Bunch selection context model

1. Overview

The selection context owns the branch-and-price policy and coordinates bunch generation with bunch compilation. The current Kotlin context is an algorithm orchestrator; it does not declare an independent variable family.

1. Dependent Contexts

Bunch generation, bunch compilation, task, rule, passenger, and aircraft.

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2. Concepts / Entities

1. Selection policy

$P$ contains reduced-cost tolerance, column limits, executor minimums, and time limits; $gen$ and $compile$ are the generation/compilation ports.

2. Shadow-price map

$\mathrm{\Pi }$ maps master symbols to dual values consumed by pricing.

3. Bunch solution

$S$ contains selected bunches and the final compilation result.

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3. Variables

No solver variables are registered by BunchSelectionContext. Branch decisions and node bounds belong to the generic branch-and-price algorithm state.

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4. Predicates

improving(b): generated bunch has acceptable reduced cost. fractional(n): node solution has a fractional selection. integral(n): selected columns pass the integer acceptance check. compatible(b,n): bunch respects node decisions.

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5. Sets

$N$: branch nodes; $B_n$: columns compatible with node n;

$D_n$: require/forbid branch decisions; ${\mathrm{\Pi }}_n$: node dual map.

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6. Intermediate Values

For node n, the queue bound and pricing result are:

$$lowerBoun{d}_n=RMP(n),pricin{g}_n=\{b\in B_n\mid r{c}_n(b)<-ϵ\}$$

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7. Assertions

Child nodes inherit immutable branch decisions and only compatible columns:

$$\mathrm{\forall }b\in B_n:compatible(b,n)=true$$

An integer candidate is accepted only after compilation and domain feasibility checks succeed.

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8. Constraints

No independent solver constraints are registered here. The policy imposes algorithmic limits on time, nodes, columns, and reduced-cost tolerance; these are termination conditions rather than mathematical rows.

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9. Objective Function (if applicable)

The context delegates the master objective to bunch compilation. It does not introduce a second objective.

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10. Algorithm References

Algorithm	Source role	Purpose
BranchAndPriceAlgorithm	selection service	process nodes, branch fractional solutions, and coordinate pricing

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11. Ubiquitous Language

Term	Symbol	Definition
branch node	n	RMP plus inherited decisions
branch decision	D_n	require/forbid assignment or link choice
incumbent	S	best accepted integer solution
lower bound	lowerBound_n	node RMP bound

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12. Design Decisions

Decision	Alternative	Rationale
keep branching in orchestration	encode branch state as domain variables	generic algorithm can reuse compilation/generation ports

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13. Change Log

Version	Change	Reason
1.0	Documented selection as orchestration context	Avoid inventing a second optimization model