A head-to-head comparison of Classical DOE vs. SDLabs Bayesian Optimization on a 6-component mixture with 4 competing objectives — demonstrating that BO finds better formulations with fewer experiments and zero design decisions.

Optically variable pigment (OVP) inks for banknote security must simultaneously deliver strong color shift (the anti-counterfeiting feature), target viscosity for the printing process, reliable adhesion to the substrate, and long-term pigment stability against sedimentation and corrosion. These objectives conflict: more pigment improves color shift but worsens stability, more binder improves adhesion but crowds out pigment, and the mixture must sum to exactly 100 wt%.

The search space includes 6 mixture components, 4 pigment types, and 3 printing methods — creating 12 categorical sub-landscapes across 4 competing objectives. This is exactly the kind of problem where classical DOE struggles and adaptive optimization excels.

Constraint: All 6 numerical components must sum to 100 wt%.

The core difference is how many decisions the scientist must make before running experiments.

With DOE, the scientist must decide upfront: which screening design (fractional factorial? Plackett-Burman?), how many runs, which factors are significant, which categorical levels to fix, which RSM design to use, desirability function shapes and bounds for each response, importance weights, and which optimization algorithm to run on the fitted model. Each decision requires expertise and can dramatically change the outcome.

Wrong desirability weights lead to the wrong optimum. Wrong screening cutoff means missing a key factor. Fixing the wrong categorical level means missing the best pigment type entirely — with no way to recover without starting over.

With SDLabs, the scientist provides: parameters and ranges, measurements, objectives (maximize/minimize/target), and constraints. That's it. No design matrix, no screening phase, no model selection, no desirability function tuning. The Gaussian Process model and acquisition function handle all experimental design decisions adaptively, informed by every result collected so far.

The classical DOE approach followed standard methodology: 16-run fractional factorial screening, then a 34-point Scheffe simplex-lattice RSM, with Derringer-Suich desirability optimization across 4 separate quadratic models.

The desirability function required setting 13 parameters (bounds, targets, shape exponents, and importance weights) before seeing any RSM data. The overall desirability at the predicted optimum was D = 0.744.

The quadratic model predicted viscosity at exactly 25.0 Pa.s, but the actual value was 31.5 — a 26% error caused by the non-linear relationship between solids content and viscosity that a quadratic polynomial cannot capture.

6 iterations of 6 formulations each, with the GP model updating after every batch:

By iteration 4 (24 experiments), BO had already surpassed DOE's 50-experiment results on Color Shift, Adhesion, and Stability. The optimizer continued improving through iteration 6, finding formulations the DOE's quadratic model could not predict.

BO wins on 3 of 4 objectives with 28% fewer experiments and zero design decisions by the scientist.

The figure below shows the Color Shift vs Viscosity landscape explored by each approach. On the left, BO progressively focuses from broad exploration (dark, early iterations) toward the high-color-shift region near the viscosity target. On the right, DOE's fixed designs cover the space uniformly but miss the optimal corner — and the DOE optimum (red star) lands above the viscosity target due to the quadratic model's prediction error.

Viscosity prediction: The quadratic model predicted exactly 25.0 Pa.s at the optimum, but the actual value was 31.5 — a 26% error. The relationship between solids content and viscosity follows a non-linear Krieger-Dougherty curve that a quadratic polynomial cannot represent. The GP model captures this non-linearity natively.

Categorical lock-in: DOE fixed the pigment type and printing method after screening with only 16 noisy data points. While it happened to choose correctly here, an unlucky screening could lock onto a suboptimal combination with no way to recover.

Desirability sensitivity: Changing the importance weight of Color Shift from 3 to 2 shifts the DOE optimum by ~5% on every response. The scientist has no way to determine the "right" weights before seeing the landscape.

No design decisions: The scientist defined parameters, objectives, and constraints. Everything else — experimental design, model selection, exploration vs exploitation — was handled automatically.

Adaptive exploration: BO explored all 12 categorical combinations naturally, discovering which pigment type and printing method worked best without a separate screening phase.

Progressive improvement: By iteration 4 (24 experiments), BO had already beaten DOE's 50-experiment result. The remaining iterations continued refining, finding formulations DOE's quadratic model could not predict.

Objective convergence — Color Shift Angle and Adhesion Score improve rapidly through iterations 1-4, then stabilize as the optimizer refines the Pareto front:

Prediction explanation — Feature importance for the best formulation found, showing which parameters most influence Color Shift Angle at the optimum:

Stability chart — The optimizer converges on a stable region of the design space, with decreasing variance in recommended formulations across later iterations: