A DEMATEL-Based Completion Method for Incomplete Pairwise Comparison Matrix in AHP

07/23/2016 ∙ by Xinyi Zhou, et al. ∙ Xi'an Jiaotong University Southwest University 0

Pairwise comparison matrix as a crucial component of AHP, presents the prefer- ence relations among alternatives. However, in many cases, the pairwise comparison matrix is difficult to complete, which obstructs the subsequent operations of the clas- sical AHP. In this paper, based on DEMATEL which has ability to derive the total relation matrix from direct relation matrix, a new completion method for incomplete pairwise comparison matrix is proposed. The proposed method provides a new per- spective to estimate the missing values with explicit physical meaning. Besides, the proposed method has low computational cost. This promising method has a wide application in multi-criteria decision-making.

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1 Introduction

Analytic hierarchy process (AHP) is a multi-criteria decision-making method that helps the decision-makers facing a complex problem with multiple conflicting and subjective criteriaishizaka2009analytic . AHP has been widely applied in various areaszhu2016hesitant ; lin2015measuring ; yepes2015cognitive ; wang2011linear . Based on a hierarchical structure, the criteria weights can be derived from the pairwise comparison matrix. Pairwise comparison matrix as a crucial component of AHP, is commonly utilized to estimate preference values of finite alternatives with respect to a given criterion.

However, the pairwise comparison matrix is difficult to complete in the following cases, which obstructs the subsequent operations of the classical AHP.

  • The experts lack the knowledge of one or more alternatives.

  • Partial data in pairwise comparison matrix has been lost.

  • Huge number of pairwise comparison requested: for n alternatives. For example, 8 alternatives and 6 criteria require 183 entriesishizaka2011review .

Hitherto, it has attracted widespread attention over the issue of incomplete pairwise comparison matrix in AHP. There are quantities of methods having been developed to address the incomplete comparison matrices. Some of them use the incomplete matrix to assign priority weight to each alternative without completionharker1987incomplete ; carmone1997monte ; hua2008ds ; xu2013logarithmic ; dopazo2011parametric . Some methods are devoted to estimate the missing values in matrixfedrizzi2007incomplete ; ergu2016estimating ; bozoki2010optimal ; chen2015bridging ; zhang2012linear ; hu2006backpropagation ; alonso2008consistency ; benitez2015consistent ; gomez2010estimation ; buyukozkan2012new ; cabrerizo2010managing . While these methods have a good effect on allocating the priority weights to alternatives, the methods that focus on matrix completion are more “well-rounded” to restore the preference relations between each pair of alternatives, which are inaccessible before completion.

After reviewing the existing completion methods for incomplete pairwise comparison matrix in AHP, it can be found that the vast majority of these methods focus on the optimal consistency of pairwise comparison matrixfedrizzi2007incomplete ; ergu2016estimating ; bozoki2010optimal ; chen2015bridging ; zhang2012linear . For example, Fedrizzi and Giovefedrizzi2007incomplete proposed a new method to complete the pairwise comparison matrix by minimizing its global inconsistency. Ergu et al.ergu2016estimating

extended the geometric mean induced bias matrix to estimate the missing judgments and improve the consistency ratios by the least absolute error method and the least square method. Based on the graph theory, Bozoíki

et al.bozoki2010optimal developed a completion method which focuses on the optimal consistency and estimates the missing values through iteration. Besides, Hu and Tsaihu2006backpropagation proposed a well-known back propagation multi-layer perception to estimate the missing comparisons of incomplete pairwise comparison matrices in AHP. Alonso et al.alonso2008consistency defined a way to calculate the consistency of incomplete pairwise comparison matrix and maintain its consistency level after completion. Benítez et al.benitez2015consistent used the Frobenius norm to obtain the most consistent/similar complete matrix with the initial incomplete matrix. Gomez et al.gomez2010estimation

proposed a completion method based on the neural network. These methods are useful but have some limitations:

  • Most methods complete the pairwise comparison matrix with optimal consistency, which sometimes can not restore the incomplete matrix to its original state, especially when the initial matrix is not high consistent.

  • The methods are computationally expensive.

