Optimizing a warranty–based sustainable product service system using game theory

Abstract

In recent years, the concepts of sustainability and product service system have been closely associated with each other. In today’s competitive markets, due to Extended Producer Responsibility and customer environmental awareness, producers shift from ‘product-seller’ towards ‘product-service provider’ for using the environmental and economic advantages of coupling a product with services. This paper proposes a novel approach to determine the optimal warranty period and the out-of-warranty replacement period, from the point of view of the producer and the customer to minimise the total cost of usage and end of life of product. As regards the fact that adopted strategies by producers and customers sometimes are in conflict and it affects choosing the optimal product usage period, a game theory model was developed in this study. Finally, a case study with data from chain of local notebook service centres was applied to demonstrate some practical aspects of the developed model.

Keywords:

• End of life management
• game theory
• stackelberg
• sustainability
• warranty

1. Introduction

Over the last decade, in the field of sustainable production, a call for ‘doing more with less material’ is made to reduce the waste during the life cycle of products (Westkämper 2000). From this perspective, combination of products and services for fulfilling customers’ requirements becomes more crucial and the concept of sustainable PSS (Mont and Tukker 2006) and eco-services (De Camillis, Raggi, and Petti 2010) have been rising in the field of sustainable production. A PSS suggests the need to link products and services for solving existing environmental problems from a systematic perspective (Mont and Tukker 2006).

The importance of services in the manufacturing industry has its origins in environmental sustainability, which highlights the need to manage the product life cycle by providing different types of services. Those activities taking place after the purchase of the product and to support customers in the usage phase, have become increasingly important as a source of differentiation and profit for manufacturers.

After sales services play an important role in customer’s decision to purchase a durable product (Kurata and Nam 2010). One of these after sales services is warranty. Warranty is one of the specific form of services that influence on enhancing demand (Xia and Gilbert 2007) since warranty serves as a signal of product reliability (DeCroix 1999). Warranty also is one of the best way to build and maintain a relationship with a customer (Hartman and Laksana 2009).

The primary role of the warranty and services is to offer a post-sale remedy for customers when a product fails to fulfil its intended performance during the warranty period. Murthy and Nguyen (1988) defined a warranty as a ‘contractual obligation incurred by a manufacturer, in connection with the sale of a product, under which the manufacturer is required to ensure proper functioning of the product, during the warranty period’. In case of product failure during warranty period, manufacturer is in charge of providing replacement/repair or this responsibility shares between the manufacturer and the customer based on the type of warranty contract (Darghouth, Chelbi, and Ait-kadi 2012).

Recently, a new function for warranty is introduced by some researchers in the area of product sustainability. Warranty can have an indirect but important effect on saving energy and material usage during product consumption phase. Product warranties are focused on repairs and replacement at the beginning of a product’s life. So it can effect waste disposal at the end of a product’s useful life by keeping product’s functionality. Because of this, the definition of ‘Green Warranty’ is introduced and researchers are going where warranties have never gone before (Warranty Week 2015). One way to reduce the consumption of resources which results in long-term sustainability of resources is product remanufacturing through free-replacement warranty policy (Chari, Diallo, and Venkatadri 2013).

Contribution and objectives of this paper are as follows:

•

A model to determine optimal warranty and out of warranty period in the PSS context from producer and customer’s point of view has been developed using Stackelberg game theory model.

•

To validate the developed model, a case study for notebook in Iranian computer market has been applied and has been optimised and the results are discussed. The goal is to determine the optimal warranty period and the optimal out-of-warranty replacement period, which minimises the total usage cost of the product and EOL 1 cost. Obviously, the model could be applied to any product and any geographical location.

At this point in time, surprisingly, only a few studies have focused on the role of warranty in the PSS context. For this reason, we have initiated a novel approach that is designed to explore services from producer and customer’s point of view as two players in Stackelberg model and it minimises the total costs during the usage and EOL phase of the product.

This paper is organised as follow. Section 2 is related to literature survey on sustainable services and the role of warranty as a design parameter in sustainability. Section 3 states the problem. Section 4 introduces modelling the problem using Stackelberg game theory model. Section 5 describes the method used to solve the proposed model. Section 6 represents the case study relating to the proposed problem and finds the optimal warranty and out of warranty period. Section 7 focuses on performance of the proposed model. Section 8 presents sensitivity analysis of parameters used in the model. Finally, a conclusion is provided in section 9 and suggestions for future works is discussed.

2. Literature survey

Based on the widespread literature of the product service system (PSS), PSS has known as a way of reaching sustainability. The concept of PSS represents a promising model to steer production and consumption systems towards sustainability. PSSs are business models based on selling performance (i.e. results) rather than products and if PSS is designed properly, it will result in providing a range of economic and competitive advantages and it will support the dematerialisation of economy and hence provide environmental benefits (Ceschin 2013). Refer to Stahel’s words: ‘Functional economy is an economy that optimizes the use of goods and services and thus, the management of existing wealth. The economic objective of the functional economy is to create the highest possible use value for the longest possible time while consuming as few material resources and energy as possible. The functional economy is therefore more sustainable, than the present economy, which is focused on production as its principal means to create wealth and material flow’ (Stahel 1997). Moreover, Sustainability means keeping an existing system operational and maintaining the ability to manufacture and ensuring that system satisfies the original requirements (Sandborn and Myers 2008). Based on Song study, there is some dependencies and correlations between PSS requirements. He has evaluated these dependencies considering different stakeholders preferences. Based on proposed rough DEMATEL-based approach in this study, the individual judgements and also the judgement distribution among all the decision-makers in same category have been considered (Song and Cao 2017).

