Identification of a threshold value for the DEMATEL method using the maximum mean de-entropy algorithm to find critical services provided by a semiconductor intellectual property mall

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Identiﬁcation of a threshold value for the DEMATEL method using the maximum

mean de-entropy algorithm to ﬁnd critical services provided by a semiconductor

intellectual property mall

Chung-Wei Li

a,*

, Gwo-Hshiung Tzeng

a,b,c,1

a_{Institute of Management of Technology, National Chiao Tung University, 33060 Hsinchu, Taiwan} b_{Department of Business and Entrepreneurial Management, Kainan University, Taoyuan, Taiwan} c

Department of Banking and Finance, Kainan University, Taoyuan, Taiwan

a r t i c l e

i n f o

Keywords: DEMATEL

Multiple criteria decision-making (MCDM) Entropy

Maximum mean de-entropy (MMDE) algorithm

a b s t r a c t

To deal with complex problems, structuring them through graphical representations and analyzing causal influences can aid in illuminating complex issues, systems, or concepts. The DEMATEL method is a meth-odology which can confirm interdependence among variables and aid in the development of a chart to reflect interrelationships between variables, and can be used for researching and solving complicated and intertwined problem groups. The end product of the DEMATEL process is a visual representation— the impact-relations map—by which respondents organize their own actions in the world. In order to obtain a suitable impact-relations map, an appropriate threshold value is needed to obtain adequate information for further analysis and decision-making. In the existing literature, the threshold value has been determined through interviews with respondents or judged by the researcher. In most cases, it is hard and time-consuming to aggregate the respondents and make a consistent decision. In addition, in order to avoid subjective judgments, a theoretical method to select the threshold value is necessary. In this paper, we propose a method based on the entropy approach, the maximum mean de-entropy algo-rithm, to achieve this purpose. Using a real case to find the interrelationships between the services of a Semiconductor Intellectual Property Mall as an example, we will compare the results obtained from the respondents and from our method, and show that the impact-relations maps from these two methods could be the same.

1. Introduction

The DEMATEL (Decision-Making Trial and Evaluation Labora-tory) method, developed by the Science and Human Affairs Pro-gram of the Battelle Memorial Institute of Geneva between 1972 and 1976, was used to research and solve complicated and inter-twined problem groups (Fontela & Gabus, 1974, 1976). DEMATEL was developed in the hope that pioneering the appropriate use of scientific research methods could improve the understanding of a specific problematique, a cluster of intertwined problems, and con-tribute to the identification of workable solutions through a hierar-chical structure. The DEMATEL method is based on graph theory, enabling us to plan and solve problems visually, so that we may di-vide the relevant factors into cause and effect groups in order to

better understand causal relationships. The methodology can con-ﬁrm interdependence among variables and aid in the development of a directed graph to reﬂect the interrelationships between variables.

The applicability of the DEMATEL method is widespread, rang-ing from analyzrang-ing world problematique decision-makrang-ing to industrial planning (Chiu, Chen, Shyu, & Tzeng, 2006;Hori & Shi-mizu, 1999; Huang, Shyu, & Tzeng, 2007; Tzeng, Chiang, & Li, 2006). The most important property of the DEMATEL method used in the multi-criteria decision-making (MCDM) field is to construct interrelations between criteria. After the interrelations between criteria were determined, the results derived from the DEMATEL method could be used for fuzzy integrals to measure the super-additive effectiveness value or for the Analytic Network Process method (ANP) (Liou, Yen, & Tzeng, 2008; Saaty, 1996; Tsai & Chou, 2009) to measure dependence and feedback relationships between certain criteria. When the DEMATEL method is used as part of a hy-brid MCDM model, the results of the DEMATEL will influence the final decision.

* Corresponding author. Tel.: +886 3 3706190.

E-mail addresses:samli0707@gmail.com(C.-W. Li),ghtzeng@mail.knu.edu.tw

(G.-H. Tzeng).

1 _{Distinguished Chair Professor.}

Contents lists available atScienceDirect

Expert Systems with Applications

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There are four steps in the DEMATEL method: (1) calculate the average matrix, (2) calculate the normalized initial direct-influence matrix, (3) derive thetotal relation matrix, and (4) set a threshold value and obtain the impact-relations map (inFig. 1, we divided Step 4 into two steps). In Step 4, an appropriate threshold value is necessary to obtain a suitable impact-relations map as well as adequate information for further analysis and decision-making. The traditional method followed to set a threshold value is con-ducting discussions with experts. The researcher sets an adequate threshold value and then outlines an impact-relations map to dis-cuss whether the impact-relations map is suitable for the structure of the problematique. If not, the threshold value is replaced by an-other value, and anan-other impact-relations map is obtained until there is a consistent opinion among the majority. Sometimes, after the researcher obtains the input data for Step 1 using question-naires, it is difficult to choose a consistent threshold value, espe-cially if there are too many experts to aggregate at the same time. When the factors of the problem are many, the work involved in obtaining a consistent threshold value becomes more complex. In order to obtain a reasonable threshold value with respect to dif-ficulty in the discussions with experts, the researcher may choose the value subjectively. The results of the threshold values may dif-fer among difdif-ferent researchers.

In contrast to the traditional method, which confronts the loop from a ‘‘set a threshold value” to obtain ‘‘the needed impact-rela-tions map”, as shown inFig. 1, we propose the maximum mean de-entropy (MMDE) algorithm to obtain a threshold value for delineating the impact-relations map. This algorithm based on the entropy approach can be used to derive a set of dispatch-nodes, the factors which strongly dispatch inﬂuences to others, and a set of receive-nodes, which are easily inﬂuenced by another factor. According to these two sets, a unique threshold value can be ob-tained for the impact-relations map.

In the numerical example, a real case is used to discover and illustrate the key services needed to attract Semiconductor Intel-lectual Property Mall (SIP) users and SIP providers to an SIP Mall. Research in the current study enabled the derivation of the interre-lated services and their structural interrelationships using the DEMATEL method, where the threshold value is selected through discussions with experts. By using the proposed MMDE algorithm to choose the threshold value, both impact-relations maps from the traditional method and the algorithm we propose are the same, although the procedures are different.

