Singapore School Mathematics Operating Manual · Chapter 44
Improving one part of a system does not always improve the whole system.
A route can become shorter while total travel time increases because congestion moves elsewhere. One machine can become faster while overall production remains unchanged because another stage is still the bottleneck. One variable can be individually optimal while violating a shared constraint. A local maximum can be lower than a distant global maximum.
This chapter develops the distinction between local optimisation and global optimisation. The central question is: are we improving the component we can see, or the objective that actually matters?
1. An objective must be defined before optimisation begins
“Best” is incomplete.
Best can mean minimum cost, minimum time, maximum area, maximum probability, maximum throughput or minimum error.
Different objectives can produce different optimal choices.
2. A local optimum is best only in a neighbourhood
A function may have a point higher than nearby points but still have a larger value elsewhere.
That point is a local maximum, not necessarily a global maximum.
3. A global optimum is best across the entire feasible domain
To claim a global maximum, every admissible candidate class must be covered.
This may require endpoints, stationary points, discrete candidates or separate branches.
4. Closed-interval optimisation needs endpoint checks
For a differentiable function on [a,b], stationary points alone are insufficient.
The global maximum or minimum can occur at an endpoint.
5. Local improvement can be blocked by a bottleneck
If one stage is already faster than the bottleneck, making it even faster may not increase system throughput.
The Bottlenecks and Limiting Constraints chapter handles this control layer.
6. Global objectives can conflict with component objectives
Minimising the time spent on one task may increase mistakes and raise total completion time.
Maximising one segment’s capacity may shift congestion downstream.
System optimisation must include interaction effects.
7. Shared constraints couple local choices
If x+y≤10, maximising x independently ignores the resource required by y.
The feasible set belongs to the combined system.
8. Trade-offs create frontiers
Improving one objective may worsen another.
For example, lower cost can come with longer time under a simple model.
There may be no single choice that is best on every dimension.
9. Multi-objective optimisation needs priorities or weights
If cost and time both matter, a rule is needed to compare them.
One approach assigns weights, but the chosen weights become part of the model.
Another approach imposes one as a constraint and optimises the other.
10. A weighted score is not neutral
Score=0.7A+0.3B privileges A.
The optimum depends on those weights.
Changing them can change the preferred solution.
11. Normalisation may be required before combining objectives
Adding dollars directly to minutes is not meaningful.
Objectives on different scales need a justified normalisation or conversion before combination.
12. A feasible solution can dominate another
If solution A is no worse than B on every objective and better on at least one, A dominates B.
B can be discarded from consideration.
This is a simple multi-objective filtering rule.
13. Non-dominated choices form a trade-off set
Several solutions may remain because each is better on one dimension and worse on another.
Mathematics can identify the frontier even when context must choose among it.
14. Greedy local choices can fail globally
Choosing the best immediate move at every step does not always produce the best total path.
A locally short edge can lead into a very long later route.
15. Dynamic programming exists because local choices can interact
In suitable problems, the best whole solution can be built from best subproblem solutions only when the state has been defined correctly.
This is an enrichment idea showing why decomposition and global structure matter together.
16. Symmetry can reduce global search
If several candidate regions are equivalent under a proven symmetry, solve one representative and transport the result.
This reduces computation without weakening global coverage.
17. Bounds can eliminate regions from global search
If an entire branch cannot beat the best current candidate, it can be discarded.
Branch-and-bound methods formalise this principle.
18. Monotonicity can make optimisation trivial
If f is increasing across the whole feasible interval, the maximum is at the upper endpoint and minimum at the lower.
No stationary-point search is needed.
19. Convexity can simplify global claims
For a convex function, any local minimum under suitable conditions is global.
This is an advanced structural idea, but it illustrates how shape can turn local information into global certainty.
20. Non-convex problems can contain several local optima
A search method can become trapped near one good solution while a better one exists elsewhere.
Starting point can matter.
21. Discrete optimisation requires admissible integer choices
A continuous optimum at x=4.6 is not a valid final answer if x counts buses or boxes.
Nearby feasible integers must be checked.
22. Rounding the continuous optimum is not always safe
If constraints or the objective are asymmetric, ordinary nearest-integer rounding can select the wrong discrete optimum.
Evaluate the relevant feasible neighbours.
23. Global optimisation includes model validity
An objective function can be optimised perfectly while representing the wrong real-world goal.
The model’s assumptions and objective definition remain part of the conclusion.
24. Robust optimisation values stability as well as peak performance
A solution with slightly lower nominal performance but much larger margin may be preferable when inputs are uncertain.
This is an extension of Decision Margins and Robustness.
25. Sensitivity can change the ranking of candidates
If one candidate’s value changes sharply under small parameter shifts while another remains stable, nominal ranking may not survive uncertainty.
26. Optimisation should be rerun after structural changes
If a bottleneck is removed, constraint is changed or objective is reweighted, the old optimum may no longer be optimal.
27. A practical optimisation audit
Ask:
What is the true objective? What is the feasible set? Are there multiple local optima? Have endpoints and discrete candidates been checked? Do component improvements affect one another? Which constraints are active? Are there trade-offs? Is the optimum robust to plausible changes?
28. Independent practice
1. Explain the difference between local and global maximum.
2. Why must endpoints be checked on a closed interval?
3. Give one reason improving one machine may not improve total throughput.
4. Why can x=4.6 be invalid as an optimum in a bus-count problem?
5. What does it mean for one solution to dominate another?
6. Why can weighted objectives change the optimum?
7. Give one example of a trade-off.
8. Why can a greedy choice fail globally?
29. Worked answers
1. Local means best among nearby feasible points; global means best across the entire feasible domain.
2. A global extremum can occur at a boundary even where the derivative condition for interior stationary points does not apply.
3. Another stage may remain the bottleneck.
4. Bus count must be a whole number.
5. It is at least as good on every objective and strictly better on at least one.
6. The weights determine the relative importance of the objectives.
7. Lower cost versus shorter time.
8. The locally best immediate move may lead to poor later options.
30. Batch 11 as one structural-decision layer
Bottlenecks and Limiting Constraints asks what currently controls performance.
Reversibility and Optionality asks which choices preserve future routes.
Coupling and Interaction Effects asks how component choices affect one another.
This chapter asks which feasible choice actually improves the whole system.
Return to the BTT Mathematics Hub for Batch 11.