Optimization of Quality, Reliability, and Warranty Policies for Micromachines under Wear Degradation

This work presents an optimization technique to determine the inspection, warranty period, and preventive maintenance policies for micromachines suffering from degradation. Specifically, wear degradation is considered, which is a common failure process for many Micro-Electro-Mechanical Systems (MEMS). The proposed mathematical model examines the impact of quality control on reliability and the duration of the warranty period given by the manufacturer or the supplier to the customer. Each of the above processes creates implementation costs. All the individual costs are integrated into a single measure, which is used to build the model and derive the optimal parameters of the quality and maintenance policies. The implementation of various levels of the quality, warranty, and maintenance policies are compared with their optimum level options to highlight their contribution to the assurance and improving product quality. To the authors’ best knowledge, the introduction of a warranty period is implemented for the first time in the open literature concerning this type of optimization model for MEMs and surely can bring additional advantages to their quality promotion strategy. The proposed optimization tool provides a comprehensive simultaneous answer to the optimal selection of all the values of the design variables determining the overall maintenance and quality management approach.


Introduction
MEMS technology exhibits excessive potential for many critical applications in aerospace, automotive, medical, nuclear, and other areas. With more extensive commercialization of MEMs, many challenging manufacturing questions are into consideration including precise dimensional control and inspection, reliability testing and modeling, avoiding stiction, and maintenance strategies [1]. These productivity, quality, and reliability questions are critical issues that influence the path of MEMS to the conventional market. Therefore, MEMS manufacturers need effective tools for optimal operational decisions. These tools can be derived from the use of equivalent tools and methods developed in a traditional industry.
Preventive maintenance refers to work carried out to maintain the equipment at the desired level of operation and to avoid failures leading to production stops. In the context of preventive maintenance includes carrying out checks and inspections of equipment and the replacement of defective units. As part of preventive maintenance, the policy of periodic maintenance is applied replacement of a unit, whereby a unit is replaced at every specific operating interval [2]. The time-based replacement policy refers to the replacement of a unit at a predefined time interval, which has been derived from the manufacturer's guidelines or existing experience. However, in many cases, the unit fails in time less than the replacement time. This case has been analyzed in [3], in which failures are divided into type I (less significant) and type II (catastrophic) failures. Type I failures require tenance, and minimal repairs providing some formulas for the average long-run cost rate of the proposed strategies.
The issue of property degradation of mechanical systems has been extensively studied and various models have been developed to describe it. The modeling of degradation has contributed to the prediction of material behavior as well as potential failure. In addition, there are many studies related to maintenance policies for mechanical systems. In this work, however, is developed a mathematical model for finding the optimal options of various quality policies such as e.g., the maintenance interval, the burn-in method, the replacement time, and the warranty time of a key component of micromachines, which is the pin joint. The pin joint of microengines wears out the more rotations the more revolutions it performs. Thus, the optimal choices of quality policies are examined which mentioned above are considered to minimize the overall costs.

