Many industrial systems involve multiple criteria and objectives, and they are very complex problems in computational science, such as task scheduling. We propose bi-criteria and bi-objective scheduling problems, which are solved by two nature-inspired evolutionary algorithms, such as Simulated Annealing (SA) and Bee Algorithm (BA). This problem is characterized by scheduling a batch of tasks on multiple machines, and it is fundamental because the solution should focus on the simultaneous optimization of two conflicting objectives: the makespan minimization and the total tardiness minimization. This problem is NP-Hard, and therefore, two evolutionary methods were used to search for solutions intelligently in this huge, very complex space. In this research, A mathematical model of the scheduling problem was developed based on the above objectives. Here, we proposed a tailored tune-up of SA and BA, both of which have been specifically developed and implemented to solve the proposed model for integrated scheduling and delivery, geared for the bifunctional nature of the problem. Quantitative results indicate that the Bee Algorithm (BA) achieves a more diverse Pareto front, with an average improvement of approximately 12–18 % in solution diversity compared to Simulated Annealing (SA). In contrast, SA converges faster, reducing computational time by about 30–40 % for large problem instances (n ≥ 80). Overall, BA provides better trade-offs between objectives, while SA offers superior computational efficiency. The results showed that both algorithms can generate solutions that are balanced and time-efficient.
Mohammed et al. (2026) studied this question.