Analytic Hierarchy Process (AHP)

The Analytic Hierarchy Process (AHP) is one of the most well-known methods of multi-criteria analysis, designed for selecting, ranking, and justifying decisions based on the subjective preferences of a decision-maker (DM). The method was developed by T. Saaty and has become widely used in problems that require consideration of multiple, often conflicting, criteria.

Essence of the Method

The Analytic Hierarchy Process is based on decomposing a complex decision problem into a hierarchical structure:

• Goal — the top of the hierarchy (what needs to be achieved).
• Criteria and sub-criteria — the aspects by which the evaluation is performed.
• Alternatives — the options from which a choice is made.

Each level of the structure is compared pairwise: the DM expresses which of two elements is more preferable with respect to an element at a higher level. These judgments form pairwise comparison matrices, from which priority weights are calculated.

Stages of Applying the Method

The Analytic Hierarchy Process is implemented in several logically sequential stages. Each stage is aimed at clarifying the DM's preferences and building a hierarchical model of the decision-making problem.

1. Building the hierarchy: decomposing the problem into a goal, criteria, sub-criteria, and alternatives.
2. Pairwise comparisons of elements: an expert assessment of preferences is performed (e.g., how much more important criterion A is than B on a scale from 1 to 9).
3. Calculating local weights: numerical weights reflecting the relative importance of elements are derived from the comparison matrices.
4. Consistency assessment: a consistency ratio is calculated to ensure the logicality and non-contradiction of the DM's preferences.
5. Aggregating weights: the final scores for the alternatives are calculated by synthesizing them across the entire hierarchy.

1. Building the Hierarchical Structure of the Problem

At this stage, the problem is decomposed into logically interconnected levels:

• Top level — the overall goal of the decision.
• Intermediate levels — criteria and, if necessary, subordinate sub-criteria.
• Bottom level — the alternative solutions.

The purpose of building the hierarchy is to reflect the logical structure of the DM's preferences and to prepare the foundation for pairwise comparisons.

2. Pairwise Comparison of Elements at Each Level

For each level (starting with the criteria relative to the goal and proceeding down the hierarchy), the DM conducts a pairwise comparison of elements regarding their influence on the element at the level above. The comparisons are performed using Saaty's scale (from 1 to 9), where:

• 1 — equal importance;
• 3 — moderate preference;
• 5 — strong preference;
• 7 — very strong preference;
• 9 — absolute preference;
• 2, 4, 6, 8 — intermediate values.

The results are recorded in a pairwise comparison matrix.

3. Calculating Local Priorities (Weights)

From each pairwise comparison matrix, a vector of local weights is calculated, reflecting the relative importance of the elements. This is typically done using the principal eigenvector method or an approximate normalized method (averaging across rows). For each element, the following are calculated:

• local weight, indicating its importance within the current matrix;
• global weight, obtained by multiplying its local weight by the weights of the parent elements.

4. Checking the Consistency of Judgments

Since the comparisons are subjective, inconsistent judgments are possible (e.g., A > B, B > C, but C > A). The AHP method includes a formal consistency check of the DM's judgments. A Consistency Index (CI) and a Consistency Ratio (CR) are calculated.

5. Aggregating Priorities and Ranking Alternatives

After calculating the weights at all levels of the hierarchy, a process of aggregation is performed up the hierarchy, starting from the alternatives. Each alternative receives an overall weight, calculated as the weighted sum of its local priorities under each criterion, multiplied by the weights of the criteria themselves. The result is a ranked list of alternatives by preference. The alternative with the highest overall priority is considered the most rational choice within the given problem structure.

Advantages of the Method

• Ease of structuring complex problems;
• Ability to handle both qualitative and quantitative criteria;
• Integration of subjective judgments into a logically rigorous procedure;
• Calculation of the degree of consistency in judgments (a reliability check);
• Applicability to both individual and group decisions.

Limitations of the Method

• Increasing complexity with a growing number of criteria and alternatives (the size of pairwise comparison matrices grows quadratically);
• Subjectivity in expressing preferences;
• Potential for distortions due to inconsistent judgments;
• Limitation to linear aggregation, which may not be suitable for complex non-linear preferences.