Operations research

Operations Research (OR) is an interdisciplinary scientific field concerned with the development and application of quantitative optimization methods based on mathematical modeling and various heuristic approaches. It serves as a tool for the preliminary quantitative justification of management decisions in complex systems of various natures: technical, economic, and organizational.

Essence and Goals

Initially, operations research was defined as a scientific method that provides managers with a quantitative basis for making decisions related to the activities of subordinate organizations. The applied nature of the discipline was emphasized, focusing on using the achievements of other sciences to analyze specific problems in improving management.

An "operation" in the context of this discipline is understood as a managed set of actions, united by a common concept and aimed at achieving a goal. The term originates from military management, where it meant a purposeful activity carried out according to a specific plan.

Operations research methods are applied in cases where it is necessary to organize a purposeful activity that can be implemented in various ways. In such situations, one must choose from among possible solutions, each of which has its own advantages and disadvantages. The goal of operations research is the preliminary quantitative justification of optimal solutions based on performance indicators. The decision-making itself is outside the scope of the discipline and falls under the competence of the decision-maker.

History and Development

Operations research as a scientific field emerged during World War II. Its formation is associated with the work of groups of scientists brought in to solve military planning problems. OR methods were used in organizing combat missions, planning naval operations, and allocating resources.

After the war, the methods began to be adapted for tasks in the civilian sector: industry, logistics, inventory management, and production reorganization. Classic works were written in the 1950s–1970s (G. Dantzig, R. Ackoff, C. Churchman, M. Arnoff).

In the USSR, operations research methods were developed primarily under the names "mathematical modeling," "mathematical programming," and "mathematical optimization methods." Key figures include L. V. Kantorovich (creator of linear programming, Nobel laureate in 1975), V. G. Gnedenko, E. S. Ventsel, and N. P. Bruslenko. Since the late 20th century, the term "production analytics" has also been used.

Methodology

Methodology in operations research includes the following stages:

1. Formalization of the initial problem;
2. Construction of a model (mathematical, simulation, etc.);
3. Solving the model (analytically or numerically);
4. Checking the adequacy of the model;
5. Implementation of the solution and sensitivity analysis.
6. A key feature of the approach is combining the manager's intuition with the results of modeling. A model is not a complete copy of reality but a tool that allows for more informed decision-making.

Objective Function and Effectiveness Criteria

Effectiveness is defined as the productivity of resource use in achieving a goal. To compare options, a quantitative criterion is introduced—the objective function. This is a formalized performance indicator that needs to be maximized (e.g., profit, productivity) or minimized (e.g., expenses, costs, time).

When multiple criteria are present, the task becomes one of multi-criteria optimization. Effective solutions in such cases are determined by the Pareto principle—as solutions that are not inferior to others across all criteria simultaneously.

Formalizability of Problems

Operations research methods are most effective for solving well-structured (formalizable) problems that allow for quantitative formulation and the construction of mathematical models. These models include variables, constraints, and an objective function. A solution is considered feasible if it satisfies all constraints; it is optimal if it also extremizes the objective function.

Mathematical Models of Operations

A mathematical model is the foundation for applying quantitative methods in operations research. It is a formalized description of a managed activity (operation) that highlights key parameters, dependencies, and goals. A model always simplifies and schematizes reality, and its accuracy is determined by the balance between the model's complexity and the available information.

Key principles of model building:

• The model should reflect the most important features of the phenomenon and account for the most significant factors.
• The model should not be overloaded with secondary details that complicate the analysis.
• There is no universal method of modeling—each model is selected individually, considering the goals, level of uncertainty, and data availability.
• It is recommended to use several models for the same phenomenon and compare the results (so-called "model comparison").

Mathematical Programming

Mathematical programming is the core of applied methods in operations research.

A problem is formulated in terms of:

• a feasible region;
• an objective function;
• constraints.

Types include linear, nonlinear, integer, and multi-criteria programming.