The rest of this paper is organized as follows: Section2brieﬂy describes the DEMATEL method. The steps of the maximum mean de-entropy algorithm will be described, explained, and discussed in Section3. In Section4, a numerical example, a real case where the goal is to ﬁnd out the interrelated services that should be pro-vided by a semiconductor intellectual properties mall and the structural interrelationship between them, is shown in order to ex-plain the proposed algorithm and discuss the results. Finally, in Section5, we draw conclusions.

2. DEMATEL method

The end product of the DEMATEL process—the impact-relations map—is a visual representation of the mind by which the respon-dent organizes his or her own action in the world. This organiza-tional process must occur for the respondent to keep internally coherent and to reach his or her personal goals. The steps of the DEMATEL method (Tzeng et al., 2006) are described as follows:

Step 1: Find the average matrix.Suppose there are h experts available to solve a complex problem and there are nfac-tors to be considered. The scores given by each expert give us a n n non-negative answer matrix Xk_{, with} 1 6 k 6 h. Thus X1, X2, . . . , Xhare the answer matrices for each of the h experts, and each element of Xk_{is an integer} denoted by xk

ij. The diagonal elements of each answer matrix Xk_{are all set to zero. We can then compute the} n n average matrix Aby averaging the h experts’ score matrices. The (i, j) element of matrix A is denoted by aij, aij¼ 1 h Xh k¼1 xk ij: ð1Þ

In application, respondents were asked to indicate the di-rect-inﬂuence that they believe each factor exerts on each of the others according to an integer scale ranging from 0 to 4. A high score from a respondent indicates a belief that greater improvement in i is required to improve j. From any group of direct matrices of respondents, it is possible to derive an average matrix A.

Step 2: Calculate the normalized initial direct-relation matrix.We then create a matrix D by using a simple matrix operation on A. Suppose we create matrix D and D = s A where s ¼ Min 1 max16i6nPnj¼1aij ; 1 max16j6nPni¼1aij " # : ð2Þ

Matrix D is called the normalized initial direct-relation ma-trix. The (i, j) element dijdenotes the direct-inﬂuence from factor xito factor xj. Suppose didenotes the row sum of the ith row of matrix D.

di¼ Xn

j¼1

dij: ð3Þ

The dishows the sum of inﬂuence directly exerted from factor xito the other factors. Suppose djdenotes the col-umn sum of the jth colcol-umn of matrix D.

dj¼ Xn

i¼1

dij: ð4Þ

Then djshows the sum of inﬂuence that factor xjreceived from the other factors. We can normalize diand djas wiðdÞ ¼ di Pn i¼1di ; ð5Þ

v

jðdÞ ¼ dj Pn j¼1dj : ð6Þ

Matrix D shows the initial inﬂuence which a factor exerts and receives from another. Each element of matrix D por-trays a contextual relationship among the elements of the system and can be converted into a visible structural mod-el—an impact-relations map—of the system with respect to that relationship. For example, as shown in Fig. 2, the respondents are requested to indicate only direct links. In the directed graph represented inFig. 2, factor i directly affects only factors j and k; while indirectly, it also affects Calculate the Average Matrix Raw Data from Experts Calculate the Normalized Initial Direct-Influence Matrix Derive the Total Relation Matrix Set a Threshold Value IsthisImpact-relations map Acceptable? Obtain an Impact-relations map The Needed Impact-relations map Yes No

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ﬁrst l, m, and n and, secondly, o and q. The digraph map helps to explain the structure of the factors.

Step 3: A continuous decrease of the indirect effects of problems along the powers of matrix D, e.g. D2_{, D}3_{, . . . , D}1_, guaran-tees convergent solutions to the matrix inversion, similar to an absorbing Markov chain matrix. Note that limm!1Dm¼ ½0nn, where [0]nnis the n n null matrix. The total relation matrix T is an n n matrix and is de-ﬁned as follows: X1 m¼1 Di¼ D þ D2þ D3þ þ Dm ¼ D I þ D þ D 2þ D3þ þ Dm1 ¼ DðI DÞ1_{ðI DÞ I þ D þ D} 2_{þ D}3_{þ þ D}m1

¼ DðI DÞ1ðI DmÞ ¼ DðI DÞ1; ð7Þ

where I is the identity matrix and T is called the total rela-tion matrix. The (i, j) element of the matrix T, tij, denotes the full direct- and indirect-inﬂuence exerted from factor xito factor xj. Like the formula(3)–(6), we can obtain ti, tj, wi(t), and

v

j(t).

Step 4: Set a threshold value and obtain the impact-relations map.

In order to explain the structural relationship among the factors while keeping the complexity of the system to a manageable level, it is necessary to set a threshold value p to ﬁlter out the negligible effects in matrix T. Using the values of wi(t) and

v

i(t) from the ma-trix of full direct/indirect-influence relations, the level of dispatch-ing and receivdispatch-ing of the influence of factor i can be defined. The interrelationship of each factor can be visualized as the oriented graphs on a two-dimensional plane after a certain threshold is set. Only those factors that have an effect in matrix T greater than the threshold value should be chosen and shown in an impact-rela-tions map.

In Step 4, the threshold value can be chosen by the decision ma-ker or through discussions with experts. If the threshold value is too low, the map will be too complex to show the necessary infor-mation for decision-making. If the threshold value is too high,

many factors will be presented as independent factors, without showing the relationships with other factors. Each time the thresh-old value increases, some factors or relationships will be removed from the map (an example based on a total relation matrix Texample is shown as formula(8)and inFig. 3). An appropriate threshold va-lue is necessary to obtain a suitable impact-relations map as well as adequate information for further analysis and decision-making.

