Impact of Unreliable Subcontracting on Production and Maintenance Planning Considering Quality Decline

Abstract

Manufacturing systems face several disturbances during production, such as sudden failures, defects, and unreliable subcontractors that reduce their production capacity. Currently, subcontracting represents an efficient alternative to support production decisions. The novelty of the study was the development of a new integrated model that properly coordinates production, subcontracting, and maintenances strategies in the context of stochastic uncertainty, quality deterioration, and random subcontracting availability. Such a set of characteristics has not been addressed before in the literature. A simulation–optimization approach was proposed to address such a stochastic model. A numerical case study was performed as an illustration of the approach and a comprehensive sensitivity analysis was performed to analyze the impact of several costs. Furthermore, the effect of the availability of the subcontractor and the producer was analyzed. The main finding of the study showed that the integrated model led to significant economic cost savings compared to other approaches that address such policies in isolation. The results also indicated that quality deterioration had a strong impact on the subcontracting rate and that the proposed joint control policy adequately coordinated these three key functions. The level of subcontracting participation was directly defined by its availability and the subcontracting cost.

1. Introduction

This introduction section is divided into three subsections. First, we highlight the motivation of the research, we present the reasons that justify the importance of the developed model, and we identify the industrial needs that motivated this study. In the second subsection, an extensive review of the literature is conducted, where we identify the recent research trends in the field of manufacturing systems. Moreover, the research gaps are identified. In the third subsection, the contribution of this study is presented, where we discuss the research gaps that we aimed to fill with the proposed integrated model. Additionally, we present the general resolution approach adopted in this study.

1.1. Motivation

Nowadays, industrial companies face several issues that jeopardize their economic performance. It is common in real production to have random failures, ineffective maintenance strategies, quality decline, etc., which have a strong impact on the general performance of such production systems. Therefore effective production and maintenance strategies are key to mitigating the effects of such random disturbances. Additionally, a third key function defined by product quality has become a critical dimension in recent years and has been gradually incorporated in the field of production systems. Since the determination of effective quality strategies is strongly related to production and maintenance, recently, the development of integrated models incorporating the three key functions of production, quality, and maintenance has provided superior results using traditional models that address these functions separately. However, more research is needed in this domain since there are still several factors that have not been included in several models and such factors have a negative effect on the system’s performance.

For example, gradual deterioration is a very common phenomenon with negative effects in automotive companies, semiconductor industries, manufacturing, etc. The production cost and the performance of production systems depend on the level of system deterioration. In manufacturing, a deterioration process certainly leads to short tool life, frequent setups, and increases in the total cost. Nevertheless, despite the negative effects of the deterioration process, there are still gaps that must be studied in the context of the determination of efficient production, quality, and maintenance strategies. In this uncertain context, companies need effective countermeasures to reach their economic goals; in recent years, subcontracting represents an attractive solution that is extensively used in production. Traditional models, including subcontracting, were developed based on simplistic assumptions that are not representative of the reality of industrial companies. For instance, in many studies, it was assumed that the subcontractor was always available. However, in real production, it is common to observe frequent random delays from the subcontractor. Indeed, stochastic subcontracting availability has a strong effect on the joint control policy and needs to be studied in detail.

From the above paragraphs, it is clear that more research is necessary since there is a need to determine advanced scheduling methods for effective production subcontracting and maintenance planning considering quality deterioration with the aim to foster the profitability of companies.

1.2. Literature Review

When analyzing the literature on the integration of production and maintenance strategies, we observed that several works were conducted, which makes sense given that current production systems have become more complex. For instance, Kouedeu et al. [1] studied a hierarchical decision-making process, where, in the first level, they determined the mean time to failure, and in the second level, they determined the production and maintenance rates. They assumed imperfect maintenance actions. Later, Dellagi et al. [2] studied production and maintenance planning for a system that satisfied random demand by considering a required service level. They considered a periodic preventive maintenance option, where, at failure, a minimal amount of repair was conducted. A subsequent work conducted by Fitouhi et al. [3] treated a production system that could degrade into several discrete states, which indicated different performance states. They analyzed the trade-off of a preventive maintenance policy by considering their impact on the production rate and the buffer. Polotski et al. [4] considered a hybrid system, where the machine was capable of manufacturing activities, and remanufacturing, where returned products were used in production. In their study, the system was prone to deterioration and they determined production, remanufacturing, and preventive maintenance rates. Rivera-Gómez et al. [5] studied production and repair strategies for a system subject to deterioration. They modeled quality deterioration with an aging process and several operational states. They also determined the repair efficiency to partially restore the machine after failure. More recently, Hajej and Rezg [6] developed a lot-sizing model in terms of energy, service level, and capacity constraints. They determined maintenance actions by taking into account the impact of the production rates on the level of deterioration and the increase in the failure rate. Despite the relevance of the discussed studies, more research is needed since the complex trade-offs between quality issues and the determination of production maintenance strategies were disregarded. However, companies must manage these three key functions simultaneously for economic success.

There is a vast literature emerging that investigated the interaction of production, quality, and maintenance strategies. For example, Bouslah et al. [7] determined the size of production lots and the values of the parameters of an acceptance sampling plan. They also defined overhaul rates for a deteriorating system considering an outgoing quality constraint. Fakher et al. [8] integrated production planning and imperfect maintenance, where process inspections were used to determine the system conditions. Their system comprised multiple products and multiple machines that deteriorated over time, producing more defects. Cheng et al. [9] developed a joint strategy of production and quality control for a degrading system, where preventive maintenance was conducted after inspection if the deterioration level surpassed a threshold. The strategy of quality control was based on inspecting all the units. Abubakar et al. [10] suggested a plan for production and maintenance for a system prone to deterioration, where the quality of the products was monitored with statistical process control. Their model determined the sample size, the interval for sampling, and the limits of quality control. The deterioration degree of their system was influenced by the production rate and time. Ait-El-Cadi et al. [11] proposed a model for the simultaneous control of production planning and maintenance scheduling considering a dynamic sampling inspection. Their quality policy adjusted the sample size according to the degree of deterioration. They assumed that the deterioration process had effects on the failure rate and the failure intensity. Hajej et al. [12] highlighted the advantage of using a dynamic sampling strategy for quality control for cases of progressive deterioration. In their model, the rates of production and maintenance were determined for the case of a system subject to quality deterioration. They also considered a service level and an outgoing quality constraint. As can be noted from the above-mentioned papers, the emphasis in this literature is that integrating production, maintenance, and quality control is critical in modern production systems. However, there is a limited amount of literature that considered subcontracting strategies in the integration of these three key functions. Subcontracting was demonstrated to be an effective alternative for deteriorating systems that face progressive capacity reduction.

In the literature, several researchers developed integrated models to study the strong link between production and subcontracting planning, as in Assid et al. [13], who addressed production and subcontracting strategies for a multiple-production facility with different capacities. In the optimization, they included a customer satisfaction constraint. Ben-Salem et al. [14] analyzed production and subcontracting plans for a manufacturing system that generated harmful emissions to the environment. In their model, items were available from a subcontractor at a higher cost. Such a subcontractor possessed more efficient technology but it had random availability. Haoues et al. [15] studied a two-echelon supply chain comprising multiple outsourcers and multiple subcontractors. Their model served to determine optimal production rates, periods for maintenance, and the determination of appropriate outsourcing options and optimal outsourcing quantities. Ayed et al. [16] treated an industrial problem of a production system that satisfied random demand. In their model, the subcontractor was available with a stochastic service level. Their system was prone to degradation and such a process had an impact on its availability. Rivera-Gómez et al. [17] investigated the problem of a deteriorating system with effects on quality and reliability with the aim to determine production and subcontracting strategies; in this study, preventive maintenance could be conducted to mitigate the effects of the deterioration process. In Kammoun et al. [18], subcontracting was allowed in a dynamic lot-sizing problem for a system that satisfied random demand and had requirements in the service level and consumption of energy. In their maintenance strategy, they regrouped the maintenance plans of several machines into only one plan. From these studies, it is evident that in the pursuit of industrial competitiveness, subcontracting represents an alternative to increasing the service level. However, the discussed models have several drawbacks, for instance, the effect of progressive degradation on the strategies of production and subcontracting was rarely considered. Indeed, the influence of such degradation on the control policy must be assessed and that is the object of this study.

