A stochastic simulation-optimization model for base-warranty and extended-warranty decision-making of under- and out-of-warranty products

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A stochastic simulation-optimization model for base-warranty and extended-warranty decision-making of under- and out-of-warranty products

Mohsen Afsahi ConceptualizationMethodologySoftwareValidationFormal AnalysisInvestigationData curationWriting Mahmood Shafiee ConceptualizationMethodologySoftwareValidationFormal AnalysisInvestigationWriting - Review PII: DOI: Reference:

S0951-8320(19)30344-8 https://doi.org/10.1016/j.ress.2019.106772 RESS 106772

To appear in:

Reliability Engineering and System Safety

Received date: Revised date: Accepted date:

4 March 2019 17 November 2019 11 December 2019

Please cite this article as: Mohsen Afsahi ConceptualizationMethodologySoftwareValidationFormal AnalysisInvestig Mahmood Shafiee ConceptualizationMethodologySoftwareValidationFormal AnalysisInvestigationWriting - Review A stochastic simulation-optimization model for base-warranty and extended-warranty decision-making of under- and out-of-warranty products, Reliability Engineering and System Safety (2019), doi: https://doi.org/10.1016/j.ress.2019.106772

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HIGHLIGHTS      

A stochastic simulation-based optimization model to jointly determine the length of base- and extendedwarranty, product price, extended warranty price, repair strategy, and the spare parts production planning; A new imperfect repair policy by combining the (p,q) rule and Kijama‘s virtual age; A hybrid teaching-learning based optimization (TLBO), Monte-Carlo simulation (MCS) and dynamic programing (DP) algorithm to solve the problem; A hybrid adaptive particle swarm optimization (APSO), Monte-Carlo simulation (MCS) and dynamic programing (DP) algorithm to validate the results; A case study involving vacuum cleaners to illustrate the developed model and solution approach; A sensitivity analysis to evaluate the influence of key parameters on the optimal solution.

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A stochastic simulation-optimization model for basewarranty and extended-warranty decision-making of under- and out-of-warranty products Mohsen Afsahi 1 , Mahmood Shafiee 2* 1

Department of Industrial Engineering, University of Science and Culture, Tehran, Iran

2

Cranfield University, College Road, Bedfordshire MK43 0AL, United Kingdom

*

Corresponding author. Email: [email protected], Tel: +44 1234 750111

Abstract In recent years, product warranties (including base-warranty and extended-warranty) have become an integral part of marketing strategy for most manufacturers as well as an important part of purchasing decision for most customers. It is crucial for manufacturers or dealers who offer warranty to their customers to optimize decisions regarding price of products, durations of base-warranty and extended-warranty, price of extended-warranty, etc. On the other side, manufacturers must decide on a cost-effective imperfect maintenance strategy and spare part inventory policy for their under-warranty and out-of-warranty products. In order to solve these decision-making problems, this study proposes a novel stochastic simulation-based optimization (SBO) model with the objective of maximizing the manufacturer‘s profit. A metaheuristic Monte-Carlo simulation algorithm integrated with a dynamic programming approach is also presented to solve the model. A case study of vacuum cleaners is provided to illustrate the developed model and its solution procedure, and the simulation results are verified with real data. Finally, a sensitivity analysis is performed in order to evaluate the impact of key parameters on the optimal solution. Our analysis shows how different planning horizons can affect warranty-related decisions for both the manufacturers and customers, providing valuable insights to corporate decision-makers.

Keywords: Base-warranty (BW); Extended-warranty (EW); Simulation-based optimization (SBO); Pricing; Spare part inventory; Imperfect repair.

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1. Introduction With rapid changes of technology and fierce competition in the market, products have become more and more complex and, therefore, customers will need an indicator enabling them to differentiate products. Warranty plays an important role in customers‘ decision-making as it is perceived as a signal of product quality. Manufacturers can increase their market share—and thereby increase their revenue—by offering long-term warranty coverage to customers. However, providing long warranty periods on products will result in high repair or replacement costs for the manufacturer or service provider. In general, these costs may vary between 2 and 10 percent of the product‘s sales price, depending on the brand of the product (Murthy, 2007). A warranty is a contractual agreement between a manufacturer (seller) and a customer (buyer) which requires the manufacturer to rectify any failures occurring during the contract period (Shafiee et al., 2014). So, warranty serves a dual role. First, it protects customers from product failures, and second, it protects manufacturers against unreasonable claims from customers (Blischke and Murthy, 1992; Shafiee et al., 2009). In the literature, researchers classify product warranties into two types of base-warranty (BW) and extended-warranty (EW) (Jack and Murthy, 2007). The BW is an integral part of product sale and its cost is factored into the sale price, whereas the EW is an optional contract which extends the period of warranty coverage after the expiration of the BW. In EW contracts, the manufacturers often offer additional services than BW for a premium cost to consumers. The EW not only provides an excessive protection for consumers but also generates profit margin for manufacturers. Nowadays, many manufacturers such as Whirlpool, Kenmore, Samsung, LG, Bosch, and General Electric offer an EW for their products such as fridge, washing machine, dishwasher, gas stove, vacuum cleaner, microwave oven, etc. There are a number of marketing and engineering decisions which must be made by manufacturers in an optimal way such that their overall profit is maximized (Chukova and Shafiee, 2013). The first important decision for a manufacturer to make is to determine the optimal length of the BW period for different product types. Although a long BW increases the willingness of customers to buy a product, it may bring significant costs to the manufacturer. The second key factor affecting customers‘ purchasing decision is the sales price. The third and fourth decision parameters are associated with the EW policy. The length and price of an EW contract should be determined in a way such that the manufacturer‘s expected profit from selling the EW is maximized. Manufacturers who offer warranty often provide also repair and maintenance services to customers to ensure products operate at optimal performance during the contract period (Shafiee and Chukova, 2013a). In general, maintenance is classified into two main types: corrective and preventive. Corrective maintenance (CM) occurs after item‘s failure to restore it to an operational state, whereas preventive maintenance (PM) is performed before an item fails in order to control its degradation and reduce the failure rate. Therefore, an important decision for manufacturers who produce repairable items and offer CM service to their customers is to determine the most effective repair policy for failed products. In many 3

practical situations, either a ―minimal‖ repair or a ―perfect‖ repair is employed by the manufacturer. Perfect repair returns the system to the ―as-good-as-new (AGAN)‖ condition, whereas minimal repair restores the system to its condition just prior to failure, i.e. ―as-badas-old (ABAO)‖. When product failures are rectified by a perfect repair, manufacturers will usually face significant repair and replacement costs. On the other side, when product failures are rectified by a minimal repair the customers may not be satisfied with the service they receive. Under such a situation, the manufacturer should adopt a CM policy by combining the minimal and perfect repairs with another kind of repair called ―imperfect‖. An imperfect repair restores the product to a condition between ABAO (minimal repair) and AGAN (perfect repair) (Shafiee et al. 2011). Therefore, the fifth and sixth decision parameters include the proportion that each type of repair (minimal, imperfect and perfect) contributes to the total number of repairs and the optimal rejuvenation level of imperfect repair actions. The last but not least decision for manufacturers to make about under-warranty and outof-warranty products is related to spare part inventory. In order to optimize the quantity of spare parts kept as inventory to support warranty claims, the manufacturers should accurately analyze the expected number of failures occurring at any given time interval of the planning horizon using time-variant reliability methods. In this study, we propose a novel stochastic simulation-based optimization (SBO) model to determine the optimal length of BW and EW periods, EW price, product‘s selling price, proportion of each type of repair action, rejuvenation level of imperfect repairs, and the demand of spare parts for under- and out-of-warranty products. A new repair policy by combining the (p , q) approach (proposed by Makis and Jardine, 1992) with the virtual age concept (proposed by Kijima, 1989) is developed. Under this policy, a failed product undergoes a perfect repair with probability P1 , an imperfect repair with probability P2 , and a minimal repair with probability P3 , where 0 ≤ P1 , P2 , P3 ≤ 1 and P1 + P2 + P3 = 1. These probabilities are all considered as decision variables under the control of the manufacturer which should be calculated so that the total expected cost is minimized. A solution approach based on a metaheuristic Monte-Carlo simulation (MCS) algorithm integrated with the dynamic programming (DP) approach is presented to solve the model. A case study of vacuum cleaners is provided to illustrate the developed model and its solution procedure, and the simulation results are verified with real data. Finally, a sensitivity analysis is performed in order to evaluate the impact of key parameters on the optimal solution. The rest of the paper is organized as follows. Section 2 reviews the existing literature of warranty policy optimization for consumable products. A description of the model, its assumptions and constraints are presented in Section 3. The optimization model with the objective of maximizing the manufacturer‘s total net profit is formulated in Section 4. The solution approach to solve the optimization problem is presented in Section 5. Section 6 provides a case study to illustrate the practicality of the model. Finally, some concluding remarks and suggestions for future research are provided in Section 7.

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2. Literature review In spite of a considerable number of studies on the warranty policy optimization for consumable products, there are a few papers considering various aspects of product marketing and engineering into an integrated decision support framework. This section briefly reviews four streams of warranty policy optimization, including: product pricing and warranty duration optimization, extended warranty optimization, warranty inventory optimization, and warranty analysis under imperfect repair policy. 2.1. Product pricing and warranty duration optimization The sale price and BW duration are two key factors affecting the manufacturers‘ profit and customers‘ willingness to buy a product. Therefore, it is necessary for manufacturers to determine the selling price and the length of the BW period for their products in a way such that the best possible balance between profit and market share is achieved. Glickman and Berger (1976) proposed a model to maximize the manufacturer‘s profit gained from products sold under warranty. The study modelled the product‘s demand as a function of two decision variables of sales price and BW duration. Menezes and Currim (1992) developed a model to optimize the duration of coverage for two types of warranty policies, including a nonrenewing free replacement warranty (NFRW) and a non-renewing pro-rate warranty (NPRW). Lin and Shue (2005) maximized the manufacturer‘s profit by determining the optimal selling price and BW length for products with different lifetime distributions and sold under a NFRW policy. Wu et al. (2006) presented a model to determine the optimal selling price and BW length for a normal lifetime distributed product sold under a renewing free replacement warranty (RFRW) policy. Huang et al. (2007a) considered product reliability, BW duration and product price as three decision variables for maximizing the manufacturer‘s profit. Kim and Park (2008) proposed a two-stage optimal control model to determine the selling price, BW duration and spare parts inventory of products sold with a NFRW policy in an integrated and optimal way. They assumed that the planning horizon has been divided into product‘s life cycle period and end-of-life period. Li et al. (2009) formulated a profit maximization model to determine simultaneously the product‘s price, BW length and production rate. They also proposed a model-driven decision support system to provide a graphical user interface (GUI) for overcoming the complexity of the analytical solution approach. Wu et al. (2009) developed a model to determine the sales price, BW duration, and production rate for products sold under a RFRW policy. Zhou et al. (2009) proposed a model for maximizing the manufacturer‘s profit by optimizing the product‘s selling price and BW length. They assumed that the manufacturer can change the length of the BW period during the product life cycle. Jalali-Naini and Shafiee (2011) proposed a decision-making model to determine the optimal selling price and upgrading strategy for used (second-hand) products sold with a warranty. Chen et al. (2012) examined the effect of various warranty pricing strategies set by a manufacturer and two competing retailers in a two-echelon supply chain on customers‘ purchasing willingness. Shafiee and Chukova (2013b) developed a model to determine the optimal upgrading strategy, warranty policy and sales price for second-hand 5

products such that the manufacturer‘s net profit was maximized. Yeh and Fang (2015) proposed a mathematical model to determine the sales price, BW duration and PM schedule for products sold with a NFRW policy. They assumed that the number of product failures follow a non-homogeneous Poisson process (NHPP). Yazdian et al. (2016) proposed a mathematical model to jointly optimize the selling price of used products, degree of their remanufacturing process and the length of their warranty period. Lei et al. (2017) proposed a model to determine the optimal price of the product (and its warranty) when customers have the choice whether or not to buy the warranty offered by the manufacturer. Giri et al. (2018) considered a two-echelon closed-loop supply chain network and developed two gametheoretic models to analyze the pricing strategies, warranty period and greening strategy. In the first model the demand was modelled as a function of product‘s selling price and BW‘s duration, while the second model considered the demand also dependent on greening level. Luo and Wu (2018) proposed a product warranty policy optimization framework to minimize the costs associated with four sources of warranty claims, including: software failures, hardware failures, software-hardware interaction failures, and human errors. Afsahi et al. (2018) developed a mathematical model to determine the optimal selling price, BW duration and spare parts inventory for under- and out-of-warranty products. They solved the model by a two-stage algorithm which was a combination of a metaheuristic algorithm and the out-ofkilter method. Zhu et al. (2018) proposed an optimization model to determine jointly the optimal product reliability, warranty policy, regular price, promotion price, and lengths of regular sales and promotions. 2.2. Extended warranty optimization There is a relatively small body of literature on extended warranty (EW) optimization. Padmanabhan (1995) proposed a model to optimize the duration of BW and EW periods for consumer durable products in monopolist markets. Lam and Lam (2001) developed a game theory model to determine the optimal EW duration for products in the case where customers are given the choice to renew the warranty at the end of free repair period. The mathematical equations for total discounted cost as well as long-run average cost per unit time were derived for both the customer and manufacturer. Jack and Murthy (2007) determined the optimal pricing strategy for EW providers and optimal maintenance and replacement strategy for customers using a game theory approach. Hartman and Laksana (2009) proposed a dynamic programming approach to evaluate different menus of EW contracts for a customer with perfect information and determine the optimal pricing strategy for the manufacturer. Wu and Longhurst (2011) formulated the expected life cycle cost of a product sold with both BW and EW from a customer‘s perspective. They assumed that the product is subject to two types of failure: minor and catastrophic. Minor failure is rectified by a minimal repair, whereas catastrophic failure is rectified by a perfect repair or replacement. Bouguerra et al. (2012) developed a mathematical model to examine different maintenance strategies under a case where customers can decide whether or not to buy an EW after the expiration of BW. Tao and Zhang (2015) determined the optimal length of EW period for products subject to PM, such that the manufacturer‘s profit was maximized. Bian et al. (2019) introduced a new EW 6

contract under which an additional trade-in service is provided during the warranty coverage. They developed a framework for manufacturers or retailers to help them select and optimize the best strategy for the proposed EW offer. 2.3. Warranty inventory optimization In order to design an optimal warranty inventory management system, the manufacturers should analyze the expected quantity of spare parts needed for under-warranty and out-ofwarranty products. This analysis has been addressed little in the warranty literature. Wang and Sheu (2003) calculated the optimal production lot size for repairable products sold under a NFRW policy. Khawam et al. (2007) formulated several models for warranty inventory management in Hitachi Global Storage Technologies (HGST). The models incorporated three sources of spare parts supply three different sources of product, including testing, remanufacturing, and new production. Huang et al. (2007b) proposed a multi-period single product inventory model for a manufacturer who supplies the parts demand from two sources, including new product and failed product under warranty. Huang et al. (2008) extended their previous study by incorporating sales as well as age information about the products into inventory management problem. Tsao et al. (2014) proposed a model for optimization of the pricing strategy as well as inventory control policy for high-tech products. Calmon (2015) developed mathematical models to optimize reverse logistics activities for products sold with BW in the consumer electronics market. The models considered a setting where there were two warranties in place: (i) a warranty offered by the retailer to the consumer, and (ii) an Original Equipment Manufacturer (OEM) warranty offered by the OEM to the retailer. 2.4. Imperfect repair Many imperfect repair models have been proposed in the literature. A summary of the basic imperfect repair models is presented in Table 1. One of the best known imperfect repair models is suggested by Brown and Proschan (1983), where at the time of each failure a perfect repair occurs with probability p and a minimal repair occurs with probability 1-p. Table 1. A summary of the basic imperfect repair models.

*Table 1* The imperfect repair models presented in Table 1 have been improved or combined with other methods in the past. Jack and Dagpunar (1994) determined the optimal number of PM actions over the BW period in a case where only a minimal repair is performed after each failure. Chukova et al. (2004) presented a framework for the analysis of the lifetime of a system subject to six types of repair, including: improved repair, complete repair, imperfect repair, minimal repair, worse repair and worst repair. Bai and Pham (2005) introduced a new warranty policy called repair-limit risk-free warranty and then determined a threshold value on the number of repairs. Yun et al. (2008) proposed two warranty servicing strategies involving minimal and imperfect repairs for repairable products. In the first strategy, the level 7

of reliability improvement under imperfect repair was dependent on the age of the product, whereas in the second strategy it was independent of the age. Park and Hoang (2010) developed two warranty cost analysis models for multi-component systems based on alteredand mixed-quasi-renewal processes. Liao (2016) developed an economic production quantity (EPQ) model for a hot standby redundant system under a PM strategy restoring the products and a NFRW policy. Zhang et al. (2018) analyzed the expected warranty costs for a twocomponent series system with stochastic dependence between components, sold under a NFRW policy. Zhao et al. (2018) considered an improvement factor for imperfect repairs and minimized the expected warranty cost for products sold with a NFRW policy. Chien et al. (2019) analyzed the cost of a free-repair warranty policy with a PM schedule for complex products. They utilized a generalized Polya process to model the failures and repairs during the BW period. Chien (2019) modeled failures and repairs by a non-homogeneous pure birth process (NHPBP) and developed an optimal PM scheme for a system under the NFRW. A summary of the literature on warranty optimization and the research gap this paper seeks to fill are given in Table 2. The existing studies on the BW and EW in terms of assumptions, modeling approaches and solving method are briefly reviewed and the differences between the current study and previous studies are demonstrated. As shown, our research integrates a number of factors which have been separately evaluated in earlier studies. Table 2. A systematic state-of-the-art review of the literature relating to warranty optimization.

*Table 2* 3. Problem definition Consider a manufacturer who offers a BW for his/her products, and the customers will have the option to purchase an EW after the BW on the products expires. Customers are typically willing to purchase the longest duration EW service with the lowest price. In such a case, the manufacturer will deal with claims originated from three sources: i) products under BW, ii) products under EW, and iii) products out-of-warranty. Three types of repair actions are considered to restore a failed product: with probability the product undergoes a perfect repair (after which the condition of the product returns to AGAN), with probability the product undergoes a minimal repair (after which the condition of the product returns to ABAO), and with probability the repair action is imperfect (after which the condition of the product is somewhere between AGAN and ABAO). On the other hand, the manufacturer is responsible to provide the spare parts required in the case of a perfect repair (or replacement with a new product). Therefore, the manufacturers must make their decision on the following factors: 1. How much should the product be priced at for sale?

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2. How long should the duration of BW be to achieve the best possible balance between the manufacturer‘s warranty costs and profits? 3. How long should the duration of EW be so that the manufacturer‘s profit from selling the products is maximized? 4. How much should the price of an EW be in order to maximize customers‘ willingness to extend their BW contracts? 5. How should the probability of each type of repair be calculated? 6. What level of rejuvenation for an imperfect repair on the product is most cost-effective? 7. How many spare parts should be ordered during the planning horizon so that the inventory cost is minimized while maximizing the customers‘ satisfaction? To answer the above-mentioned questions, this paper aims to develop a simulation-based optimization (SBO) model involving seven key variables such that the manufacturer‘s net profit is maximized. The decision variables include: (1) product‘s selling price, (2) BW duration, (3) EW duration, (4) EW price, (5) probability of each type of repair (minimal, imperfect and perfect), (6) level of product rejuvenation in imperfect repair, and (7) quantity of demand for spare parts during the planning horizon. The interactions between these variables are shown in Figure 1.

*Figure 1* Fig. 1. The interactions between seven decision variables considered in this study. 3.1. Model assumptions and notation The elements included in the formulation of the manufacturer‘s revenue function include: i) the sales volume of the product throughout its life cycle, ii) the number of sold EW contracts, and iii) the expected number of repairs associated with out-of-warranty products. The cost elements include: (1) production cost of products and spare parts, (2) repair cost of spare parts and (3) holding cost of spare parts. We further assume that:  All claims during the warranty period are valid;  Warranty policy offered to products is NFRW;  The EW begins immediately after the expiration of the BW;  The products are repairable;  The OEM is responsible for production of spare parts. The notations used in model formulation are introduced as follows: t T g Ct Et ht A CMt CPt MRt

Counter of time intervals in planning horizon Number of time intervals over the product‘s life cycle Number of time intervals for spare parts availability after the end of product‘s life cycle Production cost in tth time interval of the product‘s life cycle Variable cost of spare parts production in tth time interval Holding cost of spare parts in tth time interval Fixed production cost Minimal repair cost in tth time interval Perfect repair cost in tth time interval Minimal repair price in tth time interval for out-of-warranty products

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PRt BWmin , BWmax EWmin , EWmax Pmin , Pmax qmin , qmax PEmin , PEmax U TDt NEt NMt NIt NPt MOt IOt POt p0 q prE wB wE P1 P2 P3 DSt It

Perfect repair price in tth time interval for out-of-warranty products Lower bound and upper bound for BW duration Lower bound and upper bound for EW duration Lower bound and upper bound for product‘s selling price Lower bound and upper bound for imperfect repair level Lower bound and upper bound for EW‘s price The maximum market demand for a product The number of perfect repairs in tth time interval The number of customers who buy EW in tth time interval The number of under-warranty products undergone a minimal repair in tth time interval The number of under-warranty products undergone an imperfect repair in tth time interval The number of under-warranty products undergone a perfect repair in tth time interval The number of out-of-warranty products undergone a minimal repair in tth time interval The number of out-of-warranty products undergone an imperfect repair in tth time interval The number of out-of-warranty products undergone a perfect repair in tth time interval Initial price for a product Rejuvenation level in imperfect repair EW price BW length EW length The probability of perfect repair The probability of imperfect repair The probability of minimal repair The amount of spare parts production in tth time interval The amount of spare part inventory on-hand at the start of period t for under-warranty products

3.2. System specifications We assume that products are subject to a deterioration process and have an increasing rate of occurrence of failure represented by r(t), where t>0 denotes chronological time. In this paper, we assume that the distribution of the time to first failure follows a Weibull distribution. The Weibull distribution is commonly used as a lifetime distribution in reliability engineering applications as it generalizes the exponential distribution to cases where the rate of failure varies with time. The initial age of the product at the time of sale is set to zero. Suppose the planning horizon consists of T+g discrete time intervals, where T represents the length of the product‘s life cycle and g represents the length of the time interval for spare parts supply after the end of product‘s life. We assume that the product fails at random time epochs , where n = 1, 2,… represents the number of failures. After each failure, a CM is performed to restore the product to operation. The CM action is either a perfect repair (with probability ), an imperfect repair (with probability ), or a minimal repair (with probability ). Denote by the lengths of time between consecutive failures; therefore for any , where . Figure 2 illustrates different stages of a product during the planning horizon.

