--- title: "Operations research models" source: "https://systems-analysis.info/eng/Operations_research_models" wiki: "systems-analysis.info/eng" article: "Operations_research_models" language: "en" categories: - "Category:English" - "Category:Modeling" - "Category:Operations research" - "Category:Optimization" - "Category:Science" revision_id: 285 wiki_created_at: 2026-09-06T22:20:02Z wiki_modified_at: 2026-09-06T22:20:02Z downloaded_at: 2026-09-07T22:22:27Z --- # Operations research models **Operations Research Models** [Operations research (OR)](https://systems-analysis.info/eng/Operations_research "Operations research") uses [models](https://systems-analysis.info/eng/Model "Model") to analyze and solve problems in [management](https://systems-analysis.info/eng/Management "Management") and [decision-making](https://systems-analysis.info/eng/Decision-making "Decision-making"). An OR [model](https://systems-analysis.info/eng/Model "Model") is a simplified, formalized representation of a real operation or [system](https://systems-analysis.info/eng/System "System"), designed to study its [behavior](https://systems-analysis.info/eng/System_dynamics "System dynamics") and find the best solutions. Applying quantitative [research](https://systems-analysis.info/eng/Operations_research "Operations research") methods requires constructing a [mathematical model](https://systems-analysis.info/eng/Mathematical_model "Mathematical model") of the operation. When building a [model](https://systems-analysis.info/eng/Model "Model"), the operation is typically simplified and schematized, and this schema is described using a particular mathematical framework. A [model](https://systems-analysis.info/eng/Model "Model") of an operation is a sufficiently accurate description of the operation using a mathematical apparatus (various types of functions, equations, systems of equations, and inequalities). The effectiveness of an operation is defined as the degree of its suitability for accomplishing its task. ## General Problem Formulation and Model Structure in OR In [operations research](https://systems-analysis.info/eng/Operations_research "Operations research"), a [managerial situation](https://systems-analysis.info/eng/Management "Management") involves [goals](https://systems-analysis.info/eng/Goal "Goal") and decisions. Decisions are made to achieve these [goals](https://systems-analysis.info/eng/Goal "Goal"). The [managerial situation](https://systems-analysis.info/eng/Management "Management") is described by a [model](https://systems-analysis.info/eng/Model "Model"). A [model](https://systems-analysis.info/eng/Model "Model") contains an explicit performance measure, which determines how close a solution is to the [goal](https://systems-analysis.info/eng/Goal "Goal"). This measure depends on factors that influence the operation. All factors included in the description of an operation can be divided into two groups: - Uncontrollable (fixed) factors: External conditions or [system](https://systems-analysis.info/eng/System "System") parameters that the decision-maker cannot influence (e.g., market demand, raw material prices, weather). - Controllable (decision) factors: Parameters of the operation whose values the decision-maker can choose and change (e.g., production volume, delivery route, resource allocation). These factors are also called decision variables. Conceptually, an OR model can be represented as a "[black box](https://systems-analysis.info/eng/Black_box "Black box")", where the main focus is on defining the inputs and outputs: - Inputs: Controllable and uncontrollable variables (factors). - Model: The mathematical apparatus (functions, equations, inequalities) that describes the relationships between inputs and outputs. - Output: The performance [criterion](https://systems-analysis.info/eng/Criterion "Criterion") ([objective function](https://systems-analysis.info/eng/Objective_function "Objective function")). The performance [criterion](https://systems-analysis.info/eng/Criterion "Criterion"), expressed by some function, is called the [objective function](https://systems-analysis.info/eng/Objective_function "Objective function"). The [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") is a mathematically formulated (formalized) performance measure that needs to be maximized or minimized. ### Mathematical Definition of a Model In [OR](https://systems-analysis.info/eng/Operations_research "Operations research"), a [mathematical model](https://systems-analysis.info/eng/Mathematical_model "Mathematical model") is understood as any operator that allows for determining the output parameter values of a modeled object based on the corresponding values of its input parameters, within the set of permissible values for the input and output parameters of the modeled object. ## Typical Mathematical Model in OR Most [OR](https://systems-analysis.info/eng/Operations_research "Operations research") problems are reduced to optimization and are formulated as the following [mathematical model](https://systems-analysis.info/eng/Mathematical_model "Mathematical model"): **Maximize** (or **minimize**) the **objective function** **subject to constraints.