Module 17 · Lesson 17.2
What a weighted sum cannot see
The demonstration that a single score is not merely a simplification — it makes some good designs unreachable.
Why this matters
Collapsing several objectives into one weighted score is the most common thing done with a multi-objective problem, and it is usually defended as a practical simplification. It is worse than that.
A weighted sum does not merely pick one point on the front and ignore the rest. There are non-dominated designs that no weighting can ever select — and because they never win, nobody using a weighted score will ever see them, or know they were there.
By the end of this lesson you should be able to
- Show geometrically why a weighted sum selects only the convex hull
- Identify a design in a concave dip and confirm it is unreachable
- Explain what the weights are actually encoding
- Say what to present instead
What a weighted sum does geometrically
Minimising w·f₁ + (1−w)·f₂ means sliding a straight line of slope set by w across the objective plane until it first touches a design. Whichever design it touches wins.
So a design can only win if there is some straight line that touches it before touching anything else. That is exactly the condition for lying on the convex hull of the point set.
A design sitting in a dip — one where the front bends the wrong way — is behind the line joining its neighbours for every slope. The line always reaches one of them first. It never wins, at any weighting from 0 to 1.
The demonstration
Three designs, all non-dominated:
| Mass (t) | Carbon (tCO₂e) | |
|---|---|---|
| Left | 100 | 400 |
| Dip | 250 | 250 |
| Right | 400 | 100 |
Nothing dominates anything: each is better than the others on one axis. So all three are on the Pareto front, and the middle one is the balanced compromise — exactly the design a project looking for a middle course would want.
Sweep the weighting from 0 to 1 in forty-one steps and the middle design wins at none of them. At every weighting, one of the two extremes scores better. The library reports it as unreachable, and the lab lets you sweep the weight and watch it never appear.
Why this matters more than it looks
A project that optimises a weighted score is not getting a compromise. It is getting whichever extreme the weights happened to favour, presented as a balanced answer — and the genuinely balanced option is invisible.
And concave regions are not exotic. They arise wherever a technology changes: a timber scheme and a steel scheme sit at opposite ends of a carbon-versus-depth trade, and the hybrid between them often sits in the dip.
What the weights actually are
Ask where 40/30/30 came from and the answer is always the same: somebody chose it. That choice was the design decision — made once, quickly, invisibly, by whoever typed the numbers, and then executed with great precision by the algorithm.
Worse, weights are not interpretable. A weight of 0.4 on carbon does not mean carbon matters 40 %; it means something about the ratio of gradients at the winning point, which depends on the units and the normalisation. Change carbon from tonnes to kilogrammes and the same weights give a different answer.
What to present instead
- 1.The front, with every member's value on every objective.
- 2.What is dominated, and what is infeasible, listed separately.
- 3.The knee, offered as a prompt with its caveat.
- 4.Anything outside the model that bears on the choice — programme, contractor familiarity, planning risk.
- 5.A decision, with a reason, made by a person.
That is more work than a score and it is the work the decision actually requires.
Worked example
Sweeping the weight and watching the dip never win
Given
- The three designs above: Left (100, 400), Dip (250, 250), Right (400, 100)
- Objectives normalised to 0–1 so the weighting is meaningful
- Score = w × normalised mass + (1 − w) × normalised carbon
Find
Which design wins at each weighting
Predict first
A team optimises a weighted score of cost and carbon and reports 'the balanced optimum'. What is most likely to be true of the design they get?
Practice
Three designs have normalised objective pairs (0, 1), (0.5, 0.5) and (1, 0). At a weighting of w = 0.3, what score does the middle design achieve? Use score = w·x + (1−w)·y.
Check yourself
A design sits in a concave dip of the Pareto front. What does that mean in practice?
Check yourself
What is the practical alternative to a weighted sum for choosing among front members?
Summary
- A weighted sum slides a straight line across the objective plane, so it reaches only the convex hull
- A design in a concave dip wins at no weighting, and nobody using a score will ever see it
- Concave regions arise wherever a technology changes — they are not exotic
- Weights are not interpretable and change meaning with the units
- Present the front, the exclusions, the knee and the things outside the model — then decide
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint