Comparative Assessment of Preventive Maintenance Strategies for Enhanced Reliability, Security, and Cost Reduction in Industrial Systems

In the realm of system maintenance, traditional approaches often involve periodic replacements, where components are entirely replaced at regular intervals. However, for certain systems, this strategy may not be the most economical or practical solution. Imperfect maintenance emerges as a viable alternative, offering a more cost-effective approach to extending the lifespan of components and optimizing system reliability.

Unlike periodic replacement, imperfect maintenance focuses on performing interventions that aim to reduce the failure rate of a component without eliminating it entirely. This approach is particularly well-suited for systems that can benefit from extended service life through partial overhauls or repairs.

Consider the example of an industrial machine that undergoes periodic partial revisions. After a certain number of partial revisions, the machine receives a more comprehensive general overhaul. This exemplifies imperfect maintenance, as the interventions gradually reduce the failure rate but do not completely restore the machine to its original state.

The impact of imperfect maintenance on the system's failure rate is crucial to consider. After each maintenance action, the failure rate settles at a level between its initial state (before maintenance) and its state just before the imperfect maintenance. This dynamic behavior necessitates careful monitoring and evaluation of the maintenance strategy's effectiveness.

The Gertsbakh model provides a simplified framework for analyzing the impact of imperfect maintenance on system reliability and cost. This model assumes a constant effect from all PMs, causing the failure rate to decrease exponentially by a factor of e α (where α is a positive value) after each maintenance [16].

The average cost per unit of time, represented by the function C(T):

$\begin{aligned} & C(T)=\frac{C_c \cdot H(T)\left(1+e^\alpha+\cdots+e^{\alpha(\mathrm{K}-1)}\right)+(K-1) C_p+C_{o v}}{K T}\end{aligned}$            (1)

Underlying Assumptions for Cost Analysis:

1) Minimal repair after a failure: The assumption of minimal repair implies that after a failure, the system is restored to a condition where its failure rate remains unchanged. This is consistent with a non-homogeneous Poisson process (NHPP), where the intensity function depends on time.

Physical Meaning: Minimal repair does not improve the system’s overall reliability but merely restores it to its pre-failure state. This is realistic for complex systems where repairs address only the immediate cause of failure without eliminating underlying wear and tear.

2) Weibull distribution for system failure: The system’s failure distribution follows a Weibull model with a shape parameter β and scale parameter η. The assumption γ=0 implies that the system has no initial failure-free period (i.e., failures can occur immediately after installation).

Physical Meaning: The Weibull distribution is widely used to model systems with time-dependent failure rates. The shape parameter β determines whether the failure rate increases (β>1), decreases (β<1), or remains constant (β=1) over time [17].

The cumulative hazard function H(T) for a Weibull distribution is derived as follows:

$H(T)=\int_0^T \frac{\beta}{\eta}\left[\frac{t}{\eta}\right]^{\beta-1} d t=\frac{\beta}{\eta} \int_0^T\left[\frac{t}{\eta}\right]^{\beta-1} d t$             (2)

This represents the integral of the Weibull hazard rate over time T.

Simplification by factoring out constants.

$H(T)=\frac{\beta}{\eta} \int_0^T\left[\frac{t^{\beta-1}}{\eta^{\beta-1}}\right] d t=\frac{\beta}{\eta^\beta} \int_0^T t^{\beta-1} d t$           (3)

Integration of $t^{\beta-1}$ yields $\frac{t^\beta}{\beta}$

$H(T)=\frac{\beta}{\eta^\beta}\left[\frac{\mathrm{t}^\beta}{\beta}\right]_0^T=\frac{\beta}{\eta^\beta} *\left(\frac{T^\beta}{\beta}-0\right)$            (4)

$H(T)=\frac{T^\beta}{\eta^\beta}$       (5)

By substituting this equation into the Gertsbakh model Eq. (1), we obtain:

$\begin{aligned} & C(T) =\frac{C_c \cdot\left(T^\beta / \eta^\beta\right)\left(1+e^\alpha+\cdots+e^{\alpha(\mathrm{K}-1)}\right)+(K-1) C_p+C_{o v}}{K T}\end{aligned}$                  (6)

Separate the terms involving $T^\beta$ and the constant terms.

Hence,

$\begin{array}{r}C(T)=\frac{C_c \cdot T^\beta\left(1+e^\alpha+\cdots+e^{\alpha(\mathrm{K}-1)}\right)}{K T \eta^\beta} \\ +\frac{(K-1) C_p+C_{o v}}{K T}\end{array}$                    (7)

Therefore,

$\begin{array}{r}C(T)=\frac{C_c \cdot T^{\beta-1}\left(1+e^\alpha+\cdots+e^{\alpha(\mathrm{K}-1)}\right.}{K \eta^\beta} +\frac{(K-1) C_p+C_{o v}}{K T}\end{array}$                         (8)

This formula represents our objective function to minimize. To achieve this, we will use exact resolution methods to compare their results with those of three other algorithms.

