Optimization · Module 10 of 10
Multi-Objective and Design Optimization
Real design rarely has one goal. Light and stiff, cheap and durable, fast and efficient: these compete. Multi-objective optimization maps the trade-off so you can choose, rather than pretending one number captures the design.
Readiness check
This closing module trades competing goals. Tick only what you can do closed-notes.
- Recall an objective as a single number to minimise.
- Form a weighted sum of two quantities.
- Compare two designs on two criteria at once.
- Recall that constraints bound the feasible set.
- Recognise when improving one goal worsens another.
The core idea
With competing objectives there is no single best design, but a set of best trade-offs, the Pareto front. A design is Pareto-optimal if no other is better in one objective without being worse in another. Weighted sums pick one point on that front.
weighted sum: F = w1f1 + w2f2A dominates B if A is no worse in all and better in onePareto front = the non-dominated designsMost engineering designs balance several goals that pull against each other: a lighter part is often less stiff, a cheaper one less durable. When objectives compete, there is no single optimum. Instead there is a set of designs for which no objective can be improved without sacrificing another; these are the Pareto-optimal or non-dominated designs, and together they form the Pareto front. One design dominates another if it is at least as good in every objective and strictly better in at least one; a design is on the front if nothing dominates it. The front is the honest picture of the trade-off, and choosing among its points is an engineering judgement, not a calculation. The simplest way to select one point is the weighted-sum method: combine the objectives into F = w1f1 + w2f2 and minimise, where the weights encode the relative importance. Sweeping the weights traces out the front, and the chosen weights, made explicit, turn a multi-objective problem into a defensible single choice.
The skills, taught in order
Five skills build competing objectives, dominance, the Pareto front, and how to choose.
10.1 Competing objectives
A multi-objective problem has two or more objectives that cannot all be optimised at once, because improving one worsens another. Naming the objectives honestly, rather than collapsing them prematurely into one, is the first step of design optimization.
10.2 The weighted-sum method
The weighted-sum method combines objectives into F = Σ wifi with non-negative weights and minimises the result. The weights state the relative importance; different weights give different points, so the method converts the trade-off into a single, weight-dependent choice.
| Concept | Meaning |
|---|---|
| Dominance | better in all objectives, worse in none |
| Pareto-optimal | not dominated by any design |
| Pareto front | the set of all Pareto-optimal designs |
| Weighted sum | one point chosen by weights |
The vocabulary of trade-offs: dominance defines the front, and weights pick a point on it.
10.3 Pareto dominance
Design A dominates design B if A is no worse than B in every objective and strictly better in at least one. Dominance is the test that sorts designs: any dominated design can be improved for free and should be discarded.
10.4 The Pareto front
The Pareto front is the set of non-dominated designs, the best achievable trade-offs. Its shape shows the cost of each objective in terms of the others: a steep front means one goal is cheap to trade for the other, a flat one means it is expensive.
10.5 Choosing a design
The front narrows the choice to the trade-offs worth considering, but selecting one point is a judgement about priorities, made explicit through weights, a target on one objective, or a stakeholder decision. Optimization informs the choice; it does not replace it.
Engineering connection: every real specification, light yet stiff, cheap yet reliable, is a point chosen on a Pareto front, and stating the weights makes the decision defensible. This trade-off thinking carries into Verification, Validation, and Uncertainty Quantification, the recommended next course.
Worked example 1: choosing by weighted sum
Four candidate designs have objectives (f1, f2), both to be minimised: A (2, 8), B (4, 4), C (6, 2), D (3, 9). Choose the best design with weights (0.6, 0.4), then again with (0.8, 0.2).
- ProblemPick the best design by weighted sum for the two weight sets in Figure 1.
- Given / findDesigns A (2, 8), B (4, 4), C (6, 2), D (3, 9), both objectives minimised. Find the best for w = (0.6, 0.4) and w = (0.8, 0.2).
- AssumptionsObjectives are on comparable scales so a weighted sum is meaningful.
- ModelF = w1f1 + w2f2; choose the design with the smallest F.
- EquationsF = w1f1 + w2f2
- Solvew = (0.6, 0.4): A = 4.4, B = 4.0, C = 4.4, D = 5.4, so choose B. w = (0.8, 0.2): A = 3.2, B = 4.0, C = 5.2, D = 4.2, so choose A.
