Modern manufacturing systems operate in an increasingly uncertain and competitive environment in which production efficiency, product quality, equipment reliability, and timely delivery must be managed simultaneously. Production equipment deteriorates and fails stochastically, which can interrupt production, reduce available capacity, increase maintenance and operating costs, and adversely affect product quality. Consequently, maintenance is no longer regarded as an isolated technical activity; rather, maintenance decisions are closely interconnected with production planning, quality control, inventory management, and customer service decisions. Recent studies emphasize that jointly coordinating these decisions can provide substantial operational and economic benefits compared with treating production, maintenance, and quality as independent functions [1–3].

Preventive Maintenance (PM) is one of the principal strategies for controlling equipment deterioration and reducing the probability of unexpected failures. However, PM itself consumes time and financial resources and may temporarily reduce production capacity. Therefore, excessive maintenance may increase operating costs and cause unnecessary production interruptions, whereas insufficient maintenance can increase failure frequency, downtime, defective production, and customer-related costs. The fundamental managerial challenge is consequently to determine not only whether PM should be performed, but also when it should be performed and at what intensity. Recent research has increasingly moved from conventional age-based maintenance policies toward condition-based, dynamic, and integrated maintenance strategies that explicitly consider equipment degradation, production requirements, and product quality [2], [4], [5].

The interdependence between equipment condition and product quality makes jointly considering maintenance and quality particularly important. As production equipment deteriorates, the probability of producing nonconforming products may increase, while quality deterioration can itself accelerate equipment degradation in some manufacturing environments. Consequently, maintenance decisions influence both the reliability of the production system and the quantity and quality of its output.

Recent studies have therefore developed integrated production–maintenance–quality models in which inspection, maintenance intervention, production decisions, and quality deterioration are considered simultaneously. For example, Wang et al. [3] developed an integrated quality-control and maintenance model considering quality-dependent failures, while Wei et al. [2] incorporated imperfect maintenance and dynamic inspection into an integrated production, maintenance, and quality-control framework. These developments demonstrate the growing importance of moving beyond maintenance-only models toward integrated operational decision-making.

Another important challenge arises from the multi-product and multi-period nature of practical manufacturing systems. Manufacturing facilities frequently produce several products over a finite planning horizon, while product demand, production requirements, setup costs, processing costs, and the consequences of delayed orders may vary across periods. In such environments, maintenance decisions cannot be determined independently of production planning because a maintenance intervention reduces available production time and may influence the firm's ability to satisfy demand. Conversely, postponing maintenance to increase short-term production capacity may increase the likelihood of failure, defective production, and subsequent production losses. Recent research has therefore investigated integrating production planning and maintenance decisions in multi-product and multi-period environments. Davari et al. [4], for instance, developed an integrated simulation-fuzzy framework for PM optimization in multi-product production firms. At the same time, a more recent data-driven approach has integrated predictive maintenance with multi-period, multi-product production planning using deep learning and mathematical programming [4].

Demand uncertainty further complicates coordinating production and maintenance decisions. In real manufacturing environments, demand is rarely constant throughout the planning horizon and may fluctuate according to market conditions and customer requirements. A maintenance policy that is appropriate under low-demand conditions may be inappropriate when demand is high, and the cost of lost sales or delayed orders is substantial. Thus, maintenance intensity, inspection frequency, production quantity, inventory levels, and failure-related decisions should ideally be evaluated under alternative demand conditions rather than under a single deterministic demand assumption. Recent studies have explicitly incorporated dynamic customer demand and delivery performance into integrated production, quality, and maintenance models, highlighting the need to coordinate equipment-related decisions with market and service requirements [3]. The nature of equipment failures is another important consideration in integrated production and maintenance planning. In practice, equipment deterioration may manifest itself in different ways. A production system may move from an in-control state to an out-of-control state, producing nonconforming items without immediately stopping the production process. Such a condition may only be identified through inspection. In a more severe failure state, the equipment may become unavailable, and production may stop completely. These two failure mechanisms have different operational and economic consequences and therefore may require different maintenance and repair responses. Incorporating multiple failure modes into an integrated optimization model provides a more realistic representation of manufacturing systems than assuming a single failure mechanism. This issue is particularly relevant when inspection decisions, nonconforming production, corrective maintenance, and PM are jointly optimized.

