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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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Wang, X.-Y., Bai, T., Yang, Y., Wang, G.-F., Tian, G., & Kollányi, L. (2024) · Land · doi:10.3390/land13081330 · Cited by 7
“Subjective weights are calculated using the Analytic Hierarchy Process (AHP), a systematic decision analysis tool that employs minimal mathematical information in decision processes to provide a simple method for multi-objective, multi-criteria, or unstructured, complex decision problems [51].”
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Rai, H. P., Ogeti, P., Fadnavis, N. S., Patil, G., & Padyana, U. K. (2024) · Universal Research Reports · doi:10.36676/urr.v9.i4.1320 · Cited by 7
“• Performance: Better customer service, enhanced operations, and a flexible work environment all help businesses outperform traditional on-premise systems [6, 7].”
BackgroundAbdelkarim, A., Abdelhafez, M. H. H., Elkhayat, K., Alshenaifi, M., Alfraidi, S., Aldersoni, A., Albaqawy, G., Aldamaty, A., & Ragab, A. (2024) · Land · doi:10.3390/land13070986 · Cited by 7
Ipek, C., & Türker, U. (2024) · Environmental Earth Sciences · doi:10.1007/s12665-024-11761-5 · Cited by 7
“The association of the modified DRASTIC model estab - lished with AHP and the water quality index improved from 0.2 to 0.61 indicating moderate positive correlation between the WQI and DIV.”
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Capdevila, M., Rodrigues, K. A. P., Furlan, T., & Granollers, T. (2024) · Comput. Stand. Interfaces · doi:10.1016/j.csi.2024.103891 · Cited by 7
Asker, V. (2024) · Gospodarka Narodowa · doi:10.33119/gn/184315 · Cited by 7
“For example, the Analytical Hierarchy Process (AHP) [ Saaty, 1987 ] and Best Worst Method (BWM) [ Razaei, 2015 ] methods are based on a subjective assumption, while the VIKOR [ Opricovic, Tzeng, 2004 ], TOPSIS [ Hwang, Yoon, 1981 Despite the existence of numerous studies that examine airlines using…”
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Hardjanto, H., Safitri, Y., & Sundawati, L. (2024) · Journal of Natural Resources and Environmental Management · doi:10.29244/jpsl.14.2.220 · Cited by 7
“The three problem-solving principles stated in Saaty [18] are decomposition, comparative assessment, and priority description.”
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“This comparison was carried out using Saaty’s nine-point scale, which allowed for a comprehensive assessment and ranking of the criteria based on their significance (Saaty, 1987; Ustaoglu & Aydınoglu, 2020).”
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Was stakeholder participation in the PEFC revision process successful in Slovakia?
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Water Resources Allocation in a Transboundary River Based on a Rubinstein Bargaining Model
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“The analytic hierarchy process (AHP) can make use of less quantitative information to mathematize the thinking process of decision-making, thus providing a simple decision-making method for complex decision-making problems (Saaty 1987).”
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“AHP with some modifications (MAHP) according to Saaty (1987), and Eynizadeh and Dehghani (2020) was used with the following steps [ see Supporting Information—MAHP]: Step (1) Aligning traits and families (Figure 1).”
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Project portfolio selection and scheduling incorporating dynamic synergy
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A Hierarchy-based Analysis Approach for Blended Learning: A Case Study with Chinese Students
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“Analytic Hierarchy Process (AHP) is an effective method involving both qualitative and quantitative analysis [36].”
MethodNtawuruhunga, D., Ngowi, E., Mangi, H. O., Salanga, R. J., & Shikuku, K. (2023) · Trees, Forests and People · doi:10.1016/j.tfp.2023.100446 · Cited by 7
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“The fuzzy AHP approach was developed utilizing fuzzy logic [31] and a classical AHP method [32–34] together and is commonly used to remove ambiguity from decision-making processes [35].”
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“Using the produced criteria visualization, the AHP take place to determine the most influencing factor causing floods in the flood extent area (Saaty, 1987).”
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“, for decision-making) in relation to the highest level (the overall goal) [62].”
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“Saaty (1987) suggested a consistency ratio of the matrix less than or equal to 10%.”
