Multi-objective linguistic-neutrosophic matrix game and its applications to tourism management

Ankan Bhaumik; Sankar Kumar Roy; Gerhard Wilhelm Weber

doi:10.3934/jdg.2020031

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doi: 10.3934/jdg.2020031

Multi-objective linguistic-neutrosophic matrix game and its applications to tourism management

Ankan Bhaumik ^1, , Sankar Kumar Roy ^1,, and Gerhard Wilhelm Weber ^2,

1.	Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, India
2.	Faculty of Engineering Management, Poznan University of Technology, ul. Jacka Rychlewskiego 2, 61-138 Poznan, Poland; METU, 06800 Ankara, Turkey

^* Corresponding author: Sankar Kumar Roy

Received April 2020 Revised July 2020 Published September 2020

Game theory plays an important role in numerous decision-oriented real-life problems. Nowadays, many such problems are basically characterized by various uncertainties. Uncertainties come to happen due to decision makers' collection of data, intuition, assumption, judgement, behaviour, evaluation and lastly, due to the problem itself. Fuzzy concept with membership degree made an initialization towards the treatment of uncertainty, but it was not enough. Intuitionistic fuzzy concept was evolved concerning with both membership and non-membership degrees but failed to express reality more accurately. Then, neutrosophy logic was developed with a new degree in uncertainty, say, indeterminacy degree besides membership and non-membership degrees. Multi-objective optimization is an area of multiple-criteria decision making related with mathematical optimization problems involving more than one objective function to be optimized at the same time. Game theory (matrix game) problems with imprecise, vague information, like neutrosophic, can be formed with multiple objective functions. We develop and analyse a matrix game with multiple objectives, and solve the problem under a single-valued neutrosophic environment in linguistic approach. The main achievement of our study is that we here introduce a problem-oriented example to justify our designed methodologies with a successful real-life implications using linguistic neutrosophic data rather than crisp data as used in previous researches.

Keywords: Multi-objective optimization, uncertainty, neutrosophic set, linguistic neutrosophic number, matrix game.

Mathematics Subject Classification: Primary:91A86, 91A10;Secondary:91A80, 90B50.

Citation: Ankan Bhaumik, Sankar Kumar Roy, Gerhard Wilhelm Weber. Multi-objective linguistic-neutrosophic matrix game and its applications to tourism management. Journal of Dynamics & Games, doi: 10.3934/jdg.2020031

References:

[1]	S. Arunthavanathan, L. Goratti, L. Maggi, F. Pellegrini, S. Kandeepan and S. Reisenfield, An optimal transmission strategy in zero-sum matrix games under intelligent jamming attacks, Wireless Networks, 25 (2019), 1777-1789. doi: 10.1007/s11276-017-1629-4. Google Scholar
[2]	K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. doi: 10.1016/S0165-0114(86)80034-3. Google Scholar
[3]	A. Bhaumik, S. K. Roy and D.-F. Li, Analysis of triangular intuitionistic fuzzy matrix games using robust ranking, Journal of Intelligent and Fuzzy Systems, 33 (2017), 327-336. doi: 10.3233/JIFS-161631. Google Scholar
[4]	A. Bhaumik, S. K. Roy and G. W. Weber, Hesitant interval-valued intuitionistic fuzzy-linguistic term set approach in Prisoners' dilemma game theory using TOPSIS: A case study on Human-trafficking, Central European Journal of Operations Research, 28 (2020), 797-816. doi: 10.1007/s10100-019-00638-9. Google Scholar
[5]	A. Bhaumik and S. K. Roy, Intuitionistic interval-valued hesitant fuzzy matrix games with a new aggregation operator for solving management problem, Granular Computing, (2019), https://doi.org/10.1007/s41066-019-00191-5. doi: 10.1007/s41066-019-00191-5. Google Scholar
[6]	L. Campos, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets and Systems, 32 (1989), 275-289. doi: 10.1016/0165-0114(89)90260-1. Google Scholar
[7]	R. A. Carrasco, P. Villar, M. J. Hornos and E. Herrera-Viedma, A linguistic multicriteria decision-making model applied to hotel service quality evaluation from web data sources, International Journal of Intelligent Systems, 27 (2012), 704-731. doi: 10.1002/int.21546. Google Scholar
[8]	E. S. W. Chan and S. C. K. Wong, Hotel selection: When price is not the issue, Journal of Vacation Marketing, 12 (2006), 142-159. doi: 10.1177/1356766706062154. Google Scholar
[9]	J. L. Cohon, Multi-Objective Programming and Planning, Academic Press, New York, 1983. Google Scholar
[10]	S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359. doi: 10.1080/0233193031000149894. Google Scholar
[11]	S. Ergün, O. Palanci, S. Z. A. Gök, Ş. Nizamoǧlu and G. W. Weber, Sequencing grey games, Journal of Dynamics & Games, 7 (2020), 21-35. Google Scholar
[12]	L. Etaati and D. Sundaram, Adaptive tourist recommendation system: Conceptual frameworks and implementations, Vietnam Journal of Computer Science, 2 (2015), 95-107. doi: 10.1007/s40595-014-0034-5. Google Scholar
[13]	A. Fonseca-Morales and O. Hernández-Lerma, A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models, Journal of Dynamics & Games, 4 (2017), 195-203. doi: 10.3934/jdg.2017012. Google Scholar
[14]	P. Godinho, P. Phillips and L. Moutinho, Hotel location when competitors may react: A game-theoretic gravitational model, Tourism Management, 69 (2018), 384-396. doi: 10.1016/j.tourman.2018.06.014. Google Scholar
[15]	S. Z. A. Gök and G.-W. Weber, On dominance core and stable sets for cooperative ellipsoidal games, Optimization, 62 (2013), 1297-1308. doi: 10.1080/02331934.2013.793327. Google Scholar
[16]	F. Herrera and E. Herrera-Viedma, Linguistic decision analysis: Steps for solving decision problems under linguistic information, Fuzzy Sets and Systems, 115 (2000), 67-82. doi: 10.1016/S0165-0114(99)00024-X. Google Scholar
[17]	J. Jana and S. K. Roy, Dual hesitant fuzzy matrix games: Based on new similarity measure, Soft Computing, 23 (2019), 8873-8886. doi: 10.1007/s00500-018-3486-1. Google Scholar
[18]	J. Jana and S. K. Roy, Solution of matrix games with generalised trapezoidal fuzzy payoffs, Fuzzy Information and Engineering, 10 (2018), 213-224. doi: 10.1080/16168658.2018.1517975. Google Scholar
[19]	J. Jana and S. K. Roy, Soft matrix game: A hesitant fuzzy MCDM approach, American Journal of Mathematical and Management Sciences, (2020). doi: 10.1080/01966324.2020.1730273. Google Scholar
[20]	K. W. Lee, H. B. Kim, H. S. Kim and D. S. Lee, The determinants of factors in FIT guests' perception of hotel location, Journal of Hospitality and Tourism Management, 17 (2010), 167-174. doi: 10.1375/jhtm.17.1.167. Google Scholar
[21]	D.-F. Li, Linear Programming Models and Methods of Matrix Games with Payoffs of Triangular Fuzzy Numbers, Springer, Heidelberg, 2016. doi: 10.1007/978-3-662-48476-0. Google Scholar
[22]	D.-F. Li, Decision and Game Theory in Management with Intuitionistic Fuzzy Sets, Springer-Verlag Berlin Heidelberg (Studies in Fuzziness and Soft Computing), 308, 2014, 444 pp. doi: 10.1007/978-3-642-40712-3. Google Scholar
[23]	D.-F. Li and J.-X. Nan, A nonlinear programming approach to matrix games with payoffs of Atanassov's intuitionistic fuzzy sets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 17 (2009), 585-607. doi: 10.1142/S0218488509006157. Google Scholar
