A mean deviation based method for intuitionistic fuzzy multiple attribute decision making

A Mean Deviation Based Method for Intuitionistic Fuzzy Multiple Attribute

Decision Making

Yejun Xu

Business School

HoHai University

Nanjing, Jiangsu 210098, P R China

xuyejohn@https://www.sodocs.net/doc/cd1210462.html,

Abstract—The aim of this paper is to develop a method to determine the weights of attributes objectively under intuitionistic fuzzy environment. Based on the mean deviation, we establish an optimization model in which the information about attribute weights is completely unknown. By solving the model, we get a simple and exact formula which can be used to determine the attribute weights. After that, we utilize the intuitionistic fuzzy weighted average (IFWA) operator to aggregate the given intuitionistic fuzzy information corresponding to each alternative, and then select the most desirable alternative according to the score function and accuracy function. Finally, a practical example is given to verify the developed method and to demonstrate its practicality and effectiveness.

Keywords-Intuitionistic fuzzy set; multiple attribute decision making; mean deviation;

I.I NTRODUCTION

Intuitionistic fuzzy sets(IFS) introduced by Atanassov[1, 2] have been found to be well suited to dealing with vagueness. IFS characterized by a membership function and a non-membership function, is an extension of Zadeh’s fuzzy set[3] whose basic component is only a membership function. Since its appearance, the IFS have received more and more attention and applied it to the field of decision making. Gau and Buehrer[4] presented the concept of vague sets. Burillo and Bustince[5] showed that the notion of vague sets coincides with that of intuitionistic fuzzy sets. Based on vague sets, Chen and Tan[6], and Hong and Choi [7] utilized the minimum and maximum operations to develop some approximate technique for handling multiattr-ibute decision making problems under fuzzy environment. Szmidt and Kacprzyk [8] proposed some solution concepts such as the intuitionistic fuzzy core and consensus winner in group decision making with intuitionistic (individual and social) fuzzy preference relations, and proposed a method to aggregate the individual intuitionistic fuzzy preference relations into a social fuzzy preference relation on the basis of fuzzy majority equated with a fuzzy linguistic quantifier. Li and Cheng[9], Liang and Shi[10], Huang and Yang[11], and Wang and Xin[12] introduced some similarity measures of intuitionistic fuzzy sets and applied them to pattern recognition. Xu and Yager[13] developed some aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric(IFOWG) operator, the intuitionistic

fuzzy hybrid geometric (IFHG) operator to multiple attribute group decision making with intuitionistic fuzzy information. Xu[14] developed the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator, and the intuitionistic fuzzy hybrid averaging (IFHA) operator. However, when using these operators, the associated weighting vector is more or less determined subjectively

and the decision making information itself is not taken into consideration sufficiently. All of the above methods will be unsuitable for dealing with such situations. Therefore, it is necessary to develop a method for determining the weights objectively of the multiple attribute decision making problems under intuitionistic fuzzy environment.

In this paper, we focus our attention on developing a

method objectively named mean deviation method to determine the attribute weights under the condition that the

attribute weights are completely unknown, and the attribute

values are taking the form of intuitionistic fuzzy numbers, to overcome the above limitations. To do so, the rest of the

paper is organized as follows. In Section 2, we introduce

some basic concepts of intuitionistic fuzzy sets. In Section 3,

we establish an optimization model based on the mean deviation method. By solving this model, a simple and exact

formula is derived to determine the attribute weights. We

utilize the intuitionistic fuzzy weighted averaging (IFWA) operator to aggregate the intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to

the score function and accuracy function. In Section 4, a practical example is used to illustrate the developed models.

In Section 5, we conclude the paper and give some remarks.

II.P RELIMINARIES

In the following, we introduce some basic concepts

related to intuitionistic fuzzy sets.

In[1], Atanassov introduced a generalized fuzzy set

called intuitionistic fuzzy set, shown as follows.

Definition 1. An IFS in X is given by

{,(),()|}

A A

A x x v x x X

μ

=<>∈ (1)

which is characterized by a membership function :

A

μ

[0,1]

X→and a non-membership function :[0,1]

A

v X→,

with the condition

2010 International Conference on Artificial Intelligence and Computational Intelligence

0()()1A A x v x μ≤+≤, x X ?∈ where the numbers ()A x μ and ()A v x represent, respectively, the degree of membership and the degree of

non-membership of the element x to the set A . Definition 2. For each IFS A in X , if ()1()()A A A x x v x πμ=??,x X ?∈ (2)

is called the indeterminacy degree or hesitation degree of x

to A . Especially, if

()1()()0A A A x x v x πμ=??=,x X ?∈ (3) Then, the intuitionistic fuzzy set A is reduced to a common fuzzy set[3]. For convenience, we call (,)v αααμ= an intuitionistic fuzzy number(IFN)([15]), where [0,1]αμ∈,[0,1]v α∈, and

1v ααμ+≤.