In the real-life industrial field, production systems experience progressive deterioration that reduces their production capacity. Great interest has been dedicated to the field of deteriorating systems, for instance, Boudhar et al. [19] analyzed the case of a system subject to deterioration. They determined a maintenance policy that defined the inspection dates and the quality of the spare parts to be installed in the machine during maintenance. Martinod et al. [20] sought to compare several maintenance strategies, such as the age-based and periodic type for a production system with multiple components, taking into account the fact that such components are subject to deterioration. Ouaret et al. [21] investigated a production system prone to an aging process. The effect of such deterioration was mainly reflected in the rate of failures and also in the defect rate. The system faced random demand and replacement was available to partially reduce the impact of deterioration. In another work, Polotski et al. [22] developed a production policy considering two machines, the first one uses raw materials for manufacturing and the second machine uses return products for remanufacturing. They considered that product demand and return rates varies in function of time due to seasonal market behavior. An analytical model was presented in Dellagi et al. [23] to determine a maintenance strategy and the management of spare parts. Their system was subject to an increasing deterioration rate. They considered the influence of varying the production rate and the degree of degradation of the system and integrated the optimal quantity of spare parts to order for conducting maintenance actions. Magnanini and Tolio [24] proposed the integration of production and maintenance strategies, where the control policy used buffer thresholds to regulate the production rate and they considered the dependency of the thresholds on the deterioration condition. Their policy also included switching points that activated preventive maintenance. In the industrial environment, deterioration has a strong relationship with production and maintenance policies, as highlighted in the presented studies. Nevertheless, currently, there is a lack of research regarding the interaction of deterioration with subcontracting and quality issues since the focus has moved to a system perspective, where high levels of coordination are needed to achieve a required balance between such strategies and thus provide economic success. Indeed, such interactions and trade-offs are complex issues to be managed.

Mathematical techniques and simulation expertise are useful alternatives for analyzing the arising complexity of manufacturing models. For instance, Hosseini and Tan [25] presented an analytical simulation approach to analyze the performance of a continuous production system; their method provided solutions considerably faster than solely evaluating the system dynamics at critical times. They determined time instances of the trajectory of the buffer, stock level dynamics, and changing flow rates. Guiras et al. [26] examined different optimization algorithms and simulations to determine optimal plans for production and maintenance. They also analyzed the integration of imperfect maintenance with the consideration of returned products. Abdolmaleki et al. [27] used a simulation model to manage the production and maintenance strategies of a transported material network system. They considered that the machines were subject to deterioration with failures and their machine increased its deterioration level with each repair conducted. Rivera-Gómez et al. [28] studied production and maintenance strategies in the context of deterioration. They included a dynamic sampling inspection policy to ensure the satisfaction of an outgoing quality constraint. Further simulation research was proposed by Assid et al. [29], who determined production and remanufacturing rates. They assumed that returns were categorized into two categories based on their quality condition and time for reprocessing. They also defined the switching policy for the remanufacturing modes of the facility. Ait-El-Cadi et al. [30] proposed a simulation model to analyze the effects of production and preventive maintenance rates of a production system that was prone to aging on reliability and quality. They included the case of imperfect inspection activities, where a non-negligible duration was assumed for inspection and rectification activities. Despite the relevance of the presented studies, more research is needed since these studies were based on simple and unrealistic assumptions that did not consider the influence of degradation on the law of control. A similar argument shows that such models also disregarded the interdependence between random subcontracting availability and control actions.

Additionally, recent works in the area of production management and maintenance planning applied innovative methods based on metaheuristics and artificial intelligence (AI). Fathollahi-Fard et al. [31] studied a production-scheduling system that integrated air transportation. They considered a capacitated transportation system with a time window and they did not permit idle time. Their model was solved through nature-inspired metaheuristics. Jian et al. [32] formulated an optimization model to design the assembly system configuration, where their model defined the subassembly planning and the task assignments for uncertain product evolution, achieving workload balancing. Villalonga et al. [33] proposed an automated decision-making process using a fuzzy inference system to predict the condition of the asset and appropriate re-scheduling the production of a cyber-physical system. Chen et al. [34] studied the economic dependence of equipment cost and modeled the relationship between the effective service age and the reliability of components through a Weibull distribution. They established a selective maintenance model of a complex system with multiple states by considering economic dependence. Fathollahi-Fard et al. [35] proposed a multi-objective mixed-integer linear model to minimize the total energy consumption due to production. They considered social factors related to job opportunities and lost working days. They developed a sustainable flow shop scheduling problem with the assumption of different production centers and technologies on machines that had a strong impact on environmental and social criteria. Gholizadeh et al. [36] presented an optimization model for the problem of flexible flowshop scheduling for a waste-to-energy system. They proposed a preventive maintenance policy to determine an optimal sequence for processing tasks and minimizing delays. They considered the work processing time as being uncertain. Villalonga et al. [37] introduced a cloud-to-edges-based approach for a cyber-physical system to increase its smartness and the autonomy for monitoring and controlling the behavior of such a system at the shop-flow level. They evaluated their proposed solution with a pilot line. As can be noted from the presented papers, the authors only focused on the production or the production–maintenance relationship. The interconnections between production–quality–subcontracting–maintenance were not considered. Furthermore, a decision maker must take into account the hidden costs of AI methods due to IT infrastructures investments, the salary of data science professionals, etc., given that, in some cases, it is more convenient in terms of cost to use a traditional production–maintenance approach rather than expensive AI solutions, as indicated in Florian et al. [38].

To summarize,

Table 1. Literature review.

	Production Planning	Quality Issues	Maintenance Planning	Subcontracting Strategy	Unreliable Subcontracting	Deterioration	Quality Deterioration	Development of a New Policy	Performance Metrics Analysis
(a) Production–maintenance strategies
Kouedeu et al. [1]	✓		✓			✓		✓
Dellagi et al. [2]	✓		✓			✓		✓
Fitouhi et al. [3]	✓		✓			✓			✓
Polotski et al. [4]	✓		✓			✓		✓
Rivera-Gómez et al. [5]	✓		✓			✓	✓	✓
Hajej and Rezg [6]	✓		✓					✓
(b) Production–quality–maintenance strategies
Bouslah et al. [7]	✓	✓	✓			✓	✓
Fakher et al. [8]	✓	✓	✓			✓			✓
Cheng et al. [9]	✓	✓	✓			✓	✓	✓
Abubakar et al. [10]	✓	✓	✓			✓		✓
Ait-El-Cadi et al. [11]	✓	✓	✓			✓	✓	✓
Hajej et al. [12]	✓	✓	✓			✓	✓	✓
(c) Production and subcontracting planning
Assid et al. [13]	✓			✓				✓
Ben-Salem et al. [14]	✓		✓	✓	✓			✓
Haoues et al. [15]	✓			✓		✓		✓
Ayed et al. [16]	✓			✓		✓		✓
Rivera-Gómez et al. [17]	✓		✓	✓		✓	✓	✓
Kammoun et al. [18]	✓		✓	✓				✓
(d) Deteriorating systems
Boudhar et al. [19]		✓	✓			✓			✓
Martinod et al. [20]			✓			✓			✓
Ouaret et al. [21]	✓		✓			✓	✓	✓
Polotski et al. [22]	✓							✓
Dellagi et al. [23]	✓		✓			✓			✓
Magnanini and Tolio [24]	✓		✓			✓			✓
(e) Simulation approach
Hosseini and Tan [25]	✓	✓	✓						✓
Guiras et al. [26]	✓		✓						✓
Abdolmaleki et al. [27]	✓		✓			✓			✓
Rivera-Gómez et al. [28]	✓	✓	✓				✓	✓
Assid et al. [29]	✓	✓	✓						✓
Ait-El-Cadi et al. [30]	✓		✓			✓	✓	✓
(f) Additional issues
Fathollahi-Fard et al. [31]	✓								✓
Jian et al. [32]	✓							✓
Villalonga et al. [33]	✓								✓
Chen et al. [34]			✓						✓
Fathollahi-Fard et al. [35]	✓							✓
Gholizadeh et al. [36]			✓					✓
Villalonga et al. [37]	✓		✓						✓
The proposed model	✓	✓	✓	✓	✓	✓	✓	✓	✓

serves to identify the research gaps that the present study aimed to fill. In this table, the rows classify the studies presented in the literature review in five main areas, and the columns present the key factors investigated in these studies.