[24]	H. Liao, Z. Xu and X.-J. Zeng, Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making, Information Sciences, 271 (2014), 125-142. doi: 10.1016/j.ins.2014.02.125. Google Scholar
[25]	A. Lopez, M. J. Hornos, R. A. Carrasco and E. H. Viedma, Applying a linguistic multi-criteria decision-making model to the analysis of ICT suppliers offers, Expert Systems with Applications, 57 (2016), 127-138. Google Scholar
[26]	L. Martinez and F. Herrera, An overview on the 2-tuple linguistic model for computing with words in decision making: Extensions, applications and challenges, Information Sciences, 207 (2012), 1-18. doi: 10.1016/j.ins.2012.04.025. Google Scholar
[27]	K. Miettinen, Non-Linear Multi-Objective Optimization, Kluwer Academic Publisher, Dordrecht, 2002. Google Scholar
[28]	J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1944. Google Scholar
[29]	I. Nishizaki and M. Sakawa, Fuzzy and Multi-Objective Games for Conflict Resolution, Physica-Verlag, Heidelberg, 2001. doi: 10.1007/978-3-7908-1830-7. Google Scholar
[30]	G. Owen, Game Theory, Third Ed., Academic Press, 2001. Google Scholar
[31]	L. Rodríguez-Gallego, A. B. C. Cabrera, C. Kruk, M. Nin and A. Mauttone, Establishing limits to agriculture and afforestation: A GIS based multi-objective approach to prevent algal blooms in a coastal lagoon, Journal of Dynamics & Games, 6 (2019), 159-178. doi: 10.3934/jdg.2019012. Google Scholar
[32]	R. M. Rodriguez, L. Martinez and F. Herrera, Hesitant fuzzy linguistic term sets for decision making, IEEE Transactions on Fuzzy Systems, 20 (2012), 109-119. doi: 10.1109/TFUZZ.2011.2170076. Google Scholar
[33]	S. K. Roy, Game Theory Under MCDM and Fuzzy Set Theory, VDM Publishing, Springer, Germany, 2010. Google Scholar
[34]	S. K. Roy and A. Bhaumik, Intelligent water management: a triangular type-2 intuitionistic fuzzy matrix games approach, Water Resources Management, 32 (2018), 949-968. doi: 10.1007/s11269-017-1848-6. Google Scholar
[35]	S. K. Roy and P. Mula, Solving matrix game with rough payoffs using genetic algorithm, Operational Research: An International Journal, 16 (2016), 117-130. doi: 10.1007/s12351-015-0189-6. Google Scholar
[36]	S. K. Roy and S. K. Maiti, Reduction methods of type-2 fuzzy variables and their applications to Stackelberg game, Applied Intelligence, 50 (2020), 1398-1415. doi: 10.1007/s10489-019-01578-2. Google Scholar
[37]	S. K. Roy and C. B. Das, Fuzzy based genetic algorithm for multicriteria entropy matrix goal game, Journal of Uncertain Systems, 3 (2009), 201-209. Google Scholar
[38]	S. K. Roy and S. N. Mondal, An approach to solve fuzzy interval valued matrix game, International Journal of Operational Research, 26 (2016), 253-267. Google Scholar
[39]	V. V. Singh and A. Lisser, A second-order cone programming formulation for two player zero-sum games with chance constraints, European Journal of Operational Research, 275 (2019), 839-845. doi: 10.1016/j.ejor.2019.01.010. Google Scholar
[40]	F. Smarandache, A Unifying Field in Logics, Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press, Rehoboth, 1999. Google Scholar
[41]	H. Wang, F. Smarandache, Y. Q. Zhang and R. Sunderraman, Single valued neutrosophic sets, Multispace Multistruct, 4 (2010), 410-413. Google Scholar
[42]	H. Wang, F. Smarandache, Y.-Q. Zhang and R. Sunderraman, Interval Neutrosophic Sets and Logic: Theory and Applications in Computing, Hexis, Phoenix, AZ, 2005. Google Scholar
[43]	J. Q. Wang and X. E. Li, An application of the TODIM method with multi-valued neutrosophic set, Control and Decision, 30 (2015), 1139-1142. Google Scholar
[44]	Y. M. Wen, Y. F. Shi, G. Y. Cai, Y. Q. Miao and G. Long, A survey of personalized travel recommendation, Computer Science, (2014), Z11. Google Scholar
[45]	Z. Xu, Deviation measures of linguistic preference relations in group decision making, Omega, 33 (2005), 249-254. doi: 10.1016/j.omega.2004.04.008. Google Scholar
[46]	Z. Xu, A method based on linguistic aggregation operators for group decision making with linguistic preference relations, Information Sciences, 166 (2004), 19-30. doi: 10.1016/j.ins.2003.10.006. Google Scholar
[47]	J. Ye, Subtraction and Division Operations of Simplified Neutrosophic Sets, Information, 8 (2017), 1-7. doi: 10.3390/info8020051. Google Scholar
[48]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X. Google Scholar
[49]	L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-1, Information Sciences, 8 (1975), 199-249. doi: 10.1016/0020-0255(75)90036-5. Google Scholar
[50]	A. J. Zaslavski, Structure of approximate solutions of dynamic continuous time zero-sum games, Journal of Dynamics & Games, 1 (2014), 153-179. doi: 10.3934/jdg.2014.1.153. Google Scholar

