Enterprise’s internal control for knowledge discovery in a big data environment by an integrated hybrid model

Abstract

This research aims to (1) identify the critical risk factors that influence the governance of enterprise internal control in a big data environment, (2) depict the intertwined and complicated relationships among risk factors, and (3) yield an attainable target for performance improvement over both the short term and long term. To address these challenging issues, we propose an innovative hybrid decision architecture that combines artificial intelligence-based rule generation techniques and a multiple attribute decision making approach, called herein multiple rule-base decision making. Examining real cases, our study shows that the control environment and information technology (IT) control construction are the top dimension and criterion, respectively. This finding can be taken as a reference for managing and controlling risk factors under a big data environment. In an upcoming improvement/advancement on internal control/information technology (IT) governance, the related factors can also be viewed as essential requirements for enterprises when conducting effective internal control and audit inspection, which can help with more audit success and less lawsuit problems.

Appendices

Appendix A: ACO-FRST

ACO-FRST is briefly summarized as follows.

In the same manner that crisp equivalence is the core element of RST, fuzzy crisp equivalence is the core element of FRST [72, 73]. By taking the fuzzy similarity relation $B$ on the universe into consideration, the crisp equivalence classes can be extended to fuzzy ones. Based on the fuzzy similarity relation, the fuzzy equivalence class $\left[ g \right]_{B}$ for an instance closest to $g$ can be expressed as:

$$\mu_{{\left[ g \right]_{B} }} (h) = \mu_{B} (g,h)$$

(A.1)

The following axioms hold for a fuzzy equivalence class $F$ [74]:

$$\begin{gathered} \exists_{g} ,\begin{array}{*{20}c} {} \\ \end{array} \mu_{F} (g) = 1 \hfill \\ \mu_{F} (g) \wedge \mu_{B} (g,h) \le \mu_{F} (h) \hfill \\ \mu_{F} (g) \wedge \mu_{F} (h) \le \mu_{B} (g,h) \hfill \\ \end{gathered}$$

The first axiom is utilized to indicate that an equivalence class is non-empty. The second axiom states that the elements in $h$’s neighborhood are in the equivalence class of $h$. The last axiom denotes that any two elements in $G$ have some relation by utilizing $S$. The fuzzy $Q$-lower and $Q$-upper approximations are expressed in Eqs. (A.2) and (A.3), respectively.

$$\mu_{{\underline{Q} G}} (F_{i} ) = inf_{g} \max \left\{ {1 - \mu_{{F_{i} }} (g),\mu_{G} (g)} \right\}\begin{array}{*{20}c} {} \\ \end{array} \forall i$$

(A.2)

$$\mu_{{\overline{Q} G}} (F_{i} ) = \sup_{g} \min \left\{ {\mu_{{F_{i} }} (g),\mu_{G} (g)} \right\}\begin{array}{*{20}c} {} \\ \end{array} \forall i$$

(A.3)

where $F$ denotes as a fuzzy equivalence class that is designated as the partition of ${\mathbb{R}}$ for a given feature subset $S$.

With feature $x$ as an example, the partition of the universe by $\left\{ x \right\}$ (represented as ${\mathbb{R}}/IND\left( {\left\{ x \right\}} \right)$) is taken as fuzzy equivalence classes for that feature. In the crisp case, ${\mathbb{R}}/S$ contains sets of objects grouped together that are complicated to separate via two features $x$ and $y$. In the fuzzy case, objects may belong to numerous different classes, and so two Cartesian products of ${\mathbb{R}}/IND\left( {\left\{ x \right\}} \right)$ and ${\mathbb{R}}/IND\left( {\left\{ y \right\}} \right)$ must be considered for computing ${\mathbb{R}}/S$. The mathematical formulation is displayed as:

$${\mathbb{R}}/S = \otimes \left\{ {x \in S:{\mathbb{R}}/IND\left( {\left\{ x \right\}} \right)} \right\}$$

(A.4)

Although the universe of discourse in feature selection is finite, this is not the case in general. Therefore, $\sup$ and $\inf$ are considered in prior equations.

