An OLE Centred Production Line Maintenance Prioritisation: Guided by the criticalities of the specific characteristics of equipment operational reliability

Maintenance task prioritisation is essential for production system, especially when available resources are limited. Related studies take either system safety or productivity as the objective of system maintenance prioritisation. While since improper system maintenance can make bad equipment cooperation thus induce unstable manufacturing process and big equipment operation time loss, system process ability is also necessary to be improved in production line maintenance. Therefore, the overall line efficiency (OLE) which comprehensively evaluates production line productivity and process ability is adopted as the objective to develop maintenance prioritisation in this study. Specifically, an OLE centred maintenance prioritisation policy which assigns priorities according to the influences of the specific characteristics of equipment operational reliability on OLE is proposed: To correctly allocate limited maintenance resources to truly critical equipment and achieve good maintenance effect, equipment influences on OLE are analysed from the level of the specific performances of their operational reliability: equipment mean time between stops ( mtbs ), mean time to resume from stops ( mttr ) and buffer size N . Considering the interactions between these parameters, a surrogate model-extended Fourier amplitude sensitivity test (SM-EFAST) process is integrated to accurately and rapidly analyse the total effects of them on OLE. In the end, an application case adopting the proposed OLE centred maintenance prioritisation method achieved higher OLE and lower total maintenance cost than two current popular maintenance prioritisation methods.


Introduction
System maintenance is of great significance for production lines. Appropriate maintenance can improve system productivity and reduce production cost [1][2][3]. Therefore, rational maintenance should be arranged timely when maintenance opportunity arises, whether caused by equipment failure or production planning [4]. However, since available maintenance resources such as time, technologist and spare parts, are limited, they should be allocated orderly and scientifically in accordance with equipment criticalities to achieve good maintenance effect. For this reason, developing maintenance prioritisation for production systems is essential.
Gopalakrishnan [5] revealed the gaps in maintenance prioritisation between academic and industrial fields in his research: (i) criticality analysis methods widely adopted in practices usually lack a clear goal; (ii) most enterprises prioritise their maintenance activities according to subjective impression or experience given the unreliable results of present criticality analysis methods.
In terms of a clear goal, current studies on production line maintenance prioritisation can be classified into system risk centred [6,7] and productivity centred [8][9][10]. Since system safety and continuous production are basic principle of industrial production, risk centred prioritisations are usually adopted to establish basic rules for system operation and maintenance. In cases when system safety and continuous production has been guaranteed through these rules, its maintenance usually aims at system productivity improvement. However, the effects of risk centred maintenance prioritisations are usually limited in meeting the demand of system productivity.
Compared to risk centred prioritisations, although higher productivities were achieved in productivity centred maintenance prioritisation studies [8,[10][11][12][13], another important task of system maintenance: the improvement of system process ability, representing the cooperation of all devices and buffers in a production line, was ignored in these studies. As is known to all, improper system maintenance will make the state of each equipment greatly affected by others thus induce unstable manufacturing process, big equipment operation time loss, and low system efficiency. Therefore, the ultimate goal of production system maintenance should be its comprehensive improvement of system process ability and productivity.
Besides, Gopalakrishnan and Skoogh [17] surveyed 71 factories in Sweden and summarised that maintenance prioritisation in practices is mostly based on ABC classification, operator influence and bottleneck analysis. Chong et al. [18] reviewed production system maintenance prioritisation studies and pointed out that analytic hierarchy process (AHP), priority criterion, priority matrix and failure mode effect and criticality analysis (FMECA) are the most commonly used methods at present.
However, further analysis shows some defects of these methods in guiding production line maintenance prioritisation: ABC classification, FIS, AHP, priority criterion, and priority matrix only qualitatively classify all equipment into several priority levels, which makes it difficult to determine the specific maintenance priorities of the equipment in the same level when available resources are limited.
In comparison, bottleneck analysis, the system efficiency influence diagram and the sensitivity analysis can quantitatively analyse equipment specific maintenance priorities. But the influence of equipment on system performance were only analysed from the overall machine level, which is insufficiently specific: Even though the equipment whose reliabilities significantly affect production line efficiency are identified, whether their failure rate or repair rate is more critical to system efficiency is still unclear. This ambiguous result usually makes maintenance actions prone to bias.
Although FMEA and FMECA can provide maintenance prioritisation guidance from equipment specific critical failure modes [7,19], they are only risk centred suggestions which may be not effective enough for system comprehensive performance improvement.
Therefore, to guide specific maintenance prioritisation for production line comprehensive performance improvement, the OLE which can evaluate the overall performance of a production line integrating its productivity and process ability is adopted for the first time as the objective to make maintenance prioritisation in this paper, namely an OLE centred system maintenance prioritisation policy is proposed. Specifically, considering OLE is mainly affected by buffer capacities Ns and equipment operational reliability which is specifically shown as equipment mtbs and mttr, their total effects on OLE are accurately and efficiently analysed through an integrated SM-EFAST process, to make it clear which equipment performance should be assigned maintenance priority to get better system OLE.
The remainder of this paper is organised as follows: Section 2 introduces OLE, analyses its influencing parameters, and provides the procedure for developing OLE centred maintenance prioritisation. Section 3 integrates a SM-EFAST process to accurately and efficiently analyse parameter influences on OLE. Lastly, an application case is studied in Section 4, where the efficiency of the SM-EFAST process and the effectiveness of the OLE centred maintenance prioritisation are verified.

