Optimizing concrete mix designs
In concrete mix design, it is widely known that strength is correlated to the water / binder (w/b) ratio. Concrete professionals all know that to improve strength, you you should lower the w/b ratio of your formula.
But when you increase strength, you still have to achieve acceptable rheology, the kind that makes the concrete pumpable, or at least easily handled. Reduce water too much and you lose slump. Compensate too much with admixtures and you get viscosity or set delay problems.
Nowadays, formulators do also care about the sustainability of their formulae, which is usually translated to CO₂eq emissions. Just like the cost of formulae, CO2eq emissions are to be minimized, which limits the freedom to use more cement.
So a concrete mix designer has to balance 3 conflicting objectives: maximize strength, minimize cost and CO₂. All while keeping acceptable rheology (slump and viscosity). In other words, it's a 3-objective optimization problem with 2 constrained outcomes.
The Pareto front
In this context, you don't directly get a single metric based on which you could tell that formula A is better than formula B. Formula A may have a better mechanical strength, but a higher cost and higher CO2eq emissions.
When faced with multiple conflicting objectives, you are not looking for the absolute best formula, you are instead looking for the best trade-off. The best way to visualize it is to see where each formula sits in the the 3-dimensional space formed by our 3 objectives.
The optimal formulations sit on the so-called Pareto front, that is the best trade-offs you have so far. The ones behind it are not optimal and have been dominated by better options. When you run new experiments, you're only making progress if you push the Pareto front further. i.e. you increase the volume (or hyper-volume) it delimits.
So if you wanted to collapse your 3 objectives into one, that single objective would be to push the Pareto front as far as you can: to grow the volume it delimits.
How Alfraido handles multiple objectives and constrained outcomes
Alfraido tracks the progress of your optimization through the hyper-volume of the space bounded by the Pareto front on one side and your least-interesting acceptable results (your thresholds) on the other.
It systematically proposes the trials most likely to push the Pareto front further, without breaching the constraints you've set on the constrained outcomes. Each proposed trial comes with its predicted results, so you know what to expect before you run it. As you feed in more results, Alfraido tells you whether you've discovered a new Pareto front, and constantly shows you the best front you have so far.