Research Article  Open Access
Interval 2Tuple Linguistic Distance Operators and Their Applications to Supplier Evaluation and Selection
Abstract
With respect to multicriteria supplier selection problems with interval 2tuple linguistic information, a new decision making approach that uses distance measures is proposed. Motivated by the ordered weighted distance (OWD) measures, in this paper, we develop some interval 2tuple linguistic distance operators such as the interval 2tuple weighted distance (ITWD), the interval 2tuple ordered weighted distance (ITOWD), and the interval 2tuple hybrid weighted distance (ITHWD) operators. These aggregation operators are very useful for the treatment of input data in the form of interval 2tuple linguistic variables. We study some desirable properties of the ITOWD operator and further generalize it by using the generalized and the quasiarithmetic means. Finally, the new approach is utilized to complete a supplier selection study for an actual hospital from the healthcare industry.
1. Introduction
The ordered weighted averaging (OWA) operator [1] is a very wellknown aggregation operator, providing a parameterized family of aggregation operators which includes the maximum, the minimum, and the average. The prominent characteristic of the OWA operator is the reordering step. An interesting extension of the OWA is the use of distance measures in the OWA operator. In this respect, Xu and Chen [2] developed the ordered weighted distance (OWD) measure, which is the generalization of a variety of wellknown distance measures, such as the normalized Hamming distance, the normalized Euclidean distance, and the normalized geometric distance. The prominent characteristic of the OWD measure is that it can relieve (or intensify) the influence of unduly large or unduly small deviations on the aggregation results by assigning them low (or high) weights. MerigÃ³ and GilLafuente [3] proposed a technique for decision making using the OWA operator to calculate Hamming distance and introduced the ordered weighted averaging distance (OWAD) operator. The main advantage of this operator is that it can take into account the attitudinal character of a decision maker in the aggregation process; thus the decision maker is able to consider the decision problem more clearly according to his or her interests. MerigÃ³ et al. [4] introduced the probabilistic ordered weighted averaging distance (POWAD) operator, which uses a unified model between the probability and the OWA operator considering the degree of importance that each concept has in the aggregation. Zeng et al. [5] further extended the POWAD operator to deal with uncertain environments represented in the form of interval numbers and proposed the uncertain probabilistic ordered weighted averaging distance (UPOWAD) operator. MerigÃ³ et al. [6] studied the use of distance measures and heavy aggregations in the OWA operator and presented the heavy ordered weighted averaging distance (HOWAD) operator. It is a new aggregation operator that provides a parameterized family of aggregation operators between the minimum distance and the total distance operator.
In addition, MerigÃ³ and Casanovas [7] introduced the linguistic ordered weighted averaging distance (LOWAD) operator for linguistic decision making. Zeng and Su [8] and Zeng [9] considered the situations with intuitionistic fuzzy and intervalvalued intuitionistic information and developed some intuitionistic fuzzy weighted distance measures. Xu [10] developed some fuzzy ordered distance measures for group decision making with linguistic, interval, triangular, or trapezoidal fuzzy preference information. Xian and Sun [11] developed the fuzzy linguistic induced Euclidean ordered weighted averaging distance (FLIEOWAD) operator for group linguistic decision making, in which the criteria values take the form of fuzzy linguistic information. More recently, other different types of distance measures have been proposed in the literature, like the uncertain induced heavy ordered weighted averaging distance (UIHOWAD) operator [12], the continuous intuitionistic fuzzy ordered weighted distance (CIFOWD) measure [13], the Pythagorean fuzzy ordered weighted averaging weighted average distance (PFOWAWAD) operator [14], the fuzzy linguistic induced ordered weighted averaging Minkowski distance (FLIOWAMD) operator [15], the generalized intervalvalued 2tuple linguistic weighted distance measures [16], the generalized hesitant fuzzy linguistic weighted distance measures [17], and so on [18â€“21].
