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Article

Probabilistic Linguistic Aggregation Operators Based on Einstein t-Norm and t-Conorm and Their Application in Multi-Criteria Group Decision Making

by
Kobina Agbodah
1,2,* and
Adjei Peter Darko
1,2
1
School of Management and Economics, University of Electronic Science and Technology of China, Chengdu 610054, China
2
Center for West African Studies, University of Electronic Science and Technology of China, Chengdu 610054, China
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(1), 39; https://doi.org/10.3390/sym11010039
Submission received: 8 December 2018 / Revised: 22 December 2018 / Accepted: 24 December 2018 / Published: 2 January 2019

Abstract

:
One of the major problems of varied knowledge-based systems has to do with aggregation and fusion. Pang’s probabilistic linguistic term sets denotes aggregation of fuzzy information and it has attracted tremendous interest from researchers recently. The purpose of this article is to deal investigating methods of information aggregation under the probabilistic linguistic environment. In this situation we defined certain Einstein operational laws on probabilistic linguistic term elements (PLTESs) based on Einstein product and Einstein sum. Consequently, we develop some probabilistic linguistic aggregation operators, notably the probabilistic linguistic Einstein average (PLEA) operators, probabilistic linguistic Einstein geometric (PLEG) operators, weighted probabilistic linguistic Einstein average (WPLEA) operators, weighted probabilistic linguistic Einstein geometric (WPLEG) operators. These operators extend the weighted averaging operator and the weighted geometric operator for the purpose of aggregating probabilistic linguistic terms values respectively. Einstein t-norm and Einstein t-conorm constitute effective aggregation tools and they allow input arguments to reinforce each other downwardly and upwardly respectively. We then generate various properties of these operators. With the aid of the WPLEA and WPLEG, we originate the approaches for the application of multiple attribute group decision making (MAGDM) with the probabilistic linguistic term sets (PLTSs). Lastly, we apply an illustrative example to elucidate our proposed methods and also validate their potentials.

1. Introduction

Generally, when expressing preferences by means of linguistic information, decision-makers frequently face the challenges of uncertainties and vagueness Pang et al. [1]. To overcome this shortfall, Zadeh [2] introduced fuzzy sets (FSs) to deal with them as far as decision-making is concerned. Torra [3] subsequently proposed hesitant fuzzy sets (HFSs) to give a compelling extension of fuzzy sets to manage those situations, where a set of values are possible in the definition process of the membership of an element. However due to their limitations, Rodriguez et al. [4], introduced Hesitant fuzzy linguistic term sets (HFLTSs) to further handle vague and imprecise information whereby two or more sources of vagueness appear simultaneously. Rodriguez et al. [4], further went ahead and stated that the modelling tools of ordinary fuzzy sets are limited and besides the aforementioned tools are used to define quantitative problems. Considering the fact that mostly, uncertainty comes as a result of vagueness of explication utilized by experts in problems with qualitative nature, it will be appropriate to introduce fuzzy linguistic approach to provide tangible results. Nevertheless, in the current studies of (HFLTSs), Pang et al. [1] stated the decision makers’ proposed values cannot have the same relevance because the idea does not follow a realistic pattern. To bring some elements of clarifications, Pang et al. [1] propounded the probabilistic linguistic term sets (PLTSs). PLTSs were introduced to extend HFLTSs via the addition of probabilities without loss of the original linguistic information given by the experts. It could be mentioned that PLTSs came to light as a result of the generalization of the existing HFLTSs and HFSs models with the introduction of probabilities and hesitations. Under the decision-making environment, mentioned could be made of the useful and flexible nature of PLTSs, allowing them to depict or exhibit the qualitative judgement of experts [1]. They were introduced in the decision -making process to bring more flexibility and accuracy. Due to their relevance in dealing with uncertainties and vagueness, they are nowadays being considered as an important concept in the group decision-making domain. For instance, Pang et al. [1] form certain basic arithmetic aggregation operators, like probabilistic linguistic weighted averaging (PLWA) operator, the probabilistic linguistic weighted geometric (PLWG) operator, for aggregating PLTEs. Bai et al. [5] defined more appropriate comparison methods and institute in addition a robust way to handle PLTSs. Gou and Xu [6] established new operational laws with regards to the probabilistic information. A multi-criteria group decision-making algorithm with probabilistic interval preferences orderings was proposed by He et al. [7]. Under the probabilistic linguistic environment, Kobina et al. [8] proposed a series of probabilistic linguistic power aggregation operators manage multi-criteria decision making problems.
In decision-making, the accuracy of the final results largely depends on the information aggregation phase. For the past decade, many scholars have studied and developed numerous aggregation operators for PLTSs information [1,4,5,6,7]. It could be realized that these aggregation operators are based on the algebraic operational laws of the LTSs and PLTSs. However, the algebraic operational laws are not the only operational laws for information fusion. The Einstein operations are equally useful tools to substitute the algebraic operations [9]. Zhao et al. [10], in their research introduced Einstein product as a t-norm and Einstein sum as t-conorm. Einstein t-norm and t-conorm are successfully used for processing uncertainty and vagueness in system analysis, decision analysis, modeling and forecasting applications. For instance, Yu et al. [11] developed a family of hesitant fuzzy Einstein aggregation operators, such as the hesitant fuzzy Einstein Choquet ordered averaging operator, hesitant fuzzy Einstein Choquet ordered geometric operator, to deal with multiple attribute group decision-making under hesitant fuzzy environments. Wang and Liu [12] developed the interval-valued intuitionistic fuzzy Einstein weighted averaging (IVIFEWA) operator, demonstrated and verified their practicality and flexibility in a set of propulsion systems. Wang and Liu [13] investigated intuitionistic fuzzy weighted Einstein average (IFWEA) operator to accommodate the situations where the given arguments are AIFVs and applied IFEWA operator to MADM problem with intuitionistic fuzzy information. Yang and Yuan [14] developed the induced interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (I-IVIFEOWG) operator and applied it to deal with multiple attribute decision making under interval-valued intuitionistic fuzzy environments. Cai and Han [15] developed the induced interval-valued Einstein ordered weighted averaging operator. Wang and Sun [16] also examined the interval-valued intuitionistic fuzzy Einstein geometric Choquet integral operator. Rahman et al. [17] focused on interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator and their application to group decision making. Rahman et al. [18] proposed some interval-valued Pythagorean fuzzy Einstein weighted averaging aggregation operators. However, it seems that in the literature, there is a little investigation on aggregation techniques using the Einstein operations to aggregate probabilistic linguistic information. Hence, the aim of this paper is to explore some probabilistic linguistic aggregation operators based on the Einstein operational laws. Specifically, we develop the probabilistic linguistic Einstein average (PLEA), probabilistic linguistic Einstein geometric (PLEG), weighted probabilistic linguistic Einstein average (WPLEA) and weighted probabilistic linguistic Einstein geometric (WPLEG) aggregation operators. Taking into consideration the WPLEA and the WPLEG operators, we design a new multi-criteria group decision making (MCGDM) approach for PLTS information. The contributions of the study are as follows: (1) Our proposed methods provide more versatility in the aggregation process and they have the ability to depict the interrelationship of input arguments and the individual evaluation. (2) Considering the different situations, our proposed methods use Einstein operations with transformed PLTSs, which are more competent in handling uncertainty and vagueness than the existing PLTSs, fuzzy sets (FSs), Hesitant Fuzzy Sets (HFSs), Hesitant Fuzzy Linguistic Terms (HFLTSs). (3) The opinions of the decision-makers still remain the same in a situation where only few different linguistic terms evaluated by the DMs are considered. (4) Finally they take into consideration the probabilistic information of the input arguments and make use of the novel operational laws of PLTSs proposed by Gou et al. [6].
The remainder of the paper is structured as follows: In Section 2, we introduce certain elementary concepts and operations in relation to PLTSs and Einstein operations. Section 3 deals with Einstein operations of the transformed probabilistic linguistic term sets (PLTSs). In Section 4, we design a set of probabilistic linguistic Einstein aggregation operators (PLEA, PLEG, WPLEA, WPLEG) and then their desirable properties are also studied. In Section 5, we formulate the ways for applying MCGDM utilizing the WPLEA and WPLEG operators. In Section 6, an illustrative example is given to give an account and ascertain the proposed methods. In Section 7 we make a conclusion and we expand on future studies.

