ExtEndEd warranty of mEdical EquipmEnt subjEct to impErfEct rEpairs: an approach basEd on gEnEralizEd rEnEwal procEss and stackElbErg gamE rozszErzona gwarancja na sprzęt mEdyczny podlEgający niEpEłnym naprawom: podEjściE opartE na uogólnionym procEsiE odnowy i modElu stackElbErga

instance, it is estimated that an amount of US$ 24.83 billion will be annually spent in the medical equipment maintenance market by 2022 [20]. However, maintenance outsourcing for medical equipment is yet to be fully explored by current research (Cruz & Rincon [5]). Warranty policies define responsibilities for both parties: the OEM and hospital managers. Indeed, the OEM is responsible for repairing the devices’ eventual failures related to problems of equipment design, manufacturing and/or quality. The customers in turn should make proper use of equipment, i.e., they must comply to the specifications defined by the OEM (Rahman & Chattopadhyay [32]). Given that, a new trend has been intensified by manufacturers, selling an additional, optional coverage, which begins after the expiration of the base warranty, called Extended Warranty (EW) (Murthy


Introduction
Medical equipment plays an important role in modern healthcare institutions because they present the following purposes: diagnosis, disease prevention, monitoring and patient treatment.During the last decades, as technology has advanced, the maintenance of such equipment has become too complex to be done in-house.Therefore, this activity has been commonly assigned to the Original Equipment Manufacturer (OEM).
In this context, the importance of studying more deeply the warranty issue is reinforced by the role maintenance management plays in guaranteeing the quality of healthcare services.In diverse situations, human lives depend on the correct operation of medical devices.For João Mateus Marques de SAntAnA Rafael Luiz Velôzo SAntiAgo Márcio das Chagas MouRA isis Didier LinS ExtEndEd warranty of mEdical EquipmEnt subjEct to impErfEct rEpairs: an approach basEd on gEnEralizEd rEnEwal procEss and stackElbErg gamE rozszErzona gwarancja na sprzęt mEdyczny podlEgający niEpEłnym naprawom: podEjściE opartE na uogólnionym procEsiE odnowy i modElu stackElbErga Due to its advanced technology, maintenance services of healthcare equipment have been commonly executed by the original equipment manufacturer (OEM), which can be characterized as a monopolist.In this context, hospitals require high availability of their equipment at a reasonable servicing cost, whereas OEM aims to maximize its profit by selling extended warranty (EW) services for multiple consumers.The issue of drawing a maintenance contract between OEM and hospitals has already been treated by adopting a Stackelberg's game.However, the "as good as new" and "as bad as old" assumptions are usually considered, which are rather difficult to observe in practice, especially for healthcare institutions and their technology-intensive equipment.Thus, we here adopt generalized renewal processes (GRP) for modelling imperfect repairs, and we develop a discrete event simulation method for finding the best strategies of each player: OEM sets the prices for EW and on-demand maintenance that optimize its profit, while hospitals choose which option they should hire.We also present an application example with real data gathered from an angiography device, which is used for mapping blood vessels and diagnosing heart diseases.

