Existence and simulation of multiple solutions to an optimization model for completing incomplete fuzzy preference relations

When addressing a decision making problem with incomplete preference relations, missing information could be estimated by proposing an optimization model. The previous studies always focus on a unique solution to optimization models through a software program. But theoretical and simulation investigations of multiple solutions are seldom considered. This paper investigates the existence of multiple solutions to an optimization model and proposes a simulation procedure. First, an optimization model is recalled for completing incomplete fuzzy preference relations. It is the first time to theoretically prove the existence of multiple solutions to the optimization model under various incomplete entries. The obtained results reveal that the optimal solution to an optimization model may be an interval value, which is dependent on the number and position of missing entries in an incomplete fuzzy preference relation. The idea of multiple solutions is further extended to the situation where an intransitive fuzzy preference relation should be adjusted to a weakly transitive matrix. Second, multiple solutions to the optimization model are simulated using the quantum-behaved particle swarm optimization algorithm. It is found that multiple solutions do appear under some conditions by only running the algorithm for multiple times. Finally, the impact of multiple solutions on the optimal alternative(s) of a decision making problem is analyzed by comparing with the existing studies. The result shows that the existence of multiple solutions reflects the uncertainty of optimization models, which is of concerns to propose an effective decision-making model.