Design and pricing of maintenance service contract based on Nash non-cooperative game approach

doi:10.23919/JSEE.2024.000010

Abstract

Abstract:

Nowadays manufacturers are facing fierce challenge. Apart from the products, providing customers with multiple maintenance options in the service contract becomes more popular, since it can help to improve customer satisfaction, and ultimately promote sales and maximize profit for the manufacturer. By considering the combinations of corrective maintenance and preventive maintenance, totally three types of maintenance service contracts are designed. Moreover, attractive incentive and penalty mechanisms are adopted in the contracts. On this basis, Nash non-cooperative game is applied to analyze the revenue for both the manufacturer and customers, and so as to optimize the pricing mechanism of maintenance service contract and achieve a win-win situation. Numerical experiments are conducted. The results show that by taking into account the incentive and penalty mechanisms, the revenue can be improved for both the customers and manufacturer. Moreover, with the increase of repair rate and improvement factor in the preventive maintenance, the revenue will increase gradually for both the parties.

Key words: maintenance service contract, Nash game, incentive and penalty mechanism, corrective maintenance, preventive maintenance

Chun SU, Kui HUANG. Design and pricing of maintenance service contract based on Nash non-cooperative game approach[J]. Journal of Systems Engineering and Electronics, 2024, 35(1): 118-129.

Figures/Tables 16

Table 1

Fig 1

Table 2

Table 3

Fig 2

Table 4

Fig 3

Table 5

Table 6

Fig 4

Fig 5

Fig 6

Table 7

Fig 7

Table 8

Table 9

References 35

1	PASCUAL R, GODOY D, FIGUEROA H Optimizing maintenance service contracts under imperfect maintenance and a finite time horizon. Applied Stochastic Models in Business and Industry, 2013, 29 (5): 564- 577. doi: 10.1002/asmb.1943
2	SU C, WANG X L A two-stage preventive maintenance optimization model incorporating two-dimensional extended warranty. Reliability Engineering & System Safety, 2016, 155, 169- 178.
3	CHENG G Q, ZHOU B H, LI L Joint optimization of lot sizing and preventive maintenance threshold based on non nonlinear degradation. Computers & Industrial Engineering, 2017, 110 (8): 538- 549.
4	MA W N, ZHAO F, LI X, et al Joint optimization of inspection-based and age-based preventive maintenance and spare ordering policies for single-unit systems. Journal of Systems Engineering and Electronics, 2022, 33 (5): 1268- 1280. doi: 10.23919/JSEE.2022.000120
5	MURTHY D N P, YEUNG V Modelling and analysis of maintenance service contracts. Mathematical and Computer Modelling, 1995, 22 (10): 219- 225.
6	MURTHY D N P, ASGHARIZADEH E Optimal decision making in a maintenance service operation. European Journal of Operational Research, 1999, 116 (2): 259- 273. doi: 10.1016/S0377-2217(98)90202-8
7	ASGHARIZADEH E, MURTHY D N P Service contracts: a stochastic model. Mathematical and Computer Modelling, 2000, 31 (10/12): 11- 20. doi: 10.1016/S0895-7177(00)00068-6
8	KIM S H, COHEN M A, NETESSINE S Performance contracting in after-sales service supply chains. Management Science, 2007, 53 (12): 1843- 1858. doi: 10.1287/mnsc.1070.0741
9	JACKSON C, PASCUAL R Optimal maintenance service contract negotiation with aging equipment. European Journal of Operational Research, 2008, 189 (2): 387- 398. doi: 10.1016/j.ejor.2007.05.039
10	ISKANDAR B P, HUSNIAH H, PASARIBU U S Maintenance service contracts for equipment sold with two dimensional warranties. Quality Technology & Quantitative Management, 2014, 11 (3): 321- 333.
11	JIN T D, TIAN Z G, XIE M A game-theoretical approach for optimizing maintenance, spares and service capacity in performance contracting. International Journal of Production Economics, 2015, 161, 31- 43. doi: 10.1016/j.ijpe.2014.11.010
12	CHAGAS M M, SANTANA J M, DROGUETT E L, et al Analysis of extended warranties for medical equipment: a Stackelberg game model using priority queues. Reliability Engineering & System Safety, 2017, 168, 338- 354.
13	FENG Q, BI X, ZHAO X J, et al Heuristic hybrid game approach for fleet condition-based maintenance planning. Reliability Engineering & System Safety, 2017, 157, 166- 176.
14	JING H Y, TANG L C A risk-based approach to managing performance-based maintenance contracts. Quality and Reliability Engineering International, 2017, 33 (4): 853- 865. doi: 10.1002/qre.2062
15	PATRA P, KUMAR U D, NOWICKI D R, et al Effective management of performance-based contracts for sustainment dominant systems. International Journal of Production Economics, 2019, 208, 369- 382. doi: 10.1016/j.ijpe.2018.11.025
16	WANG W B A model for maintenance service contract design, negotiation and optimization. European Journal of Operational Research, 2010, 201 (1): 239- 246. doi: 10.1016/j.ejor.2009.02.018
17	ISKANDAR B P, HUSNIAH H Optimal preventive maintenance for a two dimensional lease contract. Computers & Industrial Engineering, 2017, 113, 693- 703.
18	SU C, CHENG L F. Two-dimensional preventive maintenance optimum for equipment sold with availability-based warranty. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 2019, 233(4): 648–657.
19	VIJAYAN V, CHATURVEDI S K. Multi-component maintenance grouping optimization based on stochastic dependency. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 2021, 235(2): 293–305.
20	XIANG Y S, ZHU Z C, COIT D W, et al Condition-based maintenance under performance-based contracting. Computers & Industrial Engineering, 2017, 111, 391- 402.
21	LI Z P, ZI Y Y, CHEN J L, et al Performance-guided maintenance policy and optimization for transmission system of shipborne antenna with multiple components. Ocean Engineering, 2020, 199, 106903. doi: 10.1016/j.oceaneng.2019.106903
22	LI J L, CHEN Y L, ZHANG Y, et al Availability modelling for periodically inspected systems under mixed maintenance policies. Journal of Systems Engineering and Electronics, 2021, 32 (3): 722- 730. doi: 10.23919/JSEE.2021.000062
23	DARGHOUTH M N, AIT-KADI D, CHELBI A Joint optimization of design, warranty and price for products sold with maintenance service contracts. Reliability Engineering & System Safety, 2017, 165, 197- 208.
24	WANG J J, ZHAO X, GUO X X Optimizing wind turbine’s maintenance policies under performance-based contract. Renewable Energy, 2019, 135, 626- 634. doi: 10.1016/j.renene.2018.12.006
25	WANG Y K, LIU Y L, CHEN J Y, et al Reliability and condition-based maintenance modeling for systems operating under performance-based contracting. Computers & Industrial Engineering, 2020, 142, 106344.
26	MIRZAHOSSEINIAN H, PIPLANI R. Compensation and incentive modeling in performance- based contracts for after market service. Proc. of the 41st International Conference on Computers & Industrial Engineering, 2011: 739–744.
27	HUSNIAH H, PASARIBU U S, ISKANDAR B P. Two dimensional maintenance contract with coordination between customer and agent. Proc. of the IEEE International Conference on Industrial Engineering and Engineering Management, 2016: 516–520.
28	HONG S, WERNZ C, STILLINGER J D Optimizing maintenance service contracts through mechanism design theory. Applied Mathematical Modelling, 2016, 40 (21/22): 8849- 8861. doi: 10.1016/j.apm.2015.07.009
29	HUSNIAH H, ANDRIANA, ISKANDAR B P Game theoretic models in fleet performance-based maintenance contracts. International Journal of Business Globalisation, 2020, 26 (1/2): 41- 56.
30	LI D, LI L Negotiation for service contracting: incentives, bargaining power, and customer risk aversion. International Transactions in Operational Research, 2022, 29 (6): 3592- 3621. doi: 10.1111/itor.13134
31	ZHANG F, JARDINE A K S Optimal maintenance models with minimal repair, periodic overhaul and complete renewal. IIE Transactions, 1998, 30 (12): 1109- 1119.
32	MACMILLAN K, SONDOW J Proofs of power sum and binomial coefficient congruences via Pascal’s identity. American Mathematical Monnthly, 2011, 118 (6): 549- 551. doi: 10.4169/amer.math.monthly.118.06.549
33	ELSAYED A E. Reliability Engineering 3rd ed. Hoboken: John Wiley & Sons, Inc., 2021
34	LI J L, CHEN Y L, ZHANG Y, et al Availability modeling for periodically inspection system with different lifetime and repair-time distribution. Chinese Journal of Aeronautics, 2019, 32 (7): 1667- 1672. doi: 10.1016/j.cja.2019.03.025
35	HAMIDI M, LIAO H, SZIDAROVSZKY F Non-cooperative and cooperative game-theoretic models for usage-based lease contracts. European Journal of Operational Research, 2016, 255 (1): 163- 174. doi: 10.1016/j.ejor.2016.04.064

