Many optimisation workflows stop at the allocation itself, but the reason behind the split often sits in the math for free. The obvious approach—scoring each channel by return per pound and funding the winners—fails because ranking assumes each choice stands alone. Budget allocation is one connected decision, so Linear Programming is the standard tool. The real value isn't the allocation but a byproduct: dual values of constraints, or shadow prices, that explain how much the objective moves if you loosen a rule by one unit. A plain LP dumps the whole budget on the single best channel. The natural fix—a binary on/off switch—turns the model into a mixed-integer program, where those shadow prices disappear. The solution is diversification without a switch: slice each channel's budget into bands with diminishing marginal yields. The first slice of a strong channel is worth a lot; its third slice may be worth less than a weaker rival's first slice, so the optimiser spreads money on its own. No binary variables means LP shadow prices remain available. Productivity is just historical KPI per pound, not a causal estimate. Naming constraints lets the split talk back: a binding budget cap is positive, a binding minimum-spend floor is negative, and a zero shadow price is a rule that never touched the decision. This turns an optimiser into a decision engine that explains itself.

A thirty-line function isn't yet a tool. Plainer layers fill the gap: input validation, scenario re-solving, forecasting with uncertainty bands, and constraint interpretation. An open-source project called CLARO wraps the optimisation with these features while keeping the LP core reusable. Everything runs on plain PuLP; CLARO is one complete implementation.

The on/off switch would have made the model quieter—and it was the one thing worth refusing. Refusing it let the plan keep explaining itself: which rules shaped it and what each one cost. That holds anywhere a fixed pool gets split under rules that bite, whether it's a budget, team, compute, or shelf space.