A linear programming model for determining ordered weighted averaging operator weights with maximal Yager’s entropy

Computers & Industrial Engineering 57 (2009) 742–747

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A linear programming model for determining ordered weighted averaging operator weights with maximal Yager’s entropy Jian Wu a,*, Bo-Liang Sun a, Chang-Yong Liang b, Shan-Lin Yang b a b

School of Business Administration, Zhejiang Normal University, Jinhua 321004, PR China Department of Management, Hefei University of Technology, Hefei 230009, PR China

a r t i c l e

i n f o

Article history: Received 26 August 2008 Accepted 5 February 2009 Available online 11 February 2009 Keywords: OWA operator Measures of information Linear objective programming Entropy

a b s t r a c t It has a wide attention about the methods for determining OWA operator weights. At the beginning of this dissertation, we provide a briefly overview of the main approaches for obtaining the OWA weights with a predefined degree of orness. Along this line, we next make an important generalization of these approaches as a special case of the well-known and more general problem of calculation of the probability distribution in the presence of uncertainty. All these existed methods for dealing these kinds of problems are quite complex. In order to simplify the process of computation, we introduce Yager’s entropy based on Minkowski metric. By analyzing its desirable properties and utilizing this measure of entropy, a linear programming (LP) model for the problem of OWA weight calculation with a predefined degree of orness has been built and can be calculated much easier. Then, this result is further extended to the more realistic case of only having partial information on the range of OWA weights except a predefined degree of orness. In the end, two numerical examples are provided to illustrate the application of the proposed approach. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The process of information aggregation has a great affect on the development of intelligent systems. Yager (1988) introduced a new information aggregation technique based on the ordered weighted averaging (OWA) operator. One key issue in the theory of the OWA operator is to determine its associated weights (Amin, 2006, 2007; Ahn, 2006; Filev & Yager, 1998; Nettleton & Torra, 2001; Wang Y, Luo, & Liu, 2007). O’Hagan (1988) proposed a maximum entropy approach, which involved a constrained nonlinear optimization problem with a predefined degree of orness as its constraint and the entropy as the objective function. Then, Fullér and Majlender (2001) transformed the maximum entropy model into a polynomial equation which can be solved analytically. Fullér and Majlender (2003) suggested a minimum variance approach to obtain the minimal variability OWA operator weights. Majlender (2005) proposed an approach for obtaining OWA operator weights based on maximal Rényi entropy for a given level of orness and pointed out that the maximum entropy approach and the minimum variance approach are its special cases, respectively. Liu and Chen (2004) proposed the PMEOWA operator and Liu (2006) proposed the MSEOWA operator. Few scholars study the nature of these models and the relationship between them. Moreover, there are two main shortcomings with in all the above approaches: (1) Generally, * Corresponding author. Tel.: +86 551 2904962. E-mail address: [email protected] (J. Wu). 0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2009.02.001

it is quite complex to acquire the solution of a constrained nonlinear optimization problem or a high-order nonlinear algebraic equation (Wang, 2005). (2) All of them are completely based on the assumption of given orness level. As a fact, it may be difficult for the decision maker (DM) to determine his/her orness in some circumstances. For instance, Xu and Da (2003) established an approach by considering the situation where partial weight information is available partially. The DM may also have other type of weight information except a predefined degree of orness (Kim & Ahn, 1999; Park & Kim, 1997). In order to simplify the complicated computation, we introduce Yager’s entropy based on Minkowski metric (Yager, 1995). By analyzing the desirable properties with this measure of entropy, we propose a novel approach to determine the weights of the OWA operator. It is a wide general method which can be specialized into many famous cases, such as the minimax model (Yager, 1993), the MSEOWA operator (Liu, 2006) and the minimum variance approach (Fullér & Majlender, 2003). We then extend it to a linear objective programming (LP) model with a predefined degree of orness. Further, we consider the (LP) model for the more realistic case, which miss a priori information for the desired orness and only has partial weight information as constraint. The dissertation is organized as follows. Section 2 gives a brief overview and makes an important generalization of the main approaches for OWA weights with a priori information for the desired orness. Section 3 proposes a LP model for OWA weights based on Yager’s entropy. In Section 4, two numerical examples are provided. Section 5 concludes the paper.

