Algebraic solution of optimal scheduling problems subject to due dates for start time of project jobs

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.