Abstract
The purpose of this paper was to develop a likelihood-based assignment method based on interval type-2 fuzzy sets and apply it to decision-making problems involving multiple criteria evaluation and the ranking/selection of alternatives. The linear assignment method is a well-known outranking method in the field of multiple criteria decision analysis. The theory of interval type-2 fuzzy sets is useful for addressing the uncertainty and imprecision associated with a subjective environment. In this paper, the key feature of the proposed method is the incorporation of the extended concept of likelihoods of fuzzy preference relations between interval type-2 trapezoidal fuzzy numbers into the main structure of the linear assignment methodology. An effective ranking procedure using the optimal membership degree determination method is proposed to determine criterion-wise preference rankings of the alternatives. The proposed method establishes the novel concepts of an (adjusted) rank frequency matrix and an (adjusted) rank contribution matrix to combine the relative performances of the alternatives in terms of each criterion. Based on a signed distance comparison approach, this paper constructs a likelihood-based assignment model to obtain an aggregate ranking of the alternatives that is in the closest agreement with the criterion-wise preferences of the alternatives. The feasibility and applicability of the proposed method are illustrated with two practical multiple criteria decision-making applications concerning the selection of landfill sites and the selection of treatment options. Finally, a comparative analysis with other relevant methods is conducted to validate the effectiveness and advantages of the current methods in decision aiding.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
1 Introduction
The concept of type-2 fuzzy sets (T2 FSs), initially introduced by Zadeh [67], is an extension of a type-1 fuzzy set (T1 FS) in which the membership function falls into a fuzzy set in the interval [0, 1] [45, 70]. Type-2 fuzzy logic has a great ability to handle higher degrees of uncertainty [50]. However, the computational complexity of T2 FSs is very high, which makes it very difficult to employ them in practical applications [16, 29, 70, 73]. Therefore, considerable concern has arisen over interval type-2 fuzzy sets (IT2 FSs) as a special case of T2 FSs in practical fields [1, 34, 70]. The computations associated with IT2 FSs are manageable because their membership values take the form of crisp intervals [17]. Additionally, IT2 FSs can model the effects of uncertainties [53, 58] and handle imprecision and imperfect information in real-world applications [13, 14, 55, 72]; thus, they are gaining increasing popularity. At present, the interval type-2 fuzzy logic theory is in a mature state and has many areas of application [31].
Decision information is often determined according to decision-makers’ opinions or assessments in multiple criteria decision-making situations [65]. Uncertainty is common in many real-life decision-making problems [46, 47] because decision-makers are not always certain about their given decision or preference information, and they often have some degree of uncertainty [48]. Accordingly, IT2 FSs have been successfully applied to tackle the issue of multiple criteria decision analysis (MCDA) in recent years [8, 16–18, 26, 30, 33, 39, 55, 59, 61, 69]. In particular, considerable studies have been conducted on fuzzy decision-making methods with interval type-2 trapezoidal fuzzy numbers (IT2 TrFNs) [3, 13–17, 20, 60, 62, 71, 72]. Furthermore, IT2 TrFNs can efficiently express linguistic evaluations or assessments by objectively transforming them into numerical variables [13, 72].
It is common for the evaluations of alternatives and the relative preference of criterion importance to be guided by decision-makers’ subjective judgments in real-world situations. To address linguistic or numerical uncertainties associated with a subjective environment, the evaluative ratings of alternatives and the importance weights of criteria used in MCDA can be appropriately expressed as IT2 TrFNs. Most existing MCDA methodologies within the IT2 TrFN environment are characterized as scoring or compromising methods; there has thus far been relatively little research on outranking methods [16]. The present study is an attempt to supplement existing MCDA models and methods based on IT2 TrFNs. Because the extended versions of outranking methods have not been thoroughly investigated in the context of IT2 TrFNs [16], this paper is primarily concerned with the development of a new outranking model using IT2 TrFNs based on the main structure of the linear assignment methodology.
The linear assignment method, introduced by Bernardo and Blin [6], is a well-known outranking model for solving multiple criteria decision-making problems. Based on a set of criterion-wise rankings and a set of criterion weights, the linear assignment method combines the criterion-wise rankings into an overall ranking that reaches a best compromise among all criterion-wise rankings in a coherent and non-heuristic framework [11]. The linear assignment methodology has been enriched by several valuable developments, such as a fuzzy linear assignment approach [49]; a combined model using the analytic hierarchy process (AHP), the technique for order preference by similarity to an ideal solution (TOPSIS), and the linear assignment method [2]; a linear assignment method for ranking materials of engineering components [38]; a fuzzy linear assignment method for an initial order [4]; an interactive fuzzy linear assignment method [5]; a fuzzy group linear assignment method for ranking electronic business process management best practices based on the perspectives of a balanced scorecard [68]; a linear assignment model for group decision making with uncertain preference information [66]; a nonlinear assignment-based model with interval-valued intuitionistic fuzzy sets under incomplete and/or inconsistent preference information [11]; a wide range of sensitivity analyses for the linear assignment method [56]; a modified assignment method considering differences among criteria values or various stochastic elements [7]; and an interval type-2 fuzzy linear assignment method based on signed distances [13].
An increasing number of theoretical developments and applications have been conducted to improve the linear assignment methodology in the decision-making field. However, relatively few studies have investigated the linear assignment methodology within the interval type-2 fuzzy environment. At present, only Chen [13] used interval type-2 fuzzy numbers to capture imprecise or uncertain decision information in the fields that require the MCDA and developed the interval type-2 fuzzy linear assignment model based on the comparisons using signed distances. Moreover, Chen illustrated and discussed the proposed method by applying it to a case in which a landfill site is selected. Chen has demonstrated the usefulness and effectiveness of extending the linear assignment method to the interval type-2 fuzzy context. On the whole, within the literature on the linear assignment methodology, comparatively little research has focused on the development of the assignment models based on IT2 TrFNs in the interval type-2 fuzzy context. Therefore, this paper attempts to promote a novel and useful assignment method to handle MCDA problems based on IT2 TrFNs for the sake of capturing more imprecise or uncertain decision information for practical applications.
The purpose of this paper is to develop a likelihood-based assignment method for addressing MCDA problems within the decision environment of IT2 TrFNs. The proposed method features prominently in incorporating the extended concept of likelihoods of fuzzy preference relations between IT2 TrFNs into the core structure of the linear assignment methods. As a whole, the proposed method makes four main contributions to the linear assignment methodology. First, the extended concepts of lower likelihoods, upper likelihoods, and mean likelihoods regarding fuzzy preference relations between IT2 TrFNs are incorporated into the linear assignment structure. It follows that processing sophisticated IT2 TrFN data becomes simple and easy via the comparison approach using the likelihoods between IT2 TrFN evaluative ratings. Second, based on the optimal membership degree determination method, an effective ranking procedure is developed to determine criterion-wise preference rankings of the alternatives. Third, an innovative method using the concepts of an (adjusted) rank frequency matrix and an (adjusted) rank contribution matrix is proposed to establish a new structure of the likelihood-based assignment method. Fourth, a signed distances approach is utilized to construct a simple linear assignment model to acquire an overall preference ranking of the alternatives. The proposed method is new and unique compared with the linear assignment methodology that has previously been developed. Furthermore, to fully utilize information contained in the evaluative ratings and fuzzy preference relations between IT2 TrFNs, this paper determines the (adjusted) rank contribution matrix based on the IT2 TrFN importance weights and the optimal degrees of membership for making the information content complementary.
As mentioned previously, the interval type-2 fuzzy linear assignment model developed by Chen [13] is currently the most relevant linear assignment methodology for addressing MCDA problems within the IT2 TrFN environment. Chen [13] examined the feasibility and applicability of her proposed methods with a practical MCDA problem of landfill site selection. To maintain methodological relevance with Chen [13], this paper chooses the same problem of landfill site selection to not only illustrate the proposed method but also to allow for comparative discussion. Alternately, the linear assignment methodology belongs to outranking models in the decision-making field. Thus, this paper also conducts comparative analyses with other well-known outranking methods, including the method of elimination and choice expressing reality (ELECTRE) and the qualitative flexible multiple criteria method (QUALIFLEX), to examine the applicability and advantages of the proposed method.
Specifically, to demonstrate the feasibility and effectiveness of the proposed likelihood-based assignment method, we present an illustrative application to the selection problem of landfill sites and compare with the interval type-2 fuzzy linear assignment method based on signed distances [13]. Furthermore, the proposed method is applied to a medical decision-making problem that addresses the selection of treatment options. Comparative analyses are also provided to examine the validity and advantages of the proposed method relative to the existing outranking methods within the interval type-2 fuzzy environment, including the extended QUALIFLEX method using a signed distance-based approach [20] and the extended ELECTRE method [20]. Finally, this paper conducts a comparative discussion with the widely used simple additive weighting (SAW) method to further examine the distinct advantage of the proposed method.
This paper is organized as follows: Sect. 2 reviews the concept of IT2 FSs and IT2 TrFNs; Sect. 3 describes an MCDA problem within an environment of IT2 TrFNs and presents the concept of likelihoods of fuzzy preference relations between IT2 TrFNs; Sect. 4 develops a likelihood-based assignment method for MCDA involving the multiple criteria evaluation/selection of alternatives; Sect. 5 demonstrates the feasibility and the applicability of the proposed methodology using two practical MCDA applications concerning the selection of landfill sites and the selection of treatment options and conducts comparisons with other relevant methods, and Sect. 6 presents the conclusions.
2 Preliminaries
This section reviews selected relevant definitions of IT2 FSs [51, 52, 54, 57] and IT2 TrFNs [9, 13–17, 62] that are used throughout the paper.
Definition 1
[51, 54, 57] Let X be a crisp set. A mapping A: X → [0, 1][0, 1] is called a T2 FS defined on the universe of discourse X, and it is denoted by:
where x indicates a primary variable, and \( A(x) \) denotes the fuzzy membership value of x in A. Note that \( A(x) \) is also called a secondary membership function or a secondary set. Additionally, \( Y_{x} (u) \) denotes the secondary membership (grade), where u indicates the primary membership (grade) of x. \( J_{x} \subseteq [0,1] \) denotes the domain of \( Y_{x} (u) \) and represents the primary membership values of \( x \in X \).
Definition 2
[51, 54] Let A be a T2 FS on X. When \( Y_{x} (u) = 1 \) for all \( u \in J_{x} \), A is known as an IT2 FS on X and can be represented by:
where \( A(x) \) is referred to as an interval membership value.
Definition 3
[51, 52, 54] Let A be an IT2 FS on X. The IT2 FS A can be fully characterized by its footprint of uncertainty (FOU), which is defined as the union of all primary memberships as follows:
Moreover, FOU(A) is a bounded region that represents the uncertainty associated with the membership grades of A.
Definition 4
[36, 51, 73, 74] Let a lower membership function (LMF) and an upper membership function (UMF) denote two type-1 membership functions that are bounds for the FOU(A) of an IT2 FS A on X. Let two T1 FSs, \( A^{ - } \): X → [0, 1] and A +: X → [0, 1], be the lower and upper fuzzy sets, respectively, with respect to A. The LMF \( A^{ - } (x) \) and the UMF \( A^{ + } (x) \) are associated with the lower bound \( FOU^{ - } (A) \) and the upper bound \( FOU^{ + } (A) \), respectively, of FOU(A), and they are defined as follows:
where \( 0 \le A^{ - } (x) \le A^{ + } (x) \le 1 \) for all \( x \in X \).
