Extension card · Hesitant
Dual Hesitant Fuzzy COPRAS (Rani, Mishra and colleagues, 2020)
This is the form of COPRAS for situations where a cell records several possible degrees of both support and rejection separately. It builds the benefit and cost sums over these dual sets, and ranks the result by a relative-importance value.
Base method
COPRAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Hesitant →
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 decision logic stays the same.
Cells. In plain hesitant COPRAS (HF-COPRAS), each cell carries a single set, that is, more than one possible degree of support. In dual hesitant COPRAS, a cell consists of two sets: a support set (h) and a separate, independent rejection set (g). Just as the question "how much do I support this supplier" can have more than one possible answer, the question "how much do I reject it" can also have more than one possible answer, independently of the first. The sum of the two sets' largest elements cannot exceed 1. When the support set stands alone and the rejection set is empty (g=∅), the cell reduces to the same information as plain hesitant COPRAS; the dual structure only adds value when there is also an independent hesitancy on the rejection side. Weights are supplied from outside; in this family DecisionMind can also derive weights from the user's own criterion ranking using SWARA, but this is a separate input route and does not change COPRAS itself.
Scale equalisation. Crisp COPRAS turns each column into a share by dividing it by its own column total. There is no separate column-sum normalisation for dual hesitant sets, because the values are already degrees between 0 and 1. If more than one expert's opinion exists, these opinions are first combined into a single collective table, weighting each expert by their own reliability (giving a higher share to an expert with less disagreement).
Distance, score and combination. For each alternative, the support–rejection pairs within the benefit criteria are combined amongst themselves, and the pairs within the cost criteria separately, using a weighted aggregation rule. This aggregation still produces a support–rejection pair, not yet a single number. This pair is then reduced to a single number by a score function before it enters crisp COPRAS's Q formula: the score is the mean of the support set minus the mean of the rejection set. A correction term, built from the cost side's scores, is added to the benefit side's score; a coefficient γ sets how this correction is shared between benefit and cost. At γ=1, only the benefit score counts; at γ=0, only the cost balance counts; DecisionMind uses γ=0.5 by default.
Result and defuzzification. Defuzzification happens in two steps. First the support–rejection pairs are combined across the criteria (still remaining a pair), and then this combined pair is reduced to a single number by the γ-weighted score function. The output is, as in crisp COPRAS, a relative-importance value and a percentage derived from it, in the same form.
DecisionMind keeps the support–rejection mean-difference score and the γ-weighted Q formula fixed in this family; the value of γ is chosen by the user and must be stated in the report.
How to Read the Output
The relative-importance value and the percentage are read exactly as in crisp COPRAS: a share relative to the best alternative, not an absolute success percentage. The difference is that this share depends on the choice of the γ coefficient. The same table can give one ranking with γ=0.5, and a different ranking once γ is shifted to another value.
Hence, instead of writing:
"According to dual hesitant COPRAS, G2 is the best supplier"
the report should read:
"With the γ=0.5 balance, G2 has the highest relative importance (θ=0.412); when γ is shifted away from the benefit side towards the cost balance (γ=0.2), first place passes to G3, so the choice of γ must be justified in the report"
When to Prefer This over the Base Method
This extension is appropriate when a criterion is expressed with more than one plausible degree of both support and rejection, independently of one another. For example: part of an expert panel strongly supports a supplier, another part holds reservations about the same supplier for independent reasons, and the wish is to preserve these two groups of opinion without collapsing them into a single shared number. If there is no independent hesitancy on the rejection side, that is, if rejection is always calculated as "1 minus support", plain hesitant COPRAS (HF-COPRAS) is sufficient and the dual structure carries no extra information. Crisp COPRAS's exit condition still holds here: if no trade-off is acceptable on a criterion, this extension is also compensatory.
Mistakes Specific to This Extension
Reporting without stating the γ coefficient. In DecisionMind's own validation example in the manifest, γ can change the top ranking between 0.2 and 0.3, and the lower rankings between 0.7 and 0.8. The γ value used must always be stated in the report.
Deriving the rejection set from the support set. Filling the g set as "1 minus h" effectively reduces the dual structure to a plain hesitant structure and erases the independent information on the rejection side.
Ignoring expert weights. Giving each expert an equal share when combining several experts' sets can give undue prominence to an expert with a high level of disagreement.
The "more advanced, therefore better" fallacy. If there is no genuinely independent source on the rejection side, opening a plain hesitant set into a dual structure adds no information; it only makes the calculation more complex.