The following equations display the new formats of fuzzy lower and upper approximations:

$$\mu_{{\underline{S} G}} (g) = \mathop {\sup }\limits_{{F \in {\mathbb{R}}/S}} \min \left( {\mu_{F} \left( g \right),\begin{array}{*{20}c} {} \\ \end{array} \mathop {\inf }\limits_{{h \in {\mathbb{R}}}} \max \left\{ {1 - \mu_{F} \left( h \right),\mu_{G} \left( h \right)} \right\}} \right)$$

(A.5)

$$\mu_{{\overline{S} G}} (g) = \mathop {\sup }\limits_{{F \in {\mathbb{R}}/P}} \min \left( {\mu_{F} \left( G \right),\begin{array}{*{20}c} {} \\ \end{array} \mathop {\sup }\limits_{{h \in {\mathbb{R}}}} \min \left( {\mu_{F} \left( g \right),\mu_{G} \left( h \right)} \right)} \right)$$

(A.6)

In real-life applications, not all $h \in {\mathbb{R}}$ are considered—only those where $\mu_{F} \left( h \right)$ is non-zero. The tuple $\left\langle {\underline{S} G,\overline{S} G} \right\rangle$ can be used to express FRST. Each set in ${\mathbb{R}}/S$ denotes an equivalence class. The magnitude of an instance that can be assigned to an equivalence class is calculated by utilizing the conjunction of constituent fuzzy equivalence classes $F_{i} ,i = 1, \ldots ,n.$

$$\mu_{{F_{1} \cap \ldots \cap F_{n} }} \left( g \right) = \min \left( {\mu_{{F_{1} }} \left( g \right),\mu_{{F_{2} }} \left( g \right), \ldots ,\mu_{{F_{n} }} \left( g \right)} \right)$$

(A.7)

By the extension of the crisp positive region in RST, the membership of an instance classified into the fuzzy positive region can be represented by:

$$\mu_{{POS_{S} \left( P \right)}} (g) = \mathop {\sup }\limits_{{G \in {\mathbb{R}}/P}} \mu_{{\underline{S} G}} (g)$$

(A.8)

Instance $g$ is not classified into the positive region only if the equivalence class it belongs to is not a constituent of the positive region. By performing the concept of the fuzzy positive region, the dependency function can be expressed as:

$$\gamma ^{\prime}_{S} (P) = \frac{{\left| {\mu_{{POS_{S} (p)}} (g)} \right|}}{{\left| {\mathbb{R}} \right|}} = \frac{{\sum\nolimits_{{G \in {\mathbb{R}}}} {\mu_{{POS_{S} (P)}} (g)} }}{{\left| {\mathbb{R}} \right|}}$$

(A.9)

The minimal subset is determined by performing Eq. (A.9) to gauge the quality of the selected subset. For a dataset with large dimensionalities, FRST becomes less suitable for minimal subset determination [75]. As identifying the minimal reduct for FRST is a classical optimization task, this study considers one swarm intelligence method, called ant colony optimization (ACO), that has proven its usefulness in the optimization task [28, 76]. The concept of ACO is a simulation of the behavior of ants foraging. During their foraging when food is discovered, as the ants transport it back and forth their body secretes a special substance known as a pheromone. By spreading these odors, ants are able to tell other ants which direction they should follow to find the food source. This concept can transform rule extraction into an optimization problem and summarize the knowledge that can be easily understood by the user to strengthen the decision maker’s judgment.

