To probe into the interrelationship among biometrics and the influential weights of dimensions/criteria when evaluating biometrics are the main issues of this paper. Therefore, to provide decision makers with a useful decision model to make the optimal decisions, a hybrid MCDM model is used to take multiple criteria into consideration at the same time. To construct the INRM for administrators of biometrics to design and make suitable biometric systems, the method of DEMATEL is adopted. Afterwards, the influential weights of criteria can be calculated. The proposed methods from the experts’ point of view mainly provide administrators of biometrics with an evaluation model to improve the systems, and thus the preferences of normal customers cannot be understood. Also, the additionally suitable criteria should be taken into consideration as the progress of technology to make the evaluation model more effective. The main stages can be introduced as follows.
Constructing the INRM by DEMATEL
The DEMATEL technique is employed to explore the interdependent and feedback problems among criteria for building the INRM25. This method has been practically utilized to decision-making problems of various fields, including solar farms, portfolio selection, online reputation, search engine optimization (SEO), and so on26,27,28,29.
The method is presented as follows: first, the influence matrix is obtained by scores. The experts are required to point out the degrees of influence among criteria in questionnaire sheet; i.e., to indicate how much the criteria affect each other. The influence matrix A can thus be acquired. Second, the normalized influence matrix H can be calculated by using Eqs. (1) and (2) to normalize A.
$$ {varvec{H}}{ = }m cdot {mathbf{A}} $$
(1)
$$ m = min left{ {frac{1}{{max_{i} sumnolimits_{j = 1}^{n} {|a_{ij} |} }},frac{1}{{max_{j} sumnolimits_{i = 1}^{n} {|a_{ij} |} }}} right} $$
(2)
Thirdly, the total influence matrix ({varvec{T}}) can be obtained by utilizing the formula, ({varvec{T}} = user2{H + H}^{2} + {varvec{H}}^{3} { + } cdots + {varvec{H}}^{q} { = }{varvec{H}}{(}{varvec{I}}{ – }{varvec{H}}{)}^{ – 1}), when (lim_{q to infty } {varvec{H}}^{q} = [0])(_{n times n} ,) where I denotes the identity matrix. The fourth step: definition of the INRM through the vectors r and d, which are defined the sum of the rows and the sum of the columns separately within the total influence matrix (user2{T = }{[}t_{ij} {]}_{n times n}) via the Eqs. (3) and (4) then
$$ {varvec{r}} = [r_{i} ]_{n times 1} = left[ {sumlimits_{j = 1}^{n} {t_{ij} } } right]_{n times 1} $$
(3)
$$ {varvec{d}} = [d_{j} ]_{n times 1} = left[ {sumlimits_{i = 1}^{n} {t_{ij} } } right]^{prime }_{1 times n} $$
(4)
where the superscript (^{prime}) represents transpose. If (r_{i}) stands for the row sum of the ith row in matrix ({varvec{T}}), then (r_{i}) displays the sum of direct and indirect influences of criterion (i) on the all other criteria. And, if (s_{j}) represents the column sum of the jth column of matrix ({varvec{T}}), then (s_{j}) shows the sum of direct and indirect receive the effects that criterion (j) is received the effects from the all other criteria. Moreover, when (i = j) the sum of the row and column aggregates ((r_{i} + d_{i} )), it exhibits the giving and received degree of influences; i.e., ((r_{i} + d_{i} )) presents the intensity of the important role that the ith criterion plays in the problem. When ((r_{i} – d_{i} )) is positive, the ith criterion affects other criteria. However, if ((r_{i} – d_{i} )) is negative, other criteria influence the ith criterion (i.e., ith criterion to be influenced from other criteria). And thus the INRM can be established for analyzing how to improve and set improvement strategies.
Obtaining criteria’s influential weights by using the DANP
Decision makers almost consider multiple criteria and determine the relative influential weights of criteria when evaluating performance30. We can use the DEMATEL technique to build the interacting relationship of normalized influential matrix ({varvec{T}}_{c}^{alpha }) by dimensions in criteria, then we can transpose the normalized influential matrix ({varvec{T}}_{c}^{alpha }); the unweighted supermatrix ({varvec{W}}{ = (}{varvec{T}}_{c}^{alpha } {)^{prime}}) can be obtained the most accurate weight of influence using the basic concept of ANP. We use the normalized of influential matrix ({varvec{T}}_{D}^{alpha }) of dimensions as weighting with unweighted supermatrix ({varvec{W}}), the weighted supermatrix ({varvec{W}}^{alpha }) can be obtained. Finally we can multiply by itself several times, until the supermatrix has converged and become a long-term stable supermatrix to a sufficiently large power z by (mathop {lim }limits_{z to infty } ;({mathbf{W}}^{alpha } )^{z}) to deal with problems of dependence and feedback among criteria for obtaining the influential weights of criteria. Thus, DANP (DEMATEL–based ANP) contains the following steps.