Design for sustainable behaviour (DfSB) is a relatively new theory related to application of design strategies which influence customer behaviour in product usage phase with the aim of achieving sustainability (Lilley 2009). Wilson et al. conducted an extensive literature survey to find considerations and limitations of this approach in changing customer behaviour (Wilson, Bhamra, and Lilley 2015).

Barquet et al. identified set of factors which contribute to make a sustainable business model from a PSS. This study shows that factors such as ‘design for Environment (DFE)’, ‘identify economic value for each stakeholder’, ‘Promote behavior change for customer and PSS provider’, ‘delineate actions to social well-being’ and ‘innovate in different levels’ could lead to create a sustainable PSS business model (Barquet et al. 2016).

Ameli et al. proposed a model to optimise selection of product design alternatives considering product environmental impacts and its life cycle costs. This model helps a producer to determine the most economical and environment-friendly design in developing a new product (Ameli, Mansour, and Ahmadi-Javid 2016a).

Ameli et al. developed a simulation-based optimisation model to maximise the total profit and minimise the product environmental impact considering regulatory restrictions, such as the waste electrical and electronic equipment directive in order to improve product EOL management through considering life cycle issues at the product design stage (Ameli, Mansour, and Ahmadi-Javid 2016b).

One logical approach to achieve sustainability is extending the product’s life by warranty and services. Customer’s decisions during the ‘usage’ phase of a product on whether to repair or throw items away affect the product life span (and thus the rate of waste generation). The E-SCOPE survey (King et al. 2006) found that 68% of respondents have cited cost as a reason why they did not get items repaired: a factor borne out by the fact that while new washing machine prices increased by only 40% during the 1980s–1990s, repair costs over this period increased by 165%. However, little research has been conducted to understand the advantages of this closed loop option with respect to sustainability.

According to customer tendency, producer can use PSS for achieving the advantage of product service sustainability. Leasing and warranty are two different kind of after sales services applied to various products. The advantage of leasing is analysed in many researches (Thurston and De La Torre 2007). But the advantage of warranties and services for sustainability is not explored yet. Also, its application in the developing countries is a relatively unexplored area and they are facing huge challenges in the management of Waste from Electrical and Electronic Equipment (WEEE) which are either internally generated or imported illegally as ‘used’ goods (Nnorom and Osibanjo 2008).

Sandborn and Myers introduced warranty as a way of achieving sustainability. Warranty was applied as a tool for keeping functionality of the product and away from reaching EOL. The simple cost model was developed to illustrate warranty cost due to product failure (Sandborn and Myers 2008).

Yu et al. developed a goal programming optimisation model based on eco-design for realisation of life cycle design for customers’ requirements variety. Extending life time of product, maintenance and EOL options were applied as decision variables (Yu, Kato, and Kimura 2001).

Kimura et al. have addressed closing the product loop with product maintenance during operation, periodical upgrade and parts/products reuse at the EOL phase and called this methodology ‘inverse manufacturing’. A computer simulation method has been developed to evaluate the product quality under deteriorated conditions during use period (Kimura, Hata, and Suzuki 1998).

Chien introduced the cost model for optimal warranty period and optimal replacement period from the perspective of the seller (manufacturer) and buyer (customer) in two steps procedure. At the first step, optimal warranty period was optimised with respect to total cost during consumption period. In the second step the optimal out of warranty period was determined with respect to optimal warranty in the first step. So in this model the trade-off between producer and customer is not modelled. Also the EOL cost of product was not considered in the developed model and only usage cost was applied (Chien 2005).

In another research by Chen, the impact of extended service contracts on retail business of durable products is analysed. Their results show how an extended service purchase is influenced by hedonic/utilitarian value of products, marketing action, and customer’s characteristics (Chen, Kalra, and Sun 2009).

Murthy determined the optimal warranty policy when a customer has an option to decide the timing to take out an extended warranty and the length it will last. Assuming a case where a manufacturer sells directly to customers, a game theoretic model proposed the optimal warranty strategies (Murthy and Nguyen 1988).

Kameshwaran showed the product service bundling and pricing for a complex durable product that will more likely be maintained than replaced. Their game theory model shows how a firm should offer an after sales programme: offering product only, product and service independently, or product and service bundled together (Kameshwaran, Viswanadham, and Desai 2009).

Aggrawal et al. described warranty as a persuasive marketing tool. They presented a two-dimensional diffusion model to obtain optimal price and warranty length for a product using exponential distribution to represent the life time distribution of a product (Aggrawal et al. 2014).