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& Djamaludin [26]).Thus, the customer decides whether to pay an extra value at the purchase epoch (Murthy & Jack [27]), whereas the OEM will correctively maintain equipment for a given period even after the ordinary warranty expires.
It is important to emphasize that EW ends up creating a conflict of interests between the owner of the equipment and the OEM.Specifically, the customer needs a high availability of its equipment at a reasonable servicing cost, whereas the manufacturer aims to maximize their profit with the addition of post-selling services.Consequently, EW affects both buyers' and manufacturer's outcomes (Ye & Murthy [41]), once the actions of one interferes in the results of the other.Due to this situation, Game Theory provides an appropriate approach to solve this problem (Forgó et al. [10]).
Among different games that can be used to model the interaction between agents, the leader-follower Stackelberg Game (SG) is a good option for drawing maintenance service contracts of medical equipment.In this context, the OEM is commonly the only party able to perform maintenance, since it has the technical knowledge, expertise, technology and spare parts for the repair execution (Rinsaka & Sandoh [33]).The hospital in turn needs its medical device available in suitable condition to provide a good service.From these two perspectives, an uneven power relationship is noted, which can be modeled via SG.The leader role is assigned to the OEM, which determines the terms of the EW, while the health institution acts as the follower, responding to actions taken by the OEM.
Quantitative studies about warranty, maintenance outsourcing and maintenance contracts are present by Kim et al. [18], Bouguerra et al. [3], Husniah et al. [14], Huang et al. [13], Moura et al. [23] and Darghouth et al. [6].However, such studies have simplifying assumptions with respect to the state of the system after a corrective maintenance (CM) intervention.In fact, those papers considered that the system returns to either an "as good as new" condition (perfect repair) or an "as bad as old" condition (minimal repair); these two situations are modeled respectively according to a Renewal Process (RP) and a Non-Homogeneous Poisson Process (NHPP); Ross [34] describes RP and NHPP in details.The use of these assumptions may yield inadequate managerial decisions, which can result in significant losses in company profits because of incorrect definition of warranty policies.
In practical terms, maintenance actions typically return the equipment to an intermediate condition between the perfect and minimal repairs, which is called imperfect repair (Kijima & Sumita [17]; Wang & Pham [37]).Kijima & Sumita [17] proposed two methods to tackle imperfect repairs: Kijima type I and Kijima type II, which gave rise to the Generalized Renewal Process (GRP) and introduced the concept of "virtual age".Furthermore, these situations generalize RP and NHPP; other approaches may be seen in Pham & Wang [31].
In this paper, we aim to join SG and GRP to model imperfect repairs -a more realistic and general assumption.This approach considers the interaction between OEM and multiple customers.The proposed model will be characterized as follows.First, we considered the OEM offers to the hospital managers two maintenance options for the period after the ordinary warranty expires: (i) an extended warranty or (ii) on-call service.EW states that for a fixed price P , the OEM should repair all failures without any additional cost over the period of the contract; if a failed unit does not get repaired before a set time τ , a penalty, which increases over time, will be incurred.For the on- call service, failures will be repaired at a fixed cost s C each, with no penalty incurred in case of delays.Given a number of hospitals that buy the equipment, this model estimates * P and * s C , which are the maximal prices hospitals accept to pay for each maintenance option.Finally, we find the optimal number of customers, i.e., number of hospitals that maximize OEM's profit.
To that end, we develop a Discrete Event Simulation (DES) based method to reproduce the GRP-queue system and obtain the perfor-mance indicators of interest.Simulation models allow for analysis of systems with complex behavior, require fewer assumptions when compared with analytical models and are used as tools to perform experimentations with systems (Marsaro & Cavalcante [21]).Due to present model's complexity, an analytical solution is unfeasible.In this sense, simulation is useful for describing equipment behavior (Ding & Kamaruddin [7]).Thus, a simulation approach is employed for obtaining a solution for the present model.
The remainder of this paper unfolds as follows.In Section 2, the theoretical background is provided, containing the adaption of the SG for the context of maintenance contracts, as well as the characteristics of GRP, emphasizing the method proposed by Yañez et al. [39].Section 3 presents the proposed model, the players' optimal strategies and the equilibrium of the game.In Section 4, a numerical example is presented, using real data from an angiograph, which is a device used for blood vessels mapping and diagnosis of organ diseases.Finally, Section 5 concludes remarks.

