Extension card · Fuzzy
Interval CODAS (Yeni and Özçelik, 2018)
Interval CODAS is the form of CODAS used when experts give both the degree of support and the degree of rejection for a criterion as intervals, and the opinions of several experts need to be combined. It directly supports a group decision and still ranks the result with a single assessment score.
Base method
CODAS →
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?
Four things change. The two-distance comparison logic does not.
Cells. In crisp CODAS every cell is a single number. Here every cell consists of four numbers: the lower and upper end of the support interval, and the lower and upper end of the rejection interval. The sum of the support upper end and the rejection upper end cannot exceed one. Pairs given separately by several experts are combined into a single group matrix with expert weights. Criterion weights, too, can be given not as a crisp number but in this same four-component structure. The method directly supports a group decision.
Scale equalisation. Crisp CODAS divides every cell by the column's largest value, or by the ratio to its smallest. There is no such division here. Every cell is multiplied by the criterion's four-component weight through a special operation: support components are multiplied among themselves, while rejection components are combined with an "either one or the other" logic (a probabilistic sum). This operation scales support and rejection together and preserves the four components' validity constraint.
Distance. In crisp CODAS the negative-ideal is built from the column's worst value. Here too, a negative-ideal pair is built for every criterion from the weighted pairs' lowest support and highest rejection components. Euclidean and Taxicab distances are computed across the four components. DecisionMind here uses a normalisation that matches the figures in Table 9 of the article that is this extension's source. This differs slightly from Park's (2008) original proposal and is a choice DecisionMind fixes for this extension.
Result and defuzzification. The assessment score is built with the same pairwise comparison rule as crisp CODAS (threshold θ, Euclidean first, Taxicab if that is not sufficient) and is already a single number. There is no separate defuzzification step; the four components only merge into a single number during the distance calculation.
How to Read the Output
The assessment score is read in the same way as in crisp CODAS. It is a relative positional measure, not a percentage, and cannot be compared with a different analysis. The difference is here: because the score comes from a four-component structure, it carries both the support-rejection balance and the interval width of each together. A criterion's interval narrowing or widening can shift the score even without the support-rejection balance itself changing.
Thus instead of writing:
"The interval CODAS score reflects the experts' shared view precisely"
the report should read:
"The score comes from combining several experts' support and rejection intervals; the width of these intervals carries separate information and should be reported alongside the score"
When to Prefer This over the Base Method
Use this extension when experts give both a support and a rejection degree for a criterion, each as an interval, and several experts' opinions need to be combined with weights. If support and rejection are adequately expressed by a single number, the simpler intuitionistic fuzzy extension may be sufficient. Measured criteria should not be brought into this extension. If the matrix must be of a single type, a measured value is written with a support interval of [1,1] and a rejection interval of [0,0]. Base CODAS's exit condition applies here exactly as it does there.
Mistakes Specific to This Extension
Failing to check the validity constraint. A pair is invalid if the sum of the support upper end and the rejection upper end exceeds one. Feeding this into the calculation without checking renders the result meaningless.
Not testing the sensitivity of the threshold value (θ). The source article tried θ between 0.01 and 0.05. Different θ values can give a different ranking; this test must not be skipped.
Silently changing the aggregation operation (IIFWA). The source article combines expert opinions with the IIFWA (weighted average) operation. Using a geometric-mean-based operation instead produces a different group matrix, and a different result, from the same inputs; this choice must be stated in the report.
Defuzzifying first and then running crisp CODAS. Reducing the four components to a single score and then running crisp CODAS is not this extension. In the illustrative example below this shortcut does not change the ranking, but it does make the scores appear larger than they are; in a closer example it could change the ranking too.
The governing principle is this:
Interval CODAS exists to carry support and rejection degrees, in interval form, combining several experts' opinions, right through to the end. If the validity constraint is not checked, the threshold is not tested, or the aggregation operation is changed without justification, the apparent precision breaks down.
Cases
The first case is a genuine literature case: Yeni and Özçelik's (2018) personnel-selection example. The second case is an illustrative construction.
1. Recruitment: Choosing personnel among four candidates (Yeni and Özçelik, 2018)
An organisation will choose one of four candidates (A1-A4) against four criteria: communication, experience, education and teamwork; all four are "higher is better." Three experts (weighted 0.40; 0.35; 0.25) arrived at interval support-rejection pairs from judgements given on a verbal scale, and the organisation combined these three opinions into a single group matrix with the IIFWA operation. The table below is this combined matrix.
