A tropical optimization problem with application to project scheduling with minimum makespan

We consider a multidimensional optimization problem in the framework of tropical mathematics. The problem is formulated to minimize a nonlinear objective function that is defined on vectors in a finitedimensional semimodule over an idempotent semifield and calculated by means of multiplicative conjugate transposition. We offer two complete direct solutions to the problem. The first solution consists of the derivation of a sharp lower bound for the objective function and the solution of an equation to find all vectors that yield the bound. The second is based on extremal properties of the spectral radius and involves the evaluation of the spectral radius of a certain matrix. We apply the result obtained to solve a real-world problem in project scheduling under the minimum makespan criterion. Key-Words: idempotent semifield, tropical mathematics, minimax optimization problem, project scheduling, minimum makespan. MSC (2010): 65K10, 15A80, 65K05, 90C48, 90B35