Chang and Lin investigated length of extended warranty for repairable products. They described preventive maintenance as an approach that seller chooses to reduce the number of product failures during the base extended warranty periods. Also seller offers a discount to extend warranty after the initial warranty period expires. They proposed a mathematical model to maximise profit from the seller viewpoint and the length of extended warranty was obtained (Chang and Lin 2012).

Liu et al. proposed a reliability-based method which specifies warranties for certain products configuration schemes. They estimated warranty costs under a minimal repair warranty. They used continuous time Markov model to measure the vulnerabilities of components involved in configuration (Liu, Liu, and Wang 2013).

Shokohyar et al. presented a model to satisfy sustainable product service system (S-PSS) requirements by considering environmental and economic impacts of the product during consumption and end of life phase from point of view of customer and manufacturer. They used a multi objective genetic algorithm to optimise service period and EOL decisions (Shokohyar, Mansour, and Karimi 2014).

Alqahtani and Gupta investigated the impact of offering warranty in a discrete-event simulation model in which the cost of manufacturing is minimised and the willingness of customer towards buying a remanufactured product is maximised simultaneously (Alqahtani and Gupta 2017).

Moneim et al., proposed a new random parameter in the failure rate of a model which aims to evaluate warranty reverse cost as a function of product age and its usage period (Moneim, Ghazy, and Hassan 2017).

Table 1 presents a brief summary of the reviewed literature. As shown in the table, several papers explore the effect of warranty by considering the producer and customer’s costs. But none of them have simultaneously considered warranty, out of warranty and EOL phase to minimise the total cost of usage and EOL phase of product from view point of customer and producer.

Table 1. Summarised literature review.

Publication	View point		Product phase consideration			Method	Findings
	Customer	Customer	Warranty period	Out-of-warranty period	EOL phase
Murthy and Nguyen (1988)		✓	✓			Linear programming	Optimal repair cost
Kimura et al. (1998)		✓	✓		✓	Simulation	Designing a rapid product life cycle
Yu, Kato, and Kimura (2001)		✓	✓		✓	Multi objective optimisation	Life cycle design
Chien (2005)	✓	✓	✓	✓		Linear Programming	Optimal Warranty and Out-of-warranty period
Sandborn and Myers (2008)		✓	✓			Linear programming	Design for sustainability
Kameshwaran, Viswanadham, and Desai (2009)	✓	✓	✓			Game theory	The best strategy for product bundling and pricing
Chen et al. (2009)	✓	✓	✓			Conceptual modelling	Examining purchases of extended service contracts
Chang and Lin (2012)		✓	✓			Linear programming	Maintenance policy and length of extended warranty
Shokohyar, Mansour, and Karimi (2014)	✓	✓	✓		✓	Genetic algorithm	Optimised service period and EOL decisions
Liu et al. (2013)		✓	✓			Markov model	Designing customised warranty offering
Aggrawal et al. (2014)		✓	✓			Non-linear programming	Optimal warranty period and price
Wilson et al. (2015)	✓					Literature review	Designing strategies influence consumer behaviour towards sustainability
Barquet et al. (2016)	✓	✓				Literature review	Identifing set of factors which contribute to make a sustainable business model from a PSS
Ameli et al. (2016)		✓			✓	Mathematical programming	Optimising selection of product design alternatives considering product environmental impacts and its life cycle costs.
Ameli et al. (2016)		✓			✓	Simulation	Improve product EOL management considering life cycle issues at the product design stage
Song and Cao (2017)		✓				Rough set theory Group DEMATEL	Identifying dependencies and correlations between PSS requirements
Alqahtani and Gupta (2017)	✓	✓	✓			Discrete-event simulation	Minimising the cost of manufacturing and maximising the willingness of consumer towards buying a remanufactured product
Moneim, Ghazy, and Hassan (2017)		✓	✓			Monte Carlo simulation	Evaluating warranty reverse cost
Current work	✓	✓	✓	✓	✓	Game theory (Stackelberg)	Determining warranty and out-of-warranty period

The original contribution of this paper is a model that integrates several considerations that many researchers have addressed separately, including optimisation of warranty and out of warranty period form producer and customer’s point of view with respect to sustainable product service system.

There are three key players in the sustainable production context, namely: producer, customer and the government and each player aims to minimise its own cost (Kameshwaran, Viswanadham, and Desai 2009). Establishing a balance between their goals, the optimised warranty and out of warranty period can be reached regarding the total cost during warranty, out of warranty and EOL phase.

3. Problem statement

In this paper, product’s usage and EOL period which is divided into three different phases namely warranty, out of warranty and EOL, have been investigated. Based on a failure-free warranty contract, the producer is responsible for all the product’s failures cost during the warranty period; any repair or replacement is free of charge for customer during the warranty period (Chien 2005). When the warranty expires, customer is responsible for product maintenance during out of warranty period. In several papers, exponential distribution is introduced as one of the best distribution functions to model product failures (Blischke 1995), (wu, chou, and huang 2009), (Shokohyar, Mansour, and Karimi 2013), (Shokohyar, Mansour, and Karimi 2014). So in the present paper, the product’s failures follow an Exponential distribution with failure rate . As the maintenance cost increases, the user has to replace the product by a new one. The only option which is considered as EOL option is landfill. Customer returns product to the producer with probability of α and producer bears a cost of C _d per product.

