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Hypersoft game theory models and their applications in multi-criteria decision making

Year 2023, Volume: 29 Issue: 7, 680 - 691, 30.12.2023

Somen Debnath Hüseyin Kamacı

Abstract

The classical game theory has been extended for soft set structures, and thus, soft game theory, fuzzy soft game theory, intuitionistic fuzzy soft game theory, neutrosophic soft game theory have been introduced. The payoff function in the soft game approaches is the set-valued function and allows the use of set operations to obtain solution, which makes it very convenient and easily applicable in practice. Also, in these game approaches, the strategies can be determined as attributes/parameters. That is, all these soft game theories are designed to manipulate parametric information using a single-attribute function. However, another powerful tool is needed to process parametric information obtained using multi-attribute function. To model such problems mathematically, the concept of hypersoft set has proposed. In this paper, a game theory model based on hypersoft set called hypersoft game theory is constructed. In this game theory, payoff function is the setvalued function and the strategies are chosen as multi-attributes. A twoperson hypersoft game is developed and different solution methods (such as hypersoft saddle point method, hypersoft elimination method, hypersoft Nash equilibrium method) are produced for such games. Also, the proposed methods are successfully applied to game theory-based decision making problems that may be encountered in real life. Finally, the two-person hypersoft game is extended to the n-person hypersoft game. Nash equilibrium of an n-person hypersoft game is described and an application for this solution method is presented.

Keywords

Hypersoft sets Hyperpayoffs two-person hypersoft games n-person hypersoft games

References

  • [1] Zadeh LA. “Fuzzy sets”. Information and Control, 8, 338-353, 1965.
  • [2] Atanassov KT. “Intuitionistic fuzzy sets”. Fuzzy Sets and Systems, 20, 87-96, 1986.
  • [3] Smarandache F. “Neutrosophic set-a generalization of the intuitionistic fuzzy set”. International Journal of Pure and Applied Mathematics, 24, 287-297, 2005.
  • [4] Kara A, Masri A, Kaya GK. “New branch location selection with AHP, ARAS and fuzzy TOPSIS: an example of a supplier company in the maritime industry”. Pamukkale University Journal of Engineering Sciences, 28(1), 148-159, 2022.
  • [5] Otay İ, Kahraman C. “A novel circular intuitionistic fuzzy AHP&VIKOR methodology: An application to a multiexpert supplier evaluation problem”. Pamukkale University Journal of Engineering Sciences, 28(1), 194-207, 2022.
  • [6] Oturakçı M, Yıldırım RB. “Analysis of supply chain risks by structural equation model and fuzzy analytical hierarchy process”. Pamukkale University Journal of Engineering Sciences, 28(1), 117-127, 2022.
  • [7] Öztayşi B, Onar SC, Gündoğdu FK, Kahraman C. “Locationbased advertisement selection using spherical fuzzy AHPVIKOR”. Journal of Multiple-Valued Logic and Soft Computing, 35, 5-23, 2020.
  • [8] Molodtsov D. “Soft set theory-first results”. Computers & Mathematics with Applications, 37, 19-31, 1999.
  • [9] Maji PK, Biswas R, Roy AR. “Soft set theory”. Computers & Mathematics with Applications, 45, 555-562, 2003.
  • [10] Çağman N, Enginoğlu S. “Soft set theory and uni-int decision making”. European Journal of Operational Research, 207, 848-855, 2010.
  • [11] Aygün E, Kamacı H. “Some generalized operations in soft set theory and their role in similarity and decision making”. Journal of Intelligent & Fuzzy Systems, 36, 6537-6547, 2019.
  • [12] Riaz M, Naeem K, Ahmad MO. “Novel concepts of soft sets with applications”. Annals of Fuzzy Mathematics and Informatics, 13(2), 239-251, 2017.
  • [13] Atagün AO, Kamacı H, Oktay O.“Reduced soft matrices and generalized products with applications in decision making”. Neural Computing and Applications, 29, 445-456, 2018.
  • [14] Kamacı H. “Similarity measure for soft matrices and its applications”. Journal of Intelligent & Fuzzy Systems, 36(4), 3061-3072, 2019.
  • [15] Kamacı H, Atagün AO, Sönmezoğlu A. “Row-products of soft matrices with applications in multiple-disjoint decision making”. Applied Soft Computing, 62, 892-914, 2018.
  • [16] Petchimuthu S, Kamacı H. “The row-products of inverse soft matrices in multicriteria decision making”. Journal of Intelligent & Fuzzy Systems, 36, 6425-6441, 2019.
  • [17] Çağman N, Karataş S. “Intuitionistic fuzzy soft set theory and its decision making”. Journal of Intelligent & Fuzzy Systems, 24(4), 829-836, 2013.
  • [18] Karaaslan F. “Neutrosophic soft sets with applications in decision making”. International Journal of Information Science and Intelligent System, 4(2), 1-20, 2015.
  • [19] Deli I. “Interval-valued neutrosophic soft sets and its decision making”. International Journal of Machine Learning and Cybernetics, 8, 665-676, 2017.
  • [20] Peng X, Yang Y, Song J, Jiang Y. “Pythagorean fuzzy soft set and its application”. Computer Engineering, 41, 224-229, 2015.
  • [21] Akçetin E, Kamacı H. “Three-valued soft set and its multicriteria group decision making via TOPSIS and ELECTRE”. Scientia Iranica E, 28(6), 3719-3742, 2021.
  • [22] Fatimah F, Rosadi D, Hakim RBF, Alcantud JCR.“N-soft sets and their decision making algorithms”. Soft Computing, 2, 3829-3842, 2018.
  • [23] Kamacı H. “Introduction to N-Soft algebraic structures”. Turkish Journal of Mathematics, 44(6), 2356-2379, 2020.
  • [24] Zhang H, Jia-Hua D, Yan C. “Multi-attribute group decisionmaking methods based on Pythagorean fuzzy N-soft sets”. IEEE Access, 8, 62298-62309, 2020.
  • [25] Riaz M, Naeem K, Zareef I, Afzal D. “Neutrosophic N-soft sets with TOPSIS method for multiple attribute decision making”. Neutrosophic Sets and Systems, 32, 1-24, 2020.
  • [26] Dalkılıç O, Demirtaş N, “VFP-soft sets and its application on decision making problems”. Journal of Polytechnic, 24(4), 1391-1399, 2021.
  • [27] Smarandache F.“Extension of soft set to hypersoft set, and then to plithgenic hypersoft set”. Neutrosophic Sets and Systems, 22, 168-170, 2018.
  • [28] Jafar MN, Saeed M, Saqlain M, Yang MS. “Trigonometric similarity measures for neutrosophic hypersoft sets with application to renewable energy source selection”. IEEE Access, 9, 129178-129187, 2021.
  • [29] Kamacı H. “On hybrid structures of hypersoft sets and rough sets”. International Journal of Modern Science and Technology, 6, 69-82, 2021.
  • [30] Martin N, Smarandache F. “Introduction to combined plithgenic hypersoft sets”. Neutrosophic Sets and Systems, 35, 503-510, 2020.
  • [31] Saeed M, Ahsan M, Siddique MK, Ahmad MR. “A study of the fundamentals of hypersoft set theory”. International Journal of Scientific and Engineering Research, 11, 320-329, 2020.
  • [32] Neumann J, Morgenstern O. Theory of Games and Economic Behavior. 1st ed. New Jersey, USA, Princeton University Press, 1944.
  • [33] Nash J. “Noncooperative games”. Annals of Mathematics, 54, 289-295, 1954.
  • [34] Campos L. “Fuzzy linear programming models to solve fuzzy matrix games”. Fuzzy Sets and Systems, 32, 275-289, 1989.
  • [35] Seikh MR, Nayak PK, Pal M. “Matrix games with intuitionistic fuzzy pay-offs”. Journal of Information and Optimization Sciences, 36, 159-181, 2015.
  • [36] Khalifa HA.“An approach for solving two-person zero-sum matrix games in neutrosophic environment”. Journal of Industrial and Systems Engineering, 12, 186-198, 2019.
  • [37] Li S, Tu G. “Probabilistic linguistic matrix game based on fuzzy envelope and prospect theory with its application”. Mathematics, 10(7), 1-30, 2022.
  • [38] Xiao H, Zhang X, Lin D, Khalifa HAEW, Edalatpanah SA. “A new methodology for solving piecewise quadratic fuzzy cooperative continuous static games”. Advances in Mathematical Physics, 222, 1-8, 2022.
  • [39] Scalzo V. “Other-regarding behaviour in fuzzy noncooperative games: Existence of altruistic equilibria”. Fuzzy Sets and Systems, 458, 118-125, 2022.
  • [40] Li S, Tu G. “Bi-matrix games with general intuitionistic fuzzy payoffs and application in corporative environmental behavior”. Symmetry, 14(4), 1-30, 2022.
  • [41] Brikaa MG, Zheng Z, Dagestani AA, Ammar E-S, AlNemer G, Zakarya M. “Ambika approach for solving matrix games with payoffs of single-valued trapezoidal neutrosophic numbers”. Journal of Intelligent and Fuzzy Systems, 42, 5139-5153, 2022.
  • [42] Deli I. Matrix Games with Simplified Neutrosophic Payoffs. Editors: Kahraman C, Otay I. Fuzzy Multi-Criteria Decision-Making Using Neutrosophic Sets, 233-246, Cham, Switzerland, Springer Nature, 2019.
  • [43] Martinez RCJ, Paucar CEP, Arboleda JIC, Lierena MAG, Caballero EG. “Neutrosophic matrix games to solve project management conflicts”. Neutrosophic Sets and Systems, 44, 10-17, 2021.
  • [44] Deli I, Çağman N. “Application of soft sets in decision making based on game theory”. Annals of Fuzzy Mathematics and Informatics, 11, 425-438, 2016.
  • [45] Deli I, Çağman N. “Fuzzy soft games”. Filomat, 29, 1901-1917, 2015.
  • [46] Mukherjee A, Debnath S. “Intuitionistic fuzzy soft game theory”. Songklanakarin Journal of Science and Technology, 40, 409-417, 2018.
  • [47] Selvakumari K, Lavanya S. “Neutrosophic fuzzy soft game”. International Journal of Engineering and Technology, 7, 667-669, 2018.
  • [48] Kamacı H. “Linguistic single-valued neutrosophic soft sets with applications in game theory”. International Journal of Intelligent Systems, 36, 3917-3960, 2021.
  • [49] Kamacı H. “Games based on simplified neutrosophic multiplicative soft sets and their applications”. Neutrosophic Sets and Systems, 47, 491-510, 2021.
  • [50] Gulistan M, Hassan N. “A generalized approach towards soft expert sets via neutrosophic cubic sets with applications in games”. Symmetry, 11(2), 1-26, 2019.
  • [51] Abbas M, Murtaza G, Smarandache F. “Basic operations on hypersoft sets and hypersoftpoint”. Neutrosophic Sets and Systems, 35, 407-421, 2020.
  • [52] Mehdi Khosrow-Pour DBA. Encyclopedia of Information Science and Technology, Forth Edition. 1st ed. Pennsylvania, USA, IGI Global Publisher, 2017.

Hiperesnek oyun teorisi modelleri ve çok kriterli karar vermede uygulamaları

Year 2023, Volume: 29 Issue: 7, 680 - 691, 30.12.2023

Somen Debnath Hüseyin Kamacı

Abstract

Klasik oyun teorisi esnek küme yapıları için genişletilmiştir ve böylece esnek oyun teorisi, bulanık esnek oyun teorisi, sezgisel bulanık esnek oyun teorisi, nötrosofik esnek oyun teorisi tanıtılmıştır. Esnek oyun yaklaşımlarında getiri fonksiyonu, küme-değerli fonksiyondur ve çözüm elde etmek için küme işlemlerinin kullanılmasına izin verir, bu da onu pratikte çok uygun ve kolay uygulanabilir kılar. Ayrıca bu oyun yaklaşımlarında stratejiler nitelikler/parametreler olarak belirlenebilir. Yani, tüm bu esnek oyun teorileri, tek-nitelikli fonksiyonları kullanarak parametrik bilgileri işlemek için tasarlanmıştır. Ancak, çok-nitelikli fonksiyon kullanılarak elde edilen parametrik bilgileri işlemek için başka bir güçlü araca ihtiyaç vardır. Bu tür problemleri matematiksel olarak modellemek için hiperesnek küme kavramı önerilmiştir. Bu makalede, hiperesnek oyun teorisi adı verilen hiperesnek kümeye dayalı bir oyun teorisi modeli oluşturulmuştur. Bu oyun teorisinde, getiri fonksiyonu küme-değerli fonksiyondur ve stratejiler çok-nitelikli olarak seçilir. İki kişilik bir hyperesnek oyun geliştirilmiş ve bu tür oyunlar için farklı çözüm yöntemleri (hiperesnek eyer noktası yöntemi, hiperesnek eliminasyon yöntemi, hiperesnek Nash dengesi yöntemi gibi) üretilmiştir. Ayrıca önerilen yöntemler, gerçek hayatta karşılaşılabilecek oyun teorisi tabanlı karar verme problemlerine başarılı bir şekilde uygulanmıştır. Son olarak, iki kişilik hiperesnek oyun n-kişilik hiperesnek oyuna genişletilmiştir. Bir n-kişilik hiperesnek oyunun Nash dengesi tanımlanmış ve bu çözüm yöntemi için bir uygulama sunulmuştur.

