grey relational analysis

Let $X\_0=\backslash left(x\_0\backslash left(1\backslash right),x\_0\backslash left(2\backslash right),\backslash dots\; ,x\_0\backslash left(n\backslash right)\backslash right)$ is an ideal data set and$X\_k=\backslash left(x\_k\backslash left(1\backslash right),x\_k\backslash left(2\backslash right),\backslash dots\; ,x\_k\backslash left(n\backslash right)\backslash right),k\backslash mathrm\{\backslash \; =\backslash \; 1,2,3,\backslash dots\; ,\}m$ are the alternative data sets of the same length. The Grey Relational Grade (GRG) between the two data sets is given by {{cite journal |last1=Javed |first1=Saad Ahmed |last2=Gunasekaran |first2=Angappa |last3=Mahmoudi |first3=Amin |date=2022 |title=DGRA: Multi-sourcing and Supplier Classification through Dynamic Grey Relational Analysis Method |journal=Computers & Industrial Engineering |volume=173 |pages=108674 |doi=10.1016/j.cie.2022.108674|s2cid=252478074 }}

$\{\backslash mathit\{\backslash Gamma\}\}\_\{0k\}=\backslash int^n\_\{j=1\}\{w\backslash left(j\backslash right)\backslash times\; \{\backslash gamma\; \}\_\{0k\}\backslash left(j\backslash right)\}$

where the Grey Relational Coefficients (GRC) is

$\{\backslash gamma\; \}\_\{0k\}\backslash left(j\backslash right)=\backslash frac\{\{\{\backslash min\}\_k\; \{\backslash min\}\_j\backslash \; \}|x\_0\backslash left(j\backslash right)-x\_k(j)|+\{\{\{\backslash xi\; \backslash left(j\backslash right)\}\_\{0k\}\backslash mathrm\{\backslash \; \}\backslash max\}\_k\; \{\backslash max\}\_j|x\_0\backslash left(j\backslash right)-x\_k(j)|\backslash \; \}\}\{|x\_0\backslash left(j\backslash right)-x\_k(j)|+\{\{\{\backslash xi\; \backslash left(j\backslash right)\}\_\{0k\}\backslash mathrm\{\backslash \; \}\backslash max\}\_k\; \{\backslash max\}\_j|x\_0\backslash left(j\backslash right)-x\_k(j)|\backslash \; \}\}$

where, $w\backslash left(j\backslash right)$ is the weight of the elements of the data sets, and is needed when the GRA method is used to solve multiple criteria decision-making problems. Here, $\backslash xi\; \backslash left(j\backslash right)\backslash in\; (0,1]$ denotes the Dynamic Distinguishing Coefficient. Thus, the GRA model defined in this way is called Dynamic Grey Relational Analysis (Dynamic GRA) model. It is the generalized form of Deng's GRA model.

GRA is an important part of grey system theory, pioneered by Deng Julong in 1982.{{Cite journal |last=Ju-Long |first=Deng |date=1982 |title=Control problems of grey systems |url=https://linkinghub.elsevier.com/retrieve/pii/S016769118280025X |journal=Systems & Control Letters |language=en |volume=1 |issue=5 |pages=288–294 |doi=10.1016/S0167-6911(82)80025-X |via=Elsevier Science Direct|url-access=subscription }} A grey system means that a system in which part of information is known and part of information is unknown. Formally, grey systems theory describes uncertainty by interval-valued unknowns called grey numbers, with the width of the interval reflecting more or less precise knowledge.{{cite journal |last1=Liu |first1=Sifeng |last2=Yang |first2=Yingjie | last3=Xie | first3=Naiming | last4=Forrest | first4=Jeffrey |date=2016 |title=New progress of Grey System Theory in the new millennium |url=http://www.ttcenter.ir/ArticleFiles/ENARTICLE/3602.pdf |journal=Grey Systems: Theory and Applications |volume=6 |issue=1 |pages=2–31 |doi=10.1108/GS-09-2015-0054 |hdl=2086/12096 |s2cid=39408530 |access-date=21 December 2021}} With this definition, information quantity and quality form a continuum from a total lack of information to complete information – from black through grey to white. Since uncertainty always exists, one is always somewhere in the middle, somewhere between the extremes, somewhere in the grey area. Grey analysis then comes to a clear set of statements about system solutions{{Specify|reason=|date=March 2020}}. At one extreme, no solution can be defined for a system with no information. At the other extreme, a system with perfect information has a unique solution. In the middle, grey systems will give a variety of available solutions. Grey relational analysis does not attempt to find the best solution, but does provide techniques for determining a good solution, an appropriate solution for real-world problems. The theory inspired many noted scholars and business leaders like Jeffrey Yi-Lin Forrest, Liu Sifeng, Ren Zhengfei and Joseph L. Badaracco, a professor at Harvard Business School.

