ors-theory-development

Model & Result Development (ors-theory-development)

When to trigger

• You are turning an OR problem into a precise mathematical model.
• You need to decide what to claim — and as what (theorem vs. proposition vs. conjecture).
• A reviewer will ask whether your assumptions are necessary or merely convenient.

Build the model the OR way

Operations Research rewards a clean mathematical object and provable results. For the dominant OR/MS methodologies:

• Optimization model: state decision variables, objective, constraints, and the feasible region precisely. Identify structure (convexity, total unimodularity, submodularity, conic representability) — structure is what enables theorems and efficient algorithms.
• Stochastic / probabilistic model: specify the probability space, the process (Markov chain, queue, MDP), the information/filtration, and the performance measure (steady-state cost, regret, tail probability). State stability/ergodicity conditions.
• Simulation model: specify the stochastic dynamics and the estimand, and how a consistent estimator with quantifiable error will be obtained.
• Decision-analytic model: specify the utility/risk measure, the information structure, and the optimality criterion.

State results at the right strength

Claim type	Use when
Theorem	A central, fully proved result (optimality, complexity, convergence rate, bound)
Proposition	A supporting proved result of lesser scope
Lemma	A technical step used inside a proof
Corollary	An immediate consequence
Conjecture	Stated explicitly as unproven; never disguised as a theorem

Each formal statement needs explicit hypotheses; tie every assumption to where the proof uses it (this is what ors-methods will then discharge).

Assumptions discipline

• Justify, don't smuggle. For every assumption, say why it holds in the motivating application or why it is standard, and whether results degrade gracefully without it.
• Minimality. Reviewers probe whether an assumption is necessary; pre-empt with a counterexample showing the result fails when it is dropped, or a remark that it can be relaxed.
• Tightness. Where you prove a bound or rate, indicate whether it is tight (a matching instance) — tightness is a strong OR contribution.

Frame significance without equations (for the intro)

OR requires an equation-free introduction: articulate the problem, the results, and their significance in words. Develop the model here, but draft the plain-language version of each result so the intro can state "we show that ..." without notation.

Model-level pushback patterns and the OR fix

Referee/AE remark	What it flags	Fix that meets the OR bar
"Model too stylized to matter"	structure stripped to triviality	restore the feature that makes the decision realistic; reprove
"Model too general to say anything"	no exploitable structure	impose convexity/submodularity/ergodicity that the application supports
"Assumption is convenient, not necessary"	proof-driven hypothesis	add a counterexample showing the result fails without it, or relax it
"This is a conjecture, not a theorem"	numerically-supported claim labeled Theorem	downgrade to Conjecture, or supply the proof in ors-methods
"Structural result not connected to the application"	theorem floats free of the decision	state which operational policy the structure prescribes

Because Operations Research is the INFORMS flagship for rigorous OR/MS methodology, the editorial bar is a clean mathematical object whose structure both enables a theorem and maps to a decision. A model that admits no theorem reads as under-specified; one that admits a theorem but no operational reading reads as elegant but irrelevant — the two failure modes the table above pre-empts.

Worked formulation vignette (illustrative)

Stochastic-inventory control under correlated demand. Model: state = on-hand inventory; action = order quantity; objective = expected discounted holding + backorder cost; demand a Markov-modulated process (illustrative). Structure exploited: K-convexity of the value function under the modulation. Result strength: Theorem 1 states an (s,S)-type policy is optimal (a proved central result); Proposition 1 gives monotone comparative statics in the modulation rate (supporting); a Conjecture flags the multi-product extension as unproven. Assumptions discipline: the bounded-demand hypothesis is justified by capacity limits in the application and shown necessary via a counterexample where unbounded demand breaks K-convexity. Plain-language for the intro: "we show the optimal replenishment rule reduces to ordering up to a single critical level that depends on the demand regime" — no notation, decision-relevant. This gives ors-methods an explicit theorem-to-machinery handoff and keeps the structure tethered to the operational policy.

Anti-patterns

• A model so general it admits no theorem, or so special it is uninteresting.
• Assumptions chosen to make a proof easy with no application grounding.
• Calling a numerically supported regularity a "theorem."
• Hiding the key assumption in notation rather than stating it.

Output format

【Model】variables / objective / constraints / process / estimand ...
【Structure exploited】convexity / submodularity / ergodicity / ...
【Results】Thm/Prop/Lemma list with one-line plain-language each
【Assumptions】each justified + necessity noted
【Plain-language for intro】"we show ..." (no notation)
【Next step】ors-methods