An Approach to Multiple Attribute Decision Making Based Preference

Proceedings of the 34th Hawaii International Conference on System Sciences - 2001

An Approach to Multiple Attribute Decision Making Based on Preference Information on Alternatives Jian Ma, Quan Zhang, Zhiping Fan, Jiazhi Liang, Duanning Zhou Department of Information Systems, City University of Hong Kong, Kowloon Tong, Hong Kong, China (isjian, isqzhang, isping, iscleve, iszdn)@is.cityu.edu.hk Abstract This paper investigates the multiple attribute decision making (MADM) problems with preference information on alternatives. A new approach is proposed to solve the MADM problems, where multiple decision makers give their preference information on alternatives in different formats. To reflect decision makers' preference information, an optimization model is constructed to assess attribute weights and then to rank the alternatives or select the most desirable one. Finally, an example is used to illustrate the proposed approach.

1. Introduction In multiple attribute decision making (MADM) problems, decision makers often need to select or rank alternatives that are associated with noncommensurate and conflicting attributes [4,14]. MADM problems arise in many real-world situations [4,7,14,18,19], for example, in production planning problems, attributes such as production rate, quality, and cost of operations are considered in the selection of the satisfactory plan. A lot of research work has been conducted on MADM problems, one of the hot research topics is the use of fuzzy set theory to solve MADM problems when imprecise information is represented in fuzzy terms [4,5,26,27]. In MADM problems, the decision makers' preference information is often used to rank alternatives or select the most desirable one. However, the decision makers' judgements vary in form and depth. A decision maker may express his/her preference on attributes or alternatives in specific style or may not indicate his/her preference at all. The decision makers' judgement skills also vary. Different decision makers may use different ways of expressing their preference information. The approaches to solving MADM problem can be classified into three categories

according to different directions of preference information given by the decision makers [4,14]: (1) the approaches without preference information [2,8,10], (2) the approaches with preference information on attributes [3,17,18,20,28], and (3) the approaches with preference information on alternatives [4-6,14,16,19,25-27]. This paper focuses on the third category, where the decision makers are able to give their preference information on alternatives. Commonly, the preference information on alternatives employs three forms: preference orderings of alternatives, utility values and fuzzy preference relations [5]. Preference orderings of alternatives, including binary preference relations between pairs of alternatives are usually used by decision makers and transformed into fuzzy preference relations [5, 6, 16, 26]. Also the utility values of alternatives are always converted into fuzzy preference relations for ranking the alternatives [5,22,27]. Of course, there are other ways of handling these preference information. For example, starting with the utility values of alternatives given by multiple decision makers, in [29], a formulized heuristic for consensus seeking is proposed, i.e., the negotiable alternatives identifier is used to locate a candidate for compromise and then to search a collective alternative. The three types of preference information from multiple decision makers are unified and fuzzy majority method with fuzzy quantifier is used to aggregate these unified preference information and to select the best acceptable alternative [5]. However, the fuzzy majority method is totally based on the decision makers' subjective preference information. In [21], weights of the criteria are evaluated to exploit the outranking relations between alternatives and to further make a certain alternative the best one. Aspired by the study in [21], this paper presents an optimization model for the MADM problem where the decision makers can also give the three types of preference information on alternatives. The optimization model is used to assess the attribute weights so as to

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Proceedings of the 34th Hawaii International Conference on System Sciences - 2001

reflect the decision makers' opinions and is also subject to the objective facts of alternatives, i.e., the decision matrix (see section 2). It is a new way of reflecting the decision makers' subjective preference information based on the objective decision information of alternatives. The organization of current paper is as follows: Section 2 presents the problem description. Section 3 proposes an approach to the MADM problem, where the three types of preference information on alternatives are given by multiple decision makers. In section 4, the example in [5] is used to illustrate the proposed approach. Conclusion is given in section 5.

