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.
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
, we need to find the equivalent transformation of PLTSs, since some probabilistic linguistic elements (PLEs) might not necessarily belong to
. Luckily Gou and Xu [
6] defined the first equivalent transformation of probabilistic linguistic term sets (PLTSs) as follows:
Definition 9 (Gou and Xu [6]). Letbe any linguistic term set.is a PLTS. The equivalent transformation function of is defined as:whereand,
.
.
Based on Definition 9, we can obtain new operational laws defined as follows:
Proposition 1. Letbe a collection of PLTSs andits equivalent transformation. Given three transformed PLTSs,, then
- (1)
- (2)
- (3)
- (4)
Based on Definition 7 and Definition 8, we give some new operations on the transformed PLTEs as follows:
Proposition 2. Letbe a linguistic term set and. Given three transformed PLTSs,and then
- (1)
- (2)
- (3)
- (4)
whereand.
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
and
then
For
, we obtain
we have
since
.
Proved as required. □
Considering operational law 1, we prove the operational law 3
Based on the operational laws (1)–(4) of
Section 3, we can easily obtain the following properties.
- (1)
.
- (2)
.
- (3)
.
- (4)
.
Proof. and .
Since and then .
Therefore
Hence, we complete the proof of Property 1. The remaining properties can easily be proved. □
5. Probabilistic Linguistic Einstein Aggregation Operators and Their Approaches to Multi-Criteria Group Decision Making
Under this section, we present a MCGDM problem where the evaluation information is likely to be expressed by the transformed PLTSs. Hence, we make use of the WPLEA or WPLEG operators to buttress our decision.
Let be a finite set of m alternatives and be a set of n attributes. Suppose that denotes the set of DMs. By using the linguistic scale each DM provides his or her linguistic evaluations over the alternative with respect to the attribute , i.e.,
Then, we determine the collective evaluations of DMs for each alternative in terms of PLTEs.
In the context of GDM, the linguistic evaluation values
with the corresponding probability
are described as the PLTS
and
is the number of linguistic terms in
. The PLTSs
denote the evaluations values over the alternatives
with respect to the attributes
where
is the
value of
and
is the probability
. In this case
and
. All the PLTSs are contained in the probabilistic linguistic decision matrix
. Hence, the result is shown as follows:
Without loss of generality, we assume that each transformed PLTS is an ordered transformed PLTS. denotes the weighting vector of the attributes and , . Based on the above results, we will use the WPLEA or WPLEG aggregation operators to develop the corresponding approach for MCGDM with probabilistic linguistic information. This approach is designed with the determination of the objective weights.
5.1. The Determination of the Objective Weights Based on Entropy Measures
Entropy method is the concept of thermodynamics, which was first introduced by Shannon into the information theory, and now is widely used in the engineering, socio-economic and other fields [
22]. Entropy measures are useful in computing weights of the criteria, and had been used widely in MCGDM problems [
23]. Shannon entropy method constitutes one of the techniques to determine the weight of the criteria when it becomes difficult to be provided by the decision-maker [
24]. Through the computation of information entropy of a proposed parameter, its weight is determined according to its relative degree of change that impact on equipment, index with larger degree of relative change as larger weight. For example, Peng et al. [
25] introduced two optimization models for the determination of the criterion weights in a multi-criteria decision-making situations . In our study, emphasis should be laid on the determination of a reasonable weight of the criteria. This has become necessary because many at times the DMs are influenced by what they have as knowledge structure, personal bias, and familiarity with the decision alternatives. Consequently, the necessity arises for us to consider the MADM problem with completely unknown weights of criteria. Therefore, there will be a need for us to establish a weight determination method on the basis of entropy technique under the probabilistic linguistic environment.
Let be a PLTE, and its equivalent transformation then the steps for determining the weights are as follows:
Step 1: Calculate the score matrix
of
Step 2: Normalize the score matrix
as follows:
where
.
Step 3: Determine the attribute weights.
