An improved whale optimization algorithm for flexible job shop scheduling problems with machine deterioration effects

Why factory timing problems matter

Behind everyday products like phones, cars, and packaged food lies a hidden puzzle: deciding which machine handles which task, and in what order. This planning challenge, known as scheduling, determines whether a factory delivers on time or keeps customers waiting. The paper explores how to plan factory work when machines slowly wear out, and introduces a whale-inspired computer algorithm that can find quicker, more realistic production plans.

How real machines age on the shop floor

In textbook scheduling, machines are treated as perfect: their processing speed never changes. Real workshops are different. As machines run for long hours, parts wear, heat builds up, and performance drops. Jobs that start later on the same machine can take longer than identical jobs processed earlier. This effect, called deterioration, is common in industries such as steel, plastics, machinery, and defense. Ignoring it makes schedules look good on paper but disappointing in practice, because actual completion times end up far longer than predicted.

Turning wear and tear into a simple rule

The authors study a flexible job shop, where each step of each job can be done on more than one machine, and they add a realistic rule for machine ageing. Instead of assuming processing time grows forever, they adopt a stepped pattern: for early use, the machine runs at basic speed; after a certain workload, extra time is added to each job as wear grows; past an upper limit, the slowdown stops increasing and stays at a fixed extra delay. Using this rule, they build a mathematical model whose goal is to minimize the total time until all jobs are finished, while respecting job order, machine choices, and the rising delays caused by deterioration.

Borrowing ideas from hunting whales

To tackle this complex planning problem, the paper improves on the Whale Optimization Algorithm, a search method inspired by how humpback whales circle and trap prey. Each “whale” in the algorithm represents one possible schedule. By repeatedly updating these schedules, the group searches for better plans. The authors redesign several parts of this process. They create a smarter way to generate the first batch of schedules, mixing global reasoning, local refinement, and randomness. They also adjust how the virtual whales shift from broad exploration early on to fine-tuning later, using a curved convergence rule and a changing “inertia” that controls step size.

Keeping the search diverse and avoiding dead ends

Standard versions of the whale method can get stuck, circling around a merely decent plan instead of finding a better one. To prevent this, the authors borrow a mutation trick from another family of algorithms, in which new candidate schedules are created by combining differences between existing ones. This random differential step injects variety back into the group. They also introduce a golden sine strategy, using smooth wave-like moves and the golden ratio to nudge the search through a wider region of the solution space while still gravitating toward promising areas. Together, these changes balance bold exploration with careful polishing.

What the tests show in practice

The team tests their improved algorithm on standard benchmark scheduling problems that have been adapted to include machine wear. They compare results against the original whale method, a classic genetic algorithm, and a grey wolf optimizer. Across most test sets, the new method finds schedules that finish earlier and does so more consistently from run to run. In a detailed example, simply adding machine deterioration nearly doubles the predicted completion time if no rescheduling is done. When the improved whale method is applied, the total time with wear included drops by about one third compared with this naive plan, showing clear gains in efficiency.

What this means for real factories

In plain terms, the study shows that planning factory work while pretending machines never slow down can seriously mislead managers. By building machine ageing into the schedule and using a refined whale-inspired search, factories can arrange tasks and machine choices in ways that better match reality, cutting waiting times and improving throughput. While the approach still assumes stable conditions and tuned settings, it points toward smarter digital tools that help workshops cope with both flexibility and wear, bringing computer-made plans closer to what actually happens on the shop floor.

Citation: Li, K., Tian, S. An improved whale optimization algorithm for flexible job shop scheduling problems with machine deterioration effects. Sci Rep 16, 14604 (2026). https://doi.org/10.1038/s41598-026-44409-4

Keywords: scheduling, optimization, manufacturing, machine wear, metaheuristics