Extension card · Fuzzy
Interval-valued intuitionistic fuzzy COPRAS (aggregative, DecisionMind derivation)
Interval-valued intuitionistic fuzzy COPRAS is the form of COPRAS for situations where the degree of support and rejection of a judgement is itself not a single number but an interval. It builds the benefit and cost sums from these four-number cells.
Base method
COPRAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Fuzzy →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Three things change; the benefit-to-cost ratio logic does not.
Cells. In crisp COPRAS every cell is a single number. Here every cell consists of four numbers: the lower and upper bound of the degree of support, and the lower and upper bound of the degree of rejection. A judgement such as "my confidence in this supplier is between 0.65 and 0.75, my reservation between 0.15 and 0.20" corresponds to these four numbers. Criterion weights are given from outside as crisp (single) numbers.
Scale equalisation. DecisionMind first reduces every four-number cell to a single score: the score is found by subtracting the sum of the lower and upper rejection bounds from the sum of the lower and upper support bounds, then dividing by two. Since this score can range between −1 and 1, 1 is added to shift it. Crisp COPRAS's column-sum normalisation is then applied to these shifted scores. This reduces the four-number cell to IF-COPRAS's support-rejection score-difference logic, but through a score that represents the interval's midpoint.
Distance / score / aggregation. The normalised and weighted scores are split into a benefit sum and a cost sum, as in crisp COPRAS; the relative significance value (Q) is built with the same ratio, using the correction term that rewards low cost.
Result and defuzzification. Defuzzification happens at the very start, not at the end: the four-number support-rejection interval descends to a single midpoint score in the very first step, and the interval's width, that is, how little is known, becomes invisible in this score. The output is directly a single benefit degree, in the same form as in crisp COPRAS.
DecisionMind fixes, in this classical form, the midpoint score function and the +1 shift; no route is followed that separately uses the interval's width as a penalty or a reward.
How to Read the Output
The output is a benefit degree and a ranking, in the same form as in crisp COPRAS; it is read the same way. The difference is here: this score rests only on the midpoint of the support-rejection interval and never shows how wide that interval is. A score coming from a narrow interval and the same score coming from a wide interval are indistinguishable in the benefit degree.
Thus instead of writing:
"According to interval-valued intuitionistic fuzzy COPRAS, A1 is the best alternative"
the report should read:
"The benefit degree rests only on the midpoint of the support-rejection interval; A1 leads with 100.00, and this does not mean 'definitively the best' without separately examining how narrow or wide A1's intervals are"
When to Prefer This over the Base Method
This extension is suitable when only an interval is known about a judgement's degree of support and rejection, that is, when an expert says "support is not exactly 0.6, but somewhere between 0.5 and 0.7." Examples: supplier audits where several assessors' support-rejection scores are summarised by an interval, or committee assessments intended to be left without reduction to a single exact ratio. If support and rejection are given as a single number (zero interval width), this extra width adds nothing, and intuitionistic fuzzy COPRAS (IF-COPRAS) is sufficient. Crisp COPRAS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also fully compensatory.
Mistakes Specific to This Extension
Allowing the lower bound to exceed the upper bound. In every interval the lower bound must be less than or equal to the upper bound, and the sum of the upper support and upper rejection bounds must not exceed 1.
Forgetting that the score function conceals the interval. DecisionMind uses the midpoint score. This is a canonical choice, but it does not report the interval's width; a narrow and a wide interval can give the same score, and this loss of information must be stated in the report.
Ignoring the shift constant (+1) and dividing the scores directly by the column sum. If negative scores are divided without shifting, the column sum can come out negative and the benefit degrees become meaningless.
Confusing this card with IVIF-COPRAS. Both use the same four-number cell shape (a support-rejection interval), but IV-COPRAS descends to a single score first and then proceeds with crisp COPRAS's division logic, whereas IVIF-COPRAS carries the four numbers through to the final step with Xu normalisation and interval-valued intuitionistic fuzzy addition (⊕), and defuzzifies only there with a λ-weighted score. These two engines follow different steps and produce different numbers; using one instead of the other changes the result.
The governing principle is this:
In interval-valued intuitionistic fuzzy COPRAS the support-rejection interval descends to a single midpoint score in the very first step. The interval's width never returns at any later step and must be stated separately in the report.
Cases
The first case is DecisionMind's validation example: since no shared IV-COPRAS application example exists in the literature, a synthetic table of three alternatives on three criteria was built, staying faithful to the formula chain. The second case is an illustrative construction.
