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Sensitivity analysis of the ordered weighted averaging operator via linear models
Accepted Manuscript Sensitivity Analysis of the Ordered Weighted Averaging Operator via Linear Models M. Khodadadi, G. Tohidi, M. Zarghami PII: DOI: Reference:
S0360-8352(17)30343-1 http://dx.doi.org/10.1016/j.cie.2017.08.001 CAIE 4848
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
31 August 2013 14 July 2017 1 August 2017
Please cite this article as: Khodadadi, M., Tohidi, G., Zarghami, M., Sensitivity Analysis of the Ordered Weighted Averaging Operator via Linear Models, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/ j.cie.2017.08.001
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Sensitivity Analysis of the Ordered Weighted Averaging Operator via Linear Models M. Khodadadi a,*, G. Tohidi b M. Zarghami c a
b
Department of Mathematics, Urmia Branch, Islamic Azad University, Urmia, Iran
Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran c
Faculty of Civil Engineering, University of Tabriz, Tabriz 51664, Iran
Corresponding author: Maryam Khodadadi, [email protected], [email protected], Tel.: +98 9141464712; fax: +98 443 3372683 Address: Maryam Khodadadi, Department of Mathematics, Urmia Branch, Islamic Azad University, Urmia, Iran.
Ghasem Tohidi: [email protected] Mahdi Zarghami: [email protected]
*
Corresponding author. Tel.: +98 9141464712; fax: +98 443 3372683
Address: Maryam Khodadadi, Department of Mathematics, Urmia Branch, Islamic Azad University, Urmia, Iran
Email addresses: [email protected], [email protected], (M.khodadadi),
[email protected] (G. Tohidi) [email protected] (M.Zarghami)
1
Sensitivity Analysis of the Ordered Weighted Averaging Operator via Linear Models Abstract The efficient computation of the weights in an ordered weighted averaging (OWA) operator plays an important role in the successful design and application of the OWA operator. In some applications, an OWA weight vector with positive and distinct components for any desired orness level is preferable. This paper proposes two complementary linear OWA (CLOWA) models for determining the weights and a compact form of the optimal weights in the general case (i.e., for any level of orness and n . The proposed models produce a unique OWA weight vector with distinct and positive components for any orness level (0,1) . In addition to a combined goodness measure, the sensitivity analysis on the outputs of the OWA operator with respect to the optimism degree of the decision maker (DM) is important. This study obtains the combined goodness measure and sensitivity analysis models for two linear methods: the minimax disparity (MD) and new proposed models. Then, to assign reliable ranks to the alternatives in the presence of two conflicting objectives—maximizing the combined goodness measure and minimizing its sensitivity to the optimism degree of the DM—the composite measure of goodness is extended. The proposed methods are applied in a water resource management problem. Keywords: Ordered weighted averaging (OWA) operator, Complementary linear OWA (CLOWA) models, Minimax disparity (MD) approach, Sensitivity analysis, Optimism degree of DM, Combined goodness measure.
1. Introduction
The ordered weighted averaging (OWA) operator, as one of the important and efficient aggregation operators, was primarily introduced by Yager (1988). Since its introduction, it has attracted great attention among researchers and has been used in a variety of fields such as multicriteria and group decision making, fuzzy system and control, database query management, data
2
mining, forecasting, approximate reasoning, neural networks, market research, expert systems, and data envelopment analysis (Xu and Da, 2003; Liu, 2011; Wang and Chin, 2011). An n-dimensional OWA operator is a mapping F :[0,1]n [0,1] , with the associated weighting vector W (w1 , w2 ,..., wn )T having the properties: w 1 w 2
w n 1; 0 w i 1, i 1,..., n .
The combined goodness measure assigned to the vector X (a1 , a2 ,
, an ) by this operator is as
follows: n
F (X ) F (a1 , a2 ,..., an ) w j b j
(1)
j 1
The components of the vector X are the evaluations of an alternative with respect to the n criteria. Before multiplying the components of the vector X by the weights, these components should be arranged in non-increasing order. Therefore, in formula (1), b j is the jth largest element among the components of the vector X . The degree of orness associated with this operator (Liu, 2012) is defined as: n j wj . j 1 n 1 n
orness( w)
(2)
A critical and main step in OWA operator research is the determination of OWA operator weights. For this purpose, Yager (1996) proposed a method for obtaining OWA weighting vectors based on fuzzy linguistic quantifiers (FLQ) (Liu, 2011). O'Hagen (1988) suggested a nonlinear model to determine OWA weights based on the measure of dispersion with a predefined degree of orness. Later, Fullér and Majlender (2003) suggested the minimum variance (MV) approach to obtain OWA weights. Yari and Chaji (2012) proposed another nonlinear OWA model for obtaining the OWA weights based on a given prior OWA vector. This model was called the maximum Bayesian entropy OWA (MBEOWA) model. By using the Lagrange multiplier method, Chaji (2017) found an analytic solution for obtaining the MBEOWA model weights.
3
Wang and Parkan (2005) proposed the first linear programming model called the minimax disparity (MD) approach to generate OWA operator weights. The weights of the MD model were obtained in a compact form by Wang and Chin (2011). Amin and Emroznejad (2006) and Wang et al. (Wang et al., 2007a, 2007b) improved the MD approach. Emrouznejad and Amin (2010) proposed an improved minimax disparity model to determine OWA operator weights that minimizes the sum of the deviation between two distinct OWA weights. Tohidi and Khodadadi (2015) obtained the optimal simplex tableau of the improved minimax disparity model in the general case (i.e., for any level of orness and n ) and then for the desired n , they introduced n 1 optimal basic feasible solutions (BFSs) of the model. Sometimes it is more preferred to use an OWA weight vector with positive and distinct components for any desired orness level (0,1) in the aggregation process. For this reason, this paper divides the interval [0,1] into [0,0.5] and [0.5,1] and proposes two linear OWA models for orness levels
[0,0.5] and [0.5,1] , separately. The proposed models generate
OWA weight vector with positive and distinct components for any orness level (0,1) . It also demonstrates that the optimal weights of the two models satisfy the dual property and that the models are complementary. Therefore, we can solve one of the models and obtain the optimal weight vectors for both of them. The paper also obtains the optimal simplex tableaux of the models in the general case (i.e., for any level of orness and n). Then, for the desired n, it introduces the unique optimal OWA weight vectors of the models. To investigate the aforementioned properties among other linear OWA models, their weights in the compact form should be available. The compact form of the OWA weights is only available for a few linear OWA models. The MD model and the improved minimax disparity model are two examples of linear models for which the compact forms of the optimal weights are available. The improved minimax disparity model produces alternative optimal OWA weights (Tohidi and Khodadadi; 2015). Under alternative optimal OWA weights, the unique aggregated value cannot be obtained for the alternatives. Since the MD model produces unique optimal OWA weights and the compact form of its optimal weights is obtained, we have selected the MD model in this study to compare it to our produced method. Furthermore, the comparison of the
4
CLOWA method and other OWA determination methods is presented as the future scope of this study. In the outcome provided by the OWA operator, two factors are important. These two factors are the combined goodness measure and the sensitivity analysis on the outputs of the OWA operator with respect to the optimism degree of the decision maker (DM). The sensitivity analysis provides deeper insight into the final decision based on risk-taking, risk-neutral, and risk-averse attitudes. Larger values of the combined goodness measure and smaller values of the sensitivity to the optimism degree of the DM are required for making robust decision on the alternatives. In this paper, we also obtain and compare the combined goodness measure and the sensitivity analysis models for two linear OWA models: the MD approach and the new CLOWA method. A similar study on this subject was performed for the FLQ approach and the MV method by Zarghami et al. (2008). In addition, to share both conflicting characteristics—the combined goodness measure and sensitivity measure—in the evaluation of alternatives, this paper extends the idea of the composite measure of goodness by using the distance from both positive and negative ideal points. The rest of this paper is organized as follows: Section 2 presents a brief description about the MD model and the compact form of its optimal weights. The two new OWA models with their weights in compact forms are given in Section 3. Section 4, which is composed of four subsections, discusses the combined goodness measure and sensitivity measure for two linear methods: the MD approach and the CLOWA method. It also compares the combined goodness measure and sensitivity measure obtained by the two methods. Finally, it extends the formula of the composite measure of goodness. Section 5 applies the proposed methods to a watershed management problem. Finally, Section 6 concludes the study. 2. Preliminaries Wang and Parkan (2005) proposed the MD model (3) to obtain OWA operator weights under the desired orness level.
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min d n
s.t.
w i 1
i
1,
n
(n i )w i 1
i
(n 1) , 0 1,
(3)
d w i w i+1 d , i 1,..., n 1, w i 0, i 1,..., n .
Model (3) minimizes the maximum disparity of the OWA operator weights. It is noteworthy that in model (3) and all of the OWA models, w1 , w2 ,..., wn are the variables to be determined after solving the models. Additionally, the degree of orness is a parameter that should be determined by the DM before solving the models. Theorem 1 presents the weights of model (3) in a compact form. Theorem 1: For a given orness degree (0.5,1) , there exists an integer k n such that
wi w1 (i 1)d for i 1,..., k
(4)
and
wi 0 for i k 1,..., n ,
(5)
where k , w1 , and d are determined by
k Min (n , INT[3n 1 3 (n 1)]),
(6)
4(k 1) 6n 6 (n 1) , k (k 1)
(7)
w1
and d
2(kw 1 1) , k (k 1)
(8)
where INT[x] is the function that rounds x down to the nearest integer. For a given orness degree (0, 0.5) , according to the dual property, the weights can be obtained (Wang and Chin, 2011).
3. New OWA Aggregation Weights
6
Some decision-making problems consider the optimistic/pessimistic view of the DM in the aggregation process. In such aggregation problems, the DM may prefer all the scores to be present in the final score for any desired orness level (0,1) . This section tries to obtain OWA weight vectors with positive and distinct components for any desired orness level (0,1) . Using these OWA operator weights for the aggregation process, all the scores will be present in the final score. To obtain the aforementioned OWA weights, they are found such that they have the properties 0 w n ... w 2 w 1 1 or 0 w 1 w 2
w n 1 . If the OWA weights have
the properties 0 w n ... w 2 w 1 1 then the orness level must be satisfied in the inequality
0.5 1 (Filev and Yager, 1995). In addition, to obtain the weights with the properties w n 1 the orness level must be satisfied in the inequality 0 0.5 (Filev
0 w 1 w 2
and Yager, 1995). To this end, the interval [0,1] is subdivided into [0,0.5] and [0.5,1] . Then, two linear models (9) and (10) are proposed for the orness levels [0,0.5] and [0.5,1] , respectively, as follows: max n
w
s.t.
i 1
i
1,
n
1 (n 1) , 0 , 2 i 1 w i 1 w i i 1,..., n 1,
(n i )w
(9)
i
w1 , w i 0, i 1,..., n .
and max n
s.t.
w i 1
i
1,
n
1 1, 2 i 1 wi wi 1 i 1,..., n 1,
(n i) w
i
(n 1) ,
(10)
wn , wi 0, i 1,..., n. 7
Models (9) and (10) generate a unique aggregation weight vector that has the maximum discrimination between the two adjacent weights. The main characteristic of models (9) and (10) is the use of simple linear models to generate a unique optimal OWA weight vector with positive and distinct components for any given level of orness. We will show that the weights obtained by models (9) and (10) satisfy the dual property. Therefore, we can solve model (9) for
[0,0.5]
and obtain the optimal weight vector of model (10) according to the dual property and vice versa. The theoretical properties of models (9) and (10) are as follows: Theorem 2: Model (9) for any (0, 0.5) and model (10) for any (0.5,1) generate a unique optimal weight vector with positive and distinct components. Proof: We prove that model (9) for n
3 generates a unique aggregation weight vector with
positive and distinct components. The proof for model (10) will be similar. First we obtain the optimal simplex tableaux of model (9). In the obtained tableaux, we will observe that z i c i 0 for all nonbasic variables. This will indicate that model (9) has a unique optimal solution (Bazarra et al., 2010). Additionally, the obtained optimal simplex tableaux will show that all the components of the optimal OWA weight vector are distinct and are in the basis with positive values. To obtain the optimal simplex tableaux, reformulate model (9) so that it is in standard format. By converting the inequality constraints of model (9) to the equality constraints, we obtain model (11): max n
s.t.
w 1, i 1
i
n
1 (n 1) , 0 , 2 i 1 wi 1 wi si 1 0 i 1,..., n 1,
(n i ) w
(11)
i
w1 s1 0, wi 0, si 0, i 1,..., n.