Let .
The attribute weight
is determined by
5.2. Probabilistic Linguistic MADM Approach
Decision-making processes comprise a series of steps: identifying the problems, constructing the preferences, evaluating the alternatives, and determining the best alternatives. With the aid of the WPLEA and the WPLEG aggregation operators, we develop a decision-making procedure for the ranking of alternatives. The detailed approach is illustrated as follows:
Step 1. In a practical decision-making problem, we determine the alternatives and a set of the attributes . Then we obtain the decision matrix provided by the decision-maker DM .
Step 2. With regards to the collective matrix , the normalization process of the entries of could be made as stated in Definition 5. The entries of the normalized matrix are arranged in a decreasing order.
Step 3. Since the operational values may exceed the boundaries of LTSs and also the PLTEs must be within the interval to satisfy the Einstein operational laws, we need to transform the PLTS to the following equivalent form .
Step 4. We determine the criteria weights. The criteria weights can be determined by way of using the following formula:
Step 5. If the DM prefers the WPLEA operator, then the aggregated value of the alternative
is determined based on (11). The result is:
If the DM prefers the WPLEG operator, then the aggregated value of the alternative
is determined based on (15). The result is:
In this case, we denote the aggregated value of the alternative as .
Step 6. Following the results of Definition 4 of
Section 2, the score and the deviation degree of
of the alternative
are computed, i.e.,
and
.
Step 7. Rank all the alternatives in accordance with the ranking results of Definition 4.
6. Illustrative Example
Information technology is has become the antidote to the numerous problems faced by health organizations to improve healthcare delivery. In order to improve healthcare delivery, the adoption of health information technology (HIT) has become vital for health organizations. Many stakeholders like the government, information technology businesses, healthcare organizations, policy makers and consumers anticipate that healthcare problems can be addressed through technological innovations [
26]. The adoption of HIT can help health administrators to reduce clinical errors, provide support to clinicians, improve patients’ information management and expand patients’ access to both remote and continuity healthcare services [
27,
28,
29,
30,
31]. Due to the relevance and involvement of various uncertainties and risks associated with healthcare process, decision makers are very often involved in the decision-making process for healthcare issues. As a global concern, the Ghanaian government will want to assess and improve the quality and safety of care as far as health information technology (HIT) innovations in public hospitals in Ghana is concerned.
Hence this problem can be considered as a multi-criteria group decision making problem (MCDM) which requires MCDM methods for an effective problem-solving. In solving real-life decision-making problems, Pang et al. [
1] stated that, decision-makers can employ linguistic terms to evaluate the performance of the alternatives, which can be used to help rank the hospitals and select the most desirable one via proper decision-making methods. To address the MCDM problem above, we adopt probabilistic linguistic values to overcome uncertainty and qualitative factors. In many situations, the preference information on attributes is uncertain and inconsistent. In order to account for the insufficiency in decision making, we present a probabilistic linguistic Einstein Aggregation (PLEA) operators to select the ideal hospital based on probabilistic linguistic values. In this section, we extend the PLMCGDM method to the healthcare environment. Government will want to assess or evaluate the performance of hospitals as far as implementation of HIT is concerned. For this, purpose, five hospitals were randomly chosen. In view of that a committee of three decision makers
was formed to select the most appropriate hospital. Therefore, we introduce a MCGDM problem in which PLTS is used to express the evaluation information. Then, we apply the WPLEA or WPLEG operator to support our decision. Let
be a finite set of five hospitals and
be the set of criteria defined for the selection process.
adopted from [
32]. The linguistic scale is
. Considering the results of [
33], the evaluations of the decision makers are shown in
Table 1,
Table 2 and
Table 3.
6.1. Decision Analysis with Our Proposed Approaches
Step 1: Following the proposed methods in
Section 5, we integrate the individual decision matrices
into a collective decision matrix by (17). Hence, the result is shown in
Table 4. For
Table 4 each PLTS
is assumed to be ordered PLTS.