The governing principle is this:
Dual hesitant COPRAS exists to hold support and rejection as two mutually independent sources; if the γ coefficient is not stated in the report, the same data can produce a different ranking.
Cases
The first case is a genuine case from the literature. It is the sustainable supplier selection example from Rani, Mishra and colleagues' (2020) paper; in this example the rejection set is empty (g=∅), that is, it is the special case of plain hesitant data within the dual structure. The second case is an illustrative construction.
1. Supply chain: Sustainable supplier selection (Rani, Mishra and colleagues, 2020)
A business will select a sustainable supply-chain partner from among five suppliers (G1–G5). Eight criteria are used: quality, cost (these two economic), production capacity, eco-friendly design, sustainable materials use, pollution (these four environmental), and sector reputation and occupational health and safety (these two social). Cost and pollution are "less is better"; the rest are "more is better". The support sets given separately by three decision experts (the procurement manager, the production manager and the quality-control manager) have been combined into a single collective table using a weighting that gives a higher share to lower disagreement (0.343; 0.349; 0.308). In this example, no expert reported a separate rejection set; the rejection set remains empty (g=∅). The criterion weights were derived from the experts' own criterion rankings using the SWARA method.
| Supplier | Quality | Cost | Capacity | Eco-design | Sustainable materials | Pollution | Reputation | OHS |
|---|---|---|---|---|---|---|---|---|
| G1 | {0.2;0.3;0.7} | {0.5;0.6;0.7} | {0.4;0.5;0.7} | {0.4;0.6;0.8} | {0.3;0.4;0.6} | {0.3;0.5;0.7} | {0.3;0.5;0.8} | {0.1;0.3;0.4} |
| G2 | {0.3;0.4;0.8} | {0.5;0.7;0.8} | {0.7;0.8;0.9} | {0.4;0.8;0.9} | {0.4;0.7;0.9} | {0.1;0.5;0.8} | {0.6;0.7;0.8} | {0.2;0.5;0.8} |
| G3 | {0.4;0.6;0.7} | {0.4;0.5;0.7} | {0.5;0.6;0.9} | {0.1;0.2;0.4} | {0.3;0.6;0.7} | {0.4;0.6;0.7} | {0.3;0.5;0.6} | {0.6;0.7;0.9} |
| G4 | {0.5;0.7;0.9} | {0.5;0.8;0.9} | {0.2;0.6;0.7} | {0.3;0.6;0.8} | {0.4;0.7;0.8} | {0.2;0.5;0.9} | {0.2;0.4;0.8} | {0.3;0.4;0.9} |
| G5 | {0.1;0.4;0.5} | {0.5;0.7;0.9} | {0.3;0.5;0.8} | {0.2;0.5;0.8} | {0.2;0.5;0.6} | {0.2;0.7;0.9} | {0.2;0.3;0.7} | {0.3;0.6;0.8} |
| Direction | more is better | less is better | more is better | more is better | more is better | less is better | more is better | more is better |
| Weight | 0.1204 | 0.1264 | 0.1099 | 0.1203 | 0.1130 | 0.1312 | 0.1359 | 0.1429 |
(In every cell the rejection set is g=∅; the sets carry only the support degree h.)
The method reduces the sets within the benefit criteria to a benefit pair, and those within the cost criteria to a cost pair, using a weighted aggregation, then converts these pairs into a single number using the support–rejection mean-difference score and builds the relative importance (θ) with the γ=0.5 balance.
| Supplier | Relative importance (θ) | Utility degree (λ) | Rank |
|---|---|---|---|
| G2 | 0.412 | 100% | 1 |
| G3 | 0.378 | 91.58% | 2 |
| G4 | 0.353 | 85.53% | 3 |
| G1 | 0.333 | 80.73% | 4 |
| G5 | 0.296 | 71.76% | 5 |
The result reads as follows. G2 holds the highest support sets on capacity (0.7-0.8-0.9), sustainable materials (0.4-0.7-0.9) and reputation (0.6-0.7-0.8); although it shows the lowest support on cost (that is, it is relatively expensive), these advantages carry it to first place. G5 has the lowest support set on quality (0.1-0.4-0.5) and comes last.
