Computational Engineering / Multi-objective design and Pareto decisions

Domination and the front

One comparison, applied everywhere — and what it leaves for the engineer to do.

Lesson review: expert verification remains open

This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

A computational engineering reviewer must check algorithms, convergence claims, statistical assumptions and numerical examples beyond the cases explicitly recorded.
Carbon and depth option values are illustrative and require their own quantity/factor validation.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

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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)

Domination

For minimised objectives, A dominates B when A is no worse on every objective and strictly better on at least one. Apply the comparison to feasible candidates. The surviving candidates form a non-dominated set; it is the true global Pareto front only if the relevant design space has been established. Identical objective vectors do not dominate each other.

The comparison removes trade-offs that are inferior on the stated objectives. It does not account for omitted criteria or uncertainty in the objective values. Those choices require engineering judgement before an option is discarded.

Report infeasible candidates separately with the breached constraints. A constraint may later change, and the reason for exclusion remains useful.

Successive non-dominated sorting assigns ranks after each front is removed. A rank indicates ordering within the candidate set, not the magnitude of the performance gap or overall design quality.

Domination

A \succ B \iff \big(\forall i:\; f_i(A) \le f_i(B)\big) \;\wedge\; \big(\exists j:\; f_j(A) < f_j(B)\big)

What it calculates: Whether one design beats another outright

f_i: 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

Every option, with its status. Shape and colour both carry the status, so neither is needed alone.
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 minimises the weighted sum for at least one weight here, including tied minima. An unsupported point on a non-convex front may have no such weight.

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.

Why this one, and what does it depend on?Decided by

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