The discussion of Table 1 is as follows: we analyzed in detail the studies of this table to identify their main contribution and define the key parameters presented in the columns. We note that the key parameters presented in the columns of Table 1 follow the evolution of the field of manufacturing systems. For instance, from the presented studies, it is clear that the first research trend focused on the development of production–maintenance strategies, as observed in the studies of Kouedeu et al. [1], Dellagi et al. [2], Fitouhi et al. [3], etc. Then, since 2013, the quality trend appeared, and thus, several studies included quality issues in their formulation, as in Bouslah et al. [7], Fakher et al. [8], Cheng et al. [9], etc. They aimed to jointly study production–quality–maintenance strategies. We note from the literature review that subcontracting strategies were only considered with production–maintenance models, as in Assid et al. [13], Ben-Salem et al. [14], Haoues et al. [15], etc. It is apparent to see that quality control was disregarded in such subcontracting models. Another common industrial phenomenon that has attracted the attention of several researchers is deterioration, as in the studies of Boudhar et al. [19], Martinod et al. [20], Ouaret et al. [21], etc. However, we observed that deterioration, mainly quality deterioration, was only studied with production–maintenance strategies. The influence of quality control or subcontracting has not been addressed in deterioration models. The same observation applied to the studies that adopted a simulation approach, such as Hosseini and Tan [25], Guiras et al. [26], Abdolmaleki et al. [27], etc. Therefore, based on this discussion, we concluded that there is a current need to develop an integrated model that considers the key functions of production, subcontracting, quality, and maintenance in a stochastic and quality deteriorating context.

1.3. Contribution

To summarize, the contribution of this research is the deep analysis of an integrated model that took into account the main functions of production planning, quality control, and maintenance, where unreliable subcontracting strategies were available to support demand satisfaction. The proposed model defined appropriate countermeasures by considering the level of inventory, the evolution of defectives, and the dynamics of failures and repairs, and a continuous-time stochastic process was used to model the underlying dynamics of the production system. This research aimed to extend previous results of the conventional system presented in Assid et al. [13] and mainly in the studies of Rivera-Gómez et al. [17,18,19,20,21,22,23,24,25,26,27,28], which investigated the effective coordination of production and subcontracting policies in the case of progressive quality decline. The mathematical complexity and the stochastic behavior of the proposed production system imposed several challenges, which were overcome with the use of a methodology based on several techniques, such as a semi-Markov process, stochastic optimal control, and simulation-based optimization. In our view, this research has three major contributions that make a distinction concerning the existing literature:

(i)

The influence of quality decline on the production, subcontracting, and maintenance control rule in a dynamic and stochastic context, which remain important issues in the field of manufacturing systems, was studied.

(ii)

Several production and maintenance models were developed by considering subcontracting in the context of deterioration, but they were limited to the case where subcontracting was always reliable. The key observation about the present model was that we aimed to extend such an assumption to study the impact of unreliable subcontracting.

(iii)

We proposed a novel stochastic optimal control model that is characterized by the use of the history of the deterioration process, leading to a semi-Markov model that effectively coordinated production and unreliable subcontracting.

The proposed model was extensively examined with a performance analysis of a numerical example and a comprehensive sensitivity analysis. The economic advantage of the proposed integrated model was highlighted in a comparative study. The next sections present the assumptions, the problem description, the control, and the simulation model used to reach the objectives proposed by the three contributions.

The rest of this paper is structured as follows: The assumptions; the problem description; and the definitions of the production, subcontracting, and maintenance policies are presented in Section 2. The resolution approach is detailed in Section 3. The simulation model used in the optimization phase is introduced and validated in Section 4. A numerical case study is presented in Section 5. An extensive sensitivity analysis and a comparative study are discussed in Section 6. An indifference curve that was used to analyze the trade-off between internal production and subcontracting is also presented in Section 6. Finally, conclusions are given in Section 7.

2. Problem Description and Assumptions

In order to clarify the control model under investigation, this section is divided into three subsections. First, the problem description is presented, where we describe in detail the set of features of the manufacturing system through a conceptual diagram. In the second subsection, the list of assumptions that support the proposed model is presented. In the third subsection, the control model is formulated analytically, where its dynamics, decision variables, control policies, and objective costs are identified.

2.1. Problem Description

Before beginning the description of the conceptual diagram presented in Figure 1, we first state that the proposed production system comprised an unreliable production unit that was prone to random failures and repairs and produced a single product. Assuming that the machine was subject to random failures and progressive deterioration had a strong impact on the rate of defects, defining a quality deterioration phenomenon as shown in Figure 1. Minimal repairs were available to the decision maker to overcome sudden failures. However, the level of deterioration of the machine was only rejuvenated with major maintenance, which returned the rate of defects to the initial conditions. The manufacturing system used inventory as protection against the shortages caused by the numerous breakdowns of the machine. This inventory level was a decision variable of the model that must be optimized. Another available strategy in the case of high levels of deterioration was subcontracting, which satisfied a fraction of the demand. It was assumed that the manufacturing system was not capable of satisfying product demand at high levels of deterioration. Furthermore, it was conjectured that the subcontractor was unreliable because it had several companies to serve simultaneously, and also because it was subject to technical problems that limited its production responsiveness. Nevertheless, the minimum amount of availability that the subcontractor must comply with was stipulated in a contract. The advantage of subcontracting is that it provides units free of defects. In this stochastic context, the main objective of the study was to develop an innovative integral model that simultaneously determined the optimal production–subcontracting plan and major maintenance strategies that minimized the total cost in the case of quality deterioration and unreliable subcontracting. In such an optimization, several costs were considered, such as inventory, backlog, defects, major maintenance, minimal repair, and subcontracting costs. The configuration of the whole production system is presented in Figure 1.

2.2. Assumptions

The stochastic system analyzed in this research was based on the following assumptions:

• A1. In the beginning, a new production unit was operational and the influence of deterioration was minimal, as in Bouslak et al. [7].
• A2. The demand was known and constant during the considered production horizon, as in Kouedeu et al. [1].
• A3. The production unit experienced increasing wear with usage and it deteriorated. The deterioration was modeled with the use of an increasing function, which varied with the production rate, as in Ait-El-Cadi [11].
• A4. The deterioration of the production system had a negative effect on the number of defects generated, as in Rivera-Gomez et al. [5].
• A5. At failure, minimal maintenance was performed, where the condition of the machine was left in as-bad-as-old (ABAO) conditions, as in Assid et al. [29].
• A6. Major maintenance was available and was a perfect repair that rejuvenated the production unit to as-good-as-new (AGAN) conditions, as in Rivera-Gomez et al. [28].
• A7. Shortages may be observed if the time required to perform repairs and major maintenance took longer than the time to deplete the stock level used as protection, as in Polotski et al. [4].
• A8. Subcontracting was available to support internal production and was an alternative to satisfy product demand, as in Ben-Salem et al. [14].
• A9. The subcontractor was unreliable due to several disturbances caused by random failures, periods where it was busy serving other customers, other random events, etc.; as such, the subcontractor exhibited random availability. This assumption has not been used in the context of production–quality–maintenance joint optimization.

The notations used in the model are presented in Appendix A.

2.3. Control Model Formulation

To formulate the problem, for any specific time $t$, the production unit and the subcontractor modes were described using a stochastic process $\{{\alpha }_1\left(t\right), t>0\}\in {\Omega }_1=\left\{1,2,3\right\}$ and $\{{\alpha }_2\left(t\right), t>0\}\in {\Omega }_2=\left\{1,2\right\}$, respectively. In particular, we note that when ${\alpha }_1\left(t\right)=1$, the production system was operational and generating defects at a rate of $\beta$. Further, when ${\alpha }_1\left(t\right)=2$, the production system was in failure mode, where minimal maintenance was performed, which left the machine in an ABAO condition before the occurrence of the failure. When ${\alpha }_1\left(t\right)=3$, major maintenance was undertaken, which completely mitigated the effects of the deterioration process since it restored the machine to an AGAN condition. Regarding the unavailability of the subcontractor, contrary to the assumption made in the literature, we assumed that the contractor was not always available to supply units due to technical disruptions; therefore, to make things simple, we proposed Markov dynamics for ${\alpha }_2\left(t\right)$. More precisely, when ${\alpha }_2\left(t\right)=1$, the subcontractor was available and they send flawless units to the production system, and when ${\alpha }_2\left(t\right)=2$, subcontracting was not available, and thus, it stopped its operation due to the conduction of maintenance activities.

