Uncertain Multi-Attribute Decision Making

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Zeshui Xu

Uncertain Multi-Attribute

Decision Making

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ISBN 978-3-662-45639-2 ISBN 978-3-662-45640-8 (eBook) DOI 10.1007/978-3-662-45640-8

Library of Congress Control Number: 2014958891

Springer Heidelberg New York Dordrecht London © Springer-Verlag Berlin Heidelberg 2015

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Zeshui Xu Business School Sichuan University Chengdu Sichuan China

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v

Preface

Multi-attribute decision making (MADM) (or called multi-objective decision mak-ing with finite alternatives) is an important component of modern decision science. The theory and methods of MADM have been extensively applied to the fields of engineering project, economy, management and military affairs, such as invest-ment decision making, venture capital project evaluation, facility location, bidding, maintenance services, military system efficiency evaluation, development ranking of industrial sectors, comprehensive evaluation of economic performance, etc. Es-sentially, MADM is to select the most desirable alternative(s) from a given finite set of alternatives according to a collection of attributes by using a proper means. It mainly consists of two stages: (1) Collect decision information. The decision information generally includes the attribute weights and the attribute values (ex-pressed as real numbers, interval numbers or linguistic labels), especially, how to determine the attribute weights is an important research topic in MADM; (2) Ag-gregate the decision information through some proper approaches. Currently, four of the most common aggregation techniques are the weighted averaging operator, the weighted geometric operator, the ordered weighted averaging operator, and the ordered weighted geometric operator.

With the increasing complexity and uncertainty of objective things and the fuzzi-ness of human thought, more and more attention has been paid to the investigation on MADM under uncertain environments, and fruitful research results have been achieved over the last decades. This book offers a systematic introduction to the methods for uncertain MADM and their applications to various practical problems. We organize the book as the following four parts, which contain twelve chapters:

Part 1 consists of three chapters (Chaps. 1–3) which introduce the methods for real-valued MADM and their applications. Concretely speaking, Chap. 1 introduces the methods for solving the decision making problems in which the information about attribute weights is completely unknown and the attribute values take the form of real numbers, and applies them to investment decision making in enter-prises and information systems, respectively, military spaceflight equipment evalu-ation, financial assessment in the institutions of higher educevalu-ation, training plane type selection, purchases of fighter planes and artillery weapons, developing new products, and cadre selection. Chapter 2 introduces the methods for MADM in

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vi Preface which the information about attribute weights is given in the form of preferences and the attribute values are real numbers, and gives their applications to the effi-ciency evaluation of equipment maintenance support systems, and the performance evaluation of military administration units. Chapter 3 introduces the methods for decision making with partial attribute weight information and exact attribute values, and applies them to the fire deployment of a defensive battle in Xiaoshan region, the evaluation and ranking of the industrial economic benefits of 16 provinces and municipalities in China, the assessment for the expansion of a coal mine, sorting the order of the enemy’s targets to attack, the improvement of old products, and the alternative selection for buying a house.

Part 2 consists of three chapters (Chaps. 4–6) which introduce the methods for interval MADM and their applications. Concretely speaking, Chap. 4 introduces the methods for the decision making problems in which the attribute weights are real numbers and the attribute values are expressed as interval numbers, and gives their applications to the evaluation of schools of a university, the exploitations of leather industry of a region and a new model of cars of an investment company, and the selection of the robots of an advanced manufacturing company. Chapter 5 intro-duces the methods for the decision making problems in which the information about attribute weights is unknown completely and the attribute values are interval num-bers. Also, these methods are applied to the purchase of artillery weapons, cadre selection of a unit, and investment decision making in natural resources. Chapter 6 introduces the methods for interval MADM with the partial attribute weight infor-mation, and applies them to determine what kind of air-conditioning system should be installed in the library, evaluate anti-ship missile weapon systems, help select a suitable refrigerator for a family, assess the investment of high technology project of venture capital firms, and purchase college textbooks, respectively.

Part 3 consists of three chapters (Chaps. 7–9) which introduce the methods for linguistic MADM and their applications. Concretely speaking, Chap. 7 introduces the methods for the decision making problems in which the information about at-tribute weights is unknown completely and the atat-tribute values take the form of linguistic labels, and applies them to investment decision making in enterprises, the fire deployment in a battle, and knowledge management performance evaluation of enterprises. Chapter 8 introduces the methods for the decision making problems in which the attribute weights are real numbers and the attribute values are linguis-tic labels, and then gives their applications to assess the management information systems of enterprises and evaluate the outstanding dissertation(s). Chapter 9 in-troduces the MADM methods for the problems where both the attribute weights and the attribute values are expressed in linguistic labels, and applies them to the partner selection of a virtual enterprise, and the quality evaluation of teachers in a middle school.

