Enhancing industrial maintenance planning: Optimization of human error reduction and spare parts management

Emroozi, Vahideh Bafandegan; Kazemi, Mostafa; Doostparast, Mahdi Article Enhancing industrial maintenance planning: Optimization of human error reduction and spare parts management Operations Research Perspectives Provided in Cooperation with: Elsevier Suggested Citation: Emroozi, Vahideh Bafandegan; Kazemi, Mostafa; Doostparast, Mahdi (2025) : Enhancing industrial maintenance planning: Optimization of human error reduction and spare parts management, Operations Research Perspectives, ISSN 2214-7160, Elsevier, Amsterdam, Vol. 14, pp. 1-24, https://doi.org/10.1016/j.orp.2025.100336 This Version is available at: https://hdl.handle.net/10419/325813 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Enhancing industrial maintenance planning: Optimization of human error reduction and spare parts management Vahideh Bafandegan Emroozi a,* , Mostafa Kazemi a , Mahdi Doostparast b a Department of Management, Faculty of Economics and Administrative Sciences, Ferdowsi University of Mashhad, Mashhad, Iran b Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran ARTICLE INFO Keywords: Human error probability Maintenance planning Inventory control of spare parts Manufacturing systems ABSTRACT Maintenance is pivotal in the industrial sector, influencing efficiency, reliability, safety, and profitability. An organized spare parts inventory supports maintenance efforts by minimizing downtime, ensuring safety, and optimizing maintenance budgets. Effective spare parts management enhances maintenance operations and improves cash flow. Conversely, human error can greatly diminish the effectiveness of maintenance efforts. This paper presents a mathematical model aimed at minimizing costs through optimized preventive maintenance (PM) planning, effective spare parts inventory control, and reduction of human error. The study provides decision-makers with crucial insights for strategically managing maintenance procedures while accounting for the effect of human error. The model is validated in real-world scenarios through sensitivity analysis, focusing on the shape parameter of the Weibull distribution, and the equipment’s effective rate. Findings reveal that as the number of periods increases, maintenance operations follow a specific, predictable cycle. Moreover, the optimal human error probability (HEP) for cost minimization is identified as 0.02. These insights guide decision-makers in recognizing factors influencing human error and implementing proactive strategies to enhance maintenance performance. 1. Introduction Maintenance operations are critical for improving organizational efficiency and ensuring reliable system performance, especially when substantial investments are made in production machinery [1–4]. Ineffective maintenance can lead to machine failures, downtime, and increased costs, including opportunity costs, reputational damage, and disruptions to production schedules [5,6]. Preventive maintenance (PM) is a widely adopted strategy to minimize unplanned breakdowns and downtime. However, poorly timed or inadequately executed PM can negatively impact efficiency and incur additional costs, such as resource allocation and temporary production halts. Effective planning and execution of maintenance are therefore essential to balance these trade-offs and optimize organizational performance [7–10]. Optimal planning of maintenance operations depends heavily on the effective management of spare parts inventory. Excessive spare parts inventory can lead to higher holding costs and unnecessary expenditures, while insufficient inventory can prolong downtime and result in greater losses due to machine inactivity [8–10]. To address these challenges, it is crucial to optimize the spare parts inventory control policy. This presents a significant challenge in balancing costs through optimal inventory management while ensuring system reliability [11,12]. Both PM and corrective maintenance (CM) significantly influence the condition and virtual age of machinery. Effective maintenance planning must account for these factors to optimize the timing and frequency of PM operations, thereby enhancing machine reliability. However, even well-planned maintenance can fail due to human error, leading to improper implementation and increased system costs. Human Error Probability (HEP) is a quantitative measure used to estimate the likelihood of human errors occurring in specific tasks or systems [13,14]. It is commonly employed in human reliability analysis (HRA) to evaluate safety in industries such as aviation, nuclear power, healthcare, and manufacturing. HEP is typically expressed as a probability value ranging from 0 (no chance of error) to 1 (certainty of error) [10,15]. HEP during PM, CM, or inspections can undermine the effectiveness of maintenance and inflate costs. To mitigate this, maintenance plans must incorporate strategies to reduce human error, ensuring both cost efficiency and improved machine reliability [16–18]. Given the significant impact of human error on maintenance operation costs and the virtual age of machines, it is essential to consider human errors when * Corresponding author. E-mail addresses: [email protected] (V. Bafandegan Emroozi), [email protected] (M. Kazemi), [email protected] (M. Doostparast). Contents lists available at ScienceDirect Operations Research Perspectives journal homepage: www.elsevier.com/locate/orp https://doi.org/10.1016/j.orp.2025.100336 Received 3 December 2024; Received in revised form 22 February 2025; Accepted 24 March 2025 Operations Research Perspectives 14 (2025) 100336 Available online 1 April 2025 2214-7160/© 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). optimizing maintenance plans. In this study, data on human error, costs, and the virtual age of machines were collected from historical data in a case study. The cost of human error and the machine’s virtual age functions were estimated using a regression method based on HEP. Thus, the proposed model quantitatively incorporates human error, highlighting its effects on costs and the virtual age of the machine. When estimating the cost function, two key challenges arise. First, the cost associated with human error in maintenance operations increases as the level of error rises. Second, improving contextual factors to mitigate human error also incurs additional costs. Contextual factors, which include environmental, organizational, and task-related elements, play a critical role in shaping HEP. By identifying, assessing, and optimizing these factors, industries can effectively reduce human errors, enhance safety measures, and improve overall system reliability. Therefore, the cost function of human error related to maintenance operations is estimated by considering both aspects. Consequently, optimizing maintenance plans and minimizing human error are essential for reducing costs and achieving economic success across various industries. Although human error significantly affects maintenance operations, previous studies have largely overlooked this aspect, focusing instead on non-human factors. While spare parts have been examined in some studies, none have specifically investigated the repercussions of human error in maintenance operations. The model presented in this paper quantitatively considers the influence of human error, contributing to a more comprehensive analysis of the problem. The research focuses on a case study of a cement company that operates continuously, utilizing heavy machinery with significant investment. Given the characteristics of the industry, proper and optimal maintenance operations are crucial for minimizing stochastic breakdowns and total costs. The model presented for this case study is multiproduct, multi-machine, and multi-condition, providing a comprehensive approach to the problem. The objective of this research is to identify the optimal values for inventory management of spare parts, as well as for planning PM and CM operations, while considering the impact of associated human errors. Accordingly, in line with previous studies in this field, we emphasize five major contributions of this paper: 1. Impact of Human Errors on Equipment Utilization and Virtual Age: The study investigates how human errors affect the effective rate of equipment utilization and the virtual age of equipment. By modeling these relationships, the research offers insights into how human errors accelerate equipment deterioration and provides strategies to minimize their impact on operational efficiency. 