Theoretical background 2.1. Generalized renewal process
RP and NHPP may be adopted to model perfect and minimal repairs respectively.According to (Lins & Droguett [19]), such methods have simplifying assumptions that may be unreal in many practical situations such as the healthcare technology intensive environment.To overcome the limitations of RP and NHPP, Kijima & Sumita [17] developed a probabilistic virtual age based model, known as the Generalized Renewal Process (GRP) that deals with all classes of maintenance actions.According to this model, q (rejuvenation parameter) may generally assume values between 0 and 1: 0 q = • represents a perfect repair (as good as new); 1 q = • corresponds to a minimal repair (as bad as old); 0 1 q << • indicates imperfect repair (better than old, worse than new).
Cases, where 0 q < and 1 q > , are also possible corresponding to better than new and worse than old conditions respectively.Generally, GRP may be classified into two types (Kijima Type I and II), according to the method used to calculate the virtual age.These types can be seen in details in Moura et al. [22] and other virtual age-based representations could be found in Guo et al. [12], Tanwar et al. [35], Ferreira et al. [9], Oliveira et al. [29] and Wang & Yang [38].
This paper uses Kijima type I so that equipment virtual age follows Eq. (1), according to which maintenance actions only act on the degradation incurred during i x , which is the time between ( ) th 1 i − repair and the th i failure: The Cumulative Distribution Function (CDF) for the time between the ( ) and th i failures can be determined from the CDF of the time until a failure conditioned on the virtual age 1 i v − as seen in Eq. ( 2): For our analysis, we consider the time to failure follows a conditioned Weibull distribution because of its flexibility to fit various deg-sciENcE aNd tEchNology radation stages.Then, Eq. ( 2) can be rewritten as Eq.(3).Note that, for 1 i = , we have the Weibull distribution itself because 0 0 v = .When there are reasonably sufficient failure data available, Maximum Likelihood Estimators (MLE) can be used to estimate GRP parameters α, β, and q.To that end, the procedure described in Yañez et al. [39] can be followed.
In the proposed model, we consider a case with multiple clients and m service channels, where maintenance teams may serve a total of M clients.If the number of failed units is greater than the service capability, a queue is formed.The waiting time in queue is decisive to the interests of both hospitals and OEM, since the former aims high availability of equipment and the latter wants to serve a higher number of customers.Therefore, a queuing formulation is employed to incorporate system behavior due to the interaction between service level and number of clients to be served.
Then, GRP governs the equipment failure process and the queue discipline follows a FCFS (first come first served) logic.We also consider times to repair follow an exponential distribution.Thus, using the conventional notation, the queue can be described as GRP/Markovian/ m / i W ∞ /M/FCFS.For this situation, analytical solutions are not available.Therefore, we proposed a DES-based algorithm is adopted to obtain the GRP-queue system measures; the DES formulation is described in Section 3.8.

Stackelberg game
SG is a non-cooperative sequential game developed by Heinrich von Stackelberg.It was originally proposed to evaluate the equilibrium of a duopoly, where competing companies decide the optimal quantity to be produced (Gibbons [11]) in a leader-follower interaction.
At the best of authors' knowledge, Murthy & Yeung [28] were the first authors to introduce SG as a tool to model maintenance service contracts.Murthy & Asgharizadeh [25] expanded the problem by creating a game between a customer and a manufacturer by assuming perfect repairs.Ashgarizadeh & Murthy [2] and Murthy & Asgharizadeh [24] incorporated multiple customers and service channels.Esmaeili et al. [8] considered a three-level service contract between a manufacturer, a customer and an independent third agent.Moura et al. [23] used priority queues to analyze the interaction among OEM and two priority classes of hospitals.All aforementioned models intend to maximize the clients' expected utility, considering parameters like risk aversion, revenue generated by the system, maintenance costs and times to repair.In such papers, the leader (OEM) provides the maintenance options for the follower (hospitals) and obtains the highest payoff, since it charges the prices that maximize profits.
Generally, in the context of complex medical equipment, the OEM has a well-trained staff, spare parts and dominates the equipment technology.Thus, OEM behaves as a leader, acts first (by defining services and respective prices) and is the only maintenance service provider.Hospitals need to guarantee minimum levels of availability for their equipment.However, they do not have expertise in the maintenance of complex equipment.Therefore, hospitals can be considered as followers, since they react to the OEM's action when choosing a service type to hire.
The SG's solution is obtained through backwards induction (Osborne and Rubinstein [30]) corresponds to a sub-game perfect Nash equilibrium of a two-stage game, with perfect information and players with different profiles (Amir [1]).Finally, it is noteworthy to say that the papers cited earlier in this Section, which adapted SG to the field of maintenance outsourcing, make simplifying assumptions about the repair structure employed.At the best of authors' knowledge, none of them adopted the imperfect repair assumption, a more realistic hypothesis, especially for complex systems.Thus, this paper aims to incorporate the GRP to model this situation and attempts to make the model more suitable to the medical environment.

Game formulation
This paper aims to determine the optimal strategies for the problem of EW for the medical context.Thus, we employ a SG formulation to model the interaction between the OEM and each customer, and the game's equilibrium will be reached by finding the reservation prices, i.e., maximum prices that customers accept to pay.
This decision problem extends the model developed by Ashgarizadeh & Murthy [2] by considering imperfect repairs.Additionally, the problem is stochastic due to the uncertainty inherent to the presence of random variables in the model.Finally, decision makers (healthcare institutions and OEM) have their own objective function that will define the respective payoffs.Both OEM and healthcare institutions are aware about their alternatives, acting rationally, and choosing strategies that maximize their respective payoffs (Osborne & Rubinstein [30]).T : Total period that the th j system is on operational state; Π Ð H : Hospital's expected profit.

