Integrated just-in-time production and imperfect maintenance management considering random quality degradation

1. Introduction

The alignment between production and maintenance strategies is paramount for attaining operational excellence and enduring performance in manufacturing systems, as denoted by several authors (Manafzadeh Dizbin & Tan, 2020; Hajej & Rezg, 2020; Jin, 2023; Liu et al., 2022). Further, Rivera-Gómez et al. (2020) introduced production thresholds and dynamic sample inspection for a degraded system. Ait El Cadi et al. (2021) developed a robust stochastic analytical framework for harmonizing production and preventive maintenance control in manufacturing systems susceptible to operational degradations. Pongha et al. (2023) devised a collaborative strategy for production, maintenance, and quality control, geared towards elevating machine availability, improving product quality, and mitigating production expenses. Assid et al. (2023) concentrated on the production planning and control of hybrid manufacturing-remanufacturing systems, proposing an integrated control scheme that governs production and disposal rates and alternates between manufacturing and remanufacturing modes to slash long-term costs.

Ensuring consistent product quality is paramount for meeting customer expectations, enhancing brand reputation, and fostering long-term business success (Rivera-Gómez et al., 2021; Ait-El-Cadi et al., 2021b; Lindström, 2023; Wan, 2023a). Moreover, Rivera-Gómez et al. (2022) contributed a stochastic model to coordinate production, subcontracting, and maintenance methods. In their model, they included stochastic uncertainty, quality degradation, and random subcontracting availability. Assid et al. (2023) integrated four major choices to coordinate manufacturing, remanufacturing, return replenishment, and quality control. Wan et al. (2023b) derived a combined production, maintenance, and statistical process control design model for continuous flow processes with various assignable causes. Shi et al. (2024) proposed the optimization of production planning, maintenance methods, and quality control in flawed manufacturing systems.

Just-in-time (JIT) strategies enable companies to reduce lead times, improve resource utilization, and enhance overall operational efficiency (Salari & Makis, 2020; Shokoufi & Rezaeian, 2020; Lyu et al., 2020; Shao et al., 2022). Moussawi et al. (2022) constructed a repairable production JIT inventory system with sporadic disturbances and continuous demand. Shabtay (2023) reported a study focusing on scheduling challenges in a distributed flow-shop scheduling JIT system. Weng et al. (2022) explained how to enhance task flow between production lines through JIT work completion in the upstream shop for the downstream shop. Xie et al. (2023) focused on scheduling JIT precast manufacturing for steel box girders.

Simulation modeling serves as a robust tool for analyzing, optimizing, and forecasting the dynamics of manufacturing systems within a controlled virtual setting, as evidenced (Mykoniatis & Harris, 2021; Ojstersek et al., 2021; Bojic et al., 2023; Chu, 2024). Additionally, Baroroh and Chu (2022) advocated for a production system simulation tailored for human operators in logistic facility design, leveraging mixed reality technology. Malega et al. (2022) delved into a large-scale computer simulation method for optimizing production processes, yielding notably efficient solutions that trim manufacturing cycles and enhance productivity. Mohammad Hadian et al. (2023) explored a stochastic model for inventory management and maintenance scheduling in an unstable manufacturing milieu. Kim et al. (2023) introduced an agent-based modeling framework for agrophotovoltaic (APV) systems, aiming to identify enduringly profitable APV configurations amid variable climatic conditions.

Deterioration models play a crucial role in understanding and mitigating the effects of equipment degradation on manufacturing system performance (Ouaret et al., 2018; Dellagi et al., 2020; Acevedo-Ojeda, 2020; Ghaleb, 2021). Similarly, Magnani and Tolio (2020) developed a threshold-based control strategy for preventive maintenance that takes inventory, maintenance, and backlog costs into account for a degrading system. Hajej et al. (2021) treated an integrated production, maintenance, and quality control plan for a degrading manufacturing system. Zhang et al. (2023) introduced the combined production-maintenance optimization issue of a deteriorating machine. Zhao et al. (2023) used a third-party logistics provider to study a coordinated production and delivery scheduling problem to improve manufacturers' competitiveness.