Texample¼ 0:0093 0:0126 0:0538 0:0523 0:0759 0:0284 0:0077 0:0292 0:0284 0:0517 0:0509 0:0729 0:0087 0:0299 0:0341 0:0313 0:0340 0:0531 0:0086 0:0752 0:0532 0:0758 0:0547 0:0532 0:0150 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 : ð8Þ

3. Maximum mean de-entropy algorithm (MMDE)

As we mentioned above, the threshold value is determined by asking experts or by the researcher (as a decision maker). Choosing a consistent threshold value is time-consuming if the impact-rela-tions maps are similar when threshold values are changed slightly. If we consider the total relation matrix as a partially ordered set, the order relation is decided by the inﬂuence value. The question about deciding a threshold value is equal to a real point set divided into two subsets: one subset provides information on the obvious inter-dependent relationships of factors but the relationships are considered not so obvious in another subset. The proposed algo-rithm is a way to choose the ‘‘cut point”.

We propose the maximum mean de-entropy (MMDE) algorithm to ﬁnd a threshold value for delineating the impact-relations map. In this algorithm, we use the approach of entropy, which has been widely applied in information science, but deﬁne another two information measures: de-entropy and mean de-entropy. In addi-tion, the proposed algorithm mainly serves to search for the threshold value by nodes (or vertices). This algorithm differs from the traditional methods through which the threshold value is decided by searching a suitable impact-relations map.

In this section, we use the symbol j as the end of a deﬁnition or a step in the proposed algorithm.

3.1. Information entropy

Entropy is a physical measurement of thermal-dynamics and has become an important concept in the social sciences (Kartam, Tzeng, & Tzeng, 1993; Zeleny, 1981). In information theory, entro-py is used to measure the expected information content of certain messages, and is a criterion for the amount of ‘‘uncertainty” repre-sented by a discrete probability distribution.

Deﬁnition 1. Let a random variable with n elements be denoted as X = {x1, x2, . . . , xn}, with a corresponding probability P = {p1, p2, . . . , pn}, then we deﬁne the entropy, H, of X as follows:

H pð 1;p2; . . . ;pnÞ ¼ X pilg pi 3 4 5 1 2 P=0.3 3 4 5 1 2 P=0.53 3 4 5 1 2 P=0.7 P=0.75 4 5 1 2

Fig. 3. Impact-relations maps based on the same total relation matrix but different threshold values.

i

k j

n m l

q o

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subject to constraints(9) and (10): Xn

i¼1

pi¼ 1; ð9Þ

pilg pi¼ 0 if pi¼ 0: j ð10Þ

By Deﬁnition 1, the value of H(p1, p2, . . . , pn) is the largest when p1= p2= = pn and we denote this largest entropy value as H 1

n;1n; . . . ;1n

. Now we will deﬁne another measure for the de-creased level of entropy: de-entropy.

Definition 2. For a given finite discrete scheme of X, the de-entropy of X is denoted as HD_{and defined as:}

HD ¼ H 1 n; 1 n; . . . ; 1 n H pð 1;p2; . . . ;pnÞ j

By Definition 2, the value of HD_{is equal to or larger than 0.} Un-like entropy, which is used for the measure of uncertainty, the HD can explain the amount of useful information derived from a spe-cific dataset, which reduces the ‘‘uncertainty” of information. We define the de-entropy for searching the threshold value in order to assess the effect of information content when adding a new node to an existing impact-relations map. By Definition 1, formula (11) can be proven (the proof can be found in (Khinchin, 1957)):

Hn¼ H 1 n; 1 n; . . . ; 1 n 6H 1 n þ 1; 1 n þ 1; . . . ; 1 n þ 1 ¼ Hnþ1 ð11Þ

Formula(11)explains that when adding a new variable to a sys-tem where all variables in the syssys-tem have the same probability, the entropy of the system will increase.

To delineate an impact-relations map, if adding a new factor to the impact-relations map can make the system less uncertain, or lead to more de-entropy, then the new factor provides worthwhile information for a decision maker. In other words, in an existing information system whose variables and corresponding probabili-ties have been ﬁxed, adding a new variable to the system will change the probability distribution; if HDnþ1>H

D

n exists, then this new variable provides useful information to avoid uncertainty for the decision maker.

3.2. The dispatch- and receive-nodes

In the DEMATEL method, the total relation matrix is the matrix used to delineate the final output of the DEMATEL method, the im-pact-relations map, after the threshold value is determined. As in the notation in Section2, an n n total relation matrix is denoted as T. The (i, j) element of the matrix T, tij, refers to the full direct-and indirect-influence exerted from factor xito factor xj. Like the ‘‘vertices” and ‘‘edges” in graph theory (Agnarsson & Greenlaw, 2007), xiand xjare vertices in the directed graph impact-relations map, and tijcan be considered as a directed edge which connects factors xiand xjwith an influence value. In an impact-relations map, every factor may influence, or be influenced by, another fac-tor, or both.

Definition 3. The (i, j) element of the matrix T is denoted as tijand refers to a directed influence relations from factor xito factor xj. For each tij, the factor xiis defined as a dispatch-node and factor xjis defined as a receive-node with respect to tij. j

By Deﬁnition 2, an n n total relation matrix T can be consid-ered as a set (set T) with n2pair ordered elements. Every subset of set T can be divided into two sets: an ordered dispatch-node set and an ordered receive-node set. For an ordered dispatch-node set (or an ordered receive-node set), we can count the frequency of

the different elements of the set. If the ﬁnite cardinality of an order dispatch-node set (or an ordered receive-node set) is m and the fre-quency of element xiis k, we assign the corresponding probability of xias pi¼mk. In this way, for an ordered set, we can assign each dif-ferent element a probability and follow Deﬁnition 1 forPn_i¼1pi¼ 1. Notation. In this paper, C(X) denotes the cardinal number of an ordered set X and N(X) denotes the cardinal number of different elements in set X. For example, if X = {1, 2, 2, 3, 1},C(X) = 5 and N(X) = 3.

