MMP.8.SD:5.4 - Preserve one unknown condition across stages

A two-stage processing route incurs costs w and 1-w, where an unknown w is fixed for the whole run and belongs to {0,1}. Choosing that route commits to both stages. A supplied alternative has total cost 1.5. The question is to minimize worst total cost, with no probability model.

The two possible cost sequences for the route are (0,1) and (1,0); both total 1. The route therefore beats the alternative. Adding the worst first-stage cost 1 to the worst second-stage cost 1 gives 2, but combines different possible runs.

For the first choice, the common total 1 already suffices. If the remaining cost is later needed and the first-stage cost is observed, that observation identifies w and determines the remaining cost.

Now the operating condition is allowed to change between stages: costs are w_1 and 1-w_2, with all four pairs (w_1,w_2) allowed. The pair (1,0) gives total cost 2. The route’s worst total is now 2, so the fixed-cost alternative 1.5 is preferable. The old conclusion fails because the admitted dependence changed.