A branch‐and‐price algorithm for identical parallel machine scheduling with multiple milestones

Abstract

This article considers an identical parallel‐machine task scheduling problem motivated by operations management of online services. A task with an integer processing time can be split into sub‐tasks with integer processing times. Each task has multiple integer milestones and at each milestone a nonnegative penalty will occur. The penalty value of a task at a milestone is a convex nonincreasing function of the completed amount by this milestone. Our objective is to determine a feasible schedule for all the tasks on given identical parallel machines, such that the sum of all tasks' total penalty at all milestones is minimized. We prove the NP‐hardness of this problem in the ordinary sense and develop a branch‐and‐price algorithm. Computational experiments utilizing data from an online service operations survey show that this algorithm is singularly efficient and promising.