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The analytic hierarchy process—what it is and how it is used
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“The number of Level 1 criteria (n = 6) remained within Saaty's [23] recommended range (7 ± 2) for cognitively tractable pairwise comparison.”
From Toward Sustainable AI Literacy in Higher Education: A Delphi–AHP Approach to Identifying Key Components of AI/Data Extracurricular Programs (2026) · cites this paper as [23]
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“The resulting six-criterion structure remained within Saaty's [23] recommended range (7 ± 2), allowing cognitively tractable pairwise comparison while preserving full domain coverage.”
From Toward Sustainable AI Literacy in Higher Education: A Delphi–AHP Approach to Identifying Key Components of AI/Data Extracurricular Programs (2026) · cites this paper as Saaty's [23]
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“The Analytic Hierarchy Process (AHP), developed by Saaty [23], is a multi-criteria decision-making technique that decomposes complex problems into hierarchical structures of goal, criteria, and alternatives, and then quantifies relative importance through pairwise comparisons.”
From Toward Sustainable AI Literacy in Higher Education: A Delphi–AHP Approach to Identifying Key Components of AI/Data Extracurricular Programs (2026) · cites this paper as [23]
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“AHP enables the conversion of qualitative expert judgments into numerical weights, with built-in consistency verification through the Consistency Ratio (CR).”
From Toward Sustainable AI Literacy in Higher Education: A Delphi–AHP Approach to Identifying Key Components of AI/Data Extracurricular Programs (2026) · cites this paper as [23]
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“The consistency ratio of 0.015 confirms acceptable logical consistency well within Saaty's [23] threshold of 0.10.”
From Toward Sustainable AI Literacy in Higher Education: A Delphi–AHP Approach to Identifying Key Components of AI/Data Extracurricular Programs (2026) · cites this paper as [23]
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“One of the most commonly used multi-criteria decision-making methods is the Analytic Hierarchy Process (AHP), which is used to determine criteria weights and rank alternatives based on expert judgment [41].”
From An Integrated AHP–Fuzzy AHP Evaluation Framework for Large Language Models in Software Engineering Education (2026) · cites this paper as [41]
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Research works printed but not linked: 12
- Saaty, 1982, Decision Making for Leaders
- Saaty, 1986, Absolute and relative measurement with the AHP. The most livable cities in the United States, Socio-econ. Plann. Sci., 20, 327, 10.1016/0038-0121(86)90043-1 · DOI
- Saaty, 1987, Stimulus—response with reciprocal kernels: the rise and fall of sensation, J. math. Sci.
- Aczél, 1983, Procedures for synthesizing ratio judgements, J. math. Psychol., 27, 93, 10.1016/0022-2496(83)90028-7 · DOI
- Hämäläinen, 1986, The analytic network process in energy policy planning, Socio-econ. Plann. Sci., 20, 399, 10.1016/0038-0121(86)90054-6 · DOI
- Harker, 1987, Alternative modes of questioning in the Analytic Hierarchy Process, Mathl Modelling, 9, 353, 10.1016/0270-0255(87)90492-1 · DOI
- Saaty, 1984, Inconsistency and rank preservation, J. math. Psychol., 28, 205, 10.1016/0022-2496(84)90027-0 · DOI
- Saaty, 1986, Axiomatic foundation of the Analytic Hierarchy Process, Mgmt Sci., 32, 841, 10.1287/mnsc.32.7.841 · DOI
- Saaty, 1987, Rank generation, preservation and reversal in the Analytic Hierarchy Process, Decis. Sci., 10.1111/j.1540-5915.1987.tb01514.x · DOI
- Aczél, 1987, Synthesizing judgements: a functional equations approach, Mathl Modelling, 9, 311, 10.1016/0270-0255(87)90487-8 · DOI
- Vargas, 1982, Reciprocal matrices with random coefficients, J. math. Modelling, 3, 69, 10.1016/0270-0255(82)90013-6 · DOI
- Zahedi, 1986, The Analytic Hierarchy Process—a survey of the method and its applications, Interfaces, 16, 96, 10.1287/inte.16.4.96 · DOI
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Sustainability of Rural Small-Scale Farmers Using a Thematic Content-Fed Analytic Hierarchy Process
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“The AHP method is a widely used technique, with a wide spectrum of applications, which includes human resources, the economy, transport administration, and sustainability in energy systems, and agriculture amongst others [81].”