Keywords

Hiperesnek kümeler Hiperesnek getiriler İki kişilik hiperesnek oyunlar n-kişilik hiperesnek oyunlar

References

  • [1] Zadeh LA. “Fuzzy sets”. Information and Control, 8, 338-353, 1965.
  • [2] Atanassov KT. “Intuitionistic fuzzy sets”. Fuzzy Sets and Systems, 20, 87-96, 1986.
  • [3] Smarandache F. “Neutrosophic set-a generalization of the intuitionistic fuzzy set”. International Journal of Pure and Applied Mathematics, 24, 287-297, 2005.
  • [4] Kara A, Masri A, Kaya GK. “New branch location selection with AHP, ARAS and fuzzy TOPSIS: an example of a supplier company in the maritime industry”. Pamukkale University Journal of Engineering Sciences, 28(1), 148-159, 2022.
  • [5] Otay İ, Kahraman C. “A novel circular intuitionistic fuzzy AHP&VIKOR methodology: An application to a multiexpert supplier evaluation problem”. Pamukkale University Journal of Engineering Sciences, 28(1), 194-207, 2022.
  • [6] Oturakçı M, Yıldırım RB. “Analysis of supply chain risks by structural equation model and fuzzy analytical hierarchy process”. Pamukkale University Journal of Engineering Sciences, 28(1), 117-127, 2022.
  • [7] Öztayşi B, Onar SC, Gündoğdu FK, Kahraman C. “Locationbased advertisement selection using spherical fuzzy AHPVIKOR”. Journal of Multiple-Valued Logic and Soft Computing, 35, 5-23, 2020.
  • [8] Molodtsov D. “Soft set theory-first results”. Computers & Mathematics with Applications, 37, 19-31, 1999.
  • [9] Maji PK, Biswas R, Roy AR. “Soft set theory”. Computers & Mathematics with Applications, 45, 555-562, 2003.
  • [10] Çağman N, Enginoğlu S. “Soft set theory and uni-int decision making”. European Journal of Operational Research, 207, 848-855, 2010.
  • [11] Aygün E, Kamacı H. “Some generalized operations in soft set theory and their role in similarity and decision making”. Journal of Intelligent & Fuzzy Systems, 36, 6537-6547, 2019.
  • [12] Riaz M, Naeem K, Ahmad MO. “Novel concepts of soft sets with applications”. Annals of Fuzzy Mathematics and Informatics, 13(2), 239-251, 2017.
  • [13] Atagün AO, Kamacı H, Oktay O.“Reduced soft matrices and generalized products with applications in decision making”. Neural Computing and Applications, 29, 445-456, 2018.
  • [14] Kamacı H. “Similarity measure for soft matrices and its applications”. Journal of Intelligent & Fuzzy Systems, 36(4), 3061-3072, 2019.
  • [15] Kamacı H, Atagün AO, Sönmezoğlu A. “Row-products of soft matrices with applications in multiple-disjoint decision making”. Applied Soft Computing, 62, 892-914, 2018.
  • [16] Petchimuthu S, Kamacı H. “The row-products of inverse soft matrices in multicriteria decision making”. Journal of Intelligent & Fuzzy Systems, 36, 6425-6441, 2019.
  • [17] Çağman N, Karataş S. “Intuitionistic fuzzy soft set theory and its decision making”. Journal of Intelligent & Fuzzy Systems, 24(4), 829-836, 2013.
  • [18] Karaaslan F. “Neutrosophic soft sets with applications in decision making”. International Journal of Information Science and Intelligent System, 4(2), 1-20, 2015.
  • [19] Deli I. “Interval-valued neutrosophic soft sets and its decision making”. International Journal of Machine Learning and Cybernetics, 8, 665-676, 2017.
  • [20] Peng X, Yang Y, Song J, Jiang Y. “Pythagorean fuzzy soft set and its application”. Computer Engineering, 41, 224-229, 2015.
  • [21] Akçetin E, Kamacı H. “Three-valued soft set and its multicriteria group decision making via TOPSIS and ELECTRE”. Scientia Iranica E, 28(6), 3719-3742, 2021.
  • [22] Fatimah F, Rosadi D, Hakim RBF, Alcantud JCR.“N-soft sets and their decision making algorithms”. Soft Computing, 2, 3829-3842, 2018.
  • [23] Kamacı H. “Introduction to N-Soft algebraic structures”. Turkish Journal of Mathematics, 44(6), 2356-2379, 2020.
  • [24] Zhang H, Jia-Hua D, Yan C. “Multi-attribute group decisionmaking methods based on Pythagorean fuzzy N-soft sets”. IEEE Access, 8, 62298-62309, 2020.
  • [25] Riaz M, Naeem K, Zareef I, Afzal D. “Neutrosophic N-soft sets with TOPSIS method for multiple attribute decision making”. Neutrosophic Sets and Systems, 32, 1-24, 2020.
  • [26] Dalkılıç O, Demirtaş N, “VFP-soft sets and its application on decision making problems”. Journal of Polytechnic, 24(4), 1391-1399, 2021.
