Optimization

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Optimal means the best under given conditions. Quality is evaluated using an optimality criterion, and the conditions are specified as constraints on additional criteria.

The desire to increase the efficiency of labor, creativity, and any purposeful activity is a natural human aspiration that has found its clear and understandable expression in the idea of optimality. The difference between a strictly scientific and a common, everyday understanding of optimality is quite small. Although expressions like "most optimal" or "achieve maximum effect with minimum cost" are mathematically incorrect, the people who use them are simply expressing a correct idea imprecisely. As soon as a concrete optimization problem arises, they quickly and easily correct their phrasing.

In mathematics, computer science, and operations research, optimization is the problem of finding the extremum (minimum or maximum) of an objective function in a certain region of a finite-dimensional vector space, constrained by a set of linear and/or nonlinear equalities and/or inequalities.

Optimization Models

An optimization model is a decision-making model that includes a performance indicator (objective function) to be optimized, subject to a set of specified constraints.

Optimization models are designed to determine the optimal (best) parameters of a modeled object from the perspective of a certain criterion, or to find the optimal (best) control policy for a process. Some model parameters are classified as control parameters; by varying them, different sets of output parameter values can be obtained. Typically, these models are built using one or more descriptive models and include a criterion that allows for the comparison of different sets of output parameter values to select the best one. Constraints in the form of equalities and inequalities, related to the specifics of the object or process under consideration, may be imposed on the range of input parameter values. The goal of optimization models is to find feasible control parameters for which the selection criterion reaches its "best value."

Optimization Models in Operations Research

The problem is formulated as a mathematical model. A typical mathematical model in operations research is stated as follows:

Maximization or minimization of an objective function, subject to constraints

Optimal solutions are those that are preferable to others based on one or more criteria. Every choice of the best option is specific, as it is based on compliance with established criteria. When speaking of an optimal option, these criteria are specified ("optimal with respect to..."). What is optimal under one criterion is not necessarily so under another.

Feasible solution — a solution that satisfies all the model's constraints. In some cases, there may be an infinite number of feasible solutions.

Optimal solution — a solution that is not only feasible but also one where the objective function reaches its maximum or minimum value.

Optimization — the maximization or minimization of an objective function.

An optimal solution — a feasible set of values for the decision variables that optimizes the objective function of an optimization model.

Optimal Choice Model

A large number of practical choice problems boil down to finding the best or most preferable options for an individual, and often to finding a single best option. Each decision-maker (DM) has their own subjective ideas about what is preferable in a specific choice situation.

There are many problems for which a mathematical model of choice can be constructed, where the concept of the best option is formalized by specifying one or more numerical performance indicators or solution quality criteria. These indicators, although set by the DM, are objective in nature, determined by the content of the problem being solved, and are expressed by some functions that depend on variables measuring the properties of the options. In such cases, the most preferable solution for the DM is considered to be the so-called optimal option, which corresponds to the extreme value of one or more performance indicators under the existing conditions.

A fundamental aspect of formulating an optimal choice problem is the ability to describe the problem situation and the DM's preferences in quantitative terms. This means, firstly, that possible solutions (alternatives, objects, courses of action) are defined by quantitative attributes (variables, parameters) measured on numerical scales. Secondly, quantitative indicators (optimality criteria, performance indicators, objective functions, value functions) must be specified, by which the quality of the chosen option is evaluated. Such situations are characteristic of well-structured problems and recurring choice situations, typical for operations research and optimal control.

To analyze possible solutions to a problem (ways to achieve a goal) and to select one or more of the best options from them, formal models of optimal choice are constructed. The model provides a simplified representation of the real problem and should reflect the most important and objectively existing dependencies and relationships between the options, their descriptive attributes, and the constraints set by controllable and uncontrollable factors. Building such a model is a task for analyst-consultants and experts, with the participation of the DM. When constructing a choice model, one must balance the adequacy and detail of the model with the required precision of the solution to the real choice problem, as well as with the amount of information needed to find the solution—both already available and to be obtained additionally.

Limitations of the Optimization Approach

Optimization problems are strictly formal mathematical problems. The practical value of the solutions to such problems directly depends on how good the initial mathematical model is. In complex systems, mathematical modeling is difficult, approximate, and imprecise. The more complex the system, the more cautiously one should approach its optimization.

From a systems analysis perspective, the attitude towards optimization can be formulated as follows: it is a powerful tool for increasing efficiency, but it should be used with increasing caution as the complexity of the problem grows.

Despite the obvious utility of the idea of optimization, practice demands a cautious approach to it. There are substantial reasons for this conclusion.

  1. The optimal solution is often unstable: seemingly insignificant changes in the problem's conditions can lead to the selection of substantially different alternatives.
  2. The system under consideration is part of a larger system, and local optimization will not necessarily lead to the same result that would be required from the subsystem when optimizing the system as a whole. This necessitates aligning the criteria of subsystems with the criteria of the overall system, often rendering local optimization unnecessary.
  3. Criteria characterize the goal only indirectly—sometimes better, sometimes worse, but always approximately. Maximizing the optimality criterion is often equated with the goal, but they are actually different things. In fact, the criterion and the goal relate to each other as a model and its original, with all the ensuing implications. Many goals are difficult or even impossible to describe quantitatively.
  4. Without specifying all the necessary constraints, we may, while maximizing the main criterion, obtain unforeseen and undesirable side effects.

See also

Literature

  • Venttsel, E. S. Operations Research: Problems, Principles, Methodology. — Moscow: Nauka, 1988. (Or a later edition)
  • Taha, Hamdy A. Operations Research: An Introduction. — Pearson. (Specify edition, e.g., 10th ed., 2017)
  • Hillier, Frederick S.; Lieberman, Gerald J. Introduction to Operations Research. — McGraw-Hill Education.
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