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Multiple attribute decision making based on fuzzy preference information on alternatives: Ranking and weighting
Fuzzy Sets and Systems 153 (2005) 331 – 346 www.elsevier.com/locate/fss
Multiple attribute decision making based on fuzzy preference information on alternatives: Ranking and weighting夡 Ying-Ming Wanga, b, c,∗ , Celik Parkand a Manchester Business School, The University of Manchester, P.O. Box 88, Manchester M60 1QD, UK b School of Public Administration, Fuzhou University, Fuzhou 350002, PR China c Center for Accounting Studies of Xiamen University, Xiamen, Fujian 361005, PR China d Department of Management, C.W. Post Campus, Long Island University, 720 Northern Blvd., Brookville, NY 11548, USA
Received 23 October 2003; received in revised form 6 January 2005; accepted 28 February 2005 Available online 18 March 2005
Abstract This paper investigates the multiple attribute decision making (MADM) problem with fuzzy preference information on alternatives and proposes an eigenvector method to rank them. Three optimization models are introduced to assess the relative importance weights of attributes in a MADM problem, which integrate subjective fuzzy preference relations and objective information in different ways. Since the decisions these models lead to are based on both subjective and objective information, they are more balanced and credible than those that consider only one type of information. An example is provided to illustrate the application of the proposed models. © 2005 Elsevier B.V. All rights reserved. Keywords: Multiple attribute decision making; Fuzzy preference relation/matrix; Weight estimation; Alternative ranking
1. Introduction In a recent paper [6] Fan et al., investigated the multiple attribute decision making (MADM) problem with fuzzy preference information on alternatives and proposed a decision aid approach to combine 夡
This research was supported by the National Natural Science Foundation of China (NSFC) under the Grant Nos. 70271056 and 70171035, and also in part by Fok Ying Tung Education Foundation under the Grant No. 71080. ∗ Corresponding author. Manchester Centre for Civil & Construction Engineering, The University of Manchester, PO Box 88, Manchester M60 1QD, UK. Tel.: +44 161 2005974; fax: +44 161 2004646. E-mail addresses: [email protected], [email protected], [email protected] (Y.-M. Wang). 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.02.018
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subjective fuzzy preference information with objective information for an overall assessment of the relative importance weights of the underlying attributes and select the best alternative. Approach Fan et al. proposed has a greater intuitive appeal than the methods that consider only one type of information. Since fuzzy preference information on alternatives involves decision maker’s (DM’s) subjective considerations, any decision making tool chosen must incorporate them when ranking alternatives. However, such a ranking could not be considered final as it would lack objectivity. To determine the final ranking of alternatives, the weights of the attributes need to be assessed using subjective fuzzy preference information as well as information that is free from any bias the decision maker (DM) may introduce through his/her subjective input. MADM problems with fuzzy preference information on alternatives are essentially problems of the importance of weighting and ranking. In this paper, an eigenvector method (EM) is proposed to derive a subjective ranking of alternatives from fuzzy preference relations, and three optimization models are introduced to integrate subjective fuzzy preference relations and objective information in different ways to assess the weights of the attributes. If the three methods produce exactly the same ranking, then that ranking is deemed to be credible. The models proposed here differ from Fan et al.’s approach in the manner they combine subjective and objective information, which involves the use of linear programming (LP) whose computational demands are much less complex. The DM’s preferences on attribute weights, if provided, can also be easily incorporated into these models, a property that makes them particularly flexible. The paper is organized as follows. Section 2 presents a brief review of the approach proposed by Fan et al. [6]. Section 3 demonstrates how the DM’s subjective ranking can be derived from the fuzzy preference relations using an EM. Section 4 computes subjective weights using the fuzzy preference relations and objective information, and develops three optimization models for integrating both subjective and objective information. All the models proposed in the paper are illustrated in Section 5 through a numerical example. Conclusions are offered in Section 6.
2. Fan et al.’s approach to MADM based on fuzzy preference information on alternatives Consider a MADM problem with n alternatives, A1 , . . . , An , and m decision attributes (criteria), G1 , . . . , Gm . Each alternative is assessed with respect to each attribute. The assessment scores assigned to the attributes are the components of a decision matrix denoted by X = (xij )n×m . Any incommensurability of the attributes is reconciled by normalizing the decision matrix X = (xij )n×m . A common method of normalization is given as zij =
zij =
xij − xjmin xjmax − xjmin xjmax − xij xjmax − xjmin
,
i = 1, . . . , n, j ∈ 1 ,
(1)
,
i = 1, . . . , n, j ∈ 2 ,
(2)
where zij is the normalized attribute value, xjmin = min1 i n {xij }, xjmax = max1 i n {xij }, and the sets 1 and 2 are, respectively, the sets of benefit attributes and cost attributes.
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Let Z = (zij )n×m be the normalized decision matrix and W =(w1 , . . . , wm )T the normalized vector of attribute relative importance weights satisfying the condition m j =1 wj = 1. According to the simple additive weighting (SAW) method [7], the overall weighted assessment value of alternative Ai , i = 1, . . . , n, is di =
m
zij wj ,
i = 1, . . . , n,
(3)
j =1
where zij represents objective information, wj , j = 1, . . . , m, are subjective weight variables. di is a linear function of weight variables and the greater the value of di the better the alternative Ai . The best alternative is the one with the greatest overall weighted assessment value. For brevity (3) can be rewritten in vector form as D = ZW,
(4)
where D = (d1 , . . . , dn )T is a vector of the overall weighted assessment values for all the alternatives with superscript T representing transpose. Suppose that the fuzzy preference information on the alternatives provided by the DM is known and given in matrix form as A1
A2 · · ·
An
p11 p12 · · · p1n p21 p22 · · · p2n .. .. .. .. , . . . . An pn1 pn2 · · · pnn
A1 A P = 2 .. .
(5)
where pij , i = 1, . . . , n and j = 1, . . . , n, are membership grades characterized by the following membership function [8,9]: 1 c ∈ (0.5, 1) pij = P (Ai , Aj ) = 0.5 f ∈ (0, 0.5) 0
if Ai is definitely preferred to Aj , if Ai is somewhat preferred to Aj , if there is no preference (i.e., indifference),
(6)
if Aj is somewhat preferred to Ai , if Aj is definitely preferred to Ai .
The matrix P is called a fuzzy preference relation [8,9] or a fuzzy preference matrix and has the following properties: (1) 0 pij 1; (2) pii = 0.5 for i = 1, . . . , n; (3) pij + pj i = 1 for i, j = 1, . . . , n, j = i.
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In order to link the fuzzy preference relation of P with the attribute weights W , Fan et al. interpreted the fuzzy preference relation P as an estimate of the following pairwise comparison matrix: d1 d1 d1 d1 + d1 d1 + d2 · · · d1 + dn A A · · · A 1 2 n d d d 2 2 2 A1 p¯ 11 p¯ 12 · · · p¯ 1n ··· d + d d + d d + d 1 2 2 2 n P¯ = A2 p¯ 21 p¯ 22 · · · p¯ 2n = 2 (7) . .. .. .. .. .. . . . . .. .. .. .. . . . . . An p¯ n1 p¯ n2 · · · p¯ nn dn dn dn ··· dn + d1 dn + d2 dn + dn If the estimate is accurate, then the following equations hold: m zik wk di = m k=1 , i, j = 1, . . . , n. pij = di + dj (z k=1 ik + zj k )wk
(8)
That is pij
m
(zik + zj k )wk −
k=1
m
zik wk = 0,
i, j = 1, . . . , n.
(9)
k=1
Due to the presence of subjectivity some of the above equations may not hold. Therefore, Fan et al. constructed the following constrained optimization model, which is a quadratic programming problem, to assess the attribute weights:
2 m n n m Min H (W ) = pij (zik + zj k )wk − zik wk s.t.
m
i=1 j =1,j =i
k=1
k=1
wk = 1,
k=1 wk 0,
(10)
k = 1, . . . , m.
The objective of this model is to find the optimum weight vector W ∗ such that the ordering of the alternatives obtained from (3) is compatible with the DM’s fuzzy preference matrix P . With the help of the Lagrange multiplier method, Fan et al. generated the following optimum weight vector [6]: W∗ =
Q−1 e , eT Q−1 e
(11)
∗ )T , e = (1, . . . , 1)T , and Q−1 is the inverse of the matrix Q = (q ) where W ∗ = (w1∗ , . . . , wm lk m×m . The elements of Q were given by
qlk =
n
n
(pj i zil − pij zj l )(pj i zik − pij zj k ),
k, l = 1, . . . , m.
