Computational Engineering / Multi-objective design and Pareto decisions
What a weighted sum cannot see
The demonstration that a single score is not merely a simplification — it makes some good designs unreachable.
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.
Read the lesson register and outstanding review notesWhy this matters
A weighted sum is one way to express preferences between objectives. It can select supported trade-offs but can miss non-dominated points in nonconvex parts of the objective set. Its scaling, weights and treatment of ties need to be stated so that the resulting choice can be reviewed.
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
Geometry of a weighted sum
Minimising w f₁ + (1 − w)f₂ selects points touched first by a supporting line with the corresponding slope. For non-negative weights this concerns the relevant lower boundary of the objective set’s convex hull. A point strictly above the line joining two trade-off neighbours can remain non-dominated yet never minimise a weighted sum.
A tie is different from an unreachable point
For Left (100,400), Middle (250,250) and Right (400,100), all three are non-dominated and collinear. At w = 0.5 they tie, so the middle design is a valid weighted-sum minimiser. It is never uniquely best. A program that returns only the first tied candidate can hide it, but that is a tie-handling issue.
Moving the middle point to (250,260) makes it strictly above the joining line while retaining non-domination. It then has no minimising positive weight. This second example demonstrates an unsupported Pareto point.
Preferences and reporting
Weights encode trade-offs and depend on objective units and normalisation. They can be meaningful when those choices are explicit; a weight of 0.4 does not by itself mean 40% importance. Show sensitivity to weights, retain ties, and present the non-dominated options and relevant constraints. A chosen score should be accompanied by its rationale and by any factors outside the model.
Worked example
Sweeping the weights in a collinear example
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 reports a weighted-sum solution as the balanced optimum. What must be checked before accepting that description?
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 express trade-offs only with stated units, scaling and preference assumptions
- Present the front, the exclusions, the knee and the things outside the model — then decide