The production line N1 of the metal packaging manufacturing company (production line) consists of nine machines arranged in series. This sequential configuration introduces specific maintenance challenges, as the failure of a single machine can significantly disrupt the entire system, leading to production downtime and high repair costs.

To address these issues, implementing a robust PM strategy is crucial to prevent machine malfunctions. Our study focuses on the critical machines within production line N1, each with distinct roles in the production process:

Rolling Band, Can-O-Mat, Strapping Machine, Ocsam Shear, Conveyor, Gear Motor, Palletizing Machine, Welding Machine and Test-O-Mat.

Each machine plays a vital role in ensuring the seamless operation of production line N1 and contributes to the manufacturing and preparation of the final products.

For the successful optimization of the PM strategy, it is essential to preprocess the available data, relying on the historical failure records of the machines in production line N1. By analyzing this data, we can design and implement a maintenance policy aimed at minimizing downtime, reducing costs, and ensuring the reliability of the production line.

The parameters in Table 1 (e.g., Cc, Cp, Cov, K, α, β, and ν) are derived from real industrial data collected from the maintenance records and operational logs of the production line N1 in a metal packaging manufacturing company.

Table 1. Optimal maintenance parameters

The shape parameter β for each machine was estimated using statistical analysis of historical failure data. Specifically, the Weibull distribution was fitted to the failure data using methods such as maximum likelihood estimation (MLE) or regression analysis.

For β<1: In the production line, most machines exhibit β<1, indicating a decreasing failure rate over time. This is typical for systems experiencing early-life failures or "infant mortality," where failures are more frequent initially but decrease as the system stabilizes. The exception is the Conveyor No. 1, which has β>1, indicating an increasing failure rate over time, consistent with wear-out failures.

To apply the presented cost model, it is essential for the parameter β to exceed 1. However, in the production line, the β value for each machine is below 1, except for the Conveyor, which satisfies all the required conditions.

The optimal maintenance interval T* and minimal cost C_min in Table 2 are calculated using a MATLAB program based on the input parameters from Table 1.

Table 2. Optimal maintenance parameters for conveyor No. 1

The results are specific to Conveyor No. 1, as it is the only machine satisfying the condition β>1, which is required for the cost model to be applicable.

Since the objective function C(T) is differentiable, we can find its minimum by solving the equation C'(T*)=0, where T* represents the optimal value of T that minimizes C(T). However, it's crucial to consider the following conditions:

$\begin{gathered}C_c>0 ; C p>0 ; C o v>0 ; T>0 ; K>0 ; \beta> 1 ; \alpha>0 \text { et } \frac{\partial}{\partial T}\left(\frac{\partial C(T)}{\partial T}\right) \geq 0\end{gathered}$                  (9)

We have:

$C(T)=\frac{C_C T^{\beta-1}\left(1+e^\alpha+\cdots+e^{\alpha(K-1)}\right)}{K \eta^\beta}+\frac{(K-1) C_p+C_{o v}}{K T}$                  (10)

So:

$\begin{gathered}C^{\prime}(T)=\frac{C_c(\beta-1) T^{\beta-2}\left(1+e^\alpha+\cdots+e^{\alpha(\mathrm{K}-1)}\right)}{K \eta^\beta} -\frac{(K-1) C_p+C_{o v}}{K T^2}\end{gathered}$                     (11)

$C^{\prime}(T)=\frac{\begin{array}{c}C_c(\beta-1) T^\beta\left(1+e^\alpha+\cdots+e^{\alpha(\mathrm{K}-1)}\right) -\eta^\beta\left((K-1) C_p+C_{o v}\right)\end{array}}{\eta^\beta T^2}$                (12)

Therefore, we derive the relationship for T such that C′(T*)=0:

$\frac{\begin{matrix}   {{C}_{\text{c}}}(\beta -1){{T}^{*\beta }}(1+{{e}^{\alpha }}+\cdots +{{e}^{\alpha (K-1)}})  \\   -{{\eta }^{\beta }}((K-1){{C}_{p}}+{{C}_{ov}})  \\\end{matrix}}{{{\eta }^{\beta }}{{T}^{*2}}}=0$                   (13)

$\begin{gathered}C_c(\beta-1) T^{* \beta}\left(1+e^\alpha+\cdots+e^{\alpha(\mathrm{K}-1)}\right)-\eta^\beta((K- \left.\text { 1) } C_p+C_{o v}\right)=0\end{gathered}$                       (14)

$\begin{gathered}C_c(\beta-1) T^{* \beta}\left(1+e^\alpha+\cdots+e^{\alpha(\mathrm{K}-1)}\right)=\eta^\beta((K-\left.1) C_p+C_{o v}\right)\end{gathered}$                (15)