• Linear programming — a branch of mathematical programming where the objective function and constraints are linear. It is used for optimization with limited resources.
• Nonlinear programming — an optimization problem in which the objective function or at least one of the constraints is nonlinear. It is applied to model complex dependencies.
• Integer programming — a type of optimization problem in which some or all of the variables must be integers. It is relevant for solving problems of a combinatorial nature.
• Multi-criteria programming — an area of optimization that considers multiple objective functions simultaneously. Solutions are chosen based on trade-offs between criteria.

Typical Problems in Operations Research

The most common classes of problems include:

• Resource Allocation Problems — Optimal allocation of limited resources among competing activities, subject to given constraints. Example: creating a production plan with limited raw materials and equipment.
• Transportation Problems — Determining the optimal transportation plan that minimizes total costs when moving products from supply points to demand points.
• Assignment Problems — Assigning agents to tasks (or equipment to operations) to minimize total costs or maximize overall effectiveness. A special case of the transportation problem.
• Queueing Problems — Modeling systems with queues (e.g., banks, warehouses, telecommunication centers) to analyze waiting times, resource utilization, and optimize the number of service channels.
• Inventory Management Problems — Determining strategies for replenishing and storing inventory to meet demand at minimum cost.
• Equipment Replacement Problems — Choosing the optimal time to replace aging or deteriorating equipment to minimize costs of repair, operation, and acquisition.
• Network Problems — Determining the critical path in project graphs, optimizing flows in networks (e.g., transportation or information), and minimizing project completion time.
• Cutting Stock and Packing Problems — Optimizing the placement of items (e.g., shapes on a sheet of material) to minimize waste.
• Game Theory Problems — Modeling conflict situations involving two or more parties with conflicting interests, and analyzing strategies in terms of payoffs and risks.
• Multi-criteria Optimization Problems — Finding solutions that are optimal with respect to several, often conflicting, criteria (e.g., quality vs. cost vs. delivery time).
• Simulation Modeling — Modeling complex systems whose behavior cannot be described accurately by analytical methods (e.g., logistics of large hubs or manufacturing systems with high uncertainty).

Each type of problem can be represented as a mathematical model containing variables, constraints, and an objective function.

Methods

• Probability theory and statistics
• Graph theory
• Game theory
• Simulation modeling
• Queueing models
• Inventory and replacement models
• Network models and critical path

Limitations of the Method

• Excessive sensitivity to initial data;
• Local optimization does not guarantee systemic optimality;
• Inadequacy of the criterion to the true goal;
• The possibility of undesirable effects arising from an incomplete consideration of constraints.

Applications

Operations research is applied in:

• logistics and inventory management;
• production planning;
• construction and capital planning;
• economics, defense, and energy;
• public and corporate governance.

Links

• Article on Operations research on Wikipedia (EN)

Literature

• Kantorovich L. V. (1939). Mathematical Methods of Organizing and Planning Production.
• Ventsel E. S. (1972). Operations Research.
• Ventsel E. S. (2004). Operations Research: Problems, Principles, Methodology. 3rd ed.
• Hillier F. S.; Lieberman G. J. (2005). Introduction to Operations Research. (trans. from English, 7th Russian ed.).
• Dantzig G. (1966). Linear Programming, Its Applications and Extensions. Trans. from English.
• Dantzig, G. B. (1963). Linear Programming and Extensions. RAND PDF.
• Kantorovich, L. V. (1960). Mathematical Methods in the Organization and Planning of Production. PDF.
• Churchman, C. W.; Ackoff, R. L.; Arnoff, E. L. (1957). Introduction to Operations Research. Archive.org.
• Hillier, F. S.; Lieberman, G. J. (2014, 10th ed.). Introduction to Operations Research. PDF.
• Winston, W. L. (2004, 4th ed.). Operations Research: Applications and Algorithms. PDFroom.
• Ford, L. R.; Fulkerson, D. R. (1956). Maximal Flow Through a Network. PDF.
• Nemhauser, G. L.; Wolsey, L. A. (1988). Integer and Combinatorial Optimization. Wiley.
• Bellman, R. (1957). Dynamic Programming. PDF.