Problem description
OEM sells a technology-intensive medical device to multiple customers (hospitals) for a cost of b C per unit.Each device, when in operational state, generates a revenue of R monetary units per time.Along with the purchase of the device, OEM provides a base warranty, during which OEM is responsible for all repairs required with no charge for the client.If a failed equipment is not returned to operational state within a period τ 1 after a failure occurs, OEM is charged with a penalty proportional to the overtime in repairing the equipment, which is the period from τ 1 to the time when the equipment returns to operation.Therefore, the penalty is θ τ > , and zero, otherwise; θ 1 is the penalty per time during over- time; ij Y is the time between the occurrence of the th i failure of equipment j and the completion of its respective repair.This penalty exists because medical equipment is vital for patients' treatment and for the hospitals' profit, and then unavailability affects their payoff and reputation.
After the expiration of the base warranty, each hospital may choose a type of repair service: i) EW or ii) on-call services.These options are described as follows: : EW -begins after the base warranty expires and has duration 2 W .The customer pays a fixed price P and the OEM repairs all failed units over 2 W at no additional cost.If a failed device is not returned to operational state within a period τ 2 � after a failure occurs, the OEM is charged a penalty.Analogously to the base warranty, the penalty is θ τ : on-call services -the OEM executes each repair at a cost of s C per intervention; no penalty is here incurred.
We also assume that option 0 A means the hospital chooses not to buy the equipment.

Hospital's decision problem
Considering the options presented by the OEM, the hospital manager decides whether to opt for either EW (A 1 ), or on-call services (A 2 ) or not to purchase the equipment (A 0 ), where the latter occurs if its expected utility is negative over the period W. Each strategy has consequences to the payoffs.Indeed, each customer's return, Π , depends on the option A k , and thus the hospitals profit can be seen in Eqs. ( 4), ( 5) and ( 6): where: For hospital (equipment) j, (1 jM ≤≤ ), j N is the total number of failures over the mission time W ; ij X ( 0 ) is the time between the th i repair and the ( ) W respectively.We consider the hospital's risk is modeled according to a utility function U , which indicates how the customer chooses among distinct options; thus, the preferred options are represented by higher utilities.The utility function considered in this model has been used in Murthy & Asgharizadeh [25] and is shown in Eq. (7), where Π represents the associated wealth: Thus, the choice k A is strongly affected by equipment availability, pricing structure and the hospital's degree of risk aversion (δ).We assume all customers are homogeneous with respect to their risk aversion and all equipment units are identical regarding their reliability.

OEM's decision problem
The OEM is considered risk neutral and its expected profit is related to the customer's optimal choice.Consequently, the OEM's payoff can be denoted as Π .In this way, the manufacturer's profits for each of the customer's possible actions k A are shown respectively in Eqs. ( 8), ( 9) and (10).Then, OEM may choose the combination P , s C , M that maximizes its expected profit, taking into account the customer's optimal strategy * A : sciENcE aNd tEchNology

Assumptions
In order to make the model manageable, we consider some assumptions: Equipment is repairable and subject to imperfect repair.The I probabilistic failure modelling is handled by GRP; The times between failures are random variables.The time to II first failure follows a Weibull distribution, as seen in Yañez et al. [39]; The times to repair are exponentially distributed with param-III eter μ; The OEM has IV m parallel service channels, i.e., in total they are capable of processing m units simultaneously (one unit per service channel); The equipment's failures are critical.Moreover, the OEM V carries out just corrective maintenance interventions; The manufacturer and the customer have complete infor-VI mation about the model's parameters, which implies that the leader is aware about the customer's risk parameter and the hospital knows the equipment reliability; If there are more failed units than the number of servers, VII a queue following a FCFS is generated.This formulation describes a queuing system with finite population M .

Players' strategies
In order to find the optimal solution for the players and the game equilibrium, we determine how the hospital's and OEM's optimal strategies are defined, as well as understand their relation, and the degree of influence between them.Then, these strategies are shown as follows.

