Optimization criterion
An optimization criterion is a feature, rule, or quantitative indicator used to evaluate and compare different alternatives (solution variants, states of a system, strategies) with the goal of selecting the best (optimal) one in tasks of optimization, operations research, and decision theory.
An optimization criterion formalizes the concept of "the best" in the context of a specific task and the goals of the decision-maker (DM).
Essence and Purpose
The purpose of an optimization criterion is to:
- Establish a measure of preference: It allows for a quantitative or qualitative determination of how much better one option is than another.
- Ensure comparability: It provides a common basis for comparing diverse alternatives.
- Guide the search for a solution: It indicates the direction of optimization—what exactly needs to be maximized or minimized.
- Formalize a goal: It translates an often qualitative task goal (e.g., "to increase efficiency") into a specific, measurable indicator.
Without a clearly defined optimization criterion, it is impossible to objectively select an optimal solution from a set of feasible ones.
Relationship with the Objective Function
In mathematical modeling and optimization, the optimization criterion is formalized as an objective function.
- An optimization criterion is a conceptual notion, a rule for selection (e.g., "minimize costs," "maximize profit").
- An Objective function is a mathematical expression (a formula) that quantitatively represents this criterion and depends on the problem's decision variables.
The optimization of the objective function (finding its extremum) is equivalent to finding the solution that is best according to the given optimization criterion.
Types of Optimization Criteria
The primary classification of optimization criteria is based on the direction of optimization:
- Maximization criteria: The goal is to find a solution where the value of the indicator is at its maximum (e.g., profit, productivity, reliability, utility).
- Minimization criteria: The goal is to find a solution where the value of the indicator is at its minimum (e.g., costs, time, risk, losses, deviation from a norm).
Criteria are also distinguished as:
- Single-criterion problems: Only one optimization criterion is used.
- Multi-criteria problems: Several, often conflicting, criteria are considered simultaneously. In this case, the search is for compromise or Pareto-optimal solutions.
Choosing an Optimization Criterion
Choosing an adequate optimization criterion is a critically important step in problem formulation. An incorrectly chosen criterion can lead to an optimal solution for the model, but an ineffective or even harmful solution for the real system or problem situation.
The choice of a criterion depends on:
- The goals of the problem and the decision-maker.
- The specifics of the system or process.
- The availability of data to calculate the indicator.
- The planning time horizon.
Often, the choice of a criterion is subjective and requires careful justification.
Role in Optimization and OR Problems
In operations research, the optimization criterion (in the form of an objective function), along with constraints, forms the basis of the mathematical model of the problem. Optimization algorithms use the objective function to evaluate and compare feasible solutions and to search for the optimal one.
Literature
- Venttsel E. S. Operations Research: Problems, Principles, Methodology. — Moscow: Nauka, 1988.
- Ackoff R. L., Sasieni M. W. Fundamentals of Operations Research. — New York: Wiley, 1968.
- Peregudov F. I., Tarasenko F. P. Introduction to Systems Analysis. — Moscow: Vysshaya Shkola, 1989.