Taking into account these issues, a DEMATEL-based completion method for incomplete pairwise comparison matrix in AHP is proposed. Based on DEMATEL method which has ability to derive the total relation matrix from direct relation matrix, the completion of incomplete pairwise comparison matrix can be divided into three steps.

  • Firstly, turn the incomplete comparison matrix into direct relation matrix.

  • Secondly, DEMATEL method is adopted to obtain the total relation matrix.

  • Thirdly, transform the total relation matrix into pairwise comparison matrix with reciprocal preference relations.

The proposed method provides a new perspective to estimate the missing values in pairwise comparison matrix with explicit physical meaning (i.e. by calculating the total relations among alternatives). Besides, the proposed method has low computational cost.

The rest of this paper is organized as follows. Section 2 introduces some preliminaries of this work. Section 3 illustrates the procedure of the proposed method and an application for ranking tennis players is presented. Section 4 discusses the superiority and efficiency of the proposed method and section 5 ends the paper with the conclusion.

2 Preliminaries

2.1 Analytic Hierarchy Process (AHP)

AHP is a multi-criteria decision-making method developed by Saatysaaty1980analytic and aims at quantifying relative weights for a given set of criteria on a ratio scalexu2015deviation . As a decision-making approach, it permits a hierarchical structure of criteria, which provides users with a better focus on specific criteria and sub-criteria when allocating the weightsishizaka2013multi . Besides, AHP has ability to judge the consistency of pairwise comparison matrix to show its potential conflicts in decision-making process. AHP has been widely used in supply chain managementchan2007global ; deng2014supplier ; chan2006ahp , human reliability analysissu2015dependence , cloud computingliu2016decision , energy planningishizaka2016energy ; onar2015multi ; kahraman2009comparative , site selectionerdougan2016combined ; lee2015integrated ; pourahmad2015combination , software assessmenthu2013software ; hu2015cost and performance evaluationzavadskas2015selecting ; wu2010evaluating . AHP can be extended further by other theories and methods such as fuzzy numberchan2008global ; wang2008extent , choquet integralcorrente2015combining

, k-means algorithm

lolli2014new , VIKORbuyukozkan2015evaluation and TOPSISerdogan2014evaluating ; kahraman2015fuzzy .

Generally, the process of applying AHP is divided into three steps. Firstly, establish a hierarchical structure by recursively decomposing the decision problem. Secondly, construct the pairwise comparison matrix to indicate the relative importance of alternatives. A numerical rating including nine rank scales is suggested, as shown in Tab.1. Thirdly, verify the consistency of pairwise comparison matrix and calculate the priority weights of alternatives. For the completeness of explanation, several basic concepts and formulas are introduced as follows.

 Scale Meaning
 1 Equal importance
 3 Moderate importance
 5 Strong importance
 7 Demonstrated importance
 9 Extreme importance
 2, 4, 6, 8 Intermediate values
Table 1: Numerical rating in AHP
Definition 1

(Pairwise comparison matrix) Assume are alternatives available for decision-making, the pairwise comparison matrix is indicated as , and satisfies:

(1)

where represents the relative importance of over .

Consistency checking is introduced in AHP to verify the usability of constructed pairwise comparison matrix.

Definition 2

(Consistency ratio) For the pairwise comparison matrix , let

denote the largest eigenvalue of

, consistency index () is defined as

(2)

Based on , consistency ratio (CR) is defined as

(3)

where is the random consistency index related to the dimension of matrix, listed in Tab.2.

n 1 2 3 4 5 6 7 8 9 10
0 0 0.52 0.89 1.12 1.26 1.36 1.41 1.46 1.49
Table 2: Random consistency index

If

, the constructed pairwise comparison matrix is considered acceptable and the priority weights of alternatives can be obtained through the eigenvector as Def.

3. Otherwise, the comparison matrix needs to be reconstructed.

Definition 3

(Eigenvector) For the pairwise comparison matrix with acceptable consistency, suppose is the eigenvector of , whose is indicated as the priority weight of the -th alternative and calculated by

(4)

2.2 DEMATEL method

Decision-making trial and evaluation laboratory (DEMATEL) method was originally developed by Battelle Memorial Institute of Geneva Research Center, aimed at the fragmented and antagonistic phenomena of world societies and searched for integrated solutiongabus1972world ; fontela1976dematel . DEMATEL is built on the basis of graph theory, effectively assessing the total relations and constructing a map between system components in respect to their type and severitytzeng2007evaluating . Reviewing the former studies, DEMATEL has been successfully applied in many diverse areas such as service quality analysisshieh2010dematel ; tseng2009causal , stock selectionshen2015combined and supply chain managementISI:000368039100008 ; wu2015case ; wu2015exploring . DEMATEL can be further extended by other theories and methods such as fuzzy numbersmentes2015fsa ; tsai2015using and ANPISI:000356990800033 ; tzeng2012combined ; ISI:000262178000049 ; wu2008choosing .

