An online joint replenishment problem combined with single machine scheduling

TL;DR

This work addresses the online joint replenishment and single-machine scheduling problem, minimizing the total replenishment cost $Kq$ plus the maximum flow time $F_{\,\max}$. It introduces a deterministic online algorithm with competitive ratio $2$ for general inputs and shows the ratio converges to $\sqrt{2}$ on $p$-bounded inputs as the number of jobs grows, supported by a structural analysis of offline optima. The paper also derives several lower bounds (e.g., $3/2$ for two jobs, $4/3$ for three or more, and about $1.015$ for long $p$-regular sequences) and provides numerical experiments validating the theoretical claims. Together, these results map the efficiency frontier for online replenishment plus scheduling and guide future improvements in online policy design and lower-bound proofs.

Abstract

This paper considers a combination of the joint replenishment problem with single machine scheduling. There is a single resource, which is required by all the jobs, and a job can be started at time point $t$ on the machine if and only the machine does not process another job at $t$, and the resource is replenished between its release date and $t$. Each replenishment has a cost, which is independent of the amount replenished. The objective is to minimize the total replenishment cost plus the maximum flow time of the jobs. We consider the online variant of the problem, where the jobs are released over time, and once a job is inserted into the schedule, its starting time cannot be changed. We propose a deterministic 2-competitive online algorithm for the general input. Moreover, we show that for a certain class of inputs (so-called $p$-bounded input), the competitive ratio of the algorithm tends to $\sqrt{2}$ as the number of jobs tends to infinity. We also derive several lower bounds for the best competitive ratio of any deterministic online algorithm under various assumptions.

Figures (5)

• Figure 1: Two feasible solutions. The arrows below the axis denote the replenishments.
• Figure 2: The inputs $I$, $D_I$ and $R_I$ for $p=3$.
• Figure 3: Competitive ratios for $\beta = 0.01, n \in \{100,200,1000\}$.
• Figure 4: Competitive ratios for $\beta = 0.001, n \in \{500,1000,5000\}$.
• Figure 5: Competitive ratios for $\beta = 0.0001, n \in \{1000,5000,10000\}$.

Theorems & Definitions (35)

• Proposition 1
• proof
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• Proposition 2
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• Lemma 1
• Lemma 2
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• Lemma 3
• proof