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“The method works with issues that have a linear hierarchical structure, such as those with criteria, sub-criteria [35], and possible solutions.”
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Planning urban EV charging stations with GIS based multi-criteria decision making
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“AHP is a multicriteria method developed by Saaty (1987) that enables informed decision-making for complex problems using mathematically pairwise criteria comparisons (Parry et al., 2018; Pramanik, 2016).”
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“It was developed by Saaty [27] and is one of the most well-known and widely used approaches for multiple-criteria decision-making.”
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“…all the tables in this article, for example, creating pairwise comparison matrices, finding priority vectors, performing consistency analysis, and determining the final priority vectors, was carried out using the methods recommended by Saaty [6–8], as explained in detail in previous studies [3,23].”
Genedi, M. A., Gouhar, N., El-Qady, G., Gaafar, I., & El-Mahmoudi, A. (2025) · Scientific Reports · doi:10.1038/s41598-025-94598-7 · Cited by 8
“Subsequently, the consistency ratio (CR) was derived by dividing the CI by the random consistency index (RI), with RI values obtained from Saaty’s standardized Table 4 59,62 .”
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“RI: random consistency index (Saaty 1987). n 1 2 3 4 5 6 7 8 9 10 RI 0 0 0.58 0.9 1.12 1.24 1.32 0.41 1.45 0.49 The acceptable consistency ratio must be less than or equal to 0.1, that is, 10% e) Weighted overlay analysis Classes for each criterion are ranked from 1 to 5 in terms of relevance…”
InfluentialDas, S., & Ghosh, T. (2024) · Earth Systems and Environment · doi:10.1007/s41748-024-00486-x · Cited by 8
“The AHP method was used based on Saaty’s method (Saaty 1987, 2001; Saaty and Vargas 2006) to determine the weight value, i.e., eigenvector, and the consistency ratio (CR) of the selected factors (Kaliraj et al. 2015; Shit et al. 2016; Ali et al. 2020; Saravanan et al. 2023; Akash et al. 2023).”
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“(1)) is constructed according to the score assigned by the experts, which reflects the relative importance of any two assessment factors. where R is a positive reciprocal matrix, the value of r ij in R can be determined by comparing r i and r j according to the 1–9 scale described in Saaty (1987).”
A modified fuzzy PFMEA model for risk-centric Six Sigma assessment under the paradigm of Quality 4.0
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“MCDM procedures such as Analytical Hierarchy Process (AHP) offer a structured framework to gauge the relative influences of several complementary and conflicting landfill evaluation criteria (R. W. Saaty, 1987).”
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“Factor weights for developing the suitability map were assigned by the three authors using the pairwise comparison method, also known as the Analytic Hierarchy Process (AHP) developed by Saaty (1987).”
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“In addition, AHP has received widespread acceptance for its usefulness in making decisions based on several factors (Saaty, 1987; Vaidya & Kumar, 2006).”
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Duxbury, D. J., Paton, N. W., & Keane, J. A. (2024) · Expert Systems with Applications · doi:10.1016/j.eswa.2024.124057 · Cited by 8
“This formula maps two normalised values ( x , y ) to the fundamental scale proposed by Saaty (Saaty 1987).”
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“The following is a problem-solving algorithm using the AHP method [19], namely: Step 1 : Determine the nature of the problem and the action required; Step 2 : Define the framework, criteria, alternatives, and overarching criteria that will form the basis for finding a solution to the problem; Step…”
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“Fundamental scale of Saaty [116].”
MethodCelik, K. T., & Kurdoğlu, B. Ç. (2024) · International Journal of Disaster Risk Reduction · doi:10.1016/j.ijdrr.2024.104439 · Cited by 8
Low/zero carbon technology diffusion and mapping for Nigeria’s decarbonization
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“A method of obtaining the weights to be used for the criteria weight calculations is based on the Analytic Hierarchical Process (AHP) with the procedure presented as follows (Balasundareshwaran et al. 2019; Saaty 1987).”