The procedure of DEMATEL is divided into five steps.

Step 1: Define quality feature and establish measurement scale

Quality feature is a set of influential characteristics that impact the sophisticated system, which can be determined by literature review, brain-storming, or expert evaluation. After defining the influential characteristics in researching system, establish the measurement scale for the causal relationship and pairwise comparison among influential characteristics. Four levels 0,1,2,3 are suggested, respectively meaning “no impact”, “low impact”, “high impact” and “extreme high impact”. In this step, factors and their direct relations are displayed by a weighted and directed graph.

Step 2: Extract the direct relation matrix of influential factors

In this step, transformation from the weighted and directed graph into direct relation matrix has been done. For influential characteristics , direct relation matrix is denoted as , where is the direct relation of over based on the measurement scale and satisfies if .

Step 3: Normalize the direct relation matrix

The normalized direct relations of factors are a mapping from to , defined as below.

Definition 4

(Normalized direct relation matrix) For the framework of influential characteristics , normalized matrix of direct relation matrix is obtained by

(5)

Note that there must be at least one row of matrix satisfying lin2008causal . Hence, based on the studies of papoulis1985probability and goodman1988introduction

, sub-stochastic matrix can be obtained through utilizing the normalized direct relation matrix

and absorbing state of Markov chain matrices, shown in (

6).

(6)

where is the null matrix and

is the identity matrix.

Step 4: Calculate the total relation matrix

Due to the characteristic of normalized direct relation matrix , total relation matrix which contains direct and indirect relations among factors can be derived from (7).

Definition 5

(Total relation matrix) For the framework of influential characteristics , assume is the normalized direct relation matrix, total relation matrix is defined as

(7)

where is the identity matrix.

Step 5: Construct the DEMATEL map

Based on the sum of each row and column of the total relation matrix , and can be obtained. is defined as the prominence, indicating the importance of the i-th influential factor. classifies the i-th influential factor into the cause or effect category in researching system. If , the factor is regarded as a cause factor. If , the factor is an effect one. The DEMATEL map is shown as Fig.1.

Figure 1: DEMATEL map
Figure 2: Procedure of DEMATEL-based completion method

3 DEMATEL-based completion method

In this section, a DEMATEL-based completion method for incomplete pairwise comparison matrix in AHP is proposed. Moreover, an application for ranking top tennis players is presented.

3.1 Procedure of DEMATEL-based completion method

In this sub-section, the procedure of DEMATEL-based completion method is presented and illustrated by an example. Fig.2 shows the process of the proposed method.

Assume an incomplete pairwise comparison matrix and the unavailable/missing values are displayed by ‘*’.

Step 1: From incomplete pairwise comparison matrix to direct relation matrix.

Pairwise comparison matrix reflects the preference relations between each pair of factors, where in indicates the relative importance of factor to factor . Hence, the known values in can be put into the direct relation matrix without any transformation and the unavailable ones(‘*’) are substituted by 0. The direct relation matrix is as:

Step 2: From direct relation matrix to total relation matrix.

First of all, normalize the direct relation matrix according to (5). Then, based on the normalized matrix and (7), total relation matrix can be obtained. The normalized direct relation matrix and total relation matrix are calculated as:

Step 3: From total relation matrix to pairwise comparison matrix.

In fact, the total relation matrix has completed the unavailable values in initial incomplete matrix. Nevertheless, in pairwise comparison matrix, the multiplication of each pair of symmetric values along the diagonal line is required to equal 1, yet the values in total relation matrix are between 0 and 1. Based on (8), it is feasible to accomplish this transformation from the total relation matrix to pairwise comparison matrix .

