Online Algorithms to Schedule a Proportionate Flexible Flow Shop of Batching Machines

TL;DR

We study online scheduling for proportionate flexible flow shops with batching machines (PFFB), motivated by on-demand production of individualized medicaments. The problem is formalized as FFs | $r_j$, $p_{ij}=p_i$, $p$-batch, $b_i$ | $f$ with four objectives: $C_{ ext{max}}$, $\, ext{sum} C_j$, $F_{ ext{max}}$, and $\, ext{sum} F_j$, in an online setting where $n$ is unknown until the end. We prove that optimal online schedules can be restricted to permutation schedules ordered by earliest release dates and introduce two online algorithms: Never-Wait, which is $2$-competitive for all four objectives, and $t$-Switch, which for the special case $s=2$ is $\,oldsymbol{\varphi}$-competitive for $C_{ ext{max}}$ and $\, ext{sum} C_j$. Lower-bound results show that the golden ratio $oldsymbol{\varphi}$ is tight for makespan and total completion time and that the $2$-competitiveness bound is tight for total flow time; the analysis extends to the non-batching special case (PFB). This work establishes the first online results for PFFB and provides a basis for improved online strategies and for understanding the gap between simple online rules and optimal offline performance.

Abstract

This paper is the first to consider online algorithms to schedule a proportionate flexible flow shop of batching machines (PFFB). The scheduling model is motivated by manufacturing processes of individualized medicaments, which are used in modern medicine to treat some serious illnesses. We provide two different online algorithms, proving also lower bounds for the offline problem to compute their competitive ratios. The first algorithm is an easy-to-implement, general local scheduling heuristic. It is 2-competitive for PFFBs with an arbitrary number of stages and for several natural scheduling objectives. We also show that for total/average flow time, no deterministic algorithm with better competitive ratio exists. For the special case with two stages and the makespan or total completion time objective, we describe an improved algorithm that achieves the best possible competitive ratio $\varphi=\frac{1+\sqrt{5}}{2}$, the golden ratio. All our results also hold for proportionate (non-flexible) flow shops of batching machines (PFB) for which this is also the first paper to study online algorithms.