*Figure 2* Fig. 2. Different stages of a product during the planning horizon. 10

3.3. Imperfect repair actions In this study, the effect of an imperfect repair on the product‘s performance is modeled by the virtual age concept. This concept was first introduced by Kijima and Sumita (1986) for repairable products. Assume that the product has the virtual age immediately after the (ith 1) repair. Then, the time to the ith failure, , exhibits the following conditional Cumulative Distribution Function (CDF): {

|

}

,

(1)

where F(xi) represents the CDF of the time to the first failure of a new product. Under this assumption that the time to first failure follows the Weibull distribution, the CDF can be rewritten as follows: [(

)

(

) ],

(2)

and the corresponding Probability Density Function (PDF) is given by: ,

*

+- ,

(3)

where α and β represent the scale parameter and shape parameter of the Weibull distribution, respectively. Under these assumptions and notations, the GRP I and GRP II models proposed by Kijima (1989) are presented in the followings: 3.3.1. GRP I model The GRP I model assumes that each repair compensates the damage incurred between two successive failures, i.e., the ith repair reduces the additional age to , where represents the rejuvenation level of each repair. Under this assumption, the respective virtual age after the ith repair is expressed by: ∑

,

(4)

where . The conditional Weibull CDF and PDF in the GRP I model are presented in Equations (5) and (6), respectively: [(

)

(

) ], ,

(5)

[

]- .

(6)

3.3.2. GRP II model The GRP II model assumes that each repair reduces cumulative damage up to ith failure. Therefore, the product‘s virtual age after the ith repair in the GRP II model is given by: ∑

,

(7)

where . The conditional Weibull CDF and PDF in the GRP II model are given by Equations (8) and (9), respectively: 11

[(

∑

)

∑

(

(

)

∑

,

) ], *(∑

(8) ∑

)

+- .

(9)

In both the GRP I and GRP II models, means that the repair restores the product to AGAN condition, whereas means that the product reverts to ABAO after a minimal repair. Imperfect repair is performed when falls within the interval between 0 and 1. In this study, is considered as a decision variable representing the quality of the repair. Figure 3 shows the relationship between the real age and the virtual age under three cases of q = 0, 0< q <1 and q = 1.

*Figure 3* Fig. 3. The relationship between ‗real age‘ and ‗virtual age‘ for three cases: (a) q = 0, (b) 0< q <1 and (c) q = 1.

4. Problem formulation In this section, we describe the mathematical model developed for this study. 4.1. Objective function The objective function is to maximize the manufacturer‘s net profit from selling the products, the EW contact, as well as some other services including repairs and spare parts provided to out-of-warranty products. Therefore, the objective function is expressed as follows: ∑

∑

∑

∑

∑

∑

∑

∑ [

]

∑ (10)

The first term in Equation (10) represents the profit obtained from selling the products, which is calculated by multiplying the net profit per product and the product demand in each time interval. The second term calculates the revenue earned from selling the EWs. The third term to the fifth term represent the costs associated with minimal, imperfect, and perfect repairs during different time intervals of planning horizon, respectively. The sixth term to the eighth term calculate the revenue earned from offering different types of repair to out-ofwarranty products. Finally, the ninth term calculates the costs associated with spare parts inventory, including their production cost and holding cost. Different elements of the objective function are expressed in the following equations: ,

(11) ,

for θ1 , θ2 >0 ,

(12)

,

(13) 12

,

(14)

.

(15)

Equation (11) shows how the product price decreases as it ages, where b>0 represents the price reduction in unit time. Equation (12) presents the product demand function, which is modelled as a function of the sales price (Pt) and the length of the BW period (wB). As can be seen, the customers‘ demand is inversely proportional to sales price but directly proportional to BW length. U represents the maximum potential market demand in each time interval, and θ1 and θ2 are respectively the price coefficient and the BW coefficient in the demand function. Equations (13) and (14) represent the effect of rejuvenation level on the cost and price of imperfect repair, respectively. Equation (15) represents the probability that a customer will purchase an EW after the BW expires, where is a constant coefficient. As can be seen, the probability that a customer will purchase an EW is an increasing function with the duration of the contract but a decreasing function with its price. This probability function can be simulated by a Monte Carlo method. The spare parts production cost is formulated as a concave function and represented as the summation of two terms: a fixed cost and a variable cost. This is given by Equation (16) as follows: {

(16)

where A is the fixed production cost and Et is the variable cost per unit production of spare part in time interval t. 4.2. Optimization model After defining different elements of the manufacturer‘s net profit function as well as the required inventory balance equations in different time intervals, the optimization problem can be formulated as follows: max (17)

z ,

(18) , (19) , (20) , (21) , (22) 13

, (23) , (24) {

,

}

(25) , (26) {

,

}

(27) Equation (17) represents the objective function that is given in Equation (10). This objective function maximizes the total net profit for a manufacturer who sells a number of products in T time intervals and then provides spare parts in g time intervals after product sale. Constrains (18)-(22) present the lower and upper bounds for product‘s initial price, imperfect repair level, BW length, EW length and EW price, respectively. Constraint (23) guarantees that the summation of all repair probabilities should be equaled to 1. Constrains (24) and (25) present the inventory balance equations in different time intervals. Constraint (26) states that the inventory of all components at the end of planning horizon must be zero. Finally, constraint (27) indicates that inventory shortage is not allowed and production rates are greater than or equal to zero.

5. Solution method In this section, an efficient and effective method is presented to solve the above-presented optimization problem. Our method involves three types of variables: 1) Type 1 variables that are set by a metaheuristic algorithm. These variables include the product‘s initial price, BW length, EW length, EW price, probability of each type of repair, and the rejuvenation level in imperfect repair. 2) Type 2 variables that are simulated by a MCS method. These variables include the number of customers who purchase an EW and the total number of minimal, imperfect and perfect repairs performed for all under- and out-of-warranty products in different time intervals over the planning horizon. 3) Type 3 variables that are optimized by a DP algorithm. These variables include the quantities of spare parts production in different time interval over the planning horizon. The pseudo code of the developed solution method is presented below. k=1 Generate randomly a population of n individuals from the first type variables While termination criteria are not met Do Calculate the second type of variables by a Monte Carlo simulation (MCS) for each individual Calculate the third type of variables by a dynamic programming (DP) algorithm for each individual Evaluate the objective function for each individual

14

Generate new population k=k+1 End While Output: Best found solution.

Firstly, the first type of the model variables are set and then the product demand in various time intervals are estimated. Secondly, the number of customers who are willing to purchase an EW is predicted by a MCS method. In this method, the number of failures are estimated based on the product reliability. Thirdly, the optimal amount of spare parts production is calculated by a DP algorithm. Lastly, the algorithm evaluates the objective function for each individual. The framework of the proposed metaheuristic-MCS-DP solution approach is graphically illustrated in Figure 4.

*Figure 4* Fig. 4. The proposed solution approach based on a hybrid metaheuristic, Monte Carlo simulation (MCS) and dynamic programming (DP) algorithm. 5.1. Estimation of the product failures using MCS MCS is a class of mathematical models that is based on generation of random events in a computer model. This generation is repeated many times to count the occurrence of a specific event. In this study, the product failures are simulated using a MCS algorithm and based on the GRP I and GRP II models. The pseudo codes of the two models are presented below. Monte-Carlo simulation algorithm for estimation of the product failures based on the GRP I model Begin Read the first type of variables set by the metaheuristic algorithm Generate a uniformly distributed number R between (0,1) [ Calculate the first failure time using the equation Generate random number If then perform minimal repair If then perform imperfect repair If then perform imperfect repair n=2, i=2 While ∑ Do Generate a random number Calculate the ith failure time using the equation

[(

)

Generate random number If then perform minimal repair If then perform imperfect repair If then perform imperfect repair n=n+1, i=i+1, End while 15

]

]

End

Monte-Carlo simulation algorithm for estimation of the product failures based on the GRP II model Begin Read the first type of variables set by the metaheuristic algorithm Generate a uniformly distributed number R between (0,1) [ Calculate the first failure time using the equation Generate random number If then perform minimal repair If then perform imperfect repair If then perform imperfect repair n=2, i=2 While ∑ Do Generate a random number Calculate the ith failure time using the equation

[(

)

]

]

Generate random number If then perform minimal repair If then perform imperfect repair If then perform imperfect repair n=n+1, i=i+1, End while End

5.2. Dynamic programming (DP) algorithm In this study, it is assumed that the cost of spare parts production is a concave function and is expressed by Equation (16). Also, we assume that the spare parts inventory level at the beginning of the first interval as well as the end of the last interval is zero, i.e. , and no inventory shortage is allowed. It is obvious that minimum of a concave function with convex constraints is attained in one of its extreme points. Here, each extreme point has a maximum of T+g non-zero variables. According to Equation (25), if , either one or both of and must take a value greater than zero. Since there are T+g constraints and T+g variables in each extreme point, therefore exactly one of variables between and should take a positive value. As a result, all extreme solutions have the following feature that . Now, if the product demand in the kth time interval is zero, both and can take a zero value. This means that in an extreme solution for some values of we can have and , but this solution cannot be optimal. If we can solve the problem through decomposition of main problem into two independent problems: planning for time intervals 1 to k-1, and planning for time intervals k+1 to T+g. The first sub-problem has k-1 constraints and k-1 non-zero variables, whereas the second sub-problem has T+g-k constraints and k-1 non-zero variables. Consequently, in the optimal solution there will be T+g-1 non-zero variables. However, the property of 16

allows us to significantly reduce the decision space and only consider the following values for : { }. When the level of inventory in a time interval becomes zero, that time interval will be a starting point for planning the production during the remaining time intervals. Therefore, we call the intervals at which the level of inventory is equal to zero as the starting point. Then, a DP algorithm is used to calculate the number of starting points over the planning horizon, because according to what was discussed, the demand between two starting points equals to quantity of spare parts production in the first time interval between them. For example if t=5 and t=7 are considered as two starting points, then we will have . th Denote by the cost of spare parts inventory control in time interval, where j=0, th 1,…, T+g-1 and k=t+1, t+2,…, T+g. The spare parts inventory in interval is assumed to cover the demands in time intervals j+1, j+2,…, k. Therefore, consists of spare parts production cost and inventory holding cost. and are calculated by the following equations: , (28) ∑

∑

{

,

}

(29) Now,

can be calculated by Equation (30): ∑

(∑

)

∑

(∑

)

.