In many situations, however, the input arguments may take the form of interval 2tuple linguistic variables [25â€“27] because of time pressure, lack of knowledge or data, and decision makersâ€™ limited attention and information processing capabilities. Furthermore, decision makers may use different linguistic term sets to express their evaluations on the established selection criteria considering their personal backgrounds, preferences, and different understanding levels to the alternatives. Therefore, it is necessary to extend the ordered weighted distance measures to accommodate the interval 2tuple linguistic environment [28â€“30], which is also the focus of this paper. For doing so, we will develop some interval 2tuple linguistic distance operators such as interval 2tuple weighted distance (ITWD), interval 2tuple ordered weighted distance (ITOWD), and interval 2tuple hybrid weighted distance (ITHWD) operators. These aggregation operators are very effective to deal with situations where the input data are expressed in interval 2tuple linguistic variables. We study some desirable properties of the ITOWD operator and further generalize it by using the generalized and the quasiarithmetic means obtaining the generalized interval 2tuple ordered weighted distance (GITOWD) and the quasiarithmetic interval 2tuple ordered weighted distance (QuasiITOWD) operators. Finally, based on the GITOWD operator, we develop an approach to group supplier evaluation and selection with interval 2tuple linguistic information and illustrate it with a numerical example.
The remainder of this paper is set out as follows. In Section 2, we introduce some basic concepts and operation laws of interval 2tuple linguistic variables. In Section 3, we develop the ITWD, the ITOWD, and the ITHWD operators and investigate some desirable properties of the ITOWD operator. In Section 4, we present an approach based on the developed interval 2tuple linguistic distance operators to multicriteria group supplier selection. A supplier selection example is given in Section 5 to verify the proposed approach and to demonstrate its feasibility and practicality. Finally, conclusions and future directions are provided in Section 6.
2. Preliminaries
2.1. 2Tuple Linguistic Variables
The 2tuple linguistic representation model was firstly presented in [31] based on the concept of symbolic translation. It is used to represent linguistic information by means of a linguistic 2tuple, , where is a linguistic term from the predefined linguistic term set and is a numerical value representing the symbolic translation. In the classical 2tuple linguistic approach, the range of is between 0 and , which is relevant to the granularity of a linguistic term set. Here, is the result of an aggregation of the indices of a set of labels assessed in the linguistic term set . To overcome this restriction, Tai and Chen [32] proposed a generalized 2tuple linguistic model and translation functions.
Definition 1. Let be a linguistic term set and let be a value representing the result of a symbolic aggregation operation. Then the generalized translation function used to obtain the 2tuple linguistic variable equivalent to is defined as follows [32]:where is the usual rounding operation, has the closest index label to , and is the value of the symbolic translation.
Definition 2. Let be a linguistic term set and let be a 2tuple. There exists a function which is able to convert a 2tuple linguistic variable into its equivalent numerical value . The reverse function is defined as follows [32]:Particularly, it is necessary to point out that the conversion of a linguistic term into a linguistic 2tuple consists of adding a value 0 as symbolic translation [31]:The comparison of linguistic information represented by 2tuples is carried out according to an ordinary lexicographic order.
Definition 3. Let and be two 2tuples; then [31, 33](1)if , then is smaller than ;(2)if , then one has the following:(a)if = , then is equal to ;(b)if , then is smaller than ;(c)if , then is bigger than .
2.2. Interval 2Tuple Linguistic Variables
Motivated by Xuâ€™s uncertain linguistic variables [22] and based on the definitions of [32], Zhang [34] proposed the interval 2tuple linguistic representation model as generalization of the 2tuple linguistic variables. Due to its characteristics and advantages, the interval 2tuple linguistic representation model has been widely applied for dealing with uncertainty in various multicriteria decision making problems [35â€“38]. It can be defined as follows.
Definition 4. Let be a linguistic term set. An interval 2tuple linguistic variable is composed of two 2tuples, denoted by , where , , and represent the linguistic label of and symbolic translation, respectively. The interval 2tuple that expresses the equivalent information to an interval value is derived by the following function [25, 34]:On the contrary, there is always a function such that an interval 2tuple can be converted into an interval value as follows:In particular, if and , then the interval 2tuple linguistic variable reduces to a 2tuple linguistic variable.
Note that the uncertain linguistic variable [32] is simpler than the interval 2tuple linguistic variable, which can also be used for dealing with group decision making with uncertain linguistic information. However, compared with the uncertain linguistic variables, Zhangâ€™s interval 2tuple linguistic variables have the following advantages [31, 34, 39]. () The interval 2tuple linguistic variable has exact characteristic in linguistic information processing, which can effectively avoid information distortion and loss in the linguistic information processing. In contrast, the uncertain linguistic variable performs the retranslation step as an approximation process to express the results in the original expression domain (initial discrete linguistic term set) provoking a lack of accuracy. () In the process of aggregating uncertain linguistic information, the operational laws of uncertain linguistic variables are not closed. Let be a linguistic term set; we have and . Clearly, the results of operation exceed the range of the linguistic term set . This problem is solved by employing the interval 2tuple linguistic variables. () Decision makers can express their preferences by the use of linguistic term sets with different granularity of uncertainty and their judgments can be better expressed with interval 2tuples from the preestablished linguistic term sets. But the uncertain linguistic variables may lead to inflexibility for managing the group decision making problem with multigranularity linguistic information.
