Cloud computing is widely used to collect data from various devices, which must be processed quickly. To manage this growing data, fog computing helps reduce delay and processing costs by assigning tasks to suitable devices. This article presents an adapted binary monarch butterfly algorithm for task offloading in a fog-cloud environment. This metaheuristic directly constructs a Pareto front, offering a solution space representation. Two versions are examined: one using random search and the other a deterministic search with crowding distance. Simulations on tasks from 40 to 500 show that the binary Monarch Butterfly algorithm can outperform state-of-the-art algorithms for cost optimization while balancing delay.