What are the most important properties of pairwise comparison matrices (pc matrices)?
Also if you can provide related applications.
Mirzakhmet Syzdykov
Explain please "They are invariant to modulus operator"? Also it will be good to provide an example for that.
Pairwise comparison matrices (PC matrices) are used in decision-making and are a way to represent and analyze the preferences of decision-makers. Here are some of the most important properties of PC matrices:
Reciprocity: In a pairwise comparison matrix, the ratio of the importance of two alternatives A and B should be the inverse of the ratio of the importance of B and A. In other words, if decision-maker X prefers alternative A to B, then decision-maker Y should prefer B to A with the same ratio.
Consistency: The consistency of a pairwise comparison matrix refers to the extent to which the judgments are internally coherent. To determine consistency, one can compute the consistency index (CI) and the consistency ratio (CR). If CR is less than 0.1, the matrix is considered to be consistent.
Normalization: In order to compare the relative importance of alternatives, the pairwise comparison matrix should be normalized so that the sum of the weights in each column is equal to 1.
Transitivity: The pairwise comparison matrix should exhibit transitivity, meaning that if alternative A is preferred over B, and B is preferred over C, then A should be preferred over C.
Non-negativity: The weights in a pairwise comparison matrix should be non-negative.
These properties are essential to ensure that the pairwise comparison matrix accurately reflects the decision-makers' preferences and can be used to make reliable decisions.
Let A = (𝑎𝑖𝑗) be n × n square matrix such that 𝑎𝑖𝑗 > 0 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑖, 𝑗 ∈ {1, 2, … , 𝑛}. Then A is said to be a pairwise comparison matrix (PC matrix) if it is expressed in the following form, A =[1 𝑎12 … 𝑎1n; (1 /𝑎12) 1 … 𝑎2𝑛;....; (1/𝑎1𝑛) (1/𝑎2𝑛) ... 1].
That is, A is a diagonal matrix with each entry of the first row is the reciprocal of each entry of the first column, and continue like this for the rest. Or
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