## COMET method - *a new quality in decision making*

Currently existing multi-criteria decision-making (MCDM) methods yield results that may be questionable and unreliable. These methods very often ignore the issue of rank reversal paradox, which is a fundamental and essential challenge of MCDM methods. In response to this challenge, the Characteristic Objects Method (COMET) was developed. The classical COMET is entirely free of the rank reversal paradox.

**COMET section**- fundamental information about COMET algorithm

**COMET software**- COMET software with manual

**References section**- the most important papers about COMET

**Contact section**- contact to the author

## COMET

This webpage presents preliminaries of the fuzzy sets theory and computational algorithm of the basic COMET method. In the Software section, COMET software and manual can be founded.

## Fuzzy sets theory: preliminaries

Definition 1The fuzzy set and the membership function

The characteristic functionμof a crisp set_{A}A ⊆ Xassigns a value of either 0 or 1 to each member ofX, as well as the crisp sets only allow a full membershipμor no membership at all_{A}(x)=1μ. This function can be generalized to a function_{A}(x)=0μso that the value assigned to the element of the universal set_{Ã }Xfalls within a specified range, i.e.,μ. The assigned value indicates the degree of membership of the element in the set_{Ã }: X → [0, 1]Ã. The function μ_{Ã }is called a membership function and the setÃ ={(x,μ, where_{Ã }(x))}x ∈ X, defined byμfor each_{Ã }(x)x ∈ X, is called a fuzzy set.

Definition 2The triangular fuzzy number (TFN)

A fuzzy set Ã, defined on the universal set of real numbersR, is told to be a triangular fuzzy number Ã(a,m,b) if its membership function has the following form:

and the following characteristics:

x_{1}, x_{2}∈ [a, b] ∧ x_{2}> x_{1}⇒ μ_{Ã}(x_{2}) > μ_{Ã}(x_{1})

x_{1}, x_{2}∈ [b, c] ∧ x_{2}> x_{1}⇒ μ_{Ã}(x_{2}) < μ_{Ã}(x_{1})

An example of triangular fuzzy number Ã(a,m,b) is presented:

Definition 3The support of a TFNÃ

The support of a TFNÃis defined as a crisp subset of theÃset in which all elements have a non-zero membership value in theÃset:

S(Ã) = {x: μ_{{Ã}}(x) > 0} = [a, b]Definition 4The core of a TFNÃ

The core of a TFNÃis a singleton (one-element fuzzy set) with the membership value equal to 1:

C(Ã) = {x: μ_{Ã}(x) = 1} = mDefinition 5The fuzzy rule

The single fuzzy rule can be based on the Modus Ponens tautology. The reasoning process uses theIF-THEN,ORandANDlogical connectives.

Definition 6The rule base

The rule base consists of logical rules determining the causal relationships existing in the system between the input and output fuzzy sets.

Definition 7The T-norm operator: product

The T-norm operator is aTfunction modeling theANDintersection operation of two or more fuzzy numbers, e.g.Ã and &Btilde;$. In the basic approach, only the ordinary product of real numbers is used as the T-norm operator:

μ_{A}(x) AND μ_{B}(y) = μ_{A}(x) · μ_{B}(y)

## The Characteristic Objects Method

Step 1.Definition of the space of the problem

The expert determines the dimensionality of the problem by selectingrcriteria,C. Then, a set of fuzzy numbers is selected for each criterion_{1}, C_{2}, ..., C_{r}C, e.g._{i}{C:_{i1}, C_{i2}, ..., C_{ici}}where care the ordinals of the fuzzy numbers for all criteria._{1},c_{2}, ...,c_{r}

Step 2.Generation of the characteristic objects

The characteristic objectsCOare obtained with the usage of the Cartesian product of the fuzzy numbers' cores of all the criteria:As a result, an ordered set of all $CO$ is obtained: where tis the count ofCOsand is equal to:

Step 3.Evaluation of the characteristic objects

The expert determines the Matrix of Expert Judgment)MEJ)by comparing theCOspairwise. The matrix is presented below:where αis the result of comparing_{ij}COand_{i}COby the expert. The function_{j}fdenotes the mental judgment function of the expert. It depends solely on the knowledge of the expert. The expert's preferences can be presented as:_{exp}After the MEJmatrix is prepared, a vertical vector of the Summed JudgmentsSJis obtained as follows:Eventually, the values of preference are approximated for each characteristic object. As a result, a vertical vector Pis obtained, where thei-throw contains the approximate value of preference forCO._{i}

Step 4.The rule base

Each characteristic object and its value of preference is converted to a fuzzy rule as follows:In this way, a complete fuzzy rule base is obtained.