The business has one hesitation. What happens if γ is shifted away from the benefit side towards the cost balance (γ=0.2)? When DecisionMind's engine was independently re-run, the θ values come out at 0.298 for G1, 0.310 for G2, 0.316 for G3, 0.257 for G4 and 0.233 for G5. First place passes from G2 to G3; G1 also overtakes G4 and rises to third place. Only G5 at the bottom stays last under both γ values.
In the report: "With the γ=0.5 balance, G2 has the highest relative importance (θ=0.412, λ=100%). When γ is shifted towards the cost balance (γ=0.2), first place passes to G3 (θ=0.316); the choice of γ and its rationale must therefore be stated in the report."
Source: Rani, Mishra and colleagues (2020), Symmetry journal, Table 4 (criteria), Table 7 (SWARA weights) and Table 8 (θ and λ values, γ=0.5). DecisionMind's dual hesitant COPRAS engine independently reproduces this table, in the special case of an empty rejection set (g=∅), to within a difference of ≤0.004. The figures for the γ=0.2 scenario were calculated separately by this card's author using the same engine, and their direction agrees with the general pattern the paper itself reports in its own Table 9 sensitivity analysis (a swap of positions at the top under low γ and at the bottom under high γ).
2. Museum Curation: A museum's selection of an artefact restoration workshop
A museum will select from among three workshops for the restoration of fragile textile artefacts. Four criteria apply: conservation expertise, turnaround time (less is better), the documentation quality of previous restorations, and fee (less is better). The museum's advisory board reported, for each workshop, both how much it trusted the workshop and, separately with its own rationale, how much reservation it held; trust and reservation come from different members, from different sets of observations, and are not derived from one another.
The method combines conservation expertise and documentation quality on the benefit side, and turnaround time and fee on the cost side, reduces both sides to a support–rejection difference score, and builds the relative importance with the γ=0.5 balance. Suppose the most experienced workshop is also the most expensive. It still comes out first, because conservation expertise is weighted more heavily than fee.
The board's hesitation is this. The reservation set for this workshop is wide; some members observed a small deviation in colour tone in a past restoration. This breadth of reservation does not show up in the relative-importance value. Given that a fragile and rare artefact is at stake, the board should look not only at the relative importance but also at the source of the reservation set.
In the report: "With the high weight given to conservation expertise, the most experienced workshop has the highest relative importance. The reservation set about this workshop is wide, and its source should be investigated further."
3. What Not to Do
Filling every cell's rejection set in the illustrative table as "1 minus the support set", for example adding G1's quality set {0.2;0.3;0.7} a rejection counterpart of {0.3;0.7;0.8}, effectively reduces the dual structure to plain hesitant COPRAS; the genuine contribution of the rejection side becomes invisible. The second error is reporting "dual hesitant COPRAS found G2 first" without stating γ; at γ=0.2, first place passes to G3, and a result written without showing this sensitivity is misleading. The third error is ignoring the expert weights (0.343; 0.349; 0.308) and averaging the three experts' sets with equal weight; this can give undue prominence to an expert with a high level of disagreement.
Formulae
The computation step by step, with the formula behind each step.
1. Step I (Rani & Mishra §3.2) - Originate the alternative set G=(G_1,…,G_m) and criteria set F=(F_1,…,F_n). Obtain l per-DE DHFE decision matrices Z=(z_{ij}^k)_{m×n}, k=1,…,l, with z_{ij}^k=⟨h_{ij}^k, g_{ij}^k⟩. The pure HFE inputs of the seminal HF-SWARA-COPRAS algorithm are the special case g_{ij}^k=∅ for all (i,j,k). Receive the criterion direction set Υ_1=benefit (max) / Υ_2=cost (min).
2. Step II (Rani & Mishra Eq.7) - Compute the crisp DE weight vector λ=(λ_1,…,λ_l). If weight_method='entropy_eq7', evaluate the HFS entropy e(ħ_k) (Eq.6 Mishra et al. 2018) on the k-th DE preference matrix and apply Eq.(7) λ_k=(1−e(ħ_k))/Σ_{k=1}^{l}(1−e(ħ_k)); the higher entropy ⇒ lower DE weight. If weight_method='direct', λ is supplied directly via expert_weight_vector additional_input.
3. Step III (Rani & Mishra Eq.8 → Zhu et al. 2012 DHFWA lift) - Aggregate the l per-DE DHFE matrices into the collective AHF-D matrix P=(ξ_{ij})_{m×n} via the Dual Hesitant Fuzzy Weighted Average (DHFWA) operator parameterised by λ_k. For pure HFE inputs (g=∅) this reduces verbatim to the HFWA Eq.(8) of Rani & Mishra; for non-empty g the non-membership component is aggregated geometrically by Zhu 2012 Eq.(4).
