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## A product service supply chain network equilibrium considering risk management in the context of COVID-19 pandemic

 1 School of Management, Jiangsu University, Zhenjiang 212013, China 2 Faculty of Management, Economy and Sciences, University Catholic of Lille, Lille 59800, France

* Corresponding author: Yongtao Peng

Received  December 2021 Revised  April 2022 Early access June 2022

Fund Project: This work was supported by the National Natural Science Foundation of China (71802099), Social Science Foundation of Jiangsu Province (21GLC005), Major Project of Philosophy and Social Science Research in Jiangsu Universities (2020SJZDA062)

This paper studies the equilibrium decision-making problem of product service supply chain (PSSC) network under the impact of COVID-19 related risks. The PSSC is composed of service-oriented transformation of manufacturing enterprises to sell product service systems (PSSs) to customers. So, under the impact of COVID-19, the network faces dual risks of products and services. This paper constructs the PSSC network of raw material suppliers, service providers, manufacturing service integrators and demand markets. Through variational inequalities, a network equilibrium model of PSSC considering risk management was established, and their decision-making problems were discussed. Three numerical examples were used to analyse the impact of risk management on the supply chain network at various levels. The results show that the risk management of upstream and downstream enterprises will have mutual influence, and the cost input of service risk management will benefit the entire PSSC network. Therefore, through the diversified development and improvement of services, the market demand for PSSs can be increased.

Citation: Yongtao Peng, Dan Xu, Eleonora Veglianti, Elisabetta Magnaghi. A product service supply chain network equilibrium considering risk management in the context of COVID-19 pandemic. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022094
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show all references

##### References:
 [1] M. B. Amor, M. Lindahl, P. Frankelius and H. B. Abdennebi, Revisiting industrial organization: Product service systems insight, J. Clean. Prod., 196 (2018), 1459-1477.  doi: 10.1016/j.jclepro.2018.05.145. [2] T. Avinadav, T. Chernonog and Y. Perlman, The effect of risk sensitivity on a supply chain of mobile applications under a consignment contract with revenue sharing and quality investment, Int. J. Prod. Econ., 168 (2015), 31-40.  doi: 10.1016/j.ijpe.2015.05.036. [3] Q. Bai and F. Meng, Impact of risk aversion on two-echelon supply chain systems with carbon emission reduction constraints, J. Ind. Manag. Optim., 16 (2020), 1943-1965.  doi: 10.3934/jimo.2019037. [4] J. E. Baz and S. Ruel, Can supply chain risk management practices mitigate the disruption impacts on supply chains' resilience and robustness? Evidence from an empirical survey in a COVID-19 outbreak era, Int. J. Prod. Econ., 233 (2021), 107972.  doi: 10.1016/j.ijpe.2020.107972. [5] A. S. Butt, Strategies to mitigate the impact of COVID-19 on supply chain disruptions: A multiple case analysis of buyers and distributors, Int. J. Logist. Manag., 2021. doi: 10.1108/IJLM-11-2020-0455. [6] P. Chowdhury, S. Paul, S. Kaisar and Md. A. Moktadir, COVID-19 pandemic related supply chain studies: A systematic review, Transport. Res. E.-Log., 148 (2021), 102271.  doi: 10.1016/j.tre.2021.102271. [7] B. Dan, H. Gao, Y. Zhang, R. Liu and S. Ma, Integrated order selection and scheduling decision making in product service supply chain with hard time windows constraints, J. Ind. Manag. Optim., 14 (2018), 165-182.  doi: 10.3934/jimo.2017041. [8] P. Daniele, Evolutionary variational inequalities and applica-tions to complex dynamic multi-level models, Transport. Res. E.