Part 4 consists of three chapters (Chaps. 10–12) which introduce the methods for uncertain linguistic MADM and their applications. In Chap. 10, we introduce the methods for the decision making problems in which the information about at-tribute weights is unknown completely and the atat-tribute values are uncertain lin-guistic variables, and show their applications in the strategic partner selection of

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vii Preface

an enterprise in the field of supply chain management. Chapter 11 introduces the methods for the decision making problems in which the attribute weights are real numbers and the attribute values are uncertain linguistic variables, and then applies them to appraise and choose investment regions in China, and the maintenance ser-vices of manufacturing enterprises. In Chap. 12, we introduce the MADM methods for the problems in which the attribute weights are interval numbers and the at-tribute values are uncertain linguistic variables, and verify their practicality via the evaluation of the socio-economic systems of cities.

This book can be used as a reference for researchers and practitioners working in the fields of fuzzy mathematics, operations research, information science, manage-ment science and engineering, etc. It can also be used as a textbook for postgradu-ate and senior undergradupostgradu-ate students. This book is a substantial extension of the book “Uncertain Multiple Attribute Decision Making: Methods and Applications” (published by Tsinghua University Press and Springer, Beijing, 2004, in Chinese).

This work was supported by the National Natural Science Foundation of China under Grant 61273209.

Zeshui Xu Chengdu October 2014

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ix

Part I

Real-Valued MADM Methods and Their

Applications

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3

Chapter 1 Real-Valued MADM with Weight Information

Unknown

Z.S. Xu, Uncertain Multi-Attribute Decision Making, DOI 10.1007/978-3-662-45640-8_1

Multi-attribute decision making (MADM) is to select the most desirable alternative(s) from a given finite set of alternatives according to a collection of attributes by using a proper means. How to make a decision under the situations where the informa-tion about attribute weights is unknown completely and the attribute values are real numbers? Aim to this issue, in this chapter, we introduce some common operators for aggregating information, such as the weighted averaging (WA) operator, the weighted geometric (WG) operator, the ordered weighted averaging (OWA) op-erator, the ordered weighted geometric (OWG) opop-erator, the combined weighted averaging (CWA) operator, and the combined weighted geometric (CWG) operator, etc. Based on these aggregation operators, we introduce some simple and practical approaches to MADM. We also introduce the MADM methods based on maximiz-ing deviations and information entropy, and with preference information on alterna-tives, respectively. Additionally, we establish a consensus maximization model for determining attribute weights in multi-attribute group decision making (MAGDM). Furthermore, we illustrate these methods in detail with some practical examples.

1.1 MADM Method Based on OWA Operator

1.1.1 OWA Operator

Yager [157] developed a simple nonlinear function for aggregating decision infor-mation in MADM, which was defined as below:

Definition 1.1 [157] Let _{OWA ℜ → ℜ, if}: n

(1.1) 1 2 1 ( , , , )n n j j j OWAω α α α ω b = … =

∑

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4 1 Real-Valued MADM with Weight Information Unknown then the function OWA is called an ordered weighted averaging (OWA) operator, where bj is the j th largest of a collection of the arguments α =i( 1,2, , )i … n, i.e.,

the arguments b jj( =1,2, ., )… n are arranged in descending order: b b1≥ 2≥ ≥... , ,bn

1 2

( , , , )n

ω= ω ω … ω is the weighting vector associated with the function OWA,

1 0, 1,2, , , n _j 1 j j j n ω ω = =

≥ = …

∑

, and ℜ is the set of all real numbers.

The fundamental aspect of the OWA operator is its reordering step. In particular, an argument ai is not associated with a particular weight ωi, but rather a weight ωi

is associated with a particular ordered position i of the arguments α =_i( 1,2, , )i … n , and thus, ωi is the weight of the position i.

Example 1.1 Let ω =(0.4,0.1,0.2,0.3) be the weighting vector of the OWA opera-tor, and ( , , , )7 18 6 2 be a collection of arguments, then

Now we introduce some desirable properties of the OWA operator:

Theorem 1.1 [157] Let ( , , , )α α₁ ₂ …α_n be a vector of arguments, and ( , , , )β β1 2 …βn

be the vector of the elements in ( , , , )α α1 2 …αn , where βj is the j th largest of ( 1,2, , )

i i n

α = … , such that β₁≥β₂ ≥ … ≥β_n, then

Proof Let

where b_j'_{is the j th largest of} _{( 1,2, , )}

i i n

β = … , andbjis thejth largest of

( 1,2, , )

i i n

α = … . Since ( , , , )β β1 2… βn is the vector in which βj(j=1,2,…n)

are arranged in descending order of the elements α =i( 1,2, , )i … n, then b'j =bj, 1,2, ,

j= … n, which completes the proof.

Theorem 1.2 [157] Let ( , , , )α α1 2 …αn and ( , ,..., )α α1' 2' α'n be two vectors of

argu-ments, such that '

i i

α α≥ , for any i, where α1≥α2 ≥ … ≥αn and α1' ≥α2' ≥ ≥... αn',

then Proof Let (7,18,6,2) 0.4 18 0.1 7 0.2 6 0.3 2 9.70 OWAω = × + × + × + × = 1 2 1 2 ( , , , )n ( , , , )n OWAω β β … β =OWAω α α …α ' 1 2 1 ( , ,..., )n n j j j OWAω β β β ω b = =