(30) is defined as the cost associated with the optimum policy for time intervals 1, 2, …, k, and is given by Equation (31): [

]

,

{

}

(31) To simplify the solution method, we define αjk as the optimum cost for periods 1, 2, …, k (the planning horizon consists of k time intervals). Given Ik=0 and j+1 is the last production period (i.e. Xj+1 > 0 and Xj+2 = Xj+3= … = Xk = 0), we have: , (32) [

]

,

(33) where F0=0. Finally, the starting points or periods that the manufacturer should produce spare parts will be determined using a backward method. Table 3 summarizes how the calculated results are reported. Table 3. A framework for the dynamic programming approach. 17

*Table 3* A numerical example. To illustrate how the DP algorithm works, a numerical example is presented. The initial inventory level is assumed to be zero. Table 4 shows the value of the parameters used to solve the DP problem. Table 4. The parameters used to solve a numerical example by the dynamic programming algorithm.

*Table 4* For solving a problem with n time intervals, the DP algorithm divides the problem to n sub-problems and calculates αij for each sub-problem. So, there are five steps for the above problem. Step 1. The sub-problem with 1 time interval M 01 = A1 + E1× NP1 = 150 + 84 × 1500 = 126,150 F1 = =F0 + M 01 = 0 + 126,150 = 126,150 Result: j*(1) = 0 , X1* = 1500 Step 2. The sub-problem with 2 time intervals M 02 = A1 + E1 .(NP1 + NP2) + h1 . NP2 = 150 + 84 × (1500 +1700) + 3 × 1700 = 274,050 M 12 = A2 + E2 NP2 = 151.5 + 85 × 1700 = 144,651.5 { Results: F2 = 270,801.5 , j*(2) = 1 , X1* = 1500 , X2* = 1700 Step 3. The sub-problem with 3 time intervals M 03 = A1 + E1 .(NP1 + NP2 + NP3) + h1 .(NP2 + NP3 ) + h2 .NP3 = 150 + 84 × (1500 + 1700 + 1850) + 3 × (1700 + 1850) + 3.2 × 1850 = 440,920 M 13 = A2 + E2 .(NP2 + NP3) + h2 .NP3 = 151.5 + 85 × (1700 + 1850) + 3.2 × 1850 = 307,821.5 M 23 = A3 + E3 .NP3 = 153 + 86 × 1850 = 159,253 { Results: F3 = 430,054.5 , j*(3) = 2 , X1* = 1500 , X2* = 1700 , X3* = 1850 Step 4. The sub-problem with 4 time intervals M 04 = A1 + E1 .(NP1 + NP2 + NP3 + NP4) + h1 .(NP2 + NP3 + NP4) + h2 .(NP3 + NP4) + h3.NP4 = 150 + 84 × (1500 + 1700 + 1850 + 1200) + 3 × (1700 + 1850 + 1200) + 3.2 × (1850 + 1200) + 3.5 × 1200 = 553,360 18

M 14 = A2 + E2 .(NP2 + NP3 + NP4) + h2 .(NP3 + NP4) + h3 .NP4 = 151.5 + 85 × (1700 + 1850 + 1200) + 3.2 × (1850 + 1200) + 3.5 × 1200 = 417,861.5 M 24 = A3 + E3 .(NP3 + NP4) + h3 .NP4 = 153 + 86 × (1850 + 1200) + 3.5 × 1200 = 266,653 M 34 = A4 + E4 .NP4 = 154.5 + 88 × 1200 = 105,754.5 { Results: F4 = 535,809 , j*(4) = 3 , X1* = 1500 , X2* = 1700 , X3* = 1850 , X4* = 1200 Step 5. The sub-problem with 5 time intervals M 05 = A1 + E1 .(NP1 + NP2 + NP3 + NP4 + NP5) + h1 .(NP2 + NP3 + NP4 + NP5) + h2 .(NP3 + NP4 + NP5) + h3 .(NP4 + NP5) + h4 .NP5 = 150 + 84 × (1500 + 1700 + 1850 + 1200 + 1100) + 3 × (1700 + 1850 + 1200 + 1100) + 3.2 × (1850 + 1200 + 1100) + 3.5 × (1200 + 1100) + 3.7 × 1100 = 660,500 M 15 = A2 + E2 .(NP2 + NP3 + NP4 + NP5) + h2 .(NP3 + NP4 + NP5) + h3 .(NP4 + NP5) + h4 .NP5 = 151.5 + 85 × (1700 + 1850 + 1200 + 1100) + 3.2 × (1850 + 1200 + 1100) + 3.5 × (1200 + 1100) + 3.7 × 1100 = 522,801 M 25 = A3 + E3 .(NP3 + NP4 + NP5) + h3 .(NP4 + NP5) + h4 .NP5 = 153 + 86 × (1850 + 1200 + 1100) + 3.5 × (1200 + 1100) + 3.7 × 1100 = 369,173 M 35 = A4 + E4 .(NP4 + NP5) + h4 .NP5 = 154.5 + 88 × (1200 + 1100) + 3.7 × 1100 = 206,624.5 M 45 = A5 + E5 .NP5 = 156 + 90 × 1100 = 99,156

{ Results: F5 = 1100.

, j*(5) = 4 , X1* = 1500 , X2* = 1700 , X3* = 1850 , X4* = 1200 , X5* =

The above results are all summarized in Table 5. Table 5. Determining the values of Fk and j*(k) from αij.

*Table 5* Now, the optimal solution is as follows: ,

19

,

.

6. Case study In this section, a case study of LG vacuum cleaners is provided in order to illustrate the developed model and its solution procedure. Goldiran (http://goldiran.ir/) is a company that produces, distributes, and provides warranty services for all LG electronics‘ products in Iran. Currently, the company offers 18 months BW and 12 months EW for his LG vacuum cleaners. The aim of this case study is twofold. First, it determines the optimal values for decision variables, and second, it evaluates how the length of the product life cycle and the duration of the spare parts supply can affect the optimal solutions. Table 6 gives the values of the model parameters for warranty policy optimization of LG VB-8720H vacuum cleaners. Table 6. The model parameters for LG VB-8720H vacuum cleaner.

*Table 6* In order to more precisely predict the costs, the time value of money is also taken into account. For this purpose, all cost parameters are multiplied by an inflation factor r. So, the future values of cost parameters are calculated using the following equations: ,

{

}

(34)

,

{

}

(35)

,

{

}

(36)

,

{

}

(37)

,

{

}

(38)

We consider each time interval equals to one month and the monthly inflation rate is assumed to be . 6.1. Solution algorithms As mentioned in section 5, our proposed algorithm consists of three stages. In the first stage, the first types of variables are set by a metaheuristic algorithm and in each iteration, the objective function is calculated using the MCS and DP algorithms. Since there is no benchmark available in the literature to validate the results, we will use two new effective population-based metaheuristics for solving the problem. These algorithms include a teaching–learning based optimization (TLBO) algorithm and a territorial particle swarm optimization (TPSO) algorithm, which are explained in following subsections: 6.1.1. Teaching–learning based optimization (TLBO) TLBO is a population-based metaheuristic algorithm that was first proposed by Rao et al. (2011) and it was later modified by Črepinšek et al. (2012). The algorithm was inspired from the teaching–learning process of a classroom. It consists of two main phases, including teaching phase and learning phase. In the teaching phase, the algorithm selects the best solution from the population and names it teacher and then tries to increase the learners‘ 20

knowledge with the help of the teacher‘s knowledge. In the learning phase, the learners help each other to improve learning. The steps of the TLBO algorithm are summarized as follows: Step 1: Generate the initial population (i.e. learners); Step 2: Select the best learner as teacher; Step 3: Calculate the difference between the current mean and the best mean using the following equations: ,

(39)

[

]

(40)

Step 4: Update the learners‘ knowledge based on the teacher‘s training according to equation (41): ,

(41)

Step 5: Update learners‘ knowledge by other learners‘ knowledge according to equations (42)-(43): (

)

(

)

( )

,

(42) ( )

,

(43) Step 6: Check the termination condition, and if the condition is not met then repeat the procedure from step 2 to 5. 6.1.2. Adaptive particle swarm optimization (APSO) Particle swarm optimization (PSO) is a population-based evolution computing algorithm inspired by social behavior of bird flocking or fish schooling. The algorithm was first proposed by Eberhart and Kennedy (1995). It generates randomly a population of solutions and iteratively tries to improve a candidate solution with regard to a given measure of quality. In the classical PSO algorithm, the algorithm updates the particle‘s position in each iteration in order to search optimal solution. The particle modifies its position based on the current velocity, the distance from its personal best-found solution (typically noted pBest), and the distance from global best-found solution (typically noted gBest). Equation (44) presents the velocity updating procedure for ith particle and (k+1)th iteration: (

)

,

(44)

where c1 and c2 are acceleration coefficients corresponding to cognitive and social behavior, ω is the inertia weight, and r1 and r2 are two independent random numbers uniformly distributed in the range of [0,1]. Thus, Equation (45) shows how the position of each particle is updated: . (45) 21

In the classical PSO algorithm, the inertia weight is constant for all the solutions in a generation, but it plays critical role in the particle‘s position movement. As a result, it will be better if the algorithm redefines inertia weight in the manner that the movement of the swarm should be controlled by the objective function. Panigrahi et al. (2008) proposed an APSO approach to update the inertia weight for each particle according to their rank. Equation (46) shows how inertia weight is calculated in each iteration: ,

(46)

where and represent the lower and upper bounds for inertia weight and is th the position of i particle among population. The steps of APSO algorithm are described as follows: Step 1: Step 2: Step 3: Step 4: follows:

Set input parameters such as: c1 , c2, , and ; Generate n numbers of population; Calculate the inertia weight according to Equation (46); Update the velocity using Equation (44) and correct it using Equation (47) as |

|

.

