{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T02:12:43Z","timestamp":1767665563778,"version":"build-2065373602"},"reference-count":52,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,11]],"date-time":"2022-01-11T00:00:00Z","timestamp":1641859200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Natural Science Foundation of Fujian Province","award":["2020J01576"],"award-info":[{"award-number":["2020J01576"]}]},{"name":"the Science and Technology Innovation Special Fund Project of Fujian agriculture and Forestry University","award":["CXZX2020110A"],"award-info":[{"award-number":["CXZX2020110A"]}]},{"name":"the Construction Fund for Digital Fujian Big Data for Agriculture and Forestry","award":["KJG18019A"],"award-info":[{"award-number":["KJG18019A"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Most linguistic-based approaches to multi-attribute group decision making (MAGDM) use symmetric, uniformly distributed sets of additive linguistic terms to express the opinions of decision makers. However, in reality, there are also some problems that require the use of asymmetric, uneven, i.e., non-equilibrium, multiplicative linguistic term sets to express the evaluation. The purpose of this paper is to propose a new approach to MAGDM under multiplicative linguistic information. The aggregation of linguistic data is an important component in MAGDM. To solve this problem, we define a chi-square for measuring the difference between multiplicative linguistic term sets. Furthermore, the linguistic generalized weighted logarithm multiple averaging (LGWLMA) operator and linguistic generalized ordered weighted logarithm multiple averaging (LGOWLMA) operator are proposed based on chi-square deviation. On the basis of the proposed two operators, we develop a novel approach to GDM with multiplicative linguistic term sets. Finally, the evaluation of transport logistics enterprises is developed to illustrate the validity and practicality of the proposed approach.<\/jats:p>","DOI":"10.3390\/sym14010136","type":"journal-article","created":{"date-parts":[[2022,1,11]],"date-time":"2022-01-11T20:33:04Z","timestamp":1641933184000},"page":"136","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Novel Approach for Multiplicative Linguistic Group Decision Making Based on Symmetrical Linguistic Chi-Square Deviation and VIKOR Method"],"prefix":"10.3390","volume":"14","author":[{"given":"Zhiwei","family":"Gong","sequence":"first","affiliation":[{"name":"Center for Discrete Mathematics and Theoretical Computer Science, Fuzhou University, Fuzhou 350116, China"},{"name":"College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou 350002, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4959-2329","authenticated-orcid":false,"given":"Jian","family":"Lin","sequence":"additional","affiliation":[{"name":"College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou 350002, China"}]},{"given":"Ling","family":"Weng","sequence":"additional","affiliation":[{"name":"College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou 350002, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.inffus.2018.10.011","article-title":"Managing incomplete preferences and consistency improvement in hesitant fuzzy linguistic preference relations with applications in group decision making","volume":"51","author":"Liu","year":"2019","journal-title":"Inf. Fusion"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.asoc.2015.09.037","article-title":"An integrated fuzzy multi criteria group decision making approach for ERP system selection","volume":"38","author":"Efe","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3213","DOI":"10.1007\/s13369-017-2986-0","article-title":"Some aggregation operators for linguistic intuitionistic fuzzy set and its application to group decision-making process using the set pair analysis","volume":"43","author":"Garg","year":"2018","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_4","first-page":"131","article-title":"Operators on single valued trapezoidal neutrosophic numbers and SVTN-group decision making","volume":"22","author":"Deli","year":"2018","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.knosys.2018.04.022","article-title":"A continuous interval-valued linguistic ORESTE method for multi-criteria group decision making","volume":"153","author":"Liao","year":"2018","journal-title":"Knowl.-Based Syst."},{"key":"ref_6","first-page":"1449","article-title":"A new method for group decision making with hesitant fuzzy preference relations based on multiplicative consistency","volume":"28","author":"Wan","year":"2019","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.knosys.2015.09.027","article-title":"Decision-making model to generate novel emergency response plans for improving coordination during large-scale emergencies","volume":"90","author":"Pan","year":"2015","journal-title":"Knowl.-Based Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.cie.2017.11.025","article-title":"A two-stage consensus method for large-scale multi-attribute group decision making with an application to earthquake shelter selection","volume":"116","author":"Xu","year":"2018","journal-title":"Comput. Ind. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1016\/j.jclepro.2018.05.275","article-title":"Proposal of a green index for small and medium-sized enterprises: A multiple criteria group decision-making approach","volume":"196","author":"Rita","year":"2018","journal-title":"J. Clean. Prod."