∑

1 2 1 ( , , , )n n j j j OWAω α α α ω b = … =

∑

' ' ' 1 2 1 2 ( , ,..., )n ( , ,..., )n OWAω α α α ≥OWAω α α α 1 2 1 ( , , , )_n n _{j j} j OWAω α α α ω b = … =

∑

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5 1.1 MADM Method Based on OWA Operator

where bj is the jth largest of α =i( 1,2, , )i … n, and bj' is the jth largest of

'_{( 1,2,..., )}

i i n

α = . Since α₁≥α₂≥ … ≥αn and α α1' ≥ '2≥ ≥... α'n, then bj =αj,

' '_, _1,2,...,

j j

b =α j= n. Also since '

i i

α α≥ , for any i , then '

j j b ≥b, j=1,2, ,… n_{. Thus,} ' 1 1 n n j j j j j j b b ω ω = = ≥

∑

, i.e., ' ' ' 1 2 1 2 ( , ,..., )n ( , ,..., )n OWAω α α α ≥OWAω α α α .

Corollary 1.1 (Monotonicity) [157] Let ( , , , )α α₁ ₂ …αn and ( , ,β β1 2 …, )βn be

any two vectors of arguments, if αi≤βi, for any i, then

Corollary 1.2 (Commutativity) [157] Let ( , , , )β β₁ ₂ … βn be any permutation of

the elements in ( , , , )α α₁ ₂ …αn , then

Theorem 1.3 (Idempotency) [157] Let ( , , , )α α₁ ₂ …αn be any vector of

argu-ments, if α_i=α, for any i, then

Proof Since 1 1 n j j ω = =

∑

, then

Theorem 1.4 [157] Let _{ω ω}₌ *₌_{(1,0, ,0)}_… _{, then} Proof According to Definition 1.1, we have

Theorem 1.5 [157] Let ω ω= _*=(0,0, ,1)… , then

' ' ' ' 1 2 1 ( , ,..., )_n n _{j j} j OWAω α α α ω b = =

∑

1 2 1 2 ( , , , )n ( , , , )n OWAω α α …α ≤OWAω β β …β 1 2 1 2 ( , , , )n ( , , , )n OWAω β β …β =OWAω α α …α 1 2 ( , , , )n OWAω β β …β =α 1 2 1 1 1 ( , , , )n n j j n j n j j j j OWAω α α α ω b ω α α ω α = = = … =

∑

=

∑

=

∑

= *( , , , ) max { }1 2 n i i OWA_ω α α … α = α * ₁ ₂ ₁ 1 ( , , , )n n j j max { }i i j OWA_ω α α α ω b b α = … =

∑

= = *( , , , ) min { }1 2 n _i i OWAω α α … α = α

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6 1 Real-Valued MADM with Weight Information Unknown Proof It follows from Definition 1.1 that

Theorem 1.6 [157] Let ω ω= Ave=__{n n}1 1, , ,… 1_n_, then

Theorem 1.7 [157] Let ( , , , )α α₁ ₂ …αn be any vector of arguments, then

Proof

which completes the proof.

Clearly, the following conclusions also hold:

Theorem 1.8 [157] If ω =j 1, ω =i 0, and i j≠ , then

where bj is the j th largest of a collection of the arguments ( , , , )α α1 2 …αn .

Especially, if j =1, then If j n= , then Theorem 1.9 [158] If ω1=α, ω =i 0, i= … −2, ,n 1, ωn = −1 α, and α ∈[0,1], then Theorem 1.10 [158] (1) If ₁ 1 n α ω = − +α, _i 1 n α ω = − , i ≠1, and α ∈[0,1], then * 1 2 1 ( , , , )n n j j n min { }i i j OWAω α α α ω b b α = … =

∑

= = 1 2 1 1 ( , , , ) Ave n n j j OWA b n ω α α α = … =

∑

*( , , , )₁ ₂ _n ( , , , )₁ ₂ _n _*( , , , )₁ ₂ _n

OWA_ω α α …α ≥OWAω α α …α ≥OWAω α α …α

* 1 2 1 1 1 2 1 1 ( , ,..., )n n j j n j ( , ,..., )n j j OWAω α α α ω b ω b b OWAω α α α = = =

∑

≤

∑

= = * 1 2 1 2 1 1 ( , , , )n n j j n j n n ( , , , )n j j OWAω α α α ω b ω b b OWAω α α α = = … =

∑

≥

∑

= = … 1 2 ( , , , )_n _j OWAω α α …α =b * 1 2 1 2 ( , , , )n ( , , , )n OWAω α α …α =OWA_ω α α …α * 1 2 1 2 ( , , , )n ( , , , )n OWAω α α …α =OWAω α α …α *( , , , ) (1₁ ₂ n ) _*( , , , )₁ ₂ n OWA_ω OWAω α α α … α + −α α α … α 1 2 ( , , , )_n OWAω α α α = …

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Especially, if α =0, then If α =1, then (2) If _i 1 n α ω = − , i n≠ , _n 1 n α ω = − +α, and α ∈[0,1], then Especially, if α =0, then If α =1, then (3) (1 ( )) n α β_n ω = − + +β, α β ∈, [0,1], and α β+ ≤1, then

Especially, if β =0, then (3) reduces to (1); If α =0, then (3) reduces to (2).

