Bidimensional dual hesitant fuzzy (BDHF) set is developed to present preferences of a decision maker or an expert, which is more objective than existing fuzzy sets such as Atanassov’s intuitionistic fuzzy set, hesitant fuzzy set, and dual hesitant fuzzy set. Then, after investigating some distance measures, we define a new generalized distance measure between two BDHF elements with parameter
Multiple attribute decision analysis (MADA) is used to deal with the problems of making an optimal choice from alternatives or generating a ranking order of alternatives in terms of attributes [
In practical MADA problems, the decision maker may feel difficulty in expressing his/her preferences exactly for the reasons of environmental uncertainty, the limitation of his/her knowledge and experience, and the urgency of time. To address such situations, many expression forms are developed to express information of a decision maker, such as intervalvalued number [
In order to address MADA problems with BDHF information, this paper proposes an extended VIKOR method. Firstly, a new generalized distance measure between BDHF sets is proposed to compare alternatives in MADA. This distance measure, as the generalization of existing distance measures, such as AIF distance measure [
The main contributions of this paper include the following:
The rest of this paper is organized as follows. Section
In this section, we review some concepts related to the proposed MADA method with BDHF information.
BDHF set can be regarded as a generalization of DHF set, as mentioned in Introduction. To make the concept of BDHF set clear, the concept of DHF set is introduced below. Here, DHF set can be considered a combination of AIF set and HF set.
Given a universe of discourse
Given
It is noted that the lengths of different HF sets may be different and thus different DHF sets may also have various lengths.
When VIKOR method is applied to analyze MADA problems, distances between decision information and ideal solutions are used to compare alternatives. As such, distance measure between BDHF information is needed for the developed method. Meanwhile, as a generalization of DHF set, BDHF set is closely related to AIF set and HF set. In order to effectively measure the distance among BDHF information, the existing generalized distance measure among AIF information and that among HF information are defined below.
Let
Let
As mentioned in Section
Optimists expect desirable outcomes and add the maximum value [
Pessimists anticipate unfavorable outcomes and may add the minimum value [
Neutrals look forward to unbiased outcomes and may add the average value [
In real life, even if the risk attitude of a decision maker is clear, the degree to which the decision maker prefers riskseeking or riskaversion may be also unknown. In other words, a decision maker with riskseeking to some extent is not certainly sure that the maximum value should be added to the shorter HF element. It is the similar case for pessimists. More importantly, any strategy has no ability to perfectly reflect the original preference of a decision maker, especially when one or more values in HF element appear several times originally.
In MADA, the entropy of decision information is usually employed to determine attribute weights. This idea is also adopted in the developed method. That is, the entropy of BDHF information is measured to create attribute weights. To do this, the entropy of AIF information is presented as necessary foundation.
A real function
Let
In this section, BDHF element is defined based on the concept of DHF element and the distance between BDHF elements is measured in order to develop the VIKOR method with BDHF information.
Reconsidering the supplier selection problem in Introduction, which is a MADA problem, we can find that when three experts give different preferences about selecting and deselecting one supplier, the union of the preferences can naturally form the DHF preference of a decision maker on the assumption that each expert is equivalently important for the decision maker. However, when two or three experts provide the same preference, the resulting DHF preference can be regarded as a transformation from dual intuitionistic fuzzy multiset to DHF set in the abstract [
Suppose a decision maker provides a DHF element
When introducing the concept of support intensity of membership and nonmembership degrees, information implied by dual intuitionistic fuzzy multiset can be effectively covered. That is, when dual intuitionistic fuzzy multiset is transformed into DHF set with support intensity of membership and nonmembership degrees, it can be recovered without information loss. Activated by this idea, DHF set is extended to be BDHF set, as defined below.
Let
For simplicity, a BDHF element is still denoted by
From Definition
Based on the original information provided by all information resources, a decision maker can provide BDHF assessments related to some alternatives on several attributes. In this process, it is required for a decision maker that
a BDHF assessment once provided by the decision maker will not be changed no matter whether other BDHF assessments are known or not;
the decision maker specifies membership degrees in a BDHF assessment without considering nonmembership degrees;
the decision maker specifies nonmembership degrees in a BDHF assessment without considering membership degrees.
Assumption
Then, under Assumption
Let
Here, under Assumption
In addition, it can be obtained from Definition
