{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T23:14:00Z","timestamp":1776294840800,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,14]],"date-time":"2020-07-14T00:00:00Z","timestamp":1594684800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"undefined  &lt;span style=&quot;color:gray;font-size:10px;&quot;&gt;undefined&lt;\/span&gt;","award":["15BTJ028"],"award-info":[{"award-number":["15BTJ028"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The paper considers the multi-criteria decision-making problem based on linguistic picture fuzzy information. Firstly, we propose the concept of linguistic picture fuzzy set(LPFS), where the positive-membership, the neutral-membership and the negative-membership are represented by linguistic variables, and its operation rules are also discussed. The linguistic picture fuzzy weighted averaging (LPFWA) operator and linguistic picture fuzzy weighted geometric (LPFWG) operator are developed based on the proposed operation rules. Secondly, we propose the generalized weighted distance measure, the generalized weighted Hausdorff distance measure, and the generalized hybrid weighted distance measure between LPFSs and discuss their properties. Thirdly, we extend the technique for order of preference by similarity to the ideal solution (TOPSIS) method and the TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method to the proposed distance measure, and the linguistic picture fuzzy entropy method is proposed to calculate the weights of the criteria. Finally, an illustrative example is given to verify the feasibility and effectiveness of the proposed methods, the comparative analysis with other existing methods and sensitivity analysis of the proposed methods are also discussed.<\/jats:p>","DOI":"10.3390\/sym12071170","type":"journal-article","created":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T05:10:30Z","timestamp":1595394630000},"page":"1170","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["The Linguistic Picture Fuzzy Set and Its Application in Multi-Criteria Decision-Making: An Illustration to the TOPSIS and TODIM Methods Based on Entropy Weight"],"prefix":"10.3390","volume":"12","author":[{"given":"Donghai","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Statistics, Hunan University of Science and Technology, Xiangtan 411201, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yan","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Statistics, Hunan University of Science and Technology, Xiangtan 411201, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zaiming","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,14]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy set","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_3","first-page":"409","article-title":"Picture fuzzy sets","volume":"30","author":"Cuong","year":"2014","journal-title":"J. Comput. Sci. Cybern."},{"key":"ref_4","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-I","volume":"8","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/0020-0255(75)90046-8","article-title":"The concept of a linguistic variable and its application to approximate reasoning-II","volume":"8","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/0020-0255(75)90017-1","article-title":"The concept of a linguistic variable and its application to approximate reasoning-III","volume":"9","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/S0165-0114(99)00024-X","article-title":"Linguistic decision analysis: Steps for solving decision problems under linguistic information","volume":"115","author":"Herrera","year":"2000","journal-title":"Fuzzy Sets Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1109\/TFUZZ.2007.896353","article-title":"A Fuzzy Linguistic Methodology to Deal With Unbalanced Linguistic Term Sets","volume":"16","author":"Herrera","year":"2008","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1016\/j.ins.2008.12.017","article-title":"Linguistic decision-making: Tools and applications","volume":"179","author":"Martinezlopez","year":"2009","journal-title":"Inf. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.ins.2013.04.006","article-title":"A group decision-making model dealing with comparative linguistic expressions based on hesitant fuzzy linguistic term sets","volume":"241","author":"Herrera","year":"2013","journal-title":"Inf. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.knosys.2014.12.009","article-title":"Qualitative decision-making with correlation coefficients of hesitant fuzzy linguistic term sets","volume":"76","author":"Liao","year":"2015","journal-title":"Knowl. Based Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1080\/18756891.2015.1061394","article-title":"An approach to multiple attribute group decision-making based on linguistic intuitionistic fuzzy numbers","volume":"8","author":"Chen","year":"2015","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_13","first-page":"66","article-title":"Picture fuzzy linguistic sets and their applications for multi-attribute group","volume":"55","author":"Ashraf","year":"2018","journal-title":"Nucleus"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1007\/s40815-017-0441-8","article-title":"On novel operational laws and aggregation operators of picture 2-tuple linguistic information for MCDM problems","volume":"20","author":"Zhang","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1007\/s12559-017-9523-z","article-title":"A novel picture fuzzy linguistic aggregation operator and its application to group decision-making","volume":"10","author":"Liu","year":"2018","journal-title":"Cogn. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1007\/s00500-016-2403-8","article-title":"Picture 2-tuple linguistic aggregation operators in multiple attribute decision-making","volume":"22","author":"Wei","year":"2018","journal-title":"Soft Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.inffus.2017.11.008","article-title":"An approach to quality function deployment based on probabilistic linguistic term sets and ORESTE method for multi-expert multi-criteria decision-making","volume":"43","author":"Wu","year":"2018","journal-title":"Inf. Fusion"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"491","DOI":"10.3846\/16111699.2016.1197147","article-title":"Picture fuzzy cross-entropy for multiple attribute decision-making problems","volume":"17","author":"Wei","year":"2016","journal-title":"J. Bus. Econ. Manag."