Faculty of Science and Natural Resources, Universiti Malaysia Sabah, 88400 Kota Kinabalu, Sabah, Malaysia
2.
School of Mathematics and Information Sciences, Numerical Simulation Key Laboratory of Sichuan Provincial Universities, Neijiang Normal University, Neijiang 641000, Sichuan Province, P. R. China
Received:
26 December 2022
Revised:
02 March 2023
Accepted:
10 March 2023
Published:
17 March 2023
Probabilistic linguistic terms set (PLTS), a new tool for expressing uncertain decision information, is composed of all possible linguistic terms (LTs) and their related probabilities. It also increases the corresponding probability of LTs in hesitant fuzzy linguistic term set (HFLTS). On the other hand, aggregation operator is an important information fusion tool, the Maclaurin symmetric mean (MSM) operator can provide more flexibility and robustness in information fusion, and make it more suitable for solving MADM problems with independent attributes. This current study adopts the merits of PLTS and MSM operator, and then a novel probabilistic linguistic decision making approach are targeted. Firstly, the operations of two PLTSs are redefined based upon Archimedean t-norm (ATN) and Archimedean t-conorm (ATC); Secondly, the probabilistic linguistic generalized MSM operator (PLGMSM) is proposed based on ATN and ATC, some properties of PLGMSM are investigated, then some special PLGMSM operators have been studied in detail when the parameters take different values and the generator of ATN takes different functions. Thirdly, the weighted probabilistic linguistic generalized MSM operator (WPLGMSM) is studied along with some properties of PLGMSM, some special WPLGMSM operators have been also investigated in detail when the parameters take different values and the generator of ATN takes different functions. Finally, on the basis of our proposed aggregation operators, the aggregated-based decision making approach is designed and an example is supplied to manifest the effectiveness of the proposed approach. Furthermore, some comparison analyses with extant decision approaches are carried out to illustrate the validity and feasibility of the proposed approach.
Citation: Ya Qin, Siti Rahayu Mohd. Hashim, Jumat Sulaiman. Probabilistic linguistic multi-attribute decision making approach based upon novel GMSM operators[J]. AIMS Mathematics, 2023, 8(5): 11727-11751. doi: 10.3934/math.2023594
Related Papers:
[1]
Shahid Hussain Gurmani, Zhao Zhang, Rana Muhammad Zulqarnain .
An integrated group decision-making technique under interval-valued probabilistic linguistic T-spherical fuzzy information and its application to the selection of cloud storage provider. AIMS Mathematics, 2023, 8(9): 20223-20253.
doi: 10.3934/math.20231031
[2]
Muhammad Akram, Sumera Naz, Feng Feng, Ghada Ali, Aqsa Shafiq .
Extended MABAC method based on 2-tuple linguistic $ T $-spherical fuzzy sets and Heronian mean operators: An application to alternative fuel selection. AIMS Mathematics, 2023, 8(5): 10619-10653.
doi: 10.3934/math.2023539
[3]
Ghous Ali, Kholood Alsager, Asad Ali .
Novel linguistic $ q $-rung orthopair fuzzy Aczel-Alsina aggregation operators for group decision-making with applications. AIMS Mathematics, 2024, 9(11): 32328-32365.
doi: 10.3934/math.20241551
[4]
Zeeshan Ali, Tahir Mahmood, Muhammad Bilal Khan .
Three-way decisions with complex q-rung orthopair 2-tuple linguistic decision-theoretic rough sets based on generalized Maclaurin symmetric mean operators. AIMS Mathematics, 2023, 8(8): 17943-17980.
doi: 10.3934/math.2023913
[5]
Muhammad Akram, Sumera Naz, Gustavo Santos-García, Muhammad Ramzan Saeed .
Extended CODAS method for MAGDM with $ 2 $-tuple linguistic $ T $-spherical fuzzy sets. AIMS Mathematics, 2023, 8(2): 3428-3468.
doi: 10.3934/math.2023176
[6]
Sumaira Yasmin, Muhammad Qiyas, Lazim Abdullah, Muhammad Naeem .
Linguistics complex intuitionistic fuzzy aggregation operators and their applications to plastic waste management approach selection. AIMS Mathematics, 2024, 9(11): 30122-30152.
doi: 10.3934/math.20241455
[7]
Abbas Qadir, Shadi N. Alghaffari, Shougi S. Abosuliman, Saleem Abdullah .
A three-way decision-making technique based on Pythagorean double hierarchy linguistic term sets for selecting logistic service provider and sustainable transportation investments. AIMS Mathematics, 2023, 8(8): 18665-18695.
doi: 10.3934/math.2023951
[8]
Wajid Azeem, Waqas Mahmood, Tahir Mahmood, Zeeshan Ali, Muhammad Naeem .
Analysis of Einstein aggregation operators based on complex intuitionistic fuzzy sets and their applications in multi-attribute decision-making. AIMS Mathematics, 2023, 8(3): 6036-6063.
doi: 10.3934/math.2023305
[9]
Tahir Mahmood, Azam, Ubaid ur Rehman, Jabbar Ahmmad .