8
We claim for 0 1
3
and 1 1 the optimal basic vectors and their optimal bases for 3 2
model (11) are, respectively, as follows:
,aw ,as ,a ], 0
1 3
1 3
1 2
xB*
(w1, w2, , w(n 1), wn ,sn , ), B1
[aw ,aw ,...,aw
xB*2
(w1, w2, , w(n 1), wn ,s1, ), B2
[aw1 ,aw2 ,...,awn 1 ,awn ,as1 ,a ],
1
1
n 1
2
n
n
where the columns of matrices B 1 and B 2 , i.e., awi , i 1,..., n , asn , a and awi , i 1,..., n , as , a , 1 are the corresponding coefficients of the basic variables xB* and xB* in the constraints matrix of 2
1
model (11). To prove this claim, the simplex tableaux of model (11) corresponding to bases B 1 and B 2 are shown in Tables 1 and 2, respectively: ----------------------------------------------Insert Table 1 Here------------------------------------------------------------------------------------------Insert Table 2 Here--------------------------------------------where I is the (n 2) (n 2) identity matrix and the columns of Tables 1 and 2 are
aSi , i
1, aS , i
1,2,..., n
i
2,..., n and b . These columns are obtained by solving the following
systems of equations (12), (13), and (14), respectively (Bazarra et al., 2010):
BaS
i
1,2,..., n
aS i
2,..., n,
i
BaS
i
Bb
aS i
i
1,
(12) (13)
b.
(14)
In addition, the values corresponding to the nonbasic variables in rows zi ci of Tables 1 and 2 are
zS
i
obtained
cS
i
cBaS
i
by
cS i i
the
zS
formulas
i
cS
i
cBaS
i
cS i i
2,..., n (Bazarra et al., 2010), where cB and cS i i
1,2,..., n 1,
,n
1
and
1, n are the
coefficients of the basic variables and nonbasic variables in the objective function of model (11). To check that B 1 and B 2 are the optimal bases of model (11), it is sufficient to show that the 9
inverses of B 1 and B 2 exist and B11b 0, B 21b 0 and z i c i 0 for all nonbasic variables (Bazarra et al., 2010). It is easy to determine that the matrices B 1 and B 2 have inverses. Tables 1 and 2 show that B11b 0, B 21b 0 and z i c i 0 for all nonbasic variables. These facts indicate that B 1 and B 2 are the optimal bases of model (11) for 0 1
3
and 1 1 , respectively. 3 2
Additionally, it can be seen from Tables 1 and 2 that z i c i 0 for all nonbasic variables; therefore, B 1 and B 2 are the unique optimal bases of model (11) (Bazarra et al., 2010). The components of vectors x B* and x B* , obtained from the last columns of Tables 1 and 2, are 1
2
shown in formulas (15) and (16). * w ,w ,...,w ,...,w ,w , s , T ; 0 1 , xB 1 2 n 1 n n i 1 3
(15)
where w 1 , w 2 2 , …, w i i , …, w n 1 (n 1) , w n n s n , s n 1 3 and
6 . n (n 1)
* w ,w ,...,w ,...,w ,w , s , T ; 1 1 , xB 1 2 n 1 n 1 i 2 3 2
(16)
where w 1 s1 , w 2 s1 2 , …, w i s1 i , …, w n 1 s1 (n 1) , s1
w n s1 n ,
6(1 2 ) 6 2 and . n (n 1) n
Formulas (15) and (16) demonstrate that w 1,w 2 ,...,w n 1 ,w n , in the components of x B* and x B* 1
2
(0, 1 3] and
are positive and distinct from each other for any given level of orness [ 1 , 1 ) , respectively. 3
2
In a similar way, we can easily indicate that the following vectors x B* and x B* and the bases 3
4
B3 and B4 are the unique optimal basic variables and the unique optimal bases of model (10) for
1 2 and 2 1, respectively. 2 3 3
10
x B* 3 (w 1 ,w 2 ,...,w i ,...,w n 1 ,w n , s n , )T ;
w1 wn (n 1) ,
where
wn sn , sn
1 2 , 2 3
(17)
w 2 w n (n 2) ,…, wi wn (n i) ,…, wn1 wn ,
4 6 6(2 1) and . n(n 1) n
x B* 4 (w 1 ,w 2 ,...,w i ,...,w n 1 ,w n , s1 , )T ;
2 1, 3
(18)
where w1 n s1 , w 2 (n 1) ,…, wi (n (i 1)) ,…, wn , s1 3 2 and
B3 [aw1 , aw2 ,..., awn1 , awn , asn , a ];
6(1 ) . n(n 1)
1 2 , 2 3
and B4 [aw1 , aw2 ,..., awn1 , awn , as1 , a ];
(19)
2 1. 3
(20)
Formulas (17) and (18) reveal that w 1,w 2 ,...,w n 1 ,w n , in the components of x B* and x B* 3
also positive and distinct from each other for any given level of orness
4
are
(0.5, 2 ] and 3
[ 2 ,1) , respectively. 3
The facts mentioned above complete the proof. Theorem 3: The OWA operator weight vector W (w 1 ,w 2 ,...,w n ) obtained by model (9) for T
0 0.5 and model (10) for 0.5 1 satisfies the following properties:
1) If orness(W )=1, then W (1,0,0,...,0)T ; 2) If orness(W )=0, then W (0,0,...,0,1)T ; 3) If orness(W )=0.5, then W
( 1 , 1 ,..., 1 )T . n
n
n
Proof: Formulas (15), (16), (17), and (18) introduce the compact forms of the optimal OWA weights for the CLOWA models.
1 into formula (18). By substituting
For orness(W )=1, it is sufficient to substitute into formula (18) we achieve W (1,0,0,...,0)T . 11
1
0 into (15), the obtained weight
For orness(W )=0, consider formula (15). By substituting vector will be W (0,0,...,0,1)T .
For orness(W )=0.5, similar to the above process, it is sufficient to substitute
0.5 into
( 1 , 1 ,..., 1 )T .
formula (16) or (17) and obtain the weight vector W
n
n
n
An OWA model satisfies the dual property, if it produces an optimal OWA weight vector
W (w 1 ,w 2 ,...,w n ) *
*
W (u1* ,u 2* ,
*
* T
for a given level of orness and an optimal OWA weight vector
,u n* )T for the level of orness 1 and vice versa, where u i* w n* i 1 for each
i 1, 2,..., n . See Emrouznejad and Amin (2010) as an example of an investigation of the dual property for improving minimax disparity model. Theorem 4 introduces the dual property for the CLOWA models. It states that models (9) and (10) satisfy the dual property with each other. This is why the models are complementary. Theorem 4: If we obtain an optimal OWA weight W given level of orness
, then W (u1* ,u 2* ,
*
(w 1 ,w 2 ,...,w n ) *
*
* T
from model (9) for a
,u n* )T will be the optimal weight vector of model
, and vice versa, where u i* w n* i 1 for each
(10) for the level of orness 1
i
1,..., n.
Proof: Consider
1 3
in the interval [0, ] and suppose W * (w 1* ,w 2* ,...,w n* )T is the optimal weight vector
of model (9) for the degree of orness model (10) for the degree of orness 1
and W (u1* ,u 2* ,
,u n* )T is the optimal weight vector of
. According to (15), the components of W
*
are as
follows:
w 1*
6 12 18 6(n 1) 6 1 3 . , w 2* , w 3* ,…, w n* 1 , w n* n (n 1) n (n 1) n (n 1) n (n 1) (n 1)
Since 1
2 3
belongs to interval [ ,1] , the components of W should be obtained from formula
(18). By substituting 1
for
in formula (18), we obtain
12
u1*
6 6(n 1) * 6(n 2) 12 6 1- 3 , u 2* , u3 ,…, u n* 1 , u n* . n (n 1) n 1 n (n 1) n (n 1) n (n 1)
By comparing the components of W and W , we find that u i* w n* i 1 for each i *
This completes the proof for the case
1,..., n.
1 3
[0, ] . The proof for the other cases is similar.
Theorems 3 and 4 show that models (9) and (10) generate valid OWA operator weights. The next section uses the OWA weight vectors produced by the MD and CLOWA methods for the sensitivity analysis of the alternatives with respect to the optimism degree of the DM.
4. Sensitivity Analysis by Linear OWA Models In the previous sections, the compact forms of the OWA weights for the MD and CLOWA models were introduced. By substituting the compact forms of the weights into (1), it is observed that for a fixed n the combined goodness measure of the alternatives is affected by two factors, the performance of the alternatives with respect to the criteria and the optimism degree of the DM. To investigate these two factors in the combined goodness measure of the alternatives, F (X ) is considered in (1) as F (X , ) . It is important to determine which of the mentioned factors has more effect on the combined goodness measure of the alternatives. To determine this, the partial derivative of the combined goodness measure is obtained with respect to the optimism degree of the DM (i.e.
F ( X , ) ) and is represented by S (X , ) . This measure
analyzes the sensitivity of the combined goodness measure of the alternatives with respect to the optimism degree of the DM. A high value of combined goodness measure and a low value of sensitivity measure for an alternative are preferable. A high value of sensitivity measure for an alternative indicates that the value of combined goodness measure of the alternative can vary wildly with different optimism degrees. If the values of the combined goodness measure of two alternatives are the same, then the alternative with the smaller value of sensitivity measure is preferable. The next subsections present the F (X , ) and S (X , ) for the CLOWA and MD methods and then compare the results of the two methods.
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4.1 Sensitivity Analysis by the CLOWA Models
Using the compact form of the OWA weights from formulas (15), (16), (17), and (18), the combined measures of goodness and the sensitivity measures for the proposed models (9) and (10), under three cases are as follows: 1. (0 1 ) 3 After substituting the compact form of the OWA weights into (1), for a fixed n, since the weights are functions of , F (X ) is considered as F (X , ) .
F (X ) F (a1 , a2 ,..., an ) w 1b1 w 2b2 w i bi F (X , ) (
6 )b1 n (n 1)
(
6 i )bi n (n 1)
(
w nb n 6 (n 1) 6 )b n 1 ( (1 3 ))b n n (n 1) (n 1)
6 i 6 ( )bi ( (1 3 ))b n (n 1) i n ( n 1) n 1
S (X , )
F (X , ) 6 ( )b1 n (n 1)
(
6i )bi n (n 1)
(
(21)
6(n 1) 6 )bn 1 ( 3)bn (22) n (n 1) (n 1)
The sensitivity measure, S (X , ) , in (22) is found through the partial derivative of F (X , ) in (21) with respect to .
The following points show the quality of dependence of S (X , ) on bi for i 1,
2 F 6i 0, i 1, bi n(n 1)
2 F 3(1 n) 0, bn (n 1)
If b1 bn , then S (X , )
,n
, n 1,
F 0 , b1 b2
bn iff S (X , )
F 0.