As regards the fact that the adopted strategies by producers and customers sometimes are in conflict and it affects the optimal product usage period. In this study, they have been considered as two rivals in game theory modelling and different strategies adopted by these two rivals in product usage (including warranty and out of warranty period) and end of life phase, have been investigated.

In the proposed model, economical aspect of sustainable development has been modelled based on stackelberg game in which the producer is the leader and determines the warranty period in order to minimise his costs. Then the customer after observing the producer’s choice, determines the out of warranty period in order to minimise his costs (follower player).

For considering the deal between producer and customer, the problem was solved backwards. Accordingly, the customer’s gain was optimised at first and optimal out of warranty period was placed in the producer’s payoff and then the optimal warranty period was determined. In order to validate the proposed model, optimal warranty and out of warranty period was determined for a case study in notebook industry using GAMS software and sensitivity analysis was performed.

Table 2 shows the two players of the proposed model and their costs and profits obtained during product usage and EOL period. The main objective is to determine the optimal warranty and out of warranty period with respect to the costs of product during usage and EOL phase from producer and customer’s point of view as the leader and follower in Stackelberg game model.

Table 2. Major elements of the proposed decision model.

	Producer	Customer
Costs	Failure costs during warranty period EOL cost	Inconvenience cost during the warranty period
		Price of obtaining the warranty
		Failure costs during out of warranty period
Profits	Warranty period profit	Out-of-warranty period profit
	Out-of-warranty period profit
Decision strategy	Warranty period	Out-of-warranty period

Producer and customer’s cost components are detailed as follows:

3.1. Producer’s costs

Producer’s costs components which are represented in Equation (2(2) ) consist of four parts as follow. Each cost components, is described in the next sub section.

3.1.1. Failure cost during warranty period

The cost of repair or replacement of a failed product during warranty period is completely the responsibility of the producer which is represented by C_pr per failure. Total failure cost depends on the reliability of a product. Some common lifetime distributions can be applied to represent the reliability of the product. The Exponential distribution is one of the most flexible distributions and the expected value of distribution function, E(x), can be determined using Equation (1(1) ) to represent the cumulative probability of units failing at time t. The total warranty cost borne by the producer is presented by Equation (3(3) ).(1)

3.1.2. Warranty period profit

Providing warranty for the product sold to the customer could have financial benefits for the producer. This gain is proportional to the warranty period. So the longer the warranty period, the more attractive the product will be to the customer. This gain is represented by Equation (5(5) ).

3.1.3. Out-of-warranty period profit

During the out-of-warranty period, customer is responsible for all of the product failure’s costs. Based on Kameshwaran et al., if producer and other independent service companies offer the same charges for repairing failed products during out-of-warranty period, customers prefer to get services from the producer as they believe producer provides better quality services (Kameshwaran, Viswanadham, and Desai 2009). In the proposed model, the probability of the customer’s returning the product to the producer for obtaining services during out-of-warranty period is presented by β. The producer gains a profit due to servicing failed products during out-of-warranty period and is represented by Equation (6(6) ).

3.1.4. EOL cost

Because of the uncertainty in return of the product at EOL phase, the probability of take-back is represented by α in the proposed model. The only option at EOL phase is landfill with the cost C _d for each product. The total EOL costs of product are shown in Equation (7(7) ).

3.2. Customer’s costs

On the marketing side, customer has his own costs. Customer’s cost components which are represented in Equation (9(9) ) consist of four parts and are as follow.

3.2.1. Warranty cost

Customer is obligated to pay more when he buys a product with warranty. This cost is directly proportional to the length of the warranty period. In this paper, the relation between warranty cost and the length of warranty period, is considered to be a linear one similar to other studies (Chien 2005; Shokohyar, Mansour, and Karimi 2014).

3.2.2. Inconvenience cost during the warranty period

Within the warranty period T _w , the producer has to maintain his product free of charge. Although the maintenance is free, the customers will experience inconvenience incurred by the product failure. That means, any failure within the warranty period not only results in the seller’s cost to provide the maintenance, but also results in a cost to the customer (e.g. handling cost, shortage cost, system down cost, waiting cost) Therefore, we have assumed that C _cw is the cost incurred by the customer due to the product failure during the warranty period.

3.2.3. Failure cost during out-of-warranty period

When the warranty expires, the customer should pay any cost due to product failures during out-of-warranty period which is represented by C _cr per failure. By increasing the maintenance cost, customer decides to end the product’s life and buy a replacement.

3.2.4. Out-of-warranty gain

When the customer keeps the product during out-of warranty period, he has gained by not paying for a replacement product. Like the producer’s warranty gain, in this paper, the relation between out-of-warranty gain and the length of out-of-warranty period, is considered to be a linear relationship.

In the next section, the mathematical formulation of the stated problem is represented.

