Multiattribute Grey Target Decision Method Based on Soft Set Theory

With respect to the Multiattribute decision-making problems in which the evaluation attribute sets are different and the evaluating values of alternatives are interval grey numbers, a multiattribute grey target decision-makingmethod in which the attribute sets are differentwas proposed.The concept of grey soft setwas defined, and its “AND”operationwas assigned by combining the intersection operation of grey number. The expression approach of new grey soft set of attribute sets considering by all decision makers were gained by applying the “AND” operation of grey soft set, and the weights of synthesis attribute were proved. The alternatives were ranked according to the size of distance of bull’s eyes of each alternative under synthetic attribute sets. The green supplier selection was illustrated to demonstrate the effectiveness of proposed model.


Introduction
As decision-making problems in social and economic life become more and more complicated, it is hard for individual decision makers of different knowledge structure and practical experience to consider all the aspects of decision problems.Most important decisions were made with participation of several specialists in order to ensure the scientificity and rationality of decisions, which is called group decision-making [1].In recent years great attention has been paid to studies on multiattribute decision-making problems under uncertain environment, and plenty achievements were gained [2,3].In the existing decision making methods, most decision systems give the same attribute sets; that is, all the decision makers give values of evaluation schemes depending on the same attribute sets.However, decision makers in practical decision-making problems usually have different backgrounds.Because they have different knowledge structures, practical experiences and preferences, the attribute sets they considered will not totally be the same.Thus decisionmaking problems in which each decision maker considers different attribute sets have caused concerns of researchers.Literature [4] proposes a rule to combine different evaluation indexes and a method to combine weight vectors.Literature [5] establishes a comparison method based on fuzzy soft set "AND" operation for multi-index schemes, considering the problems of different parameter sets by different decision makers.Literature [6] defines an "and/or" decision function based on soft set, and establishes an "and/or" group decision making method which doesn't use rough set and fuzzy set.As a mathematical tool to solve problems of uncertainty, soft set [7] overcomes the shortcomings of classical mathematical theory in dealing with problems of uncertainty; thus it has been widely used in practical decision problems [8].Because of the complexity and uncertainty of practical problems, Professor Deng Julong proposes to use gray number [9] to solve problems in which object is clear extension and unclear intension.However, when the evaluation attribute value is interval grey number, literatures using soft set theory to solve these kinds of problems are not common currently.
As a new approach to solve multiattribute decisionmaking problems, grey target decision-making [9] has got great attention since it was proposed by Professor Deng Julong.Multiattribute decision-making is usually an uncertain problem and contains multiple time points; literature [10] presents a solving model which can establish time point weights depending on grey entropy and time degree and an objective function which can aggregate distance of 2 Mathematical Problems in Engineering bull's eyes of each time point.Literature [11] puts forward a new integrated distance of bull's eyes by integrating the space projection distance from each situation to positive and negative target center.Literature [12] constructs an English learning results evaluation model which can collect teacher's information effectively based on grey target theory.Literature [13] uses multicriteria intelligent weighting grey target decision to select resettlement area for reservoir immigrants and analyzes the factors affecting the choice.
In this paper, soft set theory was introduced into grey system theory to solve multiattribute decision-making problems in which evaluation attribute sets are different and evaluation decision-making values are interval grey numbers.A multiattribute grey target decision-making method with different attribute sets was established by combining the features of grey target decision-making, and this method was applied in the choice of green supplier to illustrate the rationality and effectiveness of proposed method.
Definition 4 (see [7]).Assume that  is the initial universe set,  is a set of parameters, and () is the power set of  and  ⊆ .A pair (; ) is called a soft set over , where  is mapping given by  :  → ().
In practical problems, for parameter set  in Definition 3, each parameter in the  can be understood as an attribute probably considered by decision maker and can also be described as a certain state of attributes. is a subset of , which can be viewed as attribute set considered by a decision maker.For ∀ ∈ , () represents a set with  parameter properties, and soft set (; ) consists of set with each parameter properties of .
Through medical diagnosis we can get that ( In actual decision-making problems, object which is clear extension and unclear intension often appears.Such object has some gray nature, so using gray numbers to describe this kind of problem is of certain practical significance.As a mathematical tool to handle uncertain problems, soft sets overcome the shortcomings of classical mathematical theory, which lacks parameterization tool.Considering this, this paper proposes the definition of grey soft set.Definition 6. Assume that a grey number ⊗ is introduced on the background or universe ,  is a set of parameters, and () is the power set of set  and  ⊆ .A pair (; ) is called a grey soft set over , where  is mapping given by  :  → ().Definition 7. Assume that (; ) and (; ) are two grey soft sets over common universe , for ∀(, ) ∈  × , (, ) = () ∩ (); then (,  × ) is defined as "AND" operation of soft grey sets (; ) and (; ).By Definition 7 we can see that a new grey soft set (,  × ) can be obtained by "AND" operation of soft grey sets (; ) and (; ) and each parameter in (,  × ) is synthesized with parameter in  and , where () ∩ () represent the intersection of two grey numbers.
Example 8.A company plans to purchase product parts; it plans to select the best green supplier from the four candidates ( In the actual decision-making problems, in order to integrate evaluation information of each decision maker, "AND" operation of soft grey sets is expanded according to Definition 6. Assume that ( 1 ;  1 ), ( 2 ;  2 ), . . ., (  ;   ) are grey soft set; then we get that is, ∀( 1 ,  2 , . . .,   ) ∈  1 ×  2 × ⋅ ⋅ ⋅   ; then we get