Notation	Description	Value
C_e/$	Purchase price of the product	150000
R/$	Daily revenue of the product	400
μ	Repair rate	0.4
C_ps/$	Charge of PM carried out by the customer	500
C_p/$	Charge of PM carried out by the agent	1000
r₁	Improvement factor of PM carried out by the customer	0.5
r₂	Improvement factor of PM carried out by the agent	0.8
C_cm/$	Cost of CM carried out by the agent	1100
C_pm/$	Cost of PM carried out by the agent	700
δ/($/d)	Reward coefficient	400
α/($/d)	Penalty coefficient	300
τ₀/d	Time limit for rewards	2
τ₁/d	Time limit for punishment	3.5
T_p/d	Time for PM	1

Contract options		A₀	A₁	A₂
Optimal pricing	C_c /$	3300	5926	−
Optimal pricing	P_M /$	−	−	326270
Optimal daily expected revenue f /$		110	132	145
Optimal PM cycle N		−	11	12

L/d	Option A₀		Option A₁			Option A₂
L/d	C_c^*/$	f^*/$	C_c^*/$	f^*/$	N^*	P_M^*/$	f^*/$	N^*
1000	5050	99	8263	107	5	125 630	114	6
1500	4050	111	7062	126	8	225 966	136	9
2000	3300	110	5926	132	11	326 267	145	12
2500	2770	104	5057	133	14	426 531	150	15
3000	2383	96	4386	131	16	526 917	153	19
3500	2091	87	3877	128	19	626 793	154	20
4000	1863	76	3462	124	20	726 469	153	20

μ	Option A₀		Option A₁			Option A₂
μ	C_c^*/$	f^*/$	C_c^*/$	f^*/$	N^*	P_M^*/$	f^*/$	N^*
0.30	3133	102	5792	127	12	327712	143	13
0.35	3229	106	5854	130	11	326949	144	13
0.40	3300	110	5926	132	11	326267	145	12
0.45	3356	113	5981	133	11	325878	146	12
0.50	3400	115	5986	134	10	325582	147	12

r₁	N^*	C_c^*/$	f^*/$
0.40	10	5065	127
0.45	10	5444	129
0.50	11	5926	132
0.55	11	6460	134
0.60	12	7162	136