The shape of $7 imes7$ pin-fins heat sink is optimized numerically to obtain the minimum pressure drop and thermal resistance. In this study, the fin height (h), fin width (w), and fan-to-heat sink distance (c) are chosen as the design variables and the pressure drop $({Delta}P)$ and thermal resistance $( heta_j)$ are adopted as the objective functions. To obtain the optimum design values, we used the finite volume method for calculating the objective functions, the BFGS method for solving the unconstrained non-linear optimization problem, and the weighting method for predicting the multi-objective problem. The results show that the optimum design variables for the weighting coefficient of 0.5 are as follows: W=4.653 mm, h=59.215mm, and c=2.667mm. The objective functions corresponding to the optimal design are calculated as ${Delta}P=6.82$ Pa and $( heta_j)=0.56K/W$. The Pareto solutions are also presented for various weighting coefficients and they will offer very useful data to design the pin-fins heat sink.