Use LLMs to help define business problems, write code, and identify potential constraints. Then, hand off to a mathematical solver like Gurobi, which provides a mathematically guaranteed optimal solution that an LLM cannot, as it will never violate a hard constraint.
To determine if mathematical optimization is the right tool, check for three components: decision variables you can control (e.g., product orders), constraints that limit those decisions (e.g., budget), and a clear objective to maximize or minimize (e.g., profit).
Modern solvers like Gurobi can now directly model complex non-linear constraints, such as exponential functions. This eliminates the need for older, less accurate "linearization" techniques, increasing model realism and solving problems that were previously intractable.
When a mathematical optimization model is "infeasible," it's not a failure. It's a valuable diagnostic tool that proves a set of business rules, budgets, or constraints are fundamentally in conflict and cannot all be satisfied simultaneously. This forces clarification of priorities.
When a model is infeasible due to conflicting constraints, Gurobi's "FeasRelax" function calculates the minimum adjustment needed to find a solution. This provides a precise answer like, "If we increase the budget by 10%, the plan becomes feasible."
To successfully implement optimization, use a two-pronged approach. For leadership, focus on the bottom-line impact (cost savings, profit). For front-line staff who might feel threatened, frame it as a tool that enhances their expertise and makes their job easier, not replaces them.
If an employee with years of domain experience rejects a model's "optimal" suggestion, don't dismiss their intuition. This feedback often reveals a hidden, unwritten business rule. Their "gut feeling" should be investigated and codified as a new constraint to improve the model's realism.
Beyond optimizing race strategy, USA Cycling's Project 405 used mathematical optimization to define ideal physiological profiles for specific events. This allowed them to identify and recruit cross-training athletes who were a perfect physical match, contributing to a gold medal win.
Toronto-based fintech platform MyGoals applies mathematical optimization to personal finance. By balancing competing life goals against complex regulations, its Gurobi-powered tool increases users' after-tax retirement income by 10-20% compared to conventional methods.