OLE introduction
Due to the comprehensive evaluation of system overall efficiency, OLE is fully recognized and applied in industry [20,21]. It was proposed by Nachiappan and Anantharaman [22], and defined as: line availability line production quality performance efficiency where LT is the actual loading time of the whole production line. QN represents the number of the qualified products and CYT s is the cycle time of the system. OT 1 and OT n represents the actual operating time of the first and the last equipment in the system, which can be calculated recursively by: where PD i is the independent planned downtime of equipment i except for the system overall planned downtime, and DT i represents the total failed down time of equipment i.
Although system throughput (TH) and production rate (PR) (defined as Eq. (3) and (4), provided by Kang et al. [23]) are more common system maintenance effect evaluation indexes, there are shortcomings of them compared to OLE: system good quality + system rework quality TH = system actual order execution time system qualified products system loading time It is clear in Eq. (3) that TH expresses the number of output products per unit time thus is suitable for making production plans. While PR analyses system production efficiency which seems more suitable than TH for system maintenance effect evaluation.
However, further comparing Eq. (4) with (1), OLE covers one more factor than PR: OT n /OT 1 (production line process ability). This factor represents the system operation time loss accumulated by the operation time loss of all equipment in the production process, reflecting the connection and cooperation of all equipment and buffers in the system. Considering that improper system maintenance always makes the operation of each equipment in the system greatly affected by others thus induce big system operation time loss (low OT n /OT 1 ) and decrease system process stability and productivity, system process ability improvement is also an important task of system maintenance. This mean that OLE is more suitable than system PR to evaluate the comprehensive effect of system maintenance. Therefore, it is taken as the evaluation index and objective of system maintenance in this paper.

Improvement of OLE
For production systems without buffers, there is no doubt OTi-1 is the theoretical available operation time for equipment i, as is expressed in Eq. (2). While for production lines with buffers, part of PDi-1 and DTi-1 can be made up by the downstream buffer of equipment i-1, thus longer actual available operation time than OTi-1 are provided to equipment i.
Although this compensation is limited, it is nonignorable for short-term equipment failures and the cases when can last for a long time, where CNBDi-1 is the cumulated number of workpieces in buffer i-1 (the buffer before equipment i) when equipment i-1 stops, and CYTi represents the cycle time of equipment i. Therefore, for production lines with buffers, the important role that buffers play in maintaining the continuous production of downstream stations should be fully considered in OTi calculation: After recursive calculation, OT n and the system process ability OT n /OT 1 can be expressed as follows: Accordingly, when system maintenances are as much as possible thus improve system internal cooperation, achieve good system process ability, and finally result in good OLE. Therefore, to accomplish proper system maintenance, an OLE centred maintenance prioritisation policy is proposed in the next section.

