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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.21 No.2 pp.390-400
DOI : https://doi.org/10.7232/iems.2022.21.2.390

Optimization of Renewable Warranty by Considering System Reliability and Preventive Maintenance and Repairs

Benjarut Chaimankong, Paitoon Chetthamrongchai*
Faculty of Business Administration, Kasetsart University, Bangkok, Thailand
*Corresponding Author, E-mail: fbusptc@ku.ac.th
March 20, 2022 ; March 20, 2022 ; April 12, 2022

Abstract


One of the effective ways to ensure the reliability of the product sold is to consider service and warranty contracts. Important decision variables in warranty policies include determining the optimal warranty period and the optimal number of maintenance activities. In this research, a model has been developed with the aim of achieving the lowest expected cost rate in the life of the device and the appropriate reliability by performing the optimal number of preventive maintenance measures. The warranty policy in this study is renewable in two dimensions, in which two dimensions of repair time and failure time are considered. Any damage leading to repair is done free of charge by the manufacturer and damage leading to replacement is done jointly between the manufacturer and the consumer by agreement. As a result, two algorithms of Genetic and Imperialist Competitive Algorithm have been developed to solve the model and have been compared with numerical example solving. Furthermore, the shelfs life of the system and the number of optimal repair and maintenance measures for maintaining reliability required by the buyer were obtained.



초록


    1. INTRODUCTION

    Product reliability has become an important issue due to the everyday and rapid growth of technology and serious competition in product marketing. One of the effective methods to ensure product or service reliability is to consider aftersales services. Despite today’s fast and advanced technology, product failure is still possible (Bai et al., 2021). Therefore, many products are sold with a warranty in today’s competitive market. in general, product warranties provide consumers with a written assurance of what a manufacturer will or will not cover in the event of product failure. Product failure may occur early, which might be due to production defects and product deterioration over time. Product failure depends on its age and method of use, and most products sold with a warranty protect buyers against early product failure during a warranty period. In addition, product failure can be controlled by preventive repairs and maintenance, which is important in the cases of warranty period extension (Xu and Saleh, 2021). Moreover, the warranty offered for a product may lead to additional production costs, which could be the cost of repair (corrective repairs) of the product failed during the warranty period. However, these costs could be reduced by preventive repairs during the warranty. From the perspective of manufacturers, preventive maintenance is valuable only when the reduction in warranty service costs is greater than the additional costs incurred in preventive maintenance (Kim et al., 2004). Moreover, most manufacturing companies aim to maxim- ize their profit through the market, which includes profit flow over time. Maximization may occur through an increase of sales to increase revenue or increase of product cost to increase unit profit. Overall, customers decide based on price, which reflects the product quality. However, most customers choose warranty information to assess the product quality and use it to determine whether the product has a fair price or not. Therefore, price and warranty are two important factors for sales decision and final profit of products. In other words, price is a direct and easy sign for product assessment by customers. On the other hand, warranty and its period have a direct effect on the manufacturing costs and final product price as a sign of quality (Lin and Shue, 2005).

    Shafiee and Chukova (2013) conducted one of the new review studies in 2013, which focused on models of repair and maintenance in warranty based on cm, pm and post-warranty divisions. In addition, Park et al. (2013) proposed the ARP and BRP maintenance and repair policy in renewable warranty. Moreover, Sheu (1998) evaluated two failure models subject to shock with two ARP and BRP policies. In another study, Jung et al. (2010) studied maintenance and repair costs related to the lifecycle of products under the renewable warranty policy. Yeh and Lo (2001) evaluated preventive maintenance and repair policies and warranty based on partial repair for repairable products. In the foregoing study, the period of warranty was predetermined and the optimal postwarranty period was estimated. Moreover, Jung and Park (2016) obtained the optimal periodic preventive maintenance and repair policy during the post-warranty period after the expiration of renewable and non-renewable warranties from the perspective of the manufacturer. Park and Pham (2010) evaluated the altered quasi-renewal concepts for modeling renewable warranty costs with imperfect repairs from the perspective of manufacturers. In a study by Wang and Pham (1999), some maintenance models and availability with imperfect maintenance in production systems were studied. Moreover, Bai and Pharm (2006) carried out a cost analysis on renewable full-service warranties for multi-component systems. In another study, Zhou et al. (2007) evaluated conditionbased predictive maintenance (CBPM) scheduling for a continuously monitored system independent from warranty policies. Tang and Lin et al. (2015) evaluated a nonperiodic preventive maintenance and repair policy with reliability thresholds for complex repairable systems. In his thesis, Najafabadi presented a warranty prediction model based on neural network systems at K. N. Toosi University of Technology. Wang et al. (2015) proposed an optimal preventive maintenance strategy for repairable items under a two-dimensional warranty. Moreover, Vahdani et al. (2013) presented warranty servicing for discretely degrading items with non-zero repair time under renewing warranty. Furthermore, Bai et al. (2021) introduced adaptive reliability and implemented it in single mode failure in reliability engineering, for which they used the particle swarm algorithm (PSA). Xu and Saleh (2021) applied machine learning to estimate the reliability and safety of complex systems. In this article, we specifically focus on deep learning and introduce it as an efficient reliability increase approach.