As seen, when the degree of orness increases, any increase in bi , i 1,
, n 1 increases the
amount of S (X , ) , while any increase in bn decreases the amount of S (X , ) .
14
With a simple check, it can be shown that the compact form of the weights for the intervals [ 1 , 1 ] and [ 1 , 2 ] are the same. Thus, F (X , ) and S (X , ) for the interval [ 1 , 2 ] 3 2 2 3 3 3
are as follows: 2.
1 2 3 3
F (X ) F (a1 , a2 ,..., an ) w 1b1 w 2b2 w i bi F (X , ) (
6 2 6(1 2 ) )b1 n n (n 1)
S (X , )
F (X , ) 6 12 ( )b1 n n (n 1)
(
w nb n
6(1 2 ) 6 2 i )bi n n (n 1)
(
6 12i ( )bi n n (n 1)
6(1 2 ) 6 2 n )bn (23) n n (n 1)
6 12n ( )b n n n (n 1)
6(n 2i 1) ( )bi n (n 1) i 1 n
If i
2 F n 1 0 , then bi 2
If i
2 F n 1 0 (For odd value of n) , then bi 2
If i
2 F n 1 0 , then bi 2
If b1 bn , then S (X , ) 3.
F 0 , b1 b2
bn iff S (X , )
(24)
F 0
2 1, 3 6n (1 ) 6(n 1)(1 ) 3 2)b1 ( )b 2 n (n 1) n (n 1) 6(1 ) ( )bn n (n 1)
F (X , ) (
15
(
6( n i 1)(1 ) )bi n (n 1)
(25)
S (X , )
F (X , ) 6n 6(n 1) (3 )b1 ( )b 2 n (n 1) n (n 1)
(
6(n i 1) )bi n (n 1)
n 6 3(n 1) 6(n i 1) ( )bn ( )b1 bi n (n 1) (n 1) n (n 1) i 2
(26)
2 F 3(n 1) 0. b1 (n 1) 2 F 6(n i 1) 0, i 2, bi n(n 1) If b1 bn , then S (X , )
,n.
F 0 , b1 b2
bn iff S (X , )
F 0.
The above points indicate that when the degree of orness increases, any increase in b1 increases the value of S (X , ) , while any increase in the other bi , i 2,
, n decreases the
value of S (X , ) .
4.2 Sensitivity Analysis by the MD Method By substituting the optimal weights of model (3) into formula (1), we can obtain F (X , ) and S (X , ) for (0, 0.5) and (0.5,1) as follows: 1. [0, 0.5] 4(k 1) 6n 6(1 )(n 1) )b n k (k 1) 4(k 1) 6n 6(1 )(n 1) 3(k 1) 6n 6(1 )( n 1) ( 2i )b n i k (k 1) k (k 1)(k 1) 4(k 1) 6n 6(1 )(n 1) 3( k 1) 6n 6(1 )( n 1) ( 2(k 1) )b n k 1 k (k 1) k (k 1)(k 1) F (X , ) (
where k can be obtained as follows (Wang and Chin, 2011):
16
(27)
For i 0,1, 2,..., n 3 , if [
i i 1 n2 , ) , then k i 2 ; also, if [ , 0.5] , then 3n 3 3n 3 3n 3
k n.
S (X , )
F (X , ) k 1 6(n 1) (k 1 2i ) b n i (k 1)k (k 1) i 0
2 F k 1 0. If i , then bn i 2
2 F k 1 0 (For odd value of k ). If i , then bn i 2
If i
If
(28)
2 F k 1 0. , then bn i 2
bnk 1 bn ,
S (X , )
then
S (X , )
F (X , ) 0,
bnk 1 bnk 2
bn
iff
F (X , ) 0.
2. [0.5,1]
4(k 1) 6n 6 (n 1) )b1 k (k 1) 4(k 1) 6n 6 (n 1) 3(k 1) 6n 6 (n 1) ( 2(i 1) )bi k (k 1) k (k 1)(k 1) 4(k 1) 6n 6 (n 1) 3(k 1) 6n 6 ( n 1) ( 2(k 1) )b k k (k 1) k (k 1)(k 1) F (X , ) (
(29)
where k can be determined as follows (Wang and Chin, 2011): If [0.5,
2n 1 i i 1 ] , then k n , and for i 0,1, 2,..., n 3 , if (1 ) [ , ) , then 3n 3 3n 3 3n 3
k i 2.
By differentiating (29), we find that:
17
S (X , )
F (X , ) k 6(n 1) (k 1 2i ) bi i 1 ( k 1) k ( k 1)
(30)
Additionally, we obtain the following points:
If i
2 F k 1 0. , then bi 2
If i
2 F k 1 0 (For odd value of k ). , then bi 2
If i
2 F k 1 0. , then bi 2
If b1 bk , then S (X , )
F (X , ) 0 , b1 b2
bk iff S (X , )
F (X , ) 0.
4.3 Comparison
As mentioned previously, the combined goodness measure and the sensitivity measure can be considered as functions of the vector X and . For a fixed input vector X , functions F (X , ) and S (X , ) become functions of and are represented by FX ( ) and S X ( ) . By comparing the combined goodness measure and sensitivity measure of the
previously mentioned linear models, the following conclusions can be made.
The F (X , ) and S (X , ) of the two approaches have a linear behavior with respect to the optimism degree of the DM.
For a fixed input vector X and desired n, the number of break points of the obtained function FX ( ) by the MD model is 2n 3 , while the number of break points of the obtained function FX ( ) by the CLOWA models is 2. This indicates that FX ( ) , which is obtained by the proposed method, has a more uniform movement than the FX ( ) obtained by the MD model in [0, 1] as increases to one.
For a desired n, the value of S X ( ) cannot be calculated in 2n 3 points for the MD model and 2 points for the new method.
18
In both cases, S X ( ) does not depend on the optimism degree, and it is a discontinuous piecewise-linear function of .
Obtaining the two functions FX ( ) and S X ( ) in the MD method is more difficult than in the proposed method.
The proposed models produce a unique OWA weight vector with distinct and positive components for any orness level (0,1) . Therefore, in aggregation by the new proposed weights, none of the scores are ignored in the combined goodness measure of the alternatives, while the MD model produces some zero components for some orness levels. Hence, in the combined goodness measure obtained by this method not all the scores are present.
When
There are several DMs with different orness levels in the decision making process.
The case study is an important problem and it requires more attention by the DM.
The DM has a risk concern, etc.
In addition to F (X , ) , it is better to consider S (X , ) in the evaluation of the alternatives. The value of S (X , ) obtained by the MD model cannot be calculated for more orness degrees. According to this point and the above points for exploring the OWA weights, the proposed method is recommended. 4.4
Extending the composite measure of goodness
A realistic and robust decision on the alternatives requires the consideration of two conflicting objectives: maximizing the combined goodness measure and minimizing the sensitivity measure. Zarghami et al. (2008) proposed a measure that considers both of these objectives in the evaluation of alternatives. That measure was proposed based on the distance from negative ideal point and was named the composite measure of goodness. This section tries to extend the composite measure of goodness based on the distance from both negative and positive ideal points as follows:
19
Suppose that m alternatives are ranked with respect to the n criteria and the orness degree of the DM is
. Also, suppose that
F (X i , ) and S (X i , ) are the combined goodness
measure and the sensitivity measure of the ith alternative when the orness degree of DM is
.
Consider point ( Fmax , Smin ) as a positive ideal point where Fmax is equal to
maxF (X i , ) | i 1,..., m and S min is equal to min S (X i , ) | i 1,..., m . Also consider the point ( Fmin , Smax ) as a negative ideal point where Fmin min F (X i , ) | i 1,..., m and
S max max S (X i , ) | i 1,..., m . The combined goodness measure and the sensitivity measure of each alternative as the point (F (X i , ), S (X i , )), i 1,
, m is also considered. Then, for
simplicity, the commonly used metric L1 is used to measure the distance from the ideal points to the points (F (X i , ), S (X i , )), i 1,
, m . The extended composite measure of goodness, i ,
which is easy to calculate for each of the alternatives, is defined as follows:
i
(F (X i , ) Fmin ) (1 )(S max S (X i , )) , i 1,..., m (Fmax F (X i , )) (1 )(S (X i , ) S min )
(31)
where 0 and (1 ) 0 are the weighting coefficients for the combined goodness measure and the sensitivity measure, respectively. If the combined goodness measure of the projects is more important to the DM than the sensitivity measure, or vice versa, the weighting coefficients allow the DM to choose which measure to focus on. It is clear that i 0 . A set of alternatives can be ranked according to the ascending order of i . When one of the points, for instance, (F (X k , ), S (X k , )) , is the positive ideal point, the highest rank is assigned to this alternative and then the other alternatives are ranked via (31). In the case where 0 , if for one point S (X i , ) S min and in the case 1 , for one point Fmax F (X i , ) , then the highest rank is assigned to these alternatives and then the other
alternatives are ranked via (31). By considering the distances from positive and negative ideal points, the above measure provides a more reliable ranking of the alternatives. The next section will apply the extended composite goodness measure for ranking water resource projects. 20
5. Case Study This section applies our proposed methods to find the best project among 13 water resource projects under construction in the northwestern region of Iran. These 13 projects were evaluated and compared with each other with respect to seven criteria. The list of the projects, the criteria and the normalized evaluation of each of the projects with respect to the criteria were presented in Table IV in Zarghami et al. (2008). All the projects were in the same region and were under construction in the initial study period. The amount of water that they supplied differs; however, in the decision making process, the amount of water was represented by the economic criteria of "benefit over cost ratio" and "the amount of water allocation for prior usage". Thus, the amount of water itself did not have an impact on optimality.
5.1 Combined Measure of Goodness 0.25
0.75
the combined goodness measure 0.25
0.75
The combined measures of goodness obtained by the two approaches show that the best selection under the degree of orness 0.25 is the Shahriar project, while the best choice under the degree of orness 0.75 is the Bijar project. The correlation coefficient between the two results is 0.97 for the pessimistic view and 0.98 for the optimistic view. This demonstrates that the values obtained by the two approaches are approximately the same. 5.2 Sensitivity Measure This part provides a comparison between the projects from the viewpoint of sensitivity measure. The sensitivity measures of the 13 projects obtained using the two methods under the orness levels 0.25 and 0.75 are shown in Figure 2. -------------------------------------Insert Figures 2a and 2b Here-------------------------------------21
Figure 2, indicates that the lowest sensitivity measures according to the proposed method are given by the Germichai and Shahriar projects under the degrees of orness 0.25 and 0.75, respectively. While the MD approach has the lowest sensitivity measures for the Shahriar and Kalghan projects under the pessimistic and optimistic attitudes, respectively. The correlation coefficient between the obtained sensitivity measures using the two approaches for 0.25 is 0.726, and for 0.75 , it is 0.524. These outcomes reveal that there is a statistically significant difference between the obtained sensitivity measures using the two approaches. 5.3 General Comparison between Two Approaches A number of interesting facts can be learned by investigating the behavior of the two methods under a changing optimism degree. For this purpose, the combined goodness measure and sensitivity measure of the Sahand project, which were obtained by the two approaches, are portrayed as a function of in the following Figures 3 and 4. ------------------------------------Insert Figure 3 Here-----------------------------------------------------------------------------------------Insert Figure 4 Here-----------------------------------------------------From the detailed analysis of Figure 3, it can be seen that the combined goodness measure of the two methods increases as the degree of orness increases and the rate that the combined goodness measure goes up, decreases (in other words F
decreases) as increases. The
aggregation by the weights of the MD model for any (0,1) brings a larger value of combined goodness measure for the Sahand project than by the new proposed weights. The function FSahand ( ) (MD) has 11 break or non-differentiable points in its domain, while the function FSahand ( ) (CLOWA) has 2 break points. The break points show that the rate of increase in FSahand ( ) (MD) over its domain varies 11 times, while the rate of increase for FSahand ( ) (CLOWA) changes 2 times. In accordance with this fact, Figure 4 illustrates that the function S Sahand ( ) , which is the derivative of FSahand ( ) , cannot be defined for 11 points with the MD
method and for 2 points with the proposed method.
