A Maximization Problem in Tropical Mathematics: A Complete Solution and Application Examples

A multidimensional optimization problem is formulated in the tropical mathematics setting as to maximize a nonlinear objective function, which is defined through a multiplicative conjugate transposition operator on vectors in a finite-dimensional semimodule over a general idempotent semifield. The study is motivated by problems drawn from project scheduling, where the deviation between initiation or completion times of activities in a project is to be maximized subject to various precedence constraints among the activities. To solve the unconstrained problem, we first establish an upper bound for the objective function, and then solve a system of vector equations to find all vectors that yield the bound. As a corollary, an extension of the solution to handle constrained problems is discussed. The results obtained are applied to give complete direct solutions to the motivating problems from project scheduling. Numerical examples of the development of optimal schedules are also presented.

[1]  Abhijit Gosavi,et al.  Form Errors in Precision Metrology: A Survey of Measurement Techniques , 2012 .

[2]  A. J. Hoffman.  On abstract dual linear programs , 1963 .

[3]  Gintautas Dzemyda,et al.  Processing of Undesirable Business Events in Advanced Production Planning Systems , 2012, Informatica.

[4]  Some optimization problems with extremal operations , 1984 .

[5]  R. A. Cuninghame-Green,et al.  Describing Industrial Processes with Interference and Approximating Their Steady-State Behaviour , 1962 .

[6]  R. Weiner.  Lecture Notes in Economics and Mathematical Systems , 1985 .

[7]  Solution of generalized linear vector equations in idempotent algebra , 2006 .

[8]  Nikolai K. Krivulin.  A solution of a tropical linear vector equation , 2012, ArXiv.

[9]  S. N. N. Pandit,et al.  A New Matrix Calculus , 1961 .

[10]  Michel Minoux,et al.  Graphs, dioids and semirings : new models and algorithms , 2008 .

[11]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[12]  V. Kolokoltsov,et al.  Idempotent Analysis and Its Applications , 1997 .

[13]  Geert Jan Olsder,et al.  Max Plus at Work-Modelling and Analysis of Synchronized Systems , 2006 .

[14]  B. Ciffler.  Scheduling general production systems using schedule algebra , 1963 .

[15]  Raymond A. Cuninghame-Green,et al.  Projections in minimax algebra , 1976, Math. Program..

[16]  P. Butkovic.  Max-linear Systems: Theory and Algorithms , 2010 .

[17]  Nikolai Krivulin,et al.  Complete Solution of a Constrained Tropical Optimization Problem with Application to Location Analysis , 2013, RAMiCS.

[18]  Peter Butkovič,et al.  Introduction to max-linear programming , 2009 .

[19]  Ricardo D. Katz,et al.  Tropical linear-fractional programming and parametric mean payoff games , 2011, J. Symb. Comput..

[20]  G. Litvinov.  Maslov dequantization, idempotent and tropical mathematics: A brief introduction , 2005, math/0507014.

[21]  Raymond Cuninghame-Green,et al.  Bases in max-algebra , 2004 .

[22]  J. Golan.  Semirings and Affine Equations over Them: Theory and Applications , 2003 .

[23]  Julius Žilinskas,et al.  Multi-objective optimization aided to allocation of vertices in aesthetic drawings of special graphs , 2013 .

[24]  Nikolai Krivulin,et al.  A constrained tropical optimization problem: complete solution and application example , 2013, ArXiv.

[25]  Nikolai Krivulin,et al.  Explicit solution of a tropical optimization problem with application to project scheduling , 2013, ArXiv.

[26]  J. Bather.  Optimal decision procedures for finite Markov chains. Part II: Communicating systems , 1973, Advances in Applied Probability.

[27]  Nikolai Krivulin,et al.  A multidimensional tropical optimization problem with a non-linear objective function and linear constraints , 2013, ArXiv.

[29]  Erik Demeulemeester,et al.  Project scheduling : a research handbook , 2002 .

[30]  T.S.R. Murthy,et al.  Minimum zone evaluation of surfaces , 1980 .

[31]  K. Tam.  Optimizing and approximating eigenvectors in max-algebra , 2010 .

[32]  Karel Zimmermann,et al.  Disjunctive optimization, max-separable problems and extremal algebras , 2003, Theor. Comput. Sci..

[33]  Willi Hock,et al.  Lecture Notes in Economics and Mathematical Systems , 1981 .

[34]  Sergej Jakovlev,et al.  Comparison of Two Heuristic Approaches for Solving the production Scheduling Problem , 2011, Inf. Technol. Control..

[35]  Jean-Charles Billaut,et al.  Multicriteria scheduling , 2005, Eur. J. Oper. Res..