# Standards for the Weighting of Criteria and the Measurement of Interaction

*Standards*)

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Standards for the Individual Assessments

#### 3.2. Standards for the Initial Joint Assessments

_{iJ}for alternative i according to J will then be obtained by adding the ${C}_{ij}$ given by Equation (1) along the criteria j in J. To avoid implicitly overvaluing those criteria j with a large ${K}_{j}$, it will instead be employed thusly:

_{j∈J}${P}_{ij}$ or by 1 − Π

_{j∈J}(1 − ${P}_{ij}$). Nevertheless, counting is the easier form of evaluating the effect of joining the criteria.

_{j}).

#### 3.3. Standards for Considering Interactions

#### 3.3.1. Simplified Capacities

#### 3.3.2. Combination via the Choquet Integral

_{1}, ..., x

_{t}), of domain S = {1, ... , t} and values in R

^{+}, the Choquet Integral of x with respect to the capacity μ on S associates with x the non-negative real number

_{τ(1)}≤ x

_{τ(2)}≤ ... ≤ x

_{τ(t−1)}≤ x

_{τ(t)}and x

_{τ(0)}= 0.

## 4. Discussion

#### 4.1. First Example

#### 4.2. Second Example

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Greco, S.; Ehrgott, M.; Figueira, J. Multiple Criteria Decision Analysis: State of the Art Surveys Series; Springer: New York, NY, USA, 2016. [Google Scholar]
- Cinelli, M.; Kadziński, M.; Gonzalez, M.; Słowiński, R. How to support the application of multiple criteria decision analysis? Let us start with a comprehensive taxonomy. Omega
**2020**, 96, 102261. [Google Scholar] [CrossRef] - Sant’Anna, A.P.; Nogueira, H.D.; Rabelo, L.M. Probabilistic Composition for Fast Group Decisions. Braz. J. Oper. Prod. Manag.
**2011**, 8, 65–82. [Google Scholar] [CrossRef] - Gaviao, L.O.; Sant’Anna, A.P.; Lima, G.B.A.; Garcia, P.A.D.A.; Kostin, S.; Asrilhant, B. Selecting a Cargo Aircraft for Humanitarian and Disaster Relief Operations by Multicriteria Decision Aid Methods. IEEE Trans. Eng. Manag.
**2019**, 67, 631–640. [Google Scholar] [CrossRef] - Duarte, A.M., Jr.; Silva, H.G.A. Equity Valuation with Fuzzy Multicriteria Decision Analysis. Rev. Bras. Finanças
**2018**, 16, 221–249. [Google Scholar] - Garcia, P.A.D.A.; Sant’Anna, A.P. Vendor and logistics provider selection in the construction sector: A probabilistic preferences composition approach. Pesqui. Oper.
**2015**, 35, 363–375. [Google Scholar] [CrossRef][Green Version] - Yang, M.; Nazir, S.; Xu, Q.; Ali, S. Deep Learning Algorithms and Multicriteria Decision-Making Used in Big Data: A Systematic Literature Review. Complex.
**2020**, 2020, 2836064. [Google Scholar] [CrossRef] - Alzate-Mejía, N.; Santos-Boada, G.; de Almeida-Amazonas, J. Decision-Making under Uncertainty for the Deployment of Future Hyperconnected Networks: A Survey. Sensors
**2021**, 21, 3791. [Google Scholar] [CrossRef] - Dassonneville, R.; Tien, C. Introduction to Forecasting the 2020 US Elections. Political Sci. Politics
**2021**, 54, 47–51. [Google Scholar] [CrossRef] - Erikson, R.S.; Wlezien, C. Forecasting the 2020 Presidential Election: Leading Economic Indicators, Polls, and the Vote. Political Sci. Politics
**2021**, 54, 55–58. [Google Scholar] [CrossRef] - Heidemanns, M.; Gelman, A.; Morris, G.E. An Updated Dynamic Bayesian Forecasting Model for the US Presidential Election. Harv. Data Sci. Rev.
**2020**, 2, 4. [Google Scholar] [CrossRef] - Schofield, N.; Gallego, M. Leadership or Chaos: The Heart and Soul of Politics; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Saaty, T. The Analytic Hierarchy Process; McGraw Hill: New York, NY, USA, 1980. [Google Scholar]
- Sant’Anna, A.P.; Lima, G.B.A.; Sant’Anna, L.A.D.F.P.; Gavião, L.O. Two-Stage Composition of Probabilistic Preferences. Ann. Data Sci.
