Maintenance Planning - Revision history

Lisa Hacker at 19:16, 15 September 2023

2023-09-15T19:16:21Z

← Older revision		Revision as of 19:16, 15 September 2023
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	{{Template:~~LDABOOK_SUB~~\|Additional Reliability Analysis Tools\|Maintenance Planning}}		{{Template:LDABOOK\|Additional Reliability Analysis Tools\|Maintenance Planning}}

	For components and systems with increasing failure rates, planned replacements and inspections are often used to prevent failures in order to improve system availability and reduce overall maintenance cost. For example, the commonly used preventive maintenance strategy is to replace an aged item at a given time interval in order to minimize the long term cost per unit time. An optimal time interval exists if the failure rate increases with time and the cost of failure is much higher than the cost of a scheduled replacement.		For components and systems with increasing failure rates, planned replacements and inspections are often used to prevent failures in order to improve system availability and reduce overall maintenance cost. For example, the commonly used preventive maintenance strategy is to replace an aged item at a given time interval in order to minimize the long term cost per unit time. An optimal time interval exists if the failure rate increases with time and the cost of failure is much higher than the cost of a scheduled replacement.

John Leavitt: /* Minimize the Long Term Cost Per Unit Time */

2013-11-20T23:38:04Z

Minimize the Long Term Cost Per Unit Time

← Older revision		Revision as of 23:38, 20 November 2013
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	''M'' is the largest inspection interval where there is no overlap between the detection zones of two adjunct inspections. Any failures that occur after the M<sup>th</sup> inspection will always be detected.		''M'' is the largest inspection interval where there is no overlap between the detection zones of two adjunct inspections. Any failures that occur after the M<sup>th</sup> inspection will always be detected.

	The expected ~~replacement cycle length~~ is:		The expected long term cost per unit time is:


	<math>\begin{align}		<math>\begin{align}
	& E(\text{Cycle ~~Length~~})=\int_{0}^{x}{{{C}_{U}}f(t)dt}+\sum\limits_{k=1}^{M}{\left[ \int\limits_{kx}^{\frac{kx}{p}}{\left[ {{C}_{P}}+k{{C}_{I}} \right]}f(t)dt+\int\limits_{\frac{kx}{p}}^{(k+1)x}{\left[ {{C}_{U}}+k{{C}_{I}} \right]}f(t)dt \right]} \\		& E(\text{Cost Per Cycle})=\int_{0}^{x}{{{C}_{U}}f(t)dt}+\sum\limits_{k=1}^{M}{\left[ \int\limits_{kx}^{\frac{kx}{p}}{\left[ {{C}_{P}}+k{{C}_{I}} \right]}f(t)dt+\int\limits_{\frac{kx}{p}}^{(k+1)x}{\left[ {{C}_{U}}+k{{C}_{I}} \right]}f(t)dt \right]} \\
	& +\int\limits_{\left( M+1 \right)x}^{\frac{(M+1)x}{p}}{\left[ {{C}_{P}}+(M+1){{C}_{I}} \right]}f(t)dt+\sum\limits_{k=M+1}^{\infty }{\int\limits_{\frac{kx}{p}}^{\frac{(k+1)x}{p}}{\left[ {{C}_{P}}+(k+1){{C}_{I}} \right]f(t)dt}}		& +\int\limits_{\left( M+1 \right)x}^{\frac{(M+1)x}{p}}{\left[ {{C}_{P}}+(M+1){{C}_{I}} \right]}f(t)dt+\sum\limits_{k=M+1}^{\infty }{\int\limits_{\frac{kx}{p}}^{\frac{(k+1)x}{p}}{\left[ {{C}_{P}}+(k+1){{C}_{I}} \right]f(t)dt}}
	\end{align}</math>		\end{align}</math>


	The expected ~~cost for one~~ cycle is:		The expected replacement cycle length is:

	<math>\begin{align}		<math>\begin{align}
	& E(\text{~~Cost Per~~ Cycle})=\int_{0}^{x}{tf(t)dt}+\sum\limits_{k=1}^{M}{\left[ \int\limits_{kx}^{\frac{kx}{p}}{\left( kx \right)}f(t)dt+\int\limits_{\frac{kx}{p}}^{(k+1)x}{t}f(t)dt \right]} \\		& E(\text{Cycle Length})=\int_{0}^{x}{tf(t)dt}+\sum\limits_{k=1}^{M}{\left[ \int\limits_{kx}^{\frac{kx}{p}}{\left( kx \right)}f(t)dt+\int\limits_{\frac{kx}{p}}^{(k+1)x}{t}f(t)dt \right]} \\
	& +\int\limits_{\left( M+1 \right)x}^{\frac{(M+1)x}{p}}{\left[ (M+1)x \right]}f(t)dt+\sum\limits_{k=M+1}^{\infty }{\int\limits_{\frac{kx}{p}}^{\frac{(k+1)x}{p}}{\left[ (k+1)x \right]f(t)dt}}		& +\int\limits_{\left( M+1 \right)x}^{\frac{(M+1)x}{p}}{\left[ (M+1)x \right]}f(t)dt+\sum\limits_{k=M+1}^{\infty }{\int\limits_{\frac{kx}{p}}^{\frac{(k+1)x}{p}}{\left[ (k+1)x \right]f(t)dt}}
	\end{align}</math>		\end{align}</math>

Harry Guo: /* Optimal Inspection Interval */

2013-09-12T23:49:07Z

Optimal Inspection Interval

← Older revision		Revision as of 23:49, 12 September 2013
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	Inspection is often used to test if a device is working properly. For example, signs of a device's impending failure due to degradation, such as corrosion and crack, can be detected via inspection. The device can then be replaced before the failure actually occurs. If a failure cannot be detected during the inspection, it will be noticed immediately when the failure occurs, and the device will be replaced at the time of the failure.		Inspection is often used to test if a device is working properly. For example, signs of a device's impending failure due to degradation, such as corrosion and crack, can be detected via inspection. The device can then be replaced before the failure actually occurs. If a failure cannot be detected during the inspection, it will be noticed immediately when the failure occurs, and the device will be replaced at the time of the failure.

	Assume that a failure occurs at time <math>t\,\!</math>. If an inspection is conducted within the range of <math>[p\times t,t]\,\!\,\!</math>, the failure will be detected and the component will be replaced. The term ''p'' is a value between 0 and 1. This is illustrated in the following plot.		Assume that a failure occurs at time <math>t\,\!</math>. If an inspection is conducted within the range of <math>[p\times t,t]\,\!\,\!</math>, the failure will be detected and the component will be replaced. The term <math>p\,\!</math> is a value between 0 and 1. This is illustrated in the following plot.

	[[Image:Optimal_inspection_interval.png\|center\|650px\|]]		[[Image:Optimal_inspection_interval.png\|center\|650px\|]]

Harry Guo: /* Optimal Inspection Interval */

2013-09-12T23:48:35Z

Optimal Inspection Interval

← Older revision		Revision as of 23:48, 12 September 2013
Line 50:		Line 50:
	Inspection is often used to test if a device is working properly. For example, signs of a device's impending failure due to degradation, such as corrosion and crack, can be detected via inspection. The device can then be replaced before the failure actually occurs. If a failure cannot be detected during the inspection, it will be noticed immediately when the failure occurs, and the device will be replaced at the time of the failure.		Inspection is often used to test if a device is working properly. For example, signs of a device's impending failure due to degradation, such as corrosion and crack, can be detected via inspection. The device can then be replaced before the failure actually occurs. If a failure cannot be detected during the inspection, it will be noticed immediately when the failure occurs, and the device will be replaced at the time of the failure.

	Assume that a failure occurs at time <math>t</math>. If an inspection is conducted within the range of <math>[p\times t,t]\,\!\,\!</math>, the failure will be detected and the component will be replaced. The term ''p'' is a value between 0 and 1. This is illustrated in the following plot.		Assume that a failure occurs at time <math>t\,\!</math>. If an inspection is conducted within the range of <math>[p\times t,t]\,\!\,\!</math>, the failure will be detected and the component will be replaced. The term ''p'' is a value between 0 and 1. This is illustrated in the following plot.

	[[Image:Optimal_inspection_interval.png\|center\|650px\|]]		[[Image:Optimal_inspection_interval.png\|center\|650px\|]]

Harry Guo: /* Optimal Inspection Interval */

2013-09-12T23:47:52Z

Optimal Inspection Interval

← Older revision		Revision as of 23:47, 12 September 2013
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	==Optimal Inspection Interval==		==Optimal Inspection Interval==
	Inspection is often used to test if a device is working properly. For example, signs of a device's impending failure due to degradation, such as corrosion, can be detected via inspection. The device can then be replaced before the failure actually occurs. If a failure cannot be detected during the inspection, it will be noticed immediately when the failure occurs, and the device will be replaced at the time of the failure.		Inspection is often used to test if a device is working properly. For example, signs of a device's impending failure due to degradation, such as corrosion and crack, can be detected via inspection. The device can then be replaced before the failure actually occurs. If a failure cannot be detected during the inspection, it will be noticed immediately when the failure occurs, and the device will be replaced at the time of the failure.

