Objective function

An objective function is a mathematically formalized criterion of efficiency or preference of a decision-maker (DM), whose value is optimized (maximized or minimized) in the process of solving a choice or control problem. Typically, the value of the objective function depends on a set of decision variables, which can be adjusted in the search for an optimal solution. The objective function provides a quantitative expression of goals or preferences, which serves as the basis for making a decision choice. It plays a key role in optimization problems, systems analysis, and decision theory.

Purpose and Role in Modeling

The objective function serves to:

Formalize the goal: Translating qualitative goals (such as control, design, or planning) into a quantitatively measurable form.
Compare alternatives: Providing a single criterion for the objective comparison of different alternatives or strategies.
Optimize: Finding the best solution by identifying the extremum (maximum or minimum) of the function within the feasible region, which is defined by constraints.

Structure and Types

Mathematically, the objective function depends on a vector of decision variables. By their form, objective functions can be:

Linear: Often used in linear programming problems.
Nonlinear: Contain nonlinear dependencies (e.g., quadratic, power, exponential). Used in nonlinear programming.
Smooth or discontinuous.
Unimodal (having one extremum) or multimodal.

Problems are also distinguished as:

Single-criterion: A single objective function is optimized.
Multi-criterion: A vector of several, often conflicting, objective functions is optimized.

Examples of Objective Functions

Typical examples of objective functions include:

for maximization: profit, productivity, efficiency, utility, output volume;
for minimization: costs, expenses, execution time, losses, risk, sum of deviations from a plan.

Importance of Correct Selection

The selection of an objective function is a central step in formulating an research problem. As noted in applied literature, a correctly chosen criterion with a less precise solution is preferable to an optimal solution with a flawed criterion. An inadequate objective function will lead to a solution that does not align with the real goals of the DM or the system.

Relationship Between Minimization and Maximization

In many practical problems, minimizing one objective function can logically lead to maximizing another. For example, reducing costs or processing time in a production process can directly contribute to an increase in profit or productivity.

This principle is clearly demonstrated within the concept of lean manufacturing, where the main focus is on eliminating waste as a way to increase efficiency.

Waste in Lean Manufacturing

Overproduction;
Waiting (downtime);
Excessive transportation;
Unnecessary processing steps;
Excess inventory;
Unnecessary movement;
Defective products.

Eliminating or minimizing these types of waste forms the basis for problems in operations research and systemic management. On this basis, the following are developed:

Inventory control models;
Queueing models;
Resource allocation models.