| Candidate | Communication (support; rejection) | Experience (support; rejection) | Education (support; rejection) | Teamwork (support; rejection) |
|---|---|---|---|---|
| A1 | [0.8049; 0.8434]; [0.00; 0.1324] | [0.672; 0.7652]; [0.00; 0.1832] | [0.8722; 0.885]; [0.00; 0.1057] | [0.303; 0.4048]; [0.4003; 0.5396] |
| A2 | [0.8722; 0.885]; [0.00; 0.1057] | [0.3702; 0.4622]; [0.3617; 0.4876] | [0.4705; 0.5627]; [0.2603; 0.3869] | [0.601; 0.6935]; [0.1275; 0.2559] |
| A3 | [0.6388; 0.7342]; [0.00; 0.2122] | [0.5497; 0.6431]; [0.1737; 0.3056] | [0.6333; 0.725]; [0.10; 0.225] | [0.5101; 0.6022]; [0.2213; 0.3475] |
| A4 | [0.4705; 0.5627]; [0.2603; 0.3869] | [0.4932; 0.5885]; [0.2195; 0.3586] | [0.4859; 0.5999]; [0.00; 0.3335] | [0.90; 0.90]; [0.10; 0.10] |
| Weight | [0.5593; 0.7449]; [0.00; 0.2398] | [0.6122; 0.6688]; [0.2814; 0.3266] | [0.5669; 0.7539]; [0.00; 0.2306] | [0.3103; 0.6833]; [0.00; 0.2892] |
The method multiplies every candidate's pair by the criterion's four-component weight, builds the negative-ideal from the lowest-support, highest-rejection pair, and computes the Euclidean and Taxicab distance across the four components.
| Candidate | Assessment score | Rank |
|---|---|---|
| A1 | 0.8141 | 1 |
| A3 | -0.1490 | 2 |
| A4 | -0.2305 | 3 |
| A2 | -0.4346 | 4 |
The result reads as follows. A1 has the highest support interval and lowest rejection interval on both communication and education; it leads clearly on these two criteria. A4 has the highest support on teamwork (0.90), but its weakness on communication and experience is not fully offset by this lead.
To see how robust the decision is, the calculation was rerun with the teamwork criterion's weight interval raised and the communication criterion's correspondingly lowered. Past a certain point this redistribution swaps the positions of third-ranked A4 and second-ranked A3: A4 becomes -0.1512, A3 becomes -0.1542. A1's first place and A2's last place are not disturbed by this change.
In the report: "With the given weights, A1 has the highest assessment score (0.81). The gap between second-ranked A3 and third-ranked A4 is small, and these two candidates swap places once the weight of the teamwork criterion is increased."
Source: Yeni and Özçelik (2018), Table 6 (the combined decision matrix and weights across experts). The assessment scores were obtained by independently rerunning DecisionMind's interval CODAS engine on this matrix, and match the article's own Table 10 results (within a tolerance of 0.0001).
2. Textiles: A garment firm's choice of a new fabric supplier
A garment firm will contract with one of three fabric suppliers for a new collection. The criteria are: fabric quality, delivery-time reliability, and unit cost (the last is "lower is better"). Three of the firm's departments (design, production, purchasing) have each scored every supplier, from their own perspective, on how much they support and how much they hesitate, as intervals; these opinions have been combined with weights.
The method multiplies the three suppliers' combined pairs by the weights, builds the negative-ideal, and computes the two distances. Suppose the supplier offering the highest-quality fabric also has the highest cost. It nonetheless comes first, because the weight on the quality criterion exceeds that on cost.
The firm's hesitation is this. The difference of opinion among the three departments has produced a wide interval for the second supplier. This width shows a difference of opinion between departments and should be reported separately alongside the score.
In the report: "With the high weight given to fabric quality, the highest-quality supplier reaches the highest assessment score; the interval width for the second supplier points to a difference of opinion between departments, and this should be tracked separately."
3. What Not to Do
Had the four components in the illustrative table been reduced first to a single score ((support lower + support upper minus rejection lower minus rejection upper) divided by two) and crisp CODAS run on that, the ranking would not change (A1, A3, A4, A2), but the scores would come out larger than they should (1.03 instead of 0.81 for A1, for instance). This is not interval CODAS, and in a closer example it could change the ranking too. The second error is combining the three departments' opinions with a geometric-mean-based operation instead of IIFWA and not stating this in the report; the same inputs yield a different group matrix. The third error is leaving the threshold value (θ) at its default without ever testing it; the small gap between second-ranked A3 and third-ranked A4 is, here as in the source article, sensitive to θ.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/iv-codas
Yeni, F. B., & Özçelik, G. (2018). Interval-Valued Atanassov Intuitionistic Fuzzy CODAS Method for Multi Criteria Group Decision Making Problems. Group Decision and Negotiation, 28, 433–452. DOI: 10.1007/s10726-018-9603-9
Keshavarz Ghorabaee, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2016). A New Combinative Distance-Based Assessment (CODAS) Method for Multi-Criteria Decision-Making. Economic Computation and Economic Cybernetics Studies and Research, 50(3), 25–44. (no DOI. This article is not registered on Crossref; see the Sources section of the base CODAS card.)
Atanassov, K. T., & 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