Theorem 1.11 [158] 1. If *( , , , ) (11 2 n ) Ave( , , , )1 2 n OWA_ω OWAω α α α …α + −α α α …α 1 2 ( , , , )n OWAω α α α = … 1 2 1 2 ( , , , ) ( , , , ) Ave n n OWAω α α …α =OWAω α α …α *( , , , )1 2 n ( , , , )1 2 n OWA_ω α α … α =OWAω α α …α *( , , , ) (11 2 n ) Ave( , , , )1 2 n OWAω OWAω α α α …α + −α α α …α 1 2 ( , , , )_n OWAω α α α = … 1 2 1 2 ( , , , ) ( , , , ) Ave n n OWAω α α …α =OWAω α α …α *( , , , )1 2 n ( , , , )1 2 n OWAω α α …α =OWAω α α …α 1 (1 ( )) (1 ( )) If , _i , i 2, ,n 1, n n α β α β ω = − + +α ω = − + = … − *( , , , )1 2 n _*( , , , )1 2 n OWA_ω OWAω α α α …α +β α α …α 1 2 1 2

(1 (α β))OWAω_Ave( , , , )α α αn OWAω( , , , )α α αn

+ − + … = … 0, , 1 , , 0, , i i k k i k m m i k m ω <   = ≤ < +  ≥ + 

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8 1 Real-Valued MADM with Weight Information Unknown where k and m are integers, and k m n+ ≤ +1, then

where b_j is the j th largest of a collection of the arguments α =i( 1,2, , )i …n . 2. If

where k and m are integers, and k m n+ ≤ +1, k m≥ +1, then

where bj is the j th largest of α =i( 1,2, , )i … n.

3. If

then

where b_j is the j th largest of α =i( 1,2, , )i … n . 4. If

then

where bj is the j th largest of α =i( 1,2, , )i … n . 1 1 2 1 ( , , , )n k m j j k OWA b m ω α α α + − = … =

∑

0, , 1 , , 2 1 0, , i i k m k m i k m m i k m ω < −   = ₊ − ≤ < +  ≥ +  1 1 2 1 ( , , , ) 2 1 k m n j j k m OWA b m ω α α α + − = − … = +

∑

1 , , 0, , i i k k i k ω =  ≤  _>  1 2 1 1 ( , , , )n k j j OWA b k ω α α α = … =

∑

0, , 1 _, _, ( 1) i i k i k n k ω <   =  _≥  + −  1 2 1 ( , , , ) ( 1) n n j j k OWA b n k ω α α α = … = + −

∑

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1.1.2 Decision Making Method

Based on the OWA operator, in what follows, we introduce a method for MADM:

Step 1 For a MADM problem, let X ={ , , , }x x_{1 2} … x_n be a finite set of alternatives, 1 2

{ , , , }_m

U = u u … u be a set of attributes, whose weight information is unknown

com-pletely. A decision maker (expert) evaluates the alternative x_i with respect to the attribute u_j, and then get the attribute value a_ij. All a i =1,2, , ;_ij( … n j = 1,2,...,m)

are contained in the decision matrix A=( )a_{ij n m}_× , listed in Table 1.1.

In general, there are six types of attributes in the MADM problems, i.e., (1) benefit type (the bigger the attribute values the better); (2) cost type (the smaller the attribute values the better); (3) fixed type (the closer the attribute value to a fixed value αj the better); (4) deviation type (the further the attribute value deviates from

a fixed value αj the better); (5) interval type (the closer the attribute value to a

fixed interval __{q q}₁j, ₂j_

  (including the situation where the attribute value lies in the interval) the better); and (6) deviation interval type (the further the attribute value deviates from a fixed interval __{q q}₁j, ₂j_

  the better). Let I i =i( 1,2, ,6)… denote the

subscript sets of the attributes of benefit type, cost type, fixed type, deviation type, interval type, and deviation interval type, respectively.

In practical applications, the “dimensions” of different attributes may be different. In order to measure all attributes in dimensionless units and facilitate inter-attribute comparisons, here, we normalize each attribute value aij in the decision matrix

( )_{ij n m}

A= a _× using the following formulas:

(1.2) (1.3) or (1.2a) 1 , 1,2, , ; max { } ij ij ij i a r i n j I a = = … ∈ 2 min { } , 1,2, , ; ij i ij ij a r i n j I a = = … ∈ 1 min { } , 1,2, , ; max { } min { } ij _i ij ij ij ij i i a a r i n j I a a − = = … ∈ − u1 u2  um x₁ a₁₁ a₁₂  a1m x₂ a₂₁ a₂₂  a₂_m      xn an1 an2  anm Table 1.1 Decision matrix A

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10 1 Real-Valued MADM with Weight Information Unknown (1.3a) (1.4) (1.5) (1.6) (1.7)

and then construct the normalized decision matrix R=( )rij n m× .