5.4 Final Optimal Solutions
22
Table 3 displays the weights of the MD and CLOWA methods for n=7 and the 0.25
0.75
----------------------------------------------Insert Table 3 Here---------------------------------------This table demonstrates that the MD method produces zero components for the 0.25
0.75 . Since in this problem the DM requires all of the values to be
present in the final score of the projects, for exploring the OWA weights, the CLOWA method is selected. Table 4 displays the combined goodness measures and the sensitivity measures of the projects obtained by CLOWA method for the degree of orness 0.25. The correlation coefficient between the columns F (X ,0.25) and S (X ,0.25) is -0.36465. This demonstrates that maximizing goodness measures and minimizing their sensitivity are conflicting objectives. Therefore, formula (31) is used between these conflicting objectives. The assigned ranking to the projects by formula (31) under different values of the importance factor β is illustrated in the columns of Table 4. ----------------------------------------------Insert Table 4 Here---------------------------------------As seen from Table 4, assigning large values to the non-negative importance factor β demonstrates that the DM has no concern over the risk and will select the Shahriar project as the best project. Furthermore, assigning small values to β indicates that the DM is concerned with the risk and will choose the Germichai project as the best project. Spearman's rank correlation is used at the 0.05 level to determine the strength of the relationship between the obtained rankings from (31) for the cases β equal to 0, 0.25, 0.75 and 1. The obtained results are given in Table 5. ----------------------------------------------Insert Table 5 Here---------------------------------------As can be seen from Table 5, the rankings of the alternatives for the first three cases β = 0, 0.25 and 0.75, are approximately the same, whereas the ranks change considerably for the case β=1. These results reveal that including the sensitivity measure in the evaluation of each project can considerably affect the ranking of the projects. For a final decision on the alternatives, the exact value of β is determined through repeated interactions with the DM. After this process, the DM selected β = 0.25. According to 23
the resulted ranks for β = 0.25, the Germichai project would be the best choice. The ranks of the other projects are illustrated in column 5 of Table 4. These ranks take into account both the combined goodness and sensitivity measures and are more reliable than the obtained ranks by using only one of them.
6. Conclusions This paper proposed two complementary linear ordered weighted averaging (CLOWA) models for generating the OWA weight vectors with distinct and positive components for any orness level (0,1) . An application of the obtained OWA weights is for use in the aggregation process of the problems as the DM requires all of the values with different importance to be present in the final score of the alternatives.The optimal simplex tableaux of the models in the general case, for any level of orness and n, and the compact forms of the optimal weights were also presented. It was also noted that a realistic decision on the alternatives requires the consideration of both the combined goodness measure and the sensitivity measure. To this end, this paper obtained the combined goodness measure and the sensitivity analysis models for two linear OWA models: the MD approach and the CLOWA method. This led us to obtain further properties of the linear OWA models. It was also demonstrated that obtaining the combined goodness measure and the sensitivity analysis models in the CLOWA method were easier than in the MD approach. Therefore, in some situations, such as having several DMs with different orness levels in the decision making process, it was recommended that the CLOWA method be used for exploring the OWA weights. To share both conflicting characteristics—the combined goodness measure and sensitivity measure—in the evaluation of alternatives, this paper also extended the idea of the composite measure of goodness based on the distance from both positive and negative ideal points. Finally, the proposed methods were applied in a water resource management problem. References Amin, G.R., & Emrouznejad, A. (2006). An extended minimax disparity to determine the OWA operator weights. Computers & Industrial Engineering, 50, 312-316.
24
Bazarra, M.S., Jarvis, J.J., & Sherali, H.D. (2010). Linear Programming and Network Flows, (4th ed.). New York: John Wiley & Sons, (Chapter 3). Chaji. A.R. (2017). Analytic Approach on Maximum Bayesian Entropy Ordered Weighted Averaging Operators, Computers & Industrial Engineering, 105, 260-264. Emrouznejad, A., & Amin, G.R. (2010). Improving minimax disparity model to determine the OWA operator weights. Information Sciences,180, 1477–1485. Filev, D., & Yager, R. (1995). Analytic properties of maximum entropy OWA operators Information Sciences,85, 11–27. Fullér, R., & Majlender, P. (2003). On obtaining minimal variability OWA operator weights. Fuzzy Sets and Systems. 136, 203-215. Liu, X. (2011). A Review of the OWA Determination Methods: Classification and Some Extensions. In: Yager, R.R., Kacprzyk, J., & Beliakov, G. (Eds.), Recent Developments in the Ordered Weighted Averaging Operators, (pp. 49-90). Berlin: Springer-Verlag. Liu, X. (2012). Models to determine parameterized ordered weighted averaging operators using optimization criteria. Information Sciences, 190, 27–55. O’Hagan, M. (1988). Aggregating template rule antecedents in real-time expert systems with fuzzy set logic. In: Grove, P. (ed.), Proc. 22nd Annual IEEE Asilomar Conference on Signals, Systems and Computers, (pp. 681–689). California. Tohidi, G., & Khodadadi, M. (2015). The owa weights of improved minimax disparity model. International Journal of Intelligent Systems, 30, 781–797. Wang, Y.M., & Chin, K.S. (2011). The use of OWA operator weights for cross-efficiency aggregation. Omega. 39, 493–503. Wang, Y.M., Luo, Y., & Liu, X. (2007a). Two new models for determining OWA operator weights. Computers & Industrial Engineering, 52, 203–209. Wang, Y.M., Luo, Y., & Hua, Z. (2007b). Aggregating preference rankings using OWA operator weights. Information Sciences, 177, 3356–3363.
25
Wang, Y.M., & Parkan, C. (2005). A minimax disparity approach for obtaining OWA operator weights. Information Sciences, 175, 20-29. Xu, Z., & Da, Q.L. (2003). An overview of operators for aggregating information. International Journal of Intelligent Systems, 18, 953–969. Yager, R.R. (1988). On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Transaction on Systems, Man and Cybernetics, Part B, 18, 183–190. Yager, R.R. (1996). Quantifier guided aggregation using OWA operators. International Journal of Intelligent Systems, 11(1), 49–73. Yari. G., & Chaji. A. R (2012). Maximum Bayesian entropy method for determining ordered weighted averaging operator weights, Computers and Industrial Engineering, 63, 338–342.
Zarghami, M., Szidarovszky, F., & Ardakanian, R. (2008). Sensitivity Analysis of the OWA operator. IEEE Transaction on Systems, Man and Cybernetics, Part B, 38(2), 547- 552.
26
Tables Table 1 1 Optimal simplex tableau of model (11) (0 3 ) *
xB
zi ci
01( n 2)
sn 2
s1
s2
3n( n1) n( n1)( n1)
3( n 2)
18
n ( n 1)
n( n 1)( n 1)
1
1
sn 1
b=B b
6 n(n 1)(n 1)
6 n(n1)
w1
2 n n 1
w2
n 1
5 n
2
w3
8 n n 1
3
wn 1
2 n 4 ( n 1)
( n 1)
wn
( n 1)( n 2) 2( n 1)
n sn
sn
3n( n1) n( n1)( n1)
n
1 3
2
I ( n 2)( n 2)
3( n2) n( n1)
18 n ( n 1)( n 1)
27
6 n ( n 1)( n 1)
6 n (n 1)
Table 2 1 1 Optimal simplex tableau of model (11) ( 3 2 )
*
xB
zi ci
01( n 2)
sn 1
s3
s2 2
6
n( n1)
sn
b = B
12( n 2)
12( n 2)
6
n( n 1)( n 1)
n( n 1)( n 1)
n ( n 1)
1 b
6 n( n1)
w1
( n 2)( n 1) n ( n 1)
s1
w2
4(2n ) n ( n 1)
s1 2
w3
2(72n ) n ( n 1)
s 1 3
wn 1
2( n 5) n ( n 1)
s1 ( n 1)
wn
2( n2) n( n1)
s1 n
n 4
6 2 n
s1
n
I ( n 2)( n 2)
6 n( n1)
12( n 2) n ( n 1)( n 1)
28
12( n 2) n ( n 1)( n 1)
6 n ( n 1)
6(1 2 ) n ( n 1)
Table 3 The weights of MD and CLOWA methods for the degrees of orness 0.75 and 0.25 0.75
MD CLOWA 0.25
MD CLOWA
w1
w2
w3
w4
w5
w6
w7
0.3095
0.2524
0.195
0.138
0.080
0.0238
0
0.4375
0.16071
0.091
0.082
0.074
0.066
0.058
w1
w2
w3
w4
w5
w6
w7
0
0.0238
0.080
0.138
0.195
0.2524
0.3095
0.058
0.066
0.074
0.082
0.091
0.1607
0.4375
29
Table 4 Ranks of the projects with the new extended composite measure (31) under different values of β Measures
obtained
by
CLOWA method Projects
Different values of the non-negative importance factor β
( 0.25) F ( X ,0.25)
S ( X ,0.25)
β=0
β = 0.25
β = 0.75
β=1
Sahand
0.3973
1.189
13
13
13
4
Shahriar
0.5743
0.37
2
2
1
1
Ghalechai
0.2682
0.55
3
4
5
13
Kalghan
0.3497
0.59
4
5
4
7
Germichai
0.4785
0.31
1
1
2
2
Givi
0.2982
0.79
8
8
12
12
Taleghan
0.4275
0.59
4
3
3
3
Talvar
0.3289
0.83
10
10
9
8
Galabar
0.3249
0.81
9
9
8
9
Sanghsiah
0.314
0.73
7
7
7
10
Soral
0.3041
0.69
6
6
6
11
Siazakh
0.3917
1.04
12
12
11
5
Bijar
0.3796
0.99
11
11
10
6
30
Table 5 Result of Spearman's test at the level of 0.05 0 0.25
0
0.25
0.75
1
1
0.99
0.922
0.121
1
0.934
0.187
1
0.335
0.75
1
1
31
Combined measures of goodness
Figures 0.7 0.6 0.5 0.4 0.3 CLOWA
0.2
MD
0.1 0
Figure 1a: The combined measures of goodness obtained by the two approaches
Combined measures of goodness
(Degree of orness=0.25) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
CLOWA MD
Figure 1b: The combined measures of goodness obtained by two approaches (Degree of orness=0.75)
32
Sensitivity Measures
1.4 1.2 1 0.8 0.6 CLOWA
0.4
MD
0.2 0
Figure 2a: The sensitivity measures obtained by two approaches (Degree of orness=0.25)
Sensitivity Measures
1.4 1.2 1 0.8 0.6 0.4
CLOWA
0.2
MD
0
Figure 2b: The sensitivity measures obtained by two approaches (Degree of orness=0.75)
33
Figure 3: Combined measures of goodness for the Sahand project by the two methods
34
Figure 4: Sensitivity measures of the Sahand project by the two methods
35
Highlights
We propose two complementary linear OWA (CLOWA) models to generate the OWA weights.
The optimal simplex tableaux of CLOWA models in the general case are also obtained.
CLOWA models produce OWA weight vectors with distinct and positive components.