	Assume that a failure occurs at time <math>t</math>. If an inspection is conducted within the range of <math>[p\times t,t]\,\!\,\!</math>, the failure will be detected and the component will be replaced. The term ''p'' is a value between 0 and 1. This is illustrated in the following plot.		Assume that a failure occurs at time <math>t</math>. If an inspection is conducted within the range of <math>[p\times t,t]\,\!\,\!</math>, the failure will be detected and the component will be replaced. The term ''p'' is a value between 0 and 1. This is illustrated in the following plot.

Harry Guo: /* Minimize the Short Term Cost Per Unit Time */

2013-09-12T23:45:58Z

Minimize the Short Term Cost Per Unit Time

← Older revision		Revision as of 23:45, 12 September 2013
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	where <math>\eta </math> is the scale parameter of the Weibull distribution.		where <math>\eta \,\!</math> is the scale parameter of the Weibull distribution.

	==Optimal Inspection Interval==		==Optimal Inspection Interval==

Harry Guo: /* Optimal Preventive Replacement Interval */

2013-09-12T23:45:01Z

Optimal Preventive Replacement Interval

← Older revision		Revision as of 23:45, 12 September 2013
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	The optimal preventive maintenance interval will be the value <math>t\,\!</math> which minimizes <math>CPU{{T}_{L}}\,\!</math>. In the above equations:		The optimal preventive maintenance interval will be the value <math>x\,\!</math> which minimizes <math>CPU{{T}_{L}}\,\!</math>. In the above equations:

	*<math>f(x)\,\!</math> is the probability density function of the failure time.		*<math>f(x)\,\!</math> is the probability density function of the failure time.

Harry Guo: /* Minimize the Long Term Cost Per Unit Time */

2013-09-12T23:44:12Z

Minimize the Long Term Cost Per Unit Time

← Older revision		Revision as of 23:44, 12 September 2013
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	The expected long term cost per unit time is:		The expected long term cost per unit time is:

	<math>CPU{{T}_{L}}(t)=\frac{E(\text{Cost Per Cycle})}{E\left( \text{Cycle Length} \right)}</math>		<math>CPU{{T}_{L}}(x)=\frac{E(\text{Cost Per Cycle})}{E\left( \text{Cycle Length} \right)}</math>


	where <math>{{C}_{I}}\,\!</math> is the average cost per inspection.		where <math>{{C}_{I}}\,\!</math> is the average cost per inspection.

	===Minimize the Short Term Cost Per Unit Time===		===Minimize the Short Term Cost Per Unit Time===

Harry Guo: /* Minimize the Long Term Cost Per Unit Time */

2013-09-12T23:43:30Z

Minimize the Long Term Cost Per Unit Time

← Older revision		Revision as of 23:43, 12 September 2013
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	Therefore, the expected long term cost per unit time is:		Therefore, the expected long term cost per unit time is:

	<math>CPU{{T}_{L}}(t)=\frac{E(\text{Cost Per Cycle})}{E\left( \text{Cycle Length} \right)}=\frac{{{C}_{P}}R(x)+{{C}_{U}}\left( 1-R(x) \right)}{\int_{0}^{x}{R(t)dt}}</math>		<math>CPU{{T}_{L}}(x)=\frac{E(\text{Cost Per Cycle})}{E\left( \text{Cycle Length} \right)}=\frac{{{C}_{P}}R(x)+{{C}_{U}}\left( 1-R(x) \right)}{\int_{0}^{x}{R(t)dt}}</math>

Harry Guo: /* Minimize the Short Term Cost Per Unit Time */

2013-09-12T23:43:05Z

Minimize the Short Term Cost Per Unit Time

← Older revision		Revision as of 23:43, 12 September 2013
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	The expected cost per unit time in one replacement cycle is defined as:		The expected cost per unit time in one replacement cycle is defined as:

	<math>CPU{{T}_{S}}(t)=E\left( \frac{\text{Cost Per Cycle}}{\text{Cycle Length}} \right)=\int_{0}^{x}{\frac{{{C}_{U}}}{t}f(t)dt}+\frac{{{C}_{P}}}{x}\left( 1-F(x) \right)</math>		<math>CPU{{T}_{S}}(x)=E\left( \frac{\text{Cost Per Cycle}}{\text{Cycle Length}} \right)=\int_{0}^{x}{\frac{{{C}_{U}}}{t}f(t)dt}+\frac{{{C}_{P}}}{x}\left( 1-F(x) \right)</math>