Despite the substantial development of integrated production and maintenance models, several research opportunities remain. First, many existing studies focus on a single product, a single failure mechanism, or a limited maintenance policy. In contrast, practical manufacturing systems may simultaneously involve multiple products, multiple periods, stochastic failures, quality deterioration, and alternative maintenance levels. Second, although recent studies increasingly incorporate dynamic demand, quality deterioration, and condition-based maintenance, simultaneously considering different failure types, inspection decisions, multiple preventive-maintenance levels, and alternative demand scenarios remains relatively challenging. Third, the computational complexity of such integrated models increases substantially as the number of products, periods, scenarios, maintenance levels, and failure states increases, making efficient solution procedures necessary.

Motivated by these gaps, this study develops an integrated multi-period, multi-product production– maintenance–quality optimization model that considers two types of equipment failure. The first failure type represents a transition from an in-control to an out-of-control production state, in which nonconforming products may be generated, and the system condition can be identified through inspection. The second failure type represents a more severe equipment failure that interrupts production and requires corrective maintenance. The proposed framework further incorporates multiple levels of PM, each associated with different maintenance costs, durations, and effects on equipment age. In addition, the model considers alternative demand scenarios to represent different market conditions and their implications for production, inspection, maintenance, failures, and lost orders. The objective is to determine production, inspection, and maintenance decisions that maximize the manufacturing system's overall economic performance while accounting for interactions among equipment reliability, product quality, production capacity, and demand. To solve the resulting mixed-integer nonlinear optimization problem, this study combines an exact mathematical programming approach for smaller problem instances with a step-by-step solution procedure and a genetic algorithm for more complex instances. The proposed approach enables evaluation of alternative maintenance and inspection policies and provides managerial insights into the relationship between PM intensity, inspection frequency, production performance, failure occurrence, and profitability. In this respect, the study contributes to the literature by integrating production planning, the quality-related consequences of equipment deterioration, inspection, multiple levels of PM, and demand scenarios within a unified multi-period, multi-product framework.

The integration of production planning and maintenance decisions has received considerable attention in the operations management and maintenance literature. The main motivation behind this stream of research is that production and maintenance activities compete for the same limited resources and are therefore strongly interdependent. While increasing production rates can improve short-term revenue, intensive utilization of production equipment may accelerate deterioration and increase the probability of failure. Conversely, PM can improve equipment reliability and reduce the likelihood of unexpected breakdowns, but it requires maintenance costs and production downtime. Consequently, determining production and maintenance decisions independently may result in suboptimal operational performance.

Early studies established the importance of jointly considering production and maintenance decisions. For example, Abdolniur et al. [6] investigated the effect of maintenance policies on just-in-time production systems and demonstrated that maintenance decisions could significantly affect production-system performance. Gharbi and Kenne [7] developed production and preventive-maintenance policies for manufacturing systems with failures and emphasized the interaction between production rates and maintenance decisions. Similarly, Chelbi and Ait-Kadi [8] analyzed a production/inventory system with randomly failing production equipment subject to regular PM. These studies provided important foundations for the subsequent development of integrated production–maintenance models. The original manuscript also builds on this research stream through studies concerning production planning, PM, inventory, and unreliable production systems.

More recent studies have extended this literature by considering more realistic production environments. Davari et al. [4], for example, developed an integrated simulation-fuzzy framework for preventive-maintenance optimization in multi-product production firms. Their study demonstrates that maintenance decisions become substantially more complex when several products and production requirements are considered simultaneously. In another direction, Wang et al. [9] investigated the joint optimization of production scheduling and PM under frequent production switching, highlighting the importance of coordinating maintenance interventions with production schedules.

A common characteristic of these studies is that they do not treat maintenance as a purely technical decision about equipment replacement or repair. Rather, maintenance is increasingly viewed as an operational decision whose consequences extend to production capacity, scheduling, inventory, customer service, and overall profitability. This perspective provides an important basis for the integrated framework developed in the present study.

Although production–maintenance integration can improve system performance, considering equipment reliability without explicitly accounting for product quality may not adequately represent actual manufacturing systems. Equipment deterioration can increase the probability of producing nonconforming products, particularly when deterioration affects process stability or operating precision. Consequently, production systems may experience two simultaneous consequences of equipment degradation: reduced production availability and deterioration in product quality.

The integration of quality and maintenance decisions has therefore emerged as an important research direction. Bousaleh et al. [10], for instance, investigated joint production, quality, and maintenance control in a two-machine production line subject to operation-dependent and quality-dependent failures. Their study demonstrated the importance of considering quality-related failures when determining production and maintenance policies. Similarly, Beheshti Fakher et al. [11] developed a multi-period and multi-product profit-maximization model integrating production, maintenance, and quality decisions. This study is particularly relevant to the present research because it explicitly combines the three decision areas in a multi-period, multi-product environment. The original article identifies this study as a principal foundation for its proposed model.

Recent research has further strengthened this research stream. Lv et al. [12] investigated joint optimization of production, maintenance, and quality control while considering the effect of product-quality variation in a degraded production system. Their results emphasize that degradation should not be modeled solely through its effect on equipment availability because it may simultaneously affect the quality characteristics of manufactured products.

Wang et al. [3] also developed a joint optimization model for quality control and maintenance in a production system with quality-dependent failures. More recently, Wei et al. [2] proposed an integrated production, maintenance, and quality-control framework incorporating imperfect maintenance and dynamic inspection. These studies indicate a clear transition in the literature from traditional maintenance optimization toward integrated decision-making in which equipment condition, production performance, inspection, and product quality are considered simultaneously.

This line of research is directly related to the present study. However, the proposed model differs from many existing models by explicitly incorporating two failure mechanisms, multiple preventive-maintenance levels, inspection decisions, and alternative demand scenarios within a multi-period, multi-product framework. Inspection is an important component of manufacturing systems in which equipment deterioration may not immediately result in complete production stoppage. In such systems, a production unit may move from an in-control state to an out-of-control state and continue operating while producing a proportion of nonconforming products. Since this type of deterioration may not be directly observable, inspection provides a mechanism for identifying the system's operational and quality condition.

Several studies have investigated the relationship between inspection, maintenance, and production performance. Cai et al. [13], for example, jointly considered inspection, maintenance, and spare-parts provisioning and compared alternative inventory policies using simulation and empirical information. Azadeh et al. [14] investigated a joint quality-control and preventive-maintenance strategy and demonstrated the importance of coordinating quality inspection with maintenance decisions.

The present study incorporates this trade-off by considering inspections at different points during the planning horizon and linking inspection outcomes to the production system's operational state. When an out-of-control condition is detected, nonconforming products are separated from standard products, and the model considers an appropriate maintenance response. This structure allows the model to capture the economic consequences of delayed detection, as well as the cost of more frequent inspection.

Traditional preventive-maintenance models often assume that maintenance either completely restores equipment to an as-good-as-new condition or follows a predetermined replacement policy. Such assumptions may not accurately represent many practical manufacturing environments, where maintenance can be performed at different levels depending on the severity of deterioration, available resources, production requirements, and the desired reliability level.

Recent maintenance research increasingly considers imperfect maintenance, in which a maintenance intervention reduces equipment age or deterioration but does not necessarily restore the system completely to its original condition. This approach provides greater flexibility for modeling real manufacturing systems. The original study already adopts this perspective by assigning different costs and durations to different preventive-maintenance levels and by assuming that the maintenance level affects the reduction in the effective age of the production system.

This feature creates an additional optimization problem. A higher maintenance level may reduce the probability of future failures more effectively, but it generally requires more time and expenditure. A lower maintenance level, on the other hand, may preserve production capacity in the short term but provide less protection against future failures. Therefore, the optimal maintenance decision depends on the interaction between maintenance cost, production capacity, failure risk, product quality, and demand.

Another important dimension of the present problem is demand uncertainty. Most real manufacturing systems operate in environments where customer demand varies across time and where the consequences of failing to satisfy demand may differ depending on market conditions. Consequently, production and maintenance decisions based on a single deterministic demand level may not perform well under alternative demand conditions.

Scenario-based approaches provide a useful way to represent these differences. Instead of assuming a single demand level, alternative scenarios can be defined according to different market conditions and associated customer-service requirements. The original model follows this approach precisely by considering different demand scenarios and allowing the required production volume, number of inspections, failures, and lost orders to vary across scenarios.

This scenario-based structure is particularly relevant to the present research because it allows maintenance and production decisions to be evaluated from a broader economic perspective rather than solely on the basis of equipment reliability. In this way, the proposed model connects technical decisions concerning equipment maintenance with market-related consequences such as customer satisfaction, delayed orders, and lost revenue.

Based on the above literature, the present study develops an integrated multi-period and multi-product production, quality, inspection, and preventive-maintenance model. The main distinguishing characteristics of the proposed framework can be summarized in Table 1.

The principal contribution of the present study therefore lies not merely in adding another preventive-maintenance policy, but in integrating several interdependent operational decisions within a unified framework. Specifically, the model simultaneously determines production, inspection, and maintenance decisions while considering two types of stochastic failures, multiple preventive-maintenance levels, nonconforming production, alternative demand scenarios, and the economic consequences of delayed or lost orders.

This positioning also provides a clearer basis for evaluating the contribution of the proposed model against the most relevant studies in the literature, particularly the integrated production–maintenance–quality model of Beheshti Fakher et al. [11] and more recent integrated quality and maintenance studies such as Lv et al. [12], Wang et al. [3], and Wei et al. [2].

This study addresses a production system wherein a set of different products are produced. This system faces two types of probable failures. Failure type I occurs when the system shifts from in-control to out-of-control, while failure type II occurs when the system goes out of reach and production stops. This model is introduced with a time horizon H and several discrete periods t with a constant length L. At the beginning, the system is assumed to be under control and is producing. In type I failure, the machine is out of control and products of low quality (non-standard) are produced. This kind of failure is only determined by inspection. Type II failure is specified without inspection. In this case, the system stops and does not produce. Fig. 1 shows the system structure in different cases.

In any period t, the system is inspected, and PM is carried out if necessary, which reduces machine age. This decrease in age is due to the level of preventive operation. Each time it is determined that the system is out of control, the separation of standard and non-standard products is performed. At the end of each period, a complete system repair is done to zero.

Inspection operations are carried out on manufactured products to check their quality. If the product is detected as low quality, production is stopped, and the condition is announced as out of control. Otherwise, the PM applies to the system. The preventive level is limited, and these levels have different costs and times. There are two types of repairs for the model: complete repairs performed at the end of each period and defective repairs performed in the event of Type II failure. Type I and II failures occur with probability, and each one has a defined probability distribution function with specific parameters. Fig. 2 shows the PM and inspection in different periods.

TL=H

PM Total repairs at the end Failure type II of period

To bring this model closer to the real world, we consider different scenarios. Product demand p in each period t is not constant. Therefore, we define scenarios in high, medium, and low-level demand to examine the percentage of customer satisfaction in each case. Meanwhile, in addition to the fact that demand is different in each scenario, the volume of products that should be run by the system without an underdeveloped order also varies in different scenarios. For this purpose, the number of inspections and failures that occur in each scenario is different, and the system seeks to maximize the final profit of all these scenarios.