We obtain combined goodness measures and their sensitivity by CLOWA and MD models. The paper compares CLOWA and MD methods and extends the composite goodness
measure.
36
S0360-8352(17)30343-1 http://dx.doi.org/10.1016/j.cie.2017.08.001 CAIE 4848
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
31 August 2013 14 July 2017 1 August 2017
Please cite this article as: Khodadadi, M., Tohidi, G., Zarghami, M., Sensitivity Analysis of the Ordered Weighted Averaging Operator via Linear Models, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/ j.cie.2017.08.001
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Sensitivity Analysis of the Ordered Weighted Averaging Operator via Linear Models M. Khodadadi a,*, G. Tohidi b M. Zarghami c a
b
Department of Mathematics, Urmia Branch, Islamic Azad University, Urmia, Iran
Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran c
Faculty of Civil Engineering, University of Tabriz, Tabriz 51664, Iran
Corresponding author: Maryam Khodadadi, [email protected], [email protected], Tel.: +98 9141464712; fax: +98 443 3372683 Address: Maryam Khodadadi, Department of Mathematics, Urmia Branch, Islamic Azad University, Urmia, Iran.
Ghasem Tohidi: [email protected] Mahdi Zarghami: [email protected]
*
Corresponding author. Tel.: +98 9141464712; fax: +98 443 3372683
Address: Maryam Khodadadi, Department of Mathematics, Urmia Branch, Islamic Azad University, Urmia, Iran
Email addresses: [email protected], [email protected], (M.khodadadi),
[email protected] (G. Tohidi) [email protected] (M.Zarghami)
1
Sensitivity Analysis of the Ordered Weighted Averaging Operator via Linear Models Abstract The efficient computation of the weights in an ordered weighted averaging (OWA) operator plays an important role in the successful design and application of the OWA operator. In some applications, an OWA weight vector with positive and distinct components for any desired orness level is preferable. This paper proposes two complementary linear OWA (CLOWA) models for determining the weights and a compact form of the optimal weights in the general case (i.e., for any level of orness and n . The proposed models produce a unique OWA weight vector with distinct and positive components for any orness level (0,1) . In addition to a combined goodness measure, the sensitivity analysis on the outputs of the OWA operator with respect to the optimism degree of the decision maker (DM) is important. This study obtains the combined goodness measure and sensitivity analysis models for two linear methods: the minimax disparity (MD) and new proposed models. Then, to assign reliable ranks to the alternatives in the presence of two conflicting objectives—maximizing the combined goodness measure and minimizing its sensitivity to the optimism degree of the DM—the composite measure of goodness is extended. The proposed methods are applied in a water resource management problem. Keywords: Ordered weighted averaging (OWA) operator, Complementary linear OWA (CLOWA) models, Minimax disparity (MD) approach, Sensitivity analysis, Optimism degree of DM, Combined goodness measure.
1. Introduction
The ordered weighted averaging (OWA) operator, as one of the important and efficient aggregation operators, was primarily introduced by Yager (1988). Since its introduction, it has attracted great attention among researchers and has been used in a variety of fields such as multicriteria and group decision making, fuzzy system and control, database query management, data
2
mining, forecasting, approximate reasoning, neural networks, market research, expert systems, and data envelopment analysis (Xu and Da, 2003; Liu, 2011; Wang and Chin, 2011). An n-dimensional OWA operator is a mapping F :[0,1]n [0,1] , with the associated weighting vector W (w1 , w2 ,..., wn )T having the properties: w 1 w 2
w n 1; 0 w i 1, i 1,..., n .
The combined goodness measure assigned to the vector X (a1 , a2 ,
, an ) by this operator is as
follows: n
F (X ) F (a1 , a2 ,..., an ) w j b j
(1)
j 1
The components of the vector X are the evaluations of an alternative with respect to the n criteria. Before multiplying the components of the vector X by the weights, these components should be arranged in non-increasing order. Therefore, in formula (1), b j is the jth largest element among the components of the vector X . The degree of orness associated with this operator (Liu, 2012) is defined as: n j wj . j 1 n 1 n
orness( w)
(2)
A critical and main step in OWA operator research is the determination of OWA operator weights. For this purpose, Yager (1996) proposed a method for obtaining OWA weighting vectors based on fuzzy linguistic quantifiers (FLQ) (Liu, 2011). O'Hagen (1988) suggested a nonlinear model to determine OWA weights based on the measure of dispersion with a predefined degree of orness. Later, Fullér and Majlender (2003) suggested the minimum variance (MV) approach to obtain OWA weights. Yari and Chaji (2012) proposed another nonlinear OWA model for obtaining the OWA weights based on a given prior OWA vector. This model was called the maximum Bayesian entropy OWA (MBEOWA) model. By using the Lagrange multiplier method, Chaji (2017) found an analytic solution for obtaining the MBEOWA model weights.
3
Wang and Parkan (2005) proposed the first linear programming model called the minimax disparity (MD) approach to generate OWA operator weights. The weights of the MD model were obtained in a compact form by Wang and Chin (2011). Amin and Emroznejad (2006) and Wang et al. (Wang et al., 2007a, 2007b) improved the MD approach. Emrouznejad and Amin (2010) proposed an improved minimax disparity model to determine OWA operator weights that minimizes the sum of the deviation between two distinct OWA weights. Tohidi and Khodadadi (2015) obtained the optimal simplex tableau of the improved minimax disparity model in the general case (i.e., for any level of orness and n ) and then for the desired n , they introduced n 1 optimal basic feasible solutions (BFSs) of the model. Sometimes it is more preferred to use an OWA weight vector with positive and distinct components for any desired orness level (0,1) in the aggregation process. For this reason, this paper divides the interval [0,1] into [0,0.5] and [0.5,1] and proposes two linear OWA models for orness levels
[0,0.5] and [0.5,1] , separately. The proposed models generate
OWA weight vector with positive and distinct components for any orness level (0,1) . It also demonstrates that the optimal weights of the two models satisfy the dual property and that the models are complementary. Therefore, we can solve one of the models and obtain the optimal weight vectors for both of them. The paper also obtains the optimal simplex tableaux of the models in the general case (i.e., for any level of orness and n). Then, for the desired n, it introduces the unique optimal OWA weight vectors of the models. To investigate the aforementioned properties among other linear OWA models, their weights in the compact form should be available. The compact form of the OWA weights is only available for a few linear OWA models. The MD model and the improved minimax disparity model are two examples of linear models for which the compact forms of the optimal weights are available. The improved minimax disparity model produces alternative optimal OWA weights (Tohidi and Khodadadi; 2015). Under alternative optimal OWA weights, the unique aggregated value cannot be obtained for the alternatives. Since the MD model produces unique optimal OWA weights and the compact form of its optimal weights is obtained, we have selected the MD model in this study to compare it to our produced method. Furthermore, the comparison of the
4
CLOWA method and other OWA determination methods is presented as the future scope of this study. In the outcome provided by the OWA operator, two factors are important. These two factors are the combined goodness measure and the sensitivity analysis on the outputs of the OWA operator with respect to the optimism degree of the decision maker (DM). The sensitivity analysis provides deeper insight into the final decision based on risk-taking, risk-neutral, and risk-averse attitudes. Larger values of the combined goodness measure and smaller values of the sensitivity to the optimism degree of the DM are required for making robust decision on the alternatives. In this paper, we also obtain and compare the combined goodness measure and the sensitivity analysis models for two linear OWA models: the MD approach and the new CLOWA method. A similar study on this subject was performed for the FLQ approach and the MV method by Zarghami et al. (2008). In addition, to share both conflicting characteristics—the combined goodness measure and sensitivity measure—in the evaluation of alternatives, this paper extends the idea of the composite measure of goodness by using the distance from both positive and negative ideal points. The rest of this paper is organized as follows: Section 2 presents a brief description about the MD model and the compact form of its optimal weights. The two new OWA models with their weights in compact forms are given in Section 3. Section 4, which is composed of four subsections, discusses the combined goodness measure and sensitivity measure for two linear methods: the MD approach and the CLOWA method. It also compares the combined goodness measure and sensitivity measure obtained by the two methods. Finally, it extends the formula of the composite measure of goodness. Section 5 applies the proposed methods to a watershed management problem. Finally, Section 6 concludes the study. 2. Preliminaries Wang and Parkan (2005) proposed the MD model (3) to obtain OWA operator weights under the desired orness level.
5
min d n
s.t.
w i 1
i
1,
n
(n i )w i 1
i
(n 1) , 0 1,
(3)
d w i w i+1 d , i 1,..., n 1, w i 0, i 1,..., n .
Model (3) minimizes the maximum disparity of the OWA operator weights. It is noteworthy that in model (3) and all of the OWA models, w1 , w2 ,..., wn are the variables to be determined after solving the models. Additionally, the degree of orness is a parameter that should be determined by the DM before solving the models. Theorem 1 presents the weights of model (3) in a compact form. Theorem 1: For a given orness degree (0.5,1) , there exists an integer k n such that
wi w1 (i 1)d for i 1,..., k
(4)
and
wi 0 for i k 1,..., n ,
(5)
where k , w1 , and d are determined by
k Min (n , INT[3n 1 3 (n 1)]),
(6)
4(k 1) 6n 6 (n 1) , k (k 1)
(7)
w1
and d
2(kw 1 1) , k (k 1)
(8)
where INT[x] is the function that rounds x down to the nearest integer. For a given orness degree (0, 0.5) , according to the dual property, the weights can be obtained (Wang and Chin, 2011).
3. New OWA Aggregation Weights
6
Some decision-making problems consider the optimistic/pessimistic view of the DM in the aggregation process. In such aggregation problems, the DM may prefer all the scores to be present in the final score for any desired orness level (0,1) . This section tries to obtain OWA weight vectors with positive and distinct components for any desired orness level (0,1) . Using these OWA operator weights for the aggregation process, all the scores will be present in the final score. To obtain the aforementioned OWA weights, they are found such that they have the properties 0 w n ... w 2 w 1 1 or 0 w 1 w 2
w n 1 . If the OWA weights have
the properties 0 w n ... w 2 w 1 1 then the orness level must be satisfied in the inequality
0.5 1 (Filev and Yager, 1995). In addition, to obtain the weights with the properties w n 1 the orness level must be satisfied in the inequality 0 0.5 (Filev
0 w 1 w 2
and Yager, 1995). To this end, the interval [0,1] is subdivided into [0,0.5] and [0.5,1] . Then, two linear models (9) and (10) are proposed for the orness levels [0,0.5] and [0.5,1] , respectively, as follows: max n
w
s.t.
i 1
i
1,
n
1 (n 1) , 0 , 2 i 1 w i 1 w i i 1,..., n 1,
(n i )w
(9)
i
w1 , w i 0, i 1,..., n .
and max n
s.t.
w i 1
i
1,
n
1 1, 2 i 1 wi wi 1 i 1,..., n 1,
(n i) w
i
(n 1) ,
(10)
wn , wi 0, i 1,..., n. 7
Models (9) and (10) generate a unique aggregation weight vector that has the maximum discrimination between the two adjacent weights. The main characteristic of models (9) and (10) is the use of simple linear models to generate a unique optimal OWA weight vector with positive and distinct components for any given level of orness. We will show that the weights obtained by models (9) and (10) satisfy the dual property. Therefore, we can solve model (9) for
[0,0.5]
and obtain the optimal weight vector of model (10) according to the dual property and vice versa. The theoretical properties of models (9) and (10) are as follows: Theorem 2: Model (9) for any (0, 0.5) and model (10) for any (0.5,1) generate a unique optimal weight vector with positive and distinct components. Proof: We prove that model (9) for n
3 generates a unique aggregation weight vector with
positive and distinct components. The proof for model (10) will be similar. First we obtain the optimal simplex tableaux of model (9). In the obtained tableaux, we will observe that z i c i 0 for all nonbasic variables. This will indicate that model (9) has a unique optimal solution (Bazarra et al., 2010). Additionally, the obtained optimal simplex tableaux will show that all the components of the optimal OWA weight vector are distinct and are in the basis with positive values. To obtain the optimal simplex tableaux, reformulate model (9) so that it is in standard format. By converting the inequality constraints of model (9) to the equality constraints, we obtain model (11): max n
s.t.
w 1, i 1
i
n
1 (n 1) , 0 , 2 i 1 wi 1 wi si 1 0 i 1,..., n 1,
(n i ) w
(11)
i
w1 s1 0, wi 0, si 0, i 1,..., n.
