Performance estimation of parallel Syste under online and offline preventive maintenance

Abstract

Keywords: Cost Analysis; Mean Time to Failure; Reliability

Introduction

One of the most important characteristics used to assess system performance in engineering is reliability and is also regarded as the best starting point for any system improvement. Due to lack of adequate probabilistic knowledge, convectional reliability’s binary state i.e., performance or

failure state assumptions, are inadequate for analysing the reliability of complex industrial systems. The dubiety of each individual parameter in large industrial systems adds to the overall system reliability dubiety. As a result, a number of researchers have created various industrial systems to investigate reliability, availability, mean

*Corresponding author. *E-mail address: hamisu@csu.edu.cn, wanmath@csu.edu.cn, iyusuf.mth@buk.edu.ng, asanusi.sce@buk.edu.ng This paper was recommended for publication in revised form by Editor in Chief Ahmet Selim Dalkılıç Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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time to system failure, and cost function, as well as the impact of coverage variables, failure and repair rates. Researchers such as; [1] explored a method for evaluating the reliability and profitability of a warranty-dependent industrial system. [2] focused on evaluating the performance of serial systems with a variety of failure types and repair policies. Using Kolmogrov differential equations, [3] proposed a method for analyzing the reliability of industrial systems. [4] investigated a novel method for reliability analysis of series-parallel systems via credibility theory. [5] presented their research on the availability of standby complex systems in the event of human failure or repair. [6] presented Markov chain profit modeling and evaluation between two dissimilar systems under two forms of failures. [7] have published a paper on the performance assessment of a series-parallel industrial method. [8] presented the findings of an analysis of the multi-objective non-linear programming problem for reliability (GSA) and compared them to results computed using the practice swarm optimization (PSO) methodology. The reliability study of a two-dissimilar unit warm standby repairable system with priority in usage was addressed by [9]. [10] addressed the availability and cost analysis of an engineering system with series-connected subsystems. [11] discussed the evaluation of some reliability characteristics of a single unit system requiring two types of supporting device for operations. [12] used the Gumbel-Hougaard family copula to study the cost assessment of a complex repairable system made up of two subsystems in a series arrangement. [13] used copula distribution to offer a stochastic study of an N-unit plant with various failure kinds. Improving system performance can also be attained either by increasing the reliability of each component/subsystem in the system or by providing proper maintenance (preventive maintenance). Preventive maintenance is a type of maintenance that is carried out on a piece of equipment on a regular basis to minimize the risk of failing. It is done when the equipment is still running to keep it from breaking down unexpectedly. Cleaning, lubrication, oil changes, modifications, repairs, checking and removing components, as well as partial and full overhauls, are all part of preventive maintenance schedule in operation. Preventive maintenance is used in many industries to help maintenance managers and reliability engineers increase system reliability, availability, and revenue while lowering costs. The primary goal of every manufacturing company or factory is to increase profits by lowering production costs. In today’s competitive age, we can’t afford to lose production or have facilities fail due to increased demand. As a result, appropriate maintenance procedures must be used to achieve the best results. Online and offline preventive maintenance techniques are two of the most common and universal maintenance techniques among the many. The aim of online preventive maintenance is to inspect the system or any output on a regular basis. It is a non-interruptible operation, which means the system won’t be shut down until it’s done. This assists in cost-cutting, reducing equipment downtime,

improving asset lifespan and efficiency, and avoids unexpected expenses. Offline preventive maintenance is a type of inspection that is done manually. Regular asset scanning is sufficient for offline preventive maintenance. Many research papers on system reliability and availability in the presence of preventive maintenance have been published, including the following; [14] investigated cost-effective coal-fired power plant preventive maintenance scheduling. [15] conducted a probabilistic study of two single-unit device reliability models with preventive maintenance beyond the warranty period and deterioration. [16] examined the best preventive maintenance warranty strategy for repairable goods with growing failure rates on a regular basis. [17] proposed a stochastic study of a reheating-furnace system undergoing preventive maintenance and repair. [18] presented a report on a single unit system with preventive maintenance and repair that was subjected to the fastest possible operation and repair times. For single machine framework, [19] suggested an integrated model of production planning and incomplete preventive maintenance strategy. [20] proposed a parallel system reliability assessment with two types of preventive maintenance. According to maximum operation and repair timeframes, [21] developed a single unit system with preventive maintenance and repair. Stochastic modeling of non-identical redundant systems with priority, preventive maintenance, and Weibull failure and repair distributions was presented by [22]. The literature review above reveals that the majority of previous studies have focused on preventive maintenance and other types of repair policies. Preventive maintenance policies, both online and offline under copula distribution approach, have received little or no attention in the literature. As a result, the impact of online and offline preventive maintenance has been highlighted in this paper. The reminder of the paper is organised such that section 2 contains notations, assumptions, and description of the model under consideration. The formulation and solution of the model were captured in section 3. The model analysis was discussed in section 4. The findings are discussed in section 5, and finally the conclusion part of the paper is given in section 6.