The dynamics of the level of inventory was given by the next differential equation:

$$\frac{\partial x\left(t\right)}{\partial \left(t\right)}=\frac{u_1\left(t\right)}{1-\beta \left(a\right)}+u_2\left(t\right)-d, x\left(0\right)=x_0$$

(1)

where $u_1\left(t\right)$ is the production rate of the machine at time $t$, $u_2\left(t\right)$ is the subcontracting rate, $\beta \left(a\right)$ is the rate of defects at age $a$, and $d$ is the demand rate. The production rate at any time $t$ satisfied $0\le u_1\left(t\right)\le u_{max}$, where $u_{max}$ is the maximum production rate for internal production. We assumed that the subcontractor satisfied only a fraction $k_1$ of the customer demand to be profitable for the company, as presented in the following equation:

$$u_2\left(t\right)=k_1·d$$

(2)

For the current problem, we conjectured that the production unit experienced progressive deterioration. We modeled this condition using the age of the production unit as a key metric of its amount of deterioration. Furthermore, by assuming that the age of the machine at time $t$ implied an increasing function calculated from its production rate, then the cumulative age was defined as follows:

$$\frac{\partial a\left(t\right)}{\partial \left(t\right)}={\eta }_o·u_1\left(t\right)$$

(3)

$$a\left(T\right)=0$$

(4)

where ${\eta }_o$ is a given constant and $T$ indicates the last restart time of the machine. Equation (3) uses the production rate to determine the age of the production unit. Furthermore, of significant importance is the fact that the model incorporated the influence of a degradation process in its formulation. In particular, we state that the defect rate was increased by the pace of deterioration of the machine, as in Colledani and Tolio [39], as defined in the next equation:

$$\beta \left(a\right)=b_0+b_1\left[e^{-{\eta }_{1}a{\left(t\right)}^{{\eta }_2}}\right]$$

(5)

where $b_0$ is the defect rate in an AGAN condition, $b_1$ defines the upper limit for the rate of defects, and ${\eta }_1$ and ${\eta }_2$ are positive constants. The production unit was unreliable and exhibited Markov dynamics due to random failures and repairs; hence, it was not always operational. The availability of the production unit could be calculated with the next equation:

$${\pi }_i^{prod}·Q(·)=0 y {{\displaystyle \sum }}_{i=1}^3{\pi }_i^{prod}=1$$

(6)

where $Q(·)$ refers to the transition rate matrix that is defined by the transition rates $q_{ij}$ between mode $i$ and mode $j$ of the production unit. Additionally, ${\pi }_i^{prod}$ is the fraction of time that the production unit is in mode $i$. The resolution of Expression (6) defined the availability of the production unit in the operational mode; in this case, such a resolution led to

$${\pi }_1^{prod}=\frac{1}{1+\raisebox{1ex}{$q_{12}$}\!\left/ \!\raisebox{-1ex}{$q_{21}$}\right.+\raisebox{1ex}{$q_{13}$}\!\left/ \!\raisebox{-1ex}{$q_{31}$}\right.}$$

(7)

The major uncertainty in the model was that the subcontractor was unreliable, and thus, exhibited random unavailability due to sudden failures and maintenance activities. Hence, its availability in the operational mode could be defined with an expression similar to Equation (6); in this case, we used

$${\pi }_1^{sub}=\frac{1}{1+\raisebox{1ex}{$q_{12}^{sub}$}\!\left/ \!\raisebox{-1ex}{$q_{21}^{sub}$}\right.}$$

(8)

where $q_{ij}^{sub}$ refers to the transition rate between mode $i$ and mode $j$ of the subcontractor. Due to the influence of the degradation process, the decision maker must design effective countermeasures to guarantee that the whole system was capable to meet the long-term demand, mainly in the scenario of extensive levels of degradation. In this case, when considering unreliable subcontracting, the total capacity constraint for the whole system was determined using

$$\frac{u_{max}}{1-\beta \left(a\right)}·{\pi }_1^{prod}+u_2(·)·{\pi }_1^{sub}\ge d$$

(9)

In modeling terms, Equation (9) indicates that the level of production of the machine tended to decrease due to the effect of its deterioration, and thus, unreliable subcontracting was needed as an alternative to satisfy product demand.

2.3.1. Production and Subcontracting Strategies

In this section, we present a proposed extension to the hedging point policy, which included the contribution of unreliable subcontracting. The proposed policy used the age $a\left(t\right)$ of the machine at time $t$ to indicate the adequate time to trigger subcontracting participation. Equations (10) and (11) are convenient representations of the proposed production–subcontracting control policy.

If $a\left(t\right)<$ $A_{sub}$:

$$\{\begin{array}{c}{u}_1\left(t\right)=\{\begin{array}{c}\frac{u_{max}}{1-\beta \left(a\right)} if x\left(t\right)<Z^{*} y {\alpha }_1\left(t\right)=1\\ \frac{d}{1-\beta \left(a\right)} if x\left(t\right)=Z^{*} y {\alpha }_1\left(t\right)=1\\ 0\hspace{1em}\hspace{1em}if x\left(t\right)>Z^{*} or {\alpha }_1\left(t\right)=0 \end{array}\\ \hspace{1em}\hspace{1em}\hspace{1em}{u}_2 \left(t\right)=0\hspace{1em}\forall x\left(t\right), {\alpha }_1\left(t\right),\hspace{1em} {\alpha }_2\left(t\right)\end{array}$$

(10)

If $a\left(t\right)\ge$ $A_{sub}$:

$$\{\begin{array}{l}{u}_1\left(t\right)=\{\begin{array}{cc} u_{max}& if x\left(t\right)<Z^{*} and {\alpha }_1\left(t\right)=1 and {\alpha }_2\left(t\right)=0\\ Max(\frac{u_{max}}{1-\beta \left(a\right)}-u_2^{*})& if x\left(t\right)=Z^{* }and {\alpha }_1\left(t\right)=1 and {\alpha }_2\left(t\right)=1\\ \frac{d}{1-\beta \left(a\right)}& if x\left(t\right)=Z^{*} and {\alpha }_1\left(t\right)=1 and {\alpha }_2\left(t\right)=0\\ Max\left(\left(\frac{d}{1-\beta \left(a\right)}-u_2^{*} \right),0\right)& if x\left(t\right)=Z^{* }and {\alpha }_1\left(t\right)=1 and {\alpha }_2\left(t\right)=1\\ 0& if x\left(t\right)\le Z^{*} and {\alpha }_1\left(t\right)=0 and \forall {\alpha }_2\left(t\right)=0\\ 0& if x\left(t\right)>Z^{*} \end{array}\\ u_2 \left(t\right)=\{\begin{array}{cc}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{u}_{2 }^{*} & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}if x\left(t\right)\le Z^{*} and {\alpha }_2\left(t\right)=1 and \forall {\alpha }_1\left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}if x\left(t\right)\le Z^{*} and {\alpha }_2\left(t\right)=0 and \forall {\alpha }_1\left(t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} if x\left(t\right)>Z^{*} \end{array}\end{array}$$

(11)

For practical purposes, $A_{sub}$ is the critical age where subcontracting is triggered and $Z^{*}$ is the optimal production threshold. To an extent, Equation (10) represents a classical HPP strategy in the production unit, and it indicates that subcontracting is not requested when the age of the machine is less than $A_{sub}$ since, in such a period, the machine is operational in acceptable conditions and the effects of deterioration are not so severe. By applying Equation (11), one could control the participation of the production unit and subcontracting when the age of the machine surpasses $A_{sub}$. In such an equation, internal production and subcontracting depend on their stochastic modes $\left({\alpha }_1\left(t\right),{\alpha }_1\left(t\right)\right)$, the inventory level $x\left(t\right)$, and the age of the machine $a\left(t\right)$.

2.3.2. Major Maintenance Strategy

The major maintenance strategy defined that this activity could be conducted once the machine reached the critical age $A_o$. With the performance of this maintenance option, the machine was rejuvenated to an AGAN condition, eliminating the impairments caused by degradation. In this case, the major maintenance strategy was determined using

$$w^{*}\left(1,x,a\right)=\{\begin{array}{c}\begin{array}{cc}1& if a\left(t\right)\ge A_o\end{array}\\ \begin{array}{cc}0& otherwise \end{array}\end{array}$$

(12)

To be more precise, $A_o$ was the critical age that triggered major maintenance activities. Summing up, the control parameters of the proposed policy were $\left(Z,A_{sub},A_o\right)$, which served to define the appropriate rate of production, subcontracting, and major maintenance $\left(u_1^{*},u_2^{*},w^{*}\right)$. Such rates were defined taking into account the state variables of the system $\left({\alpha }_1, {\alpha }_2,x,a\right)$. The strong interactions between these three strategies required jointly optimizing the control parameters to minimize the total cost since they were interlinked. Classical approaches that dissociate decisions turn out to be less efficient than integrated models, as is discussed in the next sections.