8
We claim for 0 1
3
and 1 1 the optimal basic vectors and their optimal bases for 3 2
model (11) are, respectively, as follows:
,aw ,as ,a ], 0
1 3
1 3
1 2
xB*
(w1, w2, , w(n 1), wn ,sn , ), B1
[aw ,aw ,...,aw
xB*2
(w1, w2, , w(n 1), wn ,s1, ), B2
[aw1 ,aw2 ,...,awn 1 ,awn ,as1 ,a ],
1
1
n 1
2
n
n
where the columns of matrices B 1 and B 2 , i.e., awi , i 1,..., n , asn , a and awi , i 1,..., n , as , a , 1 are the corresponding coefficients of the basic variables xB* and xB* in the constraints matrix of 2
1
model (11). To prove this claim, the simplex tableaux of model (11) corresponding to bases B 1 and B 2 are shown in Tables 1 and 2, respectively: ----------------------------------------------Insert Table 1 Here------------------------------------------------------------------------------------------Insert Table 2 Here--------------------------------------------where I is the (n 2) (n 2) identity matrix and the columns of Tables 1 and 2 are
aSi , i
1, aS , i
1,2,..., n
i
2,..., n and b . These columns are obtained by solving the following
systems of equations (12), (13), and (14), respectively (Bazarra et al., 2010):
BaS
i
1,2,..., n
aS i
2,..., n,
i
BaS
i
Bb
aS i
i
1,
(12) (13)
b.
(14)
In addition, the values corresponding to the nonbasic variables in rows zi ci of Tables 1 and 2 are
zS
i
obtained
cS
i
cBaS
i
by
cS i i
the
zS
formulas
i
cS
i
cBaS
i
cS i i
2,..., n (Bazarra et al., 2010), where cB and cS i i
1,2,..., n 1,
,n
1
and
1, n are the
coefficients of the basic variables and nonbasic variables in the objective function of model (11). To check that B 1 and B 2 are the optimal bases of model (11), it is sufficient to show that the 9
inverses of B 1 and B 2 exist and B11b 0, B 21b 0 and z i c i 0 for all nonbasic variables (Bazarra et al., 2010). It is easy to determine that the matrices B 1 and B 2 have inverses. Tables 1 and 2 show that B11b 0, B 21b 0 and z i c i 0 for all nonbasic variables. These facts indicate that B 1 and B 2 are the optimal bases of model (11) for 0 1
3
and 1 1 , respectively. 3 2
Additionally, it can be seen from Tables 1 and 2 that z i c i 0 for all nonbasic variables; therefore, B 1 and B 2 are the unique optimal bases of model (11) (Bazarra et al., 2010). The components of vectors x B* and x B* , obtained from the last columns of Tables 1 and 2, are 1
2
shown in formulas (15) and (16). * w ,w ,...,w ,...,w ,w , s , T ; 0 1 , xB 1 2 n 1 n n i 1 3
(15)
where w 1 , w 2 2 , …, w i i , …, w n 1 (n 1) , w n n s n , s n 1 3 and
6 . n (n 1)
* w ,w ,...,w ,...,w ,w , s , T ; 1 1 , xB 1 2 n 1 n 1 i 2 3 2
(16)
where w 1 s1 , w 2 s1 2 , …, w i s1 i , …, w n 1 s1 (n 1) , s1
w n s1 n ,
6(1 2 ) 6 2 and . n (n 1) n
Formulas (15) and (16) demonstrate that w 1,w 2 ,...,w n 1 ,w n , in the components of x B* and x B* 1
2
(0, 1 3] and
are positive and distinct from each other for any given level of orness [ 1 , 1 ) , respectively. 3
2
In a similar way, we can easily indicate that the following vectors x B* and x B* and the bases 3
4
B3 and B4 are the unique optimal basic variables and the unique optimal bases of model (10) for
1 2 and 2 1, respectively. 2 3 3
10
x B* 3 (w 1 ,w 2 ,...,w i ,...,w n 1 ,w n , s n , )T ;
w1 wn (n 1) ,
where
wn sn , sn
1 2 , 2 3
(17)
w 2 w n (n 2) ,…, wi wn (n i) ,…, wn1 wn ,
4 6 6(2 1) and . n(n 1) n
x B* 4 (w 1 ,w 2 ,...,w i ,...,w n 1 ,w n , s1 , )T ;
2 1, 3
(18)
where w1 n s1 , w 2 (n 1) ,…, wi (n (i 1)) ,…, wn , s1 3 2 and
B3 [aw1 , aw2 ,..., awn1 , awn , asn , a ];
6(1 ) . n(n 1)
1 2 , 2 3
and B4 [aw1 , aw2 ,..., awn1 , awn , as1 , a ];
(19)
2 1. 3
(20)
Formulas (17) and (18) reveal that w 1,w 2 ,...,w n 1 ,w n , in the components of x B* and x B* 3
also positive and distinct from each other for any given level of orness
4
are
(0.5, 2 ] and 3
[ 2 ,1) , respectively. 3
The facts mentioned above complete the proof. Theorem 3: The OWA operator weight vector W (w 1 ,w 2 ,...,w n ) obtained by model (9) for T
0 0.5 and model (10) for 0.5 1 satisfies the following properties:
1) If orness(W )=1, then W (1,0,0,...,0)T ; 2) If orness(W )=0, then W (0,0,...,0,1)T ; 3) If orness(W )=0.5, then W
( 1 , 1 ,..., 1 )T . n
n
n
Proof: Formulas (15), (16), (17), and (18) introduce the compact forms of the optimal OWA weights for the CLOWA models.
1 into formula (18). By substituting
For orness(W )=1, it is sufficient to substitute into formula (18) we achieve W (1,0,0,...,0)T . 11
1
0 into (15), the obtained weight
For orness(W )=0, consider formula (15). By substituting vector will be W (0,0,...,0,1)T .
For orness(W )=0.5, similar to the above process, it is sufficient to substitute
0.5 into
( 1 , 1 ,..., 1 )T .
formula (16) or (17) and obtain the weight vector W
n
n
n
An OWA model satisfies the dual property, if it produces an optimal OWA weight vector
W (w 1 ,w 2 ,...,w n ) *
*
W (u1* ,u 2* ,
*
* T
for a given level of orness and an optimal OWA weight vector
,u n* )T for the level of orness 1 and vice versa, where u i* w n* i 1 for each
i 1, 2,..., n . See Emrouznejad and Amin (2010) as an example of an investigation of the dual property for improving minimax disparity model. Theorem 4 introduces the dual property for the CLOWA models. It states that models (9) and (10) satisfy the dual property with each other. This is why the models are complementary. Theorem 4: If we obtain an optimal OWA weight W given level of orness
, then W (u1* ,u 2* ,
*
(w 1 ,w 2 ,...,w n ) *
*
* T
from model (9) for a
,u n* )T will be the optimal weight vector of model
, and vice versa, where u i* w n* i 1 for each
(10) for the level of orness 1
i
1,..., n.
Proof: Consider
1 3
in the interval [0, ] and suppose W * (w 1* ,w 2* ,...,w n* )T is the optimal weight vector
of model (9) for the degree of orness model (10) for the degree of orness 1
and W (u1* ,u 2* ,
,u n* )T is the optimal weight vector of
. According to (15), the components of W
*
are as
follows:
w 1*
6 12 18 6(n 1) 6 1 3 . , w 2* , w 3* ,…, w n* 1 , w n* n (n 1) n (n 1) n (n 1) n (n 1) (n 1)
Since 1
2 3
belongs to interval [ ,1] , the components of W should be obtained from formula
(18). By substituting 1
for
in formula (18), we obtain
12
u1*
6 6(n 1) * 6(n 2) 12 6 1- 3 , u 2* , u3 ,…, u n* 1 , u n* . n (n 1) n 1 n (n 1) n (n 1) n (n 1)
By comparing the components of W and W , we find that u i* w n* i 1 for each i *
This completes the proof for the case
1,..., n.
1 3
[0, ] . The proof for the other cases is similar.
Theorems 3 and 4 show that models (9) and (10) generate valid OWA operator weights. The next section uses the OWA weight vectors produced by the MD and CLOWA methods for the sensitivity analysis of the alternatives with respect to the optimism degree of the DM.
4. Sensitivity Analysis by Linear OWA Models In the previous sections, the compact forms of the OWA weights for the MD and CLOWA models were introduced. By substituting the compact forms of the weights into (1), it is observed that for a fixed n the combined goodness measure of the alternatives is affected by two factors, the performance of the alternatives with respect to the criteria and the optimism degree of the DM. To investigate these two factors in the combined goodness measure of the alternatives, F (X ) is considered in (1) as F (X , ) . It is important to determine which of the mentioned factors has more effect on the combined goodness measure of the alternatives. To determine this, the partial derivative of the combined goodness measure is obtained with respect to the optimism degree of the DM (i.e.
F ( X , ) ) and is represented by S (X , ) . This measure
analyzes the sensitivity of the combined goodness measure of the alternatives with respect to the optimism degree of the DM. A high value of combined goodness measure and a low value of sensitivity measure for an alternative are preferable. A high value of sensitivity measure for an alternative indicates that the value of combined goodness measure of the alternative can vary wildly with different optimism degrees. If the values of the combined goodness measure of two alternatives are the same, then the alternative with the smaller value of sensitivity measure is preferable. The next subsections present the F (X , ) and S (X , ) for the CLOWA and MD methods and then compare the results of the two methods.
13
4.1 Sensitivity Analysis by the CLOWA Models
Using the compact form of the OWA weights from formulas (15), (16), (17), and (18), the combined measures of goodness and the sensitivity measures for the proposed models (9) and (10), under three cases are as follows: 1. (0 1 ) 3 After substituting the compact form of the OWA weights into (1), for a fixed n, since the weights are functions of , F (X ) is considered as F (X , ) .
F (X ) F (a1 , a2 ,..., an ) w 1b1 w 2b2 w i bi F (X , ) (
6 )b1 n (n 1)
(
6 i )bi n (n 1)
(
w nb n 6 (n 1) 6 )b n 1 ( (1 3 ))b n n (n 1) (n 1)
6 i 6 ( )bi ( (1 3 ))b n (n 1) i n ( n 1) n 1
S (X , )
F (X , ) 6 ( )b1 n (n 1)
(
6i )bi n (n 1)
(
(21)
6(n 1) 6 )bn 1 ( 3)bn (22) n (n 1) (n 1)
The sensitivity measure, S (X , ) , in (22) is found through the partial derivative of F (X , ) in (21) with respect to .
The following points show the quality of dependence of S (X , ) on bi for i 1,
2 F 6i 0, i 1, bi n(n 1)
2 F 3(1 n) 0, bn (n 1)
If b1 bn , then S (X , )
,n
, n 1,
F 0 , b1 b2
bn iff S (X , )
F 0.
