Recipe
Score designs by weighted sum and confirm the winner
Use when: you have several feasible designs' objective values and a stated preference between the objectives
Goal: the weighted-sum winner, confirmed to be genuinely Pareto-optimal
Taught in The weighted-sum method and its blind spot
Quick recipe
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1
Choose weights w_1,\dots,w_k with every w_i\ge0 and \sum_iw_i=1, reflecting how much each objective actually matters.
Example
Two objectives: pick w=0.4 on f_1, so the weight on f_2 is the remaining 0.6.
Why: the weights only trade off objectives consistently when they're honest shares of one whole preference, not arbitrary numbers.
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2
For every candidate, multiply each objective value by its weight and add the results. The smallest total wins.
Example
B=(2,5): s_B=0.4(2)+0.6(5)=3.8. D=(5,2): s_D=0.4(5)+0.6(2)=3.2. Since 3.2<3.8, w=0.4 selects D over B.
Why: minimizing the weighted sum is the method's whole mechanism, turning several objectives into one number that can actually be ranked.
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3
Because every weight is strictly positive, any weighted-sum minimizer is automatically Pareto-optimal. No feasible point can dominate it, so a full pairwise check against every other design isn't needed.
Example
D minimized the sum at w=0.4, so D is confirmed Pareto-optimal without recomputing D against the other six designs by hand.
Why: a point dominating the winner would also score at least as well on that same weighted sum, contradicting that the winner minimized it.
Common mistakes
- Assuming every Pareto-optimal design is reachable by SOME choice of weights — a point sitting above the chord between two frontier neighbors is invisible to every weighting.
- Using weights that are negative or don't sum to 1, which breaks the domination guarantee Step 3 relies on.
- Concluding a design isn't Pareto-optimal just because one particular weight failed to select it.