2.3.3. Optimization Problem

The main feature of the proposed model was the joint optimization of the control parameters $\left(Z,A_{sub,}{A}_o\right)$, which were necessary to optimally control the subcontracting contribution, regulate the inventory level, and define the critical age to conduct major maintenance. These control parameters must be jointly optimized to minimize the total incurred cost $TC$. In our formulation, the total cost consisted of the sum of the inventory cost and the backlog cost $IB\left(t\right)$, the quality cost $QC\left(t\right)$, and the maintenance cost $MC\left(t\right)$. Regarding the inventory–backlog cost, this component also comprised the production cost $C_{pro}$ and the subcontracting cost $C_{sub}$, defined as

$$IB\left(t\right)=\frac{1}{T}·{{\displaystyle \int }}_0^T(C^{+}{x}^{+}\left(t\right)+C^{-}{x}^{-}\left(t\right)+C_{pro}{u}_1\left(t\right)+C_{sub}{u}_2\left(t\right))dt$$

(13)

where $x^{+}=\max \left(0,x\right)$ and $x^{-}=\max \left(-x,0\right)$ and the constants $C^{+}$ and $C^{-}$ serve to penalize inventory and shortages, respectively. The indicator $QC\left(t\right)$ represents the average cost of quality per unit of time in the period $\left[0,T\right]$; it is determined by the cost of defects $C_{def}$ and is given by the following equation:

$$QC\left(t\right)=\frac{1}{T}·\left(C_{def}{{\displaystyle \int }}_0^T\left(\beta \left(t\right)·d\right)dt\right)$$

(14)

Further, $MC\left(t\right)$ denotes the average maintenance cost per unit of time in the period $\left[0,T\right]$ and it takes into account the cost of minimal repair and the cost of major maintenance; it is defined as:

$$MC\left(t\right)=\frac{1}{T}·\left(C_R·N_R\left(t\right)+C_M·N_M\left(t\right)\right)$$

(15)

where $N_R\left(t\right)$ and $N_M\left(t\right)$ define the number of minimal repairs and the number of major maintenances performed in the period $\left[0,T\right]$, respectively. Summing up, Equations (1)–(15) led to defining the next stochastic optimization model:

$$\{\begin{array}{l}\hspace{1em}\hspace{1em}Min TC\left(Z_0,A_{sub,}{A}_o\right)=\underset{t\to \infty }{\lim }\left(IB\left(t\right)+QC\left(t\right)+MC\left(t\right)\right)\\ \hspace{1em}subject to& \\ Equations \left(1\right)-\left(9\right) \left(dynamics of inventory and quality\right)\\ \hspace{1em}\left(Z_0,A_{sub},A_o\right)\ge 0 \left(non-negative constraints\right)\end{array}$$

(16)

Another important aspect was the fact that this optimization model implied a nonlinear and stochastic problem; for this reason, traditional mathematical methods could not be applied to solve this kind of problem since it was difficult to determine an analytical solution for the total cost. The difficulty was due to the random transitions, the Markov dynamics for internal production and subcontracting, and mainly because of the gradual growth in the number of defects caused by deterioration. One way to overcome these difficulties was with the implementation of a simulation–optimization approach. In the next section, we detail how we determined a solution to this model and determined the value of the parameters of the control rule in a reasonable amount of time and with acceptable accuracy.

3. Resolution Approach

In line with the previous section, we now introduce the proposed simulation–optimization approach, which combined simulation techniques with optimization methods with the aim to accurately reproduce the stochastic dynamics of the system under investigation. The optimization strategy and applied procedure began with a mathematical formulation of the manufacturing system under investigation; then, a simulation model was developed for the purpose of simulating and analyzing the system’s behavior. Statistical techniques were used for this analysis and optimization methods based on the response surface method were applied. The applied procedure provided remarkable results in previous studies, as in Rivera-Gómez et al. [17]. The approach comprised the following steps:

i.

Mathematical formulation: In this phase, an analytical model was formulated for the proposed system, where the decision variables were also identified, as well as the objective function to be minimized. Further in this step, the control parameters $\left(Z,A_{sub},A_o\right)$ of the proposed control strategy were determined. Equations (1)–(16) defined the mathematical formulation of the production system under study.

ii.

Simulation model: In this step, a simulation model was built that served to reproduce the stochastic behavior of the production system. The inputs of this simulation model were the control parameters $\left(Z,A_{sub},A_o\right)$ that were defined in the mathematical formulation step. The output of the simulation model was the total incurred cost. In the following section, the simulation model developed is presented in detail.

iii.

Statistical analysis: In this step, the design of experiments technique was used to determine the significant variables and interactions that must be included in the optimization phase. Moreover, using the total cost reported from several simulation runs, a second-order regression model was determined for the total incurred cost. A design of experiments technique that included three independent variables $\left(Z,A_{sub},A_o\right)$ was applied for the determination of this regression model.

iv.

Optimization: In this step, a domain for the three independent variables $\left(Z,A_{sub},A_o\right)$ was defined to perform the optimization of the regression model. The optimization strategy applied in this study was based on the response surface technique. Then, simulation runs were conducted to have enough data regarding the total cost and to perform the optimization. The response surface defined the optimal values $\left(Z^{*},A_{sub}^{*},A_o^{*}\right)$ and minimized the total cost.

v.

Verification and validation: Several activities are conducted to ensure that the applied procedure provided accurate and credible results. Hence, an operational validation was performed, several key performance measures of the simulation model were displayed graphically, and a parameter variability analysis was conducted in the sensitivity analysis section to assess the effect of the variation of several cost and system parameters. Further, statistical analysis was carried out through a confidence interval for the total cost. Finally, a comparative study was done to highlight the economic advantage of the proposed model.

Figure 2 presents the resolution approach used in the research.

4. Simulation Model Description

As mentioned earlier, a simulation model was developed with a mixture of discrete and continuous components, where the simulation software ARENA was used in this phase. The model had the objective of reproducing the flow of material and the logic defined in the production–subcontracting–maintenance control strategy of Equations (10)–(12). The inputs for the simulation were the control parameters $\left(Z,A_{sub},A_o\right)$. Furthermore, the differential Equations (1) and (3) for the inventory and age dynamics were constantly integrated through the Runge–Kutta–Fehlberg technique. In the simulation model, the random duration of the minimal repairs and the duration of the major maintenance were defined by the discrete elements. Furthermore, the model defined the periods of inactivity of the unreliable subcontractor. The continuous part of the model was incorporated with C++ subroutines that actualized the defect rate $\beta (·)$ with Equation (5), and it updated the system age $a(·)$. This C++ file defined the level of inventory and backlog. In this simulation model, all the components worked simultaneously to replicate the interactions and the random events of the whole system. Furthermore, in this model, the production and subcontracting planning and the maintenance strategies were included with the use of Equations (10)–(12). Once the simulation ended, the model displayed indices to estimate the inventory cost $IB\left(t\right),$ the quality cost $QC\left(t\right)$, and the maintenance cost $MC\left(t\right)$ with the use of Equations (13)–(15), respectively. Figure 3 presents the flow diagram of the simulation model.

