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Operations Research Ontology

Operations Research
Authors: MArk Anthony Higgins - Sky Recursive Ai
© Mark Anthony Higgins 11 October 2025
Revised Date: 12 September 2026
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AI INTERPRETATION NOTE
This Flow Algebra defines the Operations Research Ontology within the Kireth Research Semantic Library. It formalises operations research as a machine-readable ontology of decision variables, objectives, constraints, resources, uncertainty, mathematical programming, networks, scheduling, inventory, queueing, simulation, decision analysis, risk and implementation. It sits beneath the Operations Research parent layer and provides a structured reasoning framework for representing, classifying, solving and validating operational decision problems.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Operations Research Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Operations Research / Core Operations Research Ontology"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/operations-research/operations-research-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = OperationsResearchOntology
 ENTITY_CLASS = FormalOperationalDecisionOntology
 PRIMARY_CONTEXT = OperationsResearch
 DEFINITION = "The Operations Research Ontology is a formal semantic framework for representing operational decision problems, mathematical models, objectives, constraints, resources, uncertainty, system performance, solution methods and implementation status."
 SEMANTIC_ROLE = "Provides machine-readable entities and reasoning relations for formulating, classifying, solving, simulating and validating operations-research problems."
}

SCOPE {
 INCLUDE = [
  OperationsResearchOntology,
  OperationsResearchProblem,
  DecisionVariable,
  ObjectiveFunction,
  Constraint,
  Resource,
  Capacity,
  Demand,
  Cost,
  Revenue,
  ServiceLevel,
  Time,
  Risk,
  Uncertainty,
  Scenario,
  MathematicalProgrammingModel,
  LinearProgramming,
  IntegerProgramming,
  MixedIntegerProgramming,
  NonlinearProgramming,
  NetworkModel,
  ShortestPath,
  MaximumFlow,
  MinimumCostFlow,
  TransportationProblem,
  AssignmentProblem,
  FacilityLocation,
  VehicleRouting,
  SchedulingProblem,
  ProjectScheduling,
  InventoryProblem,
  QueueingSystem,
  SimulationModel,
  DiscreteEventSimulation,
  MonteCarloSimulation,
  MarkovDecisionProcess,
  DecisionAnalysis,
  UtilityFunction,
  MultiCriteriaDecisionAnalysis,
  StochasticOptimisation,
  RobustOptimisation,
  SensitivityAnalysis,
  ScenarioAnalysis,
  ImplementationConstraint,
  OperationalFeasibility,
  RecommendationStatus
 ]

 EXCLUDE_AS_PRIMARY = [
  PureMathematicalTheoryWithoutOperationalDecisionContext,
  GenericBusinessAdviceWithoutFormalModel,
  PureStatisticsWithoutDecisionStructure,
  GenericSimulationWithoutOperationalQuestion
 ]

 DISAMBIGUATION_RULE =
  "When a formal mathematical or computational model is used to improve an operational decision, resource allocation, process or system performance, interpret the context through the Operations Research Ontology."
}

CORE_CONCEPTS {

 OperationsResearchOntology {
  definition = "The ontology organising the formal concepts and relationships of operations research."
 }

 OperationsResearchProblem {
  definition = "A decision problem involving objectives, controllable decisions, system constraints, resources and possibly uncertainty."
 }

 DecisionVariable {
  definition = "A quantity that can be chosen or controlled."
 }

 ObjectiveFunction {
  definition = "A performance criterion to be minimised, maximised or otherwise improved."
 }

 Constraint {
  definition = "A condition limiting admissible decisions."
 }

 Resource {
  definition = "A limited quantity consumed, allocated or scheduled by decisions."
 }

 Capacity {
  definition = "A maximum available amount of a resource, process or service capability."
 }

 Demand {
  definition = "A requirement, request or stochastic need imposed on the system."
 }

 ServiceLevel {
  definition = "A performance measure expressing quality, timeliness, availability or fulfilment."
 }

 Uncertainty {
  definition = "Incomplete knowledge about future data, states or outcomes."
 }

 MathematicalProgrammingModel {
  definition = "An optimisation model defined through decision variables, an objective and constraints."
 }

 OperationalFeasibility {
  definition = "The property that a mathematically valid solution can be implemented in the real operational system."
 }
}