Definition 5
[9, 13–17, 62] Let \( A^{ - } ( = (a_{1}^{ - } ,a_{2}^{ - } ,a_{3}^{ - } ,a_{4}^{ - } ;h_{A}^{ - } )) \) and \( A^{ + } ( = (a_{1}^{ + } ,a_{2}^{ + } ,a_{3}^{ + } ,a_{4}^{ + } ;h_{A}^{ + } )) \) be the lower and upper trapezoidal fuzzy numbers, respectively, with respect to an IT2 FS A on X, where \( a_{1}^{ - } \le a_{2}^{ - } \le a_{3}^{ - } \le a_{4}^{ - } \), \( a_{1}^{ + } \le a_{2}^{ + } \le a_{3}^{ + } \le a_{4}^{ + } \), \( 0 \le h_{A}^{ - } \le h_{A}^{ + } \le 1 \), \( a_{1}^{ + } \le a_{1}^{ - } \), \( a_{4}^{ - } \le a_{4}^{ + } \), and \( A^{ - } \subseteq A^{ + } \) (if and only if \( \forall x \in X \), \( A^{ - } (x) \le A^{ + } (x) \)). The LMF \( A^{ - } (x) \) and the UMF \( A^{ + } (x) \) are lower and upper bounds, respectively, for the FOU(A) of A; they are defined as follows:
Then, A is an IT2 TrFN on X (see Fig. 1) and can be represented by:
3 Decision context based on IT2 TrFNs
This section first formulates an MCDA problem within a decision environment based on IT2 TrFNs and then presents the concept of likelihoods of fuzzy preference relations between IT2 TrFNs.
3.1 Decision environment of IT2 TrFNs
This subsection formulates an MCDA problem using IT2 TrFNs. Define \( Z = \) \( \left\{ {z_{1} ,z_{2} , \ldots ,z_{m} } \right\} \) as a set of decision alternatives, where m is the number of alternatives. Define \( C = \) \( \left\{ {c_{1} ,c_{2} , \ldots ,c_{n} } \right\} \) as a criterion set that contains the criteria by which the alternative performances are measured, where n is the number of criteria. Let x j denote the value corresponding to the criterion c j , where \( x_{j} \in X \) (i.e., the universe of discourse) for \( j = 1,2, \ldots ,n \). The set C can be generally divided into two sets, C I and C II, where C I denotes a collection of benefit criteria (i.e., larger values of x j indicate a greater preference) and C II denotes a collection of cost criteria (i.e., smaller values of x j indicate a greater preference). Note that \( C_{\text{I}} \cap C_{\text{II}} = {\emptyset } \) and \( C_{\text{I}} \cup C_{\text{II}} = C \).
The IT2 TrFN data required in an MCDA problem can be appropriately established by employing the linguistic scales with the corresponding IT2 TrFNs. Decision-makers often express their judgments using linguistic variables in many practical situations. The decision-maker’s linguistic responses can be appropriately represented by IT2 TrFNs using a specific linguistic rating system, such as three-point linguistic scales [22, 35, 37, 69], four-point linguistic scales [22], five-point linguistic scales [22, 37, 55], seven-point linguistic scales [23, 27, 33, 37, 61, 72], and nine-point linguistic scales [9, 10, 12, 14, 15, 20, 21, 63]. Most of the IT2 TrFNs corresponding to linguistic terms are nonnegative [9, 10, 12, 14, 15, 20, 21, 23, 55, 61, 63, 72]. Therefore, this paper also constructs an MCDA problem using nonnegative IT2 TrFNs.
Consider an MCDA problem wherein the ratings of alternative evaluations and the importance weights of criteria are expressed as nonnegative IT2 TrFNs. Let a nonnegative IT2 TrFN A ij denote the evaluative rating of the alternative \( z_{i} \in Z \) in terms of the criterion \( c_{j} \in C \). Let the two nonnegative T1 FSs \( A_{ij}^{ - } \): X → [0, 1] and \( A_{ij}^{ + } \): X → [0, 1] denote the lower and upper trapezoidal fuzzy numbers, respectively, with respect to A ij , where \( A_{ij}^{ - } = (a_{1ij}^{ - } ,a_{2ij}^{ - } ,a_{3ij}^{ - } ,a_{4ij}^{ - } ;h_{{A_{ij} }}^{ - } ) \) and \( A_{ij}^{ + } = (a_{1ij}^{ + } ,a_{2ij}^{ + } ,a_{3ij}^{ + } ,a_{4ij}^{ + } ;h_{{A_{ij} }}^{ + } ) \). The IT2 TrFN evaluative rating A ij is expressed as follows:
where \( 0 \le a_{1ij}^{ - } \le a_{2ij}^{ - } \le a_{3ij}^{ - } \le a_{4ij}^{ - } \), \( 0 \le a_{1ij}^{ + } \le a_{2ij}^{ + } \le a_{3ij}^{ + } \le a_{4ij}^{ + } \), \( a_{1ij}^{ + } \le a_{1ij}^{ - } \), \( a_{4ij}^{ - } \le a_{4ij}^{ + } \), \( 0 \le \) \( h_{{A_{ij} }}^{ - } \le h_{{A_{ij} }}^{ + } \le 1 \), and \( A_{ij}^{ - } \subseteq A_{ij}^{ + } \).
In a similar manner, let a nonnegative IT2 TrFN W j denote the importance weight of the criterion \( c_{j} \in C \). Let \( W_{j}^{ - } { = }(w_{1j}^{ - } ,w_{2j}^{ - } ,w_{3j}^{ - } ,w_{4j}^{ - } ;h_{{W_{j} }}^{ - } ) \) and \( W_{j}^{ + } = (w_{1j}^{ + } ,w_{2j}^{ + } , \) \( w_{3j}^{ + } ,w_{4j}^{ + } ;h_{{W_{j} }}^{ + } ) \) be the lower and upper trapezoidal fuzzy numbers, respectively, with respect to W j . Then, W j is expressed as follows:
where \( 0 \le w_{1j}^{ - } \le w_{2j}^{ - } \le w_{3j}^{ - } \le w_{4j}^{ - } \), \( 0 \le w_{1j}^{ + } \le w_{2j}^{ + } \le w_{3j}^{ + } \le w_{4j}^{ + } \), \( w_{1j}^{ + } \le w_{1j}^{ - } \), \( w_{4j}^{ - } \le w_{4j}^{ + } \), and \( 0 \le h_{{W_{j} }}^{ - } \le h_{{W_{j} }}^{ + } \le 1 \), and \( W_{j}^{ - } \subseteq W_{j}^{ + } \).
3.2 Likelihood of fuzzy preference relations between IT2 TrFNs
By extending the fuzzy preference relations [22] and the likelihoods between trapezoidal fuzzy numbers [23, 24, 40], Chen [19] and Wang et al. [62] proposed the extended concept of likelihoods of fuzzy preference relations between IT2 TrFNs. This paper adopts the likelihoods presented by Chen [19] and Wang et al. [62] to develop an approach to generating criterion-wise rankings of the alternatives.
Lee and Chen [40] developed a fuzzy decision-making method based on the likelihood-based comparison relations on intervals introduced by Xu and Da [64]. Lee and Chen [40] and Chen and Lee [24] presented the likelihood-based comparison relations using T1 FSs and IT2 FSs. Chen and Lee [24] further proposed the concepts of likelihood-based comparison relations of fuzzy sets based on the α-cut representation. Chen and Lee [22] provided a method for handling fuzzy multiple criteria hierarchical group decision-making problems based on arithmetic operations and fuzzy preference relations of IT2 FSs. Chen and Lee [23] presented a ranking method of trapezoidal IT2 FSs using the concept of likelihoods. They employed the fuzzy preference relations based on IT2 FSs to establish the upper and lower fuzzy preference matrices. Chen [19] and Wang et al. [62] extended the likelihoods proposed by Chen and Lee [22, 23] to present the modified concepts of lower and upper likelihoods for determining the likelihood of a fuzzy preference relation in the context of IT2 TrFNs.
In the decision context of IT2 TrFNs, let \( A_{ij} \) (\( = [A_{ij}^{ - } ,A_{ij}^{ + } ] = [(a_{1ij}^{ - } ,a_{2ij}^{ - } ,a_{3ij}^{ - } ,a_{4ij}^{ - } ; \) \( h_{{A_{ij} }}^{ - } ),(a_{1ij}^{ + } ,a_{2ij}^{ + } ,a_{3ij}^{ + } ,a_{4ij}^{ + } ;h_{{A_{ij} }}^{ + } )] \)) and \( A_{i'j} \) (\( = [A_{i'j}^{ - } ,A_{i'j}^{ + } ] = [(a_{1i'j}^{ - } ,a_{2i'j}^{ - } ,a_{3i'j}^{ - } ,a_{4i'j}^{ - } ;h_{{A_{i'j} }}^{ - } ),(a_{1i'j}^{ + } , \) \( a_{2i'j}^{ + } ,a_{3i'j}^{ + } ,a_{4i'j}^{ + } ;h_{{A_{i'j} }}^{ + } )] \)) denote the evaluative ratings of the alternatives \( z_{i} \) and \( z_{i'} \), respectively, with respect to the criterion \( c_{j} \in C \). As mentioned before, \( A_{ij} \) and \( A_{i'j} \) are expressed as two nonnegative IT2 TrFNs defined on X. Let a fuzzy preference relation \( A_{ij} \ge A_{i'j} \) denote the evaluative rating of the alternative \( z_{i} \) not being smaller than that of the alternative \( z_{i'} \) in regard to a specific criterion c j . Let \( L(A_{ij} \ge A_{i'j} ) \) denote the likelihood of the fuzzy preference relation \( A_{ij} \ge A_{i'j} \) for each pair of alternatives \( (z_{i} ,z_{i'} ) \). Lee and Chen [40] and Chen and Lee [22–24] determined the likelihood of a fuzzy preference relation between two lower fuzzy sets. Moreover, they calculated the likelihood for two upper fuzzy sets. The corresponding fuzzy preference matrices were separately constructed for lower and upper fuzzy sets. More specifically, they determined the lower and upper likelihoods via the relations \( A_{ij}^{ - } \ge A_{i'j}^{ - } \) and \( A_{ij}^{ + } \ge A_{i'j}^{ + } \), respectively. Next, they obtained the likelihood \( L(A_{ij} \ge A_{i'j} ) \) using the mean of the computed lower and upper likelihoods. However, the obtained likelihoods between two lower fuzzy sets (\( A_{ij}^{ - } {\text{ and }}A_{i'j}^{ - } \)) and between two upper fuzzy sets (\( A_{ij}^{ + } {\text{ and }}A_{i'j}^{ + } \)) are not true values of the lower and upper likelihoods, respectively, for a fuzzy preference relation between IT2 TrFNs.
In general, the minimal possibility of the event \( A_{ij} \ge A_{i'j} \) occurs in the comparison of the lower fuzzy set \( A_{ij}^{ - } \) and the upper fuzzy set \( A_{i'j}^{ + } \), and the maximal possibility of the event \( A_{ij} \ge A_{i'j} \) occurs in the comparison of the upper fuzzy set \( A_{ij}^{ + } \) and the lower fuzzy set \( A_{i'j}^{ - } \). Accordingly, the lower and upper likelihoods of \( L(A_{ij} \ge A_{i'j} ) \) should be determined via the relations \( A_{ij}^{ - } \ge A_{i'j}^{ + } \) and \( A_{ij}^{ + } \ge A_{i'j}^{ - } \), respectively. Therefore, based on the concept of likelihoods proposed by Lee and Chen [40] and Chen and Lee [22–24], Chen [19] and Wang et al. [62] developed a modified concept of likelihoods of fuzzy preference relations between IT2 TrFNs. Assume that at least one of \( h_{{A_{ij} }}^{ - } \ne h_{{A_{i'j} }}^{ + } \), \( a_{4ij}^{ - } \ne a_{1ij}^{ - } \), \( a_{4i'j}^{ + } \ne a_{1i'j}^{ + } \), and \( a_{\eta ij}^{ - } \ne a_{\eta i'j}^{ + } \) holds and that at least one of \( h_{{A_{ij} }}^{ + } \ne h_{{A_{i'j} }}^{ - } \), \( a_{4ij}^{ + } \ne a_{1ij}^{ + } \), \( a_{4i'j}^{ - } \ne a_{1i'j}^{ - } \), and \( a_{\eta ij}^{ + } \ne a_{\eta i'j}^{ - } \) holds, where \( \eta \in {\text{\{ }}1, 2, 3, 4{\text{\} }} \). This paper employs their modified likelihoods to present an effective ranking procedure for comparing IT2 TrFN evaluative ratings.
