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

Figures & data

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

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

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

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

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

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

Figure 1. Sensitivity analysis of .

Figure 2. Sensitivity analysis of .

Figure 3. Sensitivity analysis of β.

Figure 4. Sensitivity analysis of K ₁.

Figure 5. Sensitivity analysis of K.

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