(47) Step 5: Update the position of each particle according to Equation (48): . (48) Step 6: Check termination condition and if the condition is not met go to Step 4. 6.2. Metaheuristic parameter tuning For improving the efficiency and effectiveness of metaheuristic algorithms, their parameters have to be tuned. Parameter tuning will permit a superior flexibility and robustness but it requires a careful initialization. These parameters are usually set based on the problem structure or the search time that the user spends in solving the problem. In general, optimal parameter values that one sets for a specified metaheuristic do not exist (Talbi, 2009). In the following, the structure of the parameter tuning for TLBO and APSO algorithms has been explained. 6.2.1. Parameter tuning in TLBO-MCS-DP algorithm One of the advantages of the TLBO algorithm is the number of parameters that require to be tuned. To run the algorithm, the number of iterations and size of population need to be specified. We consider T=36 and g=54 and run the TLBO-MCS-DP algorithm for different values of the model parameters. To find the best combination of algorithm parameters, we used the Data Envelopment Analysis (DEA) method. DEA was first introduced by Charnes et al. (1978) as a mathematical programming technique for measuring the relative efficiency of homogeneous Decision Making Units (DMUs) with multiple inputs and outputs. The method 22

uses a single performance measure called the ‗relative efficiency‘, which is the sum of the weighted outputs divided by the sum of the weighted inputs. The most popular DEA technique is the Charnes-Cooper-Rhodes (CCR) model. Assume that there are k DMUs each with m inputs and n outputs to be evaluated. The CCR model measures a DMU‘s relative efficiency by comparing it to a group of other DMUs having the same set of inputs and outputs. Hence, n optimization models must be solved. Let the DMU to be individually evaluated on any trial designated as DMUo where o = 1, ... , k. The relative efficiency, Eo of DMUo with inputs of xio (i = 1, ... , m) and outputs of yro (r = 1, ... , n) is given by: ∑ Subject to: ∑ ∑

∑

.

(50)

Table 7 provides various combinations of the parameters used in TLBO-MCS-DP algorithm for 45 hours run time. Two inputs and two outputs were chosen to characterize the algorithm efficiency. The inputs include number of iterations and population size, whereas the outputs include mean objective function value for 10 times running the algorithm and reversal of mean run time. Table 7. Different combinations of the parameters used in TLBO-MCS-DP algorithm.

*Table 7* The results obtained from solving the CCR model are presented in Table 8. With regards to CCR score of DMU9, we chose the number of iterations and population size to be equal to 20 and 100, respectively. Table 8. Efficiency results for different combinations of parameters in TLBO-MCS-DP algorithm.

*Table 8* 6.2.2. Parameter tuning in APSO-MCS-DP algorithm In PSO, the parameters such as inertia weight, acceleration coefficients, population size, fitness evaluations, etc. directly influence the performance of the algorithm. The aim of this subsection is to evaluate the impact of these parameters on the quality of solutions. Here, we 23

consider the algorithm to evaluate 1600 objective functions. It uses Taguchi analysis for 3 parameters in 4 levels to obtain the best combination of APSO parameters. Table 9 provides various combinations of the parameters used in APSO-MCS-DP algorithm. Table 9. Different combinations of the parameters used in APSO-MCS-DP algorithm.

*Table 9* We used Taguchi L16 experimental design to optimize the combination of APSO-MCSDP parameters. Optimal experimental conditions were determined by calculating signal-tonoise ratios. The selected levels of the factors according to S/N results include: c1=2, c2=2, and population size which is equal to 80. Figure 5 shows the results of S/N ratio for each level.

*Figure 5* Fig. 5. The average S/N plot for factor levels. 6.3. Optimal decision variables and the manufacturer’s profit In this subsection, the case study problem is solved by the proposed algorithm with parameter values as presented in Table 6. We also analyze how the changes in planning horizon may influence the value of decision variables. We considered product‘s life cycle (T) to range from 34 to 42 months and spare parts supply period to range from 54 to 62 months. The proposed algorithm was run 10 times for each combination of T and g (in total 600 problems), using MATLAB on a Pentium 4 computer with 8GB RAM and Core i7-9700K 3.6 GHz. Table 10 and Table 11 show the optimal values for decision variables as well as the expected manufacturer‘s net profit obtained using the hybrid TLBO-MCS-DP and APSO-MCS-DP algorithms. In Table 10, the imperfect repair policy was based on Kijima I virtual age model, whereas in Table 11, Kijima II virtual age model was considered for the imperfect repair. Table 10. Optimal decision variables and the manufacturer‘s net profit under Kijima I virtual age model for imperfect repair.

*Table 10* Table 11. Optimal decision variables and the manufacturer‘s net profit under Kijima II virtual age model for imperfect repair.

*Table 11* With regards to the results shown in Table 10 and Table 11, the following important points can be concluded. First, the manufacturer should set the BW length, EW length and its price, and imperfect repair level for the entire planning horizon T and g. Second, there is a 24

little change in probabilities of repair actions when values of T and g change, therefore, it can be concluded that any change in values of T and g will have very little impact on the repair strategy. However, the repair cost is identified as a factor having an impact on the repair probabilities. Third, the average profit gained based on Kijima I model is about 8% greater than when Kijima II model is used. Forth, the manufacturer‘s net profit increases with increasing T and g. However, this decision must be made based on the company‘s strategy. Figure 6 and Figure 7 compare the manufacturer‘s net profit obtained using the hybrid TLBO-MCS-DP and APSO-MCS-DP algorithms for two imperfect repair models of Kijima I and Kijima II virtual age, respectively. The results indicate that TLBO-MCS-DP algorithm has better performance than APSO-MCS-DP algorithm.

*Figure 6* Fig. 1. Comparing the results obtained using TLBO-MCS-DP and APSO-MCS-DP algorithms under Kijima I virtual age model.

*Figure 7* Fig. 2. Comparing the results obtained using TLBO-MCS-DP and APSO-MCS-DP algorithms under Kijima II virtual age model. Figure 8 compares the manufacturer‘s net profit obtained using the hybrid TLBO-MCSDP algorithm for two imperfect repair models of Kijima I and Kijima II. The results show that imperfect repair based on Kijima I virtual age model generates more profits for the manufacturer than imperfect repair based on Kijima II virtual age model.

*Figure 8* Fig. 3. Comparing the results obtained using TLBO-MCS-DP for imperfect repair based on Kijima I and Kijima II virtual age models. 6.4. Sensitivity analysis In order to evaluate the effect of key parameters on the objective function, a comprehensive sensitivity analysis is performed. These parameters include and b, where and represent respectively the price coefficient and the BW coefficient in product demand function and b is a coefficient representing how the price of a product decreases as it ages. Table 12 shows the effect of different values of parameters and on the decision variables as well as the manufacturer‘s net profit. Larger values of cause the product‘s selling price to have more significant impact on the demand function. In this case, the algorithm proposes a longer BW period to compensate for the lower demand. Also, the results show when the impact of BW length on product demand increases, it can have a great influence on the manufacturer‘s net profit.

25

Table 12. The effect of

and

on decision variables and objective function.

*Table 12* Figure 9 shows how different values of and affect the manufacturer‘s net profit. The most interesting aspect of this graph is that if the manufacturer increases the impact of warranty on sales volume with advertising or other strategies, he will be able to increase his profit up to twice.

*Figure 9* Fig. 9. The manufacturer‘s net profit under various values of

.

Table 13 shows the effect of parameter b on decision variables as well as the manufacturer‘s net profit for two imperfect repair models of Kijima I and Kijima II. Table 13. The effect of b on decision variables and objective function.

*Table 13* It is seen from the results that the product‘s initial price has slightly changed when parameter b changed, however the imperfect repair level, BW length, EW length and EW price changed significantly with b. Furthermore, larger b values caused the price to fall more quickly, so it reduced the manufacturer‘s net profit. Figure 10 shows how the manufacturer‘s net profit decreases when b increases.

*Figure 10* Fig. 10. The manufacturer‘s net profit under various values of b.

7. Conclusion and future research This study developed a novel simulation-based optimization approach to optimize various aspects of product warranty management, such as the basic and extended warranty durations, extended warranty price, product‘s selling price, repair strategy and the spare part inventory control policy for under- and out-of-warranty products. In contrast to the earlier studies that optimized warranty decision-making parameters separately, the current study proposed an integrated approach to optimise decision variables simultaneously. Our proposed model included two key contributions: first, considering out-of-warranty products as one of the main revenue sources of the manufacturers, and second, developing new repair strategy by combining minimal, imperfect and perfect repairs. To solve the warranty decision-making problem, we developed a hybrid method based on a metaheuristic algorithm, Mont Carlo simulation and a dynamic programming approach. The hybrid method optimized the decision variables in three stages. At the first stage, the product price, the basic and extended warranty 26

durations, extended warranty price, probability of each type of repair (minimal, imperfect and perfect) and the rejuvenation level in imperfect repair model were determined by a metaheuristic algorithm. At the second stage, the expected number of failures for underwarranty and out-of-warranty products was calculated by a Monte-Carlo simulation. Finally, at the third stage, spare parts production planning was optimized by a dynamic programming algorithm. In order to illustrate the developed model and its solution approach, a case study of LG vacuum cleaners was provided and solved with a hybrid teaching–learning based optimization (TLBO), a Monte-Carlo simulation and a dynamic programing algorithm. As there was no benchmark available in the literature, a hybrid approach of adaptive particle swarm optimization (APSO), Monte-Carlo simulation and dynamic programing algorithm was developed to validate the results. Experimental analyses show that, in spite of offering the same length of base warranty and extended warranty for all types of products by the manufacturer, any changes in production planning period (T) and spare parts warranty period (g) will result in significant changes in the optimal warranty policy. As a result, the manufacturer should consider the effect of his planning horizon on key decision variables. Furthermore, we found out that imperfect repair policy had a significant impact on the manufacturer‘s profit. In this case, imperfect repair based on Kijima I virtual age model generated more profits for the manufacturer than imperfect repair based on Kijima II virtual age model. To make the model applicable to other fields such as automotive industry, the model can be extended to a two-dimensional warranty decision-making problem. In this regard, modelling the product failures in terms of age and usage during the basic warranty and extended warranty will be a challenging task. Spare parts pricing is another topic that can be further studied in this context. Other considerations such as shortage and lost sales can also be incorporated into the problem of spare parts inventory control.