Notations, Assumptions And Description Of Model

Notations Here, we provide a table with the notations and definitions used in this study. Assumptions Below are listed all the presumptions that were made for this study. i. √Firstly, all subsystems are assumed to be operational. ii. Two units from each subsystem are needed for operation. iii. When any component fails, the system’s output suffers. iv. If a subsystem unit fails, it can be repaired when it is still operational, or it can fail completely.

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Repair rate for a fully failed states of system A and system B respectively The probability that the system is in Si state at any given time Laplace transformation of state transition probability p(t)

The probability that a system is in state Si such that for i=1..., the system is under repair, and the elapsed repair time is (x, t) with x denoting repair and t denoting time.

The probability that a system is in state Si such that for i=1..., the system is under repair, and the elapsed repair time is (y, t) with y being the repair variable and t being the time variable.

The probability that a system is in state Si for i=1…, the system is under repair, and the elapsed repair time is (m, t) with repair variable being m and time variable being t.

The probability that a system is in state Si such that for i=1…, the system is under repair, and the elapsed repair time is (n, t) with n representing the repair variable and t representing the time variable.

According to the Gumbel-Hougaard family copula definition, joint probability is expressed as: . Where µ1 = ∅(x) and u2 = ex

All failure rates are expected to follow exponential distributions and are constant.

vi. Partially failed states are restored using general distribution, while entirely failed state distribution is handled by the Gumbel Hougaard family copula. vii. The repaired machine unit should function as new and the repair process should have caused no damage.

viii. The load will be ready for the system’s successful performance as soon as the failed unit is repaired. The Model Overviews In this research, we looked at a mathematical model of parallel system with two subsystems A and B, each consists of three active units in parallel that can fail in two ways: partially or fully. These two systems are separated

Is the ideal state, in which both systems are fully functional. The system works since one unit is on standby in both systems.

In this state, one unit in system A has failed, two units in both systems A and B are operational, and one unit in system B is on standby. The system is up and running.

In this state, one unit in system B has failed, two units in both systems A and B are operational, and one unit in system A is on standby. The system works.

Here, one unit in system B has previously failed, one unit in system A suddenly failed, and two units in system A are operational. The system is functional.

Previously, a unit has failed in system A, suddenly a unit failed in system B, then the system is operational.

Previously, a unit has failed in system B, suddenly a unit failed in system A, then the system is functional.

After the failure of the first unit in both systems A and B, this state denotes the online preventive maintenance state. In this state, the system is operational.

This state denotes system’s A total failure as a result of two of its unit failing. The system is not functioning.

This state represents system’s B total failure as a result of two of its unit failing. The system is not working.

This state denotes the offline preventive maintenance state where the system is neither down nor working.

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from one another and are located in different locations. When two units from both systems A and B fail, the systems continue to function, but any more failure will bring the systems to total failure. In the case of total failure, both systems A and B can be fixed simultaneously using Gumbel-Hougaard family copula, whereas general repair distribution can be used to bring back partially failed states. The systems go through online preventive

maintenance immediately after the failure of first unit with a rate of δ0 and immediately fixed with a rate of ∅(m), as well as offline preventive maintenance after the failure of the second unit with a rate of δ1 and immediately returned with a rate of ∅(n). The system has eleven states including perfect state, seven of which are operational and three of which are not. The detailed overview of the states is given below in table 2:

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MODELS’ FORMULATION AND SOLUTION Model Formulation As in Lado and Singh (2019), and Singh and Rawal (2015), we obtain the following differential difference equations as: (1)

Initial conditions P0(0) = 1, i.e., in a perfect state, the probability of a state transition at time t = 0 is zero. With the above initial condition, we can obtain Laplace transforms of equations (1) to (11) as: (22)

(8) (23) (9) (24) (10) (25) (11) The study’s boundary conditions are generated from figure 2 by analyzing the rates of exit from each state in terms of the probability of being in the initial state, P0(t): (12) (13)

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(42) The probability that the system is operational is: Solution of the models Combining equation (22) and (32) with the aid of (33) to (42), we get: (43)

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Model Analysis

Availability Analysis To make a distinction, we looked at the system’s availability in two different ways, viz:

(55) Table 3 and figure 2 illustrate the system’s availability for . copula repair distribution when

Table 3. Availability against time t with respect to copula distribution Time

Figure 2. Availability against time with respect to copula distribution.

When repair follows Gumbel-Hougaard Family copula distribution When

with all repairs set to 1, i.e., ∅(x) = ∅(y) = μ0(x) = μ0(y) = μ0(m) = μ0(n) = 1, and λ1 = 0.01, λ2 = 0.02, δ0 = 0.03, and

δ1 = 0.04 fixed. Then if we take Laplace transforms, we get:

When repair follows general distribution To obtain the availability of the system when the repair and all is through general distribution, we set repairs to 1, i.e., ∅(x) = ∅(y) = μ0(x) = μ0(y) = μ0(m) = μ0(n) = 0, and λ1 = 0.01, λ2 = 0.02, δ0 = 0.03, and δ1 = 0.04, then taking the Laplace transform of equation (54), we get the expression for availability as: (56)

Table 4 and figure 3 present the system’s availability for . general repair distribution when

Table 4. Availability against time t in terms of general repair Time

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Figure 3. Availability against time in terms of general repair.