It should be noted that the demand of main customers is only supplied with standard products, and non-standard products are sold at second-party prices in the free market and do not have the ability to respond to customers in each period. Fig. 3 shows an overview of different scenarios based on demand and response rates.

The first scenario: d=D Demand High demand-strong accountability b<0.2D Order overdue

To describe the model, the mathematical model is first described with respect to the objective function and the constraints, and then the objective and constraints will be discussed. Parameters and variables of the mathematical model are defined as below:

Indices and Parameters

Time horizon Period duration Number of durations Period indicator Product number Product indicator Number of PM levels PM levels indicator Indicator of inspection interval and PM Minimum repair cost Minimum repair time Full repair cost Once inspection cost PM factor Resettlement cost Cumulative Distribution Function of shift time (Type I Failure) Distribution Function of shift time (Type I Failure) Risk function of Type I failure Cumulative Distribution Function of Type II failure time Distribution Function of Type II failure time Risk function of Type II failure Weibull Distribution Parameters of Failure Type I Weibull Distribution Parameters of Failure Type II The length of inspection interval k in period t Demand of product p in period t for scenario s Production process cost for product p in period t Delayed order cost for product p in period t Maintenance cost for product p in period t System startup cost for product p in period t Separation unit cost for product p Production rate for product p Non-standard product rate in an out-of-control system Price of standard product p during period t Price of non-standard product p during period t PM cost at level q

PM time at level q

The considered objective Function is a profit maximization function that derives from the diversity of sales revenue and operating costs. Operating costs include production costs, maintenance costs, startup costs, order-loss costs, inspection costs, PM costs, redevelopment costs, the cost of separating standard and non-standard products, and failure type II. Product sales include standard and non-standard products and occur as machine age increases. The objective function is as follows:

(Obj.)

Constraints

(10)

(11) (12)

(13)

Bt p,s ≤ qps × dp t,s, (14) (15) (16) (17) (18) (19)

Constraints (1) and (2) show the standard and non-standard inventory of the product p during period t in scenario s, resulting from the previous period inventory plus products of this period minus the sales of this period. Constraints (4) and (5) indicate the volume of product p generated during period t based on system availability time and the production rate of product p. Constraint (6) determines that the machine age at the beginning of each period t is zero. Constraints (7) and (8) compute the age-related relationships at the beginning and end of each period after PM in each scenario s. Constraint (9) shows the number of type II failures in interval k in period t in scenario s. Constraint (10) calculates the system's availability time to produce products. Constraint (11) shows the probability of a shift in interval k of period t. Constraint (12) shows the number of non-standard production based on system age and availability in each scenario. Constraint (13) calculates the cost of separating standard and non-standard products according to system age and availability and production volume. Constraint (14) shows the maximum allowed sales in each scenario. Constraints (15)-(19) express the type of variables.

In addition, the probability distribution considered for Type I and Type II failure is the Weibull distribution. The proposed mathematical model and existing variables introduce a Mixed-integer nonlinear Programming model. This model is solved in small dimensions in GAM software using the solver MNLP, and the results are presented. Given the model's complexity, the exact solution in GAMS provides a proper upper limit for solving in smaller dimensions.

To solve the problem, two approaches are proposed.

I. A step-by-step approach that allows discussion of the objective function and obtaining the amount of profit by setting different values for the number and type of PM.

Due to the complexity of the model and the probable states available for solving the model, a method is proposed to reduce the complexity related to the nonlinearity of the model. This method consists of four basic steps, as shown in Fig. 4.

