Notes on properties of the OWA weights determination model

Computers & Industrial Engineering 52 (2007) 533–538 www.elsevier.com/locate/dsw

Notes on properties of the OWA weights determination model Gholam R. Amin

*

Department of Computer Science, Postgraduate Engineering Centre, Islamic Azad University of South Tehran Branch, Tehran 1418765663, Iran Received 4 January 2007; received in revised form 1 March 2007; accepted 2 March 2007 Available online 12 March 2007

Abstract This paper solves the recently open problem related to the OWA weights determination minimax model presented by Amin and Emrouznejad [Amin G. R., & Emrouznejad, A. (2006). An extended minimax disparity to determine the OWA operator weights. Computers & Industrial Engineering, 50, pp. 312–316]. So the contribution of this work is that it explains further the properties of the proposed OWA weights determination minimax model. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: OWA weights determination minimax model; Linear programming; Open problem

1. Introduction The concept of ordered weighted averaging (OWA) operator was introduced by Yager (1988, 1993). It has been used in applications of decision making, expert systems, neural networks, fuzzy system and control, (Filev & Yager, 1995; Pelaez & Dona, 2003; Yager & Kreinovich, 1999). It provides a general class of parametric aggregation operators that includes the min, max, and average, and has shown to be useful for modeling many different kinds of aggregation problems. Several authors have provided interesting results on group decision-making or social choice theory and multi-criteria decision-making with the help of fuzzy sets and OWA operator theory, (Chiclana, Herrera-Viedma, Herrera, & Alonso, 2007). The OWA operator affords a new information aggregation technique and has already aroused considerable research interest (Liu, 2006). For example, in machine scheduling optimization, the OWA operator is particularly useful because the amount of compensation can be adjusted freely, (Yager & Filev, 1994). An important issue related to the theory and application of OWA operators is the determination of the weights of the operators, (Amin & Emrouznejad, 2006; Fuller & Majlender, 2003; O’Hagan, 1988; Wang & Parkan, 2005). For generating the OWA operator weights Wang and Parkan (2005) proposed a linear programming model. By minimizing the maximum distance between any distinct pairs of weights instead of the adjacent weights, Amin and Emrouznejad (2006) were also contributed for determining of OWA operator weights. For further research the paper of those authors (Amin & Emrouznejad, 2006) appended an open *

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0360-8352/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.03.002

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G.R. Amin / Computers & Industrial Engineering 52 (2007) 533–538

problem. By introducing some properties of their model this study solves the open problem appeared in that article. The rest of this paper is organized as follows: Section 2 gives a brief explanation of the OWA operator weights. To prove the open problem appeared in Amin and Emrouznejad (2006), Section 3 proposes the requirement theory and methodology. Section 4 gives a numerical illustration of the method. Finally the conclusion of paper is given in Section 5. 2. OWA operator and its weight An OWA operator of dimension n is a mapping F : Rn ! R with an associated weight vector w = (w1, . . . ,wn)t such that w1 þ . . . þ wn ¼ 1; wi  0

i ¼ 1; . . . ; n

and F ða1 ; . . . ; an Þ ¼

n X

wi bi

i¼1

where bi is the ith largest of a1, . . ., an. Since the weight vector w = (w1, . . ., wn)t has the properties that wi 2 [0, 1] and w1 +    + wn = 1, it is often viewed in decision making (DM) under uncertainty as a kind of probability distribution with wi representing the probability that the DM believes bi will occur. 3. Solving the open problem Recently Amin and Emrouznejad (2006) extended the linear programming model proposed by Wang and Parkan (2005) for determination of the weights of the operators. The following model is their extended model, Amin and Emrouznejad (2006): min d s:t: n 1 X ðn  iÞwi ¼ a ¼ OrnessðwÞ n  1 i¼1