Definition 6
[19, 62] Let \( A_{ij} \) and \( A_{i'j} \) be two IT2 TrFN evaluative ratings of the alternatives \( z_{i} \) and \( z_{i'} \), respectively, with respect to the criterion \( c_{j} \in C \). The lower likelihood \( L^{ - } (A_{ij} \ge A_{i'j} ) \) and the upper likelihood \( L^{ + } (A_{ij} \ge A_{i'j} ) \) of a fuzzy preference relation \( A_{ij} \ge A_{i'j} \) are defined as follows:
The likelihood \( L(A_{ij} \ge A_{i'j} ) \) of \( A_{ij} \ge A_{i'j} \) is defined as follows:
Theorem 1
[19, 62] Let \( A_{ij} \) and \( A_{i'j} \) be two IT2 TrFN evaluative ratings. The lower likelihood \( L^{ - } (A_{ij} \ge A_{i'j} ) \) and the upper likelihood \( L^{ + } (A_{ij} \ge A_{i'j} ) \) satisfy the following properties:
- (T1.1) :
-
\( 0 \le L^{ - } (A_{ij} \ge A_{i'j} ) \le 1; \)
- (T1.2) :
-
\( 0 \le L^{ + } (A_{ij} \ge A_{i'j} ) \le 1; \)
- (T1.3) :
-
\( L^{ - } (A_{ij} \ge A_{i'j} ) + L^{ + } (A_{i'j} \ge A_{ij} ) = 1; \)
- (T1.4) :
-
\( L^{ - } (A_{ij} \ge A_{i'j} ) = 0 \) and \( L^{ + } (A_{i'j} \ge A_{ij} ) = 1 \) if \( a_{4ij}^{ + } \le a_{1i'j}^{ + } \) and \( h_{{A_{ij} }}^{ - } \le h_{{A_{i'j} }}^{ + } ; \)
- (T1.5) :
-
\( L^{ + } (A_{ij} \ge A_{i'j} ) = 0 \) and \( L^{ - } (A_{i'j} \ge A_{ij} ) = 1 \) if \( a_{1i'j}^{ - } - a_{4ij}^{ + } \ge 2\hbox{max} \left( {h_{{A_{ij} }}^{ + } - h_{{A_{i'j} }}^{ - } ,0} \right). \)
See Chen [19] for detailed proofs.
Theorem 2
[19] Let \( A_{ij} \) and \( A_{i'j} \) be two IT2 TrFN evaluative ratings. The likelihood \( L(A_{ij} \ge A_{i'j} ) \) satisfies the following properties:
- (T2.1) :
-
\( 0 \le L(A_{ij} \ge A_{i'j} ) \le 1; \)
- (T2.2) :
-
\( L(A_{ij} \ge A_{i'j} ) + L(A_{i'j} \ge A_{ij} ) = 1; \)
- (T2.3) :
-
\( L(A_{ij} \ge A_{i'j} ) = L(A_{i'j} \ge A_{ij} ) = \frac{1}{2} \) if \( L(A_{ij} \ge A_{i'j} ) = L(A_{i'j} \ge A_{ij} ); \)
- (T2.4) :
-
\( L(A_{ij} \ge A_{ij} ) = \frac{1}{2}; \)
- (T2.5) :
-
\( \sum\limits_{i = 1}^{m} {\sum\limits_{i' = 1}^{m} {L(A_{ij} \ge A_{i'j} )} } = \frac{{m^{2} }}{2}. \)
Proof
See Chen [19] for detailed proofs of (T2.1)–(T2.4). (T2.5) can be easily verified using (T2.2) and (T2.4).□
The likelihoods of pair-wise IT2 TrFN evaluative ratings with respect to each criterion \( c_{j} \in C \) can be concisely expressed in the following matrix format:
for \( j = 1,2, \cdots ,n \).
Theorem 3
Assume that L j is the likelihood matrix whose entries \( L(A_{ij} \ge A_{i'j} ) \) are given by (13). Then, L j is a fuzzy complementary judgment matrix for \( j = 1,2, \cdots ,n \).
Proof
According to (T2.1) and (T2.2), with respect to the criterion \( c_{j} \in C \), it is known that \( 0 \le L(A_{ij} \ge A_{i'j} ) \le 1 \) and \( L(A_{ij} \ge A_{i'j} ) + L(A_{i'j} \ge A_{ij} ) = 1 \) for \( i, \, i' = 1,2, \ldots ,m \). They are the conditions that a fuzzy complementary judgment matrix should satisfy [28, 32, 43]. Therefore, L j is proven to be a fuzzy complementary judgment matrix for \( j = 1,2, \ldots ,n \).□
Example 1
Consider three alternatives z 1, z 2, and z 3 with respect to a criterion x 1. Suppose that the IT2 TrFN evaluative ratings A 11 = [(0.0075, 0.0075, 0.015, 0.0525; 0.8) (0.0, 0.0, 0.02, 0.07; 1.0)], A 21 = [(0.2325, 0.255, 0.325, 0.3575; 0.8), (0.17, 0.22, 0.36, 0.42; 1.0)], and A 31 = [(0.4025, 0.4525, 0.5375, 0.5675; 0.8), (0.32, 0.41, 0.58, 0.65; 1.0)]. The likelihoods of pair-wise evaluative ratings can be computed according to (11)–(13) of Definition 6. For example, we obtain \( L^{ - } (A_{11} \ge A_{21} ) = \) \( L^{ - } (A_{11} \ge A_{31} ) = \) 0 and \( L^{ - } (A_{21} \ge A_{31} ) = \) 0.0228 using (11), \( L^{ + } (A_{11} \ge A_{21} ) = \) 0.1418, \( L^{ + } (A_{11} \ge A_{31} ) = \) 0.0269, and \( L^{ + } (A_{21} \ge A_{31} ) = \) 0.2601 using (12), and \( L(A_{11} \ge A_{21} ) = \) 0.0709, \( L(A_{11} \ge A_{31} ) = \) 0.0135, and \( L(A_{21} \ge A_{31} ) = \) 0.1415 using (13). Next, according to (14), we construct the likelihood matrix as follows:
Obviously, L 1 is a fuzzy complementary judgment matrix.
4 The likelihood-based assignment method
This section employs the extended concept of likelihoods of fuzzy preference relations between IT2 TrFNs to determine criterion-wise rankings of the alternatives. To address MCDA problems containing imprecise and uncertain type-2 fuzzy properties, this section develops a novel likelihood-based assignment method to determine an optimal preference ranking of the alternatives in accordance with the obtained criterion-wise rankings and the importance weights of criteria within the decision environment of IT2 TrFNs.
4.1 Proposed method
Based on the likelihoods of fuzzy preference relations between IT2 TrFNs, this subsection presents a ranking procedure using the optimal membership degree determination method to determine a criterion-wise preference of the alternatives. Then, a likelihood-based assignment model is constructed using the concept of an adjusted rank contribution matrix to determine the priority order of various alternatives.
Li et al. [44] and Li [41] used the likelihood of an alternative not being inferior to another alternative to construct a likelihood matrix for determining the optimal degrees of membership. Li [42, 43] employed an inclusion comparison probability to make comparisons between alternatives and proposed the optimal membership degree determination method. In a manner similar to that of Li et al. [44] and Li [41–43], we employ the optimal membership degree determination method to compare any two IT2 TrFN evaluative ratings. The m alternatives can then be ranked in terms of each criterion \( c_{j} \in C \) according to the comparison results of the optimal degrees of membership.
The concepts of relative difference indices and the relative difference matrix are proposed to determine the optimal degree of membership. Consider any two IT2 TrFN evaluative ratings \( A_{ij} \) and \( A_{i'j} \) for \( i, \, i' = 1,2, \ldots ,m \) and \( i \ne i' \). To construct the likelihood matrix L j , we first compute the lower likelihood \( L^{ - } (A_{ij} \ge A_{i'j} ) \), the upper likelihood \( L^{ + } (A_{ij} \ge A_{i'j} ) \), and the likelihood \( L(A_{ij} \ge A_{i'j} ) \) of a fuzzy preference relation \( A_{ij} \ge A_{i'j} \) for each criterion \( c_{j} \in C \). Additionally, we know that \( L(A_{ij} \ge A_{ij} ) = 0.5 \) for \( i = 1,2, \ldots ,m \) according to (T2.4). Let \( \gamma_{ii'}^{j} \) denote the relative difference index representing a linear transformation of the difference in respective sums of the likelihoods \( L(A_{ij} \ge A_{{i^{{\prime \prime }} j}} ) \) (\( i^{{\prime \prime }} = 1,2, \ldots ,m \)) in the i-th row of L j and \( L(A_{{i^{{\prime }} j}} \ge A_{{i^{{\prime \prime }} j}} ) \) (\( i^{{\prime \prime }} = 1,2, \ldots ,m \)) in the \( i^{{\prime }} \)-th row of L j . It is defined as follows:
Example 2
Let us look at Example 1 again. There are three alternatives in Example 1; thus, m = 3. According to the definition in (15), the relative difference index \( \gamma_{12}^{1} \), for example, is computed as follows:
The relative difference index \( \gamma_{ii'}^{j} \) of the pair-wise IT2 TrFN evaluative ratings \( A_{ij} \) and \( A_{i'j} \) with respect to each criterion \( c_{j} \in C \) can be concisely expressed in the following matrix format:
for \( j = 1,2, \cdots ,n \).
Theorem 4
Assume that \( \varUpsilon_{j} \) is the relative difference matrix whose entries \( \gamma_{ii'}^{j} \) are given by (15). Then, \( \varUpsilon_{j} \) is a fuzzy complementary and consistent judgment matrix for \( j = 1,2, \ldots ,n \).
Proof
According to (T2.1), (T2.2), and (T2.4), it is easily observed that
for \( i, \, i' = 1,2, \ldots ,m \). Next, it follows that
Combining (17) and (18), we obtain
for \( i, \, i' = 1,2, \cdots ,m \). It implies that \( - 1 \le \frac{{\sum\nolimits_{{i^{{\prime \prime }} = 1}}^{m} {L(A_{ij} \ge A_{ij} )} - \sum\nolimits_{{i^{{\prime \prime }} = 1}}^{m} {L(A_{{i^{{\prime }} j}} \ge A_{{i^{{\prime \prime }} j}} )} }}{m - 1} \le 1 \) because \( m \ge 1 \); furthermore,
Therefore, it is proven that \( \varUpsilon_{j} \) is fuzzy because
for \( i, \, i' = 1,2, \ldots ,m \). Next, it is easy to verify that \( \varUpsilon_{j} \) is complementary because
for \( i, \, i' = 1,2, \ldots ,m \). Furthermore, one can easily obtain
for \( i, \, i' = 1,2, \ldots ,m \). Thus, \( \varUpsilon_{j} \) is additive transitive [28, 32, 43] because \( (\gamma_{{ii^{{\prime \prime }} }}^{j} - 0.5) + (\gamma_{{i^{{\prime \prime }} i^{{\prime }} }}^{j} - 0.5) = \gamma_{{ii^{{\prime }} }}^{j} - 0.5 \). According to (19)–(21), \( \varUpsilon_{j} \) is proven to be a fuzzy complementary and consistent judgment matrix for \( j = 1,2, \ldots ,n \).□
Example 3
Continue Example 2. According to the definition in (16), the relative difference matrix \( \varUpsilon_{1} \) is constructed as follows:
Obviously, \( \varUpsilon_{1} \) is a fuzzy complementary and consistent judgment matrix.
Based on the relative difference index \( \gamma_{ii'}^{j} \) in the relative difference matrix \( \varUpsilon_{j} \), we can determine the optimal degree of membership for each IT2 TrFN evaluative rating. First, the sum of all relative difference indices of each row in the matrix \( \varUpsilon_{j} \) is computed as follows:
for \( i = 1,2, \ldots ,m. \)
Let \( \varGamma (A_{ij} ) \) denote the optimal degree of membership for the IT2 TrFN evaluative rating A ij . \( \varGamma (A_{ij} ) \) is defined by the normalized value of \( \gamma_{i}^{j} \) as follows:
for \( i = 1,2, \ldots ,m \) and \( j = 1,2, \ldots ,n. \)
Theorem 5
Let \( A_{ij} \) be an IT2 TrFN evaluative rating of the alternative \( z_{i} \) with respect to the criterion \( c_{j} \in C \) . The optimal degree of membership \( \varGamma (A_{ij} ) \) of A ij is determined by:
Proof
As defined in (23), it is obvious that
for \( i = 1,2, \ldots ,m \) and \( j = 1,2, \ldots ,n \) using (22) and (15). According to (T2.5), it is known that \( \sum\nolimits_{i = 1}^{m} {\sum\nolimits_{i' = 1}^{m} {L(A_{ij} \ge A_{i'j} )} } = {{m^{2} } \mathord{\left/ {\vphantom {{m^{2} } 2}} \right. \kern-0pt} 2} \). It follows that
Therefore, Theorem 5 is proven.□
Example 4
Continue Example 3. According to the definition in (24), the optimal degree of membership \( \varGamma (A_{21} ) \) of A 21, for example, is computed as follows:
Of course, \( \varGamma (A_{21} ) \) can also be computed using (22) and (23) as follows:
Theorem 6
The optimal degree of membership \( \varGamma (A_{ij} ) \) of the alternative \( z_{i} \in Z \) with respect to the criterion \( c_{j} \in C \) satisfies the following properties:
- (T6.1) :
-
\( \frac{1}{2m} \le \varGamma (A_{ij} ) \le \frac{3}{2m}; \)
- (T6.2) :
-
\( \sum\limits_{i = 1}^{m} {\varGamma (A_{ij} )} = 1 \) for \( c_{j} \in C. \)
Proof
Because \( 0 \le L(A_{ij} \ge A_{i'j} ) \le 1 \), \( L(A_{ij} \ge A_{ij} ) = 0.5 \), and \( L(A_{ij} \ge A_{i'j} ) + L(A_{i'j} \ge \) \( A_{ij} ) = 1 \) via Theorem 2, one can easily obtain that \( 0.5 \le \sum\nolimits_{i' = 1}^{m} {L(A_{ij} \ge A_{i'j} )} \le \) \( (m - 1) + 0.5 \). When \( \sum\nolimits_{i' = 1}^{m} {L(A_{ij} \ge A_{i'j} )} \) is equal to the lower bound 0.5, \( \varGamma (A_{ij} ) \) becomes
When \( \sum\nolimits_{i' = 1}^{m} {L(A_{ij} \ge A_{i'j} )} \) is equal to the upper bound \( (m - 1) + 0.5 \), \( \varGamma (A_{ij} ) \) becomes
According to (25) and (26), \( {1 \mathord{\left/ {\vphantom {1 {2m}}} \right. \kern-0pt} {2m}} \le \varGamma (A_{ij} ) \le {3 \mathord{\left/ {\vphantom {3 {2m}}} \right. \kern-0pt} {2m}} \) holds. Therefore, (T6.1) is valid. Additionally, it is obvious that \( \sum\nolimits_{i = 1}^{m} {\varGamma (A_{ij} )} = 1 \) because of the definition in (23). Accordingly, (T6.2) is valid.□
Example 5
Continue Example 4. Applying (24), the respective optimal degrees of membership of z 1, z 2, and z 3 with respect to x 1 are obtained as follows: \( \varGamma (A_{ 11} ){ = } \) 0.1807, \( \varGamma (A_{21} ){ = } \) 0.3451, and \( \varGamma (A_{ 31} ){ = } \) 0.4742. Obviously, the condition (T6.1) is satisfied because \( {1 \mathord{\left/ {\vphantom {1 {(2 \times 3)}}} \right. \kern-0pt} {(2 \times 3)}} \le \varGamma (A_{i1} ) \le {3 \mathord{\left/ {\vphantom {3 {(2 \times 3)}}} \right. \kern-0pt} {(2 \times 3)}} \) for i = 1, 2, 3. The condition (T6.2) is also satisfied because \( \varGamma (A_{11} ) + \varGamma (A_{21} ) + \varGamma (A_{ 31} ) = 1 \).
The m alternatives can be ranked with respect to each criterion based on the optimal degrees of membership. Specifically, in regard to each criterion \( c_{j} \in C_{\text{I}} \) (i.e., c j is a benefit criterion), the ranking order of all m alternatives can be subsequently generated according to the descending order of the \( \varGamma (A_{ij} ) \) values. Conversely, for each criterion \( c_{j} \in C_{\text{II}} \) (i.e., c j is a cost criterion), the ranking order of all m alternatives is determined according to the ascending order of the \( \varGamma (A_{ij} ) \) values.
To combine the obtained criterion-wise rankings into an overall preference ranking, we establish the main structure of the proposed likelihood-based assignment model using the concepts of an (adjusted) rank frequency matrix and an (adjusted) rank contribution matrix. In general, the MCDA problem involving multiple criteria evaluation and the ranking/selection of alternatives can be considered as that of assigning alternatives to a rank of order. Problems of this nature are assignment problems [6]. More specifically, we can give a rank frequency matrix whose entries measure the frequency that each alternative is assigned a particular rank among all criterion-wise preferences of the alternatives.
Employing the criterion-wise rankings of the m alternatives by comparing the values of \( \varGamma (A_{ij} ) \), we define a rank frequency matrix F 0 that is similar to the concept of the product-attribute matrix introduced by Bernardo and Blin [6]. F 0 is an \( m \times m \) square nonnegative matrix whose element \( f_{ik}^{0} \) (\( i, \, k = 1,2, \ldots ,m \)) represents the number of criterion-wise rankings where the alternative z i is ranked k-th, as expressed in the following matrix:
Theorem 7
The entry \( f_{ik}^{0} \) ( \( i, \, k = 1,2, \ldots ,m \) ) in the rank frequency matrix F 0 satisfies the following properties:
- (T7.1) :
-
\( \sum\nolimits_{k = 1}^{m} {f_{ik}^{0} } = n \) for \( i = 1,2, \ldots ,m; \)
- (T7.2) :
-
\( \sum\nolimits_{i = 1}^{m} {f_{ik}^{0} } = n \) for \( k = 1,2, \ldots ,m. \)
Proof
(T7.1) and (T7.2) are obvious.□
Example 6
Let the set of alternatives \( Z = \{ z_{1} ,z_{2} ,z_{3} ,z_{4} \} \) and the set of criteria \( C = \{ c_{1} ,c_{2} ,c_{3} \} \). Suppose that the criterion-wise rankings of all alternatives z i (i = 1, 2, 3, 4) based on each criterion c j (j = 1, 2, 3) are \( z_{3} \succ z_{1} \succ z_{4} \succ z_{2} \), \( z_{2} \succ z_{3} \succ z_{4} \succ z_{1} \), and \( z_{4} \succ z_{1} \succ z_{2} \succ z_{3} \). According to (27), the rank frequency matrix F 0 is constructed as follows:
Obviously, the condition (T7.1) is satisfied because \( \sum\nolimits_{k = 1}^{4} {f_{ik}^{0} } = 3 \) for i = 1, 2, 3, 4. The condition (T7.2) is also satisfied because \( \sum\nolimits_{i = 1}^{4} {f_{ik}^{0} } = 3 \) for k = 1, 2, 3, 4.
Note that equal importance is assigned to each criterion \( c_{j} \in C \) in the rank frequency matrix F 0. Nevertheless, incorporating individual subjective preferences over the criteria into the MCDA process is crucial in generating a solution that is acceptable to the decision-makers. Therefore, we define a rank contribution matrix Π0 that contains all of the information needed for an individual’s decision (i.e., the subjective weights a decision-maker place on each criterion and the criterion-wise rankings of all alternatives according to the optimal degrees of membership). Let \( c_{{j_{1} }} , \, c_{{j_{2} }} , \ldots , \) and \( c_{{j_{{f_{ik}^{0} }} }} \) represent the corresponding criteria for which the alternative z i is ranked k-th. Let the symbol \( \oplus \) represent the addition operation of IT2 TrFNs. The rank contribution matrix Π0 is determined by aggregating the products of multiplying the IT2 TrFN importance weights \( W_{{j_{1} }} , \, W_{{j_{2} }} , \ldots , \) and \( W_{{j_{{f_{ik}^{0} }} }} \) by the optimal degrees of membership \( \varGamma (A_{{ij_{1} }} ), \, \varGamma (A_{{ij_{2} }} ), \ldots , \) and \( \varGamma (A_{{ij_{{f_{ik}^{0} }} }} ) \), respectively, corresponding to \( f_{ik}^{0} \); it is defined as follows:
where the meaning of each entry \( \varPi_{ik}^{0} \) of Π0 is a measure of the concordance among all of the criteria in assigning the alternative z i a rank k, and
for \( i, \, k = 1,2, \ldots ,m \). Applying Zadeh’s extension principle [67] to the IT2 TrFN environment, we calculate \( \varPi_{ik}^{0} \) in the following manner:
Note that the operations of addition and multiplication by a nonnegative ordinary number defined on nonnegative IT2 TrFNs will obtain additional nonnegative IT2 TrFNs. Clearly, the optimal degrees of membership \( \varGamma (A_{{ij_{1} }} ) \), \( \varGamma (A_{{ij_{2} }} ), \ldots , \) and \( \varGamma (A_{{ij_{{f_{ik}^{0} }} }} ) \) are nonnegative ordinary numbers. Thus, the computation result of the \( \varPi_{ik}^{0} \) value is a nonnegative IT2 TrFN. For brevity, denote \( h_{{\varPi_{ik}^{0} }}^{ - } = \min_{\vartheta = 1}^{{f_{ik}^{0} }} h_{{W_{{j_{\vartheta } }} }}^{ - } \), \( h_{{\varPi_{ik}^{0} }}^{ + } = \min_{\vartheta = 1}^{{f_{ik}^{0} }} h_{{W_{{j_{\vartheta } }} }}^{ + } \), \( \pi_{\eta ik}^{0 - } = \sum\nolimits_{\vartheta = 1}^{{f_{ik}^{0} }} {\varGamma (A_{{ij_{\vartheta } }} ) \cdot w_{{\eta j_{\vartheta } }}^{ - } } \), and \( \pi_{\eta ik}^{0 + } = \) \( \sum\nolimits_{\vartheta = 1}^{{f_{ik}^{0} }} {\varGamma (A_{{ij_{\vartheta } }} ) \cdot w_{{\eta j_{\vartheta } }}^{ + } } \) for \( \eta = 1, 2, 3, 4 \). Then, the entry \( \varPi_{ik}^{0} \) in the rank contribution matrix Π0 is expressed as the following:
where \( 0 \le \pi_{1ik}^{0 - } \le \pi_{2ik}^{0 - } \le \pi_{3ik}^{0 - } \le \pi_{4ik}^{0 - } \), \( 0 \le \pi_{1ik}^{0 + } \le \pi_{2ik}^{0 + } \le \pi_{3ik}^{0 + } \le \pi_{4ik}^{0 + } \), \( \pi_{1ik}^{0 + } \le \pi_{1ik}^{0 - } \), \( \pi_{4ik}^{0 - } \le \pi_{4ik}^{0 + } \), and \( 0 \le h_{{\varPi_{ik}^{0} }}^{ - } \le h_{{\varPi_{ik}^{0} }}^{ + } \le 1 \).