**2018**, 7, 491–523. [Google Scholar] [CrossRef] - Keeney, R.L.; Raiffa, H.; Meyer, R.F. Decisions with Multiple Objectives: Preferences and Value Trade-Offs; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Sant’Anna, A.P. Probabilistic composition of criteria for schedule monitoring. Pesqui. Oper.
**2010**, 30, 751–767. [Google Scholar] [CrossRef] - Kojadinovic, I. Estimation of the weights of interacting criteria from the set of profiles by means of information-theoretic functionals. Eur. J. Oper. Res.
**2004**, 155, 741–751. [Google Scholar] [CrossRef] - Duarte, L.T. A novel multicriteria decision aiding method based on unsupervised aggregation via the Choquet integral. IEEE Trans. Eng. Manag.
**2017**, 65, 293–302. [Google Scholar] [CrossRef] - Santanna, A.P.; Sant’Anna, J.L. A principle of preference concentration applied to the unsupervised evaluation of the importance of multiple criteria. Pesqui. Oper.
**2019**, 39, 317–338. [Google Scholar] [CrossRef] - Capano, G.; Woo, J.J. Resilience and robustness in policy design: A critical appraisal. Policy Sci.
**2017**, 50, 399–426. [Google Scholar] [CrossRef] - Hadjimichael, A.; Gold, D.; Hadka, D.; Reed, P. Rhodium: Python Library for Many-Objective Robust Decision Making and Exploratory Modeling. J. Open Res. Softw.
**2020**, 8, 12. [Google Scholar] [CrossRef] - Sant’Anna, A.P. Rough sets analysis with antisymmetric and intransitive attributes: Classification of brazilian soccer clubs. Pesqui. Oper.
**2008**, 28, 217–230. [Google Scholar] [CrossRef][Green Version] - Skowron, A.; Dutta, S. Rough sets: Past, present, and future. Nat. Comput.
**2018**, 17, 855–876. [Google Scholar] [CrossRef][Green Version] - Choquet, G. Theory of capacities. Ann. Institut. Fourier
**1954**, 5, 131–295. [Google Scholar] [CrossRef][Green Version] - Doumpos, M.; Figueira, J.R.; Greco, S.; Zopounidis, C. New Perspectives in Multiple Criteria Decision Making—Innovative Applications and Case Studies; Springer: Berlin/Heidelberg, Germany, 2019. [Google Scholar]
- Marichal, J.-L.; Roubens, M. Determination of weights of interacting criteria from a reference set. Eur. J. Oper. Res.
**2000**, 124, 641–650. [Google Scholar] [CrossRef][Green Version] - Sant’Anna, A.P.; De Mello, J.C.C.B.S. Validating rankings in soccer championships. Pesqui. Oper.
**2012**, 32, 407–422. [Google Scholar] [CrossRef][Green Version] - Sant’Anna, A.P.; Barreto, M.F.S.S.M. Inequality Assessment by Probabilistic Development Indices. Soc. Indic. Res.
**2019**, 148, 733–746. [Google Scholar] [CrossRef] - Sant’Anna, A.P.; Martins, E.F.; Lima, G.B.A.; Da Fonseca, R.A. Beta Distributed Preferences in the Comparison of Failure Modes. Procedia Comput. Sci.
**2015**, 55, 862–869. [Google Scholar] [CrossRef][Green Version] - Martins, E.F. Instrumento Híbrido Aplicado ao Estudo da Confiabilidade Humana em Evento de Perda de Energia Elétrica Externa em Usina Nuclear. Ph.D. Thesis, Universidade Federal Fluminense, Niteroi, Brazil, 2015. [Google Scholar]

Alternative | C11 | C12 | C13 | C21 | C22 | C23 | C31 | C41 |
---|---|---|---|---|---|---|---|---|