show all references

References:

[1]	S. Arunthavanathan, L. Goratti, L. Maggi, F. Pellegrini, S. Kandeepan and S. Reisenfield, An optimal transmission strategy in zero-sum matrix games under intelligent jamming attacks, Wireless Networks, 25 (2019), 1777-1789. doi: 10.1007/s11276-017-1629-4. Google Scholar
[2]	K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. doi: 10.1016/S0165-0114(86)80034-3. Google Scholar
[3]	A. Bhaumik, S. K. Roy and D.-F. Li, Analysis of triangular intuitionistic fuzzy matrix games using robust ranking, Journal of Intelligent and Fuzzy Systems, 33 (2017), 327-336. doi: 10.3233/JIFS-161631. Google Scholar
[4]	A. Bhaumik, S. K. Roy and G. W. Weber, Hesitant interval-valued intuitionistic fuzzy-linguistic term set approach in Prisoners' dilemma game theory using TOPSIS: A case study on Human-trafficking, Central European Journal of Operations Research, 28 (2020), 797-816. doi: 10.1007/s10100-019-00638-9. Google Scholar
[5]	A. Bhaumik and S. K. Roy, Intuitionistic interval-valued hesitant fuzzy matrix games with a new aggregation operator for solving management problem, Granular Computing, (2019), https://doi.org/10.1007/s41066-019-00191-5. doi: 10.1007/s41066-019-00191-5. Google Scholar
[6]	L. Campos, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets and Systems, 32 (1989), 275-289. doi: 10.1016/0165-0114(89)90260-1. Google Scholar
[7]	R. A. Carrasco, P. Villar, M. J. Hornos and E. Herrera-Viedma, A linguistic multicriteria decision-making model applied to hotel service quality evaluation from web data sources, International Journal of Intelligent Systems, 27 (2012), 704-731. doi: 10.1002/int.21546. Google Scholar
[8]	E. S. W. Chan and S. C. K. Wong, Hotel selection: When price is not the issue, Journal of Vacation Marketing, 12 (2006), 142-159. doi: 10.1177/1356766706062154. Google Scholar
[9]	J. L. Cohon, Multi-Objective Programming and Planning, Academic Press, New York, 1983. Google Scholar
[10]	S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359. doi: 10.1080/0233193031000149894. Google Scholar
[11]	S. Ergün, O. Palanci, S. Z. A. Gök, Ş. Nizamoǧlu and G. W. Weber, Sequencing grey games, Journal of Dynamics & Games, 7 (2020), 21-35. Google Scholar
[12]	L. Etaati and D. Sundaram, Adaptive tourist recommendation system: Conceptual frameworks and implementations, Vietnam Journal of Computer Science, 2 (2015), 95-107. doi: 10.1007/s40595-014-0034-5. Google Scholar
[13]	A. Fonseca-Morales and O. Hernández-Lerma, A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models, Journal of Dynamics & Games, 4 (2017), 195-203. doi: 10.3934/jdg.2017012. Google Scholar
[14]	P. Godinho, P. Phillips and L. Moutinho, Hotel location when competitors may react: A game-theoretic gravitational model, Tourism Management, 69 (2018), 384-396. doi: 10.1016/j.tourman.2018.06.014. Google Scholar
[15]	S. Z. A. Gök and G.-W. Weber, On dominance core and stable sets for cooperative ellipsoidal games, Optimization, 62 (2013), 1297-1308. doi: 10.1080/02331934.2013.793327. Google Scholar