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“TheAnalyticHierarchy Process(AHP), proposedbySaaty 36 ,isusedtodeterminetheweightsofthecriteriaduetoits advantagesof group decision-making and fl exibility tohandlecomplexity 36 .”
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“…pairwise comparisons (Table 2), we determined the relative importance of main categories—Functional, Instructions, Output, and Creativity—using Saaty’s 1–9 scale [26], where 1 indicates equal importance, 3 moderate, 5 strong, 7 very strong, and 9 extreme importance of one criterion over another.”
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“Note: Summary of methodological characteristics, tourism-related applications, and hybridization potential of the AHP, BWM, and TOPSIS methods, adapted from R. W. Saaty (1987); Rezaei (2015, 2016); Ulkhaq et al. (2016); Kahraman and Yanık (2016); Govindan and Bouzon (2018); Pishdar et al. (2022).”
Izzati, A. N., Tumuyu, S. S., & Wardhana, Y. M. A. (2025) · Journal of Islamic Economics Lariba · doi:10.20885/jielariba.vol11.iss2.art22 · Cited by 6
“These criteria are derived from both the literature on sustainable finance and insights obtained from preliminary qualitative analysis (Canco et al., 2021; Graneheim et al., 2017; Meng & Shaikh, 2023; Saaty, 1987).”
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“Previous studies have supported this more flexible threshold in complex real-world decision-making contexts, particularly in healthcare settings where larger matrices can increase inconsistency despite stable rankings [29, 34–38].”
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Multi-Criteria Decision Making: Sustainable Water Desalination
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“In the end, Saaty’s eigenvector approach is used to construct the weight vector w = [w1, w2, w3…wn], which is a pairwise comparison matrix.”
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Harnessing satellite precision: flash flood vulnerability mapping in arid wadis
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“(Saaty 1987 ) No. of criteria 1 2 3 4 5 6 7 8 9 10 Random consistency index ( RI ) 0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49 1 3 and upper Eocene ages (includes formations such as Minia, Drunka, and Serai), and Plio-cene deposits (Moftah et al. 2022; Saleem et al. 2019 ).”
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“AHP is a multi-criteria decision making (MCDM) method, which describes a broad group of techniques designed to aid comparison of alternatives using a set of criteria, such as indicators (Saaty 1987).”
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“9) as recommended by Saaty (1987): The Random Consistency Index (RCI) could be acquired from standard tables (Saaty 1987; Alonso and Lamata 2006 ) and varies depending on the number of factors, and it is 1.12 for 5 factors.”
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Urban Flood Exposure and Vulnerability: Insights From Pendik District of Istanbul
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“This study adopted Saaty's (1987) analytic hierarchy process (AHP), which assigns weights based on relative importance.”
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“(2). where, R i is the rating of the parameter in each point; w i the weight of each parameter; N is the number of the criteria The weight for each parameter is given by the Analytical Hierar-chal Process (AHP) proposed by (Saaty 1987).”
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“AHP is a mathematical technique utilized for assigning weights to criteria through pairwise comparisons and for assessing the relative significance of each criterion [47].”
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“The Analytic Hierarchy Process (AHP) is utilised to compare the criteria and organise the criteria based on their importance [17,18]. the importance of the criteria, and a weight is considered for each criterion.”
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“The AHP, developed by (Saaty 1987), is the most popular and widely used MCE approach (Mohammed et al. 2019).”
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“Saaty (1987) proposed calculating the maximum eigenvalue ( k max ) of the comparison matrix, which is displayed in Table 5.”
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Young Generations’ Perception of Deadwood in Forest Landscapes
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“…analysis technique aimed at resolving intricate decision problems and facilitating accurate decision-making and judgment within complex systems (Saaty 1987) – was utilized to process the data gathered through Q7 and evaluate the aesthetic appreciation of different deadwood management…”
A novel Shannon entropy-based backward cloud model and cloud K-means clustering
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“In 1987, R.W. Saaty [35] introduced the analytic hierarchy process (AHP) for structured decision-making, simplifying complex problems into hierarchical structures for assessing alternatives based on multiple criteria.”
InfluentialHenao, D., Sánchez-Echeverri, L., & Tovar-Perilla, N. (2024) · Resources · doi:10.3390/resources13100139 · Cited by 6
“AHP is a structured and logical method designed to optimize complex decision-making by breaking down the problem into a hierarchical structure [30].”
Chauhan, P., Ngangom, M., & Thakkar, M. (2024) · Journal of the Indian Society of Remote Sensing · doi:10.1007/s12524-024-02014-w · Cited by 6
“…zone map involved the Analytic Hierarchy Process (AHP) technique for determining the relative importance of each cre-teria (or sub-creteria) using a scale developed by Saaty (1987), which ranges from 1 (indicating equal importance) to 9 (indicating extreme importance of one criterion over another).”
InfluentialRidolfi, R. M., Azzaro, S., Beaubien, S. E., Pra, A. D., Pontiggia, M., & Bigi, S. (2024) · International Journal of Hydrogen Energy · doi:10.1016/j.ijhydene.2024.08.291 · Cited by 6
An Operational Quality Model of Embedded Software Aligned with ISO 25000
Argotti, Y., Kenfaoui, Y., Baron, C., Abran, A., & Esteban, P. (2024) · ACM Transactions on Embedded Computing Systems · doi:10.1145/3691642 · Cited by 6
“Together with a correlation coefficient of customer satisfaction (i.e., degree of satisfaction and degree of dissatisfaction) and Saaty’s Analytic Hierarchy Process (AHP) [ 25 ], the prioritization of requirements is finalized.”
Islam, K. (2024) · Proceedings of the Indian Academy of Sciences, Earth and Planetary Sciences · doi:10.1007/s12040-024-02364-6 · Cited by 6
“AHP is the multiple-criteria decision-making (MCDM) approach introduced by Saaty (1987).”
Henderson, J., & Peeling, R. (2024) · Open Research Europe · doi:10.12688/openreseurope.18195.1 · Cited by 6
“These are Analytic Hierarchy Process (AHP) (Saaty, 1980; Saaty, 1987; Saaty, 2008 Whilst all these models for decision making (and more) are established in the literature, in practice our experience has been that MARE has proven to be the most popularly applied, probably because it is relatively…”
MethodNúñez-Acosta, A., & Sánchez-García, J. Y. (2024) · Administrative Sciences · doi:10.3390/admsci14080166 · Cited by 6
“Using an algorithm based on paired comparisons, the AHP allows for the prioritization of alternatives and assessing their relative importance (Saaty 1987).”
MethodWibowo, P. D. A. (2024) · International Journal of Current Science Research and Review · doi:10.47191/ijcsrr/v7-i7-103 · Cited by 6
“The question is whether there is a consistent theory that can address these two worlds of reality without compromising any of them (Saaty, 1987).”
BackgroundAn integrated performance assessment method for SMEs fruit processing sustainability in Indonesia
Santoso, I., Wafi'uddin, I., Lu'ayya, N. M., Choirun, A., Purnomo, M., & Sulianto, A. A. (2024) · Cogent Engineering · doi:10.1080/23311916.2024.2374941 · Cited by 6
“Analytical Hierarchy Process (AHP) is a multi-criteria decision making technique proposed by Saaty (1987).”
Kim, J., Lee, S., Arigi, A. M., & Kim, J. (2024) · Progress in Nuclear Energy · doi:10.1016/j.pnucene.2024.105209 · Cited by 9
Colombo, S., Ciotola, A., & Piazza, L. (2024) · Expert systems with applications · doi:10.1016/j.eswa.2024.124735 · Cited by 6
Kukuntod, N., & Wijitkosum, S. (2024) · Heliyon · doi:10.1016/j.heliyon.2024.e32347 · Cited by 6
“The analytical hierarchy process (AHP) is a well-known multicriteria decision analysis (MCDA) technique developed by Saaty [62].”
MethodEnhancing Solar PV Deployment: Land Suitability Assessment and Site Selection using AHP
Tryphena, B., C., P., & Vidhya, J. (2024) · 2024 International Conference on Advances in Modern Age Technologies for Health and Engineering Science (AMATHE) · doi:10.1109/amathe61652.2024.10582094 · Cited by 6
“[11] For each level in the hierarchy, construct a set of pairwise comparison matrix which establishes the priorities for the criteria, judging them in pairs by their relative importance.”
BackgroundCBS tabanlı ÇKKV-AHS yöntemi ile ekoturizm alanlarının yer seçimi: Çamlıhemşin İlçesi Örneği
Koca, C., & Kaymaz, Ç. (2024) · Geomatik · doi:10.29128/geomatik.1419623 · Cited by 6
“CR değeri sıfıra ne kadar yakın olursa karşılaştırma sonuçları da daha tutarlı olacaktır ( Saaty, 1987).”
BackgroundInfluentialBeigh, I. H., & Bukhari, S. K. (2024) · Applied Geomatics · doi:10.1007/s12518-024-00559-6 · Cited by 6
“I) value for the corresponding number of Criteria (Saaty 1987).”
Framework for Strategic Selection of Maintenance Contractors
Zubair, M. U., Farid, O., Hassan, M., Aziz, T., & Ud-Din, S. (2024) · Sustainability · doi:10.3390/su16062488 · Cited by 6
“After shortlisting factors from the literature review, a preliminary questionnaire was developed to gather expert opinions on the importance of these factors in selecting a maintenance contractor.”
MethodA Reconfigurable Cellular Remanufacturing Architecture: a multi-objective design approach
Mejía-Moncayo, C., Kenné, J., & Hof, L. A. (2024) · Journal of Remanufacturing · doi:10.1007/s13243-024-00139-2 · Cited by 6
“(1): The second method implements an analytic hierarchy process (AHP) [22].”
InfluentialZeng, S.-Z., Gao, L., & Wu, J. (2024) · Granular Computing · doi:10.1007/s41066-024-00453-x · Cited by 6
“Compared to the Analytic Hierarchy Process (AHP) (Saaty 1987), best-worst multi-criteria decision-making method (BWM) (Rezaei 2015), and full consistency method (FUCOM) (Pamucar et al. 2018), LBWA requires only the smallest n -1 comparisons, and the complexity of LBWA can be significantly reduced by increasing the numb…”
The study of children’s preferences for the design elements of learning desks based on AHP-QCA
Miao, Y.-F., Yan, S., & Xu, W. (2024) · BioResources · doi:10.15376/biores.19.2.2045-2066 · Cited by 6
“Qualitative comparative analysis (QCA) is a method of analysing data sets consisting of binary variables that was first proposed by Charles Ragin in 1987 (Ragin 2014).”
Parween, S., & Sinha, R. C. (2024) · KSCE Journal of Civil Engineering · doi:10.1007/s12205-024-1019-2 · Cited by 8
ENTROPİ VE AHP AĞIRLIKLI TOPSIS YÖNTEMİYLE FİRMALARIN PERFORMANS SIRALAMASI: BIST’TE BİR UYGULAMA
Topal, B. (2024) · Dokuz Eylül Üniversitesi Sosyal Bilimler Enstitüsü Dergisi · doi:10.16953/deusosbil.1380809 · Cited by 6
GIS-based multi-criteria approach for drought hazard modeling in the Ba river basin, Vietnam
Dang, M., & Nguyen, D. (2023) · Environmental Earth Sciences · doi:10.1007/s12665-023-11337-9 · Cited by 6
“…relevance of each parameter was determined using a comparison scale proposed by Saaty (1987), which used integer numbers from 1 to 9, where 1 represents equal importance or the least contribution and 9 indicates extreme importance or the highest contribution (Saaty 1987; Hussain et al. 2021).”
InfluentialVargas, D., Vélez, A. C., Yépez, C. J., Bravo, J. J., & Gómez, J. C. O. (2023) · Journal of Engineering · doi:10.1155/2023/1988392 · Cited by 4
A Methodology for Comprehensive Quality Evaluation of Underground Space
Guo, J., Liu, K., & Ma, Y. (2023) · Acta Geologica Sinica (English) · doi:10.1111/1755-6724.15147 · Cited by 6
“The final weight was calculated using the analytic hierarchy process (Saaty, 1987).”
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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