2. Problem statement

m (≥ 2) possible alternatives. ! attributes are known: let R = {R1 , R2 , Λ , Rn } denote a set of n (≥ 2) attributes. denote a discrete set of

weights

of

w = ( w1 , w2 , Λ n

∑w j =1

j

attributes are unknown: let T , wn ) be the vector of weights, where

= 1 , w j ≥ 0 , j = 1, Λ , n , and w j denotes

the weight of attribute

to the worst. ! utility values, or utility vector can be used by a decision maker to express his/her preference on the l l l l alternatives: U = (u1 , Λ , um ) , u i ∈ [0,1], 1 ≤ i ≤ m ,

uil represents the utility evaluation given by the decision maker el (l =1, Λ , K) to alternative S i . where

! fuzzy preference information on alternatives can be given by a decision maker. The decision maker’s preference relation is described by a binary fuzzy relation P in S, where P is a mapping S × S → [0, 1]

pik denotes the preference degree of alternative Si over S k [5,15,26-27]. We assume that P is reciprocal, by definition [5,23,26-27], (i) pik + pki = 1 and (ii) pii = − (symbol ‘−’ means that the decision

and

This paper considers the MADM problem where three types of preference information on the alternatives are given by multiple decision makers, i.e. preference ordering, utility values and fuzzy preference relations. The following assumptions or notations are used to represent the MADM problem: ! the alternatives are known: let S = {S1 , S 2 , Λ , S m }

!

i = 1, Λ , m . The alternatives are ordered from the best

maker does not need to give any preference information on alternative S i ), ∀i, k . Since the attributes are generally incommensurate, the decision matrix needs to be normalized so as to transform the various attribute values into comparable values. For the convenience of calculation and extension, the following two functions are used to calculate the degree of membership (see [9,17]):

bij =

aij − a min j a max − a min j j

,

i = 1, Λ , m , j = 1, Λ , n , for benefit criterion,

Rj .

! the decision matrix is known: let the decision matrix where

A = [aij ]m×n denote

aij ( ≥ 0 ) is the consequence

S i with respect to attribute R j , i = 1, Λ , m , j = 1, Λ , n .

bij =

a max − aij j a max − a min j j

,

i = 1, Λ , m , j = 1, Λ , n , for cost criterion,

with a numerical value for alternative

! the decision makers involved are known: let E = ( e1 ,Λ , e K ) denote the set of decision makers. Different decision makers can express their preference on the candidate alternatives in different forms, i.e., preference orderings, utility values and fuzzy preference relations. ! preference orderings, or ordered vector can be used by a decision maker el (l=1, Λ , K) to express his/her preference on the alternatives: O l = (o l (1), Λ , o l (m)) , where o l (⋅) is a permutation function over the index set {1, Λ , m} and o l (i ) represents the ranking position of alternative Si ,

where

(1)

(2)

min a max and a j are given by j

a max = max{a1 j , a2 j , Λ , amj } , j = 1, Λ , n , j

(3)

= min{a1 j , a2 j , Λ , amj } , j = 1, Λ , n .

(4)

a

min j

Then

the

decision

matrix

A = [aij ]m×n can be

transformed into the relative membership degree matrix:

B = [bij ]m×n

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(5)

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Proceedings of the 34th Hawaii International Conference on System Sciences - 2001

The problem concerned is to rank the alternatives or select the most desirable one among the finite set S , based on decision matrix A and the three types of preference information on alternatives given by the multiple decision makers. In the following section, an approach to the MADM problem is proposed, where the three types of preference information on alternatives are given by multiple decision makers. The approach is based on an optimization model which can be used to assess attribute weights and then to rank or select the most desirable alternatives.

3. An approach to the MADM problem When multiple decision makers are involved in the decision process, two phases are usually needed to attain the final solution: aggregation and exploitation. Aggregation is to combine the opinions on alternatives from different points of views; Exploitation is to rank the alternatives or to select the best one based on the collective information on the alternatives. In this section, the two types of preference information on alternatives are first converted into uniform fuzzy preference relations, then preference aggregation and approximation follow.

3.1. Preference uniformity As discussed in [5], a decision maker el (l =1, Λ , K) can use preference orderings or an ordered vector to express his/her preference on the alternatives. In this paper, the preference orderings would be transformed into uniform fuzzy preference relations between the alternatives as follows:

pijl = 12 (1 +

o l ( j ) o l (i ) − ) , 1 ≤ i ≠ j ≤ m , (6) m −1 m −1

where o ( j ) is defined as in section 2, j = 1,Λ , m . Also a decision maker el (l =1, Λ , K) can use utility vector to express his/her preference on the alternatives. The utility vector can be transformed into uniform fuzzy preference relations between the alternatives as follows [5]: l

(u il ) 2 p = l 2 , 1≤ i ≠ j ≤ m , (u i ) + (u lj ) 2 l ij

where

l i

u is defined as in section 2.

