Maintenance Decision Based on Parameter Monitoring: a Case Study

We consider a system which has an observable randomly changing increasing parameter φ(t). The observation is either continuous or discrete on a time scale with step δ. System's output on [0,t] depends on φ via the following reward functional: ∫ − = t dx x H t A 0 , ))] ( ( [ ) ( φ ψ where ψ(·)is a known increasing function. The operation and observation process can be stopped for a preventive maintenance (PM) at any instant T = K · δ, K = 1, 2, . . .. The PM costs Cp and lasts m · δ=∆. We suggest the following stopping rule for carrying out the PM: monitor the values of the functional ) /( ) ) ( ( ) ( δ δ δ η m K C K A K p + − = . Stop for PM at time δ ) 1 ( + K if ) ( ) 1 ( K K η η < + . 1. Introduction and problem formulation We consider a system which has an observable randomly changing increasing parameter φ(t). The observation may be either continuous or discrete on a time scale with observation step δ. For a given fixed trajectory φ(t), system's efficiency on [0, t] is determined by the following reward functional: ∫ − = t dx x H t A 0 , ))] ( ( [ ) ( φ ψ (1) where ψ(·) is a known increasing function. At some instant t0, when the value of the function )) ( ( ) ( x H x φ ψ α − = reaches the minimal permissible level αmin , the operation-observation process is stopped for a preventive maintenance (PM) . If we have a discrete monitoring scheme, then the PM can be carried out at the instants T = K · δ, K = 1, 2, . . .. The PM costs Cp and lasts m · δ. The goal of the PM is to achieve maximal reward per unit time. The specific feature of our model is the possibility to monitor, continuously or on a discrete time scale, the parameter φ(x), and therefore to monitor the values of A(t). The above described situation is quite common. The real-life prototype of our model is an aggregate for producing electric power. The parameter φ(t) is the randomly changing loss of the active area of pipe crossection in the power producing unit. The crossection area decreases due to the carbonization in the process of burning the coal, and the functional A(t) equals, up to a constant multiplier, to the total cost of electricity produced during the interval [0, t]. When the active area falls below some critical level, the aggregate is stopped for a PM. The PM costs Cp and its duration is ∆. After the PM, the parameter φ is brought to its initial value φ(0), and the next trajectory of φ(t) is statistically identical to the previous one. Examples of technical systems with a gradual decrease in their efficiency due to a gradual parameter changes are presented by Gaver and Mutlu (1989) . The characteristic feature of the model considered in this note is the absence of statistical information about the parameter φ(t). The only assumptions made are the following: (i) φ(0) = 0; (ii) φ(x) is strictly increasing on any [0, t]; (iii) With probability one, in the absence of PM, φ(x) crosses certain maximal admissible level φmax. Let tmax be the random moment when the parameter φ(x) reaches the level φmax. If the PM was not carried out before that moment, it will be made at this moment. Assume that we monitor the system parameter φ(x) on a discrete time with step δ. We suggest the following rule for carrying out the PM: monitor the values of the reward per unit time functional η(K): , ) ( ) ( δ δ δ η m K C K A K p + − = K = 1, 2, …; (2) Stop for PM at time (K + l)δ if η(K + 1) < η(K). In other words, we stop for PM at the instant (K + 1)δ when the reward functional η(K) starts declining. If we never observe that η(K) starts declining, we stop for the PM at the instant when we observe that the parameter φ(x) > φmax. The above maintenance rule will be termed the locally optimal stopping rule (LOSR). 2. The optimal stopping time for the PM. Put H – ψ(φ(x)) = α(x) and denote by Ω the set of all trajectories α(x). Each such trajectory has a known value H = α(0), it decreases monotonically and may be observed on the interval [0,tmax(α)], where tmax(α) is the time instant when the trajectory crosses (from above) its minimal permissible level αmin = H – ψ(φmax). Let Τ be the set of all stopping times associated with Ω. Formally, an element τ ∈Τ is a random variable such that the event τ = t belongs to the σ-algebra of the trajectories α(x)∈Ω, x ∈(0,t]. General considerations based on the regenerative nature of our preventive maintenance procedure imply that the optimal stopping time (OST) τ* is defined by the following relationship: * * 0 0 ] [ ] ) ( [ ] [ ] ) ( [ sup *