2. Development of an Integrated Optimization Model: The study proposes a novel optimization model that integrates maintenance operations, spare parts inventory control, and human error management into a unified framework. This holistic approach addresses the interdependencies between these components, enabling more efficient and cost-effective decision-making. 3. Innovative Maintenance Planning Incorporating Equipment Lifespan: The research pioneers a quantitative approach to maintenance planning by integrating equipment conditions and lifespan into the decision-making process. A novel constraint is introduced to account for equipment lifespan, allowing for the determination of threshold limits for maintenance activities in an innovative and systematic manner. 4. Consideration of Machine Setup Costs During Downtime: The study incorporates machine setup costs during production line downtime caused by PM and CM operations. This inclusion allows for more accurate budgeting and resource allocation, optimizing production efficiency and minimizing financial disruptions during maintenance periods. 5. Simultaneous Implementation of Condition-Based Maintenance (CBM) and Time-Based Maintenance (TBM): Unlike traditional PM models that rely solely on timeor usagebased scheduling, this study integrates both CBM and TBM to enhance maintenance decision-making. 6. Practical Application and Validation: The model is applied to a real-world case study of a cement factory, utilizing data from maintenance logbooks to validate its effectiveness. This practical application demonstrates the model’s ability to address real industrial challenges and improve maintenance practices in complex operational environments. 7. Quantitative Analysis of Human Error Costs in Maintenance: The research introduces a novel mathematical model to quantify the costs associated with human error in maintenance tasks. Using regression methods, the study estimates the cost function linked to the probability of human error, addressing a critical gap in existing literature. This approach provides a systematic way to evaluate and mitigate the financial impact of human errors. The remainder of this paper is organized as follows: The second section offers a concise overview of prior research on this topic. The problem statement and case study are presented in Section 3. Section 4 details the methods employed in this study. The research findings are presented in Section 5. Section 6 examines the sensitivity analysis conducted to validate the proposed model. Lastly, Section 7 presents managerial insights, while Section 8 discusses notable conclusions and offers suggestions for future studies. 2. Literature review Research conducted in this field focused on maintenance operations exclusively and ignored human error’s impact on these operations. Liu et al. [19] introduced a model that takes into account buffer inventory and imperfect PM in production system. Zheng et al. [6] highlighted the cost-effectiveness of policies based on production quantity and condition-based maintenance for managing a deteriorating production system. Lynch, et al. [20] investigated the impact of an effective maintenance system on industrial performance. Bismut et al. [21] improved the maintenance and inspection strategies for the piping systems in nuclear fuel power plants. Emami-Mehrgani et al. [22] examined the effects of human errors on repairable production systems and proposed an optimal policy to reduce production costs.. Morato et al. [23] introduced optimal maintenance planning for deteriorating structural components using a Dynamic Bayesian Network (DBN) and Markov decision process. Szpytko et al. [24] proposed a compatible and straightforward simulation approach based on a risk assessment model to optimize maintenance scheduling in different case studies. Liu et al. [25] proposed an integrated model that takes into account buffer stocks and imperfect PM in production systems. Sharifi and Taghipour [26] proposed an integrated model for production and maintenance planning. Kim et al. [27] presented a potential approach for optimizing inspection and maintenance planning. Zhang et al. [12] explored the simultaneous optimization of maintenance and spare parts inventory for a series-parallel system with dual failure modes. Nasrfard et al. [28] introduced a Petri net model that considers the state of deterioration, inspection, age-dependent repair processes, and random repair. Briˇ s and Thuy Tran [29] studied the problem of multi-objective maintenance optimization to minimize costs and maximize availability. Saleh et al. [30] introduced an intelligent Petri net algorithm to optimize maintenance operations for wind turbines. Fekri et al. [31] explored the workshop flow scheduling problem with limited multi-skilled human resources in PM. Zhu et al. [32] introduced an optimization model and algorithm for spare parts, taking into account trade-offs between cost and time as well as precedence constraints. Jiang et al. [33] optimized PM interval and the maximum inventory level, V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 2 with the ultimate goal of minimizing system downtime and inventory holding costs. Cao et al. [34] examined an optimization model for decision-making aimed at the sustainable maintenance of intricate road networks. Their work introduced a bi-level programming approach that addresses the diverse characteristics of subnetworks, varying maintenance standards, and the allocation of maintenance funds. Levitin et al. [35] focused on the optimization of CM for multistate systems with storage, particularly production-storage systems in various industries. They addressed the impact of random external shocks on system performance and introduce a corrective maintenance policy (CMP). Liu et al. [36] provided an optimal condition-based maintenance policy for leased equipment, considering hybrid PM and periodic inspection. The authors addressed the complexity of leased equipment structures due to technological advancements, posing challenges for lessors in developing maintenance policies. Lee et al. [37] developed an optimized scheduling model for railway lines using a sophisticated deep reinforcement learning algorithm. Liu et al. [38] introduced two innovative PM policies that account for substantial repair downtime. One policy is time-based, scheduling preventive replacements at one of two predetermined calendar intervals, contingent on the system’s condition. Zhou and Zheng [39] introduced a multi-objective decision optimization model for prioritizing maintenance and repair of bus failures. Mikhail et al. [40] introduced a data-driven optimization method that considers contextual conditions. They combined machine learning and reinforcement learning techniques with a reliability-based remaining useful life methodology. Wang et al. [41] introduced a dynamic predictive maintenance strategy for predicting the remaining useful life (RUL) of systems. Zheng et al. [42] explored the joint optimization of maintenance and spare part ordering from multiple suppliers for systems with multiple components. Zeng et al. [43] addressed a novel challenge in integrating PM with robot disassembly line balancing (DLB) to enhance the efficiency and stability of robotic disassembly systems. Their study focuses on optimizing both conventional disassembly scenarios and PM scenarios, while also improving the transition efficiency between these two contexts. O’Neil et al. [44] introduced a new resilience framework designed to optimize the performance of critical network infrastructures, such as power grids, telecommunications, and transportation systems. This framework tackles the challenges posed by disruptions caused by stress events and aims to enhance network resilience through efficient post-disruption restoration. Lima et al. [45] presented a novel model for managing imperfect maintenance in multi-component systems, specifically addressing the Selective Maintenance Problem (SMP) by incorporating both perfect PM and CM