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

Article Contents

2021, Volume 8, Issue 2: 101-118. Doi: 10.3934/jdg.2020031

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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
^* Corresponding author: Sankar Kumar Roy

Received: March 31, 2020

Revised: June 30, 2020

Early access: September 2020

Published: April 2021

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Abstract

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:

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

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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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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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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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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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