Multi-attribute decision-making approach and theoretic proof of its solution based on two-phase optimization

Multi-attribute decision-making problems, in which the attribute weights are partly known (interval numbers) and the attribute values are numeric are investigated. Firstly, a linear objective programming model is established and the ideal attribute weight of each alternative is obtained by solving the model. Secondly, a quadratic programming model with the constraints is established by considering the system in toto and the existence of an optimum solution to the corresponding quadratic programming model is provided. Further, the solution of the complex attribute weights is given, and thus the complex attribute value of each alternative is obtained. Based on these values, prioritizing or optimization for alternatives is made. Finally, a practical example is given to show the feasibility and availability of the developed model and method.