A1 | 97.5 | 96.5 | 97 | 94 | 93.5 | 96.5 | 99 | 96 |

A2 | 97.5 | 96.5 | 97 | 94 | 98 | 96.5 | 96 | 97 |

A3 | 97.5 | 93 | 97 | 98 | 95.5 | 96.5 | 98 | 94 |

A4 | 97.5 | 96.5 | 97 | 96 | 98 | 96.5 | 94 | 95 |

A5 | 94 | 96.5 | 93 | 94 | 93.5 | 93 | 95 | 98 |

A6 | 94 | 96.5 | 97 | 98 | 98 | 96.5 | 93 | 93 |

A7 | 94 | 96.5 | 94 | 98 | 95.5 | 96.5 | 97 | 99 |

A8 | 92 | 91.5 | 91 | 91.5 | 91.5 | 91 | 90.5 | 91.5 |

A9 | 91 | 91.5 | 91 | 91.5 | 91.5 | 91 | 90.5 | 90 |

A10 | 90 | 90 | 91 | 90 | 90 | 91 | 92 | 91.5 |

A1 | 89 | 89 | 89 | 89 | 89 | 89 | 89 | 89 |

A12/A100 | 44 | 44 | 44 | 44 | 44 | 44 | 44 | 44 |

Alternative | C1 | C2 | C3 | C4 |
---|---|---|---|---|

A1 | 0.019596 | 0.019125 | 0.020000 | 0.019394 |

A2 | 0.019596 | 0.019428 | 0.019394 | 0.019596 |

A3 | 0.019360 | 0.019529 | 0.019798 | 0.018990 |

A4 | 0.019596 | 0.019562 | 0.018990 | 0.019192 |

A5 | 0.019091 | 0.018889 | 0.019192 | 0.019798 |

A6 | 0.019360 | 0.019697 | 0.018788 | 0.018788 |

A7 | 0.019158 | 0.019529 | 0.019596 | 0.020000 |

A8 | 0.018485 | 0.018451 | 0.018283 | 0.018485 |

A9 | 0.018418 | 0.018451 | 0.018283 | 0.018182 |

A10 | 0.018249 | 0.018249 | 0.018586 | 0.018485 |

A11 | 0.017980 | 0.017980 | 0.017980 | 0.017980 |

A12 a A100 | 0.008889 | 0.008889 | 0.008889 | 0.008889 |

Criteria | L = 2 | L = 3 | Full |
---|---|---|---|

{C1} | 0.494898 | 0.331435 | 0.250323 |

{C2} | 0.497449 | 0.333144 | 0.251613 |

{C3} | 0.505102 | 0.338269 | 0.255484 |

{C4} | 0.505102 | 0.338269 | 0.255484 |

{C1,C2} | 0.988946 | 0.662301 | 0.500215 |

{C1,C3} | 1 | 0.669704 | 0.505806 |

{C1,C4} | 0.989796 | 0.662870 | 0.500645 |

{C2,C3} | 0.993197 | 0.665148 | 0.502366 |

{C2,C4} | 0.998299 | 0.668565 | 0.504946 |

{C3,C4} | 1 | 0.669704 | 0.505806 |

{C1,C2,C3} | 1 | 0.993166 | 0.750108 |

{C1,C2,C4} | 1 | 0.992597 | 0.749677 |

{C1,C3,C4} | 1 | 0.997722 | 0.753548 |

{C2,C3,C4} | 1 | 1 | 0.755269 |

Alternative | L = 2 | L = 3 | Full |
---|---|---|---|

A1 | 0.079200 | 0.078661 | 0.078132 |

A2 | 0.078377 | 0.078156 | 0.078014 |

A3 | 0.078654 | 0.078243 | 0.077684 |

A4 | 0.078299 | 0.077788 | 0.077340 |

A5 | 0.077992 | 0.077452 | 0.076988 |

A6 | 0.078086 | 0.077117 | 0.076636 |

A7 | 0.079200 | 0.078842 | 0.078301 |

A8 | 0.073938 | 0.073889 | 0.073704 |

A9 | 0.073731 | 0.073530 | 0.073334 |

A10 | 0.074143 | 0.073765 | 0.073577 |

A11 | 0.071919 | 0.071919 | 0.071919 |

A12/A100 | 0.035556 | 0.035556 | 0.035556 |

Mode of Failure | Situation |
---|---|

M1 | align replacement generator |

M2 | manually activate water feed pump |

M3 | timely activate emptying tank prevention |

M4 | restore auxiliary feedwater system |

M5 | start bleed and feed |

M6 | close motorized isolation valve |

M7 | close manual valves |

M8 | establish safety injection |

M9 | start long-term refrigeration component |

M10 | align suction from containment well |

Mode of Failure | P | F | E | O | D | C | T | W | D |
---|---|---|---|---|---|---|---|---|---|