actions. Tian et al. [46] introduced a heuristic algorithm that provides near-optimal solutions for this complex issue, focusing on efficient resource utilization and long-term system performance. They also developed a Selective Serial Maintenance Sequence Planning (SSMSP) model to optimize maintenance activities for mechanical equipment with multiple components. This model addresses inefficiencies, high costs, and resource wastage by integrating worker physical exertion and rest time into maintenance planning, thereby ensuring sustainable schedules. Additionally, it employs a multi-objective optimization approach to balance maintenance benefits, costs, and resource constraints. Zhang et al. [47] focused on enhancing the reliability and maintenance efficiency of wind-photovoltaic (PV) hybrid power systems. They developed a reliability model and a maintenance optimization model that incorporates energy complementarity strategies. This study addresses the intermittency of renewable energy systems and aims to reduce maintenance costs across different failure modes and scenarios. The proposed maintenance optimization model integrates energy complementarity strategies to optimize system performance and minimize costs. Wei and Cheng [48] developed a maintenance policy optimization framework for self-service systems aimed at maximizing profit by balancing service revenue and maintenance costs. Their model accounts for unique failure-induced demand-and-system interactions and employs a Tabu-search algorithm to optimize maintenance policies. Leppinen et al. [49] tackled the challenge of optimizing maintenance schedules for multi-component systems by considering technical structural dependencies, which significantly impact the cost-efficiency of maintenance policies. Their study introduced directed graphs as a tool to represent the economic and structural dependencies of the system, including scenarios where maintaining one component necessitates disassembling or maintaining others. The maintenance scheduling problem is modeled as a Markov Decision Process (MDP) and solved using a modified policy-iteration algorithm to determine the most cost-efficient maintenance policy. Bafandegan Emroozi et al. [50] assessed HEP in maintenance tasks using the Cognitive Reliability and Error Analysis Method (CREAM) and System Dynamics (SD) modeling. Their study identifies and quantifies factors influencing HEP, explores their interactions, and estimates associated costs using machine learning techniques. Ultimately, the research determines the optimal HEP value to minimize costs and accidents, providing managers with scenarios for effective budget allocation and improved ergonomics. Table 1 provides an overview of previous studies on maintenance. •TBM is scheduled at fixed intervals to ensure regular preventive actions and avoid unexpected failures. •CBM is incorporated through real-time condition monitoring of the equipment, allowing early detection of potential failures and enabling adaptive maintenance interventions before scheduled TBM actions. Previous studies have provided valuable insights into maintenance operations, as shown in Table 1. However, they did not consider the impact of human error on maintenance metrics. This oversight is significant because the human factor plays a critical role in maintenance operations, and neglecting it may lead to inaccurate results. Prior research has primarily focused on non-human factors, such as machine reliability and failure rates, while overlooking the effects of human error. To address this gap, our research aims to examine how HEP affects the total cost and the virtual age of machines resulting from maintenance operations. This paper conducts a quantitative analysis to determine the extent to which human error influences these costs and machine age. Additionally, we explore inventory control policies for spare parts to enhance maintenance planning. Implementing an effective inventory control policy can help reduce downtime and maintenance costs by ensuring that spare parts are readily available when needed. It is important to note that previous studies have examined maintenance operations planning based on either condition-based or time-based maintenance. However, in our study, we investigate both condition-based and time-based maintenance simultaneously. This dual approach allows us to identify which maintenance strategy is more effective for different types of machines and maintenance tasks. Overall, our research aims to provide a comprehensive understanding of the impact of human error on maintenance operations and the importance of an effective inventory control policy. 3. Methodology This section outlines a comprehensive and systematic methodology for optimizing maintenance operations, spare parts inventory control, and human error management within a cement factory. By integrating these critical components into a unified optimization model, the research aims to achieve cost-effective and efficient maintenance practices while rigorously adhering to operational constraints. The incorporation of real-world data from the cement factory not only enhances the model’s accuracy but also ensures its practical applicability and relevance to industrial settings. This approach provides a robust framework for balancing cost reduction, operational efficiency, and V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 3 equipment reliability in complex maintenance environments. 3.1. Notation This section outlines the symbols and notations used in the paper. Table 2 presents the notations and definitions used throughout the mathematical model described in this paper. 3.2. Problem statement and case study The primary purpose of this research is to identify the optimal values for decision variables related to maintenance operations (Voit,Qit,Bit,Iit,bBit,bIit), spare parts inventory control (zkjt,ajt, ψ cjt,ξcj,ξʹ cj,t[m] kjt ,t[p] jt ), and human error (p, pk), aiming to reduce costs while adhering to various constraints. It is important to note that carrying out PM and CM operations can lead to production line downtime and disrupt organizational processes. In this model, the timing and frequency of each level of PM operations are determined based on machine age, cost constraints, and other organizational limitations. In this study, focusing on the case of a cement factory, maintenance operations are examined for two critical pieces of equipment: a rotary kiln and a clinker silo. The analysis includes four key spare parts associated with this equipment: bearings, a side motor, seals, and a pair of brushes. Additionally, the study considers three different levels of PM operations and one feature to assess the condition of the equipment (i.e., noise). It is worth mentioning that the model is specifically designed for repairable mechanical equipment that undergoes a deterioration process and operates independently. The data used for the research comes from the maintenance logbooks of the plant and is presented in Tables 1A and 2A in the Appendix. Simultaneously, the availability of spare parts is crucial for carrying out maintenance effectively and promptly, making spare parts inventory control a critical component of the process. This paper optimizes the order quantity of spare parts. Furthermore, the optimal level of human error is identified based on the conditions that influence human resource errors (i.e., Common Performance Conditions (CPCs)), as well as its impact on machine age and associated costs. The cost of human error in maintenance tasks and its effect on machine age are considered as a function of human error. Fig. 1 illustrates the structure and logic of the model through a detailed diagrammatic representation. 