Prioritisation based on OLE influencing parameters
In practice, allocating limited resources to really critical objects according to the criticality of OLE influencing factors is very important for system maintenance. For this purpose, production line OLE influencing factors are analysed in Fig. 1 on the basis of Eq. (1) and (7).
The green blocks in Fig. 1 are system maintenance related factors while the grey ones are out of the responsibility of system maintenance. For the reason that equipment degradations are always shown as reactive maintenance (RM) and preventive maintenance (PM) in practice, only equipment PM, RM, as well as buffer size adjustment are recognised as system maintenance related OLE influencing factors in this paper.
Furthermore, since equipment PM and RM are specifically shown as equipment stop and maintenance, their characteristics can be uniformly expressed as mtbs i and mttr i : where tbs ij is the time period between the (j-1) th and j th stop of equipment i, ttr ij represents the time period of equipment i to resume from its j th stop, and n is the total stop times of equipment i.
To be noted, different from mtbf (mean time between failures) in equipment inherent reliability, mtbs is a characteristic of equipment operational reliability: the former stipulates that only equipment relevant failures can be considered, while the latter includes non-relevant failures and PM caused equipment stops besides that, which are also meaningful in equipment practical stage thus should be contained in equipment operational reliability evaluation.
In conclusion, equipment mtbs i , mttr i and buffer size N i are system maintenance related OLE influenc- ing parameters to be studied in this research to make maintenance prioritisations. The specific procedure of the OLE centred maintenance prioritisation policy is shown in Fig. 2. The main procedure of this method: parameter influence analysis is detailed in the following section. Fig. 2 The procedure of the OLE centred maintenance prioritisation policy

Parameter total effect on OLE
Considering the status of each equipment in the system can affect the production of upstream and downstream equipment, the impact of any equipment on OLE is not independent, but affected by the performance of other equipment [24]. Therefore, interactions among all equipment mtbss, mttrs and buffer capacities Ns should be considered: the actual influence of a parameter on OLE should be its total effect. Taking a system with three factors (k, l and h) as an example, the total effect of k on OLE should be: (10) where S k represents the main effect of k on OLE, and I k is the total interaction of k with all other parameters, including the interactions between k and h, k and l, and k, h and l. Apparently, it is the total effects of each mtbs i , mttr i and N i on production line OLE that should be recognised as their actual influences on OLE.

The SM-EFAST process
As a typical global sensitivity analysis method available for systems with a great number of influencing factors [25][26][27][28], the extended Fourier amplitude sensitivity test (EFAST) is adopted to analyse the total effect of each parameter k (mtbs i , mttr i and N i ) on production line OLE. The conventional process of parameter total effect analysis with EFAST is shown in Fig. 3 (a). Specifically, EFAST experiments should first be designed for each parameter k with Eq. (11): Assign an integer frequency ω k to each parameter k and select a transformation function G k for it (Details please refer to [27]). Subsequently, the experimental sample points x k for parameter k can be calculated by: As s varies uniformly between −π and π, the experimental sample points x k for each parameter k can be generated. After performing the designed EFAST experiments on the simulation model of the target production line, outputs (OLEs) at the experimental sample points can be calculated with Eq. (1). Then the total effect of parameter k on OLE can be approximately calculated with the generated outputs by: where f(s) represents the experimental output corresponding to each s, namely, the OLE at each experimental sample point x k (s); M usually takes 4 [27]; max(ω~k) is the largest ω assigned to other parameters (the complementary set) except for k; N is the number of sampling points required to analyse the where ω max should satisfy: According to [27], the EFAST experimental sample size required for a complete EFAST to analyse the total effects of all parameters of a system is as follows: (17) where l is the number of OLE influencing parameters, and N r is the total number of transformation functions adopted for all parameters.
It is required in [27] that: 1) the step between the frequencies of the complementary set (ω~i) must be as large as possible and 2) the number of factors to which the same frequency is assigned must be as low as possible. These rules make the EFAST experimental sample size for systems with a large number of parameters enormous.
However, the number of OLE influencing parameters in a production line is unfortunately large, which may inevitably cause great EFAST experiment time if the experiments are performed on a production line OLE simulation evaluation model (OLESEM), making the parameter influence analysis process inefficient. Therefore, an SM-EFAST process is integrated in Fig. 3 (b) to analyse parameter influences on OLE more efficiently.
The highlighted processes and the red words are the differences of the SM-EFAST process compared with the conventional process: For the quick response characteristic of surrogate models (SM) [29,30], EFAST experiment time can be greatly reduced by conducting the experiments on the SM of the OLESEM instead of the OLESEM itself.
However, from the view of the whole parameter influence analysis process, although EFAST experiment time can be greatly reduced by conducting experiments on the SM, the SM-EFAST process adds a SM training step to the conventional process. To effectively improve the efficiency of the whole process, it is necessary to minimise the time required for this additional step.
In the process of SM training, two essential elements are required [31]: sample size and model algorithm. To avoid blind exploration time wasted on these two elements, next section analyses clear SMtraining rules to efficiently and quickly train an accurate SM for an OLESEM.

