Module 17 · Lesson 17.1
Domination and the front
One comparison, applied everywhere — and what it leaves for the engineer to do.
Why this matters
Every module in Stage D so far has minimised one number. Real design does not have one number. It has mass, cost, carbon, depth, programme, buildability, robustness and adaptability, and they compete.
The temptation is to collapse them into a single score with weights. Module 2 called that hiding the decision inside the weights; this module shows numerically what else it costs — there are good designs a weighted score can never reach, and nobody who uses one will ever see them.
By the end of this lesson you should be able to
- State the domination test precisely, including why 'strictly better in at least one' is needed
- Separate infeasible designs from dominated ones and say why the distinction matters
- Build a front and assign ranks to what is left
- Read a front as a set of legitimate answers
What you should already know
- Optimisation as objective, variables and constraints (Module 15)
- Search methods and their evaluation cost (Module 16)
The one comparison
Design A dominates B if A is no worse than B in every objective and strictly better in at least one.
That is the whole definition, and both halves are load-bearing.
No worse in every objective — if A beats B on carbon and loses on depth, neither dominates. They are a trade-off, and no arithmetic resolves it.
Strictly better in at least one — without this, identical designs would dominate each other, and the front would be empty.
Apply the test to every pair. What survives — the designs nothing dominates — is the Pareto front.
What the front means
Every design on the front is a legitimate answer. Moving from one to another always costs you something: that is what non-dominated means.
Every design off the front is beaten outright. There is a design that is at least as good on everything and better on something, so nothing recommends it. Removing them is the only pruning that needs no judgement at all.
The front narrows the choice without making it. That is exactly the right division of labour: arithmetic does the part that is arithmetic, and the engineer does the part that is not.
Infeasible is not dominated
A design that violates a hard constraint is not a poor trade-off. It is not a design. It must be removed before the front is built, and it must be reported separately rather than dropped.
The reason is that constraints get revisited. A scheme excluded for being 60 mm too deep is worth knowing about when the floor-to-floor height is under discussion — and if it vanished silently, nobody will remember it existed.
Ranks
Strip the front away and the remainder has its own front. That is rank 1. Strip that and you get rank 2, and so on.
Ranks are useful when the front is very small and you want the next tier to look at, or when a search needs to prefer better solutions without discarding everything else. They are not a quality score: a rank-1 design may be very close behind the front or a long way off it.
What it calculates: Whether one design beats another outright
- fi
- The i-th objective, oriented so smaller is better (varies)
- A ≻ B
- A dominates B (—)
This assumes
- Every objective is oriented consistently — maximisation is handled by negating
- Both designs are feasible; an infeasible one is removed before the test is applied
- The objectives are the ones that matter, which is an engineering judgement the arithmetic cannot check
In plain terms: No weights appear anywhere. That is the point: domination is the strongest statement you can make about two designs without deciding how to trade one objective against another, and it is therefore the only pruning that needs no judgement.
Try it
Pareto explorer
Four scheme options for the same building, on any two objectives. The tool builds the front, marks what is dominated, and refuses to name a winner.
Option set
The second is three designs contrived so one sits in a dip.
Horizontal axis
Vertical axis
| Option | Embodied carbon | Structural depth | Status | Crowding |
|---|---|---|---|---|
| CLT and glulam hybrid | 447 | 465 | on the front | ∞ |
| Concrete band beam and slab | 666 | 374 | on the front | 2.00 |
| Composite steel frame | 782 | 480 | dominated (rank 1) | — |
| Reinforced concrete flat slab | 866 | 318 | on the front | ∞ |
- On the front
- 3 of 4
- Dominated
- 1
- Infeasible
- 0
- Reachable by a weighted sum
- 3 of 3
- Unreachable at any weighting
- none
Every front member is reachable by some weighting here, because this front happens to be convex. Switch to the concave demonstration and one design becomes unreachable at every weighting.
Knee: Concrete band beam and slab
- The knee is where the trade-off curve bends most sharply. It is a good place to start a conversation and a bad place to end one: it depends on the axes you chose and their scaling, and it knows nothing about who is paying or what they value.
Your decision
The tool will not choose. Pick an option and say why — a rejection or a selection with no reason cannot be reviewed and cannot be revisited.
Still missing: chosen, reason, decidedBy. Six months from now the choice will be a fact and the reason will be gone unless it is written down.
Try this
- On the four real schemes, plot carbon against depth: three are on the front and the composite steel frame is dominated by the concrete band beam on both axes.
- Now switch the vertical axis to mass. The front collapses to a single design — the CLT hybrid dominates everything. Nothing about the schemes changed; the question did.
- Switch to the concave demonstration. All three designs are non-dominated, and the balanced middle one wins at none of the 41 weightings.
What this shows: Everything on the front is a legitimate answer — and a weighted sum cannot reach a design that sits in a concave dip.
Worked example
Four schemes, and the one that is dominated
Given
- The four schemes from Module 20's carbon comparison, at a 9 m grid over 8 000 m² and six storeys
- Objectives: embodied carbon (tCO₂e) and structural depth (mm), both minimised
- CLT hybrid 447 t, 465 mm · Band beam 666 t, 374 mm · Composite steel 782 t, 480 mm · Flat slab 866 t, 318 mm
Find
Which schemes are on the front, and what that says
Practice
Design A has mass 210 t and carbon 340 tCO₂e. Design B has mass 260 t and carbon 410 tCO₂e. Both are minimised. How many objectives does A beat B on?
Check yourself
A scheme is excluded from a Pareto comparison because its structural depth exceeds the agreed floor zone by 60 mm. How should it be reported?
Worked example
Why a design can leave the front without changing
Given
- Four schemes are compared on carbon and depth; one is dominated
- The comparison is then repeated on carbon and mass
Find
Why the front changes when none of the designs did
Practice
Five designs are compared and two are dominated. How many lie on the Pareto front?
Check yourself
What does it mean for a design to be on the Pareto front?
Check yourself
Why should infeasible designs be reported separately rather than dominated away?
Summary
- Domination: no worse everywhere, strictly better somewhere
- The front is every design nothing dominates, and all of them are legitimate
- Removing dominated designs is the only pruning that needs no judgement
- Infeasible designs are removed first and reported separately, because constraints get revisited
- The front narrows the choice without making it
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