The rank contribution matrix Π0 provides a simple approach for effectively measuring the contribution of each alternative to the overall performance (i.e., overall agreement with the criterion-wise rankings) if it is assigned a particular rank. Nevertheless, a troublesome issue might occur in situations where some alternatives are tied with respect to a criterion, resulting in confusing computations of the \( \varPi_{ik}^{0} \) values. To overcome this difficulty, the initial ranking must be separated into φ! equalized rankings when the φ alternatives are tied in terms of a specific criterion [11, 13]. Additionally, each of these equalized rankings is weighted 1/φ!.
Let φ j represent the number of tied alternatives regarding a specific criterion c j . Note that, if no tied alternatives are found with respect to c j , we assume that φ j = 1. Let \( f_{ik} \) denote the frequency with which the alternative z i is ranked as the k-th criterion-wise ranking after separating the initial criterion-wise ranking into φ j ! equalized rankings for all \( c_{j} \in C \). Then, we obtain the adjusted rank frequency matrix F as follows:
Theorem 8
The entry \( f_{ik} \) \( (i, \, k = 1,2, \ldots ,m) \) in the adjusted rank frequency matrix F satisfies the following properties:
- (T8.1) :
-
\( \sum\limits_{k = 1}^{m} {f_{ik} } = n \quad for \quad i=1,2,\ldots, m; \)
- (T8.2) :
-
\( \sum\limits_{i = 1}^{m} {f_{ik} } = n \quad for \quad k=1,2, \ldots,m. \)
Proof
(T8.1) and (T8.2) are obvious. □
Let \( c_{{j_{1} }} , \, c_{{j_{2} }} , \ldots , \) and \( c_{{j_{{f_{ik} }} }} \) be the criteria for which the alternative z i is ranked k-th. The adjusted rank contribution matrix \( \varPi \) is expressed as follows:
where
for \( i, \, k = 1,2, \ldots ,m \). Similarly, applying Zadeh’s extension principle to the IT2 TrFN environment, we calculate \( \varPi_{ik} \) in the following manner:
Because \( 1/(\varphi_{{j_{1} }} !), \, 1/(\varphi_{{j_{2} }} !), \ldots ,{\text{ and }}1/(\varphi_{{j_{{f_{ik} }} }} !) \) are nonnegative ordinary numbers, the computation results of the \( \varPi_{ik} \) values are also IT2 TrFNs. For brevity, denote \( h_{{\varPi_{ik} }}^{ - } = \min_{\vartheta = 1}^{{f_{ik} }} h_{{W_{{j_{\vartheta } }} }}^{ - } \), \( h_{{\varPi_{ik} }}^{ + } = \min_{\vartheta = 1}^{{f_{ik} }} h_{{W_{{j_{\vartheta } }} }}^{ + } \), \( \pi_{\eta ik}^{ - } = \sum\nolimits_{\vartheta = 1}^{{f_{ik} }} {{{\varGamma (A_{{ij_{\vartheta } }} ) \cdot w_{{\eta j_{\vartheta } }}^{ - } } \mathord{\left/ {\vphantom {{\varGamma (A_{{ij_{\vartheta } }} ) \cdot w_{{\eta j_{\vartheta } }}^{ - } } {\varphi_{{j_{\vartheta } }} !}}} \right. \kern-0pt} {\varphi_{{j_{\vartheta } }} !}}} \), and \( \pi_{1ik}^{ + } = \sum\nolimits_{\vartheta = 1}^{{f_{ik} }} {{{\varGamma (A_{{ij_{\vartheta } }} ) \cdot w_{{1j_{\vartheta } }}^{ + } } \mathord{\left/ {\vphantom {{\varGamma (A_{{ij_{\vartheta } }} ) \cdot w_{{1j_{\vartheta } }}^{ + } } {\varphi_{{j_{\vartheta } }} !}}} \right. \kern-0pt} {\varphi_{{j_{\vartheta } }} !}}} \) for \( \eta = 1, 2, 3, 4 \). Then, the entry \( \varPi_{ik} \) in the adjusted rank contribution matrix \( \varPi \) is expressed as the following:
where \( 0 \le \pi_{1ik}^{ - } \le \pi_{2ik}^{ - } \le \pi_{3ik}^{ - } \le \pi_{4ik}^{ - } \), \( 0 \le \pi_{1ik}^{ + } \le \pi_{2ik}^{ + } \le \pi_{3ik}^{ + } \le \pi_{4ik}^{ + } \), \( \pi_{1ik}^{ + } \le \pi_{1ik}^{ - } \), \( \pi_{4ik}^{ - } \le \pi_{4ik}^{ + } \), and \( 0 \le h_{{\varPi_{ik} }}^{ - } \le h_{{\varPi_{ik} }}^{ + } \le 1 \).
Based on the rank contribution matrix Π0 or the adjusted rank contribution matrix \( \varPi \), we attempt to obtain an aggregate ranking that can effectively combine the relative performance of each alternative with respect to each criterion. In other words, this aggregate ranking is an overall ranking that is in the closest agreement with the criterion-wise rankings. Therefore, when the alternative z i is assigned to the k-th overall rank, we can employ \( \varPi_{ik}^{0} \) or \( \varPi_{ik} \) to measure the contributions of z i to the overall ranking. Additionally, the larger the contribution indicated by \( \varPi_{ik}^{0} \) or \( \varPi_{ik} \), the more concordance from assigning z i to the k-th overall rank will result.
However, the values of \( \varPi_{ik}^{0} \) and \( \varPi_{ik} \) are IT2 TrFNs and cannot be easily compared. Thus, we use a signed distance-based method presented by Chen [10, 12, 14] and Chen et al. [20] to obtain comparable values of \( \varPi_{ik}^{0} \) (or \( \varPi_{ik} \)) for each \( i, \, k = 1,2, \ldots ,m \). A signed distance (i.e., an oriented distance or directed distance) has often been employed to determine the rankings of fuzzy numbers [14]. The concept of signed distances has been extended to the decision environment of IT2 TrFNs, and the signed distance-based method has been successfully employed to develop several useful MCDA methods [10, 12–15, 20]. The signed distance-based method can use both positive and negative values to define the ordering of IT2 TrFNs, an approach that considerably differs from ordinary distance measures. Signed distances are real numbers; thus, they satisfy linear ordering. It follows that the signed distance based on IT2 TrFNs satisfies the law of trichotomy. Therefore, the concept of signed distances is suggested for acquiring comparable values of \( \varPi_{ik}^{0} \) in the rank contribution matrix Π0 or \( \varPi_{ik} \) in the adjusted rank contribution matrix \( \varPi \).
In reference to Chen [10, 12, 15] and Chen et al. [20], the signed distance \( d_{ik}^{0} \) from \( \varPi_{ik}^{0} \) to the level-1 fuzzy number map onto the vertical axis at the origin of the coordinates is expressed as follows:
for all \( i, \, k = 1,2, \ldots ,m \). The signed distance \( d_{ik} \) from \( \varPi_{ik} \) to the level-1 fuzzy number map onto the vertical axis at the origin of the coordinates is expressed as follows:
for all \( i, \, k = 1,2, \ldots ,m \).
Computing the signed distances \( d_{ik}^{0} \) or \( d_{ik} \) for all \( i, \, k = 1,2, \ldots ,m \), we can then find the alternative z i for each k-th rank that maximizes \( \sum\nolimits_{k = 1}^{m} {d_{ik}^{0} } \) or \( \sum\nolimits_{k = 1}^{m} {d_{ik} } \). This problem is an m! comparison problem. A pure integer linear programming model can be established in the case of a large m to solve the m! comparison problem. Let a permutation matrix P denote an \( m \times m \) square matrix whose element P ik = 1 if the alternative z i is assigned to the overall rank k; otherwise, P ik = 0. In general, we prefer the overall ranking in which the value of \( d_{ik}^{0} \cdot P_{ik} \) or \( d_{ik} \cdot P_{ik} \) is the largest. This is because the overall ranking with the highest \( d_{ik}^{0} \cdot P_{ik} \) or \( d_{ik} \cdot P_{ik} \) produces the best compromise among all of the criterion-wise rankings of the alternatives. Accordingly, all of the conceivable rankings must be examined, and the one that yields the largest value of \( d_{ik}^{0} \cdot P_{ik} \) or \( d_{ik} \cdot P_{ik} \) will be chosen. Because the alternative z i can be assigned to only one rank in the overall ranking, we know that \( \sum\nolimits_{k = 1}^{m} {P_{ik} } = 1 \). Similarly, a given rank k can only have one alternative assigned to it, and thus \( \sum\nolimits_{i = 1}^{m} {P_{ik} } = 1 \).
Consider the situations in which no tied alternatives exist when comparing the \( \varGamma (A_{ij} ) \) values with respect to each criterion \( c_{j} \in C \), i.e., φ j = 1 for all \( j = 1,2, \ldots ,n \). The proposed likelihood-based assignment model can be expressed using the following pure integer linear programming format:
Alternately, consider the situations in which some tied alternatives exist when comparing the \( \varGamma (A_{ij} ) \) values, i.e., \( \varphi_{j} \ne 1 \) for at least one \( j = 1,2, \ldots ,n \). Then, the proposed likelihood-based assignment model in tied situations can be expressed as follows:
The signed distances \( d_{ik}^{0} \) and \( d_{ik} \) can be obtained via \( \varPi_{ik}^{0} \) in the rank contribution matrix Π0 and \( \varPi_{ik} \) in the adjusted rank contribution matrix \( \varPi \), respectively, from the criterion-wise rankings, whereas P ik is unknown and has yet to be determined by the likelihood-based assignment model. Using a branch-and-bound algorithm, we can solve the pure integer linear programming problem [M1] or [M2] to acquire the optimal permutation matrix P *. The optimal ordering can then be sequentially obtained by multiplying Z by P *.
4.2 Proposed algorithm
The proposed likelihood-based assignment method for solving an MCDA problem within the IT2 TrFN decision environment comprises three phases: problem formulation and data input; likelihood-based comparisons and the ranking procedure; and construction of the linear assignment model. Figure 2 depicts a flowchart of the proposed likelihood-based assignment method.
The three phases can be summarized by the following series of steps.
4.2.1 Steps 1 and 2: problem formulation and input stage
- Step 1:
-
Formulate an MCDA problem. Specify the criterion set \( C = \left\{ {c_{1} ,c_{2} , \ldots ,c_{n} } \right\} \) and the alternative set \( Z = \left\{ {z_{1} ,z_{2} , \ldots ,z_{m} } \right\}. \)
- Step 2:
-
Select appropriate linguistic variables or other data collection tools to establish the importance weight W j in (10) of the criterion \( c_{j} \in C \) and the evaluative rating A ij in (9) of the alternative \( z_{i} \in Z \) with respect to the criterion \( c_{j} \in C \) based on IT2 TrFNs.
