A Method for Multi-attribute Decision-making with Complementary Preference Information on Alternatives

Hong-an Zhou

School of Science, Xi’an Technological University, Xi’an 710032, P. R. China

E-mail:honganzhou@sohu.com

Keywords: Multi-Attribute Decision-Making; Goal Programming; Weight; Priority.

Abstract. The multi-attribute decision making (MADM) problem, in which the information about attribute weights are known partly and the decision maker (DM) has fuzzy complementary preference relation on alternatives, is investigated in this paper. Firstly, The objective decision-making information based on the subjective fuzzy complementary preference information on alternatives is uniformed by using a translation function. Secondly, a goal programming model is established. The attribute weights are obtained by solving the model, thus the overall values of the alternatives are gained by using the additive weighting method. Based on these values, the ranking priorities or selecting the best on alternatives are processed. The method can sufficiently utilize the objective information of alternatives and meet the subjective requirements of the DM as much as possible, and it is also characterized by simple operation and easy to implement on a computer. Finally, a practical example is illustrated to show the feasibility and availability of the developed method.

Introduction

The MADM problems, whose theories and methods have been widely used in many fields such as fuzzy system optimization, economic estimation and selection of mining address, etc., is taking up an important status in the modern decision-making science. Up to now, a lot of good results[1-10] have been obtained on the researches of the MADM problems without preference information on alternatives. However, in the real process of decision-making, the decision maker (DM) often has the subjective preference information on alternatives[11-16]. This is what is called the MADM problems with preference information on alternatives (MADM-PA, for short). As a result, the MADM-PA problems have been investigated and some methods of solving theses problems have been proposed recently. The MADM-PA problems that the information about attribute weights is unknown completely, the attribute values are numbers and the DM’s preference relation on alternatives take the form of fuzzy complementary judgment matrix[11-12] or reciprocal judgment matrix[13-14] are studied in Refs [11-14]. A quadratics programming method is presented. The MADM-PA problem, in which the attribute weights are interval numbers and the DM’s preference relation on alternatives is in the fuzzy complementary judgment matrix and reciprocal judgment matrix is investigated in [15]. The objective decision-making information is transformed the multiplicative complementary judgment matrix and reciprocal judgment matrix, the linear programming approach is presented by using minimum modem both the subjective and objective information. The MADM-PA problem that the information about attribute weights is known partly and the DM has ordering preference on alternatives is studied in [16]. The method based on sensitivity analysis is presented. The MADM problem that the information about attribute weights are known partly and the DM’s preference information on alternatives take the form of fuzzy complementary judgment matrix is investigated in this paper. In our method, Section 2 gives simple representation of the studied problem. The principle and approach of decision-making are given in section 3. An application example is given in section 4. Finally, some concluding remarks are included.

Advanced Materials Research Vol. 658 (2013) pp 541-545Online available since 2013/Jan/25 at www.scientific.net© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.658.541

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Representation of Problem

The basic model of the MADM-PA that will be studied in this paper can be represented as follow:

� Let 1 2{ , , , }nX x x x= � be a set of n ( 2≥ ) feasible alternatives, and denote {N = 1, 2, , }n� ;

� Let 1 2{ , , , }mS s s s= � be a set of m ( 2≥ ) attributes, and denote {1,2, , }M m= � ;

� Let 1 2( , , , )Tmω ω ω ω= � be the vector of weights, such that 0R L

i i iω ω ω≥ ≥ ≥ , 1

1,m

L

i

i

ω=

≤∑

1 1

1, 1m m

R

i i

i i

ω ω= =

≥ =∑ ∑ , where R

iω and L

iω are the upper and lower bounds of iω , respectively;

� Let ( )lj n nB b ×= be the DM’s fuzzy complementary preference judgment matrix on

alternatives, where ljb denotes a ratio of preference degree for the alternative lx over jx , and such

that 1lj jlb b+ = , ,l j N∈ ;

� Let ( )ij m nA a ×= be a decision matrix, where ija is the attribute value for the alternative jx

with respect to attribute is , ,i M j N∈ ∈ .

In general, there are benefit attribute values 1I , cost attribute values 2I , steady attribute values 3I

and interval values 4I in the MADM and the different attribute values may be the different dimension.