(12)
i=1 j =1,j =i
The weight vector determined by (11) is significant only when it satisfies the nonnegativity constraint W ∗ 0; otherwise, the quadratic programming model (10) should be solved. For example, for the fuzzy
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preference matrix P and the normalized decision matrix Z 0 1 5/6 1/3 0.50 0.44 0.64 0.54 0.56 0.50 0.69 0.60 0 , , Z = 5/12 3/5 1 P = 1 0.36 0.31 0.50 0.40 0 0 1 2/3 2/5 2/3 2/3 0.46 0.40 0.60 0.50 we can derive using (12) Q = (qlk )m×m
2.1058 −1.0603 −0.9361 1.9288 −1.0603 0.6739 0.4438 −0.7281 , = −0.9361 0.4438 0.5169 −0.9173 1.9288 −0.7281 −0.9173 2.1903
whose inverse is Q−1
45903 45640 −3383 −26667 45640 45382 −3367 −26515 . = −3383 −3367 261 1969 −26667 −26515 1969 15494
By (11), W ∗ = (0.7463, 0.7421, −0.0549, −0.4335)T , which is neither nonnegative nor the optimum solution of (10). As this example illustrates, the use of (11) to assess the attribute weights is rather limited. The approach of Fan et al. [6] is to calculate the overall weighted assessment value of each alternative after the assessment of the optimum attribute weights by (3) or (4), and rank the alternatives on the basis of their overall weighted assessment values. This approach assumes that the DM is able to provide the fuzzy preference relation P , and P can be approximated by the matrix P¯ defined by (7). In the next two sections, different means of combining both the subjective and objective information will be explored and the fuzzy preference information on the alternatives will be used to reach the DM’s final decision. 3. Deriving a subjective ranking from a fuzzy preference matrix The fuzzy preference matrix P provides not only information on the pairwise comparisons of the alternatives, but also some useful indirect information. The DM’s subjective decision making and ranking can be derived directly from the fuzzy preference matrix. There are a number of AHP-based priority methods [13], which can be used to derive the priorities from a multiplicative comparison matrix that satisfies pij · pj i = 1 rather than pij + pj i = 1. Wang [15] summarized 21 priority methods based on AHP. In our opinion, most of them can be configured to derive a subjective ranking from a fuzzy preference matrix. Fan and Li [4] suggested the use of the least squares method (LSM) [13] to derive a ranking from a fuzzy preference matrix. Fan et al. [5] suggested a goal programming (GP) [2] method for ranking alternatives. Xu and Da [17] utilized the least deviation method (LDM) [14,15] to obtain the priority vector of a fuzzy preference relation. Xu [16] also employed the GP method [2] to obtain the priority vector of incomplete fuzzy preference relations. In this section, we propose an EM, which is an extension of the method investigated by Lipovetsky and Conklin [10] within the framework of AHP, to obtain the DM’s subjective ranking from his/her fuzzy preference matrix.
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Let, again, D = (d1 , . . . , dn )T be the overall weighted assessment value vector for n alternatives. In order to rank the alternatives it is not necessary to estimate the absolute value of D; its normalized relative value is sufficient. We will assume from this point on that D is a normalized overall assessment value vector, which satisfies eT D = 1.
(13)
The following equations hold: d1 d1 d1 (d1 + d2 ) + (d1 + d3 ) + · · · + (d1 + dn ) = (n − 1)d1 , d1 + d 2 d1 + d 3 d1 + dn d2 d2 d2 (d2 + d1 ) + (d2 + d3 ) + · · · + (d2 + dn ) = (n − 1)d2 , d2 + d 1 d2 + d3 d2 + dn .. . dn dn dn (dn + d1 ) + (dn + d2 ) + · · · + (dn + dn−1 ) = (n − 1)dn . dn + d2 dn + dn−1 dn + d 1
(14)
If P is the accurate estimate of the pairwise comparison matrix P¯ = (p¯ ij )n×n in (7), then the above system of equations can be rewritten as p12 (d1 + d2 ) + p13 (d1 + d3 ) + · · · + p1n (d1 + dn ) = (n − 1)d1 , p21 (d2 + d1 ) + p23 (d2 + d3 ) + · · · + p2n (d2 + dn ) = (n − 1)d2 , .. . pn1 (dn + d1 ) + pn2 (dn + d2 ) + · · · + pn,n−1 (dn + dn−1 ) = (n − 1)dn .
(15)
or as n p1j d1 + p12 d2 + · · · + p1n dn = (n − 1)d1 , j =2
p21 d1 +
n
p2j d2 + · · · + p2n dn = (n − 1)d2 ,
j =1,j =2
.. .
n−1 pn1 d1 + pn2 d2 + · · · + pnj dn = (n − 1)dn . j =1
(16)
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Letting
n
p1j
j =2 p 21 A= .. . pn1
p12 n j =1,j =2
···
p1n
p2j · · ·
p2n
.. .
.. .
pn2
···
n−1 j =1
.. . pnj
337
.
(17)
Eq. (16) is rewritten in matrix form as AD = (n − 1)D.
(18)
D may be viewed as the principal right-eigenvector of matrix A. Thus, Eq. (18) can further be written in the form of an eigenvalue problem as AD = max D.
(19)
We solve (19) to derive an estimate of D just as we would with the AHP [13]. In order to distinguish the DM’s subjective estimate from D, we rewrite (19) as ˆ ADˆ = max D,
(20)
where Dˆ = (dˆ1 , . . . , dˆn )T represents the DM’s subjective estimate of D, which is referred to as the subjective priority vector. The above method is referred to as the EM. For the eigenvalue problem (20), we propose the following theorems: Theorem 1. In (20), max ≡ n − 1. Proof. Let > 0 be a nonnegative eigenvalue of matrix A. We have n p1j dˆ1 + p12 dˆ2 + · · · + p1n dˆn = dˆ1 , j =2
p21 dˆ1 +
n
p2j dˆ2 + · · · + p2n dˆn = dˆ2 ,
j =1,j =2
.. .
n−1 pn1 dˆ1 + pn2 dˆ2 + · · · + pnj dˆn = dˆn .
(21)
j =1
Side by side summation of the above equations gives n (p1j + pj 1 )dˆ1 + j =2
n j =1,j =2
(p2j + pj 2 )dˆ2 + · · · +
n−1 n (pnj + pj n )dˆn = dˆi j =1
i=1
(22)
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or (n − 1)dˆ1 + (n − 1)dˆ2 + · · · + (n − 1)dˆn =
n
dˆi .
(23)
i=1
It is evident that as long as ni=1 dˆi = 0 and is nonnegative, = n − 1. Since A is a positive matrix, according to the Perron–Frobenius theorem [12], its maximal eigenvalue, max , and the corresponding eigenvector Dˆ are both positive. Therefore, max ≡ n − 1.
(24)
Theorem 2. Let Dˆ be the principal right-eigenvector of matrix A corresponding to max = n − 1, then Dˆ > 0. This theorem is a direct result of the Perron–Frobenius theorem [12]. The above two theorems show that for any fuzzy preference matrix P we can always find a positive ˆ where the matrix A is determined by (17). The normalized eigenvector Dˆ satisfying ADˆ = (n − 1)D, principal right-eigenvector Dˆ is calculated by the following iterative algorithm: Step 1: Set k = 1 and Dˆ (0) = (1/n)e and = 10−5 , where is a specified precision coefficient and (0) (0) (0) Dˆ = (dˆ1 , . . . , dˆn )T . (k) (k) Step 2: Compute Dˆ (k) = ADˆ (k−1) and normalize it, where Dˆ (k) = (dˆ1 , . . . , dˆn )T is a normalized vector. Step 3: If Dˆ (k) − Dˆ (k−1) , then Dˆ = Dˆ (k) , where Dˆ (k) − Dˆ (k−1) is a vector norm, which will take one of the three forms: absolute, square or minimax. Otherwise, let k = k + 1 and go to Step 2. An alternate method for obtaining the principal right-eigenvector Dˆ is to solve the following LP model [15]: Min J = eT E [A − (n − 1)e]Dˆ − E = 0, s.t. eT Dˆ = 1, ˆ E 0, D,
(25)
where E = (1 , . . . , n )T is a nonnegative vector of deviation variables. It is easy to prove that the optimum value of J , J ∗ = 0. In fact, by Theorems 1 and 2, for any fuzzy ˆ Such an eigenvector preference matrix we can always find an eigenvector Dˆ such that ADˆ = (n − 1)D. ∗ ∗ T ∗ ∗ always leads to E = 0 and J = e E = 0. So, if J = 0, then it cannot be optimum. 4. Computing subjective weights using fuzzy preference relation and decision matrix It has been recognized in the literature that the DM’s subjective preference information should be considered along with objective information when searching for a decision. How to incorporate these two types of information is an issue that has not yet found a universally acceptable solution. There are various approaches that can be adopted and Fan et al.’s is one possibility, which is applicable for the case when
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the DM’s preference information is available in the form of a fuzzy preference matrix. In this section, we propose three new approaches for combining subjective and objective decision information to assess the weights of attributes. 4.1. LDM for integrating subjective fuzzy preference relations and objective decision matrix information, LDM-1 This approach combines the DM’s subjective fuzzy preference matrix directly with the objective decision matrix in a way that is quite different from Fan et al.’s. It is known from Theorem 1 that for any fuzzy ˆ preference matrix P we can always find a positive normalized eigenvector Dˆ such that ADˆ = (n − 1)D, ˆ where the matrix A is determined by (17) and D is an estimate of D. If the estimate is accurate then there must exist AD = (n − 1)D. Substituting (4) in the equation, we get AZW = (n − 1)ZW,
(26)
where Z is the normalized decision matrix and W is the normalized weight vector satisfying eT W = 1. Due to the existence of fuzziness and subjectivity, however, an accurate estimate is nearly impossible. So (26) cannot hold exactly in most cases. In view of this, we define the deviation vector E = [AZ − (n − 1)Z]W,
(27)
where E = (1 , . . . , n )T and find the attribute weights that minimize the sum of the absolute values of the components of E by the following LP: Min J =
s.t.
n i=1
|i |
[AZ − (n − 1)Z]W − E = 0, eT W = 1, W 0.
(28)
Letting + i =
i + | i |
2 i can be written as − i = + i − i ,
and − i =
−i + |i | , 2
i = 1, . . . , n,
i = 1, . . . , n,
(29)
(30)
− where the scalar product + i · i = 0. Now, the optimization model (28) can be written as an LP model:
Min J = eT (E + + E − ) [AZ − (n − 1)Z]W − E + + E − = 0, s.t. eT W = 1, W, E + , E − 0,
(31)
− ∗ + T − − T ∗ ∗ T where E + = (+ 1 , . . . , n ) and E = (1 , . . . , n ) . The optimum weight vector, W = (w1 , . . . , wm ) , can be easily obtained by solving (31), which incorporates both subjective and objective information. With
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the attribute weights obtained by (31), the overall weighted assessment values for the alternatives can be computed by either (3) or (4), and the final decision can be reached. We call the method described above the LDM for integrating subjective fuzzy preference relations and objective decision matrix information or LDM-1.
4.2. LDM for integrating subjective priority vector and objective decision matrix information, LDM-2 This approach uses the subjective fuzzy preference matrix information in an indirect way. It combines the DM’s subjective estimates of the overall weighted assessment values for alternatives with objective decision matrix information. Let Dˆ = (dˆ1 , . . . , dˆn )T be the DM’s subjective estimate of D = (d1 , . . . , dn )T , which can be estimated using the EM that was explained earlier, in Section 3. It is expected that Dˆ and D would be nearly the same. Dˆ is a normalized vector of assessment values but D is not. Therefore, m
di
dˆi ≈ n
k=1
dk
=
j =1 zij wj n m k=1 j =1 zkj wj
m = m
j =1
j =1
n
zij wj
k=1
, zkj wj
i = 1, . . . , n,
(32)
which can be equivalently expressed as m
zij wj − dˆi
j =1
n m j =1
zkj wj ≈ 0,
i = 1, . . . , n.
(33)
k=1
Let i =
m
zij − dˆi
n
j =1
zkj wj ,
i = 1, . . . , n.
(34)
k=1
Applying the same argument as earlier, the following optimization model is therefore constructed to minimize the sum of the absolute values of the components of E: n | i | Min J = i=1 m n ˆ zij − di zkj wj − i = 0,
s.t.
j =1 m j =1
k=1
i = 1, . . . , n,
(35)
wj = 1,
wj 0,
j = 1, . . . , m.
With the help of the variable transformation of (29), the above optimization model can be expressed as the following LP model, which we call the LDM for integrating subjective priority vector and objective
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decision matrix information or LDM-2: n − Min J = (+ i + i ) i=1 m n − zkj wj − + zij − dˆi i + i = 0,
s.t.
j =1 m j =1
k=1
i = 1, . . . , n,
(36)
wj = 1,
− wj , + i , i 0,
j = 1, . . . , m,
i = 1, . . . , n.
4.3. LDM for integrating subjective ranking and objective decision matrix information, LDM-3 Another way of using the subjective fuzzy preference matrix on the alternatives indirectly is to utilize its ranking order. Suppose the ranking order obtained from the fuzzy preference matrix is Ai1 Ai2 · · · Ain , implying “preferred to”, which can be obtained by means of the EM or other methods and can also be characterized in terms of the subjective overall weighted assessment values as dˆi1 dˆi2 · · · dˆin . It is desirable that a ranking order that is characterized by di1 di2 · · · din is produced based on the information embedded in an objective decision matrix. However, such a ranking order may not be exactly the same as the one that is characterized by dˆi1 dˆi2 · · · dˆin , in which case the ranking order characterized by di1 di2 · · · din is not valid. We introduce the nonnegative deviation variables i , . . . , in−1 such that di1 − di2 + i1 0, di2 − di3 + i2 0, .. . din−1 − din + in−1 0.
(37)
Substituting (4) in (37), we obtain m (zi1 j − zi2 j )wj + i1 0, j =1 m (zi2 j − zi3 j )wj + i2 0, j =1
.. . m (zin−1 j − zin j )wj + in−1 0. j =1
(38)
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Obviously all the deviation variables should be made as small as possible, which is accomplished by solving the following LP model: Min J = i1 + i2 + · · · + in−1 m (zil j − zil+1 j )wj + il 0, s.t.
j =1 m j =1
l = 1, . . . , n − 1, (39)
wj = 1,
wj , il 0, j = 1, . . . , m; l = 1, . . . , n − 1.
We call this method the LDM for integrating subjective ranking and objective decision matrix information or LDM-3. If the optimum objective function value of (39) is not equal to zero, that is J ∗ > 0, then the optimum weight vector, W ∗ , can be uniquely determined. Usually, when the number of alternatives is greater than the number of attributes, n > m, it is very likely that J ∗ > 0. If, on the other hand, J ∗ = 0, there may exist multiple optimum solutions to the LP (39). In this case, W ∗ cannot be determined uniquely but can be characterized by an interval vector. This is often the case when n m. The following pair of LP models can be used to reveal the weight intervals: Min/max wi m s.t.
(zil j − zil+1 j )wj 0,
j =1 m j =1
l = 1, . . . , n − 1, (40)
wj = 1,
wj 0,
j = 1, . . . , m.
The optimum solutions to the above pair of LP models form the weight intervals is denoted by [wiL , wiU ], i = 1, . . . , m, based on which a partial order may be generated for n alternatives. It is argued that interval weights and partial order would cause making a decision to be difficult because of the existence of uncertainty. Therefore, a reasonable point estimate for the weight vector is preferable. Let m (zil j − zil+1 j )wj 0, l = 1, . . . , n − 1, j =1 m (41) = wj = 1, j =1 wj 0, j = 1, . . . , m. is a convex polygon and its centroid is the most appropriate point estimate. We refer to the weights obtained in this manner as centroid weights. In order to compute the centroid weights, we need to know all the vertices of , which can be obtained by solving models (40) for all wi , i = 1, . . . , m. Suppose (k) (k) that has K0 vertices, denoted by W (k) = (w1 , . . . , wm )T , k = 1, . . . , K0 . Then its centroid weights can be calculated by [1]
W
(C)
K0 1 = W (k) . K0 k=1
(42)
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Since each vertex of satisfies the DM’s preference ranking Ai1 Ai2 · · · Ain , the centroid weights will also lead to the same preference ranking.
5. Numerical example In this section, Fan et al.’s illustrative example is re-examined to show how the DM’s subjective decision making and ranking can be derived from a fuzzy preference matrix and how they can be combined with objective decision matrix information in different ways to give an overall assessment of each alternative. Example. A potential buyer, the DM, intends to buy a house. He has four alternatives to choose from: A1 , . . . , A4 . The attributes (factors) s/he considers include price in $s (G1 ), size in m2 (G2 ), distance to work in kms (G3 ) and environmental characteristics (G4 ). Among the four attributes, G2 and G4 are benefit attributes, which receive high values for desirable alternatives, and G1 and G3 are cost attributes, which receive low values for desirable alternatives. The decision matrix for this MADM problem is given by G 1 G2 A1 3.0 100 X = A2 2.5 80 A3 1.8 50 A4 2.2 70
G3 G4 10 7 8 5 . 20 11 12 9
The potential buyer (DM) provides his/her fuzzy preference matrix on the four alternatives as follows: 0.50 0.44 0.64 0.54 0.56 0.50 0.69 0.60 P = . 0.36 0.31 0.50 0.40 0.46 0.40 0.60 0.50 First, we normalize the decision matrix X using (1) and (2). The normalized decision matrix is 0 1 5/6 1/3 5/12 3/5 1 0 Z= . 1 0 0 1 2/3 2/5 2/3 2/3 Next, we derive the buyer’s subjective decision making and ranking from his/her fuzzy preference matrix using the EM. From the fuzzy preference matrix P and (17), we have 1.62 0.44 0.64 0.54 0.56 1.85 0.69 0.60 A= , 0.36 0.31 1.07 0.40 0.46 0.40 0.60 1.46
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whose maximum eigenvalue is max = 3 and the corresponding principal right-eigenvector is Dˆ = (dˆ1 , dˆ2 , dˆ3 , dˆ4 )T = (0.2713, 0.3446, 0.1537, 0.2304)T , which ranks the four alternatives as A2 A1 A4 A3 . We can obtain exactly the same result by solving the LP model (25): Min J = 1 + 2 + 3 + 4 −1.38dˆ1 + 0.44dˆ2 + 0.64dˆ3 + 0.54dˆ4 − 1 = 0, 0.56dˆ1 − 1.15dˆ2 + 0.69dˆ3 + 0.60dˆ4 − 2 = 0, 0.36dˆ1 + 0.31dˆ2 − 1.93dˆ3 + 0.40dˆ4 − 3 = 0, s.t.