As seen, when the degree of orness increases, any increase in bi , i 1,
, n 1 increases the
amount of S (X , ) , while any increase in bn decreases the amount of S (X , ) .
14
With a simple check, it can be shown that the compact form of the weights for the intervals [ 1 , 1 ] and [ 1 , 2 ] are the same. Thus, F (X , ) and S (X , ) for the interval [ 1 , 2 ] 3 2 2 3 3 3
are as follows: 2.
1 2 3 3
F (X ) F (a1 , a2 ,..., an ) w 1b1 w 2b2 w i bi F (X , ) (
6 2 6(1 2 ) )b1 n n (n 1)
S (X , )
F (X , ) 6 12 ( )b1 n n (n 1)
(
w nb n
6(1 2 ) 6 2 i )bi n n (n 1)
(
6 12i ( )bi n n (n 1)
6(1 2 ) 6 2 n )bn (23) n n (n 1)
6 12n ( )b n n n (n 1)
6(n 2i 1) ( )bi n (n 1) i 1 n
If i
2 F n 1 0 , then bi 2
If i
2 F n 1 0 (For odd value of n) , then bi 2
If i
2 F n 1 0 , then bi 2
If b1 bn , then S (X , ) 3.
F 0 , b1 b2
bn iff S (X , )
(24)
F 0
2 1, 3 6n (1 ) 6(n 1)(1 ) 3 2)b1 ( )b 2 n (n 1) n (n 1) 6(1 ) ( )bn n (n 1)
F (X , ) (
15
(
6( n i 1)(1 ) )bi n (n 1)
(25)
S (X , )
F (X , ) 6n 6(n 1) (3 )b1 ( )b 2 n (n 1) n (n 1)
(
6(n i 1) )bi n (n 1)
n 6 3(n 1) 6(n i 1) ( )bn ( )b1 bi n (n 1) (n 1) n (n 1) i 2
(26)
2 F 3(n 1) 0. b1 (n 1) 2 F 6(n i 1) 0, i 2, bi n(n 1) If b1 bn , then S (X , )
,n.
F 0 , b1 b2
bn iff S (X , )
F 0.
The above points indicate that when the degree of orness increases, any increase in b1 increases the value of S (X , ) , while any increase in the other bi , i 2,
, n decreases the
value of S (X , ) .
4.2 Sensitivity Analysis by the MD Method By substituting the optimal weights of model (3) into formula (1), we can obtain F (X , ) and S (X , ) for (0, 0.5) and (0.5,1) as follows: 1. [0, 0.5] 4(k 1) 6n 6(1 )(n 1) )b n k (k 1) 4(k 1) 6n 6(1 )(n 1) 3(k 1) 6n 6(1 )( n 1) ( 2i )b n i k (k 1) k (k 1)(k 1) 4(k 1) 6n 6(1 )(n 1) 3( k 1) 6n 6(1 )( n 1) ( 2(k 1) )b n k 1 k (k 1) k (k 1)(k 1) F (X , ) (
where k can be obtained as follows (Wang and Chin, 2011):
16
(27)
For i 0,1, 2,..., n 3 , if [
i i 1 n2 , ) , then k i 2 ; also, if [ , 0.5] , then 3n 3 3n 3 3n 3
k n.
S (X , )
F (X , ) k 1 6(n 1) (k 1 2i ) b n i (k 1)k (k 1) i 0
2 F k 1 0. If i , then bn i 2
2 F k 1 0 (For odd value of k ). If i , then bn i 2
If i
If
(28)
2 F k 1 0. , then bn i 2
bnk 1 bn ,
S (X , )
then
S (X , )
F (X , ) 0,
bnk 1 bnk 2
bn
iff
F (X , ) 0.
2. [0.5,1]
4(k 1) 6n 6 (n 1) )b1 k (k 1) 4(k 1) 6n 6 (n 1) 3(k 1) 6n 6 (n 1) ( 2(i 1) )bi k (k 1) k (k 1)(k 1) 4(k 1) 6n 6 (n 1) 3(k 1) 6n 6 ( n 1) ( 2(k 1) )b k k (k 1) k (k 1)(k 1) F (X , ) (
(29)
where k can be determined as follows (Wang and Chin, 2011): If [0.5,
2n 1 i i 1 ] , then k n , and for i 0,1, 2,..., n 3 , if (1 ) [ , ) , then 3n 3 3n 3 3n 3
k i 2.
By differentiating (29), we find that:
17
S (X , )
F (X , ) k 6(n 1) (k 1 2i ) bi i 1 ( k 1) k ( k 1)
(30)
Additionally, we obtain the following points:
If i
2 F k 1 0. , then bi 2
If i
2 F k 1 0 (For odd value of k ). , then bi 2
If i
2 F k 1 0. , then bi 2
If b1 bk , then S (X , )
F (X , ) 0 , b1 b2
bk iff S (X , )
F (X , ) 0.
4.3 Comparison
As mentioned previously, the combined goodness measure and the sensitivity measure can be considered as functions of the vector X and . For a fixed input vector X , functions F (X , ) and S (X , ) become functions of and are represented by FX ( ) and S X ( ) . By comparing the combined goodness measure and sensitivity measure of the
previously mentioned linear models, the following conclusions can be made.
The F (X , ) and S (X , ) of the two approaches have a linear behavior with respect to the optimism degree of the DM.
For a fixed input vector X and desired n, the number of break points of the obtained function FX ( ) by the MD model is 2n 3 , while the number of break points of the obtained function FX ( ) by the CLOWA models is 2. This indicates that FX ( ) , which is obtained by the proposed method, has a more uniform movement than the FX ( ) obtained by the MD model in [0, 1] as increases to one.
For a desired n, the value of S X ( ) cannot be calculated in 2n 3 points for the MD model and 2 points for the new method.
18
In both cases, S X ( ) does not depend on the optimism degree, and it is a discontinuous piecewise-linear function of .
Obtaining the two functions FX ( ) and S X ( ) in the MD method is more difficult than in the proposed method.
The proposed models produce a unique OWA weight vector with distinct and positive components for any orness level (0,1) . Therefore, in aggregation by the new proposed weights, none of the scores are ignored in the combined goodness measure of the alternatives, while the MD model produces some zero components for some orness levels. Hence, in the combined goodness measure obtained by this method not all the scores are present.
When
There are several DMs with different orness levels in the decision making process.
The case study is an important problem and it requires more attention by the DM.
The DM has a risk concern, etc.
In addition to F (X , ) , it is better to consider S (X , ) in the evaluation of the alternatives. The value of S (X , ) obtained by the MD model cannot be calculated for more orness degrees. According to this point and the above points for exploring the OWA weights, the proposed method is recommended. 4.4
Extending the composite measure of goodness
A realistic and robust decision on the alternatives requires the consideration of two conflicting objectives: maximizing the combined goodness measure and minimizing the sensitivity measure. Zarghami et al. (2008) proposed a measure that considers both of these objectives in the evaluation of alternatives. That measure was proposed based on the distance from negative ideal point and was named the composite measure of goodness. This section tries to extend the composite measure of goodness based on the distance from both negative and positive ideal points as follows:
19
Suppose that m alternatives are ranked with respect to the n criteria and the orness degree of the DM is
. Also, suppose that
F (X i , ) and S (X i , ) are the combined goodness
measure and the sensitivity measure of the ith alternative when the orness degree of DM is
.
Consider point ( Fmax , Smin ) as a positive ideal point where Fmax is equal to
maxF (X i , ) | i 1,..., m and S min is equal to min S (X i , ) | i 1,..., m . Also consider the point ( Fmin , Smax ) as a negative ideal point where Fmin min F (X i , ) | i 1,..., m and
S max max S (X i , ) | i 1,..., m . The combined goodness measure and the sensitivity measure of each alternative as the point (F (X i , ), S (X i , )), i 1,
, m is also considered. Then, for
simplicity, the commonly used metric L1 is used to measure the distance from the ideal points to the points (F (X i , ), S (X i , )), i 1,
, m . The extended composite measure of goodness, i ,
which is easy to calculate for each of the alternatives, is defined as follows:
i
(F (X i , ) Fmin ) (1 )(S max S (X i , )) , i 1,..., m (Fmax F (X i , )) (1 )(S (X i , ) S min )
(31)
where 0 and (1 ) 0 are the weighting coefficients for the combined goodness measure and the sensitivity measure, respectively. If the combined goodness measure of the projects is more important to the DM than the sensitivity measure, or vice versa, the weighting coefficients allow the DM to choose which measure to focus on. It is clear that i 0 . A set of alternatives can be ranked according to the ascending order of i . When one of the points, for instance, (F (X k , ), S (X k , )) , is the positive ideal point, the highest rank is assigned to this alternative and then the other alternatives are ranked via (31). In the case where 0 , if for one point S (X i , ) S min and in the case 1 , for one point Fmax F (X i , ) , then the highest rank is assigned to these alternatives and then the other
alternatives are ranked via (31). By considering the distances from positive and negative ideal points, the above measure provides a more reliable ranking of the alternatives. The next section will apply the extended composite goodness measure for ranking water resource projects. 20
5. Case Study This section applies our proposed methods to find the best project among 13 water resource projects under construction in the northwestern region of Iran. These 13 projects were evaluated and compared with each other with respect to seven criteria. The list of the projects, the criteria and the normalized evaluation of each of the projects with respect to the criteria were presented in Table IV in Zarghami et al. (2008). All the projects were in the same region and were under construction in the initial study period. The amount of water that they supplied differs; however, in the decision making process, the amount of water was represented by the economic criteria of "benefit over cost ratio" and "the amount of water allocation for prior usage". Thus, the amount of water itself did not have an impact on optimality.
5.1 Combined Measure of Goodness 0.25
0.75
the combined goodness measure 0.25
0.75
The combined measures of goodness obtained by the two approaches show that the best selection under the degree of orness 0.25 is the Shahriar project, while the best choice under the degree of orness 0.75 is the Bijar project. The correlation coefficient between the two results is 0.97 for the pessimistic view and 0.98 for the optimistic view. This demonstrates that the values obtained by the two approaches are approximately the same. 5.2 Sensitivity Measure This part provides a comparison between the projects from the viewpoint of sensitivity measure. The sensitivity measures of the 13 projects obtained using the two methods under the orness levels 0.25 and 0.75 are shown in Figure 2. -------------------------------------Insert Figures 2a and 2b Here-------------------------------------21
Figure 2, indicates that the lowest sensitivity measures according to the proposed method are given by the Germichai and Shahriar projects under the degrees of orness 0.25 and 0.75, respectively. While the MD approach has the lowest sensitivity measures for the Shahriar and Kalghan projects under the pessimistic and optimistic attitudes, respectively. The correlation coefficient between the obtained sensitivity measures using the two approaches for 0.25 is 0.726, and for 0.75 , it is 0.524. These outcomes reveal that there is a statistically significant difference between the obtained sensitivity measures using the two approaches. 5.3 General Comparison between Two Approaches A number of interesting facts can be learned by investigating the behavior of the two methods under a changing optimism degree. For this purpose, the combined goodness measure and sensitivity measure of the Sahand project, which were obtained by the two approaches, are portrayed as a function of in the following Figures 3 and 4. ------------------------------------Insert Figure 3 Here-----------------------------------------------------------------------------------------Insert Figure 4 Here-----------------------------------------------------From the detailed analysis of Figure 3, it can be seen that the combined goodness measure of the two methods increases as the degree of orness increases and the rate that the combined goodness measure goes up, decreases (in other words F
decreases) as increases. The
aggregation by the weights of the MD model for any (0,1) brings a larger value of combined goodness measure for the Sahand project than by the new proposed weights. The function FSahand ( ) (MD) has 11 break or non-differentiable points in its domain, while the function FSahand ( ) (CLOWA) has 2 break points. The break points show that the rate of increase in FSahand ( ) (MD) over its domain varies 11 times, while the rate of increase for FSahand ( ) (CLOWA) changes 2 times. In accordance with this fact, Figure 4 illustrates that the function S Sahand ( ) , which is the derivative of FSahand ( ) , cannot be defined for 11 points with the MD
method and for 2 points with the proposed method.