Definition 3[13]. Let (,)v αααμ=be an intuitionistic fuzzy

number, a score function S of an intuitionistic fuzzy number can be represented as follows:

()S v αααμ=? (4)

where ()[1,1]S α∈?. For an IFN (,)v αααμ=, it is clear that if the deviation

between αμ and v α gets greater, which means the value αμ

gets bigger and the value v α gets smaller, then the IFN α

gets greater.

Definition 4[13]. Let (,)v αααμ= be an intuitionistic fuzzy number, an accuracy function H to evaluate the degree of accuracy of the intuitionistic fuzzy number can be represented as follows:

()H v αααμ=+ (5)

where ()[0,1]H α∈. The larger the value of ()H α, the higher the degree of accuracy of the degree of membership of the IFN α.

Xu[13] introduced an order relation between two intuitionistic fuzzy numbers in the following. Definition 5. Let (,)v αααμ= and (,)v βββμ= be two

intuitionistic fuzzy numbers, ()S v αααμ=?and (

)S ββμ= v β? be the scores of α and β, respectively, and let ()H α

v ααμ=+ and ()H v βββμ=+ be the accuracy degrees of α and β, then

? If ()()S S αβ<, then α is smaller than β, denoted by αβ<.

? If ()()S S αβ=, then

(1) If ()()H H αβ=, then αand β represent the same information, i.e., αβμμ=, v v αβ=, denoted by αβ=; (2) If ()()H H αβ<, then αis smaller than β, denoted by αβ<.

To aggregate intuitionistic preference information, Xu

[39] defined the following operations.

Definition 6[16]. Let (,)v αααμ= and (,)v βββμ= be two

intuitionistic fuzzy numbers, then

(1) (,)v v αβαβαβαβμμμμ+=+???; (2) (,)v v v v αβαβαβαβμμ?=?+??; (3) (1(1),)v λλ

ααλαμ=??,0λ>; (4) (,1(1))v λλ

λα

ααμ=??,0λ>. Definition 7[15]. Let (,)v αααμ=,(,)v βββμ=be two intuitionistic fuzzy numbers, then we call

()1

(,)||||2

d v v αβαβαβαβμμ=?=?+? (6) th

e deviation between α and β. III. M EAN DEVIATION METHOD

The multiple-attribute decision-making problems under study can be described in detail as follows.

Let 12{,,...,}m X x x x =(2m ≥) be a discrete set of m feasible alternatives, 12{,,...,}n U u u u =be a finite set of attributes. For each alternative i x X ∈, the decision maker gives his/her preference value ij r with respect to attribute j u U ∈, where ij r takes the form of intuitionistic fuzzy

numbers, that is (,)ij ij ij r v μ=, [0,1]ij μ∈, [0,1]ij v ∈, and 1ij ij v μ+≤, 1,2,...,i m =, 1,2,...,j n =, then all the preference values of the alternatives consists the decision matrix ()ij m n R r ×=.

Definition 8[14]. Let ()ij m n

R r ×= be the intuitionistic fuzzy

decision matrix, 12(,,...,)i i i in r r r r = be the vector of attribute values corresponding to the alternative i x , 1,

2,...,i m =, then we call

121122()IF WA (,,...,)...i w i i in i i n in z w r r r w r w r w r ==+++ 111(1),()j j n n w w ij ij

j j v μ==??=??????∏∏ (7) the overall value of the alternative i x , where 12(,,w w w =

...,)T n w is the weighting vector of attributes.

In the situation where the information about attribute

weights is completely known, i.e., each attribute weight can

be provided by the expert with crisp numerical value, we can aggregate all the weighted attribute values

corresponding to each alternative into an overall one by

using (7). Based on the overall attribute values ()i z w of the

alternatives i x (1,2,...,i m =), we can rank all these

alternatives and then select the most desirable one(s). The greater ()i z w , the better the alternative i x will be. However, in this paper, we consider the attribute weight information

about the attribute is completely unknown, thus, we need to determine the attribute weight firstly.

