Keywords

96.1 Introduction

The Interpretative Structural Modeling (ISM) is a common structure modeling technology and widely used in many fields (Li 2011; Thakkar et al. 2004, 2007; Singh 2007; Kanungo and Bhatnagar 2002; Bolaños et al. 2005). In terms of the workflow of ISM, when directed graph is obtained, the establishment of reachable matrix is a quite important step (Wang 1998). Usually, the main methods for establishing reachable matrix are formula method (Wang 2000), experience dialogue method (Zhenkui 1998) and Warshall algorithm (Lipschutz and Lipson 2002; Wang and Ge 1996) and so on. In this paper, the relations tracking method for reachable matrix is presented, compared with the other methods, this method can avoid complex matrix operations.

96.2 Relations Tracking Method

96.2.1 Definition

Relations tracking method, just as its name implies, the relationships among many factors in a system should be tracked firstly and then the reachable matrix may be established. In this paper we use Fig. 96.1 as an example to illustrate.

Fig. 96.1
figure 1

Directed connection diagram

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96.2.2 Steps

The first step is to find out the direct reachable set of each node.

Direct reachable set is the element set that a node can reach directly not including the node itself, expressed with D<i>. For example, for Fig. 96.1, the direct reachable set of node S2 is D<2> = {3, 4}. Similarly, D<4> = Ø. All the direct reachable sets are shown in Table 96.1.

Table 96.1 The direct reachable sets
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The second step is to find out the tracking reachable set of each node.

Tracking reachable set is the element set that each node can reach, whether directly or indirectly, which includes the node itself, expressed with R<i>. For reachable matrix, this step is quite important.

The core idea of relations tracking method is as follows. Each node can be viewed as a source node and the direct reachable set of the node will be obtained, then each node in the direct reachable set can be viewed as a branch node which can be used as the next level branch node, thus the tree branch of each node can be obtained and all the nodes of tree branch constitute the tracking reachable set of each node. If a direct reachable set is an empty set, then the tracking reachable set is the node itself. For example, in Table 96.1, the direct reachable set of S4 is empty, then R<4> = {4}.

The important principles: in the process of branching, if a node is repeated, then the node should be omitted, that is the node is no longer continue to branch.

For example, for the tracking reachable set of S1, the tree branch obtained is shown in Fig. 96.2.

Fig. 96.2
figure 2

The tree branch of S1

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Thus, R < 1> = {1, 2, 3, 4, 5}, similarly, R < 2> = {1, 2, 3, 4, 5}, R < 3> = {1, 2, 3, 4, 5}, R < 5> = {1, 2, 3, 4, 5}, R < 6> = {1, 2, 3, 4, 5, 6}, the tree branches of S2, S3, S5 and S6 are shown in Fig. 96.3.

Fig. 96.3
figure 3

The tree branches of S2, S3, S5, S6

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The third step is to write out the reachable matrix.

The reachable matrix M is shown below.

$$ {\text{M}} = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array} } \right] $$

This method can avoid fundamentally complex matrix operations, and it just needs to track the relations among nodes.

96.2.3 The Explanation of Relations Tracking Method

The relations tracking method reflects the essence of reachable matrix. In terms of this method, the tracking reachable set can be obtained directly based on the directed graph, so the reachable matrix obtained from relations tracking method is just requested. From reachable matrix M it can be known that the elements in row 1, 2, 3 and 5 are same, which suggest that S1, S2, S3, S5 may form loops.

The relations tracking method can avoid the repeated searching because a repeated node will be no longer continue to branch. Breadth First Search is a method looking for the shortest path between two nodes in a directed graph (Wang et al. 1994; Lu and Feng 2006; Yuan and Wang 2011), although the repeated searching can be avoided, it requires that a clear hierarchical relation be established firstly. However, in the ISM process, the hierarchical relationship is just obtained after reachable matrix.

96.3 Comparative Analysis of Several Methods

96.3.1 Formula Method

With formula method, the reachable matrix is obtained in terms of (A + I)r = (A + I)r−1 = M. A is adjacency matrix, I is unit matrix, M is reachable matrix. A can be obtained based on directed graph.

Usually, in terms of the formula, after the sequential operation, Ai can be obtained, that is A1≠A2≠…≠Ar−1 = Ar = Ar+1, Ar = (A + I)r, r ≤ n−1, n is order number. Because M = (A + I)r, then M = Ar = Ar−1. Also taken Fig. 96.1 as an example, the operation process is shown below.

$$ {\text{A}} = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ \end{array} } \right] $$
$$ \begin{gathered} {\text{A}}_{1} = {\text{A}} + {\text{ I}} \hfill \\ = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 \\ \end{array} } \right] \hfill \\ \end{gathered} $$
$$ \begin{gathered} {\text{A}}_{ 2} { = }\left( {\text{A + I}} \right)^{ 2} {\text{ = A}}_{ 1}^{ 2} \hfill \\ = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array} } \right] \hfill \\ \end{gathered} $$
$$ \begin{gathered} {\text{A}}_{ 3} { = }\left( {\text{A + I}} \right)^{ 3} {\text{ = A}}_{ 2} . {\text{A}}_{ 1} \hfill \\ = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array} } \right] \hfill \\ \end{gathered} $$
$$ \begin{gathered} {\text{A}}_{ 4} { = }\left( {\text{A + I}} \right)^{ 4} {\text{ = A}}_{ 3} . {\text{A}}_{ 1} {\text{ = A}}_{ 3} {\text{ = M}} \hfill \\ = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array} } \right] \hfill \\ \end{gathered} $$

Formula method is a traditional method and used widely, but the biggest drawback of this method is the complicated matrixes calculation which is only acceptable when elements are quite few. Actually a system is often large with many elements, and the relationships among elements are quite complicated. Therefore complicated matrix calculation decreases the practicality of this method (Tian and Wang 2003).

96.3.2 Warshall Algorithm

The steps of this algorithm are shown below. ①P ← A. ②k ← 1. ③i ← 1. ④if pik = 1, then pij ← pij∨pkj, j = 1,2,…,n. ⑤i ← i+1. If i ≤ n, then turn to step ④. ⑥k ← k+1. If k ≤ n, then turn to step ③, otherwise stop. A is adjacency matrix, P is reachable matrix.

Among these steps, the step ④ is crucial. For Fig. 96.1, the operation process is shown as follows.

Because the calculation process is quite tedious, so the partial steps (k = 4 and 5) are omitted.

When k = 6, the comparing result is as follows.

$$ \begin{gathered} \begin{array}{*{20}c} {{\text{k}} \leftarrow 6} & {{\text{i}} \leftarrow 1 {\text{ p}}_{ 1 6} { = 0}} \\ {} & {{\text{i}} \leftarrow 2 {\text{ p}}_{ 2 6} { = 0}} \\ {} & {{\text{i}} \leftarrow 3 {\text{ p}}_{ 3 6} { = 0}} \\ {} & {{\text{i}} \leftarrow 4 {\text{ p}}_{ 4 6} { = 0}} \\ {} & {{\text{i}} \leftarrow 5 {\text{ p}}_{ 5 6} { = 0}} \\ {} & {{\text{i}} \leftarrow 6 {\text{ p}}_{ 6 6} { = 0}} \\ \end{array} \hfill \\ \hfill \\ \end{gathered} $$

The essence of Warshall algorithm is matrix comparing and matrix updating based on the comparisons, at last the reachable matrix can be obtained. The comparing scope of formula method is between matrix and matrix, while the comparing scope of Warshall algorithm is limited in a matrix. However, Warshall algorithm also involves a large number of repeated comparing. From the operations above it can be known that when k = 3, i = 5, matrix P do not change, but according to this algorithm, the comparing of remainder must go on.

Another problem of Warshall algorithm is, reachable matrix obtained from Warshall algorithm cannot reflect the circumstance that a node gets to itself. However, the reachable matrix obtained from formula method can reflect the circumstance that each node gets to itself. The reason is that the operation of formula method is made after the adjacency matrix and unit matrix added together. This paper argues that when reachable matrix obtained from Warshall algorithm plus unit matrix, the problem can be solved effectively.

Compared with formula method and Warshall algorithm, relations tracking method take the form of tree branch to track the reachable set, thus the complicated operations can be avoided. Of course, with computer programming the establishment of reachable matrix may be more efficient.

96.4 Conclusion

When establishing ISM, the calculation of reachable matrix is always quite crucial and tedious, in order to solve this problem, the relations tracking method for reachable matrix is presented, compared with the other methods, this method can avoid complex matrix operations and consequently enhance the practical operability of ISM.