This paper contributes a unified simulation-optimization framework that jointly optimizes JIT timing/hedging decisions and imperfect-maintenance scheduling/efficiency, while explicitly modeling random quality deterioration via an age-dependent Ornstein-Uhlenbeck process. Unlike prior work that treats JIT or maintenance as separate problems, our model couples production, quality, and imperfect maintenance decisions. The rest of the paper is outlined as follows. Section 2 presents the problem statement of the system under study. The methodology adopted in the study has been presented in Section 3. Then Section 4 describes the simulation model used in the optimization. Section 5 presents a numerical example. An extensive sensitivity analysis has been conducted in Section 6. Section 7 presents a comparative study, and Section 8 concludes the paper.

2. Problem Statement

The system comprises three states, denoted by Ω = {1, 2, 3}, and $q_{\alpha {\alpha }^{\prime }}$ signifies the transition rate from state α to α′. At mode α(t) = 1, the production unit operates normally. At α(t) = 2, the unit experiences a failure mode, where minimal repairs occur. When α(t) = 3, the unit undergoes imperfect maintenance, partially restoring its condition. The dynamics of the inventory level are modeled with the following formula:

For modeling defect generation, a stochastic differential equation is employed, denoting an Ornstein-Uhlenbeck process.

$Z_{\beta }(0)=Z_{\beta }^0=0$

Although the initial condition $Z_{\beta }^0$ is initialized deterministically with ( $Z_{\beta }^0=0$ ), the presence of the drift coefficient b, the diffusion term $\sigma (a)$ , and the Brownian increment $d{W}_1(t)$ ensures that the resulting OU process is stochastic. Consequently, the defect rate $\beta (a(t))$ in Equation (3) evolves as a random process.

The age of the system can be determined by solving the next differential equation.

The model employs two evolving averages, the defect rate ${\mu }_{\beta }(a)$ and the diffusion coefficient $\sigma (a)$ , both represented as functions of system age.

The constants β₀, β₁, σ₀, σ₁, ${\theta }_1$ , and ${\theta }_2$ can be derived from historical system data collected during manufacturing operations. We adopt Equations 5 and 6 for the age dependence because they flexibly capture non-linear deterioration patterns (early-life, inflection, accelerating failure) while remaining straightforward in their parameters for estimation. Furthermore, the steady state availability of the production unit in its operational state, denoted as ${\pi }_1$ , can be computed as follows.

To ensure the viability of the production system, the following feasibility criterion must be fulfilled to ensure demand satisfaction despite high levels of deterioration.

Here, $\beta (a)$ represents the defective rate process, modeled as an Ornstein-Uhlenbeck process with finite variance over the infinite horizon. The JIT strategy is integrated with the hedging point policy. This policy determines the production rate at any given time t through the next specific expression.

The JIT production strategy is adopted when the production unit's age is below a specified threshold, termed $B_{JIT}$ . Regarding the imperfect maintenance policy, it is initiated when the production unit's age surpasses a critical value, denoted as $A_0$ .

Additionally, Equation (10) is supplemented by the next additional expression.

The variable Φ signifies the maintenance efficiency, ranging from 0 to 1. The system's age before and after imperfect maintenance is denoted as $a\left(t_m^{-}\right)$ and $a\left(t_m^{+}\right)$ , respectively. Maintenance is imperfect at each maintenance time, the age is updated according to Equation 11. The inventory cost per unit time during the interval [0, T] defined as IB(t), can be calculated as follows.

With $x^{+}=max\left(0,x\right)$ and $x^{-}=max\left(-x,0\right).$ . The constants $C^{+}$ and $C^{-}$ penalize the inventory cost and unit shortage, respectively. Moreover, the quality cost per unit time QC(t) within the time frame [0, T] is defined as the cumulative cost of defects.

The total maintenance expenses incurred over a specified period can be obtained as follows.

The indices $N_R\left(t\right)$ and $N_M\left(t\right)$ signify the count of minimum repairs and imperfect maintenance occurring within the time span [0, T]. The objective of this optimization model is to minimize the total incurred cost.

Subject to:

Equations (1)–(8) (inventory and quality dynamics)

Equations (9)–(11) (control policy)

The current problem presents itself as nonlinear and profoundly stochastic, rendering it challenging to tackle using conventional mathematical programming approaches. Given these complexities, an alternative method based on simulation optimization is suggested as a viable approach to ascertain the optimal solution.

3. Materials and Methods

The methodology adopted in this paper is based on a simulation-optimization approach and comprises the following steps:

• 1.