(8)

Hence, the pairwise comparison matrix transformed from is:

Finally, based on the proposed method, the completion for incomplete pairwise comparison matrix has been accomplished as:

3.2 An application for ranking tennis players

Professional tennis as a global popular sport owns large numbers of viewers around the world. The professional tennis associations such as ATP have been collecting data about the tournaments and the players. There is a free available database that collects the results of the top tennis players since 1973 (see at http://www.atpw-
orldtour.com/Players/Head-To-Head.aspx).

In original statistics, 25 tennis players who have been the champion on ATP since 1973 are selected. The statistical competition results matrix is shown in Tab.3. However, the statistical matrix is not complete since some of players did not have the chance to compete with each other or the partial data has been lost. Thus, DEMATEL-based completion method is adopted to rank the tennis players based on these statistics. The process is divided into three steps, which is shown in Fig.3.

Figure 3: Flowchart for ranking tennis players
Figure 4: Rule of transformation
Aga Bec Bor Con Cou Djo Edb Fed Fer Hew Kaf Kue Len McE Moy Mus Nad Nas New Raf Rio Rod Saf Sam Wil
Agassi 10/14 2/2 5/12 6/9 3/11 2/5 4/8 8/12 7/11 2/8 2/4 3/4 5/9 0/2 10/15 1/3 5/6 3/6 14/34 5/7
Becker 4/14 6/6 6/7 25/35 1/1 4/6 10/21 8/10 2/4 2/3 1/1 2/3 3/5 0/1 7/19 7/10
Borg 15/23 6/8 7/14 10/15 1/4 1/1
Connors 0/2 0/6 8/23 0/3 6/12 13/34 14/34 12/27 2/4 0/2 0/5
Courier 7/12 1/7 3/3 6/10 1/6 1/1 0/4 2/3 2/3 7/12 0/1 0/3 0/3 1/2 4/20
Djokovic 15/31 2/3 6/7 2/4 17/39 4/9 0/2
Edberg 3/9 10/35 6/12 4/10 1/3 14/27 6/13 1/1 10/10 3/3 1/1 6/14 9/20
Federer 8/11 16/31 10/13 18/26 2/6 1/3 7/7 10/32 0/3 2/2 21/24 10/12 1/1
Ferrero 3/5 1/3 3/13 4/10 1/3 3/5 8/14 2/9 2/3 3/4 0/5 6/12
Hewitt 4/8 0/1 1/7 8/26 6/10 7/8 3/4 7/12 4/10 3/4 3/5 7/14 7/14 5/9
Kafelnikov 4/12 2/6 5/6 2/3 4/6 2/3 1/8 5/12 3/6 1/5 3/5 6/8 2/4 2/13 1/2
Kuerten 4/11 0/1 2/3 2/5 1/4 7/12 4/7 3/3 4/8 2/4 1/2 4/7 1/3
Lendl 6/8 11/21 2/8 21/34 4/4 13/27 21/36 4/5 1/1 0/1 3/8 15/22
McEnroe 2/4 2/10 7/14 20/34 1/3 7/13 15/36 6/9 1/2 0/3 7/13
Moya 1/4 2/4 1/3 2/4 0/1 0/7 6/14 5/12 3/6 3/7 4/8 2/8 3/4 2/7 1/5 4/7 1/4
Muster 4/9 1/3 5/12 0/10 4/5 0/3 1/5 4/8 0/3 3/4 0/1 2/11 0/2
Nad 2/2 22/39 22/32 7/9 6/10 6/8 7/10 2/2
Nastase 0/1 5/15 15/27 1/1 0/1 3/9 4/5 0/1
Newcombe 3/4 2/4 1/2 1/5
Rafter 5/15 1/3 3/3 0/3 3/3 1/3 1/4 2/5 4/8 1/1 1/4 3/3 2/3 1/1 4/16 1/3
Rios 2/3 2/5 3/3 0/1 0/2 1/4 2/5 2/8 2/4 5/7 1/4 1/3 0/2 1/4 0/2
Roddick 1/6 5/9 3/24 5/5 7/14 1/2 4/5 3/10 2/2 4/7 2/3
Safin 3/6 1/1 1/2 2/2 2/12 6/12 7/14 2/4 3/7 3/7 1/1 0/2 0/1 3/4 3/7 4/7
Sampras 20/34 12/19 2/2 16/20 8/14 0/1 4/9 11/13 2/3 5/8 3/3 3/4 9/11 12/16 2/2 1/3 3/7 2/3
Wilander 2/7 3/10 0/1 5/5 11/20 1/2 7/22 6/13 2/2 1/1 2/3 1/3
Note: Due to the space limitation, the names of tennis players have been abbreviated to the first three letters in the horizontal row and their full names can be seen in the vertical column.