3.3. Assumption related to the presented model The following assumptions underpin the model presented: Constrained Time Horizon The model operates within a finite time horizon, with all operations, costs, and activities analyzed over this duration. This ensures the model Table 1 Previous studies on maintenance. No Reference Maintenance Strategy Subcategory Distribution of failure function Function estimation related to HEP Inventory management of spare parts Solution approach Cost Reductionn coefficient 1 [20] PM TBM –✖ ✖ ✔ Genetic algorithm (GA) 2 [33] PM, CM TBM Weibull ✖ ✖ ✔ Monte Carlo Simulation 3 [16] PM, CM TBM Non-homogenous Markov processes ✖ ✖ ✖ Hamilton–Jacobi–Bellman (HJB) 4 [19] PM, CM TBM Uniform ✖ ✖ ✖ Kushner and Dupuis’ method and value iteration or policy iteration algorithms 6 [27] PM, CM TBM Log-Normal ✖ ✖ ✖ SAW, TOPSIS, ELECTRE 7 [25] PM, CM TBM Gamma ✖ ✖ ✔ A Lagrangian relaxation-based heuristic approach 8 [42] PM, CM CBM Weibull ✖ ✖ ✖ The policy-iteration algorithm in the semiMarkov decision process (SMDP) 9 [26] PM, CM TBM Weibull ✖ ✖ ✖ GA, simulated annealing (SA) algorithm, and a teaching–learning-based optimization (TLBO) 10 [24] PM, CM TBM Weibull ✖ ✖ ✖ Petri net algorithm 11 [21] PM, CM TBM Gamma ✖ ✖ ✖ Heuristic parameters optimization 12 [23] PM, CM CBM Weibull ✖ ✖ ✖ the POMDP dynamics 14 [12] PM, CM CBM Gamma ✖ ✖ ✔ Monte Carlo simulation 15 [32] PM, CM CBM uniform ✖ ✖ ✔ Heuristic-based on a standard critical path method. 16 [28] PM, CM CBM Weibull ✖ ✖ ✖ Petri net algorithm 17 [29] PM, CM TBM Exponential ✖ ✖ ✖ Innovative and updated calculation methodology by MATLAB software 19 [31] PM TBM –✖ ✖ ✖ Metaheuristic method (GA) 20 [34] PM, CM CBM Exponential ✖ ✖ ✖ Metaheuristic method (GA) 21 [35] CM CBM –✖ ✖ ✖ Metaheuristic method (GA) 22 [40] PM, CM CBM Kaplan-Meier (KM) ✖ ✖ ✖ machine learning and reinforcement learning 23 [37] PM, CM CBM Exponential ✖ ✖ ✖ Deep reinforcement learning 24 [36] PM, CM CBM Gamma ✖ ✖ ✖ Metaheuristic method 25 [38] PM, CM CBM Exponential ✖ ✖ ✖ Mathematical model 26 [41] PM CBM –✖ ✖ ✔ CNN 27 [42]– – Exponential ✖ ✖ ✔ Hybrid deep reinforcement learning algorithm (HDRL) 28 [45] PM, CM SMP –✖ ✖ ✖ Heuristic algorithm 29 [46] PM, CM TBM –✖ ✖ ✖ Enhanced metaheuristic (brainstorming optimization +large neighborhood search) 30 [49] PM, CM TBM Exponential ✖ ✖ ✖ MDP model, Modified policy-iteration algorithm 31 [50] PM, CM – – ✔ ✔ ✖ SD modeling and machine learning 32 This study PM, CM TBM, CBM Weibull ✔ ✔ ✔ Mathematical model (GAMS) Condition-Based Maintenance: CBM, Time-Based Maintenance: TBM, Selective Maintenance Problem (SMP). V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 4 aligns with practical planning periods, such as financial years or project lifespans. Spare Parts Shortage Costs The cost implications of spare parts shortages are modeled based on the cumulative shortage over the planning horizon. This approach accounts for variations in demand and inventory levels while ensuring that downtime costs resulting from spare part unavailability are comprehensively captured. Maintenance Cost Dependencies Spare part costs for maintenance are influenced by dynamic inventory levels and service-level requirements, incorporating penalties for stockouts and overstocking. Cost of Corrective and Preventive Actions The cost of replacing parts post-failure is explicitly higher than preventive replacement costs due to unplanned downtime and potential secondary damages. This difference is dynamically calculated based on severity and time-of-failure scenarios. Failure Rate Distribution The failure rate exhibits an increasing trend over time, modeled using a Weibull distribution with a shape parameter (β>1). The initial parameter estimates were informed by expert judgment, ensuring alignment with domain knowledge, and subsequently refined and valiTable 2 Notations. Sets MThe set of periods represented by the index t; KThe set of different level types of PM represented by the index k; JThe set of equipment indexed by j; IThe set of spare parts indexed by i; CThe set of equipment conditions indexed by c; Decision variables p[total]Human error probability. pkHuman error probability associated with conducting kthlevel of PM operations. zkjt The binary variable equals 1 if a PM operation is performed on the machine j at the kth level in period t; Otherwise, it equals 0. ajt The virtual age of the machine j in period t. t[p] jt The available time on machine j for carrying out production operations in period t. t[m] kjt Preventive maintenance time on the machine j at the level k in period t. ψ cjt The level of implementation of PM operations on machine j under condition c in period t. Voit The binary variable for ordering or not ordering spare parts i in period t. bIit The binary variable equals 1 if there is inventory available for ith spare parts in period t; Otherwise, it equals 0. bBit The binary variable equals 1 if there is a shortage for ith spare parts in period t. Otherwise, it equals 0. Bit The amount of shortage of ith spare parts in period t. Iit The amount of holding of ith spare part in period t. Q it The order quantity of ith spare part in period t. Parameters Shit The cost of shortage of ith spare part in period t (each unit). hit The cost of holding of ith spare part in period t (each unit). Csetjt Setup cost after machine downtime resulting from PM and CM operations on jthmachine in period t. dpikjt The demand for ith spare parts for the implementation of PM operations at kthlevel on machine j in period t. dcijt The demand for ith spare parts for implementation of CM operations on machine j in period t. CO it The fixed ordering cost for ithspare part in period t. LT[E] iThe lead time of emergency orders for ith spare parts. MTTRjt CM time on jth machine in period t. Warit The warehouse capacity for ith spare parts in period t. Nmkjt The number of technicians needed to perform PM operations on the jth machine at kthlevel in period t. Nrjt The number of technicians needed to perform CM operations on jthmachine in period t. Nspijt The number of ith spare parts to carry out CM operation on jthmachine in period t. Hmkjt The cost associated with human resources for PM operations on the jthmachine at kthlevel in period t. Hrjt The cost associated with human resources for CM operations on machine j in period t. Cljt The cost of the lost opportunity of production line downtime on machine j in period t. Cuit Price of each unit of ith spare part in period t. ACjt The noise of jthmachine in period t. COit Ordering cost of ith spare part in period t. cdit purchasing cost of each unit of ith spare part in period t. Q[max] it The maximum quantity for ordering ith spare parts in period t. ϑcjt The optimal level of PM operations on jthmachine under cthcondition in period t. ξʹ cj The upper threshold for cthcondition on machine j. ξcj The lower threshold for cthcondition on machine j. rThe learning coefficient of human resources in the implementation of PM operations on machine j. μ The number of levels of PM operations. HLength of the planning horizon. βWeibull distribution shape parameter. η Weibull distribution scale parameter. α kThe effective rate of virtual age through the executionkthlevel of PM operation. lThe interval length. AEjThe minimum accessibility*of jth machine. TB The maximum allocated budget. pcurrent The probability of human error in the current state. * In this paper, machine’s accessibility is considered as the available and operational time of the equipment. V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 5 dated using Maximum Likelihood Estimation (MLE) for greater accuracy and reliability. Learning Curve Effects on PM Times Over time, personnel improve in performing PM, reducing the time required per task. This improvement follows a refined learning curve, incorporating plateau points where skill improvements diminish. Corrective Maintenance (CM) Costs Stability CM costs are assumed constant per unit task, but additional costs such as logistical delays, material surcharges, or overtime penalties are considered in sensitivity analyses. Exclusion of Opportunistic Maintenance Opportunistic maintenance actions are excluded; however, the model provides the flexibility to integrate them in future expansions. Immediate CM Operations CM operations are executed without delays upon fault detection. The fault detection system is assumed to be robust, with negligible lag between failure and response. Preventive Maintenance Levels PM is categorized into three levels—minimal, imperfect, and perfect. Each level’s effectiveness is probabilistically modeled, incorporating both human error and equipment improvement factors. This provides realistic variations in outcomes based on effort and expertise. Impact of CM on Failure Rate CM is modeled as minimal, assuming it restores equipment functionality without altering its inherent failure