Optimization Model
Devices at the microscale can be classified into four classes [23] according to their operational interactions. Class I devices have no free moving parts, like accelerometers, pressure sensors, or strain gauges; Class II devices have moving parts with no friction or contact surfaces, such as gyros, resonators, and filters; Class III devices have moving parts with contact surfaces such as relay and valve pump; Class IV equipment has moving parts with friction and contact surfaces such as shutters, scanners, optical switches. The first three classes can achieve high reliability if adequately manufactured and packaged. For class IV devices, where frictional surfaces cannot be avoided, failure analysis and reliability assessment must be performed to drive the robust commercialization of micromachines [24].
The failure modes in a micromachine can be wear, friction, fracture, contamination, stiction, etc. The micromachine used in this study is the electrostatically driven microactuator (microengine) developed at Sandia National Laboratories [25] (National Technology and Engineering Solutions of Sandia, LLC., Albuquerque, New Mexico, USA). The micromachine consists of orthogonal linear comb drive actuators that are mechanically connected to a rotating gear as seen in Figure 1. By applying voltages, the linear displacement of the comb drives is transformed into circular motion. The linkage arms are connected to the gear via a pin joint. The gear rotates about a hub, which is anchored to the substrate [25]. The dominant failure mechanism was mainly identified as visible wear on the friction surface, often resulting in a seized micromotor or a micromotor with a damaged shaft seal [25,26]. Wear can be defined as the removal of material from a solid surface by mechanical action. Abrasion degradation is a very complex phenomenon, involving both the mechanical and chemical properties of the objects in contact, as well as the pressure and surface speed with which the objects are in contact.
Micromachines 2022, 13, 1899 3 of 16 mechanism of system failure, Hashemi et al. [22] proposed optimal age-based and block preventive maintenance models by considering the costs of preventive maintenance, corrective maintenance, and minimal repairs providing some formulas for the average longrun cost rate of the proposed strategies. The issue of property degradation of mechanical systems has been extensively studied and various models have been developed to describe it. The modeling of degradation has contributed to the prediction of material behavior as well as potential failure. In addition, there are many studies related to maintenance policies for mechanical systems. In this work, however, is developed a mathematical model for finding the optimal options of various quality policies such as e.g., the maintenance interval, the burn-in method, the replacement time, and the warranty time of a key component of micromachines, which is the pin joint. The pin joint of microengines wears out the more rotations the more revolutions it performs. Thus, the optimal choices of quality policies are examined which mentioned above are considered to minimize the overall costs.

Optimization Model
Devices at the microscale can be classified into four classes [23] according to their operational interactions. Class I devices have no free moving parts, like accelerometers, pressure sensors, or strain gauges; Class II devices have moving parts with no friction or contact surfaces, such as gyros, resonators, and filters; Class III devices have moving parts with contact surfaces such as relay and valve pump; Class IV equipment has moving parts with friction and contact surfaces such as shutters, scanners, optical switches. The first three classes can achieve high reliability if adequately manufactured and packaged. For class IV devices, where frictional surfaces cannot be avoided, failure analysis and reliability assessment must be performed to drive the robust commercialization of micromachines [24].
The failure modes in a micromachine can be wear, friction, fracture, contamination, stiction, etc. The micromachine used in this study is the electrostatically driven microactuator (microengine) developed at Sandia National Laboratories [25] (National Technology and Engineering Solutions of Sandia, LLC., Albuquerque, New Mexico, U.S.A.). The micromachine consists of orthogonal linear comb drive actuators that are mechanically connected to a rotating gear as seen in Figure 1. By applying voltages, the linear displacement of the comb drives is transformed into circular motion. The linkage arms are connected to the gear via a pin joint. The gear rotates about a hub, which is anchored to the substrate [25]. The dominant failure mechanism was mainly identified as visible wear on the friction surface, often resulting in a seized micromotor or a micromotor with a damaged shaft seal [25,26]. Wear can be defined as the removal of material from a solid surface by mechanical action. Abrasion degradation is a very complex phenomenon, involving both the mechanical and chemical properties of the objects in contact, as well as the pressure and surface speed with which the objects are in contact.  The pin joint is a critical area for micromotors in which either less significant soft failures due to wear occur or where severe failures (hard failures) due to material breakage [27]. In the first case, the wear of the pin joint material exceeds a critical value but continues to operate while in severe failures the system stops operating. In addition, environmental conditions affect the degree of wear of the components as it has been shown that with a reduction in the humidity of the operating environment, there is an increase in the rate of wear [25,26].
Rubbing creates friction and often leads to the creation of abrasive materials or debris. The configuration of this material can lead to several different failure mechanisms. This is due to equipment-associated particle contamination, third-body abrasive particles that alter the motion tolerance, particle contamination that prevents or interferes with the movement and adhesion of surfaces, or frictional contact [28]. The wear mechanism may depend on the temperature achieved during the friction process. The material can be rubbed on the contact surface; surface materials can oxidize-Then fade, etc. Many parameters must be examined to determine the root cause of the wear, which makes analysis simple but time consuming.
To simultaneously improve quality, reliability and the warranty plan over the lifetime of micromachines, a systematic inspection, preventive replacement procedure, and warranty model have been developed, as described in Figure 2. The pin joint is a critical area for micromotors in which either less significant soft failures due to wear occur or where severe failures (hard failures) due to material breakage [27]. In the first case, the wear of the pin joint material exceeds a critical value but continues to operate while in severe failures the system stops operating. In addition, environmental conditions affect the degree of wear of the components as it has been shown that with a reduction in the humidity of the operating environment, there is an increase in the rate of wear [25,26].
Rubbing creates friction and often leads to the creation of abrasive materials or debris. The configuration of this material can lead to several different failure mechanisms. This is due to equipment-associated particle contamination, third-body abrasive particles that alter the motion tolerance, particle contamination that prevents or interferes with the movement and adhesion of surfaces, or frictional contact [28]. The wear mechanism may depend on the temperature achieved during the friction process. The material can be rubbed on the contact surface; surface materials can oxidize-Then fade, etc. Many parameters must be examined to determine the root cause of the wear, which makes analysis simple but time consuming.
To simultaneously improve quality, reliability and the warranty plan over the lifetime of micromachines, a systematic inspection, preventive replacement procedure, and warranty model have been developed, as described in Figure 2. A post-manufacturing burn-in process for micromachines is used to identify and remove defective and early failed components. Burn-in is a crucial method for achieving reliable parts and systems, but it also exposes them to stress. For burn-in components, non-destructive inspection is applied to separate the fraction of units whose wear A post-manufacturing burn-in process for micromachines is used to identify and remove defective and early failed components. Burn-in is a crucial method for achieving reliable parts and systems, but it also exposes them to stress. For burn-in components, non-destructive inspection is applied to separate the fraction of units whose wear surpasses the specific specification threshold. The screened units, with a high level of quality, are then released for field operation until they reach the time of periodic replacement when the cost of a forthcoming failure makes it economically advantageous to replace them with new ones. The preventive replacement procedure is used to avoid failure due to the wear of standard operating units. The introduction of a warranty period can bring additional advantages to the quality promotion strategy and could therefore be adopted.
For a simultaneous optimal selection of the values of the design variables determining the overall maintenance and quality management strategy, a comprehensive methodology is developed in the following sections.

