In this paper, we first propose a multi-objective task-scheduling optimization problem that minimizes both the makespans and total costs in a fog-cloud environment. Then, we suggest an optimization model based on a Discrete Non-dominated Sorting Genetic Algorithm II (DNSGA-II) to deal with the discrete multi-objective task-scheduling problem and to automatically allocate tasks that should be executed either on fog or cloud nodes. The NSGA-II algorithm is adapted to discretize crossover and mutation evolutionary operators, rather than using continuous operators that require high computational resources and not able to allocate proper computing nodes. In our model, the communications between the fog and cloud tiers are formulated as a multi-objective function to optimize the execution of tasks. The proposed model allocates computing resources that would effectively run on either the fog or cloud nodes. Moreover, it efficiently organizes the distribution of workloads through various computing resources at the fog. Several experiments are conducted to determine the performance of the proposed model compared with a continuous NSGA-II (CNSGA-II) algorithm and four peer mechanisms. The outcomes demonstrate that the model is capable of achieving dynamic task scheduling with minimizing the total execution times (i.e. makespans) and costs in fog-cloud environments