered} {\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(t\right)}\right)=\mathit{exp}\left(-{\widehat{\lambda }}_i^0-\sum _{l=1}^{k_i}{\widehat{\lambda }}_i^l({\xi }_{{\sigma }^{\left(i\right)}\left(t\right)i}{)}^{q_{il}}\right), \\ t=1, 2, 3, 4; {\sigma }^{\left(i\right)}\in S_4,i=1, 2, 3. \end{gathered}$$

The results are presented in the

Table 2. The insufficient associated probabilities class (IAPC)$\{{\widehat{P}}_{{\sigma }^{\left(i\right)}}{\}}_{i=1}^3$.

$\mathrm{For} \mathrm{the} \mathrm{expert} e_1$: ${\mathit{\sigma }}^{\left(\mathbf{1}\right)}=\left(\mathbf{3},\mathbf{2},\mathbf{1},\mathbf{4}\right)$	${\widehat{\mathit{P}}}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}}\left({\mathit{x}}_{\mathbf{3}}\right)$	${\widehat{\mathit{P}}}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}}\left({\mathit{x}}_{\mathbf{2}}\right)$	${\widehat{\mathit{P}}}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}}\left({\mathit{x}}_{\mathbf{1}}\right)$	${\widehat{\mathit{P}}}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}}\left({\mathit{x}}_{\mathbf{4}}\right)$
$\mathrm{Values} \mathrm{of} {\widehat{\mathit{P}}}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}}\left({\mathit{x}}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}(.)}\right)$	0.209243	0.230296	0.253462	0.306999
$\mathrm{For} \mathrm{the} \mathrm{expert} e_2$: ${\mathit{\sigma }}^{\left(\mathbf{2}\right)}=\left(\mathbf{4},\mathbf{1},\mathbf{3},\mathbf{2}\right)$	${\widehat{P}}_{{\sigma }^{\left(2\right)}}\left(x_4\right)$	${\widehat{P}}_{{\sigma }^{\left(2\right)}}\left(x_1\right)$	${\widehat{P}}_{{\sigma }^{\left(2\right)}}\left(x_3\right)$	${\widehat{P}}_{{\sigma }^{\left(2\right)}}\left(x_2\right)$
$\mathrm{Values} \mathrm{of} {\widehat{\mathit{P}}}_{{\mathit{\sigma }}^{\left(\mathbf{2}\right)}}\left({\mathit{x}}_{{\mathit{\sigma }}^{\left(\mathbf{2}\right)}(.)}\right)$	0.188734	0.244374	0.256719	0.310173
$\mathrm{For} \mathrm{the} \mathrm{expert} e_3$: ${\mathit{\sigma }}^{\left(\mathbf{3}\right)}=\left(\mathbf{4},\mathbf{2},\mathbf{1},\mathbf{3}\right)$	${\widehat{P}}_{{\sigma }^{\left(3\right)}}\left(x_4\right)$	${\widehat{P}}_{{\sigma }^{\left(3\right)}}\left(x_2\right)$	${\widehat{P}}_{{\sigma }^{\left(3\right)}}\left(x_1\right)$	${\widehat{P}}_{{\sigma }^{\left(\mathbf{3}\right)}}\left(x_{\mathbf{3}}\right)$
$\mathrm{Values} \mathrm{of} {\widehat{\mathit{P}}}_{{\mathit{\sigma }}^{\left(3\right)}}\left({\mathit{x}}_{{\mathit{\sigma }}^{\left(3\right)}(.)}\right)$	0.17978	0.223298	0.271998	0.324924

. It should be noted that the probabilities associated with the generated values after each new attempt (for the same initial Table 1) differ by less than $0.5\times {10}^{-4}$ in value.

Step 3. Calculation of the dual Choquet’s second order capacities $(\widehat{g},{\widehat{g}}^{\ast })$ by the formulas:

$$\widehat{g}(A)=\underset{e_i\in E}{\mathrm{m}\mathrm{i}\mathrm{n}}{\widehat{P}}_{{\sigma }^{\left(i\right)}}(A),\hspace{1em}{\widehat{g}}^{\ast }(A)=\underset{e_i\in E}{\mathrm{m}\mathrm{a}\mathrm{x}}{\widehat{P}}_{{\sigma }^{\left(i\right)}}(A).$$

The results are presented in the

Table 3. Values of dual Choquet’s second order capacities $(\widehat{g},{\widehat{g}}^{*})$.