Step 2 Utilize the OWA operator to aggregate all the attribute values r j =ij( 1,2,...,m) of the alternative xi, and get the overall attribute value zi( )ω :

where bij is the j th largest of r lil( 1,2, , )= …m , ω=( , , ,ω ω1 2 …ωm) is the

weight-ing vector associated with the OWA operator, ω ≥j 0, j=1,2, ,… m, 1 1 m j j ω = =

∑

,

which can be obtained by using a proper method presented in Sect. 1.1, or by the normal distribution (Gaussian distribution) based method) [126, 160]:

2 max { } , 1,2, , ; max { } min { } ij ij i ij ij ij i i a a r i n j I a a − = = … ∈ −

{

}

3 1 , 1,2, , ; max ij j ij ij j i a r i n j I a α α − = − = … ∈ −

{

}

{

}

{

}

4 min , 1,2, , ; max min ij j i ij ij j ij j _i ij j i a r a i n j I a a α α α α − = − − = … ∈ − − −

{

}

{

1 2

}

1 2 1 2 1 2 5 max , 1 , ,

max min{ }, max{ }

1, , 1,2,..., ; j j ij ij _j _j i ij j j ij ij ij _i _i _i j j ij q a a q a q q q a a q r a q q i n j I  − −  − _{∉ } _  ₋ ₋ =      _{∈ } _  = ∈

{

}

{

1 2

}

1 2 1 2 1 2 6 max , , ,

max min{ }, max{ }

1, , 1,2,..., ; j j ij ij _j _j i ij j j ij ij ij _i _i _i j j ij q a a q a q q q a a q r a q q i n j I  − −  _{∉ } _  ₋ ₋ =      _{∈ } _  = ∈ 1 2 1 ( ) ( , , , ) m i i i im j ij j z ω OWA r rω r ω b = = … =

∑

2 2 2 2 ( ) 2 ( ) 2 1 , 1,2, , m m m m j j _i m i e _j _m e µ σ µ σ ω − − − − = = = …

∑

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where

The prominent characteristic of the method above is that it can relieve the influ-ence of unfair arguments on the decision result by assigning low weights to those “false” or “biased”’’ ones.

Step 3 Rank all the alternatives x ii( 1,2, , )= … n according to the values zi( )ω

(i = 1,2,...,n) in descending order.

1.1.3 Practical Example

Example 1.2 Consider a MADM problem that an investment bank wants to invest

a sum of money in the best option of enterprises (alternatives), and there are four enterprises x ii( = 1 2 3 4 to choose from. The investment bank tries to evaluate , , , )

the candidate enterprises by using five evaluation indices (attributes) [60]: (1) u1: output value (10 4_{$); (2)}_u

2: investment cost (10 4$); (3) u3: sales volume (10 4$); (4) u4: proportion of national income; (5) u5: level of environmental contamina-tion. The investment bank inspects the performances of last four years of the four companies with respect to the five indices (where the levels of environmental con-tamination of all these enterprises are given by the related environmental protec-tion departments), and the evaluaprotec-tion values are contained in the decision matrix

A=( )aij 4 5× , listed in Table 1.2:

Among the five indices u jj( = 1 2 3 4 5 , u, , , , ) 2 and u5 are of cost type, and the others are of benefit type. The weight information about the indices is also unknown completely.

Considering that the indices have two different types (benefit and cost types), we first transform the attribute values of cost type into the attribute values of benefit type by using Eqs. (1.2) and (1.3), then A is transformed into R=( )rij 4 5×, shown in Table 1.3.

Then we utilize the OWA operator (1.1) to aggregate all the attribute val-ues r jij( = 1 2 3 4 5 of the enterprise x, , , , ) i, and get the overall attribute value

( ) i

z ω (without loss of generality, we use the method given in Theorem 1.10

to determine the weighting vector associated with the OWA operator, and get (0.36,0.16,0.16,0.16,0.16) ω = , here, we take α =0.2): 2 1 1₍₁ _), 1 ₍ ₎ 2 m m m m i m i m µ σ µ = = + =

∑

− 1( ) ( , , , , )11 12 13 14 15 0.36 1.0000 0.16 0.9343 0.16 0.7647 0.16 0.7591 0.16 0.6811 0.8618 z ω =OWA r r r r rω = × + × + × + × + × =

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12 1 Real-Valued MADM with Weight Information Unknown

Finally, we rank all the enterprises x ii( = 1 2 3 4 according to , , , ) zi( )ω (i = 1 2 3 4, , , )

in descending order:

where “  ” denotes “be superior to”, and thus, the best enterprise is x4.

2( ) ( , , , , )21 22 23 24 25 0.36 1.0000 0.16 1.0000 0.16 0.7926 0.16 0.7246 0.16 0.6777 0.8712 z ω =OWA r r r r rω = × + × + × + × + × = 3( ) ( , , , , )31 32 33 34 35 0.36 1.0000 0.16 1.0000 0.16 0.8667 0.16 0.7195 0.16 0.6189 0.8728 z ω =OWA r r r r rω = × + × + × + × + × = 4( ) ( , , , , )41 42 43 44 45 0.36 0.9904 0.16 0.9871 0.16 0.9024 0.16 0.8749 0.16 0.4643 0.8731 z ω =OWA r r r r rω = × + × + × + × + × = x4x3x2 x1

