Analytic Hierarchy Process (AHP)

The Analytic Hierarchy Process (AHP) is a structured technique developed by Thomas L. Saaty in the 1970s for organizing and analyzing complex decisions. It is widely used in decision-making processes that involve multiple criteria, especially when both qualitative and quantitative aspects must be considered. AHP helps decision makers to model a problem in the form of a hierarchy, allowing for rational and consistent pairwise comparisons between decision elements.

AHP structures a decision into a multi-level hierarchical tree. At the top of the hierarchy is the overall goal of the decision. This goal is then broken down into sub-goals or criteria, and each criterion may further be subdivided into sub-criteria. The bottom level of the hierarchy consists of the various alternatives to be evaluated. This hierarchical model enables users to understand how individual elements contribute to the overall objective.

One of the most powerful aspects of AHP is its use of pairwise comparisons. Each element in a level is compared to every other element in the same level with respect to their importance or preference towards an element in the level above. These comparisons are usually made using Saaty's 1–9 scale, which reflects the intensity of preference. The results of the comparisons are represented in a reciprocal matrix, from which the priority vector (relative weights) is derived, typically through an eigenvalue method or a geometric mean.

The consistency of judgments is an important feature of the AHP. A consistency ratio (CR) is calculated to indicate whether the pairwise comparisons are logically coherent. A CR below 0.1 is generally considered acceptable. If not, decision makers are advised to re-evaluate their comparisons to reduce inconsistency.

The final outcome of the AHP process is a set of priority weights for the alternatives, indicating the most preferable option based on the decision hierarchy. AHP can accommodate both individual and group decision-making and is applicable in various domains such as business strategy, resource allocation, policy analysis, project selection, and engineering design.

For more information, you can consult the Wikipedia article on AHP. A detailed formal description of the methodology is also provided in the foundational paper by Thomas L. Saaty: "Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process" (Operations Research, 2001) .

Tutorial

Create Your AHP

In the "Create your AHP" section, you can build a new Analytic Hierarchy Process (AHP) model using an interactive tree editor.

To add a node to your tree, click on the "+" symbol. You can rename any node by double-clicking on its label. Leaf nodes can be defined as one of three types:

• Saaty: a node where you will enter a Saaty pairwise comparison matrix.
• Metric_min: a node associated with a metric to be minimized.
• Metric_max: a node associated with a metric to be maximized.

Non-leaf nodes automatically represent Saaty matrices. When using metric-based leaves, intermediate Saaty matrices are generated during the resolution process.

To remove a node, select it and press the Delete key. You can also create a new alternative by double-clicking outside the current tree area.

Below the editor, you can choose the resolution method:

• Arithmetic: the simplest resolution method.
• Geometric: a more robust method using geometric mean.
• Eigenvalue: the most precise and complex method, based on eigenvector computation.

Use "Export Your Tree" to download the structure in JSON format. This export does not include matrix values and is not saved to the database. You can later import it into the solver tool.

Alternatively, use "Export Your Tree with Matrix" to export both the structure and default matrix values, which can be edited before solving.

Manage Your AHP

Paste your tree in JSON format into the text area below and click on "Import" to load it.

The panel on the right displays two visualizations based on the currently selected node of the AHP tree:

• Radar Chart: shows the priority of each alternative based on the sub-criteria of the selected node (only available for non-leaf nodes).
• History Chart: illustrates how each alternative is evaluated with respect to the selected node.

The currently selected node is indicated by the icon. You can select a different node by clicking on the icon next to it.

When a node is selected, the panel displays the priority values of its direct descendant nodes.

To modify the matrix values of a node, click the icon. Depending on the node type, a corresponding interface will expand:

• Metric matrix (leaf node only): a table will appear where you can directly edit the metric values.
• Saaty matrix: select the node, and a slider interface will be shown. Choose two sub-criteria (or alternatives if the node is a leaf). The most recently selected node will be highlighted in red, and the previously selected one in orange. You can then use the slider to adjust their relative importance, or manually enter a value. The red item is interpreted as being "value times" more important than the orange one. The reciprocal value is automatically applied in the opposite direction.

To close the matrix editing interface (which is required before editing another node), click on the icon.

Each alternative node displays its global priority value, computed with respect to the entire tree structure.