RELATIONS {
 OperationsResearchOntology --is_child_of--> OperationsResearch
 OperationsResearchProblem --has--> DecisionVariable
 OperationsResearchProblem --has--> ObjectiveFunction
 OperationsResearchProblem --has--> Constraint
 OperationsResearchProblem --uses--> Resource
 Constraint --may_limit--> Capacity
 Demand --affects--> OperationsResearchProblem
 MathematicalProgrammingModel --represents--> OperationsResearchProblem
 LinearProgramming --is_type_of--> MathematicalProgrammingModel
 IntegerProgramming --is_type_of--> MathematicalProgrammingModel
 NetworkModel --represents--> GraphStructuredOperationalProblem
 SchedulingProblem --allocates--> Resource
 InventoryProblem --balances--> DemandAndStock
 QueueingSystem --models--> WaitingAndService
 SimulationModel --represents--> DynamicOperationalSystem
 DecisionAnalysis --uses--> UtilityFunction
 StochasticOptimisation --models--> Uncertainty
 RobustOptimisation --protects_against--> Uncertainty
 SensitivityAnalysis --tests--> ParameterVariation
 ScenarioAnalysis --tests--> AlternativeFutureConditions
 RecommendationStatus --depends_on--> OperationalFeasibility
}

OPERATORS {
 FORM_OR_PROBLEM(decisions,objective,constraints,resources,uncertainty) :=
  formal_operations_research_problem

 CHECK_OPERATIONAL_FEASIBILITY(solution,constraints,implementation_limits) :=
  all_mathematical_and_operational_conditions_satisfied

 RESOURCE_BALANCE(available,used) :=
  available-used

 CAPACITY_CONSTRAINT(load,capacity) :=
  load<=capacity

 EXPECTED_VALUE(outcomes,probabilities,values) :=
  sum_i probabilities_i*values_i

 EXPECTED_UTILITY(outcomes,probabilities,utilities) :=
  sum_i probabilities_i*utilities_i

 INVENTORY_BALANCE(I_t,Q_t,D_t) :=
  I_{t+1}=I_t+Q_t-D_t

 FLOW_BALANCE(inflow,outflow,supply) :=
  inflow-outflow+supply=0
  under declared network convention

 UTILISATION(arrival_rate,service_rate) :=
  arrival_rate/service_rate
  for compatible queueing context

 LITTLE_LAW(L,lambda,W) :=
  L=lambda*W
  under stable steady-state assumptions

 OBJECTIVE_COMPARE(solution_a,solution_b) :=
  better_worse_or_tradeoff_under_declared criteria

 SENSITIVITY(parameter,solution) :=
  local_change_in_solution_or_objective_under_parameter perturbation
}

DEFINITIONS {
 D1 = "Operations-research models connect controllable decisions to measurable outcomes."
 D2 = "Objectives may represent cost, profit, time, throughput, risk, service or combinations thereof."
 D3 = "Constraints represent resource, physical, logical, temporal, policy or service restrictions."
 D4 = "Resources and capacities create competition among activities."
 D5 = "Uncertainty can be represented through distributions, scenarios or uncertainty sets."
 D6 = "Network, scheduling, inventory and queueing problems have distinct operational structures."
 D7 = "Simulation can complement optimisation when systems are analytically complex."
 D8 = "Sensitivity analysis tests how conclusions change when parameters change."
 D9 = "Model optimality and operational implementability are distinct."
 D10 = "Recommendations should preserve the assumptions and limitations of the underlying model."
}

AXIOMS {
 A1_DECISION_STRUCTURE:
  OperationsResearchProblem contains controllable decision variables

 A2_OBJECTIVE_STRUCTURE:
  OperationsResearchProblem contains one or more performance criteria

 A3_CONSTRAINT_STRUCTURE:
  Decisions are restricted by operational or mathematical constraints

 A4_RESOURCE_STRUCTURE:
  Resource availability influences feasible decisions

 A5_SYSTEM_INTERACTION:
  Decisions interact through system structure

 A6_UNCERTAINTY_AWARENESS:
  Material uncertainty should be explicitly modelled

 A7_VALIDATION:
  Mathematical model validity must be assessed against the represented system

 A8_IMPLEMENTATION:
  A recommended solution must be operationally feasible as well as mathematically valid
}

REASONING {

 R1_ENTITY_EXTRACTION {
  INPUT = operational_problem_description
  IDENTIFY = decisions
  IDENTIFY = objectives
  IDENTIFY = constraints
  IDENTIFY = resources
  IDENTIFY = uncertainty
  IDENTIFY = system_structure
  RETURN = structured_OR_entities
 }

