Operations research models

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Operations Research Models

Operations research (OR) uses models to analyze and solve problems in management and decision-making. An OR model is a simplified, formalized representation of a real operation or system, designed to study its behavior and find the best solutions.

Applying quantitative research methods requires constructing a mathematical model of the operation. When building a model, the operation is typically simplified and schematized, and this schema is described using a particular mathematical framework. A 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, a managerial situation involves goals and decisions. Decisions are made to achieve these goals. The managerial situation is described by a model.

A model contains an explicit performance measure, which determines how close a solution is to the 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 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", 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 (objective function).

The performance criterion, expressed by some function, is called the objective function. The objective function is a mathematically formulated (formalized) performance measure that needs to be maximized or minimized.

Mathematical Definition of a Model

In OR, a 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 problems are reduced to optimization and are formulated as the following mathematical model:

Maximize (or minimize) the objective function subject to constraints.

  • Objective Function: Quantitatively expresses the criterion by which a solution is evaluated (e.g., profit, cost, time). The choice of the objective function is a central and crucial aspect of the research. It is better to find a non-optimal solution with a correctly chosen criterion than an optimal solution with an incorrect one.
  • Constraints: Mathematical expressions (in the form of equalities or inequalities) that the model's 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. The set of all feasible solutions forms the feasible region. There can be an infinite number of such solutions.
  • Optimal solution: A feasible solution at which the objective function reaches its extreme (maximum or minimum) value. The optimal solution (if one exists) is always located within the feasible region.
    • In some cases, an optimal solution may not exist (e.g., if the feasible region is empty or the objective function is unbounded on the 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 ("optimal with respect to...").

Classification of Operations Research Models

Models in OR 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 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 and/or constraints are nonlinear functions.
  • Convex programming models: A special case of NLP where a convex objective function is minimized (or a concave one is maximized) over a convex 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.
  • 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, and the search for optimal strategies.

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)

See also

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