3.3. Maximum mean de-entropy algorithm

Based on a calculated total relation matrix T, the steps of the proposed maximum mean de-entropy algorithm for determining a threshold value are described as follows:

Step 1: Transforming the n n total relation matrix T into an ordered set T, {t11,t12, . . . , t21,t22, . . . , tnn}, rearranging the ele-ment order in set T from large to small, and transforming to a corresponding ordered triplets (tij, xi,xj) set denotes T*. j Every element of set T, tij, can also be considered as an ordered triplet (tij, xi,xj) as (inﬂuence value, dispatch-node, receive-node). As the matrix Texample_{of the example} men-tioned above, the transformed and rearranged set, Texample_, is {0.0759, 0.0758, 0.0752, . . . , 0.0077}. The ordered triplets set is {(0.0759,1, 5), (0.0758, 5, 2), (0.0752,4, 5), . . . , (0.0077, 2, 2)} and the cardinal number of T*example_, C(T*example_{), is 25.}

Step 2: Taking the second element, the dispatch-node, from the ordered triplets of the set T*_{and then obtaining a new} ordered dispatch-node set, TDi_. _j

According to the set T*_{, we can derive the corresponding} ordered dispatch-node set. As the set T*example _{of the} example in Step 1, the ordered dispatch-node set TDi_is {1, 5, 4, . . . , 2} and C(TDi_{) is also 25.}

Step 3: Taking the ﬁrst t elements of TDi_{as a new set T}Di t, assign the probability of different elements, and then calculate the HDof the set TDit ;H

Di

t. We can calculate the mean de-entropy by MDEDi t ¼ HDi t NðTDi tÞ

. At ﬁrst, the t is set as 1, then of value of t is determined by raising the value from 1 to C(TDi_{) in increments of 1. j}

Why we use HDit NðTDi tÞ

as ‘‘mean de-entropy” rather than HDit CðTDi tÞ must be clariﬁed. Regardless of how many times a dis-patch-node repeats in a set TDi

t , this dispatch-node will show in the impact-relations map only once if we use this TDit to draw the impact-relations map. The H

Di t is the de-entropy of NðTDi

t Þ dispatch-nodes in the impact-relations map, not CðTDi

t Þ dispatch-nodes. In this step, we can obtain C(TDi_{) mean de-entropy values. As the} set T*example_{, we will obtain 25 mean de-entropy values.} Step 4: In C(TDi_{) mean de-entropy values, select the maximum}

mean de-entropy and its corresponding TDi_t. This dis-patch-node set, with the maximum mean de-entropy, is denoted as TDi

max. j

Step 5: Similar to Steps 2–4, an ordered receive-node set TRe_and a maximum mean de-entropy receive-node set TRemaxcan be derived. j

Step 6: Taking the ﬁrst u elements in T*_{as the subset, T}Th_{, which} includes all elements of TDimaxin the dispatch-node and all elements of TRe

maxin the receive-node, the minimum inﬂu-ence value in TTh_{is the threshold value, and formula}₍₁₂₎ holds

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In Step 6, the elements of TDi

maxare the ‘‘more important” factors which provide more information about inﬂuence dispatching for a decision maker than other factors. The elements of TRemax provide information on which are easily inﬂuenced. If we use the ordered triplets TTh_;_TDi

max, and T Re

max in the structured directed graphs GðTThÞ; GðTDimaxÞ and GðT

Re

maxÞ, formula(13)holds. GðTTh

Þ ¼ GðTDimaxÞ [ GðT Re

maxÞ ð13Þ

with the property of G TDimax ¼ G TRemax or G TDimax #G TRemax or G TDimax G TRemax : If G TDi max ¼ G TRe max

, then G(TTh_{) is the perfect directed graph} for the impact- relations map with both the maximum mean de-en-tropy dispatch-node set and receive-node set. If G TDimax

#G TRemax or G TDi max G TRemax

, then the structured G(TTh_{) is the minimum} impact-relations map which includes the necessary maximum mean de-entropy dispatch- and receive-node sets.

Based on Texample, the results from Steps 1 to 6 are shown in Ta-ble 1.

4. Numerical case of the semiconductor intellectual property mall

In this section, a real case is shown by using the maximum mean de-entropy algorithm to set the threshold value. The original threshold value was determined through discussions with experts. By using the maximum mean de-entropy algorithm, the threshold value is different from the original result, but the impact-relations maps from these two threshold values are similar.

4.1. The semiconductor intellectual property mall case

SIP design, a new industry, is rapidly growing, which challenges both providers and users to develop infrastructure and standard

interfaces. Establishing an SIP mall to provide a full array of SIP business services is a new concept used to promote growth of the SIP industry. Many foundries and governments have been in-volved in setting up SIP malls; however, the major services needed for an SIP mall to attract SIP providers and users must be clariﬁed. After we discussed and revised the questionnaire with experts, eighteen interrelated services, denoted from x1 to x18, were in-cluded in the ﬁnal questionnaire. Twenty-four companies agreed to answer the questionnaire and discuss their responses. These 24 companies were experienced as licensees and licensors in the SIP business and had extensive knowledge about SIP trading and licensing. The DEMATEL method was used to discover and illus-trate the key services needed to attract SIP users and providers to an SIP mall. Next, a total relation matrix was obtained from the nineteen 18 18 weighted matrices, shown inFig. 4.

Based on the matrix T, the maximum threshold value that al-lowed all services to be displayed on the impact-relations map was 0.36. When the threshold value increased to 0.45, only two di-rect relationships existed. The threshold value was determined by raising the threshold value from 0.36 to 0.45 in increments of 0.01 and conferring with experts in order to determine the optimal va-lue to sufﬁciently display the interrelationships among these ser-vices. The threshold value was then set at 0.42, and the structured impact-relations map is shown inFig. 5.

4.2. Maximum mean de-entropy algorithm results

Following the steps in Section 3.3, we obtained the results shown below:

Step 1: After transforming the total relation matrix T, shown in

Fig. 4, the ordered triplets set T* _{was obtained as} {(0.4612, 13, 14), (0.4587, 1, 14), (0.4489, 15, 14), (0.4357, 1, 13), (0.4355, 13, 17), . . . , (0.2051, 2, 2)}.