Multiattribute Grey Target Decision Method Based on Soft Set
Let  = { 1 ,  2 , . . .,   } be the scheme set,  = { (b) for cost-type target where as the normalized effect evaluation vector of scheme   under the attribute    , which is interval gray number on [0, 1], and the normalized effect evaluation vector of scheme   under the attribute sets   is and then we get the normalized decision matrix of various alternatives by expert In order to get the comprehensive effect evaluation value of each scheme, and simultaneously considering the different attribute set considered by the decision makers, according to the attribute sets   considered by experts   and standardization decision matrix, the evaluation information of each scheme on various attributes is expressed as in the form of grey soft sets ( 1 ;  1 ), ( 2 ;  2 ), . . ., (  ;   ); according to the formula (12), we get According to Definition 6, we do "AND" operation of soft grey set ( 1 ;  1 ), ( where Definition 10.
is called optimal effect vector of grey-target decision-making; it is also called positive bull's-eye.
then  +  is referred to as positive bull's-eye distance of scheme   .
Sort the programs according to the order of  +  , and the smaller the  +  is, the better the program is.

Example Analysis
A company plans to purchase product parts; it plans to select the best green supplier from four candidates  = { 1 , Represent the evaluation information as grey soft sets ( 1 ;  1 ), ( 2 ;  2 ) according to formula (12), then operate  1.After "AND" operation of grey soft sets ( 1 ;  1 ) and ( 2 ;  2 ), the obtained decision matrix is shown in Table 2.
Weight  *  of each attribute can be obtained after combination according to formula (13) and is shown in Table 3.
Positive bull's-eye distance obtained through formula ( 16) is So the schemes can be ranked as  2 ≻  4 ≻  3 ≻  1 and decision makers can take supplier  2 into prior consideration.
In order to further illustrate the method proposed in this paper, then we will use traditional gray target decision method to solve the above problems.First, the original data is normalized; then we can obtain the bull's-eye distance of each scheme according to formulas ( 16)-( 17):  + 1 = 0.9892,  + 2 = 0.6843,  + 3 = 0.8953,  + 4 = 0.9466, so the schemes are ranked as  2 ≻  3 ≻  4 ≻  1 , and  2 is the best scheme.Comparison between the two methods shows that the order between  3 and  4 has changed.This is because this paper has taken into account the situation in which different decision makers consider different attribute sets, and attribute values of each schemes are not simply weighting summed, but we can fuse the attribute sets considered by different specialists firstly, and then fusion attribute values are weighting summed.The change is reasonable through analysis of the original decision matrix data, and this proves that the method proposed in this paper is reasonable and effective and provides a more rational approach to multiattribute selecting problems that involves multiparty participation and decision makers consider different attribute sets.
= 1, 2, . . ., ;  = 1, 2, . . .,   ;  = 1, 2, . . ., ) as the effect evaluation value of scheme   under the attribute    by expert   , and the effect evaluation value is an interval grey number; then the effect evaluation value of scheme   under the attribute    by expert   is  = 1, 2, . . ., ,  = 1, 2, . . .,   ;  = 1, 2, . . ., ), and  = (   ) × is the decision matrix of various alternatives by expert   .Considering frequently different polarity of different attributes, in order to eliminate diversity on the dimension unit and increase comparability of the effect evaluation value of each scheme under different attributes, and comparing between the attributes, decision matrix is standard.Attributes are classified as cost-type target and benefit-type target; we introduce the following grey poor transformation formula:(a) for benefit-type target th attribute which is considered by expert   , and  = 1, 2, . . .,   ,   ⊆ ,   ⊆ ,   ≤ .

Table 3 :
Weight of each attribute after combination.is obtained through combination of attributes in  1 and  2 , and the composition of each attribute by combination is shown in Table "AND" operation to (  ;   ) ( = 1, 2) according to Definition 6, and obtain the new grey soft set (,  1 ×  2 ); after combination, there will be  = 3 × 3 = 9 attributes.Suppose the attribute set after combination is  * = { * 1 ,  * 2 , . . .,  * 9 }, every attribute in  *