Application of Rough Classification of Multiobjective Extension Group Decisionmaking under Uncertainty
ZHU Jiajun[1]
ZHENG Jianguo[2]
QIN Chaoyong[3]
Abstract: On account
of the problem of incomplete information system in classification of extension
group decisionmaking, this paper studies attribution reduction with
decisionmaking function based on the group interaction and individual
preferences assembly for achieving the goal of rough classification of
multiobjective extension group decisionmaking under uncertainty. Then, this
paper describes the idea and operating processes of multiobjective
extension classification model in
order to provide decisionmakers with more practical, easy to operate and
objective classification. Finally, an example concerning practical problem is
given to demonstrate the classification process. Combining by extension
association and rough reduction, this method not only takes the advantages of
dynamic classification in extension decisionmaking, but also achieves the
elimination of redundant attributes, conducive to the promotion on the accuracy
and the reliability of the classification results in multiobjective extension
group decisionmaking.
Key
words: extension group decisionmaking; matterelement analysis; extension
association; rough set; attribution reduction
1. Introduction
Extenics is a new science, which studies the extension possibility and extension laws of things, and explores means for extension and innovation. The cognition in basic concept and theoretical frame is deepening step by step. As an important component of extenics, extension decisionmaking is a new subdiscipline which integrates scientific thinking, systems science and mathematics through the correlation function and the extension transformation to seek satisfaction in the decisionmaking space. Extension decisionmaking analyses the various subsystem compatibility with correlation function based on a mathematical tool of extension set, and through matterelement transformation to change contradictory issue into a compatibility issue in order to extend relevant decisionmaking strategy. Matterelement theory has a good adaptability and feasibility in description and analysis of natural language to achieve dynamic and systematic decisionmaking based on extension transformation, it makes artificial intelligence based on matterelement extension decision has a broader use of space.
Developing rapidly, extenics
acquires quite a great progress in basic theory and application research. Based
on the concept of ndimensional matter element extension set (CAO, YANG. 2006), gives the concepts of multilayer
multidimensional matter element system extension set and its positive field,
negative field, zero boundary and its extension field as well as its stable
field in order to study contradictory problems of multilayer multidimensional
complex systems. By using knowledge presentation and reasoning technique in extension
theory (CAO, PENG,
2006), established
intelligent decision support system based on extension expert system (ZHANG, WANG,2000). develops fuzzy gray matterelement space and fuzzy extension
economic space which is combined with newly emerging fields such as fuzzy sets
and fuzzy systems, extension sets, gray system and set pair analysis, and then
some fuzzy extension mathematical models are suggested, several sets of fuzzy
decision support systems based on the extension theory are presented applied to
the large scale systems. Based on extension matterelement theory
(SHENG, ZHAO, 2006), presents an automatic online measuring method of distributed
production plan track using the multisensor and a new extension measurement
method which can realize the right time to finish the production plan and to
supply data guarantee for the production plan and control in core enterprise
under supply chain. According to limitation of FGESDSS (YANG,ZHANG,
2007), puts forward a new approach for decisionmaking that is called Set
Pair Extension Space Decision Support System based on set pair analysis and
extension theory, the model can characterize both the favoring evidence and the
opposing evidence for every scheme. Based on extension theory and extension engineering methods (LIU,
LIU, 2007), brings forward a new kind of machinelearning method that is
called extension machine method which can pile up experience
in the continually use and obtain the exact knowledge about decision, corrects
its parameter and ameliorate the arithmetic of itself, thus improving its
capability of selflearning (Wang, Tseng, 2009). presents
a novel classified method that is called Extension Genetic Algorithm (EGA)
which combines extension theory and genetic algorithm (GA), is extremely
innovative, in order to eliminate try and error adjustment of modeling
parameters and increase accuracy of the classification..
In addition, the extension method
also applies to the land development and consolidation project management (ZHANG, WEI, 2007), decisionmaking of risk investment (BAI, 2008), comprehensive
evaluation (XIE, LI,
2008; ZHAO, ZHU,
2008),
intelligent control (CHAO, LEE, YEN, 2008; ZHANG, CHENG, 2007), data mining (CHEN, 2003), fault diagnosis (JIN, CHEN,
2006; YE, 2009), pattern recognition (HUNG, FENG,
2008), etc.
Based on matterelement extension
theory and rough
set
theory, this paper makes a study of multiobjective classification optimization
of extension group decisionmaking. Through studying the extension transformation
under uncertainty, this paper analyzes advantages and disadvantages of
extension classification, thus, the attribute reduction methods of rough set is
introduced to improve the effect of extension classification under uncertainty.