wj  wi þ d  0; wi  wj þ d  0 n X wi ¼ 1

0a1 i ¼ 1; . . . ; n  1; j ¼ i þ 1; . . . ; n

i¼1

wi  0

i ¼ 1; . . . ; n

ð1Þ

As the first property of model (1) the following Lemma shows that under which condition over the Orness (w) = a all the optimal operator weights are equal. Lemma. Orness(w) = a = 0.5 if and only if w ¼ ðw1 ; . . . ; wn Þ ¼ ðn1 ; . . . ; n1 Þ. Proof. Assume d* = 0, that is all the optimal weights are equal, wi ¼ p for i = 1, . . ., n. It implies that n n X X wi ¼ p 1 ¼ pn ¼ 1 i¼1

i¼1

1

So p = n . The other constraints of model (1) conclude wi  wj þ d ¼ p  p þ d ¼ d  0; wj  wi þ d ¼ p  p þ d ¼ d  0 n n X X nðn  1Þ ¼ ðn  1Þa ðn  iÞwi ¼ p ðn  iÞ ¼ pððn  1Þ þ ðn  2Þ þ    þ 2 þ 1Þ ¼ p 2 i¼1 i¼1

G.R. Amin / Computers & Industrial Engineering 52 (2007) 533–538

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Hence having the assumption wi ¼ p for each i = 1, . . ., n the only possible Orness(w) = a is 0.5. Conversely suppose a = 0.5. In this case it is easy to see that the only feasible solution of model (1) is w = (w1, . . ., wn) = (n1, . . ., n1) which is the optimal solution of the model. Now we impose the assumption n = 2k, where k is a positive integer. Denote wi = l1 and wk+i = l2 for i = 1, . . ., k. We are looking for a condition over Orness(w) = a for which w ¼ ðw1 ; . . . ; wk ; wkþ1 ; . . . ; w2 k¼n Þ ¼ ðl1 ; . . . ; l1 ; l2 ; . . . ; l2 Þ is an optimal solution of model (1). We have n X

wi ¼

i¼1

k X

wi þ

i¼1

k X

wkþi ¼ 1

i¼1

whereby the following equation is concluded l1 þ l2 ¼

1 k

ð2Þ

The other constraints of the model give 8  > < l1  l1 þ d if i 2 I 1 ; j 2 I 1 wi  wj þ d ¼ l2  l2 þ d if i 2 I 2 ; j 2 I 2 > : l1  l2 þ d if i 2 I 1 ; j 2 I 2 and 8  > < l1  l1 þ d wj  wi þ d ¼ l2  l2 þ d > : l2  l1 þ d 

if i 2 I 1 ; j 2 I 1 if i 2 I 2 ; j 2 I 2 if i 2 I 1 ; j 2 I 2

Where I1 = {i:1  i  k} and I2 = {i:1 + k  i  2k}. To satisfy the above constraints it is enough to impose d   l1  l2 & d   l2  l1

ð3Þ

The last constraint of model (1) is given n X

ðn  iÞwi ¼ ðn  1Þa ¼ ðn  1ÞOrnessðwÞ

i¼1 n X

ðn  iÞwi ¼

i2I 1

i¼1

l1

X

X X ðn  iÞwi þ ðn  iÞwi ¼ ðn  1Þa

ðn  iÞ þ l2

i2I 1

X

i2I 2

ðn  iÞ ¼ ðn  1Þa

i2I 2

P P Denote u1 ¼ i2I 1 ðn  iÞ and u2 ¼ i2I 2 ðn  iÞ. Using a Calculus software, like Mathematica, we can compute the quantities u1 and u 2 as shown below X u1 ¼ ðn  iÞ ¼ ðn  1Þ þ ðn  2Þ þ . . . ðn  kÞ ¼ 12kð3k  1Þ i2I 1

X ðn  iÞ ¼ ðn  k  1Þ þ . . . 2 þ 1 ¼ 12kðk  1Þ u2 ¼ i2I 2

Therefore l1u1 + l2u2 = (n  1)a epitomizes l1 12kð3k  1Þ þ l2 12kðk  1Þ ¼ ðn  1Þa