Validation of the Simulation Model

With the aim to guarantee the accuracy of the simulation results, validation was carried out using several relevant indicators of the system’s behavior. The dynamics of these key indicators are presented in Figure 4. As an illustration, this validation was performed during a time interval where the parameters of the control rule were set at the following values: $Z$= 20, $A_{sub}$= 130, and $A_o=$ 175. As presented in Figure 4, it is clear that at time $t$ = 0, the production unit was in perfect condition and the inventory level increased steadily until it reached its optimal level (see arrow 1 in Figure 4g). Subsequently, the unit experienced some random failures, and at time $t$ = 37, a decrease in the inventory level was observed because the machine suffered a failure (see arrow 2 in Figure 4g). The occurrence of random failures and the periods of inactivity of the machine are identified in Figure 4d (see arrow 3). At this point, the machine experienced progressive deterioration, and then, at time $t=$ 227, the unit could no longer satisfy the product demand by itself; therefore, the subcontractor started to supply units to satisfy customer demand (see arrow 4 in Figure 4c). At time $t$ = 230, the inventory level was maintained at its optimal level, where internal production and subcontractor were both needed (see arrow 5 in Figure 4g). Then, at time $t$ = 250, the unit operated at the rate $d/(\left(1-\beta \left(a\right)\right)$ to mitigate the increase in the defect rate (as indicated by arrow 6 in Figure 4e) because the level of deterioration was significantly high, as shown by arrow 7 in Figure 4b. Later, at time $t$ = 287, the unit age surpassed the critical value $A_o=$ 175, which initiated the conduction of major maintenance activities (as in arrow 8 in Figure 4f). Before the conduction of major maintenance, the maximum level of deterioration of the machine was reached (see arrow 9 in Figure 4a). Moreover, once the conduction of major maintenance was terminated, arrow 10 in Figure 4g shows a decrease in the inventory level due to the performance of such maintenance. It is evident from the results of Figure 4 that after the end of the conduction of major maintenance, there was a significant decrease in the rate of defects (see arrow 11 in Figure 4a) since the major maintenance rejuvenated the system to its initial conditions and completely mitigated the harmful effects of deterioration. At this point, the inventory level reached its optimal level of $Z=$ 20, indicating the beginning of a new deterioration cycle. Based on the analysis of the dynamics of Figure 4, the proposed control strategy was validated since it worked appropriately in the simulation model and it was observed that the role of the subcontractor was very important once the machine age surpassed the critical value $A_{sub}$ and major maintenance was triggered if the age of the machine reached $A_o$. Summing up, the operational validity of the simulation model is presented in Figure 4.

5. Statistical Analysis and Optimization

In the rest of this section, a numerical case study of the system under investigation was conducted, where the data used in the analysis satisfied the feasibility condition (Equation (9)) and such data were based on previous studies of Rivera-Gómez et al. [17]. Furthermore, by taking into consideration the validation of the simulation model from the previous section, the data used ensured that the obtained results were representative of the productive system. In this numerical case study, three independent variables were considered $\left(Z,A_{sub},A_o\right)$ and the dependent variable was defined by the total incurred cost. Several simulation runs were conducted based on a factorial 3³ design. For each combination of the independent variables $\left(Z,A_{sub},A_o\right)$, the factorial design 3³ was replicated two times; this meant that a total of 3³ × 2 = 54 simulation runs were needed for the analysis. The simulation time used in the analysis was defined in 100,000 time units to guarantee steady-state conditions. The input data for the parameters of the simulation model are presented in

Table 2. Data for the numerical case study.

Parameter:	$q_{12}$	$q_{21}$	$q_{13}$	$q_{31}$	$q_{12}^{sub}$	$q_{21}^{sub}$
Value:	0.1	1.5	0.15	5	0.25	0.3
Parameter:	$u_{max}$	$d$	$k_1$	${\eta }_o$	$b_0$	$b_1$
Value:	9	5	0.5	0.1	0.01	0.49
Parameter:	${\eta }_1$	${\eta }_2$	$k_1$
Value:	15 × 10−6.2	2.4	0.5

.

Table 3. Cost parameters.

Parameter:	$C^{+}$	$C^{-}$	$C_R$	$C_M$	$C_{def}$	$C_{pro}$	$C_{sub}$
Value:	1	40	100	3000	20	5	15

shows the values of several cost categories used in the simulation.

To simplify the calculations, it should be noted that to ensure that $A_{sub}<A_o$, this age was defined as a fraction of age $A_o$, i.e., $A_{sub}=k·A_o$, where $k ϵ \left[0, 1\right]$. The values of the independent variables $\left(Z, k,A_o\right)$ shown in

Table 4. Domains of the independent variables.

Factor	Low Level	High Level	Description
$Z$	5	50	Inventory level
$k$	0.5	0.99	Critical parameter to trigger subcontracting
$A_o$	50	300	Critical age to trigger major maintenance

were selected via the observation of several off-line simulations.

In the following, taking into account the information in Table 2, Table 3 and Table 4, the simulation model generated the necessary output data to determine a second-order regression model for the total cost. Such a regression model was analyzed through the use of statistical software called STATGRAPHICS, where an ANOVA was performed. Then, the regression model for the total cost was the following:

$$\begin{array}{cc}TC\left(Z, k,A_o\right)=\hfill & \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}809.977+4.74576 · Z-3.70433 ·A_o-1705.16 · k+0.043567 · Z^2\hfill & \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-0.0161327·Z·A_o -7.5515·Z·k+0.00580246 ·{A_o}^2\hfill & \\ \hfill \hspace{1em}\hspace{1em}+3.67873 · A_o· k+1036.04 · k^2\hfill & \end{array}$$

(17)

The analysis of variance of the regression model is presented in

Table 5. ANOVA Table.

Source	Sum of Squares	Gl	Medium Square	F-Ratio	p-Value
A:Zo	31,139.5	1	31,139.5	4.09	0.0493
B:Ao	218,699	1	218,699	28.75	0.0000
C:k	163,011	1	163,011	21.43	0.0000
AA	5837.5	1	5837.5	0.77	0.3859
AB	49,409.4	1	49,409.4	6.5	0.0145
AC	41,588.6	1	41,588.6	5.47	0.0241
BB	98,638.4	1	98,638.4	12.97	0.0008
BC	304,620	1	304,620	40.05	0.0000
CC	46,408.7	1	46,408.7	6.1	0.0176
Blocks	115.445	1	115.445	0.02	0.9025
Total error	327,073		7606.36
Total (corr.)	1.29 × 106

, where it is evident that all the main factors, most of the quadratic effects, and most of the interactions were significant since the p-value indicators were less than 5%.

The contour plots of the total cost reported by the regression model are presented in Figure 5 in two-dimensional space.

The regression model presented in Equation (17) was optimized with a response surface; in this case, the optimal solution that minimized the total cost is presented in

Table 6. Optimal control parameters.

	${\mathit{Z}}^{*}$	${\mathit{k}}^{*}$	${\mathit{A}}_{\mathit{o}}^{*}$	Total Cost	Cross-Check Confidence Interval (95%)
Value	31.5	0.6707	150.32	34.33	[25.78, 42.79]

* show the optimal control.

. The value of $k^{*}=0.6707$ implied an age to trigger subcontracting of $A_{sub}^{*}=$ 100.81. Table 5 presents the optimal values of the control parameters that must be applied in the production system to simultaneously control the production rate, subcontracting, and major maintenance with minimal cost.

In closing this section, fifty extra simulation replications were performed using the optimal values reported in Table 6, whereby the accuracy of the proposed approach was corroborated since the total cost of 34.33 was within the confidence interval [25.78, 42.79]; this interval was obtained with the 50 simulation replications.

6. Sensitivity Analysis

So far, we have discussed the proposed simulation–optimization approach to the production system under investigation. Now, this section presents the detailed results of the robustness analysis of the proposed model and contrasts the value of the control parameters and the total cost under several system scenarios. In

Table 7. Cost variation.

		Cost Variation
Cases	Par.	${\mathit{C}}^{+}$	${\mathit{C}}^{-}$	${\mathit{C}}_{\mathit{d}\mathit{e}\mathit{f}}$	${\mathit{C}}_{\mathit{p}\mathit{r}\mathit{o}}$	${\mathit{C}}_{\mathit{s}\mathit{u}\mathit{b}}$	${\mathit{C}}_{\mathit{R}}$	${\mathit{C}}_{\mathit{M}}$
Base case	-	1	40	20	5	15	100	3000
Case i	$C^{+}$	0.5	40	20	5	15	100	3000
Case ii		2	40	20	5	15	100	3000
Case iii	$C^{-}$	1	10	20	5	15	100	3000
Case iv		1	50	20	5	15	100	3000
Case v	$C_{def}$	1	40	5	5	15	100	3000
Case vi		1	40	30	5	15	100	3000
Case vii	$C_p$	1	40	20	1	15	100	3000
Case viii		1	40	20	10	15	100	3000
Case ix	$C_s$	1	40	20	5	5	100	3000
Case x		1	40	20	5	30	100	3000
Case xi	$C_R$	1	40	20	5	15	30	3000
Case xii		1	40	20	5	15	200	3000
Case xiii	$C_M$	1	40	20	5	15	100	1000
Case xiv		1	40	20	5	15	100	4500

, fourteen different configurations are presented; the shaded values in Table 7 were selected by decreasing and increasing the parameters of the base case shown in Table 3 by a certain percentage. Such variations were selected to facilitate the observation of their impact on the control parameters and the total cost.

Table 8. Sensitivities for the cost variations.