4. Mathematical formulation

In this section, a mathematical model is developed in order to minimise the total costs during usage and EOL phase from the producer and customer’s point of view as two players in a game theory. Based on the dynamic nature of the deal between producer and customer, in which the producer determines the warranty period at the time of selling of the product and after that customer decides about keeping the product after expiring the warranty period (out-of-warranty period), Stackelberg game theory model is one of the best approaches of modelling problem in which the producer is the leader player and the customer is the follower player (Gibbons 1992a, 1992b; Rasmusen 1994). The nomenclature of the model is given in Table 3. The proposed model is divided into two parts; one part is from producer’s point of view and the other from customer’s point of view.

Table 3. The nomenclature of the proposed model.

Decision variables
T w	Warranty period
T ow	Out-of-warranty period
	Total producer’s costs
	Total customer’s costs
Cost pw	Producer’s costs during warranty period
Profit pow	Producer’s profit of servicing failed products during out-of-warranty period
Cost pEOL	Producer’s costs during EOL phase
Profit pw	Producer’s profit of offering warranty
Cost cw	Customer’s cost during warranty period
Cost cow	Customer’s cost during out-of-warranty period
Profit cow	Customer’s profit during out-of-warranty period
α	Probability of product return by customer at EOL
C d	Costs of product disposal
C pf	Warranty cost per failure paid by producer
C cw	Customer’s inconvenience cost per failed product during warranty period
C cf	Repair cost per failure paid by the customer during out of warranty period
X w	Number of product failures during warranty period
X ow	Number of product failures during post-warranty period
K 1	Coefficient of producer’s profit for offering warranty
K 2	Coefficient of customer’s costs for using warranty
K	Coefficient of customer’s profit for keeping `product during out-of-warranty period
	Exponential function rate parameter during warranty period
	Exponential function rate parameter during post-warranty period ()
β	Percentage of customers that require services from the producer during out of warranty period

4.1. Mathematical formulation from producer’s point of view

As mentioned above, producer’s costs are divided into four parts which are shown in Equation (2(2) ):(2)

In Equation (2(2) ), Cost _pw is producer’s cost during warranty period. If the failure cost per product is represented by C _pr , then the producer’s cost during warranty period (Cost _pw ) is given by Equation (3(3) ).(3)

Equation (3(3) ) can be simplified as follows:(4)

Producer profit, , due to offering warranty can be calculated as shown in Equation (5(5) ).(5)

Profit _pow , is producer’s profit during out-of-warranty period. If β% of customers get services from the producer during out-of-warranty period, therefore, the producer’s profit during out-of-warranty period is given by Equation (6(6) ).(6)

Cost _pEOL is the cost borne by the producer to landfill the product at EOL period and it is represented by Equation (7(7) ). It should be mentioned that product disposal is the only option which has been considered in this model. So in case of product landfill, there is no salvage value for the producer because it has been assumed that the product will be used by the customer until the end of it’s useful life and will be returned to the producer by customer in situation that there is not any considerable salvage value for it.(7)

By substituting Equations (4(4) )–(7(7) ) in Equation (2(2) ), the total producer’s costs can be calculated as shown by Equation (8(8) ).(8)

4.2. Mathematical formulation from customer’s point of view

As mentioned above, customer’s costs are divided into three parts which are represented by Equation (9(9) ).(9)

Cost _CW is customer’s cost during warranty period which is made up of customer’s inconvenience cost per product failure during warranty period and the cost of obtaining warranty which is proportional to the length of the warranty period. Equation (10(10) ) represents customer’s cost during warranty period.(10)

Equation (10(10) ) can be simplified as follows:(11)

Cost _cow is customer’s cost during out-of-warranty period and is shown by Equation (12(12) ). The cost per product failure for the customer during out-of-warranty period is C _cf .(12)

Equation (12(12) ) can be simplified as follows:(13)

Profit _COW is customer’s profit due to keeping the product during out-of-warranty period which based on Equation (14(14) ) is proportional to out-of-warranty period.(14)

By substituting Equations (11(11) ), (13(13) ) and (14(14) ) in Equation (9(9) ), the total customer’s costs can be calculated by Equation (15(15) ).(15)

5. Solving the proposed model using backward induction method

Backward induction method is one of the best techniques in solving dynamic game theory models (Gibbons 1992a), (Osborne and Rubinstein 1994). So in this paper, the Stackelberg game theory model of producer-customer in determining the optimal usage period of the product, has been solved using backward induction method. Based on this technique, the problem is solved from the end and after determining the Nash equilibrium of the sub games and eliminating them, the total Nash equilibrium of the model is determined (Gibbons 1992b).

In the proposed model, the producer is the leader in Stackelberg game theory model and as the first player, determines the warranty period when he sells his product. In the next stage, the customer as the follower player in the Stackelberg game theory model, determines the optimal out-of-warranty period after observing the offered warranty period by the leader. To solve the dynamic Stackelberg game theory model of producer-customer, the following steps are applied:

•	Step 1: In order to optimise the customer’s profit, the derivative of Equation (15(15) ) with respect to TOW is obtained and is equated to zero.
•	Step 2: from the previous step, equation representing the relation between T OW and T W is obtained.
•	Step 3: By substituting the relation between T OW and T W (obtained in the previous step) in Equation (8(8) ), the producer’s profit is converted in to a univariate non-linear model.
•	Step 4: to obtain the optimal T W , the GAMS software was used to solve the univariate non-linear model of step 3.
•	Step 5: To achieve the optimal T OW , the calculated T W from step 4, is substituted in equation obtained in step 2.