Equation (A.10) represents the probability that an ant moves from node i to node j at time t.

$$l_{ij}^{m} (t) = \frac{{\left[ {\alpha_{ij} (t)} \right]^{\beta } \cdot \left[ {\eta_{ij} } \right]^{\lambda } }}{{\sum\nolimits_{{p \in J_{i}^{m} }} {\left[ {\alpha_{ip} (t)} \right]^{\beta } \cdot \left[ {\eta_{ip} } \right]^{\lambda } } }}$$

(A.10)

Here, $m$ denotes the number of ants, $J_{i}^{m}$ represents the set of ant $m$’s unvisited node, $\eta_{ij}$ expresses the heuristic desirability of selecting node $j$ when at node $i$, and $\alpha_{ij} (t)$ indicates the amount of pheromone on edge $\left( {i,j} \right)$. Here, $\beta$ and $\lambda$ are determined manually. The pheromone on each edge can be updated by utilizing the following equation.

$$\alpha_{ij} (t + 1) = (1 - \rho ) \cdot \alpha_{ij} (t) + \Delta \alpha_{ij} (t)$$

(A.11)

Here, $\Delta \alpha_{ij} (t) = \sum\limits_{m = 1}^{n} {\left( {\gamma ^{\prime}\left( {G^{m} } \right)/\left| {G^{m} } \right|} \right)}$.

The term ρ represents the evaporation coefficient of the pheromone, and $G^{m}$ is the feature subset determined by ant $m$. The pheromones are updated based on the goodness of the feature subset ($\gamma ^{\prime}$) in FRST. More detailed illustrations of ACO-FRST can be seen in [28, 76, 77].

Appendix B: D-ANP

DEMATEL can be arranged briefly by the following procedures.

• Procedure 1: Creating the average relation matrix Z

The direct influence is evaluated by each expert, and the degree of mutual influence is identified using a pairwise comparison based on the measurement scale of absolutely no influence (0), low influence (1), medium influence (2), high influence (3), and very high influence (4) to form a direct relation matrix. The average relation matrix Z is displayed in Eq. (B.1).

$${\varvec{Z}} = \left[ {\begin{array}{*{20}c} {z_{11}^{{}} } & \cdots & {z_{1j}^{{}} } & \cdots & {z_{1n}^{{}} } \\ \vdots & {} & \vdots & {} & \vdots \\ {z_{i1}^{{}} } & \cdots & {z_{ij}^{{}} } & \cdots & {z_{in}^{{}} } \\ \vdots & {} & \vdots & {} & \vdots \\ {z_{n1}^{{}} } & \cdots & {z_{nj}^{{}} } & \cdots & {z_{nn}^{{}} } \\ \end{array} } \right]$$

(B.1)

• Procedure 2: Calculating the normalized direct influence matrix H. The normalized direct influence matrix H is calculated using Eqs. (B.2) and (B.3):

  $${\varvec{H}} = \phi \cdot {\varvec{Z}}$$

  (B.2)
  $$\phi = \min \left\{ {\frac{1}{{\max_{1\; \le i \le n} \sum\nolimits_{j = 1}^{n} {z_{ij} } }},\frac{1}{{\max_{1\; \le j \le n} \sum\nolimits_{i = 1}^{n} {z_{ij} } }}} \right\}$$

  (B.3)

• Procedure 3: Computing the total relationship matrix T. The total relationship matrix, which involves direct and indirect effects, can be obtained by applying Eq. (B.4).

  $${\varvec{T}} = {\varvec{M}} + {\varvec{M}}^{2} + {\varvec{M}}^{3} + \ldots + {\varvec{M}}^{l} = {\varvec{M}}({\varvec{I}} - {\varvec{M}})^{ - 1} \quad {\text{when}}\quad \mathop {\lim }\limits_{l \to \infty } {\varvec{M}}^{l} = \left[ {\mathbf{0}} \right]_{n \times n}$$

  (B.4)

  where I is the identity matrix.

• Procedure 4: Drawing IINRM. We use the summation of row vector ${\varvec{r}} = (r_{i} )_{n \times 1}$ ($\left[ {\sum\nolimits_{j = 1}^{n} {t_{ij} } } \right]_{n \times 1} = (r_{1} \ldots ,r_{i} \ldots ,r_{n} )^{\prime}$) and column vector ${\varvec{s}} = (s_{j} )_{n \times 1}$ ($\left[ {\sum\nolimits_{i = 1}^{n} {t_{ij} } } \right]^{\prime }_{1 \times n} = (s_{1} \ldots ,s_{j} \ldots ,s_{n} )^{\prime}$) in the relationship matrix T to illustrate the degree of influence among criteria and dimensions; when $i = j$, IINRM is constructed. The casual graph of the influential network relationship can be obtained by mapping the values of $(r_{i} + s_{i} )$ and $(r_{i} - s_{i} )$, and the results are utilized to improve the favor value of each dimension and criterion.

