Flight budget pacing

Each flight’s budget is first spread evenly across its days (the plan, p). The blend then finds daily budgets x that stay as close to the plan as possible in percentage terms:

c is the max daily change, and s is how far a day goes past it. λ is the break penalty: higher values push the solver to break the cap on as few days as possible. This runs in two passes. The first finds the days where the cap has to break; the second re-solves with the cap lifted on just those days, so an unavoidable break is taken in one clean step instead of being shrunk by bending the days around it. When a break lands within 3 days of a flight change, the second pass also tries putting it on the flight’s first day, where the plan itself changes, and keeps whichever stays closer to plan. A final pass then eases into each break: the flights on either side of it may move by the ramp-in allowance, so budgets start stepping down in the days before instead of holding flat and dropping all at once. Every day stays between the lowest and highest flight levels around it, so a budget never rises just to fall again. μ is the smoothness setting: it penalizes sharp changes in direction, so budgets curve into and out of each change. Smoothing is applied in the final pass and skips the days right around a cap break. Every flight total stays within the flight tolerance of plan, the campaign total matches the plan exactly, and no day goes below $0.

Budgets are rounded to cents without adding new breaks. The solver aims slightly inside the flight tolerance so rounding doesn’t push flights over; on flights under about $20, cents can still add a few tenths of a percent, shown in amber.