Then, the basic operations of BDHF elements can be defined in the following.
Let
As mentioned in Introduction, the computation of distance measure between BDHF elements is a key step in the proposed VIKOR method. From Definition
Let
From Definition
Let
Because the distance measure between BDHF elements consists of all combinations of AIF numbers with their support intensity, then, combining Definition
Let
As analyzed in Section
Then, in order to demonstrate the process of calculating the distance measure in Definition
Let
Obviously, it can be seen from Example
Three special cases of the generalized distance measure in Definition
When
When
When
The generalized distance measure in Definition
Given two BDHF elements,
Boundedness is
Commutativity is
Conditional reflexivity is
Theorem
As mentioned in Definition
If a function
Based on Definitions
Let
In this equation, it is obvious that
The numerator of
To achieve the mentioned work simply, we firstly design a new function denoted by
Suppose that
Lemma
Suppose that
Here,
Suppose that
Theorem
Let
The key characteristic of the distance measure proposed in this paper is to combine membership or nonmembership degrees with their support intensity. Now, if we do not consider support intensity of membership or nonmembership degrees, could different distance measures between BDHF elements be obtained? It can be calculated in Example
After obtaining the generalized distance measure between two BDHF elements, the generalized distance measure between two BDHF sets could also be defined in the following.
Let
Similar to the generalized distance measure in Definition
Suppose that
Boundedness is
Commutativity is
Conditional reflexivity is
Theorem
In this section, we introduce the proposed method, mainly including the modeling of MADA problems under BDHF environment, determination of attribute weights, and the process of extended VIKOR method.
Differing from conventional MADA approaches, fuzzy decision matrix with BDHF assessments is constructed in this section.
Suppose that a MADA problem includes
It can be known from Definition
Weight, as a useful technique for reflecting relative importance of objectives, has been widely applied in MADA. To date, a lot of methods for determining weights are proposed such as AHP, maximizing deviation method, the CRITIC (criteria importance through intercriteria correlation) method [
Let
Through observing the features of entropy measure in Definition
Given a function
when
when
Theorem
Based on Definition
A real function
Theorem
After obtaining the new entropy measure, the weight of attribute
According to entropy theory, the smaller the entropy value for each attribute is, the more the useful information the decision maker can acquire. Therefore, the attribute should be assigned a bigger weight and vice versa.
VIKOR method is used to find the compromise solution which is the closest to the ideal solution, and a compromise means an agreement established by mutual concessions [
Then, the procedure of the VIKOR method in BDHF context to deal with MADA problems is shown in Figure
The procedure of the proposed method.
Identify
Construct a BDHF decision matrix in which BDHF assessments are provided by a decision maker.
Create attribute weights based on the extended entropy measure.
Determine the positive ideal solution (PIS)
Compute the values
Calculate the values
Rank the alternatives, sorting by the values
Propose a compromise solution in which the alternative
If one of these conditions is not satisfied, then a set of compromise solutions is proposed, which consists of
alternative
alternative
In this section, an assessment problem of people’s livelihood project is analyzed by the proposed fuzzy VIKOR method to demonstrate its validity and applicability.
Premier Li in China, on behalf of the State Council, in the Report on the Work of the Government 2015 pointed out that part of the important work of the government in 2015 was to strengthen safeguards for people’s standard of living which is the basis of achieving steady and sound economic development and ensuring social harmony and stability. Establishing people’s livelihood project is the product of the mentioned work. Here, people’s livelihood is a concept reflecting specific characteristic of China. The regions which have different performances in people’s livelihood project can be considered to be assigned different resources and opportunities to enhance their development. So, how to assess people’s livelihood of different regions is a hot topic attracting wide range of attention, which can be regarded as a MADA problem.
In this section, taking seven regions in East China as an example, we investigate how to help a decision maker which is an official from Ministry of Civil Affairs of the People’s Republic of China to select several relatively better ones of these regions to be demonstration zones which can be given top priority to obtain resources and support. The decision maker firstly identifies seven regions including Shanghai
In order to help the decision maker generate BDHF assessments, five experts are invited from Department of Housing and UrbanRural Development of Anhui Province, Department of Civil Affairs of Anhui Province, and Hefei University of Technology to independently and anonymously evaluate the seven regions on each attribute. Then, the decision maker combines the experts’ opinions and his preference to provide a BDHF decision matrix
BDHF decision matrix for the problem of evaluating people’s livelihood projects.
Attributes     