$\mathit{A}\subseteq \mathit{X}$	$\widehat{\mathit{g}}\left(\mathit{A}\right)$	${\widehat{\mathit{g}}}^{\mathit{*}}\left(\mathit{A}\right)$
$\left\{x_1\right\}$	0.244374	0.271998
$\left\{x_2\right\}$	0.223298	0.310173
$\left\{x_3\right\}$	0.209243	0.324924
$\left\{x_4\right\}$	0.17978	0.306999
$\{x_1,x_2\}$	0.483758	0.554547
$\{x_1,x_3\}$	0.462705	0.596922
$\{x_1,x_4\}$	0.433108	0.560461
$\{x_2,x_4\}$	0.403078	0.537295
$\{x_3,x_4\}$	0.445453	0.516242
$\{x_1,x_2,x_3\}$	0.693001	0.82022
$\{x_1,x_2,x_4\}$	0.675076	0.790757
$\{x_2,x_3,x_4\}$	0.728002	0.755626
$\{x_1,x_2,x_3,x_4\}$	1.0	1.0
$\{\varnothing \}$	0.0	0.0

.

Step 4. Calculation of the associated possibility distributions ${\pi }_{{\sigma }^{\left(i\right)}}$ on $X$ by the following formula (

Table 4. Associated possibility distributions $\left\{{\pi }_{{\sigma }^{\left(i\right)}}\right\},i=1, 2, 3$, on $X$.

$\mathrm{For} \mathrm{the} \mathrm{expert} e_1$: ${\mathit{\sigma }}^{\left(1\right)}=\left(\mathbf{3},\mathbf{2},\mathbf{1},\mathbf{4}\right)$	${\mathit{\pi }}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}}\left({\mathit{x}}_{\mathbf{3}}\right)$	${\mathit{\pi }}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}}\left({\mathit{x}}_{\mathbf{2}}\right)$	${\mathit{\pi }}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}}\left({\mathit{x}}_{\mathbf{1}}\right)$	${\mathit{\pi }}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}}\left({\mathit{x}}_{\mathbf{4}}\right)$
$\mathrm{Values} \mathrm{of} {\mathit{\pi }}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}}\left({\mathit{x}}_{{\mathit{\sigma }}^{\left(\mathbf{1}\right)}(.)}\right)$	0.836972	0.900131	0.946463	1.000000
$\mathrm{For} \mathrm{the} \mathrm{expert} e_2$: ${\mathit{\sigma }}^{\left(\mathbf{2}\right)}=\left(\mathbf{4},\mathbf{1},\mathbf{3},\mathbf{2}\right)$	${\pi }_{{\sigma }^{\left(2\right)}}\left(x_4\right)$	${\pi }_{{\sigma }^{\left(2\right)}}\left(x_1\right)$	${\pi }_{{\sigma }^{\left(2\right)}}\left(x_3\right)$	${\pi }_{{\sigma }^{\left(2\right)}}\left(x_2\right)$
$\mathrm{Values} \mathrm{of} {\mathit{\pi }}_{{\mathit{\sigma }}^{\left(\mathbf{2}\right)}}\left({\mathit{x}}_{{\mathit{\sigma }}^{\left(\mathbf{2}\right)}(.)}\right)$	0.754936	0.921856	0.946546	1.000000
$\mathrm{For} \mathrm{the} \mathrm{expert} e_3$: ${\mathit{\sigma }}^{\left(\mathbf{3}\right)}=\left(\mathbf{4},\mathbf{2},\mathbf{1},\mathbf{3}\right)$	${\pi }_{{\sigma }^{\left(3\right)}}\left(x_4\right)$	${\pi }_{{\sigma }^{\left(3\right)}}\left(x_2\right)$	${\pi }_{{\sigma }^{\left(3\right)}}\left(x_1\right)$	${\pi }_{{\sigma }^{\left(3\right)}}\left(x_3\right)$
$\mathrm{Values} \mathrm{of} {\mathit{\pi }}_{{\mathit{\sigma }}^{\left(\mathbf{3}\right)}}\left({\mathit{x}}_{{\mathit{\sigma }}^{\left(\mathbf{3}\right)}(.)}\right)$	0.71912	0.849674	0.947074	1.000000

):

$${\pi }_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(l\right)}\right)=\sum _{j=1}^n\min \left\{{\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(l\right)}\right),{\widehat{P}}_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)\right\}, l=1,\dots ,n.$$