2. Preliminaries

2.1. Probabilistic Linguistic Term Sets (PLTSs)

The theory of PLTSs Pang et al. [1] is an extension of the concepts of HFLTSs. In what follows, we present some basic concepts of PLTSs and the corresponding operations.
Definition 1 (Pang et al. [1]).
Let S = { s t / t = 0 , 1 , , τ } be a linguistic term set. Then a probabilistic linguistic term set (PLTS) is defined as:
L ( p ) = { L ( k ) ( p ( k ) ) / L ( k ) S , r ( k ) t , p ( k ) 0 , k = 1 , 2 , , # L ( p ) , k = 1 # L ( p ) p k 1 } ,
where L ( k ) ( p k ) is the linguistic term L ( k ) associated with the probability p ( k ) , r ( k ) is the subscript of L ( k ) and # L ( p ) is the number of all linguistic terms in L ( p ) .
In a PLTSs, the positions of elements can be swapped arbitrarily. To make sure the operational results are straightforwardly ascertained, Pang et al. [1] proposed the ordered PLTS. It is described as:
Definition 2 (Pang et al. [1]).
Given a PLTS L ( p ) = { L k ( p ( k ) ) k = 1 , 2 , , L ( p ) } and r ( k ) is the subscript of linguistic term L ( k ) . L ( p ) is called an ordered PLTS, if the linguistic terms L ( k ) ( p k ) are arranged according to the values of r ( k ) p ( k ) in descending order.
With regards to comparing the PLTSs, Pang et al. [1] defined the scores and the deviation degree of a PLTS:
Definition 3 (Pang et al. [1]).
Let L ( p ) = { L k ( p ( k ) ) k = 1 , 2 , , L ( p ) } be a PLTS, and r ( k ) is the subscript of linguistic term L ( k ) . Then, the score of L ( p ) is defined as follows:
E ( L ( p ) ) = s α ¯ ,
where α ¯ = k = 1 L ( p ) r ( k ) p ( k ) / k = 1 L ( p ) p ( k ) . The deviation degree of L ( p ) is:
σ ( L ( p ) ) = ( k = 1 L ( p ) ( ( p ( k ) ( r ( k ) α ¯ ) ) 2 ) 0.5 ) / k = 1 L ( p ) p ( k ) .
Based on the score and the deviation degree of a PLTS, Pang et al. [1] further proposed the following laws to compare them:
Definition 4 (Pang et al. [1]).
Given two PLTSs L 1 ( p ) and L 2 ( p ) . E ( L 1 ( p ) ) and E ( L 2 ( p ) ) are the scores of L 1 ( p ) and L 2 ( p ) , respectively.
(1) 
If E ( L 1 ( p ) ) > E ( L 2 ( p ) ) , t h e n L 1 ( p ) i s b i g g e r t h a n L 2 ( p ) , d e n o t e d b y L 1 ( p ) > L 2 ( p ) ;
(2) 
If E ( L 1 ( p ) ) < E ( L 2 ( p ) ) , t h e n L 1 ( p ) i s s m a l l e r t h a n L 2 ( p ) , d e n o t e d b y L 1 ( p ) < L 2 ( p ) ;
(3) 
E ( L 2 ( p ) ) = E ( L 2 ( p ) ) , t h e n w e n e e d t o c o m p a r e t h e i r d e v i a t i o n deg r e e
(a) 
if σ ( L 1 ( p ) ) = σ ( L 2 ( p ) ) , t h e n L 1 ( p ) i s e q u a l t o L 2 ( p ) , d e n o t e d b y L 1 ( p ) ~ L 2 ( p ) ;
(b) 
if σ ( L 1 ( p ) ) > σ ( L 2 ( p ) ) , t h e n L 1 ( p ) i s s m a l l e r t h a n L 2 ( p ) , d e n o t e d b y L 1 ( p ) < L 2 ( p ) ;
(c) 
if σ ( L 1 ( p ) ) < σ ( L 2 ( p ) ) , t h e n L 1 ( p ) i s g r e a t e r t h a n L 2 ( p ) , d e n o t e d b y L 1 ( p ) > L 2 ( p ) .
A careful examination of the comparison laws of PLTSs may reveal that the number of their corresponding linguistic terms may be unequal. In order to address this problem, Pang et al. [1] normalized the PLTSs by increasing the numbers of linguistic terms for PLTSs. Hence, the normalization of PLTSs is defined as follows:
Definition 5 (Pang et al. [1]).
Let L 1 ( p ) = { L 1 k ( p 1 ( k ) ) k = 1 , 2 , , # L 1 ( p ) } be a probabilistic linguistic term set and let L 2 ( p ) = { L 2 k ( p 2 ( k ) ) k = 1 , 2 , , # L 2 ( p ) } be another PLTS. # L 1 ( p ) and # L 2 ( p ) are the numbers of linguistic terms in L 1 ( p ) and L 2 ( p ) . If # L 1 ( p ) > # L 2 ( p ) , then we will add # L 1 ( p ) # L 2 ( p ) linguistic terms to L 2 ( p ) so that the numbers of linguistic terms in L 1 ( p ) and L 2 ( p ) are identical. The added linguistic terms are the smallest ones in L 2 ( p ) and their probabilities are zero. Analogously, L 1 ( p ) < L 2 ( p ) , we can use the similar method.
Definition 6 (Pang et al. [1]).
Let S = { s t / t = 0 , 1 , , τ } be a linguistic term set. Given three PLTSs, L ( p ) , L 1 ( p ) and L 2 ( p ) their basic operations are summarized as follows:
(1) 
L 1 ( p ) L 2 ( p ) = L ( k ) 1 L 1 ( p ) , L ( k ) 2 L 2 ( p ) { ( p ( k ) 1 L ( k ) 1 p ( k ) 2 L ( k ) 2 ) } ;
(2) 
L 1 ( p ) L 2 ( p ) = L ( k ) 1 L 1 ( p ) , L ( k ) 2 L 2 ( p ) { ( L ( k ) 1 ) p 1 ( k ) ( L ( k ) 2 ) p 2 ( k ) } ;
(3) 
λ ( L ( p ) ) = λ L ( k ) L ( p ) { λ p ( k ) L ( k ) } a n d λ 0 ;
(4) 
( L ( p ) ) λ = λ L ( k ) L ( p ) { ( L ( k ) ) λ p ( k ) } a n d λ 0 ;