Degradation Model
To derive the model describing the degradation of the characteristic, it is first necessary to determine the factors which influence the way affects it. The wear of the pin joint due to the rotation of the gear during the operation of a micromachine increases with the number of revolutions it performs. Therefore, the key factor in the model is the operating time, t [19]. The wear of the gear depends also on three other factors. These factors are the radius, r, of the pin, the coefficient, c, relating to the wear and hardness of the material, and the force, F, developed between the pin and the gear. This leads to a linear model in which all factors influence proportionally the wear of the pin, X, and the relationship is For various microengines, some of the factors are constant, while others are considered random. The radius of the pin varies for each and is considered a random factor with a mean value of µ r and a standard deviation of σ r . The force depends on the rotation frequency of the gear and consequently on the input voltage of the drive. It is therefore a random factor with mean value µ F and standard deviation σ F . Thus, at any time, for any value of radius and force, the wear X(t) is considered to follow the normal distribution with a mean value and standard deviation Here, we notice that more complex models considering simultaneously other failure modes could be developed. Such models may be useful for cases where the dominant failure modes have been demonstrated to be more than one.

Effect of Quality Control on Reliability
After the production stage of the microengines, the burn-in technique is applied to identify the defective units. Then, the non-destructive testing of all the units produced follows to remove those whose wear has exceeded the failure threshold H. To demonstrate the importance of quality control, it is necessary to calculate the reliability when 100% inspection of the units is applied and compare it with the reliability when none is applied.
When non-destructive testing is not applied after the burn-in process, the defective units are not detected and the reliability at any time, t, will be equal to the probability that the pin joint wear does not exceed the threshold failure threshold H When post-burn-in testing is applied then only those units whose wear is less than the failure threshold are marketed. Thus, reliability will be a bound probability and will be equal to the probability that the wear X(t) at any time does not exceed the failure threshold, H, given that it has successfully passed the burn-in procedure for a time t 0 . Mathematically, the reliability function will be given by where t 0 < t < τ, and τ is the replacement time.
Comparing the previous equations, we observe that the reliability with the application of non-destructive testing is equal to the reliability without testing divided by a number less than unity (the probability that the post-burn-in decay is less than the failure threshold). Therefore, the application of the non-destructive control increases the reliability value.