3.2. Preference aggregation

Since multiple decision makers are involved in the evaluation and selection process, after their preference information on the alternatives are transformed into uniform fuzzy preference relations, the next step is to aggregate these uniform fuzzy preference relations. In [5], the fuzzy majority method with the fuzzy linguistic quantifier "at least half" and "as many as possible" are used to find the collective fuzzy preference relations. However, in this paper, the most common method "simple additive weighting method" [4,14] is adopted to aggregate these fuzzy preference relations from the multiple decision makers, i.e., K

g ik = ∑ hl pikl , 1 ≤ i ≠ j ≤ m ,

(8)

l =1

where

hl represents the relative importance assigned to

the decision maker el, l=1, …, K.

3.3. Preference approximation Using the "simple additive weighting method" [4,14], the overall value of alternative S i can be expressed as n

d i = ∑ bij w j , i=1,Λ, m,

(9)

j =1

where

d i is an explicit function of the variables w j (j

=1,Λ,n). Based on the overall values, the ranking results of alternatives can be obtained. The greater the ranking value d i is, the better the corresponding alternative Si will be. In order to make the information consistent, the overall values of alternatives can be transformed into fuzzy preference relations. Thus, by using equation (9), g ik is defined as,

g ik =

di , di + dk n

(7)

∑b w ij

=

j

j =1

n

∑ (b

ij

+ bkj ) w j

1≤ i ≠ k ≤ m,

(10)

j =1

where the significance of

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g ik is similar to that of g ik .

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Proceedings of the 34th Hawaii International Conference on System Sciences - 2001

The difference between

λ is the Lagrangian multiplier. Differentiating equation (14) with respect to w j and λ respectively, the

g ik and g ik is given by

where

f ik ( w) = g ik − g ik

following equations are obtained: m

n

∑b w ij

= g ik −

j

j =1

n

∑ (b

ij

+ bkj ) w j

, 1 ≤ i ≠ k ≤ m . (11)

m

n

∑∑{∑[ g ik (bij + bkj ) − bij ]w j }[ g ik (bij + bkj ) − bij ] i =1 k =1 k ≠i

j =1

+ λ =0, j =1,Λ,n, n

∑w

j =1

j =1

j

= 1.

(15) (16)

f ij (w) is an explicit function of w j (j =1,Λ,

Apparently,

n). For the convenience of calculation, the above deviation degree may be changed into the following form: n

n

j =1

j =1

Equations (15) and (16) can be expressed in the following matrix form

Q  T e

uik ( w) = g ik ∑ (bij + bkj ) w j − ∑ bij w j , 1 ≤ i ≠ k ≤ m (12)

e   w  0   =  , 0   λ   1 

(17)

e = (1, 1,Λ ,1)1T×n , Q = (qlj ) n×n . The elements in matrix Q are,

where To reflect the decision maker's preference information, we can minimize f ik (w) or u ij (w) by assessing the attribute weights. So we can construct the following constrained optimization model:

m

m

qlj = ∑∑ [ g ik (bil + bkl ) − bil ][ g ik (bij + bkj ) − bij ] i =1 k =1 k ≠i

Minimize

m

m

m

n

n

U (w) = ∑∑[ gik ∑(bij + bkj )w j − ∑bij w j ] , (13a) 2

i =1 k =1 k ≠i

j =1

j =1

m

= ∑∑ [ g ik bkl − g ki bil ][ g ik bkj − g ki bij ] , i =1 k =1 k ≠i

l , j = 1,Λ , n .

(18)

subject to n

∑w

By solving equation (17), we have

= 1,

(13b)

w j ≥ 0 , j =1,Λ,n.