AUTHOR DECLARATION We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). He is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author and which has been configured to accept email from [email protected] Signed by all authors as follows: Mohsen Afsahi , Mahmood Shafiee , 17th November 2019

AUTHORS STATEMENT Mohsen Afsahi: Conceptualization, Methodology, Software, Validation, Formal Analysis, Investigation, Data curation, Writing - Original Draft, Visualization, Project administration. Mahmood Shafiee: Conceptualization, Methodology, Software, Validation, Formal Analysis, Investigation, Writing - Review & Editing, Visualization, Supervision.

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generalized Polya process repairable products under free-repair warranty‘, European Journal of Operational Research 279(1), 68–78. Chukova, S., Arnold, R. and Wang, D.Q. (2004) ‗Warranty analysis: An approach to modeling imperfect repairs‘, International Journal of Production Economics 89(1), 57–68. Chukova, S. and Shafiee, M. (2013) ‗One-dimensional warranty cost analysis for second-hand items: An overview‘, International Journal of Quality and Reliability Management, 30(3) 239–255. Črepinšek, M., Liu, S.-H. and Mernik, L. (2012) ‗A note on teaching–learning-based optimization algorithm‘, Information Sciences 212, 79–93. Eberhart, R. and Kennedy, J. (1995) ‗A new optimizer using particle swarm theory‘, In: Proceedings of the Sixth International Symposium on Micro Machine and Human Science, 4-6 October, Nagoya, Japan, pp. 39–43. Finkelstein, M. (2015) ‗On the optimal degree of imperfect repair‘, Reliability Engineering & System Safety 138, 54–58. Giri, B.C., Mondal, C. and Maiti, T. (2018) ‗Analysing a closed-loop supply chain with selling price, warranty period and green sensitive consumer demand under revenue sharing contract‘, Journal of Cleaner Production 190, 822–837. Glickman, T.S. and Berger, P.D. (1976) ‗Optimal price and protection period decisions for a product under warranty‘, Management Science 22(12), 1381–1390. Hartman, J.C. and Laksana, K. (2009) ‗Designing and pricing menus of extended warranty contracts‘, Naval Research Logistics 56(3), 199–214. Huang, H.-Z., Liu, Z.-J. and Murthy, D.N.P. (2007a) ‗Optimal reliability, warranty and price for new products‘, IIE Transactions 39(8), 819–827. Huang, W., Kulkarni, V. and Swaminathan, J.M. (2007b) ‗Coordinated inventory planning for new and old products under warranty‘, Probability in the Engineering and Informational Sciences 21(02), 261–287. Huang, W., Kulkarni, V. and Swaminathan, J. M. (2008) ‗Managing the inventory of an item with a replacement warranty‘, Management Science 54(8), 1441–1452. Jack, N. and Dagpunar, J.S. (1994) ‗An optimal imperfect maintenance policy over a warranty period‘, Microelectronics Reliability 34(3), 529–534. Jack, N. and Murthy, D.N.P. (2007) ‗A flexible extended warranty and related optimal strategies‘, Journal of the Operational Research Society, 58(12), 1612–1620. Jalali-Naini, S.G. and Shafiee, M. (2011) ‗Joint determination of price and upgrade level for a warranted second-hand product‘, The International Journal of Advanced Manufacturing Technology 54(9–12), 1187–1198. Khawam, J., Hausman, W.H. and Cheng, D.W. (2007) ‗Warranty inventory optimization for Hitachi global storage technologies, Inc.‘, Interfaces 37(5), 455–471. Kijima, M. (1989) ‗Some results for repairable systems with general repair‘, Journal of Applied Probability 26(01), 89–102. Kijima, M. and Sumita, U. (1986) ‗A useful generalization of renewal theory: counting processes governed by non-negative Markovian increments‘, Journal of Applied Probability 23(01), 71–88. Kim, B. and Park, S. (2008) ‗Optimal pricing, EOL (end of life) warranty, and spare parts manufacturing strategy amid product transition‘, European Journal of Operational Research 188(3), 723–745. Lam, Y. and Lam, P.K.W. (2001) ‗An extended warranty policy with options open to consumers‘, European Journal of Operational Research 131(3), 514–529. Lei, Y., Liu, Q. and Shum, S. (2017) ‗Warranty pricing with consumer learning‘, European Journal of Operational Research, 263(2), 596–610. Liao, G.-L. (2016) ‗Optimal economic production quantity policy for a parallel system with repair, rework, free-repair warranty and maintenance‘, International Journal of Production Research 54(20), 6265–6280. Lin, P.-C. and Shue, L.-Y. (2005) ‗Application of optimal control theory to product pricing and warranty with free replacement under the influence of basic lifetime distributions‘, Computers & Industrial Engineering 48(1), 69–82.

Lin, P.-C., Wang, J. and Chin, S.-S. (2009) ‗Dynamic optimisation of price, warranty length and production rate‘, International Journal of Systems Science 40(4), 411–420. Luo, M. and Wu, S. (2018) ‗A comprehensive analysis of warranty claims and optimal policies‘, European Journal of Operational Research 276(1), 144–159. Makis, V. and Jardine, A.K.S. (1992) ‗Optimal replacement in the proportional hazards model‘, INFOR: Information Systems and Operational Research 30, 172-83. Menezes, M.A.J. and Currim, I.S. (1992) ‗An approach for determination of warranty length‘, International Journal of Research in Marketing 9(2), 177–195. Murthy, D.N.P. (2007) ‗Product reliability and warranty: an overview and future research‘, Produção 17(3), 426–434. Padmanabhan, V. (1995) ‗Usage heterogeneity and extended warranties‘, Journal of Economics & Management Strategy 4(1), 33–53. Panigrahi, B.K., Pandi, V.R. and Das, S. (2008) ‗Adaptive particle swarm optimization approach for static and dynamic economic load dispatch‘, Energy Conversion and Management 49(6), 1407– 1415. Park, M. and Hoang, P. (2010) ‗Altered quasi-renewal concepts for modeling renewable warranty costs with imperfect repairs‘, Mathematical and Computer Modelling 52(9–10), 1435–1450. Rao, R.V., Savsani, V.J. and Vakharia, D.P. (2011) ‗Teaching–learning-based optimization: A novel method for constrained mechanical design optimization problems‘, Computer-Aided Design 43(3), 303–315. Shafiee, M. and Chukova, S. (2013a) ‗Maintenance models in warranty: A literature review‘, European Journal of Operational Research 229(3), 561–572. Shafiee, M. and Chukova, S. (2013b) ‗Optimal upgrade strategy, warranty policy and sale price for second-hand products‘, Applied Stochastic Models in Business and Industry 29(2), pp. 157–169. Shafiee, M., Chukova, S. and Yun, W.Y. (2014) ‗Optimal burn-in and warranty for a product with post-warranty failure penalty‘, The International Journal of Advanced Manufacturing Technology, 70(1-4), 297–307. Shafiee, M., Finkelstein, M. and Chukova, S. (2011) ‗Burn-in and imperfect preventive maintenance strategies for warranted products‘, Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 225(2), 211–218. Shafiee, M., Saidi-Mehrabad, M. and Jalali-Naini, S.G. (2009) ‗Warranty and sustainable improvement of used products through remanufacturing‘, International Journal of Product Lifecycle Management, 4(1-3), 68–83. Talbi, E.-G. (2009) Metaheuristics: From Design to Implementation. John Wiley & Sons. ISBN: 9780-470-27858-1, 624 Pages. Tao, N. and Zhang, S. (2015) ‗The optimal extended warranty length of durable-goods-based preventive maintenance behaviour‘, Systems Science & Control Engineering 3(1), 472–477. Tsao, Y.-C. Tend, W.-G., Chen, R.-S. and Chou, W.-Y. (2014) ‗Pricing and inventory policies for Hitech products under replacement warranty‘, International Journal of Systems Science 45(6), 1255– 1267. Wang, C.-H. and Sheu, S.-H. (2003) ‗Optimal lot sizing for products sold under free-repair warranty‘, European Journal of Operational Research 149(1), 131–141. Wu, C.-C., Chou, C.-Y. and Huang, C. (2009) ‗Optimal price, warranty length and production rate for free replacement policy in the static demand market‘, Omega 37(1), 29–39. Wu, C.-C., Lin, P.-C. and Chou, C.-Y. (2006) ‗Determination of price and warranty length for a normal lifetime distributed product‘, International Journal of Production Economics 102(1), 95– 107. Wu, S. and Longhurst, P. (2011) ‗Optimising age-replacement and extended non-renewing warranty policies in lifecycle costing‘, International Journal of Production Economics 130(2), 262–267. Yazdian, S.A., Shahanaghi, K. and Makui, A. (2016) ‗Joint optimisation of price, warranty and recovery planning in remanufacturing of used products under linear and non-linear demand, return and cost functions‘, International Journal of Systems Science 47(5), 1155–1175. Yeh, C.-W. and Fang, C.-C. (2015) ‗Optimal decision for warranty with consideration of marketing

and production capacity‘, International Journal of Production Research 53(18), 5456–5471. Yun, W.Y., Murthy, D.N.P. and Jack, N. (2008) ‗Warranty servicing with imperfect repair‘, International Journal of Production Economics 111(1), 159–169. Zhang, N., Fouladirad, M. and Barros, A. (2018) ‗Evaluation of the warranty cost of a product with type III stochastic dependence between components‘, Applied Mathematical Modelling 59, 39–53. Zhao, X., He, S. and Xie, M. (2018) ‗Utilizing experimental degradation data for warranty cost optimization under imperfect repair‘, Reliability Engineering & System Safety 177, 108–119. Zhou, Z. Li, Y. and Tang, K. (2009) ‗Dynamic pricing and warranty policies for products with fixed lifetime‘, European Journal of Operational Research 196(3), 940–948. Zhu, X., Jiao, C. and Yuan, T. (2018) ‗Optimal decisions on product reliability, sales and promotion under nonrenewable warranties‘, Reliability Engineering & System Safety, https://doi.org/10.1016/j.ress.2018.09.017 (in print).

Spare part Warehouse

Products situations in the market

Corrective Maintenance

Spare part demand

Manufacturer produces products and spare parts

Claims related to products under Base Warranty

Perfect Repair P1

Claims related to products under Extended Warranty

Repair Center

P2

Products Consumers

Imperfect Repair

P3 Claims related to out-ofwarranty products

Minimal Repair

Fig. 1. The interactions between seven decision variables considered in this study.

Products enter to market

𝑡

Products failure

...

𝑡

𝑃

Pe ect ep

𝑃

I pe ec ep

𝑃

M

𝑤𝐵

Base Warranty period

ep

𝑡

𝑤𝐵

Customers choose Extended Warranty with probability Λ 𝑤𝐸 𝑝𝑟𝐸

Manufacturer repair products with specific prices Time

...

𝑡

𝑤𝐸

...