M1 | 0.09 | 0.08 | 0.08 | 0.20 | 0.07 | 0.06 | 0.05 | 0.09 | 0.09 |

M2 | 0.03 | 0.03 | 0.03 | 0.10 | 0.09 | 0.05 | 0.04 | 0.03 | 0.03 |

M3 | 0.27 | 0.06 | 0.08 | 0.08 | 0.11 | 0.05 | 0.05 | 0.27 | 0.02 |

M4 | 0.06 | 0.03 | 0.03 | 0.13 | 0.09 | 0.05 | 0.04 | 0.06 | 0.03 |

M5 | 0.12 | 0.23 | 0.16 | 0.08 | 0.09 | 0.08 | 0.05 | 0.12 | 0.09 |

M6 | 0.06 | 0.04 | 0.04 | 0.06 | 0.14 | 0.19 | 0.28 | 0.06 | 0.26 |

M7 | 0.12 | 0.08 | 0.08 | 0.08 | 0.11 | 0.25 | 0.23 | 0.12 | 0.21 |

M8 | 0.07 | 0.12 | 0.19 | 0.10 | 0.11 | 0.16 | 0.15 | 0.07 | 0.21 |

M9 | 0.06 | 0.15 | 0.16 | 0.06 | 0.09 | 0.05 | 0.08 | 0.06 | 0.02 |

M10 | 0.12 | 0.19 | 0.16 | 0.08 | 0.11 | 0.06 | 0.06 | 0.12 | 0.04 |

Mode of Failure | P | F | E | O | U | C | T | W | D |
---|---|---|---|---|---|---|---|---|---|

M1 | 0.17 | 0.10 | 0.10 | 0.22 | 0.06 | 0.03 | 0.00 | 0.17 | 0.17 |

M2 | 0.06 | 0.06 | 0.06 | 0.22 | 0.19 | 0.17 | 0.14 | 0.06 | 0.06 |

M3 | 0.21 | 0.08 | 0.13 | 0.13 | 0.17 | 0.04 | 0.04 | 0.21 | 0.00 |

M4 | 0.15 | 0.03 | 0.03 | 0.22 | 0.19 | 0.11 | 0.08 | 0.15 | 0.03 |

M5 | 0.15 | 0.22 | 0.19 | 0.04 | 0.10 | 0.04 | 0.00 | 0.15 | 0.10 |

M6 | 0.08 | 0.01 | 0.01 | 0.08 | 0.14 | 0.17 | 0.22 | 0.08 | 0.19 |

M7 | 0.13 | 0.03 | 0.03 | 0.03 | 0.08 | 0.22 | 0.19 | 0.13 | 0.17 |

M8 | 0.01 | 0.11 | 0.19 | 0.06 | 0.08 | 0.17 | 0.14 | 0.01 | 0.22 |

M9 | 0.08 | 0.19 | 0.22 | 0.08 | 0.17 | 0.03 | 0.14 | 0.08 | 0.00 |

M10 | 0.15 | 0.22 | 0.19 | 0.08 | 0.11 | 0.04 | 0.04 | 0.15 | 0.00 |

Factor | L = 2 | L = 3 | L = 4 | L = 5 | L = 6 | L = 7 | L = 8 | L = 9 | Full |
---|---|---|---|---|---|---|---|---|---|

P | 0.01840 | 0.01736 | 0.01700 | 0.01648 | 0.01582 | 0.01544 | 0.01494 | 0.01422 | 0.01366 |

F | 0.02222 | 0.02095 | 0.01925 | 0.01793 | 0.01702 | 0.01644 | 0.01561 | 0.01465 | 0.01408 |

E | 0.02083 | 0.02037 | 0.02021 | 0.01917 | 0.01828 | 0.01763 | 0.01673 | 0.01570 | 0.01508 |

O | 0.02222 | 0.02222 | 0.02047 | 0.01884 | 0.01785 | 0.01725 | 0.01650 | 0.01564 | 0.01508 |

U | 0.01944 | 0.01852 | 0.01791 | 0.01716 | 0.01652 | 0.01615 | 0.01568 | 0.01518 | 0.01477 |

C | 0.01944 | 0.01852 | 0.01762 | 0.01602 | 0.01490 | 0.01435 | 0.01364 | 0.01291 | 0.01247 |

T | 0.02049 | 0.01829 | 0.01753 | 0.01646 | 0.01524 | 0.01467 | 0.01399 | 0.01305 | 0.01249 |

W | 0.01840 | 0.01736 | 0.01700 | 0.01648 | 0.01582 | 0.01544 | 0.01494 | 0.01422 | 0.01366 |

D | 0.02066 | 0.01933 | 0.01839 | 0.01685 | 0.01545 | 0.01441 | 0.01354 | 0.01262 | 0.01208 |

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Sant’Anna, A.P. Standards for the Weighting of Criteria and the Measurement of Interaction. *Standards* **2021**, *1*, 105-116.
https://doi.org/10.3390/standards1020010

**AMA Style**

Sant’Anna AP. Standards for the Weighting of Criteria and the Measurement of Interaction. *Standards*. 2021; 1(2):105-116.
https://doi.org/10.3390/standards1020010

**Chicago/Turabian Style**

Sant’Anna, Annibal Parracho. 2021. "Standards for the Weighting of Criteria and the Measurement of Interaction" *Standards* 1, no. 2: 105-116.
https://doi.org/10.3390/standards1020010