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 "title": "TODIM method",
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 "excerpt": "TODIM (TOmada de Decisão Interativa e Multicritério) is a multi-criteria decision-making method, proposed in the early 1990s, that ranks alternatives by gains and losses relative to a reference point.",
 "snippet": "TODIM (TOmada de Decisão Interativa e Multicritério) is a multi-criteria decision-making method, proposed in the early 1990s, that ranks alternatives by gains and losses relative to a reference point.",
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 "markdown": "# TODIM method\n\nTODIM (Portuguese: TOmada de Decisão Interativa e Multicritério, Interactive and Multicriteria Decision Making) is a discrete multi-criteria decision-making method that ranks alternatives by measuring the gains and losses of each alternative against every other, relative to a reference point, in the spirit of prospect theory (behavioral economics theory: people judge outcomes as gains/losses from a reference point).<sup>[1](https://www.scielo.br/j/pope/a/XrxSHqDfKqjVCDGKKr7bkTv/?lang=en)</sup> It is used for choice and ranking problems with several criteria, such as selecting suppliers, service plans, materials, or trading rules.<sup>[1](https://www.scielo.br/j/pope/a/XrxSHqDfKqjVCDGKKr7bkTv/?lang=en)</sup>\n\n| Key fact | Detail |\n|---|---|\n| Acronym | TOmada de Decisão Interativa e Multicritério (Interactive and Multicriteria Decision Making)<sup>[1](https://www.scielo.br/j/pope/a/XrxSHqDfKqjVCDGKKr7bkTv/?lang=en)</sup> |\n| Theoretical basis | Prospect theory and its cumulative (nonlinear) form; gains and losses measured from a reference point<sup>[2](https://doi.org/10.1007/s10479-013-1345-0)</sup> |\n| Core output | A global dominance score \\( \\xi_{i} \\) per alternative, standardized between 0 and 1, that orders the alternatives<sup>[3](http://www.ijmp.jor.br/index.php/ijmp/article/view/1468)</sup> |\n| Key parameter | θ, the attenuation factor of losses, typically varied between 1 and 10 in sensitivity analysis<sup>[4](https://ideas.repec.org/a/spr/annopr/v211y2013i1p531-54810.1007-s10479-013-1345-0.html)</sup> |\n| Weights handling | Relative weights \\( w_{cr} = w_{c}/w_{r} \\) computed against the criterion of maximum weight<sup>[5](https://www.mdpi.com/2227-9717/10/3/609)</sup> |\n| Known failure modes | Two weight paradoxes (weight consistency, weight monotonicity) in the original formulation; uncertain measurements, weights, and θ<sup>[6](https://doi.org/10.1016/j.ejor.2018.02.054)</sup><sup> • </sup><sup>[7](https://dl.acm.org/doi/10.1016/j.cie.2017.10.006)</sup> |\n\n## How it works\n\nTODIM evaluates alternatives pairwise. For each criterion, the difference between two alternatives is read as a gain if positive, a loss if negative, and as the reference point if zero; these differences are the gains and losses of prospect theory's value function.<sup>[8](https://bibliotekanauki.pl/articles/578558.pdf)</sup> The method's value function has the same shape as the gains/losses function of cumulative prospect theory, and gains and losses are always established with respect to a reference point.<sup>[2](https://doi.org/10.1007/s10479-013-1345-0)</sup>\n\nLosses count less than gains, and the attenuation factor θ controls by how much. In the dominance function, a favorable difference contributes the relative weight times the normalized difference, an equal value contributes zero, and an unfavorable difference contributes the weighted difference divided by θ:<sup>[9](https://doi.org/10.3390/su11185057)</sup>\n\n\\[ \\phi_{j}(A_{i},A_{f})=\\begin{cases} \\dfrac{\\bar{w}_{jr}}{\\sum_{j=1}^{n}\\bar{w}_{jr}}\\, d(\\tilde{g}_{ij},\\tilde{g}_{fj}) & \\text{if } \\tilde{g}_{ij}>\\tilde{g}_{fj} \\\\ 0 & \\text{if } \\tilde{g}_{ij}=\\tilde{g}_{fj} \\\\ -\\dfrac{1}{\\theta}\\,\\dfrac{\\bar{w}_{jr}}{\\sum_{j=1}^{n}\\bar{w}_{jr}}\\, d(\\tilde{g}_{ij},\\tilde{g}_{fj}) & \\text{if } \\tilde{g}_{ij}<\\tilde{g}_{fj} \\end{cases} \\]\n\nwhere θ is the attenuation factor of losses and d is the distance between the two evaluations. Different choices of θ produce different shapes of the value function in the negative quadrant.<sup>[10](https://reference-global.com/download/article/10.2478/v10209-011-0001-1.pdf)</sup> The per-criterion contributions are summed into an additive difference function \\( \\delta(i,j) = \\sum_{c} \\phi_{c}(i,j) \\), which establishes the dominance of one alternative over another.<sup>[11](https://proceedings.informingscience.org/IS2002Proceedings/papers/Costa118Prior.pdf)</sup>\n\n## How it is done\n\nA practitioner runs the following sequence, from decision matrix to ranking:<sup>[3](http://www.ijmp.jor.br/index.php/ijmp/article/view/1468)</sup>\n\n1. Normalize the decision matrix by dividing each value by the sum of the values in its criterion.\n2. Compute relative weights \\( w_{cr} = w_{c}/w_{r} \\), where the reference criterion \\( C_{r} \\) is the criterion of maximum weight (\\( w_{r} = \\max\\{w_{c}\\} \\)).<sup>[5](https://www.mdpi.com/2227-9717/10/3/609)</sup>\n3. Build one partial dominance matrix \\( O_{c} \\) per criterion, with entries \\( \\phi_{c} \\) computed with the chosen θ, commonly varied in [1, 10].<sup>[8](https://bibliotekanauki.pl/articles/578558.pdf)</sup><sup> • </sup><sup>[4](https://ideas.repec.org/a/spr/annopr/v211y2013i1p531-54810.1007-s10479-013-1345-0.html)</sup>\n4. Sum the partial matrices into the global dominance \\( \\delta(A_{i}, A_{j}) \\) for each pair.\n5. Compute each alternative's performance \\( \\xi_{i} \\) as the sum of its dominance degrees, standardized between zero and 1.0 (a min-max normalization of \\( \\sum_{k} \\delta(A_{i}, A_{k}) \\)).<sup>[12](https://link.springer.com/article/10.1007/s44196-025-00806-7)</sup>\n6. Order the alternatives by \\( \\xi_{i} \\).\n\nThe reference point can be set in two ways: coordinates of zero gain and zero loss for every criterion, or the status quo; the zero-gain/zero-loss option is considered most promising when no additional information exists.<sup>[4](https://ideas.repec.org/a/spr/annopr/v211y2013i1p531-54810.1007-s10479-013-1345-0.html)</sup><sup> • </sup><sup>[10](https://reference-global.com/download/article/10.2478/v10209-011-0001-1.pdf)</sup> Value judgments may be expressed on a cardinal or a verbal scale.<sup>[11](https://proceedings.informingscience.org/IS2002Proceedings/papers/Costa118Prior.pdf)</sup>\n\n## Origin\n\nThe TODIM method was proposed by Luiz Flávio Autran Monteiro Gomes and Changsok Lima at the beginning of the 1990s through two articles published in European journals; a later study applied the method to the multicriteria rental evaluation of residential properties.<sup>[10](https://reference-global.com/download/article/10.2478/v10209-011-0001-1.pdf)</sup><sup> • </sup><sup>[22](http://www.din.uem.br/sbpo/sbpo2012/pdf/arq0207.pdf)</sup> Its intellectual precursor is prospect theory, published by [Daniel Kahneman](https://www.edgepedia.org/daniel-kahneman) and [Amos Tversky](https://www.edgepedia.org/amos-tversky) in [Econometrica](https://www.edgepedia.org/econometrica) in 1979.<sup>[13](https://doi.org/10.2307/1914185)</sup> Earlier work by Salminen, extending Korhonen, Moskowitz, and Wallenius on linear prospect theory in multi-criteria problems, preceded TODIM; TODIM departs from that line by being founded on the original, nonlinear prospect theory.