. Based on the Entropy Shannon measure determined in Step 4, the weighting vector of the attributes
is
. We use WPLEA and WPLEG aggregation operators to analyze the results of
Table 5. With respect to the above results and the proposed methods in
Section 5, the detail steps are shown as follows.
Step 2. Considering the collective matrix
, we normalize the entries of
as stated in Definition 5. Then, the normalized entries are arranged in decreasing order. The results are presented in the
Table 5.
Step 3. Since the operational values may exceed the boundaries of LTSs and also the PLTEs must be within the interval
to satisfy the Einstein operational laws, we need to transform the normalized PLTS
to the following equivalent form
. The results are given in
Table 6.
Step 4. We derive the criteria weights by utilizing (20) and the following is the weight vector of
Step 5. Aggregate the probabilistic linguistic values for each alternative by the WPLEA (or WPLEG) operator.
If the decision-maker chooses the WPLEA operator, then the aggregated value of the alternatives
is determined based on (11).
In case the decision -maker considers the WPLEG operator, then the aggregated value of the alternative
is determined based on (15)
. In the same way, we denote the aggregated value of the alternative
as
. The results are:
Step 6. On the basis of the results of Definition 10, the scores of the alternatives can be computed, i.e., . If the DM uses WPLEA operator to calculate the decision formation, the scores are determined below:
; ; ; ; .
If the DM uses WPLEG operator to aggregate the decision formation, the scores are determined as follows:
; ; ; ; .
Step 7. If the DM uses WPLEA operator, we can determine the ranking of the scores of the alternatives based on the results of Step 6. It is shown as follows:
Hence, the ordering of the alternatives is:
If the DM uses WPLEG operator, we can determine the ranking of the scores of the alternatives based on the results of Step 6. It is shown as follows:
In this case, the ordering of the alternatives is:
6.2. Comparison Analysis
Under the probabilistic linguistic information, Ref. [
1] developed an aggregation-based method for MAGDM. In order to validate the effectiveness of our proposed methods, we carefully compute the decision results for Ref. [
1] and Ref. [
33]. The decision results are demonstrated in
Table 7.
Considering the linguistic power weighted average (LPWA) of Ref. [
33] the rank is
. Also, that of Ref. [
1] is
. For both methods the ranking is the same but different from our rankings, thus WPLEA:
and WPLEG:
. However, the optimal decision result conforms to our best alternative which is also
for both methods, thus WPLEA and WPLEG.
On the MCGDM problems under probabilistic linguistic environment, we introduced our model to improve upon the existing techniques. Unlike the existing model of Ref. [
33] considered in this paper, our model yields better results because since probabilities were not considered in their approach the accuracy of preference information of the DMs might be questionable and the ignorance of the probabilistic information may lead to erroneous decision results. Pang et al. [
1] stated that completely ignoring the importance and probability of possible linguistic term sets in GDM with linguistic information is not rational. In addition, without the PLTSs, it might not be easy for the DMs to provide several possible linguistic values over an alternative or an attribute. This situation exposes the shortcomings of the model proposed in Ref. [
33], in spite of the power average involvement in the aggregation process.
Now comparing our results with that of Ref. [
1], our model yields better results. The reason being that since PLTSs itself as a theory has some limitations, it encounters the relationship phenomenon between inputs arguments. Luckily our proposed model provides more versatility in the aggregation process and it has the ability to depict the interrelationship of input arguments. To throw more light on the weaknesses, Gou et al. [
6] stated that the existing operational laws of linguistic term and the extended linguistic terms are very unreasonable because occasionally their operational values exceed the bounds of linguistic term sets (LTSs). Besides, their novel operational laws can reduce the computational complexity experienced in [
1] and also keep the probability information complete after operations. In general, WPLEG applies to the average of the ratio data and is mainly used to calculate the average growth (or change) rate of the data. From the characteristics of
Table 7, the WPLEA is much better than WPLEG.