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Quality investment strategies in a complementary supply chain with an unreliable supplier

  • *Corresponding author: Huaming Song

    *Corresponding author: Huaming Song 

The first author is supported by [Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX21_0359)]

Abstract Full Text(HTML) Figure(12) / Table(7) Related Papers Cited by
  • This study investigates investment strategy in a supply chain that comprises one manufacturer and two complementary suppliers - a reliable supplier and an unreliable supplier. The unreliable supplier's quality improvement capacity is uncertain. Where the manufacturer determines to invest in which suppliers' quality improvement activities and suppliers decide the quality improvement levels of their components, respectively. We demonstrate three potential strategies to highlight the manufacturer's and suppliers' optimal choices: investing in an unreliable supplier, investing in a reliable supplier, and investing in both. Investing in two suppliers results in higher quality improvement levels and profits for the manufacturer, and the optimal level of product quality improvement is not monotonically related to the efficiency rate. The unreliable supplier can benefit the most from the investment strategy, while the manufacturer profits the least. The uncertainty of an unreliable supplier is more likely to affect a reliable supplier than himself. There are two effects: mutual hold-up and spillover effects result in counter-intuitive findings. Lastly, we relax our assumptions to examine their impacts on the manufacturer's strategy choice.

    Mathematics Subject Classification: Primary: 91A80, 91A35, 90B06, 90B50.

    Citation:

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  • Figure 1.  The four investment strategies.

    Figure 2.  The sequence of events.

    Figure 3.  Impact of $ \beta $ and $ \zeta $ on optimal investment level.

    Figure 4.  Impact of $ \beta $ and $ \zeta $ on optimal investment level.

    Figure 5.  Impact of $ \beta $ and $ \zeta $ on wholesale price and retail price.

    Figure 6.  Impact of $ \beta $ and $ \zeta $ on he optimal investment strategy of suppliers.

    Figure 7.  Impact of $ \beta $ and $ \zeta $ on the optimal manufacturer strategy.

    Figure 8.  Impact of $ k_i $ on optimal investment level.

    Figure 9.  Impact of $ k_i $ on the product quality's optimal strategy.

    Figure 10.  Impact of $ k_i $ on optimal wholesale price and retail price.

    Figure 11.  Impact of $ k_i $ on optimal investment strategy of suppliers.

    Figure 12.  Impact of $ k_i $ on the manufacturer's optimal strategy.

    Table 1.  Contributions of some authors related to our model.

    Authors Quality improvement Unreliable supplier Quality uncertainty Complementary supply chain Strategy
    Sim and Kim (2020) Outsourcing
    Chakraborty et al.(2019) Cost-sharing
    Golmohammadi et al.(2022) Ordering timing
    Fan et al. (2020) Cost-sharing
    Yoo et al. (2020) Incentive
    He and Zhao (2016) Vendor managed inventory
    Lee et al. (2013) Quality-compensation
    Dong et al. (2021) Procurement
    Liu et al. (2018) Sale effort commitmen
    Fu et al. (2021) Investment
    Pan et al. (2022) Backup sourcing
    Our work Investment
     | Show Table
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    Table 2.  Notations

    Indexes Description
    Indices
    j the manufacturer's investing strategy: no investment, invest in an unreliable supplier, invest in a reliable supplier, invest in both, where $ j \in \left\{ B,U,R,T \right\} $
    i ith component, where $ i \in \left\{ 1,2 \right\} $, when $ i=1 $ means the unreliable supplier's component, when $ i=2 $ means the reliable supplier's component
    Parameters
    $ \zeta $ consumers' quality sensitivity to the product
    $ \beta $ consumers' retail price sensitivity to the product
    $ k_i $ quality improvement cost parameter for $ i $th component
    $ e $ a random variable with mean $ \mu $ and variance $ \sigma^2 $
    Decision variables
    $ q_{i}^{j} $ quality improvement level of $ i $th component in $ j $ scenario
    $ w_{i}^{j} $ wholesale price of $ i $th component in $ j $ scenario
    $ p^{j} $ retail price of the product in $ j $ scenario
    $ \phi^{j} $ the manufacturer's investment share in suppliers in $ j $ scenario
    Outcomes
    $ Q^{j} $ quality improvement level of the product in $ j $ scenario
    $ D^{j} $ consumer demand in $ j $ scenario
    $ \pi_{M}^{j} $ the manufacturer's profit in $ j $ scenario
    $ \pi_{Si}^{j} $ the supplier $ i $'s profit in $ j $ scenario
     | Show Table
    DownLoad: CSV

    Table B.1.  The Equilibrium solutions of product quality, demand and profits in B strategy.

