Integer Programming Model¶
Method — instantiates Discrete Commitment Optimization
A formal declarative statement of a discrete decision — its binary and whole-number variables, objective, and constraints — written to be handed to a solver.
An Integer Programming Model is the formulation — the precise mathematical statement of a discrete commitment problem. It names the decision variables (some restricted to 0/1, some to whole numbers), writes the objective as a linear expression to maximize or minimize, and writes every rule as a linear constraint the variables must satisfy. What makes it this mechanism, and not a solver, is that it declares rather than searches: it says what a legal choice is and what makes one choice better than another, and stops there. It is a specification, deliberately silent about how a solution will be found. That separation — model as the thing that states the problem, solver as the thing that answers it — is the single distinction that keeps it apart from its near-twin.
Example¶
A telecom operator is planning which of 40 candidate cell sites to upgrade to 5G this year, and how many spectrum blocks to license per upgraded site. Each site is a yes/no decision; the spectrum count is a whole number between zero and four. The analyst writes it out formally: a binary variable per site, an integer variable per site's spectrum blocks, an objective that maximizes projected coverage-weighted subscribers, and constraints — total capital under the annual budget, spectrum only licensable at an upgraded site, and no more than three upgrades in any single region so crews aren't overstretched.
Written this way, the decision becomes a self-contained artifact: forty binaries, forty integers, one objective row, a handful of constraint rows. The analyst has not yet chosen anything — she has stated, unambiguously, what choosing means. That statement is what she can now hand to a solver, circulate for review, or edit when the budget changes, without re-deriving the whole problem. The value of the model is exactly this: it turns a fuzzy planning debate into a precise, inspectable object.
How it works¶
- Declare the variables. One decision per unit — binary for select/don't, integer for counts, with explicit domains.
- Write the objective. A single linear expression capturing what "good" means (subscribers, profit, cost), to be maximized or minimized.
- Encode every rule as a linear constraint. Budgets, prerequisites (spectrum implies upgrade), mutual exclusions, and per-region caps become inequalities over the variables.
- Leave it unsolved. The model's job ends at a complete, consistent formulation. Its quality is judged by fidelity — does it faithfully represent the real decision? — not by any answer, because it produces none.[n1]
Tuning parameters¶
- Variable granularity — one binary per site, or finer variables per site-and-technology. Finer models capture more options but grow the formulation and the eventual solve time.
- Linearization choices — how non-linear realities (economies of scale, either/or logic) are recast as linear constraints with auxiliary variables. Cleaner linearizations solve faster but can distort the real relationship.
- Constraint tightness — how close the written constraints hug the true feasible region. Tighter formulations solve far faster; loose ones are easier to write but punish the solver.
- Objective composition — single objective, or a weighted blend of competing goals. Blending is convenient but buries value trade-offs inside weights.
When it helps, and when it misleads¶
Its strength is discipline: forcing a messy discrete decision into variables, an objective, and constraints surfaces hidden assumptions and makes the whole thing reviewable and reusable. A well-built model outlives any one run — change a budget number and the same formulation answers a new question.
Its failure mode is that the model is an abstraction, and an abstraction can be faithful or false. Everything the formulation omits — a real prerequisite left out, a cost approximated as linear when it isn't — produces an "optimal" answer to the wrong problem, with all the authority of mathematics behind it. The classic misuse is objective laundering: burying a contested value judgment inside an innocuous-looking weight so the trade-off never gets debated. The guarding discipline is to review the formulation as a stated argument — is every constraint real, is every omission acceptable, does the objective encode a defensible priority — before any solver is trusted to answer it.
How it implements the components¶
binary_decision_variable— declares the select/don't-select variables at the heart of the formulation.integer_constraint— restricts count variables (spectrum blocks) to whole numbers, forbidding fractional answers.objective_function— the explicit linear expression the formulation is written to optimize.coupling_constraint— the linear inequalities (budget, prerequisites, mutual exclusions, regional caps) that tie the variables together.
It does not implement selection_rule, combinatorial_feasible_set, or feasibility_audit — searching the formulation for an answer and certifying that answer is the job of its near-twin, Integer Programming Solver. The model states the problem; the solver solves it.
Related¶
- Instantiates: Discrete Commitment Optimization — it is the formal-representation variant.
- Sibling mechanisms: Integer Programming Solver (consumes this model) · Assignment Model · Constraint Satisfaction Search · Crew Scheduling Model · Facility Location Model · Project Selection Matrix · Selection Review Board · Solver Dashboard
Editorial Notes¶
Form Classification¶
Form family: Representation, Specification & Plan
Rationale: Integer Programming Model operates as a non-executable information artifact that externalizes static or prospective structure because it a formal declarative statement of a discrete decision — its binary and whole-number variables, objective, and constraints — written to be handed to a solver
Independent corroboration: The frozen evidence defines Integer Programming Model as 'A formal declarative statement of a discrete decision — its binary and whole-number variables, objective, and constraints — written to be handed to a solver', so its operative form is Representation, Specification & Plan.
Review outcome: Independent reviewer agreement; high confidence.
Origin Attribution¶
Primary origin: Operations Research
Origin pattern: Convergent development
Present-day reach: Multi-domain
Rationale: Declarative objectives, constraints, and integral decision variables are the canonical modeling language of integer programming in operations research.
Related originating lineages:
- Mathematics — Discrete optimization and polyhedral theory materially provide the model's formal foundations.
Review resolution: Both independent reviews place the primary lineage in operations_research. The queued differences (origin_mode_disagreement, domain_reach_disagreement) concern secondary metadata rather than primary provenance. The final retains mathematics only where a reviewer supplied a formative-lineage rationale; this does not convert downstream applicability into origin. origin_mode=convergent because the reviewers document independently established or materially co-developing traditions. domain_reach=multi_domain records application breadth separately from provenance.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
The Assignment, Crew Scheduling, and Facility Location models are all specializations of this general formulation — each is an integer program with a characteristic structure (a compatibility grid, a set-covering skeleton, a siting-plus-coverage linkage). This page is the generic parent; reach for a specialized sibling when the problem fits its named shape, and for this one when it doesn't.
[n1] The LP relaxation — dropping the integrality requirement and letting variables take fractional values — bounds an integer program's best achievable objective and is the workhorse a solver exploits. The model defines the relaxation implicitly; exploiting it is the solver's concern, underscoring the model/solver split. ↩