- CheckWhen f2 carries more weight, the balanced design B wins; when f1 dominates, the f1-light design A wins. The weights, not the math, drove the change.
- ConclusionThe weighted sum selects a single design, but the answer depends on the weights. Making them explicit turns a value judgement into a defensible choice.
Worked example 2: the Pareto front
For the same four designs, A (2, 8), B (4, 4), C (6, 2), D (3, 9), with both objectives minimised, find which are Pareto-optimal and which are dominated.
- ProblemFind the Pareto-optimal designs among those in Figure 2.
- Given / findA (2, 8), B (4, 4), C (6, 2), D (3, 9), both minimised. Identify the front and any dominated design.
- AssumptionsBoth objectives are minimised; dominance requires no-worse in all and better in one.
- ModelDesign P dominates Q if P1 ≤ Q1 and P2 ≤ Q2 with at least one strict.
- EquationsA (2, 8) vs D (3, 9): 2 < 3 and 8 < 9 ⇒ A dominates D
- SolveCheck each pair. A, B, C trade off (as f1 rises, f2 falls), so none dominates another: all three are Pareto-optimal. D (3, 9) is worse than A (2, 8) in both objectives, so D is dominated. Pareto front = {A, B, C}.
- CheckAlong A, B, C each objective improves only by worsening the other, the signature of a front. D sits above and to the right of A, confirming domination.
- ConclusionThe Pareto front is {A, B, C}; D should be discarded. The front is the honest menu of trade-offs from which a design is chosen.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Choosing a dominated design | A design beaten in every objective | "Is anything better in all objectives?" | Discard dominated designs before choosing. |
| Hidden weights | A choice presented as objective | "What weights did this assume?" | State the weights; they are a value judgement. |
| Summing incomparable scales | One objective swamps the sum | "Are the objectives normalised?" | Scale objectives before a weighted sum. |
| Expecting a single optimum | Insisting on one best design | "Do the objectives compete?" | With competing goals, report the front, then choose. |
Practice ladder
For design B (4, 4) with weights (0.5, 0.5), find the weighted-sum value.
Show answer
F = 0.5(4) + 0.5(4) = 4.0.
Does A (2, 8) dominate C (6, 2), both minimised?
Show answer
No: A is better in f1 (2 < 6) but worse in f2 (8 > 2). Neither dominates; both are trade-offs on the front.
A new design E (5, 5) is added to A (2, 8), B (4, 4), C (6, 2). Is E on the Pareto front?
Show answer
B (4, 4) has f1 = 4 < 5 and f2 = 4 < 5, so B dominates E. E is not on the front.
You must choose between a light-but-flexible bracket and a stiff-but-heavy one. Explain how a Pareto front and stated weights would justify your choice.
What good work looks like
Plot mass against compliance for candidate designs, keep only the non-dominated ones, then choose a point by weights that reflect the application (weight-critical versus stiffness-critical), stating those weights so the decision is transparent and defensible.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Take a real design with two competing objectives, plot the Pareto front of candidates, and choose one point with explicitly stated weights and a justification.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. When is there no single optimum?
When two or more objectives genuinely compete.
2. Define Pareto dominance.
A dominates B if A is no worse in every objective and better in at least one.
3. What is the Pareto front?
The set of non-dominated designs, the best achievable trade-offs.
4. What does the weighted-sum method do?
Combines objectives with weights into one number to pick a single design.
5. Why state the weights?
They encode a value judgement, so making them explicit keeps the choice defensible.
Textbook mapping
This module draws on Nocedal and Wright, Numerical Optimization, 2nd edition for the underlying single-objective machinery, with the multi-objective and Pareto framing from standard engineering design-optimization practice.
| Topic in this module | Where to read more |
|---|---|
| Single-objective methods behind each solve | Nocedal & Wright, Chapters 2 to 17 |
| Weighted sums and scalarisation | Engineering design-optimization references |
| Pareto dominance and fronts | Engineering design-optimization references |
The numerical methods follow Nocedal and Wright; the multi-objective and Pareto framing uses standard engineering design-optimization practice. The concepts are standard across references.