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Understanding the behavior of the Thin Lightly-Reinforced Concrete Wall (TLRCW) building system
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“Whitaker (1987) and Vargas and Saaty (2011) proposed a scale for paired comparison matrices, a square matrix that complies with reciprocal properties and homogeneity.”
MethodMulti-Criteria Decision-Making Tools for Project Selection by International Conglomerates
Lee, H.-Y., Yi, P.-H., Selvanathan, K., Chandrahasan, P., & Chemmangattuvalappil, N. (2023) · Process Integration and Optimization for Sustainability · doi:10.1007/s41660-023-00376-1 · Cited by 8
“Developed by Thomas L. Saaty during the 1970s, the analytic hierarchy process (AHP) is a basic decision-making methodology (Saaty 1987).”
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“This process involves utilizing expert ratings obtained from the questionnaire and calculating the weights using the Analytic Hierarchy Process (AHP) method [35].”
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“This method scores criteria by pairwise comparisons (Saaty, 1987).”
MethodWei, S., Li, Y., Yang, H. C., Xie, M., & Wang, Y. (2023) · Transport Policy · doi:10.1016/j.tranpol.2023.08.009 · Cited by 8
Calvo-Saad, M. J., Solís-Chaves, J. S., & Murillo-Arango, W. (2023) · Heliyon · doi:10.1016/j.heliyon.2023.e19874 · Cited by 8
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“AHP provides significant convenience to the users regarding mathematical operations and intelligibility [49] while reducing complex problems to a simple and very flexible model, providing an easily applicable decision-making methodology [50].”
BackgroundModrego-Monforte, I., Barrena-Herrán, M., & Grijalba, O. (2023) · Land · doi:10.3390/land12081551 · Cited by 8
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“Note that the 0.10 threshold, as recommended by Saaty [30], reflects an order-of-magnitude separation between the priority of consistency and the tolerable measurement error, preserving sensible accuracy while allowing minor inconsistency, allowing to capture evolving knowledge and judgment…”
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“The AHP is a structured and systematic approach to decision-making, particularly suited for complex problems involving multiple criteria [49].”
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“The Analytical Hierarchy Process (AHP), developed by Thomas L. Saaty in 1977 and further detailed in 1987 [12], is a multi-criteria decision-making tool for complex problems with conflicting criteria [13].”
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“In Stage The construction of a comprehensive evaluation indicator system for power enterprise procurement performance requires systematic consideration of the unique characteristics and operational requirements of power enterprises within competitive electricity markets 24 .”
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“The AHP is a decision-making method that combines qualitative and quantitative analysis (Saaty, 1987), suitable for addressing complex systemic problems.”
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“The AHP method does not handle ambiguous data as well as FAHP, but its computations are simpler (Saaty 1987).”
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“The AHP is a versatile tool that uses numerical values derived from data and expert judgement to compare and evaluate elements and is particularly useful for complex problems requiring collaboration and human insight (Saaty 1987); Bhushan and Rai 2004).”
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“…matrices to determine the importance value of each indicator, and the result shows that the consistency of the pairwise comparison matrices at all levels met the consistency test criterion, with CR 0.10 [37], suggesting acceptable logical consistency and supporting scientific rigor and reliability.”
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“The rating system was based on Saaty's scale [ 67].”
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“After determining the nitrogen regulation intervals for different growth stages, the rationality of each interval was validated using the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) model based on the entropy weight method [36].”
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“Developed by Thomas L. Saaty in 1977 and further refined in 1987 [34], the AHP is a multi-criteria decision-making method designed to facilitate complex decisions involving multiple conflicting criteria [35].”
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Plastic waste management in agriculture through a GIS-based territory design approach
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Barbosa, C. R., Vedana, L. A., Belladona, R., & de Vargas, T. (2025) · Journal of South American Earth Sciences · doi:10.1016/j.jsames.2025.105595 · Cited by 7
A comprehensive flood risk mapping in Gangetic interfluvial flood plain region
Baral, M. R. R. C. A. D. P. G. G., Chakraborty, A., Mukherjee, K., Bhadra, T., Dash, P., & Sarkar, R. (2025) · Discover Geoscience · doi:10.1007/s44288-025-00142-5 · Cited by 7
“This method is developed by Saaty in the 1970s [59].”
