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A Generalized DEMATEL Theory with a Shrinkage Coefficient for an Indirect Relation Matrix

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A Generalized DEMATEL Theory with a Shrinkage Coefficient for an Indirect Relation Matrix MATEC Web of Conferences 119, 01020 (2017) DOI: 10.1051/ matecconf/201711901020 IMETI 2016 A generalized DEMATEL theory with a shrinkage coefficient for an indirect relation matrix Hsiang-Chuan Liu1, Ben-Chang Shia2, Yih-Chang Ou3,a and Huan-Wen Su4 1Department of Biomedical Informatics, Asia University, Taichung 41354, Taiwan 2Big Data Research Center & School of Management School of Health Care Administration, Taipei Medical University, Taipei 11031, Taiwan 3Department of Finance, Ling Tung University, Taichung 40852, Taiwan 4 Graduate School of Business Administration, Fu Jen Catholic University, Xinzhuang Dist., New Taipei City 24205, Taiwan Abstract. In this paper, a novel decision-making trial and evaluation laboratory (DEMATEL) theory with a shrinkage coefficient of indirect relation matrix is proposed, and a useful validity index, called Liu’s validity index, is also proposed for evaluating the performance of any DEMATEL model. If the shrinkage coefficient of an indirect relation matrix is equal to 1, then this new theory is identical to the traditional theory; in other words, it is a generalization of the traditional theory. Furthermore, the indirect relation is always considerably greater than the direct one in traditional DEMATEL theory, which is unreasonable and unfair because it overemphasizes the influence of the indirect relation. We prove in this paper that if the shrinkage coefficient is equal to 0.5, then the indirect relation is less than its direct relation. Because the shrinkage coefficient belongs to [0.5, 1], according to Liu’s validity index, we can find a more appropriate shrinkage coefficient to obtain a more efficient DEMATEL method. Some crucial properties of this new theory are discussed, and a simple example is provided to illustrate the advantages of the proposed theory. 1 Introduction A decision-making trial and evaluation laboratory (DEMATEL) was developed by Gabus and Fontela [1] and Fontela and Gabus [2] for resolving complex and difficult problems. To date, DEMATEL has been widely used as one of the most effective tools for solving cause–effect relationships among evaluation factors [3-10], but the indirect relation of a DEMATEL model is always markedly greater than its direct relation, which is unreasonable and unfair because it overemphasizes the influence of the indirect relation. In this paper, to overcome this drawback, we propose an external shrinkage coefficient for constructing a reduced indirect relation matrix, such that the new indirect relation matrix is less than its direct relation matrix, and for evaluating the performance of any DEMATEL model. We also propose a useful validity index, called Liu’s validity index, for evaluating the performance of any DEMATEL model. Subsequently, a generalized DEMATEL is derived and shown to be more useful and effective than previous DEMATEL models. a Corresponding author : [email protected] © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). MATEC Web of Conferences 119, 01020 (2017) DOI: 10.1051/ matecconf/201711901020 IMETI 2016 2 Traditional DEMATEL model The procedure of the traditional DEMATEL method is briefly introduced as follows: Step 1: Calculate the initial direct relation matrix Q If n factors are asked by N experts to evaluate the degree of direct influence between two factors through pair-wise comparison, then the degree to which expert e perceives factor i as affecting factor j is expressed as e e + qeij , 1,2,.., N. qij 0,1, 2,3, 4 , ij, 1,2,..., n (1) For each expert e, an individual direct relation matrix is constructed as FVee Qqeij, e 1,2,.., Nq , ii 0, i 1,2,..., n (2) HXnn and we can obtain their average direct relation matrix, which is called the initial direct relation matrix Q; 11NN QqFV BB Qq,, qe ij, 1,2,..., n (3) HXijnn e ij ij NNee11 Step 2: Calculate the direct relation matrix A IYnn 1 FV AaHXij Q,max,JZBB q ij q ij (4) nn 1,ij n K[ji11 where aiii0, 1,2,..., n , 0 aijijn ij 1, , , 1,2,..., nn and 0BBaaij , ij 1, ij , 1,2,..., n (5) ij11 Step 3: Calculate the indirect relation matrix B and the total relation matrix T On the basis of Markov chain theory, we have k limA 0nn (6) k FV FV23 k 21 BbHXij limHX AA ... A AIA , (7) nn k FV FV TtHXij ABab ij ij (8) nn HXnn Step 4: Calculate the relation degree and prominence degree of each factor nn rtctiiijikiBB, , 1,2,..., n (9) jk11 Where ri indicates the total dispatch of both direct and indirect effects that factor i has on the other factors, and ci indicates the total direct and indirect effects that factor i has on the other factors. 