What is mathematical optimization, and what can it do for your business?

Have you ever built a schedule late at night by manually moving data around in Excel with poor visibility of constrained resources?

Mathematical optimization is a technology that sounds like a mere university topic until you have seen it in action.

Whether you are working in the railways, for an airline, or onsite at an industrial plant, planning and scheduling is made difficult not by any single decision, but by how those individual decisions interact. We will start with shutdown scheduling, where you have many thousands of activities, potentially hundreds of crews with different competencies, a handful of cranes, complex isolations, permit windows, and a fixed production deadline. Nobody can hold thousands of interacting activities in their head and know their schedule is the best one possible.

We tend to anchor to past strategies, rules of thumb, and even leave unnecessary buffers to accommodate the complexity. The result is a schedule that works but carries unnecessary deadline risk and quietly leaves value on the table.

How does mathematical optimization help

Mathematical optimization is a way of describing a decision problem precisely. We write down what is controllable, what constraints must be respected, and what "good" means. Then we employ sophisticated solvers that use cutting-edge algorithms to search all feasible schedules for the best schedule.

The key words are "all feasible" and "best". This is a far cry from simulation methods that try a range of scenarios on for size. Modern solvers, like Gurobi Optimization, can consider more candidate schedules than there are atoms in the universe. More on what a solver actually does.

Mathematical optimization models are comprised of three ingredients:

• Decision variables: the things you control and can influence. For example, in shutdowns these are when we commence specific work activities, and how we use our critical resources.
• Constraints: the rules of physics and the rules of your business. For shutdowns these are our safety procedures, our resource availability and complex system interactions.
• An objective: what you are trying to achieve. Whether it is reducing cost or maximizing revenue, you will have competing business priorities that you want to balance and weight to see their impact on what "optimal" looks like. For shutdowns, this is often a series of resource availability conformance questions against finishing before the plant production resume deadlines.

Solving the unknown

Most organizations know what a workable plan looks like, but its logic lives across spreadsheets, procedures, and operators' heads. The first job of an optimization expert is to bring that knowledge together and translate it into a precise model: what can change, which rules can and cannot be broken, and the trade-off discussions worth having. A model that is too simple may overlook critical operational detail, and a model that is too complex might be difficult to reason with and even miss the big picture. It takes experience to know the right path to take.

However, once the results of an optimized model are validated, ratified, and trusted within your organization, planning processes can be integrated so teams can rapidly compare options, quantify trade-offs, and respond to new conditions. For as long as the inputs to the planning problem change over time, the model continues to pay dividends.

Challenges

The biggest challenges that mathematical optimization faces stem from trust and confidence in the models. To the uninitiated it can be viewed as a method to automate human jobs and eliminate intuition and instinct from the equation. In practice, a good model enables a planner to spend less time mechanically punching numbers and more time adding value through critical thinking: evaluating trade-offs, providing concrete advice to managers, and reasoning through complex problems. Optimization does not replace people, but it can greatly improve the 80% of time spent manually resolving clashes. The best solutions come from an experienced planner equipped with a robust optimization model, not from either one on their own.

AI vs optimization

A unique feature of mathematical optimization is that it can robustly prove how far a schedule is from the theoretical best - generative AI methods yearn for this level of precision. It is of course not magic. Optimization does not guess and does not hallucinate like ChatGPT or Claude. It does, however, rely on the quality of your data and the quality of the model, and it is difficult to build a model that accurately captures your operations and business priorities.

Where to start?

Pick one planning problem where the value-add is high, the constraints are understood, and today's process is manual. Then ask "how do we know this is a good plan?"

If the answer is "it's feasible and we've always done it this way," there is a gap between feasible and optimal.

Get in touch today to close this gap with mathematical precision.