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Computer simulations can be intensive as is the case in Computational Fluid Dynamics (CFD) and Finite Element Analysis (FEA). The computational cost can become prohibitive when using these simulations with multiobjective design optimization. One way to address this issue is to replace a computationally intensive simulation by an approximation which allows for a quick evaluation of a large number of design alternatives as needed by an optimizer. This dissertation proposes an approach for multiobjective design optimization when combined with computationally expensive simulations for heat exchanger design problems. The research is performed along four research directions. These are: (1) a new Online Approximation Assisted Multiobjective Optimization (OAAMO) approach with a focus on the expected optimum region, (2) a new approximation assisted multiobjective optimization with global and local metamodeling that always produces feasible solutions, (3) a framework that integrates OAAMO with multiscale simulations (OAAMOMS) for design of heat exchangers at the segment and heat exchanger levels, and (4) applications of OAAMO combined with CFD for shape design of a header for a new generation of heat exchangers using Non-Uniform Rational B-Splines (NURBS). The approaches developed in this thesis are also applied to optimize a coldplate used in electronic cooling devices and different types of plate heat exchangers. In addition many numerical test problems are solved by the proposed methods. The results of these studies show that the proposed online approximation assisted multiobjective optimization is an efficient approach that can be used to predict optimum solutions for a wide class of problems including heat exchanger design problems while reducing significantly the computational cost when compared with existing methods.