The DANP consists of four steps. In the first step the influence network structure based can be constructed on DEMATEL technique. Secondly, obtain the unweighted supermatrix. The total influence matrix (T_{{{}_{c}}}^{{}}) displayed in Eq. (5) is derived from DEMATEL.
(5)
Use the total degree of influence to normalize each level of TC for obtaining TCα by Eq. (6).
(6)
where ({mathbf{T}}_{c}^{alpha 11}) can be calculated via Eqs. (7) and (8); by the same way ({mathbf{T}}_{c}^{alpha nn}) can be acquired.
$$ d_{i}^{11} = sumlimits_{j = 1}^{{m_{1} }} {t_{{_{C} ij}}^{11} } ,;i = 1,2, ldots ,m_{1} $$
(7)
$$ {mathbf{T}}_{{{}_{C}}}^{alpha 11} = left[ {begin{array}{*{20}c} {t_{{_{C} 11}}^{11} /d_{1}^{11} } & cdots & {t_{{_{C} 1j}}^{11} /d_{1}^{11} } & cdots & {t_{{_{C}^{{1m_{1} }} }}^{11} /d_{1}^{11} } \ vdots & {} & vdots & {} & vdots \ {t_{{_{C} i1}}^{11} /d_{i}^{11} } & cdots & {t_{{_{C} ij}}^{{{}_{{}}11}} /d_{i}^{11} } & cdots & {t_{{_{C}^{{im_{1} }} }}^{{{}_{{}}11}} /d_{i}^{11} } \ vdots & {} & vdots & {} & vdots \ {t_{{_{C}^{{m_{1} 1}} }}^{11} /d_{{m_{1} }}^{11} } & cdots & {t_{{_{C}^{{m_{1} j}} }}^{11} /d_{{m_{1} }}^{11} } & cdots & {t_{{_{C}^{{m_{1} m_{1} }} }}^{11} /d_{{m_{1} }}^{11} } \ end{array} } right] = left[ {begin{array}{*{20}c} {t_{{_{C} 11}}^{alpha 11} } & cdots & {t_{{_{C1j} }}^{alpha 11} } & cdots & {t_{{_{C}^{{1m_{1} }} }}^{alpha 11} } \ vdots & {} & vdots & {} & vdots \ {t_{{_{C} i1}}^{alpha 11} } & cdots & {t_{{_{C} ij}}^{alpha 11} } & cdots & {t_{{_{C}^{{im_{1} }} }}^{alpha 11} } \ vdots & {} & vdots & {} & vdots \ {t_{{_{C}^{{m_{1} 1}} }}^{alpha 11} } & cdots & {t_{{_{C}^{{m_{1j} }} }}^{alpha 11} } & cdots & {t_{{_{C}^{{m_{1} m_{1} }} }}^{alpha 11} } \ end{array} } right] $$
(8)
The unweighted supermatrix can be obtained by utilizing the interdependent relationship in group to array ({mathbf{T}}_{{{}_{C}}}^{alpha }) by Eq. (9).
(9)
where ({mathbf{W}}^{11}) is displayed by Eq. (10), and ({mathbf{W}}^{nn}) in the same way. A blank space or 0 in the matrix presents independence of the group of criteria or a single criterion in relation to other criteria.
$$ W^{{11}} = ({T}^{{11}} )^{{prime }} = begin{array}{*{20}c} {begin{array}{*{20}c} {c_{{11}} } & cdots & {c_{{1i}} } & cdots & {c_{{1m_{{_{{1}} }} }} } \ end{array} } \ {begin{array}{*{20}l} {c_{{11}} } hfill \ vdots hfill \ {c_{{1j}} } hfill \ vdots hfill \ {c_{{1m_{{_{{1}} }} }} } hfill \ end{array} left[ {begin{array}{*{20}c} {t_{{c{11}}}^{{alpha {11}}} } & cdots & {t_{{ci{1}}}^{{alpha {11}}} } & cdots & {t_{{cm_{{_{{1}} }} 1}}^{{alpha {11}}} } \ vdots & {} & vdots & {} & vdots \ {t_{{c1j}}^{{alpha {11}}} } & cdots & {t_{{cij}}^{{alpha {11}}} } & cdots & {t_{{cm_{{_{{1}} }} j}}^{{alpha {11}}} } \ vdots & {} & vdots & {} & vdots \ {t_{{c{1}m_{{_{{1}} }} }}^{{alpha {11}}} } & ldots & {t_{{cim_{{_{{1}} }} }}^{{alpha {11}}} } & cdots & {t_{{cm_{{_{{1}} }} m_{{_{{1}} }} }}^{{alpha {11}}} } \ end{array} } right]} \ end{array} $$
(10)
The third step is to obtain the weighted supermatrix. The total influence matrix of dimensions ({mathbf{T}}_{D}) is computed by Eq. (11). Use the total degree of influence to normalize each level of ({mathbf{T}}_{D}) by Eq. (12) to obtain ({mathbf{T}}_{D}^{alpha }).