Step 2: According to the results of Step 1, the ordered dispatch-node set TDi _can _be _derived _as _{{13, 1, 15, 1, 13,} 16, . . . , 7, 4, 6, 7, 7, 2}.

Table 1

The results from Step 1 to Step 6.

Item Data

Step 1: The ordered triplets set T*example

{(0.0759, 1, 5), (0.0758, 5, 2), (0.0752, 4, 5), (0.0729, 3, 2), (0.0547, 5, 3), (0.0538, 1, 3), (0.0532, 5, 1), (0.0532, 5, 4), (0.0531, 4, 3), (0.0523, 1, 4), (0.0517, 2, 5), (0.0509, 3, 1), (0.0341, 3, 5), (0.0340, 4, 2), (0.0313, 4, 1), (0.0299, 3, 4), (0.0292, 2, 3), (0.0284, 2, 1), (0.0284, 2, 4), (0.0150, 5, 5), (0.0126, 1, 2), (0.0093, 1, 1), (0.0087, 3, 3), (0.0086, 4, 4), (0.0077, 2, 2)} Step 2: Dispatch-node set, TDi {1, 5, 4, 3, 5, 1, 5, 5, 4, 1, 2, 3, 3, 4, 4, 3, 2, 2, 2, 5, 1, 1, 3, 4, 2} Step 3.1: TDi t sets and MDEDi t values TDi

1 ¼ f1g; MDEDi1 ¼ 0; TDi2 ¼ f1; 5g; MDEDi2 ¼ 0; TDi3 ¼ f1; 5; 4g; MDEDi3 ¼ 0; TDi4 ¼ f1; 5; 4; 3g; MDEDi4 ¼ 0; TDi5 ¼ f1; 5; 4; 3; 5g; MDEDi5 ¼ 0:0135; . . . ; TDi 25¼ f1; 5; 4; 3; 5; 1; 5; 5; 4; 1; 2; 3; 3; 4; 4; 3; 2; 2; 2; 5; 1; 1; 3; 4; 2g; MDEDi25¼ 0; Step 3.2: Set of 25 MDEDit values {0, 0, 0, 0, 0.0135, 0.0142, 0.0273, 0.0433, 0.0283, 0.0266, 0.0283, 0.0185, 0.0169, 0.0145, 0.0160, 0.0165, 0.0060, 0.0019, 0.0012, 0.0025, 0.0009, 0.0012, 0.0012, 0.0007, 0} Step 4.1: Maximum MDEDi t 0.0433 Step 4.2: Dispatch-node set of maximum MDEDi t {1, 5, 4, 3, 5, 1, 5, 5} = {1, 3, 4, 5} Step 5.1:Receive-node set, TRe {5, 2, 5, 2, 3, 3, 1, 4, 3, 4, 5, 1, 5, 2, 1, 4, 3, 1, 4, 5, 2, 1, 3, 4, 2} Step 5.2: Set of 25 MDERe t values {0, 0, 0.0283, 0, 0.0146, 0, 0.0086, 0.0099, 0.0173, 0.0105, 0.0126, 0.0041, 0.0089, 0.0071, 0.0045, 0.0015, 0.0020, 0.0019, 0.0012, 0.0025, 0.0009, 0.0012, 0.0012, 0.0007, 0} Step 5.3:Maximum MDERe t 0.0283 Step 5.4: Receive-node

set of the maximum MDERet

{5, 2, 5}= {2, 5}

Step 6.1:TDi

max {(0.0759, 1 , 5), (0.0758, 5 , 2), (0.0752, 4 , 5), (0.0729, 3 , 2)} (the nodes in shaded box is the needed dispatch-nodes shown at ﬁrst time in the ordered

set) Step 6.2: TRe

max {(0.0759, 1, 5 ), (0.0758, 5, 2 )} (the nodes in shaded box is the needed receive-nodes shown at ﬁrst time in the ordered set)

Step 6.3: TTh

{(0.0759, 1, 5), (0.0758, 5, 2), (0.0752, 4, 5), (0.0729, 3, 2)} Step 6.4: Threshold

value

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Step 3: Based on the set TDi_{, a collection of sets T}Di

t, in which t is from 1 to 324, can be obtained. After we calculate all of the HD_{values of the sets T}Di

t, we can obtain a set with 324 mean de-entropy values, {0, 0, 0, 0.0196, 0.0146, 0.0142, . . . , 0, 0, 0}, shown inFig. 6.

Step 4: Within the set obtained in Step 3, the maximum mean de-entropy value is 0.0485 and the corresponding dis-patch-node set is {13, 1, 15, 1, 13, 16, 15, 1, 13, 13, 15, 15, 13, 13, 1, 9, 15, 3, 1, 1, 1}.

Step 5: Similar to Steps 2–4, the ordered receive-node set TRe, the de-entropy value set of TRe_{, a maximum mean de-entropy} value, and corresponding receive-node set TRe

max are shown inFig. 7andTable 2.

Step 6: According to the results of Steps 4 and 5, the elements {1, 3, 9, 13, 15, 16} must be the dispatch-nodes and the elements {13, 14, 17} must be the receive-nodes in the impact-relations map. Based on these two constraints, the needed subset, TTh_{, of the ordered set T}* _is {(0.4612,13,14), (0.4587, 1 , 14), (0.4490,15, 14), (0.4357, 1,13), (0.4355, 13,17), (0.4351,16, 14), (0.4329, 15, 17), (0.4328, 1, 15), (0.4313, 13, 15), (0.4312, 13, 9), (0.4304, 15, 13), (0.4266, 15, 16), (0.4265, 13, 16), (0.4263, 13, 11), (0.4252, 1, 9), (0.4222,9 , 14), (0.4213, 15, 9), (0.4201, 3 , 13)}. In above

set TTh_{, the nodes in the shaded box are the needed} dis-patch-nodes shown the first time in the ordered set TTh_, the nodes in the non-shaded box are the needed dis-patch-nodes shown the first time in the ordered set TTh_, and the minimum influence value in TTh_{is the threshold} value, 0.4201.