characteristics, which remain Weibull-distributed with stable parameters. Single Level of CM Operations The study assumes a uniform CM approach. Future work may explore differentiated CM levels based on failure severity. Inventory Control System Spare parts for PM follow a fixed-order interval (FOI) system, with reorder points optimized for minimum total cost, considering order size, holding costs, and stockout penalties. Under this approach (FOI), orders are placed at predetermined, regular intervals, rather than being triggered by a reorder point. The order quantity is determined based on the inventory position at the review time to ensure sufficient stock until the next review period. Alignment of Inspection and PM Periods Inventory inspections coincide with PM periods, optimizing scheduling and resource utilization. Dual Spare Parts Ordering System Spare parts are procured through regular and emergency orders, with emergency orders incurring higher costs but ensuring service continuity during unexpected demand spikes. Fig. 1. Representation of the mathematical model’s structure. V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 6 The virtual age of a machine The virtual age of a machine is defined as the measure of its effective age, based on factors such as operational history, maintenance, and usage, rather than its actual chronological age. Homogeneous Weibull Parameters Across Equipment This study assumes identical scale ( η ) and shape (β) parameters for the Weibull distribution across all equipment types, implying consistent failure characteristics under similar operational conditions. 3.4. Mathematical model 3.4.1. The objective function The objective function of this research is so as to reduce inventory control costs associated with spare parts, downtime, human error, and CM and PM operation costs Inventory control The inventory control system ensures that spare parts are available when needed for maintenance operations. The model incorporates safety stock levels and reorder points to prevent stockouts.If the inventory level of spare parts falls below the safety stock level, an emergency order is triggered. Emergency orders are expedited to minimize downtime, but they incur higher costs compared to regular orders. This ensures that maintenance operations can proceed without significant delays, even in cases of unexpected spare parts shortages. Inventory holding cost of spare parts Holding costs are determined by the spare parts ordered in excess of the demand at the end of each time period. Since the demand for these spare parts is influenced by machine failures, its quantity is estimated using the demand function for machine failure, making it a random variable. In reality, the demand for spare parts derives from the overall demand for CM and PM operations that require spare parts replacement. Hence, the demand for spare parts is also a random variable, and the following equations are employed to calculate the average inventory and shortages. Iit =Qit +Ii(t−1)−dcit −∑k∈Kdpikt −Bi(t−1)(1) Bit = − Qit −Ii(t−1)+dcit +∑k∈Kdpikt +Bi(t−1) Qit =Q[max] it −Ii(t−1)(2) Based on the stochastic behavior model proposed by Xiang et al., 2018 [51] inventory and shortage are formulated as a backorder with a loss function in Eq. (3). l(ϑ,w) = E[max (ϑ−w,0)] (3) E represents the expected value of the random variable ω and the scalar variable ϑ. As previously mentioned, demand for spare parts is a random variable, and the order quantity for spare parts, inventory, and shortage are scalar variables. Therefore, Eqs. (4) and 5 can be presented for nonlinear estimation of inventory and shortage. Iit =l(Qit +Ii(t−1)−∑k∈Kdpikt −Bi(t−1),dcit) =E[max(Qit +Ii(t−1)−dcit −∑k∈Kdpikt −Bi(t−1),0)] =E(Qit +Ii(t−1)−dcit −∑k∈Kdpikt −Bi(t−1))+ (4) Bit =l(−Qit −Ii(t−1)+∑k∈Kdpikt +Bi(t−1),−dcit) =E[max(−Qit −Ii(t−1)+dcit +∑k∈Kdpikt +Bi(t−1),0)] =E(−Qit −Ii(t−1)+dcit +∑k∈Kdpikt +Bi(t−1))+ (5) Consequently, Eqs. (6) and 7 can be formulated as follows: E[Qit +Ii(t−1)−Bi(t−1)−∑k∈Kdpikt −dcit]≤It ⇒Qit +Ii(t−1)−Bi(t−1)−∑k∈Kdpikt −E[dcit] ≤ Iit (6) E[∑k∈Kdpikt −dcit −Qit −Ii(t−1)+Bi(t−1)]≤Bit ⇒∑k∈Kdpikt −E[dcit] − Qit −Ii(t−1)+Bi(t−1)≤Bit (7) where E[dcit] = ∑ j∈J Nscijt[((ajt +l)β−aβ jt) η β] Therefore, given the stochastic nature of spare parts demand, the inventory average in each period is multiplied by the holding cost per unit according to Eq. (8). If the random demand surpasses the number of spare parts ordered for that period, the cost of shortages is included in the total cost. In this model, the total shortage cost is calculated by multiplying the shortage by the cost of shortage in each unit, as specified in Eq. (9). ∑i∈I∑t∈MhitbIit(Qit +Ii(t−1)−Bi(t−1)−∑k∈Kdpikt −E[dcit]) +∑i∈I∑t∈MShitbBit(∑k∈Kdpikt −E[dcit] − Qit −Ii(t−1)+Bi(t−1))(8) where E[dcit] = ∑ j∈J Nscijt[((ajt +l)β−aβ jt) η β](9) Eq. (4) represents the expected inventory average as the difference between the ordered amount of spare parts and their demand. This equation indicates that if the ordered quantity of spare parts exceeds the random demand, the expression will have a positive value, and if the demand for spare parts exceeds the ordered quantity of spare parts, the expected inventory will be zero. Similarly, Eq. (5) shows the expected shortage average. Eq. (10) is always non-negative It is evident that these two equations cannot simultaneously assign values to themselves. In other words, if there is inventory in any period, there will be no shortage, and vice versa. Therefore, Eq. (9) represents this fact as follows: bBit +bIit =0⇒bBit +bIit ≤1 (10) Ordering cost of spare parts The ordering costs in this study comprise of two different scenarios, and the decision regarding the type of ordering strategy depends on the inventory level. If the inventory level of spare parts is lower than the reorder point but higher than the safety stock level at the inspection point, the ordering timing will follow the usual procedure. Nevertheless, if the inventory level of spare parts falls below the safety stock level, the ordering will be done in an emergency state, leading to a reduction in ordering timing and an increase in costs compared to the usual situation. Therefore, emergency orders are a critical component of the V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 7 maintenance policy. When spare parts are unavailable in the inventory and fall below the safety stock level, the system initiates an emergency procurement process. This process reduces the lead time for spare parts delivery but increases the associated costs. The decision to place an emergency order is based on the urgency of the maintenance task and the criticality of the equipment. For example, CM tasks that require immediate attention are prioritized for emergency orders to minimize production downtime. Fig. 2 clearly and comprehensively illustrates the process of ordering spare parts and its relationship with PM and CM operations. Thus, to calculate the ordering cost according to the type of ordering strategy, we can exploit Eq. (11) that: COit =⎧ ⎨ ⎩ 0if ROPit <Iit NOit if SSit ≤Iit ≤ROPit EOit if Iit <SSit ⇒COit =⎧ ⎨ ⎩ 0if ROPit +1≤Iit NOit if SSit ≤Iit ≤ROPit EOit if Iit ≤SSit −1 (11) In order to calculate the ordering cost, it needs to be multiplied by the decision variable that is related to ordering or not ordering in each period. Consequently, the total cost for different periods can be calculated using Eq. (12), where the ordering cost is assumed to be a fixed cost in this study. Besides, the cost of purchasing spare parts is also dependent on the binary variable of ordering or not ordering. The cost of purchasing spare parts in each period is equal to the multiplication of the purchasing cost per unit of spare parts and the ordered quantity. If an order is placed in a specific period, this cost will be added to the model. Otherwise, no cost will be included in the model for that period. The order quantity in each period is obtained through the difference between the inventory level and the maximum order quantity. TOit =∑i∈I∑t∈MVOit.COit +∑i∈I∑t∈MCuitQitVOit (12) Qit =Q[max] it −Ii(t−1)(13) Eq. (14) defines the constraint that is related to the demand for spare parts and its dependence on the binary variable of ordering or not ordering. This equation clarifies that if the binary variable VOt equals