Improved algorithms for single machine serial-batch scheduling to minimize makespan and maximum cost

TL;DR

Addresses the bicriteria scheduling problem of minimizing makespan $C_{\max}$ and maximum cost $f_{\max}$ when scheduling $n$ jobs on a serial-batch machine with batch capacity $b$ and setup time $s$, considering both bounded ($b<n$) and unbounded ($b\ge n$) models and two precedence types. Proposes a CSF-based framework with auxiliary problems (AUX1/AUX2) and Pareto-driven main procedures (MAIN1/MAIN2) to enumerate all Pareto-optimal points $\Omega(\mathcal{J})$ in $O(n^3)$ time. Improves the previous best-known running time from $O(n^4)$ for both the bounded and unbounded formulations and provides schedules corresponding to each Pareto point. Experimental validation on randomly generated instances demonstrates practical efficiency and corroborates the theoretical time bounds.

Abstract

This paper studies the bicriteria problem of scheduling $n$ jobs on a serial-batch machine to minimize makespan and maximum cost simultaneously. A serial-batch machine can process up to $b$ jobs as a batch, where $b$ is known as the batch capacity. When a new batch starts, a constant setup time is required for the machine. Within each batch, the jobs are processed sequentially, and thus the processing time of a batch equals the sum of the processing times of its jobs. All the jobs in a batch have the same completion time, namely, the completion time of the batch. The main result is an $O(n^3)$-time algorithm which can generate all Pareto optimal points for the bounded model ($b<n$) without precedence relation. The algorithm can be modified to solve the unbounded model ($b\ge n$) with strict precedence relation in $O(n^3)$ time as well. The results improve the previously best known running time of $O(n^4)$ for both the bounded and unbounded models.

Theorems & Definitions (11)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• Lemma 5
• proof
• Theorem 6
• proof