In order to optimise the customer payoff, derivative of the Equation (15(15) ) is obtained which is shown in Equation (16(16) ).(16)

In order to obtain the optimal relationship between T _W and T _OW in Equation (16(16) ), it should be converted in to a separable function of T _W and T _OW . The only part of the Equation (16(16) ) which is not separable is Ln(T _W  + T _OW ). One of the common methods in order to convert a function in to a separable function, is using linear approximation (Cooper 1974). Linear approximation of Ln(T _W  + T _OW ) based on regression method is obtained from Equation (17(17) ).(17)

As it is shown in Table 4, the error rate of the linear approximation of Equation (17(17) ) for T _W  + T _OW  > 5 is less than 1 month. By considering the fact that variables of the proposed model are warranty and out-of-warranty replacement periods which would be satisfied by accuracy of 1 month, this linear approximation would be still efficient.

Table 4. Error rate of the linear approximation.

T w + T Ow	Ln(T w + T Ow )	0.021 (T w + T Ow ) + 2.745	Error rate (in month)
1	0	2.766	2.766
5	1.694	2.850	1.241
10	2.303	2.955	0.652
15	2.760	3.060	0.352
20	2.996	3.165	0.169
25	3.219	3.270	0.051
30	3.401	3.375	−0.026
35	3.555	3.480	−0.075
40	3.689	3.585	−0.104
45	3.807	3.690	−0.117
50	3.912	3.795	−0.117
55	4.007	3.900	−0.107
60	4.094	4.005	−0.089
65	4.174	4.110	−0.064
70	4.248	4.215	−0.033
75	4.317	4.320	0.002
80	4.382	4.425	0.042
85	4.443	4.530	0.087
90	4.500	4.635	0.135
95	4.554	4.740	0.186
100	4.605	4.845	0.240

The relation between T _OW and T _W is obtained by substituting Equation (17(17) ) in Equation (16(16) ) and it is shown in Equation (18(18) ).(18)

By substituting Equation (18(18) ) in producer’s payoff from Equation (8(8) ), the producer’s payoff turns in to a univariate non-linear model which is shown in Equation (19(19) ).(19)

6. Illustrative case study

A case study, using data for the notebook computer market in Iran represented in Shokohyar, Mansour, and Karimi (2014) has been adopted to validate the proposed model. Table 5 includes all of the cost parameters for keyboard parts which have been obtained from Shokohyar, Mansour, and Karimi (2014) which has been provided by Iranrahjoo chain of service centres which is the exclusive representative of Sony laptops in Iran and have many years of experience in this field. According to these experts, when a notebook breaks down, it will need replacement with probability of 20% and repairing with probability of 80%. The probability of requiring replacement doubles during out-of-warranty period.

Table 5. Cost parameters for keyboard parts.

Part name	During warranty period		During out-of-warranty period		Disposal
	Replacing ($)	Repairing ($)	Replacing ($)	Repairing ($)	C d ($)
Electronic function unit	7.56	3.30	9.87	4.32	6.96
Base	2.70	1.02	3.42	1.43	2.34
Key foil	7.62	4.08	10.21	5.41	6.42
Keys	14.70	5.82	19.26	7.62	11.58
Total	32.58	14.22	42.76	18.78	27.3
	C pf  = 0.2 (32.58) + 0.8 (14.22) = 17.892		C cf  = 0.4 (42.76) + 0.6 (18.78) = 28.372

The other parameters involved in the proposed model which have been chosen based on the experts’ opinion of personnel at Iranrahjoo chain of service centres are shown in Table 6.

Table 6. The other cost parameters.

K	K 1	K 2	β	α			C cw
8	4	4	0.5	0.5	0.01	0.01	0.1 C pf

The proposed model was applied to the case study and the optimal solution was obtained by solving the producer payoff from Equation (19(19) ) in GAMS 23.6 software. The optimal solution obtained from GAMS indicated that the optimal proposed warranty period offered by producer is equal to 21.423 months with the profit of 79.743 US dollars.

By substituting () in Equation (18(18) ), the optimal out-of-warranty is obtained as follows:

By substituting () in Equation (15(15) ), the optimal customer’s cost is obtained as follows:

The optimal out-of-warranty replacement period acceptable by the customer is equal to 32.596 months with the cost of 63.149 US dollars.

7. Model performance

Current paper presents a novel game theory model in which optimal warranty and out of warranty period are determined in the PSS context from producer and customer points of view. Considering the fact this model benefits from a novel approach to the problem, performance of the proposed model is assessed in two parts as follows: the first part focuses on logic, parameters and components of the proposed model, and the second part focuses on the results of the proposed model.