zero, no order will be placed, and as a result, the value of Qt will be zero. However, if VOit equals one, the value of Qt can vary from zero to its maximum value (Q[max]), depending on the current inventory level. Qit ≤Q[max] it VOit (14) Downtime cost Machine downtime can occur due to several reasons, and this study focuses on three significant factors. First, it involves the unavailability of spare parts during maintenance operations. Second, it concerns the average time required to perform CM to restore the machine to production operations. Third, it pertains to the execution of PM operations at various levels, which necessitates temporary halts in the production line. Notably, the time required for these operations will decrease as employees gain experience from performing them more frequently. To calculate the cost of machine downtime, we begin by determining the cost per unit of time for production line downtime and then multiply it by the duration of the machine downtime. The costs associated with machine downtime, specifically the second and third types, will be Fig. 2. The inventory control system of spare parts. V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 8 zkt,VOit,bIit,bBit,y[1] it ,y[2] it ,y[3] it ∈ {0,1} 0.00005 ≤pk,pCM,pins,p[total]≤pcurrent ψ [total] t, ψ ct ∈Int (44) As a result, the research model is formulated as follows: subject to Qit +Ii(t−1)−Bi(t−1)−∑k∈Kdpikt −E[dcit] ≤ Iit ∀i,t(46) ∑k∈Kdpikt +E[dcit] − Qit −Ii(t−1)+Bi(t−1)≤Bit ∀i,t(47) bIit +bBit ≤1∀i,t(48) Qit ≤Q[max] it VOit ∀i,t(49) Qit =(Q[max] it −Ii(t−1))VOit ∀i,t(50) t[m] kjt =γkjzkjt ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1+∑ l∈M l≤M−1 ʹ zkjl ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ lnr ln2∀k,j,t(51) ajt =(aj(t−1)+l)(1−∑k∈Kzkjt α k(1−pk))∀j,t(52) ξʹ cj −ξʹ cj −ξcj μ −2( ψ cjt −1)≤ϑcjt ≤ξʹ cj −ξʹ cj −ξcj μ −2( ψ cjt −2), μ >2∀c,j,t (53) ∑k∈Kzkjt =1∀j,t(54) ψ [total] jt =∑k∈Kkkzkjt ∀j,t(55) ψ [total] jt ≤ ψ cjt ∀c,j,t(56) Qit +Ii(t−1)−Bi(t−1)−dpikt −E(dcit) ≤ Warit ∀i,t(57) Qit +Ii(t−1)−Bi(t−1)≤dpikt +E(dcit) ∀i,t(58) t[p] jt =l−∑ k∈K zkjtt[m] kjt −MTTRjt[((ajt +l)β−aβ jt) η β] −LT[Etotal] tbBi(t−1)[((ajt +l)β−aβ jt) η β]∀j,t(59) LT[E] ibBit ≤LT[Etotal] t∀i,t(60) ∑k∈Kdpikt =∑k∈KNspiktzkt ∀i,t(61) −Q[max] it ∑j∈Jzkjt ≤∑k∈Kdpikt ≤ − Q[max] it ∑j∈Jzkjt ∀k,j,t(62) ∑ t∈M t[p] jt ≥AEj∀j(63) VOit ≥1−y[3] it ∀i,t(64) (ROPit +1) − My[1] it ≤Iit ≤Q[max] i∀i,t(65) SSit −My[2] it ≤Iit ≤ROPit +My[2] it ∀i,t(66) Iit ≤ (SSit −1) + My[3] it ∀i,t(67) y[1] it +y[2] it +y[3] it =2∀i,t(68) EOit(1−y[3] it )+NOit(1−y[2] it )=COit ∀i,t(69) −Q[max] i(bBit) ≤ Iit ≤ (1−bBit)Q[max] i∀i,t(70) −Q[max] i(bIit) ≤ Bit ≤ (1−bIit)Q[max] i∀i,t(71) min∑ i∈I∑ t∈M hitbIit(Qit +Ii(t−1)−Bi(t−1)−∑ k∈K dpikt −E[dcit]) +∑ i∈I∑ t∈M ShitbBit(∑ k∈K dpikt −E[dcit] − Qit −Ii(t−1)+Bi(t−1)) +∑ i∈I∑ t∈M VOit.COit +∑ i∈I∑ t∈M CuitQitVOit +∑ i∈I∑ j∈J∑ t∈M(CljtLT[Etotal] i+EOit)bBi(t−1)[((ajt +l)β−aβ jt) η β] +∑ i∈I∑ kʹ,k∈K kʹ<k∑ j∈J∑ t∈M(CljtLT[E] i+EOit)bBi(t−1)zkʹjt +f(p) +∑ k∈K∑ j∈J∑ t∈M((CLjt +HmkjtNmkjt)t[m] kjt +Csetjt)zkjt +∑ j∈J∑ t∈M((Cljt +HrjtNrjt)MTTRjt +Csetjt)[((ajt +l)β−aβ jt) η β] (45) V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 15 p[total]=1−(∏k∈K(1−pk))(1−pcorrective)(1−pins)(72) Total cost ≤TB (73) zkjt,VOit,bIit,bBit,y[1] it ,y[2] it ,y[3] it ∈ {0,1}(74) 0.00005 ≤pk,pcorrective,pins,p[total]≤pcurrent (75) ψ [total] jt , ψ cjt ∈Int (76) Qit,Iit,Bit,dpikt,dcit,at,COit,t[p] jt ,t[m] kjt ≥0 (77) 4. Findings The model is formulated as a mixed-integer nonlinear programming (MINLP) problem and solved using GAMS software. The computations are carried out on a system equipped with an AMD Ryzen 32200U processor running at 2.5 GHz, 8 GB of RAM, and a 64-bit operating system. The results obtained from solving the model, which utilizes data from a case study involving multiple machines, spare parts, conditions, and three different levels of PM operations across various periods, are presented in Table 5. In this model, the PM approach is designed such that different levels of maintenance operations, each with varying effectiveness rates, influence the equipment’s lifespan. Based on the overall results, the most effective level of PM operations is selected. Since one of the primary objectives of PM is to prevent system failures and unexpected production line shutdowns, the model incorporates costs related to production line disruptions that may occur due to the implementation of specific levels of PM and CM operations, as well as shortages of certain spare parts. The results obtained from solving the model indicate the optimal levels of PM operations for different periods, taking into account both costs and the effective rate of the equipment. Implementing PM operations at the lowest level (Level 3) requires less cost, time, and resources, such as manpower and spare parts, compared to higher levels of PM operations. Conversely, executing PM operations at the highest level (Level 1) demands the most time, cost, and resources. Based on data from the case study company, performing Level 1 PM operations to restore equipment to an “as good as new” condition involves significant expenses. Therefore, as observed in the model results, Level 2 or Level 3 PM operations have been selected for various periods. As the number of periods increases, it becomes evident that maintenance operations begin to follow a specific and predictable cycle. This cyclic pattern allows for more efficient planning and allocation of resources, ultimately enhancing the overall maintenance strategy. The results of solving the problem for the variables of PM operations are presented in Table 5. Furthermore, the types of spare parts required for each level of PM operations are detailed in Table 5. As evident from the results, no spare parts are needed for unselected levels of PM operations. For the selected levels in each period, the appropriate spare parts are determined based on the chosen level of maintenance. The time required for executing PM operations is influenced by the learning curve effect, leading to a reduction in time for each subsequent execution of the same PM level. Consequently, as illustrated in Table 5, the duration for performing Level 2 and Level 3 PM operations decreases over time. Specifically, the time for Level 2 PM operations decreases from 0.083 h to 0.031 h, while the time for Level 3 PM operations decreases from 0.025 h to 0.011 h. This reduction reflects the efficiency gained through experience, improved techniques, and the cumulative impact of the learning curve. These findings highlight the importance of considering learning effects in maintenance planning to optimize resource utilization and enhance Table 6 The results of sensitivity analysis ( η ). No. Δzcost η Δ η 1−0.10285 36 0.285714 2−0.10296 35 0.25 3−0.001206 34 0.232143 4−0.000980 33 0.178571 5−0.000815 32 0.142857 6−0.000727 31.5 0.125 7−0.000540 30.5 0.089286 8−0.000421 30 0.071429 9−0.000338 29.5 0.053571 10 −0.000230 29 0.035714 11 −0.000118 28.5 0.017857 12 0 28 0 13 0.000123 27.5 −0.01786 14 0.000252 27 −0.03571 15 0.224561 26 −0.07143 16 0.653825 24 −0.14286 Fig. 8. The impact of changes in η parameters on the objective function. Fig. 9. The ratio of changes in total maintenance costs to changes in the scale parameter ( η ). V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 16 operational efficiency. The results obtained from solving the first model for binary variables related to spare parts indicate that inventory is consistently available across all examined periods, with no shortages recorded. This outcome is logical, as the cost of maintaining spare parts is typically lower than the cost associated with shortages, which is notably higher in the context of the cement factory case study. Although PM operations are pre-scheduled, and auxiliary lines and side motors are utilized to prevent production stoppages and kiln shutdowns, significant costs can still arise if spare parts are not available in the required quantities and the lead time for procurement exceeds the availability of these auxiliary resources. Furthermore, the total cost of maintenance operations over the 36-month period amounts to 3501,281,000 units of currency. Additionally, the optimal HEP for minimizing human error-related costs is determined to be 0.02. These findings underscore the importance of maintaining adequate spare parts inventory and optimizing maintenance strategies to mitigate costs and ensure operational efficiency. 