Prioritization and selection of operating system by employing geometric aggregation operators based on Aczel-Alsina t-norm and t-conorm in the environment of bipolar complex fuzzy set. AIMS Mathematics, 2023, 8(10): 25220-25248.
doi: 10.3934/math.20231286
[10]
Misbah Rasheed, ElSayed Tag-Eldin, Nivin A. Ghamry, Muntazim Abbas Hashmi, Muhammad Kamran, Umber Rana .
Decision-making algorithm based on Pythagorean fuzzy environment with probabilistic hesitant fuzzy set and Choquet integral. AIMS Mathematics, 2023, 8(5): 12422-12455.
doi: 10.3934/math.2023624
Abstract
Probabilistic linguistic terms set (PLTS), a new tool for expressing uncertain decision information, is composed of all possible linguistic terms (LTs) and their related probabilities. It also increases the corresponding probability of LTs in hesitant fuzzy linguistic term set (HFLTS). On the other hand, aggregation operator is an important information fusion tool, the Maclaurin symmetric mean (MSM) operator can provide more flexibility and robustness in information fusion, and make it more suitable for solving MADM problems with independent attributes. This current study adopts the merits of PLTS and MSM operator, and then a novel probabilistic linguistic decision making approach are targeted. Firstly, the operations of two PLTSs are redefined based upon Archimedean t-norm (ATN) and Archimedean t-conorm (ATC); Secondly, the probabilistic linguistic generalized MSM operator (PLGMSM) is proposed based on ATN and ATC, some properties of PLGMSM are investigated, then some special PLGMSM operators have been studied in detail when the parameters take different values and the generator of ATN takes different functions. Thirdly, the weighted probabilistic linguistic generalized MSM operator (WPLGMSM) is studied along with some properties of PLGMSM, some special WPLGMSM operators have been also investigated in detail when the parameters take different values and the generator of ATN takes different functions. Finally, on the basis of our proposed aggregation operators, the aggregated-based decision making approach is designed and an example is supplied to manifest the effectiveness of the proposed approach. Furthermore, some comparison analyses with extant decision approaches are carried out to illustrate the validity and feasibility of the proposed approach.
1.
Introduction
Generally speaking, multi-criteria evaluation refers to the evaluation conducted under multiple criteria that cannot be replaced by each other. In the specific evaluation process, information is often missing because of the wide range of evaluation criteria. Some information is available but very inaccurate; Some can only give a rough range based on empirical judgment. Therefore, in the process of scheme evaluation, quantitative calculation shall be carried out for those criteria that can be accurately quantified; For those criteria that are difficult to accurately quantify or cannot be quantified, it is necessary to make a rough estimation or invite relevant experts to conduct qualitative analysis and hierarchical semi quantitative description. On account of the complexity and uncertainty of objective things, as well as the fuzziness brought by human cognitive level and thinking mode, it is difficult for experts to give accurate and quantitative information in evaluation process. Therefore, how to realize mutual transformation between qualitative and quantitative as well as reflect the soft reasoning ability in linguistic expression has always been a research hot-spot in uncertain system evaluation and decision-making.
For example, the fuzzy set [1] shows the relationship between scheme and criterion in a quantitative way, which has been recognized by many scholars. Since then, quantitative decision tools, that is, various extensions of fuzzy set [2,3,4,5], have been emerged to show decision information. However, with the ceaseless advancement and change of decision-making environment, it is difficult for decision-makers to use a set of quantitative and specific values to describe the decision-making information of a scheme under a certain criterion. To solve this deficiency, Zadeh proposed linguistic variable (LV) [6], which qualitatively displays the decision information of decision-makers in a non-numerical way for the first time. Then, In line with LV, decision tools such as uncertain LV [7,8,9], hesitant fuzzy linguistic term set (HFLTS) [10,11], terms with weakening modifiers [12] appeared to help decision-makers give qualitative decision information. In the face of complex MADM problems, owing to the influence of complex information as well as uncertain factors of group cognition, people sometimes use a single linguistic term (LT) to describe attribute evaluation information, but sometimes need to use several LTs to express decision information at the same time [13,14]. For example, when students evaluate the quality of class teaching, they may use both "good" and "very good" to evaluate it at the same time. Inspired by hesitant fuzzy sets [15] and linguistic term set (LTS) [6], Rodriguez et al [10] defined HFLTS in 2012, which allows decision-makers to use several possible LTs to evaluate attributes simultaneously.
Although HFLTSs can meet the needs of decision-makers to express information by multiple LTs, the HFLTSs are assumed that the weight of all possible LTs are equal. Obviously, this assumption is too idealistic and inconsistent with actual situation. Because although decision-makers shilly-shally about several possible LTs, they may tend to use some of them under certain circumstances. Therefore, different LTs should possess different weights. In line with this reality, Pang et al. (2016) Proposed probabilistic linguistic term sets (PLTSs), which is composed of possible LTs and associated with their probabilities [16]. On the one hand, probability linguistic contains several LTs to show decision-makers view in decision-making, which retains the good nature of HFLTS. On the other hand, it reflects the corresponding weights of several LTs. This way of displaying decision information by combining qualitative and quantitative information well reflects decision-makers decision information, which will not lose linguistic evaluation information, so they can make decision evaluation more in line with the reality. Although it is difficult to give a definite decision-making view on the problems that needs to decide, it will give the weight of corresponding view and give relatively clear decision-making information as much as possible to help experts solve decision-making problems. Briefly, the merit of PLTSs is that it can express information more completely and accurately. Hence, PLTSs could be utilized to solve practical decision-making problems.