The mean deviation method is proposed by Wang[17] to deal with MADM problems with numerical information. The authors of this paper[18] also used this method to deal with the linguistic group multiple attribute decision making problems, in which the information about the attribute weights are completely unknown and the attributes values are in the forms of linguistic variables. Its main ideal is as follows. For the MADM problems, we need to compare the collective preference values to rank the alternatives, the larger the ranking value ()i z w , the better the corresponding alternative i x is. If the performance values of each alternative have little differences under an attribute, it shows that such an attribute plays a small important role in the priority procedure. Contrariwise, if some attribute makes the performance values among all the alternatives have obvious differences, such an attribute plays an important role in choosing the best alternative. So to the view of sorting the alternatives, if one attribute has similar attribute values across alternatives, it should be assigned a small weight; otherwise, the attribute which makes larger deviations should be assigned a bigger weight, in spite of the degree of its own importance. Especially, if all available alternatives score about equally with respect to a given attribute, then such an attribute will be judged unimportant by most experts. In other word, such an attribute should be assigned a very small weight. Wang[17] suggests that zero should be assigned to the attribute of this kind. The difference of attribute values can be measured using mean deviation. In the following, we will propose the mean deviation method to deal with the group decision making problem under intuitionistic fuzzy environment.

For the attribute j u , the mean deviation of alternative i x to all the other alternatives can be expressed as follows:

111

111(,)m m m

j j ij tj j ij j i t i V w r r w d r r m m m ====?=∑∑∑,1,2,...,j n =

(8)

where 1111111(1),()m

m m m m j tj tj tj t t t r r v m μ===??==??????∑∏∏ denotes the

mean value of the attribute j u , (

1

(,)12

ij j ij d r r μ=

?+ 1111(1)()m m

m m tj ij tj t t v v μ==??+???∏∏denotes the deviation of mean value j r to the attribute value ij r of the alternative x i for the attribute j u . So j V denotes the mean deviation for the attribute j u .

Based on the aforementioned analysis, we have to choose the weight vector w to maximize all the mean deviation values for all the attributes. To do so, we can construct the model as follows:

(M-1) max 11

11()(,)n n

m j j ij j j j i F w V w d r r m ===??

==????∑∑∑ (9)

s.t. 21

1n

j j w ==∑,0j w ≥. (10)

Let

1

1(,)m

j ij j i d r r m δ==

∑ (11) Then, the above model can be transformed into the following model (M-2)

(M-2) 121max (

)s.t. 1,0,1,...,,

n

j j

j n

j j

j F w w w w j n δ==?

=??

??=≥=??

∑∑ To solve the above model, we construct the Lagrange function

2111(,)12n

n j j j j j L w w w λδλ==??

=+?????

∑∑ (12)

where λis the Lagrange multiplier. Since both functions ()F w and (,)L w λ are differentiable for j w , 1,2,...,j n = differentiating (12) with respect to j w , 1,2,...,j n = and setting the partial derivatives equal to zero, we get the following set of equations:

0j j j

L

w w δλ?=+=?,1,2,...,j n = (13) 211102n j j L w λ=??

?=?=?????

∑ (14) Solving this model, we get

j w =

Thus (15) is the extreme point of model(M-2).

By normalizing j w to let the sum of j w , 1,2,...,j n =

be a unit, we have *11

j j j n n j j

j j w w w δδ====∑∑, 1,2,...,j n = (16) As a mater of fact, j δ represents the mean deviation of all alternatives for the attribute j u . Because the larger j δ, the more important the attribute j u is, Eq.(16) is obtained directly by using each j δ

divide the sum of j δ. The theoretic foundation of this method is based on information

theory, that is, the attribute providing more information should be assigned a bigger weight.

Based on the above models, we develop a practical method for solving the multiple attribute decision making problems, in which the information about attribute weights is completely unknown, and the attribute values take the form of intuitionistic fuzzy values. The method involves the following steps:

Step 1. For each alternative i x X ∈, the decision maker

gives his/her preference value ij r

with respect to attribute j u U ∈, where ij r takes the form of intuit- ionistic fuzzy numbers, that is (,)ij ij ij r v μ=, ij μ∈ [0,1],[0,1]ij v ∈, and 1ij ij v μ+≤, 1,2,...,i m =, j =

1,2,...,n , then all the preference values of the alte- rnatives consists the decision matrix ()ij m n R r ×=.

Step 2. If the information about the attribute weights is

completely unknown, we solve the model (M-2) to

obtain the optimal weighting vector ***1

2(,,w w w = *...,)T n

w .

