---
title: "Decision problem"
source: "https://systems-analysis.info/eng/Decision_problem"
wiki: "systems-analysis.info/eng"
article: "Decision_problem"
language: "en"
categories:
  - "Category:Decision theory"
  - "Category:Decision-making"
  - "Category:English"
  - "Category:Science"
revision_id: 117
wiki_created_at: 2026-09-06T22:17:36Z
wiki_modified_at: 2026-09-06T22:17:36Z
downloaded_at: 2026-09-07T22:21:08Z
---

# Decision problem

A **decision problem** is a formalized description of a problematic situation that requires selecting one or more preferred options (alternatives) from a set of possible ones, taking into account given goals, constraints, and evaluation criteria. Such problems are a central object of study in decision theory, systems analysis, management, economics, and other disciplines.

## Formal Problem Statement

A decision problem is generally described by a quintuple:

**D = (F, A, X, G, R)**

Where:

- **F** — the problem formulation, which includes a description of the problem, goals, and requirements for the outcome. It may include a model of the problem (e.g., a mathematical one).
- **A** — the set of alternatives from which the choice is made. Alternatives can be predefined, generated during the process, or emerge at later stages.
- **X** — the set of features (attributes) that characterize the alternatives. It includes objective measurable parameters and subjective assessments.
- **G** — the system of constraints that defines the feasible region of solutions. Constraints can be logical, quantitative, or expert-based.
- **R** — the preferences of the Decision-Maker (DM), used to evaluate alternatives and make the final decision. They can be expressed in the form of utility functions, ranks, criteria weights, etc.

## Elements of a Decision Problem

### Alternatives (A)

- Predefined;
- Generated during the solution process;
- Emerging after analysis.

### Features and Assessments (X)

- Quantitative (e.g., cost, speed);
- Qualitative (convenience, reliability);
- Composite (aggregated indicators).

### Constraints (G)

- Boundary conditions;
- Parameter compatibility;
- Logical dependencies;
- Constraints can simplify or complicate the solution.

### Preferences (R)

- Individual or group;
- Reflect the subjective importance of features;
- Can be partially included in constraints (G) to simplify analysis.

## Factors Influencing the Problem

Factors can be conditionally divided into:

- **Controllable** — goals, alternatives, subjective assessments;
- **Uncontrollable** — objective features, external environment.

By degree of certainty:

- **Deterministic** — parameters are known precisely;
- **Stochastic** — probabilistic characteristics are given;
- **Uncertain** — parameters are unknown or represented as ranges, expert assessments, or fuzzy sets.

## Classification of Decision Problems

### By Frequency of Occurrence

- Unique — new, non-standard situations;
- Recurring — typical, well-studied problems.

### By Solution Implementation Timeframe

- Short-term (operational);
- Medium-term (tactical);
- Long-term (strategic).

### By Type of Outcome

- Selecting the best alternatives;
- Ranking all options;
- Classifying options into groups.

### By Number of Alternatives

- Few (units, tens);
- Numerous (hundreds, thousands);
- Infinite (continuous solution spaces).

### By Dependence Between Alternatives

- Independent — they do not influence each other;
- Dependent — there are connections or constraints between them.

### By Number of Decision-Makers (DMs)

- Individual;
- Collective (independent DMs with different interests);
- Organizational (dependent DMs united by a common goal).

### By DM Involvement

- Without DM involvement (automated procedures);
- Limited involvement (at the final stage);
- Full involvement (at all stages).

### By Method of Preference Representation

- Holistic choice;
- Criteria-based choice.

### By Number and Type of Criteria

- Single-criterion;
- Multi-criteria:
  - Independent;
  - Dependent.

### By Information

**By type:**

- Quantitative;
- Qualitative;
- Mixed.

**By source:**

- Objective (measurements, calculations);
- Subjective (opinions of DMs, experts).

**By dynamics:**

- Static (does not change);
- Dynamic (is updated).

**By degree of certainty:**

- Deterministic;
- Probabilistic (risk);
- Uncertain (complete uncertainty).

## Importance of Problem Formalization

Formalizing a decision problem allows for:

- clearly defining the goal and success criteria;
- identifying the set of feasible solutions;
- selecting a solution method appropriate for the problem's structure;
- increasing the validity and reliability of the decisions made.

## Example of a Problem Formulation

Assessing a patient's condition can be framed as a problem of choosing a diagnosis based on features such as body temperature, blood pressure, and the presence of pain. Constraints define the permissible combinations of symptoms, and the physician's preferences can influence the choice of a diagnostic hypothesis.

## See also

- [Decision-making](https://systems-analysis.info/eng/Decision-making "Decision-making")
- [Optimal decision](https://systems-analysis.info/eng/Optimal_decision "Optimal decision")
- [Rational choice](https://systems-analysis.info/eng/Rational_choice_(decision_theory) "Rational choice (decision theory)")
- [Collective choice](https://systems-analysis.info/eng/Collective_choice "Collective choice")
- [Systems analysis](https://systems-analysis.info/eng/Systems_analysis "Systems analysis")