Approach for Multiattribute Decision-Making with an Interval Grey Number Based on Bonferroni Mean
Aiming at the multiattribute decision-making problem in the form of interval grey number, considering the interrelationship between the scheme attributes, this study defines the grey number weighted geometric Bonferroni mean operator for aggregating information. The feasibility of the proposed method is verified through a comparison of the example analysis and methods.
Multiattribute decision-making problems often occur in daily life; thus, they have attracted much attention with rapid development, and their methods are very important. Information forms are also diverse. Decision information with “poor information and few data” of grey uncertain characteristics  can often be expressed by grey number. A grey number has a certain range of values but cannot determine its specific value, and it can be operated to generate meaningful information.
Using the grey number as the basic element, it has become increasingly extensive to establish the model under the grey information. The grey model has many applications in forecasting [2, 3] and decision-making  and has achieved good results. Scholars have attempted to analyze the grey number problems from multiple perspectives when solving the decision-making problems. There has been some discussion on the basic definition of kernel and grey degree. Dang et al.  established the fixed weight clustering model, and Liu et al.  proposed a decision method for grey correlation comprehensive closeness and similarity. An integration operator is used to process the information in the decision problem, which is beneficial for obtaining a better decision choice. For the information in the scheme, in , which reduced the influence of interrelationships of attributes based on the Choquet integral, the dynamic order weighted average operator was used for aggregation to reduce the influence of subjective factors . In , the intuitionistic grey number was defined with its operator, and the grey decision model on the general grey number was studied. Although the integration operator is used in these decision models to try to reduce the influence of the relationship between attributes, the problem of correlation between attributes is not solved. The Bonferroni mean operator considers the interrelationship between attributes.
So far, few studies have used the Bonferroni mean operator in grey decision information , but there have been many studies in uncertain linguistic and fuzzy information environments. Bonferroni  first proposed the Bonferroni mean (BM) operator, which can be used for the integration between the operators. Subsequently, the aggregation property [12–14] of the Bonferroni mean operator was explored and applied to uncertain information . Dutta and Guha  made a decision regarding the scheme by aggregating the partition Bonferroni mean operator with partition attributes. In recent years, the development trend of the Bonferroni mean operator has been good in the hesitant and fuzzy environments. To overcome the limitation that operators consider only the real number and cannot handle other types of data, more systematic research results have been achieved on the theory and properties of Bonferroni mean operators, such as fuzzy Bonferroni mean operators [17, 18], fuzzy weighted average Bonferroni mean operators, weighted geometric Bonferroni mean operators, and generalized intuition fuzzy Bonferroni mean operators . Meanwhile, that has proposed the generalized hesitation fuzzy Bonferroni mean operator , the interval hesitation fuzzy Bonferroni mean operator , and the interval hesitation fuzzy geometric Bonferroni mean operator  in a hesitation fuzzy environment. It is also feasible to use the Bonferroni mean operator to solve multiattribute decision-making problems [23–25].
Under the excitation of the ideal Bonferroni mean and geometric Bonferroni mean, the study extends several Bonferroni mean operators under grey number information. Considering the possibility of interrelationship between the attributes, the operators are used to aggregate information on the alternatives. The scheme is selected according to the evaluation value.
2. Basic Concepts and Theories
In this section, we recall the concepts and definitions of interval grey numbers, and we reviewed the concepts of , which are the foundations of the correlation operator methods and the decision-making methods we proposed.
2.1. The Concepts of Grey Number
Definition 1 (see ). Let be an unknown real number within a union set of closed or open grey intervals, where ; if , is an integer and , , and , for any grey interval , then is called as a general grey number. and are called the lower and upper limits of , respectively, where the interval grey number and real number are special cases of general grey number.
Definition 2 (see ). A grey number which has both upper bounds and lower bounds is referred to as an interval grey number. For any interval number , it is obvious that . Assuming that the emergence background or universe of grey number is , is the measurement of the number field of grey number . Then, we haveThe degree of greyness of the grey number satisfies properties given as follows:(1)(2) when .(3) is directly proportional to and inversely proportional to .
Definition 3 (see ). Assume that thenThe union of the grey numbers and , and their intersection, is
2.2. The Bonferroni Mean (BM) Operator
Definition 4 (see ). If and is a collection of a non-negative real numbers. Then the aggregation functionsis called the Bonferroni mean (BM) operator.
3. The Grey Number Bonferroni Mean Operator
3.1. The Concept of Grey Number Bonferroni Mean Operator
The BM operator usually operates in a real number environment and that under the interval grey number is defined as follows.
Definition 5. Assume that , is a set of interval grey numbers, for any , ifthat the is called the grey number Bonferroni mean operator.
The operator has excellent properties such as idempotency, boundedness, and commutativity.
Theorem 1 (Idempotency). Let be a set of interval grey numbers. If all are equal, for all , then
Theorem 2 (Boundedness). Let be a set of interval grey numbers, and letThen
Theorem 3 (Commutativity). Let and be two interval grey numbers, where is any permutation of , , thenNow, we discuss some special cases of the with respect to the parameters and :
Case 1. If , then from the , we getwhich we call as the interval grey number generalized mean operator.
Case 2. If and , then by the , we havewhich we call as the interval grey numbers square mean operator.
Case 3. If and , then the reduces to interval grey numbers, which means the operator iswhich we call as the interval grey number mean operator.
Case 4. If and , then by the , we havewhich we call as the interval grey number square mean operator.
3.2. The Concepts of GNWBM and GNGBM Operators
Considering that the input arguments may have different importance, we define the interval grey numbers weighted Bonferroni mean operator.
