Heat sink is part of a thermal convertor that cools the convertor by dissipation of heat into the surrounding air. In order to obtain the optimal design of any heat sink, the rate of entropy generation must be minimized. However, reduced generation of entropy is accompanied by an increase in the cost, and hence these two contradictory objectives must be simultaneously optimized. In the present study, Multi-objective Optimization Evolutionary Algorithm Based On Decomposition (MOEA/D) has been used for optimization. The complete mathematical model governing the heat sink has been investigated and its basic experimental and computational data have been derived. According to the results, all the four considered parameters in the optimization had a relatively linear relation to the cost, but the number of fins and the air flow speed had a more significant effect on the entropy generation rate. In sum, when the cost is considered, a better evaluation of the results is obtained. With regard to the optimization process in this work, the objective functions of price and generation of entropy have linearly entered the numerical equations. The values for the distance and the length of the fins have been obtained using RSM and decision-maker methods. Besides, the optimized values of the objective functions have been expressed for each type of the flows related to the heat sink.