SM-training rules
The difficulty of training an accurate SM is related to four factors: the complexity, dimension and breadth of the original model, as well as the algorithm adopted to train the SM. For an OLESEM, the first three factors respectively refer to the structure of the production line, the number of OLE influencing parameters and the range where OLE may reach when system equipment are operated under different maintenance policies (hereafter called OLE range).
To study how these factors affect SM training and explore the rules to quickly train an accurate SM for an OLESEM, six experimental production lines are designed and analysed (detailed in Appendix). The SM-training rules are summarised as follows: 1) For algorithm selection, preference should be provided to the radial basis function (RBF) or the response surface method (RSM); 2) Compared with production lines without buffers, those with buffers require less samples to train accurate SMs, because buffers can maintain more stable production process; 3) In terms of the sample size for accurate SM training, it should be set according to production line OLE range, specific suggestions are shown in Table 1. With the SM-EFAST process provided in Fig. 3 (b) and the SM training rules explored above, it is easier to efficiently and accurately analyse the total effects of mtbs i , mttr i and N i on OLE.

Case introduction
The proposed OLE centred maintenance prioritization policy is applied on a real auto spare part processing line in simulation. Basic information for this processing line is shown in Table 2.
Equipment mtbs, mttr and buffer N ranges shown in Table 2 are statistically analysed with the tbs and ttr data recorded in the PLCs of the CNC machine tools, production line computerized maintenance management system, as well as worker experience. Since each parallel workstation is equipped with the same machine model, and the processing task and working environment of each machine at a parallel station are the same, their performances are basically the same. Therefore, the range of the same parameter of each machine tool at a parallel station is the same, for example, mtbs 11 =mtbs 12 =mtbs 1 and mttr 11 =mttr 12 =mttr 1 .

Results and corresponding strategy
Influences of equipment mtbs, mttr and buffer size N on the OLE of this line are analysed with the SM-EFAST process: With the parameter ranges in Table  2, the OLE range of this processing line is evaluated to be 4.54%. According to the SM-Training rules, an accurate SM of this processing line is trained with RBF by 1000 samples. After conducting the EFAST experiments (designed by Eq. (11)) on the obtained SM, the OLEs at the experimental points are obtained and finally used to analyse the total effect of each parameter on the OLE of this processing line (by Eq. (12)-(16)). Results are shown in Fig. 4.
It is obvious from Fig. 4(a) that the SM-EFAST process can greatly reduce the EFAST experiment time compared with the conventional process and efficiently analyse the total effects of mtbs i , mttr i and N i on the OLE of the case line. This improvement not only owes to the replacement of the OLESEM by the SM, but also strongly supported by the SM training rules, which clearly guided the SM training process without blindness thus guaranteed the efficiency and accuracy of the whole SM training process.  Besides, the analysis results displayed in Fig. 4(b) show that mttr 3 has the most significant total effect on OLE, followed by N 3 and mttr 4 . The impact of N 4 , mttr 2 and mttr 5 can be classified to the third echelon. As to the other parameters, the impact of mtbs 3 , mtbs 5 , N 2 , mtbs 2 , mtbs 4 on OLE decreases in turn. While M 1 and N 1 have little impact within their available ranges.
Accordingly, guided by the generated parameter total effects, the maintenance prioritisation strategy for this processing line is formulated as follows: 1) Reduce the maintenance time of equipment whose mttr is more critical to OLE (mttr critical equipment): a) In cases when multiple equipment fail at the same time, maintenance priority (technologists, accessories etc.) should be set to mttr critical equipment; b) Quickly respond to the failures of mttr critical equipment thus reduce their maintenance response time; c) Strengthen the maintenance skill training of the mttr critical equipment.
2) Since N 3 and N 4 have the second and the fourth significant influence on OLE, their capacities are adjusted to 30 and 25 respectively, comprehensively considering their significances, cost of roller setting, and the space adequacy for online workers to operate and place tools； 3) Fig. 4 Fig.4 (b).