<sup>[10](https://reference-global.com/download/article/10.2478/v10209-011-0001-1.pdf)</sup>\n\n## Variants\n\nMost variants adapt TODIM to uncertain or structured input information:\n\n- **Fuzzy and intuitionistic fuzzy TODIM.** These versions replace crisp evaluations with fuzzy numbers so that uncertain MCDM problems can be handled when the original method cannot treat the uncertainty.<sup>[14](https://www.sciencedirect.com/science/article/abs/pii/S0950705113002645)</sup> An extension to intuitionistic linguistic multiple attribute decision making appeared in Symmetry in 2017 by Shuwei Wang and [Jia Liu](https://www.edgepedia.org/jia-liu).<sup>[15](https://doi.org/10.3390/sym9060095)</sup>\n- **Interval type-2 fuzzy TODIM**, applied to green supplier selection, was published in the European Journal of Operational Research in 2016 by Jindong Qin, Xinwang Liu, and Witold Pedrycz.<sup>[16](https://doi.org/10.1016/j.ejor.2016.09.059)</sup>\n- **IVIF-TODIM** works with interval-valued intuitionistic fuzzy numbers and aggregates heterogeneous assessments (crisp numbers, interval numbers, triangular fuzzy numbers) into a collective matrix before applying the dominance calculation.<sup>[9](https://doi.org/10.3390/su11185057)</sup>\n- **Probabilistic linguistic TODIM** handles probabilistic linguistic term sets; a version based on similarity measures and entropy was published by Cun Wei and Jiang Wu in 2019,<sup>[17](https://doi.org/10.3233/jifs-191164)</sup> and a 2025 extension for group decision making combines CRITIC and best-worst-method weights with a new distance measure, reporting that the ordering of alternatives stays the same across different parameters ρ and λ.<sup>[12](https://link.springer.com/article/10.1007/s44196-025-00806-7)</sup>\n- **Choquet-integral TODIM** rewrites the dominance measures through the [Choquet integral](https://www.edgepedia.org/choquet-integral) to capture interactions between criteria; it appeared in Annals of Operations Research in 2013 by Luiz Flavio Autran Monteiro Gomes, Maria Augusta Soares Machado, and Luis Alberto Duncan Rangel.<sup>[2](https://doi.org/10.1007/s10479-013-1345-0)</sup>\n- **SMAA-TODIM** applies stochastic multiobjective acceptability analysis to explore the uncertainties in criteria measurements, weights, and θ simultaneously.<sup>[7](https://dl.acm.org/doi/10.1016/j.cie.2017.10.006)</sup>\n- **Generalized TODIM**, analyzed by Bonifacio Llamazares in the European Journal of Operational Research in 2018, is a simplified and generalized formulation that avoids two weight paradoxes of the original method.<sup>[6](https://doi.org/10.1016/j.ejor.2018.02.054)</sup>\n- **Evidential reasoning TODIM.** A 2024 study combined heterogeneous evidential reasoning with TODIM, designing a loss-aversion parameter that takes different values depending on the decision maker's risk attitude.<sup>[18](http://www.jorms.net/EN/10.12005/orms.2024.0324)</sup>\n\n## Applications\n\nDocumented applications include evaluation of broadband [Internet access](https://www.edgepedia.org/internet-access) plans, where sensitivity runs over θ from 1 to 10 were performed;<sup>[1](https://www.scielo.br/j/pope/a/XrxSHqDfKqjVCDGKKr7bkTv/?lang=en)</sup> supplier selection, including a real furniture-industry case where fuzziness was added to capture decision-maker bias that classical methods may miss;<sup>[19](https://link.springer.com/article/10.1080/18756891.2015.1001954)</sup> sustainable and green supplier evaluation;<sup>[16](https://doi.org/10.1016/j.ejor.2016.09.059)</sup><sup> • </sup><sup>[9](https://doi.org/10.3390/su11185057)</sup> selection of technical trading rules in a prospect-theory-based trading system;<sup>[5](https://www.mdpi.com/2227-9717/10/3/609)</sup> and material selection.