Hospital's optimal strategy
The customer's expected utility U is derived from two random variables ( ij X , ij Y ), the customer's decision k A , and the pricing structure ( max P ; max s C ) imposed by the OEM.Given the assumptions of Section 3.6, the expected utilities for each decision are given in Eqs. ( 11), ( 12) and (13) obtained by using Eqs.( 4), ( 5) and ( 6).For a pair ( P , s C ) determined by the manufacturer, the customers analyze their expected utilities and choose the option that returns the highest payoff. ) The reservation prices depend on equipment expected number of failures j EN     , and the number of units sold M .Then, they are fairly influenced by the repair assumption.For instance, by considering imperfect repair ( 0 1 q ≤≤ ), it is expected that devices with the same shape β 1 β > and scale α 0 α > fail less frequently than in the context presented by Moura et al. [23], who considered minimal repairs.Thus, the change in repair hypothesis can modify the expected payoffs, and consequently a change of strategies.The mathematical approach used to reproduce the imperfect repair is the GRP, a which makes use of DES and allows for modeling imperfect, perfect and minimal repair assumptions, adapting to a broader range of scenarios.Another point to emphasize corresponds to the reservation prices ( max P , max s C ), which are defined by Varian [36] as the highest prices a consumer is willing to pay.These prices affect the decision for what strategy k A is chosen.Thus, determine the hospital's reservation prices is essential to find the OEM's expected profit, once the pricing structure has strong influence on the healthcare's institutions decision.Indeed, given the hospital's reservation prices and the pricing structure imposed by the OEM, the EW model can be seen on its extensive form, as a sequential game tree in Figure 1a, which shows all possible decisions for the players.
In the case the prices charged by the OEM are higher than the reservation prices ( A should be chosen.Figure 1b presents the s P C − plan, and shows that the customer's optimal choice is characterized by three regions Ω 0 , Ω 1 and Ω 2 .In Ω 0 , * The curve Γ is obtained by equalizing the expected utilities for options A , and represents indifference between the maintenance options since the same payoff is returned to the hospital.

OEM's optimal strategy
Since the OEM is considered to be risk neutral, its optimal strategy corresponds to the pricing structure that maximizes its expected profit ( max P , max s C ).To choose its optimal strategy, OEM compares the expected profit for each type of service provided, varying the number of customers M , and then the optimal number of customers is determined.
Considering assumption VI in Section 3.6 (complete information), we conclude the OEM knows the hospital's reservation prices.Thus, OEM builds a structure that captures all the consumer surplus, which implies in the maximization of the producer profit.[25] sciENcE aNd tEchNology

the customer's maximum willingness to pay for the EW and on-call maintenance interventions respectively; b) Customer's optimal options adapted from Murthy & Asgharizadeh
Given that, we use Eqs.( 12) and ( 13) to obtain the reservation prices.Indeed, we equalize Eq. ( 12) to zero, and then isolate P in order to determine max P max P . Thus, the reservation price of the EW can be given in (14): Now, we calculate max s C by using Eq. ( 13).However, since it's impossible to isolate s C , we use a numerical method to find its expected value.Eq. (15) shows max s C equilibrium equation: Notice that in Eqs. ( 14) and ( 15 N , which are random variables.Therefore, we developed a DES-based algorithm, which is explained in more detail in the following Section, to obtain these values.

