Relative Membership Degree Method for Fuzzy Multi attribute Decision making with Preference Information on Alternatives

2017 2nd International Conference on Computer, Network Security and Communication Engineering (CNSCE 2017) ISBN: 978-1-60595-439-4

Relative Membership Degree Method for Fuzzy Multi-attribute

Decision-making with Preference Information on Alternatives

Hong-an ZHOU

and Ke-xin ZHOU

1

School of Science, Xi’an Technological University, Xi’an 710021, P. R. China 2

School of mathematics and statistics, Xidian University, Xi’an 710126, P. R. China

*Corresponding author

Keywords: Fuzzy multi-attribute decision-making, Weight, Relative membership degree, Priority.

Abstract. The fuzzy multi-attribute decision-making (FMADM) problem that the attribute weights are interval numbers and the decision maker (DM) has avail preference information on alternatives is investigated. Firstly, a quadratic programming model based on the minimum sum of deviation squares between the subjective and objective decision-making preference information on alternatives is established, the attribute weights are obtained by solving the model and thus the overall values of every alternative are gained by using the additive weighting method. Secondly, The fuzzy positive ideal solution (FPIS) and fuzzy negative ideal solution (FNIS) of alternatives are introduced. Based on fuzzy sets theory, the relative membership degree of every alternative is obtained by means of the distance to which an alternative corresponds to the FPIS and FNIS, the alternatives are ranked by using the relative membership. The method not only avoids comparing for fuzzy numbers, but also has the advantages of simple operation and easy calculation. Finally, a practical example is illustrated to show the feasibility and availability of the developed method.

Introduction

presented to rank alternatives. In this paper, we propose a method for the FMADM problem that the attribute weights are interval numbers and DM has avail preference information on alternatives is investigated. a quadratic programming model based on the minimum sum of deviation squares between the subjective and objective decision-making preference information on alternatives is firstly established to determine the attribute weights, then the overall values of alternatives are gained. Then the FPIS and FNIS of alternatives aer introduced. Based on fuzzy sets theory, the relative membership degree to which an alternative corresponds to the best alternative is obtained by means of the synthetically weighted distance between the overall values of every alternative and the ideal solution, the alternatives are ranked by using the relative membership. A practical example is lastly illustrated to show the feasibility and availability of the developed method.

Preliminary Knowledge

For the sake of convenience, we first introduce some operational laws of triangular fuzzy numbers, which will be useful in the later part of this paper. Let a=[a aL, M,aM] and b=[b bL, M,bR], where

0<aL ≤aM ≤aR, 0<bL M R

b b

≤ ≤ . Their operational laws are listed as follow in Ref [10]:

1)a=b iff aL =b aL, M =bM,aR =bR;

2)_{a+ =b} [_aL_{+b a}L, M _+bM,_aR_+bR]_;

3)_{a− =b} [_aL _−bR,_aM _−bM,_aR_−bL]_; 4)λa=[λaL,λaM,λaR], whereλ >0;

5)_ab=[_{a b a b}L L, M M,_{a b}R R]_{; 6) [ , ]}

L M R

R M R

b b b b

a = a a a

.

Definition 1 Leta=[a aL, M,aR] and b=[b bL, M,bR],

then 1( ) 3

L L M M R R

a b= a b +a b +a b is called the multiply between a and b .Specially, if a=b , then

2 2 2

1

[( ) ( ) ( ) ] 3

L M R

aa = a + a + a a is called the module of a in Ref [10].

Definition2 Leta=[a aL, M,aR] and b=[b bL, M,bR], then

2 2 2

1

( , ) [( ) ( ) ( ) ]

3

L L M M U U

d a b = a −b + a −b + a −b

(1) is called the distane between aandb in Ref [10].

The basic model of the FMADM problem which will be studied in this paper can be represented as follows:

LetX ={ ,x x1 2,,xn}_{be a set of} n₍≥2_{) feasible alternatives, and denote} N ={1, 2,, }n _. LetS ={ ,s s1 2,,sm} _{be a set of} m₍≥2_{) attributes, and denote} M ={1, 2,, }m _.

