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

  • * Corresponding author: Yongtao Peng

    * Corresponding author: Yongtao Peng 

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)

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  • 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.

    Mathematics Subject Classification: Primary: 90B06.


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  • Figure 1.  The PSSC network construct

    Figure 2.  Influence of the raw material suppliers' flexibility input on equilibrium conditions

    Figure 3.  Influence of service providers' ability on equilibrium conditions

    Figure 4.  Influence of the transaction security level of manufacturing service integrators on equilibrium conditions

    Table 1.  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
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    Table 2.  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 $
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