GRP-queue model simulation
As seen in Section 2.1, the presented formulation describes a queuing system with finite population ( M ), where times until failures follow a GRP.According to assumption III in Section 3.6, times to repair are exponentially distributed.If the number of failed units is greater than the number of servers ( m ), a queue is formed, and follows a FCFS rule.
In order to find the optimal reservation prices max P and max s C , it is necessary to simulate the alternating failure-repair process considering a GRP/Markovian/ m / i W ∞ /M//FCFS queuing system.We adopted a DES-based approach to represent the behavior of the system of interest, allowing us for modeling and solving problems that would otherwise be considered intractable or too complex (Zio [42]).
The DES algorithm we developed for queue simulation is shown in Figure 2.This proposed algorithm is similar to what was presented by Moura et al. [23], with two main changes: (i) our simulation model covers both base and extended periods, while the aforementioned paper only simulates the extended period; (ii) our model is formulated for a single customer class, while the aforementioned work allows for two priority classes, which are not implemented here.Our formulation is explained as follows.
At 0 t = , all M devices are considered new, the time to first failure for each equipment follows a Weibull distribution, queue is empty, and the first failure immediately begins its respective repair.Next, we track if the future events are failures or repairs based on the min ( a t , d t ), where a t is the next time of arrival (occurrence of a failure) and d t is the time of departure (next repair completion).Note that d t is initially set to infinity in step 1.2, since there is no equipment being repaired; as a consequence, it will never be smaller than a t .If a failure occurs when all service crews are busy, i.e., there are at least m failed units, the just failed device waits in queue.If there is at least one available service crew when a failure occurs, this failed device will immediately begin its repair.This procedure will continue over the total period ( W ). We also consider that if a failure occurs during 1 W , but its respective repair is completed during 2 W with overtime, the penalty charged is θ 1 .During simulation, all events of interest (times of failures and repairs, number of failures for each device, downtime and overtime of each device) are logged so that they can be used to obtain information about system availability, number of failures ( j N ), downtime of equipment ( dt j T ) and overtime ( ov j T ).These are the main outputs of the simulation (step 3), used later to obtain the prices of EWs and on-call services.
After the queue is simulated, customer's optimal strategy is defined.To that end, we estimate the reservation prices for each option offered by the OEM, which are given by Eqs. ( 14) and ( 15), using data from many replications ( rep n ) of the simulation.The complete process used to find the optimal number of customers and their respective reservation prices is given in Figure 3, and is more detailed in Figure 4.Note that the prices that the customers are willing to pay for each option depends on their risk aversion δ.
Initially, the population size (number of devices) is 1 M = , and then the DES is repeated for the defined number of replications, resulting in estimates for max P and max s C .Since the OEM has limited service capability (the number of service crews), M cannot be increased indefinitely because high values of M result in longer waiting times in queue, thus reducing operational time and increasing the occurrence of penalties.Therefore, M is increased and this process is repeated until the OEM's profits decrease for both repair service strategies (so that optimal number of hospitals served is guaranteed to be found); this is possible since, increasing M beyond its optimal value results in considerable increases in queue waiting times and OEM penalties (Ashgarizadeh & Murthy [2]).Finally, the optimal number of hospitals and the respective service prices are found by choosing the number of devices that result in the highest values for OEM's profit.

. Model's parameters estimation
An application example is here presented by using a failure database of an angiography device, which is technology-intensive, and supports the treatment and diagnosis of cardiovascular diseases.The angiogram, used for visualization of arteries based on x-rays, begins with introduction of an iodine contrast material injection into blood vessels though a catheter.X-ray angiogram provides anatomical information about blood vessels (Çimen et al. [4]).By watching the flow of the contrast fluid, the doctor can identify obstructions and narrowing, proceeding with treatment.
Angiography failures can result in incorrect diagnosis and inappropriate patient treatment, having negative consequences in the patient's health and hospital's reputation.Thus, angiographies are fundamental for hospitals profitability, and their unavailability represents a great loss of revenue, resulting in a negative economic impact.
Table 1 shows 38 times between critical failures and their respective times to repair.Considering this device is subject to imperfect repairs, we need to use the simulation-based solution proposed in the previous Section to reproduce the GRP-queue system.To that end, we first obtain the MLEs for the GRP parameters by using the procedure described in Yañez et al. [39].
We also obtained the MLE's for parameters α and β when the assumptions of perfect ( ) 0 q = and minimal ( ) repairs are adopted.By restricting q to fixed values, it is expected that MLE's result in inferior likelihood, or at most as good as that of obtained by imperfect repair assumption, since parameter search space is restricted.The MLE's for a , β and q for each repair hypothesis are given in Table 2, which also shows the mean squared error (MSE) for simulation data with each repair assumption in relation to observed data.Note that the lowest MSE is obtained when we considered imperfect repairs, which attests that imperfect repair is the most suitable assumption for this case.
Next, we estimate the expected number of failures j EN     by using the procedure described by Yañez et al. [39], and we compared the results against the observed failure data in Table 1. Figure 5 shows this comparison under assumptions of imperfect repairs.
For the application example, we will use the parameters shown in Table 3.The inputs given above were used to feed the simulation algorithm described in Figure 2. In order to find the results, the procedures given in Figure 3 and By using those parameters and GRP MLE's, the optimal number of equipment, and hospital's reservation prices obtained were respectively: * 47 M = , max P = $ 183,991 and max s C = $ 8,550.As it can be  1.Failure dataset of an Angiography device seen, the price of hiring an EW is near the range between 5% and 12% of the purchase price b C , which is consistent to what happens in the clinical environment (Murthy & Djamaludin [26]).The OEM's expected profit for the options ( 1A , 2 A ) are respectively ( ) , returning an expected profit of $ 2,979,171.Finally, the optimal choice for the client is 1 A and the expected utility for the customer is zero, as the OEM extracts all the consumer's surplus.

