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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

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 1 The fuzzy set and the membership function
The characteristic function μ_A of a crisp set A ⊆ X assigns a value of either 0 or 1 to each member of X, as well as the crisp sets only allow a full membership μ_A(x)=1 or no membership at all μ_A(x)=0. This function can be generalized to a function μ_à so that the value assigned to the element of the universal set X falls within a specified range, i.e., μ_à : X → [0, 1]. The assigned value indicates the degree of membership of the element in the set à . The function μ_à is called a membership function and the set à ={(x,μ_à (x))}, where x ∈ X, defined by μ_à (x) for each x ∈ X, is called a fuzzy set.

Definition 2 The triangular fuzzy number (TFN)
A fuzzy set Ã, defined on the universal set of real numbers R, is told to be a triangular fuzzy number Ã(a,m,b) if its membership function has the following form:
and the following characteristics:
x₁, x₂ ∈ [a, b] ∧ x₂ > x₁ ⇒ μ_Ã(x₂) > μ_Ã(x₁)
x₁, x₂ ∈ [b, c] ∧ x₂ > x₁ ⇒ μ_Ã(x₂) < μ_Ã(x₁)

An example of triangular fuzzy number Ã(a,m,b) is presented:
Definition 3 The support of a TFN Ã
The support of a TFN à is defined as a crisp subset of the à set in which all elements have a non-zero membership value in the à set:
S(Ã) = {x: μ_{Ã}(x) > 0} = [a, b] Definition 4 The core of a TFN Ã
The core of a TFN Ã is a singleton (one-element fuzzy set) with the membership value equal to 1:
C(Ã) = {x: μ_Ã(x) = 1} = m Definition 5 The fuzzy rule
The single fuzzy rule can be based on the Modus Ponens tautology. The reasoning process uses the IF-THEN, OR and AND logical connectives.

Definition 6 The 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 7 The T-norm operator: product
The T-norm operator is a T function modeling the AND intersection 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 selecting r criteria, C₁, C₂, ..., C_r. Then, a set of fuzzy numbers is selected for each criterion C_i, e.g. {C_i1, C_i2, ..., C_ici}: where c₁,c₂, ...,c_r are the ordinals of the fuzzy numbers for all criteria.

Step 2. Generation of the characteristic objects
The characteristic objects CO are 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 t is the count of COs and is equal to:

Step 3. Evaluation of the characteristic objects
The expert determines the Matrix of Expert Judgment )MEJ) by comparing the COs pairwise. The matrix is presented below: where α_ij is the result of comparing CO_i and CO_j by the expert. The function f_exp denotes the mental judgment function of the expert. It depends solely on the knowledge of the expert. The expert's preferences can be presented as: After the MEJ matrix is prepared, a vertical vector of the Summed Judgments SJ is obtained as follows: Eventually, the values of preference are approximated for each characteristic object. As a result, a vertical vector P is obtained, where the i-th row contains the approximate value of preference for CO_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.: A_i={a_1i, a_2i, ..., a_ri} This set corresponds to the criteria C₁, C₂, ..., C_r. Mamdani's fuzzy inference method is used to compute the preference of the i-th alternative. The rule base guarantees that the obtained results are unequivocal. The COMET is completely free of rank reversal.

• DSS COMET software PL
• DSS COMET software ENG
• Manual

Projects

NCN Preludium

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

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.

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