Trapezoidal Linguistic Cubic Fuzzy TOPSIS Method and Application in a Group Decision Making Program

1 Introduction

Multiple attribute decision making (MADM) issues are the essential research used in choice hypotheses. As the question assets are fuzzy and uncertain, the complicated properties in the choice issues are not consistently expressed as fresh numbers and some of them additionally can be discerned by fuzzy numbers, e.g. interval number, linguistic variable, IFN and so on. In such conditions, the acquisition office can exhibit an essential part of the cost decrease and provider determination is a standout amongst the most active elements of achieveing the best administration [13]. In a typical multi criteria decision making (MCDM) problem, different and normally contrary criteria are quickly set up to settle on a choice. A wide-ranging review and organization of the MCDM approach for vendor selection were carried out in [5]. In reality, the rating models of choices, and also rank weights of criteria, regularly have different sorts of ambiguity, imprecision or subjectiveness and one cannot always utilize the conventional basic leadership frameworks for these issues. Typically, no inimitable optimal solution exists for such problems and it is necessary to use a decision maker’s performance to differentiate between solutions. MCDM has been an active area of research since the 1970s. Different methods have been offered by many researchers, including the analytic hierarchy process (AHP), the technique for order preference by similarity to ideal solution (TOPSIS) and MCDM. One of the most broadly used multi-criteria decision exploration methods is the TOPSIS method, which was proposed by Hwang and Yoon in 1981 [15], and extended by Yoon in 1987 [45], as well as by Hwang et al. in 1993 [16]. In the TOPSIS method, the optimal alternative is nearest to the positive ideal solution (PIS) and the farthest from the negative ideal solution (NIS). It controls inaccurate and imprecise data, mostly modeling social judgments. Numerous fuzzy TOPSIS methods and applications have been established since the 1990s, e.g. for supplier selection [35], [44], finance [2], [29], the power industry [48], [50], and negotiation problems [28]. In our study, we discuss a fuzzy extension of the TOPSIS method as presented by Chen [3]. The TOPSIS developed by Hwang and Yoon is a practical technique to solve MCDM problems.

Xu et al. [41] proposed a new aggregation method to solve a heterogeneous multiple attribute group decision-making (MAGDM) problem which involves real numbers, interval numbers, triangular fuzzy numbers (TFNs), trapezoidal fuzzy numbers (TrFNs), linguistic values and Atanassov’s intuitionistic fuzzy numbers (AIFNs). Li et al. [19] proposed a method which can effectively avoid the failure caused by the use of inconsistent decision information and provides a decision-making idea for the case where “the truth be held in minority”. Krishankumar et al. [18] presented a new two-tier decision-making framework with linguistic preferences for scientific decision making. Dwivedi et al. [7] proposed a generalized fuzzy TOPSIS method as a versatile evaluation model. The model is suitable for different types of fuzzy or interval-valued numbers, with or without subjective weights of criteria being defined by evaluators. Li et al. [20] developed a prospect value determination method based on multiple reference points (mRPs) under a trapezoidal intuitionistic fuzzy environment.