5.4 Final Optimal Solutions
22
Table 3 displays the weights of the MD and CLOWA methods for n=7 and the 0.25
0.75
----------------------------------------------Insert Table 3 Here---------------------------------------This table demonstrates that the MD method produces zero components for the 0.25
0.75 . Since in this problem the DM requires all of the values to be
present in the final score of the projects, for exploring the OWA weights, the CLOWA method is selected. Table 4 displays the combined goodness measures and the sensitivity measures of the projects obtained by CLOWA method for the degree of orness 0.25. The correlation coefficient between the columns F (X ,0.25) and S (X ,0.25) is -0.36465. This demonstrates that maximizing goodness measures and minimizing their sensitivity are conflicting objectives. Therefore, formula (31) is used between these conflicting objectives. The assigned ranking to the projects by formula (31) under different values of the importance factor β is illustrated in the columns of Table 4. ----------------------------------------------Insert Table 4 Here---------------------------------------As seen from Table 4, assigning large values to the non-negative importance factor β demonstrates that the DM has no concern over the risk and will select the Shahriar project as the best project. Furthermore, assigning small values to β indicates that the DM is concerned with the risk and will choose the Germichai project as the best project. Spearman's rank correlation is used at the 0.05 level to determine the strength of the relationship between the obtained rankings from (31) for the cases β equal to 0, 0.25, 0.75 and 1. The obtained results are given in Table 5. ----------------------------------------------Insert Table 5 Here---------------------------------------As can be seen from Table 5, the rankings of the alternatives for the first three cases β = 0, 0.25 and 0.75, are approximately the same, whereas the ranks change considerably for the case β=1. These results reveal that including the sensitivity measure in the evaluation of each project can considerably affect the ranking of the projects. For a final decision on the alternatives, the exact value of β is determined through repeated interactions with the DM. After this process, the DM selected β = 0.25. According to 23
the resulted ranks for β = 0.25, the Germichai project would be the best choice. The ranks of the other projects are illustrated in column 5 of Table 4. These ranks take into account both the combined goodness and sensitivity measures and are more reliable than the obtained ranks by using only one of them.
6. Conclusions This paper proposed two complementary linear ordered weighted averaging (CLOWA) models for generating the OWA weight vectors with distinct and positive components for any orness level (0,1) . An application of the obtained OWA weights is for use in the aggregation process of the problems as the DM requires all of the values with different importance to be present in the final score of the alternatives.The optimal simplex tableaux of the models in the general case, for any level of orness and n, and the compact forms of the optimal weights were also presented. It was also noted that a realistic decision on the alternatives requires the consideration of both the combined goodness measure and the sensitivity measure. To this end, this paper obtained the combined goodness measure and the sensitivity analysis models for two linear OWA models: the MD approach and the CLOWA method. This led us to obtain further properties of the linear OWA models. It was also demonstrated that obtaining the combined goodness measure and the sensitivity analysis models in the CLOWA method were easier than in the MD approach. Therefore, in some situations, such as having several DMs with different orness levels in the decision making process, it was recommended that the CLOWA method be used for exploring the OWA weights. To share both conflicting characteristics—the combined goodness measure and sensitivity measure—in the evaluation of alternatives, this paper also extended the idea of the composite measure of goodness based on the distance from both positive and negative ideal points. Finally, the proposed methods were applied in a water resource management problem. References Amin, G.R., & Emrouznejad, A. (2006). An extended minimax disparity to determine the OWA operator weights. Computers & Industrial Engineering, 50, 312-316.
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Bazarra, M.S., Jarvis, J.J., & Sherali, H.D. (2010). Linear Programming and Network Flows, (4th ed.). New York: John Wiley & Sons, (Chapter 3). Chaji. A.R. (2017). Analytic Approach on Maximum Bayesian Entropy Ordered Weighted Averaging Operators, Computers & Industrial Engineering, 105, 260-264. Emrouznejad, A., & Amin, G.R. (2010). Improving minimax disparity model to determine the OWA operator weights. Information Sciences,180, 1477–1485. Filev, D., & Yager, R. (1995). Analytic properties of maximum entropy OWA operators Information Sciences,85, 11–27. Fullér, R., & Majlender, P. (2003). On obtaining minimal variability OWA operator weights. Fuzzy Sets and Systems. 136, 203-215. Liu, X. (2011). A Review of the OWA Determination Methods: Classification and Some Extensions. In: Yager, R.R., Kacprzyk, J., & Beliakov, G. (Eds.), Recent Developments in the Ordered Weighted Averaging Operators, (pp. 49-90). Berlin: Springer-Verlag. Liu, X. (2012). Models to determine parameterized ordered weighted averaging operators using optimization criteria. Information Sciences, 190, 27–55. O’Hagan, M. (1988). Aggregating template rule antecedents in real-time expert systems with fuzzy set logic. In: Grove, P. (ed.), Proc. 22nd Annual IEEE Asilomar Conference on Signals, Systems and Computers, (pp. 681–689). California. Tohidi, G., & Khodadadi, M. (2015). The owa weights of improved minimax disparity model. International Journal of Intelligent Systems, 30, 781–797. Wang, Y.M., & Chin, K.S. (2011). The use of OWA operator weights for cross-efficiency aggregation. Omega. 39, 493–503. Wang, Y.M., Luo, Y., & Liu, X. (2007a). Two new models for determining OWA operator weights. Computers & Industrial Engineering, 52, 203–209. Wang, Y.M., Luo, Y., & Hua, Z. (2007b). Aggregating preference rankings using OWA operator weights. Information Sciences, 177, 3356–3363.
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Wang, Y.M., & Parkan, C. (2005). A minimax disparity approach for obtaining OWA operator weights. Information Sciences, 175, 20-29. Xu, Z., & Da, Q.L. (2003). An overview of operators for aggregating information. International Journal of Intelligent Systems, 18, 953–969. Yager, R.R. (1988). On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Transaction on Systems, Man and Cybernetics, Part B, 18, 183–190. Yager, R.R. (1996). Quantifier guided aggregation using OWA operators. International Journal of Intelligent Systems, 11(1), 49–73. Yari. G., & Chaji. A. R (2012). Maximum Bayesian entropy method for determining ordered weighted averaging operator weights, Computers and Industrial Engineering, 63, 338–342.
Zarghami, M., Szidarovszky, F., & Ardakanian, R. (2008). Sensitivity Analysis of the OWA operator. IEEE Transaction on Systems, Man and Cybernetics, Part B, 38(2), 547- 552.
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Tables Table 1 1 Optimal simplex tableau of model (11) (0 3 ) *
xB
zi ci
01( n 2)
sn 2
s1
s2
3n( n1) n( n1)( n1)
3( n 2)
18
n ( n 1)
n( n 1)( n 1)
1
1
sn 1
b=B b
6 n(n 1)(n 1)
6 n(n1)
w1
2 n n 1
w2
n 1
5 n
2
w3
8 n n 1
3
wn 1
2 n 4 ( n 1)
( n 1)
wn
( n 1)( n 2) 2( n 1)
n sn
sn
3n( n1) n( n1)( n1)
n
1 3
2
I ( n 2)( n 2)
3( n2) n( n1)
18 n ( n 1)( n 1)
27
6 n ( n 1)( n 1)
6 n (n 1)
Table 2 1 1 Optimal simplex tableau of model (11) ( 3 2 )
*
xB
zi ci
01( n 2)
sn 1
s3
s2 2
6
n( n1)
sn
b = B
12( n 2)
12( n 2)
6
n( n 1)( n 1)
n( n 1)( n 1)
n ( n 1)
1 b
6 n( n1)
w1
( n 2)( n 1) n ( n 1)
s1
w2
4(2n ) n ( n 1)
s1 2
w3
2(72n ) n ( n 1)
s 1 3
wn 1
2( n 5) n ( n 1)
s1 ( n 1)
wn
2( n2) n( n1)
s1 n
n 4
6 2 n
s1
n
I ( n 2)( n 2)
6 n( n1)
12( n 2) n ( n 1)( n 1)
28
12( n 2) n ( n 1)( n 1)
6 n ( n 1)
6(1 2 ) n ( n 1)
Table 3 The weights of MD and CLOWA methods for the degrees of orness 0.75 and 0.25 0.75
MD CLOWA 0.25
MD CLOWA
w1
w2
w3
w4
w5
w6
w7
0.3095
0.2524
0.195
0.138
0.080
0.0238
0
0.4375
0.16071
0.091
0.082
0.074
0.066
0.058
w1
w2
w3
w4
w5
w6
w7
0
0.0238
0.080
0.138
0.195
0.2524
0.3095
0.058
0.066
0.074
0.082
0.091
0.1607
0.4375
29
Table 4 Ranks of the projects with the new extended composite measure (31) under different values of β Measures
obtained
by
CLOWA method Projects
Different values of the non-negative importance factor β
( 0.25) F ( X ,0.25)
S ( X ,0.25)
β=0
β = 0.25
β = 0.75
β=1
Sahand
0.3973
1.189
13
13
13
4
Shahriar
0.5743
0.37
2
2
1
1
Ghalechai
0.2682
0.55
3
4
5
13
Kalghan
0.3497
0.59
4
5
4
7
Germichai
0.4785
0.31
1
1
2
2
Givi
0.2982
0.79
8
8
12
12
Taleghan
0.4275
0.59
4
3
3
3
Talvar
0.3289
0.83
10
10
9
8
Galabar
0.3249
0.81
9
9
8
9
Sanghsiah
0.314
0.73
7
7
7
10
Soral
0.3041
0.69
6
6
6
11
Siazakh
0.3917
1.04
12
12
11
5
Bijar
0.3796
0.99
11
11
10
6
30
Table 5 Result of Spearman's test at the level of 0.05 0 0.25
0
0.25
0.75
1
1
0.99
0.922
0.121
1
0.934
0.187
1
0.335
0.75
1
1
31
Combined measures of goodness
Figures 0.7 0.6 0.5 0.4 0.3 CLOWA
0.2
MD
0.1 0
Figure 1a: The combined measures of goodness obtained by the two approaches
Combined measures of goodness
(Degree of orness=0.25) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
CLOWA MD
Figure 1b: The combined measures of goodness obtained by two approaches (Degree of orness=0.75)
32
Sensitivity Measures
1.4 1.2 1 0.8 0.6 CLOWA
0.4
MD
0.2 0
Figure 2a: The sensitivity measures obtained by two approaches (Degree of orness=0.25)
Sensitivity Measures
1.4 1.2 1 0.8 0.6 0.4
CLOWA
0.2
MD
0
Figure 2b: The sensitivity measures obtained by two approaches (Degree of orness=0.75)
33
Figure 3: Combined measures of goodness for the Sahand project by the two methods
34
Figure 4: Sensitivity measures of the Sahand project by the two methods
35
Highlights
We propose two complementary linear OWA (CLOWA) models to generate the OWA weights.
The optimal simplex tableaux of CLOWA models in the general case are also obtained.
CLOWA models produce OWA weight vectors with distinct and positive components.
We obtain combined goodness measures and their sensitivity by CLOWA and MD models. The paper compares CLOWA and MD methods and extends the composite goodness
measure.
36