# Analytic hierarchy process

Analytic hierarchy process |
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**Analytic hierarchy process** (AHP) is a decision-making technique used in management that allows complex decisions to be made by breaking them down into hierarchical levels. It can be used to determine the relative importance of different criteria in a decision, or to determine which of several options is the best choice. AHP involves decomposing a decision problem into a hierarchy of smaller and more easily manageable sub-problems, with each sub-problem corresponding to a level in the hierarchy. AHP uses pairwise comparison of criteria and sub-criteria to determine their relative importance. It then combines the evaluations to reach a consensus decision.

## Example of analytic hierarchy process

- Analytic hierarchy process (AHP) can be used for employee selection and evaluation. For example, consider a company looking to hire a new sales representative. The company could use AHP to compare candidates in terms of criteria such as experience, education, communication skills, and customer service. By comparing each candidate to the others, the company can determine which candidate is the best fit for the job.
- AHP can also be used to prioritize projects. For example, a company that has several projects on the go can use AHP to determine which project should be given the highest priority. The company could compare the projects in terms of criteria such as impact, cost, time to completion, and customer need. By comparing each project to the others, the company can determine which project should be given the highest priority.
- AHP can also be used for resource allocation. For example, a company with limited resources can use AHP to determine how best to allocate those resources. The company could compare projects in terms of criteria such as impact, cost, and time to completion. By comparing each project to the others, the company can determine which project should be given the most resources.

## Formula of analytic hierarchy process

The Analytic Hierarchy Process (AHP) is a powerful tool for decision making that enables complex decisions to be broken down into a hierarchy of smaller, more manageable sub-problems. The AHP uses pairwise comparisons to determine the relative importance of criteria and sub-criteria in a decision. The overall process is based on the following formulas:

**The pairwise comparison matrix**:

The pairwise comparison matrix is a square matrix with m rows and m columns, where m is the number of criteria or sub-criteria to be compared. Each row and column corresponds to one criterion or sub-criterion and the elements of the matrix represent the relative importance of the criteria or sub-criteria. For example, if we are comparing three criteria A, B and C, then the pairwise comparison matrix would be:

\begin{bmatrix} 1 & \text{A/B} & \text{A/C}\\ \text{B/A} & 1 & \text{B/C}\\ \text{C/A} & \text{C/B} & 1 \end{bmatrix}

**The normalized matrix**:

The normalized matrix is obtained from the pairwise comparison matrix by normalizing the rows such that each row sums to 1. This is achieved by dividing each element of the matrix by the sum of its row. For example, if we are comparing three criteria A, B and C, then the normalized matrix would be:

\begin{bmatrix} \frac{1}{\text{A/B} + \text{A/C}} & \frac{\text{A/B}}{\text{A/B} + \text{A/C}} & \frac{\text{A/C}}{\text{A/B} + \text{A/C}}\\ \frac{\text{B/A}}{\text{B/A} + \text{B/C}} & \frac{1}{\text{B/A} + \text{B/C}} & \frac{\text{B/C}}{\text{B/A} + \text{B/C}}\\ \frac{\text{C/A}}{\text{C/A} + \text{C/B}} & \frac{\text{C/B}}{\text{C/A} + \text{C/B}} & \frac{1}{\text{C/A} + \text{C/B}} \end{bmatrix}

**The eigenvector matrix**:

The eigenvector matrix is obtained by multiplying the normalized matrix by its transpose. This gives us a matrix with m rows and m columns, where m is the number of criteria or sub-criteria being compared. Each row and column corresponds to one criterion or sub-criteria and the elements of the matrix represent the relative importance of the criteria or sub-criteria. For example, if we are comparing three criteria A, B and C, then the eigenvector matrix would be:

\begin{bmatrix} \frac{(\text{A/B})^2 + (\text{A/C})^2}{\text{A/B} + \text{A/C}} & \frac{\text{A/B} \times \text{B/A} + \text{A/C} \times \text{C/A}}{\text{A/B} + \text{A/C}} & \frac{\text{A/B} \times \text{B/C} + \text{A/C} \times \text{C/B}}{\text{A/B} + \text{A/C}}\\ \frac{\text{B/A} \times \text{A/B} + \text{B/C} \times \text{C/A}}{\text{B/A} + \text{B/C}} & \frac{(\text{B/A})^2 + (\text{B/C})^2}{\text{B/A} + \text{B/C}} & \frac{\text{B/A} \times \text{A/C} + \text{B/C} \times \text{C/B}}{\text{B/A} + \text{B/C}}\\ \frac{\text{C/A} \times \text{A/B} + \text{C/B} \times \text{B/A}}{\

## When to use analytic hierarchy process

Analytic hierarchy process (AHP) is an effective decision-making tool that is used to systematically evaluate complex decisions by breaking them down into hierarchical levels. It can be used in a wide range of applications, including:

- Strategic planning - AHP can be used to evaluate criteria such as objectives, objectives weighting, strategies, and resource allocation in order to identify the best approach to achieving a company’s goals.
- Project management - AHP can help to prioritize tasks and identify the most efficient use of resources for each project.
- Product design - AHP can be used to evaluate the relative importance of different characteristics and features of a product.
- Risk management - AHP can be used to identify and prioritize risks and determine the best risk mitigation strategies.
- Vendor selection - AHP can be used to evaluate vendors based on criteria such as quality, cost, and delivery speed.
- Business process optimization - AHP can be used to identify and prioritize process improvement opportunities.