Reliability Analysis

By reducing all repair rates to zero i.e., ∅(x) = ∅(y) = μ0(x) = μ0(y) = μ0(m) = μ0(n) = 0, and taking λ1 = 0.01, λ2 = 0.02, δ0 = 0.03, and δ1 = 0.04 in equation (54), then using the inverse Laplace transform, the system’s reliability can be expressed as:

Mean Time to Failure (MTTF) Analysis The expression for MTTF can be obtained by setting all repairs to zero and restricting as s reaches zero:

Reliability can be calculated using equations (57) for different values of time t = 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100, as shown in Table 5 and Figure 4.

Setting λ1 = 0.01, λ2 = 0.02, δ0 = 0.03, δ1 = 0.04 and varying λ1, λ2, δ0 and δ1 one as 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09 respectively in equation (58), one may

Table 5. Variation of reliability corresponding to time t Time

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Table 6. MTTF variation in terms of failure rates Failure Rate

obtain table 6 and figure 5 which show the variation of MTTF in terms of failure rates. Sensitivity Analysis MTTF sensitivity can be determined using the partial derivative of MTTF with respect to failure rates in equation (58). Table 7 and Figure 6 display the MTTF sensitivity for fixed values of failure rates as λ1 = 0.01, λ2 = 0.02, δ0 = 0.03, δ1 = 0.04. Cost Function Analysis Case 1: Expression for cost function/Expected profit when the repair follows copula distribution If the service facility is open at all times, the formula below will determine the estimated profit for the interval [0, t). (59)

Where K1, K2 and K3 in the interval [0, t) are the revenue generated, service cost per unit time, and, service cost due to online and offline maintenance, respectively. Equation (60) can be obtained for the same set of parameters in equation (55).

Results And Discussion

For investigating the effects of online and offline preventive maintenance, we calculate the system availability, reliability, mean time to failure (MTTF), MTTF sensitivity, and cost function for the established models numerically. When failure rates are set at various values, figure 2 shows how availability changes over time. The figure depicts how the availability of repairable systems declines over time, but eventually stabilizes at zero after a sufficiently long interval of time. The graphical representation of the model can predict the future behavior of a complex system for a given set of parameters at any time. The values of availability when repair follows a general distribution are similar to the values of availability when repair follows the GumbelHougaard family copula distribution; this can be seen in figure 2 and 3. This is attributable to the system’s online and offline preventive maintenance. However, there is no denying that using copula repair increases system’s availability over general repair. Figure 4 shows the difference in reliability over time. When compared to system availability, it can be seen that system reliability plummets. This is due to lack of system repairs. The general consensus is that the less the repairs, the lower the reliability. This illustrates what failure to manage the structure/system entails. Figure 5 illustrates the differences in the value of the system’s mean time to failure (MTTF) for failure rates λ1, λ2, δ0 and δ1 respectively, fixing other parameters constant. The MTTF values with respect to δ0 and δ1 is much higher than the MTTF values corresponding to λ1 and λ2. According to this sensitivity study, online and offline preventive maintenance failure rates are more responsible for proper system functioning. Figure 5 illustrates this. The information on the sensitivity analysis conducted in this paper is shown in figure 6. Figures 7 and 8 demonstrate the difference in effective benefit over time when the repair follows a copula distribution or a general distribution, respectively, with revenue cost per unit time is set at 1.0, cost of online and offline maintenance set at 0.1, the service cost is varied and the failure rates constant. Figures 7 and 8 show that when the repair follows the copula distribution, the expected gain is higher than when the repair follows the general distribution. Because of online and offline preventive maintenance, the expected benefit when repairing with copula distribution is comparable to the expected profit when repairing with general repair. According to this sensitivity review,

the importance of using copula repair would be decreased if online and offline preventive maintenance is mandated. However, copula repair is still recommended for the system’s proper operation.

Conclusion

The reliability metrics for different failure values and repair rates are critically analyzed in order to measure the performance of the systems under consideration. The system’s transient probabilities and reliability metrics such as availability, reliability, mean time to failure and cost are calculated using the Markovian process, Laplace transformation, and supplementary variable approaches. The influence of time and other system characteristics on reliability metrics was simulated using MATLAB. Numerical experiments were used to obtain and validate the fundamental expressions such as availability, reliability, mean time to failure, sensitivity, and cost function. On the basis of the numerical results obtained for a particular case in figures 2-8 and tables 1-7, it is clear that the value, i.e., the importance of copula repairs has decreased with online and offline preventive maintenance. This means by implementing online and offline preventive maintenance, system failure would be greatly reduced, resulting in improved output and revenue generation. It is generally known that system failure will reduce production efficiency and may even result in a tragedy. This research paper will help system engineers, programmers, and maintenance managers design more important systems and maintain them in the best interests of humanity. Future iterations of this work may combine the Gumbel-Hougaard family Copula with the Bivariate Gumbel-Hougaard family Copula.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

Sigma J Eng Nat Sci, Vol. 42, No. 4, pp. 1088−1100, August, 2024

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