  ANP can be arranged briefly by the following procedures.

• Procedure 1: Formulating the unweighted super-matrix W. First, the total-influence relation matrix ${\varvec{T}}_{C}$ is obtained from DEMATEL as:

  $${\varvec{T}}_{C} = \begin{array}{*{20}c} \begin{gathered} D_{1} \hfill \\ \hfill \\ \hfill \\ \vdots \hfill \\ \hfill \\ \end{gathered} \\ \begin{gathered} \hfill \\ D_{i} \hfill \\ \hfill \\ \hfill \\ \vdots \hfill \\ \end{gathered} \\ \begin{subarray}{l} \\ \\ \end{subarray} \\ {D_{m} } \\ \end{array} \begin{array}{*{20}c} {c_{11} } \\ {c_{12} } \\ \vdots \\ {c_{{1m_{1} }} } \\ \vdots \\ {ci1} \\ {c_{i2} } \\ \vdots \\ {c_{{im_{i} }} } \\ \vdots \\ {c_{m1} } \\ {c_{m2} } \\ \vdots \\ {c_{{mm_{m} }} } \\ \end{array} \mathop {\mathop {\left[ {\begin{array}{*{20}c} {{\varvec{T}}_{C}^{11} } & \cdots & {{\varvec{T}}_{C}^{1j} } & \cdots & {{\varvec{T}}_{C}^{1m} } \\ \vdots & {} & \vdots & {} & \vdots \\ {{\varvec{T}}_{C}^{i1} } & \cdots & {{\varvec{T}}_{C}^{ij} } & \cdots & {{\varvec{T}}_{C}^{im} } \\ \vdots & {} & \vdots & {} & \vdots \\ {{\varvec{T}}_{C}^{m1} } & \cdots & {{\varvec{T}}_{C}^{mj} } & \cdots & {{\varvec{T}}_{C}^{mm} } \\ \end{array} } \right]}\limits^{{\begin{array}{*{20}c} {\;\;\;\;c_{11 \cdots } c_{{1m_{1} }} \;\;...\;\;} & {c_{j1 \cdots } c_{{jm_{j} }} \;} & \cdots \\ \end{array} \;\quad \;c_{m1 \cdots } c_{{mm_{m} }} \,\;\;}} }\limits^{{\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\quad \quad \;D_{1} \quad \;\;\quad \;} & {} & {\quad D_{j} \;} \\ \end{array} } & {} & {} \\ \end{array} } & {} & {} \\ \end{array} } & {D_{m} } & {} \\ \end{array} \;\;}}_{{m_{j} \times m_{j} |m < n,\;\sum\nolimits_{j = 1}^{m} {m_{j} = n} }}$$

  (B.5)

Normalizing the total-influence relation matrix ${\varvec{T}}_{C}$, we next derive a new matrix ${\varvec{T}}_{C}^{\rho }$:

$${\varvec{T}}_{C}^{\rho } = \begin{array}{*{20}c} \begin{gathered} D_{1} \hfill \\ \hfill \\ \hfill \\ \vdots \hfill \\ \hfill \\ \end{gathered} \\ \begin{gathered} \hfill \\ D_{i} \hfill \\ \hfill \\ \hfill \\ \vdots \hfill \\ \end{gathered} \\ \begin{subarray}{l} \\ \\ \end{subarray} \\ {D_{m} } \\ \end{array} \begin{array}{*{20}c} {c_{11} } \\ {c_{12} } \\ \vdots \\ {c_{{1m_{1} }} } \\ \vdots \\ {ci1} \\ {c_{i2} } \\ \vdots \\ {c_{{im_{i} }} } \\ \vdots \\ {c_{m1} } \\ {c_{m2} } \\ \vdots \\ {c_{{mm_{m} }} } \\ \end{array} \mathop {\mathop {\left[ {\begin{array}{*{20}c} {{\varvec{T}}_{C}^{\rho 11} } & \cdots & {{\varvec{T}}_{C}^{\rho 1j} } & \cdots & {{\varvec{T}}_{C}^{\rho 1m} } \\ \vdots & {} & \vdots & {} & \vdots \\ {{\varvec{T}}_{C}^{\rho i1} } & \cdots & {{\varvec{T}}_{C}^{\rho ij} } & \cdots & {{\varvec{T}}_{C}^{\rho im} } \\ \vdots & {} & \vdots & {} & \vdots \\ {{\varvec{T}}_{C}^{\rho m1} } & \cdots & {{\varvec{T}}_{C}^{\rho mj} } & \cdots & {{\varvec{T}}_{C}^{\rho mm} } \\ \end{array} } \right]}\limits^{{\begin{array}{*{20}c} {\;\;\;\;c_{11 \cdots } c_{{1m_{1} }} \;\;...\;\;} & {c_{j1 \cdots } c_{{jm_{j} }} \;} & \cdots \\ \end{array} \;\quad \;c_{n1 \cdots } c_{{mm_{m} }} \,\;\;}} }\limits^{{\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\quad \quad \;D_{1} \quad \;\;\quad \;} & {} & {\quad D_{j} \;} \\ \end{array} } & {} & {} \\ \end{array} } & {} & {} \\ \end{array} } & {D_{m} } & {} \\ \end{array} \;\;}}_{{m_{j} \times m_{j} |m < n,\;\sum\nolimits_{j = 1}^{m} {m_{j} = n} }}$$

(B.6)

Transposing the normalized total-influence relation matrix ${\varvec{T}}_{C}^{\rho }$, we yield the unweighted super-matrix ${\varvec{W}} = ({\varvec{T}}_{C}^{\rho } )^{^{\prime}}$ as:

$${\varvec{W}} = \left( {{\varvec{T}}_{C}^{\rho } } \right)^{\prime } = \begin{array}{*{20}c} \begin{gathered} D_{1} \hfill \\ \hfill \\ \hfill \\ \vdots \hfill \\ \hfill \\ \end{gathered} \\ \begin{gathered} \hfill \\ D_{j} \hfill \\ \hfill \\ \hfill \\ \vdots \hfill \\ \end{gathered} \\ \begin{subarray}{l} \\ \\ \end{subarray} \\ {D_{m} } \\ \end{array} \begin{array}{*{20}c} {c_{11} } \\ {c_{12} } \\ \vdots \\ {c_{{1m_{1} }} } \\ \vdots \\ {cj1} \\ {c_{j2} } \\ \vdots \\ {c_{{jm_{j} }} } \\ \vdots \\ {c_{m1} } \\ {c_{m2} } \\ \vdots \\ {c_{{mm_{m} }} } \\ \end{array} \mathop {\mathop {\left[ {\begin{array}{*{20}c} {{\varvec{W}}^{11} } & \cdots & {{\varvec{W}}^{i1} } & \cdots & {{\varvec{W}}^{m1} } \\ \vdots & {} & \vdots & {} & \vdots \\ {{\varvec{W}}^{1j} } & \cdots & {{\varvec{W}}^{ij} } & \cdots & {{\varvec{W}}^{mj} } \\ \vdots & {} & \vdots & {} & \vdots \\ {{\varvec{W}}^{1m,} } & \cdots & {{\varvec{W}}^{im} } & \cdots & {{\varvec{W}}^{mm} } \\ \end{array} } \right]}\limits^{{\begin{array}{*{20}c} {\;\;\;\;c_{11 \cdots } c_{{1m_{1} }} \;\;...\;\;} & {c_{j1 \cdots } c_{{im_{i} }} \;} & \cdots \\ \end{array} \;\quad \;c_{n1 \cdots } c_{{mm_{m} }} \,\;}} }\limits^{{\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\quad \quad \;D_{1} \quad \;\;\quad \;} & {} & {\quad D_{i} \;\;} \\ \end{array} } & {} & {} \\ \end{array} } & {} & {} \\ \end{array} } & {D_{m} \,} & {} \\ \end{array} \;}}_{{m_{j} \times m_{j} |m < n,\;\sum\nolimits_{j = 1}^{m} {m_{j} = n} }}$$