(d_{i} = sumlimits_{j = 1}^{n} {t_{D}^{ij} }), (i = 1,2,…,n)
$$ {mathbf{T}}_{D} {text{ = }}left[ {begin{array}{*{20}c} {t_{{_{D} }}^{{11}} } & cdots & {t_{{_{D} }}^{{1j}} } & cdots & {t_{{_{D} }}^{{1n}} } \ vdots & {} & vdots & {} & vdots \ {t_{{_{D} }}^{{i1}} } & cdots & {t_{{_{D} }}^{{ij}} } & cdots & {t_{{_{D} }}^{{in}} } \ vdots & {} & vdots & {} & vdots \ {t_{{_{D} }}^{{n1}} } & cdots & {t_{{_{D} }}^{{nj}} } & cdots & {t_{{_{D} }}^{{nn}} } \ end{array} } right] $$
(11)
$$ {mathbf{T}}_{D}^{alpha } = left[ {begin{array}{*{20}c} {t_{{_{D} }}^{11} /d_{1} } & cdots & {t_{{_{D} }}^{1j} /d_{1} } & cdots & {t_{{_{D} }}^{1n} /d_{1} } \ vdots & {} & vdots & {} & vdots \ {t_{{_{D} }}^{i1} /d_{i} } & cdots & {t_{{_{D} }}^{ij} /d_{i} } & cdots & {t_{{_{D} }}^{in} /d_{i} } \ vdots & {} & vdots & {} & vdots \ {t_{{_{D} }}^{n1} /d_{n} } & cdots & {t_{{_{D} }}^{nj} /d_{n} } & cdots & {t_{{_{D} }}^{nn} /d_{n} } \ end{array} } right];; = left[ {begin{array}{*{20}c} {t_{{_{D} }}^{alpha 11} } & cdots & {t_{{_{D} }}^{alpha 1j} } & cdots & {t_{{_{D} }}^{alpha 1n} } \ vdots & {} & vdots & {} & vdots \ {t_{{_{D} }}^{alpha i1} } & cdots & {t_{{_{D} }}^{alpha ij} } & cdots & {t_{{_{D} }}^{alpha in} } \ vdots & {} & vdots & {} & vdots \ {t_{{_{D} }}^{alpha n1} } & cdots & {t_{{_{D} }}^{alpha nj} } & cdots & {t_{{_{D} }}^{alpha nn} } \ end{array} } right] $$
(12)
The weighted supermatrix ({varvec{W}}^{alpha }) can be derived from normalizing ({mathbf{T}}_{D}^{alpha }) into the unweighted supermatrix ({varvec{W}}) for normalized supermatrix ({varvec{W}}^{alpha }) exhibited in Eq. (13).
$$ {varvec{W}}^{alpha } = {varvec{T}}_{D}^{alpha } {varvec{W}} = left[ {begin{array}{*{20}c} {t_{D}^{{alpha {11}}} times {varvec{W}}^{{{11}}} } & cdots & {t_{D}^{{alpha i{1}}} times {varvec{W}}^{i1} } & cdots & {t_{D}^{alpha n1} times {varvec{W}}^{n1} } \ vdots & {} & vdots & {} & vdots \ {t_{D}^{alpha 1j} times {varvec{W}}^{1j} } & cdots & {t_{D}^{alpha ij} times {varvec{W}}^{ij} } & cdots & {t_{D}^{alpha nj} times {varvec{W}}^{nj} } \ vdots & {} & vdots & {} & vdots \ {t_{D}^{alpha 1n} times {varvec{W}}^{1n} } & cdots & {t_{D}^{alpha in} times {varvec{W}}^{in} } & cdots & {t_{D}^{alpha nn} times {varvec{W}}^{nn} } \ end{array} } right] $$
(13)
Fourthly, obtain the influential weights of DANP. The weighted supermatrix ({varvec{W}}^{alpha }) multiplies by itself many times to calculate the limit supermatrix based on the concept of Markov Chain. The influential weight of each criterion can thus be obtained by (lim_{z to infty } ({mathbf{W}}^{alpha } )^{z}). The influential weights of DANP are obtained by the limit supermatrix application ({mathbf{W}}^{alpha }) with power z, an adequately large integer, until the supermatrix ({mathbf{W}}^{alpha }) has converged and becomes a long-term stable supermatrix to obtain the global influential weight.
Consequently, the hybrid MCDM model combined the DEMATEL technique with basic concept of ANP (DEMATEL-based ANP, called DANP) can handle problems with interrelationship (interdependence and feedback) among dimensions/criteria in real world.