Quality and Reliability
Joint optimization of quality and reliability requires the definition of the measure to be used for this purpose and the factors to be used in this model. The factors considered in finding the optimal model are the time of the burn in process, b, the value of the wear, H', below which the system is considered to fail and the replacement time, τ. The measure that can be used consists of the sum of three different costs divided by the expected time of use. These costs are the quality cost, the expected failure cost, and the replacement costs.

Quality Cost
The unit cost of quality is given by where C Q (η) is the loss of quality, C S (η) is the cost of the rejected units, and C I is the cost of the inspection of each unit, which is usually fixed. The loss of quality can be calculated from Taguchi's loss function, in which there are three different types depending on the nature of the feature under consideration. These types are the smaller the better (S-type), the larger the better (L-type), and the better the target value (T-type). In the case of the pin joint the ideal is for the wear value to be as low as possible. Therefore, the relationship that characterizes the S-Type case is chosen which is where k is the coefficient used to convert the deviations into economic values. The quality loss can be calculated using the expected value of L(X(t 0 )) and will be given by where f X(t 0 ) is the probability density function at the end of the burn-in application. If a unit exceeds the failure limit, it shall either be repaired or rejected. If q(η) is the percentage of units that successfully pass the burn-in stage, then this percentage will be given by Moreover, if the scrap/reworked cost per unit is s then the scrapped portion is (1-q(η)) and the total expected scrapped cost will be Inspection costs, C I , are fixed and independent of the failure threshold η.

Failure Cost-Preventive Replacement Cost
The failure cost of each unit f C is fixed and s-independent of the time of failure and can be estimated by a one-year warranty cost or a one-time repair cost. Thus, the total Micromachines 2022, 13, 1899 7 of 15 failure cost depends on the reliability value at the time of replacement and is given by the relation In the case where we have no failure before the replacement time, the replacement based on policy will be done at that time at a replacement cost, RC. However, in case we have a failure before the scheduled time of replacement, then an additional replacement cost, RC, arises. In the more general case, the average expected failure cost will be given by the sum of the failure cost and the replacement cost. Otherwise, if it has not failed by τ, it should be replaced based on economic considerations, and the cost is RC. Thus, the expected total failure plus replacement cost at τ is FC(τ) + RC.

Expected Time of Use
The expected time of use is a function of the burn-in time, t 0 , and the replacement time, τ, and will be given by the following equation where f T (t) is the probability density function of the failure time for a Bernstein distribution with two parameters and will be given by the equation where and This is the pdf for a two-parameter Bernstein distribution.

Effect of the Warranty
A warranty is a contractual obligation of a manufacturer for the sale of its products. Manufacturers essentially declare that repair any damage that occurs for the duration of the warranty. But this creates an additional cost.
A product that has successfully passed the burn-in process is promoted to the market at the time, t 0 . Assuming that the micromachines are given a warranty period, w. If the product fails during its use at time t w (where t w < w) then under the supplier's contract with the customer, the supplier must replace it, and this produces an additional cost c 2 (t w ).
The expected cost for a product that comes onto the market after a burn-in period, t 0 , and has a warranty period, w, it is given by the relation where r b (t w ) is the reliability function after application of the burn-in procedure. However, if the failures occurring after the application of the burn-in process are repaired, the reliability of the product will be

Total Cost and Optimization
The expected total system cost per unit expected usage time including warranty is described as In fact, the upper bound of the replacement interval is usually specified and is denoted as B τ . To minimize this cost, the optimal options of burn-in time, t 0 , the replacement time, τ, the failure threshold, η, and the warranty interval, w, are sought. The constrained optimization model is expressed as subjected to The Sequential Quadratic Programming (SPQ) method [29] is employed to solve the optimization problem. This method is a technique used to find the optimal solution to non-linear problems consisting of several subproblems. The algorithm starts from an initial hypothesis and stops after a series of iterations when the criteria set are satisfied. Calculates the optimal solution of the subproblems for which a quadratic model of the function of the original problem.