Sensitivity analysis
For better understanding on how sensitive the model is regarding parameters and repair assumptions, we present a sensitivity analysis.Note that in the tables presented throughout this analysis, base results are highlighted in grey.

Effect of q variation
First, in order to identify possible output changes, we disregard the MLE value found for q in the previous Section, and vary the rejuvenation parameter q as shown in Table 4.As seen in Section 2.1, the rejuvenation parameter measures the quality of repair.When q approaches zero, the quality of repair increases, returning the failed unit almost to "as good as new".Thus, the wear of equipment is reduced, and it is reasonable to expect a lower failure rate.As q increases in turn the quality of repair decreases, and equipment suffers higher wear over time.Queue length, overtime ( dt j i T W ) and amount of penalty incurred rise under option A 1 , making the OEM serves fewer clients, which reduces their profit.For  ≥ 0.16, the increased number of failures and amount of penalties are so high so that result in a change of strategy.

Variation on the model parameters
Generally, consumers with high risk-aversion are less willing to pay for the services, reducing OEM's profit.In this situation, the OEM tends to sell equipment to fewer customers.Table 5 shows how the players' optimal strategies change for different risk-aversion parameter values.For δ δ ≤ 0.08, strategy A 2 is chosen since customers become more tolerant to risk, thus not choosing EW anymore.For this case, OEM can perform maintenance for a greater number of hospitals, thus increasing waiting time in queue and downtime.However, profit is still increased, since customers are not as risk-averse as in the other tested cases, accepting to pay more for services.For δ 0.08 δ > , the hospital's optimal strategy is to hire the EW (A 1 ).The variations on results due to changes in the device characteristic life α are given in Table 6.Higher values of α result in longer times to failures, which in turn decreases the expected number of failures j EN     .Consequently, devices have increased availability and generated revenue, also increasing the OEM's profits.Yet, the number of sold units M increases along with α, and due to the occurrence of fewer failures, customers pay considerably more for oncall repairs, since the total value they are willing to pay for repair services is now spread across fewer payments.â max P = also increases with α increment, however at a lowest rate, because while devices fail less frequently, the OEM serves more devices as α increases, which causes more time spent in queues.For α α ≤ 1,140, strategy 2 A is selected, due to increased failure frequency and, consequently, higher amount of penalties to the OEM.
Changes in optimal strategies due to variations in the shape parameter β can be seen in Table 7.For higher values of β, equipment wears EN     , which in turn causes greater unavailability.Thus, the probability of penalty being incurred also increases, and the OEM chooses to serve fewer customers.In fact, as β ≥ β ≥1.75, EW is no longer advantageous, and A 2 becomes the optimal strategy; notice that the significant increase in M for β ≥1.75 occurs because the OEM pays no penalty at all for strategy A 2 , which allows it for serving more customers.The increase in max P when β ≥1.75 is due to the number of customers being served, which increases the likelihood of overtime, and the customers' willingness to pay for services.Notice in Eq. ( 14) that max P increases along with the expected overtime in repairing the device E ov j T     , which is higher as more customers are served, resulting in longer waiting times in queue.Table 8 presents how optimal strategies change due to variation in the service rate μ.With higher values of μ, the service crew can repair a greater number of failed devices per time unit, decreasing the queue size.Therefore, the OEM can deal with a greater number of customers, increasing its own profit.On the other hand, when μ decreases, the service crew can repair fewer items per time unit.Notice that for μ ≤ 0.175, the service crew is too slow to repair the equipment by τ, resulting in a large amount of penalty under the option A 1 .Then, the OEM's optimal strategy changes to set max PP > and max s s CC = , which induces the customer to choose A 2 , i.e., on-call services.
Table 9 shows how the optimal solution changes due to the maximum times to repair τ, which influences how long equipment waits (in queue or being repaired) without penalizing the OEM.The higher the value of τ, less penalty is incurred, because the frequency of those delays in returning the equipment to operational state will be lower.In other words, the OEM's payoff increases as τ increases.When the acceptable tolerance τ is smaller, overtime and penalty increase under EW option.Then, for values of τ 10 τ ≤ , it is best for the manufacturer to sell on-call repairs.
Table 10 presents the behavior of optimal solution due to revenue R variations.Notice that with the increase of monetary incomes to the hospital, they are willing to pay considerably more for each option.Since hospitals generate more revenue with operation of their devices, they become willing to pay significantly more for maintenance services.Notice, however, that hospital's decision did not change with variations in R .Finally, Table 11 and Table 12 show the changes in optimal solution due to variations on penalty parameters θ 1 and θ 2 respectively.Even though θ and τ both influence the total amount of penalties, their effects are distinct.High values for θ 1 increases the importance of repairing the equipment by τ, since delays will be severely penalized.
For smaller values of θ 1 , the OEM is able to serve a greater number of hospitals due to the decrease in penalties.However, when the number of hospitals increases, the penalty during 2 W also increases (since the reduction in penalty rate is only for 1 W , in this case), which may cause strategy 2 A to be selected.This happens for θ 1 ≤ 0.7.In the case of θ 2 , when it rises too much, strategy 2 A is chosen, since penalties during 2 W increase considerably.This occurs for θ 2 ≥ 3.3.

