The aim of this paper is to propose a satisfactory degree by using a nonlinear programming method for solving multi-attribute decision making (MADM) problems, in which rating of alternatives on attributes is expressed via interval-valued intuitionistic fuzzy (IVIF) sets and preference information on attributes is complete. Concretely, a nonlinear programming model is firstly explored to determine the satisfactory degree which is the ratio of the square of the weight Eucildean distance between an alternative and the IVIF negative ideal solution (IVIFNIS) to the sum of the square of the weight Eucildean distance between the IVIF negative ideal solution and the IVIF positive ideal solution (IVIFPIS). Another nonlinear programming model is also developed to obtain satisfactory intuitionistic fuzzy sets, and then the general satisfactory degrees of the satisfactory intuitionistic fuzzy sets are used to generate the ranking order of the alternatives. Finally, a real example of the investment company selection problem is employed to verify the applicability of the proposed approach and to illustrate its practicality and effectiveness.

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