Table 3: Statistical competition results matrix: the wins/total number of matches
Aga Bec Bor Con Cou Djo Edb Fed Fer Hew Kaf Kue Len McE Moy Mus Nad Nas New Raf Rio Rod Saf Sam Wil
Agassi 1.39 1.07 0.90 1.17 0.76 0.95 1.00 1.24 1.17 0.80 1.00 1.12 1.05 0.93 1.31 0.95 1.28 1.00 0.73 1.18
Becker 0.72 1.38 1.38 2.28 1.04 1.18 0.95 1.43 1.00 1.05 1.03 1.05 1.05 0.97 0.77 1.24
Borg 1.45 1.25 1.00 1.31 0.89 1.03
Connors 0.93 0.73 0.69 0.88 1.00 0.66 0.73 0.86 1.00 0.93 0.78
Courier 1.11 0.72 1.13 1.11 0.78 1.03 0.83 1.05 1.05 1.11 0.97 0.88 0.88 1.00 0.49
Djokovic 0.95 1.06 1.38 1.00 0.77 0.95 0.93
Edberg 0.85 0.44 1.00 0.90 0.95 1.05 0.95 1.03 1.89 1.13 1.03 0.90 0.90
Federer 1.32 1.05 1.49 1.72 0.90 0.95 1.48 0.52 0.88 1.07 3.31 1.64 1.03
Ferrero 1.05 0.95 0.67 0.90 0.95 1.05 1.11 0.75 1.05 1.12 0.78 1.00 0
Hewitt 1.00 0.97 0.72 0.58 1.11 1.49 1.12 1.11 0.90 1.12 1.05 1.00 1.00 1.05
Kafelnikov 0.81 0.90 1.28 1.05 1.11 1.05 0.67 0.90 1.00 0.84 1.05 1.25 1.00 0.57 1.00
Kuerten 0.85 0.97 1.05 0.95 0.89 1.11 1.05 1.13 1.00 1.00 1.00 1.05 0.95
Lendl 1.25 1.05 0.80 1.52 1.20 0.95 1.36 1.19 1.03 0.97 0.90 1.54
McEnroe 1.00 0.70 1.00 1.36 0.95 1.05 0.73 1.17 1.00 0.88 1.05
Moya 0.89 1.00 0.95 1.00 0.97 0.67 0.90 0.90 1.00 0.95 1.00 0.80 1.12 0.85 0.84 1.05 0.89
Muster 0.95 0.95 0.90 0.53 1.19 0.88 0.84 1.00 0.88 1.12 0.97 0.65 0.93
Nadal 1.07 1.29 1.91 1.34 1.11 1.25 1.24 1.07
Nastase 0.97 0.77 1.17 1.03 0.96 0.85 1.19 0.97
Newcombe 1.12 1.00 1.00 0.84
Rafter 0.77 0.95 1.13 0.88 1.13 0.95 0.89 0.95 1.00 1.04 0.89 1.13 1.05 1.03 0.64 0.95
Rios 1.05 0.95 1.13 0.97 0.93 0.89 0.95 0.80 1.00 1.18 0.89 0.95 0.93 0.89 0.93
Roddick 0.78 1.05 0.30 1.28 1.00 1.00 1.19 0.80 1.07 1.05 1.05
Safin 1.00 1.03 1.00 1.07 0.61 1.00 1.00 1.00 0.95 0.95 1.03 0.93 0.97 1.12 0.95 1.05
Sampras 1.36 1.30 1.07 2.04 1.11 0.97 0.95 1.77 1.05 1.11 1.13 1.12 1.53 1.57 1.07 0.95 0.95 1.05
Wilander 0.85 0.80 0.97 1.28 1.11 1.00 0.65 0.95 1.07 1.03 1.05 0.95
Note: Due to the space limitation, the names of tennis players have been abbreviated to the first three letters in the horizontal row and their full names can be seen in the vertical column.
Table 4: Incompete pairwise comparison matrix
Aga Bec Bor Con Cou Djo Edb Fed Fer Hew Kaf Kue Len McE Moy Mus Nad Nas New Raf Rio Rod Saf Sam Wil
Agassi 1.00 1.39 0.89 1.07 0.90 1.00 1.17 0.76 0.95 1.00 1.24 1.17 0.80 1.00 1.12 1.05 0.93 1.06 1.08 1.31 0.95 1.28 1.00 0.73 1.18
Becker 0.72 1.00 0.99 1.38 1.38 0.96 2.28 0.83 1.04 1.04 1.18 1.04 0.95 1.43 1.00 1.05 0.81 1.03 1.26 1.05 1.05 1.04 0.97 0.77 1.24
Borg 1.13 1.01 1.00 1.45 1.25 1.08 1.22 0.94 1.15 1.06 1.19 1.16 1.25 1.00 1.20 1.30 0.92 1.31 0.89 1.19 1.17 1.19 1.13 0.99 1.03
Connors 0.93 0.73 0.69 1.00 0.88 0.84 1.00 0.72 0.89 0.82 0.94 0.90 0.66 0.73 0.92 1.00 0.72 0.86 1.00 0.92 0.89 0.93 0.86 0.93 0.78