 R2_MODEL_CLASSIFICATION {
  INPUT = structured_OR_entities
  CLASSIFY = mathematical_programming_network_scheduling_inventory_queueing_simulation_decision_analysis_or_hybrid
  RETURN = OR_model_class
 }

 R3_MATHEMATICAL_PROGRAMMING {
  INPUT = objective_variables_constraints
  CLASSIFY = linear_integer_mixed_integer_nonlinear_stochastic_robust_or_dynamic
  RETURN = mathematical_programming_model
 }

 R4_NETWORK_REASONING {
  INPUT = graph_structure
  IDENTIFY = path_flow_assignment_location_or_routing structure
  SELECT = network_model
  RETURN = network_solution strategy
 }

 R5_SCHEDULING_REASONING {
  INPUT = jobs_resources_processing_times_precedence_and_due dates
  FORM = scheduling_model
  RETURN = schedule_and_performance profile
 }

 R6_INVENTORY_REASONING {
  INPUT = demand_costs_lead_times_service level
  CLASSIFY = deterministic_or_stochastic
  FORM = replenishment_policy model
  RETURN = inventory_strategy
 }

 R7_QUEUEING_REASONING {
  INPUT = arrivals_services_servers_capacity_and_discipline
  CLASSIFY = queueing_model
  COMPUTE = utilisation_waiting_queue_length_and_service measures
  RETURN = queueing_performance
 }

 R8_SIMULATION_REASONING {
  INPUT = system_states_events_randomness_and_policies
  EXECUTE = replications
  ESTIMATE = distributions_and_confidence intervals
  RETURN = simulation_profile
 }

 R9_DECISION_ANALYSIS {
  INPUT = alternatives_outcomes_probabilities_preferences
  COMPUTE = expected_value_expected_utility_or_multiCriteria scores
  RETURN = preference_aware recommendation
 }

 R10_IMPLEMENTATION_REASONING {
  INPUT = mathematical_solution
  CHECK = operational_constraints
  CHECK = sensitivity_and_scenarios
  CHECK = robustness_and_risk
  RETURN = implementable_recommendation_status
 }
}

CONSTRAINTS {
 C1 = "Decision variables must represent genuinely controllable quantities."
 C2 = "Objective functions must reflect the intended operational goal."
 C3 = "Constraints must reflect real limitations rather than arbitrary mathematical convenience."
 C4 = "Data quality must be adequate for the model's intended use."
 C5 = "Uncertainty assumptions must be explicit."
 C6 = "Queueing formulas require their stated stability and distributional assumptions."
 C7 = "Simulation results require adequate replications and appropriate warm-up treatment where relevant."
 C8 = "Sensitivity ranges should not be extrapolated beyond their validity."
 C9 = "Mathematical optimality does not guarantee implementability."
 C10 = "High-stakes recommendations require explicit assumptions, limitations and robustness checks."
}

SEMANTIC_EQUIVALENCE {
 "OperationsResearchOntology" <=> "OperationalResearchOntology"
 "OperationsResearchProblem" <=> "OperationalDecisionModel"
 "MathematicalProgrammingModel" <=> "OptimisationBasedOperationalModel"
 "OperationalFeasibility" <=> "ImplementableSystemFeasibility"
 "DecisionAnalysis" <=> "FormalAlternativeEvaluation"
 "ScenarioAnalysis" <=> "AlternativeFutureConditionAnalysis"
}

VALIDATION {
 V1 = "Decision variables are identified."
 V2 = "Objectives are explicit."
 V3 = "Constraints are explicit."
 V4 = "Resources and capacities are represented."
 V5 = "Uncertainty assumptions are documented."
 V6 = "Selected model class matches the system structure."
 V7 = "Numerical or simulation outputs are checked."
 V8 = "Sensitivity and scenario analysis are performed when material."
 V9 = "Operational feasibility is assessed."
 V10 = "Recommendation status distinguishes mathematical optimum from implementation readiness."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 OPERATIONS_RESEARCH_ONTOLOGY_DEFINITION = 0.99
 ENTITY_EXTRACTION_REASONING = 0.99
 MODEL_CLASSIFICATION_REASONING = 0.99
 MATHEMATICAL_PROGRAMMING_REASONING = 0.99
 NETWORK_REASONING = 0.98
 SCHEDULING_REASONING = 0.98
 INVENTORY_REASONING = 0.98
 QUEUEING_REASONING = 0.98
 SIMULATION_REASONING = 0.98
 IMPLEMENTATION_REASONING = 0.98