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2. The overview and generalization of the methods for OWA weights An OWA operator of dimension n is a mapping, OWA : Rn ! R, that has an associated n vector w ¼ ðw1 ; w2 ; . . . ; wn ÞT such that P wj 2 ½0; 1 and nj¼1 wj ¼ 1. Furthermore,

OWAða1 ; a2 ; . . . ; an Þ ¼

n X

wj bj

ð1Þ

j¼1

where bj is the jth largest element of the collection of the aggregated objects a1 ; a2 ; . . . ; an . Clearly, one key point of the OWA operator theory is to determine its associated weights. O’Hagan (1988) suggested a maximum entropy approach, which requires the solution of the following constrained nonlinear optimization problem:

Max dispðWÞ ¼ 

n X

Max Hðp1 ; . . . ; pn Þ ¼  wi ln wi

ð2Þ

i¼1 n X ni s:t: ornessðWÞ ¼ wi ¼ a; n 1 i¼1 n X

wi ¼ 1;

06a61

n X

n X

pi ¼ 1;

wi ln wi

ð3Þ

To resolve model (6), we introduce Lagrangian Multipliers k0 ; k1 ; . . . ; kr ; . . . ; km , in the usual way, and obtain the result

pi ¼ exp f½k0 þ k1 f1 ðxi Þ þ    þ km fm ðxi Þg in which the constants k0 ; k1 ; . . . ; kr ; . . . ; km are determined from

Zðk1 ; . . . ; km Þ ¼

n X

@ ln Z ðk1 ; . . . kr . . . ; km Þ @kr k0 ¼ ln Z ðk1 ; . . . kr . . . ; km Þ

wi 2 ½0; 1; i ¼ ð1; . . . ; nÞ

where xi > xj for i < jði; j ¼ 1; 2; . . . ; nÞ and xi > c > xj . Fullér and Majlender (2003) suggested a minimum variance approach, which minimizes the variance of OWA operator weights under a given level of orness. A set of OWA operator weights with minimal variability could then be generated. Their approach requires the solution of the following mathematical programming problem:

2 n  n 1X 1 1 X 1 Min D2 ðWÞ ¼ wi  ¼ w2  n i¼1 n n i¼1 i n n X i¼1 n X

wi ¼ 1;

ni wi ¼ a; n1

! ð4Þ

06a61

wi 2 ½0; 1; i ¼ ð1; . . . ; nÞ

Recently, Majlender (2005) proposed an approach for obtaining OWA operator weights based on maximal Rényi entropy for a given level of orness. This approach is based on the solution of the following parametric mathematical programming problem: n n X X 1 Max Hc ðWÞ ¼ wci ¼ log2 wci log2 1c i¼1 i¼1

n X

wi ¼ 1;

ð9Þ ð10Þ

Max Hðp1 ; . . . ; pn Þ ¼ 

n X

pi ln pi

ð11Þ

i¼1

s:t: hf1 ðxÞi ¼

n X

pi f1 ðxi Þ

i¼1 n X

pi ¼ 1;

pi P 0; i ¼ 1; . . . ; n

i¼1

i¼1

n X ni s:t: ornessðWÞ ¼ wi ¼ a; n 1 i¼1

ð8Þ

Especially when fr ðxi Þ ¼ 0; r ¼ 2; . . . ; m; i ¼ 1; . . . ; n, we can get

i¼1

s:t: ornessðWÞ ¼

exp f½k1 f1 ðxi Þ þ    þ km fm ðxi Þg

hfr ðxÞi ¼ 

xi wi ¼ c;

wi ¼ 1;

ð7Þ

i¼1

i¼1 n X

pi fr ðxi Þ

pi P 0; r ¼ 1; . . . ; m; i ¼ 1; . . . ; n

i¼1 n X

ð6Þ

i¼1

The weight resolved from model (2) is called that maximum entropy OWA (MEOWA) weights. Liu and Chen (2004) proposed the PMEOWA operator. The problem can be formulated as:

s:t:

pi ln pi

i¼1

i¼1

Max dispðWÞ ¼ 

n X i¼1

s:t: hfr ðxÞi ¼

wi 2 ½0; 1; i ¼ ð1; . . . ; nÞ

n X

To discuss the relationships in the above models, Wu, Liang, and Huang, 2007 considered the models (2)–(5) as a method for setting up faire probability distributions on the basis of partial information. For analyzing the probability distributions, we shall introduce the principle of maximum entropy proposed by Jaynes (1957), which has the important property that no possibility is ignored and assigns positive weight to every situation that is not absolutely excluded by the given information. The quantityx is capable of assuming the discrete values xi ði ¼ 1; 2; . . . ; nÞ. We are not given the corresponding probabilities pi ; all we know is hfr ðxÞi, which is the expectation value of the function fr ðxÞ. To determine the probabilities pi fairly, Jaynes proposed the optimization problem:

!1=ð1cÞ

Obviously, model (11) includes models (2) and (4) as its special ni and f1 ðxi Þ ¼ xi , respectively. So we can find that cases for f1 ðxi Þ ¼ n1 model (2) and model (4) satisfy the principle of maximum entropy. But the above approaches are quite complex for requiring the solution of a constrained nonlinear optimization problem. Furthermore, the computational burden of models (2)–(5) will increase when the DM put more information into these models. To resolve this problem, this paper will propose a linear objective programming (LP) model for determining OWA operator weights by utilizing the Minkowski metric-based measures of entropy introduced by Yager. 3. LP model for OWA weights based on Yager’s entropy

ð5Þ

06a61

wi 2 ½0; 1; i ¼ ð1; . . . ; nÞ

i¼1

From strict monotonicity of the logarithm function, Majlender proved that model (5) includes models (2) and (4) as its special cases for c ¼ 1 and c ¼ 2, respectively.

3.1. Yager’s entropy based on minkowski metric and its properties Yager (1995) developed a measure of neg-entropy as:

Z !1=Z    n  X  1  p  1 ¼ Q Z ðPÞ ¼ D P; ; Z P 1; i ¼ 1; 2;. .. ;n i  n n i¼1

ð12Þ

where the vector ½1=n is the probability distribution pi ¼ 1=n and ½Ij  denotes the probability distribution with pj ¼ 1. It is easy to see that Q Z ðIi Þ ¼ Q Z ðIj Þ ¼ Q Z ðIÞ, the distance to any singleton is the

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J. Wu et al. / Computers & Industrial Engineering 57 (2009) 742–747

same and the maximal value for Q Z ðPÞ occurs when P ¼ ½I. In this case, where P ¼ ½I, we get

 Z !1=Z 1 Q Z ð½IÞ ¼ þ ðn  1Þ n !1=Z n1 1 1þ ¼ ZP1 n ðn  1ÞZ1 

n1 n

Z

H1 ðPÞ ¼ Q 1 ðIÞ  Q 1 ðPÞ ¼

ð13Þ

HZ ðPÞ ¼ Q Z ðIÞ  Q Z ðPÞ

ð14Þ

and then get: rIf Z ! 1, then Q Z ðPÞ ¼ Maxi ½pi   1 and Q Z ðIÞ ¼ 1  1, so n n

HZ ðPÞ ¼ 1  Max ½pi 

ð15Þ

i

Pn

2 i¼1 pi

 1n

1=2

and Q 2 ðIÞ ¼ ð1  1n Þ1=2 ,

so

H2 ðPÞ ¼

!1=2  1=2 n X 1 1 1  p2i  n n i¼1

 n   2ðn  1Þ X pi  1    n n i¼1

ð20Þ

This alternative measure of entropy H1 ðPÞ has the following desirable properties: Theorem 1. Suppose H1 ðPÞ is a measure of entropy and ðrð1Þ; rð2Þ; . . . ; rðnÞÞ is any permutation of ð1; 2; . . . ; nÞ, we can have

So Yager defined a class of measures of entropy as

sIf Z ¼ 2, then Q 2 ðPÞ ¼

If we let z ¼ 1, then the Minkowski metric based entropy H1 ðPÞ can be reformulated as

ð16Þ

H1 ðprð1Þ ; prð2Þ ; . . . ; prðnÞ Þ ¼ H1 ðp1 ; p2 ; . . . ; pn Þ

ð21Þ

This property essentially exhibits the kind of Symmetry, which requires that the value of H1 ðPÞ be indifferent to the indexing of the individual probability values. Theorem 2. The entropy measure H1 ðPÞ should uniquely take its Maximum 2ðn  1Þ=n when all the outcomes in the space are equally likely (pi ¼ 1=n for all i). Proof. Since Q 1 ðPÞ ¼