2.2. Einstein Operations

The concept of a triangular norm was introduced by Klement et al. [19] in order to generalize the triangular inequality of a metric. The existing notion of a t-norm and its dual operator (t-conorm) originated from Schweizer and Sklar [20]. These two operations can be applied as a generalization of the Boolean logic connectives to multi-valued logic. The t-norms generalize the conjunctives ‘AND’ operator and the t-conorms generalize the disjunctive ‘OR’ operator. This situation allows them to be used to define the intersection and union operations in fuzzy logic. Einstein operations include the Einstein product and Einstein sum, which are examples of t-norm and t-conorm, respectively are defined as follows.
Definition 7 ([21]).
Einstein product as, a t-norm is a function T : [ 0 , 1 ] × [ 0 , 1 ] [ 0 , 1 ] such that
x ε y = x y 1 + ( 1 x ) ( 1 y ) ,    ( x , y ) [ 0 , 1 ] 2
Definition 8 ([21]).
Einstein sum as, a t c o n o r m is also a function S : [ 0 , 1 ] × [ 0 , 1 ] [ 0 , 1 ] such that
x ε y = x + y 1 + x y ( x , y ) [ 0 , 1 ] 2

3. Einstein Operations of Transformed Probabilistic Linguistic Term Sets

Since Einstein operational laws need to obey some conditions before they can be carried out, thus the values of the individual arguments must be within the interval [ 0 , 1 ] , we need to find the equivalent transformation of PLTSs, since some probabilistic linguistic elements (PLEs) might not necessarily belong to [ 0 , 1 ] . Luckily Gou and Xu [6] defined the first equivalent transformation of probabilistic linguistic term sets (PLTSs) as follows:
Definition 9 (Gou and Xu [6]).
Let S = { s t / t = τ , , 1 , 0 , 1 , , τ } be any linguistic term set. L ( p ) is a PLTS. The equivalent transformation function of L ( p ) is defined as:
g ( L ( p ) ) = { [ r ( k ) 2 τ + 1 2 ] ( p ( k ) ) } = L γ ( p )
where g : [ τ , τ ] [ 0 , 1 ] and γ = g ( L ( k ) ) , γ [ 0 , 1 ] . g ( L ( k ) ) = ( r ( k ) 2 τ + 1 2 ) = γ .
Based on Definition 9, we can obtain new operational laws defined as follows:
Proposition 1.
Let L ( p ) = { L ( k ) i ( p i ( k ) ) / k = 1 , 2 , , # L i ( p ) } ( i = 1 , 2 , , n ) be a collection of PLTSs and g ( L ( p ) ) its equivalent transformation. Given three transformed PLTSs g ( L ( p ) ) , g ( L 1 ( p ) ) , g ( L 2 ( p ) ) then
(1) 
g ( L 1 ( p ) ) g ( L 2 ( p ) ) = g ( L 1 ( k ) ) g ( L 1 ( p ) ) , g ( L 2 ( k ) ) g ( L 2 ( p ) ) { ( g ( L 1 ( k ) ) + g ( L 2 ( k ) ) g ( L 1 ( k ) ) g ( L 2 ( k ) ) ) ( p 1 ( k ) p 2 ( k ) ) }
(2) 
g ( L 1 ( p ) ) g ( L 2 ( p ) ) = g ( L 1 ( k ) ) g ( L 1 ( p ) ) , g ( L 2 ( k ) ) g ( L 2 ( p ) ) { ( g ( L 1 ( k ) ) × g ( L 2 ( k ) ) ) ( p 1 ( k ) p 2 ( k ) ) }
(3) 
λ g ( L ( p ) ) = g ( L ( k ) ) g ( L ( p ) ) { ( 1 ( 1 g ( L ( k ) ) ) λ ) ( p ( k ) ) } ; λ 0
(4) 
g ( L ( p ) ) λ = g ( L ( k ) ) g ( L ( p ) ) { g ( L ( k ) ) λ ( p ( k ) ) } ; λ 0
Based on Definition 7 and Definition 8, we give some new operations on the transformed PLTEs as follows:
Proposition 2.
Let S = { s t / t = 0 , 1 , , τ } be a linguistic term set and λ > 0 . Given three transformed PLTSs g ( L ( p ) ) , g ( L 1 ( p ) ) and g ( L 2 ( p ) ) then
(1) 
g ( L 1 ( p ) ) ε g ( L 2 ( p ) ) = g ( L 1 ( k ) ) g ( L 1 ( p ) ) , g ( L 2 ( k ) ) g ( L 2 ( p ) ) { p 1 ( k ) g ( L 1 ( k ) ) + p 2 ( k ) g ( L 2 ( k ) ) 1 + [ p 1 ( k ) g ( L 1 ( k ) ) ] [ p 2 ( k ) g ( L 2 ( k ) ) ] }
(2) 
g ( L 1 ( p ) ) ε g ( L 2 ( p ) ) = g ( L 1 ( k ) ) g ( L 1 ( p ) ) , g ( L 2 ( k ) ) g ( L 2 ( p ) ) { [ p 1 ( k ) g ( L 1 ( k ) ) ] [ p 2 ( k ) g ( L 2 ( k ) ) ] 1 + [ 1 p 1 ( k ) g ( L 1 ( k ) ) ] [ 1 p 2 ( k ) g ( L 2 ( k ) ) ] }
(3) 
λ . ε g ( L ( p ) ) = g ( L ( k ) ) g ( L ( p ) ) { [ 1 + p ( k ) g ( L ( k ) ) ] λ [ 1 p ( k ) g ( L ( k ) ) ] λ [ 1 + p ( k ) g ( L ( k ) ) ] λ + [ 1 p ( k ) g ( L ( k ) ) ] λ }
(4) 