-Log., 46 (2010), 855-880.  doi: 10.1016/j.tre.2010.03.005. [9] Y. Daultani, S. Kumar, O. S. Vaidya and M. K. Tiwari, A supply chain network equilibrium model for operational and opportunism risk mitigation, Int. J. Prod. Res., 53 (2015), 5685-5715.  doi: 10.1080/00207543.2015.1056325. [10] V. Dohale, P. Ambilkar, A. Gunasekaran and P. Verma, Supply chain risk mitigation strategies during COVID-19: Exploratory cases of "make-to-order" handloom saree apparel industries, Int. J. Phys. Distrib, 52 (2022), 109-129.  doi: 10.1108/IJPDLM-12-2020-0450. [11] Z. Feng, Z. Wang and Y. Chen, The equilibrium of closed-loop supply chain supernetwork with time-depen-dent parameters, Transport. Res. E.-Log., 64 (2014), 1-11.  doi: 10.1016/j.tre.2014.01.009. [12] H. Gebauer, M. Paiola and N. Saccani, Characterizing service networks for moving from products to solutions, Ind. Mark. Manag., 42 (2013), 31-46.  doi: 10.1016/j.indmarman.2012.11.002. [13] R. W. Grubbström, The dependence of the incremental risk rate of interest on absolute risk aversion-Applying the Laplace transform to risk preference evaluation, Int. J. Prod. Econ., 212 (2019), 51-59.  doi: 10.1016/j.ijpe.2019.01.031. [14] Z. Guan, X. Zhang, M. Zhou and Y. Dan, Demand information sharing in competing supply chains with raw material spplier-provided service, Int. J. Prod. Econ., 220 (2020), 107450.  doi: 10.1016/j.ijpe.2019.07.023. [15] R. B. Handfield, G. Graham and L. Burns, Corona virus, tariffs, trade wars and supply chain evolutionary design, Int. J. Oper. Prod. Manag., 40 (2020), 1649-1660.  doi: 10.1108/IJOPM-03-2020-0171. [16] Y. He, Supply risk sharing in a closed-loop supply chain, Int. J. Prod. Econ., 183 (2017), 39-52.  doi: 10.1016/j.ijpe.2016.10.012. [17] J. E. Hobbs, Food supply chains during the COVID-19 pandemic, Can. J. Agr. Econ., 68 (2020), 171-176.  doi: 10.1111/cjag.12237. [18] D. Ivanov, Lean resilience: AURA (Active Usage of Resilience Assets) framework for post-COVID-19 supply chain management, Int. J. Logist. Manag., 2021. doi: 10.1108/IJLM-11-2020-0448. [19] D. Ivanov and A. Das, Coronavirus (COVID-19/SARS-CoV-2) and supply chain resilience: A research note, Int. J. Integrated Supply Manag., 13 (2020), 90-102.  doi: 10.1504/IJISM.2020.107780. [20] D. Ivanov and A. Dolgui, Viability of intertwined supply networks: Extending the supply chain resilience angles towards survivability, A position paper motivated by COVID-19 outbreak, Int. J. Prod. Res., 58 (2020), 2904-2915.  doi: 10.1080/00207543.2020.1750727. [21] M. S. S. Jajja, K. A. Chatha and S. Farooq, Impact of supply chain risk on agility performance: Mediating role of supply chain integration, Int. J. Prod. Econ., 205 (2018), 118-138.  doi: 10.1016/j.ijpe.2018.08.032. [22] M. Johnson and C. Mena, Supply chain management for servitised products: A multi-industry case study, Int. J. Prod. Econ., 114 (2008), 27-39.  doi: 10.1016/j.ijpe.2007.09.011. [23] D. J. Ketchen and C. W. Craighead, Research at the intersection of entrepreneurship, supply chain management, and strategic management: Opportunities highlighted by COVID-19, J. Manage., 46 (2020), 1330-1341.  doi: 10.1177/0149206320945028. [24] Z. Liu and A. Nagurney, Supply chain networks with global outsourcing and quick-response production under demand and cost uncertainty, Ann. Oper. Res., 208 (2013), 251-289.  doi: 10.1007/s10479-011-1006-0. [25] Z. Liu and J. Wang, Supply chain network equilibrium with strategic supplier investment: A real options perspective, Int. J. Prod. Econ., 208 (2019), 184-198.  doi: 10.1016/j.ijpe.2018.11.010. [26] B. Malmir and C. W. Zobel, An applied approach to multi-criteria humanitarian supply chain planning for pandemic response, Journal of Humanitarian Logistics and Supply Chain Management, 11 (2021), 320-346.  doi: 10.1108/JHLSCM-08-2020-0064. [27] A. Mandel and V. Veetil, The economic cost of COVID lockdowns: An out-of-equilibrium analysis, Economics of Disasters and Climate Change, 4 (2020), 431-451.  