Step 3. Utilize the weighting vector *

*

**1

2

(,,...,)T n

w w w w =

and by (7), we can obtain the overall values *()i z w (1,2,...,i m =) of the alternatives i x (1,2,...,i m =).

Step 4. Calculate the scores ()i S z of the overall intuitionistic

fuzzy preference value *()i z w (1,2,...,i m =) to rank all the alternatives x i (1,2,...,i m =) and then to select the best one(s)(if there is no difference between two

scores (

)i S z and ()j S z , then we need to calculate the accuracy degrees ()i H z and ()j H z of the overall intuitionistic fuzzy values i z and j z , respectively, and then rank the alternatives i x and j x in accordance with the accuracy degrees ()i H z and ()j H z . Step 5. Rank all the alternatives i x (1,2,...,i m =) and select

the best one(s) in accordance with the ()i S z and ()i H z (1,2,...,i m =).

Step 6. End.

IV. I LLUSTRATIVE E XAMPLE

In this section, we discuss a problem concerning with a manufacturing company, searching the best global supplier for one of its most critical parts used in assembling process (adapted from[19]). The attributes which are considered here in selection of five potential global suppliers i x (1,...,5i =) are (1) u 1: Overall cost of the product; (2) u 2: Quality of the product; (3) u 3: Service performance of supplier; (4) u 4: Supplier’s profile; and (5) u 5: Risk factor. The expert represents the characteristics of the potential global suppliers i x (1,...,5i =) by the IFNs ij r (,1,2,...,5i j =) with respect

to the attributes j u (1,2,...,5j =), list in Table 1.(i.e. intuitionistic fuzzy decision matrix 55()ij R r ×=).

TABLE I. I NTUITIONISTIC FUZZY DECISION MATRIX R =( R ij )5×5

u 1

u 2

u 3

u 4

u 5

x 1 (0.4,0.5) (0.5,0.2) (0.6,0.2) (0.8,0.1) (0.7,0.3) x 2 (0.6,0.2) (0.7,0.2) (0.3,0.4) (0.5,0.1) (0.8,0.2) x 3 (0.7,0.3) (0.8,0.1) (0.5,0.5) (0.3,0.2) (0.6,0.3) x 4 (0.3,0.4) (0.7,0.1) (0.6,0.1) (0.4,0.3) (0.9,0.1)

x 5 (0.8,0.1) (0.3,0.4) (0.4,0.5) (0.7,0.2) (0.5,0.2)

Step 1. Assume the weighting vector of the attribute is completely unknown, by applying (16), we get the optimal weighting vector *

T =(0.2358,0.1945,0.2158,0.1994,0.1545)w

Step 2. Utilize the weighting vector ****125(,,...,)T

w w w w =

and (7) to calculate the overall values *()i z w

(1,2,...,5i =) of the alternatives i x

(1,2,...,5i =).

*1()(0.6171,0.2302)z w =, *2()(0.5991,0.2023)z w =, *3()(0.6168,0.2495)z w =, *4()(0.6223,0.1726)z w =,

*

5()(0.5960,0.2368)z w =.

Step 3. Utilize (4) to calculate the score of scores ()i S z of

the overall intuitionistic fuzzy preference values *()i z w

(1,2,...,5i =).

1()0.3869S z =, 2()0.3968S z =, 3()0.3673S z =, 4()0.4497S z =, 5()0.3592S z =. thus

42135()()()()()S z S z S z S z S z ;;;;

Step 4. Utilize the scores ()i S z (1,2,...,5i =) to rank the

alternatives x i

(1,2,...,5i =). 42135x x x x x ;;;;

and then the most desirable global supplier is x 4.

V.

C ONCLUSIONS

In this paper, we study the multiple attribute decision making problems, in which the information about attribute weights is completely unknown and the attribute values are expressed in intuitionistic fuzzy numbers(IFNs). In order to get the optimal attribute weights, we establish an optimization model based on the mean deviation method. By solving the model, we get a simple and exact formula which can be used to determine the attribute weights. After that, we utilize the intuitionistic fuzzy weighted average (IFWA) operator to aggregate the given intuitionistic fuzzy numbers decision information, and then select the most desirable alternative according to the score function and accuracy function. Finally, a practical example is given to verify the developed method and to demonstrate its practicality and effectiveness. And also, the method can be extended to the group intuitionisitic fuzzy decision making easily.

