3.3 Reachability matrix and level partitioning
The SSIM is transformed into a reachability matrix as described in step (4) of the ISM methodology. After incorporating the transitivities, the final reachability matrix is achieved which is presented in Table II (1* denotes transitivity).
The final reachability matrix depicts the driving and dependence power of each risk. Driving power of each risk is the total number of risks (including itself) which it affects, i.e. the sum of interactions in the rows. Conversely, dependence power of each risk is the total number of risks (including itself) by which it is affected, i.e. the sum of interactions in the columns. Depending on their driving and dependence power, the risks will later be classified into autonomous, dependent, linkage and independent risks.
The final reachability matrix leads to the reachability and antecedent set for each risk. The reachability set R(si) of the element si is the set of elements defined in the columns that contain 1 in row si. Similarly, the antecedent set A(si) of the element si is the set of elements defined in the rows which contain 1 in the column si. In the present case, the risks along with their reachability, antecedent and intersection set as well as resulting levels are shown in Table III. The process (as described in step (4) of ISM methodology) is completed in ten iterations.
3.4 Development of digraph and formation of ISM
Based on the reachability matrix, a conical matrix (lower triangular format) is developed by arranging the elements according to their levels (Table IV).
Based on the conical reachability matrix, the initial digraph including transitive links is obtained. After removing indirect links, the final digraph is obtained. Next, the elements descriptions are written in the digraph to call it the ISM (Figure 3).
The developed ISM has no cycles or feedbacks. Elements are related in pure hierarchical pattern.
3.5 MICMAC analysis
Identification and classification of the various supply chain risks are essential to develop the ISM under study. Comparing the hierarchy of risks in the various classifications (direct, indirect, potential) leads to rich source of information. MICMAC is an indirect classification method to critically analyze the scope of each element. The objective of the MICMAC analysis is to assess the driving power and dependence of supply chain risks (Mandal and Deshmukh, 1994; Saxena and Sushil, 1990). In Table II, the sum along the rows and the columns indicates the driving power and dependence, respectively.
All elements are divided into four groups of risks (autonomous, dependent, linkage and independent). Group I includes autonomous elements that have weak driver power and weak dependence. Group II consists of dependent elements that have weak driver power and strong dependence. The third group includes linkage elements that have both strong driving and dependence power. In group IV, all independent elements are clustered that have strong driving power, but poor dependence power. Figure 4 shows the classification of the analysed risks based on their driving power and dependence.
3.6 Fuzzy MICMAC analysis
The analysis can be further improved by considering the strength of relationships instead of the mere consideration of relationships so far. By strength of relationship, we mean the strength of risk i's impact (given its occurrence) on risk j's probability of occurrence. This strength of impact can be defined by qualitative consideration on a 0‐1 scale, as shown in Table V.
These values are superimposed on the initial reachability matrix from step (4) in ISM methodology. The resulting fuzzy direct relationship matrix is shown in Table VI.
According to Zimmermann (1991), there are three types of fuzzy compositions in order to determine the strength of the fuzzy indirect relation from element i to j: max‐min, max‐product and max‐average. In the context of this research, the max‐min composition is the most suitable, since the fuzzy relations represent the strength of relations. That means, that the minimal strength has to be the maximum of all possible minimal impacts from i to j. If the fuzzy relations represent the probability of relations, the max‐product approach seems to be the most suitable. In order to obtain indirect relationships, the fuzzy direct relationship matrix is modified based on the computational steps given in Yenradee and Dangton (2000). The resulting direct and indirect fuzzy relationship matrix with driving power and dependence is given in Table VII.