Crisp DEMATEL influence model

1. Define the factors and the meaning of a rating

A factor should have a stable definition shared by everyone supplying a judgment. In the fictional engine-room maintenance example, planning, spares, work readiness and execution are four distinct aspects of a process. Entry aᵢⱼ means the assessed direct influence from row factor i to column factor j. It is not the reverse influence, a correlation coefficient, a measured probability or a time-series causal effect. State the period and operating context before collecting ratings. Duplicate or overlapping factors can create artificial feedback. This workbench accepts a completed single matrix; it does not collect or validate expert opinions.

2. Keep the scale and the diagonal policy explicit

The declared scale is 0 to 4: zero means no perceived direct influence, followed by low, medium, high and very high judgments. Fractional values are allowed for an already-aggregated assessment; every cell is still required. Blank is not synonymous with no influence. The direct diagonal must be exactly zero, a modeling policy that excludes direct self-influence at entry. Indirect paths can nevertheless return to their starting point, so the total-relation diagonal is generally positive. Those feedback terms are retained in every row and column sum even though self-loops are omitted from the drawing.

3. Normalize with one named rule

Compute every row sum and every column sum of A. The divisor s is the maximum of all of them; D=A/s when s is positive. This is the max-row/max-column version specified in the cited 2011 paper, not the row-only alternative. Multiplying every rating by the same positive constant leaves D unchanged, provided the new ratings remain within the accepted scale. Consequently the normalized output does not preserve the absolute overall strength of uniformly rescaled judgments. Scaling one entry or one row can change s and therefore every normalized entry. The raw matrix and s are always shown to make this dependency inspectable.

4. Ask whether feedback actually decays

D describes direct propagation, D² describes two-step propagation and later powers describe longer walks, including repeated visits. The series D+D²+… exists when the spectral radius is below one. Merely bounding row and column sums by one does not guarantee that condition: the two-factor matrix with ratings 4 in both directions normalizes to an undiminished cycle. Its inverse formula is invalid. This implementation seeks an outward-rounded upper bound on the infinity norm of a power Dᵏ that is strictly below one. Such contraction certifies decay of subsequent blocks of powers for the stored matrix.

5. Verify the inverse before reporting T

The check examines powers 1, 2, 4 and so on up to 65536, with at most seventeen norm checks. It is a bounded numerical acceptance policy, so a valid but very slowly contracting model can be refused. After the convergence gate, pivoted elimination computes (I−D)⁻¹. Pivots below 10⁻¹², infinity-norm condition numbers above 10⁸, inverse residuals above 10⁻⁹ or scaled total-equation residuals above 10⁻¹² stop the calculation. No epsilon is added to the diagonal and no damping factor is inserted into the ratings. If the model is rejected, any revision must be explicit and justified by the person responsible for it.

6. Interpret totals without claiming causal discovery

The total matrix is T=D(I−D)⁻¹. Row sum Rᵢ is the model’s total outgoing influence from factor i; column sum Cᵢ is the total incoming influence to that factor. Prominence R+C summarizes involvement in the supplied network. Relation R−C describes a net transmitter or net receiver within that network. These names summarize an algebraic consequence of judgments; they do not show that changing a factor will cause a measured outcome. Near-zero relation signs can be numerical artifacts, especially when the inverse is ill-conditioned. A conditioning- and scale-aware guard marks these cases as numerically unresolved; raw values remain visible. This guard is a conservative display policy, not a confidence interval or rigorous error bound. A large prominence is not a maintenance priority, an importance weight, a risk estimate or a budget allocation. External evidence and a separate decision framework are needed for those uses.

7. Keep the graph separate from the calculation

The directed network displays an off-diagonal edge only when its total influence is strictly greater than τ. A high threshold gives a simpler drawing and may hide substantial connections; a low threshold can create a dense picture. The full T matrix, row and column totals, prominence and relation always use every entry regardless of τ. Node positions are a fixed circular layout, not a proximity measure. The second figure locates factors by prominence and relation, and its horizontal zero line separates positive and negative net relations. Both figures have complete numeric tables so the drawing is not the only way to inspect the result.

8. Handle absence of information honestly

If every entry is zero, s is zero and A/s is undefined. The explicit no-information branch reports zero D, zero T, identity inverse and zero summaries instead of dividing by zero or inventing a weak network. This branch does not imply that the real system has no relationships; it only records what the submitted matrix contains. The worked example is entirely synthetic and demonstrates a valid contracting system. Loading it restores all factor labels, the objective, sixteen ratings and τ=0.5. Export records the scale, normalizer, all intermediate matrices, checks and displayed edges, so a later reader can distinguish the model from its presentation.

A complete synthetic maritime example

All names, ratings and performance figures below were invented for this lesson. They are not observations from a vessel, port or company.

Decision or study question: Explore fictional engine-room maintenance influences.

Factor summaries from the full T matrix

These are consequences of supplied perceived influence ratings. A positive relation is not proof of empirical causality, and prominence is not a safety priority or an importance weight.

s = 4

B = (I−D)⁻¹. Scaled residual = raw residual / max(1, ‖T‖∞‖I−D‖∞ + ‖D‖∞). Outward rounding is used only for the contraction check; D and A are not altered.

Tables display up to 10 significant digits as formatting, not guaranteed accuracy. Computation and CSV use unrounded JavaScript numbers. Numerical tolerance is not an operational acceptance criterion.

Exact result for this example: T = (1/139) × [[85,208,160,132], [96,109,148,136], [52,88,57,120], [56,52,40,33]].

What the result does and does not establish

• Direct-influence ratings are subjective inputs. The words transmitter and receiver refer only to this supplied model, not established empirical causes.
• Uniform rescaling cancels during normalization. The method cannot compare absolute overall influence between uniformly scaled matrices.
• The max-row/max-column normalizer does not guarantee convergence. Singular, nearly singular and numerically unverified inputs are rejected.
• The 0–4 scale, zero diagonal, bounded dimensions and numerical gates are declared implementation policies. No fuzzy, revised or damped DEMATEL variant is silently substituted.
• Correlation, experiments, domain evidence and temporal mechanisms require separate examination. An influence picture alone cannot approve a vessel operation or select a safe maintenance strategy.

Primary sources and implementation scope

Chen & Tzeng (2011), Assessment Model for Improving Educational Curriculum Materials Based on the DANP Technique with Grey Relational Analysis, CSEDU, 299–308. p.302, (1)–(2): max-row/max-column normalization and total relation; row/column summaries on the same page. Only its crisp DEMATEL component is implemented here, not DANP or grey relational analysis.

Lee et al. (2013), Revised DEMATEL: Resolving the Infeasibility of DEMATEL, Applied Mathematical Modelling 37, 6746–6757. Cited for the warning that convergence cannot be assumed; this tool does not implement the paper’s revised normalization.

Original teaching text and synthetic examples. Numerical limits, tie handling and the no-information branch are explicit implementation policies. dematel v1.0.0.