The increase of heat generated in integrated circuit because of the miniaturization of electronic components requires more aggressive cooling solutions in order to minimize this high heat flux and address the temperature non-uniformity. In this paper, a manifold microchannel heat sinks has been investigated. In order to enhance the heat transfer performance of the microchannel, an improved version of the augmented epsilon constraint method is adopted for the optimization of the device. Four non-dimensional design variables have been used to describe the geometry of the manifold microchannel heat sinks. The thermal performance and the pumping power have been incorporated in the mathematical programming formulation as indicators of the thermal performance. A surrogate-based approximation based on the Response Surface Approximation has been utilized to evaluate these two objectives. This new mathematical approach has been implemented in the General Algebraic Modelling Systems (GAMS). Details about single and multi-objective optimization formulation of the problem will be disclosed. Optimal solutions describing the best geometrical configuration of the device will be computed. The implications of the geometrical configuration on the performance the manifold microchannel heat sinks will form part of the main contribution of this study.