4. Step IV (Rani & Mishra §3.2 Steps IV-A..IV-F, Eqs.9-11) - Compute SWARA criterion weights w=(w_1,…,w_n). (IV-A) Defuzzify each AHF-D cell via the DHFE score s(ξ_{ij})=mean(h)−mean(g) (Zhu 2012 Def. 2; reduces to Mishra Eq.(2) S(ħ)=mean(h) when g=∅). (IV-B) DE ranks criteria highest→lowest by significance. (IV-C) DE supplies comparative significance s_j for j>1. (IV-D) Comparative coefficient k_j: k_1=1, k_j=s_j+1 for j>1 (Eq.9). (IV-E) Recalculated weight p_j: p_1=1, p_j=p_{j−1}/k_j for j>1 (Eq.10). (IV-F) Normalised criterion weight w_j=p_j/Σ_{j=1}^{n} p_j (Eq.11), Σw_j=1.
5. Step V-prep (Rani & Mishra 2020 §3.2 identity tag) - In the seminal HF-SWARA-COPRAS recipe the cost direction is *not* normalised by an explicit complement; Eqs.12-13 aggregate the raw benefit and cost cells of P and the direction inversion is performed analytically by the (1−γ)·(Σ S(v_k))/(S(v_i)·Σ 1/S(v_k)) term in Eq.15 (Step VI). Therefore F5 is an identity pass-through P'≡P preserved as a structural anchor in the engine's F-sequence; downstream Step V (F6) uses P' verbatim and reproduces paper Table 8 values for S(σ_i) and S(v_i) within ≤0.005 (paper reports 3-decimal precision).
6. Step V (Rani & Mishra Eqs.12-13) - Compute the per-alternative benefit-aggregate σ_i (Eq.12) and cost-aggregate v_i (Eq.13) as DHFE-valued weighted sums via Zhu 2012 Eqs.(5,7) (scalar multiplication λh and addition ⊕). σ_i aggregates over benefit columns Υ_1 of P; v_i aggregates over cost columns Υ_2 of P (using the direction-normalised P' from F5 when DHFE complement applies). The criterion weights w_j come from F4.
7. Step VI-prep - Defuzzify σ_i and v_i into crisp scalars S(σ_i) and S(v_i) via the DHFE score function s(d)=mean(h)−mean(g) (Zhu 2012 Def. 2). For pure HFE inputs g=∅ both σ_i and v_i carry empty non-membership and S reduces to Mishra Eq.(2). For DHFE inputs the score may be negative (s(d)∈[−1,1]); document the convention and ensure subsequent Q-formula handles S(v_i)≤0 via the |·| convention or via adding a positive offset (see L.deviations_from_seminal).
8. Step VI (Rani & Mishra Eqs.14-15) - Compute the relative weight θ_i of each alternative via the COPRAS compound ratio θ_i=γ·S(σ_i)+(1−γ)·(Σ_i S(v_i))/(S(v_i)·Σ_i (1/S(v_i))), γ∈[0,1]. Eq.(14) is the unweighted form (γ=1 limit), Eq.(15) is the strategy-parameterised form (γ=0.5 default). The compound ratio is the literature-standard Zavadskas-Kaklauskas 1996 COPRAS Q-formula applied on DHFE scores.
9. Steps VII-VIII (Rani & Mishra Eqs.16-17) - Determine the priority order and degree of utility. Step VII: the optimal alternative is G^*=arg max_i θ_i (Eq.16); alternatives are sorted descending by θ_i. Step VIII: degree of utility λ_i=(θ_i/θ_max)×100% (Eq.17), λ_i∈[0,100]%, λ_{i^*}=100%.
Sources
Rani, P., Mishra, A. R., Krishankumar, R., Mardani, A., Cavallaro, F., Ravichandran, K. S., & Balasubramanian, K. (2020). Hesitant Fuzzy SWARA-Complex Proportional Assessment Approach for Sustainable Supplier Selection. Symmetry, 12(7), 1152. DOI: 10.3390/sym12071152
Zhu, B., Xu, Z., & Xia, M. (2012). Dual hesitant fuzzy sets. Journal of Applied Mathematics, 2012, 1–13. DOI: 10.1155/2012/879629
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)
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418