𝑡

𝑇

𝑔

𝑡

Extended Warranty period

Total Product‘s Warranty period

Out-of-warranty period

Fig. 2. Different stages of a product during the planning horizon.

𝑇

𝑔

𝑎

𝑏

Virtual

Virtual

𝑣

𝑣 𝑣 𝑣

𝑣 𝑣 𝑦

𝑦

𝑦

Actual

𝑦

𝑞

𝑦

𝑦 𝑞

𝑐

Actual

1

Virtual 𝑣 𝑣 𝑣 𝑦

𝑦 𝑞

𝑦

Actual

1

Fig. 3. The relationship between ‗real age‘ and ‗virtual age‘ for three cases: (a) q = 0, (b) 0< q <1 and (c) q = 1.

Generating population by first type variables

Adding spare part lot-sizing information to each individual

⋮

⋮

Calculating second type variables by Monte Carlo simulation

Evaluate each individual

Calculating third type variables by solving spare part inventory control problem by dynamic programming algorithm

Generate new population by first type variables

Sort population

No

Stopping criteria met

Yes Stop

Fig. 4. The proposed solution approach based on a hybrid metaheuristic, Monte Carlo simulation (MCS) and dynamic programming (DP) algorithm.

Main Effects Plot for SN ratios Data Means

C1

134.7

C2

134.4

Mean of SN ratios

134.1 133.8 133.5 1.00

1.50

1.75

2.00

80

100

1.00

1.50

1.75

2.00

PS

134.7 134.4 134.1 133.8 133.5 40

60

Signal-to-noise: Larger is better

Fig. 5. The average S/N plot for factor levels.

TLBO_MCS_DP

APSO_MCS_DP

MANUFACTURER'S PROFIT (×106)

$80 $75 $70 $65 $60 $55 $50 $45 $40 34_54 34_58 34_62 36_54 36_58 36_62 38_54 38_58 38_62 40_54 40_58 40_62 42_54 42_58 42_62

T-G

Fig. 6. Comparing the results obtained using TLBO-MCS-DP and APSO-MCS-DP algorithms under Kijima I virtual age model.

MANUFACTURER'S PROFIT (×106 )

$7 $7 $6 $6 $5 $5 $4 3 4 _ 5 43 4 _ 5 83 4 _ 6 23 6 _ 5 43 6 _ 5 83 6 _ 6 23 8 _ 5 43 8 _ 5 83 8 _ 6 24 0 _ 5 44 0 _ 5 84 0 _ 6 24 2 _ 5 44 2 _ 5 84 2 _ 6 2

T-G TLBO_MCS_DP

APSO_MCS_DP

Fig. 7. Comparing the results obtained using TLBO-MCS-DP and APSO-MCS-DP algorithms under Kijima II virtual age model.

MANUFACTURER'S PROFIT (×106)

TLBO_MCS_DP with Kijima I

TLBO_MCS_DP with Kijima II

$80 $75 $70 $65 $60 $55 $50 $45 $40 34_54 34_58 34_62 36_54 36_58 36_62 38_54 38_58 38_62 40_54 40_58 40_62 42_54 42_58 42_62

T-G

Fig. 8. Comparing the results obtained using TLBO-MCS-DP for imperfect repair based on Kijima I and Kijima II virtual age models.

MANUFACTURER'S PROFIT (×106)

Kijima I

Kijima II

$110 $100 $90 $80 $70 $60 $50 $40 $30 0.5-1

0.5-1.5

0.5-2

0.5-2.5

1-1

1-1.5

1-2

1-2.5

1.5-1

1.5-1.5

Fig. 9. The manufacturer‘s net profit under various values of

Kijima I

1.5-2

1.5-2.5

.

Kijima II

MANUFACTURER'S PROFIT (×106)

$60

$55 $50 $45 $40 $35

$30 0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Fig. 10. The manufacturer‘s net profit under various values of b.

0.85

0.9

0.95

1

Table 1. A summary of the basic imperfect repair models. Imperfect repair model (

) rule

Improvement factor Virtual age Quasi-renewal process Shock model

description The type of repair is perfect with probability p(t) and is minimal with probability q(t), where t is the system age. The failure rate of system changes after repair in a state between as good as new and as bad as old Consider a virtual age which is a positive function of system‘s real age. After each repair, the product‘s virtual age reduces relative to the degree of repair After each repair, the system‘s life is reduced to a fraction of where In this approach, the system is suffered from a random damage due to shocks, then it is rectified if cumulative shock exceeds from a predetermined level

Reference Makis and Jardine (1992) Malik (1979) Kijima (1989) Wang and Pham (1996) Kijima and Nakagawa (1991)

Table 2. A systematic state-of-the-art review of the literature relating to warranty optimization. Warranty policy Chukova et al. (2004) Lin and Shue (2005) Bai and Pham (2005) Wu et al. (2006) Huang et al. (2007) Jack and Murthy (2007) Kim and Park (2008) Huang et al. (2008) Yun et al. (2008) Lin et al. (2009) Wu et al. (2009) Hartman and Laksana (2009) Park and Hoang (2010) Wu and Longhurst (2011) Bouguerra et al. (2012) Tsao et al. (2014) Yeh and Fang (2015) Tao and Zhang (2015) Yazdian et al. (2016) Lei et al. (2017) Afsahi et al. (2018) Zhao et al. (2018) Lou and Wu (2018) Zhu et al. (2018) Chien et al. (2019) Bian et al. (2019) Chien (2019)

This study

*

NFRW NFRW RFRW RFRW NFRW NFRW NFRW RFRW / RPRW NFRW RFRW RFRW NFRW RFRW NFRW NFRW NFRW NFRW / NPRW NFRW NFRW NFRW NFRW NFRW NFRW NFRW / NPRW NFRW NFRW NFRW NFRW *

Inventory control Length Length Price Dynamic Static Repairable Non-repairable Minimal Imperfect Perfect *MP SP BW

                 

EW

product pricing

    -

   -

 

 

          

      -

Type of product

Type of repair

         



      -

      

              

       -

     

   -

    

  

    

-

-

    -

   

Product situation * UW *OW                            

 

*BW: Base Warranty; EW: Extended Warranty; MP: Main Product; SP: Spare part; UW: Under Warranty; OW: Out-of-Warranty; NFRW: Non-Renewing Free Replacement Warranty; RFRW: Renewing Free Replacement Warranty

Table 3. A framework for the dynamic programming approach. The planning horizon consists of T+g time intervals 1 2 3 … T+g … α01 α02 α03 α0,T+g … α12 α13 α1,T+g … α23 α2,T+g . . .

αT+g-1,T+g

F1 [

] is the optimum previous restart point

j*(1)

F2

F3

j*(2)

j*(3)

… …

FT+g j*(T+g)

Table 4. The parameters used to solve a numerical example by the dynamic programming algorithm. Time interval

Spare parts demand in each time interval (NPt)

Fixed cost for producing spare parts in tth time interval (At)

Procurement cost per unit of spare part in tth time interval (Et)

Holding cost in tth time interval (ht)

1 2 3 4 5

1500 1700 1850 1200 1100

150 151.5 153 154.5 156

84 85 86 88 90

3 3.2 3.5 3.7 4

Table 5. Determining the values of Fk and j*(k) from αij. k

1

2

3

4

5

0 1 2 3 4

126,150*

275,400 270,801.5*

440,920 433,971 430,054.5*

553,360 554,011.5 547,214.5 535,809*

Fk

126,150 0

270,801.5 1

430,054.5 2

535,809 3

660,500 648,951 639,974.5 636,679.5 634,956* 634,956 4

j

j * (k )

Table 6. The model parameters for LG VB-8720H vacuum cleaner. 24

16

15

9

$ 245

$ 260

0.1

0.9

$ 14

$ 22

3000

$ 200

8

$ 84

3

150

$ 82

0.1

0.55

0.5

2

50.22

3.7

1.1,1.1

Table 7. Different combinations of the parameters used in TLBO-MCS-DP algorithm. Inputs

outputs

DMUj

Number of iterations

Population size

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10 DMU11 DMU12

40 30 20 40 30 20 40 30 20 40 30 20

40 54 80 45 60 90 50 67 100 55 74 110

Mean objective function value 4,524,606 4,659,059 4,640,026 4,785,514 4,805,198 4,906,166 5,063,487 5,114,583 5,401,852 5,402,062 5,401,949 5,402,392

reversal of mean run time 0.0435 0.0400 0.0417 0.0385 0.0357 0.0370 0.0345 0.0303 0.0278 0.0244 0.0233 0.0222

Table 8. Efficiency results for different combinations of parameters in TLBO-MCS-DP algorithm. DMUj DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10 DMU11 DMU12

CCR score 0.987 0.979 0.818 0.995 0.943 0.786 0.994 0.931 1 0.986 0.917 0.794

Rank 4 6 10 2 7 12 3 8 1 5 9 11

Table 9. Different combinations of the parameters used in APSO-MCS-DP algorithm. APSO parameters

c1 c2 Population size (PS)

Level 1 1 1 40

Level 2 1.5 1.5 60

Level 3 1.75 1.75 80

Level 4 2 2 100

Table 10. Optimal decision variables and the manufacturer‘s net profit under Kijima I virtual age model for imperfect repair.

T

34

36

38

40

42

g

P0

54

260

58 62

Optimal solution with TLBO-MCS-DP algorithm q wB wE PrE mean

Best sol

P0

q

Optimal solution with APSO-MCS-DP algorithm wB wE PrE mean

Best sol

0.58

22

13

22

0.01,0.82,0.17

4,720,800

4,871,866

258

0.32

21

13

20

0.01,0.67,0.32

4,579,176

4,716,551

260

0.2

24

12

22

0.02,0.55,0.43

5,235,300

5,408,065

260

0.27

23

12

21

0.01,0.65,0.34

5,130,594

5,284,512

260

0.22

22

10

22

0.01,0.6,0.39

5,596,900

5,792,792

259

0.26

21

12

21

0.01,0.71,0.38

5,652,869

5,822,455

54

260

0.21

24

14

22

0.01,0.55,0.44

5,398,700

5,571,458

260

0.24

24

14

21.7

0.01,0.75,0.24

5,290,726

5,449,448

58

260

0.48

24

15

21.4

0.01,0.67,0.32

5,523,100

5,705,362

259.5

0.28

24

15

22

0.02,0.57,0.41

5,357,407

5,518,129

62

260

0.2

24

11

22

0.01,0.67,0.32

5,899,600

6,100,186

260

0.25

24

13

20

0.03,0.57,0.40

5,781,608

5,955,056

54

260

0.21

24

14

22

0.01,0.50,0.49

5,446,302

5,620,584

260

0.31

24

14

22

0.02,0.64,0.34

5,718,617

5,690,176

58

260

0.2

22

15

14

0.02,0.46,0.52

5,819,200

6,011,234

260

0.26

22

10

21.5

0.02,0.66,0.32

5,644,624

5,813,963

62

259.7

0.2

21

14

14.7

0.001,0.78,0.20

6,116,355

6,330,427

259.7

0.28

21

13

21

0.02,0.58,0.40

5,994,027

6,173,848

54

260

0.2

23

13

22

0.01,0.51,0.48

6,149,735

6,352,676

260

0.3

23

13

22

0.02,0.69,0.28

6,026,740

6,207,542

58

260

0.28

20

11

18.9

0.01,0.52,0.47

6,150,860

6,359,989

258.7

0.35

20

14

22

0.03,0.62,0.35

6,258,403

6,352,155

62

260

0.30

20

11

22

0.01,0.69,0.3

6,485,702

6,719,187

260

0.39

20

13

22

0.02,0.56,0.42

6,420,844

6,613,469

54

260

0.39

23

11

14.9

0.02,0.74,0.24

6,090,573

6,285,471

259.1

0.27

23

11

22

0.01,0.61,0.38

5,968,761

6,202,824

58

260

0.2

24

9

21.9

0.02,0.67,0.31

6,745,315

6,974,656

260

0.2

24

13

22

0.03,0.55,0.42

6,610,408

6,902,720

62

260

0.2

24

10

22

0.01,0.48,0.51

7,419,600

7,671,866

260

0.24

24

14

22

0.01,0.68,0.31

7,271,208

7,601,344

Table 11. Optimal decision variables and the manufacturer‘s net profit under Kijima II virtual age model for imperfect repair.