Based on the operations of uncertain linguistic variables [22], Zhang [34] further gave some basic operational laws of interval 2tuples and proposed the interval 2tuple weighted average (ITWA) operator.
Definition 5. Consider any three interval 2tuples, , , and , and let ; then their operations are defined as follows [34]:(1);(2).
Definition 6. Let be a set of interval 2tuples and let be their associated weights, with . The ITWA operator is defined as [34]Inspired by the distance measure in uncertain linguistic environment [40], the distance between interval 2tuples can be defined below.
Definition 7. Let and be any two interval 2tuples; thenis called the interval 2tuple distance between and . Particularly, if the interval 2tuples and are degenerated to 2tuples and , then the interval 2tuple distance will become the 2tuple distance [41].
3. Interval 2Tuple Linguistic Distance Operators
The ordered weighted averaging distance (OWAD) operator [3] is an extension of the traditional Hamming distance by using the OWA operator, which provides a parameterized family of aggregation operators ranging from the minimum to the maximum distance. For two real numbers sets and , the OWAD operator is defined as follows.
Definition 8. An OWAD operator of dimension is a mapping OWAD: which has an associated weighting vector , with and , such thatwhere represents the th largest of the individual distance .
Further, Xu and Chen [2] developed an ordered weighted distance (OWD) measure, which generalizes a variety of wellknown distance measures and aggregation operators.
Definition 9. An OWD measure of dimension is a mapping OWD: which has an associated weighting vector , with and , such thatwhere is the th largest of the individual distance . Ifâ€‰â€‰, then the OWD measure is reduced to the OWAD operator; ifâ€‰â€‰ and , then the OWD measure is reduced to the normalized Hamming distance.
3.1. Interval 2Tuple Linguistic Distance Operators
The OWAD and the OWD operators have only been used in the situations in which the input arguments are exact values. However, judgments of people depend on personal psychological aspects such as experience, learning, situation, and state of mind. It is more suitable for decision makers to provide their preferences by means of linguistic variables rather than numerical ones.
For convenience, let be the set of all interval 2tuples, let be the set of all 2tuples, and let and be two sets of interval 2tuples. Based on (7), we define an interval 2tuple weighted distance (ITWD) operator as follows.
Definition 10. An ITWD operator of dimension is a mapping ITWD: , which has an associated weight vector with and , such thatwhere is the interval 2tuple distance between and .
In particular, if , then the ITWD becomes the interval 2tuple normalized distance (ITND) operator:If the sets of interval 2tuples and are degenerated to the sets of 2tuples and , then the ITWD is reduced to the 2tuple weighted distance (TWD) operator:where is the 2tuple distance between and .
Based on the OWA and the ITWD operators, we define an interval 2tuple ordered weighted distance (ITOWD) operator as follows.
Definition 11. An ITOWD operator of dimension is a mapping ITOWD: , which has an associated weight vector , with and , such thatwhere is the th largest of the interval 2tuple distance .
Particularly, if there is a tie between and , then we replace each of and by their average in the process of aggregation. If items are tied, then we replace these by replicas of their average. If , then the ITOWD becomes the ITND; if the position of is the same as the ordered position of , then the ITWD is obtained. Moreover, if and are degenerated to and , then the ITOWD is reduced to the 2tuple ordered weighted distance (TOWD) operator:where is the th largest of the 2tuple distance .
Similar to the OWAD operator, the ITOWD operator is commutative, monotonic, idempotent, and bounded. These properties can be shown with the following theorems.
Theorem 12 (commutativityOWA aggregation). Assume that is the ITOWD operator. If is any permutation of the arguments , then
Theorem 13 (commutativitydistance measure). Assume that is the ITOWD operator; then
Theorem 14 (monotonicity). Assume that is the ITOWD operator. If , for all , then
Theorem 15 (idempotency). Assume that is the ITOWD operator. If , for all , then
The proofs of the above theorems are straightforward and thus are omitted.