0.46dˆ1 + 0.40dˆ2 + 0.60dˆ3 − 1.54dˆ4 − 4 = 0, dˆ1 + dˆ2 + dˆ3 + dˆ4 = 1, dˆ1 , dˆ2 , dˆ3 , dˆ4 0, 1 , 2 , 3 , 4 0.
The above overall weighted assessment value-vector and ranking order are purely subjective. The final decision making still needs to consider the objective decision matrix information. Using Fan et al.’s approach and (11), we obtain W = (0.7463, 0.7421, −0.0549, −0.4335)T . The corresponding objective function value is H (W ) = 0.1220372 × 10−4 . Since the weights w3 and w4 do not satisfy the nonnegativity constraint, we solve the quadratic programming problem (10) to obtain W ∗ = (0.3266, 0.2962, 0.3772, 0)T . The corresponding optimum objective function value is H (W ∗ ) = 0.2068575 × 10−1 . We have been unable to validate Fan et al.’s result, namely, W ∗ = (0.2914, 0.0955, 0.5787, 0.0344)T , which corresponds to an objective function value of H (W ∗ ) = 0.3236951 × 10−1 . It seems to us that there is a mistake in Fan et al.’s computations; with their results the objective function fails to achieve its minimum and also leads to the inconsistent ranking of A2 A4 A1 A3 , which contradicts the DM’s subjective ranking of A2 A1 A4 A3 . Although inconsistent ranking orders are allowed and sometimes unavoidable, as far as this example is concerned, inconsistency is not at all inevitable. Now we explore the use of the LDMs developed in Section 4. The application of LDM-1, through the solution of (31), produces W ∗ = (0.3413, 0.3512, 0.3435, 0)T , which incorporates both the subjective fuzzy preference relations and the objective decision matrix information. The corresponding overall weighted assessment value vector is D = (d1 , d2 , d3 , d4 )T = (0.6014, 0.6748, 0.3413, 0.5826)T . After normalization, it becomes D = (0.2734, 0.3067, 0.1551, 0.2648)T , which is very close to the DM’s subjective estimate Dˆ = (0.2713, 0.3446, 0.1537, 0.2304)T and both lead to the same ranking order: A2 A1 A4 A3 . On the other hand, the application of LDM-2, through (36), produces W ∗ = (0.3397, 0.2965, 0.3638, T 0) . The corresponding overall weighted assessment value vector turns out to be D = (0.5996, 0.6832, 0.3397, 0.5876)T , which, after normalization, becomes D = (0.2713, 0.3091, 0.1537, 0.2659) that is also very close to the DM’s subjective estimate and leads to the same ranking order: A2 A1 A4 A3 . Based on the subjective ranking order A2 A1 A4 A3 , which is derived from the DM’s fuzzy preference matrix using the EM, the following convex polygon is constructed to assess the weights of the
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Table 1 The vertices of polygon and the weights Weight
w1 w2 w3 w4
Vertices #1
#2
#3
#4
0 0 1 0
0 0 0.6667 0.3333
0 0.2941 0.7059 0
0.4580 0.4962 0.0458 0
Interval weight
Centroid weight
[0, 0.4580] [0, 0.4962] [0.0458, 1] [0, 0.3333]
0.1145 0.1976 0.6046 0.0833
four attributes: 5 2 1 1 12 w1 − 5 w2 + 6 w3 − 3 w4 0, − 2 w + 3 w + 1 w − 1 w 0, 3 1 5 2 6 3 3 4 = − 1 w1 + 2 w2 + 2 w3 − 1 w4 0, 3 5 3 3 w1 + w2 + w3 + w4 = 1, w1 , w2 , w3 , w4 0. The polygon has four vertices, as shown in Table 1. The corresponding interval weights and the centroid weights are shown in the last two columns of Table 1. The overall weighted assessment value vector for the four alternatives effected by this set of centroid weights is D = (0.7292, 0.7709, 0.1978, 0.6140)T , which yields exactly the same ranking: A2 A1 A4 A3 . In this example, all the three LDMs produce the same ranking that is also consistent with the DM’s subjective ranking, which means that the DM’s subjective decision making and ranking is fully supported by the objective decision matrix information.
6. Concluding remarks In this paper, MADM problems with fuzzy preference information on alternatives have been investigated. An eigenvector method has been developed to derive the subjective ranking from a fuzzy preference relation, and three optimization models and methods have been proposed to combine the DM’s subjective preferences on alternatives and objective decision matrix information in three different ways to assess the attribute weights. A numerical example provided by Fan et al. has been revisited to demonstrate the proposed methods. It has been shown that the methods developed in this paper are rational, feasible and effective. In order to better assess the weights of attributes and avoid assigning unrealistic weights, additional preference information on the weights may be elicited from the DM. For example, a threshold value, wL , for the weights may be set to avoid the occurrence of zero weight. A partial order for the weights may also be sought. Any additional preference information can be easily incorporated into the methods that have been proposed in this paper. Compared with Fan et al.’s approach, the LDMs introduced here provide greater flexibility for solving MADM problems with preference information on alternatives and/or attributes.
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Finally, we must point out that in this paper the fuzzy preference relations on alternatives are assumed to be known. This is a strong assumption. While how to elicit fuzzy preference relations is an important issue, this is a topic that falls outside the purpose of the present paper. The DM may formulate his/her preferences on alternatives with respect to each attribute individually and then aggregate them using one of the available approaches; for example, the ordered weighted averaging (OWA) operators [18]. The interested reader is referred to [3] for more on this topic. Acknowledgments The authors would like to express their sincere thanks to the Editors-in-Chief and three anonymous referees for helping to improve this paper with their constructive comments and suggestions. References [1] A. Arbel, L.G. Vargas, Preference simulation and preference programming: robustness issues in priority derivation, European J. Oper. Res. 69 (1993) 200–209. [2] N. Bryson, A goal programming method for generating priority vectors, J. Oper. Res. Soc. 46 (1995) 641–648. [3] F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Sets and Systems 97 (1998) 33–48. [4] Z.P. Fan, H.Y. Li, A ranking method for alternatives based on fuzzy preference relation, J. Northeast. Univ. (Natur. Sci.) 20 (6) (1999) 651–653 (in Chinese). [5] Z.P. Fan, H.Y. Li, G.F. Hu, Fuzzy judgement matrix and the goal programming method for ranking alternatives, J. Northeast. Univ. (Natur. Sci.) 21 (1) (2000) 60–62 (in Chinese). [6] Z.P. Fan, J. Ma, Q. Zhang, An approach to multiple attribute decision making based on fuzzy preference information on alternatives, Fuzzy Sets and Systems 131 (2002) 101–106. [7] C.L. Hwang, K. Yoon, Multiple Attribute Decision Making: Methods and Applications, Springer, Berlin, 1981. [8] J. Kacprzyk, Group decision making with a fuzzy majority, Fuzzy Sets and Systems 18 (1986) 105–118. [9] J. Kacprzyk, M. Fedrizzi, H. Nurmi, Group decision making and consensus under fuzzy preferences and fuzzy majority, Fuzzy Sets and Systems 49 (1992) 21–31. [10] S. Lipovetsky, W.M. Conklin, Robust estimation of priorities in the AHP, European J. Oper. Res. 137 (2002) 110–122. [12] C.R. Rao, M.B. Rao, Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientific, Singapore, 2001. [13] T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980. [14] Y.M. Wang, A new generalized least deviation priority method of group comparison matrix, Systems Eng.—Theory & Practice 14 (9) (1994) 63–68 (in Chinese). [15] Y.M. Wang, An overview of priority methods of comparison matrix, J. Decision Making and Decision Support Systems 5 (3) (1995) 101–114 (in Chinese). [16] Z.S. Xu, Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation, Internat. J. Approx. Reason. 36 (2004) 261–270. [17] Z.S. Xu, Q.L. Da, A least deviation method to obtain a priority vector of a fuzzy preference relation, European J. Oper. Res. 164 (2005) 206–216. [18] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Systems Man Cybernet. 18 (1988) 183–190.
