Intuitionistic fuzzy CODAS (Daami Remadi & Moalla Frikha, 2020)

Four suppliers (A1-A4) are scored linguistically on four criteria: g1 service level (more is better), g2 quality (more is better), g3 price (less is better), g4 environmental management system score (more is better). Scores are taken from the seven-term scale (VP…VG); this fixture uses a single already-combined score table (expert weights are taken as equal, λ is not broken out). Criterion weights are equal (0.25).

The method converts linguistic terms into triangular intuitionistic numbers using the seven-term scale, applies complementation on the cost criterion (g3), divides by the column's largest modal value, multiplies by the weights, builds the negative-ideal, reduces with the score function, and computes Euclidean and Taxicab distance. Because every pairwise Euclidean difference among the four alternatives exceeds the threshold (τ=0.02), Taxicab distance does not come into play in any comparison in this example.

The result reads as follows. A4 scores close to the top two degrees on two criteria, "good" on g1 and "very good" on g2. Its "very poor" (MP) degree on g3 (price, less is better) turns into an advantage after complementation. The reason is that a low raw score means something good on a cost criterion. This strong position on three criteria more than compensates for its middling ("not bad", F) degree on g4, and puts A4 well ahead of the others. A3, by contrast, drops to last place through its "very good" (VG) degree on g3, which after complementation means the worst price performance; it also receives a "very poor" (MP) degree on g4.

The decision carries one hesitation. If A4's degree on g3 (price) had been "very good" (VG) instead of "very poor" (MP), that is, if the supplier's price were actually high, A4 would drop from first place (5.01) to third (0.74), and A1 would lead (1.17). A4's first place is extremely sensitive to the linguistic degree of a single criterion (g3). By contrast, changing the criterion weights alone, for example raising a single criterion from 0.25 to 0.55, did not disturb A4's first place.

In the report: "With equal weights (0.25) and a τ=0.02 threshold, A4 reaches the highest assessment score (5.01); this result is comparatively robust to the weight distribution but extremely sensitive to A4's linguistic degree on the price criterion (MP). If this degree is corrected to 'very good' (VG), A4 drops to third place and A1 leads."

Source: DecisionMind's validation fixture for the TIF-CODAS engine; the problem theme is drawn from the green-supplier-selection example of Daami Remadi & Moalla Frikha (2020), based on the Banaeian et al. (2016) data set, but the cell values do not come from the article's published Table X; they come from DecisionMind's synthetic validation fixture (manifest note: "paper data unavailable/anonymized"). The ranking and figures have been independently recomputed and verified through the engine by this card's author, and do not match the article's own Table X figures (or its reported result of "best alternative A3"). This discrepancy is recorded in the review notes.

A municipality's environmental services department will choose one of three vehicle models to renew its fleet. Criteria: fuel efficiency, ease of maintenance, purchase price (less is better) and emissions performance. Three technical experts (the fleet maintenance chief, an environmental engineer, a procurement specialist) have scored each model on the seven-term linguistic scale; the department has given the experts different weights (λ), because the maintenance chief's opinion has been judged more decisive from the standpoint of operating cost.

The method combines the three experts' scores with the λ weights, applies complementation on the cost criterion, reduces with the weighted score function, and computes both distances to the negative-ideal. Suppose the model with the highest fuel-efficiency score also has the highest purchase price. It still comes first, because the combined weight of fuel efficiency and emissions performance exceeds that of price. The Euclidean gap between the second and third models fell below the threshold, so the ranking between them was decided by Taxicab distance.

The department also had a hesitation. The maintenance chief's weight has been kept markedly higher than the other two experts'; whether the first place changes if the maintenance chief's score shifts by one term should be tested separately. The choice of the most expensive model should also be justified on budget grounds.

In the report: "With the high weight given to fuel efficiency and emissions performance, one model reaches the highest assessment score; this result is sensitive to the maintenance chief's score and to the distribution of expert weights (λ), and this sensitivity should be shown separately in the report."

The first mistake is reducing the illustrative example's linguistic terms to a single number with the score function from the start, and then running crisp CODAS. In that case A4 still comes first, but the ranking of A1 and A3 changes (in correct TIF-CODAS, A1 is third and A3 fourth; under the defuzzify-first shortcut, A3 is third and A1 fourth). This shortcut therefore does not merely inflate the scores; it also reverses the second and third places. The second mistake, as in the local-government example, is presenting results as a "shared expert view" without determining or showing the expert weights (λ) in the report; the weight distribution can change the outcome. The third mistake is skipping complementation on a cost criterion such as price and applying the benefit criterion's normalisation rule directly; this penalises the low-priced, good alternative and reverses the ranking.