  Definition and scope of the problem: The intricacies of the manufacturing system, including its constituent elements, operational processes, and overarching objectives, have been delineated in Section 2.

• 2.

  Review of existing literature: A rigorous exploration of the extant body of scholarly work pertaining to production planning, simulation methodologies, and associated techniques has been undertaken in Section 1.

• 3.

  Collection of data and model development: An intricate simulation model is meticulously crafted utilizing sophisticated software platforms, such as Arena, in Section 4. For a broader adoption of the model, a detailed pseudocode has been provided in Appendix.

• 4.

  Verification and Validation Mechanisms: The veracity of the simulation model has been substantiated through exhaustive comparison vis-à-vis anticipated outcomes and theoretical postulations in Section 4.

• 5.

  Analysis of Results: The synthesized simulation results have been subjected to an incisive analytical scrutiny in an extensive sensitivity analysis in Section 5.

4. Simulation model

A validation analysis is conducted in the simulation model that tracks key indicators representing system behavior. Fig. 1 depicts the dynamics of the production system over time, with control parameters set at $Z_p=20$ , $B_{JIT}=100$ , $A_o=200$ , and $\varphi$ Fig. 1 evaluation confirms the efficacy of production and imperfect maintenance policies, where safety stock is warranted at $B_{JIT}$ , and imperfect maintenance is triggered when the unit's age exceeds $A_o$ . Quantitative validation complements Fig. 1 across 108 simulation runs, since the model achieves $R^2=$ 0.902 as will be presented in the next section.

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Fig. 1.

Graphical validation of the simulation model

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5. Results and Discussion

We consider three independent variables ( $Z_p$ , $B_{JIT}$ , $A_o$ ) to ascertain a dependent variable denoted as the total incurred cost. We employ a three factorial design for simulation runs, which is replicated four times, resulting in a total of (3³x4) = 108 simulations. Each simulation spans 100,000 time units to achieve steady-state conditions. Table 1 showcases the parameter values utilized in this numerical example.

Table 2 presents the cost parameters employed in the calculation.

Baseline parameter values are consistent with prior studies in the literature (Ouaret et al. 2018) and were calibrated where necessary to reproduce their reported statistics. To facilitate the analysis, we defined $B_{JIT}$ as k ⋅ Ao, with k ranging between 0 and 1. The simulation outcomes undergo analysis utilizing the STATGRAPHICS statistical program to conduct an ANOVA study. From the ANOVA Table 3, it is evident that all principal components, quadratic effects, and interrelationships are statistically significant with $R^2=$ 0.902. From the ANOVA analysis, we can derive the optimal value of the control parameters presented in the sensitivity analysis.

6. Sensitivity Analysis

Table 4 presents the results derived from the sensitivity analysis. These results offer an opportunity for analysis and interpretation, providing valuable insights into the data to understand the impact of varying the different costs considered in the model. The results presented in Table 4 are logical and far from being exhaustive. We discuss the impact of the following costs:

• •

  Variation in the cost of imperfect maintenance: an escalation in the cost of imperfect maintenance C_M (case VIII) extends the maintenance age $A_o$ , requiring the unit to deteriorate further before maintenance is deemed necessary. This prolonged operational duration enhances customer demand fulfillment, curbing inventory needs and so reducing the threshold $Z_P$ . Moreover, extended operational periods facilitate longer JIT policy implementations, thereby increasing the critical age $B_{JIT}$ .

• •

  Defective cost variation: As the cost associated with defective units $C_{def}$ (case X) rises, it's expected to witness increased maintenance frequency to swiftly restore units and minimize the delivery of defective units to end customers, which reduces the age $A_o$ . Furthermore, we note a tendency to elevate inventory levels $Z_P$ , ensuring customers receive defect-free units. This strategy is reinforced by a reduction in the critical age $B_{JIT}$ that promotes the existence of more units.

Additionally, the sensitivity analysis of the parameters of the stochastic process has produced results outlined in Table 5.

• •

  Parameter variation of $b$ : It is observed that reducing the parameter $b$ (case XI) prompts more frequent imperfect maintenance to mitigate defective product output, which reduces the age $A_o$ . Because more defectives are generated in the short term. Further, we note a rise in inventory levels to ensure fulfillment of customer demand with defect-free units, thereby elevating the value of $Z_P$ . This analysis is complemented by a decrease in the critical age $B_{JIT}$ .

