A best possible algorithm for an online scheduling problem with position-based learning effect

Abstract

<p style= text-indent:20px; >In this paper, we focus on an online scheduling problem with position-based learning effect on a single machine, where the jobs are released online over time and preemption is not allowed. The information about each job <inline-formula><tex-math id= M1 >\\begin{document}$ J_j $\\end{document}</tex-math></inline-formula>, including the basic processing time <inline-formula><tex-math id= M2 >\\begin{document}$ p_j $\\end{document}</tex-math></inline-formula> and the release time <inline-formula><tex-math id= M3 >\\begin{document}$ r_j $\\end{document}</tex-math></inline-formula>, is only available when it arrives. The actual processing time <inline-formula><tex-math id= M4 >\\begin{document}$ p_j $\\end{document}</tex-math></inline-formula> of each job <inline-formula><tex-math id= M5 >\\begin{document}$ J_j $\\end{document}</tex-math></inline-formula> is defined as a function related to its position <inline-formula><tex-math id= M6 >\\begin{document}$ r $\\end{document}</tex-math></inline-formula>, i.e., <inline-formula><tex-math id= M7 >\\begin{document}$ p_j = p_j(\\alpha-r\\beta) $\\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id= M8 >\\begin{document}$ \\alpha $\\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id= M9 >\\begin{document}$ \\beta $\\end{document}</tex-math></inline-formula> are both nonnegative learning index. Our goal is to minimize the sum of completion time of all jobs. For this problem, we design a deterministic polynomial time online algorithm <i>Delayed Shortest Basic Processing Time</i> (DSBPT). In order to facilitate the understanding of the online algorithm, we present a relatively common and simple example to describe the execution process of the algorithm, and then by competitive analysis, we show that online algorithm DSBPT is a best possible online algorithm with a competitive ratio of 2.</p>