# Optimal Scheduling for Two Criteria for a Single Machine with Arbitrary Due Dates of Tasks

• Michael Z. Zgurovsky
• Alexander A. Pavlov
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 173)

## Abstract

We consider the problem of constructing a feasible (in which all tasks complete before their due dates) schedule for a single machine with arbitrary due dates and maximum start time of the machine or minimum total earliness of the tasks completion times in relation to their due dates. It is shown that for the criterion of maximum start time of the machine the problem is polynomially solvable, we give a polynomial algorithm for its solving. With a fixed start time of the machine the problem is polynomially solvable for the criterion of minimizing the total earliness of the completion times of the tasks if qualitatively proven and statistically significant properties of an optimal solution (Heuristics 1 and 2) are met. The problem with an arbitrary start time of the machine to construct an optimal schedule minimizing the total earliness of the tasks completion times is intractable: an exact polynomial algorithm for its solving is not known. For the case if the Heuristics 1 and 2 are true for an arbitrary start time of the machine, we develop an efficient PSC-algorithm. For the opposite case, this PSC-algorithm is an efficient approximation algorithm for the problem solving.

## References

1. 1.
Pavlov, A.A., Misura, E.B., Khalus, E.A.: Polinomial’nyi algoritm polucheniya dopustimogo raspisaniya s maksimal’no pozdnim momentom nachala vypolneniya odnim priborom nezavisimyh zadaniy proizvol’noi dlitel’nosti s raznymi direktivnymi srokami, pri kotorom vse zadaniya ostayutsya nezapazdyvayuschimi (Полиномиальный алгоритм получения допустимого расписания с максимально поздним моментом начала выполнения одним прибором независимых заданий произвольной длительности с разными директивными сроками, при котором все задания остаются незапаздывающими; A polynomial algorithm for obtaining a feasible schedule with the latest processing start time on a single machine for independent tasks with arbitrary processing times and different due dates when all tasks remain non-tardy). Paper presented at the 1st international conference Іnformacіinі tehnologії yak іnnovacіynyi shlyah rozvitku Ukrainy u XXI stolіttі, Transcarpathian State University, Uzhhorod, 6–8 Dec 2012 (in Russian)Google Scholar
2. 2.
Pavlov, A.A., Misura, E.B., Khalus, E.A.: Issledovanie svoistv zadachi kalendarnogo planirovaniya dlya odnogo pribora po kriteriyu minimizacii summarnogo operejeniya zadaniy pri uslovii dopustimosti raspisaniya (Исследование свойств задачи календарного планирования для одного прибора по критерию минимизации суммарного опережения заданий при условии допустимости расписания; Properties’ research of the scheduling problem for a single machine by minimizing the total earliness of tasks with the condition of the schedule feasibility). Visnyk NTUU KPI Inform. Oper. Comput. Sci. 56, 98–102 (2012) (in Russian)Google Scholar
3. 3.
Pavlov, A.A., Khalus, E.A.: Sostavlenie dopustimogo raspisaniya vypolneniya rabot na odnom pribore, optimal’nogo po kriteriyu minimizacii summarnogo operejeniya rabot (Составление допустимого расписания выполнения работ на одном приборе, оптимального по критерию минимизации суммарного опережения работ; Drawing up a feasible schedule of jobs on one machine in order to minimize the total earliness of jobs). Visnyk NTUU KPI Inform. Oper. Comput. Sci. 61, 27–34 (2014) (in Russian)Google Scholar
4. 4.
Pavlov, A.A., Misura, E.B., Khalus, E.A.: Skladannya rozkladu vikonannya zavdan’ na odnomu priladі z metoyu mіnіmіzacіi sumarnogo vyperedjennya ta znahodjennya maksimal’nogo pіzn’ogo momentu pochatku vikonannya zavdan’ v dopustimomu rozkladі (Складання розкладу виконання завдань на одному приладі з метою мінімізації сумарного випередження та знаходження максимального пізнього моменту початку виконання завдань в допустимому розкладі; Single machine scheduling to minimize the total earliness and find the latest start time of tasks in a feasible schedule). Paper presented at the 21st international conference on automatic control Automatics-2014, National Technical University of Ukraine, Kyiv, 23–27 Sept 2014 (in Ukrainian)Google Scholar
5. 5.
Tanaev, V.S., Shkurba, V.V.: Vvedenie v Teoriju Raspisaniy (Введение в теорию расписаний; Introduction to Scheduling Theory). Nauka, Moscow (1975) (in Russian)Google Scholar
6. 6.
Zgurovsky, M.Z., Pavlov, A.A.: Prinyatie Resheniy v Setevyh Sistemah s Ogranichennymi Resursami (Принятие решений в сетевых системах с ограниченными ресурсами; Decision Making in Network Systems with Limited Resources), Naukova dumka, Kyiv (2010) (in Russian)Google Scholar