    2. STATEMENT OF THE PROBLEM

    It is crucial to study the relationship between warranty and repair and maintenance. Incomplete net preventive activities of the Effective age of the system are reduced by considering maintenance and repair, thereby improving reliability (Abedi et al., 2020;Afanasyev et al., 2021). In fact, net preventive activities make the system younger but do not change the degradation rate. After the original warranty period, preventive maintenance and repair activities are considered non-periodically. During the original warranty period, corrective measures are taken in case of product failure. First, the product is repaired but the process will be terminated and the product will be replaced if the repair time exceeds the predetermined limit. The warranty is not renewed at the time of repair, but the replaced product is considered new and the warranty will be awarded and renewed to the same extent as the original warranty period. moreover, expected costs are considered to be fixed during the maintenance activities. Therefore, the policy of renewed warranty of the combined free repair-shared exchange model is applied. Notably, the manufacturer uses a combination of two FRW and PRW warranties during the warranty period, such that each corrective measure followed FRW and all repair costs are paid by the producer, and each replacement will follow PRW policy in the form of PRO_RATA, and the cost will be divided between the producer and the buyer at the time of selling the product and according to age and based on the pre-agreed agreement. It is worth noting that repair is incomplete in the warranty period, which is an advantage for getting closer to the real world. In addition, while repair and maintenance are incomplete in the postwarranty period, any repair will be partial in that period.

    Furthermore, incomplete repair in the warranty period, which is below the specified repair limit, follows the qrp process and operates with a βi coefficient, which is called the age-reducing factor. Moreover, a twodimensional warranty model is considered, which is completely different from the conventional two-dimensional warranty model, which uses product age (actual time of age) and consumption (calendar time) as two factors affecting warranty policy. In addition, the number and time of repairs are used as the main criteria in the warranty policy. The lack of use of the two-dimensional warranty model is the inability to easily estimate the consumption rate in many systems. For instance, while determining the age of electronic devices, such as laptops, computers and refrigerators, or a specific device such as a nuclear reactor, is fairly easy, documenting their actual consumption rate is extremely difficult. In such situations, the use of a twodimensional warranty is not applicable. On the other hand, information is easily obtained based on the time of failure and time of repair for such systems to adopt a new twodimensional warranty policy. Therefore, the usual twodimensional issues (considering the two dimensions of age and time of use) are applicable for renewable issues that are renewed by changing the warranty. Since maintenance and repair are based on conditions in the postwarranty periods, conditions are considered as a variable, meaning that reliability will be added to the model as a problem decision variable and is obtained such that the lowest expected cost is obtained in a certain period.

    3. METHODOLOGY

    The problem model includes two warranty and postwarranty sections, and costs are evaluated from the customer’s perspective. In the first section, costs include replacement costs. However, since these costs are shared between the two parties, the part of costs paid by the consumer must also be estimated. In addition to the replacement costs, costs of system failure exist in the warranty period, which will lead to a cost for the consumer during the correction operations due to system failure. Costs in the post-warranty period include the incomplete implementation of preventive maintenance and repair and costs of partial repair and system failure. At the end of the period, the system will generally be replaced at the expense of the consumer.