[16]	F. Herrera and E. Herrera-Viedma, Linguistic decision analysis: Steps for solving decision problems under linguistic information, Fuzzy Sets and Systems, 115 (2000), 67-82. doi: 10.1016/S0165-0114(99)00024-X. Google Scholar
[17]	J. Jana and S. K. Roy, Dual hesitant fuzzy matrix games: Based on new similarity measure, Soft Computing, 23 (2019), 8873-8886. doi: 10.1007/s00500-018-3486-1. Google Scholar
[18]	J. Jana and S. K. Roy, Solution of matrix games with generalised trapezoidal fuzzy payoffs, Fuzzy Information and Engineering, 10 (2018), 213-224. doi: 10.1080/16168658.2018.1517975. Google Scholar
[19]	J. Jana and S. K. Roy, Soft matrix game: A hesitant fuzzy MCDM approach, American Journal of Mathematical and Management Sciences, (2020). doi: 10.1080/01966324.2020.1730273. Google Scholar
[20]	K. W. Lee, H. B. Kim, H. S. Kim and D. S. Lee, The determinants of factors in FIT guests' perception of hotel location, Journal of Hospitality and Tourism Management, 17 (2010), 167-174. doi: 10.1375/jhtm.17.1.167. Google Scholar
[21]	D.-F. Li, Linear Programming Models and Methods of Matrix Games with Payoffs of Triangular Fuzzy Numbers, Springer, Heidelberg, 2016. doi: 10.1007/978-3-662-48476-0. Google Scholar
[22]	D.-F. Li, Decision and Game Theory in Management with Intuitionistic Fuzzy Sets, Springer-Verlag Berlin Heidelberg (Studies in Fuzziness and Soft Computing), 308, 2014, 444 pp. doi: 10.1007/978-3-642-40712-3. Google Scholar
[23]	D.-F. Li and J.-X. Nan, A nonlinear programming approach to matrix games with payoffs of Atanassov's intuitionistic fuzzy sets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 17 (2009), 585-607. doi: 10.1142/S0218488509006157. Google Scholar
[24]	H. Liao, Z. Xu and X.-J. Zeng, Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making, Information Sciences, 271 (2014), 125-142. doi: 10.1016/j.ins.2014.02.125. Google Scholar
[25]	A. Lopez, M. J. Hornos, R. A. Carrasco and E. H. Viedma, Applying a linguistic multi-criteria decision-making model to the analysis of ICT suppliers offers, Expert Systems with Applications, 57 (2016), 127-138. Google Scholar
[26]	L. Martinez and F. Herrera, An overview on the 2-tuple linguistic model for computing with words in decision making: Extensions, applications and challenges, Information Sciences, 207 (2012), 1-18. doi: 10.1016/j.ins.2012.04.025. Google Scholar
[27]	K. Miettinen, Non-Linear Multi-Objective Optimization, Kluwer Academic Publisher, Dordrecht, 2002. Google Scholar
[28]	J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1944. Google Scholar
[29]	I. Nishizaki and M. Sakawa, Fuzzy and Multi-Objective Games for Conflict Resolution, Physica-Verlag, Heidelberg, 2001. doi: 10.1007/978-3-7908-1830-7. Google Scholar
[30]	G. Owen, Game Theory, Third Ed., Academic Press, 2001. Google Scholar
[31]	L. Rodríguez-Gallego, A. B. C. Cabrera, C. Kruk, M. Nin and A. Mauttone, Establishing limits to agriculture and afforestation: A GIS based multi-objective approach to prevent algal blooms in a coastal lagoon, Journal of Dynamics & Games, 6 (2019), 159-178. doi: 10.3934/jdg.2019012. Google Scholar
[32]	R. M. Rodriguez, L. Martinez and F. Herrera, Hesitant fuzzy linguistic term sets for decision making, IEEE Transactions on Fuzzy Systems, 20 (2012), 109-119. doi: 10.1109/TFUZZ.2011.2170076. Google Scholar
[33]	S. K. Roy, Game Theory Under MCDM and Fuzzy Set Theory, VDM Publishing, Springer, Germany, 2010. Google Scholar
[34]	S. K. Roy and A. Bhaumik, Intelligent water management: a triangular type-2 intuitionistic fuzzy matrix games approach, Water Resources Management, 32 (2018), 949-968. doi: 10.1007/s11269-017-1848-6. Google Scholar
[35]	S. K. Roy and P. Mula, Solving matrix game with rough payoffs using genetic algorithm, Operational Research: An International Journal, 16 (2016), 117-130. doi: 10.1007/s12351-015-0189-6. Google Scholar
[36]	S. K. Roy and S. K. Maiti, Reduction methods of type-2 fuzzy variables and their applications to Stackelberg game, Applied Intelligence, 50 (2020), 1398-1415. doi: 10.1007/s10489-019-01578-2. Google Scholar
[37]	S. K. Roy and C. B. Das, Fuzzy based genetic algorithm for multicriteria entropy matrix goal game, Journal of Uncertain Systems, 3 (2009), 201-209. Google Scholar
[38]	S. K. Roy and S. N. Mondal, An approach to solve fuzzy interval valued matrix game, International Journal of Operational Research, 26 (2016), 253-267. Google Scholar
[39]	V. V. Singh and A. Lisser, A second-order cone programming formulation for two player zero-sum games with chance constraints, European Journal of Operational Research, 275 (2019), 839-845. doi: 10.1016/j.ejor.2019.01.010. Google Scholar
[40]	F. Smarandache, A Unifying Field in Logics, Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press, Rehoboth, 1999. Google Scholar
[41]	H. Wang, F. Smarandache, Y. Q. Zhang and R. Sunderraman, Single valued neutrosophic sets, Multispace Multistruct, 4 (2010), 410-413. Google Scholar
[42]	H. Wang, F. Smarandache, Y.-Q. Zhang and R. Sunderraman, Interval Neutrosophic Sets and Logic: Theory and Applications in Computing, Hexis, Phoenix, AZ, 2005. Google Scholar
[43]	J. Q. Wang and X. E. Li, An application of the TODIM method with multi-valued neutrosophic set, Control and Decision, 30 (2015), 1139-1142. Google Scholar
[44]	Y. M. Wen, Y. F. Shi, G. Y. Cai, Y. Q. Miao and G. Long, A survey of personalized travel recommendation, Computer Science, (2014), Z11. Google Scholar
[45]	Z. Xu, Deviation measures of linguistic preference relations in group decision making, Omega, 33 (2005), 249-254. doi: 10.1016/j.omega.2004.04.008. Google Scholar
[46]	Z. Xu, A method based on linguistic aggregation operators for group decision making with linguistic preference relations, Information Sciences, 166 (2004), 19-30. doi: 10.1016/j.ins.2003.10.006. Google Scholar
[47]	J. Ye, Subtraction and Division Operations of Simplified Neutrosophic Sets, Information, 8 (2017), 1-7. doi: 10.3390/info8020051. Google Scholar
[48]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X. Google Scholar
[49]	L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-1, Information Sciences, 8 (1975), 199-249. doi: 10.1016/0020-0255(75)90036-5. Google Scholar
[50]	A. J. Zaslavski, Structure of approximate solutions of dynamic continuous time zero-sum games, Journal of Dynamics & Games, 1 (2014), 153-179. doi: 10.3934/jdg.2014.1.153. Google Scholar