  • [27] Smarandache F.“Extension of soft set to hypersoft set, and then to plithgenic hypersoft set”. Neutrosophic Sets and Systems, 22, 168-170, 2018.
  • [28] Jafar MN, Saeed M, Saqlain M, Yang MS. “Trigonometric similarity measures for neutrosophic hypersoft sets with application to renewable energy source selection”. IEEE Access, 9, 129178-129187, 2021.
  • [29] Kamacı H. “On hybrid structures of hypersoft sets and rough sets”. International Journal of Modern Science and Technology, 6, 69-82, 2021.
  • [30] Martin N, Smarandache F. “Introduction to combined plithgenic hypersoft sets”. Neutrosophic Sets and Systems, 35, 503-510, 2020.
  • [31] Saeed M, Ahsan M, Siddique MK, Ahmad MR. “A study of the fundamentals of hypersoft set theory”. International Journal of Scientific and Engineering Research, 11, 320-329, 2020.
  • [32] Neumann J, Morgenstern O. Theory of Games and Economic Behavior. 1st ed. New Jersey, USA, Princeton University Press, 1944.
  • [33] Nash J. “Noncooperative games”. Annals of Mathematics, 54, 289-295, 1954.
  • [34] Campos L. “Fuzzy linear programming models to solve fuzzy matrix games”. Fuzzy Sets and Systems, 32, 275-289, 1989.
  • [35] Seikh MR, Nayak PK, Pal M. “Matrix games with intuitionistic fuzzy pay-offs”. Journal of Information and Optimization Sciences, 36, 159-181, 2015.
  • [36] Khalifa HA.“An approach for solving two-person zero-sum matrix games in neutrosophic environment”. Journal of Industrial and Systems Engineering, 12, 186-198, 2019.
  • [37] Li S, Tu G. “Probabilistic linguistic matrix game based on fuzzy envelope and prospect theory with its application”. Mathematics, 10(7), 1-30, 2022.
  • [38] Xiao H, Zhang X, Lin D, Khalifa HAEW, Edalatpanah SA. “A new methodology for solving piecewise quadratic fuzzy cooperative continuous static games”. Advances in Mathematical Physics, 222, 1-8, 2022.
  • [39] Scalzo V. “Other-regarding behaviour in fuzzy noncooperative games: Existence of altruistic equilibria”. Fuzzy Sets and Systems, 458, 118-125, 2022.
  • [40] Li S, Tu G. “Bi-matrix games with general intuitionistic fuzzy payoffs and application in corporative environmental behavior”. Symmetry, 14(4), 1-30, 2022.
  • [41] Brikaa MG, Zheng Z, Dagestani AA, Ammar E-S, AlNemer G, Zakarya M. “Ambika approach for solving matrix games with payoffs of single-valued trapezoidal neutrosophic numbers”. Journal of Intelligent and Fuzzy Systems, 42, 5139-5153, 2022.
  • [42] Deli I. Matrix Games with Simplified Neutrosophic Payoffs. Editors: Kahraman C, Otay I. Fuzzy Multi-Criteria Decision-Making Using Neutrosophic Sets, 233-246, Cham, Switzerland, Springer Nature, 2019.
  • [43] Martinez RCJ, Paucar CEP, Arboleda JIC, Lierena MAG, Caballero EG. “Neutrosophic matrix games to solve project management conflicts”. Neutrosophic Sets and Systems, 44, 10-17, 2021.
  • [44] Deli I, Çağman N. “Application of soft sets in decision making based on game theory”. Annals of Fuzzy Mathematics and Informatics, 11, 425-438, 2016.
  • [45] Deli I, Çağman N. “Fuzzy soft games”. Filomat, 29, 1901-1917, 2015.
  • [46] Mukherjee A, Debnath S. “Intuitionistic fuzzy soft game theory”. Songklanakarin Journal of Science and Technology, 40, 409-417, 2018.
  • [47] Selvakumari K, Lavanya S. “Neutrosophic fuzzy soft game”. International Journal of Engineering and Technology, 7, 667-669, 2018.
  • [48] Kamacı H. “Linguistic single-valued neutrosophic soft sets with applications in game theory”. International Journal of Intelligent Systems, 36, 3917-3960, 2021.
  • [49] Kamacı H. “Games based on simplified neutrosophic multiplicative soft sets and their applications”. Neutrosophic Sets and Systems, 47, 491-510, 2021.
  • [50] Gulistan M, Hassan N. “A generalized approach towards soft expert sets via neutrosophic cubic sets with applications in games”. Symmetry, 11(2), 1-26, 2019.
  • [51] Abbas M, Murtaza G, Smarandache F. “Basic operations on hypersoft sets and hypersoftpoint”. Neutrosophic Sets and Systems, 35, 407-421, 2020.
  • [52] Mehdi Khosrow-Pour DBA. Encyclopedia of Information Science and Technology, Forth Edition. 1st ed. Pennsylvania, USA, IGI Global Publisher, 2017.
There are 52 citations in total.