    3.1 Mathematical Symbols

    A: Indices

    • i = the number of maintenance and repair services during the post-warranty period

    B: Parameters

    • CD The unit cost of failure in w and pw:

    • CR : The unit cost of replacement in w

    • βi : Age reduction factor in w period

    • r0 : Repair time threshold in w period

    • a1,a2 : The scale parameter in the density function in w period for the failure time and repair time, respectively

    • b1, b2 : The figure parameter in the density function in w period for the failure time and repair time, respectively

    • Na : Number of failures repaired in w period

    • Nb : Number of failures replaced in w period

    • Cm : The unit cost of incomplete maintenance and repair in pw period

    • C0 : Cost of replacement at the end of pw period

    • Cr : The unit cost of partial repair in pw period

    • δ : Age reduction factor in pw period

    • Yi : Effective age of the system before net activity in pw period

    • α : Scale parameter in density function in pw period

    • Beta : Failure rate in density function in pw period

    C: Variables

    • W : Warranty period

    • Nm : Number of scheduled repairs and maintenance in pw period

    • ri : Reliability threshold in the second scenario in pw period

    D: Definitions of some abbreviations

    • Cdf & pdf : Probability density function and cumulative distribution function

    • pm : Preventive maintenance and repair

    • NHPP : Non-homogeneous Poisson Process

    • ECR : Expected cost rate

    • Y,T : Time of failure and time of repair, respectively

    • F(t) & F(t),f (t), h(t) : Density function, pdf and cdf and reliability function of T failure time, respectively

    • G(y)&G(y), g(y) : Pdf and cdf and reliability function of Y repair time, respectively

    • FRW : Free repair Warranty

    • FPRW : Renewing Pro-Rata Replacement Warranty

    • E(na) : The number of expected repairs in w period

    • E(nb) : The number of expected replacements in w period

    • Pmf : Probability mass function

    3.2 Distribution of Number of System Failures

    N is the number of system failures, and N distribution and different statistical features caused by warranty cost function are estimated in time unit of sold products.

    In this section, pmf of the number of system failures is obtained by Equation 1 under replacement operation which is the same as with a complete repair.

    P [ N = n ] = [ F s ( w ) ] n R s ( w ) for n = 1 , 2 , 3 ,   ..
    (1)

    Each Fi,s (w) is different under each incomplete repair. Therefore, pmf of the number of system failures is obtained by Equation 2:

    P [ N = n ] = ( i = 1 n ( F is ( w ) ) ) ( R ( n + 1 ) s ( w ) )
    (2)

    3.3 Proposition of Warranty Modeling under Modified QRP

    In the main QRP, each failure has a pattern similar to X 2 = a . X 1 , X 3 = a . X 2 , X 4 = a . X 3 , ... , X n = a n 1 . X 1 .

    In this pattern, X1 represents the first failure within range, meaning that the repair time is decremental by α deduction, and α is smaller than one. QRP, which has the following pattern, is more explained below:

    Definition: a counting process {N(t), t>0}with F distribution and > 0 β n , β n random process and is fixed, and QRP is modified if X n = β n . X 1 and n = 2,3,..while β n وX 1 ~ F when n = 2,3,.. , and βn are necessarily not equal. In corrected QRP, pdf and cdf modified for n = 2,3,4,.. are in the form of equations 3 and 4:

    F n ( x ) = F 1 ( 1 β n    X )
    (3)

    f n (x)= 1 β n f 1 ( 1 β n    X)
    (4)

    The reliability function for i failure by using modified QRP is in the form of Equation 5:

    R is = 1 F is ( w ) = 1 f is ( x ) dx 1 1 β is    f 1 ( 1 β is    X ) dx
    (5)

    The system reliability function with n failures is based on equations 6 and 7:

    R s ( w ) = i = 1 n ( R is ( w ) )   ( 1 f 1 s ( x ) dx ) i = 2 n ( 1 1 β is    f 1 ( 1 β is    X ) dx) 
    (6)

    = ( f 1 s ( x ) dx ) ( i = 2 n ( 1 β is    f 1 ( 1 β is    X ) dx  ) ) ( 1 1 β ( n + 1 ) s    f 1 ( 1 β ( n + 1 ) s    X ) dx  )
    (7)