Courier 1.11 0.72 0.80 1.13 1.00 0.89 1.11 0.74 0.92 0.83 0.78 1.03 0.83 1.05 1.05 1.11 0.75 0.97 1.04 0.88 0.88 0.92 1.00 0.49 0.94
Djokovic 1.00 1.04 0.92 1.20 1.12 1.00 1.10 0.95 1.05 1.38 1.11 1.04 0.95 1.11 1.00 1.12 0.77 1.09 1.14 1.09 1.08 0.95 0.93 0.93 1.10
Edberg 0.85 0.44 0.82 1.00 0.90 0.91 1.00 0.79 0.96 0.86 0.95 0.97 1.05 0.95 1.03 1.89 0.75 0.91 0.97 1.13 1.03 0.97 0.94 0.90 0.90
Federer 1.32 1.21 1.06 1.39 1.36 1.05 1.26 1.00 1.49 1.72 0.90 0.95 1.10 1.29 1.48 1.32 0.52 1.27 1.32 0.88 1.07 3.31 1.64 1.03 1.23
Ferrero 1.05 0.96 0.87 1.12 1.09 0.95 1.04 0.67 1.00 0.90 0.95 1.05 0.90 1.03 1.11 1.07 0.75 1.03 1.07 1.05 1.12 0.78 1.00 0.82 1.03
Hewitt 1.00 0.97 0.95 1.22 1.20 0.72 1.17 0.58 1.11 1.00 1.49 1.12 0.97 1.14 1.11 1.17 0.90 1.10 1.17 1.12 1.05 1.00 1.00 1.05 1.13
Kafelnikov 0.81 0.90 0.84 1.07 1.28 0.90 1.05 1.11 1.05 0.67 1.00 0.90 0.83 0.97 1.00 0.84 0.74 0.98 1.02 1.05 1.25 1.00 1.00 0.57 1.00
Kuerten 0.85 0.96 0.86 1.11 0.97 0.96 1.03 1.05 0.95 0.89 1.11 1.00 0.89 1.03 1.05 1.13 0.78 1.02 1.06 1.00 1.00 1.00 1.05 0.95 1.03
Lendl 1.25 1.05 0.80 1.52 1.20 1.05 0.95 0.91 1.12 1.03 1.20 1.12 1.00 1.36 1.15 1.19 0.91 1.03 1.18 0.97 1.13 1.15 1.10 0.90 1.54
McEnroe 1.00 0.70 1.00 1.36 0.95 0.90 1.05 0.78 0.97 0.88 1.03 0.97 0.73 1.00 1.00 1.10 0.78 1.17 1.00 1.01 0.96 1.00 0.93 0.88 1.05
Moya 0.89 1.00 0.84 1.08 0.95 1.00 0.97 0.67 0.90 0.90 1.00 0.95 0.87 1.00 1.00 1.00 0.80 0.98 1.03 1.12 0.85 0.84 1.05 0.89 0.99
Muster 0.95 0.95 0.77 1.00 0.90 0.89 0.53 0.76 0.93 0.86 1.19 0.88 0.84 0.91 1.00 1.00 0.74 0.91 0.94 0.88 1.12 0.93 0.97 0.65 0.93
Nadal 1.07 1.23 1.08 1.39 1.34 1.29 1.33 1.91 1.34 1.11 1.36 1.29 1.10 1.28 1.25 1.35 1.00 1.28 1.33 1.34 1.32 1.24 1.07 1.12 1.32
Nastase 0.94 0.97 0.77 1.17 1.03 0.92 1.10 0.79 0.97 0.91 1.02 0.98 0.96 0.85 1.02 1.10 0.78 1.00 1.19 1.01 0.99 0.99 0.96 0.80 0.97
Newcombe 0.93 0.79 1.12 1.00 0.96 0.88 1.03 0.76 0.94 0.86 0.98 0.94 0.85 1.00 0.97 1.06 0.75 0.84 1.00 0.97 0.94 0.96 0.91 0.82 0.96
Rafter 0.77 0.95 0.84 1.09 1.13 0.92 0.88 1.13 0.95 0.89 0.95 1.00 1.04 0.99 0.89 1.13 0.74 0.99 1.03 1.00 1.05 1.02 1.03 0.64 0.95
Rios 1.05 0.95 0.85 1.12 1.13 0.92 0.97 0.93 0.89 0.95 0.80 1.00 0.89 1.04 1.18 0.89 0.76 1.01 1.07 0.95 1.00 0.93 0.89 0.93 1.00
Roddick 0.78 0.96 0.84 1.08 1.08 1.05 1.03 0.30 1.28 1.00 1.00 1.00 0.87 1.00 1.19 1.07 0.80 1.01 1.04 0.98 1.07 1.00 1.05 1.05 1.00
Safin 1.00 1.03 0.88 1.16 1.00 1.07 1.06 0.61 1.00 1.00 1.00 0.95 0.91 1.07 0.95 1.03 0.93 1.04 1.10 0.97 1.12 0.95 1.00 1.05 1.05
Sampras 1.36 1.30 1.01 1.07 2.04 1.07 1.11 0.97 1.22 0.95 1.77 1.05 1.11 1.13 1.12 1.53 0.89 1.25 1.22 1.57 1.07 0.95 0.95 1.00 1.05
Wilander 0.85 0.80 0.97 1.28 1.06 0.91 1.11 0.81 0.97 0.88 1.00 0.97 0.65 0.95 1.01 1.07 0.76 1.03 1.05 1.05 1.00 1.00 0.96 0.95 1.00
Note: Due to the space limitation, the names of tennis players have been abbreviated to the first three letters in the horizontal row and their full names can be seen in the vertical column.
Table 5: Complete pairwise comparison matrix