Based on the subset obtained in Step 6, the threshold value could be determined as 0.4201 and then the impact-relations map can be structured. In this case, the impact-relations map de-rived from the MMDE algorithm is same as that shown inFig. 5. 4.3. Discussion

The premise of the DEMATEL method is that the factors are not totally pair-wise independent. One important reason for using the DEMATEL method to solve a speciﬁc problematique is to under-stand the interrelations between factors and expressing the rela-tionships in a directed graph. If all information in the total relation matrix is displayed in the impact-relations map, then the impact-relations map is deﬁned as a ‘‘complete graph” in graph

0.3416 0.2903 0.3741 0.3153 0.3167 0.3221 0.2748 0.3195 0.3562 0.3336 0.3609 0.3177 0.4116 0.3395 0.4075 0.3815 0.3205 0.3656 0.2946 0.2051 0.2745 0.2693 0.2488 0.2331 0.2178 0.2547 0.2713 0.2561 0.2765 0.2517 0.3088 0.2762 0.2951 0.2979 0.2582 0.2653 0.3938 0.2991 0.3136 0.3099 0.2980 0.2879 0.2815 0.3392 0.3646 0.3102 0.3496 0.3022 0.4023 0.3231 0.3902 0.3648 0.3098 0.3539 0.3161 0.2688 0.3129 0.2399 0.2663 0.2567 0.2428 0.2739 0.3039 0.2779 0.3203 0.2636 0.3420 0.2984 0.3194 0.3174 0.2774 0.2946 0.3927 0.3025 0.3604 0.3263 0.2637 0.2863 0.2709 0.3172 0.3585 0.3311 0.3348 0.3077 0.3997 0.3443 0.3890 0.3685 0.3208 0.3595 0.3936 0.2889 0.3454 0.3067 0.2932 0.2491 0.2754 0.3035 0.3294 0.3204 0.3456 0.2953 0.3820 0.3293 0.3746 0.3688 0.2995 0.3346 0.3230 0.2546 0.3103 0.2781 0.2558 0.2570 0.2069 0.2629 0.2889 0.2813 0.2960 0.2513 0.3356 0.2807 0.3340 0.3373 0.2600 0.2944 0.3736 0.2929 0.3714 0.3164 0.2965 0.2869 0.2743 0.2760 0.3460 0.3138 0.3346 0.2993 0.3816 0.3319 0.3824 0.3664 0.3072 0.3392 0.4252 0.3411 0.4148 0.3556 0.3411 0.3210 0.3107 0.3732 0.3349 0.3511 0.3776 0.3280 0.4312 0.3754 0.4213 0.3950 0.3434 0.3867 0.3969 0.3124 0.3708 0.3362 0.3132 0.3105 0.2981 0.3335 0.3683 0.2900 0.3509 0.3120 0.4067 0.3548 0.4022 0.3807 0.3305 0.3535 0.4188 0.3223 0.3990 0.3645 0.3346 0.3190 0.3206 0.3511 0.3883 0.3501 0.3209 0.3194 0.4263 0.3565 0.4060 0.3856 0.3298 0.3683 0.3984 0.3175 0.3735 0.3401 0.3192 0.3153 0.2879 0.3392 0.3700 0.3361 0.3493 0.2705 0.4029 0.3511 0.3932 0.3700 0.3268 0.3550 0.4357 0.3369 0.4201 0.3579 0.3363 0.3338 0.3139 0.3736 0.4029 0.3594 0.3876 0.3434 0.3774 0.3639 0.4304 0.3971 0.3449 0.3869 0.4587 0.3644 0.4162 0.3846 0.3535 0.3477 0.3333 0.3865 0.4222 0.3823 0.4092 0.3535 0.4612 0.3410 0.4490 0.4351 0.3702 0.4028 0.4328 0.3169 0.4056 0.3447 0.3389 0.3278 0.3122 0.3650 0.3909 0.3678 0.3764 0.3350 0.4313 0.3641 0.3672 0.4116 0.3483 0.3851 0.4192 0.3211 0.3809 0.3338 0.3213 0.3259 0.3191 0.3643 0.3777 0.3636 0.3767 0.3286 0.4265 0.3797 0.4266 0.3440 0.3423 0.3723 0.4185 0.3342 0.3941 0.3602 0.3422 0.3266 0.3111 0.3619 0.3892 0.3651 0.3717 0.3414 0.4355 0.3831 0.4329 0.4061 0.2985 0.3766 0.3791 0.2947 0.3690 0.3058 0.3011 0.2911 0.2728 0.3308 0.3454 0.3227 0.3396 0.3014 0.3826 0.3287 0.3964 0.3553 0.3037 0.2936

T =

Fig. 4. The total relation matrix of the SIP mall case.

x1 x3 x15 x13 x11 x9 x16 x17 x14 0.4328 0.4252 0.4201 0.4357 0.4587 0.4263 0.4312 0.4612 0.4222 0.4265 0.4266 0.4351 0.4490 0.4329 0.4355

Fig. 5. Impact-relations map based on the threshold value p = 0.42.

50 100 150 200 250 300 0.01 0.02 0.03 0.04 Maximum Mean De - Entropy of Dispatch Node = 0.0485

Fig. 6. 324 mean de-entropy values with a maximum mean de-entropy value of 0.0485.

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theory, and every distinct vertex is connected by an edge. For a decision maker, this is no better than a situation which has no information. Another purpose for using DEMATEL is to avoid too much useless information. Selecting an adequate threshold value to judge whether a relation is obvious is a key question for the DEMATEL method. The proposed MMDE algorithm has some prop-erties that differ from the traditional method to make the thresh-old value, as discussed below.