2 MATEC Web of Conferences 119, 01020 (2017) DOI: 10.1051/ matecconf/201711901020 IMETI 2016 The relation degree of factor i is expressed as xrciiii, 1,2,..., n (10) The prominence degree of factor i is expressed as yrciiii, 1,2,..., n (11) The relation–prominence matrix is expressed as n xy, (12) iii1 Step 5: Set the threshold value (α) To eliminate some minor effect elements in matrix T to derive the impact–relations map, Ou Yang et al. [4] suggested the following threshold value: nn 1 Yij2 BBt (13) n ij11 Li and Tzeng [5] suggested a more informative threshold value, M , on the basis of their maximum mean de-entropy algorithm. Step 6: Build a cause–effect relationship diagram If tortij Y, ij M , then factor i is a net dispatch node of factor j, and factor j is a net receive node of factor i, which is expressed as xyii,,,,, x j y j orxy ii x j y j. (14) n The graph of xy, , which includes the net direct edges, can present a cause–effect relationship iii1 diagram. 3 Critical properties of the generalized DEMATEL model In this paper, we consider the critical properties of the traditional DEMATEL model to derive a more effective DEMATEL model, and then propose a generalized DEMATEL model. The critical properties of the generalized DEMATEL model are briefly described as follows. Theorem 1. Critical properties of the generalized DEMATEL model If nn (a) AaFV FV a12,, A21 FV a FVBB aa Am F amm 1 V F aa V HXij HX IJ HX IJ11 ik kj H IJ X ik kj (15) nn nn nnHXkknn nn H X where nn ai0, 1,2,..., naijn , 0, , 1,2,..., ,BB a 1, a 1 (16) ii ijji11 ij ij IYnn I n n Y 1 mmm (b) maxJZBBaaij , ij 1, max J B a ij , B am ij Z , 2 (17) 1,ij n 1, ij n K[ji11 K j 1 i 1 [ 3 MATEC Web of Conferences 119, 01020 (2017) DOI: 10.1051/ matecconf/201711901020 IMETI 2016 IYnn FV 2 1 (c) BbHXij AIA,max,JZBB b ij b ij (18) nn 1,ij n K[ji11 IYnn +FVd 2 1 !" dd (d) Bbdij dAIdAd,0.5,1,max, dJZBB b ijij b (19) HXnn 1,ij n K[ji11 then (i) mm1 1,m 2 (20) (ii) 22#;11,mm # mN + (21) m m m ss1 (iii) lim B , d lim Bd (22) m m s2 s2 ;# (iv) d 0.5 00.5 1 (23) 1 ; (v) d 0.5 d 1 (24) 1 2 Proof nnnnnnn2 (i) BBBBBBBaaaaaaa 1 jjkkjkj1111111ij ik kj kj ik ik ij nn nn ;BBaa2 1, in general,1BB amm1 a jj11ij ij jj11ij ij nnmm1 Similarly, BBaa1, we can obtainmm1 1,m 2. ii11ij ij nnnnnnnn3222 (ii) BBBBBBBBaaaaaaaa 1 jjkkjjkj11111111ij ik kj kj ik kj ik ij nnnnn nnn BBBBBBBBaaaaaaaa42222222 1 jjkkj11111ij ik kj kj ik jkk 111 kj ik ik nn2 In general, BBaamm#1, we can obtainmm2 #+1,mN . jj11ij ij IYnn I n n Y s mmss (iii) maxJZBBbbij , ij lim B max J B a ij , B a ij Z lim B 1,ij nmm ks11 1, ij n K[ji11 K j 1 i 1 [ m ss1 Similarly, we have d lim Bd . m s2 mm ss11s 22 d (iv) We know that d limBBdd lim . mm ss221 d m # 2 d s1s 2 d Because 01, if 1, then d limBd 1; m 1 d s2 1 d 1 that is, d ;#01. (25) 2 d 0.5 4 MATEC Web of Conferences 119, 01020 (2017) DOI: 10.1051/ matecconf/201711901020 IMETI 2016 m s1s 2 (v) Because d limBd , is an increasing function of d and and m s2 m # 2 d s1s 2 d 01,1, then d limBd 1, m 1 d s2 1 d 2 dd11 ; ; let 10.51, d then 0.5 d 1. (26) 11dd221 4 Method of the generalized DEMATEL model Because the indirect relation of a traditional DEMATEL model is always markedly greater than its direct relation, it overemphasizes the influence of the indirect relation. To overcome this drawback, given the aforementioned properties, an external shrinkage coefficient, d, is provided for constructing a reduced indirect relation matrix, and a useful validity index, called Liu’s validity index, is also proposed for evaluating the performance of any DEMATEL model. Subsequently, a generalized DEMATEL model is derived and shown to be more useful and effective than previous DEMATEL models. FV +FVd 2 1 1 BbdijHX dAIdAd,,1GW (27) nn HX2 FV +FVddFV 1 TtdHX ij ABab dHX ij ij ,,1 dGW (28) nn HX2 The relation–prominence of the DEMATEL AB, d is defined as nn dd d dd d RAB(,dii ) x , y r iiii c , r c (29) ii11 nn dddd where rtctiniijiijBB, , 1,2,..., (30) jk11 nn 1 d Yij2 BBt (31) n ij11 dd If tortij Y, ij M , then factor i is a net dispatch node of factor j, and factor j is a net receive node of factor i , which can be expressed as dd dd xyii,, xy jj (32) n The graph of xy, , which includes the net direct edges, can present a cause–effect relationship iii1 diagram.
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