Aggregation operator (AO) is an important tool for information fusion. Most AOs are built on the special triangle t-norm. Archimedeans t-norm (ATN) and Archimedeans t-conorm (ATC) are composed of t-norm (TN) and t-conorm(TC) families. They can deduce some basic algorithms of fuzzy sets. Linguistic scaling functions can define different semantics for LTs in different linguistic environments. At the same time, the significant advantage of Muirhead operator is that it can reflect the relationship between any parameters. Liu et al. [17] defined the algorithm for PLTS based on ATN and linguistic scaling function, and then combined the Muirhead average operator with the PLTSs to propose the Archimedeans Muirhead average operator and Archimedeans weighted Muirhead average operator of probabilistic linguistic, Archimedean dual Muirhead average operator of probability linguistic and Archimedean weighted dual Muirhead average operator of probability linguistic. After that, more and more attention has been paid to various aggregation operators [18,19,20]. In real MADM, it is rare that the evaluation attributes of various alternatives are independent with each other. For example, there is a positive correlation between teaching quality and lesson design, that is, the better the curriculum design, the higher the quality of teaching. For capturing these dependencies, Maclaurin [21] initially proposed Maclaurin symmetric mean (MSM) operator, which can consider the relationship between multiple attribute values at the same time, and MSM has an adjusting parameter k. The MSM operator is a flexible operator that can consider the relationship between several attribute values. Therefore, it is essential to develop some MSM operators [22,23,24,25,26,27,28,29,30,31,32,33,34] in different polymerization environment; Besides, the operation laws are essential in the process of aggregation, and they can generate many operation laws based on certain ATTs and ATCs. Although MSM operators have attracted a lot of attentions since it's appearance, MSM operator has some disadvantages. The main disadvantage of MSM operator is that it only focus on the overall relationship, ignoring heterogeneity among individuals. To address this handicap, Detemple and Robertson [35] proposed generalized Maclaurin symmetric mean (GMSM), which is considered as a new generalization of MSM. GMSM can not only reflect the relationship of the whole, but also consider the importance level of individuals. Besides, compared with MSM, GMSM can avoid information loss. Because the polymerization process increased equality constraints. Therefore, GMSM is extensively employed in information fusion.
Since PLTSs are introduced by integrating the LTSs and the HFSs, PLTSs can successfully express random and fuzzy information. Therefore, it is necessary to develop a novel important probabilistic linguistic information fusion tool (that is, PLGMSM operator) which can not only combine the merits of PLTSs and MSM, but also reflect the relationship of the whole and consider the importance level of individuals. These considerations lead us to lock the main targets that follow from this work:
(1) To introduce new probabilistic linguistic GMSM operators along with investigate some properties as well as some special situations;
(2) To construct an MADM algorithm based upon the proposed PLGMSM operators;
(3) To manifest an example based on probabilistic linguistic information to prove the availability of the proposed MADM approach;
(4) To analyze the sensitivities of parameters in the proposed aggregation operators.
To achieve the above objective, some probabilistic linguistic GMSM operators are introduced for PLTSs based on ATN and ATC in current work. The structure of this work is arranged as: In Sect.2, some related basic concepts are presented, for instance PLTS, MSM operators, ATN and ATC, etc. In Sect.3, the PLGMSM are introduced based upon the ATN and ATC, some properties of the PLGMSM operators and special situations of PLGMSM operators are also given. In Sect.4, the weighted PLGMSM (WPLGMSM) is introduced based upon the ATN and ATC, some properties of the PLGMSM and special situations of WPLGMSM are also listed in this section. Section. 5 constructs a MADM method for evaluating quality of classroom teaching. Some comparisons are carried out in Sect.6. and a conclusion is made in Sect.7.
2.
Related knowledge
Some basic concepts will be reviewed in this part, including linguistic term set (LTS), probabilistic linguistic term set (PLTS), ATN and ATC.
2.1. PLTS
Definition 2.1.[36] Suppose S={sv|v=−τ,…,−1,0,1,…,τ} be a LTS, where sv expresses a possible value of a LV, and τ is a positive integer. For any two LVs sα,sβ∈S, it satisfies: if α>β, then sα>sβ.
Definition 2.2.[16] Suppose S={sv|v=−τ,…,−1,0,1,…,τ} be a LTS, a PLTS is defined as following:
(2) if E(ℓ1(p))=E(ℓ2(p)), when σ(ℓ1(p))>σ(ℓ2(p)), then ℓ1(p)<ℓ2(p);
(3) if E(ℓ1(p))=E(ℓ2(p)), when σ(ℓ1(p))=σ(ℓ2(p)), then ℓ1(p)=ℓ2(p).
In order to calculate the PLTSs more conveniently, the transformation function g was introduced by Gou et al. [37]. Suppose there is a LTS S and a PLTS ℓ(p), the g and g−1 are defined as:
AOs are very important tools for information fusion in some decision-making problems. However, most of AOs are defined based on TNs and TCs. So it is essentially to review TNs and TCs before the operations of PLTS are given.