4.3.1. The MMDE mainly serves to decide the ‘‘node” rather than the ‘‘map”

Using traditional methods, the main issue of discussion is whether a ‘‘map” is suitable to the problematique after a threshold value is set. In traditional methods, the researcher set a subject adequate threshold to draw the impact-relations map and

dis-cussed it with experts to obtain a consistent opinion. If experts are not in agreement on the results, the researcher increases or de-creases the threshold value to create another impact-relations map and again discusses it with experts until a consistent impact-rela-tions map is accepted by experts and the final threshold value is set. In the SIP mall example, the traditional method involves find-ing a ‘‘suitable” threshold value, 0.36, and then raisfind-ing the thresh-old value from 0.36 to 0.45 in increments of 0.01 and conferring with experts about the impact-relations map corresponding to each value, finally determining the optimal value to be 0.42.

In the proposed MMDE, the main issue is about whether it is suitable to add a new ‘‘node”. If adding a new node can improve the ‘‘mean de-entropy”, then adding it can be helpful to understand a problematique by decreasing the uncertainty of information. Using MMDE, we first decide that the nodes 1, 3, 9, 13, 15 and 16 have to be the dispatch-nodes in the impact-relations map while the nodes 13, 14, and 17 have to be the receive-nodes in the map, and then set the threshold value at 0.4201. The processes and results for these two methods are different, but the impact-relations maps are same. This means that MMDE is a suitable method to determine a threshold value in the first, or the final, step in order to discuss the adequacy of the impact-relations map. 4.3.2. The MMDE considers the properties of both the dispatch and receive influences of a factor

In the DEMATEL method, after a suitable map is obtained, the focus of the problem can be shown by analyzing the values w(i and

v

i, as formulas(5) and (6), of the factors in the map. The value (wi+

v

i)—the index representing the strength of the inﬂuence of both dispatched and received—shows the central role that factor i plays in the problem. Commonly, a higher (wi+

v

i) value means that the factor has a stronger connection with the other factors and plays a central role. A higher (wi

v

i) value means that this factor has a stronger inﬂuence on other factors than the inﬂuence it receives from them. If (wi

v

i) is positive, then factor i is

50 100 150 200 250 300 0.02 0.04 0.06 Maximum Mean De -Entropy of Receive Node = 0.0770

Fig. 7. 324 mean entropy values of receive-nodes set with a maximum mean de-entropy value of 0.0770.

Table 2

The results derived from Steps 2 to 6 using the MMDE algorithm.

Item Data Receive-node set, TRe {14, 14, 14, 13, 17, 14, 17, 15, 15, 9, 13, 16, 16, 11, 9, 14, 9, 13, 16, 11,17, 14, 9, 1, 15, 14, 1, 10, 17, 15, 11, 13, 3, 12, 14, 10, 5, 11, 12, 18, 10, 13, 9, 6, 3, 17, 5, 12, 15, 3, 17, 5, 11, 13, 9, 13, 14, 15, 11, 14, 17, 8, 18, 14, 6, 1, 8, 10, 16, 16, 18, 9, 16, 16, 17, 15, 9, 6, 13, 1, 13, 9, 12, 8, 16, 8, 17, 10, 14, 12, 12, 18, 6, 5, 10, 15, 11, 8, 1, 11, 17, 15, 3, 14, 3, 16, 15, 16, 15, 13, 17, 1, 17, 5, 5, 13, 5, 13, 11, 1, 9, 10, 18, 12, 3, 14, 10, 14, 12, 11, 9, 10, 11, 3, 12, 15, 14, 6, 8, 18, 6, 13, 5, 15, 13, 9, 17, 16, 4, 17, 9, 16, 9, 12, 3, 18, 12, 8, 1, 15, 1, 7, 14, 13, 10, 13, 12, 7, 15, 11, 6, 8, 7, 17, 5, 13, 1, 14, 10, 16, 9, 8, 5, 18, 10, 11, 6, 6, 18, 16, 15, 9, 12, 5, 17, 16, 1, 18, 7, 3, 11, 11, 4, 5, 16, 16, 9, 6, 1, 16, 12, 11, 11, 1, 4, 11, 1, 12, 4, 1, 5, 8, 15, 12, 4, 1, 13, 8, 10, 4, 10, 15, 10, 3, 17, 9, 10, 7, 3, 3, 3, 2, 5, 8, 6, 18, 4, 18, 6, 5, 3, 18, 18, 8, 6, 3, 4, 10, 2, 3, 8, 17, 7, 6, 2, 18, 7, 4, 2, 6, 8, 18, 18, 1, 6, 7, 12, 10, 3, 8, 5, 7, 3, 7, 7, 4, 4, 2, 2, 6, 8, 1, 2, 8, 4, 18, 2, 5, 2, 4, 12, 4, 2, 4, 7, 5, 7, 2, 7, 4, 2, 7, 7, 2, 2, 7, 2, 6, 4, 4, 2, 2, 7, 2}