Furthermore, Wang and Li [34] proposed intuitionistic linguistic sets, intuitionistic linguistic numbers, intuitionistic two semantics and the Hamming distance between two intuitionistic two-semantics, and rank the alternatives by calculating the comprehensive membership degree to the ideal solution for each alternative. So, Chen and Liu [4] proposed the linguistic intuitionistic fuzzy numbers (LIFNs) which can combine linguistic variable (LV) and intuitionistic fuzzy numbers (IFNs). Li et al. [21] proposed some new operational laws including subtraction, division, then the entropy of LIFNs and the extended VIKOR method were proposed to deal with uncertain information with LIFNs. Liu and Wang [26] proposed some improved linguistic intuitionistic operational laws to overcome the weaknesses in linguistic operations. In addition, some aggregation operators for LIFNs were developed for different goals [23], [24], [25]. The intuitionistic linguistic variables are more correct for fast fuzzy data than the uncertain linguistic variables. The linguistic variables are relaxed to deal with qualitative data, in general, of necessity we allocate a linguistic set which comprises some linguistic terms, but in the applied application, we might have the choice for a linguistic term from the linguistic set to accurately express the assessment data for an impartial assessment. We can complete it using the indeterminate linguistic variables, but this is not accurate. We can then use the intuitionistic linguistic variables to increase the accuracy, by selecting a linguistic term, which is neighboring the evaluation data, from the linguistic set. Then we give a membership degree and a non-membership degree to this linguistic term. This creates an intuitionistic linguistic number. For example, we can appraise the appearence of a car using the linguistic set S = (s₁, s₂, s₃, s₄, s₅, s₆, s₇) = {extremely poor, very poor, poor, slightly poor, fair, slightly good, good, very good, extremely good}. We may reflect the performance estimate result as better than “good” (s₅) and lower than “very good” (s₆), however, we can use the uncertain linguistic number [s₅, s₆] to express this evaluation result, but this is not accurate, because it merely provides a range. In this situation, we can use an intuitionistic linguistic number to evaluate the car; firstly, we can give a linguistic term (s₆), then we give the membership degree (0.8) and non-membership degree (0) to s₆. So, the intuitionistic linguistic number is s₆, (0.8, 0.0). Of course, if we use a linguistic term (s₅), it will not fully fast the assessment data, because the membership degree will be 1 and non-membership degree is 0. If we use linguistic term (s₇), the membership degree to s₇ may be 0.7 and non-membership degree to s₇ may be 0.1, etc. So, we think the intuitionistic linguistic variables can express the assessment data more accurate than the uncertain linguistic variables. Yue [47] developed a new methodology for GDM problems in an intuitionistic fuzzy environment. In this model, the weights of decision makers are determined by using an extended TOPSIS technique. Ou et al. [27] proposed the linguistic intuitionistic fuzzy set TOPSIS method for LMCDMs, compared with the traditional TOPSIS method, which is different from the positive ideal solution, the negative ideal solution and the relative closeness degrees of alternatives, in addition, we designed an algorithm to finish the linguistic intuitionistic fuzzy set TOPSIS method for LMCDMs. Ren et al. [12] presented the positive (optimistic) and negative (pessimistic) information of each criterion provided by each decision maker and aggregate these by using weights of decision makers to obtain the hesitant fuzzy linguistic positive and negative ideal solutions. Pei et al. [51] developed the fuzzy linguistic multiset TOPSIS method for linguistic decision making, the method mainly consists of transformation, aggregation and exploitation phases.

Xu [38] developed a system based on the uncertain linguistic ordered weighted averaging (ULOWA) and the uncertain linguistic hybrid aggregation (ULHA) operators. Xu et al. [39] developed some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator.

Dong et al. [6] introduced overcoming the drawback of using the single real number to represent membership degree and non-membership degree for intuitionistic linguistic set (ILS) and the concept of the interval-valued intuitionistic linguistic set (IVILS) through representing the membership degree and non-membership degree with intervals for ILS. Wan et al. [30] developed the construct of a novel bi-objective interval-valued intuitionistic fuzzy (IVIF) mathematical programming of minimizing the inconsistency index and meanwhile maximizing the consistency index, which solved by the technically developed linear goal programming approach. Wan et al. [31] developed the fuzzy mathematical programming method for solving heterogeneous multiattribute decision-making problems based on the linear programming technique for multidimensional analysis of preference. Wan et al. [33] derived the PIS, NIS and the criteria weights simultaneously, a new four-objective hesitant fuzzy mathematical programming model by minimizing the hesitant fuzzy positive ideal group inconsistency index (HFPGICI) and hesitant fuzzy negative ideal group inconsistency index (HFNGICI) as well as maximizing the HFPGCI and HFNGCI. Wan et al. [32] proposed a new general method to aggregate the attribute value vector into interval-valued intuitionistic fuzzy numbers (IVIFNs) under a heterogeneous MAGDM environment utilizing the relative closeness in the technique for order preference by its similarity to the ideal solution (TOPSIS). Xu et al. [40] developed as crisp numbers, intervals, intuitionistic fuzzy sets (IFSs), linguistic variables and hesitant fuzzy sets (HFSs).