Step 5. Identification of the associated fuzzy subsets, ${\tilde{\xi }}^{{\sigma }^{\left(i\right)}}$, by the associated possibility distributions $\left\{{\pi }_{{\sigma }^{\left(i\right)}}\right(x_1),\dots ,{\pi }_{{\sigma }^{\left(i\right)}}(x_n\left)\right\}$ as follows:

$$\left\{{\mu }_{{\sigma }^{\left(i\right)}\left(j\right)}^i\equiv {\mu }_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right)={\pi }_{{\sigma }^{\left(i\right)}}\left(x_{{\sigma }^{\left(i\right)}\left(j\right)}\right);j=1,\dots ,4;{\sigma }^{\left(i\right)}\in S_4\right\},i=1, 2, 3.$$

Step 6. Calculation of expertons on the membership levels of associated fuzzy subsets ${\tilde{\xi }}^{{\sigma }^{\left(i\right)}}$ by the following formula:

$$E_{{\sigma }^{\left(i\right)}\left(j\right)}^i=\frac{1}{L}\sum _{l=1}^L{I}_{\left[a_l;1\right]}\left({\mu }_{{\sigma }^{\left(i\right)}\left(j\right)}^i\right), i=1,\dots ,s;\hspace{0.33em}j=1,\dots ,n,$$

where, in our case, ${n=4, s=3, L=10, I}_{\left[a_l;1\right]}\left(\cdot \right)$ is an indicator function, and

$a_1$	$a_2$	$a_3$	$a_4$	$a_5$	$a_6$	$a_7$	$a_8$	$a_9$	$a_{10}$
0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0

The results are presented in

Table 5. Expertons $E_{{\sigma }^{\left(i\right)}\left(j\right)}^i$ on the membership levels of associated fuzzy subsets $\left\{{\tilde{\xi }}^{{\sigma }^{\left(i\right)}}\right\}$.

${\mathit{E}}_{{\mathit{\sigma }}^{\left(\mathit{i}\right)}\left(\mathit{j}\right)}^{\mathit{i}}/\mathit{i}$	1	2	3
1	0.9	0.9	0.9
2	0.9	1	0.8
3	0.8	0.9	1
4	1	0.7	0.7

.

Step 7. Calculation of the weighted averaging expertons as membership levels of the image $\tilde{\xi }$ on $X$ as follows:

$${\mu }_j=\sum _{i=1}^s\left(w_i{E}_j^i\right),\hspace{1em}j=1,\dots n; n=4, s=3.$$

Step 8. Identification of IEDs image $\tilde{\xi } \left(n=4\right)$ as follows:

$$\tilde{\xi }=⟨\begin{array}{ccccc}{x}_1& x_2& x_3& x_4& X\\ 0.9& 0.9& 0.9& 0.8& \mu \end{array}⟩.$$

Step 9. Identification of the permutation $\tau =\{{\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4\}\in S_4$ for which ${\mu }_{\tau \left(1\right)}\le {\mu }_{\tau \left(2\right)}\le {\mu }_{\tau \left(3\right)}\le {\mu }_{\tau \left(4\right)}$. We receive $\tau =\{4,3,2,1\}$.

Step 10. Construction of a consonant structure $K_j=\left\{x_{\tau \left(j\right)},\dots ,x_{\tau \left(n\right)}\right\},\hspace{1em}j=1,\dots ,n,$ and calculation values of the dual Choquet’s second-order capacities $(\widehat{g},{\widehat{g}}^{\ast })$ as follows (

Table 6. $K_j=\{x_{\tau \left(j\right)},\dots ,x_{\tau \left(n\right)}\},\hspace{1em}{g}_j=\widehat{g}\left(K_i\right), g_j^{*}={\widehat{g}}^{*}\left(K_i\right), j=1, 2, 3, 4$.

${\mathit{K}}_{\mathit{i}}$	${\mathit{K}}_1=\{{\mathit{x}}_{\mathit{\tau }\left(1\right)},{\mathit{x}}_{\mathit{\tau }\left(2\right)},{\mathit{x}}_{\mathit{\tau }\left(3\right)},{\mathit{x}}_{\mathit{\tau }\left(4\right)}\}$	${\mathit{K}}_2=\{{\mathit{x}}_{\mathit{\tau }\left(2\right)},{\mathit{x}}_{\mathit{\tau }\left(3\right)},{\mathit{x}}_{\mathit{\tau }\left(4\right)}\}$	${\mathit{K}}_3=\{{\mathit{x}}_{\mathit{\tau }\left(3\right)},{\mathit{x}}_{\mathit{\tau }\left(4\right)}\}$	${\mathit{K}}_3=\left\{{\mathit{x}}_{\mathit{\tau }\left(4\right)}\right\}$
$g_i$	1.0	0.693001	0.483758	0.244374
$g_i^{*}$	1.0	0.82022	0.554547	0.271998