Obviously, 1 2 3 4M I I I I= ∪ ∪ ∪ . For the convenience of decision-making, we need to deal with all

attributes in dimensionless units and normalize each attribute value. This can be achieved by

normalizing ija in the matrix ( )ij m nA a ×= into a corresponding element ijr in the matrix ( )ij m nR r ×=

with the following formulas [7] .

min

max min

ij ijj

ij

ij ijjj

a ar

a a

−=

−, 1,i I j N∈ ∈ (1)

max

max min

ij ij

j

ij

ij ijjj

a a

ra a

−

=−

, 2 ,i I j N∈ ∈ (2)

*

*1

max

ij i

ij

ij ij

a ar

a a

−= −

−, 3,i I j N∈ ∈ ，where *

ia the best steady value of

(3)

max{ , }1 , [ , ]

max{ min ,max }

1, [ , ]

L R

i ij ij i L R

ij i iL R

i ij ij iij j j

L R

ij i i

q a a qa q q

q a a qr

a q q

− −− ∉ − −=

∈

, 4 ,i I j N∈ ∈ (4)

Then using the simple additive weighting method [1] , the overall value of alternative jx can be

expressed as 1

( ) ,m

j i ij

i

z r j Nω ω=

= ∈∑ (5)

Principle and Approach of Decision-Making

Considering that the overall value ( )jz ω is derived from the decision matrix ( )ij m nA a ×= , and thus,

we here regard it as the objective preference values. In order to reflect to the DM’s subjective

preference on alternatives, we can transform ( )jz ω into fuzzy complementary preference relation

( )lj n nC c ×= , ,l j N∈ , where ljc can be defined as follow:

1 1 1

1 1 1[ ( ) ( ) 1] [ 1] ( 1), ,

2 2 2

m m m

lj l j i il i ij i il ij

i i i

c z z r r r r l j Nω ω ω ω ω= = =

= − + = − + = − + ∈∑ ∑ ∑ ， (6)

is

542 Materials and Manufacturing Research

Obviously, the meaning of ljc in the matrix ( )lj n nC c ×= is similar to that of ljb in the DM’s

preference matrix ( )lj n nB b ×= . If the fuzzy complementary preference relations C and B are

consistent, then C B= , i.e., , ,lj ljb c l j N= ∈ ，, hence we have

1 1

1( 1) ( 1) 2 0

2

m m

lj i il ij i il ij lj

i i

b r r r r bω ω= =

= − + ⇔ − + − =∑ ∑ ， ,l j N∈ (7)

However, there always exist some deviations between the subjective preference values and the corresponding objective preference values on alternatives in the real life. This is, (7) does not hold in

the general case. Hence, we introduce the deviation item ljf , i.e. we let

1

( 1) 2m

lj i il ij lj

i

f r r bω=

= − + −∑ , ,l j N∈ (8)

It is easy to know that we can minimize all above deviations to obtain the reasonable attribute

weights 1 2( , , , )Tmω ω ω ω= � . Therefore, the following multi-objective optimization model can be

established from (8):

(M1) 1

min ( 1) 2m

lj i il ij lj

i

f r r bω=

= − + −∑ ．

1 1 1

. . 0, 1, 1, 1, ,m m m

R L L R

i i i i i i

i i i

s t l j Nω ω ω ω ω ω= = =

≥ ≥ ≥ ≤ ≥ = ∈∑ ∑ ∑ .

Since all above objective functions are expected to be equal to zero, we can transform the model M1 into the following objective programming model M2:

(M2) 1 1

m in ( )n n

lj lj lj lj

l j

J s d t d+ −

= =

= +∑ ∑ ．

1

. . ( 1) 2 0m

i il ij lj lj lj

i

s t r r b d dω + −

=

− + − − + =∑ ;

0, 0, 0, ,lj lj lj ljd d d d l j N− + + −≥ ≥ = ∀ ∈ ;

0R L

i i iω ω ω≥ ≥ ≥ ,1

1,m

L

i

i

ω=

≤∑1

1,m

R

i

i

ω=

≥∑1

1m

i

i

ω=

=∑ , , ,i M l j N∈ ∈ .

where ljd+ and

ljd− denote the upper and lower deviation variables of

1

( 1) 2m

i il ij lj

i

r r bω=

− + −∑ ， whose

expected value is equal to zero, respectively. ljs And ljt denote the weighted coefficient, respectively.

