Scheduling jobs with position and sum-of-processing-time based processing times

Abstract

a b s t r a c tThe paper is devoted to some single-machine scheduling problems with variable job processing times. The objectives are to minimize the makespan (i.e., the maximum completion time of all jobs), and to minimize the total completion time. For some special cases, we show that these problems can be solved in polynomial time. For some another special cases of the makespan and the total completion time minimization problems, we prove that an optimal schedule has an V-shape property in terms of processing times. We also propose a heuristic algorithm by utilizing the V-shape property.

Refined conditions for V-shaped optimal sequencing on a single machine to minimize total completion time under combined effects

“…For problems under a combination of a cumulative concave normalized effect and a positional effect, we also derive conditions which hold for a wide range of problems. However, they do not hold for the problem in which both cumulative and positional effects are polynomial, which contradicts the claim made by Lu et al (2015).…”

“…Consider first problem 1| p j (r ) = p j f (P r /P)g(r )| C j , where f (P r /P) is a normalized polynomial function (27) with 0 < A < 1 and g(r ) = r a for a > 0. As mentioned earlier, Lu et al (2015) claim that for that problem, an optimal permutation is V-shaped, although no rigorous proof has been given.…”

“…This subsection is aimed at resolving the status of problem 1| p j (r ) = p j f (P r /P)g(r )| C j for a wide range of functions that define the combined effect, including those functions that are considered in Lu et al (2015). Recall that Lu et al (2015) address problem 1| p j (r ) = p j f (P r /P)g(r )| C j and claim that for f (P r /P) given by a normalized polynomial function (27) with 0 < A < 1 and g(r ) = r a for a > 0, an optimal permutation is V-shaped. However, the proof technique used in Lu et al (2015) is based on Mosheiov (2005) and is therefore incorrect.…”

“…Recall that Lu et al (2015) address problem 1| p j (r ) = p j f (P r /P)g(r )| C j and claim that for f (P r /P) given by a normalized polynomial function (27) with 0 < A < 1 and g(r ) = r a for a > 0, an optimal permutation is V-shaped. However, the proof technique used in Lu et al (2015) is based on Mosheiov (2005) and is therefore incorrect. That leaves the status of the problem open.…”

“…Although the conditions hold for a wide range of problems, they do not hold for the model in which both functions f and g are polynomial. The latter problem has been studied by Lu et al (2015), where relying on a wrong proof technique, the authors claim that the problem admits an optimal V-shaped sequencing policy.…”