The COMET Method
Characteristic Objects Method - computational algorithm and fuzzy sets theory
Fuzzy Sets Theory: Preliminaries
Definition 1: The fuzzy set and the membership function
The characteristic function μA 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 μA(x)=1 or no membership at all μA(x)=0. This function can be generalized to a function μà so that the value assigned to the element of the universal set X falls within a specified range, i.e., μÃ: X → [0, 1]. The assigned value indicates the degree of membership of the element in the set Ã.
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:
where a and b define the boundaries of the support (the range where membership is non-zero), and m represents the core (the point where membership equals 1). The triangular shape is created by linear interpolation between these three points.
An example of triangular fuzzy number Ã(a,m,b) is presented:
Why TFNs are important in COMET: Triangular fuzzy numbers allow the method to handle uncertainty and vagueness in decision-making. They enable smooth transitions between different preference levels and make the evaluation process more flexible than using crisp values alone.
An example of triangular fuzzy number Ã(a,m,b) is presented:
Definition 3: The support of a TFN
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:
S(Ã) = {x: μÃ(x) > 0} = [a, b]
Definition 4: The core of a TFN
The core of a TFN Ã is a singleton (one-element fuzzy set) with the membership value equal to 1:
C(Ã) = {x: μÃ(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. In the basic approach, only the ordinary product of real numbers is used as the T-norm operator:
μA(x) AND μB(y) = μA(x) · μB(y)
Application in COMET: The T-norm operator is essential for combining multiple fuzzy criteria during the inference process. When evaluating an alternative against the fuzzy rule base, the T-norm determines how well the alternative matches each rule's conditions, enabling the method to produce meaningful preference values through Mamdani's fuzzy inference mechanism.
The Characteristic Objects Method Algorithm
1
Definition of the space of the problem
The expert determines the dimensionality of the problem by selecting r criteria, C1, C2, ..., Cr. Then, a set of fuzzy numbers is selected for each criterion Ci:
where c1, c2, ..., cr are the ordinals of the fuzzy numbers for all criteria.
Important considerations: The characteristic values should define the minimum and maximum bounds of the decision problem domain. Typically, three characteristic values (minimum, maximum, and a point between them) are sufficient for most problems. These values can be uniformly distributed, selected based on statistics, or based on expert knowledge and Expected Solution Point.
Important considerations: The characteristic values should define the minimum and maximum bounds of the decision problem domain. Typically, three characteristic values (minimum, maximum, and a point between them) are sufficient for most problems. These values can be uniformly distributed, selected based on statistics, or based on expert knowledge and Expected Solution Point.
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:
3
Evaluation of the characteristic objects
The expert determines the Matrix of Expert Judgment (MEJ) by comparing the COs pairwise. The MEJ matrix contains values of 0, 0.5, or 1, where:
Efficiency note: Because the matrix is symmetric and diagonal elements equal 0.5, only the upper triangle needs to be identified, resulting in t(t-1)/2 pairwise comparisons.
After the MEJ matrix is prepared, a vertical vector of the Summed Judgments SJ is obtained:
Finally, preference values Pi are calculated by assigning uniformly distributed values in the range [0, 1] to each unique value of SJ.
- 1.0 - if the first CO is better than the second
- 0.5 - if both COs are equally preferred
- 0.0 - if the first CO is worse than the second
After the MEJ matrix is prepared, a vertical vector of the Summed Judgments SJ is obtained:
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.
5
Inference and the final ranking
Each alternative is presented as a set of crisp numbers:
Why this matters: The fuzzy rule base enables evaluation of any alternative within the defined decision problem domain. Each alternative is evaluated independently based on the complete decision model, not in comparison to other alternatives. This independence guarantees that the obtained results are unequivocal and makes the COMET method completely free of rank reversal—adding or removing alternatives never affects the ranking of other alternatives.
The final preference values lie in the range [0, 1], where 1 indicates the best possible alternative and 0 the worst, enabling clear and interpretable decision-making.
Ai = {a1i, a2i, ..., ari}
This set corresponds to the criteria C1, C2, ..., Cr. Mamdani's fuzzy inference method is used to compute the preference of the i-th alternative.
Why this matters: The fuzzy rule base enables evaluation of any alternative within the defined decision problem domain. Each alternative is evaluated independently based on the complete decision model, not in comparison to other alternatives. This independence guarantees that the obtained results are unequivocal and makes the COMET method completely free of rank reversal—adding or removing alternatives never affects the ranking of other alternatives.
The final preference values lie in the range [0, 1], where 1 indicates the best possible alternative and 0 the worst, enabling clear and interpretable decision-making.
Simple Numerical Example
Problem Statement
Consider a simple two-criterion decision problem where we want to rank alternatives based on:
- C1: Profit (scale 0-10, maximize)
- C2: Cost (in USD, 0-100, minimize)
Step 1: Characteristic Values
We select three characteristic values for each criterion:
- Profit: [0, 8, 10]
- Cost: [0, 50, 100]
Step 2: Characteristic Objects
Cartesian product generates 9 COs:
CO1 = [0, 0]
CO2 = [0, 50]
CO3 = [0, 100]
CO4 = [8, 0]
CO5 = [8, 50]
CO6 = [8, 100]
CO7 = [10, 0] ← Best
CO8 = [10, 50]
CO9 = [10, 100]
CO2 = [0, 50]
CO3 = [0, 100]
CO4 = [8, 0]
CO5 = [8, 50]
CO6 = [8, 100]
CO7 = [10, 0] ← Best
CO8 = [10, 50]
CO9 = [10, 100]
Step 3: Expert Evaluation
The expert compares COs pairwise in the MEJ matrix. After summing judgments and converting to preferences, we get:
Preference Values:
P1 = 0.25, P2 = 0.125, P3 = 0.0
P4 = 0.75, P5 = 0.625, P6 = 0.375
P7 = 1.0, P8 = 0.875, P9 = 0.5
P1 = 0.25, P2 = 0.125, P3 = 0.0
P4 = 0.75, P5 = 0.625, P6 = 0.375
P7 = 1.0, P8 = 0.875, P9 = 0.5
Step 4-5: Evaluation Results
Using Mamdani inference on sample alternatives:
| Ai | C1 | C2 | Pi | Ri |
|---|---|---|---|---|
| A1 | 3 | 20 | 0.388 | 4 |
| A2 | 9 | 60 | 0.688 | 1 |
| A3 | 5 | 40 | 0.463 | 3 |
| A4 | 7 | 50 | 0.563 | 2 |
Key Insight
Alternative A2 ranks first with the highest profit (9), even though it has moderate cost. The COMET method evaluated it independently using the fuzzy rule base—if we add or remove other alternatives, A2's preference value and relative ranking remain unchanged. This demonstrates complete resistance to rank reversal, a critical advantage over traditional MCDA methods.