g ( L ( p ) ) ε λ = g ( L ( k ) ) g ( L ( p ) ) { 2 [ p ( k ) g ( L ( k ) ) ] λ [ 2 p ( k ) g ( L ( k ) ) ] λ + [ p ( k ) g ( L ( k ) ) ] λ } where g ( L ( k ) ) p ( k ) [ 0 , 1 ] and g ( L i ( p ) ) = g ( L i ( k ) ) p i ( k ) .
Since the operational law 1 and 2 are straightforward, we will prove operational laws 3 and 4.
Proof. 
In the following we firstly prove operational law 4 on the basis of operational law 2.
Based on Definition 7 and Definition 8, let x = g ( L 1 ( p ) ) and y = g ( L 2 ( p ) ) then
g ( L 1 ( p ) ) ε g ( L 2 ( p ) ) = g ( L 1 ( p ) ) g ( L 2 ( p ) ) 1 + ( 1 g ( L 1 ( p ) ) ) ( 1 g ( L 2 ( p ) ) ) = g ( L 1 ( p ) ) g ( L 2 ( p ) ) 1 + 1 g ( L 1 ( p ) ) g ( L 2 ( p ) ) + g ( L 1 ( p ) ) g ( L 2 ( p ) ) = g ( L 1 ( p ) ) g ( L 2 ( p ) ) 2 g ( L 1 ( p ) ) g ( L 2 ( p ) ) + g ( L 1 ( p ) ) g ( L 2 ( p ) )
For g ( L 1 ( p ) ) = g ( L 2 ( p ) ) = g ( L ( p ) ) , we obtain
g ( L ( p ) ) ε 2 = g ( L ( p ) ) 2 2 2 g ( L ( p ) ) + g ( L ( p ) ) 2
g ( L ( p ) ) ε 2 = 2 g ( L ( p ) ) 2 ( 2 g ( L ( p ) ) ) 2 + g ( L ( p ) ) 2 λ R
we have g ( L ( p ) ) λ = 2 g ( L ( p ) ) λ ( 2 g ( L ( p ) ) ) λ + g ( L ( p ) ) λ g ( L ( p ) ) ε λ = g ( L ( k ) ) g ( L ( p ) ) { 2 [ p ( k ) g ( L ( k ) ) ] λ [ 2 p ( k ) g ( L ( k ) ) ] λ + [ p ( k ) g ( L ( k ) ) ] λ } since g ( L ( p ) ) = g ( L ( k ) ) p ( k ) .
Proved as required. □
Considering operational law 1, we prove the operational law 3
g ( L 1 ( p ) ) ε g ( L 2 ( p ) ) = g ( L 1 ( p ) ) g ( L 2 ( p ) ) 1 + g ( L 1 ( p ) ) g ( L 2 ( p ) )
If
g ( L 1 ( p ) ) = g ( L 2 ( p ) ) = g ( L ( p ) )
then
2 ε g ( L ( p ) ) = 2 g ( L ( p ) ) 1 + g ( L ( p ) ) 2   so   λ R
we have
λ ε g ( L ( p ) ) = λ g ( L ( p ) ) 1 + g ( L ( p ) ) λ = ( 1 + g ( L ( p ) ) ) λ ( 1 g ( L ( p ) ) ) λ ( 1 + g ( L ( p ) ) ) λ + ( 1 g ( L ( p ) ) ) λ
λ . ε g ( L ( p ) ) = g ( L ( k ) ) g ( L ( p ) ) { [ 1 + p ( k ) g ( L ( k ) ) ] λ [ 1 p ( k ) g ( L ( k ) ) ] λ [ 1 + p ( k ) g ( L ( k ) ) ] λ + [ 1 p ( k ) g ( L ( k ) ) ] λ }
since
g ( L ( p ) ) = g ( L ( k ) ) p ( k )
Based on the operational laws (1)–(4) of Section 3, we can easily obtain the following properties.
(1)
g ( L 1 ( p ) ) ε g ( L 2 ( p ) ) = g ( L 2 ( p ) ) ε g ( L 1 ( p ) ) .
(2)
( g ( L 1 ( p ) ) ε g ( L 2 ( p ) ) ) ε g ( L 3 ( p ) ) = g ( L 1 ( p ) ) ε ( g ( L 2 ( p ) ) ε g ( L 3 ( p ) ) ) .
(3)
λ . ε ( g ( L 1 ( p ) ) ε g ( L 2 ( p ) ) ) = λ . ε g ( L 1 ( p ) ) ε λ . ε g ( L 2 ( p ) ) .
(4)
λ 1 . ε ( λ 2 . ε g ( L ( p ) ) ) = ( λ 1 λ 2 ) . ε g ( L ( p ) ) .
Proof. 
g ( L 1 ( p ) ) ε g ( L 2 ( p ) ) = p 1 ( k ) g ( L 1 ( k ) ) p 2 ( k ) g ( L 2 ( k ) ) 1 p 1 ( k ) g ( L 1 ( k ) ) p 2 ( k ) g ( L 2 ( k ) ) and g ( L 2 ( p ) ) ε g ( L 1 ( p ) ) = p 2 ( k ) g ( L 2 ( k ) ) p 1 ( k ) g ( L 1 ( k ) ) 1 + p 2 ( k ) g ( L 2 ( k ) ) p 1 ( k ) g ( L 1 ( k ) ) .
Since p 1 ( k ) g ( L 1 ( k ) ) p 2 ( k ) g ( L 2 ( k ) ) = p 2 ( k ) g ( L 2 ( k ) ) p 1 ( k ) g ( L 1 ( k ) ) and p 1 ( k ) g ( L 1 ( k ) ) p 2 ( k ) g ( L 2 ( k ) ) = p 2 ( k ) g ( L 2 ( k ) ) p 1 ( k ) g ( L 1 ( k ) ) then 1 + p 1 ( k ) g ( L 1 ( k ) ) p 2 ( k ) g ( L 2 ( k ) ) = 1 + p 2 ( k ) g ( L 2 ( k ) ) p 1 ( k ) g ( L 1 ( k ) ) .
Therefore p 1 ( k ) g ( L 1 ( k ) ) p 2 ( k ) g ( L 2 ( k ) ) 1 + p 1 ( k ) g ( L 1 ( k ) ) p 2 ( k ) g ( L 2 ( k ) ) = p 2 ( k ) g ( L 2 ( k ) ) p 1 ( k ) g ( L 1 ( k ) ) 1 + p 2 ( k ) g ( L 2 ( k ) ) p 1 ( k ) g ( L 1 ( k ) )
g ( L 1 ( p ) ) ε g ( L 2 ( p ) ) = p 1 ( k ) g ( L 1 ( k ) ) p 2 ( k ) g ( L 2 ( k ) ) 1 + p 1 ( k ) g ( L 1 ( k ) ) p 2 ( k ) g ( L 2 ( k ) ) + p 2 ( k ) g ( L 2 ( k ) ) p 1 ( k ) g ( L 1 ( k ) ) 1 + p 2 ( k ) g ( L 2 ( k ) ) p 1 ( k ) g ( L 1 ( k ) ) = p 2 ( k ) g ( L 2 ( k ) ) p 1 ( k ) g ( L 1 ( k ) ) 1 + p 2 ( k ) g ( L 2 ( k ) ) p 1 ( k ) g ( L 1 ( k ) ) + p 1 ( k ) g ( L 1 ( k ) ) p 2 ( k ) g ( L 2 ( k ) ) 1 + p 1 ( k ) g ( L 1 ( k ) ) p 2 ( k ) g ( L 2 ( k ) ) = g ( L 2 ( p ) ) ε g ( L 1 ( p ) )
Hence, we complete the proof of Property 1. The remaining properties can easily be proved. □