Table 1.2 Decision matrix A

u₁ u₂ u₃ u₄ u₅

x₁ 8350 5300 6135 0.82 0.17

x₂ 7455 4952 6527 0.65 0.13

x₃ 11,000 8001 9008 0.59 0.15

x₄ 9624 5000 8892 0.74 0.28

Table 1.3 Decision matrix R

u₁ u₂ u₃ u₄ u₅

x₁ 0.7591 0.9343 0.6811 1.0000 0.7647

x₂ 0.6777 1.0000 0.7246 0.7926 1.0000

x₃ 1.0000 0.6189 1.0000 0.7195 0.8667

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13 1.2 MAGDM Method Based on OWA and CWA Operators

1.2 MAGDM Method Based on OWA and CWA

Operators

1.2.1 CWA Operator

Definition 1.2 [38, 147] Let WA_{: ℜ → ℜ , if}n

(1.8) where w=( , , , )w w1 2 … w_n is the weight vector of a collection of the arguments

( 1,2, , ) i i n α = … , wj ≥ 0, j=1 2, , , , and … n wj j n =

∑

= 1

1, then the function WA is called a weighted averaging (WA) operator.

Clearly, the basic steps of the WA operator are that it first weights all the given arguments by a normalized weight vector, and then aggregates these weighted argu-ments by addition.

Example 1.3 Let ( , , , )7 18 6 2 be a collection of arguments, and w = ( . , . , . , . )0 4 0 1 0 2 0 3 be their weight vector, then

Definition 1.3 [109] Let CWA: ℜ → ℜ, ifn

where ω=( , , , )ω ω1 2 …ωn is the weighting vector associated with the CWA

opera-tor, ω ∈j [0,1],j=1,2, ,…n,

1 1

n j j= ω =

∑

, bj is the j th largest of a collection of the

weighted arguments n wi iα =( 1,2, , )i … n; w=( , , , )w w1 2 … wn is the weight vector

of the arguments α =i( 1,2, , )i … n, wi ∈[ , ],0 1 i=1 2, , , , … n wi i n = =

∑

1 1 , n is the bal-ancing coefficient. Then we call the function CWA a combined weighted averaging (CWA) operator.

Example 1.4 Let ω =(0.1,0.4,0.4,0.1) be the weighting vector associated with the CWA operator, ( , , , ) (7,18,6,2)α α α α =1 2 3 4 be a collection of arguments, whose weight vector is w = ( . , . , . , . )0 2 0 3 0 1 0 4 , then

1 2 1 ( , , , ) n w n j j j WA α α α wα = … =

∑

WA_w( , , , )7 18 6 2 =0 4 7 0 1 18 0 2 6 0 3 2 6 4. × + . × + . × + . × = . , 1 2 1 ( , , , ) n w n j j j CWA ω α α α ω b = … =

∑

1 1 2 2 3 3 4 4 4wα =5.6, 4w α =21.6, 4w α =2.4, 4w α =3.2

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14 1 Real-Valued MADM with Weight Information Unknown from which we get

Therefore,

Theorem 1.12 [109] The WA operator is a special case of the CWA operator. Proof Let 1 1, , ,1

n n n

ω =__ … __, then

which completes the proof.

Theorem 1.13 [109] The OWA operator is special case of the CWA operator. Proof Let w

n n n

=_1 1, , ,… 1_, then nwi iα =αi( 1,2, , )i= …n , thus the weighted version of the arguments, n wi iα =( 1,2, , )i … n, are also themselves. Therefore,

We can see from Theorems 1.12 and 1.13 that the CWA operator generalizes both the WA and OWA operators. It considers not only the importance of each argument itself, but also the importance of its ordered position.

1.2.2 Decision Making Method

For a complicated decision making problem, it usually involves multiple decision makers to participate in the decision making process so as to reach a scientific and rational decision result. In the following, we introduce a MAGDM method based on the OWA and CWAA operators [109]:

Step 1 Let X and U be the sets of alternatives and attributes, respectively, and the

information about the attribute weights is unknown. Let D={ , , , }d d1 2 … d_t be the

set of decision makers (experts), whose weight vector is λ=( , , , )λ λ1 2 … λt , where 0 k λ ≥ , k=1 2, , , , and … t 1 1 t k k λ = =

∑

. The decision maker d_k∈ provides his/her D

preference (or called attribute value) aij( )k over the alternative xi∈ with respect X

to the attribute u_j∈U. All the attribute values a_ij( )k ₍i₌_{1 2}_{, , , ;}_… n _j₌_{1 2}_{, , , ) are}_… _m

1 21.6, 2 5.6, 3 3.2, 4 2.4 b = b = b = b = , ( , ,..., ) 0.1 21.6 0.4 5.6 0.4 3.2 0.1 2.41 2 4 5.92 w CWA ω α α α = × + × + × + × = , 1 2 1 1 1 1 2 1 ( , ,..., ) ( , ,..., ) n n n w n j j j j j j i w n CWA b b w n WA ωα α α ω α α α α = = = = = = =

∑

, ( , , , )1 2 ( , , , )1 2 w n n CWA ω α α …α =OWAω α α …α

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contained in the decision matrix Ak. If the “dimensions” of the attributes are

differ-ent, then we need to normalize each attribute value aij( )k in the decision matrix Ak

using the formulas (1.2)–(1.7) into the normalized decision matrix Rk =(rij( )k )n m× .