Cubic sets introduced by Jun et al. [17], are the generalizations of fuzzy sets and intuitionistic fuzzy sets, in which there are two representations, one is used for the degree of membership and other is used for the degree of non-membership. The membership function is held in the form of an interval while non-membership is thought to be the normal fuzzy set. A fuzzy number is a quantity whose value is imprecise, rather than exact as is the case with “ordinary” (single-valued) numbers.

Any fuzzy number can be thought of as a function whose domain is a specified set (usually the set of real numbers), and whose range is the span of non-negative real numbers between, and including, 0 and 1000. Each numerical value in the domain is assigned a specific “grade of membership” where 0 represents the smallest possible grade, and 1000 is the largest possible grade. The technique of positive ideal and negative ideal points easily produces satisfactory results which are composed of the overall best criteria values and overall worst criteria values attainable. Linguistic and subjective evaluations take place in questionnaire form. Each linguistic variable has its own numerical value in the predefined scale. In classical trapezoidal linguistic cubic fuzzy numbers, these numerical values are exact numbers whereas in the fuzzy AHP method they are intervals between two numbers with the most likely value. Due to the nature of human beings, linguistic values can change from person to person. In these circumstances, taking the fuzziness into account will provide less risky decisions.

We are interested in GDM situations defined in linguistic contexts, that is, it is assumed that decision makers use linguistic values in the pairwise comparison of alternatives in preference relations. For instance, linguistic values as being “Low” or “High” could be used. It should be pointed out that the linguistic values may be organized in a linear fashion, as there is an apparent linear order among them. In any case, a quantification of the linguistic values is required in order to use them. Finally, we should point out that a joint treatment of the linguistic values, coming from the decision makers involved in the GDM problem, is considered here. On the one hand, this allows us to deal with the linguistic values in a unified fashion. On the other hand, it allows us to reconcile the semantics of the linguistic values in such a way that the individual consistencies are made comparable and, therefore, could be aggregated to arrive at the joint view at the optimization criterion.

Fahmi et al. [11] defined the triangular cubic fuzzy number and operational laws. Fahmi et al. [10] defined the weighted average operator of triangular cubic fuzzy numbers and the Hamming distance of the trapezoidal cubic fuzzy numbers (TCFN). In [8], Fahmi et al. defined the aggregation operators for triangular cubic linguistic hesitant fuzzy sets which includes the generalized triangular cubic linguistic hesitant fuzzy weighted averaging (GTCLHFWA) operator, the generalized triangular cubic linguistic hesitant fuzzy weighted geometric (GTCHFWG) operator, the generalized triangular cubic linguistic hesitant fuzzy ordered weighted average (GTCLHFOWA) operator, the generalized triangular cubic linguistic hesitant fuzzy ordered weighted geometric (GTCLHFOWG) operator, the generalized triangular cubic linguistic hesitant fuzzy hybrid averaging (GTCLHFHA) operator and the generalized triangular cubic linguistic hesitant fuzzy hybrid geometric (GTCLHFHG) operator. Fahmi et al. [9] proposed the cubic TOPSIS method and cubic gray relation analysis (GRA) method. Finally, the proposed method is used for selection in sol-gel synthesis of titanium carbide nanopowders.