4. Probabilistic Linguistic Aggregation Operators

Based on the probabilistic linguistic environment, we treat the input arguments as PLTSs and we deeply investigate the extension of Einstein t-norm and Einstein t-conorm aggregation operators.

4.1. Probabilistic Linguistic Einstein Average (PLEA) Aggregation Operators

In this section, we discuss the extension of Einstein t-conorm aggregation operators to accommodate the probabilistic linguistic environment. Specifically, we propose some probabilistic linguistic Einstein average operators, i.e., Probabilistic Linguistic Einstein Average (PLEA) and Weighted Probabilistic Linguistic Einstein Average (WPLEA) which allows the input arguments to reinforce and support each other during the aggregation process.

4.1.1. PLEA

Based on the results of Definitions 1 and operational law (1) of Proposition 1 we present the definition of the PLEA aggregation operator as follows:
Definition 10.
Let L ( p ) = { L ( k ) i ( p i ( k ) ) / k = 1 , 2 , , # L i ( p ) } ( i = 1 , 2 , , n ) be a collection of PLTSs and g ( L ( p ) ) its equivalent transformation. A probabilistic linguistic Einstein Average (PLEA) operator is a mapping g ( L n ( p ) ) g ( L ( p ) ) , such that
P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = ε i = 1 n 1 n g ( L i ( p ) )
Theorem 1.
Let L ( p ) = { L ( k ) ( p ( k ) ) / L ( k ) S , r ( k ) t , p ( k ) 0 , k = 1 , 2 , , # L ( p ) , k = 1 # L ( p ) p k 1 } , ( i = 1 , 2 , , n ) be a collection of PLTSs and g ( L ( p ) ) its equivalent transformation, then their aggregated value by using PLEA operator is also a PLTE and
P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = g ( L i ( k ) ) g ( L i ( p ) ) i = 1 , 2 , , n { i = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) 1 n i = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) 1 n i = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) 1 n + i = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) 1 n }
Proof. 
We proved (8) by using mathematical induction on n. For n = 2 , according to the operational law (3) of Definition 10, we have
( 1 2 ) ε g ( L 1 ( p ) ) = g ( L 1 ( k ) ) g ( L 1 ( p ) ) { [ 1 + p 1 ( k ) g ( L 1 ( k ) ) ] 1 2 [ 1 p 1 ( k ) g ( L 1 ( k ) ) ] 1 2 [ 1 + p 1 ( k ) g ( L 1 ( k ) ) ] 1 2 + [ 1 p 1 ( k ) g ( L 1 ( k ) ) ] 1 2 }
( 1 2 ) ε g ( L 2 ( p ) ) = g ( L 2 ( k ) ) g ( L 2 ( p ) ) { [ 1 + p 2 ( k ) g ( L 2 ( k ) ) ] 1 2 [ 1 p 2 ( k ) g ( L 2 ( k ) ) ] 1 2 [ 1 + p 2 ( k ) g ( L 2 ( k ) ) ] 1 2 + [ 1 p 2 ( k ) g ( L 2 ( k ) ) ] 1 2 }
then
P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) ) = ( 1 2 ) ε g ( L 1 ( p ) ) ε ( 1 2 ) ε g ( L 2 ( p ) )
( 1 + p 1 ( k ) g ( L 1 ( k ) ) ) 1 2 ( 1 + p 2 ( k ) g ( L 2 ( k ) ) ) 1 2 ( 1 p 1 ( k ) g ( L 1 ( k ) ) ) 1 2 ( 1 p 2 ( k ) g ( L 2 ( k ) ) ) 1 2 ( 1 + p 1 ( k ) g ( L 1 ( k ) ) ) 1 2 ( 1 + p 2 ( k ) g ( L 2 ( k ) ) ) 1 2 + ( 1 p 1 ( k ) g ( L 1 ( k ) ) ) 1 2 ( 1 p 2 ( k ) g ( L 2 ( k ) ) ) 1 2
And
( 1 2 ) ε g ( L 1 ( p ) ) ( 1 2 ) ε g ( L 2 ( p ) ) = g ( L 1 ( k ) ) g ( L 1 ( p ) ) , g ( L 2 ( k ) ) g ( L 2 ( p ) ) i = 1 2 ( 1 + p i ( k ) g ( L i ( k ) ) ) 1 2 i = 1 2 ( 1 p i ( k ) g ( L i ( k ) ) ) 1 2 i = 1 2 ( 1 + p i ( k ) g ( L i ( k ) ) ) 1 2 + i = 1 2 ( 1 p i ( k ) g ( L i ( k ) ) ) 1 2
That is for n = 2 (8) holds. If Equation (8) holds for n = m , i.e.,
i = 1 m ( 1 m g ( L i ( p ) ) ) = P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L m ( p ) ) ) = g ( L i ( k ) ) g ( L i ( p ) ) i = 1 , 2 , , m i = 1 m ( 1 + p i ( k ) g ( L i ( k ) ) ) 1 m i = 1 m ( 1 p i ( k ) g ( L i ( k ) ) ) 1 m i = 1 m ( 1 + p i ( k ) g ( L i ( k ) ) ) 1 m + i = 1 m ( 1 p i ( k ) g ( L i ( k ) ) ) 1 m
Then, for n = m + 1 based on Definition 11 and the operational laws of Definition 10, we have
i = 1 m + 1 ( 1 n g ( L i ( p ) ) ) = i = 1 m ( 1 n g ( L i ( p ) ) ) ( 1 n g ( L m + 1 ( p ) ) ) = P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L m + 1 ( p ) ) ) = g ( L i ( k ) ) g ( L i ( p ) ) ( i = 1 , 2 , , m ) { i = 1 m ( 1 + p i ( k ) g ( L i ( k ) ) ) 1 m i = 1 m ( 1 p i ( k ) g ( L i ( k ) ) ) 1 m i = 1 m ( 1 + p 1 ( k ) g ( L i ( k ) ) ) 1 m + i = 1 m ( 1 p i ( k ) g ( L i ( k ) ) ) 1 m } = g ( L i ( k ) ) g ( L ( p ) ) ( i = 1 , 2 , , n ) i = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) 1 n i = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) 1 n i = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) 1 n + i = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) 1 n
i.e., for n = m + 1 , (8) holds, which completes the proof of Theorem 1. □
Illustrative example to demonstrate the validity of the operational laws in Definition 10.
Considering [6] let L 1 ( p ) = { s 1 ( 0.3 ) , s 2 ( 0.2 ) , s 3 ( 0.5 ) } ; L 2 ( p ) = { s 1 ( 0.2 ) , s 0 ( 0.3 ) } and λ = 1 2 . After normalization we obtained L 2 ( p ) = { s 1 ( 0.4 ) , s 0 ( 0.6 ) } . Given that g ( L i ( p ) ) = ( r i ( k ) 2 τ + 1 2 ) ( p i ( k ) ) , we obtained g ( L 1 ( p ) ) = { 2 3 ( 0.3 ) , 5 6 ( 0.2 ) , 1 ( 0.5 ) , } and g ( L 2 ( p ) ) = { 1 3 ( 0.4 ) , 1 2 ( 0.6 ) } .
Considering operational law 1 and operational law 3 we obtain
1 2 g ( L 1 ( p ) ) ε 1 2 g ( L 2 ( p ) ) = { ( 1 + 0.6667 × 0.3 ) 1 2 ( 1 0.6667 × 0.3 ) 1 2 ( 1 + 0.6667 × 0.3 ) 1 2 + ( 1 0.6667 × 0.3 ) 1 2 + ( 1 + 0.5 × 0.3333 ) 1 2 ( 1 0.5 × 0.3333 ) 1 2 ( 1 + 0.5 × 0.3333 ) 1 2 + ( 1 0.5 × 0.3333 ) 1 2 1 + ( 1 + 0.6667 × 0.3 ) 1 2 ( 1 0.6667 × 0.3 ) 1 2 ( 1 + 0.6667 × 0.3 ) 1 2 + ( 1 0.6667 × 0.3 ) 1 2 × ( 1 + 0.5 × 0.3333 ) 1 2 ( 1 0.5 × 0.3333 ) 1 2 ( 1 + 0.5 × 0.3333 ) 1 2 + ( 1 0.5 × 0.3333 ) 1 2 , .......................... }
P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) ) = ( 0.1668 , 0.2506 , 0.1500 , 0.2344 , 0.3290 , 0.4048 )
With respect to Definition 11 and Theorem 1, it can easily be proven that the PLEA aggregation operator has the following desirable properties. □
Property 1 (Idempotency).
Let g ( L i ( p ) ) ( i = 1 , 2 , , n ) be a collection of transformed PLTSs. If all g ( L i ( p ) ) ( i = 1 , 2 , , n ) are equal, i.e., g ( L i ( p ) ) = g ( L ( p ) ) then
P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = g ( L ( p ) )
Proof. 
If g ( L i ( p ) ) = g ( L ( p ) ) for all i , then P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) is computed as follows:
P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = ε i = 1 n 1 n g ( L i ( p ) ) = ε i = 1 n 1 n g ( L ( p ) )
Property 2 (Boundedness).
Let g ( L i ( p ) ) ( i = 1 , 2 , , n ) be a collection of PLTSs, then we have:
min i = 1 n min k = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) g ( L ) max i = 1 n max k = 1 # L i ( p ) p i ( k ) g ( L i ( k ) )
where g ( L ) P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) .
Proof. 
According to the result of Theorem 1, P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) is computed as:
P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = ε i = 1 n 1 n g ( L i ( p ) ) .
Then, we can deduce the following relationships:
min i = 1 n min k = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) p i ( k ) g ( L i ( k ) ) max i = 1 n max k = 1 # L i ( p ) p i ( k ) g ( L i ( k ) )
By using the result of Theorem 1, we can easily conclude the proof of Property 2. □
Property 3 (Monotonicity).
Let g ( L i ( p ) ) and g ( L i ( p ) ) be two sets of PLTSs and the numbers of linguistic terms in g ( L i ( p ) ) and g ( L i ( p ) ) are identical ( i = 1 , 2 , , n ) . If g ( L i ( k ) ) ( p i ( k ) ) g ( L i ( k ) ) ( p i ( k ) ) for all i , i.e., g ( L i ( p ) ) g ( L i ( p ) ) , then
P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) .
Property 4 (Commutativity).
Let ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) and ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) be a collection of PLTSs and let be any permutation of ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) then
P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) )
Proof. 
Because ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) is any permutation of ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) , by utilizing the results of Theorem 1 we can easily finish the proof of Property 4. □