Concluding remarks
In this paper, a decision model for an Extended Warranty involving hospitals and OEM was proposed.For modelling this situation and determining the players' optimal strategies, a SG formulation was employed, with the OEM being the leader and the hospital the follower.This situation is commonly found in the market of technologyintensive equipment, which is characterized by a greater bargaining power for the manufacturer, which is the only part capable of performing maintenance interventions adequately.
In order to approximate the problem to a more realistic context, we considered the equipment is subject to imperfect repairs, and to model this issue, two approaches were joined: GRP and queueing theory.Additionally, an application example was presented with real failure data of an angiograph to determine the optimal strategies for each player and demonstrate applicability of the model.Furthermore, we perform a series of sensitivity analyses by showing how model results and players' strategies behave under different scenarios.
Some limitations of the presented approach may also be pointed out.In real-world situations, customers do not present homogeneous risk behavior, consequently, different customers often choose different strategies.Also, different agents commonly have access to different levels of information (asymmetric information), so that it is difficult to predict the actions of other players.Based on these limitations, and also intending to extend the present model, the following features could be implemented: Consideration of information asymmetry by employing a princi-• pal-agent formulation (Jiang et al. [15], Jin et al. [16]).Analysis of consumer usage rate, along with the definition of a • two-dimensional warranty policy (Yang et al. [40]).
A dynamic SG with a greater time horizon to analyze the pos-• sibilities for renewal of the extended warranty, analyzing the behavior of the players during longer periods.Incorporation of a heterogeneous market, with different profiles • of customers, represented by different risk-aversion parameters.

kA:N 2 W
Decision variable of the hospital; b C : Purchase price of the equipment; r C : OEM's mean cost to repair a failed unit; s C : Price charged by the OEM per repair; max s C : Hospital's reservation price for the on-call service; M : Number of customers; * M : Optimal number of customers; rep n : Number of Monte Carlo replications; : Number of failures occurred over the intervals 1W and , respectively, for the th j device ( th j device is held by the th j customer (hospital)); j N : Total number of failures during W for the th j device;

T : Period that the th j system is on operational state during 1 W
and 2 W respectively; op j

Fig. 1 .
Fig. 1. a) The game tree -max P and max s C represent the customer's maximum willingness to pay for the EW and on-call maintenance interventions respectively; b) Customer's optimal options adapted from Murthy & Asgharizadeh[25]

Fig. 2 .
Fig. 2. DES algorithm used to simulate the queue system

Fig. 3 .Fig. 5 .
Fig. 3. Diagram of the optimization process Nash Equilibrium occurs when the OEM sets P = max P and s C > max s C is the time between the last failure and W ;

Table 3 .
Parameters for the application example

Table 4 .
Optimal solution changes due to q variations

Table 5 .
Optimal solution changes due to δ variations

Table 6 .
Optimal solution changes due to α variations

Table 7 .
Optimal solution changes due to β variations

Table 8 .
Optimal solution changes due to μ variations

Table 9 .
Optimal solution changes due to τ variations

Table 10 .
Optimal solution changes due to R variations

Table 11 .
Optimal solution changes due to θ 1 variations

Table 12 .
Optimal solution changes due to θ 2 variations