Abstract
We address single-machine scheduling problems for which the actual processing times of jobs are subject to various effects, including a positional effect, a cumulative effect and their combination. We review the known results on the problems to minimize the makespan, the sum of the completion times and their combinations and identify the problems for which an optimal sequence cannot be found by simple priority rules such as Shortest Processing Time (SPT) and/or Longest Processing Time (LPT). Typically, these are problems to minimize the sum of the completion times under a deterioration effect, and we verify under which conditions for these problems an optimal permutation is V-shaped (an LPT subsequence followed by an SPT subsequence). We demonstrate that previously used techniques for proving that an optimal sequence is V-shaped are not properly justified. We use the corrected method to describe a wide range of problems with a pure positional effect and a combination of a cumulative effect with a positional effect for which an optimal sequence is V-shaped. On the other hand, we show that even the refined approach has its limitations.
Highlights
Since the early 1990s, there has been a considerable interest in enhanced scheduling models in which the processing times of jobs are affected by their locations in the schedule
The case of a combined effect (4), provided that function f is concave and the array g(r ), 1 ≤ r ≤ n, is non-decreasing, is not fully symmetric to that presented in Theorem 4, and only the makespan Cmax can be minimized by a priority rule, this time Longest Processing Time (LPT)
We refine the proof technique previously employed for proving the existence of an optimal V-shaped sequencing policy for a range of scheduling problems with various time-changing effects such as positional, cumulative and their combination
Summary
Since the early 1990s, there has been a considerable interest in enhanced scheduling models in which the processing times of jobs are affected by their locations in the schedule. 3, we review the problems 1| p j (r ) = p j f (Pr ) g(r )| with a combined effect While some of these problems accept an optimal sequencing policy based on either the SPT or LPT rule, problem 1| p j (r ) = p j f (Pr )| C j with a pure cumulative deterioration effect given by a concave function f , including a polynomial function f (Pr ) = (1 + Pr )A, 0 < A < 1, is not solvable by a priority rule. Refining this result, we show that for this problem, an optimal permutation is not even V-shaped. It only implies that more advanced methods have to be used for proving or disproving the V-shapeness of an optimal permutation
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