Let ( 1, 2, , )

T m

ω= ω ω ω

be the vector of weights, such that

1 1 1

0, 1, 1, 1,

m m m

R L L R

k k k k k k

k k k

ω ω ω ω ω ω

= = =

≥ ≥ ≥

∑

∑

∑

L k

ω

and ω_kRare the left and right bounds of ω_k, respectively. And ω_k is the weight of attributes_k.

Let A=(akj m n) ×

be the fuzzy decision matrix, where [ , , ]

L M R

kj kj kj kj

a = a a a

is an attribute value, which takes the form of triangular fuzzy numbers, for the alternative xj with respect to attributesk_, k∈M j, ∈N _.

In general, there are cost attribute values and benefit attribute values in the FMADM and the different attribute values may be the different dimension. 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 a_kj in the matrix A= (a_{kj m n}) × into a corresponding elements rkjin

the matrix R=( )r_{kj m n}× by using the following formula in Ref [10]:

2 1 2 1 2 1 / ( ) / ( ) / ( ) n L L

kj kj kj

j n

M M

kj kj kj

j n

R R

kj kj kj

j

r a a

r a a

r a a

= = =  =     =     ₌ 

∑

∑

∑

, for benefit attributes_k,

,

k∈M j∈N ₍₂₎

2 1 2 1 2 1

(1/ ) / ( 1/ )

(1/ ) / ( 1/ )

(1/ ) / ( 1/ )

n

L R

kj kj kj

j n

M M

kj kj kj

j n

R L

kj kj kj

j

r a a

r a a

r a a

= = =  =     =     ₌ 

∑

∑

∑

, for cost attributes_k,

,

k∈M j∈N ₍₃₎

Where [ L, M, R]

kj kj kj kj

r = r r r , and the normalized attribute values r_kjcan be regard as the objective decision values given by the DM.

Then using the simple additive weighting method, the fuzzy overall value of alternatives can be expressed as :

1

( ) ,

m

j k kj

k

z ω ω r j N

=

=

∑

(4)

Principle and Model of Decision-Making

In order to rank the alternatives, we must determine the attribute weights at first. Because there always exist some differences between the DM’s the subjective preference values v_j and the

corresponding fuzzy overall values z_j( )ω _{of alternatives in the real life, we can minimize the sum}

of deviation squares between the subjective preference value and the corresponding objective preference values. Therefore, the following quadratic programming model can be established:

(M1)

2 2

1 1

1 1 1

min ( ) ( )

. . 0, 1, 1, 1.

n m

k kj j

j k

m m m

R L L R

k k k k k k

k k k

d r v

st

ω

ω

ω

ω

ω

ω

ω

ω

= = = = =  = −     _{≥ ≥ ≥ ≤ ≥ =} 

∑ ∑

∑

∑

∑

alternatives, we give the following concepts.

Definition3 Let r (r1,r2, ,rm)T

+ + +

+

=

and [ L,

k _k

r+ = r+ M, R]

k k

r+ r+ [max _kjL, max _kjM, max _kjR]

j r j r j r

= , then

( )

z+ ω

= 1 m k k k r ω + =

∑

Analogously, Let r (r1,r2, ,rm)T

− − −

−

=

and [ L,

k _k

r− = r− M, R]

k k

r− r− [min L, min M, min R]

kj kj kj

j r j r j r

= ,

thenz−( )ω

= 1 m k k k r ω − =

∑

Because the FPIS and FNIS are inexistent in the real life. Based on fuzzy sets theory, we can calculate the synthetically weighted distance between the overall values of every alternative and the ideal solution by using formula (1), then the relative membership degreeµ_j to which an alternative corresponds to the best alternative is obtained by using the following formula (5).

Let _{min ( )} 2 2_{( ( ), ( ))}

j j j

L µ µ d z ω z+ ω

= _{(1 )}2 2_{( ( ), ( ))}

j d zj z

µ ω − ω

+ − .

2 2

( )

2 ( ( ), ( )) 2(1 ) ( ( ), ( ))

j

j j j j

j dL

d z z d z z

d

µ

µ ω ω µ ω ω

µ + − = − − 0 = _. Then 2 2 2 ( ( ), ( )) ( ( ), ( )) ( ( ), ( )) j j j j

d z z

d z z d z z

ω ω

µ

ω ω ω ω

− + − = + 2 1 2 ( ( ), ( )) [1 ] ( ( ), ( )) j j

d z z

d z z

ω ω ω ω + − − = + (5) It is easy to know that the greater the value ofµ_j, the better the alternativex_j.Therefore, we can

rank all the alternatives and find the best alternative according to the values of µ_j(j∈N).