Definition 2.3.[37] If the function ς:[0,1]2→[0,1] meets the following four requirements for all α,β,δ∈[0,1], it was named as a TN:
(1) ς(α,β)=ς(β,α);
(2) ς(α,ς(β,δ))=ς(ς(α,β),δ);
(3) ς(α,β)≤ς(α,δ), if β≤δ;
(4) ς(1,α)=α.
A TC ς∗[38] is a mapping from [0,1]2 to [0,1], if ς∗ meets the following four requirements for all α,β,δ∈[0,1]:
(1) ς∗(α,β)=ς∗(β,α);
(2) ς∗(α,ς∗(β,δ))=ς∗(ς∗(α,β),δ);
(3) ς∗(α,β)≤ς∗(α,δ), if β≤δ;
(4) ς∗(0,α)=α.
The TN ς and TC ς∗ are dual, that is, ς∗(α,β)=1−ς(1−α,1−β).
A TN ς is Archimedean t-norm (ATN), if there exists an integral n, such that ς(a,⋯,a⏟ntimes)<b for any (a,b)∈[0,1]2. A TC ς∗ is Archimedean t-conorm (ATC), if there is an integral n, such that ς∗(a,⋯,a⏟ntimes)>b for any (a,b)∈[0,1]2. Specially, if ς and ς∗ satisfy the three given requirements:
(1) ς and ς∗ are continuous;
(2) ς and ς∗ and are strictly increasing;
(3) for all α∈[0,1], and ς∗(α,α)>α,
then ς and ς∗ are strict ATN and strict ATC respectively.
Assuming there is an additive generator J:[0,1]→[0,∞). A strict ATN ς(α,β) can be defined by:
ς(α,β)=J−1(J(α)+J(β)),
(2.7)
where J−1 is the inverse of J. Similarly, its ATC ς∗(α,β) also can be generated by its additive generator J∗:
ς∗(α,β)=(J∗)−1(J∗(α)+J∗(β)),
(2.8)
where J∗(α)=J(1−α), (J∗)−1(α)=1−J−1(α) and (J∗)−1 is the inverse of J∗.
Moreover, we can also derive ς∗(α,β) as:
ς∗(α,β)=1−J−1(J(1−α)+J(1−β)).
(2.9)
2.3. Generalized MSM operators
MSM operator [21] was originally proposed by Maclaurin and then further generalized by Detemple [35], it's merit is that it can reflect the relationship between multiple input parameters. The MSM is defined as follows:
Definition 2.4.[21] Let ξ1,ξ2,⋯,ξn be n nonnegative real numbers, and m=1,…,n. A MSM operator will be expressed as
MSM(m)(ξ1,ξ2…,ξn)=(∑1≤i1<⋯<im≤nm∏j=1ξijCmn)1m,
(2.10)
where (i1,i2,…,in) is a permutation of (1,2,…,n).
Definition 2.5.[35] Let ξ1,ξ2,⋯,ξn be n nonnegative real numbers, and μj≥0. A GMSM operator can be expressed as
Remark 3.1.In Definition 3.1, J is a generator of ATN, when J takes different function which satisfies the condition of generators, we can obtain different operations of two PLTSs. Therefore, Definition 3.1 can be regarded as a unified expression of some existing operations of PLTSs.
3.2. PLGMSM operators based on ATN and ATC
In what follows, ℓi(p)={ℓ(t)i(p(t)i)|t=1,2,…,#ℓi(p)} if not specifically stated. Based upon operational laws of PLTSs defined in Defintion 3.1, PLGMSM operator can be proposed and listed as follows.
Definition 3.2.Let ℓ1(p),⋯,ℓn(p) be a group of PLTSs, the probabilistic linguistic generalized MSM operator (PLGMSM) based on ATN and ATC is a function PLGMSM:Ωn→Ω and
Proof. The proofs is similar to Property 3 in [24]. So, the details are omitted.
3.3. Some special PLGMSM operators based on different generators
In this section, some special PLGMSMS operators will be investigated when parameters take different values and the generator takes different functions.
3.3.1. When parameters takes different values
(a) When m=1, the PLGMSM operator based on ATN and ATC will reduce to
(c) If J(x)=lnε+(1−ε)xx (ε>0), then it has J−1(x)=εex+ε−1. We can get probabilistic linguistic Archimedean Hamacher GMSM (PLAHGMSM) operators as follows.
4.
Weighted probabilistic linguistic generalized MSM operators based upon ATN and ATC
Due to each individual's different background knowledge and preference, the importance should be different. Hence, it is essential to consider the individual weight information to make the decision results are more reasonable and scientific. In this section, the weighted probabilistic linguistic generalized MSM operators based on ATN and ATC will be introduced.
Definition 4.1.Let ℓ1(p),⋯,ℓn(p) be n PLTSs and wi be the weight of ℓi(p) with wi∈[0,1], ∑ni=1wi=1. The WPLGMSM is a function WPLGMSM(m, u1, u2, …, um):Ωn→Ω, if
Proof. The proofs of this theorem is similar to Property 3 in [24]. So, the details are omitted.
4.1. Some special WPLGMSM operators based on different generators
In this section, some special PLGMSMS operators will be investigated when the parameters take different values and the generator takes different function.
4.1.1. When parameters take different values
(a) When m=1, the WPLGMSM operator based on ATN and ATC will reduce to
In the what follows, the special situations of the WPLGMSM based on ATN and ATC will be discussed.
