Check how a weighted option ranking depends on its weights

A weighted total can conceal a preference choice inside an apparently objective number. Hold the inputs fixed and check how the result changes with the weights. This manual exercise uses invented cardinal scores on the same declared scale. It does not measure productivity, estimate delivery, rate colleagues, or choose your team's policy.

Separate feasibility from the scoring model

Suppose two fictional workshop formats, A and B, already satisfy the same mandatory requirements. Give A toy scores of four on the first dimension and one on the second; give B two and three. Higher is better on both dimensions, and the exercise explicitly treats the scores as cardinal and comparable.

Record those assumptions before calculating. Real ordinal labels such as acceptable and good do not automatically support this arithmetic. A missing hard requirement cannot be compensated for by a large total either. Establish feasibility and score interpretation separately from the preference weight.

Write the exact weighted expressions

Let w be the first dimension's weight, between zero and one. The second dimension receives one minus w, so the two weights sum to one. A's score is four times w plus one times one minus w, which simplifies to one plus three times w.

B's score is two times w plus three times one minus w, which simplifies to three minus w. Keep these expressions beside the raw inputs. An unexplained total hides whether the weights add correctly or whether one dimension was accidentally counted twice.

Compare two weights and locate the tie

At w equal to one quarter, A scores 1.75 and B scores 2.75. At w equal to three quarters, A scores 3.25 and B scores 2.25. The preferred option reverses even though no input score changed. Only the declared preference weight changed.

Set one plus three times w equal to three minus w. Four times w equals two, so the tie occurs at one half. Below one half, B has the higher toy total; above one half, A does. At one half both score 2.5. Preserve that equality instead of rounding it into an invented winner.

Discuss the preference rather than hiding it

Ask the actual decision owner which weighting assumptions are acceptable, without selecting a weight after inspecting which option it favors. If reasonable participants prefer weights on opposite sides of the crossover, record that disagreement. A single selected total does not make the competing preference disappear.

Keep omitted considerations visible. A newly identified mandatory requirement may invalidate the initial feasibility assertion. An incomparable scale may invalidate the weighted-sum model. In either case, repair the assumptions before discussing the numerical ranking, rather than adjusting weights until the preferred option wins.

Leave a conditional ranking record

Retain the toy scores, dimension directions, formulas, checked weights, crossover, and unresolved preferences in an ordinary document. Ask another reader to explain why the result reverses and what stays fixed. They should distinguish a preference-sensitive result from new evidence about the options.

Unlike dominance screening, this exercise intentionally examines a particular total-score model; it does not claim that model is universally appropriate. The finish is a conditional result with its assumptions exposed, not a declaration of best overall or a promise that the chosen format will succeed.