Step 5.Inference and the final ranking

Each alternative is presented as a set of crisp numbers, e.g.:This set corresponds to the criteria A_{i}={a_{1i}, a_{2i}, ..., a_{ri}}C. Mamdani's fuzzy inference method is used to compute the preference of the_{1}, C_{2}, ..., C_{r}i-thalternative. The rule base guarantees that the obtained results are unequivocal. The COMET is completely free of rank reversal.

## Software

## Projects

### NCN Preludium

*A new method using reference objects to support decision-making process in multi-criteria problems under uncertainty*

The objective of the proposed research is to develop a new method using reference objects to support decision-making in multi-criteria problems under uncertainty.
The motivation for the proposed research is the fact that in many areas of science, including, behavioral economics, sustainable development or the management, we are dealing more and more with multi-criteria problems whose solution is sought in the conditions of uncertainty. It means that important decision problems involving mostly a lot of contradictory criteria are also considered using imprecise or uncertain data and information.The project is supported by the National Science Centre, the agreement no.

**UMO-2016/23/N/HS4/0193**

#### List of publications:

- Multicriteria Selection of Online Advertising Content for the Habituation Effect Reduction (in press)
- Sałabun, W., Karczmarczyk, A., Wątróbski, J., & Jankowski, J. (2018, November). Handling Data Uncertainty in Decision Making with COMET. In 2018 IEEE Symposium Series on Computational Intelligence (SSCI) (pp. 1478-1484). IEEE.
- Sałabun, W., Karczmarczyk, A., & Wątróbski, J. (2018, November). Decision-Making using the Hesitant Fuzzy Sets COMET Method: An Empirical Study of the Electric City Buses Selection. In 2018 IEEE Symposium Series on Computational Intelligence (SSCI) (pp. 1485-1492). IEEE.
- Sałabun, W., & Karczmarczyk, A. (2018). Using the comet method in the sustainable city transport problem: an empirical study of the electric powered cars. Procedia computer science, 126, 2248-2260.
- Faizi, S., Sałabun, W., Rashid, T., Wątróbski, J., & Zafar, S. (2017). Group decision-making for hesitant fuzzy sets based on characteristic objects method. Symmetry, 9(8), 136.
- Wątróbski, J., Sałabun, W., Karczmarczyk, A., & Wolski, W. (2017, September). Sustainable decision-making using the COMET method: An empirical study of the ammonium nitrate transport management. In 2017 Federated Conference on Computer Science and Information Systems (FedCSIS) (pp. 949-958). IEEE.
- Bashir, Z., Wątróbski, J., Rashid, T., Sałabun, W., & Ali, J. (2017). Intuitionistic-fuzzy goals in zero-sum multi criteria matrix games. Symmetry, 9(8), 158.