This improved extension classification model can help decisionmakes to observe the effect of
classification from the dynamic point of view,
and to identify the main factors which impact program’s classification changes
under different decisionmaking preferences. As a result,
systematic classification problems of multiobjective extension group
decisionmaking under uncertainty can be solved.
2. extension classification and transformation of extension group decisionmaking
Let
means schemes , means decisionmakers of , the value of is ,. Then the composite matterelement of
multidimensional group decisionmaking is.
Definition 1: let R R is the composite element set of group decisionmaking, is the extension set, then a matterelement extension set of group decision making in R is as follows:
R (1)
Among them, is the joint field of the matterelement
extension set, is the joint field which is composed of standard things and
things which can be transformed into standard things, in other words, it is the
range of evaluation value of joint field about decisionmakers . is the classical
field of the matterelement
extension set, is the standard object which means the range of evaluation value of standard
object about
decisionmakers ,,.
The association degree between value and interval of assessment as follows (YANG, ZHANG, CAI, 2002):
(2)
means the distance between and limited interval of classical field and means the distance between and limited interval of joint field. The formula of the distance between point and limited interval is:
(3)
Thus, the integrated association degree based on weights of decisionmaker is:
(4)
Based on the extended association degree, the
evaluation value of scheme about
decisionmakers can be judge whether it is belong to type.
(5)
Through to summary the judgment results of
decisionmakers about scheme , then:
(6)
If , means that the evaluation results of the
decisionmakers on the scheme is belong to type. Otherwise,
if , means that the evaluation results of is not belong to type.
Furthermore, based on the integrated association
degree, the evaluation value of scheme can be judge whether it is belong to type.
(7)
Through to summary the judgment results about scheme , then:
(8)
If , means that the
evaluation results of the scheme is belong to type. But, if , means that the evaluation results of the scheme is not belongs to type.
3. Multiobjective conversion and standardization of extension group decisionmaking based on decisionmaking preferences
Definition 2 (Cai, 1999): let matterelement _{} and _{}. “And” refers to both get_{}and_{}, call _{}.“Or” means taking either _{}or_{}, call _{}.All appearance:
_{} (9)
_{} (10)
Definition 3: let matterelement _{}. If _{}, _{}, call _{}is a non matterelement of _{}, _{}; if _{}, that _{}, _{}, _{}means "Not" operation which change matterelement _{} to _{}.
Inference 1: the rules of logic operation under the matterelement
with same matter:
_{} (11)
_{} (12)
Inference 2: the rules of logic
operation under the matterelement with same features:
_{} (13)
_{} (14)
Matterelement combines the thing, its characteristics and feature values into one set. For a multiple dimension matterelement can describe multiple aspects of a thing, it is possible to build a modal which can describe systematic decisionmaking problems of multiobjective conversion and multiindex evaluation in group decisionmaking by matterelement.
Let _{},_{}, _{} and _{}，_{}, _{}, _{} means _{}decisionmakers of _{}, _{} of
field _{} is _{}_{} _{}, then the composite matterelement
of multiobjective and multidimensional group decisionmaking is _{}.
Due to differences goals would affect the outcome of the
decisionmaking, through the composite matterelement should to be
standardization in order to meet the needs of data processing under the
multiobjective matterelement with same matter or same features. According to
Definition 3, let multiobjective group decisionmaking matterelement _{}. The smaller the better for the
composite matterelement is_{},_{} , _{} means that _{} is able to change to the bigger the
better for the composite matterelement under target _{},_{}，_{}.
_{}_{} =_{} (15)
The same principle, as well as the object that is changed
from the bigger the better to the smaller the better for the composite
matterelement.
According to Definition 2 and Inference 1, based on target
conformity under decisionmaking preference _{}, a correlation matrix _{}of the target _{} is established with _{} of _{} about _{}.
_{}_{} (16)
If_{}, then _{}, which is pessimistic decisionmaking method; if_{}, then _{}, which is optimistic decisionmaking method; if _{}, _{}, which is compromise decisionmaking method. Then
_{}
(17)
_{}
The comprehensive association degree is :
_{} (18)
which of _{} about _{},_{}is the weight factor of _{}.