doi: 10.1007/s41885-020-00066-z. [28] S. Modgil, S. Gupta, R. Stekelorum and I. Laguir, AI technologies and their impact on supply chain resilience during COVID-19, International Journal of Physical Distribution & Logistics Management, 52 (2022), 130-149.  doi: 10.1108/IJPDLM-12-2020-0434. [29] V. Muerza, E. Larrodé and J. M. Moreno-Jiménez, Identification and selection of ICTs for freight transport in product service supply chain diversification, Ind. Manag. Data Syst., 117 (2017), 1469-1484.  doi: 10.1108/IMDS-09-2016-0375. [30] M. Munir, M. S. S. Jajja and K. A. Chatha, Supply chain risk management and operational performance: The enabling role of supply chain integration, Int. J. Prod. Econ., 227 (2020), 107667.  doi: 10.1016/j.ijpe.2020.107667. [31] A. Nagurney, Supply chain game theory network modeling under labor constraints: Applications to the Covid-19 pandemic, Eur. J. Oper. Res., 293 (2021), 880-891.  doi: 10.1016/j.ejor.2020.12.054. [32] A. Nagurney, Optimization of supply chain networks with inclusion of labor: Applications to COVID-19 pandemic disruptions, Int. J. Prod. Econ., 235 (2021), 108080. [33] A. Nagurney, P. Daniele and S. Shukla, A supply chain network game theory model of cybersecurity investments with nonlinear budget constraints, Ann. Oper. Res., 248 (2017), 405-427.  doi: 10.1007/s10479-016-2209-1. [34] A. Nagurney, J. Dong and D. Zhang, A supply chain network equilibrium model, Transport. Res. E.-Log., 38 (2002), 281-303.  doi: 10.1016/S1366-5545(01)00020-5. [35] K. Nikolopoulos, S. Punia, A. Schäfers, C. Tsinopoulos and C. Vasilakis, Forecasting and planning during a pandemic: COVID-19 growth rates, supply chain disruptions, and governmental decisions, Eur. J. Oper. Res., 290 (2020), 99-115.  doi: 10.1016/j.ejor.2020.08.001. [36] Y. Peng, D. Xu, Y. Li and K. Wang, A product service supply chain network equilibrium model considering capacity constraints, Math. Probl. Eng., (2020), Art. ID 1295072, 15 pp. doi: 10.1155/2020/1295072. [37] M. M. Queiroz, D. Ivanov, A. Dolgui and S. F. Wamba, Impacts of epidemic outbreaks on supply chains: Mapping a research agenda amid the COVID-19 pandemic through a structured literature review, Ann. Oper. Res., (2020), 1-38.  doi: 10.1007/s10479-020-03685-7. [38] V. H. Remko, Research opportunities for a more resilient post-COVID-19 supply chain - closing the gap between research findings and industry practice, Int. J. Oper. Prod. Manag., 40 (2020), 341-355.  doi: 10.1108/IJOPM-03-2020-0165. [39] S. Rezapour, J. K. Allen and F. Mistree, Reliable flow in forward and after-sales supply chains considering propagated uncertainty, Transport. Res. E.-Log., 93 (2016), 409-436.  doi: 10.1016/j.tre.2016.04.016. [40] M. Rizou, I. M. Galanakis, T. M. S. Aldawood and C. M. Galanakis, Safety of foods, food supply chain and environment within the COVID-19 pandemic, Trends Food Sci. Technol., 102 (2020), 293-299.  doi: 10.1016/j.tifs.2020.06.008. [41] J. Sarkis, Supply chain sustainability: learning from the COVID-19 pandemic, Int. J. Oper. Prod. Manag., 41 (2021), 63-73.  doi: 10.1108/IJOPM-08-2020-0568. [42] A. Sharma, A. Adhikary and S. B. Borah, Covid-19's impact on supply chain decisions: Strategic insights from NASDAQ 100 firms using Twitter data, J. Bus. Res., 117 (2020), 443-449.  doi: 10.1016/j.jbusres.2020.05.035. [43] T. Shu, F. Yang, S. Chen, S. Wang, K. K. Lai and L. Gan, Contract coordination in dual sourcing supply chain under supply disruption risk, Math. Probl. Eng., (2015), Art. ID 473212, 10 pp. doi: 10.1155/2015/473212. [44] R. Sreedevi and H. Saranga, Uncertainty and supply chain risk: The moderating role of supply chain flexibility in risk mitigation, Int. J. Prod. Econ., 193 (2017), 332-342.  