Analytic Hierarchy Process (AHP) with Algorithm Implementation

The Analytic Hierarchy Process (AHP) is a classic algorithm in mathematical modeling used to solve complex decision-making problems, particularly suitable for systems analysis involving multiple criteria and objectives. Its core principle involves decomposing decision problems into hierarchical structures and quantifying the comparative importance of various elements to derive optimal solutions through systematic pairwise comparisons.

Algorithm Implementation Workflow Hierarchical Model Construction: Break down the problem into objective layer (top), criteria layer (middle), and alternatives layer (bottom), forming a tree structure. For example, when selecting a tourist destination, the objective layer is "best destination," criteria layer may include cost, scenery, etc., and alternatives layer contains specific cities. In code implementation, this can be represented using nested dictionaries or object-oriented programming classes. Judgment Matrix Formation: Perform pairwise comparisons of elements at the same hierarchical level using the 1-9 scale method (1 for equal importance, 9 for extreme importance) to quantify subjective judgments. In Python, this can be implemented using numpy arrays where each element a_ij represents the relative importance of element i over j. Consistency Validation: Calculate the consistency ratio (CR) of the matrix. If CR < 0.1, the logical consistency is acceptable; otherwise, judgment values need adjustment. The implementation involves computing eigenvalues and consistency index using numpy.linalg.eig() function. Weight Calculation: Derive element weights through eigenvector method or geometric mean method, then aggregate weights across hierarchy levels to obtain final alternative rankings. Code implementation typically uses numpy for matrix operations and weight normalization functions.

Application Scenarios Suitable for decision problems requiring trade-offs among qualitative factors such as resource allocation, risk assessment, and supplier selection. Its strength lies in structuring complex problems, though it relies on expert experience and requires attention to subjective bias mitigation through sensitivity analysis.

Advanced Considerations Can be combined with entropy weight method or fuzzy mathematics to optimize weight calculation and enhance objectivity. In mathematical modeling competitions, AHP is frequently integrated with methods like TOPSIS to solve comprehensive evaluation problems through multi-method hybrid algorithms.