Algebraic solution of optimal sheduling problems subject to due dates for start time of project jobs
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St Petersburg State University
Abstract
A direct solution is proposed for problems of drawing up the optimal schedule for jobs in a
project based on using models and methods of tropical optimization. The problems of the
development of an optimal plan are reduced to problems of tropical optimization, which
consist in minimizing the objective function under given strict constraints on the start and
finish time of jobs. As optimality criteria of the plan the maximum deviation from the due
dates of the start of the jobs of the project is taken, which needs to be minimized. Strict
time constraints for the jobs are given in the form of precedence relations and bounds for
the start and finish times of the jobs. Such tasks arise if it is necessary for one reason or
another (for example, due to technological limitations or safety requirements) to start jobs
at specified time. In the article constraints and objective functions are first described in
terms of ordinary mathematics and then the considered problems of project sheduling are
set. Elements of tropical mathematics are discussed which are necessary for the presentation
of the problems of drawing up the optimal schedule in a tropical form. Then the
sheduling problems are formulated in terms of the idempotent mathematics and reduced to
the problem of tropical optimization. Solution of the problems are presented in an explicit
analytical form, which well suited for both formal analysis and numerical computations.
An explanatory numerical example is provided at the end of the article.
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Gubanov S.A. Algebraic solution of optimal sheduling problems subject to due dates for start time of project jobs. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2022, vol. 9 (67), issue 4, pp. 602–611. https://doi.org/10.21638/spbu01.2022.403 (In Russian)