Abstract
The solution of the following problems is offered. Suppose a multiset J (¦J¦=p) is given. For each pair of elementsα andβ εJ, a number 1 ≤ χ∝β≤P is given. Moreover, if 1 < x∝β<p then x∝β is undefined. If x∝β=1, then x∝β=p. Problem 1. Find the permutation λ1...λF of elements of the multiset J satisfying the following conditions. Let λi≐α, λi=β. If i,j < x∝β, thenj <i. If i,j > x∝β, then i<j. Such a permutation is called a PC-schedule. Problem 2. Find a PC-schedule in which the following property holds: if i < x∝β < j, λi=α, λj=β, then
. Such a PC-schedule is called an SC-schedule. The conditions under which these problems have solutions are studied. For their solution an algorithm of shifts is used with the complexity O(¦B(J)¦2¦J¦).
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Literature cited
K. V. Shakhbazyan and N. B. Lebedinskaya, “Efficient methods of optimizing schedules for one machine (survey),” J. Sov. Math.,24, No. 1 (1984).
D. E. Knuth, The Art of Computer Programming. Vol. 3, Sorting and Search, Addison-Wesley (1973).
K. V. Shakhbazyan, “Algorithm of shifts for optimal structural schedules,” J. Sov. Math. (this issue).
V. S. Tanaev and V. V. Shkruba, Introduction to Scheduling Theory [in Russian], Nauka, Moscow (1975).
T. E. Safonova, “On stability of compatible schedules,”J. Sov. Math. (this issue).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 44–72, 1983.
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Shakhbazyan, K.V. Solution of two sequencing problems. J Math Sci 29, 1677–1699 (1985). https://doi.org/10.1007/BF02105040
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DOI: https://doi.org/10.1007/BF02105040