<p>This paper presents a two-stage optimization replacement policy considering reliability and economy for a system with <i>N</i> multiple spare units. We assume that the failure time of each unit is stochastic, and the operating system is replaced at failure or at the planned replacement time intervals <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10603_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$T_{k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10603_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">$k=1,2,\ldots ,N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>) from its installation, whichever occurs first. In stage 1, we maximizes mean time to failure (MTTF) to obtain the optimal replacement time intervals <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10603_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">$T_{1}^{},T_{2}^{},\ldots ,T_{N}^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>T</mi> <mrow> <mn>1</mn> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>,</mo> <mmultiscripts> <mi>T</mi> <mrow> <mn>2</mn> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mmultiscripts> <mi>T</mi> <mrow> <mi>N</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </mrow> </math></EquationSource> </InlineEquation>. In stage 2, the aim is to find the optimal number of spare units <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10603_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$N^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>N</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation> which minimizes the expected cost rate, and the existing condition of the unique optimal solution is given. Finally, numerical examples and sensitivity analysis are carried out to demonstrate the presented policy.</p>

A two-stage optimization replacement policy with multiple spare units

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发表日期：

2025年3月1日

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Chunxiao Zhang,
Hejiao Shen,
Yizhou Bai,
Chao Yan

摘要

This paper presents a two-stage optimization replacement policy considering reliability and economy for a system with N multiple spare units. We assume that the failure time of each unit is stochastic, and the operating system is replaced at failure or at the planned replacement time intervals $T_{k}$ $T_{k}$ ( $k=1,2,\ldots ,N$ $k = 1, 2, \dots, N$ ) from its installation, whichever occurs first. In stage 1, we maximizes mean time to failure (MTTF) to obtain the optimal replacement time intervals $T_{1}^{*},T_{2}^{*},\ldots ,T_{N}^{*}$ $T_{1}^{*}, T_{2}^{*}, \dots, T_{N}^{*}$ . In stage 2, the aim is to find the optimal number of spare units $N^{*}$ $N^{*}$ which minimizes the expected cost rate, and the existing condition of the unique optimal solution is given. Finally, numerical examples and sensitivity analysis are carried out to demonstrate the presented policy.