H1 ðPÞ ¼

 Pn  1 i¼1 pi  n P 0, we can obtain

 n   2ðn  1Þ X p  1 6 2ðn  1Þ  i   n n n i¼1

Utilizing the metric-based measures of entropy HZ ðPÞ and the principle of maximum entropy, we propose an approach for obtaining OWA operator weights, which requires the solution of the following mathematical programming problem:

So maximal entropy Max H1 ðPÞ ¼ 2ðn1Þ occurs only when pi ¼ 1=n n for all i. h

Max HZ ðWÞ ¼ Q Z ðIÞ  Q Z ðWÞ n X xi wi ¼ c; s:t:

Theorem 3. The entropy measure H1 ðPÞ should uniquely take its Minimum 0 when P ¼ ½Ii ; ð½Ii  denotes the probability distribution with pi ¼ 1Þ.

ð17Þ

i¼1 n X

wi ¼ 1;

wi 2 ½0; 1; i ¼ ð1; . . . ; nÞ

i¼1

  P  ¼  ni¼1 pi  1n þ n1 . Proof. Let P ¼ ½Ii , so Q 1 ðIÞ ¼ 1  1n þ n1 n n Assume ðrð1Þ; rð2Þ; . . . ; rðnÞÞ is a permutation of ð1; 2; . . . ; nÞ such that prði1Þ P prðiÞ and from Theorem 1, we can get that

where xi > xj for i < jði; j ¼ 1; 2; . . . ; nÞ and xi > c > xj . In reality, model (17) includes some existed approach as its special cases for different parameter z. For an example, Yager (1993) expressed a measure of entropy as 1  Maxi ½wi , and then proposed a minimax model:

    X n n 1 n  1 X 1 n  1  Q 1 ðIÞ ¼  pi   þ prðiÞ   þ ¼  i¼1  i¼1 n n n n

Min :Maxi ðwi Þ

    n n X 1 n  1  1 X n1  Q 1 ðIÞ ¼  prðiÞ þ prðjÞ   þ prðiÞ þ ¼ prðjÞ   þ  i–j n n n n i–j

n X ni wi ¼ a; s:t: ornessðWÞ ¼ n 1 i¼1 n X

wi ¼ 1;

ð18Þ 06a61

wi 2 ½0; 1; i ¼ ð1; . . . ; nÞ

Without generalization, prðjþ1Þ ; . . . ; prðnÞ 6 1=n, thus

P

i¼1 ni When let Z ! 1 and xi ¼ n1 ; i ¼ 1; 2; . . . ; n, we can get HðwÞ ¼ 1  Maxi ½wi . Obviously, model (17) is equivalent to model (18). In the following, we can see that model (17) include MSEOWA operator and the minimum variance approach when z ¼ 2. Liu (2006) proposed the MSEOWA operator. The problem can be formulated as:

Min D2 ðWÞ ¼ s:t:

n X

! n 1 X 1 w2i  n i¼1 n

xi wi ¼ c;

wi ¼ 1;

prð1Þ ; prð2Þ ; . . . ; prðjÞ P 1=n;

  X j j  n  X X   1 1 p  1 ¼ Q 1 ðPÞ ðprðiÞ  Þ þ  prðiÞ ¼  rðiÞ n n n i¼1 i¼1 i¼jþ1

So Max Q 1 ðPÞ ¼ Q 1 ðIÞ ¼ 2ðn1Þ , then we have n

H1 ðPÞ ¼ Q 1 ðIÞ  Q 1 ðPÞ P Q 1 ðIÞ  MaxQ 1 ðPÞ P 0: This property associated with H1 ðPÞ is that its minimal value occurs at the probability distribution that provides the most certainty amount in a situation. h

ð19Þ

i¼1 n X

suppose

wi 2 ½0; 1; i ¼ ð1; . . . ; nÞ

i¼1

where xi > xj for i < jði; j ¼ 1; 2; . . . ; nÞ and xi > c > xj . P When z ¼ 2, then we have HðWÞ ¼ 1  ni¼1 w2i . So model (17) is equivalent to model (19). We also can find that model (17) and ni ; i ¼ 1; 2; . . . ; n. model (4) are equivalent when z ¼ 2 and xi ¼ n1 So model (17) includes the MSEOWA operator (19) and the minimum variance approach (4), as its special case for z ¼ 2.