$T^{* \beta}=\frac{\eta^\beta\left((K-1) C_p+C_{o v}\right)}{C_c(\beta-1)\left(1+e^\alpha+\cdots+e^{\alpha(K-1)}\right)}$                      (16)

$T^*=\sqrt[\beta]{\frac{\eta^\beta\left((K-1) C_p+C_{o v}\right)}{C_c(\beta-1)\left(1+e^\alpha+\cdots+e^{\alpha(K-1)}\right)}}$                        (17)

MATLAB program developed to determine the optimal maintenance interval (T*) in both hours and days, alongside the corresponding minimal cost. This program processes a detailed dataset, which includes variables such as specific maintenance costs, the number of partial revisions (K), and the maintenance efficiency factor (α). These variables are integrated into the program to yield precise and actionable insights for optimizing maintenance schedules and reducing overall costs. Table 1 provides a summary of these data points for a specific piece of equipment, "Conveyor N°1".

To apply the presented cost model, it is essential for the parameter β to exceed 1. However, in the production line, the β value for each machine is below 1, except for the Conveyor, which satisfies all the required conditions.

The MATLAB program uses these inputs to calculate the optimal maintenance interval (T*) and the associated minimal cost, providing a structured approach to maintenance scheduling that enhances efficiency and cost-effectiveness. The program results are summarized in Table 2.

Table 2 indicates that performing maintenance on Conveyor No. 1 every 197.86 hours (or about every 8.24 days) will result in the lowest possible maintenance cost of 81,345.84 USD per time period. This optimized schedule helps in minimizing downtime and maximizing the efficiency and cost-effectiveness of the maintenance operations. Figure 1 graphically shows the evolution of cost C as a function of T.

The graph illustrates the relationship between maintenance periodicity (T) and the corresponding cost (C(T)). The curve rapidly decreases as T increases, indicating that frequent maintenance (low T values) incurs extremely high costs. As T increases, the cost drops sharply and then levels off. The red dot marks the optimal maintenance interval (T*), approximately 197.86 hours, where the cost is minimized. This optimal point highlights that very frequent maintenance is costly due to unnecessary downtime and labor, while infrequent maintenance increases operational costs and potential equipment failures. Thus, the graph emphasizes the importance of selecting the optimal maintenance interval to balance the frequency of activities with associated costs, ensuring a cost-effective maintenance strategy.

To determine which optimization method is most effective for improving PM interventions, we will compare the performance of SA, GA, and ACO. This comparison will consider several factors, including the optimal maintenance interval (T*), minimum cost per ton (C_min), number of iterations, execution time, and number of evaluations for Conveyor No. 1. By analyzing these metrics, we aim to identify the strengths and weaknesses of each algorithm in terms of accuracy, efficiency, and computational demand, providing insights into which method offers the best balance for enhancing PM strategies.

Table 6 provides a general comparison of three optimization methods-SA, GA, and ACO-for improving PM interventions on Conveyor No. 1. The exact mathematical calculation serves as a benchmark with an optimal maintenance interval (T*) of 197.86 hours (8.24 days) and a minimum cost (C_min) of 81,345.84 USD/t. The exact Calculation presented in Table 6 represents the optimal solution obtained through a brute-force enumeration method. This approach systematically evaluates all possible maintenance schedules within the defined constraints. Specifically, for Conveyor No. 1, we generated and evaluated every possible combination of maintenance intervals, considering the underlying assumptions of minimal repair after failure and a Weibull distribution for system failure (γ=0). The cost function (C_min) was calculated for each schedule, and the schedule that yielded the minimum cost was identified as the optimal solution. This exhaustive search method serves as a benchmark to assess the performance and accuracy of the heuristic optimization algorithms (ACO, GA, SA). Due to the computational complexity of brute-force enumeration, this method is only feasible for relatively small-scale problems, such as the single conveyor system considered in this study. The optimization algorithms used in this study ACO, GA, and SA each have distinct strengths and weaknesses. ACO excels in exploring complex search spaces and escaping local optima through pheromone trails but is computationally expensive and sensitive to parameter choices. GA is robust and effective for global optimization, maintaining diversity through crossover and mutation, but it can be computationally intensive and prone to premature convergence if parameters are poorly tuned. SA is simple to implement and computationally efficient, with the ability to escape local optima by accepting worse solutions early in the search, but its performance depends heavily on the cooling schedule and may not explore the search space as thoroughly as ACO or GA. While ACO and GA are better suited for global exploration, SA is more efficient for local refinement. These trade-offs highlight the importance of selecting the appropriate algorithm based on problem complexity, computational resources, and desired outcomes.

Table 6. Comparison of optimization algorithms for Conveyor No. 1