4.1.2. WPLEA

In this section, we mainly consider the aggregation of the probabilistic linguistic information on the basis of the operational laws defined in Section 3. In the following, we present the weighted probabilistic linguistic Einstein average (WPLEA) operator based on the weighted arithmetic mean.
Definition 11.
Let L i ( p ) be a collection of PLTSs, w = ( w 1 , w 2 , , w n , ) T denotes the weighting vector of L i ( p ) and w i [ 0 , 1 ] , i = 1 n w i = 1 . Given the value of the weight w = ( w 1 , w 2 , , w n , ) T , we define weighted probabilistic linguistic Einstein average (WPLEA) as follows:
W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = i = 1 n w i g ( L i ( p ) )
Especially, if w = ( 1 / n , 1 / n , , 1 / n , ) T , then the WPLEA operator reduces to the probabilistic linguistic Einstein average (PLEA) operator:
P L E A ( L 1 ( p ) , L 2 ( p ) , , L n ( p ) ) = 1 n g ( L 1 ( p ) ) ε 1 n g ( L 2 ( p ) ) ε ε 1 n g ( L n ( p ) )
In light of the operations of the PLTSs described in Definition 10, we can deduce the following theorem.
Theorem 2.
Suppose that g ( L i ( p ) ) ( i = 1 , 2 , , n ) is a collection of transformed PLTSs, then their aggregated values by using the WPLEA operator is also a transformed PLTS, and:
W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = g ( L i ( k ) ) g ( L i ( p ) ) { i = 1 n ( 1 + g ( L i ( k ) ) p i ( k ) ) w i i = 1 n ( 1 g ( L i ( k ) ) p i ( k ) ) w i i = 1 n ( 1 + g ( L i ( k ) ) p i ( k ) ) w i + i = 1 n ( 1 g ( L i ( k ) ) p i ( k ) ) w i }
where w = ( w 1 , w 2 , , w n , ) T is the weight vector of g ( L i ( p ) ) ( i = 1 , 2 , , n ) with w i [ 0 , 1 ] and i = 1 n w i = 1 .
Proof. 
In the following, we first prove (11), by using mathematical induction on n:
For n = 2 : Since
w 1 g ( L 1 ( p ) ) = g ( L 1 ( k ) ) g ( L 1 ( p ) ) { ( ( 1 + p 1 ( k ) g ( L 1 ( k ) ) ) w 1 ( 1 p 1 ( k ) g ( L 1 ( k ) ) ) w 1 ( 1 + p 1 ( k ) g ( L 1 ( k ) ) ) w 1 + ( 1 p 1 ( k ) g ( L 1 ( k ) ) ) w 1 ) }
w 2 g ( L 2 ( p ) ) = g ( L 2 ( k ) ) g ( L 2 ( p ) ) { ( ( 1 + p 2 ( k ) g ( L 2 ( k ) ) ) w 2 ( 1 p 2 ( k ) g ( L 2 ( k ) ) ) w 2 ( 1 + p 2 ( k ) g ( L 2 ( k ) ) ) w 2 + ( 1 p 2 ( k ) g ( L 2 ( k ) ) ) w 2 ) }
Then
w 1 g ( L 1 ( p ) ) w 2 g ( L 2 ( p ) ) = g ( L 1 ( k ) ) g ( L 1 ( p ) ) , g ( L 2 ( k ) ) g ( L 2 ( p ) ) , { ( 1 + p 1 ( k ) g ( L 1 ( k ) ) ) w 1 ( 1 p 1 ( k ) g ( L 1 ( k ) ) ) w 1 ( 1 + p 1 ( k ) g ( L 1 ( k ) ) ) w 1 + ( 1 p 1 ( k ) g ( L 1 ( k ) ) ) w 1 + ( 1 + p 2 ( k ) g ( L 2 ( k ) ) ) w 2 ( 1 p 2 ( k ) g ( L 2 ( k ) ) ) w 2 ( 1 + p 2 ( k ) g ( L 2 ( k ) ) ) w 2 + ( 1 p 2 ( k ) g ( L 2 ( k ) ) ) w 2 1 + ( 1 + p 1 ( k ) g ( L 1 ( k ) ) ) w 1 ( 1 p 1 ( k ) g ( L 1 ( k ) ) ) w 1 ( 1 + p 1 ( k ) g ( L 1 ( k ) ) ) w 1 + ( 1 p 1 ( k ) g ( L 1 ( k ) ) ) w 1 . ( 1 + p 2 ( k ) g ( L 2 ( k ) ) ) w 2 ( 1 p 2 ( k ) g ( L 2 ( k ) ) ) w 2 ( 1 + p 2 ( k ) g ( L 2 ( k ) ) ) w 2 + ( 1 p 2 ( k ) g ( L 2 ( k ) ) ) w 2 } = g ( L i ( k ) ) g ( L i ( p ) ) ( i = 1 , 2 ) { i = 1 2 ( 1 + p i ( k ) g ( L i ( k ) ) ) w i i = 1 2 ( 1 p i ( k ) g ( L i ( k ) ) ) w i i = 1 2 ( 1 + p i ( k ) g ( L i ( k ) ) ) w i + i = 1 2 ( 1 p i ( k ) g ( L i ( k ) ) ) w i }
If (11) holds for n = m , that is
i = 1 m w i g ( L i ( p ) ) = g ( L i ( k ) ) g ( L i ( p ) ) ( i = 1 , 2 , , m ) { i = 1 m ( 1 + p i ( k ) g ( L i ( k ) ) ) w i i = 1 m ( 1 p i ( k ) g ( L i ( k ) ) ) w i i = 1 m ( 1 + p i ( k ) g ( L i ( k ) ) ) w i + i = 1 m ( 1 p i ( k ) g ( L i ( k ) ) ) w i }
Then, when n = m + 1 , by the operations of PLTEs, we have:
i = 1 m + 1 w i g ( L i ( p ) ) = i = 1 m w i g ( L i ( p ) ) w m + 1 g ( L m + 1 ( p ) ) = g ( L i ( k ) ) g ( L i ( p ) ) ( i = 1 , 2 , , m ) { i = 1 m ( 1 + p i ( k ) g ( L i ( k ) ) ) i = 1 m ( 1 p i ( k ) g ( L i ( k ) ) ) i = 1 m ( 1 + p i ( k ) g ( L i ( k ) ) ) + i = 1 m ( 1 p i ( k ) g ( L i ( k ) ) ) } g ( L m + 1 ( k ) ) g ( L m + 1 ( p ) ) { ( 1 + p i ( k ) g ( L i ( k ) ) ) w m + 1 ( 1 p i ( k ) g ( L i ( k ) ) ) w m + 1 ( 1 + p i ( k ) g ( L i ( k ) ) ) w m + 1 + ( 1 p i ( k ) g ( L i ( k ) ) ) w m + 1 } = g ( L i ( k ) ) g ( L i ( p ) ) ( i = 1 , 2 , , m + 1 ) { i = 1 m + 1 ( 1 + p i ( k ) g ( L i ( k ) ) ) w i i = 1 m + 1 ( 1 p i ( k ) g ( L i ( k ) ) ) w i i = 1 m + 1 ( 1 + p i ( k ) g ( L i ( k ) ) ) w i + i = 1 m + 1 ( 1 p i ( k ) g ( L i ( k ) ) ) w i }
i.e., (11) holds for n = m + 1 . Thus, Equation (11) holds for all n . Then
W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = g ( L i ( k ) ) g ( L i ( p ) ) ( i = 1 , 2 , , n ) { i = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) w i i = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) w i i = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) w i + i = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) w i }
This completes the proof of Theorem 2. □
Example 1.
Considering [6] let L 1 ( p ) = { s 1 ( 0.3 ) , s 2 ( 0.2 ) , s 3 ( 0.5 ) } ; L 2 ( p ) = { s 1 ( 0.2 ) , s 0 ( 0.3 ) } and w = ( 0.5 , 0.5 ) T . After normalization we obtained L 2 ( p ) = { s 1 ( 0.4 ) , s 0 ( 0.6 ) } . Given that g ( L i ( p ) ) = ( r i ( k ) 2 τ + 1 2 ) ( p i ( k ) ) , we obtained g ( L 1 ( p ) ) = { 2 3 ( 0.3 ) , 5 6 ( 0.2 ) , 1 ( 0.5 ) , } and g ( L 2 ( p ) ) = { 1 3 ( 0.4 ) , 1 2 ( 0.6 ) } .
W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) ) = g ( L i ( k ) ) g ( L i ( p ) ) ( i = 1 , 2 ) { i = 1 2 ( 1 + p i ( k ) g ( L i ( k ) ) ) w i i = 1 2 ( 1 p i ( k ) g ( L i ( k ) ) ) w i i = 1 2 ( 1 + p i ( k ) g ( L i ( k ) ) ) w i + i = 1 2 ( 1 p i ( k ) g ( L i ( k ) ) ) w i } = { ( 1 + 0.2 ) 0.5 ( 1 + 0.13333 ) 0.5 ( 1 0.2 ) 0.5 ( 1 0.13333 ) 0.5 ( 1 + 0.2 ) 0.5 ( 1 + 0.13333 ) 0.5 + ( 1 0.2 ) 0.5 ( 1 0.13333 ) 0.5 , ( 1 + 0.2 ) 0.5 ( 1 + 0.3 ) 0.5 ( 1 0.2 ) 0.5 ( 1 0.3 ) 0.5 ( 1 + 0.2 ) 0.5 ( 1 + 0.3 ) 0.5 + ( 1 0.2 ) 0.5 ( 1 0.3 ) 0.5 , ( 1 + 0.16667 ) 0.5 ( 1 + 0.13333 ) 0.5 ( 1 0.16667 ) 0.5 ( 1 0.13333 ) 0.5 ( 1 + 0.16667 ) 0.5 ( 1 + 0.13333 ) 0.5 + ( 1 0.16667 ) 0.5 ( 1 0.13333 ) 0.5 , ( 1 + 0.16667 ) 0.5 ( 1 + 0.3 ) 0.5 ( 1 0.16667 ) 0.5 ( 1 0.3 ) 0.5 ( 1 + 0.16667 ) 0.5 ( 1 + 0.3 ) 0.5 + ( 1 0.16667 ) 0.5 ( 1 0.3 ) 0.5 , ( 1 + 0.5 ) 0.5 ( 1 + 0.13333 ) 0.5 ( 1 0.5 ) 0.5 ( 1 0.13333 ) 0.5 ( 1 + 0.5 ) 0.5 ( 1 + 0.13333 ) 0.5 + ( 1 0.5 ) 0.5 ( 1 0.13333 ) 0.5 , ( 1 + 0.5 ) 0.5 ( 1 + 0.3 ) 0.5 ( 1 0.5 ) 0.5 ( 1 0.3 ) 0.5 ( 1 + 0.5 ) 0.5 ( 1 + 0.3 ) 0.5 + ( 1 0.5 ) 0.5 ( 1 0.3 ) 0.5 , } W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) ) = ( 0.1668 , 0.2506 , 0.1500 , 0.2344 , 0.3290 , 0.4048 )