4.2.2 Steps 3–6: likelihood-based comparison and ranking stage
- Step 3:
-
Apply (11) and (12) to compute the lower and upper likelihoods, \( L^{ - } (A_{ij} \ge A_{i'j} ) \) and \( L^{ + } (A_{ij} \ge A_{i'j} ) \), respectively, of a fuzzy preference relation \( A_{ij} \ge A_{i'j} \) for each criterion \( c_{j} \in C \) and each pair of alternatives \( (z_{i} ,z_{i'} ) \), where \( z_{i} , \, z_{i'} \in Z \) and \( i \ne i'. \)
- Step 4:
-
Calculate the likelihood \( L(A_{ij} \ge A_{i'j} ) \) using (13) for each criterion \( c_{j} \in C \) and each pair of alternatives \( (z_{i} ,z_{i'} ) \), where \( z_{i} , \, z_{i'} \in Z \) and \( i \ne i' \). Moreover, \( L(A_{ij} \ge A_{ij} ) = 0.5 \) for \( i = 1,2, \ldots ,m. \)
- Step 5:
-
Employ (24) to determine the optimal degree of membership \( \varGamma (A_{ij} ) \) for \( z_{i} \in Z \) and \( c_{j} \in C. \)
- Step 6:
-
Rank the m alternatives with respect to each criterion \( c_{j} \in C_{\text{I}} \) and \( c_{j} \in C_{\text{II}} \) according to the descending order and the ascending order, respectively, of the \( \varGamma (A_{ij} ) \) values. If the φ j alternatives are tied with respect to a specific criterion c j , then the φ j ! equalized rankings should be listed separately.
4.2.3 Steps 7–10: linear assignment modeling stage
- Step 7:
-
Compute \( \varPi_{ik}^{0} \) using (30) and construct the rank contribution matrix Π0 when φ j = 1 for all \( j = 1,2, \ldots ,n \). Alternately, compute \( \varPi_{ik} \) using (35) and establish the adjusted rank contribution matrix \( \varPi \) when \( \varphi_{j} \ne 1 \) for at least one \( j = 1,2, \ldots ,n. \)
- Step 8:
-
Use (37) or (38) to derive the signed distances \( d_{ik}^{0} \) or \( d_{ik} \), respectively, for each \( i, \, k = 1,2, \ldots ,m. \)
- Step 9:
-
Define the permutation matrix P as an \( m \times m \) square matrix. Construct a linear assignment model in [M1] or [M2] according to the \( d_{ik}^{0} \) values or the \( d_{ik} \) values, respectively.
- Step 10:
-
Solve [M1] or [M2] using a branch-and-bound algorithm and obtain the optimal permutation matrix P *. Next, apply the permutation matrix P * to Z to obtain the optimal ranking order of the m alternatives.
5 Applications and comparative analyses
This section illustrates and discusses the proposed likelihood-based assignment method by applying it to two practical MCDA applications: the selection problem of landfill sites introduced by Chen [13] and the medical decision-making problem presented by Chen et al. [20]. Furthermore, comparative discussions are subsequently devoted to validate the results of the proposed method with the results from relevant outranking methods. Finally, additional comparisons are conducted to investigate the effectiveness and advantages of the proposed method relative to a widely used scoring method.
5.1 Illustrative application
Only a small number of attempts to extend the linear assignment method to the interval type-2 fuzzy decision environment or allow it to cope with IT2 TrFN data have thus far been conducted. Chen [13] developed an interval type-2 fuzzy linear assignment method based on signed distances to handle multiple criteria decisions and implemented the proposed method to a practical example involving a landfill site selection problem in Kaohsiung City. Chen’s interval type-2 fuzzy linear assignment method and our proposed method belong to the linear assignment methodology. Thus, Chen’s illustrative application of landfill site selection can provide a common basis for implementing a comparative analysis.
The illustrative application presented by Chen [13] assesses the characteristics of candidate landfill sites to determine whether the candidate sites are qualified in regard to the interests and the rights of the stakeholders and general public using seven criteria: transportation convenience (c 1), terrain suitability (c 2), community equity (c 3), environmental impact (c 4), ecological impact (c 5), construction cost (c 6), and historic impact (c 7). Among the criteria for evaluating candidate sites, c 1, c 2, and c 3 are the benefit criteria, whereas all of the others are cost criteria. Given the dense population of Kaohsiung City, the number of suitable landfill locations is limited. There are four candidate locations: z 1 (located in the Cijin district), z 2 (located at the side of an embankment in the Nanzih district), z 3 (located on a hillside by the sea in the Gushan district), and z 4 (located on a hillside in the Zuoying district).
The computational procedure of the proposed likelihood-based assignment method is illustrated as follows. In Step 1, there are four candidate landfill sites in the MCDA problem; the set of all candidate sites is denoted as \( Z = \{ z_{1} ,z_{2} ,z_{3} ,z_{4} \} \). The set of criteria is denoted as \( C = \{ c_{1} ,c_{2} , \ldots ,c_{7} \} \), with \( C_{\text{I}} = \{ c_{1} ,c_{2} ,c_{3} \} \) and \( C_{\text{II}} = \{ c_{4} ,c_{5} ,c_{6} ,c_{7} \} \). In Step 2, the city government evaluated the four landfill sites based on the seven criteria and the importance weights of the criteria using the linguistic rating system in Table 1 [13]. The details of the linguistic evaluations are outlined in Table 2. These linguistic terms were subsequently converted into the IT2 TrFN evaluative ratings A ij of \( z_{i} \in Z \) with respect to \( c_{j} \in C \) and the IT2 TrFN importance weights W j of \( c_{j} \in C \).
In Step 3, we calculated \( L^{ - } (A_{ij} \ge A_{i'j} ) \) and \( L^{ + } (A_{ij} \ge A_{i'j} ) \) for each pair of alternatives \( (z_{i} ,z_{i'} ) \) (\( z_{i} , \, z_{i'} \in Z \) and \( i \ne i' \)) with respect to each criterion \( c_{j} \in C \). Then, we computed the likelihood \( L(A_{ij} \ge A_{i'j} ) \) according to Step 4. Note that \( L(A_{ij} \ge A_{ij} ) = 0.5 \) for \( i = 1, 2, 3, 4 \). The computation results of the likelihoods for all the fuzzy preference relations between IT2 TrFNs are presented in Table 3.
In Step 5, we applied (24) to obtain the optimal degree of membership \( \varGamma (A_{ij} ) \) for \( z_{i} \in Z \) and \( c_{j} \in C \). For example, \( \varGamma (A_{23} ) \) is computed as follows:
Next, we applied Step 6 to rank the four alternatives in terms of each benefit criterion and each cost criterion according to the decreasing order and the increasing order, respectively, of the \( \varGamma (A_{ij} ) \) values. Because the alternatives z 2 and z 4 are tied with respect to the criterion c 3, φ 3 = 2, and the original ranking \( z_{2} \sim z_{4} \succ z_{3} \succ z_{1} \) is equalized as two corresponding rankings, \( z_{2} \succ z_{4} \succ z_{3} \succ z_{1} \) and \( z_{4} \succ z_{2} \succ z_{3} \succ z_{1} \). Additionally, the alternatives z 3 and z 4 are tied with respect to the criterion c 6; thus, φ 6 = 2, and the original ranking \( z_{2} \succ z_{3} \sim z_{4} \succ z_{1} \) is equalized as two corresponding rankings, \( z_{2} \succ z_{3} \succ z_{4} \succ z_{1} \) and \( z_{2} \succ z_{4} \succ z_{3} \succ z_{1} \). The other φ j = 1 for \( j = 1,2,4,5,7 \). The results of the optimal degrees of membership and criterion-wise rankings are indicated in Table 4.
Because \( \varphi_{3} , \, \varphi_{6} \ne 1 \), we employed (35) to determine \( \varPi_{ik} \) for i, k = 1, 2, 3, 4 according to Step 7. Take the argument \( \pi_{142}^{ - } \) of \( \varPi_{42} \) (\( = [(\pi_{142}^{ - } ,\pi_{242}^{ - } ,\pi_{342}^{ - } ,\pi_{442}^{ - } ;h_{{\varPi_{42} }}^{ - } ), \) \( (\pi_{142}^{ + } ,\pi_{242}^{ + } ,\pi_{342}^{ + } ,\pi_{442}^{ + } ;h_{{\varPi_{42} }}^{ + } )] \)) for example. As indicated in (36), we have
Then, in Step 8, we used (38) to calculate the signed distance \( d_{ik} \) for i, k = 1, 2, 3, 4. Consider Π 31 (= [(0.30, 0.31, 0.34, 0.34; 0.80), (0.27, 0.30, 0.35, 0.37; 1.00)]) for example. The signed distance d 31 of Π 31 is computed as follows:
The computation results of the adjusted rank contribution matrix \( \varPi \) and signed distances are revealed in Table 5.
In Step 9, we defined the permutation matrix P as a \( 4 \times 4 \) square matrix, with element P ik for i, k = 1, 2, 3, 4. According to [M2], we established a linear assignment model with \( d_{ik} \) values as follows:
Finally, in Step 10, we solved the model in (41) using a branch-and-bound algorithm. The optimal objective value is 4.73. Additionally, we obtained the optimal permutation matrix P * as follows:
Next, we determined the ordering (z 1, z 2, z 3, z 4) by multiplying Z by P *. That is,\( \left( {z_{1} ,z_{2} ,z_{3} ,z_{4} } \right) \cdot \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{array} } \right] = \left( {z_{1} ,z_{2} ,z_{4} ,z_{3} } \right) \).
The optimal ranking order of the four candidate sites is \( z_{1} \succ z_{2} \succ z_{4} \succ z_{3} \). Thus, z 1 (located in the Cijin district) is the best choice. This ranking result is the same as that obtained using the interval type-2 fuzzy linear assignment method based on signed distances [13].
5.2 Comparative discussions with outranking methods
The linear assignment methodology belongs to outranking models in the MCDA field. Accordingly, this paper further selected other well-known outranking methods (e.g., ELECTRE and QUALIFLEX) to conduct a comparative analysis and examine the applicability and advantages of the proposed method.
Chen et al. [20] developed an extended QUALIFLEX method for handling MCDA problems based on IT2 FSs. They also extended the ELECTRE method to the decision environment of IT2 FSs. Chen et al. [20] presented a medical decision-making problem of selecting a suitable treatment method for acute inflammatory demyelinating disease. Their case comes from the Department of Neurology, Chang Gung Memorial Hospital in Taiwan. They applied the extended QUALIFLEX method and the extended ELECTRE method to the medical decision-making problem. Additionally, Chen and Tsui [25] developed the intuitionistic fuzzy QUALIFLEX method based on optimistic and pessimistic estimations. Although Chen and Tsui’s proposed method is suitable for the intuitionistic fuzzy decision environment, they also applied it to the same medical decision-making problem.
In addition to a comparative discussion regarding the selection problem of landfill sites using the interval type-2 fuzzy linear assignment method [13], we further applied the proposed method to a medical decision-making problem introduced by Chen et al. [20] to facilitate comparative analyses with other outranking methods. With regard to the medical decision-making problem, this paper compared the solution result via the proposed likelihood-based assignment model with the results yielded by the extended QUALIFLEX method [20], the intuitionistic fuzzy QUALIFLEX method [25], and the extended ELECTRE method [20].