Lawmchullova, I., Lalrinawma, J., Rinkimi, L., Lalngaihawma, J., Rao, C. U. B., & Biswas, B. (2025) · Environmental Earth Sciences · doi:10.1007/s12665-025-12244-x · Cited by 7
“The analytic hierarchy process (AHP) was used to determine the weights of selected criteria (Saaty 1987).”
Tripathi, S. K., Kant, R., & Shankar, R. (2025) · Management of environmental quality · doi:10.1108/meq-08-2024-0326 · Cited by 7
Grimmel, P., Niemeyer, J. F., Tan, C. F., Sun, Y., Zhao, Y., Schöling, N., Yeo, Z., Mennenga, M., Carlow, V. M., & Herrmann, C. A. (2025) · Journal of Industrial Ecology · doi:10.1111/jiec.70015 · Cited by 10
“The weights can be determined using an analytical hierarchy process, which evaluates and prioritizes the relative importance of the features systematically (Saaty, 1987).”
Watanabe, W. C., Patitad, P., Ransikarbum, K., & Janmontree, J. (2025) · Journal of Industrial and Production Engineering · doi:10.1080/21681015.2025.2487496 · Cited by 5
“Originally developed for military resource allocation [33], it is now applied in fields like health [34], engineering [35], and education [36].”
Ucak-Ozkaya, G. (2025) · Food Science & Nutrition · doi:10.1002/fsn3.70122 · Cited by 7
Meng, J.-T., Fang, X., Xu, J., & Zhang, Z. (2025) · Buildings · doi:10.3390/buildings15071067 · Cited by 8
Chauhan, P., Ngangom, M., & Thakkar, M. (2025) · Modeling Earth Systems and Environment · doi:10.1007/s40808-025-02339-6 · Cited by 7
“Using the AHP method, we calculated the weights by determining the eigenvalues (λ) for each factor, as outlined by Saaty (1987), and ensured consistency through a consistency ratio (CR) check.”
SWOT and PESTEL analyses in assessing the sustainability of In-stream wetlands in Egypt
Mostafa, H. (2025) · Water Science · doi:10.1080/23570008.2025.2463743 · Cited by 6
“Factors are ranked in importance on a 1 to 5 scale; 1 represents the least important, and 5 is the most important (Saaty, 1987; White et al., 2015), as shown in Table 2.”
Valera, K., & Sharifi, A. (2025) · International Journal of Disaster Risk Reduction · doi:10.1016/j.ijdrr.2025.105362 · Cited by 7
Unbiased criteria identification for two-sided matching: An environment-based design approach
Tozlu, B., Akgunduz, A., & Zeng, Y. (2025) · Expert systems with applications · doi:10.1016/j.eswa.2025.127233 · Cited by 7
Sizing and Siting of Energy Storage Systems for Mitigating Forecast Mismatch in Transmission Grid
Tricarico, G., Gonzalez-Longatt, F., Marasciuolo, F., Ishchenko, O., Dicorato, M., & Forte, G. (2025) · IEEE transactions on industry applications · doi:10.1109/tia.2025.3532580 · Cited by 7
“1) AHP Analysis: The AHP [36], [37] is based on the determination of the h × h pairwise comparison matrix C i , for each criterion i , whose element c represents the prevalence of the x -th alternative compared to the y -th one, if c > 1, in a scale of values up to 9 according to importance…”
Aprigliano, V., Toro, C., Rojas, G., Bastías, I., Cardoso, M., Santos, T., da Silva, M. A. V., Bustos, E., de Oliveira, U. R., & Seriani, S. (2025) · ISPRS Int. J. Geo Inf. · doi:10.3390/ijgi14010038 · Cited by 7
A strategic evaluation of Indonesia’s policy on mineral value addition
Firmanto, A. B., Wibisono, D., Siallagan, M., & Mubarok, M. Z. (2025) · Mineral Economics · doi:10.1007/s13563-024-00484-5 · Cited by 5
“The creation of a hierarchical structure primarily serves to categorize the objectives, considerations, and decision-making elements into top, middle, and bottom levels based on their mutual relationship (Saaty 1987).”