    E(nb) is the number of replacements in w. By using an expected value, the number of replacements is calculated based on Equation 8:

    E ( nb ) = nb = 1 nb [ ( i = 1 nb ( 0 w f is ( y ) dy ) ( 1 0 w f is ( y )  dy ) ) ]
    (8)

    E(na) the number of repairs in w. Using an expected value, the number of repairs is estimated based on Equation 9:

    E ( na ) = na = 1 na ( 0 w f 1 s ( x ) dx ) [ i = 2 na ( 0 w 1 β is f is ( 1 β is x ) dx ) ]   ( ( 0 w 1 β ( n + 1 ) s    f 1 s ( 1 β ( n + 1 ) s    x ) dx ) )
    (9)

    3.4 Life Cycle Duration during the Warranty Period (Original Warranty and Renewed Warranty)

    Since the warranty is renewed after each repair in the TL sections, each replacement is effective in the life cycle period and will be later explained based on Equation 10.

    IEMS-21-2-390_EQ10.gif
    (10)

    The number of replacement services is calculated by Equation 11:

    IEMS-21-2-390_EQ11.gif
    (11)

    By estimating the expected value on a conditional expected value corresponding to Nb, the expected life cycle is achieved based on the combined warranty model.

    IEMS-21-2-390_EQ12.gif
    (12)

    3.5 Estimation of System Costs during Warranty Period

    Costs of this section are divided into replacement and failure costs, respectively. In addition, the combined warranty model encompasses two policies of FRW (free repair warranty) and RPRW.

    3.5.1 Replacement Costs during Warranty Period

    As a part of the combined policy, replacements are carried out under the PMR policy, and costs are shared jointly between the two parties. The replacement costs from the perspective of consumers can follow the product age-dependent function of Tnb.

    C r = j = 1 Nb C r T nb W
    (13)

    IEMS-21-2-390_EQ14.gif
    (14)

    EC(w) = E ( E ( C ( w ) | Nb = nb ) ) = nb = 0 { E ( E ( C ( w ) | Nb = nb ) P ( Nb = nb ) }
    (15)

    IEMS-21-2-390_EQ16.gif
    (16)

    3.5.2 System Failure Costs during Warranty Period

    Based on this part of the policy, the repair costs must be paid by the manufacturer and the consumer will incur no costs for repair. As mentioned above, the product will be replaced if the repair time exceeds the allowed repair time, and the warranty will be renewed by replacing the product. This continues until reaching w age. If F(t) is the age distribution function and f(t) is the probability density function of t, Fi,s (t) will be the cumulative distribution function (cdf) of system failure times after (i-1) repairs during w period. Each Fi,s (w) will be different by considering incomplete repairs. The number of repairs and replacements must be determined to calculate the failure costs. This is mainly due to the fact that the system stops for a certain time with each number of repairs. In addition, while the repairs are not time-consuming, because the repair time spent as much as the amount of time is not counted in the number of repairs performed for the number of replacements made, the calculation method will be, as follows, where the number of failures (repair + replacement) is calculated. Therefore, failure costs will be equal to:

    Total Expected Downtime Cos = ( E ( na ) + E ( nb ) ) × CD
    (17)

    3.6 Estimation of Post-Warranty Period Costs

    In this section, costs include incomplete preventive maintenance and repair costs and costs of doing partial repair and system replacement costs at the end of the period.

    3.6.1 Costs of Incomplete Preventive Maintenance and Repair Implementation

    In the post-warranty period, the maintenance and repair measures reduce the effective age of the system, thereby improving system reliability. Therefore, the system becomes younger by performing pm measures but the degradation rate cannot be reduced. The system’s age will reduce by different age reduction factors φ with incomplete maintenance and repair activities. It is assumed that the PAS model can be performed to describe system failure status following pm measures based on the knowledge and experience of failure engineers, and the φ reduction factor has an incremental sequence. This is used to reflect failure conditions in the system that become worse, and pm measures will be used after worsening of the conditions if pm measures are performed with similar costs. This assumption is logical since spending similar costs in each pm measure can cause the system in a repairable system to have reduced recovery rates. Accordingly, the effective age of the system is obtained before the i-th pm measure using the following equation:

    y i = w + x i + φ i 1 y i 1 = w + x i + φ i 1 x i 1 + + φ i 1 φ i 2 φ 1 x 1
    (18)

    when 0 = φ 1 < ... < φ Nm 1 < 1 and i =1, 2,…, Nm , assuming that y0 = w and y1 = w + x1, therefore, xi is the period between the i-th and (i-1)-th pm measure φiyi and the effective age representative of the system immediately after the i-th pm measure. It is assumed that in postwarranty period, the failure process follows NHPP given the negligible time for corrective measures, and the system fails with an equal density function between pm measures. The expected number of failures during the life cycle of the system can be achieved using Equation 19:

    N r = w y 1 λ(t)dt + φ 1 y 1 y 2 λ(t)dt + + φ Nm 1 y Nm 1 y i λ(t)dt = [ w β + i = 1 Nm y i β ( φ i 1 y i 1 ) β ]
    (19)

    Therefore, the total expected value in time unit required for performing pm measures is obtained based on Equation 20:

    C ( Nm ) = 1 T L [ C 0 + C r N r + ( Nm 1 ) C m ]
    (20)

    Cm(Nm1) is the cost of performing pm measures during the post-warranty period.

    In addition, the system’s lifetime is estimated in post-warranty mode using the total time intervals for pm cycle based on Equation 21:

    T L = i = 1 Nm x i = i = 1 Nm y i i = 1 Nm φ i 1 y i 1 = y Nm + i = 1 Nm 1 ( 1 φ i ) y i
    (21)

    It is notable that the post-warranty costs include the repair costs, pm activity performance costs, and total system replacement costs at the end of the period. These costs are fixed and can be obtained by assessing failures by the engineering section. Therefore, it is logical to assume that C0> Cm.

    3.7 System Reliability as a Variable Condition

    As discussed before, it is logical to consider the conditions to be variable to achieve reliability when pm models are based on conditions. In general, reliability can be the possibility of the system working during some of the t periods. If a positive continuous random variable T is defined as the system failure time, the system reliability can be obtained according to the following equations:

    R ( t ) = Pr { T t }
    (22)

    When lim t R(t) = 0 , R(0) = 1 , R ( t ) 0 . Since the failure process has equal density function λ(t) = αβt β 1 based on NHPP assumption

    R ( t ) = Pr { N ( t ) = 0 } = exp [ 0 t λ ( u ) du ] = exp ( αt β )
    (23)

    R(t) is the system reliability without pm measures, and R m ( i ) ( t ) represents system reliability immediately after the i-th pm activity. It is clear that R m ( 0 ) ( t ) = R ( t ) = exp ( αt β ) when 0≤ t < y1.

    Moreover, the system has a y1 age after the first pm measure. In addition, under the first pm measure at y1 age, reliability of the system is equal to:

    R m 1 ( t ) = R ( y 1   ) R m ( t | y 1   ) = exp [ ( w y 1 λ ( u ) du + φ 1 y 1 t λ ( u ) du ) ] = exp { α [ w β + t β + ( 1 φ 1 β ) y 1 β   ] }
    (24)

    For φ1y1 ≤ t < y2, when R(y1) can remain intact (without failure) until the first pm measure.

    R m ( t ) = exp [ ( φ 1 y 1 t λ ( u ) du ) ]
    (25)

    The possibility of remaining intact for the additional time is t−φ1y1, where the system is maintained until y1. The system’s effective age can be reduced immediately to φ1y1 after the first pm measure with the effect of maintenance and repair, and system reliability is the possibility of remaining intact during the y1+(t−φ1y1) period. In addition, system reliability immediately after the (i-1)-th pm measure is obtained, as follows:

    R m ( i 1 ) ( t ) = R ( y 1 ) R m ( y 2 | y 1 ) R m ( t | y i 1 ) = exp { α [ w β + t β + j = 1 i 1 ( 1 φ j β ) y j β ] }
    (26)