Based on Definition 11 and Theorem 2, we can deduce the following desirable properties for the WPLEA aggregation operator.
Property 5 (Idempotency).
Let g ( L i ( p ) ) ( i = 1 , 2 , , n ) be a collection of PLTSs. If all g ( L i ( p ) ) ( i = 1 , 2 , , n ) are equal, i.e., g ( L i ( p ) ) = g ( L ( p ) ) , then
W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = g ( L ( p ) )
Proof. 
If g ( L i ( p ) ) = g ( L ( p ) ) for all i , then W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) is computed as follows:
W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = ε i = 1 n w i g ( L i ( p ) ) = ε i = 1 n w i g ( L ( p ) ) = i = 1 n w i g ( L ( p ) ) = g ( L ( p ) )
Property 6 (Boundedness).
Let g ( L i ( p ) ) ( i = 1 , 2 , , n ) be a collection of PLTSs, then we have
min i = 1 n min k = 1 # g ( L i ( p ) ) p i ( k ) g ( L i ( k ) ) g ( L ) max i = 1 n max k = 1 # g ( L i ( p ) ) p i ( k ) g ( L i ( k ) )
where g ( L ) W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) .
Proof. 
Let f ( x ) = 1 x 1 + x , 0 x 1 , and suppose that x 1 x 2 , then f ( x 1 ) f ( x 2 ) = 1 x 1 1 + x 1 1 x 2 1 + x 2 = 2 x 2 x 1 ( 1 + x 1 ) ( 1 + x 2 ) 0 , and thus, the function f ( x ) is decreasing.
Because min i = 1 n min k = 1 # g ( L i ( p ) ) p i ( k ) g ( L i ( k ) ) g ( L ) max i = 1 n max k = 1 # g ( L i ( p ) ) p i ( k ) g ( L i ( k ) ) for every
g ( L i ( k ) ) g ( L i ( p ) ) 1 max i = 1 n max i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) 1 + max i = 1 n max i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) 1 p i ( k ) g ( L i ( k ) ) 1 + p i ( k ) g ( L i ( k ) ) 1 min i = 1 n min i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) 1 + min i = 1 n min i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) )
Since ( i = 1 , 2 , , n ) and ( k = 1 , 2 , , # g ( L i ( p ) ) ) then, for w i 0 , we have
( 1 max i = 1 n max i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) 1 + max i = 1 n max i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) ) w i ( 1 p i ( k ) g ( L i ( k ) ) 1 + p i ( k ) g ( L i ( k ) ) ) w i ( 1 min i = 1 n min i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) 1 + min i = 1 n min i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) ) w i ( i = 1 , 2 , , n ) g ( L i ( k ) ) g ( L i ( p ) ) ( i = 1 , 2 , , n ) { i = 1 n ( 1 max i = 1 n max i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) 1 + max i = 1 n max i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) ) w i } g ( L i ( k ) ) g ( L i ( p ) ) ( i = 1 , 2 , , n ) { i = 1 n ( 1 p i ( k ) g ( L i ( k ) ) 1 + p i ( k ) g ( L i ( k ) ) ) w i } g ( L i ( k ) ) g ( L i ( p ) ) ( i = 1 , 2 , , n ) { i = 1 n ( 1 min i = 1 n min i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) 1 + min i = 1 n min i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) ) w i } ( 2 1 + max i = 1 n max i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) ) 1 + g ( L i ( k ) ) g ( L i ( p ) ) ( i = 1 , 2 , , n ) { i = 1 n ( 1 p i ( k ) g ( L i ( k ) ) 1 + p i ( k ) g ( L i ( k ) ) ) w i } ( 2 1 + min i = 1 n min i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) ) min i = 1 n min i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) 2 1 + g ( L i ( k ) ) g ( L i ( p ) ) { i = 1 n ( 1 p i ( k ) g ( L i ( k ) ) 1 + p i ( k ) g ( L i ( k ) ) ) w i } 1 max i = 1 n max i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) min i = 1 n min i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) g ( L i ( k ) ) g ( L i ( p ) ) i = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) w i i = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) w i i = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) w i + i = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) w i max i = 1 n max i = 1 # L i ( p ) p i ( k ) g ( L i ( k ) ) .
Hence, the proof of Property 6 is completed. □
Property 7 (Monotonicity).
Let g ( L i ( p ) ) and g ( L i ( p ) ) be two collections of PLTSs and the numbers of linguistic terms in g ( L i ( p ) ) and g ( L i ( p ) ) are identical ( i = 1 , 2 , , n ) . Assume that w = ( w 1 , w 2 , , w n ) T is an associated weighting vector with w i [ 0 , 1 ] and i = 1 n w i = 1 . If g ( L i ( k ) ) ( p i ( k ) ) g ( L i ( k ) ) ( p i ( k ) ) for all i , i.e., g ( L i ( p ) ) g ( L i ( p ) ) then,
W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) .
Proof. 
Let W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = g ( L ( p ) ) and W P L E A ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = g ( L ( p ) ) . Let f ( x ) = 1 + x 1 x , x [ 0 , 1 ] ; then, it is an increasing function. If g ( L i ( p ) ) g ( L i ( p ) ) for all i , then f ( g ( L ( p ) ) ) f ( g ( L ( p ) ) ) , i.e., 1 + g ( L ( p ) ) 1 g ( L ( p ) ) 1 + g ( L ( p ) ) 1 g ( L ( p ) ) , for all i . Therefore, we have:
g ( L ( p ) ) = g ( L i ( k ) ) g ( L i ( P ) ) ( i = 1 , 2 , , m ) { j = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) w i j = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) w i j = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) w i + j = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) w i } = 1 2 1 + g ( L i ( k ) ) g ( L i ( P ) ) ( i = 1 , 2 , , m ) { i = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) 1 p i ( k ) g ( L i ( k ) ) ) w i } 1 2 1 + g ( L i ( k ) ) g ( L i ( P ) ) ( i = 1 , 2 , , m ) { i = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) 1 p i ( k ) g ( L i ( k ) ) ) w i } = g ( L i ( k ) ) g ( L i ( P ) ) ( i = 1 , 2 , , m ) { j = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) w i j = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) w i j = 1 n ( 1 + p i ( k ) g ( L i ( k ) ) ) w i + j = 1 n ( 1 p i ( k ) g ( L i ( k ) ) ) w i } = g ( L ( p ) )
Therefore, we complete the proof of Property 7. □