Multiple attribute decision making based on fuzzy preference information on alternatives: Ranking and weighting夡 Ying-Ming Wanga, b, c,∗ , Celik Parkand a Manchester Business School, The University of Manchester, P.O. Box 88, Manchester M60 1QD, UK b School of Public Administration, Fuzhou University, Fuzhou 350002, PR China c Center for Accounting Studies of Xiamen University, Xiamen, Fujian 361005, PR China d Department of Management, C.W. Post Campus, Long Island University, 720 Northern Blvd., Brookville, NY 11548, USA
Received 23 October 2003; received in revised form 6 January 2005; accepted 28 February 2005 Available online 18 March 2005
Abstract This paper investigates the multiple attribute decision making (MADM) problem with fuzzy preference information on alternatives and proposes an eigenvector method to rank them. Three optimization models are introduced to assess the relative importance weights of attributes in a MADM problem, which integrate subjective fuzzy preference relations and objective information in different ways. Since the decisions these models lead to are based on both subjective and objective information, they are more balanced and credible than those that consider only one type of information. An example is provided to illustrate the application of the proposed models. © 2005 Elsevier B.V. All rights reserved. Keywords: Multiple attribute decision making; Fuzzy preference relation/matrix; Weight estimation; Alternative ranking
1. Introduction In a recent paper [6] Fan et al., investigated the multiple attribute decision making (MADM) problem with fuzzy preference information on alternatives and proposed a decision aid approach to combine 夡
This research was supported by the National Natural Science Foundation of China (NSFC) under the Grant Nos. 70271056 and 70171035, and also in part by Fok Ying Tung Education Foundation under the Grant No. 71080. ∗ Corresponding author. Manchester Centre for Civil & Construction Engineering, The University of Manchester, PO Box 88, Manchester M60 1QD, UK. Tel.: +44 161 2005974; fax: +44 161 2004646. E-mail addresses: [email protected], [email protected], [email protected] (Y.-M. Wang). 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.02.018
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subjective fuzzy preference information with objective information for an overall assessment of the relative importance weights of the underlying attributes and select the best alternative. Approach Fan et al. proposed has a greater intuitive appeal than the methods that consider only one type of information. Since fuzzy preference information on alternatives involves decision maker’s (DM’s) subjective considerations, any decision making tool chosen must incorporate them when ranking alternatives. However, such a ranking could not be considered final as it would lack objectivity. To determine the final ranking of alternatives, the weights of the attributes need to be assessed using subjective fuzzy preference information as well as information that is free from any bias the decision maker (DM) may introduce through his/her subjective input. MADM problems with fuzzy preference information on alternatives are essentially problems of the importance of weighting and ranking. In this paper, an eigenvector method (EM) is proposed to derive a subjective ranking of alternatives from fuzzy preference relations, and three optimization models are introduced to integrate subjective fuzzy preference relations and objective information in different ways to assess the weights of the attributes. If the three methods produce exactly the same ranking, then that ranking is deemed to be credible. The models proposed here differ from Fan et al.’s approach in the manner they combine subjective and objective information, which involves the use of linear programming (LP) whose computational demands are much less complex. The DM’s preferences on attribute weights, if provided, can also be easily incorporated into these models, a property that makes them particularly flexible. The paper is organized as follows. Section 2 presents a brief review of the approach proposed by Fan et al. [6]. Section 3 demonstrates how the DM’s subjective ranking can be derived from the fuzzy preference relations using an EM. Section 4 computes subjective weights using the fuzzy preference relations and objective information, and develops three optimization models for integrating both subjective and objective information. All the models proposed in the paper are illustrated in Section 5 through a numerical example. Conclusions are offered in Section 6.
2. Fan et al.’s approach to MADM based on fuzzy preference information on alternatives Consider a MADM problem with n alternatives, A1 , . . . , An , and m decision attributes (criteria), G1 , . . . , Gm . Each alternative is assessed with respect to each attribute. The assessment scores assigned to the attributes are the components of a decision matrix denoted by X = (xij )n×m . Any incommensurability of the attributes is reconciled by normalizing the decision matrix X = (xij )n×m . A common method of normalization is given as zij =
zij =
xij − xjmin xjmax − xjmin xjmax − xij xjmax − xjmin
,
i = 1, . . . , n, j ∈ 1 ,
(1)
,
i = 1, . . . , n, j ∈ 2 ,
(2)
where zij is the normalized attribute value, xjmin = min1 i n {xij }, xjmax = max1 i n {xij }, and the sets 1 and 2 are, respectively, the sets of benefit attributes and cost attributes.
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Let Z = (zij )n×m be the normalized decision matrix and W =(w1 , . . . , wm )T the normalized vector of attribute relative importance weights satisfying the condition m j =1 wj = 1. According to the simple additive weighting (SAW) method [7], the overall weighted assessment value of alternative Ai , i = 1, . . . , n, is di =
m
zij wj ,
i = 1, . . . , n,
(3)
j =1
where zij represents objective information, wj , j = 1, . . . , m, are subjective weight variables. di is a linear function of weight variables and the greater the value of di the better the alternative Ai . The best alternative is the one with the greatest overall weighted assessment value. For brevity (3) can be rewritten in vector form as D = ZW,
(4)
where D = (d1 , . . . , dn )T is a vector of the overall weighted assessment values for all the alternatives with superscript T representing transpose. Suppose that the fuzzy preference information on the alternatives provided by the DM is known and given in matrix form as A1
A2 · · ·
An
p11 p12 · · · p1n p21 p22 · · · p2n .. .. .. .. , . . . . An pn1 pn2 · · · pnn
A1 A P = 2 .. .
(5)
where pij , i = 1, . . . , n and j = 1, . . . , n, are membership grades characterized by the following membership function [8,9]: 1 c ∈ (0.5, 1) pij = P (Ai , Aj ) = 0.5 f ∈ (0, 0.5) 0
if Ai is definitely preferred to Aj , if Ai is somewhat preferred to Aj , if there is no preference (i.e., indifference),
(6)
if Aj is somewhat preferred to Ai , if Aj is definitely preferred to Ai .
The matrix P is called a fuzzy preference relation [8,9] or a fuzzy preference matrix and has the following properties: (1) 0 pij 1; (2) pii = 0.5 for i = 1, . . . , n; (3) pij + pj i = 1 for i, j = 1, . . . , n, j = i.
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In order to link the fuzzy preference relation of P with the attribute weights W , Fan et al. interpreted the fuzzy preference relation P as an estimate of the following pairwise comparison matrix: d1 d1 d1 d1 + d1 d1 + d2 · · · d1 + dn A A · · · A 1 2 n d d d 2 2 2 A1 p¯ 11 p¯ 12 · · · p¯ 1n ··· d + d d + d d + d 1 2 2 2 n P¯ = A2 p¯ 21 p¯ 22 · · · p¯ 2n = 2 (7) . .. .. .. .. .. . . . . .. .. .. .. . . . . . An p¯ n1 p¯ n2 · · · p¯ nn dn dn dn ··· dn + d1 dn + d2 dn + dn If the estimate is accurate, then the following equations hold: m zik wk di = m k=1 , i, j = 1, . . . , n. pij = di + dj (z k=1 ik + zj k )wk
(8)
That is pij
m
(zik + zj k )wk −
k=1
m
zik wk = 0,
i, j = 1, . . . , n.
(9)
k=1
Due to the presence of subjectivity some of the above equations may not hold. Therefore, Fan et al. constructed the following constrained optimization model, which is a quadratic programming problem, to assess the attribute weights:
2 m n n m Min H (W ) = pij (zik + zj k )wk − zik wk s.t.
m
i=1 j =1,j =i
k=1
k=1
wk = 1,
k=1 wk 0,
(10)
k = 1, . . . , m.
The objective of this model is to find the optimum weight vector W ∗ such that the ordering of the alternatives obtained from (3) is compatible with the DM’s fuzzy preference matrix P . With the help of the Lagrange multiplier method, Fan et al. generated the following optimum weight vector [6]: W∗ =
Q−1 e , eT Q−1 e
(11)
∗ )T , e = (1, . . . , 1)T , and Q−1 is the inverse of the matrix Q = (q ) where W ∗ = (w1∗ , . . . , wm lk m×m . The elements of Q were given by
qlk =
n
n
(pj i zil − pij zj l )(pj i zik − pij zj k ),
k, l = 1, . . . , m.