(1)	parameters and components of the proposed model:

Validation of some of the parameters and components of the proposed model based on some previous works are as follows:

•	Chien (2005)

Chien model and the proposed model are similar considering following parameters and components:

◦	Expected value of product breakdowns effects in the cost of the warranty period
◦	Linear relationship between the length of the warranty period and the producer’s profit for offering warranty
◦	Customer’s inconvenience cost per failed product during warranty period
◦	Expected value of product breakdowns effects in the cost of the out-of-warranty period.
•	Subramanian, Gupta, and Talbot (2005)
◦	Similarity in modelling product EOL costs.
•	Kameshwaran, Viswanadham, and Desai (2009)
◦	Considering the probability of requesting service from the producer during out of warranty period
•	Shokohyar, Mansour, and Karimi (2013)

Their model and the proposed model are similar in considering following parameters and components:

◦	Expected value of product breakdowns effects in the cost of the warranty period
◦	Customer’s inconvenience cost per failed product during warranty period
◦	Costs of product disposal at the EOL phase
•	(Shokohyar, Mansour, and Karimi 2014)

Their model and the proposed model are similar in considering following parameters and components:

◦	Expected value of product breakdowns effects in the cost of the warranty period
◦	Linear relationship between the length of the warranty period and the producer’s profit for offering warranty
◦	Customer’s inconvenience cost per failed product during warranty period
◦	Costs of product disposal at the EOL phase

(2)	Results of the proposed model:

The proposed model is not any extension of previous models and based on the surveyed literature, there is not any specified model which its results could exactly be compared with the proposed model. However, a comparison has been carried out between this paper and two other papers regarding some comparable aspects as follows:

•

Kameshwaran et al. investigated the integration of offering services with product considering two market structures: monopoly and duopoly. They modelled the strategic interactions between competitors. The results of this paper showed that either in monopoly or duopoly market, entering to the market of offering services beside the product, leads firms to obtain non-zero profit (Kameshwaran, Viswanadham, and Desai 2009). Results of the analyzed case study based on proposed model are also in line with Kameshwaran paper and confirms that entering to the market of PSS not only has economic benefits for the producer but also the customer benefits from these economic advantages.

•

Shokohyar et al. presented a model to satisfy sustainable PSS requirements considering environmental and economic impacts of the product during consumption and end of life phase from point of view of customer and manufacturer using a case study of a notebook market which its data are somehow similar to the proposed model. The case study and related data that were used by Shokohyaer et al. were used as input data in this research. They used simulation-based optimization approach to investigate their results, while in this paper, a game theory approach was adopted, where similar input data but different modeling and solution approach. Compared to Shokohyar et al., the proposed model in PSS literature, warranty and out of warranty period increased from 17 and 21 months (Shokohyar, Mansour, and Karimi 2014) to 21 and 32 months, respectively, in this paper. This result is aligned with regulatory framework for product design and related lifecycle issues which is used in many countries like European ones in which there are many policies which either directly or indirectly affect the promotion of extending the life of products (Montalvo, Peck, and Rietveld 2016).

8. Sensitivity analysis

In order to validate the proposed model, a sensitivity analysis is performed to investigate the effects of changing some of the parameters in the optimal solution of the model.

and are shape parameters of the exponential distribution of product failures during T _W and T _OW , respectively. The sensitivity analysis of these two parameters is shown in Figures 1 and 2.

Figure 1. Sensitivity analysis of .

Figure 2. Sensitivity analysis of .

As shown in Figure 1(a), by increasing , optimal warranty period is decreased. As determines the failure behaviour of the product during warranty period, it’s increase will result in increased number of failures during warranty period and as shown in Figure 1(b) cost(p) will be increased and hence, it’s not economical for the producer to offer long-term warranty.

Also, as it is shown in Figure 1(a), by increasing , the optimal out-of-warranty period is increased. Due to the fact that is the shape parameter of exponential distribution during warranty period, it doesn’t affect the optimal out-of-warranty period directly. But since the usage period is made up of warranty and out-of-warranty period, by decreasing warranty period, there is a chance of increasing out-of-warranty period and it will result in more profit for the customer.

As shown in Figure 2(a), by increasing , optimal warranty period is decreased slightly. As is the shape parameter of exponential distribution during out-of-warranty period, it doesn’t affect the optimal warranty period directly. But due to the fact that in Stackelberg model, producer’s and customer’s payoffs were optimised simultaneously, varying one of the player’s payoff, could affect the other player’s optimal solution.

Also, as shown in Figure 2(a), by increasing , the optimal out-of-warranty period is decreased because cost(p) will be increased by increasing as shown in Figure 2(b). determines the failure behaviour of the product during the out-of-warranty period. Its increase result in increased number of failures during out-of-warranty period and therefore, as shown in Figure 2(b) it is not economical for the customer to keep the product for long-term during the out-of-warranty period.

β is the parameter representing the percentage of customers which prefer to obtain services from the producer during out-of-warranty period. Figure 3 shows the sensitivity analysis of β.

Figure 3. Sensitivity analysis of β.