Thus, the comprehensive association degree of _{}about decision makers _{}and weight factors _{} is :
_{} (19)
To make
matterelement extension set _{}and to give transform _{} under field _{}, call:
_{}
_{}
_{} (20)
Based on changing classical field and preferences of
extension group decisionmaking, we are able to observe the changes of optimal
scheme from the dynamic point of view and compare optimal scheme with other
schemes under different conditions in order to obtain a optimal classification
under no preference. However, this classification remains in a simple
classification can not analyze the decisionmakers on the impact of the
decisionmaking options and can not reflect the correlation between schemes. In
addition, sometimes the judgment result of some policy makers or schemes could
be belong to more categories based on extension transformation, thus the
formation of incomplete decisionmaking situations, it also adds uncertainty to
the scheme classification of decisionmaking.
4. Extension group decisionmaking attribute reduction and classification under Uncertainty
Rough set theory is a mathematical tool to deal with ambiguous and uncertainties data (PAWLAK, 1982), which has been used in various fields such as machine learning, pattern recognition, knowledge discovery,etc. Attribute reduction is a core part of rough set theory which is used to eliminate redundant attributes in the decisionmaking table.
Definition 4 (JELONEK, 1995): let _{} is a decisionmaking information systems, _{}, if _{}, call _{} is partition consistent set, if any real subset of _{} is not partition consistent set, then _{} is partition reduction set.
If _{},then
_{},
the results of _{} which are classified by attribute
_{} and _{} are identical. Thus, the object set
described by _{} also can be described by partition consistent set _{}and partition consistent set _{}.
Definition 5: let _{} is a decisionmaking information systems,
_{}，
_{}
(21)
call_{}is
partition discrimination set of _{} and_{}，then _{}is
the partition discrimination matrix of decisionmaking information systems.
_{} (22)
Theorem 1, let _{} is a decisionmaking
information systems, for any _{}, partition discrimination
set has the following properties:
（1）_{}；
（2）_{}
（3）_{}
_{}
Based on the attribute reduction method of rough set, incomplete decisionmaking system which is produced after extension transformation can be further classified, it is to added extension and improvement of the classification.
Definition 6, if the extension of group decisionmaking _{} changes to _{} after extension transformation, any _{} and _{} are the only established, then known as the perfect extension group decisionmaking information system, otherwise known as the incomplete information system.
Let _{} is a incomplete extension group decisionmaking information system, _{}, _{}is decision attribute, then recorded as:
_{}
_{} (23)
which _{} is the inclusion degree on _{}.
_{}
(24)
which is means similar type of _{},_{}
_{}
(25)
which expressed the similar relationship on _{},
_{}
(26)
is the decisionmaking
function _{}. if any _{} there is _{} set up, then _{} is the largest distribution consistent set of _{}. If _{} is the largest distribution consistent set,
and any
really subset of _{} is not the largest distribution consistent set of _{}, then _{} is the largest distribution reduction set
of _{}.
Let _{} is one of all options in a incomplete
extension group decisionmaking information system _{}, _{} is the largest distribution reduction set
of _{}, _{} is the smallest set of the largest distribution reduction set of
all options, _{},_{}. If _{}, then _{} is the partition set of core decisionmakers _{}, _{} is the partition set of relative necessary decisionmakers _{}; _{} is the partition set of unnecessary decisionmakers _{}.
According to
the smallest principle which expresses the sum
of deviation absolute value between evaluation values _{}of
schemes _{} and comprehensive evaluation value _{} of the schemes, to determine the the root attribute of
classification.
_{} (27)
If _{},
through the smallest principle which expresses
the sum of deviation absolute value between _{} and comprehensive evaluation value _{} of the schemes, to determine the subattributes of
classification.
_{} (28)
Among them, _{},_{}. First of all, using the root
attribute _{} to create the beginning nodes of
classification and to create branches based on each value of the root attribute.
Secondly, let the the minimum set _{} of the largest distribution reduction set of all options as a
subset attributes to leads branches, in order to achieve the division of the
sample.
5. The framework and the steps of multiobjective extension rough classification of group decisionmaking
5.1 The framework and ideas of model
Based on combining extension group decisionmaking with
classification of rough set method, attribute reduction is introduced to
improve the extension classification, so as to enhance the classification
results of extension group decisionmaking categories under uncertainty.
The core of the model is that through the rough reduction
to solve the uncertainties of extension classification, and to realize multiobjective
extension classification under decisionmaking preferences, so as to enhance
the applicability and reliability of the extension classification. There are
two major parts of extension rough classification model of group
decisionmaking: Firstly, through the correlation function to achieve extension
transformation, in order to achieve dynamic classification of the
decisionmaking schemes; Secondly, through the attribute reduction and
decisionmaking function to improve extension classification under uncertainty,
and to analyze the impact of decisionmaking preference, decisionmaking
relevance upon multiobjective classification results.