Step 2 Utilize the OWA operator (1.1) to aggregate all the attribute values rij( )k (j=1 2, , , )… m in the i th line of the decision matrix Rk, and then get the overall

attribute value zi( )k ( )ω of the alternative xi corresponding to the decision maker dk:

where ω=( , , ,ω ω₁ ₂ …ωm), ω ≥j 0, j=1 2, , , , … m 1 1 m j j=ω =

∑

, and bij( )k is the j th largest of r_il( )k ₍l₌_{1 2}_{, , , )}_… m _.

Step 3 Aggregate all the overall attribute values ( )k ( )( 1,2, , ) i

z ω k= … t of the

alter-native xi corresponding to the decision makers d kk( =1 2, , , )…t by using the CWA

operator, and then get the collective overall attribute value zi( , ')λ ω :

where ' ' ' '

1 2

( , ,..., )t

ω = ω ω ω is the weighting vector associated with the CWA

opera-tor, ' ' 1 0, 1,2, , , t 1 k k k k t ω ω =

≥ = …

∑

= , bi( )k is the k th largest of a collection of the

weighted arguments t zλl i( )l ( )( 1,2, , ),ω l= … t and t is the balancing coefficient. Step 4 Rank all the alternatives x i_i( =1 2, , , )… n according to _{( , )}'

i

z λ ω (i=1 2, , , )… n, and then select the most desirable one.

The method above first utilizes the OWA operator to aggregate all the attri-bute values of an alternative with respect to all the attriattri-butes given by a decision maker, and then uses the CWA operator to fuse all the derived overall attribute val-ues corresponding to all the decision makers for an alternative. Considering that in the process of group decision making, some individuals may provide unduly high or unduly low preferences to their preferred or repugnant alternatives. The CWA operator can not only reflect the importance of the decision makers themselves, but also reduce as much as possible the influence of those unduly high or unduly low arguments on the decision result by assigning them lower weights, and thus make the decision results more reasonable and reliable.

( ) ( ) ( ) ( ) 1 2 ( ) 1 ( ) ( , ,..., ) , 1,2,..., , 1,2,..., k k k k i i i im m k j ij j z OWA r r r b i n k t ω ω ω = = =

∑

= =

(

)

' ' (1) (2) ( ) , ' ( ) 1 ( , ) ( ), ( ),..., ( ) , 1,2,..., t i i i i t k k i k z CWA z z z b i n λ ω λ ω ω ω ω ω = = =

∑

=

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1.2.3 Practical Example

Example 1.5 Let’s consider a decision making problem of assessing aerospace

equipment [10]. The attributes (or indices) which are used here in assessment of four types of aerospace equipments x ii( = 1 2 3 4 are: (1) u, , , ) 1: missile early-warning capacities; (2) u2: imaging detection capability; (3) u3: communications support capability; (4) u4: electronic surveillance capacity; (5) u5: satellite map-ping capability; (6) u6: navigation and positioning capabilities; (7) u7: marine mon-itoring capacity; and (8) u8: weather forecasting capability. The weight information about the attributes is completely unknown, and there are four decision makers

d k_k( = 1 2 3 4 , whose weight vector is , , , ) λ =(0.27,0.23,0.24,0.26). The decision makers evaluate the aerospace equipments x ii( = 1 2 3 4 with respect to the attri-, , , )

butes u jj( =1 2, , , )…8 by using the hundred-mark system, and then get the

attri-bute values contained in the decision matrices R_k ₌ r_ijk k₌

×

( ( )) (4 8 1 2 3 4 , which , , , ) are listed in Tables 1.4, 1.5, 1.6, and 1.7, respectively.

Since all the attributes u jj( =1 2, , , )…8 are of benefit type, the normalization

is not needed.

In what follows, we utilize the method given in Sect. 1.2.2 to solve the problem, which involves the following steps:

Step 1 Utilize the OWA operator (here, we use the method given in Theorem

1.10 (1) to determine the weighting vector associated with the OWA operator, (0.3,0.1,0.1,0.1,0.1,0.1,0.1,0.1)

ω = , when α =0.2) to aggregate all the attribute

values in the i th line of the decision matrix Rk, and then get the overall attribute

value ( )k _{( )} i

z ω of the decision maker dk:

Similarly, we have

Step 2 Aggregate all the overall attribute values ( )k _{( )(} _1,2,3,4) i

z ω k= of the

aero-space equipment xi corresponding to the decision makers d kk( = 1 2 3 4 by using , , , ) (1) (1) (1) (1) 1 ( ) ( ,11 12 ,...,18 ) 0.3 95 0.1 90 0.1 90 0.1 85 0.1 85 z ω =OWA r rω r = × + × + × + × + × 0.1 80 0.1 70 0.1 60 84.50 × + × + × = (1) (1) (1) (2) (2) 2 ( ) 82, 3 ( ) 83, 4 ( ) 79, 1 ( ) 79, 2 ( ) 79 z ω = z ω = z ω = z ω = z ω = (2) (2) (3) (3) (3) 3 ( ) 82.5, 4 ( ) 74.5, 1 ( ) 75, 2 ( ) 80, 3 ( ) 89.5 z ω = z ω = z ω = z ω = z ω = (3) (4) (4) (4) 4 ( ) 73.5, 1 ( ) 80, 3 ( ) 87.5, 4 ( ) 76 z ω = z ω = z ω = z ω =

