FCILOS (Fuzzy Criterion Impact LOSs)

FCILOS is not a ranking method; it does not rank alternatives, it produces criterion weights. It carries CILOS's same "impact loss" idea into situations where cells are given not as a single number but as a three-cornered fuzzy number (a lower, middle and upper end). If your data comes from expert judgement and is "approximate," you can use this three-ended range directly rather than the single number that crisp CILOS requires.

Podvezko, Zavadskas and Podviezko proposed it in 2020, the same team behind CILOS and IDOCRIW. Their paper runs classical CILOS's five steps separately on the lower, middle and upper component of the triangular fuzzy number and then proposes averaging the three results; FCILOS is a direct implementation of this recipe.

FCILOS's underlying question is the same as CILOS's: had you chosen the alternative that is best in one criterion, how much would you have lost on the other criteria? Its difference is that it asks this question three times: once at data's most pessimistic (lower) end, once at its most likely (middle) value, and once at its most optimistic (upper) end. Each end produces a separate loss calculation and a separate weight; FCILOS averages these three weights into a single reportable weight.

This design carries an important consequence. If the weights at the three ends come out close to one another, they agree on which criterion costs the most even though the data is uncertain, and the result is reliable. If the ends are far apart, the uncertainty makes the weight itself uncertain, and the final (averaged) figure conceals this internal conflict. Choosing FCILOS means accepting the view that "performance is known only approximately, and this approximation itself must be factored into the calculation."

The method proceeds through five steps, each applying CILOS's counterpart to a three-cornered fuzzy number.

First, validation and cost conversion. Every cell is checked for being an ordered triangular fuzzy number (lower ≤ middle ≤ upper) and for being positive. Cost criteria are inverted by swapping the lower and upper ends, so that a "lower is better" criterion is also read as "higher is better."

Second, division by the fuzzy column sum. Every column is divided by its own fuzzy total. In this division, the lower end is divided by the upper end and the upper end by the lower end, so that uncertainty grows or shrinks in the correct direction.

Third, building three separate best profiles. For each of the lower, middle and upper components separately, the best-performing alternative on each criterion is found, and three separate "best profile" tables are built for these three components.

Fourth, three separate loss systems and their solution. For each component, a loss matrix is built as in CILOS and the homogeneous system is solved; the result is three separate weight vectors, for the lower, middle and upper components. These three vectors need not be one another's small-medium-large counterpart; each is the result of its own independent calculation.

Fifth, averaging. The three weight vectors are averaged alternative by alternative and rescaled so that they sum to 1. This is the single weight that is reported.

The formulas behind each step and the intermediate tables are given on the DecisionMind method page; this card carries no formulas.

The FCILOS weight carries the same meaning as the CILOS weight: it measures how expensive it turns out, within this set of alternatives, to give up a criterion. The difference is that this single figure is in fact the average of three separate calculations (pessimistic, likely, optimistic). If the three calculations sit close together, the average offers a reliable summary. If they are far apart, and one may even be rising while another falls, the average conceals this conflict and it must be flagged separately in the report.

You should therefore check whether a criterion's lower, middle and upper component weights move in the same direction. If the lower component is the lowest, the middle sits in the middle and the upper is the highest, that is, if they move in the same direction, uncertainty is only enlarging or shrinking the weight. If one component comes out highest and another lowest, that is, if the direction does not hold, data uncertainty is making the weight itself contestable.

Thus instead of writing:

"FCILOS analysis showed that price is the most important criterion"

the report should read:

"In this fuzzy data set, price's average weight came out highest; but the lower-middle-upper component weights do not move in the same direction, so the ranking is sensitive to data uncertainty"

FCILOS works with triangular fuzzy numbers (TFN): every cell consists of three values (lower, middle, upper) and the condition lower ≤ middle ≤ upper must hold. All values must be strictly positive. DecisionMind's CILOS family has three members alongside the base method: crisp CILOS, triangular fuzzy FCILOS, and scenario-based SCENARIO-FUZZY-CILOS. If your data can be expressed with a single crisp number, FCILOS is not needed; crisp CILOS suffices.

You need: alternatives in rows, criteria in columns, an ordered triangular fuzzy number in every cell, no empty cells. Every criterion needs a "higher is better or lower is better" direction. No weight is entered; the method produces the weight. A minimum of two alternatives and two criteria is required.

FCILOS is appropriate if your data comes from expert judgement and the expert can give a range, such as "roughly this much, at worst this, at best this." Using fuzzy data when a precise measurement is possible adds an unnecessary layer of uncertainty; crisp CILOS should be preferred in that case.

The situations where it should not be used follow from its philosophy. If a component is zero or negative, the method is left undefined, because every step involves a division. If a single alternative is best on every criterion at once (the same problem as in crisp CILOS), the loss system becomes singular and the weights become undefined. If the lower-middle-upper component weights systematically move in opposite directions, reporting a single averaged figure can be misleading; in that case the three components should be reported separately.

Data is approximate/expert-estimated, three-ended values can be given → FCILOS

Data is measured precisely → CILOS

Both opportunity cost and dispersion (entropy) should enter together, data is fuzzy → FUZZY-IDOCRIW

Uncertainty is given as scenarios (several possible tables) → SCENARIO-FUZZY-CILOS

FCILOS's greatest strength is that it does not reduce uncertainty to a single number too early and lose it; the lower, middle and upper components are carried separately right to the end of the calculation and are only averaged at the final step. This makes visible how "robust" a criterion's weight is, that is, how closely its three components agree. It preserves CILOS's opportunity-cost logic, so it can be interpreted the same way as crisp CILOS. Being an extension published by the same team (Zavadskas, Podvezko and co-authors), it is directly comparable with crisp CILOS.

Its limitations are largely inherited from CILOS and compounded by fuzziness. First, the three components' weights need not be one another's small-medium-large counterpart; Podvezko, Zavadskas and Podviezko's (2020) own paper states this explicitly, and DecisionMind does not force this ordering to be corrected. Second, the fuzzy Entropy and fuzzy IDOCRIW extensions in the same family as the source paper are not implemented in DecisionMind because, in the authors' own words, they can produce negative weights; this shows that some members of the family are mathematically fragile. Third, every fragility present in crisp CILOS (such as one alternative coming out best on every criterion, or dependence on the alternative set) is repeated here three times over. Fourth, for users who must collect triangular fuzzy numbers, the data-collection burden triples compared with crisp data.

The most common mistake is assuming the three components' weights should be ordered from smallest to largest; a criterion's lower-component weight can come out higher than its upper-component weight, and this is not an error but a natural result of the method.

A second mistake is reporting only the average weight and never checking how closely the three components agree; in that case, a weight that looks solid may in fact be the average of three conflicting calculations. A third is not entering the lower-middle-upper values in order (for example, lower > middle); the method refuses to run in that case. A fourth is artificially fuzzifying crisp data (writing the same number three times) and running FCILOS; this adds unnecessary computational load and gives no information different from crisp CILOS. A fifth is trying to ignore the undefined-result error that arises when a single alternative comes out best on every component at once.

The governing principle is this:

The single weight FCILOS reports is the average of three independent calculations (pessimistic, likely, optimistic); if these three calculations do not move in the same direction, the weight is not robust, and the report must show this clearly.