Results and Discussion
This section presents the implementation of the proposed methods using numerical examples. Firstly, selecting suitable data, the levels of the decision variables that minimize the total cost are calculated, and then the effect of the decision variables on the total cost is examined.

Numerical Data
To apply the model and extract the desired results and conclusions, the numerical data are presented in Table 1. It has been assumed that the surface wear of the microengines follows the normal distribution. The source of the data is given by Tanner et al. [30] considering the coefficient c in Equation (1), the radius of the pin joint r, the nominal value of the force applied between rubbing surfaces F, the quality loss factor, the failure per unit cost f c , the replacement cost RC, and the rejection cost. Here, the replacement cost within warranty c 2 is also considered.   (20) and (21) applying the SQP method. Here, we notice that this test case and its results are directly comparable only with the ones presented in [1]. If we apply the values of the example in [1] and we simplify the model neglecting the warranty policies to be the same as the corresponding one in [1], the obtained results are identical.

Comparison of Quality Policies
In order to examine the effect of quality control after the application of the burn-in process, initially in this section the different values of the total cost without quality control, i.e., without a failure limit (η) of the defective units, will be calculated.
Firstly, the numerical model is applied, after having reset to zero the rejection threshold h, changing each time the value of the time interval of the burn-in application. The constant values selected are a replacement time τ = 50,000 revolutions, and a warranty period w = 25,000 revolutions. The burn-in time t 0 varies from 0 to 5000 revolutions, while the failure threshold is η = 0 µm 3 , due to the absence of quality control. The results are summarized in Figure 3a. We note that the application of a longer time interval of the burn-in process reasonably increases the total costs. In addition, we find a first insight into the values of the total costs incurred in the absence of quality control.
First, the values of the decision variables from which the model starts are given to calculate the cost values until the optimal solution are reached. The initial values selected are burn-in time = 200 revolutions, failure threshold η = 0.001 μm 3 , replacement time τ = 25,000 revolutions, and warranty period w = 10,000 revolutions. The values of the variables that minimize the total cost, which are derived from the application of the proposed model, are burn-in time * = 236 revolutions, failure threshold η * = 5.81 × 10 −3 μm 3 , replacement time τ * = 58,945 revolutions, warranty period w * = 11,340 revolutions. The total cost for the optimal solution is TC = 0.0404 €/unit. It is noticed that the above values obtained from the solution of the constrained optimization model expressed in Equations (20) and (21) applying the SQP method. Here, we notice that this test case and its results are directly comparable only with the ones presented in [1]. If we apply the values of the example in [1] and we simplify the model neglecting the warranty policies to be the same as the corresponding one in [1], the obtained results are identical.