## References

#### Journals

- Sałabun, W., & Karczmarczyk, A. (2018). Using the comet method in the sustainable city transport problem: an empirical study of the electric powered cars. Procedia computer science, 126, 2248-2260.
- Faizi, S.; Sałabun, W.; Rashid, T.; Wątróbski, J.; Zafar, S. Group Decision-Making for Hesitant Fuzzy Sets Based on Characteristic Objects Method. Symmetry 2017, 9, 136.
- Faizi, S., Rashid, T., Sałabun, W., Zafar, S., Wątróbski, J. (2017).
*Decision Making with Uncertainty Using Hesitant Fuzzy Sets*. International Journal of Fuzzy Systems, 1-11. - Sałabun, W., Piegat, A. (2016).
*Comparative analysis of MCDM methods for the assessment of mortality in patients with acute coronary syndrome*. Artificial Intelligence Review, 1-15. - Sałabun, W., Napierała, M., Bykowski, J. (2015).
*The Identification of Multi-Criteria Model of the Signicficance of Drainage Pumping Stations in Poland*. Acta Scientiarum Polonorum. Formatio Circumiectus, 14(3), 147-163. - Sałabun, W. (2015).
*Assessing the 10-year risk of hard arteriosclerotic cardiovascular disease events using the characteristic objects method.*Studies & Proceedings Polish Association for Knowledge Management, 77, 65-76. - Sałabun, W. (2015).
*Fuzzy Multi-Criteria Decision-Making Method: the Modular Approach in the Characteristic Objects Method*. Studies & Proceedings of Polish Association for Knowledge Management, 77, 54-64. - Sałabun, W. (2015).
*The Characteristic Objects Method: A New Distance‐based Approach to Multicriteria Decision‐making Problems*. Journal of Multi-Criteria Decision Analysis, 22(1-2), 37-50. - Sałabun, W. (2014).
*Reduction in the number of comparisons required to create matrix of expert judgment in the comet method*. Management and Production Engineering Review, 5(3), 62-69. - Sałabun, W. (2014).
*Application of the fuzzy multi-criteria decision-making method to identify nonlinear decision models*. Interantional Journal of Computer Applications, 89(15), 1-6. - Piegat, A., Sałabun, W. (2014).
*Identification of a multicriteria decision-making model using the characteristic objects method*. Applied Computational Intelligence and Soft Computing, 2014, 14. - Piegat, A., Sałabun, W. (2012).
*Nonlinearity of human multi-criteria in decision-making*. Journal of Theoretical and Applied Computer Science, 6(3), 36-49. - Sałabun, W. (2012).
*The use of fuzzy logic to evaluate the nonlinearity of human multi-criteria used in decision making*. Przegląd Elektrotechniczny, 88(10b), 235-238.

#### Chapters

- Sałabun, W., Karczmarczyk, A., Wątróbski, J., & Jankowski, J. (2018, November). Handling Data Uncertainty in Decision Making with COMET. In 2018 IEEE Symposium Series on Computational Intelligence (SSCI) (pp. 1478-1484). IEEE.
- Sałabun, W., Karczmarczyk, A., & Wątróbski, J. (2018, November). Decision-Making using the Hesitant Fuzzy Sets COMET Method: An Empirical Study of the Electric City Buses Selection. In 2018 IEEE Symposium Series on Computational Intelligence (SSCI) (pp. 1485-1492). IEEE.
- Wątróbski, J., Sałabun, W., Karczmarczyk, A., Wolski, W. (2017).
*Sustainable Decision-Making using the COMET Method: An Empirical Study of the Ammonium Nitrate Transport Management*. In press. - Jankowski, J., Sałabun, W., Wątróbski, J. (2017).
*Identification of a multi-criteria assessment model of relation between editorial and commercial content in web systems*. In Multimedia and Network Information Systems (pp. 295-305). Springer International Publishing. - Sałabun, W., Wątróbski, J., & Piegat, A. (2016, June).
*Identification of a Multi-criteria Model of Location Assessment for Renewable Energy Sources*. In International Conference on Artificial Intelligence and Soft Computing (pp. 321-332). Springer, Cham. - Sałabun, W., Ziemba, P., Wątróbski, J. (2016).
*The Rank Reversals Paradox in Management Decisions: The Comparison of the AHP and COMET Methods*. In Intelligent Decision Technologies 2016 (pp. 181-191). Springer International Publishing. - Watróbski, J., Sałabun, W. (2016).
*The characteristic objects method: a new intelligent decision support tool for sustainable manufacturing*. In Sustainable Design and Manufacturing 2016 (pp. 349-359). Springer International Publishing. - Sałabun, W., Ziemba, P. (2016).
*Application of the Characteristic Objects Method in Supply Chain Management and Logistics*. In Recent Developments in Intelligent Information and Database Systems (pp. 445-453). Springer International Publishing. - Piegat, A., Sałabun, W. (2015, June).
*Comparative analysis of MCDM methods for assessing the severity of chronic liver disease*. In International Conference on Artificial Intelligence and Soft Computing (pp. 228-238). Springer, Cham.