Theorem 16 (bounded). Assume that is the ITOWD operator; then
Proof. Let , and let ; thenTherefore,
Another important issue is the determination of the weighting vector associated with the ITOWD operator. In the literature, various methods have been suggested for the OWA weights generation, which can also be implemented for the ITOWD operator, such as the normal distribution based method [42], the maximum Bayesian entropy method [43], and the least squares based method [44]. Inspired by [8, 45], in the following, we give three ways to determine the ITOWD weights.(1)Let â€‰then , and .(2) Let â€‰then , and .(3)Let â€‰then we defineâ€‰from which we get , and .
Note that the weight vector derived from (22) is a monotonic decreasing sequence, the weight vector derived from (23) is a monotonic increasing sequence, and the weight vector derived from (25) combines the above two cases; that is, the closer the value to the mean , the larger the weight .
Clearly, the fundamental characteristic of the ITWD operator is that it considers the importance of each given interval 2tuple distance, whereas the fundamental characteristic of the ITOWD operator is the reordering step, and it weights all the ordered positions of the interval 2tuple distances instead of weighting the given interval 2tuple distances themselves. Motived by the idea of the linguistic hybrid geometric averaging (LHGA) operator [42], in the following, we develop an interval 2tuple hybrid weighted distance (ITHWD) operator that weights both the given interval 2tuple distances and their ordered positions.
Definition 17. An ITHWD operator of dimension is a mapping ITHWD: , which has an associated weight vector , with and , such thatwhere is the th largest of the weighted interval 2tuple distance , is the weight vector of , with and , and is the balancing coefficient.
In particular, if , then the ITHWD is reduced to the ITOWD operator; if , then the ITHWD is reduced to the ITWD operator. Moreover, if and are degenerated to and , then the ITHWD is reduced to the 2tuple hybrid weighted distance (THWD) operator:where is the th largest of the weighted 2tuple distance , is the weight vector of , with and , and is the balancing coefficient.
3.2. Generalizations of the ITOWD Operator
In what follows, generalizations of the ITOWD operator are presented by using the generalized and the quasiarithmetic means.
Definition 18. A generalized interval 2tuple ordered weighted distance (GITOWD) operator of dimension is a mapping GITOWD: , which has an associated weight vector , with and , such thatwhere is the th largest of the interval 2tuple distance and is a parameter such that .
Similar to the OWA and the GOWA operators [1, 46], the GITOWD operator has many desirable properties, such as commutativity, monotonicity, boundedness, and idempotency. Particularly, if there are ties between interval 2tuple distances, as in the case of the ITOWD operator, we replace each of the tied arguments by their generalized mean in the process of aggregation. If and are degenerated to and , then we can get the generalized 2tuple ordered weighted distance (GTOWD) operator. The GITOWD operator provides a parameterized family of aggregation operators. In order to study this family, we can analyze the weighting vector or the parameterâ€‰â€‰. By choosing a different manifestation of the weighting vector in the GITOWD operator, we are able to obtain different types of distance operators:(i)The interval 2tuple maximum distance is found if and , for all .(ii)The interval 2tuple minimum distance is found if and , for all .(iii)The generalized interval 2tuple normalized distance (GITND) operator is formed when , for all .(iv)The generalized interval 2tuple weighted distance (GITWD) operator is obtained when the position of is the same as the ordered position of .Some special cases can also be obtained with the change of the parameterâ€‰â€‰:(i)If , then the GITOWD is reduced to the ITOWD operator.(ii)If , then the GITOWD is reduced to the interval 2tuple ordered weighted geometric distance (ITOWGD) operator.(iii)If , then the GITOWD is reduced to the interval 2tuple ordered weighted harmonic distance (ITOWHD) operator.(iv)If , then the GITOWD is reduced to the interval 2tuple ordered weighted Euclidean distance (ITOWED) operator.(v)If , then the GITOWD is reduced to the interval 2tuple ordered weighted cubic distance (ITOWCD) operator.
Definition 19. A quasiarithmetic interval 2tuple ordered weighted distance (QuasiITOWD) operator of dimension is a mapping QuasiITOWD: , which has an associated weight vector , with and , such thatwhere is the th largest of the interval 2tuple distance and is a general continuous strictly monotone function.