Comparison of Quality Policies
In order to examine the effect of quality control after the application of the burn-in process, initially in this section the different values of the total cost without quality control, i.e., without a failure limit (η) of the defective units, will be calculated.
Firstly, the numerical model is applied, after having reset to zero the rejection threshold h, changing each time the value of the time interval of the burn-in application. The constant values selected are a replacement time τ = 50,000 revolutions, and a warranty period w = 25,000 revolutions. The burn-in time varies from 0 to 5,000 revolutions, while the failure threshold is η = 0 μm 3 , due to the absence of quality control. The results are summarized in Figure 3a. We note that the application of a longer time interval of the burn-in process reasonably increases the total costs. In addition, we find a first insight into the values of the total costs incurred in the absence of quality control.   To draw conclusions on the impact of quality control on the total cost, the proposed model is applied by considering the quality control expressed through the failure threshold. A failure threshold η = 0.001 µm 3 is chosen to calculate the total cost. The constant values selected are a replacement time τ = 50,000 revolutions, and a warranty period w = 25,000 revolutions. The burn-in time t 0 varies from 0 to 5000 revolutions, while the failure threshold is η = 0.001 µm 3 , due to the quality control process. The results are summarized in Figure 3b. From the above results we notice that the total cost increases with the increase of the burn-in time. However, we observe that while doubling the time of the burn-in process there is no meaningful change in the value of total cost. Specifically, by increasing by 1000 revolutions the burn-in time the total cost increases less than 5% in all cases. Moreover, we observe that, in this case, the absence of quality control can produce ten times (1000%) greater total cost than the one with quality control in most cases.
For a more detailed analysis of the results and to allow comparison of the results, the box plots for total costs as a function of the change in burn-in time when quality control is or is not applied are presented in Figure 4. We observe a sharp decrease in the cost values with application of quality control. Furthermore, the results obtained without quality control show a considerably larger dispersion. model is applied by considering the quality control expressed through the failure threshold. A failure threshold η = 0.001 μm 3 is chosen to calculate the total cost. The constant values selected are a replacement time τ = 50,000 revolutions, and a warranty period w = 25000 revolutions. The burn-in time varies from 0 to 5,000 revolutions, while the failure threshold is η = 0.001 μm 3 , due to the quality control process. The results are summarized in Figure 3b. From the above results we notice that the total cost increases with the increase of the burn-in time. However, we observe that while doubling the time of the burn-in process there is no meaningful change in the value of total cost. Specifically, by increasing by 1,000 revolutions the burn-in time the total cost increases less than 5% in all cases. Moreover, we observe that, in this case, the absence of quality control can produce ten times (1,000%) greater total cost than the one with quality control in most cases.
For a more detailed analysis of the results and to allow comparison of the results, the box plots for total costs as a function of the change in burn-in time when quality control is or is not applied are presented in Figure 4. We observe a sharp decrease in the cost values with application of quality control. Furthermore, the results obtained without quality control show a considerably larger dispersion. The same procedure is then repeated keeping all variables constant except for the replacement time interval, τ. The constant values selected are a burn-in time = 1,000 revolutions, and a warranty period w = 25,000 revolutions. The replacement time, τ, varies from 25,000 to 150,000 revolutions, while the failure threshold is η = 0 μm 3 , due to the absence of quality control. The results are summarized in Figure 5a. It is obvious that increasing the replacement interval leads the reduction of overall costs when no quality control is applied. The same procedure is then repeated keeping all variables constant except for the replacement time interval, τ. The constant values selected are a burn-in time t 0 = 1000 revolutions, and a warranty period w = 25,000 revolutions. The replacement time, τ, varies from 25,000 to 150,000 revolutions, while the failure threshold is η = 0 µm 3 , due to the absence of quality control. The results are summarized in Figure 5a. It is obvious that increasing the replacement interval leads the reduction of overall costs when no quality control is applied.
A similar procedure is repeated keeping all the factors of the model constant, each time changing the replacement time applying quality control. The constant values selected are a burn-in time t 0 = 1000 revolutions, a warranty period w = 25,000 revolutions, and a failure threshold η = 0.001 µm 3 . The replacement time varies again from 25,000 to 150,000 revolutions. The results are illustrated in Figure 5b. We can observe that with the increase of the replacement time the total cost as the replacement cost decreases. Furthermore, increasing the replacement time, it contributes significantly to reducing costs. Near at the optimal value of 58,945 rotations the total cost is about 0.85 €/item, while when the value is under sampled the total cost increases and approaches 7.5 €/item. This is because the sooner a component is replaced the less likely it will fail. These results suggest that increasing the preventive replacement time has a significant effect on the reduction of the total cost. The rate of reduction decreases with increasing replacement time.
The boxplot diagram in Figure 6 is then assembled from the results of previous analysis for the various values of the replacement time. In this case as well, we observe a reduction in total cost when quality control is applied as well as a smaller dispersion of the results appears. Moreover, we observe that, in this case, the absence of quality control can produce 100 times greater total cost than the one with quality control in some cases.  Figure 5b. We can observe that with the increase of the replacement time the total cost as the replacement cost decreases. Furthermore, increasing the replacement time, it contributes significantly to reducing costs. Near at the optimal value of 58,945 rotations the total cost is about 0.85 €/item, while when the value is under sampled the total cost increases and approaches 7.5 €/item. This is because the sooner a component is replaced the less likely it will fail. These results suggest that increasing the preventive replacement time has a significant effect on the reduction of the total cost. The rate of reduction decreases with increasing replacement time.
The boxplot diagram in Figure 6 is then assembled from the results of previous analysis for the various values of the replacement time. In this case as well, we observe a reduction in total cost when quality control is applied as well as a smaller dispersion of the results appears. Moreover, we observe that, in this case, the absence of quality control can produce 100 times greater total cost than the one with quality control in some cases.  Furthermore, the application of the model without quality control is implemented by varying this time the length of the warranty period provided by the manufacturer. The constant values selected are a burn-in time = 1,000 revolutions, and a replacement time τ = 50,000 revolutions. The warranty period, w, varies from 10,000 to 100,000 revolutions, while the failure threshold is η = 0 μm 3 , due to the absence of quality control. The results obtained are shown in Figure 7a. Ιt can be observed that the total cost increases with the increase in the total warranty period provided by the manufacturer.  Furthermore, the application of the model without quality control is implemented by varying this time the length of the warranty period provided by the manufacturer. The constant values selected are a burn-in time t 0 = 1000 revolutions, and a replacement time τ = 50,000 revolutions. The warranty period, w, varies from 10,000 to 100,000 revolutions, while the failure threshold is η = 0 µm 3 , due to the absence of quality control. The results obtained are shown in Figure 7a. It can be observed that the total cost increases with the increase in the total warranty period provided by the manufacturer.
Furthermore, the application of the model without quality control is implemented by varying this time the length of the warranty period provided by the manufacturer. The constant values selected are a burn-in time = 1,000 revolutions, and a replacement time τ = 50,000 revolutions. The warranty period, w, varies from 10,000 to 100,000 revolutions, while the failure threshold is η = 0 μm 3 , due to the absence of quality control. The results obtained are shown in Figure 7a. Ιt can be observed that the total cost increases with the increase in the total warranty period provided by the manufacturer.  The warranty period is also examined for its influence on total costs when quality control is applied. The constant values selected are a replacement time τ = 50,000 revolutions, and a burn-in period of t 0 = 1000 revolutions. The warranty time w varies from 10,000 to 100,000 revolutions, while the failure threshold is η = 0.001 µm 3 , due to the quality control process. The results are summarized in Figure 7b. From the presented results we additionally conclude that with an increase in the warranty period the total cost increases as more failures occur during this period which must be replaced by the manufacturer.
To further compare the effect of period of the warranty on the total cost, a corresponding boxplot is presented in Figure 8. And in this case, we come to the same conclusion, i.e., the noticeable reduction in total costs and in the less dispersion of results by applying quality control. The warranty period is also examined for its influence on total costs when quality control is applied. The constant values selected are a replacement time τ = 50,000 revolutions, and a burn-in period of t0 = 1,000 revolutions. The warranty time varies from 10,000 to 100,000 revolutions, while the failure threshold is η = 0.001 μm 3 , due to the quality control process. The results are summarized in Figure 7b. From the presented results we additionally conclude that with an increase in the warranty period the total cost increases as more failures occur during this period which must be replaced by the manufacturer.
To further compare the effect of period of the warranty on the total cost, a corresponding boxplot is presented in Figure 8. And in this case, we come to the same conclusion, i.e., the noticeable reduction in total costs and in the less dispersion of results by applying quality control. The effect of the quality control threshold, i.e., the different values of the failure threshold, on the total cost is finally examined. The constant values selected are a replacement time τ = 50,000 revolutions, a burn-in time 1,000 revolutions, and a warranty The effect of the quality control threshold, i.e., the different values of the failure threshold, on the total cost is finally examined. The constant values selected are a replacement time τ = 50,000 revolutions, a burn-in time t 0 = 1000 revolutions, and a warranty period w = 25,000 revolutions. The failure threshold varies from 0.0005 to 0.0100 µm 3 . The results are depicted in Figure 9.  From the above results we observe that as the failure threshold decreases the total cost increases, which is since the lower the failure threshold, the more units are rejected after the burn-in stage and are considered to fail.
An important observation concerns the dispersion of results. We observe that when there is no quality control the results show a large dispersion from the mean value and a large range from the minimum and maximum values, while when there is a quality control for all three decision variables it is observed that the results are clustered around the mean value. We also observe that increasing the time of application of the burn-in procedure only slightly increases the total cost with values close to 0.885 €/unit, since its application for a longer period contributes to the detection of more defective units which are not passed on to the consumer.