Step 1. Select the probability distributions for Types I and II

Step 2. Determine the number of inspections and PM actions to undertake in each period

Step 3. For each combination of the PM decision variables, evaluate the expected profit, and solve the resulting LSP

Step 4. Compare the resulting optimal solutions of all the optimization models solved in step 3, and select the final optimal solution

Step 1. Probability distributions of Type I and II failures are indicated. In this paper, the probability of failure follows a Weibull distribution, which increases the probability of failure and produces non-standard parts as the functionality and age of the machine increase. The probability density function of type I failure is as in Eqs. (20)-(25).

f(u) = λφuφ−1e−λuφ, u > 0, φ ≥ 1, λ > 0. (20)

where λ and φ are Weibull parameters.

Its corresponding cumulative distribution function is:

F(t) = 1 − e−λuφ, u > 0, φ ≥ 1, λ > 0. (21)

Its hazard rate is equal to:

r(u) = λφuφ−1, u > 0, φ ≥ 1, λ > 0. (22)

The probability density function of Type II failures is also Weibull with parameters θ and ρ as follows:

g(u) = θρuρ−1e−θuρ, u > 0, ρ ≥ 1, θ > 0. (23)

Its corresponding cumulative distribution function is:

G(u) = 1 − e−θuρ, u > 0, ρ ≥ 1, θ > 0. (24)

Its hazard rate is equal to:

ζ(u) = θρuρ−1, u > 0, ρ ≥ 1, θ > 0. (25)

Step 2. After determining the type of failure distributions, the number of inspections required in each period and the type of PM activity are specified. The specified inspections are carried out during the period and are performed at given intervals to simplify the problem. The length of the time interval required for inspection is obtained from the following formula:

It = , t = 1, … , T, k = 1, … , mt. (26)

mt + 1 Step 3. After determining failure rates, in this step equations 6 to 9 are solved for each PM level. From the solution of these equations, the following result is obtained, which leads to an estimation of the system age at the beginning and the end of each inspection period.

[θ × (yt k)ρ × TMR] + yt k = wt k + lk t − (TPM × Mt k) + [θ × (wt k)ρ × TMR]. (27)

At this stage, the start and end age and failure distributions can be obtained for different types of PM modes. After placing the specified values, decision variables of inventory amount, shortage amount, production amount of standard and non-standard products, and sales amount of each product should be calculated. The nonlinear model transforms into a simple linear model that can be solved with any software.

Step 4. Finally, compare results from different PM modes to determine the optimal mode for the system. II. The genetic metaheuristic method uses different PM modes, inspection, and production volume as elements of the population.

A genetic algorithm as a metaheuristic implementation needs to decide how to represent solutions in an efficient way in the search space [15], [16]. Representation should be easy to decode and calculate to reduce the run time of the algorithm. In the considered problem, the first part of deriving the answer includes variables related to production volume in each period, the second part includes startup variables, and the third part includes PM types in each inspection period. Fig 5 shows initial answer determining.

After setting parameters and constructing the initial response, the genetic parameters are determined. Final parameters are chosen based on the problem structure and several consecutive solving problems.

The GA is a population-based metaheuristic, so it needs an initial population of individuals to start. This population can either be produced randomly or based on some other algorithms. In this study, each chromosome of the initial population is generated randomly [16], [17].

After generating the initial population and new generations, the value of the objective function is calculated and considered as the fitness value of each solution (fv(x)). In this way, the solutions are sorted according to the objective value.

The parent selection method determines how to choose the chromosomes from the current population for the crossing procedure. In this study, the parents are selected based on the roulette-wheel rule. After calculation, the fitness value (fv(x)) for each solution, the roulette-wheel operation selects the parents. In general, the crossover operator generates a new chromosome (offspring) with higher fitness. After that, the mutation operator is applied to maintain population diversity across successive generations and to further exploit the solution space. We tested different crossover methods and found that two-point crossover (2PX) performed better than the others. Here, three different operators, including the Swap operator, Reversion operator, and Insertion operator, are applied randomly in each iteration to enhance the performance of the proposed algorithm.