Dossow, M., Chen, M., Spliethoff, H., & Fendt, S. (2025) · MethodsX · doi:10.1016/j.mex.2025.103194 · Cited by 7
“The AHP was first developed by Thomas L. Saaty [ 43 ].”
InfluentialChen, M., Dossow, M., Spliethoff, H., & Fendt, S. (2025) · Journal of Cleaner Production · doi:10.1016/j.jclepro.2024.144615 · Cited by 7
Ganie, P. A., Posti, R., Bharti, V., Sehgal, V. K., Sarma, D., & Pandey, P. K. (2024) · Environmental Monitoring & Assessment · doi:10.1007/s10661-024-13240-7 · Cited by 7
“The AHP method includes measures for handling judgment uncertainties through the calculation of the principal eigenvalue ( λ max ) and consistency index (CI), with the latter introduced by Saaty as a way to measure the level of consistency or deviation in assessments (Saaty, 1987).”
Pamucar, D. S., Šimić, V., Dabić-Miletić, S., Zaidan, A. A., Petrović, N., & Narayanamoorthy, S. (2024) · Engineering Applications of Artificial Intelligence · doi:10.1016/j.engappai.2024.109254 · Cited by 10
Matching supply chain management challenges with blockchain applications
Berneis, M., & Winkler, H. (2024) · Supply Chain Management · doi:10.1108/scm-12-2023-0615 · Cited by 7
Li, J., Zong, R., Wang, Y., & Deng, W. (2024) · Applied Sciences · doi:10.3390/app142210451 · Cited by 10
Sela, S., Dobermann, A., Cerri, C. E., Svoray, T., van Es, H., Amsili, J., Biradar, S., Luzon, U., & Katz, S. (2024) · npj Sustainable Agriculture · doi:10.1038/s44264-024-00031-3 · Cited by 7
“Among the available quantitative methodologies for prioritization, the Analytical Hierarchy Process (AHP) is a widely used multicriteria assessment (MCA) methodology 26,27 .”
Hao, J., Dong, S., & Li, J. (2024) · BMC Public Health · doi:10.1186/s12889-024-20406-y · Cited by 6
“The AHP is highly applicable to multi-level indicator systems, such as the one in this study, where weights are derived by constructing judgment matrices for each level of indicators and conducting pairwise comparisons [37].”
MethodAssessing Tram-Train feasibility via multicriteria approach: The case of Brescia (Italy)
Assefa, N. G., Carra, M., Ventura, R., Richiedei, A., & Barabino, B. (2024) · Research in Transportation Business & Management · doi:10.1016/j.rtbm.2024.101178 · Cited by 7
Metadata
- Authors
- Saaty, R. W.
- Corresponding author
- Saaty, R.W.
- Venue
- Mathematical Modelling
- Publisher
- Elsevier BV
- Published
- 1987
- Volume
- 9
- Issue
- 3-5
- Pages
- 161-176
- ISSN
- 0270-0255 (ISSN-L)
- OpenAlex
- W2046078150
- Cited by
- 5,380
- References
- 15
- Citations by year
- 2026: 623 · 2025: 735 · 2024: 697 · 2023: 662 · 2022: 518 · 2021: 481 · 2020: 382 · 2019: 284 · 2018: 226 · 2017: 173 · 2016: 124 · 2015: 107 · 2014: 62 · 2013: 52 · 2012: 39
- FWCI
- 4.84
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Mathematical Modelling (publishedVersion)
- Topics
- Social Sciences › Decision Sciences › Management Science and Operations Research › Multi-Criteria Decision Making
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Game Theory and Voting Systems
- Social Sciences › Business, Management and Accounting › Strategy and Management › Strategic Planning and Analysis
- Keywords
- analytic hierarchy process, ratio scale, measurement theory, axioms, inconsistency, rank preservation, reversal, absolute measurement, relative measurement, multi-criteria decision making
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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