The theory has been applied in various fields of engineering and management. Initially, the grey method was adapted to effectively study air pollution{{Cite journal|last1=Pai|first1=Tzu-Yi|last2=Hanaki|first2=Keisuke|last3=Chiou|first3=Ren-Jie|date=2013-03-27|title=Forecasting Hourly Roadside Particulate Matter in Taipei County of Taiwan Based on First-Order and One-Variable Grey Model|journal=CLEAN - Soil, Air, Water|volume=41|issue=8|pages=737–742|doi=10.1002/clen.201000402|bibcode=2013CSAW...41..737P }} and subsequently used to investigate the nonlinear multiple-dimensional model of the socio-economic activities’ impact on the city air pollution.{{Cite journal |last1=Xiaolu |first1=Li |last2=Zheng |first2=Wenfeng |last3=Yin |first3=Lirong |last4=Yin |first4=Zhengtong |last5=Song |first5=Lihong |last6=Tian |first6=Xia |date=2017-08-10 |title=Influence of Social-economic Activities on Air Pollutants in Beijing, China |journal=Open Geosciences |volume=9 |issue=1 |pages=314–321 |bibcode=2017OGeo....9...26L |doi=10.1515/geo-2017-0026 |doi-access=free}} It has also been used to study the research output and growth of countries.{{Citation|title=Predicting the research output/growth of selected countries: application of Even GM (1, 1) and NDGM models |journal=Scientometrics|volume=115|pages=395–413|doi=10.1007/s11192-017-2586-5 |year = 2018|last1 = Javed|first1 = Saad Ahmed|last2=Liu|first2=Sifeng|s2cid=3686413}}

In the world, there are many universities, associations and societies promoting grey system theory e.g., International Association of Grey Systems and Decision Sciences (IAGSUA), Chinese Grey System Association (CGSA), Grey Systems Society of China (GSSC), Grey Systems Society of Pakistan (GSSP), Polish Scientific Society of Grey Systems (PSGS), Grey Systems Committee (IEEE Systems, Man, and Cybernetics Society), Centre for Computational Intelligence (De Montfort University), etc.{{Cite web|url=http://www.iagsua.org/|title=International Association of Grey Systems and Uncertainty Analysis {{!}} International Association of Grey Systems and Uncertainty Analysis|language=en-US|access-date=2019-04-10}}{{Cite web|url=http://greysys.org/projects/grey-systems-society-pakistan/|title=Grey Systems Society of Pakistan (GSSP)|website=GreySys Foundation|language=en-US|access-date=2019-04-10}}{{Cite web|url=http://grey.org.tw/web/index.php|title=Chinese Grey System Association|website=grey.org.tw|access-date=2019-04-10}}{{Cite web|url=http://www.ieeesmc.org/technical-activities/systems-science-and-engineering/grey-systems|title=Grey Systems - IEEE SMC|website=www.ieeesmc.org|access-date=2019-04-10}}{{Cite web|url=https://www.dmu.ac.uk/research/research-faculties-and-institutes/technology/cci/centre-of-computational-intelligence.aspx|title=Centre of Computational Intelligence|website=www.dmu.ac.uk|access-date=2019-04-10}}

There are several journals dedicated to grey systems research and studies e.g., "The Journal of Grey System" (UK),{{Cite web|url=http://researchinformation.co.uk/grey.php|title=RIL: The Journal of Grey System|website=researchinformation.co.uk|access-date=2019-04-10}}{{Cite web|url=https://www.scimagojr.com/journalsearch.php?q=19700172903&tip=sid|title=Journal of Grey System|website=www.scimagojr.com|access-date=2019-04-10}} "Grey Systems Theory and Application" (Emerald Group Publishing),{{Cite web|url=http://emeraldgrouppublishing.com/products/journals/journals.htm?id=gs|title=Emerald {{!}} Grey Systems information|website=emeraldgrouppublishing.com|access-date=2019-04-10}} "International Journal of Grey Systems" (USA),{{Cite web|title=International Journal of Grey Systems|url=http://publish.thescienceinsight.com/index.php/ijgs/index|access-date=2021-03-25|website=publish.thescienceinsight.com}} "Journal of Grey System" (Taiwan),{{Cite web|url=http://grey.org.tw/web/docitem.php?cD0xMiZnPTI2JmxuPUVuZw==|title=Chinese Grey System Association|website=grey.org.tw|access-date=2019-04-10}} "The Grey Journal",{{Cite web|url=http://www.greynet.org/thegreyjournal.html|title=Grey Literature - TGJ, An International Journal on Grey Literature|website=www.greynet.org|access-date=2019-04-10}} Journal of Intelligent and Fuzzy Systems,{{Cite web|url=https://www.iospress.nl/html/10641246.php|title=Journal of Intelligent & Fuzzy Systems|website=www.iospress.nl|access-date=2019-04-10}} Kybernetes, etc.

{{Reflist}}

• Chan WK and Tong TKL, (2007), Multi-criteria material selections and end-of-life product strategy: Grey relational analysis approach, Materials & Design, Volume 28, Issue 5, Pages 1539-1546

• The Encyclopedia of Grey System Theory http://greysystem.org/
• Free Multi-criteria Decision Aiding (MCDA) Tools for Research Students http://sites.google.com/site/mcdafreeware/

{{DEFAULTSORT:Grey Relational Analysis}}

Category:Information theory

Category:Group decision-making

Category:Multiple-criteria decision analysis