April  2021, 8(2): 101-118. doi: 10.3934/jdg.2020031

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

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  April 2021 Early access  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.

Citation: Ankan Bhaumik, Sankar Kumar Roy, Gerhard Wilhelm Weber. Multi-objective linguistic-neutrosophic matrix game and its applications to tourism management. Journal of Dynamics and Games, 2021, 8 (2) : 101-118. doi: 10.3934/jdg.2020031
References:
[1]

S. ArunthavanathanL. GorattiL. MaggiF. PellegriniS. 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.

[2]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.  doi: 10.1016/S0165-0114(86)80034-3.

[3]

A. BhaumikS. 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.

[4]

A. BhaumikS. 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.

[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.

[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.

[7]

R. A. CarrascoP. VillarM. 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.

[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.

[9] J. L. Cohon, Multi-Objective Programming and Planning, Academic Press, New York, 1983. 
[10]

S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359.  doi: 10.1080/0233193031000149894.

[11]

S. ErgünO. PalanciS. Z. A. GökŞ. Nizamoǧlu and G. W. Weber, Sequencing grey games, Journal of Dynamics & Games, 7 (2020), 21-35. 

[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.

[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.

[14]

P. GodinhoP. 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.

[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.

[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.

[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.

[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.

[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.

[20]

K. W. LeeH. B. KimH. 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.

[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.

[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.

[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.

[24]

H. LiaoZ. 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.

[25]

A. LopezM. J. HornosR. 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. 

[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.

[27]

K. Miettinen, Non-Linear Multi-Objective Optimization, Kluwer Academic Publisher, Dordrecht, 2002.

[28] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1944. 
[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.

[30] G. Owen, Game Theory, Third Ed., Academic Press, 2001. 
[31]

L. Rodríguez-GallegoA. B. C. CabreraC. KrukM. 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.

[32]

R. M. RodriguezL. 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.

[33]

S. K. Roy, Game Theory Under MCDM and Fuzzy Set Theory, VDM Publishing, Springer, Germany, 2010.

[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.

[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.

[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.

[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. 

[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. 

[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.

[40] F. Smarandache, A Unifying Field in Logics, Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press, Rehoboth, 1999. 
[41]

H. WangF. SmarandacheY. Q. Zhang and R. Sunderraman, Single valued neutrosophic sets, Multispace Multistruct, 4 (2010), 410-413. 

[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.

[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. 

[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.

[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.

[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.

[47]

J. Ye, Subtraction and Division Operations of Simplified Neutrosophic Sets, Information, 8 (2017), 1-7.  doi: 10.3390/info8020051.

[48]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.

[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.

[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.

show all references

References:
[1]

S. ArunthavanathanL. GorattiL. MaggiF. PellegriniS. 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.

[2]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.  doi: 10.1016/S0165-0114(86)80034-3.

[3]

A. BhaumikS. 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.

[4]

A. BhaumikS. 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.

[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.

[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.

[7]

R. A. CarrascoP. VillarM. 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.

[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.

[9] J. L. Cohon, Multi-Objective Programming and Planning, Academic Press, New York, 1983. 
[10]

S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359.  doi: 10.1080/0233193031000149894.

[11]

S. ErgünO. PalanciS. Z. A. GökŞ. Nizamoǧlu and G. W. Weber, Sequencing grey games, Journal of Dynamics & Games, 7 (2020), 21-35. 

[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.

[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.

[14]

P. GodinhoP. 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.

[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.

[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.

[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.

[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.

[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.

[20]

K. W. LeeH. B. KimH. 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.

[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.

[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.

[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.

[24]

H. LiaoZ. 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.

[25]

A. LopezM. J. HornosR. 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. 

[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.

[27]

K. Miettinen, Non-Linear Multi-Objective Optimization, Kluwer Academic Publisher, Dordrecht, 2002.

[28] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1944. 
[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.

[30] G. Owen, Game Theory, Third Ed., Academic Press, 2001. 
[31]

L. Rodríguez-GallegoA. B. C. CabreraC. KrukM. 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.

[32]

R. M. RodriguezL. 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.

[33]

S. K. Roy, Game Theory Under MCDM and Fuzzy Set Theory, VDM Publishing, Springer, Germany, 2010.

[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.

[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.

[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.

[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. 

[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. 

[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.

[40] F. Smarandache, A Unifying Field in Logics, Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press, Rehoboth, 1999. 
[41]

H. WangF. SmarandacheY. Q. Zhang and R. Sunderraman, Single valued neutrosophic sets, Multispace Multistruct, 4 (2010), 410-413. 

[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.

[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. 

[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.

[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.

[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.

[47]

J. Ye, Subtraction and Division Operations of Simplified Neutrosophic Sets, Information, 8 (2017), 1-7.  doi: 10.3390/info8020051.

[48]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.

[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.

[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.

Figure 1.  Maximizing player with strategies and objective values
Figure 2.  Minimizing player with strategies and objective values
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) $
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
$ (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
$ (\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
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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