5. Sensitive analysis Sensitivity analysis is a widely used technique to evaluate the impact of variations in one or more input parameters on one or more desired outputs. It plays a crucial role in enhancing the understanding of a model’s behavior and results, providing insights into its precision and effectiveness. In this section, a sensitivity analysis method is employed to assess the robustness of the designed model. This is accomplished by systematically varying key parameters to create a range of scenarios, including both reductions and increases in their values. Specifically, a comprehensive sensitivity analysis was conducted on two critical model parameters: 1. The parameter η of the Weibull distribution: The scale parameter ( η ) is often referred to as the characteristic life or lifetime characteristic of the equipment. 2. The effective rate of equipment By analyzing these parameters under different scenarios, the sensitivity analysis provides valuable insights into how changes in these inputs affect the model’s outputs, such as maintenance costs, optimal PM Table 7 The changes in total maintenance operation costs for various effective rates of equipment. Effective rate α kΔ α kΔzcost K ¼1(0.825, 0.5, 0) −0.175 0.0721044 (0.85, 0.5, 0) −0.150 0.0613767 (0.875, 0.5, 0) −0.125 0.057943 (0.9, 0.5, 0) −0.100 0.0403553 (0.925, 0.5, 0) −0.075 0.0300581 (0.95, 0.5, 0) −0.050 0.01999009 (0.975, 0.5, 0) −0.025 0.009882 (1, 0.5, 0) 0 0 K ¼2(1, 0.25, 0) −0.500 0.033876 (1, 0.3, 0) −0.400 0.027060 (1, 0.35, 0) −0.300 0.020264 (1, 0.4, 0) −0.200 0.013488 (1, 0.45, 0) −0.100 0.006734 (1, 0.5, 0) 0 0 (1, 0.55, 0) 0.100 −0.006713 (1, 0.6, 0) 0.200 −0.111557 (1, 0.65, 0) 0.300 −0.115236 (1, 0.7, 0) 0.400 −0.118907 (1, 0.75, 0) 0.500 −0.122570 K ¼3(1, 0.5, 0) 0 0 (1, 0.5, 0.05) 0.050 −0.00427 (1, 0.5, 0.1) 0.100 −0.00852 (1, 0.5, 0.15) 0.150 −0.012777 (1, 0.5, 0.2) 0.200 −0.01701 (1, 0.5, 0.25) 0.250 −0.02125 (1, 0.5, 0.3) 0.300 −0.02547 (1, 0.5, 0.35) 0.350 −0.02969 (1, 0.5, 0.4) 0.400 −0.03389 (1, 0.5, 0.45) 0.450 −0.03809 (1, 0.5, 0.5) 0.500 −0.04228 (1, 0.5, 0.55) 0.550 −0.04647 Fig. 10. The changes in costs relative to the effective equipment rate for Level 1 PM. Fig. 11. The changes in costs relative to the effective equipment rate for Level 2 PM. Fig. 12. The changes in costs relative to the effective equipment rate for Level 3 PM. V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 17 levels, and inventory management strategies. This process helps validate the model’s reliability and adaptability to varying operational conditions. 5.1. Sensitivity analysis with respect to the scale parameter ( η ) The scale parameter η , often referred to as the lifetime characteristic, is directly associated with the mean time to failure (MTTF), which is calculated as MTTF= η Γ(1 +1/β). An increase in the scale parameter η leads to a longer MTTF, meaning the equipment can remain operational for a more extended period, thereby delaying the occurrence of failures. As a result, both CM and PM costs decrease with a higher η , as the need for maintenance interventions is reduced. Table 6 provides detailed information on the variations of the scale parameter η and its impact on maintenance costs. Additionally, Fig. 8 illustrates these variations and their influence on maintenance costs, offering a visual representation of how changes in η affect the overall maintenance strategy. This analysis highlights the importance of the scale parameter in determining equipment reliability and maintenance planning, demonstrating that higher values of η contribute to reduced maintenance frequency and costs. Fig. 9 illustrates not only the inverse relationship between the scale parameter η and total maintenance costs but also emphasizes a more significant reduction in costs as the values of η increase. The figure clearly depicts cost variations within the range 27.5< η <32.5, encompassing all observed cost values. Additionally, the figure reveals that cost changes are less pronounced for higher values of η compared to lower values, indicating a diminishing marginal effect as η increases. The circles in Fig. 9 highlight shifts in the optimal solution for specific values of the scale parameter η . These shifts demonstrate how changes in η influence the model’s outcomes, particularly in terms of cost optimization and maintenance strategy adjustments. This visualization underscores the importance of the scale parameter in determining maintenance costs and provides valuable insights into the sensitivity of the model to variations in η . 5.2. Sensitivity analysis with respect to the effective rate of equipment As described, various levels of PM operations have varying effects on the lifespan of equipment. Specifically, each PM level impacts the virtual lifespan of the system differently, reflecting their distinct influences on equipment durability. This study investigates how the effectiveness of PM operations ranges from minimal to perfect. Each PM level affects the effective rate of the equipment in a unique way, thereby altering the system’s virtual lifespan accordingly. The changes in total maintenance operation costs for various effective equipment rates are illustrated in Table 7. Based on the results obtained from solving the model, as shown in Table 7, increasing the effective equipment rate leads to a reduction in costs. This outcome aligns with the assumptions of the problem and is entirely logical. An increase in the effective equipment rate typically results in a decrease in the equipment’s virtual age, which in turn reduces the equipment’s failure rate and, consequently, the associated maintenance costs. Figs. 10–12 illustrate this relationship: as the effective equipment rate increases, the virtual lifespan of the equipment decreases, and vice versa. In other words, a longer effective lifespan of the equipment shifts PM from a minimal to a more comprehensive level of implementation. This transition enhances the equipment’s lifespan and reduces its failure rate, further contributing to lower maintenance costs. These findings highlight the importance of optimizing the effective equipment rate to achieve cost-efficient and reliable maintenance strategies. It is crucial to note that variations in the effective rate for any maintenance level can influence the optimal solution. For example, if the effective rate for Level 2 PM increases from 0.5 (imperfect) to 1 (perfect), the optimal solution may change. Instead of performing Level 2 maintenance in just one period, it might become optimal to apply this level of maintenance across multiple periods. This is because the costeffectiveness of Level 2 maintenance, given its cost and effectiveness, could be more beneficial compared to other maintenance levels when extended over a greater number of periods. As shown in contour Figs. 10–12, regions with uneven boundary curvature indicate where the optimal solution values change. Specifically, the optimal solutions for PM operations vary within the following effective equipment rate ranges: •First level of PM: Changes occur when α 1<0.85. •Second level of PM: Adjustments are observed within 0.5< α 2≤0.6. •Third level of PM: Variations are evident within 0.3< α 3≤0.5. Even if the optimal solutions remain stable in other ranges, variations in the effective equipment rate can still impact the optimal value of the objective function (cost). This is because the equipment’s lifespan affects the failure rate, which consequently influences the overall maintenance costs. 