Mastering the Analytic Hierarchy Process: A Step‑by‑Step Guide

1 Analytic Hierarchy Process

The Analytic Hierarchy Process (AHP), proposed by Professor Thomas L. Saaty in the 1980s, is a practical multi‑criteria decision‑making method that combines qualitative and quantitative factors by structuring the decision problem into a hierarchy and quantifying the relative importance of each element.

Since its introduction to China in 1982, AHP has been widely applied in energy system analysis, urban planning, economic management, scientific research evaluation, and many other socio‑economic fields due to its ability to integrate qualitative and quantitative information and its systematic, flexible, and simple nature.

Basic idea of AHP: First decompose the problem into hierarchical levels, then synthesize the results. The problem is broken down from the overall goal to criteria, sub‑criteria, and finally alternatives, forming a multi‑level analysis structure that yields relative weights or rankings for the lowest‑level elements.

The modeling process generally follows four steps:

Establish a recursive hierarchical structure model.

Construct judgment matrices for each level.

Perform local ranking and consistency verification.

Obtain global ranking and conduct overall consistency checks.

2 Steps

Step 1: Build the hierarchy. In this stage, the decision problem and objectives are broken down into relevant decision elements (criteria and alternatives). The team creates a hierarchical diagram that reflects the structure of the alternatives.

Step 2: Construct pairwise comparison matrices. Elements are compared two‑by‑two on a scale of 1 to 9, where 1 indicates equal importance and 9 indicates extreme importance of one element over another. The corresponding scale values are shown in the accompanying table.

The data‑processing steps include extracting the pairwise comparison matrix, calculating the maximum eigenvalue and its eigenvector, and normalizing the eigenvector to obtain the weight vector for each factor.

Step 3: Consistency test. To validate the AHP results, compute the consistency ratio (CR) using the consistency index (CI) derived from the maximum eigenvalue. The CI is calculated as (λ_max‑n)/(n‑1), where n is the matrix order. The RI values depend on n, as shown in the table below. A CR below 0.10 indicates acceptable consistency.

3 Summary

This article briefly introduced the concept and basic procedure of the Analytic Hierarchy Process.