As we have discussed earlier that cubic sets are the generalization of intuitionistic fuzzy sets and a powerful tool to deal with fuzziness. Also, trapezoidal linguistic intuitionistic fuzzy sets are suitable to deal with fuzziness. However, there may be a situation where the decision maker may provide the degree of membership and nonmembership of a particular attribute in such a way that the membership degree is a trapezoidal linguistic interval fuzzy number and the non-membership degree is a trapezoidal linguistic fuzzy number. Therefore, to overcome this shortcoming we generalize the concept of trapezoidal linguistic intuitionistic fuzzy sets and introduce the concept of trapezoidal linguistic cubic fuzzy sets which are very suitable to be used for depicting uncertain or fuzzy information. If we take only one element in the membership degree of the trapezoidal linguistic cubic fuzzy number, i.e. instead of an interval we take a fuzzy number, than we get trapezoidal linguistic intuitionistic fuzzy numbers, similarly if we take the membership degree as a fuzzy number and the non-membership degree as beingequal to zero, than we get trapezoidal linguistic fuzzy numbers. Thus, motivated by the idea proposed by Li et al. [19], in this paper we first proposed a trapezoidal linguistic cubic fuzzy weighted geometric (TrLCFWG) operator and a trapezoidal linguistic cubic fuzzy TOPSIS method for a multi-attribute decision-making problem.

Despite having much related literature on the problem under consideration, the following aspects related to trapezoidal cubic linguistic fuzzy sets and their aggregation operators motivated the researchers to carry out an in-depth inquiry into the current study.

1. The main advantages of the proposed operators are that these aggregation operators provided more accurate and precious results compared to the mentioned operators.

2. We generalized the concept of the trapezoidal linguistic cubic fuzzy number, the trapezoidal linguistic intuitionistic fuzzy sets and introduce the concept of the trapezoidal linguistic cubic fuzzy number. If we take only one element in the membership degree of the trapezoidal linguistic cubic fuzzy number, i.e. instead of interval we take a fuzzy number, then we get trapezoidal linguistic intuitionistic fuzzy numbers, similarly if we take the membership degree as a fuzzy number and non-membership degree as being equal to zero, then we get trapezoidal linguistic fuzzy numbers.

3. The objectives of the study include:

   Proposing a trapezoidal linguistic cubic fuzzy number, operational laws, Hamming distance.

   Establishing a MADM program approach based on the trapezoidal linguistic cubic fuzzy TOPSIS method.

   Providing illustrative examples of the MADM program.

4. In order to testify to the application of the developed method, we apply the trapezoidal linguistic cubic fuzzy number in the decision making.

5. The initial decision matrix is composed of LVs. In order to fully consider the randomness and ambiguity of the linguistic term, we convert LVs into the trapezoidal linguistic cubic fuzzy number, and the decision matrix is transformed into the trapezoidal linguistic cubic fuzzy decision matrix.

6. The operator can fully express the uncertainty of the qualitative concept and trapezoidal linguistic cubic fuzzy operators can capture the interdependencies among any multiple inputs or attributes by a variable parameter. The aggregation operators can take into account the importance of the attributes weights. Nevertheless, sometimes, for some MAGDM problems, the weights of the attributes are important factors for the decision process.

In Section 2, we first introduce some basic definitions of the fuzzy set, cubic set, trapezoidal linguistic cubic fuzzy number, and Hamming distance. In Section 3, we propose a trapezoidal linguistic cubic fuzzy TOPSIS method and a numerical example. In Section 4, we compare this with a different method. The paper is concluded in Section 5.

2 Preliminaries

1. s1⊕s2=[sα1,sβ1]⊕[sα2,sβ2]=[sα1⊕sα2,sβ1⊕sβ2]=[sα1+α2,sβ1+β2],

2. λs=λ[sα,sβ]=[λsα,λsβ]=[sλα,sλβ], where λ∈[0,1];

3. s1⊕s2=s2⊕s1,

4. λ(s1⊕s2)=λs1⊕λs2, where λ∈[0,1];

5. (λ1+λ2)s=λ1s+λ2s, where λ,1λ2∈[0,1].