6. Managerial insights The results of this study provide valuable insights into determining the most appropriate timing and intensity of PM operations, considering their impact on equipment lifespan, conditions, and associated expenditures. Overall, this paper presents a cost-reduction strategy for industries, aimed at improving equipment availability and reducing the HEP. The findings of this study assist managers in making informed decisions regarding the optimal level of HEP, taking into account the costs associated with human error, resource allocation for human error reduction, and the organization’s overall budget. Managers can plan organizational and personnel conditions in a way that maintains the level of error at a desired and optimal threshold. In other words, they can establish desirable thresholds for each of the Common Performance Conditions (CPCs) corresponding to human errors. Furthermore, the results of the proposed model significantly contribute to enhancing maintenance operations planning by incorporating human error into the decision-making process. This, in turn, leads to optimal decision-making regarding various levels of PM operations and improved management of spare part inventory. As a result, the costs associated with CM operations and overall organizational expenses can be managed in a cost-effective manner. The presented model offers significant benefits to organizations where PM plays a crucial role, and the costs associated with human error and equipment failure are substantial. Additionally, this model provides considerable advantages for organizations that prefer to perform PM operations based on both equipment age and conditions. The simultaneous execution of PM operations based on both conditions and time can greatly enhance decision-making outcomes. This approach proves to be a reliable strategy for organizations that must conduct PM operations while considering time constraints and equipment conditions, ensuring a balance between cost efficiency and operational reliability. V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 18 7. Conclusions and suggestions This paper is notable for its innovative approach in estimating the cost function related to human errors in maintenance tasks by leveraging regression analysis and historical data. The study highlights the significant effect of HEP on the efficiency and effectiveness of maintenance operations. For the first time, it demonstrates how human error can influence the effective rate of equipment lifespan, providing a novel perspective on the interplay between human factors and equipment reliability. In this study, a limited selection of spare parts has been analyzed based on the equipment under consideration. The research focuses exclusively on evaluating the spare parts typically utilized for both PM and CM operations for two specific machines. Establishing prioritized rankings and selecting the most essential spare parts—guided by factors such as required quantity, significance for operational objectives, lead time, supplier accessibility, repairability, pricing, and other relevant considerations—can significantly enhance the efficiency and effectiveness of spare parts inventory management. Furthermore, future investigations could delve into exploring how personnel training impacts not only PM but also CM operations, as improved training may reduce human errors and enhance maintenance outcomes. Additionally, the study assumes a constant cost for CM and PM operations across all time periods. It is suggested that future research endeavors should account for real-world dynamics by incorporating factors such as inflation rates, fluctuations in spare parts inventory levels, and other economic variables to better reflect actual conditions. Such refinements would further improve the model’s applicability and accuracy in real-world industrial settings. This study focuses primarily on determining the optimal value for HEP, without proposing specific strategies for enhancing and mitigating human errors across various CPCs. To address this limitation, it is advisable to develop a novel mathematical model that dynamically adjusts based on the optimal HEP. Such a model would provide a more comprehensive and impactful approach to reducing human errors and their associated costs. In practical terms, this would involve implementing enhancements in the CPCs, taking into account the current organizational context, financial limitations, and the influence of these factors on human errors. This approach should facilitate the identification of optimal interventions for modifying each sub-condition and overall performance criteria, ensuring a more targeted and effective reduction in human errors. Furthermore, for future research endeavors, it is recommended to incorporate the consideration of variable variance in random factors during problem-solving processes. By establishing threshold limits for these variances, researchers can offer more practical and actionable insights for addressing human errors in real-world scenarios. This would not only enhance the robustness of the model but also provide decisionmakers with clearer guidelines for implementing error-reduction strategies in dynamic and uncertain environments. A valuable direction for future research would be to develop a more comprehensive model that integrates both learning and forgetting processes in the context of PM operations. This framework would quantify the rate at which skills are acquired and lost over multiple maintenance cycles, considering factors such as task complexity, frequency of practice, and workforce experience. By exploring how learning-forgetting dynamics influence maintenance scheduling, HEP, and associated costs, this research could provide actionable insights into optimizing workforce training programs and operational efficiency. To further advance the understanding of human error in maintenance operations and its economic implications, a future study could focus on developing a dynamic cost-benefit optimization model for HEP reduction across diverse maintenance domains. This model would integrate a detailed classification of human errors (e.g., slips, lapses, mistakes, violations) and evaluate the impact of specific contextual factors—such as training programs, procedural improvements, humanmachine interface (HMI) design, and environmental conditions—on HEP and associated costs. The study could also explore domain-specific HEP thresholds, identifying the point at which further investments in error reduction yield diminishing returns. By incorporating real-world case studies and simulation-based scenarios, the research would provide a decisionsupport framework to help organizations prioritize cost-effective strategies for minimizing human errors in inspection, PM, and CM activities. Furthermore, the study could investigate the role of emerging technologies, such as AI-driven predictive maintenance systems and augmented reality (AR) tools, in reducing human errors and optimizing maintenance efficiency. These technologies could enhance decisionmaking, improve task execution, and reduce the cognitive load on maintenance personnel. By integrating these tools into the model, organizations could align their maintenance operations with predefined targets while balancing the trade-offs between error reduction costs and system reliability. Such a framework would offer practical, data-driven insights for enhancing maintenance performance across industries, enabling organizations to achieve higher reliability, lower costs, and improved safety standards. By addressing the interplay between human factors, technological advancements, and economic considerations, this research would contribute significantly to the field of maintenance optimization and human error management. CRediT authorship contribution statement Vahideh Bafandegan Emroozi: Conceptualization, Methodology, Software, Writing – original draft, Visualization. Mostafa Kazemi: Supervision, Data curation. Mahdi Doostparast: Writing – review & editing, Validation. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 19 Appendix The data related to this paper, derived from the case study, is presented in Tables 1A to 2A. Table 1 A. Data related to case study. (continued on next page) V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 20 Table 1 (continued) V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 21 Table 2 A. Data related to case study. (continued on next page) V. Bafandegan Emroozi et al. Operations Research Perspectives 14 (2025) 100336 22 Data availability Data will be made available on request. References [1] Alaswad S, Xiang Y. A review on condition-based maintenance optimization models for stochastically deteriorating system. Reliab Eng Syst Saf 2017;157: 54–63. https://doi.org/10.1016/j.ress.2016.08.009. Jan. [2] Guo W, Jin J(Judy), Hu SJ. Allocation of maintenance resources in mixed model assembly systems. 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