(1) If J(x)=−lnx, then J−1(x) = e−x. The weighted probabilistic linguistic Archimedean Algebraic GMSM (WPLAAGMSM) operators will be obtained as follows:
(2) If J(x)=ln 2−xx, it has J−1(x) = 2ex+1. Then the weighted probabilistic linguistic Archimedean Einstein GMSM (WPLAEGMSM) operators will be obtained as follows:
(3) If J(x)=ln ε+(1−ε)xx(ε>0), then J−1(x) = εex+ε−1, the weighted probabilistic linguistic Archimedean Hamacher GMSM (WPLAHGMSM) operators will be obtained as follows:
In the following, we will continue to give some examples to testify different aggregation operators, and discuss some special situations for diverse parameters.
Example 4.1.Let ℓ1={s - 1(0.3),s1(0.7)}, ℓ2={s1(1)}, ℓ3={s0(0.25),s2(0.75)} be three PLTSs. Suppose w=(0.35,0.25,0.4) is the weight vector of ℓ1,ℓ2,ℓ3. Set τ=3, then, with the function g, ℓ1,ℓ2,ℓ3 will converted into g(ℓ1(p))={0.33(0.3),0.67(0.7)}, g(ℓ2(p))={0.67(1)}, g(ℓ3(p))={0.5(0.25),0.83(0.75)}, respectively.
Then will use the WPLGMSM operator based on ATN and ATC to fuse ℓ1,ℓ2,ℓ3. set m=2,u1=1,u2=2, with different additive generators, the aggregated results are obtained as follows:
5.
Decision-making approach based upon WPLGMSM and application in teaching quality evaluation
5.1. Aggregation-based decision-making approach
Before giving the decision-making approach, a formal description of a MADM problem with probabilistic linguistic information will be given. Suppose A={A1,…,Ak} be a set of diverse alternatives, CR={CR1,CR2,…,CRl} be the set of different attributes, and w={w1,w2,…,wl} be the weight vector of attributes CRi with wi and ∑li=1wi=1. A probabilistic linguistic decision matrix can be expressed as M = (ℓ(p)ij)k×l, where ℓ(p)ij = {ℓ(t)ij(p(t)ij)|t=1,2,…,#ℓ(p)ij} is a PLTS, and ℓ(p)ij expresses the evaluation value of alternatives Aj(j=1,2,…,k) for the attributes CRi(i=1,…,l).
In line with the given above-mentioned description of MADM problem, the proposed aggregated operators will adopted to address some actual issues and find an ideal alternative. Some main procedures are listed as follows:
Step 1. Standardize the attribute values by the following ways:
Step 2. Transform all attributes values ℓ(p)ij of each alternative to probabilistic hesitant fuzzy element r(p)ij.
Step 3. Aggregate all attributes values r(p)ij of each alternative to the comprehensive values r(p)j.
Step 4. Transform r(p)j into PLTS ℓ(p)j.
Step 5. Calculate the score function and the deviation degree of Aj(j=1,…,k) by Eq (2.3) and Eq (2.4).
Step 6. Rank all alternatives and then choose the desirable one.
5.2. Teaching quality evaluation in universities
This section will discuss the decision making option based upon the given WPLGMSM with experimental cases.
Example 5.1.At present, continuous improvement of education quality has been placed at an important position in colleges and universities, and the evaluation of teaching quality is the baton for the healthy development of education, as well as an essential part of education mechanism. Exploring the evaluation index system of teaching quality, building a scientific evaluation model, and forming a reasonable education evaluation system will help to improve the teaching quality and promote the high-quality development of education. In this study four main factors will be used as teaching quality evaluation indexes, they are CR1: teaching content, CR2: teaching method, CR3: teaching effect and CR4: teaching attitude. It is assumed that the weight of four indexes are w=(0.2,0.3,0.4,0.1), and four teachers' (A1−A4) teaching course Bayesian formula and its application in Probability Theory and Mathematical Statistics will be evaluated. At the same time, through the questionnaire survey on teaching experience of some graduated students and teachers, they are required to evaluate with the following linguistic variables through their own experience:
The following task is to make decision by using the procedure in Section 5.1:
Step 1. Standardize the attribute values ℓ(p)ij. As all criteria are benefit-type, so it is not necessary to standardize.
Step 2. Transformed all attributes values of each alternative to probabilistic hesitant fuzzy element r(p)ij. We set τ=3 and use the function of g. Probabilistic linguistic information will transformed into probabilistic hesitant fuzzy element and listed r(p)ij in Table 2.
Table 2.
Probabilistic hesitant fuzzy element decision information matrix.
Step 3.-Step 4. We choose the aggregation operator based on algebraic generator to fuse decision information. As the vast numbers, the results not listed here.
Step 5. Calculate the expected value and listed as follows:
Step 6. Determine the desirable alternative according to the expected values. From the calculated results of Step 5, we have A4≻A1≻A2≻A3. Therefore, A4 is the desirable one.
Meanwhile, we use other three proposed aggregation operators to fuse above decision information, the results are listed in Table 3.
Table 3.
The ranking based three proposed aggregation operators when u1 = 1, u2 = 2.