    
 
    
 
    
 
    
 
    
 
    
 
    
 
Attributes     
 
    
 
    
 
    
 
    
 
    
 
    
 
    

Due to the nature of the problem of evaluating people’s livelihood projects, there is no information available for the decision maker to subjectively assign attribute weights. As a result, the entropy method in Section
When the attribute weights for the problem are known, the distance between each alternative and positive ideal solution and that between positive and negative ideal solutions can be measured and further applied to generate
Distance measure between each alternative and positive ideal solution on each attribute.
Attributes        

 0.3  0.36  0.26  0.18  0.18  0.54  0.28 
 0.32  0.24  0.3  0.34  0.28  0.32  0.26 
 0.32  0.42  0.18  0.26  0.34  0.24  0.42 
 0.54  0.3  0.56  0.56  0.38  0.36  0.46 
 0.62  0.66  0.44  0.42  0.5  0.52  0.42 
 0.46  0.46  0.5  0.5  0.34  0.4  0.56 
 0.18  0.2  0.36  0.32  0.38  0.52  0.34 
Because the parameter
The ranking order of the regions by
Attributes        

By  0.287  0.295  0.312  0.454  0.511  0.459  0.323 
Ranking  1  2  3  5  7  6  4 
By  0.058  0.055  0.062  0.091  0.099  0.082  0.063 
Ranking  2  1  3  6  7  5  4 
By  0.028  0.017  0.142  0.787  1  0.685  0.167 
Ranking  2  1  3  6  7  5  4 
In order to further obtain compromise solution, the two conditions in Step
From Table
Similar to VIKOR method, TOPSIS (technique for order preference by similarity to ideal solution) is also a distancebased method. TOPSIS method proposed by Hwang and Yoon [
Determine the positive ideal solution
Calculate the separation measures using the proposed distance measure in Definition
Calculate the closeness coefficient of each alternative.
Rank the alternatives through the value of closeness coefficient as
The ranking order of alternatives using VIKOR method is slightly different from that using TOPSIS method, especially in the selection of the appropriate alternatives. In the TOPSIS method, a solution is selected with the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution, while VIKOR method focuses on finding a compromise solution considering the balance between total satisfaction and individual regret. As the TOPSIS method ignores the relative importance of the mentioned distances [
Thus, from the application of the proposed method in evaluating people’s livelihood project of several regions and the above comparative analysis, the main advantages of the proposed method are demonstrated as follows from both practical and theoretical perspectives:
It proposes BDHF set to help a decision maker express their preference in solving a MADA problem which is more objective and scientific than many existing fuzzy sets. From the practical perspective, BDHF set is more suitable in solving a practical MADA problem such as the evaluation of people’s livelihood project of several regions in China.
It defines a new generalized distance measure between BDHF assessments to avoid missing or changing much original information provided by the decision maker, which is more suitable for the most realworld decision making, for example, the evaluation problem of people’s livelihood project of several regions in China.
It determines attribute weights by using a new entropy measure with BDHF information, which can avoid subjective and arbitrary judgment of the decision maker.
It generates a compromise solution instead of an optimum solution to help the decision maker achieve decision goal, which is more suitable for addressing a practical MADA problem, such as the evaluation problem of people’s livelihood project of several regions in China.
In addition, the complexity of the proposed method is discussed from the following three aspects:
Overall, the proposed method is easy to be implemented in solving practical MADA problems.
Combining the advantages of AIF information with that of HF information, BDHF information is defined in this paper to present the preferences of a decision maker or an expert. In order to solve MADA problems with BDHF information, a new VIKOR method is proposed. Here, attribute weights are determined by a new entropy measure with BDHF information. In VIKOR method, the definition of distance measure is a key step. However, the effect of parameter
Although the proposed method in this paper provides a way to find solutions to MADA problems with BDHF assessments, it cannot analyze more complex problems where intervalvalued BDHF assessments are given by a group of decision makers. To address such challenge, we intend to extend the method to analyze group decision making problems with intervalvalued BDHF assessments.
It can be deduced from Definition
Besides, it can be inferred that
Thus, property
Definition
If
Suppose that
On the assumption that
The assumption of
Therefore, Lemma
From the discussion after Lemma
Therefore, it can be inferred from
Thus, Theorem
Firstly, taking the partial derivative of
Secondly, one can find the critical point of
Finally, Theorem
When each BDHF element reduced to a crisp set, it can be inferred that
If
Because
Definition
Thus, Theorem
The authors declare that they have no competing interests.
This research was supported by the Foundation for Innovative Research Groups of the Natural Science Foundation of China (no. 71521001), the National Key Basic Research Program of China (no. 2013CB329603), and the National Natural Science Foundation of China (nos. 71571060, 71201043, 71303073, 71501054, and 71131002).