(B.7)

• Procedure 2: Obtaining the weighted super-matrix ${\varvec{W}}^{\rho }$. We derive the total-influence relation matrix for dimensions ${\varvec{T}}_{D}$ using the DEMATEL technique as:

  $$\user2{\rm T}_{D}^{{}} = \left[ {\begin{array}{*{20}c} {t_{D}^{11} } & \cdots & {t_{D}^{1j} } & \cdots & {t_{D}^{1m} } \\ \vdots & {} & \vdots & {} & \vdots \\ {t_{D}^{i1} } & \cdots & {t_{D}^{ij} } & \cdots & {t_{D}^{im} } \\ \vdots & {} & \vdots & {} & \vdots \\ {t_{D}^{m1} } & \cdots & {t_{D}^{mj} } & \cdots & {t_{D}^{mm} } \\ \end{array} } \right]_{m \times m}$$

  (B.8)

The normalized influential matrix for clusters ${\varvec{T}}_{D}^{\rho }$ is formed through each element divided by the sum of corresponding rows $d_{i} = \sum\limits_{j = 1}^{n} {t_{ij}^{D} }$ in the calculation process:

$$\user2{\rm T}_{{\text{D}}}^{\rho } = \left[ {\begin{array}{*{20}c} {t_{D}^{11} /{\text{d}}_{1} } & \cdots & {t_{D}^{1j} /d_{1} } & \cdots & {t_{D}^{1m} /d_{1} } \\ \vdots & {} & \vdots & {} & \vdots \\ {t_{D}^{i1} /d_{i} } & \cdots & {t_{D}^{ij} /d_{i} } & \cdots & {t_{D}^{im} /d_{i} } \\ \vdots & {} & \vdots & {} & \vdots \\ {t_{D}^{m1} /d_{m} } & \cdots & {t_{D}^{mj} /d_{m} } & \cdots & {t_{D}^{mm} /d_{m} } \\ \end{array} } \right]_{m \times m} = \left[ {\begin{array}{*{20}c} {t_{11}^{\rho D} } & \cdots & {t_{1j}^{\rho D} } & \cdots & {t_{1m}^{\rho D} } \\ \vdots & {} & \vdots & {} & \vdots \\ {t_{i1}^{\rho D} } & \cdots & {t_{ij}^{\rho D} } & \cdots & {t_{im}^{\rho D} } \\ \vdots & {} & \vdots & {} & \vdots \\ {t_{m1}^{\rho D} } & \cdots & {t_{mj}^{\rho D} } & \cdots & {t_{mm}^{\rho D} } \\ \end{array} } \right]_{m \times m}$$

(B.9)

Finally, we obtain the weighted super-matrix ${\varvec{W}}^{\rho }$ for the normalized total-relation matrix as:

$${\varvec{W}}^{\rho } = {\varvec{T}}_{D}^{\rho } \times {\varvec{W}} = \begin{array}{*{20}c} \begin{gathered} D_{1} \hfill \\ \hfill \\ \hfill \\ \vdots \hfill \\ \hfill \\ \end{gathered} \\ \begin{gathered} \hfill \\ D_{j} \hfill \\ \hfill \\ \hfill \\ \vdots \hfill \\ \end{gathered} \\ \begin{subarray}{l} \\ \\ \end{subarray} \\ {D_{m} } \\ \end{array} \begin{array}{*{20}c} \begin{gathered} c_{11} \hfill \\ c_{\begin{subarray}{l} 12 \\ \vdots \end{subarray} } \hfill \\ c_{\begin{subarray}{l} 1m_{1} \\ \vdots \end{subarray} } \hfill \\ c_{j1} \hfill \\ c_{\begin{subarray}{l} j2 \\ \vdots \end{subarray} } \hfill \\ \end{gathered} \\ \begin{gathered} c_{\begin{subarray}{l} jm_{j} \\ \vdots \end{subarray} } \hfill \\ c_{m1} \hfill \\ c_{\begin{subarray}{l} m2 \\ \vdots \end{subarray} } \hfill \\ c_{{mm_{m} }} \hfill \\ \end{gathered} \\ \end{array} \mathop {\mathop {\left[ {\begin{array}{*{20}c} {t_{11}^{\rho D} \times {\varvec{W}}^{11} } & \cdots & {t_{i1}^{\rho D} \times {\varvec{W}}^{i1} } & \cdots & {t_{m1}^{\rho D} \times {\varvec{W}}^{m1} } \\ \vdots & {} & \vdots & {} & \vdots \\ {t_{1j}^{\rho D} \times {\varvec{W}}^{1j} } & \cdots & {t_{ij}^{\rho D} \times {\varvec{W}}^{ij} } & \cdots & {t_{mj}^{\rho D} \times {\varvec{W}}^{mj} } \\ \vdots & {} & \vdots & {} & \vdots \\ {t_{1m}^{\rho D} \times {\varvec{W}}^{1m} } & \cdots & {t_{im}^{\rho D} \times {\varvec{W}}^{im} } & \cdots & {t_{mm}^{\rho D} \times {\varvec{W}}^{mm} } \\ \end{array} } \right]}\limits^{{\begin{array}{*{20}c} {\;\;c_{11 \cdots } c_{{1m_{1} }} \quad \;\,\quad \;\;...\quad \quad \quad \quad } & {\;\;c_{i1 \cdots } c_{{im_{i} }} \;\;\;} & {\;\quad \; \cdots } \\ \end{array} \;\quad \quad \quad \;\;c_{m1 \cdots } c_{{mm_{m} }} \,\;}} }\limits^{{\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\quad \quad \;\;\;D_{1} \quad \quad \;\;\quad \;} & {\quad \;} & {\quad \quad \;\quad \quad D_{i} \;} \\ \end{array} } & \quad & {} \\ \end{array} } & {} & {} \\ \end{array} \quad \quad } & {D_{m} } & {} \\ \end{array} \quad \;\;}}$$

(B.10)

where $t_{ij}^{\rho D} = {{t_{D}^{ij} } \mathord{\left/ {\vphantom {{t_{D}^{ij} } {d_{i} }}} \right. \kern-\nulldelimiterspace} {d_{i} }}$.

• Procedure 3: Providing the global weights. Limiting the weighted super-matrix based on the Markov Chain approach for reaching a long-term stable super-matrix $\mathop {\lim }\limits_{\varphi \to \infty } ({\varvec{W}}^{\rho } )^{\varphi }$, we arrive at the D-ANP influence weights $(w_{1} \ldots ,w_{j} \ldots ,w_{n} )$.

Appendix C: Modified-VIKOR approach

The modified-VIKOR can be arranged briefly by the following procedures.

• Procedure 1: Normalizing the performance gap calculation. As $f_{j}^{aspiration} = 10$ ${( }{\varvec{f}}^{aspiration} = (f_{1}^{aspiration} ,...,f_{j}^{aspiration} ,...,f_{n}^{aspiration} ))$ and $f_{j}^{worst} = 0$ (${ (}{\varvec{f}}^{worst} = (f_{1}^{worst} ,...,f_{j}^{worst} ,...,f_{n}^{worst} ))$ are the positive ideal solution (aspiration level) and negative ideal solution (worst level) among criteria j = 1,2,…,n, respectively, we utilize the concept of the “aspiration-worst” to determine the normalized rating matrix of the internal control practices of alternative k for criterion j. Accordingly, we obtain the performance rating ratios $\left[ {r_{gj} } \right]_{G \times n}$ through a normalized performance matrix $\left[ {f_{gj} } \right]_{G \times n}$, as shown in Eq. (C.1).