Theorem 4. Assume distribution P has pi ¼ 1=n for all i and _ _ _ distribution p has p j ¼ 1=m for allj, then H1 ðp Þ > H1 ðPÞ if m > n. _

Proof. Let pi ¼ 1=n and p > H1 ðPÞj ¼ 1=m, then we have _ 2ðn  1Þ 2 2ðm  1Þ ¼ 2  andH1 ðp Þ > H1 ðPÞ ¼ n n m 2 ¼ 2  ; respectively m

H1 ðPÞ ¼

Since m > n,

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J. Wu et al. / Computers & Industrial Engineering 57 (2009) 742–747 _

Thus H1 ðp Þ > H1 ðPÞ. This property essentially exhibits the kind of Monotonicity, which requires that the entropy value of H1 ðPÞ should increase as n increases. h

Proof. If weighting vector W  ¼ ðw1 ; . . . ; wn ÞT is optimal to LP (24) for a given level of ornessðW  Þ ¼ a, then we will get

Min J ¼

n X ðeþj þ ej Þ

ð25Þ

j¼1

3.2. LP model under a predefined degree of orness Based the measure of entropy H1 ðPÞ and the minimal distance approach (17), we obtain the following multi-objective programming model (MOP):

 n   2ðn  1Þ X wi  1 Max H1 ðWÞ ¼    n n i¼1 n X ni s:t: ornessðWÞ ¼ wi ¼ a; n 1 i¼1 n X

wi ¼ 1;

1  eþj þ ej ¼ 0; eþj P 0; ej P 0; n n X nj  ornessðW  Þ ¼ w ¼ a; 0 6 a 6 1; n 1 j j¼1

s:t: wj 

n X

ð22Þ

wj ¼ 1;

 ^  Þ ¼ 1 Pn ðn  jÞw Because ornessðW j¼1 njþ1 ¼ 1  ornessðW Þ, n1 so we can have

06a61

Min J ¼

wi 2 ½0; 1; i ¼ ð1; . . . ; nÞ

n X ðeþj þ ej Þ

Thus we can obtain the following multi-objective programming model for minimizing the distance its vector from the vector of maximal entropy, pi ¼ 1=n, which is equivalent to model (22).

Min DðWÞ ¼

 n  X  wi  1  n

ð23Þ

i¼1

s:t: ornessðWÞ ¼

n X i¼1

wi ¼ 1;

ni wi ¼ a; n1

1  eþj þ ej ¼ 0; eþj P 0; ej P 0; n n X nj  ^ Þ ¼ 1  ornessðW w ¼ 1  a; 0 6 a 6 1; n 1 j j¼1

s:t: wj 

n X

wj ¼ 1;

wj 2 ½0; 1; j ¼ ð1;    ; nÞ

j¼1

06a61

^  ¼ ðw ; w ; . . . ; w ÞT is an optimal to LP (25) for Obviously, W n n1 1 ornessðWÞ ¼ 1  a. h

wi 2 ½0; 1; i ¼ ð1; . . . ; nÞ

i¼1

3.3. LP model with partial weight information as constraint

Solution to the above minimization problem is found by solving the following linear objective programming (LP) model: n X Min J ¼ ðeþj þ ej Þ

ð24Þ

j¼1

1  eþj þ ej ¼ 0; eþj P 0; ej P 0 n n X nj ornessðWÞ ¼ wj ¼ a; 0 6 a 6 1 n 1 j¼1 s:t: wj 

n X

ð26Þ

j¼1

i¼1

n X

wj 2 ½0; 1; j ¼ ð1; . . . ; nÞ

j¼1

wj ¼ 1;

wj 2 ½0; 1; j ¼ ð1; . . . ; nÞ

j¼1  where, eþ j ; ej are the upper and lower deviation variables of 1=n, respectively. From the LP model (24), we have the following theorems.

