Template:Economical life model rsa

Economical Life Model

One consideration in reducing the cost to maintain repairable systems is to establish an overhaul policy that will minimize the total life cost of the system. However, an overhaul policy makes sense only if [math]\displaystyle{ \beta \gt 1 }[/math] . It does not make sense to implement an overhaul policy if [math]\displaystyle{ \beta \lt 1 }[/math] since wearout is not present. If you assume that there is a point at which it is cheaper to overhaul a system than to continue repairs, what is the overhaul time that will minimize the total life cycle cost while considering repair cost and the cost of overhaul?
Denote [math]\displaystyle{ {{C}_{1}} }[/math] as the average repair cost (unscheduled), [math]\displaystyle{ {{C}_{2}} }[/math] as the replacement or overhaul cost and [math]\displaystyle{ {{C}_{3}} }[/math] as the average cost of scheduled maintenance. Scheduled maintenance is performed for every [math]\displaystyle{ S }[/math] miles or time interval. In addition, let [math]\displaystyle{ {{N}_{1}} }[/math] be the number of failures in [math]\displaystyle{ [0,t] }[/math] and let [math]\displaystyle{ {{N}_{2}} }[/math] be the number of replacements in [math]\displaystyle{ [0,t] }[/math] . Suppose that replacement or overhaul occurs at times [math]\displaystyle{ T }[/math] , [math]\displaystyle{ 2T }[/math] , [math]\displaystyle{ 3T }[/math] . The problem is to select the optimum overhaul time [math]\displaystyle{ T={{T}_{0}} }[/math] so as to minimize the long term average system cost (unscheduled maintenance, replacement cost and scheduled maintenance). Since [math]\displaystyle{ \beta \gt 1 }[/math] , the average system cost is minimized when the system is overhauled (or replaced) at time [math]\displaystyle{ {{T}_{0}} }[/math] such that the instantaneous maintenance cost equals the average system cost. The total system cost between overhaul or replacement is:

[math]\displaystyle{ TSC(T)={{C}_{1}}E(N(T))+{{C}_{2}}+{{C}_{3}}\frac{T}{S} }[/math]

So the average system cost is:

[math]\displaystyle{ C(T)=\frac{{{C}_{1}}E(N(T))+{{C}_{2}}+{{C}_{3}}\tfrac{T}{S}}{T} }[/math]

The instantaneous maintenance cost at time [math]\displaystyle{ T }[/math] is equal to:

[math]\displaystyle{ IMC(T)={{C}_{1}}\lambda \beta {{T}^{\beta -1}}+\frac{{{C}_{3}}}{S} }[/math]

The following equation holds at optimum overhaul time [math]\displaystyle{ {{T}_{0}} }[/math] :

[math]\displaystyle{ \begin{align} & {{C}_{1}}\lambda \beta T_{0}^{\beta -1}+\frac{{{C}_{3}}}{S}= & \frac{{{C}_{1}}E(N(T))+{{C}_{2}}+{{C}_{3}}\tfrac{T}{S}}{T} \\ & = & \frac{{{C}_{1}}\lambda T_{0}^{\beta }+{{C}_{2}}+{{C}_{3}}\tfrac{{{T}_{0}}}{S}}{{{T}_{0}}} \end{align} }[/math]

Therefore:

[math]\displaystyle{ {{T}_{0}}={{\left[ \frac{{{C}_{2}}}{\lambda (\beta -1){{C}_{1}}} \right]}^{1/\beta }} }[/math]

When there is no scheduled maintenance, Eqn. (ecolm) becomes:

[math]\displaystyle{ {{C}_{1}}\lambda \beta T_{0}^{\beta -1}=\frac{{{C}_{1}}\lambda T_{0}^{\beta }+{{C}_{2}}}{{{T}_{0}}} }[/math]

The optimum overhaul time, [math]\displaystyle{ {{T}_{0}} }[/math] , is the same as Eqn. (optimt), so for periodic maintenance scheduled every [math]\displaystyle{ S }[/math] miles, the replacement or overhaul time is the same as for the unscheduled and replacement or overhaul cost model.