(12)
i=1 j =1,j =i
The weight vector determined by (11) is significant only when it satisfies the nonnegativity constraint W ∗ 0; otherwise, the quadratic programming model (10) should be solved. For example, for the fuzzy
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preference matrix P and the normalized decision matrix Z 0 1 5/6 1/3 0.50 0.44 0.64 0.54 0.56 0.50 0.69 0.60 0 , , Z = 5/12 3/5 1 P = 1 0.36 0.31 0.50 0.40 0 0 1 2/3 2/5 2/3 2/3 0.46 0.40 0.60 0.50 we can derive using (12) Q = (qlk )m×m
2.1058 −1.0603 −0.9361 1.9288 −1.0603 0.6739 0.4438 −0.7281 , = −0.9361 0.4438 0.5169 −0.9173 1.9288 −0.7281 −0.9173 2.1903
whose inverse is Q−1
45903 45640 −3383 −26667 45640 45382 −3367 −26515 . = −3383 −3367 261 1969 −26667 −26515 1969 15494
By (11), W ∗ = (0.7463, 0.7421, −0.0549, −0.4335)T , which is neither nonnegative nor the optimum solution of (10). As this example illustrates, the use of (11) to assess the attribute weights is rather limited. The approach of Fan et al. [6] is to calculate the overall weighted assessment value of each alternative after the assessment of the optimum attribute weights by (3) or (4), and rank the alternatives on the basis of their overall weighted assessment values. This approach assumes that the DM is able to provide the fuzzy preference relation P , and P can be approximated by the matrix P¯ defined by (7). In the next two sections, different means of combining both the subjective and objective information will be explored and the fuzzy preference information on the alternatives will be used to reach the DM’s final decision. 3. Deriving a subjective ranking from a fuzzy preference matrix The fuzzy preference matrix P provides not only information on the pairwise comparisons of the alternatives, but also some useful indirect information. The DM’s subjective decision making and ranking can be derived directly from the fuzzy preference matrix. There are a number of AHP-based priority methods [13], which can be used to derive the priorities from a multiplicative comparison matrix that satisfies pij · pj i = 1 rather than pij + pj i = 1. Wang [15] summarized 21 priority methods based on AHP. In our opinion, most of them can be configured to derive a subjective ranking from a fuzzy preference matrix. Fan and Li [4] suggested the use of the least squares method (LSM) [13] to derive a ranking from a fuzzy preference matrix. Fan et al. [5] suggested a goal programming (GP) [2] method for ranking alternatives. Xu and Da [17] utilized the least deviation method (LDM) [14,15] to obtain the priority vector of a fuzzy preference relation. Xu [16] also employed the GP method [2] to obtain the priority vector of incomplete fuzzy preference relations. In this section, we propose an EM, which is an extension of the method investigated by Lipovetsky and Conklin [10] within the framework of AHP, to obtain the DM’s subjective ranking from his/her fuzzy preference matrix.
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Let, again, D = (d1 , . . . , dn )T be the overall weighted assessment value vector for n alternatives. In order to rank the alternatives it is not necessary to estimate the absolute value of D; its normalized relative value is sufficient. We will assume from this point on that D is a normalized overall assessment value vector, which satisfies eT D = 1.
(13)
The following equations hold: d1 d1 d1 (d1 + d2 ) + (d1 + d3 ) + · · · + (d1 + dn ) = (n − 1)d1 , d1 + d 2 d1 + d 3 d1 + dn d2 d2 d2 (d2 + d1 ) + (d2 + d3 ) + · · · + (d2 + dn ) = (n − 1)d2 , d2 + d 1 d2 + d3 d2 + dn .. . dn dn dn (dn + d1 ) + (dn + d2 ) + · · · + (dn + dn−1 ) = (n − 1)dn . dn + d2 dn + dn−1 dn + d 1
(14)
If P is the accurate estimate of the pairwise comparison matrix P¯ = (p¯ ij )n×n in (7), then the above system of equations can be rewritten as p12 (d1 + d2 ) + p13 (d1 + d3 ) + · · · + p1n (d1 + dn ) = (n − 1)d1 , p21 (d2 + d1 ) + p23 (d2 + d3 ) + · · · + p2n (d2 + dn ) = (n − 1)d2 , .. . pn1 (dn + d1 ) + pn2 (dn + d2 ) + · · · + pn,n−1 (dn + dn−1 ) = (n − 1)dn .
(15)
or as n p1j d1 + p12 d2 + · · · + p1n dn = (n − 1)d1 , j =2
p21 d1 +
n
p2j d2 + · · · + p2n dn = (n − 1)d2 ,
j =1,j =2
.. .
n−1 pn1 d1 + pn2 d2 + · · · + pnj dn = (n − 1)dn . j =1
(16)
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Letting
n
p1j
j =2 p 21 A= .. . pn1
p12 n j =1,j =2
···
p1n
p2j · · ·
p2n
.. .
.. .
pn2
···
n−1 j =1
.. . pnj
337
.
(17)
Eq. (16) is rewritten in matrix form as AD = (n − 1)D.
(18)
D may be viewed as the principal right-eigenvector of matrix A. Thus, Eq. (18) can further be written in the form of an eigenvalue problem as AD = max D.
(19)
We solve (19) to derive an estimate of D just as we would with the AHP [13]. In order to distinguish the DM’s subjective estimate from D, we rewrite (19) as ˆ ADˆ = max D,
(20)
where Dˆ = (dˆ1 , . . . , dˆn )T represents the DM’s subjective estimate of D, which is referred to as the subjective priority vector. The above method is referred to as the EM. For the eigenvalue problem (20), we propose the following theorems: Theorem 1. In (20), max ≡ n − 1. Proof. Let > 0 be a nonnegative eigenvalue of matrix A. We have n p1j dˆ1 + p12 dˆ2 + · · · + p1n dˆn = dˆ1 , j =2
p21 dˆ1 +
n
p2j dˆ2 + · · · + p2n dˆn = dˆ2 ,
j =1,j =2
.. .
n−1 pn1 dˆ1 + pn2 dˆ2 + · · · + pnj dˆn = dˆn .
(21)
j =1
Side by side summation of the above equations gives n (p1j + pj 1 )dˆ1 + j =2
n j =1,j =2
(p2j + pj 2 )dˆ2 + · · · +
n−1 n (pnj + pj n )dˆn = dˆi j =1
i=1
(22)
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or (n − 1)dˆ1 + (n − 1)dˆ2 + · · · + (n − 1)dˆn =
n
dˆi .
(23)
i=1
It is evident that as long as ni=1 dˆi = 0 and is nonnegative, = n − 1. Since A is a positive matrix, according to the Perron–Frobenius theorem [12], its maximal eigenvalue, max , and the corresponding eigenvector Dˆ are both positive. Therefore, max ≡ n − 1.
(24)
Theorem 2. Let Dˆ be the principal right-eigenvector of matrix A corresponding to max = n − 1, then Dˆ > 0. This theorem is a direct result of the Perron–Frobenius theorem [12]. The above two theorems show that for any fuzzy preference matrix P we can always find a positive ˆ where the matrix A is determined by (17). The normalized eigenvector Dˆ satisfying ADˆ = (n − 1)D, principal right-eigenvector Dˆ is calculated by the following iterative algorithm: Step 1: Set k = 1 and Dˆ (0) = (1/n)e and = 10−5 , where is a specified precision coefficient and (0) (0) (0) Dˆ = (dˆ1 , . . . , dˆn )T . (k) (k) Step 2: Compute Dˆ (k) = ADˆ (k−1) and normalize it, where Dˆ (k) = (dˆ1 , . . . , dˆn )T is a normalized vector. Step 3: If Dˆ (k) − Dˆ (k−1) , then Dˆ = Dˆ (k) , where Dˆ (k) − Dˆ (k−1) is a vector norm, which will take one of the three forms: absolute, square or minimax. Otherwise, let k = k + 1 and go to Step 2. An alternate method for obtaining the principal right-eigenvector Dˆ is to solve the following LP model [15]: Min J = eT E [A − (n − 1)e]Dˆ − E = 0, s.t. eT Dˆ = 1, ˆ E 0, D,
(25)
where E = (1 , . . . , n )T is a nonnegative vector of deviation variables. It is easy to prove that the optimum value of J , J ∗ = 0. In fact, by Theorems 1 and 2, for any fuzzy ˆ Such an eigenvector preference matrix we can always find an eigenvector Dˆ such that ADˆ = (n − 1)D. ∗ ∗ T ∗ ∗ always leads to E = 0 and J = e E = 0. So, if J = 0, then it cannot be optimum. 4. Computing subjective weights using fuzzy preference relation and decision matrix It has been recognized in the literature that the DM’s subjective preference information should be considered along with objective information when searching for a decision. How to incorporate these two types of information is an issue that has not yet found a universally acceptable solution. There are various approaches that can be adopted and Fan et al.’s is one possibility, which is applicable for the case when
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the DM’s preference information is available in the form of a fuzzy preference matrix. In this section, we propose three new approaches for combining subjective and objective decision information to assess the weights of attributes. 4.1. LDM for integrating subjective fuzzy preference relations and objective decision matrix information, LDM-1 This approach combines the DM’s subjective fuzzy preference matrix directly with the objective decision matrix in a way that is quite different from Fan et al.’s. It is known from Theorem 1 that for any fuzzy ˆ preference matrix P we can always find a positive normalized eigenvector Dˆ such that ADˆ = (n − 1)D, ˆ where the matrix A is determined by (17) and D is an estimate of D. If the estimate is accurate then there must exist AD = (n − 1)D. Substituting (4) in the equation, we get AZW = (n − 1)ZW,
(26)
where Z is the normalized decision matrix and W is the normalized weight vector satisfying eT W = 1. Due to the existence of fuzziness and subjectivity, however, an accurate estimate is nearly impossible. So (26) cannot hold exactly in most cases. In view of this, we define the deviation vector E = [AZ − (n − 1)Z]W,
(27)
where E = (1 , . . . , n )T and find the attribute weights that minimize the sum of the absolute values of the components of E by the following LP: Min J =
s.t.
n i=1
|i |
[AZ − (n − 1)Z]W − E = 0, eT W = 1, W 0.