4.2. Probabilistic Linguistic Einstein Geometric (PLEG) Aggregation Operators

In this section, we explore the fusion of Einstein t-norm (EG) aggregation operators under the probabilistic linguistic environment. We proposed the probabilistic linguistic Einstein geometric (PLEG) and the weighted probabilistic linguistic Einstein geometric (WPLEG) operators.

4.2.1. PLEG

On the basis of the operational laws (4) of Definition 10 and (5) of Definition 8, we present the definition of the PLEG aggregation operator as follows:
Definition 12.
Let L ( p ) = { L ( k ) i ( p i ( k ) ) / k = 1 , 2 , , # L i ( p ) } ( i = 1 , 2 , , n ) be a collection of PLTSs and g ( L i ( p ) ) its equivalent transformation. A probabilistic linguistic Einstein Average (PLEG) operator is a mapping g ( L n ( p ) ) g ( L ( p ) ) such that
P L E G ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = ε i = 1 n ( g ( L i ( p ) ) ) ^ ( 1 n )
Theorem 3.
Let L ( p ) = { L ( k ) i ( p i ( k ) ) / k = 1 , 2 , , # L i ( p ) } ( i = 1 , 2 , , n ) be a collection of PLTSs and g ( L ( p ) ) its equivalent transformation, then their aggregated value by using PLEG operator is also a PLTE and
P L E G ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) , , g ( L n ( p ) ) ) = g ( L 1 ( k ) ) g ( L 1 ( p ) ) , g ( L 2 ( k ) ) g ( L 2 ( p ) ) , , g ( L n ( k ) ) g ( L n ( p ) ) { 2 i = 1 n ( p i ( k ) g ( L i ( k ) ) ) 1 n i = 1 n ( 2 p i ( k ) g ( L i ( k ) ) ) 1 n + i = 1 n ( p i ( k ) g ( L i ( k ) ) ) 1 n }
where g ( L i ( k ) ) = ( r ( k ) 2 τ + 1 2 ) .
We proved (13) by using mathematical induction on n. For n = 2 , according to the operational law (4) of Definition 10, we have
g ( L 1 ( p ) ) ( 1 n ) = g ( L 1 ( k ) ) g ( L 1 ( p ) ) { ( p 1 ( k ) g ( L 1 ( k ) ) ) 1 n ( 2 p 1 ( k ) g ( L 1 ( k ) ) ) 1 n + ( p 1 ( k ) g ( L 1 ( k ) ) ) 1 n }
g ( L 2 ( p ) ) ( 1 n ) = g ( L 2 ( k ) ) g ( L 2 ( p ) ) { ( p 2 ( k ) g ( L 2 ( k ) ) ) 1 n ( 2 p 2 ( k ) g ( L 2 ( k ) ) ) 1 n + ( p 2 ( k ) g ( L 2 ( k ) ) ) 1 n }
Then
P L E G ( g ( L 1 ( p ) ) , g ( L 2 ( p ) ) ) = g ( L 1 ( p ) ) ( 1 n ) ε g ( L 2 ( p ) ) ( 1 n ) = 2 ( p 1 ( k ) g ( L 1 ( k ) ) ) 1 n