Figure 1. Maximizing player with strategies and objective values

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Figure 2. Minimizing player with strategies and objective values

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Table 1. Relations between linguistic terms and neutrosophic sets.

Linguistic terms	Semantic neutrosophic sets	Neutrosophic terms
Good	$ (s_{0.75}, s_{0.25}, s_{0.45}) $	$ (0.75, 0.25, 0.45) $
Slightly Good	$ (s_{0.70}, s_{0.30}, s_{0.25}) $	$ (0.70, 0.30, 0.25) $
Very Good	$ (s_{0.90}, s_{0.20}, s_{0.70}) $	$ (0.90, 0.20, 0.70) $
Very Poor	$ (s_{0.05}, s_{0.58}, s_{0.85}) $	$ (0.05, 0.58, 0.85) $
Slightly Poor	$ (s_{0.30}, s_{0.45}, s_{0.35}) $	$ (0.30, 0.45, 0.35) $
Fair	$ (s_{0.50}, s_{0.45}, s_{0.37}) $	$ (0.50, 0.45, 0.37) $

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Linguistic terms	Semantic neutrosophic sets	Neutrosophic terms
Good	$ (s_{0.75}, s_{0.25}, s_{0.45}) $	$ (0.75, 0.25, 0.45) $
Slightly Good	$ (s_{0.70}, s_{0.30}, s_{0.25}) $	$ (0.70, 0.30, 0.25) $
Very Good	$ (s_{0.90}, s_{0.20}, s_{0.70}) $	$ (0.90, 0.20, 0.70) $
Very Poor	$ (s_{0.05}, s_{0.58}, s_{0.85}) $	$ (0.05, 0.58, 0.85) $
Slightly Poor	$ (s_{0.30}, s_{0.45}, s_{0.35}) $	$ (0.30, 0.45, 0.35) $
Fair	$ (s_{0.50}, s_{0.45}, s_{0.37}) $	$ (0.50, 0.45, 0.37) $

Table 2. Tabular values of strategies and gain-floors for maximizing player.

$ (w_{1}, w_{2}, w_{3}) $	$ (y_{1}, y_{2}) $	$ (\mu, \nu, \tau) $	objective value
(0.40, 0.30, 0.30)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.0698889
(0.40, 0.50, 0.10)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.1003333
(0.50, 0.30, 0.20)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.1862222
(0.30, 0.30, 0.40)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	-0.0464444
(0.30, 0.10, 0.60)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	-0.0768889
(0.70, 0.20, 0.10)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.4036667
(0.65, 0.25, 0.10)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.3531111
(1.00, 0.00, 0.00)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.7222222
(0.00, 1.00, 0.00)	(0.8333333, 0.1666667)	(0.0000000, 0.2833333, 0.4366667)	-0.2833333
(0.00, 0.00, 1.00)	(0.6226415, 0.3773585)	(0.0000000, 0.3254717, 0.4198113)	-0.4198113
(0.25, 0.25, 0.50)	(0.8333333, 0.1666667)	(0.7083333, 0.2833333, 0.4366667)	-0.1120833
(0.50, 0.25, 0.25)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.1786111
(0.37, 0.33, 0.30)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.0395556
(0.47, 0.23, 0.30)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.1406667
(0.30, 0.20, 0.50)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	-0.0616667

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$ (w_{1}, w_{2}, w_{3}) $	$ (y_{1}, y_{2}) $	$ (\mu, \nu, \tau) $	objective value
(0.40, 0.30, 0.30)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.0698889
(0.40, 0.50, 0.10)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.1003333
(0.50, 0.30, 0.20)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.1862222
(0.30, 0.30, 0.40)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	-0.0464444
(0.30, 0.10, 0.60)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	-0.0768889
(0.70, 0.20, 0.10)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.4036667
(0.65, 0.25, 0.10)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.3531111
(1.00, 0.00, 0.00)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.7222222
(0.00, 1.00, 0.00)	(0.8333333, 0.1666667)	(0.0000000, 0.2833333, 0.4366667)	-0.2833333
(0.00, 0.00, 1.00)	(0.6226415, 0.3773585)	(0.0000000, 0.3254717, 0.4198113)	-0.4198113
(0.25, 0.25, 0.50)	(0.8333333, 0.1666667)	(0.7083333, 0.2833333, 0.4366667)	-0.1120833
(0.50, 0.25, 0.25)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.1786111
(0.37, 0.33, 0.30)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.0395556
(0.47, 0.23, 0.30)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	0.1406667
(0.30, 0.20, 0.50)	(0.8888889, 0.1111111)	(0.7222222, 0.2888889, 0.4411111)	-0.0616667

Table 3. Tabular values of strategies and loss-ceilings for minimizing player.

