Abstract
Max-plus algebra has been applied to several fields like matrix algebra, cryptography, transportation, manufacturing, information technology and study of discrete event systems like subway traffic networks, parallel processing systems, telecommunication networks for many years. In this paper, we discuss various optimization problem using methods based on max-plus algebra, which has maximization and addition as its basic arithmetic operations. We present some sub-classes of mathematical optimization problems like linear programming, convex quadratic programming problem, fractional programming problem, bimatrix game problem and some classes of stochastic game problem in max algebraic framework and discuss various connections between max-plus algebra and optimization.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Akian, M., Bapat, R.B., Gaubert, S.: Max-plus algebra. Hogben, L., Brualdi, R., Greenbaum, A., Mathias, R. (eds) Handbook of linear algebra, discrete mathematics and its applications, Vol. 39, 2nd edn, Chapman and Hall (2014)
Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.-P.: Synchronization and linearity. Wiley, New York (1992)
Bapat, R.B., Stanford, D., van den Driessche, P.: The eigenproblem in max algebra, DMS-631-IR, University of Victoria, British Columbia (1993)
Bapat, R.B.: Max algebra and graph theory. In: Krishnamurthy, Ravichandran, N. (eds) Proceedings of the Advance Workshop and Tutorial in Operations Research, Allied Publisher Pvt Ltd. (2012)
Bapat, R.B.: Pattern properties and spectral inequalities in max algebra. SIAM J. Matrix Anal. Appl. 16, 964–976 (1995)
Bouquard, J.-L., Lent, C., Billaut, J.-C.: Application of an optimization problem in max-plus algebra to scheduling problems. Discrete Appl. Math. 154, 2064–2079 (2006)
Burkard, R.E., Butkovic, P.: Max algebra and the linear assignment problem. Math. Program. Ser. B 98, 415–429 (2003)
Butkovic, P.: Max-algebra: the linear algebra of combinatorics? Linear Algebra Appl. 367, 313–335 (2003)
Butkovic, P.: Max-linear systems: theory and algorithms. Springer, Berlin (2010)
Cuninghame-Green, R.A.: Minimax algebra, lecture notes in economics and mathematical systems (1979)
Cuninghame-Green, R.A.: Minimax algebra. Lecture notes in economics and mathematical systems, pp. 166. Springer, Berlin, New Yotk (1979)
De Schutter, B., De Moor, B.: The extended linear complementarity problem. Math. Program. 71(3), 289–325 (1995)
Filar, J.A.: Orderfield property for stochastic games when the player who controls transitions changes from state to state. JOTA 34, 503–515 (1981)
Filar, J.A., Vrieze, O.J.: Competitive markov decision processes. Springer, New York (1997)
Fink, A.M.: Equilibrium in a stochastic \(n\)-person game. J. Sci. Hiroshima Univ. Ser. A 28, 89–93 (1964)
Gillette, D.: Stochastic game with zero step probabilities. In: Tucker, A.W., Dresher, M., Wolfe, P. (eds) Theory of games. Princeton University Press, Princeton, New Jersey (1957)
Heidergott, B., Jan Olsder, G., van der Woude, J.: Max plus at work modeling and analysis of synchronized systems: a course on max-plus algebra and its applications, Princeton University Press (2006)
Mohan, S.R., Raghavan, T.E.S.: An algorithm for discounted switching control games. OR Spektrum 9, 41–45 (1987)
Mohan, S.R., Neogy, S.K., Parthasarathy, T.: Pivoting algorithms for some classes of stochastic games: a survey. Int. Game Theory Rev. 3, 253–281 (2001)
Raghavan, T.E.S., Filar, J.A.: Algorithms for stochastic games, a survey. Zietch. Oper. Res. 35, 437–472 (1991)
Shapley, L.S.: Stochastic games. Proc. Natl. Acad. Sci. 39, 1095–1100 (1953)
Sobel, M.J.: Noncooperative stochastic games. Ann. Math. Stat. 42, 1930–1935 (1971)
Vrieze, O.J., Tijs, S.H., Raghavan, T.E.S., Filar, J.A.: A finite algorithm for the switching controller stochastic game. OR Spektrum 5, 15–24 (1983)
Zimmermann, U.: Linear and combinatorial optimization in ordered algebraic structures. North-Holland, Amsterdam (1981)
Acknowledgements
The authors would like to thank the anonymous referees for their constructive suggestions which considerably improve the overall presentation of the paper. The authors would like to thank Professor R. B. Bapat, Indian Statistical Institute, Delhi Centre for his valuable comments and suggestions. The first author wants to thank the Science and Engineering Research Board, Department of Science & Technology, Government of India, for financial support for this research.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Dubey, D., Neogy, S.K., Sinha, S. (2020). Max Plus Algebra, Optimization and Game Theory. In: Roy, P., Cao, X., Li, XZ., Das, P., Deo, S. (eds) Mathematical Analysis and Applications in Modeling. ICMAAM 2018. Springer Proceedings in Mathematics & Statistics, vol 302. Springer, Singapore. https://doi.org/10.1007/978-981-15-0422-8_29
Download citation
DOI: https://doi.org/10.1007/978-981-15-0422-8_29
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-0421-1
Online ISBN: 978-981-15-0422-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)