):

$$g_j=\widehat{g}\left(K_i\right),\hspace{1em}{g}_j^{\ast }={\widehat{g}}^{\ast }\left(K_i\right),\hspace{1em}j=1,2,3,4$$

Step 11. Calculation of the extreme FEVs $F,F^{\ast }$ by the Sugeno’s integral as follows:

$$\begin{gathered} F={\int }_X{\mu }_{\tilde{\xi }}\left(x\right)\circ \widehat{g}(\cdot )=\underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}\{{\mu }_{\tau \left(i\right)}\wedge g_i\}=0.80000, \\ {F}^{*}={\int }_X{\mu }_{\tilde{\xi }}\left(x\right)\circ {\widehat{g}}^{*}(\cdot )=\underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}\{{\mu }_{\tau \left(i\right)}\wedge g_i^{\ast }\}=0.82022. \end{gathered}$$

Step 12. Finding the predictive alternatives as the MTV of image $\tilde{\xi }$, as follows:

$$MTV=\{x\in X/F\le {\mu }_{\tilde{\xi }}(x)\le F^{\ast }\}.$$

The prediction of the MGDM is alternative $x_4$, because $MTV=\left\{x_4\right\}$.

7. Sensitivity and Comparative Analysis

Sensitivity Analysis. The results of our approach represent the optimal predicted value recommendations for alternatives. It depends on the several initial values or values from intermediate calculations. Therefore, we conducted a sensitivity analysis with respect to the predicted values in order to determine the stability of the method in relation to certain changes in the initial data. We took the case of the previous example and considered it as expert data, i.e., the “disturbed” values of the estimates of Table 1 with accuracy $\epsilon$ (see

Table 7. Experts’ evaluations, “disturbed” by the accuracy $\epsilon >0$ on the variable $\xi$ (${\xi }_{ji}\equiv {\xi }^i\left(x_j\right),j=1,2,3,4; i=1,2,3)$ and IMMEDs $\{{\widehat{E}}_{{\sigma }^{\left(i\right)}}^i{\}}_{i=1}^3$.

	${\mathit{e}}_1\leftrightarrow {\mathit{\xi }}^1$	${\mathit{e}}_2\leftrightarrow {\mathit{\xi }}^2$	${\mathit{e}}_3\leftrightarrow {\mathit{\xi }}^3$
$x_1$	0.65 + $\epsilon$	0.75 + $\epsilon$	0.45 + $\epsilon$
$x_2$	0.75 + $\epsilon$	0.65 + $\epsilon$	0.55 + $\epsilon$
$x_3$	0.85 + $\epsilon$	0.73 + $\epsilon$	0.35 + $\epsilon$
$x_4$	0.45 + $\epsilon$	0.85 + $\epsilon$	0.65 + $\epsilon$
${\sigma }^{(i)}=({\sigma }^{(i)}(1),{\sigma }^{(i)}(2),{\sigma }^{(i)}(3),{\sigma }^{(i)}(4))$	${\sigma }^{(1)}=(3,2,1,4)$	${\sigma }^{(2)}=(4,1,3,2)$	${\sigma }^{(3)}=(4,2,1,3)$
${\overline{\xi }}_{{\sigma }^{\left(i\right)}}^1$	0.70 + $\epsilon$	0.7 + $\epsilon$	0.50 + $\epsilon$
${\overline{\xi }}_{{\sigma }^{\left(i\right)}}^2$	0.45 + $\epsilon$	0.50 + $\epsilon$	0.35 + $\epsilon$

).

For each small value of $\epsilon >0$ from the data in Table 7, we went through steps 1–12 of the scheme discussed above. Computational results, as epsilon-dependent results, are given in the diagrams below. As an epsilon, we took “disturbed” values with a step of 0.002 from the following set: $\epsilon \in \{0.000,\dots , 0.010, 0.012,\dots , 0.020,\dots , 0.030\}$. We obtained the following results, which we divided into six sections and presented the results in the form of diagrams (Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6).

We obtain approximately similar, stable dependences on $\epsilon$ for the rest of the experts as well, where the agreement of the stable values of the associated probabilities is guaranteed with an accuracy of one tenth.

We obtain approximately similar, stable dependences on $\epsilon$ for the subsets of the rest of the alternatives, where the coincidence of the phase-size stable values is guaranteed within a hundredth for accuracy.

