Explicit solution of a tropical optimization problem with application to project scheduling
Provides a theoretical solution for a class of tropical optimization problems with potential application to scheduling, but the practical impact is limited to specific formulations.
The authors solve a new multidimensional tropical optimization problem by deriving a sharp lower bound and explicit solution in compact vector form, and apply it to just-in-time scheduling problems.
A new multidimensional optimization problem is considered in the tropical mathematics setting. The problem is to minimize a nonlinear function defined on a finite-dimensional semimodule over an idempotent semifield and given by a conjugate transposition operator. A special case of the problem, which arises in just-in-time scheduling, serves as a motivation for the study. To solve the general problem, we derive a sharp lower bound for the objective function and then find vectors that yield the bound. Under general conditions, an explicit solution is obtained in a compact vector form. This result is applied to provide new solutions for scheduling problems under consideration. To illustrate, numerical examples are also presented.