Methods · Objective weighting
Fuzzy IDOCRIW
Fuzzy IDOCRIW carries classical IDOCRIW's "both spread and opportunity cost" logic into fuzzy data by running it separately on the lower, middle and upper end of the triangular fuzzy number and averaging the three results.
Base method's data type: Fuzzy
What Is the Method?
Fuzzy IDOCRIW is not a ranking method; it does not order alternatives, it produces criterion weights. Classical IDOCRIW combines two objective-weighting ideas: Entropy's question of "how much does this criterion separate the alternatives" and CILOS's question of "what does it cost to miss out on being best on this criterion." The two answers are multiplied together and normalised. Fuzzy IDOCRIW, when your data is given as a triangular fuzzy number (lower, middle, upper end), computes this product three times, separately at each end, and then averages.
Zavadskas and Podvezko proposed classical IDOCRIW in 2016, and the same team published a fuzzy extension of CILOS and IDOCRIW in 2020. An important distinction applies here: DecisionMind's Fuzzy IDOCRIW does not implement that 2020 paper's own proposed fuzzy IDOCRIW formula. The paper's authors themselves report that their fuzzy IDOCRIW and fuzzy Entropy branches can produce negative weights in some cases; DecisionMind has therefore not implemented that branch, using instead its own engineering profile, which runs the classical IDOCRIW calculation separately at the three ends and averages them. This is stated explicitly further on in the card.
The Philosophy Behind It
The idea behind classical IDOCRIW is not to trust a single signal. Entropy looks only at how much the alternatives differ on a given criterion; CILOS looks only at what it costs to miss out on being best on that criterion. IDOCRIW multiplies the two together: a criterion that both separates the alternatives well and is expensive to sacrifice receives a high weight. Fuzzy IDOCRIW keeps the same idea, but when the data is uncertain it carries out this multiplication separately at the pessimistic, most-likely and optimistic ends.
One consequence of this design is that either of the two signals can end up "signal-free" in fuzzy data. If a single alternative happens to be best on every criterion at once, CILOS's opportunity-cost calculation flattens out, and the weight falls entirely to Entropy's spread signal. DecisionMind's own verification example runs into exactly this situation, as shown in Case 1 of this card.
How It Works
The method proceeds through three steps.
First, validation. Every cell is checked to be an ordered triangular fuzzy number (lower ≤ middle ≤ upper), and the "more is better or less is better" direction is confirmed for every criterion.
Second, three separate classical IDOCRIW calculations. The lower, middle and upper component of the triangular fuzzy number are each treated as though they were a separate crisp decision table. Entropy weight and CILOS weight are computed separately for each, multiplied together and normalised. The result is three separate weight vectors, one for each component.
Third, averaging and reporting. The three weight vectors are averaged criterion by criterion and rescaled so that they sum to 1. At this step DecisionMind also records how much the fuzzy envelope (the lower–upper spread) changes the result: when the three components sit close together, fuzziness has little effect on the weight; when they are far apart, the effect is larger.
The formulas behind each step and the intermediate tables are given on the DecisionMind method page; this card carries no formulas.
How to Read the Output
The fuzzy IDOCRIW weight tries to measure the same two things as classical IDOCRIW: a criterion's power to separate the alternatives and the cost of sacrificing it. The difference is that this measurement is the average of three separate (pessimistic, most likely, optimistic) calculations. Which component the weight comes from, Entropy or CILOS, cannot be read off without looking at the balance between the two signals.
One important point deserves attention here: if a single alternative is best on every criterion at once, CILOS's opportunity-cost signal is zeroed out and the weight falls entirely back to Entropy's spread signal. In that case, saying "the fuzzy IDOCRIW weight" is, in effect, the same as saying "the fuzzy Entropy weight," and the report must state this distinction.
Thus instead of writing:
"The fuzzy IDOCRIW analysis showed that C2 is the most important criterion"
the report should read:
"In this fuzzy dataset, C2 came out with the highest weight; but because one alternative in this alternative set is best on every criterion at once, the CILOS component carried no signal, so the result is in fact identical to the fuzzy Entropy weight"
Data Type and Inputs
Fuzzy IDOCRIW works with triangular fuzzy numbers (TFNs): every cell consists of three values (lower, middle, upper), and lower ≤ middle ≤ upper must hold. In DecisionMind's IDOCRIW family there are two members alongside the base method: classical IDOCRIW and this fuzzy extension; the family has no other extensions, only these two members. If your data can be expressed as a single crisp number, classical IDOCRIW is sufficient.
You need the following: alternatives in rows, criteria in columns, an ordered triangular fuzzy number in every cell, and no empty cells. Direction information is required for every criterion. Weights are not entered; the method produces them. The alternative set must let at least one criterion distinguish between more than one alternative; otherwise the CILOS component carries no signal.