T

34

36

38

40

42

Optimal solution with TLBO-MCS-DP algorithm q wB wE PrE mean

g

P0

54

259

0.27

18

10

58

260

0.37

24

62

260

0.27

24

54

260

0.2

58

260

62

Best sol

P0

q

Optimal solution with APSO-MCS-DP algorithm wB wE PrE mean

Best sol

18.3

0.007,0.96,0.033

4,469,683

4,648,470

258

0.2

20

9

19

0.01,0.74,0.25

4,380,289

4,489,796

14

22

0.006,0.98,0.009

4,776,464

4,967,523

260

0.25

22

12

21

0.01,0.84,0.15

4,728,699

4,894,203

9

18.84

0.007,0.80,0.18

4,932,877

5,179,521

259

0.3

23

10

22

0.01,0.82,0.17

4,834,219

4,955,074

24

11

20.51

0.007,0.82,0.17

4,980,029

5,129,430

260

0.2

22

11

21

0.02,0.81,0.16

4,830,628

4,951,394

0.2

24

10

22

0.004,0.64,0.34

5,132,903

5,312,554

259

0.27

22

12

21

0.02,0.75,0.23

4,978,916

5,103,389

260

0.24

23

14

22

0.012,0.87,0.11

5,159,887

5,366,283

260

0.22

23

12

21.8

0.02,0.79,0.19

5,056,689

5,183,106

54

260

0.4

24

11

22

0.001,0.59,0.399

5,178,518

5,437,444

260

0.31

21

13

22

0.01,0.68,0.31

5,074,948

5,252,571

58

260

0.3

18

11

15.50

0.001,0.97,0.029

5,240,496

5,423,913

260

0.2

22

12

18

0.02,0.76,0.22

5,345,306

5,532,392

62

260

0.2

16

9

22

0.001,0.899,0.1

5,250,224

5,460,233

259

0.3

22

14

21

0.02,0.69,0.27

5,145,220

5,273,851

54

260

0.28

22

9

15.17

0.001,90,0.099

5,494,932

5,742,204

260

0.29

21

10

18

0.02,0.79,0.18

5,714,729

5,914,745

58

260

0.2

24

9

22

0.01,0.87,0.12

5,796,982

5,999,876

259

0.2

23

14

20

0.01,0.82,0.17

5,681,042

5,823,068

62

260

0.37

23

11

22

0.01,0.84,0.15

5,900,137

6,136,142

260

0.2

22

13

21

0.01,0.77,0.22

5,959,138

6,167,708

54

260

0.2

24

9

14

0.02,0.71,0.27

5,732,685

5,961,992

259

0.25

22

11

18

0.02,0.69,0.29

5,847,339

6,051,996

58

260

0.2

24

9

20.86

0.01,0.93,0.06

6,190,412

6,438,028

260

0.25

22

11

20

0.01,0.75,0.22

6,004,700

6,154,818

62

260

0.2

24

10

21.67

0.01,0.82,0.17

6,335,465

6,620,561

260

0.26

24

13

22

0.01,0.78,0.21

6,018,692

6,169,159

Table 12. The effect of

and

on decision variables and objective function.

Optimal value with TLBO-MCS-DP algorithm with considering Kijima I virtual age model P0 q wB wE PrE mean Best sol

Optimal value with TLBO-MCS-DP algorithm with considering Kijima II virtual age model P0 q wB wE PrE mean Best sol

1

260

0.22

16

15

20.9

0.02,0.5,0.48

4,442,018

4,579,400

259.65

0.2

20

10

21.4

0.01,0.87,0.12

3,871,949

3,991,700

1.5

260

0.2

20

9

20.1

0.01,0.57,0.42

4,472,379

4,610,700

260

0.2

18

9

22

0.01,0.86,0.13

4,033,946

4,261,800

2

260

0.2

24

9

18.3

0.01,0.79,0.22

5,390,678

5,557,400

260

0.20

19

13

22

0.03,0.53,0.43

4,532,526

4,775,800

2.5

260

0.69

24

13

22

0.02,0.85,0.13

10,006,763

10,037,900

260

0.83

24

15

22

0.01,0.82,0.14

8,588,775

8,957,500

1

259.86

0.40

16

10

22

0.03,0.61,0.36

4,066,628

4,192,400

260

0.2

16

9

14.3

0.04,0.78,0.18

3,457,967

3,771,100

1.5

260

0.2

20

10

21.8

0.02,0.64,0.34

4,256,651

4,388,300

260

0.2

16

13

17.8

0.01,0.92,0.07

3,695,132

4,015,600

2

260

0.23

23

9

14

0.02,0.63,0.35

5,016,355

5,171,500

259.69

0.2

24

9

20.6

0.02,0.46,0.52

4,103,793

4,436,900

2.5

260

0.27

24

10

16.4

0.01,0.82,0.17

9,930,086

10,023,800

260

0.31

24

10

22

0.01,0.93,0.06

8,612,651

9,188,300

1

260

0.26

20

13

22

0.01,0.64,0.35

3,834,313

3,952,900

260

0.2

16

11

17.7

0.04,0.60,0.36

3,278,129

3,585,700

1.5

260

0.48

22

13

21.4

0.02,0.76,0.22

3,829,172

3,947,600

260

0.2

16

14

20

0.03,0.68,0.29

3,384,829

3,695,700

2

259.4

0.2

24

13

22

0.02,0.71,0.27

4,759,596

4,906,800

260

0.2

24

10

19.5

0.01,0.89,0.1

4,181,677

4,414,100

2.5

260

0.60

24

15

16.7

0.01,0.92,0.07

9,328,296

9,616,800

259.88

0.2

24

11

20

0.02,0.75,0.23

8,513,510

8,983,000

0.5

1

1.5

Table 13. The effect of b on decision variables and objective function. Optimal value with TLBO-MCS-DP algorithm with considering Kijima I virtual age model P0 q wB wE PrE mean Best sol 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

260 260 260 260 260 260 260 259.9 260 260 260 260 260 260 260 260 260 260 260

0.2 0.22 0.42 0.2 0.35 0.27 0.45 0.2 0.2 0.2 0.33 0.28 0.2 0.2 0.25 0.25 0.2 0.2 0.2

16 16 24 23 24 24 21 22 21 16 23 23 24 22 23 18 16 16 15.3

15 12 9 10 14 12 11 9 9 9 12 12 14 15 9 9 10 15 10

20.2 15.4 17.9 22 21.2 22 16.7 19.8 16.3 14 20.7 16.4 22 22 17.6 21.6 15.3 20.2 16

0.04,0.47,0.49 0.01,0.59,0.4 0.02,0.76,0.22 0.01,0.74,0.25 0.01,0.59,0.4 0.02,0.51,0.47 0.01,0.83,0.16 0.03,0.74,0.23 0.01,0.68,0.31 0.02,0.65,0.33 0.03,0.66,0.31 0.02,0.41,0.57 0.01,0.56,0.43 0.02,0.47,0.52 0.01,0.75,0.24 0.03,0.81,0.16 0.03,0.67,0.3 0.02,0.47,0.51 0.01,0.67,0.32

5,181,158 5,073003 5,238873 5,281747 5,108796 5,021884 4,698001 4,826138 4,630004 4,369462 4,658619 4,376058 4,691211 4,473543 4,380/52 3,966136 4,002608 4,001,158 3,954,100

5,341,400 5,229,900 5,400,900 5,445,100 5,266,800 5,177,200 4,843,300 4,975,400 4,773,200 4,504,600 4,802,700 4,511,400 4,836,300 4,611,900 4,516,000 4,088,800 4,126,400 4,111,400 4,126,400

Optimal value with TLBO-MCS-DP algorithm with considering Kijima II virtual age model P0 q wB wE PrE mean Best sol 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260

0.2 0.2 0.2 0.34 0.22 0.22 0.2 0.2 0.2 0.2 0.33 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.27

21 24 24 23 24 19 24 18 24 16 19 24 20 24 24 24 22 18 22

10 11 11 15 9 14 9 9 12 9 10 9 11 9 9 10 9 11 11

17.6 14 22 22 22 22 16.4 22 19.9 16.8 19.7 22 19 22 17 22 18 22 23

0.01,0.88,0.11 0.01,0.93,0.06 0.03,0.40,0.57 0.02,0.50,0.48 0.03,0.56,0.41 0.01,0.80,0.19 0.02,0.43,0.55 0.01,0.92,0.07 0.02,0.66,0.32 0.01,0.91,0.08 0.01,0.87,0.12 0.02,0.67,0.31 0.01,0.92,0.07 0.01,0.85,0.14 0.02,0.71,0.27 0.04,0.45,0.51 0.01,0.92,0.06 0.01,0.83,0.26 0.02,0.52,0.46

4,811,898 4,511,724 4,785,046 4,532,598 4,542,986 4,343,262 4,356,100 4,218,802 4,320,820 3,862,180 3,908,338 4,026,232 3,925,194 3,883,348 3,765,356 3,746,442 3,593,366 3,404,814 3,144,050

4,910,100 4,603,800 4,882,700 4,625,100 4,635,700 4,431,900 4,445,000 4,304,900 4,409,000 3,941,000 3,988,100 4,108,400 4,005,300 3,962,600 3,842,200 3,822,900 3,666,700 3,474,300 3,445,000