fpf.tools
Library / Mathematical Modeling DPF
Jump to passage
In this reading
Source · lines 531–546

Link to current text

Published source confirmed at last check

Source changed 2026-10-03 07:42:37 UTC · snapshot created 2026-10-03 07:43:27 UTC · last check 2026-10-03 07:50:10 UTC

MMP.10:4.4 - Make the answer correspond to the question

Define how to recover the requested object or quantity from a satisfying assignment. Then work in both directions: represent an intended admissible case, and interpret an allowed assignment. Use the construction and its conditions to establish the reach of this correspondence. A small case can expose a mistake; a claim about every case needs the corresponding argument.

Match that reach to the requested result:

  • To use a witness, its recovered object must satisfy the original requirements.
  • To conclude that no intended object exists from inconsistency of the formulation, every intended object must have a representation in it.
  • To infer all possible values, translate the quantity as well as the admissible cases; C.16.IR supplies the projection question.
  • To optimize, translate the objective and preference as well as feasibility. Distinguish a bound from a value attained by an object.
  • To count or sample objects, account for multiple representations of the same object. One representation per object is one solution; weighting or grouping representations can be another.

Auxiliary variables can change what a returned number means. Suppose a finite nonempty set of finish times f_i is determined by the other variables. Introduce a real auxiliary T used only in T >= f_i and the objective of minimizing T. Lowering T to max_i f_i then preserves feasibility, so at an attained optimum T equals the latest finish. If T must instead be an integer and the latest finish is 1/2, its minimum is 1. A merely feasible intermediate T can also exceed the latest finish. Recover the actual latest finish as max_i f_i; infer equality with T only when its domain and other conditions permit that lowering.

Choose an obtaining method for the constructed question. Manual substitution may suffice; another problem needs a numerical method, symbolic derivation or search. C.29.2 separates the required mathematical result from the procedure and its execution. Use that method’s actual conclusion: finding no case within a time budget differs from establishing inconsistency. Preserve any restriction or approximation when returning the result to the original question through C.29.1.