Neutrosophic SPOTIS (Abdel-aziem, Mohamed & Abdelhafeez, 2023)

An investor will decide among ten portfolio options (IPA1–IPA10) on nine criteria; seven of the criteria are "higher is better" (IPC1–IPC6, IPC8), and two are "lower is better" (IPC7, IPC9, for example risk and cost indicators). Every cell is a neutrosophic triple (T, I, F) derived from expert scores.

(The full matrix has ten rows; IPA1 and IPA2 are shown as examples. The T-I-F values for the remaining eight portfolios are recorded exactly as they appear in the DecisionMind manifest and in the source article's Table 2, p. 36.)

The method reduces every cell to a score with the formula s = (1+T−2I−F)/2, builds every criterion's fixed bound (in this case, since the article does not publish the weight and bound information numerically) from the ten portfolios' own observed lowest/highest score, and sums the scaled distances with equal weight (1/9 for each criterion).

This ranking matches, exactly, the ranking from the article's own Table 2 result (IPA1 > IPA3 > IPA10 > IPA8 > IPA4 > IPA2 > IPA5 > IPA7 > IPA6 > IPA9). However, the absolute distance value reported by the article (0.490 for IPA1, for example) differs from the value computed here (0.255), because the article's actual weights and fixed bounds have not been published numerically and DecisionMind has fallen back to equal weights and data-derived bounds.

The result can be read as follows. IPA1 is clearly first; the second-, third- and fourth-placed portfolios, IPA3, IPA10 and IPA8, are very close to one another (0.471 – 0.479 – 0.483, a difference of under 0.01).

The investor has a hesitation, precisely at this closeness. If the weight on IPC1 is doubled and the other eight criteria are shrunk proportionately, IPA10 takes second place from IPA3 (IPA10 at 0.471, IPA3 at 0.484), and IPA8 falls from fourth to eighth place (0.535). IPA1 stays first under every scenario.

In the report: "With the source article's decision matrix and criterion directions, the ranking computed with equal weights and this set's own observed bounds matches the article's reported ranking exactly; IPA1 is clearly first. However, because the article does not publish its actual weights and fixed bounds numerically, the very small difference (under 0.01) between the second-, third- and fourth-placed portfolios (IPA3, IPA10, IPA8) is quite sensitive to the criterion weights, and this uncertainty should be stated explicitly in the report."

Source: Abdel-aziem, A. H., Mohamed, H. K., & Abdelhafeez, A. (2023), Table 2, p. 36; the decision matrix and criterion directions are taken exactly from the article. Because the article's reported weight and fixed-bound values are not published numerically (see the approval notes), the distance values above were computed independently in Python by this card's author, using equal weights and a data-derived bound assumption; DecisionMind's kernel test also compares only the ranking (not the absolute values) against the article, and this ranking matches exactly.

A municipality will choose among three contractors for waste collection, and will repeat this comparison in every tender period against fixed minimum/maximum standards defined in regulation. Three criteria are evaluated: "this contractor delivers the service as contracted" (C1, higher is better), "this contractor complies with environmental regulation" (C2, higher is better), and "this contractor has a high complaint rate" (C3, lower is better). The degrees of support, indeterminacy and falsity are derived from past audit reports; the fixed bounds (on the scored scale) are taken from the minimum and maximum standards defined in regulation.

The method reduces every cell to a score (Y1-C1: 0.525 · Y1-C2: 0.550 · Y1-C3: 0.175 · and similarly for the others), builds the ideal from the fixed bounds set in regulation (the upper bound for C1 and C2, the lower bound for C3), and sums the scaled distances with the weights.

The result can be read as follows. Y3 has the highest score on contractual compliance (0.625) and the lowest score on complaint risk (0.125); its difference from Y1 is very small (0.379 against 0.386). Y2 has the best score on environmental compliance, but its weak score on contractual compliance raises its total distance.

The municipality has a hesitation, precisely at the closeness between Y1 and Y3. If Y1's complaint record worsens, that is, if its T value rises from 0.30 to 0.45 and its score therefore rises from 0.175 to 0.325, Y1's distance rises to 0.472 and it drops to third place, behind Y2 as well; Y3 stays first. In other words, Y1's hold on second place depends on its complaint record not worsening.

In the report: "Under the fixed bounds set in regulation, Y3 is closest to the ideal (distance 0.379); its difference from Y1 is very small (0.386), and Y1 could drop to third place if its complaint record worsens. As long as the fixed bounds are not changed along with the regulation, this comparison can be repeated on the same ruler in future tender periods."

The first error, in the investment example, is ignoring the under-0.01 difference between the second-, third- and fourth-placed portfolios and reporting "IPA3 is clearly second." Because the article does not publish the actual weights, this ranking is fragile and the report must say so.

The second error is presenting DecisionMind's equal-weight default result (distance 0.255) as if it were the article's own Table 2 value (0.490). These are two different calculations; only their rankings match.

The third error, in the local-government example, is ignoring Y1's high F (falsity) value of (0.30;0.20;0.55) on complaint risk and deciding prematurely that "Y1 is the best contractor" by looking only at its contractual-compliance score. The total distance combines all three criteria; it cannot be read from a single one.