As shown in Figure 3(a), increasing β results in decreasing the optimal warranty period and increasing the optimal out-of-warranty period. The reason is that by increasing the number of customers receiving services from the producer during out-of-warranty period, the producer’s profit during this period reduces such that he prefers offering a short-term warranty and hence benefiting from servicing product’s failures. If there are more competitions in the market for providing after sales services, the percentage of customers (β) will reduce and hence, producer will be motivated to offer longer warranty periods. As it is shown in Figure 3(b) cost(p) and cost(c) will be decreased by increasing β.

K ₁ is the producer’s profit margin of offering warranty and its sensitivity analysis is shown in Figure 4.

Figure 4. Sensitivity analysis of K ₁.

By increasing K ₁ (Figure 4(a)) warranty period is increased resulting in an increase in the producer’s profit which result in decreased cost(p) as shown in Figure 4(b).

As shown in Figure 4(a), by increasing K ₁, the optimal out-of-warranty period is decreased. Since the usage period is made up of warranty and out-of-warranty period, by decreasing warranty period, there is a chance of increasing out-of-warranty period and it will result in more profit for the customer.

K is the customer’s profit margin for keeping the product during the out-of-warranty period and its sensitivity analysis is shown in Figure 5.

Figure 5. Sensitivity analysis of K.

As shown in Figure 5(a), increasing K, doesn’t affect the optimal warranty period but it increases the optimal out-of-warranty period. Considering the fact that K is the customer’s profit margin of keeping the product during out-of-warranty period, it is clear by increasing the parameter K, it becomes more economical for the customer to keep the product longer during this period as shown in Figure 5(b). This is particularly true if the product costs more.

9. Conclusions and suggestion for future work

The environmental service design is an important issue which needs optimisation in many developing countries, such as Iran. If adequate services are provided for customers, it will delay products in reaching their EOL phase and consequently there will be less material and waste generation. Therefore, the relation between warranty and out-of-warranty period and EOL decisions with respect to environmental and economic impacts is getting more and more attention. This paper proposed a Stackelberg game theory model aiming to find optimised warranty and out-of-warranty period by minimising total cost of product during usage and EOL phase with regard to producer and customer’s point of view. Data from a major notebook service centre in Iran was used to validate the proposed model. Results of our analysis indicated that the optimal proposed warranty period offered by producer is equal to 21.423 months with the profit of 79.743 US dollars. The optimal out-of-warranty replacement period acceptable by the customer is equal to 32.596 months with the cost of 63.149 US dollars.

Offering warranty and after sales services by producer, is a way of convincing customer to keep the product in use for a longer period. Based on obtained result of this model, offering warranty by the producer, not only optimise producer’s profit and customer costs, but also encourages the customer to make the product usage period longer and hence serves as a way of achieving environmental objectives.

This model is applicable to all industries in which the product needs some after sales services and it is economically feasible for the producer to offer warranty beside its product. Industries like automobile manufacturing, computer manufacturing and smart phone manufacturing are the best candidates for using this model.

Some of the future research directions that can be derived from the work are presented here:

•	Due to some limitations on availability of data, parameters such as demand and production costs are excluded. Future works can focus on adding these parameters to the model.
•	Government plays a key role in sustainable production by implementing rules forcing producer to follow sustainable development objectives. Because of many constraints such as the lack of appropriate data, authors have overlooked the role of government. There is every possibility to develop the model in future research by considering the government as the third player. Environmentalist and NGOs’ role in achieving sustainable development aims, also can be investigated in future researches.
•	Landfill was the only EOL option used in this work. However, other options namely reuse, remanufacture and recycle could also be investigated.
•	Due to the importance of warranty in automotive market, future works can focus on this industry.

Notes on contributors

Mahsa Arabi received her MSc in Industrial Engineering from Amirkabir University of Technology (AUT) in Iran. Her research interests include sustainable development, game theory, organizational transformation and supply chain management.

Saeed Mansour received his BSc (Honors) and PhD in Production Engineering and Production Management, Nottingham University, England. He is an currently associate professor at Department of Industrial Engineering, Amirkabir University of Technology (AUT) and the Vice Principal (Academics) of the department. His research interests include: Sustainable Development, Green manufacturing, Waste Management, New Product Development, Additive Manufacturing, design and manufacture. The author is reviewer for many international journals and has published numerous papers in well-known international journals and proceedings of refereed international conferences. Some of his previous publications have appeared in: Journal of Cleaner Production, journal of Waste Management & Research, International Journal of Resources, Conservation and Recycling, Journal of Environment, Development and Sustainability, International Journal of Life Cycle Assessment, Journal of Intelligent Manufacturing, Journal of Advanced Manufacturing Technology, International Journal of Computer Integrated Manufacturing.

Sajjad Shokouhyar has been an assistant professor at the Department of Management and Accounting in Shahid Beheshti University (SBU) in Iran since January 2014. He received his PhD in Industrial Engineering from the Polytechnic of Tehran in Iran. He received his MSc and BSc in Industrial Engineering from the Amirkabir University of Technology (AUT) in Iran. His research interests include data mining, soft computing, supply chain management and industrial issues.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. End of life.