{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T18:04:22Z","timestamp":1781633062134,"version":"3.54.5"},"reference-count":55,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,3,31]],"date-time":"2018-03-31T00:00:00Z","timestamp":1522454400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Determination of subjective weights, which are based on the opinions and preferences of decision-makers, is one of the most important matters in the process of multi-criteria decision-making (MCDM). Step-wise Weight Assessment Ratio Analysis (SWARA) is an efficient method for obtaining the subjective weights of criteria in the MCDM problems. On the other hand, decision-makers may express their opinions with a degree of uncertainty. Using the symmetric interval type-2 fuzzy sets enables us to not only capture the uncertainty of information flexibly but also to perform computations simply. In this paper, we propose an extended SWARA method with symmetric interval type-2 fuzzy sets to determine the weights of criteria based on the opinions of a group of decision-makers. The weights determined by the proposed approach involve the uncertainty of decision-makers\u2019 preferences and the symmetric form of the weights makes them more interpretable. To show the procedure of the proposed approach, it is used to determine the importance of intellectual capital dimensions and components in a company. The results show that the proposed approach is efficient in determining the subjective weights of criteria and capturing the uncertainty of information.<\/jats:p>","DOI":"10.3390\/sym10040091","type":"journal-article","created":{"date-parts":[[2018,4,2]],"date-time":"2018-04-02T12:32:20Z","timestamp":1522672340000},"page":"91","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":48,"title":["An Extended Step-Wise Weight Assessment Ratio Analysis with Symmetric Interval Type-2 Fuzzy Sets for Determining the Subjective Weights of Criteria in Multi-Criteria Decision-Making Problems"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0362-1633","authenticated-orcid":false,"given":"Mehdi","family":"Keshavarz-Ghorabaee","sequence":"first","affiliation":[{"name":"Department of Industrial Management, Faculty of Management and Accounting, Allameh Tabataba\u2019i University, Tehran 1489684511, Iran"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maghsoud","family":"Amiri","sequence":"additional","affiliation":[{"name":"Department of Industrial Management, Faculty of Management and Accounting, Allameh Tabataba\u2019i University, Tehran 1489684511, Iran"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3201-949X","authenticated-orcid":false,"given":"Edmundas","family":"Zavadskas","sequence":"additional","affiliation":[{"name":"Department of Construction Management and Real Estate, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5835-9388","authenticated-orcid":false,"given":"Zenonas","family":"Turskis","sequence":"additional","affiliation":[{"name":"Department of Construction Management and Real Estate, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1734-3216","authenticated-orcid":false,"given":"Jurgita","family":"Antucheviciene","sequence":"additional","affiliation":[{"name":"Department of Construction Management and Real Estate, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1111\/itor.12102","article-title":"A study of decision process in MCDM problems with large number of criteria","volume":"22","author":"Liu","year":"2015","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1016\/0305-0548(94)00059-H","article-title":"Determining objective weights in multiple criteria problems: The CRITIC method","volume":"22","author":"Diakoulaki","year":"1995","journal-title":"Comput. Oper. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/0749-5978(90)90041-7","article-title":"Judgments of relative importance in decision making: Global vs local interpretations of subjective weight","volume":"47","author":"Goldstein","year":"1990","journal-title":"Organ. Behav. Hum. Decis. Process."},{"key":"ref_4","unstructured":"Simos, J. (1990). Evaluer L\u2019impact sur L\u2019environnement: Une Approche Originale par L\u2019analyse Multicrit\u00e8re et la N\u00e9gociation, Presses Polytechniques et Universitaires Romandes."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Zardari, N.H., Ahmed, K., Shirazi, S.M., and Yusop, Z.B. (2014). Weighting Methods and Their Effects on Multi-Criteria Decision Making Model Outcomes in Water Resources Management, Springer.","DOI":"10.1007\/978-3-319-12586-2"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"243","DOI":"10.3846\/jbem.2010.12","article-title":"Selection of rational dispute resolution method by applying new step-wise weight assessment ratio analysis (SWARA)","volume":"11","author":"Kersuliene","year":"2010","journal-title":"J. Bus. Econ. Manag."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1016\/j.catena.2015.07.020","article-title":"A new hybrid model using step-wise weight assessment ratio analysis (SWARA) technique and adaptive neuro-fuzzy inference system (ANFIS) for regional landslide hazard assessment in Iran","volume":"135","author":"Dehnavi","year":"2015","journal-title":"CATENA"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"49","DOI":"10.15388\/Informatica.2016.76","article-title":"The framework for the selection of personnel based on the SWARA and ARAS methods under uncertainties","volume":"27","author":"Karabasevic","year":"2016","journal-title":"Informatica"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"831","DOI":"10.3846\/13923730.2016.1189457","article-title":"Model for rapid assessment of vulnerability of office buildings to blast using SWARA and SMART methods (a case study of swiss re tower)","volume":"22","author":"Nakhaei","year":"2016","journal-title":"J. Civ. Eng. Manag."},{"key":"ref_10","first-page":"120","article-title":"An integrated decision making approach for ERP system selection using SWARA and PROMETHEE method","volume":"3","author":"Shukla","year":"2016","journal-title":"Int. J. Intell. Enterp."},{"key":"ref_11","first-page":"140","article-title":"A new integrated decision making approach based on SWARA and OCRA methods for the hotel selection problem","volume":"8","year":"2016","journal-title":"Int. J. Adv. Oper. Manag."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2401","DOI":"10.1007\/s00170-016-9880-x","article-title":"Sustainable third-party reverse logistic provider selection with fuzzy SWARA and fuzzy MOORA in plastic industry","volume":"91","author":"Mavi","year":"2017","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1007\/s12517-017-3290-8","article-title":"Application of stepwise weight assessment ratio analysis (SWARA) for copper prospectivity mapping in the Anarak region, central Iran","volume":"10","author":"Panahi","year":"2017","journal-title":"Arab. J. Geosci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"599","DOI":"10.3846\/16111699.2017.1327455","article-title":"New group decision-making ARCAS approach based on the integration of the SWARA and the ARAS methods adapted for negotiations","volume":"18","author":"Stanujkic","year":"2017","journal-title":"J. Bus. Econ. Manag."},{"key":"ref_15","first-page":"15","article-title":"An approach to criteria weights determination by integrating the Delphi and the adapted SWARA methods","volume":"22","year":"2017","journal-title":"Manag. J. Theory Pract. Manag."},{"key":"ref_16","first-page":"39","article-title":"A new approach for selecting alternatives based on the adapted weighted sum and the SWARA methods: A case of personnel selection","volume":"51","author":"Stanujkic","year":"2017","journal-title":"Econ. Comput. Econ. Cybern. Stud. Res."},{"key":"ref_17","first-page":"75","article-title":"An approach to personnel selection in the tourism industry based on the SWARA and the WASPAS methods","volume":"51","author":"Urosevic","year":"2017","journal-title":"Econ. Comput. Econ. Cybern. Stud. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"70","DOI":"10.15240\/tul\/001\/2017-3-005","article-title":"Analysing larg supply chain management competitive strategies in Iranian cement industries","volume":"20","author":"Jamali","year":"2017","journal-title":"E+M Ekon. Manag."},{"key":"ref_19","first-page":"117","article-title":"Integrated multi-criteria evaluation of house\u2019s plan shape based on the EDAS and SWARA methods","volume":"9","author":"Turskis","year":"2017","journal-title":"Eng. Struct. Technol."},{"key":"ref_20","first-page":"67","article-title":"Comparison of Max100, SWARA and pairwise weight elicitation methods","volume":"7","author":"Tayyar","year":"2017","journal-title":"Int. J. Eng. Res. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"524","DOI":"10.3846\/13923730.2017.1281842","article-title":"Hybrid SWARA-COPRAS method for risk assessment in deep foundation excavation project: An Iranian case study","volume":"23","author":"Valipour","year":"2017","journal-title":"J. Civ. Eng. Manag."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.acme.2017.04.011","article-title":"A new hybrid fuzzy MCDM approach for evaluation of construction equipment with sustainability considerations","volume":"18","author":"Amiri","year":"2018","journal-title":"Arch. Civ. Mech. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1002\/hfm.20713","article-title":"Competency-based IT personnel selection using a hybrid SWARA and ARAS-G methodology","volume":"28","author":"Dahooie","year":"2018","journal-title":"Hum. Factors Ergon. Manuf. Serv. Ind."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.asoc.2017.03.045","article-title":"A systematic review and meta-analysis of SWARA and WASPAS methods: Theory and applications with recent fuzzy developments","volume":"57","author":"Mardani","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Liu, Z., Qin, K., and Pei, Z. (2017). A method for fuzzy soft sets in decision-making based on an ideal solution. Symmetry, 9.","DOI":"10.3390\/sym9100246"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Alcantud, J., Rambaud, S., and Torrecillas, M. (2017). Valuation fuzzy soft sets: A flexible fuzzy soft set based decision making procedure for the valuation of assets. Symmetry, 9.","DOI":"10.3390\/sym9110253"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Wang, Z.-X., and Li, J. (2017). Correlation coefficients of probabilistic hesitant fuzzy elements and their applications to evaluation of the alternatives. Symmetry, 9.","DOI":"10.3390\/sym9110259"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Liu, P. (2017). Multiple attribute decision-making methods based on normal intuitionistic fuzzy interaction aggregation operators. Symmetry, 9.","DOI":"10.3390\/sym9110261"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Liu, P., Mahmood, T., and Khan, Q. (2017). Multi-attribute decision-making based on prioritized aggregation operator under hesitant intuitionistic fuzzy linguistic environment. Symmetry, 9.","DOI":"10.3390\/sym9110270"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Ren, F.L., Kong, M.M., and Pei, Z. (2017). A new hesitant fuzzy linguistic TOPSIS method for group multi-criteria linguistic decision making. Symmetry, 9.","DOI":"10.3390\/sym9120289"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Garcia, N., Puente, J., Fernandez, I., and Priore, P. (2018). Suitability of a consensual fuzzy inference system to evaluate suppliers of strategic products. Symmetry, 10.","DOI":"10.3390\/sym10010022"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0020-0255(75)90036-5","article-title":"The concept of a linguistic variable and its application to approximate reasoning\u2014I","volume":"8","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1109\/91.873577","article-title":"Interval type-2 fuzzy logic systems: Theory and design","volume":"8","author":"Liang","year":"2000","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1109\/TFUZZ.2006.879986","article-title":"Interval type-2 fuzzy logic systems made simple","volume":"14","author":"Mendel","year":"2006","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1109\/TFUZZ.2006.881441","article-title":"Type-2 fuzzistics for symmetric interval type-2 fuzzy sets: Part 1, forward problems","volume":"14","author":"Mendel","year":"2006","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_37","first-page":"48","article-title":"Multi-criteria group decision-making using an extended EDAS method with interval type-2 fuzzy sets","volume":"20","author":"Amiri","year":"2017","journal-title":"E+M Ekon. Manag."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"269","DOI":"10.15388\/Informatica.2017.129","article-title":"Interval type-2 fuzzy c-control charts: An application in a food company","volume":"28","author":"Senturk","year":"2017","journal-title":"Informatica"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.oceaneng.2016.11.010","article-title":"Application of AHP and VIKOR methods under interval type 2 fuzzy environment in maritime transportation","volume":"129","author":"Soner","year":"2017","journal-title":"Ocean Eng."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1016\/j.cie.2017.08.017","article-title":"A new multi-criteria model based on interval type-2 fuzzy sets and EDAS method for supplier evaluation and order allocation with environmental considerations","volume":"112","author":"Amiri","year":"2017","journal-title":"Comput. Ind. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1016\/j.ejor.2016.09.059","article-title":"An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment","volume":"258","author":"Qin","year":"2017","journal-title":"Eur. J. Oper. Res."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.eswa.2016.11.001","article-title":"Development of an interval type-2 fuzzy sets based hierarchical MADM model by combining DEMATEL and TOPSIS","volume":"70","year":"2017","journal-title":"Expert Syst. Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.jairtraman.2016.11.013","article-title":"Airline new route selection based on interval type-2 fuzzy MCDM: A case study of new route between Turkey-North American region destinations","volume":"59","author":"Deveci","year":"2017","journal-title":"J. Air Transp. Manag."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1016\/j.asoc.2017.04.001","article-title":"An ELECTRE I-based multi-criteria group decision making method with interval type-2 fuzzy numbers and its application to supplier selection","volume":"57","author":"Zhong","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"3051","DOI":"10.1007\/s00500-015-1990-0","article-title":"A fuzzy multi-criteria group decision making based on ranking interval type-2 fuzzy variables and an application to transportation mode selection problem","volume":"21","author":"Kundu","year":"2017","journal-title":"Soft Comput."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3207","DOI":"10.1007\/s00500-015-2004-y","article-title":"A multiple attribute interval type-2 fuzzy group decision making and its application to supplier selection with extended LINMAP method","volume":"21","author":"Qin","year":"2017","journal-title":"Soft Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3469","DOI":"10.3233\/JIFS-16608","article-title":"GRP method for multiple attribute group decision making under trapezoidal interval type-2 fuzzy environment","volume":"33","author":"Ju","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1809","DOI":"10.3233\/JIFS-161188","article-title":"Interval type-2 fuzzy information measures and their applications to attribute decision-making approach","volume":"33","author":"Yao","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Liu, K., Liu, Y., and Qin, J. (2018). An integrated ANP-VIKOR methodology for sustainable supplier selection with interval type-2 fuzzy sets. Granul. Comput.","DOI":"10.1007\/s41066-017-0071-4"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1016\/j.ins.2017.12.006","article-title":"An interval type-2 fuzzy TOPSIS model for large scale group decision making problems with social network information","volume":"432","author":"Wu","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/j.knosys.2015.06.004","article-title":"A comprehensive review of multi criteria decision making approaches based on interval type-2 fuzzy sets","volume":"85","author":"Celik","year":"2015","journal-title":"Knowl.-Based Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/S0020-7373(81)80051-X","article-title":"The IF THEN ELSE statement and interval-valued fuzzy sets of higher type","volume":"15","author":"Hisdal","year":"1981","journal-title":"Int. J. Man Mach. Stud."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/S0165-0114(98)00176-6","article-title":"A new approach for defuzzification","volume":"111","author":"Ma","year":"2000","journal-title":"Fuzzy Sets Syst."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.rcim.2015.04.007","article-title":"Developing an MCDM method for robot selection with interval type-2 fuzzy sets","volume":"37","year":"2016","journal-title":"Robot. Comput.-Integr. Manuf."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.ijar.2006.07.005","article-title":"Prioritization of organizational capital measurement indicators using fuzzy AHP","volume":"44","author":"Bozbura","year":"2007","journal-title":"Int. J. Approx. Reason."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/4\/91\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:59:14Z","timestamp":1760194754000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/4\/91"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,31]]},"references-count":55,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,4]]}},"alternative-id":["sym10040091"],"URL":"https:\/\/doi.org\/10.3390\/sym10040091","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,3,31]]}}}