When to Use It, When Not To
Fuzzy IDOCRIW is appropriate when your data comes from expert judgement, when you want both the spread signal and the opportunity-cost signal taken into account together, and when this data can be expressed as a three-point interval. If crisp data is available, classical IDOCRIW is sufficient; the extra fuzzy layer is unnecessary.
The cases where it should not be used follow from its own philosophy. If a single alternative is best on every criterion at once, the CILOS component carries no signal and the method effectively runs only Entropy; in that case, using fuzzy Entropy directly is the more honest choice. If what is wanted is exactly the fuzzy IDOCRIW formulation the literature itself proposes (Podvezko, Zavadskas and Podviezko, 2020), DecisionMind's profile is not suitable, because it does not implement that formulation; this point is stated separately below.
Data is fuzzy, both spread and opportunity cost should enter together → Fuzzy IDOCRIW
Data is crisp → IDOCRIW
Only the spread signal is needed, data is fuzzy → FUZZY-ENTROPY
Only the opportunity-cost signal is needed, data is fuzzy → FCILOS
Strengths
Fuzzy IDOCRIW's chief strength is that it combines two distinct objective weighting signals, spread and opportunity cost, in a single fuzzy calculation. It carries uncertainty through three separate components and averages only at the last step, which lets DecisionMind record how much difference each component shows, that is, how much the fuzzy envelope changes the result. Because it preserves classical IDOCRIW's same two-signal logic, it remains comparable with the crisp result.
Weaknesses
Its limitations are both inherited and specific to itself. First, a single alternative coming out best on more than one criterion leaves the CILOS component without a signal; in that case the result quietly reverts to pure Entropy, and the user may not notice. Second, and most importantly, this profile does not implement the literature's published fuzzy IDOCRIW formulation (Podvezko, Zavadskas and Podviezko, 2020); because that formulation, by the authors' own admission, can produce negative weights, DecisionMind instead uses its own profile, which runs the crisp calculation separately at the three ends and averages them. This is a defensible engineering choice, but "fuzzy IDOCRIW in the literature" and "fuzzy IDOCRIW in DecisionMind" are not the same thing. Third, all of classical IDOCRIW's fragilities, dependence on the alternative set, the choice of cost conversion, are repeated here threefold.
Common Mistakes
The most common mistake is assuming this method is a direct implementation of the "Fuzzy IDOCRIW" paper in the literature; DecisionMind here uses its own TFN-component profile, not the paper's formula.
A second mistake is reporting the weight as reflecting "both spread and opportunity cost" without noticing that the CILOS component carried no signal; in that case the weight reflects only spread. A third is reporting only the average without checking how consistent the lower, middle and upper components are with one another. A fourth is artificially fuzzifying crisp data to run this method; it gives no information different from classical IDOCRIW.
The governing principle is this:
The weight fuzzy IDOCRIW reports is the average of the spread and opportunity-cost signals, each calculated separately at three ends; if the opportunity-cost signal has been zeroed out, the weight in fact reflects only spread, and the report must state this.
Cases
Each case opens with a fuzzy decision table, describes in words what the method does to that table, and shows how to read the resulting weights. The first case is DecisionMind's own source-code verification example; it is not drawn from a publication. The remaining cases are illustrative constructions.
1. Illustrative example: A three-criterion fuzzy table (DecisionMind source-code verification example)
This example is not a case from the literature. It is a small table built to check whether DecisionMind's engine code produces exactly the expected result; it is not drawn from any publication, and DecisionMind records this explicitly. Four alternatives are evaluated on three criteria with triangular fuzzy numbers; C1 and C3 are "more is better," C2 is "less is better."
| Alternative | C1 (lower·middle·upper) | C2 | C3 |
|---|---|---|---|
| A1 | 1·2·3 | 4·5·6 | 7·8·9 |
| A2 | 2·3·5 | 7·8·10 | 3·4·6 |
| A3 | 5·6·7 | 2·3·4 | 8·9·10 |
| A4 | 4·5·6 | 6·7·8 | 1·2·3 |
The method runs a separate Entropy and a separate CILOS calculation for each component. A notable situation arises here: A3 is the best alternative on all three criteria and at all three components (lower, middle, upper) at once. This flattens CILOS's opportunity-cost calculation at every component; the CILOS weight for all three components comes out at exactly 0.333 · 0.333 · 0.333, adding no extra information to any criterion.
| Criterion | Entropy (average) | CILOS (at each component) | Fuzzy IDOCRIW (average) |
|---|---|---|---|
| C1 | 0.323 | 0.333 (no signal) | 0.323 |
| C2 | 0.358 | 0.333 (no signal) | 0.358 |
| C3 | 0.319 | 0.333 (no signal) | 0.319 |
The result reads as follows. Because the CILOS component assigns the same value (0.333) to all three criteria, IDOCRIW's multiplication step does not favour any criterion; the final weights come entirely from Entropy, which is why the last two columns are identical. This shows that the claim "two signals are being combined" does not hold in this SPECIFIC table; the table never tests the CILOS half of the method.