## Types of analytic hierarchy process

Analytic hierarchy process (AHP) is a decision-making technique used in management that allows complex decisions to be made by breaking them down into hierarchical levels. There are several types of AHP that are commonly used, including:

- Multi-attribute decision making (MADM) - This type of AHP is used to compare multiple criteria in a decision-making process. It involves evaluating the relative importance of each criterion to reach a consensus decision.
- Pairwise comparison - This type of AHP uses pairwise comparison to determine the relative importance of criteria in a decision problem. It involves comparing each criterion to every other criterion in order to determine which criteria are more important.
- Hierarchical decision making (HDM) - This type of AHP is used to prioritize different criteria in a decision problem. It uses a hierarchy of criteria to determine which criteria are more important than others.
- Monte Carlo simulation - This type of AHP is used to simulate different scenarios and determine the optimal decision. It involves running multiple simulations and evaluating the most favorable outcomes.

## Steps of analytic hierarchy process

**Step 1**: Define the problem: Identify the decision to be made and define the criteria to be used in making the decision.**Step 2**: Construct the hierarchy: Break the problem down into hierarchical levels, with each level corresponding to a sub-problem.**Step 3**: Analyze the criteria: Analyze the criteria and sub-criteria to understand their relative importance.**Step 4**: Create pairwise comparison matrices: Create pairwise comparison matrices to compare the criteria and sub-criteria to each other.**Step 5**: Calculate the weights: Calculate the relative importance of each criterion by summing the pairwise comparison matrices.**Step 6**: Reach a consensus: Combine the weights to reach a consensus decision.

## Advantages of analytic hierarchy process

One of the main advantages of using the Analytic Hierarchy Process (AHP) is its ability to account for the complexity of a decision problem and to effectively identify the most important criteria and sub-criteria. AHP can offer several advantages over traditional decision-making techniques:

- AHP provides a systematic and structured way to identify and prioritize criteria and sub-criteria in a decision-making process.
- AHP is a quantitative method that relies on pairwise comparisons and mathematical calculations to determine the relative importance of each criterion and sub-criterion.
- AHP can accommodate multiple stakeholders and their preferences in a decision-making process.
- AHP can help provide a more transparent decision-making process as stakeholders can understand the AHP model and the calculations used to reach the final decision.
- AHP can also be used to integrate qualitative and quantitative data in the decision-making process.
- AHP can be easily adapted to address different types of decision-making problems.

## Limitations of analytic hierarchy process

Analytic hierarchy process (AHP) is a powerful decision-making technique used in management that allows complex decisions to be made by breaking them down into hierarchical levels. However, AHP is not without its limitations. The following are some of the main limitations of AHP:

**Subjectivity**: AHP relies on subjective judgments, which means that the decision-making process can be affected by personal biases.**Complexity**: AHP requires a high level of expertise in decision-making, which can make the process difficult for those with less experience.**Over-Reliance on Pairwise Comparisons**: AHP relies heavily on pairwise comparisons, which can be time-consuming and may not reflect the complexity of the decision-making process.**Difficulty of Modeling**: AHP can be difficult to model and can require significant resources to develop and maintain.**Lack of Flexibility**: AHP is not very flexible and may not be suitable for rapidly changing environments.

The following approaches are related to the Analytic Hierarchy Process:

**Analytic Network Process (ANP)**: This approach involves the use of networks to represent the decision-making problem and uses mathematical programming to find the best solution.**Multi-Criteria Decision Analysis (MCDA)**: This approach involves the use of multiple criteria to evaluate the different options in a decision-making problem. It uses a combination of mathematical models, graphical methods, and expert opinion to make the best decision.**Multi-Objective Decision Making (MODM)**: This approach involves the use of multiple objectives to evaluate the different options in a decision-making problem. It uses a combination of mathematical models and graphical methods to find the most optimal solution.**Group Decision Making (GDM)**: This approach involves the use of a group of individuals to make a decision. It uses a combination of structured techniques such as brainstorming, Delphi technique, and nominal group technique to make the best decision.

## Suggested literature

- Vaidya, O. S., & Kumar, S. (2006).
*Analytic hierarchy process: An overview of applications*. European Journal of operational research, 169(1), 1-29. - Vargas, L. G. (1990).
*An overview of the analytic hierarchy process and its applications*. European journal of operational research, 48(1), 2-8. - Saaty, T. L. (1988).
*What is the analytic hierarchy process? (pp*. 109-121). Springer Berlin Heidelberg.