(29)

the CWA operator (suppose that its weighting vector ' 1 1 1 1_{, , ,} 6 3 3 6

ω _{= }_ __). To do that, we first use λ, t and ( )k _{( )( ,} _1,2,3,4)

i z ω i k= to derive ( )k ( ) k i t zλ ω ( ,i k =1,2,3,4): (1) (1) (1) 1 1 1 2 1 3 4λz ( ) 91.26, 4ω = λ z ( ) 88.56, 4ω = λ z ( ) 89.64ω = (1) (2) (2) 1 4 2 1 2 2 4λ z ( ) 85.32, 4ω = λ z ( ) 72.68, 4ω = λ z ( ) 72.68ω = (2) (2) (3) 2 3 2 4 3 1 4λ z ( ) 75.9, 4ω = λ z ( ) 68.54, 4ω = λ z ( ) 72ω = (3) (3) (3) 3 2 3 3 3 4 4λ z ( ) 76.8, 4ω = λ z ( ) 85.92, 4ω = λ z ( ) 70.56ω =

Table 1.5 Decision matrix R₂

u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈

x₁ 60 75 90 65 70 95 70 75

x₂ 85 60 60 65 90 75 95 70

x₃ 60 65 75 80 90 95 90 80

x₄ 65 60 60 70 90 85 70 65

Table 1.6 Decision matrix R₃

u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈

x₁ 60 75 85 60 85 80 60 75

x₂ 80 75 60 90 85 65 85 80

x₃ 95 80 85 85 90 90 85 95

x₄ 60 65 50 60 95 80 65 70

Table 1.7 Decision matrix R₄

u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈

x₁ 70 80 85 65 80 90 70 80

x₂ 85 70 70 80 95 70 85 85

x₃ 90 85 80 80 95 85 80 90

x₄ 65 70 60 65 90 85 70 75

Table 1.4 Decision matrix R₁

u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈

x₁ 85 90 95 60 70 80 90 85

x₂ 95 80 60 70 90 85 80 70

x₃ 65 75 95 65 90 95 70 85

(30)

and thus, the collective overall attribute values of the aerospace equipment

x i_i( = 1 2 3 4 are, , , )

Step 3 Rank the aerospace equipments x i =i( 1,2,3,4) according to z1(λ ω, ′)

( 1,2,3,4)i = :

and then, the best aerospace equipment is x₃.

1.3 MADM Method Based on the OWG Operator

1.3.1 OWG Operator

Definition 1.4 [43, 144] Let OWG_{: (}_ℜ+₎n _{→ ℜ}+_{, if}

(1.9) where ω=( , , , )ω ω₁ ₂ …ωn is the exponential weighting vector associated with the

OWG operator, ω ∈_j [0,1], j=1 2, , , , … n 1 1 n i j ω = =

∑

, and b_j is the j th largest of a collection of the arguments α =i( 1,2, , )i … n, ℜ+ is the set of positive real numbers, then the function OWG is called an ordered weighted geometric (OWG) operator.

(4) (4) (4) 4 1 4 2 4 3 4λ z ( ) 83.20, 4ω = λ z ( ) 86.32, 4ω = λ z ( ) 91ω = (4) 4 4 4λ z ( ) 79.56ω = ' 1( , ) 1₆ 91.26 1₃ 83.20 1₃ 72.68 1₆ 72 79.17 z λ ω = × + × + × + × = ' 2 1 1 1 1 z ( , ) 88.56 86.32 76.8 72.68 79.87 6 3 3 6 λ ω = × + × + × + × = ' 3( , ) 1₆ 91 1₃ 89.64 1₃ 85.92 1₆ 75.90 86.34 z λ ω = × + × + × + × = ' 4( , ) 1₆ 85.32 1₃ 79.56 1₃ 70.56 1₆ 68.54 75.68 z λ ω = × + × + × + × = 3 2 1 4 x x  x x 1 2 1 ( , , , ) n j n j j OWGω α α α bω = … =

Uncertain Multi-Attribute Decision Making

Zeshui Xu

Uncertain Multi-Attribute

Decision Making

Preface

Contents

Part I

Real-Valued MADM Methods and Their

Applications

Chapter 1

Real-Valued MADM with Weight Information

Unknown

1.1 MADM Method Based on OWA Operator

1.1.1 OWA Operator

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

1.1.2 Decision Making Method

∑

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

∑

∑

1.1.3 Practical Example

∑

1.2 MAGDM Method Based on OWA and CWA

Operators

1.2.1 CWA Operator

∑

∑

∑

∑

∑

1.2.2 Decision Making Method

∑

∑

∑

∑

∑

∑

∑

(

)

∑

1.2.3 Practical Example

1.3 MADM Method Based on the OWG Operator