(13c)

j =1

j

Model (13a)-(13c) is a quadratic programming problem. To solve this model, we first ignore the nonnegative constraint (13c), and then set up the following Lagrangian function, m

m

n

n

j =1

j =1

L = ∑∑ [ g ik ∑ (bij + bkj ) w j − ∑ bij w j ]2 i =1 k =1 k ≠i

n

+ 2λ (∑ w j − 1) , j =1

(14)

w∗ = Q −1e eT Q −1e ,

(19)

λ∗ = − 1 eT Q −1e ,

(20)

where Q is positive definite and invertible (See Appendix for detailed proof). ∗ If equation (9) is substituted with the w j ( j = 1, Λ , n ), the overall values of every alternatives can be obtained. The weight vector determined by equation (19) has practical meanings only if it satisfies the constraint (13c), ∗ ∗ i.e., w ≥ 0 . In practical MADM problems, the w obtained by equation (19) may satisfy the nonnegative constraint (13c). However, if the weight vector does not satisfy the nonnegative constraint, or if for all pairs of i

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Proceedings of the 34th Hawaii International Conference on System Sciences - 2001

and k, 1 ≤ i ≠ k ≤ m ,

n

n

j =1

j =1

g ik ∑ w j bkj = g ki ∑ w j bij , and

Q is not invertible, then model (13a)-(13c) should be solved by using a quadratic programming method [12].

4. Illustrative example Purchasing a house is a traditional MADM problem where the proposed approach can be nest used. A potential buyer intends to select a house from four alternatives (i.e. S1 , S 2 , S 3 and S 4 ). When making a decision, the attributes considered include: 1) R1 : house price ($10,000),

R2 : dwelling area (m2), 3) R3 : distance between every house and the work 2)

locality (km), 4) R4 : natural environment (assessment value). Among the four attributes,

R2 and R4 are of benefit

R1 and R3 are of cost type. The decision matrix with the four attributes ( R1 , R2 , R3 and R4 ) and the four alternatives ( S1 , S 2 , S 3 and S 4 ) is presented as

type,

 3.0 100 10 7     2.2 70 12 9  A= , 2.5 80 8 5     1.8 50 20 11   B by using

1 5 6 1 3  0    2 3 2 5 2 3 2 3 B= . 5 12 3 5 1 0     1  0 0 1   Suppose six persons (i.e., decision makers) e1, e2, e3, e4, e5, e6 help the buyer to give their opinions in terms of ordered vectors, utility vectors and fuzzy preference relations as follows [5]:

O1 ={3, 1, 4, 2}, e2: O 2 ={3, 2, 1, 4}, 3 4 e3: U ={0.5, 0.7, 1, 0.1}, e4: U ={0.7, 0.9, 0.6, 0.3}, e1:

 − 0.5 0.7 1     0.5 − 0.8 0.6  6 e6 : P =  . 0.3 0.2 − 0.8     0 0.4 0.2 −    To unify the preference information, the transformation functions in (6) and (7) are used and the results are as follows [5]:

 − 16 23 13  5  2 6 − 1 3 1 P =1 , 0 − 16  3    2 1 5 − 3 3 6 

 − 13 16 23  2  1 1 3 − 3 6 2 P =5 2 , − 1 6 3    1 5 0 − 3 6 

25 25 0.2  − 74 26   49  49 0.98  −  74 149 3 P = , 100  0.8 100 − 149 101    1 0.02 100 −  101  26

follows:

which can be normalized into matrix equations (1)-(3) and (4) as follows

 − 0.1 0.6 0.7     0.9 − 0.8 0.4  5 , e5 : P =  0.4 0.2 − 0.9     0.3 0.6 0.1 −   

−  81  130 4 P =  36  85  9  58

  0.9  − . 36 − 0.8  117  0.1 0.2 −  49 130

49 85

49 58

81 117

Suppose the decision makers are equally important. Using the "simple additive weighting method" [4,14] to aggregate above six decision makers' fuzzy preference information, we have

0.3025 0.4850 0.7511   −   0.6591 0.7300  −  0.6975 G = 0.5150 0.3409 0.7761  −    0.2489 0.2700 0.2239  −  

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Proceedings of the 34th Hawaii International Conference on System Sciences - 2001