In the medical decision-making problem presented by Chen et al. [20], the attending physician assessed the patient’s medical history and her current physical conditions and provided the following treatment options: steroid therapy (z 1), plasmapheresis (z 2), and albumin immune therapy (z 3). Nine criteria were used to evaluate the alternatives, including survival rate (c 1), severity of the side effects (c 2), severity of the complications (c 3), probability of a cure (c 4), discomfort index of the treatment (c 5), cost (c 6), number of days of hospitalization (c 7), probability of a recurrence (c 8), and self-care capacity (c 9). According to the IT2 TrFN data in Chen et al. [20], we applied the proposed method to analyze the same medical decision-making problem and constructed the following linear assignment model:
We solved the model in (42) and obtained \( P_{13}^{*} = P_{21}^{*} = P_{32}^{*} = 1 \) and the remaining \( P_{ik}^{*} = 0 \) in the optimal permutation matrix P *. Therefore, the optimal ranking order of the three treatment options is \( z_{2} \succ z_{3} \succ z_{1} \), which is the same as those obtained using the extended QUALIFLEX method [20] and the intuitionistic fuzzy QUALIFLEX method [25]. Additionally, the extended ELECTRE method renders the outranking relationships \( z_{2} \succ z_{1} \) and \( z_{3} \succ z_{1} \) [20], which are also in agreement with the result obtained via our proposed method.
A summary of the results obtained by applying the proposed method and the comparative methods to the selection problem of landfill sites and the medical decision-making problem of treatment options is presented in Table 6.
Compared with the interval type-2 fuzzy linear assignment method [13], the computations associated with our proposed method are simple and effective. More specifically, we have no need to normalize IT2 TrFN evaluative ratings and importance weights when using the proposed likelihood-based assignment method. However, Chen’s [13] method must use a scale normalization method to establish the normalized decision matrix and the normalized importance weights of criteria. Consider the application of the landfill site selection problem. Although the procedure of the proposed method and the developed linear assignment model in (41) are easy to implement, the illustrative application still reveals a valid and reasonable result using the proposed method.
Conversely, when employing Chen’s [13] interval type-2 fuzzy linear assignment method to solve MCDA problems, it can be observed that no attention has been paid to incorporating the IT2 TrFN evaluative ratings into the weighted rank frequency matrix. More specifically, the weighted rank frequency matrix presented by Chen [13] only contains the information of the rank frequency and the criterion importance. In contrast, our proposed method considers all of the useful information contained in the evaluative ratings and importance weights. To make the information content complementary, we determine the (adjusted) rank contribution matrix by aggregating the products of multiplying the IT2 TrFN importance weights by the optimal degrees of membership. This approach can not only consider the criterion importance but also fully utilize information contained in the evaluative ratings and fuzzy preference relations between IT2 TrFNs.
Consider the second application of the medical decision-making problem. Note that the priority orders of z 2 and z 3 cannot be differentiated via the extended ELECTRE results even though z 1 is dominated by z 2 and z 3. In contrast, the proposed likelihood-based assignment method can differentiate the priority orders of z 2 and z 3, rendering the outranking relationship of \( z_{2} \succ z_{3} \). Therefore, the obtained result yielded by the proposed method provides more important information than that of Chen et al. [20] extended ELECTRE method. Compared with the extended QUALIFLEX method, the intuitionistic fuzzy QUALIFLEX method, and the extended ELECTRE method, the computations associated with the proposed method are simple and effective, and the linear assignment model in (42) is easy to solve for the optimal solution.
As a whole, the comparative results indicate that the solution result using the proposed method is valid and credible even though the computation procedure of the proposed method is much simpler than that of the comparative methods (i.e., the interval type-2 fuzzy linear assignment method, the extended QUALIFLEX method, the intuitionistic fuzzy QUALIFLEX method, and the extended ELECTRE method). The proposed likelihood-based assignment method fully utilizes the extended concept of likelihoods regarding fuzzy preference relations between IT2 TrFNs. This method does not require a complicated computation procedure and produces actionable and valid results that aid the MCDA process.
5.3 Additional comparative discussions with scoring methods
This subsection compares the results of the current likelihood-based assignment method with those of the widely used scoring model. The scoring model selects the alternative that has the highest score (or the maximum utility/value). Over the past few decades, the most traditionally representative method of the scoring model is the SAW method. Thus, this paper conducted a comparative analysis with the well-known SAW method to further examine the distinct advantage of the proposed method.
In general, some tied alternatives possibly exist in the solution results when employing most of the existing decision-making methods to solve an MCDA problem. For example, an alternative ties with another alternative because of the same overall evaluation values if the SAW method is used or because of the same closeness coefficients if the TOPSIS method is used. In particular, the result of more than one best alternative is a sticky problem for the decision-maker because he/she must make a hard decision among the tied best alternatives. Consider a simple MCDA problem with two similar alternatives and three benefit criteria in the context of IT2 TrFNs. Assume that the three criteria are of equal importance. Because equal importance is assigned to the three criteria, let the IT2 TrFN importance weights \( W_{1} = W_{2} = W_{3} = [(1, 1, 1, 1; 1), (1, 1, 1, 1; 1)]\) for convenience. Suppose that the IT2 TrFN evaluative ratings are given as follows: \( A_{11} = \) [(0.78, 0.82, 0.89, 0.91; 0.80), (0.74, 0.78, 0.92, 0.97; 1.00)], \( A_{21} = \) [(0.78, 0.82, 0.89, 0.91; 0.80), (0.71, 0.78, 0.92, 0.97; 1.00)], \( A_{12} = A_{22} = \) [(0.95, 0.99, 0.99, 0.99; 0.80), (0.93, 0.98, 1.00, 1.00; 1.00)], and \( A_{13} = A_{23} = \) [(0.09, 0.12, 0.16, 0.18; 0.80), (0.04, 0.10, 0.18, 0.23; 1.00)]. Note that the IT2 TrFN evaluative ratings of the alternatives z 1 and z 2 are the same except for \( a_{111}^{ + } \) (=0.74) in A 11 and \( a_{121}^{ + } \) (=0.71) in A 21. It directly follows that the fuzzy preference relation \( A_{11} > A_{21} \) holds. Accordingly, \( z_{1} \succ z_{2} \) is the expected solution result.
Employing the SAW method, we calculated the weighted average value EV i (\( =\! [EV_{i}^{ - } ,EV_{i}^{ + } ] \)) of each \( z_{i} \in Z \) by multiplying A ij by W j and then summing these products over all of the criteria in the following manner:
The computation results in the example were as follows: \( EV_{1} = \) [(1.82, 1.93, 2.04, 2.08; 0.80), (1.71, 1.86, 2.10, 2.20; 1.00)] and \( EV_{2} = \) [(1.82, 1.93, 2.04, 2.08; 0.80), (1.68, 1.86, 2.10, 2.20; 1.00)].
Next, we employed the signed distance-based method to acquire the overall evaluation value of each alternative as follows:
In the example, the overall evaluation values were derived as follows: \( d(EV_{1} ) = \) 3.94 and \( d(EV_{2} ) = \) 3.94. The result of \( z_{1} \sim z_{2} \) was acquired via the SAW method.
Instead, we applied the proposed method to the same decision-making problem. The following linear assignment model was constructed:
We solved the above model and acquired \( P_{11}^{*} = P_{22}^{*} = 1 \) and \( P_{12}^{*} = P_{21}^{*} = 0 \) in the optimal permutation matrix P *. Thus, the ranking order of the alternatives is \( z_{1} \succ z_{2} \). Therefore, the priority orders of z 1 and z 2 can be clearly differentiated via the proposed method.
As revealed in the respective solution results using SAW and the proposed method, the likelihood-based assignment method can indeed differentiate the priority orders of similar alternatives and provide a definite complete ranking of all alternatives for decision support. This paper establishes the likelihood-based assignment model to determine the priority order of various alternatives. In general, the tied situation among alternatives hardly occurs by virtue of using the permutation matrix in the current method. Specifically, the proposed likelihood-based assignment method is expressed using a pure integer linear programming model. The solution result of the optimal permutation matrix can clearly differentiate the priority orders of all alternatives and render the outranking relationship of alternatives that have similar overall evaluation values (or closeness coefficients if the TOPSIS method is used). Therefore, the proposed method is appropriate for addressing the MCDA problem if the decision-maker cannot accept the final result of tied alternatives or would like to obtain a crisp complete ranking order of various alternatives. In short, differentiation of the priority orders among similar alternatives is a distinct advantage of the proposed method.
Combining the application results regarding the selection problem of landfill sites and the medical decision-making problem, the usefulness of the proposed likelihood-based assignment method for practical applications has been demonstrated. In brief, the proposed method has various significant advantages over the existing relevant methods, including simple and effective computations, full utilization of the information contained in the evaluative ratings and importance weights, a pure integer linear programming model that is easy to solve, and differentiation of the priority orders among all alternatives.
6 Conclusions
In this paper, we have proposed a new approach for evaluating and selecting the alternatives for solving MCDA problems in the context of IT2 TrFNs. This approach has been developed based on the linear assignment method with some significant modifications. First, we have established an effective ranking procedure using the concept of likelihoods to compare IT2 TrFN evaluative ratings. This paper has employed novel concepts of lower and upper likelihoods proposed by Chen [19] and Wang et al. [62] to determine the likelihood of a fuzzy preference relation in the context of IT2 TrFNs. Based on the obtained likelihoods, we have employed a ranking procedure using the optimal membership degree determination method to identify criterion-wise preference rankings of the decision alternatives. Consider the situation that no tied alternatives are found when comparing the optimal degrees of membership with respect to each criterion. We have established the concepts of a rank frequency matrix and a rank contribution matrix to combine the relative performances of the alternatives in terms of each criterion. Alternately, concerning the fact that some tied alternatives exist when comparing the optimal degrees of membership in regard to at least one criterion, we have proposed the concepts of an adjusted rank frequency matrix and an adjusted rank contribution matrix. To obtain an aggregate ranking of the alternatives, this paper has separately employed the rank contribution matrix and the adjusted rank contribution matrix to construct two likelihood-based assignment models using a signed distances approach. This aggregate ranking is an overall ranking that is in the closest agreement with the criterion-wise preferences of the alternatives. Furthermore, this paper has provided an algorithmic procedure consisting of three phases, including problem formulation, likelihood-based comparisons and ranking, and linear assignment modeling, to implement the proposed likelihood-based assignment method for handling MCDA problems in the context of IT2 TrFNs. The comparative results with other relevant methods also indicate that the effectiveness and the applicability of the proposed likelihood-based assignment method were validated within the decision environment of IT2 TrFNs.