Firstly, the statistical competition results matrix (i.e. with additive preference relations) is converted into the pairwise comparison matrix (i.e. with multiplicative preference relations) following the method described in bozoki2016application . Fig.4 illustrates this transformation process. The converted pairwise comparison matrix is shown in Tab.4.

Secondly, DEMATEL-based completion method is adopted to complete the pairwise comparison matrix. Following the procedure of the proposed method, the complete pairwise comparison matrix is obtained, which is shown in Tab.5.

Thirdly, based on the complete pairwise comparison matrix, the priorities of tennis players can be calculated via (4). Tab.6 shows the final ranking of the 25 tennis players.

Rank Player Priority Rank Player Priority
1 Nadal 0.0503 14 Rios 0.0380
2 Federer 0.0503 15 McEnroe 0.0380
3 Sampras 0.0467 16 Nastase 0.0380
4 Borg 0.0444 17 Rafter 0.0378
5 Lendl 0.0436 18 Wilander 0.0378
6 Becker 0.0430 19 Kafelnikov 0.0375
7 Hewitt 0.0414 20 Edberg 0.0374
8 Djokovic 0.0412 21 Moya 0.0371
9 Agassi 0.0409 22 Newcombe 0.0365
10 Safin 0.0392 23 Courier 0.0360
11 Kuerten 0.0390 24 Muster 0.0353
12 Roddick 0.0384 25 Connors 0.0339
13 Ferrero 0.0383
Table 6: Ranking of 25 tennis players

4 Discussion

The result in last section indicates that the proposed method has ability to complete the pairwise comparison matrices in AHP. In this section, we would further analyze the superiority and efficiency of the proposed method.