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2909","DOI":"10.3233\/JIFS-191443","article-title":"GRA method for waste incineration plants location problem with probabilistic linguistic multiple attribute group decision making","volume":"39","author":"Lei","year":"2020","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2154","DOI":"10.1016\/j.jclepro.2016.11.067","article-title":"A cloud decision framework in pure 2-tuple linguistic setting and its application for low-speed wind farm site selection","volume":"142","author":"Wu","year":"2017","journal-title":"J. Clean. Prod."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1007\/s40815-020-00885-y","article-title":"Medical supplier selection with a group decision-making method based on incomplete probabilistic linguistic preference relations","volume":"23","author":"Liao","year":"2021","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.ins.2014.07.034","article-title":"Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems","volume":"288","author":"Wang","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"816","DOI":"10.1016\/j.knosys.2011.03.003","article-title":"On compatibility of uncertain additive linguistic preference relations and its application in the group decision making","volume":"24","author":"Chen","year":"2011","journal-title":"Knowl.-Based Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1016\/j.asoc.2014.08.035","article-title":"Linguistic group decision making with induced aggregation operators and probabilistic information","volume":"24","author":"Merigo","year":"2014","journal-title":"Appl. Soft Comput. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4000","DOI":"10.1016\/j.eswa.2009.11.016","article-title":"A method for group decision-making based on multi-granularity uncertain linguistic information","volume":"37","author":"Fan","year":"2009","journal-title":"Expert Syst. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.ins.2003.10.006","article-title":"A method based on linguistic aggregation operators for group decision making with linguistic preference relations","volume":"166","author":"Xu","year":"2003","journal-title":"Inf. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.ins.2004.02.003","article-title":"Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment","volume":"168","author":"Xu","year":"2004","journal-title":"Inf. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"115088","DOI":"10.1016\/j.eswa.2021.115088","article-title":"Best-Worst method and Hamacher aggregation operations for intuitionistic 2-tuple linguistic sets","volume":"181","author":"Faizi","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1006","DOI":"10.1109\/TFUZZ.2013.2239650","article-title":"Linguistic computational model based on 2-tuples and intervals","volume":"21","author":"Dong","year":"2013","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/s00500-014-1259-z","article-title":"Multi-criteria group decision making method based on interval 2-tuple linguistic information and Choquet integral aggregation operators","volume":"19","author":"Wang","year":"2015","journal-title":"Soft Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"5256","DOI":"10.1016\/j.apm.2014.04.022","article-title":"A new method for multiple criteria group decision making with incomplete weight information under linguistic environment","volume":"38","author":"Ju","year":"2014","journal-title":"Appl. Math. Model."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.omega.2015.12.005","article-title":"Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations","volume":"65","author":"Wu","year":"2016","journal-title":"Omega"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1142\/S0218488504003211","article-title":"EOWA and EOWG operators for aggregating linguistic labels based on linguistic preference relations","volume":"12","author":"Xu","year":"2004","journal-title":"Fuzziness Knowl.-Based Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1007\/s10726-006-9034-x","article-title":"A practical procedure for group decision making under incomplete multiplicative linguistic preference relations","volume":"15","author":"Xu","year":"2006","journal-title":"Group Decis. Negot."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1016\/j.ins.2021.05.056","article-title":"Group decision making based on consistency and consensus analysis of dual multiplicative linguistic preference relations","volume":"572","author":"Meng","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"75041","DOI":"10.1109\/ACCESS.2019.2919505","article-title":"Expanding grey relational analysis with the comparable degree for dual probabilistic multiplicative linguistic term sets and its application on the cloud enterprise","volume":"7","author":"Xie","year":"2019","journal-title":"IEEE Access"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1611","DOI":"10.1111\/itor.12222","article-title":"An approach to multiplicative linguistic group decision making based on possibility degrees","volume":"25","author":"Xia","year":"2018","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1037","DOI":"10.3233\/JIFS-17365","article-title":"A consistency-based approach to group decision making with uncertain multiplicative linguistic fuzzy preference relations","volume":"35","author":"Tang","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1016\/j.dss.2004.08.011","article-title":"An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations","volume":"41","author":"Xu","year":"2004","journal-title":"Decis. Support Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"19848","DOI":"10.1109\/ACCESS.2019.2892239","article-title":"A novel group decision making method under uncertain multiplicative linguistic environment for information system selection","volume":"7","author":"Lin","year":"2019","journal-title":"IEEE Access"},{"key":"ref_33","first-page":"604","article-title":"Uncertain linguistic information based C-OWA and C-OWG operators and their application","volume":"6","author":"Zhang","year":"2005","journal-title":"J. Pla Univ. Sci. Technol."