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1007\/s40815-016-0266-x","article-title":"Picture 2-Tuple Linguistic Bonferroni Mean Operators and Their Application to Multiple Attribute Decision Making","volume":"19","author":"Wei","year":"2017","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_20","first-page":"477","article-title":"Some geometric aggregation operators based on picture fuzzy sets and their application in multiple attribute decision-making","volume":"37","author":"Wang","year":"2017","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.asoc.2017.12.014","article-title":"Picture fuzzy normalized projection-based VIKOR method for the risk evaluation of construction project","volume":"64","author":"Wang","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"591","DOI":"10.3233\/IFS-141338","article-title":"Correlation coefficients for picture fuzzy sets","volume":"28","author":"Singh","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"713","DOI":"10.3233\/JIFS-161798","article-title":"Picture fuzzy aggregation operators and their application to multiple attribute decision-making","volume":"33","author":"Wei","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/j.asoc.2016.05.009","article-title":"Generalized picture distance measure and applications to picture fuzzy clustering","volume":"46","author":"Son","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s10700-016-9249-5","article-title":"Measuring analogousness in picture fuzzy sets: From picture distance measures to picture association measures","volume":"16","author":"Son","year":"2017","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.asoc.2018.10.021","article-title":"Picture fuzzy Dombi aggregation operators: Application to MADM process","volume":"74","author":"Jana","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2128","DOI":"10.1016\/j.ins.2011.01.028","article-title":"Distance and similarity measures for hesitant fuzzy sets","volume":"181","author":"Xu","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2063","DOI":"10.1016\/j.patrec.2005.03.018","article-title":"Distance measure between intuitionistic fuzzy sets","volume":"26","author":"Wang","year":"2005","journal-title":"Pattern Recognit. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1572","DOI":"10.1002\/int.22108","article-title":"Some cosine similarity measures and distance measures between q-rung orthopair fuzzy sets","volume":"34","author":"Liu","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1002\/int.21934","article-title":"Distance measure of Pythagorean fuzzy sets","volume":"33","author":"Li","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/j.asoc.2019.03.043","article-title":"Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis","volume":"79","author":"Xiao","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_32","unstructured":"Yoon, K., and Hwang, C.L. (1981). TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) Ca Multiple Attribute Decision Making, Springer."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"267","DOI":"10.3233\/JIFS-171164","article-title":"Extension of TOPSIS method base on Choquet integral under interval-valued Pythagorean fuzzy environment","volume":"34","author":"Abdullah","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"878","DOI":"10.1002\/int.22079","article-title":"Fermatean fuzzy linguistic set and its application in multicriteria decision-making","volume":"34","author":"Liu","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/j.ins.2019.05.074","article-title":"Fostering linguistic decision-making under uncertainty: A proportional interval type-2 hesitant fuzzy TOPSIS approach based on Hamacher aggregation operators and andness optimization models","volume":"500","author":"Chen","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_36","first-page":"113","article-title":"TODIM: Basics and apllication to multicriteria ranking of projects with environmental impacts","volume":"16","author":"Gomes","year":"1991","journal-title":"Found. Control Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.ejor.2007.10.046","article-title":"An application of the TODIM method to the multicriteria rental evaluation of residential properties","volume":"193","author":"Gomes","year":"2009","journal-title":"Eur. J. Oper. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.knosys.2014.02.006","article-title":"The TODIM analysis approach based on novel measured functions under hesitant fuzzy environment","volume":"61","author":"Zhang","year":"2014","journal-title":"Knowl. Based Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s00521-016-2436-z","article-title":"A projection-based TODIM method under multi-valued neutrosophic environments and its application in personnel selection","volume":"29","author":"Ji","year":"2018","journal-title":"Neural Comput. Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2807","DOI":"10.1002\/int.22162","article-title":"Distance measure for Fermatean fuzzy linguistic term sets based on linguistic scale function: An illustration of the TODIM and TOPSIS methods","volume":"34","author":"Liu","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_41","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":"2004","journal-title":"Inf. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/j.omega.2005.08.004","article-title":"Global supplier development considering risk factors using fuzzy extended AHP-based approach","volume":"35","author":"Chan","year":"2007","journal-title":"Omega"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.knosys.2013.08.028","article-title":"IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision-making","volume":"53","author":"Krohling","year":"2013","journal-title":"Knowl. Based Syst."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"11363","DOI":"10.1016\/j.eswa.2009.03.039","article-title":"A multi-criteria intuitionistic fuzzy group decision-making for supplier selection with TOPSIS method","volume":"36","author":"Boran","year":"2009","journal-title":"Expert Syst. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1170\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:51:25Z","timestamp":1760176285000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1170"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,14]]},"references-count":44,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["sym12071170"],"URL":"https:\/\/doi.org\/10.3390\/sym12071170","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,14]]}}}