5.2 The steps and content of model
Step1: To establish a multiobjective extension group decisionmaking information system which includes expert set , scheme set and target set in order to obtain the multiobjective extension matterelement set _{};
Step2: To set the weights _{} of decisionmakers _{} and decisionmaking preference _{};
Step3: To input data, when the target is a negative index in decisionmaking, data of this target must be transformed with (15);
Step4: To achieve the goal of multiobjective conversion
and standardization in order to gain a comprehensive matrix of multiobjective
extension matterelement set under decisionmaking preference _{};
Step5: To determine joint field _{}and classical field_{}, and to set gradelevel _{}of extension classification;
Step6: Based on correlation function (18), to achieve extension conversion in order to carry out the initial classification of schemes, and to calculate the evaluation value _{} and comprehensive evaluation value _{}of schemes based on (2) to (8) for constituting a extension group decisionmaking system.;
Step7: Based on extension classification, to use of attribute reduction for reclassification under decisionmaking preference _{};
Step8: To compare initial classification result with reclassification of classification result, if the classification results can meet the needs of classification goals, go to the last step; otherwise go to the next step.
Step9: If the model need to update data, then go to step 3 to continue classification after reenter the data , otherwise go to step 5 to continue classification after reset the classical domain;
Step10: To output classification results, the classification ends
here.
6. A Case Study
The upcoming 2010 Shanghai World
Expo and the 2010 Guangzhou Asian Games have brought tremendous business
opportunities to many domestic enterprises. A large toy and gift manufacturers
in Wuxi hope to upgrade the production plans so as to expand the production
capacity and product scale. On the basis of sales forecasts, premarket
research and verification of the expert group, this company studied out a
specific combination production plan. A multiobjective matterelement
extension group decisionmaking information model _{} is established so that classify and
evaluate the new plans of creative projects.(Table 1)
Among them, experts set is , schemes set is , targets set is , said that “income” (positive index), said that “cost”(negative index) and said that “production efficiency”(positive index). Set the weights of the experts =(0.2,0.2,0.2,0.2,0.2), and set decisionmaking preference ,and .
Furthermore, gradelevel of extension classification and the initial level variable _{}should be determined. _{} , _{} said that “Eligible”, _{} said that “Middling”, _{} said that “Good” and _{} said that “Excellent”. And to determine joint field _{} and the initial classical field _{}.
Because _{} is a negative index, therefore, it should be translated into positive index with (15). Then, according to steps of model, we can establish a multiobjective composite matterelement matrix under different preference in order to achieve rough classification of schemes.
If _{} which means optimistic decisionmaking method, we can obtain the following Table 2.
According to Definition 6, _{}is an incomplete information system, so the smallest set of the largest distribution reduction set of all options is _{}, and we can obtain the root attribute of classification is _{}based on formula (27) and (28). If _{}3, then we can obtain rough classification as follow Figure 2; if _{}4, then we can gain rough classification as follow Figure 3.
The same way,
if_{}, we obtain the table
of extension evaluation value of group decisionmaking as Table
3.
Then, we can gain rough classification as follow Figure 4.
If, we obtain the table of extension evaluation value of group
decisionmaking as table 4:
If _{}3, we can obtain rough classification as follow Figure 5; if _{}4, then we can gain Figure 6.
From different preferences point of view we can see:
if , the
uncertainty of decisionmaking data does not affect the classification results
of schemes, belong to “Excellent”;
if , the
uncertainty of decisionmaking data has affected the classification results of which is belongs to not only “Middling” but also “Good” ; if , only is belongs to
“Middling”. Therefore, based on multiobjective rough extension classification,
the best scheme is , the worst scheme is , the smallest affected by the preferences is .
7. Conclusion
Through changing the classical
field，extension group decisionmaking achieves extension transformation,
thereby extension group decisionmaking information systems is established in
order to achieves the goal of dynamic classification and analysis for data and
programs. However, incomplete information decisionmaking systems often
generates after extension change, which has brought uncertainty to
classification. Therefore, this paper combines extension group decisionmaking
with rough set classification method to achieve dynamic classification through
extension transformation, and on this basis uses the method of attribute
reduction to achieve reclassification, thus improving the rationality and the
practicability of classification.
Multiobjective rough extension classification is an
important aspect of data analysis, knowledge extraction of extension group
decisionmaking, which can achieve the goal of multiproject classification,
multiobjective
assessment based on the group interaction and
individual preferences assembly. It can be applied to investment planning,
project management, risk control etc. under uncertainty.
Acknowledgment
The authors owe a lot to the
funding from the Colleges and Universities’ Philosophy and Social Science Fund
Project of Education Department of Jiangsu Province (09SJD790055).