1. Illustrative example: Three alternatives assessed on three criteria (DecisionMind validation example)
Three alternatives (A1, A2, A3) are assessed on three criteria (C1, C2 higher is better; C3 lower is better) with interval support-rejection pairs. The weights are C1=0.40, C2=0.35, C3=0.25.
| Alternative | C1 | C2 | C3 (cost) |
|---|---|---|---|
| A1 | support [0.65;0.75] / rejection [0.15;0.20] | support [0.45;0.55] / rejection [0.30;0.40] | support [0.55;0.65] / rejection [0.20;0.30] |
| A2 | support [0.75;0.85] / rejection [0.05;0.10] | support [0.55;0.65] / rejection [0.20;0.30] | support [0.35;0.45] / rejection [0.40;0.50] |
| A3 | support [0.55;0.65] / rejection [0.20;0.30] | support [0.65;0.75] / rejection [0.15;0.20] | support [0.45;0.55] / rejection [0.30;0.40] |
The method swaps support and rejection on C3, reduces every cell to a midpoint score and shifts it by adding 1, normalises against the column sum and multiplies by the weight. It splits off the benefit sum and cost sum and computes the relative significance value (Q).
| Alternative | Benefit degree | Rank |
|---|---|---|
| A1 | 100.00 | 1 |
| A2 | 98.47 | 2 |
| A3 | 97.84 | 3 |
The result reads as follows. The gap between the three alternatives is small: only 2.16 points separate the highest and the lowest benefit degree. A1 does not have the highest support-rejection score on C2 (that belongs to A3), but it carries a strong score on C1 and sits in a balanced position on C3 (cost, once reversed); this combination across the three criteria carries it into first place by a narrow margin.
The decision's hesitation is this: if the weight is shifted a small step towards C2, redistributed as C1=0.35, C2=0.40, C3=0.25, the benefit degrees (via Q, computed by independently rerunning DecisionMind's engine in Python) come out at 0.3352 for A1, 0.3303 for A2, and 0.3345 for A3. A1 keeps first place, but A2 and A3 swap positions: A3 becomes second, A2 third. This shows that the preference between the bottom two alternatives is sensitive even to a small weight shift.
In the report: "With the given weights (0.40; 0.35; 0.25), A1 has the highest benefit degree (100.00), but the gap between the three alternatives is small (2.16 points). Once the weight is shifted a small step towards C2 (0.35; 0.40; 0.25), A2 and A3 swap places; the ranking below A1's first place should therefore be treated as sensitive to the weight distribution."
Source: DecisionMind's validation example for the interval-valued intuitionistic fuzzy COPRAS engine; since no shared IV-COPRAS application table exists in the literature, this was built synthetically, and the expected results were derived from the manifest's own formula chain (midpoint score, +1 shift, column-sum normalisation, benefit/cost split, Q formula). The figures for the weight-change scenario were independently recomputed on the same engine by this card's author.
2. Theatre: Choosing a stage costume workshop
A state theatre will choose, among three candidates, the workshop to sew the new season's costumes. There are three criteria: sewing quality (higher is better), delivery time (lower is better) and fabric-supply reliability (higher is better). The costume designer and the production manager have reported, for each workshop, both how much they trust it and how much reservation they have, as an interval with a lower and upper bound that also reflects the difference of opinion between them.
The method compares the three workshops: it swaps support and rejection on delivery time, reduces every cell to a midpoint score and normalises it, multiplies by the weights and sums. The highest weight (0.5) has been given to sewing quality. Suppose the workshop with the highest-quality sewing is also the slowest to deliver; it nonetheless comes first, because the weight on sewing quality exceeds that on delivery time.
The production manager's hesitation is this: this workshop's delivery-time interval is wide, say between 3 and 6 weeks. If the premiere date is fixed, this width carries a risk of delaying the premiere in the worst case. The manager should look not only at the benefit degree but also at the upper end of the delivery-time interval.
In the report: "With the high weight given to sewing quality, the highest-quality workshop comes out ahead. This workshop's delivery-time interval is wide; since the premiere date is fixed, the worst-case scenario should be separately assessed."
3. What Not to Do
In the illustrative example, averaging the four bounds down to a single support-rejection pair and then running intuitionistic fuzzy COPRAS (IF-COPRAS): this erases from the outset the "how little is known" information carried by the interval's width and makes a separate calculation pointless. The second error is presenting the 100.00-against-98.47 gap between A1 and A2 as a fixed lead without ever noting its sensitivity to the weights; A2 and A3 already swap places under a small weight shift. The third error is comparing IV-COPRAS's result directly with IVIF-COPRAS's; although the two engines use the same cell shape, they follow different normalisation and aggregation steps, and their numbers are not interchangeable.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/iv-copras
Atanassov, K., & Gargov, G. (1989). Interval valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 31(3), 343–349. DOI: 10.1016/0165-0114(89)90205-4
Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. DOI: 10.1016/S0165-0114(86)80034-3
Zavadskas, E. K., & Kaklauskas, A. (1996). Determination of an efficient contractor by using the new method of multicriteria assessment. In International Symposium for the Organization and Management of Construction: Shaping Theory and Practice, Vol. 2, 94–104. (no DOI)
Moore, R. E. (1966). Interval Analysis. Prentice-Hall. (no DOI)