Conclusions
In this paper, a mathematical model with economic cost factors was developed based on the degradation of an important component of micromotors, the pin. This model could be applied to other components of micromotors, whether electrical or mechanical. The wear of micromotors increases with the number of revolutions of the pin and depends on three other factors, namely the radius of the pin, the hardness coefficient of the material and the force developed between the gear and the pin. Based on the calculation of the degradation relationship, which is exhibited by the micromotors and the rotation pin which is the contact point between the actuator and the gear, the various costs were calculated. Quality costs, failure costs, replacement costs and warranty costs were considered. Deriving the total cost expression, the values of the decision parameters that minimize the total cost were calculated using computational techniques.
Using a parametric analysis to study the sensitivity of the solution on the decision variables, the following conclusions are arisen:  the presence of quality control significantly reduces the total costs for all the decision variables. Concerning the numerical example, the total cost is reduced more than ten times (1000%) when quality control is applied in most cases. From the above results we observe that as the failure threshold decreases the total cost increases, which is since the lower the failure threshold, the more units are rejected after the burn-in stage and are considered to fail.
An important observation concerns the dispersion of results. We observe that when there is no quality control the results show a large dispersion from the mean value and a large range from the minimum and maximum values, while when there is a quality control for all three decision variables it is observed that the results are clustered around the mean value. We also observe that increasing the time of application of the burn-in procedure only slightly increases the total cost with values close to 0.885 €/unit, since its application for a longer period contributes to the detection of more defective units which are not passed on to the consumer.

