Models of inspection, routine service, and replacement for a serviceable one-component system
Introduction
As one of the key operational support elements of production, preventive maintenance (PM) scheduling always plays an important role in production planning and scheduling, [1], [2]. Though condition-based maintenance has become more popular in recent years, a combination of periodical and condition-based PM is always better than a single maintenance policy alone, [3], [4]. PM is often defined as the maintenance activities which are performed before the system failure in order to avoid catastrophic failure consequences. These activities include lubricating, cleaning, calibrating, inspection and preventive replacement of defective systems. Inspections can be done manually only by technicians or by using some condition monitoring devices taking relevant measurements with manual inspection if necessary. Since both types are used in this paper so we use inspection to refer manual inspections and monitoring to refer condition-based monitoring. Further, we define in this paper those non-inspection and non-replacement activities of PM as routine service (RS) to distinguish them from the inspection and replacement activities of PM.
For any PM program, inspection is always a necessary activity, as it provides information on the status of the system checked to facilitate the determination and execution of repair and replacement decisions. However, there have been very few papers in literature that addressed the RS activities of PM. A typical example is a car service where a 72-point inspection and the change of oil and filters (RS activities) are performed along with any repair or replacement of defective systems.
It is well known that most systems' failures do not occur suddenly and this is why inspection is necessary. To investigate why inspection is an important part of a PM program, Christer proposed the delay-time concept to justify the necessity of inspection activities for a system PM program, [5]. The delay-time concept considers the system failure process as a two-stage process, from new to an initial point that a defect can be first identified by an inspection, and then from that point to failure if the defect is unattended. The time period from the initial point of an identifiable defect to failure is called the failure delay-time of the system or simply the delay-time. If such time is zero then it means that failures occur suddenly without any identifiable prior symptom or without any delay from the point of starting to fail. But as said earlier, most systems' failures will not occur immediately without any pre-warning so some delay-time exists. If an inspection is carried out during the delay-time, the defect present should or could be identified and removed immediately or subsequently, depending on the quality of the inspection and defect removal. The models established using the delay-time concept explicitly model the relationship between the number of failures and inspection intervals within a PM scheme, and therefore, can be used within a cost or downtime model to optimize the inspection interval. See [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] for further details of the delay-time concept and modeling applications. However, most previous delay-time based models only modeled the effect of inspection and ignored the influence of RS. There are two types of delay-time-based models, [19], that is, a model to describe a complex system with many components and a model for a one-component system. In this paper, we only analyze a one-component system where only one dominant failure mode exists and the repair action is replacement if the system is identified to be in the delay-time stage. Such a one-component system is non-repairable, but serviceable as defined in this paper, which extends the definition in [20] for a one-component system. For short we call such a serviceable one-component system as a system in this paper subsequently. Since the two-stage failure process essentially defines a two-state state variable for the system at one time before failure, namely normal or defective, so we also use the state at times to refer the status of the system revealed by inspection at one time.
A hypothesis, which can be validated empirically if data exist, is that RS can prolong the life of the system, [13]. This is actually what underpins the necessity of RS activities. We take this and further assume that RS can only prolong the life when the system is in the normal stage. This is because when the system is identified to be in the failure delay-time stage, the system should be repaired or replaced immediately as we assumed later. There are many existing methods to model the maintenance effect such as age-reduction models, [21], [22], geometric processes, [23], [24] and proportional hazard models, [25], [26]. However, most of them either modeled the effect of failure-based maintenance without PM or modeled the effect of PM broadly without differentiating whether the PM is a RS or an inspection with replacement. It is a fact that inspections will not change the state of the system if the system is normal working, but RS will somehow prolong the life of the system even it is healthy. This is why a routine change of engine oil is highly recommended. To model the effect of RS we borrow the concept of age-reduction models, [21], [22], to alter the hazard function of the system. In this way, the effect of RS is manipulated through the changed remaining normal life by the change of the associated hazard function.
The new contributions of this paper are: (1) it differentiates and models four different types of PM activities in two models; (2) it considers particularly the effect of RS for a serviceable one-component system; (3) it considers monitoring as part of inspection with full manual inspection as a supplement action if required; and finally (4) both an analytical and a hybrid of simulation with analytical calculation are proposed for solutions.
The remaining parts of the paper are organised as follows. Section 2 presents the assumptions and notation of the proposed model. Section 3 focuses on modelling development. Section 4 presents numerical examples and Section 5 concludes the paper.
Section snippets
Assumptions and notation
We propose the following assumptions for the sake of model building.
- 1.
A system is subjected to a single failure mode. This is a common assumption used in previous researches for one component systems, [9], [10], [19]. If more failure modes are involved, one has to model each of them with different probability density functions and there could be possibly correlations among them. This issue has been addressed in [27] but will substantially increase the complexity of the model, which is not the
The models
The key relationship we need to establish is the effect of RS on the hazard rate function. We borrow the idea of age reduction models, [21], [22], to change the virtual age of the system by letting . This is a form of the age reduction model and there are other forms used in literature such as the exponential and power law forms, [22]. As which form to be used in practice, we suggest to use a goodness of fit measure such as AIC or BIC, [30], to choose the appropriate
A numerical example
The data is partially taken from [33]. All distributions are assumed to be Weibull though there is no restriction on the distribution form to be used since the calculations were all carried out numerically. The distribution forms, distribution and cost parameters are shown in Table 1, Table 2, Table 3. All time measurement units are in days including the inspection and monitoring intervals.
Fig. 2 shows the hazard rate functions with two values of a; one is a=1.00 which implies that RS has no
Conclusions
RS as defined in this paper is a common practice adopted in industry as a way to prolong the life of the system, but is often ignored in maintenance modelling. We model in this paper the four common maintenance techniques used in PM programmes, that is, RS, inspection, monitoring and preventive replacement. The effect of RS is modelled by altering the hazard rate function with an age reduction factor after each RS. The effect is clearly shown in Fig. 2, Fig. 3. Of course, other approaches can
Notation
- t
PM interval
- tj
jth PM epoch since new
- j
Sequence of the PM epochs
- Xj
The remaining time to the delay-time stage after the jth PM with pdf , cdf and survival function
- r0(x)
Baseline hazard rate function without PM
- rj(x)
Hazard rate function at x after the jth PM
- Y
Delay-time with pdf and cdf
- Hx
Health index at time x
- Ha
Threshold of the probability for the system to be in the normal state
- Cs
Inspection cost
- Cm
Monitoring cost
- Crm
Routine service cost
- Cp
Preventive replacement cost
- Cf
Acknowledgement
This work is partially supported by the National Natural Science Foundation of China under grant number 71231001, by the PhD supervisor fund of MOE and by the Fundamental Research Funds for the Central Universities of China, FRF-SD-12-020A, FRF-MP-13-009A, FRF-SD-13-004B.
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