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Tables and Figures
Table 1: Composite matterelement matrix
of extension classification
Scheme 
_{} 
_{} 
_{} 
_{} _{} _{} _{} _{} 
_{} _{} _{} _{} _{} 
_{} _{} _{} _{} _{} 

_{} 
8.4 8.2 7.9 8.1 9.2 
1.7 1.6 1.5 2.3 1.9 
9.6 8.5 9.1 8.2 8.7 
_{} 
9.3 9.8 8.5 9.1 8.7 
2.5 2.1 1.8 2.1 3.4 
8.3 7.7 8.8 9.2 8.9 
_{} 
9.1 7.1 9.4 8.4 9.1 
3.3 2.2 1.3 1.0 2.2 
9.1 8.6 9.4 7.6 8.8 
_{} 
9.2 8.9 8.7 8.8 9.4 
1.3 0.9 1.2 1.5 0.9 
8.8 9.1 8.9 8.7 9.2 
_{} 
8.3 8.7 7.9 8.6 8.5 
1.9 1.8 0.9 1.1 1.6 
9.1 8.3 8.4 8.3 8.2 
_{} 
7.5 8.7 9.1 8.3 7.8 
2.2 2.1 1.2 2.2 1.7 
9.2 7.9 8.2 8.3 7.8 
_{} 
8.1 7.5 8.3 9.3 8.5 
1.6 2.4 1.8 1.3 1.9 
8.9 8.9 7.8 8.1 8.5 
_{} 
8.3 8.4 8.2 7.2 8.1 
2.9 2.6 1.9 1.8 3.1 
8.0 8.3 8.2 8.3 8.0 
Table 2: The table of extension
evaluation value of group decisionmaking under _{}
Scheme 
_{} 
_{} 

_{} 
_{} 
_{} 
_{} 
_{} 

_{} 
4 
3 
4 
3 
4 
4 
_{} 
4 
4 
3 
4 
3 
4 
_{} 
4 
3 
4 
3,4 
4 
4 
_{} 
4 
4 
3 
3 
4 
4 
_{} 
4 
3 
4 
3 
3 
3 
_{} 
4 
3 
4 
3 
3 
3 
_{} 
3 
3 
3 
4 
3 
3 
_{} 
2 
3 
3 
3 
3 
3 
Table 3: The table of extension evaluation value of group decisionmaking under _{}
Scheme 
_{} 
_{} 

_{} 
_{} 
_{} 
_{} 
_{} 

_{} 
3 
3 
2 
2 
3 
2,3 
_{} 
2 
2 
3 
2 
1 
2 
_{} 
1 
2 
3 
2 
2 
2 
_{} 
3 
3 
3 
3 
4 
3 
_{} 
3 
3 
2 
3 
3 
3 
_{} 
2 
2 
3 
2 
2 
2 
_{} 
3 
2 
2 
3 
3 
2 
_{} 
2 
2 
3 
2 
1 
2 
Table 4: The table of extension evaluation value of group decisionmaking under _{}
Scheme 
_{} 
_{} 

_{} 
_{} 
_{} 
_{} 
_{} 

_{} 
3 
3 
3 
2 
3 
3 
_{} 
3 
3 
3 
3 
2 
3 
_{} 
2 
2 
4 
3 
3 
3 
_{} 
3 
3,4 
3 
3 
4 
3 
_{} 
3 
3 
3 
3 
3 
3 
_{} 
3 
3 
3 
3 
3 
3 
_{} 
3 
3 
3 
3 
2 
3 
_{} 
2 
2 
3 
2 
2 
2 
Figure 1: Operation
process of the rough classification
model of multiobjective extension group decisionmaking
Figure 2: Extension rough classification under _{}(Ⅰ)
Figure 3: Extension rough classification under _{}(Ⅱ)
Figure 4: Extension rough classification under _{}
Figure 5: Extension rough classification under _{}(Ⅰ)
Figure 6: Extension rough classification under _{}(Ⅱ)
[1] School of Business & Management, Donghua University. Department of Accountancy & Finance, Wuxi Institute of Commerce, Shanghai, P.R.China.
Email: zhujiajun@mail.dhu.edu.cn
[2] School of Business & Management, Donghua University, Shanghai, P.R.China.
Email: ZJG@dhu.edu.cn
[3] School of Business & Management, Donghua University, School of Mathematics & Information Science, Guangxi University, Nanning, P.R.China.
Email: qcy@mail.dhu.edu.cn
* Received 12 July 2009; accepted 19 August 2009
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