Caveat: this example was built to verify DecisionMind's engine against a reference calculation, and a table that distinctively tests the CILOS component still stands as an open item in DecisionMind's own records. It would therefore be wrong to present the weights from this table as "proof that IDOCRIW balances its two signals."
In the report: "Criterion weights were computed with fuzzy IDOCRIW; because one alternative (A3) in this alternative set is best on every criterion and at every fuzzy component at once, the opportunity-cost signal was zeroed out, and the reported weights are in fact identical to the fuzzy Entropy weights."
Source: DecisionMind fuzzy IDOCRIW engine, source-code verification example (not from a published paper, but from the engine's own engineering record). DecisionMind's own note records that this example does not test the CILOS component, and that a distinctive example is awaited as an open item.
2. Telecommunications: Fuzzy weighting for base-station site selection
A telecommunications operator wants to derive criterion weights from data before choosing among four candidate base-station sites. Field engineers assessed three criteria with triangular fuzzy numbers: coverage area, installation cost and proximity to existing infrastructure. Installation cost is "less is better."
The method runs a separate Entropy and CILOS calculation for each component, multiplies them, and averages the three components. Suppose no single candidate site turns out best on all three criteria at once; in that case the CILOS component carries a genuine signal, and coverage area, showing both high spread and high opportunity cost, receives the highest weight.
The operator's hesitation: it wants the report to show separately whether coverage area's weight comes from both signals, spread and opportunity cost, being high, or from only one of them; otherwise it is not clear which signal dominates.
In the report: "Criterion weights were computed with fuzzy IDOCRIW; coverage area's high weight comes both from its strong separation of the alternatives and from the high cost of missing out on being best on this criterion."
3. Food safety: Fuzzy weighting for supply-chain inspection-point selection
A food safety authority wants to weight four criteria before deciding which supply-chain points to direct its limited inspection capacity towards: number of past violations, volume of product handled, dependence on the cold chain, and time elapsed since the last inspection. Inspectors assessed these four criteria with triangular fuzzy numbers.
The method runs a separate Entropy and CILOS calculation for each component. Suppose the number of past violations both separated the alternatives strongly and carried a high cost for ignoring the worst-performing point on this criterion; once these two signals combine, number of past violations receives, by a clear margin, the highest weight.
The authority's hesitation: the cold-chain dependence criterion received a low weight, but this does not mean the criterion is unimportant for food safety; it means the alternatives did not separate sufficiently on it across these four points and the opportunity cost came out low. The authority decides to support this criterion with a separate threshold rule, automatically flagging for inspection any point above a given dependence level.
In the report: "Criterion weights were computed with fuzzy IDOCRIW; the low weight on cold-chain dependence reflects the data pattern across these four points, not the criterion's unimportance; this criterion will additionally be monitored through a threshold rule."
4. What Not to Do
In the illustrative example, if A3 coming out best on every criterion and at every component were reported unnoticed as "fuzzy IDOCRIW balanced two signals," the fact that only Entropy was actually operating would be concealed. A second error is assuming this method is a direct implementation of the fuzzy IDOCRIW formula published by Podvezko, Zavadskas and Podviezko (2020); DecisionMind here uses a different, its own TFN-component profile. A third error is presenting only the average weight without ever checking how consistent the lower, middle and upper components are with one another.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/fuzzy-idocriw
Zavadskas, E. K., & Podvezko, V. (2016). Integrated determination of objective criteria weights in MCDM. International Journal of Information Technology & Decision Making, 15(2), 267–283. DOI: 10.1142/S0219622016500036
Podvezko, V., Zavadskas, E. K., & Podviezko, A. (2020). An extension of the new objective weight assessment methods CILOS and IDOCRIW to fuzzy MCDM. Economic Computation and Economic Cybernetics Studies and Research, 54(2), 59–75. DOI: 10.24818/18423264/54.2.20.04 (The profile described by DecisionMind in this card does not implement this paper's own fuzzy IDOCRIW formula; the paper reports that this formula can produce negative weights.)
Keshavarz-Ghorabaee, M., Amiri, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2021). Determination of objective weights using a new method based on the removal effects of criteria (MEREC). Symmetry, 13(4), 525. DOI: 10.3390/sym13040525
Odu, G. O. (2019). Weighting methods for multi-criteria decision making technique. Journal of Applied Sciences and Environmental Management, 23(8), 1449–1457. DOI: 10.4314/jasem.v23i8.7