Considering all above object functions are fair and the DM has no preference to them. Therefore,

1lj ljs t= = , ,l j N∀ ∈ ．Thus the model M2 can be transformed into the following liner programming

model M3:

(M3) 1 1

min ( )n n

lj lj

l j

J d d+ −

= =

= +∑∑ ．

1

. . ( 1) 2 0m

i il ij lj lj lj

i

s t r r b d dω + −

=

− + − − + =∑ ;

0, 0, 0, ,lj lj lj ljd d d d l j N− + + −≥ ≥ = ∀ ∈ ;

0R L

i i iω ω ω≥ ≥ ≥ ,1

1,m

L

i

i

ω=

≤∑1

1,m

R

i

i

ω=

≥∑1

1m

i

i

ω=

=∑ , , ,i M l j N∈ ∈ .

The vector of attribute weightsω can be obtained by solving the model M3. Substituting ω into

equation (5), we can get the overall value ( )jz ω of the alternative jx . Obviously, the bigger the vales

of ( )jz ω is, the better the corresponding alternative jx .

Advanced Materials Research Vol. 658 543

Based on the above discussion, we give the solving processes to show the feasibility and availability of the proposed method by a practical example in the next section.

Practical Example

An investment company is planning to exploit a new product and there are four investment

alternatives ( 1, 2,3, 4,5)jx j = to be considered. The main attributes of evaluating the alternatives are

listed as 1s : investment amount, 2s : expected net profit amount, 3s : venture profit amount； 4s :

venture loss amount, respectively. Among these four attribute, 2s and 3s are of benefit attributes, 1s

and 4s are of cost attributes. The attribute values of each alternative are listed in Table1 (attribute unit

is10000$), and the weighted information ( 1,2,3,i iω = 4) is known partly as 10.1 0.5ω≤ ≤ ;

20.1 0.3ω≤ ≤ ； 30.01 0.2ω≤ ≤ ；0.25 ≤ 4 0.45ω ≤ .

Table1 Decision matrix A�

Step1 By（1）and（2），we can get the normalized decision matrix R from Table1 as follow:

4 5

0 1 0.010 0.926 0.287

0.492 1 0.153 0.508 0( )

0.407 0 0.784 0.071 1

1 0 1 0.254 0.707

ijR r ×

= =

0.5 0.6 0.5 0.9 0.7

0.4 0.5 0.4 0.7 0.5

0.5 0.6 0.5 0.7 0.2

0.1 0.3 0.3 0.5 0.1

0.3 0.5 0.8 0.9 0.5

=

B

Step2 Suppose that the DM gives his/her fuzzy complementary preference relation B on alternatives as top:

Then, by solving the model (M3), the weight vector of the attributes is obtained as

(0.102,0.300,0.200,0.398)Tω = .

Step3 By (5), the overall value ( )jz ω of the alternative jx as follow:

1 0.627z = ； 2 0.402z = ； 3 0.602z = ； 4 0.362z = ； 5 0.511z = ．

Thus the ranking priorities of the corresponding alternatives is 4 2 5 3 1x x x x x≺ ≺ ≺ ≺ , This is, the

best alternative is 1x ．

Conclusion

In this paper，a new method based on objective programming is proposed with regard to the MADM

problems, in which the information about weights is partly known and the DM’s preference information on alternatives take the form of fuzzy complementary judgment matrix. And the proposed method is applied the project evaluation in the field of venture investment. Not only the method can sufficiently utilize the objective information of alternatives, meet the subjective requirements of the DM and implement the unification between the subjective and objective information, hence make the decision-making results to be more reasonable. But also it has the advantages of simple operation and easy to implement on a computer.

1x 2x 3x 4x 5x

1s 5.2 10.08 5.25 9.72 6.60

2s 5.2 6.70 4.20 5.25 3.75

3s 4.73 5.71 3.82 5.54 3.30

4s 0.473 1.599 0.473 1.313 0.803

544 Materials and Manufacturing Research

Acknowledgment

This work was supported by the Science Research Foundation of Education Committee of Shaanxi Province (11JK0501)

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Materials and Manufacturing Research 10.4028/www.scientific.net/AMR.658 A Method for Multi-Attribute Decision-Making with Complementary Preference Information on

Alternatives 10.4028/www.scientific.net/AMR.658.541