A pricing engine picks the largest applicable discount tier first, then the next largest, to maximise total savings on an order — and it produces a demonstrably suboptimal answer on a specific combination of tiers, even though the same greedy approach worked correctly on every case anyone tested. How do you tell, in general, whether a problem like this can be solved greedily, and if it can't, what's the actual fix?

Greedy isn't a technique you apply to a problem — it's a property a problem either has or doesn't, and the test is a specific question: does taking the locally best step always leave a situation from which the overall best answer is still reachable? The classic counterexample is coin change with denominations 25, 10 and 1 making 30 — greedy takes the 25 first, then five 1-cent coins, for six coins total, when three 10s would have been optimal; taking the 25 destroyed the 10-10-10 answer, and nothing in the greedy algorithm itself can detect that it just made an irrecoverable mistake. The pricing engine's tier-stacking is almost certainly the same shape: taking the largest tier first can make a combination of two smaller tiers unreachable, even though that combination would have saved more overall. When greedy doesn't provably hold, the fix isn't a smarter greedy rule — it's dynamic programming, which considers all the choices rather than committing to one, but avoids the exponential blowup of trying literally everything by never solving the same subproblem twice.