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Chodha, V., Dubey, R., Kumar, R., Singh, S., and Kaur, S. (2021). Selection of industrial arc welding robot with TOPSIS and Entropy MCDM techniques. Mater. Today Proc.","DOI":"10.1016\/j.matpr.2021.04.487"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Dhara, A., Kaur, G., Kishan, P.M., Majumder, A., and Yadav, R. (2021). An efficient decision support system for selecting very light business jet using CRITIC-TOPSIS method. Aircr. Eng. Aerosp. Technol.","DOI":"10.1108\/AEAT-04-2021-0135"},{"key":"ref_36","first-page":"1","article-title":"Priority ranking of road pavements for maintenance using analytical hierarchy process and VIKOR method","volume":"28","author":"Chundi","year":"2022","journal-title":"Innov. Infrastruct. Solut."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4899","DOI":"10.1007\/s00500-019-04244-4","article-title":"Multi-criteria decision making approach based on PROMETHEE with probabilistic simplified neutrosophic sets","volume":"24","author":"Altun","year":"2020","journal-title":"Soft Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1016\/j.cie.2019.07.052","article-title":"An extended MULTIMOORA method for probabilistic linguistic multi-criteria group decision-making based on prospect theory","volume":"136","author":"Liu","year":"2019","journal-title":"Comput. Ind. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/exsy.12593","article-title":"An integrated probabilistic linguistic projection method for MCGDM based on ELECTRE III and the weighted convex median voting rule","volume":"37","author":"Chen","year":"2020","journal-title":"Expert Syst."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Sa\u0142abun, W., Watr\u00f3bski, J., and Shekhovtsov, A. (2020). Are MCDA methods benchmarkable? A comparative study of TOPSIS, VIKOR, COPRAS, and PROMETHEE II methods. Symmetry, 12.","DOI":"10.3390\/sym12091549"},{"key":"ref_41","unstructured":"Patel, N. (2013). A Comparative Analysis of TOPSIS & VIKOR Methods in the Selection of Industrial Robots. [Ph.D. Dissertation, National Institute of Technology]."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"4578","DOI":"10.1016\/j.procs.2021.09.236","article-title":"Comparison of Fuzzy TOPSIS, Fuzzy VIKOR, Fuzzy WASPAS and Fuzzy MMOORA methods in the housing selection problem","volume":"192","author":"Kizielewicz","year":"2021","journal-title":"Procedia Comput. Sci."},{"key":"ref_43","unstructured":"Opricovic, S. (1998). Multicriteria Optimization of Civil Engineering Systems. [Ph.D. Thesis, Faculty of Civil Engineering]. Volume 2."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/S0377-2217(03)00020-1","article-title":"Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS","volume":"156","author":"Opricovic","year":"2004","journal-title":"Eur. J. Oper. Res."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1080\/03081070500422836","article-title":"On generalized induced linguistic aggregation operators","volume":"35","author":"Xu","year":"2006","journal-title":"Int. J. Gen. Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s10700-005-4913-1","article-title":"A direct approach to group decision making with uncertain additive linguistic preference relations","volume":"5","author":"Xu","year":"2006","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1109\/TFUZZ.2003.822679","article-title":"Quantitative weights and aggregation","volume":"12","author":"Calvo","year":"2004","journal-title":"IEEE Trans Fuzzy Syst."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1420","DOI":"10.1016\/j.fss.2009.05.012","article-title":"Aggregation functions based on penalties","volume":"161","author":"Calvo","year":"2010","journal-title":"Fuzzy Sets Syst."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ins.2010.08.043","article-title":"Aggregation functions: Means","volume":"181","author":"Grabisch","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/0165-0114(93)90194-M","article-title":"Families of OWA operators","volume":"59","author":"Yager","year":"1993","journal-title":"Fuzzy Sets Syst."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s10726-011-9267-1","article-title":"Generalized multiple averaging operators and their applications to group decision making","volume":"22","author":"Zhou","year":"2013","journal-title":"Group Decis. Negot."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1020","DOI":"10.1016\/S1001-0742(06)60032-6","article-title":"Entropy method for determination of weight of evaluating indicators in fuzzy synthetic evaluation for water quality assessment","volume":"18","author":"Zou","year":"2006","journal-title":"J. Environ. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/136\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:02:02Z","timestamp":1760364122000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/136"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,11]]},"references-count":52,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["sym14010136"],"URL":"https:\/\/doi.org\/10.3390\/sym14010136","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,1,11]]}}}