  $$[r_{gj} ]_{G \times n} = {{[\left( {\left| {f_{j}^{aspiration} - f_{gj} } \right|} \right)} \mathord{\left/ {\vphantom {{[\left( {\left| {f_{j}^{aspiration} - f_{gj} } \right|} \right)} {\left( {\left| {f_{j}^{aspiration} - f_{j}^{worst} } \right|} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\left| {f_{j}^{aspiration} - f_{j}^{worst} } \right|} \right)}}]_{G \times n}$$

  (C.1)

• Procedure 2: Determining the maximum group utility $S_{g}$ and the minimum individual regret gap $Q_{g}$. We calculate these gap values [70] using the following two equations:

  $$L_{g}^{\varphi = 1} = S_{g} = \sum\limits_{j = 1}^{n} {w_{j} r_{gj} } = \sum\limits_{j = 1}^{n} {w_{j} (} |f_{j}^{aspiration} - f_{gj} |)/(|f_{j}^{aspiration} - f_{j}^{worst} |),g = 1,2, \ldots ,n.$$

  (C.2)
  $$L_{g}^{\varphi = \infty } = Q_{g} = \max_{j} (r_{gj} |g = 1,2,...,n)$$

  (C.3)

  where $w_{j}$ represents the weights of the j^th criterion (called D-ANP influential weights), and $f_{gj}$ indicates the real performance value in each criterion j of each alternative. The performance gap-ratio ($r_{gj}$) reduction of each alternative can thus be achieved.

• Procedure 3: Computing the comprehensive index values $R_{g}$. We calculate the integrated values from the following equation:

  $$R_{g} = v(S_{g} - S^{aspiration} )/(S^{worst} - S^{aspiration} ) + (1 - v)(Q_{g} - Q^{aspiration} )/(Q^{worst} - Q^{aspiration} )$$

  (C.4)

where $S^{aspiration} = \max_{g} S_{g}$, $S^{worst} = \min_{g} S_{g}$, $Q^{aspiration} = \max_{g} Q_{g}$, $Q^{worst} = \min_{g} Q_{g}$, and $0 \le \nu \le 1$.

Appendix D: K-means approach

Cluster analysis groups a set of instances by some characteristics in a multi-dimensional space so that the instances in the same group are more similar to each other than to those in other groups [78, 79]. With easy computation and intuitiveness, the K-means clustering approach is one of the most well-known types of clustering. Its fundamental concept is to identify K centroids for K clusters, and these centroids should be as far as possible from each other. The basic illustration of K-means runs as follows.

For a dataset $G = \left\{ {x_{1} , \ldots ,x_{N} } \right\},x_{i} \in R^{g}$, K-means partitions the data into K disjoint groups $\left( {B_{1}^{*} , \ldots ,B_{k}^{*} } \right)$. The quality of a cluster outcome is then evaluated by a normalized intra-cluster variance that is represented by:

$$\frac{1}{N}\sum\limits_{j = 1}^{k} {\sum\limits_{{x_{i} \in B_{j}^{ * } }} {\left\| {x_{i} - B_{j} } \right\|_{2} } }$$

(D.1)

where $B_{j}$ denotes the centroid of cluster $B_{j}^{*}$; the smaller the variance value is, the more superior is the clustering quality. The algorithm starts by randomly selecting $k$ points as initial centroids by the centroid selection approach. The quality of the centroids is then adjusted iteratively until the centroids do not change. In each iteration, the approach traverses whole instances, assigns the instances to the nearest cluster, and then updates the centroid of each cluster.

$$B_{j}^{t} = \frac{{\sum\nolimits_{{x_{i} \in B_{j}^{*} }} {x_{i}^{t} } }}{{\left| {B_{j}^{*} } \right|}},\forall t \in \left\{ {1, \ldots ,g} \right\}$$

(D.2)

where the $t$ th dimension of the $j$ th centroid is represented as $B_{j}^{t}$, and the $t$ th dimension of $x_{i}$ is denoted by $x_{i}^{t}$. For a more detailed illustration, one can refer to Ni et al. [80] and Saha and Mukherjee [81].