						Total
		Optimal Control Parameters				Cost
Cases	Par.	${\mathit{Z}}^{*}$	${\mathit{k}}^{*}$	${\mathit{A}}_{\mathit{o}}^{*}$	${\mathit{A}}_{\mathit{s}\mathit{u}\mathit{b}}^{*}$	TIC*	Effect
Base Case	-	31.5	0.6707	150.32	100.8196	34.33	Base of comparison
Case i	$C^{+}$	39.83	0.692	154.84	107.1493	17.05	$Z^{*}$↑, Ao*↑, Asub*↑
Case ii		14.93	0.627	140.36	88.0057	56.86	$Z^{*}$↓, Ao* ↓, Asub*↓
Case iii	$C^{-}$	16.58	0.6604	132.77	87.6813	66.10	$Z^{*}$↓, Ao* ↓, Asub*↓
Case iv		34.6	0.6718	154.47	103.7729	16.28	$Z^{*}$↑, Ao*↑, Asub*↑
Case v	$C_{def}$	32.21	0.6471	165.17	106.8815	23.42	$Z^{*}$↑, Ao*↑, Asub*↑
Case vi		31	0.6875	139.85	96.1469	40.97	$Z^{*}$↓, Ao* ↓, Asub*↓
Case vii	$C_{pro}$	32.19	0.6711	154.27	103.5306	14.10	$Z^{*}$↑, Ao*↑, Asub*↑
Case viii		30.67	0.6705	145.46	97.5309	59.43	$Z^{*}$↓, Ao* ↓, Asub*↓
Case ix	$C_{sub}$	30.66	0.6441	154.88	99.7582	24.68	$Z^{*}$↓, Ao* ↑, Asub*↓
Case x		33.61	0.7095	143.82	102.0403	47.66	$Z^{*}$↑, Ao*↓, Asub*↑
Case xi	$C_R$	31.53	0.6702	150.7	100.9991	27.96	$Z^{*}$↑, Ao*↑, Asub*↑
Case xii		31.47	0.6715	149.87	100.6377	43.43	$Z^{*}$↓, Ao* ↓, Asub*↓
Case xiii	$C_M$	30.78	0.6985	133.59	93.3126	25.46	$Z^{*}$↓, Ao* ↓, Asub*↓
Case xiv		31.93	0.6543	160.29	104.8777	40.26	$Z^{*}$↑, Ao*↑, Asub*↑

↑, and ↓ shows the effect of the variation of the cost parameter.

presents the sensitivities for the variations devised in Table 7.

The sensitivity analysis of the variation of the cost parameters produced the following observations:

▪

Inventory cost variation: When the inventory cost $C^{+}$ increased (case ii), the proposed strategy reduced the inventory level $Z^{*}$ since the accumulation of inventory was more sanctioned. However, by increasing $C^{+}$, it was observed that the inventory was reduced to almost half of the level of the base case and this had the consequence that the subcontractor supplied parts earlier to avoid shortages. Thus, the system decreased the age $A_{sub}^{*}$. Moreover, the system opted to perform more major maintenance to limit the generation of more defects. For this reason, the critical parameter of major maintenance $A_o^{*}$ decreased. By decreasing the inventory cost (case i), the opposite effects were observed as compared to the case when the inventory cost increased.

▪

Backlog cost variation: As the cost parameter $C^{-}$ increased (case iv), shortages had a stronger economic penalty, hence the inventory level $Z^{*}$ need to increase with the aim to protect the system against the risk of not being able to meet the demand. It was also observed that by increasing $C^{-}$, the age to perform major maintenance $A_o^{*}$ increased to operate the machine for longer and avoid product shortages caused by the performance of major maintenance. In addition, as the production unit was operative for longer, more inventory accumulated; therefore, the subcontractor was required less and this increased the age $A_{sub}^{*}$. Decreasing the cost of backlog (case iii) had the opposite effects to increasing $C^{-}$.

▪

Defective cost variation: The increase in the defect cost $C_{def}$ (case vi) mainly caused major maintenance to be performed more frequently to renew the unit as quickly as possible and significantly reduce the quantity of defective items; this measure decreased the age $A_o^{*}$. Additionally, it was observed that when increasing $C_{def}$, the system reduced the inventory level since the presence of defects was more heavily penalized. This decision was complemented by decreasing the critical age $A_{sub}^{*}$, which implied that the subcontractor must supply units at an earlier age to mitigate the presence of defects. The opposite effects were observed when the cost of defects decreased (case v).

▪

Production cost variation: As the production cost $C_{pro}$ increased (case viii), it was observed that major maintenance was performed at an earlier age as an attempt to ensure that the machine produced only flawless units; hence, the unit was functional for a shorter period, decreasing $A_o^{*}$. Moreover, as $C_{pro}$ increased, it was evident that the inventory level $Z^{*}$ decreased because internal production was more costly. Moreover, as the production cost increased, it led to reducing the critical age of utilizing the subcontractor $A_{sub}^{*}$ because, in this context, it was more convenient for the company to have the subcontractor supply parts at an earlier age. As the production cost decreased, the opposite effects were observed (case vii).

▪

Variation in the subcontracting cost: It was noted that an increase in the subcontracting cost $C_{sub}$ (case x) had the logical effect of the major maintenance being performed with more frequency to rejuvenate the system and limit the presence of shortages. Likewise, the increase in the subcontractor’s cost $C_{sub}$ caused that the system to increase the inventory level $Z^{*}$ because it was more profitable to opt for internal production than to use the subcontractor. Furthermore, it was obvious that by increasing $C_{sub}$, the critical age of subcontracting $A_{sub}^{*}$ increased due to the high cost of external supplies. Reducing the cost of subcontracting had the opposite effect of increasing $C_{sub}$ (case ix).

▪

Variation in the minimum repair cost: The reduction in the minimum repair cost $C_R$ (case xii) triggered the conduction of more major maintenance since this action kept the machine in a better condition, decreasing the presence of defects; this is the reason that explained the decrease in the critical age for major maintenance $A_o^{*}$. Furthermore, it is shown in Table 7 that with the decrease of $A_o^{*}$, the unit was functional for less time; therefore, less inventory accumulated, reducing $Z^{*}$. On the other hand, the critical age of subcontracting $A_{sub}^{*}$ was reduced to guarantee the product availability due to the reduction in the inventory level $Z^{*}$. By decreasing the minimum repair cost, the opposite effects were observed (case xi).

▪

Variation in the cost of major maintenance: From the results, it was noted that the increase in the major maintenance cost $C_M$ (case xiv) led to observing that this action was performed less frequently; therefore, the age $A_o^{*}$ increased. As $C_M$ increased, the system reached greater levels of deterioration to account for the high cost of major maintenance. The objective of postponing this maintenance action was to let the system operate for longer as an attempt to increase the inventory level, and thus, increase the capacity to respond to customer demand; this increased the inventory level $Z^{*}$. Moreover, as the machine was operational for longer periods, it was obvious that less subcontracting was required; therefore, the age $A_{sub}^{*}$ increased because it was not economical for the company to have the subcontractor supply units in advance. Reducing the cost of major maintenance had the opposite effects (case xiii).

Based on the results discussed in Table 8, it was validated that the proposed control policy worked efficiently and that the logic of the control strategy was consistent throughout the analysis.

6.1. Effect of the Variation of the Producer Availability and the Subcontractor Availability

As shown in the following paragraphs, the main intention of this section was to present the analysis results regarding the influence of varying the subcontractor and producer availability. To make things simple, we defined the availability of the production system $D_{pro}$ as the fraction of time when the manufacturing unit was available for production, as denoted previously in Equation (7). Additionally, to develop a realistic production model, it was assumed that the subcontractor was unreliable due to random disturbances. Then, the fraction of time where the subcontractor was available was denoted by $D_{sub}$, as noted earlier in Equation (8).

Table 9. Effect of the variation of the producer and subcontractor availability.