It is obviously the ranking results are consistent with different aggregated operators. Meanwhile, we feed back the ranking results of this paper to some evaluators. Most of them state that the results are in line with their selection order, which also demonstrates the rationalities and effectiveness of the proposed method in Section 5.1.
6.
Comparative analyses
To further justify the validity and robustness of our proposed decision-making method, more comparisons will be carried out in this section.
6.1. Comparison with the method of possibility degree matrix
B. Fang, et al.[40] proposed an improved possibility degree formula to assess the education and teaching quality in military academies with probabilistic linguistic MCDM method, some main concepts are reviewed as follows.
Definition 6.1.[40] Assume S={sυ|υ=−τ,…,−1,0,1,…,τ} be a LTS, for any two PLTSs ℓ1(p) and ℓ2(p), the possibility degree of ℓ1(p)⩾ℓ2(p) could be defined as follows:
in which, r(t)1 and r(t)2 are the subscript of ℓ(t)1 and ℓ(t)2, p(t)1 and p(t)2 are the corresponding probability, respectively.
Step 1. Calculate comprehensive possibility degree matrix. For each attribute CRj, wj is the weight of CRj and n∑j=1wj=1. If Pjik=P(ℓij≥ℓkj), then the possibility degree matrix of will be defined as follows:
Pj=(Pj11Pj12⋯Pj1nPj21Pj22⋯Pj2n⋮⋮⋱⋮Pjn1Pj2n⋯Pjnn).
Step 2. Then with the method of weighted arithmetic average, the comprehensive possibility degree matrix will be calculated by
P=n∑j=1wjPj.
Step 3. Calculate the ranking results of alternatives Set δ=(δ1,δ2,⋯,δm)T is the ordering vector of matrix P, and 0≤δi≤1 with m∑i=1δi=1. Then the alternatives can be ranked with the values of δi. The higher value of δi, the better of the alternative, in which,
δi=1m(m∑k=1Pik+1)−0.5,
with this method, the teaching quality in Example 5.1 could be ranked as follows.
Firstly, calculating the possibility degree matrix with Step 1.
According to the above results, the ranking order is
A4≻A1≻A2≻A3,
which is the same as our proposed method.
6.2. Comparison with Zhao's method
We take the example of Zhao [39]. To evaluate four cities (Λ1: Nanchang, Λ2: Ganzhou, Λ3: Jiujiang, Λ4: Jingdezhen) intelligent transportation system. Taking four factors into consideration (CR1: Traffic data collection, CR2: Convenient transportation, CR3: Accident emergency handling capacity, CR4: Traffic signal equipment) to improve the rationality of evaluation. There we set w=(0.2,0.35,0.25,0.2)T are the weight of CRi(i=1,3,4), and the LTS is:
It has Λ3≻Λ1≻Λ2≻Λ4. Hence, the optimal intelligent transportation system is Λ3 (Jiujiang), which is the same as the answer in Zhao [39].
6.3. Comparison with Liu's method
Peide Liu et al. [17] proposed probabilistic linguistic Archimedean Muirhead mean operators to rank the alternatives. For the case of maximization profit problems, we make a comparison between our methods and Archimedean Muirhead Mean operators' methods.
For four potential projects Λi(i=1,2,3,4), directors need to choose the desirable one through four attributes (Λ1: Financial perspective, Λ2: Customers satisfaction, Λ3: Internal business process, Λ4: Learning and growth) to improve the rationality of evaluation. Suppose the weight of attributes are w=(0.2,0.3,0.3,0.2), and the LTS is:
To assure the calculated results more accurate, we do not normalize the probabilities. Using the function of g, we get the probabilistic hesitant fuzzy matrix and listed in Table 7.
According to the ranking order, the optimal project is Λ1, which is the same obtained by other extant decision making approach. This also shows the reliability and effectiveness of the method.
Although the proposed decision-making approach integrated the advantages of PLTSs and GMSM operators, there is a limitation of the proposed approach, that is, as the number of elements increases, the complexity of calculation will increase, corresponding.
7.
Conclusions
Classical MSM operators have been attracted great attention of many scholars and widely been used in the field of information fusion due to their biggest merit that they can reflect the relationship between multiple input arguments. On this basis, Wang generalized the traditional MSM operators and introduced the generalized MSM operators. On the other hand, PLTS, a new tool for describing uncertain decision information, can better reflect the actual decision-making problems such as the hesitation of decision-makers, the relative importance of linguistic variables. Combining the merits of PTLS and GMSM operators, PLGMSM and WPLGMSM based on ATN and ATC are proposed and their properties are also investigated. Meanwhile, some special situations are discussed when parameters take different values and the generators of ATN take different function. Besides, this proposed method is applied in teaching quality evaluation in universities, to evaluate an ideal classroom teaching in four alternatives. At last, several comparison analysis are adopted to ensure the validity of the decision-making results, which further verify the reasonability of our proposed decision-making approach.
In future studies, we will continue the current work in expanding and applying the current operators into other contexts. Also some novel MADM approaches will be developed to address some decision-making problems with probabilistic linguistic information. The proposed MADM problem could also be used to other complicated issues.
Acknowledgments
The research was funded by the General Program of Natural Funding of Sichuan Province (No: 2021JY018), Scientific Research Project of Neijiang Normal University (2022ZD10, 18TD08, 2021TD04.)
