Abstract
Recently, the TODIM (an acronym in Portuguese for Interactive Multi-criteria Decision Making) approach, which can characterize the decision makers’ psychological behaviors under risk, has been introduced to handle multiple attribute decision making (MADM) problems. Moreover, 2-tuple linguistic term set is an effective tool for depicting uncertainty of the MADM problems. In this paper, we will extend the TODIM method to the MADM with the 2-tuple linguistic information. Firstly, the definition and distance of 2-tuple linguistic information are briefly introduced, and the steps of the classical TODIM method for MADM problems are presented. Then, on the basis of the classical TODIM method, the extended TODIM method is proposed to deal with MADM problems in which the attribute values are in the 2-tuple linguistic information, and its significant characteristic is that it can fully consider the decision makers’ bounded rationality which is a real action in decision making. Finally, a numerical example for evaluating the service quality of boutique tourist scenic spot is proposed to verify the developed approach and its practicality and effectiveness.
Keywords
Introduction
Multi-criteria group decision-making (MCGDM) problems are wide-spread in real-life decision-making situations, especially with the increasing complexity of the socio-economic environment [1, 2]. In reality, decision-making information is usually uncertain or fuzzy, due to the complexity of things and in recognition of the limitations of decision makers. In order to thoroughly describe fuzzy information, Herrera and Martinez [3] proposed the 2-tuple linguistic model, composed of a linguistic term and a real number, to represent assessment information in a way that can effectively avoid information loss. Consequently, 2-tuple linguistic MCGDM problems have captured the attention of many researchers in recent years [4–6]. 2-tuple linguistic information can describe the fuzziness of decision-making information, while it seems imperfect and inaccurate to deal with information in terms of randomness. In fact, randomness and fuzziness are the most important and fundamental of all kinds of uncertainty [7, 8]. For instance, for a linguistic decision-making problem, decision maker A may think that 75% fulfillment of a task is “good”, while decision maker B may hold that less than 80% fulfillment of the same task cannot be considered to be “good” with the same linguistic term scale. So in such a way, when considering the degree of certainty of an element belonging to a qualitative concept in a specific universe, it is more feasible to allow a stochastic disturbance of the membership degree encircling a determined central value than to allow a fixed number [9, 10]. Fortunately, the cloud model can easily overcome this weakness and make decision-making processes more realistic. The cloud model, which is a quantitative and qualitative uncertainty conversion model proposed by Professor Deyi Li based on traditional fuzzy set theory and probability statistics [11], has distinct advantages in terms of dealing with vague and random decision-making information. It not only easily characterizes the concept of uncertainty in the natural language, but also reflects the intrinsic connection between randomness and fuzziness. Therefore, the cloud model can be used to depict the randomness of 2-tuple linguistic information. To the best of the authors’ knowledge, however, research on converting 2-tuple linguistic variables into clouds has not been reported in the existing literature. The decision information in some practical MAGDM situations may be unquantifiable due to its nature, or cannot be precisely assessed in a quantitative form, but may be assessed in a qualitative one. Thus, it may take the form of linguistic variables [12], such as “poor”, “fair”, and “very good”. To utilize linguistic variables, a pre-defined linguistic assessment set is needed. Unfortunately, the traditional linguistic assessment set is discrete. So in many cases, the decision information provided by DMs may not match any of the original linguistic phrases in the linguistic assessment sets, resulting in loss of information. To overcome these limitations, Herrera and Martinez [3] introduced the 2-tuple linguistic representation model of which the significant advantage is to be continuous in its domain. Therefore, it can express any counting of information in the universe of the discourse. Recently, the 2-tuple linguistic model has been widely studied. Dong, et al. [13] developed two different models based on linguistic 2-tuples to address term sets that are not uniformly and symmetrically distributed. Truck [14] stressed a comparison between the 2-tuple semantic model and the 2-tuple symbolic model, and then proved that links can be made between them. Zhu et al. [15] utilized two 2-tuples in a 2-dimension linguistic lattice implication algebra to represent a 2-dimension linguistic label for more precise computing and aggregating 2-dimension linguistic information. Xu et al. [16] proposed a four-way procedure to estimate missing preference values when dealing with acceptable incomplete 2-tuple fuzzy linguistic preference relations in group decision-making. Gong et al. [17] established an optimization model of group consensus of 2-tuple linguistic preferential relations. In addition, the 2-tuple linguistic variable has been applied to many practical MCGDM problems such as supplier selection [18, 19], material selection [20], site selection [21], emergency response capacity evaluation [22] and in-flight service quality evaluation [23].
In this paper, we will extend the TODIM method to the MADM with the 2-tuple linguistic information. Firstly, the definition and distance of 2-tuple linguistic information are briefly introduced, and the steps of the classical TODIM method for MADM problems are presented. Then, on the basis of the classical TODIM method, the extended TODIM method is proposed to deal with MADM problems in which the attribute values are in the 2-tuple linguistic information, and its significant characteristic is that it can fully consider the decision makers’ bounded rationality which is a real action in decision making. Furthermore, we extend the above results to interval neutrosophic environment. Finally, a numerical example for evaluating the service quality of boutique tourist scenic spot is proposed to verify the developed approach and its practicality and effectiveness.
Preliminaries
Herrera [24, 25] first introduced the 2-tuple fuzzy linguistic approach for overcoming the drawback of the classical computational models, which include the semantic model and symbolic model. The 2-tuple linguistic model is a kind of new information processing method. It takes 2-tuple to represent linguistic assessment information and carry out operation. The basic concept of linguistic 2-tuple is symbolic translation. The 2-tuple linguistic representation and computational model has received more and more attention since its appearance.
In the following, we shall introduce the definition of the 2-tuple linguistic representation and computational model.
Let S ={ s i |i = 1, 2, ⋯ , t } be a linguistic term set with odd cardinality. Any label, s i represents a possible value for a linguistic variable, and it should satisfy the following characteristics [24, 25]:
(1) The set is ordered: s
i
> s
j
, if i > j; (2) Max operator: max(s
i
, s
j
) = s
i
, if s
i
≥ s
j
; (3) Min operator: min(s
i
, s
j
) = s
i
, if s
i
≤ s
j
. For example, S can be defined as
Herrera and Martinez [24, 25] developed the 2-tuple fuzzy linguistic representation model based on the concept of symbolic translation. It is used for representing the linguistic assessment information by means of a 2-tuple (s i , α i ), where s i is a linguistic label from predefined linguistic term set S and α i is the value of symbolic translation, and α i ∈ [- 0.5, 0.5) .
From Definitions 1 and 2, we can conclude that the conversion of a linguistic term into a linguistic 2-tuple consists of adding a value 0 as symbolic translation:
If k < l then (s
k
, a
k
) is smaller than (s
l
, a
l
); If k = l then if a
k
= a
l
then (s
k
, a
k
), (s
l
, a
l
) represents the same information; if a
k
< a
l
then (s
k
, a
k
) is smaller than (s
l
, a
l
); if a
k
> a
l
then (s
k
, a
k
) is bigger than (s
l
, a
l
).
if d (δ1, δ2) = 0, then δ1 = δ2; d (δ1, δ2) = d (δ2, δ1); d (δ1, δ2) + d (δ2, δ3) ≥ d (δ1, δ3); 0 ≤ d (δ1, δ2) ≤ 1.
That means δ1 = δ2, so if d (δ1, δ2) =0, then δ1 = δ2 is right. d (δ1, δ2) = d (δ2, δ1) d (δ1, δ2) = d (δ2, δ1) is hold. d (δ1, δ2) + d (δ2, δ3) ≥ d (δ1, δ3)
So we complete the proof.□
So we complete the proof. (P3) d (δ1, δ2) + d (δ2, δ3) ≥ d (δ1, δ3) is hold.
So we complete the proof. (P4) 0 ≤ d (δ1, δ2) ≤ 1 is hold.
TODIM method [26] which based on prospect theory (PT) is to consider the subjectivity of DM’s behaviors, which provides the dominance of each alternative over others by some particular operation formulas; can be more reasonable and scientific in the application of MAGDM problems [27–29].
Assume that {l1, l2, … l
m
} be a group of alternatives, {c1, c2, … c
n
} be a list of criteria with weighting vector be {w1, w2, … w
n
}, thereby satisfying w
i
∈ [0, 1] and
For benefit attributes:
For cost attributes:
Assume that {l1, l2, … l
m
} be a group of alternatives, {d1, d2, … d
λ
} be a list of experts with weighting vector be {v1, v2, … v
t
}, and {c1, c2, … c
n
} be a list of criteria with weighting vector be {w1, w2, … w
n
}, thereby satisfying w
i
∈ [0, 1] , v
i
∈ [0, 1] and
Consider both the 2-tuple linguistic information and traditional TODIM method which based on prospect theory (PT), we try to propose a 2-tuple linguistic TODIM method to solve MAGDM problems effectively. The model can be depicted as follows:
For benefit attributes:
For cost attributes:
Next we construct a matrix model of dominance degree
The overall dominance δ (l
i
, l
t
) matrix can be constructed by formula (25) as follows:
Calculating steps based on MAGDM problems
In this section, we provide a numerical example for evaluating the service quality of boutique tourist scenic spot by using 2-tuple linguistic TODIM method. Assume that five possible boutique tourist scenic spots l i (i = 1, 2, 3, 4, 5) to be selected and four criteria to assess these boutique tourist scenic spots: ding172c1 is the human factors in construction projects; ding173c2 is the building materials and equipment factors; ding174c3 is the management factors; ding175c4 is the environmental factors. The five possible boutique tourist scenic spots φ i (i = 1, 2, 3, 4, 5) are to be evaluated with 2-tuple linguistic information with the four criteria by three experts d k (criteria weight ω1 = (0.4, 0.1, 0.3, 0.2), experts weight ω2 = (0.3, 0.4, 0.3) .), which are listed in Tables 1–3.
2-tuple linguistic information decision matrix by d1
2-tuple linguistic information decision matrix by d1
2-tuple linguistic information decision matrix by d2
2-tuple linguistic information decision matrix by d3
For expert d1, the dominance degree
For expert d2, the dominance degree
For expert d3, the dominance degree
In our article, we proposed the 2-tuple linguistic TODIM method based on the fundamental theories of 2-tuple linguistic sets and origin TODIM model. Firstly, we briefly introduce the definition, operation laws and the distance calculating method of 2-tuple linguistic model. Then, the calculating steps of the origin TODIM model are simply presented. Thereafter, we extend the origin TODIM model to the2-tuple linguistic TODIM method, in our proposed method; it’s more reasonable and scientific for considering the subjectivity of DM’s behaviors and the dominance of each alternative over others. Finally, a numerical example for evaluating the service quality of boutique tourist scenic spot has been proposed to illustrate the new method and some comparisons are also conducted to further illustrate advantages of the new method.
In the future, the application of the proposed models and methods of 2-tuple linguistic can be investigated in the MAGDM problems, risk analysis and many other uncertain and fuzzy environments [30–43].
Footnotes
Acknowledgment
This work is sponsored by the Social Science Foundation of Shaanxi Education Department, A Study of Optimization of Interpretive System in Tourist Destinations in Shaanxi (2013JK0301).
