Ideal solutions of TOPSIS based on vague sets and their application to landmark preference

Ideal solutions are an important part of the technique for order preference by similarity to ideal solution (TOPSIS) based on vague sets. In order to expand and supplement the ideal solutions of TOPSIS based on vague sets, and considering the potential influence on the degree of truth-membership and the degree of false-membership by the degree of unknown in vague sets, three potential ideal solutions are proposed by looking for new ideal solution forms between the maximal ideal solutions and the actual ideal solutions. Proof in theoretical shows that all the proposed potential ideal solutions are vague sets, which means that they all can apply to the calculation of TOPSIS based on vague sets. The proposed potential ideal solutions are three types of supplementary forms for the ideal solutions of TOPSIS based on vague sets. Several properties of the proposed potential ideal solutions are discussed, and it shows that the proposed potential ideal solutions can be converted to the maximal ideal solutions and the actual ideal solutions under certain conditions. The proposed potential ideal solutions are applied to landmark preference based on TOPSIS, which effectiveness and feasibility are illustrated, and a new way for landmark preference is provided in the meanwhile.