Details

Primary Language English
Subjects Decision Support and Group Support Systems
Journal Section Research Article
Authors

Somen Debnath This is me

Hüseyin Kamacı

Publication Date December 30, 2023
Published in Issue Year 2023 Volume: 29 Issue: 7

Cite

APA Debnath, S., & Kamacı, H. (2023). Hypersoft game theory models and their applications in multi-criteria decision making. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi, 29(7), 680-691.
AMA Debnath S, Kamacı H. Hypersoft game theory models and their applications in multi-criteria decision making. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi. December 2023;29(7):680-691.
Chicago Debnath, Somen, and Hüseyin Kamacı. “Hypersoft Game Theory Models and Their Applications in Multi-Criteria Decision Making”. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi 29, no. 7 (December 2023): 680-91.
EndNote Debnath S, Kamacı H (December 1, 2023) Hypersoft game theory models and their applications in multi-criteria decision making. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi 29 7 680–691.
IEEE S. Debnath and H. Kamacı, “Hypersoft game theory models and their applications in multi-criteria decision making”, Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi, vol. 29, no. 7, pp. 680–691, 2023.
ISNAD Debnath, Somen - Kamacı, Hüseyin. “Hypersoft Game Theory Models and Their Applications in Multi-Criteria Decision Making”. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi 29/7 (December 2023), 680-691.
JAMA Debnath S, Kamacı H. Hypersoft game theory models and their applications in multi-criteria decision making. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi. 2023;29:680–691.
MLA Debnath, Somen and Hüseyin Kamacı. “Hypersoft Game Theory Models and Their Applications in Multi-Criteria Decision Making”. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi, vol. 29, no. 7, 2023, pp. 680-91.
Vancouver Debnath S, Kamacı H. Hypersoft game theory models and their applications in multi-criteria decision making. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi. 2023;29(7):680-91.

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