Based on the above discussion, we will give the solving processes of the proposed method by a practical example in the following section.

Practical Example

An investment company is planning to exploit a new product and there are four investment alternatives x_j(j=1, 2, 3, 4) to be evaluated. In the decision making process, there are five main

attributes to be considered, that is, s₁: investment amount;s₂: expected net profit amount; s₃: venture profit amount; s₄: investment real profit amount; s₅: venture loss amount, respectively. Obviously, s₂,s₃ and s₄ are of benefit attributes, s₁ands₅are of cost attributes. We suppose that the DM has avail preference information on alternatives as follows:

1 [0.50,0.55,0.60], 2 [0.40,0.45,0.50],

v = v = v₃=[0.35,0.40,0.50],v₄=[0.55,0.57,0.60].

Both the attribute values of every alternative and the weighted information ω_k(k =1, 2, 3, 4, 5) are listed in Table 1 (attribute unit is10000$):

Table1. Decision matrix A and weighted information.

k

s ωk x1 x2 x3 x4

1

s _{[0.25,0.39] [0.80,0.85,0.90] [0.90,0.95,1.00] [0.88,0.91,0.95] [0.86,0.89,0.95]}

2

s _{[0.12,0.25] [0.90,0.92,0.95] [0.89,0.90,0.93] [0.84,0.86,0.90] [0.90,0.92,0.95]}

3

s _{[0.09,0.12] [0.91,0.94,0.95] [0.90,0.92,0.95] [0.91,0.94,0.97] [0.90,0.95,0.97]}

4

Step 1 By (2) and (3), we can get the normalized decision matrixR from Table1 as follows:

5 4

[0.44,0.47,0.50] [0.50,0.53,0.55] [0.49,0.50,0.53] [0.48,0.49,0.53]

[0.50,0.51,0.53] [0.49,0.50,0.51] [0.46,0.47,0.50] [0.50,0.51,0.53]

( )_kj [0.49,0.50,0.51] [0.48,0.49,0.51] [0.49,0.50,0.52] [0.48,0.51,

R= r × = 0.52]

[0.49,0.51,0.52] [0.50,0.51,0.52] [0.48,0.50,0.51] [0.48,0.49,0.50]

[0.49,0.50,0.51] [0.48,0.49,0.51] [0.49,0.51,0.53] [0.47,0.49,0.51]

 

 

 

 

 

 

 

 

Step 2 By solving the model M1, the weight vector of the attributes is obtained as follows: (0.35, 0.24, 0.10, 0.14, 0.17)T

ω= .

Step 3 By (4) and the normalized decision matrixR, the overall value z_j( )ω of the alternative x_j

(j∈N), the FPIS z+( )ω

and FNIS z−( )ω

can be calculated as follows:

1( ) [0.475, 0.493, 0.513]

z ω =

;

2( ) [0.492, 0.509, 0.525]

z ω =

;

3( ) [0.481, 0.496, 0.519]

z ω =

;

4( ) [0.483, 0.497, 0.521]

z ω =

; ( ) [0.497, 0.517, 0.535]

z+ ω

=

; ( ) [0.459, 0.478, 0.505]

z− ω

=

.

Step4 by (5), we calculate the relative membership degreeµ_j to which an alternative corresponds

to the best alternative as follows:µ₁=0.261;µ₂=0.928;µ₃=0.513;µ₄ =0.601.

Since µ₁<µ₃<µ₄<µ₂ , so the ranking priorities of the corresponding alternatives isx₁≺x₃≺x₄≺x₂, This is, the best alternative isx₂.

Conclusion

In this paper, a new method based on quadratic programming and relative membership degree is proposed with regard to the FMADM problems, in which the attribute weights are interval numbers and DM has avail preference information on alternatives. The developed method not only avoids comparing and ranking triangular fuzzy numbers, but also reflects both the objective information and the DM’s subjective preferences on alternative. Moreover, the theoretic analysis and numerical results show that the developed method is feasible and efficient.

Acknowledgment

In this paper, the research was sponsored by the Natural Science Research Foundation of Shaanxi Province (2015JQ 1007)

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