    B Solutions
    $ \phi $ /
    $ \mathbb{E}Q $ $ \frac{\zeta\left( \left( \mu^{2} + \sigma^{2} \right)k_{1} + \mu k_{2} \right)}{\left( {\mu^{2} + \sigma^{2}} \right)k_{1}\left( {9k_{2} - \zeta^{2}} \right) - \zeta^{2}\mu^{2}k_{2}} $
    $ \mathbb{E}D $ $ \frac{3\left( \mu^{2} + \sigma^{2} \right)k_{1}k_{2}}{2\left( \left( {\mu^{2} + \sigma^{2}} \right)k_{1}\left( {9k_{2} - \zeta^{2}} \right) - \zeta^{2}\mu^{2}k_{2} \right)} $
    $ {\mathbb{E}\pi}_{M} $ $ \frac{9\left( \mu^{2} + \sigma^{2} \right)^{2}k_{1}^{2}k_{2}^{2}}{4\left( \left( \mu^{2} + \sigma^{2} \right)k_{1}\left( \zeta^{2} - 9k_{2} \right) + \zeta^{2}\mu^{2}k_{2} \right)^{2}} $
    $ {\mathbb{E}\pi}_{S1} $ $ \frac{\left( \mu^{2} + \sigma^{2} \right)k_{1}\left( 9\left( \mu^{2} + \sigma^{2} \right)k_{1} - \zeta^{2}\mu^{2} \right)k_{2}^{2}}{2\left( \left( \mu^{2} + \sigma^{2} \right)k_{1}\left( \zeta^{2} - 9k_{2} \right) + \zeta^{2}\mu^{2}k_{2} \right)^{2}} $
    $ {\mathbb{E}\pi}_{S2} $ $ \frac{\left( \mu^{2} + \sigma^{2} \right)^{2}k_{1}^{2}\left( \zeta^{2} - 9k_{2} \right)k_{2}}{{2\left( \left( {\mu^{2} + \sigma^{2}} \right)k_{1}\left( {9k_{2} - \zeta^{2}} \right) - \zeta^{2}\mu^{2}k_{2} \right)}^{2}} $
     | Show Table
    DownLoad: CSV

    Table B.2.  The Equilibrium solutions of product quality, demand and profits in U strategy.

    U Solutions
    $ \phi $ $ \frac{\zeta^{2}\left( \left( k_{1} + k_{2} \right)\mu^{2} + k_{1}\sigma^{2} \right)}{k_{1}\left( 18k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right)} $
    $ \mathbb{E}Q $ $ \frac{\zeta\left( k_{2}\zeta^{2}\mu(1 + \mu) - 18k_{1}k_{2}\left( \mu^{2} + 1 \right) + 2k_{1}\zeta^{2}\left( \mu^{2} + 1 \right) - 18{k_{2}}^{2}\mu \right)}{k_{2}\zeta^{2}\left( 27k_{2} - 2\zeta^{2} \right)\mu^{2} - 2k_{1}\left( - 9k_{2} + \zeta^{2} \right)^{2}\left( \mu^{2} + 1 \right)} $
    $ \mathbb{E}D $ $ \frac{3k_{2}\left( 2k_{1}\left( 9k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) - k_{2}\zeta^{2}\mu^{2} \right)}{2k2\zeta^{2}\left( 2\zeta^{2} - 27k_{2} \right)\mu^{2} + 4k_{1}\left( \zeta^{2} - 9k_{2} \right)^{2}\left( \mu^{2} + \sigma^{2} \right)} $
    $ {\mathbb{E}\pi}_{M} $ $ \frac{{k_{2}}^{2}\left( \zeta^{2}\mu^{2} + 18k_{1}\left( \mu^{2} + \sigma^{2} \right) \right)}{4k_{2}\zeta^{2}\left( 2\zeta^{2} - 27k_{2} \right)\mu^{2} + 8k_{1}\left( \zeta^{2} - 9k_{2} \right)^{2}\left( \mu^{2} + \sigma^{2} \right)} $
    $ {\mathbb{E}\pi}_{S1} $ $ \frac{{k_{2}}^{2}\left( 2k_{1}\left( 9k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) - k_{2}\zeta^{2}\mu^{2} \right)\left( \zeta^{2}\left( \zeta^{2} - 27k_{2} \right)\mu^{2} + 18k_{1}\left( 9k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) \right)}{2\left( k_{2}\zeta^{2}\left( 2\zeta^{2} - 27k_{2} \right)\mu^{2} + 2k_{1}\left( \zeta^{2} - 9k_{2} \right)^{2}\left( \mu^{2} + \sigma^{2} \right) \right)^{2}} $
    $ {\mathbb{E}\pi}_{S2} $ $ \frac{k_{2}\left( 9k_{2} - \zeta^{2} \right)\left( k2\zeta^{2}\mu^{2} - 2k_{1}\left( 9k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) \right)^{2}}{2\left( k_{2}\zeta^{2}\left( 2\zeta^{2} - 27k_{2} \right)\mu^{2} + 2k_{1}\left( \zeta^{2} - 9k_{2} \right)^{2}\left( \mu^{2} + \sigma^{2} \right) \right)^{2}} $
     | Show Table
    DownLoad: CSV

