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Modified ISM/TISM Process with Simultaneous Transitivity Checks for Reducing Direct Pair Comparisons

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Abstract

Both interpretive structural modeling (ISM) and total interpretive structural modeling (TISM) are pair comparison methods to evolve hierarchical relationships among a set of elements. These methods help to convert ill-structured mental models into well-articulated models that act as base for conceptualization and theory building. One major challenge in applications of these methods is the number of pair comparisons to be made that increase exponentially with the increase in number of elements. Another challenge is the transitivity check on reachability matrix. This paper proposes a modified ISM/TISM process that addresses both these challenges by simultaneously carrying out transitivity checks along with the successive pair-wise comparisons. The pairs having transitive relationships in the process need not to be compared further. This reduces the expert-based pair comparisons drastically and provides the fully transitive reachability matrix in one go, thereby enabling easy implementation. It provides a complete flow chart for the same and is illustrated with already reported examples in past literature.

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Authors and Affiliations

  1. Department of Management Studies, Indian Institute of Technology Delhi, Vishwakarma Bhawan, Shaheed Jeet Singh Marg, New Delhi, 110016, India

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Appendices

Appendix 1: Sample Questionnaire for 5 Element Problem with Transitivity Checks Side-by-Side

  1. * Transitive links; ** decision by expert
  2. i–j—Element i influencing element j. j–i—Element j influencing element i. i=j—Both elements i and j influence each other. 0—No relationship between elements i and j

Appendix 2: Select Extreme Cases

Exhibit II.1: All Immediate Comparisons Have Direct Forward Links

(a) Reachability Matrix with Transitive Links

figure b

(b) Digraph Exhibiting Direct Comparisons and Transitive Links

figure c

Exhibit II.2: All Immediate Comparisons Have Direct Backward Links

(a) Reachability Matrix with Transitive Links

figure d

(b) Digraph Exhibiting Direct Comparisons and Transitive Links

figure e

Exhibit II.3: All Immediate Comparisons Have Both Way Relationships

(a) Reachability Matrix with Transitive Links

figure f

(b) Digraph Exhibiting Direct Comparisons and Transitive Links

figure g

Exhibit II.4: All Concerned Pairs n(n − 1)/2 Have No Relationship

(a) Reachability Matrix with Nil Transitive Links

figure h

(b) Digraph Exhibiting No Direct and Transitive Links

figure i

Appendix 3: Example 1 (Sushil 2017): Final Correct Matrix from Original Paper Taken as Basis

Exhibit III.1: Filled Questionnaire with Transitivity Checks

  1. * Transitive links; ** decision by expert (as given in Sushil 2016)
  2. Bold entry gives the selection (blue color for direct comparison and green color for transitive comparison)
  3. i–j—Element i influencing element j. j–i—Element j influencing element i. i=j—Both elements i and j influence each other. 0—No relationship between elements i and j

Exhibit III.2: Step-by-Step Building of Transitive Reachability Matrix and Digraph

Step 1: Pair Comparison 1=2

figure j

Step 2: Pair Comparison 2–3

figure k

Step 3: Transitivity Check 1–3

figure l

Step 4: Pair Comparison 3=4

figure m

Step 5: Transitivity Check 2–4

figure n

Step 6: Transitivity Check 1–4

figure o

Step 7: Pair Comparison 4–5

figure p

Step 8: Transitivity Check 3–5

figure q

Step 9: Transitivity Check 2–5

figure r

Step 10: Transitivity Check 1–5

figure s

Exhibit III.3: Fully Transitive Reachability Matrix

figure t

Exhibit III.4: Hierarchical Digraph Obtained by Level Partitioning (Sushil 2016)

figure u

Appendix 4: Example 2 (Sushil 2012): Final Reachability Matrix Taken as Base

Exhibit IV.1: Reachability Matrix with Transitive Links Following the Modified Process

figure v

Exhibit IV.2: Digraph with Direct and Transitive Links As Per Modified Process

figure w

Exhibit IV.3: Digraph After Hierarchical Partitioning (Sushil 2012)

figure x

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Sushil Modified ISM/TISM Process with Simultaneous Transitivity Checks for Reducing Direct Pair Comparisons. Glob J Flex Syst Manag 18, 331–351 (2017). https://doi.org/10.1007/s40171-017-0167-3

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