** - [Objective Function](https://systems-analysis.info/eng/Objective_function "Objective function"): Quantitatively expresses the criterion by which a solution is evaluated (e.g., profit, cost, time). The [choice of the objective function](https://systems-analysis.info/eng/Objective_function "Objective function") is a central and crucial aspect of the [research](https://systems-analysis.info/eng/Operations_research "Operations research"). It is better to find a non-optimal solution with a correctly chosen [criterion](https://systems-analysis.info/eng/Criterion "Criterion") than an optimal solution with an incorrect one. - Constraints: Mathematical expressions (in the form of equalities or inequalities) that the [model's](https://systems-analysis.info/eng/Model "Model") variables must satisfy. They reflect real-world limits on resources, technological requirements, planned targets, and other conditions. Constraints narrow the set of possible solutions. ## Solutions in OR Models - Feasible solution: Any set of variable values that satisfies **all** the constraints of the [model](https://systems-analysis.info/eng/Model "Model"). The set of all feasible solutions forms the [feasible region](https://systems-analysis.info/eng/Feasible_region "Feasible region"). There can be an infinite number of such solutions. - Optimal solution: A feasible solution at which the [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") reaches its extreme (maximum or minimum) value. The optimal solution (if one exists) is always located within the [feasible region](https://systems-analysis.info/eng/Feasible_region "Feasible region"). - In some cases, an optimal solution may not exist (e.g., if the [feasible region](https://systems-analysis.info/eng/Feasible_region "Feasible region") is empty or the [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") is unbounded on the [feasible region](https://systems-analysis.info/eng/Feasible_region "Feasible region")). - An optimal solution is one that is preferable to others according to a given optimization criterion. - Optimality is always relative to a [criterion](https://systems-analysis.info/eng/Criterion "Criterion") ("optimal with respect to..."). ## Classification of Operations Research Models [Models](https://systems-analysis.info/eng/Model "Model") in [OR](https://systems-analysis.info/eng/Operations_research "Operations research") can be classified according to various characteristics, particularly by the mathematical apparatus used and the type of problem: ### By Mathematical Structure - Linear programming (LP) models: The [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") and all constraints are linear functions of the variables. - Integer LP models: Some or all variables must take integer values. - Nonlinear programming (NLP) models: The [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") and/or constraints are nonlinear functions. - Convex programming models: A special case of NLP where a convex [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") is minimized (or a concave one is maximized) over a convex [feasible region](https://systems-analysis.info/eng/Feasible_region "Feasible region"). - Dynamic programming (DP) models: Used for problems where decisions are made sequentially over time, and the optimality criterion is expressed through recurrence relations. - Heuristic models: Applied when finding the exact optimum is infeasible due to high computational complexity. Heuristic methods are used to find a "good enough" solution. ### By Typical OR Problems - Network planning and management problems: Optimization of the time and cost for completing complex projects (e.g., critical path method). - Queueing problems (Queueing theory): Analysis and optimization of systems with queues (determining the number of service channels, service time). - Inventory control problems: Determining optimal inventory levels and order sizes to minimize costs while satisfying demand. - Resource allocation problems: Optimal assignment of limited resources among competing operations or [activities](https://systems-analysis.info/eng/Activity "Activity"). - Equipment repair and replacement problems: Determining optimal times for repairing or replacing equipment, taking into account wear and aging. - Game theory models: Analysis of conflict situations with multiple parties pursuing different [goals](https://systems-analysis.info/eng/Goal "Goal"), and the search for optimal strategies. ## External links - Operations research — Wikipedia ## See also - [Operations research](https://systems-analysis.info/eng/Operations_research "Operations research") - [Mathematical model](https://systems-analysis.info/eng/Mathematical_model "Mathematical model") - [Optimization](https://systems-analysis.info/eng/Optimization "Optimization") - [Objective function](https://systems-analysis.info/eng/Objective_function "Objective function") - [Constraints](https://systems-analysis.info/eng/Constraints "Constraints") - [Optimal solution](https://systems-analysis.info/eng/Optimal_solution_(optimization) "Optimal solution (optimization)") - [Linear programming](https://systems-analysis.info/eng/Linear_programming "Linear programming") - [Dynamic programming](https://systems-analysis.info/eng/Dynamic_programming "Dynamic programming") - [Network model (operations research)](https://systems-analysis.info/eng/Network_model_(operations_research) "Network model (operations research)") - [Decision theory](https://systems-analysis.info/eng/Decision_theory "Decision theory") - [Simulation modeling](https://systems-analysis.info/eng/Simulation_modeling "Simulation modeling") - [Systems analysis](https://systems-analysis.info/eng/Systems_analysis "Systems analysis") ## Literature - *Venttsel, E. S.* Operations Research: Problems, Principles, Methodology. — Moscow: Nauka, 1988. - *Ackoff, Russell L., and Sasieni, Maurice W.* Fundamentals of Operations Research. — New York: John Wiley & Sons, 1968. - *Taha, Hamdy A.* Operations Research: An Introduction. — Pearson. (10th ed., 2017)