    Table B.3.  The Equilibrium solutions of product quality, demand and profits in R strategy.

    R Solutions
    $ \phi $ $ \frac{\zeta^{2}\left( \left( k_{1} + k_{2} \right)\mu^{2} + k_{1}\sigma^{2} \right)}{k_{2}\zeta^{2}\mu^{2} - 18k_{1}k_{2}\left( \mu^{2} + \sigma^{2} \right)} $
    $ \mathbb{E}Q $ $ \frac{\zeta\left( 2k_{2}\zeta^{2}\mu^{3} + k_{1}\mu\left( - 18k_2 + \zeta^{2}(1 + \mu) \right)\left( \mu^{2} + 1 \right) - 18{k_{1}}^{2}\left( \mu^{2} + 1 \right)^{2} \right)}{2k_{1}\zeta^{2}\left( {18k_{2} - \zeta^{2}} \right)\mu^{2}\left( {\mu^{2} + 1} \right) - 27{k_{1}}^{2}\left( 6k_{2} - \zeta^{2} \right)\left( \mu^{2} + 1 \right)^{2} - 2k_{2}\zeta^{4}\mu^{4}} $
    $ \mathbb{E}D $ $ \frac{3k_{1}\left( \mu^{2} + \sigma^{2} \right)\left( k_{1}\left( 18k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) - 2k_{2}\zeta^{2}\mu^{2} \right)}{4k_{2}\zeta^{4}\mu^{4} + 4k_{1}\zeta^{2}\left( \zeta^{2} - 18k_{2} \right)\mu^{2}\left( \mu^{2} + \sigma^{2} \right) + 54{k_{1}}^{2}\left( 6k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right)^{2}} $
    $ {\mathbb{E}\pi}_{M} $ $ \frac{k_{1}\left( \mu^{2} + \sigma^{2} \right)\left( \zeta^{2}\mu^{2} - 9k_{1}\left( \mu^{2} + \sigma^{2} \right) \right)\left( 2k_{2}\zeta^{2}\mu^{2} - k_{1}\left( 18k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) \right)^{2}}{2\left( 2k_{2}\zeta^{4}\mu^{4} + 2k_{1}\zeta^{2}\left( \zeta^{2} - 18k_{2} \right)\mu^{2}\left( \mu^{2} + \sigma^{2} \right) + 27{k_{1}}^{2}\left( 6k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right)^{2} \right)^{2}} $
    $ {\mathbb{E}\pi}_{S1} $ $ \frac{k_{1}\left( \mu^{2} + 1 \right)\left( 9k_{1}\left( \mu^{2} + 1 \right) - \zeta^{2}\mu^{2} \right)\left( k_{1}\left( 18k_{2} - \zeta^{2} \right)\left( \mu^{2} + 1 \right) - 2k_{2}\zeta^{2}\mu^{2} \right)^{2}}{2\left( 2k_{2}\zeta^{4}\mu^{4} + 2k_{1}\zeta^{2}\left( \zeta^{2} - 18k_{2} \right)\mu^{2}\left( \mu^{2} + 1 \right) + 27{k_{1}}^{2}\left( 6k_{2} - \zeta^{2} \right)\left( \mu^{2} + 1 \right)^{2} \right)^{2}} $
    $ {\mathbb{E}\pi}_{S2} $ $ \frac{{k_{1}}^{2}\left( {\mu^{2} + 1} \right)^{2}\left( \zeta^{2}\left( 18k_{2} - \zeta^{2} \right)\mu^{2} + 27k_{1}\left( 6k_{2} - \zeta^{2} \right)\left( \mu^{2} + 1 \right) \right)\left( 2k_{2}\zeta^{2}\mu^{2} - k_{1}\left( 18k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) \right)}{2\left( 2k_{2}\zeta^{4}\mu^{4} - 2k_{1}\zeta^{2}\left( 18k_{2} + \zeta^{2} \right)\mu^{2}\left( \mu^{2} + \sigma^{2} \right) + 27{k_{1}}^{2}\left( 6k_2 - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right)^{2} \right)^{2}} $
     | Show Table
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    Table B.4.  The Equilibrium solutions of product quality, demand and profits in Tstrategy.