(28)
Letting + i =
i + | i |
2 i can be written as − i = + i − i ,
and − i =
−i + |i | , 2
i = 1, . . . , n,
i = 1, . . . , n,
(29)
(30)
− where the scalar product + i · i = 0. Now, the optimization model (28) can be written as an LP model:
Min J = eT (E + + E − ) [AZ − (n − 1)Z]W − E + + E − = 0, s.t. eT W = 1, W, E + , E − 0,
(31)
− ∗ + T − − T ∗ ∗ T where E + = (+ 1 , . . . , n ) and E = (1 , . . . , n ) . The optimum weight vector, W = (w1 , . . . , wm ) , can be easily obtained by solving (31), which incorporates both subjective and objective information. With
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the attribute weights obtained by (31), the overall weighted assessment values for the alternatives can be computed by either (3) or (4), and the final decision can be reached. We call the method described above the LDM for integrating subjective fuzzy preference relations and objective decision matrix information or LDM-1.
4.2. LDM for integrating subjective priority vector and objective decision matrix information, LDM-2 This approach uses the subjective fuzzy preference matrix information in an indirect way. It combines the DM’s subjective estimates of the overall weighted assessment values for alternatives with objective decision matrix information. Let Dˆ = (dˆ1 , . . . , dˆn )T be the DM’s subjective estimate of D = (d1 , . . . , dn )T , which can be estimated using the EM that was explained earlier, in Section 3. It is expected that Dˆ and D would be nearly the same. Dˆ is a normalized vector of assessment values but D is not. Therefore, m
di
dˆi ≈ n
k=1
dk
=
j =1 zij wj n m k=1 j =1 zkj wj
m = m
j =1
j =1
n
zij wj
k=1
, zkj wj
i = 1, . . . , n,
(32)
which can be equivalently expressed as m
zij wj − dˆi
j =1
n m j =1
zkj wj ≈ 0,
i = 1, . . . , n.
(33)
k=1
Let i =
m
zij − dˆi
n
j =1
zkj wj ,
i = 1, . . . , n.
(34)
k=1
Applying the same argument as earlier, the following optimization model is therefore constructed to minimize the sum of the absolute values of the components of E: n | i | Min J = i=1 m n ˆ zij − di zkj wj − i = 0,
s.t.
j =1 m j =1
k=1
i = 1, . . . , n,
(35)
wj = 1,
wj 0,
j = 1, . . . , m.
With the help of the variable transformation of (29), the above optimization model can be expressed as the following LP model, which we call the LDM for integrating subjective priority vector and objective
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decision matrix information or LDM-2: n − Min J = (+ i + i ) i=1 m n − zkj wj − + zij − dˆi i + i = 0,
s.t.
j =1 m j =1
k=1
i = 1, . . . , n,
(36)
wj = 1,
− wj , + i , i 0,
j = 1, . . . , m,
i = 1, . . . , n.
4.3. LDM for integrating subjective ranking and objective decision matrix information, LDM-3 Another way of using the subjective fuzzy preference matrix on the alternatives indirectly is to utilize its ranking order. Suppose the ranking order obtained from the fuzzy preference matrix is Ai1 Ai2 · · · Ain , implying “preferred to”, which can be obtained by means of the EM or other methods and can also be characterized in terms of the subjective overall weighted assessment values as dˆi1 dˆi2 · · · dˆin . It is desirable that a ranking order that is characterized by di1 di2 · · · din is produced based on the information embedded in an objective decision matrix. However, such a ranking order may not be exactly the same as the one that is characterized by dˆi1 dˆi2 · · · dˆin , in which case the ranking order characterized by di1 di2 · · · din is not valid. We introduce the nonnegative deviation variables i , . . . , in−1 such that di1 − di2 + i1 0, di2 − di3 + i2 0, .. . din−1 − din + in−1 0.
(37)
Substituting (4) in (37), we obtain m (zi1 j − zi2 j )wj + i1 0, j =1 m (zi2 j − zi3 j )wj + i2 0, j =1
.. . m (zin−1 j − zin j )wj + in−1 0. j =1
(38)
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Obviously all the deviation variables should be made as small as possible, which is accomplished by solving the following LP model: Min J = i1 + i2 + · · · + in−1 m (zil j − zil+1 j )wj + il 0, s.t.
j =1 m j =1
l = 1, . . . , n − 1, (39)
wj = 1,
wj , il 0, j = 1, . . . , m; l = 1, . . . , n − 1.
We call this method the LDM for integrating subjective ranking and objective decision matrix information or LDM-3. If the optimum objective function value of (39) is not equal to zero, that is J ∗ > 0, then the optimum weight vector, W ∗ , can be uniquely determined. Usually, when the number of alternatives is greater than the number of attributes, n > m, it is very likely that J ∗ > 0. If, on the other hand, J ∗ = 0, there may exist multiple optimum solutions to the LP (39). In this case, W ∗ cannot be determined uniquely but can be characterized by an interval vector. This is often the case when n m. The following pair of LP models can be used to reveal the weight intervals: Min/max wi m s.t.
(zil j − zil+1 j )wj 0,
j =1 m j =1
l = 1, . . . , n − 1, (40)
wj = 1,
wj 0,
j = 1, . . . , m.
The optimum solutions to the above pair of LP models form the weight intervals is denoted by [wiL , wiU ], i = 1, . . . , m, based on which a partial order may be generated for n alternatives. It is argued that interval weights and partial order would cause making a decision to be difficult because of the existence of uncertainty. Therefore, a reasonable point estimate for the weight vector is preferable. Let m (zil j − zil+1 j )wj 0, l = 1, . . . , n − 1, j =1 m (41) = wj = 1, j =1 wj 0, j = 1, . . . , m. is a convex polygon and its centroid is the most appropriate point estimate. We refer to the weights obtained in this manner as centroid weights. In order to compute the centroid weights, we need to know all the vertices of , which can be obtained by solving models (40) for all wi , i = 1, . . . , m. Suppose (k) (k) that has K0 vertices, denoted by W (k) = (w1 , . . . , wm )T , k = 1, . . . , K0 . Then its centroid weights can be calculated by [1]
W
(C)
K0 1 = W (k) . K0 k=1
(42)
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Since each vertex of satisfies the DM’s preference ranking Ai1 Ai2 · · · Ain , the centroid weights will also lead to the same preference ranking.
5. Numerical example In this section, Fan et al.’s illustrative example is re-examined to show how the DM’s subjective decision making and ranking can be derived from a fuzzy preference matrix and how they can be combined with objective decision matrix information in different ways to give an overall assessment of each alternative. Example. A potential buyer, the DM, intends to buy a house. He has four alternatives to choose from: A1 , . . . , A4 . The attributes (factors) s/he considers include price in $s (G1 ), size in m2 (G2 ), distance to work in kms (G3 ) and environmental characteristics (G4 ). Among the four attributes, G2 and G4 are benefit attributes, which receive high values for desirable alternatives, and G1 and G3 are cost attributes, which receive low values for desirable alternatives. The decision matrix for this MADM problem is given by G 1 G2 A1 3.0 100 X = A2 2.5 80 A3 1.8 50 A4 2.2 70
G3 G4 10 7 8 5 . 20 11 12 9
The potential buyer (DM) provides his/her fuzzy preference matrix on the four alternatives as follows: 0.50 0.44 0.64 0.54 0.56 0.50 0.69 0.60 P = . 0.36 0.31 0.50 0.40 0.46 0.40 0.60 0.50 First, we normalize the decision matrix X using (1) and (2). The normalized decision matrix is 0 1 5/6 1/3 5/12 3/5 1 0 Z= . 1 0 0 1 2/3 2/5 2/3 2/3 Next, we derive the buyer’s subjective decision making and ranking from his/her fuzzy preference matrix using the EM. From the fuzzy preference matrix P and (17), we have 1.62 0.44 0.64 0.54 0.56 1.85 0.69 0.60 A= , 0.36 0.31 1.07 0.40 0.46 0.40 0.60 1.46
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whose maximum eigenvalue is max = 3 and the corresponding principal right-eigenvector is Dˆ = (dˆ1 , dˆ2 , dˆ3 , dˆ4 )T = (0.2713, 0.3446, 0.1537, 0.2304)T , which ranks the four alternatives as A2 A1 A4 A3 . We can obtain exactly the same result by solving the LP model (25): Min J = 1 + 2 + 3 + 4 −1.38dˆ1 + 0.44dˆ2 + 0.64dˆ3 + 0.54dˆ4 − 1 = 0, 0.56dˆ1 − 1.15dˆ2 + 0.69dˆ3 + 0.60dˆ4 − 2 = 0, 0.36dˆ1 + 0.31dˆ2 − 1.93dˆ3 + 0.40dˆ4 − 3 = 0, s.t.