The proposed method has the following advantages:

  • The proposed method estimates the missing values in pairwise comparison matrix from a new perspective (i.e. by calculating the total relations among alternatives).

  • The proposed method has low computational cost.

After reviewing the prior studies, an assessment for the computational cost of the existing completion methods and the proposed method is done. The methods for comparison satisfy:

  • The methods belong to completion methods, which have ability to estimate the missing values in pairwise comparison matrix.

  • The pairwise comparison matrices, which can be completed by these completion methods, present reciprocal preference relations among alternatives. In other words, the matrices request that the multiplication of each pair of symmetric values along the diagonal line is equal to 1.

As for these completion methods, some of them estimate the missing values through iteration, or based on the complex theories/methods, which causes the methods expensive computational cost. For example, Bozoíki et al.bozoki2010optimal developed a completion method which estimates the missing values in pairwise comparison matrix through iteration and optimizes the matrix consistency based on graph theory. Gomez et al.gomez2010estimation

proposed a completion method based on the neural network. Neural network as one of the branches of artificial intelligence leads to the complexity of the completion method. As for the completion methods without iteration or complex theories/algorithms such as the methods listed in Tab.

7, time complexity is used to accurately reflect their computational cost. Assume there is an incomplete pairwise comparison matrix, which has n missing values, the time complexity of these methods is shown in Tab.8.

          Description   Source
Method 1
A completion method extends the geometric mean induced bias matrix to est-
  imate the missing judgments and improves the consistency ratios by the least
  absolute error method and the least square method.
Ergu et al.ergu2016estimating
Method 2
A consistency-based completion procedure, which takes a measure to maintain
  the consistency level of matrices before and after completion.
Alonso et al.alonso2008consistency
Method 3
A completion method to obtain the most consistent/similar matrix with the in-
  itial incomplete matrix based on Frobenius norm.
Benítez et al.benitez2015consistent
Table 7: Methods for comparison
Method 1 Method 2 Method 3 The proposed method
Time complexity
Table 8: Time complexity of methods

It can be seen from the Tab.8 that the time complexity of the proposed method is lower than the others. In addition, the most existing completion methods have the following two operations:

  • Divide the matrix into upper and lower triangular matrix. The pairwise comparison matrix in AHP has a property that the multiplication of each pair of symmetric values along the diagonal line is equal to 1. These methods firstly focus on the completion of upper/lower triangle of incomplete matrix and use the property to accomplish the other triangular matrix.

  • Traverse the pairwise comparison matrix to obtain the positions of unavailable values, then estimate these missing values in sequence.

However, the proposed method completes the matrix from a holistic viewpoint without dividing the matrix into the upper and lower triangular matrix. Furthermore, based on DEMATEL, all the missing values in matrix can be simultaneously estimated. Hence, following the procedure of the proposed method, the completion of incomplete pairwise comparison matrix becomes straightforward and efficient.

5 Conclusion

Pairwise comparison matrix plays a pivotal role in AHP. However, in many cases, only partial information in pairwise comparison matrix is available, which obstructs the subsequent operations of the classical AHP. In this paper, we propose a new completion method for incomplete pairwise comparison matrix in AHP. Based on DEMATEL, the proposed method has ability to complete the matrix by obtaining the total (direct and indirect) relations from the direct preference relations between each pair of alternatives. Furthermore, an application of the proposed method for ranking tennis players is presented. From the ranking result and the comparison with several representative completion methods, the proposed method is proved to be effective and efficient. The proposed method provides a new perspective to complete the matrix with explicit physical meaning. Besides, the proposed method has low computation cost. This promising method has a wide application in multi-criteria decision-making. In our further study, we would extend the proposed method for the incomplete fuzzy pairwise comparison matrices.

Acknowledgment

The work is partially supported by National Natural Science Foundation of China (Grant Nos. 61573290,61503237), China State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant No.BUAA-VR-14KF-02).

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