The maximum-entropy distribution may be asserted for the positive reason that it is uniquely determined as the one which is maximally noncommittal with regard to missing information, instead of the negative one that there was no reason to think otherwise. As previously pointed, the ornessa is a kind of information of the decision maker. But, the DM may have the other partial weight information in real decision. For an example, Xu and Da (2003) considered the situation where the weight information is available partially and suggested an approach based on the following mathematical programming problem:

Max dispðWÞ ¼  s:t: ornessðWÞ ¼

n X i¼1 n X i¼1

Theorem 5. For an OWA operator weight vector W ¼ ðw1 ; . . . ; wn ÞT determined by the LP (24)

n X

wi ¼ 1;

wi ln wi ni wi ¼ a; n1

ð27Þ 0 6 a 6 1;

wi 2 ½0; 1; i ¼ ð1; . . . ; nÞ;

i¼1

(1) If ornessðWÞ ¼ 1, then W ¼ ð1; 0; . . . ; 0ÞT ; (2) If ornessðWÞ ¼ 0, then W ¼ ð0; 0; . . . ; 1ÞT ; (3) If ornessðWÞ ¼ 0:5, then W ¼ ð1=n; 1=n; . . . ; 1=nÞT . Proof. When ornessðWÞ ¼ 1; W ¼ ð1; 0; . . . ; 0ÞT is the only feasible and therefore optimized solution, which satisfies the orness Pn 1 constraint: ornessðWÞ ¼ n1 i¼1 ðn  iÞwi ¼ 1. The proof of (2) is similar as(1). When ornessðWÞ ¼ 0:5; W ¼ ð1=n; 1=n; . . . ; 1=nÞT is the feasible solution of LP (24) and J ¼ 0, which is the lower bound of the value objective function. Obviously, W ¼ ð1=n; 1=n; . . . ; 1=nÞT is the only optimized solution of LP (24) for ornessðWÞ ¼ 0 and J ¼ 0. h Theorem 6. If weighting vector W  ¼ ðw1 ; . . . ; wn ÞT is optimal to LP ^  ¼ ðw ; w ; . . . ; (24) for a given level of ornessðWÞ ¼ a, then W n n1  T w1 Þ is an optimal to LP (24) for ornessðWÞ ¼ 1  a.

wi 2 H where H is the set of all known weight information. The types of H provided by group members are linearly unequal constraints, which can be constructed by the following forms, (1) (2) (3) (4) (5)

A weak ranking: fwi P wj g; A strict ranking: fwi  wj P ai g; A ranking with multiples: fwi P ai wj g; An interval form: fai 6 wi 6 ai þ ei g; A ranking of differences: fwi  wj P wk  wl g; for j–k–l.

where ai and ei are nonnegative constants. But, model (27) requires the solution of a constrained nonlinear optimization problem, which bears large computational burden. Furthermore, it is difficult for the DM to determine his/her orness in some circumstance. But, the DM may have the partial weight

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J. Wu et al. / Computers & Industrial Engineering 57 (2009) 742–747

information such as w 2 H, where H is the set of all known weight information. To resolve the above problem, we propose the following linear objective programming (LP) model:

Min J ¼

n X ðeþj þ ej Þ

ð28Þ

Example 2. Suppose the DMs have partial weight information considering two cases: Case 1. 0 6 w1 6 0:2; 0:1 6 w2 6 0:3; 0:3 6 w3 6 0:4; 0:1 6 w4 6 0:3; 0 6 w5 6 0:2So,the LP (27) can be written as:

j¼1

Min J ¼

1 s:t: wj   eþj þ ej ¼ 0; eþj P 0; ej P 0 n n X wj ¼ 1; wj 2 ½0; 1; j ¼ ð1; . . . ; nÞ

5 X ðeþj þ ej Þ j¼1

s:t: w1  0:2  eþ1 þ e1 ¼ 0 w2  0:2  eþ2 þ e2 ¼ 0

j¼1

w3  0:2  eþ3 þ e3 ¼ 0

wj 2 H

w4  0:2  eþ4 þ e4 ¼ 0

 where, eþ j ; ej are the upper and lower deviation variables of 1=n, respectively. In the following, we will give two numerical example illustrates the application of the proposed approach.

w5  0:2  eþ5 þ e5 ¼ 0 0 6 w1 6 0:2 0:1 6 w2 6 0:3 0:3 6 w3 6 0:4

4. Numerical example

0:1 6 w4 6 0:3 Example 1. Suppose that the DM give different level of orness:a ¼ 1; 0:9; . . . ; 0:1; 0 and n ¼ 5. We need to determine OWA operator weights with different level of orness. Let a ¼ 0:8, the LP (24) is written as:

Min J ¼

5 X

ðeþj þ ej Þ

j¼1

s:t: w1  0:2  eþ1 þ e1 ¼ 0

0 6 w5 6 0:2 w1 þ w2 þ w3 þ w4 þ w5 ¼ 1 eþj P 0;

So we can obtain the optimum solution of the above LP:

w1 ¼ 0:1696;

w2 ¼ 0:1804;

w5

orness ¼ 0:5:

¼ 0:1696;

w3  0:2  eþ3 þ e3 ¼ 0 w4  0:2  eþ4 þ e4 ¼ 0

Min J ¼

w5  0:2  eþ5 þ e5 ¼ 0 3 2 1 w1 þ w2 þ w3 þ w4 ¼ 0:8 4 4 4 w1 þ w2 þ w3 þ w4 þ w5 ¼ 1 wj P 0; eþj P 0; ej P 0; j ¼ ð1; . . . ; nÞ

¼ 0:5333;

¼ 0:2;

w3

¼ 0:2;

w4

w2  0:2  eþ2 þ e2 ¼ 0 w3  0:2  eþ3 þ e3 ¼ 0 w4  0:2  eþ4 þ e4 ¼ 0

¼ 0:0667;

w5

w5  0:2  eþ5 þ e5 ¼ 0

¼ 0:

0:18 6 w1 6 0:5 0:25 6 w2 6 0:55 0:16 6 w3 6 0:4 0:15 6 w4 6 0:25 0:12 6 w5 6 0:2 w1 þ w2 þ w3 þ w4 þ w5 ¼ 1

Table 1 The OWA operator weights generated by LP model (25). 1.0

0.9

orness ðWÞ ¼ a w1 1.0 0.7 0.0 0.2 w2 w3 0.0 0.1 0.0 0.0 w4 0.0 0.0 w5

5 X ðeþj þ ej Þ j¼1

For different level of orness:a ¼ 1; 0:9; . . . ; 0:1; 0, the OWA operator weights generated by our minimal distance approach are shown in Table 1.We also provide the OWA operator weights generated by maximum entropy approach in Table 2.

W

w4 ¼ 0:1804;

s:t:w1  0:2  eþ1 þ e1 ¼ 0

So we can obtain the optimum solution of the above LP:

w2

w3 ¼ 0:3;

Case 2. 0:18 6 w1 6 0:5; 0:25 6 w2 6 0:55; 0:16 6 w3 6 0:4; 0:15 6 w4 6 0:25; 0:12 6 w5 6 0:2So, the LP (28) can be written as:

w2  0:2  eþ2 þ e2 ¼ 0

w1

ej P 0; j ¼ ð1; . . . ; nÞ

eþj P 0;

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.5333 0.200 0.10 0.0667 0.0000

0.4 0.2 0.2 0.2 0.0

0.3 0.2 0.2 0.2 0.1

0.2 0.2 0.2 0.2 0.2

0.1 0.2 0.2 0.2 0.3

0.0 0.2 0.2 0.2 0.4

0.0000 0.0667 0.10 0.20 0.5333

0.0 0.0 0.1 0.2 0.7

0 0 0 0 1

ej P 0; j ¼ ð1; . . . ; nÞ

So we can obtain the optimum solution of the above LP:

w1 ¼ 0:1939;

w2 ¼ 0:25;

w5

orness ¼ 0:5242:

¼ 0:1748;

w3 ¼ 0:1900;

w4 ¼ 0:1913;

Certainly, the DM can give out other types of weight information according his/her knowledge, which also can be calculated by our linear objective programming model (28).

Table 2 The OWA operator weights generated by maximum entropy approach. W

1.0

orness ðWÞ ¼ a w1 1.0 0.0 w2 0.0 w3 0.0 w4 0.0 w5

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.7105 0.2068 0.0602 0.0175 0.0050

0.5307 0.2565 0.1240 0.0599 0.0289

0.2 0.2574 0.0672 0.1086 0.0706

0.2884 0.2353 0.1920 0.156 0.1277

0.2 0.2 0.12 0.2 0.2

0.1277 0.1566 0.1920 0.2353 0.2884

0.0706 0.1086 0.0672 0.2574 0.3962

0.0289 0.0599 0.1240 0.2565 0.5307

0.0050 0.0175 0.0602 0.2068 0.7105

0 0 0 0 1

J. Wu et al. / Computers & Industrial Engineering 57 (2009) 742–747

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