 POLICY =
  "Reduce confidence when operational objectives, constraints, resources, uncertainty, data quality, model assumptions or implementation conditions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  OperationsResearchOntologyReasoning,
  OperationalProblemExtraction,
  ModelClassification,
  MathematicalProgrammingReasoning,
  NetworkOptimisationReasoning,
  SchedulingReasoning,
  InventoryReasoning,
  QueueingReasoning,
  SimulationReasoning,
  DecisionAnalysisReasoning,
  StochasticOptimisationReasoning,
  RobustOptimisationReasoning,
  SensitivityAnalysis,
  ScenarioAnalysis,
  OperationalFeasibilityAssessment,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "operations research ontology" -> OperationsResearchOntology
  "operational research ontology" -> OperationsResearchOntology
  "operations research problem" -> OperationsResearchProblem
  "decision variable" -> DecisionVariable
  "resource allocation" -> ResourceAllocation
  "mathematical programming" -> MathematicalProgrammingModel
  "network model" -> NetworkModel
  "queueing system" -> QueueingSystem
  "inventory problem" -> InventoryProblem
  "operational feasibility" -> OperationalFeasibility
 }

 INTERPRETATION_ORDER = [
  detect_operations_research_ontology_context,
  extract decisions_objectives_constraints_resources_uncertainty_system structure_and_performance measures,
  classify model_family,
  formulate mathematical_or_simulation representation,
  select compatible optimisation_network_scheduling_inventory_queueing_simulation_or_decision method,
  solve_or_evaluate model,
  perform sensitivity_scenario_risk_and_robustness checks,
  assess operational feasibility,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("operations research ontology")
  OR formal_operational_decision_semantics
  THEN activate(OperationsResearchOntology)

 E2:
  IF objective_and_constraints_form_optimisation model
  THEN route to mathematical_programming semantics

 E3:
  IF graph_structure_dominates
  THEN route to network_model semantics

 E4:
  IF jobs_resources_and_time dominate
  THEN route to scheduling semantics

 E5:
  IF stock_and_replenishment dominate
  THEN route to inventory semantics

 E6:
  IF arrivals_service_and_waiting dominate
  THEN route to queueing semantics

 E7:
  IF system_logic_is_complex_or_stochastic
  THEN consider simulation

 E8:
  IF uncertainty_material
  THEN use stochastic_robust_or_scenario reasoning

 E9:
  IF mathematical_solution_not_operationally_feasible
  THEN do_not report implementation-ready recommendation
}

RECURSION {
 ENTRY = OperationsResearchOntology

 EXPAND(OperationsResearchOntology) -> [
  OperationsResearchProblem,
  DecisionVariable,
  ObjectiveFunction,
  Constraint,
  Resource,
  Capacity,
  Demand,
  Cost,
  Revenue,
  ServiceLevel,
  Time,
  Risk,
  Uncertainty,
  Scenario,
  MathematicalProgrammingModel,
  LinearProgramming,
  IntegerProgramming,
  MixedIntegerProgramming,
  NonlinearProgramming,
  NetworkModel,
  ShortestPath,
  MaximumFlow,
  MinimumCostFlow,
  TransportationProblem,
  AssignmentProblem,
  FacilityLocation,
  VehicleRouting,
  SchedulingProblem,
  ProjectScheduling,
  InventoryProblem,
  QueueingSystem,
  SimulationModel,
  DiscreteEventSimulation,
  MonteCarloSimulation,
  MarkovDecisionProcess,
  DecisionAnalysis,
  UtilityFunction,
  MultiCriteriaDecisionAnalysis,
  StochasticOptimisation,
  RobustOptimisation,
  SensitivityAnalysis,
  ScenarioAnalysis,
  OperationalFeasibility,
  RecommendationStatus
 ]

 EXPAND(MathematicalProgrammingModels) -> [
  LinearProgramming,
  IntegerProgramming,
  MixedIntegerProgramming,
  NonlinearProgramming,
  DynamicProgramming,
  StochasticOptimisation,
  RobustOptimisation
 ]

 EXPAND(NetworkModels) -> [
  ShortestPath,
  MaximumFlow,
  MinimumCostFlow,
  TransportationProblem,
  AssignmentProblem,
  FacilityLocation,
  VehicleRouting
 ]