The new generation Pt+1 is selected from the parents and offspring, not based solely on fitness values. In each generation of the proposed GA, a new population is generated by replacing the second 50% of the current population with the first 50% of the new population (offspring). Based on this Procedure, there is no guarantee that the inserted best solutions are always better than the worst solutions in the current population. This approach encourages the algorithm to search various regions and avoids premature convergence during the algorithm's progress. The flow chart of the proposed GA is shown in Fig. 6.

In this paper, the initial population is 10, mutation size is 2, and the number of elements for crossover is 18. Stop conditions are investigated in three modes: maximum iterations equal to 100, earning a negative profit, and having 20 consecutive moves without improvement.

After each iteration, the profit from each population is computed. These populations are arranged by performance improvement, and the 3 populations with the least profit are replaced using mutation and crossover operators. For the crossover operator, the first two populations with the best profit improvement are selected, and a number of their elements are moved randomly. This operator produces two children, which are placed in the eighth and ninth population positions, and creates the third child using the mutation operator. This operator randomly modifies several members and tries to create a different child. This child will replace the tenth population.

The best answer for PM types in the three inspection periods is (1 1 4, 4 1 3, 4 1 4), with a profit function of 4.9275. If the production quantity in each period is kept constant for each product (4 4 4.4 4 4.4 4 4), it has the worst profit function because no PM occurs and the number of failures and non-standard production increases. Fig. 7 presents a sample answer of the genetic algorithm:

The next, and very important, discussion is the number of inspections required in the system. This number of inspections is investigated based on the cost of each inspection, the PM of each period, and failure costs. According to parameters specified in the model, profit increases as the number of inspections increases. Therefore, based on the site in the industry, the number of necessary inspections is selected. Tables 2 and 3 present the input parameters and production data from Beheshti Fakher et al. [11].

Using the values in Tables 2 and 3, Fig. 7 shows the profit function by number of inspections. The highest profit is obtained when the number of inspections is between 16 and 18.

In this paper, we proposed a profit-maximizing model integrated with quality, production, and maintenance decisions. Despite many existing models, this model considers two types of failures, I and II, for the system’s failure probability. Therefore, production involves two types of standard and non-standard products, and an inspection factor is used to identify non-standard products manufactured in type I failure. In the case of type II failure, the system stops, and there is no production. In addition, two types of repairs are applied to the system. One is the complete repair at the end of each period, and the other is an incomplete repair performed when Type II occurs. These two types of failure are probable, and therefore, statistical distributions are used in this model. Another characteristic of this research, which is less likely to be found in other studies, is that it considers different scenarios. Demand and response rates vary in each period due to different real-world situations, so three scenarios are considered. Because this problem is NP-hard, a metaheuristic method is used to solve it. The method used is genetic, and the software used is GAMS.

Given the highest profit (objective function), the best response is obtained in three inspection periods. On the other hand, the worst case that gives the lowest profit is specified, and this lowest profit is because no PM occurs, which increases the number of failures and non-standard products manufactured.

Another important issue in this study is determining the number of inspections and PMs needed to maximize profits, since inspections and PMs incur costs. Instead, carrying out inspections and PM activities will reduce or even eliminate non-standard products. The results show that, according to the parameters specified in the paper’s model, increasing the number of inspections increases profit. Therefore, the number of necessary inspections is selected based on the industry site.

For developing the topic used in this paper, several studies can be done, including:

I. Extending the proposed model in terms of several machines.

II. Using different metaheuristic methods, comparing results, and identifying the best method for solving this problem.

III. Defining the objective function to minimize costs, and consequently changing the constraints.

IV. Determining optimal buffer level as one of the problem variables due to the difference in demand and response rate in different scenarios.

The authors sincerely appreciate everyone who contributed to completing this research.

This research was conducted without financial support from any public, private, or non-profit organization.

The data used in this study are available from the corresponding author upon reasonable request.