Mean de-entropy value set of TRe {0, 0, 0, 0.0654, 0.0494, 0.0770, 0.0476, 0.0433, 0.0283, 0.0277, 0.0187, 0.0193, 0.0133, 0.0141, 0.0099, 0.0156, 0.0152, 0.0143, 0.0130, 0.0078, 0.0063, 0.0098, 0.0099, 0.0077, 0.0119, 0.0149, 0.0106, 0.0133, 0.0132, 0.0117, 0.0114, 0.0135, 0.0133, 0.0153, 0.0163, 0.0139, 0.0147, 0.0145, 0.0125, 0.0128, 0.0119, 0.0128, 0.0130, 0.0133, 0.0115, 0.0123, 0.0116, 0.0101, 0.0102, 0.0095, 0.0099, 0.0092, 0.0093, 0.0096, 0.0101, 0.0103, 0.0111, 0.0112, 0.0114, 0.0121, 0.0124, 0.0110, 0.0120, 0.0128, 0.0115, 0.0102, 0.0095, 0.0091, 0.0087, 0.0085, 0.0078, 0.0079, 0.0080, 0.0083, 0.0084, 0.0087, 0.0088, 0.0090, 0.0082, 0.0078, 0.0070, 0.0073, 0.0069, 0.0071, 0.0072, 0.0075, 0.0070, 0.0067, 0.0072, 0.0069, 0.0067, 0.0062, 0.0058, 0.0053, 0.0051, 0.0051, 0.0051, 0.0053, 0.0050, 0.0047, 0.0048, 0.0050, 0.0045, 0.0042, 0.0042, 0.0045, 0.0046, 0.0048, 0.0049, 0.0050, 0.0052, 0.0050, 0.0046, 0.0048, 0.0046, 0.0047, 0.0046, 0.0048, 0.0048, 0.0046, 0.0047, 0.0043, 0.0042, 0.0040, 0.0037, 0.0037, 0.0039, 0.0041, 0.0041, 0.0040, 0.0041, 0.0041, 0.0041, 0.0039, 0.0039, 0.0040, 0.0042, 0.0039, 0.0035, 0.0033, 0.0030, 0.0031, 0.0032, 0.0031, 0.0031, 0.0032, 0.0033, 0.0033, 0.0034, 0.0058, 0.0057, 0.0058, 0.0059, 0.0060, 0.0062, 0.0061, 0.0059, 0.0058, 0.0056, 0.0055, 0.0054, 0.0055, 0.0075, 0.0076, 0.0078, 0.0077, 0.0077, 0.0070, 0.0071, 0.0072, 0.0071, 0.0071, 0.0069, 0.0067, 0.0068, 0.0063, 0.0064, 0.0064, 0.0064, 0.0063, 0.0065, 0.0064, 0.0063, 0.0061, 0.0061, 0.0061, 0.0060, 0.0059, 0.0058, 0.0058, 0.0058, 0.0059, 0.0059, 0.0060, 0.0059, 0.0060, 0.0058, 0.0055, 0.0054, 0.0054, 0.0053, 0.0054, 0.0054, 0.0055, 0.0055, 0.0055, 0.0055, 0.0055, 0.0054, 0.0048, 0.0047, 0.0048, 0.0043, 0.0043, 0.0044, 0.0045, 0.0045, 0.0045, 0.0041, 0.0041, 0.0041, 0.0041, 0.0040, 0.0037, 0.0038, 0.0038, 0.0038, 0.0038, 0.0037, 0.0036, 0.0034, 0.0034, 0.0034, 0.0034, 0.0034, 0.0035, 0.0035, 0.0032, 0.0031, 0.0030, 0.0030, 0.0052, 0.0051, 0.0051, 0.0050, 0.0049, 0.0047, 0.0047, 0.0046, 0.0046, 0.0046, 0.0045, 0.0045, 0.0045, 0.0044, 0.0044, 0.0044, 0.0043, 0.0043, 0.0043, 0.0038, 0.0038, 0.0035, 0.0035, 0.0031, 0.0031, 0.0028, 0.0027, 0.0025, 0.0025, 0.0025, 0.0025, 0.0025, 0.0025, 0.0025, 0.0025, 0.0023, 0.0023, 0.0023, 0.0023, 0.0023, 0.0023, 0.0022, 0.0021, 0.0020, 0.0019, 0.0018, 0.0016, 0.0014, 0.0014, 0.0014, 0.0014, 0.0012, 0.0012, 0.0011, 0.0011, 0.0010, 0.0010, 0.0010, 0.0009, 0.0008, 0.0008, 0.0007, 0.0007, 0.0006, 0.0006, 0.0005, 0.0004, 0.0004, 0.0003, 0.0003, 0.0002, 0.0002, 0.0002, 0.0001, 0, 0, 0, 0, 0, 0, 0, 0, 0}

Maximum mean de-entropy value

0.0770 The receive-node set of

the maximum mean de-entropy value

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inﬂuencing the other factors. Different (wi+

v

i) and (wi

v

i) values will be explained along with the structure of the factors’ effects.

Using the proposed MMDE, we search the nodes, including dis-patch- and receive-nodes, simultaneously. The MMDE not only considers the factors which strongly influence others, but also the factors which are easily influenced by other factors. The results obtained through the proposed algorithm follow the goals of the DEMATEL in finding out the interrelationships of ‘‘important” fac-tors for allocating resources efficiently.

4.3.3. The MMDE can obtain a unique threshold value

To create a total relation matrix, the threshold value is deter-mined through discussions with respondents or subjectively by the researcher, so the threshold value may differ if the experts or the researcher change. In the traditional method, the researcher may determine the threshold value by decreasing the value (this will change the impact-relations map from simple to complex) or by increasing the value (this will change the impact-relations map from complex to simple), so the results of these two methods may different. If too many factors are included, the problematique becomes too complex. Using the MMDE, a researcher can obtain a unique threshold value, which is helpful to solve the problem a re-searcher confronts in regards to selecting a consistent threshold value.

5. Conclusions

In the DEMATEL process, an appropriate threshold value is important in order to obtain adequate information to delineate the impact-relations map for further analysis and decision-making. Until now, the threshold value has been determined through dis-cussions with respondents or chosen subjectively by researchers. It is time-consuming to make a consistent decision on the thresh-old value, especially when the number of factors in the problema-tique makes it too difﬁcult to discuss the adequacy of an impact-relations map. If the threshold is determined by the researcher alone, it is important to clarify how to choose the speciﬁc value. A theoretical method to aid in deciding the threshold value is necessary.

This paper proposed an MMDE algorithm to determine the threshold value. The MMDE uses the approach of entropy, but also uses two other measures for the stability of information: ‘‘de-en-tropy” and ‘‘mean de-en‘‘de-en-tropy”. MMDE is mainly used to decide whether a node is suitable to express in the impact-relations map. With this method, a unique threshold value can be obtained, solving the problem of choosing the threshold value in the tradi-tional way. In the numerical example, we show that the results from the MMDE are the same as the traditional method.

In future research, we will aim to apply this algorithm to other areas in information science and data mining in order to measure ‘‘adequate information”, especially when faced with concerns about ‘‘too much information to make a decision”.

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