doi: 10.1016/j.ijpe.2017.07.024. [45] D. D. P. Thompson and R. Anderson, The COVID-19 response: considerations for future humanitarian supply chain and logistics management research, Journal of Humanitarian Logistics and Supply Chain Management, 11 (2021), 157-175.  doi: 10.1108/JHLSCM-01-2021-0006. [46] W. Wang, P. Zhang, J. Ding, J. Li, H. Sun and L. He, Closed-loop supply chain network equilibrium model with retailer-collection under legislation, J. Ind. Manag. Optim., 15 (2019), 199-219.  doi: 10.3934/jimo.2018039. [47] Z. Xu, A. Elomri, Q. Zhang, C. Liu and L. Shi, Status review and research strategies on product-service supply chain, Proc. Inst. Mech. Eng. B J. Eng. Manuf., 234 (2020), 1075-1086.  doi: 10.1177/0954405420905199. [48] G.-F. Yang, Z.-P. Wang and X.-Q. Li, The optimization of the closed-loop supply network, Transport. Res. E.-Log., 45 (2009), 16-28.  doi: 10.1016/j.tre.2008.02.007. [49] J. Yang, H. Xie, G. Yu and M. 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The PSSC network construct
Influence of the raw material suppliers' flexibility input on equilibrium conditions
Influence of service providers' ability on equilibrium conditions
Influence of the transaction security level of manufacturing service integrators on equilibrium conditions
Key notations
 Notation Definition $L$ Number of raw material suppliers $N$ Number of service providers $M$ Number of manufacturing service integrators $K$ Number of demand markets $q_{lm}$ Quantity of products sold by raw material suppliers to manufacturing service integrators $q_l$ The production output of the raw material supplier $s_n$ The service activities that service provider $n$ $s_{nm}$ The transaction volume of $n$ and $m$ $q_{mk}^s$ The product service systems $q_m^s$ All the product service systems produced by the manufacturing service integrator $p_k$ The price of the product service system purchased by the demand market $\tau$ The flexibility level of raw material suppliers $\theta$ The possibility of risk occurrence $\omega$ The unit profit loss of risk occurrence during trading ${\gamma _l}$ The Lagrangian multipliers of constraints (8) ${\delta _l}$ The Lagrangian multipliers of constraints (9) ${\gamma ^1}$ The vectors that correspond to ${\gamma _l}$ ${\delta ^1}$ The vectors that correspond to ${\delta _l}$ $\varepsilon$ The Lagrangian multipliers of constraints (13) and the vectors that correspond to $\varepsilon$ $\lambda$ The Lagrangian multipliers of constraints (16) and the vectors that correspond to $\lambda$ $\mu$ The Lagrangian multipliers of constraints (17) and the vectors that correspond to $\mu$ $\vartheta$ The service provider's ability to cope with risks $\alpha$ The proportions of products $q$ in a complete product service system $\beta$ The proportions of service $s$ in a complete product service system $q$ Products in a complete product service system $s$ Service in a complete product service system $r$ The safe transaction level between the manufacturing service integrator and demand market $d$ The customer's demand for a product $a$ The potential demand of consumer market $b$ The coefficient of price $p$ The product price
 Notation Definition $L$ Number of raw material suppliers $N$ Number of service providers $M$ Number of manufacturing service integrators $K$ Number of demand markets $q_{lm}$ Quantity of products sold by raw material suppliers to manufacturing service integrators $q_l$ The production output of the raw material supplier $s_n$ The service activities that service provider $n$ $s_{nm}$ The transaction volume of $n$ and $m$ $q_{mk}^s$ The product service systems $q_m^s$ All the product service systems produced by the manufacturing service integrator $p_k$ The price of the product service system purchased by the demand market $\tau$ The flexibility level of raw material suppliers $\theta$ The possibility of risk occurrence $\omega$ The unit profit loss of risk occurrence during trading ${\gamma _l}$ The Lagrangian multipliers of constraints (8) ${\delta _l}$ The Lagrangian multipliers of constraints (9) ${\gamma ^1}$ The vectors that correspond to ${\gamma _l}$ ${\delta ^1}$ The vectors that correspond to ${\delta _l}$ $\varepsilon$ The Lagrangian multipliers of constraints (13) and the vectors that correspond to $\varepsilon$ $\lambda$ The Lagrangian multipliers of constraints (16) and the vectors that correspond to $\lambda$ $\mu$ The Lagrangian multipliers of constraints (17) and the vectors that correspond to $\mu$ $\vartheta$ The service provider's ability to cope with risks $\alpha$ The proportions of products $q$ in a complete product service system $\beta$ The proportions of service $s$ in a complete product service system $q$ Products in a complete product service system $s$ Service in a complete product service system $r$ The safe transaction level between the manufacturing service integrator and demand market $d$ The customer's demand for a product $a$ The potential demand of consumer market $b$ The coefficient of price $p$ The product price
Functions and parameters used in the calculation
 Function name Raw material supplier 1 Raw material supplier 2 Production cost ${f_1}\left( {{Q_1}} \right) = 2.5q_1^2 + {q_1}{q_2} + 2{q_1}$ ${f_2}\left( {{Q_2}} \right) = 3q_2^2 + 1.5{q_1}{q_2} + 2{q_2}$ Transaction cost $c_m^l\left( {{q_{lm}}} \right) = 0.2q_{lm}^2 + {q_{lm}}$ Function name Service provider 1 Service provider 2 Service activity cost ${f_1}\left( {{S_1}} \right) = 2.5s_1^2/2$ ${f_2}\left( {{S_2}} \right) = 3s_2^2/2$ Transaction cost $c_m^n\left( {{s_{nm}}} \right) = 0.2s_{nm}^2 + {s_{nm}}$ Function name Integrator 1 Integrator 2 Integration cost ${f_1}\left( {Q_1^S} \right) = 5/2q_1^{s2} + q_1^sq_2^s + 6.6q_1^s$ ${f_2}\left( {Q_2^S} \right) = 6/2q_2^{s2} + 1.5q_1^sq_2^s + 6.6q_2^s$ Transaction cost 1 $c_l^m\left( {{q_{lm}}} \right) = 0.2q_{lm}^2 + {q_{lm}}$ Transaction cost 2 $c_n^m\left( {{s_{nm}}} \right) = 0.2s_{nm}^2 + {s_{nm}}$ Transaction cost 3 $c_k^m\left( {q_{mk}^s} \right) = 0.2q_{mk}^{s2} + q_{mk}^s$ Function name Demand market 1 Demand market 2 Market demand function ${d_1}\left( {{p_1}, {p_2}} \right) = 1000 - 2{p_1} + 1.5{p_2}$ ${d_2}\left( {{p_2}, {p_1}} \right) = 1000 - 2{p_2} + 1.5{p_1}$ Transaction cost $c_m^k\left( {q_{mk}^s} \right) = 0.2q_{mk}^{s2} + q_{mk}^s$ Flexibility improving costs $f\left( \tau \right) = 3{\tau ^2} + 0.5\tau + 4.25$ Unit loss cost $\omega = 6$ Probability of risk $\theta = 0.3$
 Function name Raw material supplier 1 Raw material supplier 2 Production cost ${f_1}\left( {{Q_1}} \right) = 2.5q_1^2 + {q_1}{q_2} + 2{q_1}$ ${f_2}\left( {{Q_2}} \right) = 3q_2^2 + 1.5{q_1}{q_2} + 2{q_2}$ Transaction cost $c_m^l\left( {{q_{lm}}} \right) = 0.2q_{lm}^2 + {q_{lm}}$ Function name Service provider 1 Service provider 2 Service activity cost ${f_1}\left( {{S_1}} \right) = 2.5s_1^2/2$ ${f_2}\left( {{S_2}} \right) = 3s_2^2/2$ Transaction cost $c_m^n\left( {{s_{nm}}} \right) = 0.2s_{nm}^2 + {s_{nm}}$ Function name Integrator 1 Integrator 2 Integration cost ${f_1}\left( {Q_1^S} \right) = 5/2q_1^{s2} + q_1^sq_2^s + 6.6q_1^s$ ${f_2}\left( {Q_2^S} \right) = 6/2q_2^{s2} + 1.5q_1^sq_2^s + 6.6q_2^s$ Transaction cost 1 $c_l^m\left( {{q_{lm}}} \right) = 0.2q_{lm}^2 + {q_{lm}}$ Transaction cost 2 $c_n^m\left( {{s_{nm}}} \right) = 0.2s_{nm}^2 + {s_{nm}}$ Transaction cost 3 $c_k^m\left( {q_{mk}^s} \right) = 0.2q_{mk}^{s2} + q_{mk}^s$ Function name Demand market 1 Demand market 2 Market demand function ${d_1}\left( {{p_1}, {p_2}} \right) = 1000 - 2{p_1} + 1.5{p_2}$ ${d_2}\left( {{p_2}, {p_1}} \right) = 1000 - 2{p_2} + 1.5{p_1}$ Transaction cost $c_m^k\left( {q_{mk}^s} \right) = 0.2q_{mk}^{s2} + q_{mk}^s$ Flexibility improving costs $f\left( \tau \right) = 3{\tau ^2} + 0.5\tau + 4.25$ Unit loss cost $\omega = 6$ Probability of risk $\theta = 0.3$
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