3 Trapezoidal Linguistic Cubic Fuzzy TOPSIS Method

The trapezoidal linguistic cubic fuzzy TOPSIS is a decision making technique. It is a goal based approach for finding the alternative that is closest to the ideal solution.

In this section, we apply a trapezoidal cubic fuzzy set to a linguistic TOPSIS method. We define a new extension of the trapezoidal linguistic cubic fuzzy TOPSIS method by using a trapezoidal cubic fuzzy set.

Step 1: Suppose that a trapezoidal linguistic cubic fuzzy TOPSIS method decision-making problem under multiple attributes has m students and n decision attributes. The framework of the trapezoidal linguistic cubic decision matric can be exhibit as follows:

Step 2: Construct a normalized trapezoidal linguistic cubic fuzzy TOPSIS decision matrix R = [β_ij]. The normalized value β_ij is calculated as:

Step 3: Construct the weighted normalized trapezoidal linguistic cubic fuzzy TOPSIS decision matrix by multiplying the normalized trapezoidal linguistic cubic decision matrix by its associated weights. The weight vector W=(w1,w2,…wn) collected of the isolated weights wj(j=1,2,3,…n) for each attribute C_j satisfying ∑j=1nWj=1. The weighted normalized value is deliberate by Equation (3)

where 0 ≤ B_j ≤ 1 i = 1, 2, 3…., m and j = 1, 2, 3…, n.

Step 4: Determine the positive ideal solution (α+) and negative ideal solution (α−). The trapezoidal linguistic cubic fuzzy TOPSIS method positive-ideal solution (TrLCFPIS, α⁺) and the trapezoidal linguistic cubic fuzzy negative-ideal solution (TrLCFNIS, α⁻) are shown as

Step 5: Calculate the separation measures, using the n-dimensional Euclidean distance. The separation of each candidate from the trapezoidal linguistic cubic positive ideal solution qi∗⟨[B−,B+],η⟩ is given as

The separation of each candidate from the trapezoidal linguistic cubic negative ideal solution qi−⟨[B−,B+],η⟩ is given as

Step 6: Calculate the relative closeness to the ideal solution. This progression comprehends the relative closeness to an ideal solution by the equation

3.1 Decision Support Algorithm

In this subsection, we present the MAGDM approach based on an extended trapezoidal cubic linguistic fuzzy TOPSIS method using TCFN. In the MAGDM, decision makers should firstly determine the preferences of alternatives on the criteria. Based on these preference values, decision makers can select the best alternative. Because the information available for decision makers is vague and imprecise under uncertain environments, the decision makers cannot easily use an exact value to estimate their preferences.

Referring to Figure 1, the decision support algorithm will be as follows:

Figure 1:

Flow Chart of the Extended Trapezoidal Linguistic Cubic Fuzzy TOPSIS Method.

1. Suppose that a trapezoidal linguistic cubic fuzzy TOPSIS method decision-making problem under multiple attributes has m students and n decision attributes. After all the TrLCFN calculation provided by decision-makers of Eq. (1).

2. Determining normalized value r_ij by using Eq. (2)

3. The weight vector W=(w1,w2,…wn) collected of the isolated weights wj(j=1,2,3,…n) for each attribute C_j satisfying ∑j=1nWj=1. The weighted normalized value is calculate of criteria using Eq. (3)

4. Identifying the trapezoidal linguistic cubic fuzzy TOPSIS method positive-ideal solution (TrLCFPIS, α⁺) and the trapezoidal linguistic cubic fuzzy negative-ideal solution (TrLCFNIS, α⁻) by using Eqs. (4)–(5)

5. Calculate distance from the trapezoidal linguistic cubic positive ideal solution qi+⟨[B−,B+],η⟩ and trapezoidal linguistic cubic negative ideal solution qi−⟨[B−,B+],η⟩ separation measures by using Eqs. (6)–(7)

6. For each alternative the relative closeness to the ideal solution Z_i is computed by Eq. (8).

7. Ranking the alternatives according to the relative closeness coefficient.