By equations (18) and (19), the weight vector on the attributes can be obtained as follows: w∗ = (0.1725, 0.3475, 0.2774, 0.2026) T . The overall values of the four alternatives can be obtained by using equation (9), i.e., d1 =0.6462, d 2 =0.5740, d 3 =0.5578, d 4 =.3751. Thus the ranking result of the alternatives is

S1 φ S 2 φ S 3 φ S 4 . 5. Summary This paper proposes a new approach to solve the MADM problem with three types of preference information on alternatives. An optimization model is used to assess attribute weights and then to rank alternatives. In the proposed approach, the different types of preference information are firstly transformed into uniform fuzzy preference relations and aggregated. Then ranking or selection of the alternatives reflects the decision makers' subjective preference based on the objective decision information (decision information matrix A). The proposed approach is also an extension of the fuzzy majority method with fuzzy linguistic quantifier [5]. The proposed approach is not without limitations. It only considers the three types of preference information on the alternatives, i.e., preference ordering, utility values and fuzzy preference relations. Further research would be conducted to investigate the situations where the preference information on alternatives can also be given in the forms of linguistic variables [11,13] and fuzzy selected subset of all the alternatives [30].

Acknowledgement This research is partly supported by the Competitive Earmarked Research Grant (CERG) of Hong Kong (Project No. 9040375), National Natural Science Foundation of China (NSFC) (Project No. 79600006), and Research Grant Council of Hong Kong and NSFC joint research scheme (Project No. 9050137).

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[22] K. Nakamura, "Preference relations on set of fuzzy utilities as a basis for decision making", Fuzzy Sets and Systems, 20 (1986) 147-162. [23] S.A. Orlovski, "Decision-making with a fuzzy preference relation", Fuzzy Sets and Systems, 1 (1978) 155-167. [24] Redfern D. and C. Campbell, The MATLAB 5 Handbook, Springer, New York, 1998. [25] V. Srinivasan, and A.D. Shocker, "Linear programming techniques for multidimensional analysis of preference", Psychometrika, 38 (1973) 337-369. [26] T. Tanino, "Fuzzy preference orderings in group decision making", Fuzzy Sets and Systems, 12 (1984) 117-131. [27] Tanino, T., On group decision making under fuzzy preference, in: J. Kacprzyk and M. Fedrizzi, Eds., Multiperson Decision Making Using Fuzzy Sets and Possibility Theory, Kluwer Academic publishers, Dordrecht, 1990. [28] M. Weber, "Decision making with incomplete information", European Journal of Operational Research, 112 (1999) 44-57. [29] J. Yen and T.X. Bui, "The negotiable alternatives identifier for group negotiation support", Applied Mathematics and Computation, 104(1999) 259-276.

n

n

∑w q

W QW = T

2 j

jj

j =1

n

+∑

∑w w q l

m

m

n

∑∑∑ w

=2

j

lj

l =1 j =1, j ≠ l

2 j

( pik bkj − p ki bij ) 2

+2

i =1 k >i j =1

m

m

n

n

∑∑∑∑ 2w w ( p l

j

ik

bkl − p ki bil )( pik bkj − p ki bij )

i =1 k >i l =1 j >l

m

=2

m

n

∑∑[∑ w ( p j

i =1 k >i

ik

bkj − p ki bij )]2 .

j =1

If there exist at least a pair of i and k, 1 ≤ i ≠ k ≤ m , such

that

W T QW >0.

n

n

j =1

j =1

pik ∑ w j bkj ≠ p ki ∑ w j bij

,

then

Therefore Q is positive definite and

invertible.

[30] Zhou, D.N., Fuzzy Group Decision Support System Approach to Group Decision Making Under Multiple Criteria. Dissertation of Doctor of Philosophy, City University of Hong Kong, January, 2000.

Appendix: Proof of Q is positive definite and invertible By equation (15), we have m

m

qlj = 2∑∑ ( pik bkl − pki bil )( pik bkj − p ki bij ) , i =1 k >i

m

l , j = 1,Λ , n, l ≠ j

m

q jj = 2∑∑ ( pik bkj − p ki bij ) 2 , i =1 k >i

Given any

j = 1,Λ , n .

W = ( w1 , w1 ,Λ , wn ) T ≠ 0 , we have

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