A CKNOWLEDGMENT

This work was supported by Hohai University "the Fundamental Research Funds for the Central Universities (2009B04514) "

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Tab1-4 一、tabstat fpm fpm_hat , by(pop_cat) stat(mean count) format(%9.2f) 1、命令tabstat --汇总统计表（主要用于描述性统计） 2、语法：tabstat varlist [if] [in] [weight] [, options] 3、fpm fpm_hat：这两个是输出组；by(pop_cat)：变量组； stat(mean count):stat()展示统计结果，mean count 非遗漏观测计数的平均 值;format(%9.2f)：设置变量的输出格式（%9.2f：字符串数据格式） Tab5-6 一、xi: reg fpm fpm_hat pop pop_2 pop_3 i.term i.regions,r cluster(id_city) 1、命令regress--线性回归（简单线性回归）xi: reg生成虚拟变量回归 2、语法：regress depvar [indepvars] [if] [in] [weight] [, options] 3、depvar为被解释变量：fpm； indepvars为解释变量：pm_hat pop pop_2 pop_3 i.term i.regions 二、foreach与forevalue区别： 对foreach的几种具体形式进行讲解。 —①foreachlnamein anylist{ —该种形式允许一般形式的列表（list），列表中的各个元素用空格分开。 —例如， —foreach x in mpg weight-turn { —summarize `x' —} —这时，循环会执行两次，即令局部宏x依次为mpg和weight-turn，来计算其描述统计量。— —②foreachlnameof local lmacname { —或 foreachlnameof global gmacname { —这里，第一种是对局部宏lmacname中的各项进行循环，第二种是对全局宏gmacname中的各项进行循环。因为很多时候，我们事先并不知道具体的要循环的元素，而是将这些元素存储在宏中，所以这种形式很常见。此外，在所有的循环方式中，这两种方式的执行速度最快。 —③foreachlnameof varlistvarlist{ —这里，of和第一个varlist是命令格式的一部分，第二个varlist是具体的变量列表。该种形式表示我们按照变量的方式来对第二个varlist进行解读。例如，我们输入下面的语句：—foreachx of varlistmpg weight-turn { —summarize `x' —} —这里，循环会执行四次，依次对mpg、weight、length和turn进行。这里，weight-turn表示从weight到turn的变量，对于“usaauto.dta”的数据，即包括变量weight、length和turn。— —④foreachlnameof newlistnewvarlist{ —这里，foreach…of newlist…是命令格式的一部分，lname是局部宏的名称，newvarlist是新变量列表。Stata会检查指定的新变量名是否有效，但Stata并不自动将其生成。例如，我们

佳能5D2相机固件更新说明书 中文版

- S.Chinese - EOS 5D Mark II 固件更新步骤

固件更新步骤 下列说明中的x.x.x.代表当前的固件版本或更新的固件版本。 (1) 准备更新固件所需的项目。 1.机身 2.专用电池（电池必须完全充满电）或专用交流电适配器套装（选购） 3.CF卡（64MB或更大，64GB或更小） 4. 固件更新文件（可从佳能网站下载。） (2) 创建固件更新文件。 1.从佳能网站下载压缩的自解压文件。 2.解压下载文件，并创建固件更新文件。 如何解压固件更新文件 Windows 双击下载文件时，将出现以下屏幕。单击[确定]，将解压下载文件并生成固件更新文件。 Macintosh 下载的文件会自动解压并生成固件更新文件。如果下载文件没有自动解压，请双击下载文件。 3.检查固件更新文件的大小。 如果文件大小不匹配，请再次下载固件更新文件。 如何确认固件更新文件的大小 Windows 右键单击固件更新文件的图标，并从弹出的菜单中选择[属性]。 Macintosh 选择固件更新文件的图标，然后从[文件(File)]菜单中选择[Get Info(获得信息)]。 4. 固件更新文件的名称和尺寸可以在网站上查到。

如果使用CF读卡器，请从第(3)步开始操作。如果不使用CF读卡器，请从第(4-1)步开始操作。 (3) 将固件更新文件复制到CF卡。 1.将通过相机格式化的CF卡插入CF读卡器。 2.将固件更新文件复制到打开CF卡时(根目录)出现的第一个窗口中。 3.将CF卡从读卡器中取出。 *取出CF卡时，请务必按照计算机或读卡器说明中所述步骤操作。 *如果固件更新文件被放在CF卡的子文件夹下，则相机无法找到它。 4.旋转模式转盘选择

模式(或除全自动模式外的其他某个模式)。 5.将带固件的CF卡插入相机。 6.打开电源开关，然后按下