We obtain approximately similar, stable dependences on $\epsilon$ for the rest of the experts as well, where the stable values of the associated abilities are guaranteed to match within a tenth for accuracy.

We obtain stable dependences on $\epsilon$ in the estimations of the agreement levels of alternatives, where stable values are guaranteed to coincide within hundredths for accuracy (Figure 4).

We obtain stable dependences of the “forecast band”$, [F,F^{\ast }]$, interval estimates on $\epsilon$, where stable values are guaranteed to coincide within hundredths for accuracy (Figure 5).

The stability of the methodology is evident in the presented example of decision-making. A similar analysis would be performed for any multi-attribute decision-making problem if the approach developed in the present article was chosen as the decision-making method.

Comparative analysis. In order to compare the results obtained by the constructed approach, the calculations obtained at the level of the presented example are compared with the results calculated by other classical statistical aggregation operators. These operators are Weighted Averaging (WA), Weighted Geometric (WG), Ordered Weighted Averaging (OWA), and Ordered Weighted Geometric (OWG) [44] operators. The aggregation operators mentioned in our notation take the following form with the following values (all weights are 1/3):

$$\begin{gathered} WA(x_j)=\sum _{i=1}^s{w}_i{\xi }_{ji},\hspace{1em}j=1,\dots ,4; s=3; \\ WG(x_j)=\prod _{i=1}^s({\xi }_{ji}{)}^{w_i},\hspace{1em}j=1,\dots ,4; s=3; \\ OWA(x_j)=\sum _{i=1}^s{w}_i{\xi }_{j\tau \left(i\right)},j=1,\dots ,4; s=3; \\ OWG(x_j)=\prod _{i=1}^s({\xi }_{j\tau \left(i\right)}{)}^{w_i},j=1,\dots ,4; s=3, \end{gathered}$$

(49)

where, ${\xi }_{j\tau \left(1\right)}\ge {\xi }_{j\tau \left(2\right)}\ge \dots \ge {\xi }_{j\tau \left(s\right)}, n=4, s=3$. ${\xi }_{ji}, j=1,\dots ,4; i=1,2,3$. Data are taken from Table 7. The aggregation results that were calculated using these operators are presented in

Table 8. Aggregation results of experts’ evaluations (Table 7) by the operators from Equation (46). Ranking of alternatives.

WA	WG	OWA	OWG
$WA(x_1)$ = 0.6166	$WG(x_1)$ = 0.6031	$OWA(x_1)$ = 0.6167	$OWG(x_1)$ = 0.6031
$WA(x_2)$ = 0.6500	$WG(x_2)$ = 0.6448	$OWA(x_2)$ = 0.6500	$OWG(x_2)$ = 0.6448
$WA(x_3)$ = 0.6433	$WG(x_3)$ = 0.6011	$OWA(x_3)$ = 0.6433	$OWG(x_3)$ = 0.6011
$WA(x_4)$ = 0.6500	$WG(x_4)$ = 0.6288	$OWA(x_4)$ = 0.6500	$OWG(x_4)$ = 0.6288
The ranking of alternatives
$x_2=x_4\succ x_3\succ x_1\hspace{0.17em}\hspace{0.17em}$	$x_2\succ x_4\succ x_1\succ x_3\hspace{0.17em}\hspace{0.17em}$	$x_2=x_4\succ x_3\succ x_1\hspace{0.17em}\hspace{0.17em}$	$x_2\succ x_4\succ x_1\succ x_3\hspace{0.17em}\hspace{0.17em}$

.

We also used the aggregation results calculated by the well-known TOPSIS [47] approach:

$$\begin{gathered} TOPSIS\left(x_1\right)=0.5000; TOPSIS\left(x_2\right)=0.5954, \\ TOPSIS\left(x_3\right)=0.4740; TOPSIS\left(x_4\right)=0.5224. \end{gathered}$$

The ranking of alternatives according to the TOPSIS approach is as follows:

$$x_2\succ x_4\succ x_1\succ x_3.$$

In multi-attribute decision-making models with aggregated values, the alternatives are ranked by aggregation operators from the highest aggregated value to the lowest aggregated value; also, for the selection with the Pareto optimal, the recommended alternative with the largest aggregated value. It becomes clear that as a result of aggregation with classical operators from Equation (48) and the TOPSIS approach, $\left\{x_2\right\}$ is an optimal alternative. $\left\{x_4\right\}$ is only in the second position, which represents the best recommended alternative according to the approach developed here. What should we look for in the difference?