Comparison analysis with other strategies
The strategy provided above is simulated on the simulation model of this processing line. The resulted OLE, system PR, and maintenance total cost are compared with those resulted from the original firstcome-first-served without prioritisation policy (FCFS), a risk centred policy, and a productivity centred policy (Table 3). To be specific, the most popular methods of the latter two policies: FMECA and active period-based bottleneck analysis [9] are adopted. It is obvious that from both the perspective of system performance and maintenance total cost, the application results generated from the three prioritisation strategies are significantly better than those from the original FCFS strategy. It is also apparent that the OLE centred prioritisation results better than the risk centred and the productivity centred ones. This is due to its reasonable resource allocation: FMECA analysed critical failure modes to system safety and continuous production, which have already been given full attention and strict regulations to in the original PM plan and operation specification. Under such circumstances, only system safety and reliability rather than system efficiency will be further improved when resources are further tilted to these aspects, because the equipment performances that have important contribution to system efficiency are ignored and did not get timely maintenance.
As for bottleneck analysis, it only detects system critical equipment thus suggests to improve both the mttr and mtbs of the critical equipment as much as possible. While in cases when either mtbs or mttr of some critical equipment are less important than some parameters of lower critical equipment, these actions may lead not only to waste of resources but also worse effect, for allocating limited resources to less important equipment performance whilst ignoring the actual critical equipment performance.
In comparison, the SM-EFAST process analyses equipment influences on OLE from the level of the specific characteristics of their operational reliability, thus provide clear and specific prioritisation suggestions to system maintenance and correctly allocate the limited resources to really important equipment, avoiding the waste of resources and getting better maintenance effect meanwhile. This is the most fundamental reason why the system maintenance prioritisation strategy guided by the SM-EFAST process can result better from both the aspects of system performance and total cost. In addition, comparing the values of OLE and system PR shown in Table 3, the former is significantly lower. This is resulted from the system process ability OT n /OT 1 covered by OLE: It is impossible to perfectly make up for all equipment PD and DT losses no matter how perfect a maintenance strategy is, thus OT n /OT 1 <1. In terms of OT n /OT 1 under different strategies, it is obvious the one under the OLE centred strategy is the best, indicating this strategy can guide better system maintenance, improve system internal cooperation and achieve good OLE.

Conclusions
An OLE centred maintenance prioritisation policy is proposed in this research, where OLE is adopted as the objective of system maintenance to improve system comprehensive performance. Moreover, with an integrated SM-EFAST process which can accurately and efficiently analyse the total effects of equipment mtbs, mttr and buffer size N on OLE, this policy can provide clear and specific suggestions to system maintenance task prioritisation from the level of the specific characteristics of equipment operational reliability: mtbs and mttr. With these two novelties introduced to the OLE centred maintenance prioritisation policy, limited maintenance resources can be rationally allocated to truly critical equipment, so as to achieve good maintenance effect and low maintenance cost.
Advantages of this OLE centred maintenance prioritisation policy is verified through its application on a real auto spare part processing line: Analysis results show that OLE is more appropriate for production line maintenance effect evaluation than system PR, for its comprehensive and objective. Besides, thanks to the replacement of OLESEM with SM in EFAST experiments and the explored SMtraining rules, the efficiency of the SM-EFAST process is evidently better than that of the conventional EFAST process. More importantly, with the prioritisation guidance provided from the level of the specific characteristics of equipment operational reliability, maintenance resources can be correctly allocated to the truly critical equipment performance, thus result in higher OLE and lower total maintenance cost than the popular and widely used bottleneck analysis and FMECA maintenance prioritisation methods.
For future direction, with the development of equipment prognostic and health management, equipment online and prognostic performance will be further considered in equipment maintenance task prioritisation and decision making.

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2) Compared with the production line without buffers, that with buffers requires less samples to train accurate SM, for the reason that buffers can cause more stable production process; 3) The SM-training sample size for OLESEMs (with buffers) should be set according to the OLE range of the production line: If RBF and RSM are selected, only 300 sample points are sufficient to train a wellfitted SM when the OLE range is under 2%, 500 sample points are required when 2% < OLE range < 4%, and approximately 800 sample points are adequate to train an accurate SM when 4% < OLE range < 5%.  On the basis of these findings, the sample sizes needed to train SMs for production lines (with buffers) with larger OLE ranges are further explored: 4%-6%:1000; 6%-8%:1300; 8%-10%:1800.