Conclusions
In this paper, a mathematical model with economic cost factors was developed based on the degradation of an important component of micromotors, the pin. This model could be applied to other components of micromotors, whether electrical or mechanical. The wear of micromotors increases with the number of revolutions of the pin and depends on three other factors, namely the radius of the pin, the hardness coefficient of the material and the force developed between the gear and the pin. Based on the calculation of the degradation relationship, which is exhibited by the micromotors and the rotation pin which is the contact point between the actuator and the gear, the various costs were calculated. Quality costs, failure costs, replacement costs and warranty costs were considered. Deriving the total cost expression, the values of the decision parameters that minimize the total cost were calculated using computational techniques.
Using a parametric analysis to study the sensitivity of the solution on the decision variables, the following conclusions are arisen: • the presence of quality control significantly reduces the total costs for all the decision variables. Concerning the numerical example, the total cost is reduced more than ten times (1000%) when quality control is applied in most cases. • a large dispersion from the mean with a large range from the minimum and maximum values exists without a quality control, while when a quality control is implemented, the results are clustered around the mean for all the decision variables. • the increase in the burn-in time slightly increases the overall cost, while its application for longer time contributes to the detection of more defective units which are not passed on to the consumer. Concerning the numerical example, increasing by 1000 revolutions the burn-in time the total cost increases by less than 5% in all cases. • the increase in the replacement interval is observed to contribute significantly to the reduction of total cost since the sooner a component is replaced, the less likely it is to fail. • the increase in the warranty period provided by the manufacturer increases the total cost significantly.