As we can see, the GITOWD operator is a particular case of the QuasiITOWD operator when . If and are degenerated to and , then we can get the quasiarithmetic 2tuple ordered weighted distance (QuasiTOWD) operator. Note that all properties and particular cases commented in the GITOWD operator can also be discussed in this generalization.
4. The Proposed Multicriteria Group Supplier Selection Method
In this section, we develop an approach based on the proposed interval 2tuple linguistic distance operators for solving multicriteria group supplier selection problems.
Suppose that a group supplier selection problem has decision makers , alternatives , and decision criteria . Each decision maker is given a weight satisfying to reflect his/her relative importance in the supplier selection process. Let be the linguistic decision matrix of the th decision maker, where is the linguistic information provided by on the assessment of with respect to . In addition, decision makers may use different linguistic term sets to express their assessment values.
Next, we apply the ITWA and the GITOWD operators for multicriteria group supplier selection under interval 2tuple linguistic environment.
Step 1. Convert the linguistic decision matrix into the interval 2tuple linguistic decision matrix , where and .
Suppose that provides his assessments in a set of five linguistic terms : The linguistic information provided in the decision matrix can be converted into corresponding interval 2tuple linguistic assessments according to the following ways:(i)A certain grade such as poor, which can be written as (ii)An interval such as poormedium, which means that the assessment of an alternative concerning the criterion under consideration is between poor and medium. This can be expressed as .
Remark 20. In particular supplier selection problems, there exist many situations where information may be unquantifiable due to its nature, or the precise quantitative information may be unavailable or the cost for its computation is too high. Thus, it is more reasonable and natural for decision makers to make their judgments by using linguistic expressions. Generally, three main methods have been introduced for dealing with qualitative assessments [31, 39, 47]. The first method is based on membership functions [48], which converts linguistic information into fuzzy numbers by means of a membership function. However, this method led to a certain degree of information loss in the transformation process. The second method is based on linguistic symbols [49] that made computations on the subscripts of linguistic terms and was easy to operate. However, this approach may lead to inflexibility for different semantics. The third method is based on linguistic 2tuples [29], which can avoid the information distortion and loss in linguistic information processing and has been widely utilized for managing linguistic MCDM problems. Therefore, to deal with linguistic information more reasonably and accurately, the decision maskersâ€™ assessments on alternative suppliers are first transformed into interval 2tuples in the proposed approach.
Step 2. Utilize the ITWA operator to aggregate all the interval 2tuple linguistic decision matrices into a collective interval 2tuple linguistic decision matrix .
Step 3. Determine the ideal level of each criterion in order to characterize the collective ideal alternative , where
Step 4. Calculate the separation measure of each alternative from the ideal alternative by using the GITOWD operator:where is the th largest of the interval 2tuple distance ; is the weighting vector of the GITOWD operator such that and . Note that it is possible to consider a wide range of GITOWD operators such as those described in Section 3.
Step 5. Rank all the alternatives and select the best one(s) according to the increasing order of their separation measures.
Step 6. End.
5. An Illustrative Example
5.1. Example Illustration
In this section, we develop an illustrative example of the new approach in a group decision making problem of supplier selection. Suppose that a tertiary care hospital desires to select the most appropriate supplier for one of the key medical devices in the general anesthesia process. After preliminary screening, six suppliers, , have remained as alternatives for further evaluation. In order to evaluate the alternative suppliers and select the best one, an expert committee of three decision makers, , and , has been formed. The selection decision is made on the basis of one objective and five criteria . These criteria, which are critical for the supplier selection, are defined as follows:â€‰: technical capabilityâ€‰: delivery performanceâ€‰: product qualityâ€‰: flexibilityâ€‰: price/cost
The three decision makers employ different linguistic term sets to assess the suitability of the suppliers with respect to the above selection criteria. Specifically, provides his assessments by using the linguistic term set ; provides his assessments using ; provides her assessments using . These linguistic term sets are denoted as follows:The linguistic assessments of the six alternatives on each criterion provided by the three decision makers are presented in Table 1.

With this information, we can make an aggregation in order to make a decision. First, we convert the linguistic decision matrix shown in Table 1 into the interval 2tuple linguistic decision matrix , which is depicted in Table 2. Then, we aggregate the information of the three experts to obtain a collective interval 2tuple linguistic decision matrix. We use the ITWA operator to obtain this matrix while assuming that . The results are shown in Table 3.