<sup>[20](https://pubs.aip.org/aip/acp/article/2484/1/030007/2879576/Generalized-TODIM-method-and-its-application-in)</sup>\n\n## Limitations and alternatives\n\nThree types of TODIM inputs are usually uncertain: the criteria measurements, the criteria weights, and the attenuation factor of the losses; these uncertainties can coexist.<sup>[7](https://dl.acm.org/doi/10.1016/j.cie.2017.10.006)</sup> The original formulation is vulnerable to two paradoxes affecting the weights of the model, weight consistency and weight monotonicity; Llamazares's generalization establishes conditions under which they are avoided, and a 2024 material-selection study confirms the generalized method satisfies both properties.<sup>[6](https://doi.org/10.1016/j.ejor.2018.02.054)</sup><sup> • </sup><sup>[20](https://pubs.aip.org/aip/acp/article/2484/1/030007/2879576/Generalized-TODIM-method-and-its-application-in)</sup>\n\nPublished comparisons of θ sensitivity are reassuring but limited. In the broadband study, the ranking obtained with θ = 1 varied very little from rankings with θ up to 10;<sup>[1](https://www.scielo.br/j/pope/a/XrxSHqDfKqjVCDGKKr7bkTv/?lang=en)</sup> in the trading-rule study, changing θ over {0.1, 0.25, 0.5, 0.75, 0.9} altered the selected rule in only 10% of scenarios relative to the nominal θ = 0.5.<sup>[5](https://www.mdpi.com/2227-9717/10/3/609)</sup> The meaning of θ itself is reported inconsistently: one study reads θ < 1 as risk-averse behavior and θ > 1 as more attenuated risk preferences,<sup>[5](https://www.mdpi.com/2227-9717/10/3/609)</sup> while a 2025 paper states that θ > 1 indicates a risk-averse decision maker and that larger θ means a higher degree of loss avoidance.<sup>[12](https://link.springer.com/article/10.1007/s44196-025-00806-7)</sup> An open question is this conflicting interpretation of θ across published studies.\n\nTODIM is described as non-compensatory, meaning advantages on one criterion cannot be traded off against disadvantages on another, and its embedded normalization is said to minimize rank reversal.<sup>[5](https://www.mdpi.com/2227-9717/10/3/609)</sup> In a sensitivity experiment with random weight changes of up to 20%, ELECTRE III, PROMETHEE II, and TODIM showed no internal ranking inconsistency, while only TOPSIS failed to maintain its best alternative across iterations, with five changes.<sup>[21](https://www.scielo.br/j/pope/a/KXsVppH73BVRjNQMBfdcfvf/?lang=en)</sup> Against PROMETHEE II specifically, TODIM differs in two ways: it splits the partial dominance equation into conditional branches for gain, indifference, and loss, and it incorporates the mitigation factor θ on losses.<sup>[5](https://www.mdpi.com/2227-9717/10/3/609)</sup> Under certain hypotheses, SAW and PROMETHEE II can be obtained as specific cases of the generalized TODIM method.<sup>[6](https://doi.org/10.1016/j.ejor.2018.02.054)</sup> A comparative study of TODIM's adherence to prospect theory concluded that its variations still do not bring the benefits of the consolidated theory to decision-aiding contexts.<sup>[3](http://www.ijmp.jor.br/index.php/ijmp/article/view/1468)</sup>\n\n## References\n\n1. [An application of the TODIM method to the evaluation of Broadband Internet plans](https://www.scielo.br/j/pope/a/XrxSHqDfKqjVCDGKKr7bkTv/?lang=en)\n2. [Luiz Flavio Autran Monteiro Gomes, Maria Augusta Soares Machado, Luis Alberto Duncan Rangel (2013). Behavioral multi-criteria decision analysis: the TODIM method with criteria interactions. Annals of Operations Research.](https://doi.org/10.1007/s10479-013-1345-0)\n3. [Comparative analysis of the TODIM method adherence to prospect theory](http://www.ijmp.jor.br/index.php/ijmp/article/view/1468)\n4. [Behavioral multi-criteria decision analysis: the TODIM method with criteria interactions (Annals of Operations Research, 2013)](https://ideas.repec.org/a/spr/annopr/v211y2013i1p531-54810.1007-s10479-013-1345-0.html)\n5. [A Multicriteria Decision Trading System Based on Prospect Theory: A Risk Return Analysis of the TODIM Method](https://www.mdpi.com/2227-9717/10/3/609)\n6. [Bonifacio Llamazares (2018). An analysis of the generalized TODIM method. European Journal of Operational Research.](https://doi.org/10.1016/j.ejor.2018.02.054)\n7. [The SMAA-TODIM approach (Computers and Industrial Engineering)](https://dl.acm.org/doi/10.1016/j.cie.2017.10.006)\n8. [Passos, Gomes, TODIM classification paper](https://bibliotekanauki.pl/articles/578558.pdf)\n9. [Ren-Jie Mao and colleagues (2019). A Heterogeneous MCDM Framework for Sustainable Supplier Evaluation and Selection Based on the IVIF-TODIM Method. Sustainability.](https://doi.org/10.3390/su11185057)\n10. [Behavioral multi-criteria decision analysis: further elaborations on the TODIM method](https://reference-global.com/download/article/10.2478/v10209-011-0001-1.pdf)\n11. [Priorities Assignment for Information Systems Based on Todim Multicriteria Method](https://proceedings.informingscience.org/IS2002Proceedings/papers/Costa118Prior.pdf)\n12. [Improved TODIM Method for Probabilistic Linguistic MAGDM Based on New Distance Measure](https://link.springer.com/article/10.1007/s44196-025-00806-7)\n13. [Daniel Kahneman, Amos Tversky (1979). Prospect Theory: An Analysis of Decision under Risk. Econometrica.](https://doi.org/10.2307/1914185)\n14. [IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making](https://www.sciencedirect.com/science/article/abs/pii/S0950705113002645)\n15. [Shuwei Wang, Jia Liu (2017). Extension of the TODIM Method to Intuitionistic Linguistic Multiple Attribute Decision Making. Symmetry.](https://doi.org/10.3390/sym9060095)\n16. [Jindong Qin, Xinwang Liu, Witold Pedrycz (2016). An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment. European Journal of Operational Research.](https://doi.org/10.1016/j.ejor.2016.09.059)\n17. [Cun Wei, Jiang Wu (2019). TODIM method for probabilistic linguistic multiple attribute group decision making based on the similarity measures and entropy. Journal of Intelligent & Fuzzy Systems.](https://doi.org/10.3233/jifs-191164)\n18. [Heterogeneous Evidential Reasoning Decision Making Method Based on TODIM](http://www.jorms.net/EN/10.12005/orms.2024.0324)\n19. [A Fuzzy TODIM Approach for the Supplier Selection Problem](https://link.springer.com/article/10.1080/18756891.2015.1001954)\n20. [Generalized TODIM method and its application in material selection process (AIP Conference Proceedings, 2024, vol. 2484)](https://pubs.aip.org/aip/acp/article/2484/1/030007/2879576/Generalized-TODIM-method-and-its-application-in)\n21. [Considerations regarding the choice of ranking multiple criteria decision making methods (Cadernos de Saúde Pública / SciELO)](https://www.scielo.br/j/pope/a/KXsVppH73BVRjNQMBfdcfvf/?lang=en)\n22. [Arq0207 (din.uem.br)](http://www.din.uem.br/sbpo/sbpo2012/pdf/arq0207.pdf)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n",
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 "speakable": "TODIM is a multi-criteria decision-making method, proposed in the early 1990s, that ranks alternatives by gains and losses relative to a reference point."
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