Definition 6. Let be a set of interval grey numbers and , be the weight vector of , where indicates the degree of importance of , satisfying and . Ifthen is called as the interval grey numbers weighted Bonferroni mean operator.
Definition 7. Let and , be collections of non-negative real numbers. Subsequently, the aggregation functionsis called as the geometric Bonferroni mean operator.
However, the geometric Bonferroni mean (BM) operator has usually been used in situations where the input arguments are non-negative real numbers. We shall extend the GBM operators to accommodate the situations where the input arguments are interval grey numbers information. In this section, we shall investigate the GBM operator under interval grey numbers environments. Based on Definition 7, we define the interval grey numbers geometric Bonferroni mean operator as follows:
Definition 8. Let be a set of interval grey numbers and let . Ifthen is called the grey numbers geometric Bonferroni mean operator.
Similar to the operator, the operator also has properties such as idempotent, boundedness, and commutativity; therefore, it will not be detailed here.
Now, we discuss some special cases of the with respect to the parameters and : Case 1. If , then from the , we get which we call as the interval grey numbers generalized geometric mean operator. Case 2. If and , then by the , we have which we call as the interval grey numbers square geometric mean operator. Case 3. If and , then the reduces to interval grey numbers, which means the operator is which we call as the interval grey number geometric mean operator. Case 4. If and , then by the , we havewhich we call as the interval grey numbers square geometric mean operator.
3.3. The Concept of GNWGBM Operator
Considering that the input arguments may have different importance, we define the interval grey numbers weighted geometric Bonferroni mean operator.
Definition 9. Let be a set of interval grey numbers and , be the weight vector of , where indicates the importance degree of , satisfying and . Ifthen is called the interval grey numbers weighted geometric Bonferroni mean operator.
4. Approach for Multiattribute Decision-Making with an Interval Grey Number Based on Bonferroni Mean
A multiattribute evaluation system usually has different dimensions and orders of magnitude owing to different properties of each evaluation index. Dimensionless data processing mainly deals with the comparability of the data. Therefore, it is necessary to standardize the raw index data to ensure the reliability of the results and eliminate its dimensionality so that a comprehensive evaluation and analysis can be carried out.
Definition 10 (see ). Let be the matrix of interval grey number matrix, where is transformed into a dimensionless normalized matrix in which the standardized method of is given as follows.
For benefit attributes, For cost attributes,
Definition 11 (see ). Let and be two interval grey numbers, and let then the degree of possibility of is defined asFrom the Definition 10, we can easily get the results as follows:(1)(2)where especially
In this section, we utilize the interval grey numbers weighted Bonferroni mean operator (interval grey numbers weighted geometric Bonferroni mean operator) to develop an approach for multiple attribute decision-making problems with interval grey numbers information, which can be described as follows:
Step 1. Utilize the decision information given in matrix , and the operator (in general, we can take )
Step 2. Normalizing the interval grey numbers
In general, owing to the different attributes and representative meanings of information data, we normalize the attribute information to maintain unified dimensions and avoid evaluation criteria. We performed different treatments for the different types of attribute information.
Step 3. Integrating the normalization information matrix, we utilize the normalization matrix within the and the operators asto derive the overall interval grey numbers preference values of the alternative .
Step 4. For simplicity, we let and then we develop a complementary matrix as , where Then, we haveWe then rank the collective overall preference values in the descending order in accordance with the values of .
Step 5. Get the best scheme according to the sort.
5. Application Example
Example 1. (Housing purchase problem) There are four types of residential options, denoted by . House buyers mainly consider the following indicators: housing area , equipment (level), environmental (level) and price (per unit area: yuan ), and distance (mile). Among them, is the benefit index and is the cost index. The weight of each indicator is where The case data are selected from reference . Step 1: To construct the decision matrix interval grey number in Table 1 Step 2: Normalize the decision matrix of the interval grey number using formulas (22) and (23) in Table 2 Step 3: Utilize the normalization matrix and the operator using formula (14) (in general, we can take ) as Alternatively, the operator using formula (21) (in general, we can take ) is Step 4: The possibility is calculated from the overall preference values using formula (24). Then we have The possibility is calculated from the preference values obtained by the operator: Possibility is calculated from the preference values obtained by the operator: We know that the order we get from both operators is , namely, . Step 5: Get the best scheme according to the sort.So , namely, the scheme is the best one.
It can be seen from the above analysis in Table 3 that the order of the two results is exactly consistent. The optimal scheme is .
Example 2. (The example is taken from ) Assume that a company needs to hire a branch manager and that there are currently four candidates . This study selected the first three interviews. In order to recruit the most suitable candidates, the decision-makers mainly consider the following aspects: confidence , interpersonal communication , work enthusiasm , and expected salary . The interview scores are presented in Table 4. The weight of each attribute is the Step 1: To construct the decision matrix interval grey number in Table 4 Step 2: To normalize the decision matrix of interval grey number by formulas (22) and (23) in Table 5 Step3, Step4, and Step5 are the same as in Example 1, and the possibility is calculated from the preference values obtained by the operator: So , namely, the scheme is the best one.This paper is the same as the best candidate obtained at the different moments of the three interviews in the literature .
The method used in this study is simpler, and it is reasonable. It can be observed that the proposed method is effective.
In the case of information acquisition, the interval grey number is used to describe the decision information, which is more in line with the actual situation of management, engineering, and so on. Considering the mutual relationship between the attributes of schemes is also necessary for the development of practical problems. This study defines the interval grey number weighted geometric Bonferroni mean operator, which is used to aggregate grey number information. Two examples are provided to demonstrate the effectiveness of the proposed method, and its practicability and effectiveness are also proved.
The data used to support the findings of this study have been placed in the HowNet repository.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
The research was funded by the National Natural Science Foundation of China (11871118).
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