Scheduling two types of jobs with minimum makespan

TL;DR

This paper presents an efficient algorithm for scheduling two job types on multiple machines to minimize the makespan, considering quadratic and linear processing times for batches of jobs.

Contribution

It introduces an $O(n^2p\log(n))$ dynamic programming algorithm for quadratic batch processing times, applicable also to linear cases, optimizing job scheduling.

Findings

Algorithm efficiently minimizes makespan for quadratic batch processing.

Applicable to linear batch processing with same complexity.

Provides optimal scheduling solution for two job types on multiple machines.

Abstract

We consider scheduling two types of jobs (A-job and B-job) to $p$ machines and minimizing their makespan. A group of same type of jobs processed consecutively by a machine is called a batch. For machine $v$, processing $x$ A-jobs in a batch takes $k^A_vx^2$ time units for a given speed $k^A_v$, and processing $x$ B-jobs in a batch takes $k^B_vx^2$ time units for a given speed $k^B_v$. We give an $O(n^2p\log(n))$ algorithm based on dynamic programming and binary search for solving this problem, where $n$ denotes the maximal number of A-jobs and B-jobs to be distributed to the machines. Our algorithm also fits the easier linear case where each batch of length $x$ of $A$-jobs takes $k^A_v x$ time units and each batch of length $x$ of $B$-jobs takes $k^B_vx$ time units. The running time is the same as the above case.