I. B. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets Syst., 20 (1986), 191–210. https://doi.org/10.1016/0165-0114(86)90077-1 doi: 10.1016/0165-0114(86)90077-1
L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Inf. Sci., 8 (1975), 199–249. https://doi.org/10.1016/0020-0255(75)90036-5 doi: 10.1016/0020-0255(75)90036-5
[7]
M. Yazdi, Linguistic methods under fuzzy information in system safety and reliability analysis, In: Studies in Fuzziness and Soft Computing, Springer Cham, 414 (2022). https://doi.org/10.1007/978-3-030-93352-4
[8]
H. Li, M. Yazdi, Advanced decision-making methods and applications in system safety and reliability problems, In: Studies in Systems, Decision and Control, Springer Cham, 211 (2022). https://doi.org/10.1007/978-3-031-07430-1
[9]
Z. S. Xu, Induced uncertain linguistic OWA operators applied to group decision making, Inform. Fusion, 7 (2006), 231–238. https://doi.org/10.1016/j.inffus.2004.06.005 doi: 10.1016/j.inffus.2004.06.005
[10]
R. M. Rodriguez, L. Martinez, F. Herrera, Hesitant fuzzy linguistic term sets for decision making, IEEE T. Fuzzy Syst., 20 (2012), 109–119. https://doi.org/10.1109/TFUZZ.2011.2170076 doi: 10.1109/TFUZZ.2011.2170076
[11]
H. Wang, Extended hesitant fuzzy linguistic term sets and their aggregation in group decision making, Int. J. Comput. Intell. Syst., 8 (2015), 14–33. https://doi.org/10.1080/18756891.2014.964010 doi: 10.1080/18756891.2014.964010
[12]
H. Wang, Z. S. Xu, X. J. Zeng, Linguistic terms with weakened hedges: A model for qualitative decision making under uncertainty, Inform. Sci., 433–434 (2018), 37–54. https://doi.org/10.1016/j.ins.2017.12.036 doi: 10.1016/j.ins.2017.12.036
[13]
S. Karimi, K. N. Papamichail, C. P. Holland, The effect of prior knowledge and decision-making style on the online purchase decision-making process: A typology of consumer shopping behaviour, Decis. Support Syst., 77 (2015), 137–147. https://doi.org/10.1016/j.dss.2015.06.004 doi: 10.1016/j.dss.2015.06.004
[14]
S. Heblich, A. Lameli, G. Riener, The effect of perceived regional accents on individual economic behavior: A lab experiment on linguistic performance, cognitive ratings and economic decisions, Plos One, 10 (2015), e0113475. https://doi.org/10.1371/journal.pone.0113475 doi: 10.1371/journal.pone.0113475
Q. Pang, H. Wang, Z. S. Xu, Probabilistic linguistic term sets in multi-attribute group decision making, Inform. Sci., 369 (2016), 128–143. https://doi.org/10.1016/j.ins.2016.06.021 doi: 10.1016/j.ins.2016.06.021
[17]
P. D. Liu, F. Teng, Multiple attribute group decision making methods based on some normal neutrosophic number Heronian Mean operators, J. Intell. Fuzzy Syst., 32 (2017), 2375–2391. https://doi.org/10.3233/JIFS-16345 doi: 10.3233/JIFS-16345
[18]
M. Qiyas, S. Abdullah, Y. Liu, M. Naeem, Multi-criteria decision support systems based on linguistic intuitionistic cubic fuzzy aggregation operators, J. Amb. Intel. Hum. Comp., 12 (2022), 8285–8303.
[19]
M. Qiyas, T. Madrar, S. Khan, S. Abdullah, T. Botmart, Decision support system based on fuzzy credibility Dombi aggregation operators and modified TOPSIS method, AIMS Mathematics, 10 (2022), 19057–19082. https://doi.org/10.3934/math.20221047 doi: 10.3934/math.20221047
[20]
M. Qiyas, N. Khan, M. Naeem, S. Abdullah, Intuitionistic fuzzy credibility Dombi aggregation operators and their application of railway train selection in Pakistan, AIMS Mathematics, 8 (2023), 6520–6542. https://doi.org/10.3934/math.2023329 doi: 10.3934/math.2023329
[21]
C. Maclaurin, A second letter to Martin Folkes, Esq.; concerning the roots of equations, with the demonstration of other rules in algebra, Phil. Transactions, 36 (1729), 59–96. https://doi.org/10.1098/rstl.1729.0011 doi: 10.1098/rstl.1729.0011
[22]
D. J. Yu, Hesitant fuzzy multi-criteria decision making methods based on Heronian mean, Technol. Econ. Dev. Eco., 23 (2017), 296–315. https://doi.org/10.3846/20294913.2015.1072755 doi: 10.3846/20294913.2015.1072755
[23]
J. D. Qin, X. W. Liu, Approaches to uncertain linguistic multiple attribute decision making based on dual Maclaurin symmetric mean, J. Intell. Fuzzy Syst., 29 (2015), 171–186. https://doi.org/10.3233/IFS-151584 doi: 10.3233/IFS-151584
[24]
J. D. Qin, X. W. Liu, W. Pedrycz, Hesitant fuzzy Maclaurin symmetric mean operators and its application to multiple-attribute decision making, Int. J. Fuzzy Syst., 17 (2015), 509–520.