ð4Þ

Finally taken under account the open problem’s assumptions the following simple linear equations are obtained

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1 k d  l1  l2 &d  l2  l1 l1 þ l2 ¼ l1 12kð3k

 1Þ þ

l2 12kðk

ð5Þ

 1Þ ¼ ðn  1Þa:

Now consider the following linear programs amin ¼ min a s:t: 1 k d  l1 þ l2  0

l1 þ l2 ¼

d þ l1  l2  0  1Þl1 þ 12kðk  1Þl2  ð2k  1Þa ¼ 0

1 kð3k 2

0a1

ð6Þ

and amax ¼ max a s:t: 1 k d  l1 þ l2  0

l1 þ l2 ¼

d þ l1  l2  0  1Þl1 þ 12kðk  1Þl2  ð2k  1Þa ¼ 0

1 kð3k 2

0  a  1:

ð7Þ

Note that the decision variables of the models are l1, l2, d and a. Also amin > 0; amax > 0. Substituting l1 = k1  l2 in the last constraint of the above models gives us 3k  1  k 2 l2 ¼ ð2k  1Þa: 2 So the optimal values of models (6) and (7) occur when l2 ¼ k 1 and l2 ¼ 0, respectively. Therefore the optimal values of the models are amin ¼

k1 2ð2k  1Þ

amax ¼

3k  1 ; 2ð2k  1Þ

and

respectively. Note that the minimum and maximum possible values for the Orness of OWA operator weights are given above without the need to solve any model. So the following theorems are proved. h Theorem 1. For the given a 2 ðamin ; amax Þ the optimal solution of model (1) can be computed as the compact form w ¼ ðw1 ; . . . ; wk ; wkþ1 ; . . . ; w2k¼n Þ with wi ¼ l1 ; wkþi ¼ l2 and i = 1, . . ., k. Besides if a ¼ amin then l1 = 0, l2 = k1 and if a ¼ amax then l1 = k1, l2 = 0. Theorem 2. Limit amin ¼ 14 and Limit amax ¼ 34. k!þ1

k!þ1

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Table 1 The optimal values of models (6) and (7) n 6 8 10 12 14 16

amin

amax

0.2 0.2143 0.2222 0.2273 0.2308 0.2333

0.8 0.7857 0.7778 0.7727 0.7692 0.7667

4. Numerical illustration In order to illustrate the methodology of obtaining the closed optimal form we give a numerical example containing for n = 6 until n = 16. For example for n = 6 the corresponding LPs are given amin ¼ min a s:t: 1 3 d  l1 þ l2  0 d þ l1  l2  0

l1 þ l2 ¼

12l1 þ 3l2  5a ¼ 0 0a1 l1  0; l2  0 and amax ¼ max a s:t: 1 3 d  l1 þ l2  0

l1 þ l2 ¼

d þ l1  l2  0 12l1 þ 3l2  5a ¼ 0 0a1 l1  0; l2  0: The optimal solutions are l1 ¼ 0; l2 ¼ d ¼ 0:3333; amin ¼ 0:2 and l1 ¼ d ¼ 0:3333, l2 ¼ 0; amax ¼ 0:8, respectively. Table 1 shows the optimal values of the other cases which are obtained without solving the corresponding models. Using Theorem 1, for any convex combination a ¼ kamin þ ð1  kÞamax where 0 < k < 1, the optimal solution of model (1) has a compact mathematical form. This is significant from the computational point of view. 5. Conclusion remarks This study investigated some properties of the extended minimax model corresponding to the ordered weighted averaging (OWA) operator weights. It also solved the open problem of the extended OWA minimax model which is recently proposed. To illustrate the methodology of the paper a numerical example is shown. Acknowledgement The author expresses his appreciation for the suggestions of two anonymous referees and the editor of this journal.

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G.R. Amin / Computers & Industrial Engineering 52 (2007) 533–538

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