Our approach is based only on the use of expert information in decision-making models. As is known, in such cases, expert data is incomplete and this incompleteness is reflected in bipolar characteristics, fuzzy uncertainty and imprecision, which are always in a certain conflicting resistance when considering specific problems. Solving the fuzzy uncertainty and imprecision modeling problem is an important guarantee for creating a less risky and more reliable decision-making environment. Note that the solution of this problem becomes essential for those decision-making models in which high-quality interactions between attributes are observed. We were able to partially solve this problem in the presented article, where the properties of the probabilistic class associated with monotonic measures, reflected in the applications of the Choquet and Sugeno integrals, were used to identify the most typical recommended alternatives. Classical aggregation operators in Equation (48) lack the ability to use both poles; the same refers to the case with the TOPSIS approach. As always, they only use the phenomenon of inaccuracy.

In conclusion we note that our research is concerned with information analysis of the complex uncertainty and imprecision modeling, and IED use in the modelling requires more precise decisions with minimal decision risks from the point of view of systems research.

8. Conclusions

This study presents a new approach to MGDM modeling with interacting alternatives. The main direction concerns the presentation of expert assessments and the analysis of insufficient data. As is known, the phenomenon of interaction of alternatives in MGDM modeling is best described by such integral operators, which take into account both poles of insufficient expert data, uncertainty, and imprecision. The new approach, developed in this paper, uses two such well-known integral operators, the Choquet and Sugeno integrals. The obtained results can be schematically and briefly described as follows. At the beginning, for the first pole of expert insufficient data, the index of uncertainty, is considered. Using the uncertainty index, the fuzzy measure identification scheme is constructed, taking into account the IED. The concept of a complete class of associated moments, based on the IED, is introduced. The concept of the monotone t-th power moment is defined as the value of the Choquet finite integral on the values of the t-th power of the expert estimates with respect to the fuzzy measure. This definition is consistent with a certain associated moment. Insufficient monotone moment expert data (IMMED)m associated with the experts E, is defined. For the estimation of insufficient fuzzy measures associated probabilistic classes, the principle of maximum entropy is used, taking into account IMMED. A non-linear mathematical programming problem is built, the realization of which is reduced to the solution of the system of transcendental equations. The next step concerns the second pole of insufficient expert data, the description of the data imprecision as a data image and a fuzzy subset of the set of alternatives. A consonant structure of alternatives is constructed over the set of all alternatives for each expert. In the evidence theory approximation the focal probabilities of the focal elements are related to the insufficient associated probabilities. An associated probability distribution is created, which for a given expert data is equated to a data image, or a certain fuzzy subset of the set of alternatives. In the experton theory approach, the images of the experts are condensed into a single image that represents the final image as a descriptor of the second pole of expert incompleteness, or data imprecision. The next step deals with the tool for the joint aggregation of the first and second poles of insufficient expert data in the form of the Sugeno integral. The result of the aggregation is connected to the definition of the most typical values, MTVs, as predictive and recommends alternatives to the set of alternatives, taking into account insufficient expert data. The constructed recommendation system is formally described for the case of the general MAGDM model. Finally, a simple demonstration example for the MGDM model is constructed to illustrate the application of the new approach. The example uses the scheme of the approach developed in this article. An illustrative example of this scheme is realized by obtaining predictive alternatives. The identification of the recommended optimal alternatives of the approach depends on several initial parameters and several values of intermediate calculations. Therefore, within the framework of the example, a sensitivity analysis is performed to determine the stability of the method. The obtained results are acceptable. In order to compare the constructed approach, the results obtained at the level of the presented example are compared with other classical statistical aggregation operators, the WA, WG, OWA and OWG operators, as well as with the aggregation results of the well-known TOPSIS approach. There is some inconsistency between the aggregation results and the identifications of the recommended alternatives. This is due to two things. First, both poles of insufficient expert data are taken into account in the new approach, unlike classical aggregations. Second, the phenomenon of interaction of arguments is also taken into account in the Choquet and Sugeno integral aggregations used in the new approach. Of course, the new approach developed in the study is limited in the sense that if the expert data are presented in modern semantic forms of fuzzy sets, such as orthopair values of a dual nature, Atanassov intuitionistic numbers, Yager’s Pythagorean numbers, or q-rang orthopair numbers, then the methodology will not be applicable anymore. Even if assessments by experts will not be presented in monotone expectations, but in certain transactions, then we will have to slightly modify the method. In our future research, a new approach will be developed for just such cases.