References
Abdullah L, Najib L (2014) A new type-2 fuzzy set of linguistic variables for the fuzzy analytic hierarchy process. Expert Syst Appl 41(7):3297–3305
Amiri M, Zandieh M, Soltani R, Vahdani B (2009) A hybrid multi-criteria decision-making model for firms competence evaluation. Expert Syst Appl 36(10):12314–12322
Baležentis T, Zeng S (2013) Group multi-criteria decision making based upon interval-valued fuzzy numbers: an extension of the MULTIMOORA method. Expert Syst Appl 40(2):543–550
Bashiri M, Badri H (2011) A group decision making procedure for fuzzy interactive linear assignment programming. Expert Syst Appl 38(5):5561–5568
Bashiri M, Badri H, Hejazi TH (2011) Selecting optimum maintenance strategy by fuzzy interactive linear assignment method. Appl Math Model 35(1):152–164
Bernardo JJ, Blin JM (1977) A programming model of consumer choice among multi-attributed brands. J Consum Res 4(2):111–118
Borovička A (2013) Possible modifications of the multiple criteria assignment method. Czech Econ Rev 7(1):55–67
Celik E, Bilisik ON, Erdogan M, Gumus AT, Baracli H (2013) An integrated novel interval type-2 fuzzy MCDM method to improve customer satisfaction in public transportation for Istanbul. Transp Res E Logist Transp Rev 58:28–51
Chen T-Y (2011) An integrated approach for assessing criterion importance with interval type-2 fuzzy sets and signed distances. J Chin Inst Ind Eng 28(8):553–572
Chen T-Y (2011) Signed distanced-based TOPSIS method for multiple criteria decision analysis based on generalized interval-valued fuzzy numbers. Int J Inf Technol Decis Mak 10(6):1131–1159
Chen T-Y (2012) Nonlinear assignment-based methods for interval-valued intuitionistic fuzzy multi-criteria decision analysis with incomplete preference information. Int J Inf Technol Decis Mak 11(4):821–855
Chen T-Y (2012) Multiple criteria group decision-making with generalized interval-valued fuzzy numbers based on signed distances and incomplete weights. Appl Math Model 36(7):3029–3052
Chen T-Y (2013) A linear assignment method for multiple-criteria decision analysis with interval type-2 fuzzy sets. Appl Soft Comput 13(5):2735–2748
Chen T-Y (2013) An interactive method for multiple criteria group decision analysis based on interval type-2 fuzzy sets and its application to medical decision making. Fuzzy Optim Decis Mak 12(3):323–356
Chen T-Y (2013) A signed-distance-based approach to importance assessment and multi-criteria group decision analysis based on interval type-2 fuzzy set. Knowl Inf Syst 35(1):193–231
Chen T-Y (2014) A PROMETHEE-based outranking method for multiple criteria decision analysis with interval type-2 fuzzy sets. Soft Comput 18(5):923–940
Chen T-Y (2014) An ELECTRE-based outranking method for multiple criteria group decision making using interval type-2 fuzzy sets. Inf Sci 263:1–21
Chen T-Y (2015) An interval type-2 fuzzy PROMETHEE method using a likelihood-based outranking comparison approach. Inf Fusion 25:105–120
Chen T-Y (2015) Likelihoods of interval type-2 trapezoidal fuzzy preference relations and their application to multiple criteria decision analysis. Inf Sci 295:303–322
Chen T-Y, Chang C-H, Lu J-R (2013) The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making. Eur J Oper Res 226(3):615–625
Chen S-M, Chen J-H (2009) Fuzzy risk analysis based on similarity measures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operators. Expert Syst Appl 36(3–2):6309–6317
Chen S-M, Lee L-W (2010) Fuzzy multiple criteria hierarchical group decision-making based on interval type-2 fuzzy sets. IEEE Trans Syst Man Cybern A Syst Hum 40(5):1120–1128
Chen S-M, Lee L-W (2010) Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. Expert Syst Appl 37(1):824–833
Chen S-M, Lee L-W (2010) Fuzzy decision-making based on likelihood-based comparison relations. IEEE Trans Fuzzy Syst 18(3):613–628
Chen T-Y, Tsui C-W (2012) Intuitionistic fuzzy QUALIFLEX method for optimistic and pessimistic decision making. Adv Inf Sci Serv Sci 4(14):219–226
Chen S-M, Wang C-Y (2013) Fuzzy decision making systems based on interval type-2 fuzzy sets. Inf Sci 242:1–21
Chen S-M, Yang M-W, Lee L-W, Yang S-W (2012) Fuzzy multiple attributes group decision-making based on ranking interval type-2 fuzzy sets. Expert Syst Appl 39(5):5295–5308
Chiclana F, Herrera-Viedma E, Alonso S, Herrera F (2009) Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity. IEEE Trans Fuzzy Syst 17(1):14–23
Cortes-Rios JC, Gomez-Ramirez E, Ortiz-De-La-Vega HA, Castillo O, Melin P (2014) Optimal design of interval type 2 fuzzy controllers based on a simple tuning algorithm. Appl Soft Comput 23:270–285
Das S, Chowdhury SR, Saha H (2012) Accuracy enhancement in a fuzzy expert decision making system through appropriate determination of membership functions and its application in a medical diagnostic decision making system. J Med Syst 36(3):1607–1620
De Los Angeles Hernández M, Melin P, Méndez GM, Castillo O, López-Juarez I (2015) A hybrid learning method composed by the orthogonal least-squares and the back-propagation learning algorithms for interval A2-C1 type-1 non-singleton type-2 TSK fuzzy logic systems. Soft Comput 19(3):661–678
Garmendia L, Recasens J (2009) How to make T-transitive a proximity relation. IEEE Trans Fuzzy Syst 17(1):200–207
Gilan SS, Sebt MH, Shahhosseini V (2012) Computing with words for hierarchical competency based selection of personnel in construction companies. Appl Soft Comput 12(2):860–871
Greenfield S, Chiclana F (2013) Defuzzification of the discretised generalised type-2 fuzzy set: experimental evaluation. Inf Sci 244:1–25
Han S, Mendel JM (2012) A new method for managing the uncertainties in evaluating multi-person multi-criteria location choices, using a perceptual computer. Ann Oper Res 195(1):277–309
Hong DH, Lee S (2002) Some algebraic properties and a distance measure for interval-valued fuzzy numbers. Inf Sci 148(1):1–10
Hosseini MB, Tarokh MJ (2011) Interval type-2 fuzzy set extension of DEMATEL method. In: Das VV, Thankachan N (eds) Communications in Computer and Information Science 250 CIIT 2011. Springer, Berlin, pp 157–165
Jahan A, Ismail MY, Mustapha F, Sapuan SM (2010) Material selection based on ordinal data. Mater Des 31(7):3180–3187
Lai H-L, Chen T-Y (2015) Client acceptance method for audit firms based on interval-valued fuzzy numbers. Technol Econ Dev Econ 21(1):1–27
Lee L-W, Chen S-M (2009) A new method for fuzzy decision-making based on likelihood-based comparison relations. In: Proceedings of the Eighth International Conference on Machine Learning and Cybernetics, Baoding, China, 12–15 July 2009, pp. 3021–3025
Li D-F (2010) Linear programming method for MADM with interval-valued intuitionistic fuzzy sets. Expert Syst Appl 37(8):5939–5945
Li D-F (2010) TOPSIS-based nonlinear-programming methodology for multiattribute decision making with interval-valued intuitionistic fuzzy sets. IEEE Trans Fuzzy Syst 18(2):299–311
Li D-F (2011) Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information. Appl Soft Comput 11(4):3402–3418
Li D-F, Wang Y-C, Liu S, Shan F (2009) Fractional programming methodology for multi-attribute group decision-making using IFS. Appl Soft Comput 9(1):219–225
Li C, Yi J, Wang M, Zhang G (2013) Monotonic type-2 fuzzy neural network and its application to thermal comfort prediction. Neural Comput Appl 23(7–8):1987–1998
Liao H, Xu ZS (2014) Multi-criteria decision making with intuitionistic fuzzy PROMETHEE. J Intell Fuzzy Syst 27(4):1703–1717
Liao H, Xu ZS, Zeng X-J (2015) Hesitant fuzzy linguistic VIKOR method and its application in qualitative multiple criteria decision making. IEEE Trans Fuzzy Syst 23(5):1343–1355
Liu B, Shen Y, Chen X, Chen Y, Wang X (2014) A partial binary tree DEA-DA cyclic classification model for decision makers in complex multi-attribute large-group interval-valued intuitionistic fuzzy decision-making problems. Inf Fusion 18(1):119–130
Liu H-T, Wang W-K (2008) An integrated fuzzy approach for provider evaluation and selection in third-party logistics. Expert Syst Appl 36(3):4387–4398
Melin P, Castillo O (2014) A review on type-2 fuzzy logic applications in clustering, classification and pattern recognition. Appl Soft Comput 21:568–577
Mendel JM (2001) Uncertain rule-based fuzzy logic systems: introduction and new directions. Prentice-Hall, Upper-Saddle River
Mendel JM, John RIB (2002) Type-2 fuzzy sets made simple. IEEE Trans Fuzzy Syst 10(2):117–127
Méndez GM, De Los Angeles Hernández M (2013) Hybrid learning mechanism for interval A2-C1 type-2 non-singleton type-2 Takagi-Sugeno-Kang fuzzy logic systems. Inf Sci 220:149–169
Moharrer M, Tahayori H, Livi L, Sadeghian A, Rizzi A (2015) Interval type-2 fuzzy sets to model linguistic label perception in online services satisfaction. Soft Comput 19(1):237–250
Ngan S-C (2013) A type-2 linguistic set theory and its application to multi-criteria decision making. Comput Ind Eng 64(2):721–730
Poorzahedy H, Rezaei A (2013) Peer evaluation of multi-attribute analysis techniques: case of a light rail transit network choice. Sci Iran 20(3):371–386
Takáč Z (2013) Inclusion and subsethood measure for interval-valued fuzzy sets and for continuous type-2 fuzzy sets. Fuzzy Sets Syst 224:106–120
Tavoosi J, Badamchizadeh MA (2013) A class of type-2 fuzzy neural networks for nonlinear dynamical system identification. Neural Comput Appl 23(3–4):707–717
Tung WL, Quek C (2007) A brain-inspired fuzzy semantic memory model for learning and reasoning with uncertainty. Neural Comput Appl 16(6):559–569
Wang J-C, Chen T-Y (2014) A closeness coefficient-based multiple criteria decision-making method using interval type-2 fuzzy sets and its application to watershed site selection. J Ind Prod Eng 31(1):1–16
Wang W, Liu X, Qin Y (2012) Multi-attribute group decision making models under interval type-2 fuzzy environment. Knowl Based Syst 30:121–128
Wang J-C, Tsao C-Y, Chen T-Y (2015) A likelihood-based QUALIFLEX method with interval type-2 fuzzy sets for multiple criteria decision analysis. Soft Comput 19(8):2225–2243
Wei S-H, Chen S-M (2009) Fuzzy risk analysis based on interval-valued fuzzy numbers. Expert Syst Appl 36(2–1):2285–2299
Xu ZS, Da QL (2003) A likelihood-based method for priorities of interval judgment matrices. Chin J Manag Sci 11(1):63–65
Xu ZS, Liao H (2015) A survey of approaches to decision making with intuitionistic fuzzy preference relations. Knowl Based Syst 80:131–142
You T, Fan Z, Yu Z (2012) An assignment method for group decision making with uncertain preference ordinals. J Syst Sci Syst Eng 21(2):174–183
Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Inf Sci 8(3):199–249
Zandi F, Tavana M (2011) A fuzzy multi-objective balanced scorecard approach for selecting an optimal electronic business process management best practice (e-BPMBP). Bus Process Manag J 17(1):147–178
Zhai D, Mendel JM (2011) Uncertainty measures for general type-2 fuzzy sets. Inf Sci 181(3):503–518
Zhang Z (2013) On characterization of generalized interval type-2 fuzzy rough sets. Inf Sci 219:124–150
Zhang X, Jin F, Liu P (2013) A grey relational projection method for multi-attribute decision making based on intuitionistic trapezoidal fuzzy number. Appl Math Model 37(5):3467–3477
Zhang Z, Zhang S (2013) A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets. Appl Math Model 37(7):4948–4971
Zhao T, Xiao J (2014) General type-2 fuzzy rough sets based on & #x03B1;-plane representation theory. Soft Comput 18(2):227–237
Zhao T, Xiao J, Li Y, Deng X (2014) A new approach to similarity and inclusion measures between general type-2 fuzzy sets. Soft Comput 18(4):809–823
Acknowledgments
The author acknowledges the assistance of the respected editor and the anonymous referees for their insightful and constructive comments, which have helped to improve the overall quality of the paper. The author is grateful for grant funding support from the Taiwan Ministry of Science and Technology (MOST 102-2410-H-182-013-MY3) and Chang Gung Memorial Hospital (BMRP 574) during the completion of this study.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, TY. A likelihood-based assignment method for multiple criteria decision analysis with interval type-2 fuzzy information. Neural Comput & Applic 28, 4023–4045 (2017). https://doi.org/10.1007/s00521-016-2288-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00521-016-2288-6