    In mode of φi1yi−1 ≤ t < yi at the design and planning stage, a suitable strategy that includes the assessment of a system failure behavior is required. Relative costs of maintenance and repair are critical for the decision-maker. The basic assumptions of the study were discussed in the previous section of the research. Two pm models based on reliability are proposed for repairable perishable systems and are assessed according to optimal solutions and appropriate features. In this section, by taking more pm measures to keep the system in the highest state of reliability, it is clear that costs will increase because with performing each pm measure, the system will be out of reach and iteration of operations will be costly. Therefore, it is important for decision-makers to achieve these variables at the highest reliability level, where incurred costs could be managed as well. The goal is to find optimal reliability and the optimal number of maintenance and repair measures in order to minimize the total expected costs. In this regard, the main objective for the decision-maker is to establish a maintenance and repair strategy post-warranty after finishing the warranty periods. Using equation – and considering that reliability is equal to rc in this scenario:

    y i = ( ln r α ) + w beta beta C m ( N m 1 ) + C 0 + ( C r + CD ) × ( w beta + i = 1 N m [ y i beta ( δ i 1 y i 1 ) beta ] )
    (27)

    W + ( ( lnr α ) + w beta beta ) i = 1 N m 1 ( 1 δ i )
    (28)

    • A) Objective Function Numerator:

      ( E ( na ) + ( E ( nb ) ) × CD ) + ( CR W × ( 0 w xf ( x ) dx 1 ( ( 0 w f ( x ) dx ) ( r 0 f ( y ) dy ) ) ) ) + C m ( N m 1 ) + C 0 + ( C r + CD ) ( ln ( r ) + αw beta ) ( i = 0 N m 1 ( 1 δ i beta ) )
      (29)

    • B) Objective Function Denominator:

      0 w xf ( x ) dx 1 ( ( 0 w f ( x ) dx ) ( r 0 f ( y ) dy ) ) + 0 w xf ( x ) dx 1 ( ( 0 w f 1 s ( x ) dx ) [ i = 2 na ( 0 w 1 β is f is ( 1 β is x ) dx ) ] ( 0 r 0 f ( y ) dy ) ) + w + ( ( lnr α ) + w beta beta ) i = 1 N m 1 ( 1 δ i )
      (30)

    3.8 Introduction of Functions Used in Two Warranty and Post-warranty Periods

    The functions used in the warranty period are Weibull distribution and the function used in the postwarranty period is the power law density function. It is assumed that system failure time follows Weibull distribution with different parameters. Product reliability is affected by Weibull distribution parameters that are extensively used in reliability engineering since other distributions (e.g., exponential, Rayleigh and normal) are specific examples of Weibull distribution. Moreover, since its reliability allows it to be an accurate representation of a variety of life distributions, each function parameter is calculated by the maximum likelihood estimation (MLE) method, and the age reduction factor is calculated using the equation below:

    φ i = i ( 2 *i + 1 )
    (31)

    According to the mentioned equations, the PAS model is used for failure modeling based on failure engineering knowledge and experience, which can be used for system failure status after each pm measure. In addition, the age reduction factor, which is an increasing trend for reflecting conditions where system return becomes worse, will be suitable for after each pm measures that are performed with equal costs. This assumption is logical since spending equal costs in each pm measure reduces the returning degrees of repairable perishable system.

    5. RESULTS AND DISCUSSION

    The model presented in the previous section is of MINLP type, which generally includes the simultaneous optimization of discrete and continuous values and is of NP-hard problems due to including a high number of parameters that have strong and non-linear interactions with each other. For such problems, as the number of variables increases, the response space grows exponentially. Conventional solution methods such as branch and bound are not adequate for real-size problems that have multiple variables since their solution time is not justifiable. Due to the high solution time and the lack of an optimal global solution to large-scale problems, this section presents a genetic algorithm (GA) and imperialist competitive algorithm to solve the model since the complexity of the problem will have no significant impact on the performance of metaheuristics methods. The mentioned algorithms are commonly used as optimization functions. Using the Taguchi method and trial and error method, the values of the GA parameters are estimated, as follows (Table 1):

    Given the use of imperialist competitive algorithm along with GA, the values of the number of iterations and the number of generations are the same in order to compare the two algorithms fairly. Other parameters of the algorithm, which are presented below, are considered equal in different examples (Table 2):

