Compare tradeoffs without inventing one universal score

Try

Compare three synthetic transport plans by cost and travel time, both to be minimized: A=(2,8), B=(4,5), C=(7,3). None is better than another in both coordinates. Add D=(5,7): B is cheaper and faster, so D is dominated. The efficient alternatives form a tradeoff set. Mathematics can identify domination without deciding how much time is worth one unit of cost; that preference requires a stated decision criterion.

Predict

Predict whether minimizing a positive weighted sum can return a dominated point. It cannot, because a dominating point would strictly lower the sum. Predict whether every efficient point can be obtained that way. In a nonconvex attainable objective set, some efficient points are unsupported and no positive weighting selects them. A weighted score is useful, but it is not a complete description of all tradeoffs in every discrete or nonlinear problem.

Experiment

Use a three-point attainable set for minimizing two objectives. Compare A=(0,2), B=(1,3/2), C=(2,0). All three are nondominated.

Point	Weighted score with weights u,v>0
A	2v
B	u+3v/2
C	2u

For B to beat A, u≤v/2 is necessary. For B to beat C, u≥3v/2 is necessary. These are incompatible for positive v. Thus B is efficient but unsupported. Draw the three points and the segment joining A to C; B lies above that segment, explaining why a linear supporting objective cannot select it from this discrete set.

Understand

A feasible decision is Pareto efficient if no other feasible decision is no worse in every objective and strictly better in at least one. Weak efficiency excludes only points strictly improved in every objective, a weaker requirement. If every weight is strictly positive, a global minimizer of the weighted sum is Pareto efficient: a dominating point would produce a strictly smaller sum. With zero weights, improvement in an ignored objective need not change the score, so only weaker conclusions follow without tie-breaking.

For convex attainable upper images, separation theorems relate efficient boundary points to supporting nonnegative weights under suitable hypotheses. Positive weights and proper efficiency require further qualifications; do not turn this into an unrestricted claim that every efficient point is uniquely generated by positive weights. The finite counterexample demonstrates why convexity matters. Scaling an objective also changes the implied tradeoff: multiplying travel time by sixty to change hours into minutes requires adjusting its weight if the underlying preferences are meant to stay the same.

Apply

An ε-constraint approach minimizes the first objective subject to the second being at most a chosen threshold. It can expose unsupported points: in the three-point example, require the second objective at most 3/2, then minimizing the first chooses B over C. However, an optimizer may tie in the first objective while another feasible point improves the second. A secondary objective or suitable uniqueness condition is needed before claiming full Pareto efficiency from an arbitrary tied optimizer.

For continuous objectives f₁(x)=x and f₂(x)=(1−x)² on [0,1], increasing x worsens f₁ but improves f₂, so every point is efficient. The equally weighted sum has derivative 2x−1 and chooses x=1/2. This is one preference-dependent selection from an entire efficient curve. It does not prove the other points are mathematically inferior for every stakeholder.

Continue

Report objectives, units, dominance rules, weights or thresholds and any tie-breaking convention. Avoid hiding a value judgment inside an unexplained composite score. Continue to Uncertainty and robust design to examine decisions whose coefficients are not known exactly. Robust feasibility will add another explicit modeling choice: which uncertain scenarios the decision is required to survive.