[25]
R. F. Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proc. Edinb. Math. Soc., 21 (1902), 144–162. https://doi.org/10.1017/S001309150003460X doi: 10.1017/S001309150003460X
[26]
P. D. Liu, D. F. Li, Some muirhead mean operators for intuitionistic fuzzy numbers and their applications to group decision making, PloS One, 12 (2017), e0168767. https://doi.org/10.1371/journal.pone.0168767 doi: 10.1371/journal.pone.0168767
[27]
P. D. Liu, F. Teng, Some muirhead mean operators for probabilistic linguistic term sets and their applications to multiple attribute decision-making, Appl. Soft Comput. J., 68 (2018), 396–431 https://doi.org/10.1016/j.asoc.2018.03.027 doi: 10.1016/j.asoc.2018.03.027
[28]
G. Beliakov, S. James, J. Mordelová, T. Rückschlossová, R. R. Yager, Generalized Bonferroni mean operators in multi-criteria aggregation, Fuzzy Sets Syst., 161 (2010), 2227-2242. https://doi.org/10.1016/j.fss.2010.04.004 doi: 10.1016/j.fss.2010.04.004
[29]
Y. D. He, Z. He, H. Y. Chen, Intuitionistic fuzzy interaction Bonferroni means and its application to multiple attribute decision making, IEEE Trans. Cybernetics, 45 (2015), 116–128. https://doi.org/10.1109/TCYB.2014.2320910 doi: 10.1109/TCYB.2014.2320910
[30]
Z. S. Xu, R. R. Yager, Intuitionistic fuzzy Bonferroni means, IEEE Trans. Syst. Man Cy. B, 41 (2011), 568–578. https://doi.org/10.1109/TSMCB.2010.2072918 doi: 10.1109/TSMCB.2010.2072918
[31]
P. D. Liu, S. M. Chen, Group decision making based on Heronian aggregation operators of intuitionistic fuzzy numbers, IEEE Trans. Cybernetics, 47 (2016), 2514–2530. https://doi.org/10.1109/TCYB.2016. 2634599 doi: 10.1109/TCYB.2016.2634599
[32]
J. D. Qin, X. W. Liu, An approach to intuitionistic fuzzy multiple attribute decision making based on Maclaurin symmetric mean operators, J. Intell. Fuzzy Syst., 27 (2014), 2177–2190.
[33]
P. D. Liu, X. H. Zhang, Some Maclaurin symmetric mean operators for single-valued trapezoidal neutrosophic numbers and their applications to group decision making, Int. J. Fuzzy Syst., 20 (2018), 45–61 https://doi.org/10.1007/s40815-017-0335-9 doi: 10.1007/s40815-017-0335-9
[34]
Y. B. Ju, X. Y. Liu, D. W. Ju, Some new intuitionistic linguistic aggregation operators based on Maclaurin symmetric mean and their applications to multiple attribute group decision making, Soft Comput., 20 (2016), 4521–4548.
[35]
D. Detemple, J. Robertson, On generalized symmetric means of two variables, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 634 (1979), 236–238 https://doi:10.1002/anie.200704684 doi: 10.1002/anie.200704684
[36]
F. Herrera, E. Herrera-Viedma, Linguistic decision analysis: steps for solving decision problems under linguistic information, Fuzzy Sets Syst., 115 (2000), 67–82. https://doi.org/10.1016/S0165-0114(99)00024-X doi: 10.1016/S0165-0114(99)00024-X
[37]
X. Gou, Z. Xu, H. Liao, Multiple criteria decision making based on Bonferroni means with hesitant fuzzy linguistic information, Soft Comput., 21 (2017), 6515–6529.
K. X. Zhao, X. J. Huang, Probabilistic linguistic hesitant fuzzy set and its application in multiple attribute decision-making, J. Nanchang Univ. (Natural Sci.), 41 (2017), 511–518. https://doi.org/10.3969/j.issn.1006-0464.2017.06.001 doi: 10.3969/j.issn.1006-0464.2017.06.001
[40]
B. Fang, B. Han, D. Y. Xie, Probabilistic linguistic multi-attribute decision-making method based on possibility degree matrix, Control Decis., 37 (2022), 2149–2156. https://doi.org/10.13195/j.kzyjc.2021.0350 doi: 10.13195/j.kzyjc.2021.0350
This article has been cited by:
1.
Shizhou Weng, Zhengwei Huang, Yuejin Lv,
Probability numbers for multi-attribute decision-making,
2024,
46,
10641246,
6109,
10.3233/JIFS-223565
2.
Zeynep Tuğçe Kalender,
Olasılıklı dil terimi kümeleri yaklaşımı kullanılarak akıllı ev teknolojilerinin benimsenmesinde tüketici dinamiklerinin incelenmesi,
2025,
40,
1300-1884,
1099,
10.17341/gazimmfd.1396803
3.
Jing Guo, Xianjun Zhu, Wenfeng Li, Hui Li,
A noval approach for dual probabilistic linguistic multi-criteria decision making based on prioritized Maclaurin symmetric mean operators and VIKOR,
2025,
48,
1064-1246,
447,
10.3233/JIFS-224232