$ (\omega_{1}, \omega_{2}, \omega_{3}) $	$ (z_{1}, z_{2}) $	$ (\sigma, \rho, \eta) $	objective value
(0.40, 0.30, 0.30)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	0.0937736
(0.40, 0.50, 0.10)	(0.4444444, 0.5555556)	(0.7222222, 0.2777778, 0.3388889)	0.1161111
(0.50, 0.30, 0.20)	(0.4444444, 0.5555556)	(0.7222222, 0.2777778, 0.3388889)	0.2100000
(0.30, 0.30, 0.40)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	-0.0224528
(0.30, 0.10, 0.60)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	-0.0549057
(0.70, 0.20, 0.10)	(0.4444444, 0.5555556)	(0.7222222, 0.2777778, 0.3388889)	0.4161111
(0.65, 0.25, 0.10)	(0.4444444, 0.5555556)	(0.7222222, 0.2777778, 0.3388889)	0.3661111
(1.00, 0.00, 0.00)	(0.4444444, 0.5555556)	(0.7222222, 0.0000000, 0.0000000)	0.7222222
(0.00, 1.00, 0.00)	(0.3333333, 0.6666667)	(0.7666667, 0.2833333, 0.0000000)	-0.2833333
(0.00, 0.00, 1.00)	(0.8490566, 0.1509434)	(0.7424528, 0.0000000, 0.4198113)	-0.4198113
(0.25, 0.25, 0.50)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	-0.0886792
(0.50, 0.25, 0.25)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	0.2018868
(0.37, 0.33, 0.30)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	0.0637736
(0.47, 0.23, 0.30)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	0.1637736
(0.30, 0.20, 0.50)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	-0.0386792

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$ (\omega_{1}, \omega_{2}, \omega_{3}) $	$ (z_{1}, z_{2}) $	$ (\sigma, \rho, \eta) $	objective value
(0.40, 0.30, 0.30)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	0.0937736
(0.40, 0.50, 0.10)	(0.4444444, 0.5555556)	(0.7222222, 0.2777778, 0.3388889)	0.1161111
(0.50, 0.30, 0.20)	(0.4444444, 0.5555556)	(0.7222222, 0.2777778, 0.3388889)	0.2100000
(0.30, 0.30, 0.40)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	-0.0224528
(0.30, 0.10, 0.60)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	-0.0549057
(0.70, 0.20, 0.10)	(0.4444444, 0.5555556)	(0.7222222, 0.2777778, 0.3388889)	0.4161111
(0.65, 0.25, 0.10)	(0.4444444, 0.5555556)	(0.7222222, 0.2777778, 0.3388889)	0.3661111
(1.00, 0.00, 0.00)	(0.4444444, 0.5555556)	(0.7222222, 0.0000000, 0.0000000)	0.7222222
(0.00, 1.00, 0.00)	(0.3333333, 0.6666667)	(0.7666667, 0.2833333, 0.0000000)	-0.2833333
(0.00, 0.00, 1.00)	(0.8490566, 0.1509434)	(0.7424528, 0.0000000, 0.4198113)	-0.4198113
(0.25, 0.25, 0.50)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	-0.0886792
(0.50, 0.25, 0.25)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	0.2018868
(0.37, 0.33, 0.30)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	0.0637736
(0.47, 0.23, 0.30)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	0.1637736
(0.30, 0.20, 0.50)	(0.8490566, 0.1509434)	(0.7424528, 0.2575472, 0.4198113)	-0.0386792

Table 4. Selected studies on hotel recommendation through optimization processes.

References	Topic	Used methods	Study space
Carrasco et al. [7]	Hotel-service quality evaluation	Linguistic MCDM method	Linguistic web data
Chan and Wong [8]	Hotel selection	Statistical methods	Real-life survey data
Etaati and Sundaram [12]	Adaptive tourism recommendation system	ATRS & MCDM models with double loop learning model	Survey data
Godinho et al. [14]	Hotel-location through competitor's reactions	Game-theoretic gravitational method	Game and location problem related data
Lee et al. [20]	FIT guests' perception of hotel location	Cronbach's alpha-test, ANOVA	Location problem oriented data
Our paper	Hotel recommendation	Game-theoretic multi-objective programming	Linguistic-neutrosophic data

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References	Topic	Used methods	Study space
Carrasco et al. [7]	Hotel-service quality evaluation	Linguistic MCDM method	Linguistic web data
Chan and Wong [8]	Hotel selection	Statistical methods	Real-life survey data
Etaati and Sundaram [12]	Adaptive tourism recommendation system	ATRS & MCDM models with double loop learning model	Survey data
Godinho et al. [14]	Hotel-location through competitor's reactions	Game-theoretic gravitational method	Game and location problem related data
Lee et al. [20]	FIT guests' perception of hotel location	Cronbach's alpha-test, ANOVA	Location problem oriented data
Our paper	Hotel recommendation	Game-theoretic multi-objective programming	Linguistic-neutrosophic data

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