0.46dˆ1 + 0.40dˆ2 + 0.60dˆ3 − 1.54dˆ4 − 4 = 0, dˆ1 + dˆ2 + dˆ3 + dˆ4 = 1, dˆ1 , dˆ2 , dˆ3 , dˆ4 0, 1 , 2 , 3 , 4 0.
The above overall weighted assessment value-vector and ranking order are purely subjective. The final decision making still needs to consider the objective decision matrix information. Using Fan et al.’s approach and (11), we obtain W = (0.7463, 0.7421, −0.0549, −0.4335)T . The corresponding objective function value is H (W ) = 0.1220372 × 10−4 . Since the weights w3 and w4 do not satisfy the nonnegativity constraint, we solve the quadratic programming problem (10) to obtain W ∗ = (0.3266, 0.2962, 0.3772, 0)T . The corresponding optimum objective function value is H (W ∗ ) = 0.2068575 × 10−1 . We have been unable to validate Fan et al.’s result, namely, W ∗ = (0.2914, 0.0955, 0.5787, 0.0344)T , which corresponds to an objective function value of H (W ∗ ) = 0.3236951 × 10−1 . It seems to us that there is a mistake in Fan et al.’s computations; with their results the objective function fails to achieve its minimum and also leads to the inconsistent ranking of A2 A4 A1 A3 , which contradicts the DM’s subjective ranking of A2 A1 A4 A3 . Although inconsistent ranking orders are allowed and sometimes unavoidable, as far as this example is concerned, inconsistency is not at all inevitable. Now we explore the use of the LDMs developed in Section 4. The application of LDM-1, through the solution of (31), produces W ∗ = (0.3413, 0.3512, 0.3435, 0)T , which incorporates both the subjective fuzzy preference relations and the objective decision matrix information. The corresponding overall weighted assessment value vector is D = (d1 , d2 , d3 , d4 )T = (0.6014, 0.6748, 0.3413, 0.5826)T . After normalization, it becomes D = (0.2734, 0.3067, 0.1551, 0.2648)T , which is very close to the DM’s subjective estimate Dˆ = (0.2713, 0.3446, 0.1537, 0.2304)T and both lead to the same ranking order: A2 A1 A4 A3 . On the other hand, the application of LDM-2, through (36), produces W ∗ = (0.3397, 0.2965, 0.3638, T 0) . The corresponding overall weighted assessment value vector turns out to be D = (0.5996, 0.6832, 0.3397, 0.5876)T , which, after normalization, becomes D = (0.2713, 0.3091, 0.1537, 0.2659) that is also very close to the DM’s subjective estimate and leads to the same ranking order: A2 A1 A4 A3 . Based on the subjective ranking order A2 A1 A4 A3 , which is derived from the DM’s fuzzy preference matrix using the EM, the following convex polygon is constructed to assess the weights of the
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Table 1 The vertices of polygon and the weights Weight
w1 w2 w3 w4
Vertices #1
#2
#3
#4
0 0 1 0
0 0 0.6667 0.3333
0 0.2941 0.7059 0
0.4580 0.4962 0.0458 0
Interval weight
Centroid weight
[0, 0.4580] [0, 0.4962] [0.0458, 1] [0, 0.3333]
0.1145 0.1976 0.6046 0.0833
four attributes: 5 2 1 1 12 w1 − 5 w2 + 6 w3 − 3 w4 0, − 2 w + 3 w + 1 w − 1 w 0, 3 1 5 2 6 3 3 4 = − 1 w1 + 2 w2 + 2 w3 − 1 w4 0, 3 5 3 3 w1 + w2 + w3 + w4 = 1, w1 , w2 , w3 , w4 0. The polygon has four vertices, as shown in Table 1. The corresponding interval weights and the centroid weights are shown in the last two columns of Table 1. The overall weighted assessment value vector for the four alternatives effected by this set of centroid weights is D = (0.7292, 0.7709, 0.1978, 0.6140)T , which yields exactly the same ranking: A2 A1 A4 A3 . In this example, all the three LDMs produce the same ranking that is also consistent with the DM’s subjective ranking, which means that the DM’s subjective decision making and ranking is fully supported by the objective decision matrix information.
6. Concluding remarks In this paper, MADM problems with fuzzy preference information on alternatives have been investigated. An eigenvector method has been developed to derive the subjective ranking from a fuzzy preference relation, and three optimization models and methods have been proposed to combine the DM’s subjective preferences on alternatives and objective decision matrix information in three different ways to assess the attribute weights. A numerical example provided by Fan et al. has been revisited to demonstrate the proposed methods. It has been shown that the methods developed in this paper are rational, feasible and effective. In order to better assess the weights of attributes and avoid assigning unrealistic weights, additional preference information on the weights may be elicited from the DM. For example, a threshold value, wL , for the weights may be set to avoid the occurrence of zero weight. A partial order for the weights may also be sought. Any additional preference information can be easily incorporated into the methods that have been proposed in this paper. Compared with Fan et al.’s approach, the LDMs introduced here provide greater flexibility for solving MADM problems with preference information on alternatives and/or attributes.
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Finally, we must point out that in this paper the fuzzy preference relations on alternatives are assumed to be known. This is a strong assumption. While how to elicit fuzzy preference relations is an important issue, this is a topic that falls outside the purpose of the present paper. The DM may formulate his/her preferences on alternatives with respect to each attribute individually and then aggregate them using one of the available approaches; for example, the ordered weighted averaging (OWA) operators [18]. The interested reader is referred to [3] for more on this topic. Acknowledgments The authors would like to express their sincere thanks to the Editors-in-Chief and three anonymous referees for helping to improve this paper with their constructive comments and suggestions. References [1] A. Arbel, L.G. Vargas, Preference simulation and preference programming: robustness issues in priority derivation, European J. Oper. Res. 69 (1993) 200–209. [2] N. Bryson, A goal programming method for generating priority vectors, J. Oper. Res. Soc. 46 (1995) 641–648. [3] F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Sets and Systems 97 (1998) 33–48. [4] Z.P. Fan, H.Y. Li, A ranking method for alternatives based on fuzzy preference relation, J. Northeast. Univ. (Natur. Sci.) 20 (6) (1999) 651–653 (in Chinese). [5] Z.P. Fan, H.Y. Li, G.F. Hu, Fuzzy judgement matrix and the goal programming method for ranking alternatives, J. Northeast. Univ. (Natur. Sci.) 21 (1) (2000) 60–62 (in Chinese). [6] Z.P. Fan, J. Ma, Q. Zhang, An approach to multiple attribute decision making based on fuzzy preference information on alternatives, Fuzzy Sets and Systems 131 (2002) 101–106. [7] C.L. Hwang, K. Yoon, Multiple Attribute Decision Making: Methods and Applications, Springer, Berlin, 1981. [8] J. Kacprzyk, Group decision making with a fuzzy majority, Fuzzy Sets and Systems 18 (1986) 105–118. [9] J. Kacprzyk, M. Fedrizzi, H. Nurmi, Group decision making and consensus under fuzzy preferences and fuzzy majority, Fuzzy Sets and Systems 49 (1992) 21–31. [10] S. Lipovetsky, W.M. Conklin, Robust estimation of priorities in the AHP, European J. Oper. Res. 137 (2002) 110–122. [12] C.R. Rao, M.B. Rao, Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientific, Singapore, 2001. [13] T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980. [14] Y.M. Wang, A new generalized least deviation priority method of group comparison matrix, Systems Eng.—Theory & Practice 14 (9) (1994) 63–68 (in Chinese). [15] Y.M. Wang, An overview of priority methods of comparison matrix, J. Decision Making and Decision Support Systems 5 (3) (1995) 101–114 (in Chinese). [16] Z.S. Xu, Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation, Internat. J. Approx. Reason. 36 (2004) 261–270. [17] Z.S. Xu, Q.L. Da, A least deviation method to obtain a priority vector of a fuzzy preference relation, European J. Oper. Res. 164 (2005) 206–216. [18] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Systems Man Cybernet. 18 (1988) 183–190.