Singapore School Mathematics Operating Manual · Chapter 47
Organise multi-objective choices using dominance and Pareto frontiers without hiding value judgements inside a single weighted score.
This chapter extends the operating manual as a cross-topic control layer. It is designed to complement, not replace, existing BTT owners for constraints, optimisation, error diagnosis, modelling, proof and examination craft.
1. Multiple objectives can conflict
The key discipline is to make multiple objectives can conflict explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
2. Dominance removes clearly inferior choices
The key discipline is to make dominance removes clearly inferior choices explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
3. Dominated candidates need no further trade-off debate
The key discipline is to make dominated candidates need no further trade-off debate explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
4. Non-dominated does not mean uniquely best
The key discipline is to make non-dominated does not mean uniquely best explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
5. The Pareto frontier contains non-dominated feasible choices
The key discipline is to make the pareto frontier contains non-dominated feasible choices explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
6. Feasibility comes before Pareto analysis
The key discipline is to make feasibility comes before pareto analysis explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
7. Objective direction must be explicit
The key discipline is to make objective direction must be explicit explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
8. Units can differ across objectives
The key discipline is to make units can differ across objectives explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
9. Weighted sums choose one compromise rule
The key discipline is to make weighted sums choose one compromise rule explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
10. Extreme weights emphasise one objective
The key discipline is to make extreme weights emphasise one objective explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
11. Constraints can convert an objective into a requirement
The key discipline is to make constraints can convert an objective into a requirement explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
12. Threshold preferences create satisficing
The key discipline is to make threshold preferences create satisficing explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
13. Robustness can be another objective
The key discipline is to make robustness can be another objective explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
14. Frontier shape reveals trade-off rate
The key discipline is to make frontier shape reveals trade-off rate explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
15. Knee points are descriptive, not automatic winners
The key discipline is to make knee points are descriptive, not automatic winners explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
16. More objectives make dominance rarer
The key discipline is to make more objectives make dominance rarer explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
17. Bad objectives create bad frontiers
The key discipline is to make bad objectives create bad frontiers explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
18. Mathematics separates structure from choice
The key discipline is to make mathematics separates structure from choice explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
19. A Pareto audit
Ask what the problem requires, what can fail, what evidence is decisive, and which earlier operating-manual control should be invoked before calculation continues.
20. Worked mini-cases
Case A. A candidate satisfies every preference but violates a domain restriction. The candidate remains inadmissible because validity is checked before preference.
Case B. Two calculations disagree. Return to the last trusted common input, separate the branches, and identify whether the discrepancy comes from a shared dependency or a branch-specific step.
Case C. A result lies close to a threshold. Preserve precision and compare the uncertainty interval with the decision margin before classifying it.
21. Independent practice
1. Identify the admissibility condition in a problem of your choice.
2. Give one independent check that could catch a wrong intermediate result.
3. Explain why a mathematically attractive candidate may still be invalid.
4. Describe one situation where uncertainty changes the strength of a conclusion.
5. State what evidence would justify stopping the calculation.
22. Worked answers
1. The admissibility condition is the rule every candidate must satisfy before comparison.
2. Examples include substitution, a unit check, a bound, an alternative representation or an independent method.
3. Objective performance cannot compensate for violation of a hard mathematical condition.
4. A rounded or measured value near a threshold may have a plausible interval crossing both decision regimes.
5. A complete proof, verified candidate set, guaranteed bound, tolerance certificate or exhausted feasible search can justify stopping.
23. Continue through Batch 12
Return to the BTT Mathematics Hub for Batch 12 and use the neighbouring chapters as a connected control system.