Incomplete preference matrix with elements from an Alo-group and its application to ranking of alternatives

A preference matrix is the result of pairwise comparison and is a powerful method in multi-criteria optimization. When comparing two elements, the decision maker assigns the value from a given scale which is a linearly ordered Abelian group (Alogroup) to any pair of alternatives representing the element of the preference matrix (P-matrix). The well known multiplicative, additive and fuzzy preference matrices are generalized. Some situations where elements of the P-matrix are missing are focused on and a general method for completing a P-matrix with missing elements called the extension of the P-matrix is proposed. Two important particular cases of fuzzy P-matrix with missing elements are discussed. Some illustrative numerical examples are given.