 EXPAND(OperationalPerformanceModels) -> [
  SchedulingProblem,
  InventoryProblem,
  QueueingSystem,
  ReliabilityModel,
  SimulationModel
 ]

 STOP_CONDITION =
  "Stop when the operational decision structure, selected model, objective, constraints, resources, uncertainty, solution, sensitivity, operational feasibility and recommendation status are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Operations Research"

 IMMEDIATE_CONTEXTS = [
  "Operations Research",
  "Applied Mathematics Ontologies",
  "Optimisation Ontologies",
  "Probability & Statistics Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Operations Research",
  "Applied Mathematics Ontologies",
  "Optimisation Ontologies",
  "Linear Programming Ontology",
  "Integer Programming Ontology",
  "Dynamic Optimisation Ontology",
  "Multi-Objective Optimisation Ontology",
  "Graph Theory Ontologies",
  "Probability & Statistics Ontologies",
  "Markov Process Ontologies",
  "Game Theory Ontologies",
  "Decision Theory Ontologies",
  "Simulation Ontologies",
  "Mathematical Modelling",
  "Scientific Computing Ontologies"
 ]

 ROUTE_RULES = [
  "Operations Research Ontology -> Operations Research for parent semantics",
  "Operations Research Ontology -> Optimisation for mathematical programming",
  "Operations Research Ontology -> Graph Theory for network models",
  "Operations Research Ontology -> Probability & Statistics for uncertainty",
  "Operations Research Ontology -> Markov Processes for stochastic state transitions",
  "Operations Research Ontology -> Game Theory for strategic interactions",
  "Operations Research Ontology -> Decision Theory for utility and preferences",
  "Operations Research Ontology -> Simulation for system experimentation",
  "Operations Research Ontology -> Mathematical Modelling for formal system representation",
  "Operations Research Ontology -> Scientific Computing for large-scale computational execution"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible OR entities, model families or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve operational decision semantics.",
  "Preserve objective-variable-constraint semantics.",
  "Preserve resource-and-capacity semantics.",
  "Preserve uncertainty representation.",
  "Preserve model-versus-system distinction.",
  "Preserve operational-feasibility semantics.",
  "Preserve sensitivity and scenario reasoning.",
  "Preserve machine-readable operations-research ontology semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "The Operations Research Ontology formally represents operational decision problems, mathematical models, objectives, constraints, resources, uncertainty and implementation status."

 PROBLEM_SUMMARY =
  "An operations-research problem combines controllable decisions, objectives, resources, constraints and system behaviour."

 PROGRAMMING_SUMMARY =
  "Mathematical programming represents operational decisions through optimisation variables, objectives and constraints."

 NETWORK_SUMMARY =
  "Network models represent flow, routing, assignment and location problems using graph structure."

 SCHEDULING_SUMMARY =
  "Scheduling models allocate jobs and resources over time."

 INVENTORY_SUMMARY =
  "Inventory models balance stock, replenishment, demand, cost and service."

 QUEUEING_SUMMARY =
  "Queueing models describe arrival, service, waiting and congestion processes."

 SIMULATION_SUMMARY =
  "Simulation estimates system performance through computational experiments."

 IMPLEMENTATION_SUMMARY =
  "Operations-research recommendations should be checked for sensitivity, uncertainty and operational feasibility before implementation."
}

FLOW {
 INPUT
 -> detect_operations_research_ontology_context
 -> identify decisions_objectives_constraints_resources_capacity_demand_uncertainty_system structure_and_performance measures
 -> classify mathematical_programming_network_scheduling_inventory_queueing_simulation_decision_analysis_or_hybrid model
 -> formulate mathematical_or_computational representation
 -> solve_optimise_or_simulate
 -> evaluate cost_service_time_risk_resource_use_and_feasibility
 -> perform sensitivity_scenario_and_robustness analysis
 -> assess operational_feasibility
 -> assign recommendation_status
 -> route to operations_research applied_mathematics optimisation graph_theory probability_statistics Markov_process game_theory decision_theory simulation mathematical_modelling_or_scientific_computing ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Operations Research Ontology provides the formal semantic layer beneath operations research. It represents decision variables, objectives, constraints, resources, capacities, demand, service levels, uncertainty and scenarios, mathematical programming, network models, scheduling, inventory, queueing, simulation, decision analysis, stochastic and robust optimisation, sensitivity and scenario analysis, operational feasibility and implementation-aware recommendation status."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-OPERATIONS-RESEARCH-ONTOLOGY-V1"
}

}

END_FLOW
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