Constraints

A constraint is a condition, rule, or factor that limits the set of possible states, decisions, or actions of a system. Constraints can be physical, economic, logical, legal, social, or of another nature, and they play a key role in modeling, planning, management, and decision-making.

• in systems analysis — a constraint defines the boundaries for the behavior of a system or its components.
• in economics — these are resource limits (financial, time, labor, etc.).
• in law — these are frameworks for permissible behavior established by regulations.
• in operations research — these are mathematical expressions that define the feasible region of solutions.

Constraints in Operations Research Problems

Constraints are one of the key elements in operations research problems, determining the feasibility of solutions within a mathematical model. The optimality of a solution is determined with mandatory consideration of its compliance with the established constraints.

A constraint is a mathematical expression in the form of an equality or inequality that the model's variables must satisfy. Constraints narrow the set of feasible solutions. In some cases, given a system of constraints, an optimal solution may not exist.

Constraints are used to formally describe real-world conditions and can include:

• quotas and standards;
• carrying capacity of vehicles;
• volume of a planned task;
• weight or dimensional characteristics of equipment;
• limitations on available resources (material, time, financial, etc.).

Changing the configuration of constraints can change the optimal solution. In reality, constraints can be physical, economic, technological, or political in nature and may not always be strictly formalizable.

A specific solution is considered the best only within a given model and with a specific system of constraints. The more accurately the model reflects the real situation, the closer the found solution is to the true optimum.

Types of Mathematical Constraints

In mathematical modeling, especially in optimization and OR, constraints are usually represented as:

• Inequality Constraints: These set an upper or lower bound for some combination of variables. They indicate that a certain quantity (e.g., resource consumption) must not exceed a given limit or, conversely, must be no less than a certain threshold value.
• Equality Constraints: These require a specific combination of variables to be strictly equal to a given value. They are often used to describe balance relationships (e.g., the volume of production must exactly match the plan) or technological requirements.
• Sign Constraints: These specify the permissible range of values for the variables themselves. The most common is the requirement of non-negativity for variables, meaning their values cannot be less than zero. This reflects the physical or economic meaning of many quantities (e.g., production volumes, resource quantities, and time cannot be negative).

Role of Constraints in Defining the Feasible Region

The set of all constraints in a problem defines the Feasible Region — a subset of the variable space that contains all alternatives satisfying the given conditions. The search for an optimal solution is conducted entirely within the feasible region. If the feasible region is empty (i.e., there is no set of variables that satisfies all constraints simultaneously), then the problem has no solution.

Active and Inactive Constraints

At the point of an optimal solution:

• An Active Constraint is an inequality constraint that is satisfied as a strict equality (i.e., it holds with zero slack). It directly affects the optimal value of the objective function; relaxing it could improve the result.
• An Inactive Constraint is an inequality constraint that is satisfied with a strict inequality (i.e., with a margin or slack). Small changes to such a constraint will generally not affect the optimal solution.

Analyzing the activity of constraints is important for the sensitivity analysis of the model.

Literature

• Venttsel, E. S. Operations Research: Problems, Principles, Methodology. — Moscow: Nauka, 1988.
• Ackoff, R. L., & Sasieni, M. W. Fundamentals of Operations Research. — Moscow: Mir, 1971.
• Hillier, Frederick S.; Lieberman, Gerald J. Introduction to Operations Research. — McGraw-Hill Education. (11th ed., 2021)
• Volkova, V. N., & Denisov, A. A. Systems Theory and Systems Analysis: a textbook for universities. Moscow: Yurayt Publishing, 2025.

See also

• Operations Research
• Systems Analysis
• Optimization
• Mathematical Model
• Objective Function
• Feasible Region
• Linear Programming
• Decision Theory