• •

  Parameter variation of_ ${\sigma }_1$ : an upsurge in ${\sigma }_1$ results (case XVI) in heightened imperfect maintenance occurrences to minimize defect presence. This stems from the fact that defect generation peaks escalate with an increase in the ${\sigma }_1$ parameter. Additionally, the system opts to augment the inventory level $Z_P$ to ensure customer demand fulfillment with defect-free units. Complementing these adjustments, the critical age $B_{JIT}$ is decreased.

7. Comparative Study

In this section, we perform a thorough comparative analysis aimed at elucidating the potential cost-saving benefits associated with the implementation of our proposed control policy. The analysis focuses on evaluating the performance of our proposed policy (denoted as Policy-I), in contrast to several alternative policies that are denoted as follows:

• •

  Policy-II: This policy does not incorporate a JIT production strategy. Instead, it relies solely on determining a constant inventory level $Z_p$ throughout the entire time horizon. Notably, Policy-II disregards the control parameter $B_{JIT}$ , focusing solely on ( $Z_p$ , $A_o$ ), while also accounting for the occurrence of random defects.

• •

  Policy-III: In this case, the control strategy does not optimize imperfect maintenance parameters, resulting in the parameter $A_o$ being set to a fixed value. Similar to Policy-II, Policy-III incorporates a random generation of defects into its analysis.

• •

  Policy IV: This option shares similar control parameters with Policy-I, such as $Z_p$ , $B_{JIT}$ , and $A_o$ . However, Policy-IV disregards random defect generation and does not employ the Ornstein-Uhlenbeck process to simulate defect production.

• •

  Policy-V: In this option, the timing for implementing the JIT strategy is not optimized, indicating that the parameter B_JIT is not part of the optimization process. The control parameters for this policy solely consist of the inventory level and the age for imperfect maintenance ( $Z_p$ , $A_o$ ). Moreover, in this policy, there is no utilization of a stochastic process to model random defect generation.

• •

  Policy-VI: This option does not include the optimization of the timing for starting imperfect maintenance in its formulation. The control parameters for this policy only involve the inventory level and the timing for implementing the JIT strategy ( $Z_p$ , $A_o$ ). Additionally, the Ornstein-Uhlenbeck process is not utilized to simulate random defect production within this policy.

The comparative analysis results presented in Table 6 indicate that Policy-II incurs a 47.89% higher cost compared to Policy-I. This cost differential stems from various factors. Initially, Policy-II accrues inventory from the outset of the time horizon, resulting in additional inventory expenses. In contrast, Policy-I adopts a JIT strategy with zero inventory during a specific period, resulting in substantial cost savings. Additionally, the postponement of imperfect maintenance in Policy-II contributes to an elevated defect cost, as a greater number of defects are generated due to the heightened critical age $A_o$ . The findings extracted from Table 6 underscore the superior economic benefits offered by the proposed control policy compared to other examined alternatives (Policies II-VI) that dissociate production, maintenance, and quality decisions. Under all the analyzed cases, our Policy-I provided the lowest total cost, further confirming that the results are robust and not an artifact of the chosen parameter set.

8. Conclusions

This study presents a novel control policy integrating JIT production strategies and imperfect maintenance policies to optimize system performance while minimizing costs. Through extensive numerical simulations and sensitivity analysis, we note how fluctuations in inventory costs, shortage costs, minimal repair costs, imperfect maintenance costs, and defective costs can impact inventory levels, maintenance policies, and overall costs. Additionally, we explore the sensitivity of Ornstein-Uhlenbeck process parameters and their influence on defect generation and system performance. By implementing our proposed control policy, organizations can enhance operational efficiency, reduce costs, and maintain high product quality in dynamic production environments. One limitation of the current work is that it considers a single-product, single-machine system. While this setup enables a clearer analysis of the control policy and defect dynamics, extending the framework to multi-product and multi-machine environments is a natural next step that could enhance the generalizability of the findings. This is left as a direction for future research. Although the proposed framework is validated through simulation-based experiments rather than an actual industrial case, the numerical example and sensitivity analysis rely on data extracted from several peer-reviewed articles published in high-impact JCR journals. This ensures that the parameters used are realistic and based on industrial practices. Future work will focus on the validation of the model in collaboration with industrial partners to further highlight its practical relevance.