    While the convergence of the two algorithms is suitable, the reliability value is rounded by the imperialist competitive algorithm since it is a decimal number and its precision is of paramount importance. However, rounding does not lead to a precise value. Nevertheless, the two algorithms yield very close solutions. According to Table 3, the reliability value (r), the number of maintenance and repair measures (nm) and duration of post-warranty (w) are estimated in seven warranty periods and the results are compared. Expected cost rates of the warranty period have a decreasing trend in 0.2-0.75 periods and an uptrend in 0.75-1.5 periods and reliability have different values according to the warranty period. As the warranty period increases, it is expected that the number of maintenance and repair activities and the duration of the postwarranty period will decrease in order to reduce costs, which is established in all periods except for the maximum warranty period (i.e., 1.5), which, contrary to what is expected, has the longest post-warranty period and the lowest cost rate with the lowest reliability that is achieved at the highest intervals of incomplete preventive maintenance planning and maintenance. When the costs of maintenance and repair measures decrease in the post-warranty period, the cost rates are much less than cm=10 . The optimal cost rate occurs in a 0.4 period, where the number of maintenance and repair activities have the highest possible value and reliability has the highest value. If the 1.5 period is not considered, it will have a high postwarranty period compared to other periods.

    According to Figure 1, the objective function extremely increases with an increase in the weight value. In addition, the imperialist competitive algorithm always presents a lower value, compared to the GA. According to Figure 2, reliability decreases with the increase in weight. In addition, the imperialist competitive algorithm yields higher reliability values, which shows the superiority of this algorithm compared to the GA. The sensitivity analysis of the costs of maintenance and repair measures in post-warranty is carried out in Tables 4. The rate of costs is much less than cm=10 when there is a reduction in the cost. In addition, the optimal cost rate occurs in 0.4 period, where there is the highest number of maintenance and repair activities and reliability has the highest value. If the 1.5 period is not considered, it will have a higher post-warranty period compared to other periods. The total expected cost increases with an increase in the maintenance and repair costs. In addition, the lowest rate is observed in the 0.75 warranty period, which has the highest reliability and the lowest number of activities and postwarranty period.

    The sensitive analysis of the cost of replacement in the warranty period (CR) is carried out in Tables 5 and 6. The expected cost rate is less than other periods when the mentioned value is lower. In addition, the lowest cost is observed in the 0.75 period, and its reliability is higher than warranty periods less than 0.75. The increase of these costs leads to a slight increase in the expected cost rate, and the lowest rate is observed in the 0.75 period.

    5. CONCLUSION

    Equipment reliability is increasingly important for buyers to reduce sudden failure costs, increase the lifespan of pieces, increase competitiveness, timely meeting of demands, and maintain corporate credibility. Failure may occur due to technical defects, poor performance, equipment lifespan, or environmental and operational stresses. On the other hand, the manufacturer's warranty guarantees the buyer against product failure. In fact, the manufacturer can minimize the effects of such failures by ensuring after-sales service by determining the appropriate warranty period and service contracts. However, inappropriate determining of the warranty period can lead to manufacturer or buyer’s loss instead of profit. The present study evaluated costs from buyers' perspectives, and attempts were made to determine the system’s shelf life.

    Given that the seller will bear a part of device repair costs during the sales period, the buyer will benefit from the highest efficiency of the system with the lowest costs. The presented model was solved for numerical examples using GA and imperialist competitive algorithm in the current study. The number of repairs and maintenance activities were assessed in warranty periods 0.2-1.5. The system's shelf life and the number of optimal repair and maintenance measures for maintaining reliability required by the buyer were obtained to minimize the total expected costs.

    Figures

    IEMS-21-2-390_F1.gif

    A comparison of GA and imperialist competitive algorithm in terms of objective function.

    IEMS-21-2-390_F2.gif

    A comparison of GA and imperialist competitive algorithm in terms of reliability.

    Tables

    GA parameter values

    Imperialist competitive algorithm parameter values

    Evaluation of expected cost rate in different warranty periods

    Evaluation of expected cost rate in different warranty periods in cm = 15

    Evaluation of the expected cost rate in different warranty periods in CR = 30

    Evaluation of expected cost rate in different warranty periods in CR = 50

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