    T Solutions
    $ \phi $ $ \frac{1}{18}\zeta^{2}\left( \frac{1}{k_{2}} + \frac{\mu^{2}}{k_{1}\left( \mu^{2} + \sigma^{2} \right)} \right) $
    $ \mathbb{E}Q $ $ \frac{2\zeta\left( k_{2}\mu + k_{1}\left( \mu^{2} + \sigma^{2} \right) \right)}{3k_{1}\left( 6k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) - 3k_{2}\zeta^{2}\mu^{2}} $
    $ \mathbb{E}D $ $ \frac{k_{2}\zeta^{2}\mu^{2} - k_{1}\left( 18k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right)}{18\left( k_{2}\zeta^{2}\mu^{2} - k_{1}\left( 6k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) \right)} $
    $ {\mathbb{E}\pi}_{M} $ $ \frac{k_{2}\zeta^{2}\mu^{2} + k_{1}\left( 18k_{2} + \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right)}{108\left( k_{1}\left( 6k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) - k_{2}\zeta^{2}\mu^{2} \right)} $
    $ {\mathbb{E}\pi}_{S1} $ $ \frac{\left( k_{1}\left( 18k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) - 3k_{2}\zeta^{2}\mu^{2} \right)\left( k_{1}\left( 18k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) - k_{2}\zeta^{2}\mu^{2} \right)}{162\left( k_{2}\zeta^{2}\mu^{2} - k_{1}\left( 6k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) \right)^{2}} $
    $ {\mathbb{E}\pi}_{S2} $ $ \frac{\left( k_{2}\zeta^{2}\mu^{2} - 3k_{1}\left( 6k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) \right)\left( k_{2}\zeta^{2}\mu^{2} - k_{1}\left( 18k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) \right)}{162\left( k_{2}\zeta^{2}\mu^{2} - k_{1}\left( 6k_{2} - \zeta^{2} \right)\left( \mu^{2} + \sigma^{2} \right) \right)^{2}} $
     | Show Table
    DownLoad: CSV

    Table B.5.  Threshold values in Proposition 4.6-5.2.

    $ \sigma^{A} $ $ \frac{1}{2}\sqrt{\frac{64 - 112\kappa + 49\kappa^{2}}{\kappa}} $
    $ \sigma^{B} $ $ \frac{\sqrt{\kappa}(119 + 103\kappa)}{144 + \kappa(245 + 103\kappa)} $
    $ \kappa^{A} $ $ the \ root \ of \ Q^{U*} = Q^{R*} $
    $ \kappa^{B} $ $ the \ root \ of \ E\pi_{S2}^{U*} = E\pi_{S2}^{R*} $
    $ \kappa^{C} $ 0.31
    $ \kappa^{D} $ 3.77
    $ \kappa^{E} $ 0.46
    $ \kappa^{F} $ 3.31
    $ \kappa^{G} $ 1.03
    $ \kappa^{H} $ 1.08
    $ \kappa^{I} $ 3.00
    $ \kappa^{J} $ 3.34
    $ \kappa^{K} $ 0.79
     | Show Table
    DownLoad: CSV
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