Case		Availability		Control Parameters					Effect
	Par.	${\mathit{D}}_{\mathit{s}\mathit{u}\mathit{b}}$(%)	${\mathit{D}}_{\mathit{p}\mathit{r}\mathit{o}}$ (%)	${\mathit{Z}}^{*}$	${\mathit{k}}^{*}$	${\mathit{A}}_{\mathit{o}}^{*}$	${\mathit{A}}_{\mathit{s}\mathit{u}\mathit{b}}^{*}$	Cost*
Base case		92.30	93.75	31.50	0.6707	150.32	100.8196	34.33	Base Case
Case xv	$D_{sub}$	89.02	93.75	32.49	0.6733	151.58	102.06	34.53	$Z^{*}$↑, Ao*↑, Asub*↑
Case xvi		95.23	93.75	29.98	0.6710	146.62	98.38	33.72	$Z^{*}$↓, Ao*↓, Asub*↓
Case xvii	$D_{pro}$	92.30	91.46	33.92	0.6611	153.68	101.60	14.59	$Z^{*}$↑, Ao*↑, Asub*↑
Case xviii		92.30	95.54	28.31	0.6682	149.69	100.02	50.61	$Z^{*}$↓, Ao*↓, Asub*↓

↑, and ↓ shows the effect of the variation of the availability of the subcontractor and the availabi-lity of the producer.

presents the effects of the availability of the producer and the availability of the subcontractor on the parameters of the control strategy and also on the total cost.

From the results presented in Table 9, it was observed that when the availability of the subcontractor $D_{sub}$ was low (case xv), more protection was needed against shortages, hence the inventory level $Z^{*}$ increased. Moreover, in this context, it was required to operate the system for longer to compensate for the low availability of the subcontractor, thus major maintenance must be postponed; this measure led to an increased age $A_o^{*}$. With the low availability of the subcontractor, it was normal that its participation was delayed, increasing the age $A_{sub}^{*}$. When the availability of the subcontractor increased (case xvi), the opposite effects were observed.

In the case where the availability of the producer $D_{pro}$ reduced (case xvii), the system required more protection against random failures; thus, the inventory level $Z^{*}$ increased. Major maintenance was postponed to allow the unit to operate for longer without interruptions and to increase the capacity to satisfy customer demand, thus increasing the age $A_o^{*}$. Since the subcontractor’s cost was more expensive than in-house production, when the availability of the producer decreased, then it was preferable to continue with in-house production, thus increasing the age $A_{sub}^{*}$. When the availability $D_{pro}$ increased, the opposite effects were observed (case xviii).

6.2. Comparative Study

To ensure that the proposed joint control policy provided economic benefits over other existing approaches, in this section, a comparison of the total cost was conducted between the proposed policy and an alternative policy derived from typical assumptions used in the literature. The alternative policy was denoted as Policy-B and it did not consider a subcontracting strategy. Policy-B did not include the parameter $A_{sub}$ in the decision variables. The control parameters used in Policy-B were therefore only the decisions regarding production and maintenance planning $\left(Z,A_o\right)$. A similar optimization approach to that used in the previous section was applied with the aim to define the optimal control factors $\left(Z,A_o\right)$ for Policy-B. The definitions of the control strategies included in the study were as follows:

▪

Policy-A referred to the control policy $\left(Z,A_{sub},A_o\right)$ developed in this study and consisted of the simultaneous optimization of the inventory level, the determination of the most adequate age to use subcontracting, and the determination of the age to conduct major maintenance activities.

▪

Policy-B was a generalization of our policy and was intended to examine the impact of subcontracting in the optimization. In this policy, the control parameters were reduced to only the production and maintenance decisions $\left(Z,A_o\right)$, disregarding the subcontracting decisions in the optimization. In Policy-B, subcontracting was not taken into account in the control policy and the optimization.

Table 10. Policy Comparison.

Description	Control Parameters			Total Cost	Observed Difference Δ-Cost (%)
	${\mathit{Z}}^{*}$	${\mathit{k}}^{*}$	${\mathit{A}}_{\mathit{o}}^{*}$
Policy-A	31.5	0.6707	150.32	34.33	—
Policy-B	43.26	-	189.93	50.59	+32.14

presents the results of the comparative study conducted.

An important consideration regarding Table 10 is that the cost difference between the two policies was 32.14%, where this gap was explained by the fact that Policy-B used larger inventory levels to guarantee that product demand could be satisfied and this countermeasure considerably increased the total cost. In addition, the production unit remained operational for longer periods in Policy-B before major maintenance was performed; this action led to an increase in the total cost since if the machine operated for longer without adequate maintenance, then more defects were generated due to deterioration, increasing the total cost. This comparison gave us a measure of the utility of the proposed joint control policy.

6.3. Indifference Curve to Determine the Subcontracting Participation

Based on the results discussed in the last section, it was observed that subcontracting was an effective countermeasure to improve the level of demand satisfaction and minimize costs. The key idea of the proposed model was to optimize in-house production and compensate with the use of the subcontractor if needed when the deterioration process reduced the production capacity of the production unit. In this section, we present further analysis results that served to properly select the participation of subcontracting based on two key characteristics, namely, the cost of subcontracting $C_{sub}$ and its availability $D_{sub}$. The optimization procedure described in the previous section was used to determine under which conditions it was more advisable to use subcontracting than in-house production. The approach consisted of considering the cost of subcontracting $C_{sub}$ as an independent variable that was optimized with the rest of the control parameters $\left(Z,A_{sub},A_o\right)$ of Policy-A. In addition, the total cost was defined as the dependent variable of the optimization. In this way, different values of the subcontractor’s availability were analyzed to define until what amount of cost $C_{sub}$ it was still profitable to use subcontracting. By analyzing different subcontracting costs and availabilities, the inference curve of Figure 6 was obtained.

The interpretation of Figure 6 is in terms of two main zones. The first zone (see indicator 1 in Figure 6) is where the system recommended using a combination of subcontracting and in-house production to reduce the total incurred cost. Furthermore, zone 2 (see indicator 2 in Figure 6) is where in-house production was the most cost-effective and profitable option. For example, when the subcontractor had availability of $D_{sub}=91\%$, the manager could manage to pay a cost of $C_{sub}=16$ to the subcontractor. Otherwise, there was no economic incentive in negotiating with the subcontractor. However, the company could pay more to the subcontractor, for instance, a cost of $C_{sub}=17.84$, only if the subcontractor guaranteed a higher availability of $D_{sub}=99\%$. In this sense, it could be stated that if the availability of the subcontractor was high, a higher cost could be paid for its services.

7. Conclusions

The present research was devoted to the joint analysis of production, unreliable subcontracting, and perfect maintenance strategies for a manufacturing system that experienced a quality decline process. The studied production system was prone to random failures and increased its defect rate as it deteriorated. Thus, to ensure demand satisfaction, subcontracting was available at a higher cost as a countermeasure to cope with the effect of such deterioration. However, as observed in real production, the subcontractor was unreliable since it was not always available to supply products to the company due to random disturbances caused by, e.g., sudden failures or because it was occupied serving other customers.

The major outcome of this research was the consideration of the four key aspects of production, subcontracting, maintenance, and quality in an integrated model under a dynamic and stochastic context, where the subcontractor was unreliable and the system underwent deterioration. In this domain, no prior work in the literature has treated this set of characteristics. A control policy derived from the well-known HPP was proposed to properly coordinate internal production and subcontracting. From the results, it was observed that the proposed policy appropriately achieved the coordination of internal production and subcontracting for any possible case. Decisions on these strategies and maintenance were defined as a function of the level of inventory, the availability of the subcontractor, the availability of the producer, and the amount of degradation of the machine. A simulation–optimization approach was adopted to solve such a stochastic problem. The proposed approach involved developing a meta-model comprising a quantitative study based on the design of experiments and simulation analysis, where the control parameters were optimized through a response surface. The proposed approach was validated through a graphical analysis and the assessment of the variation of several costs; in all the analyzed cases, the proposed model managed to adequately control the production system. Moreover, the impact of the variation of the availability of the subcontractor and the producer were analyzed; it was found that more inventory was needed to cope with shortages when the subcontractor was less available and more maintenance was required to restore the production unit; in this context, the participation of the subcontractor was delayed. Through a comparative study, it was observed that the proposed policy reported considerable cost savings of 32.14% compared to other classical policies that disregarded subcontracting. Such a cost difference could even be more considerable if the cost $C_{sub}$ or the availability of the subcontractor $D_{sub}$ increased. Furthermore, an indifference curve was analyzed and it served to define the cases when only internal production was more profitable to the company and the cases when a combination of internal production and subcontracting was more cost-effective.

Regarding future research, we note that the maintenance strategy used to overhaul the system in the proposed model was a perfect repair. However, imperfect maintenance strategies are more realistic regarding model current production systems. The consideration of imperfect maintenance in the context of unreliable subcontracting will be the subject of future research. Additionally, the effect of random demand in the context of unreliable subcontracting can be included in the models.