[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

Optimal pricing and sourcing strategies in the presence of supply uncertainty and competition

  • Published:
Journal of Intelligent Manufacturing Aims and scope Submit manuscript

Abstract

The implementation of cost leadership strategy can enable enterprises to obtain a lasting competitive advantage. In this paper, we construct a supply chain with two competing suppliers and two competing manufacturers. The suppliers act as the Stackelberg leaders, selling components to the follower manufacturers. The manufacturers use different sourcing strategies, one of which only uses the reliable supplier, while another adopts contingent dual sourcing. We derive the analytical form of the equilibrium solution of order quantities of manufacturers and wholesale prices of suppliers in different scenarios and compare the decision differences when reliable supplier stays in different game positions. We further investigate the impact of different cooperation contract between the manufacturer who adopts dual sourcing and unreliable supplier on the procurement and pricing strategies by numerical experiments. The results illustrate that reliable supplier acts as the Stackelberg leader are more beneficial to suppliers. Manufacturer with dual sourcing can work with the unreliable supplier by revenue sharing contract to achieve a win–win situation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Data availability

All data underlying the findings described in our manuscript are fully available without any restriction and all relevant data are within the paper.

References

  • Abginehchi, S., & Farahani, R. Z. (2010). Modeling and analysis for determining optimal suppliers under stochastic lead times. Applied Mathematical Modelling, 34(5), 1311–1328.

    Article  Google Scholar 

  • Burke, G. J., Carrillo, J. E., & Vakharia, A. J. (2007). Single versus multiple supplier sourcing strategies. European Journal of Operational Research, 182(1), 95–112.

    Article  Google Scholar 

  • Cachon, G. P., & Lariviere, M. A. (2005). Supply chain coordination with revenue-sharing contracts: Strengths and limitations. Management Science, 1(51), 30–44.

    Article  Google Scholar 

  • Chakraborty, T., Chauhan, S. S., & Vidyarthi, N. (2015). Coordination and competition in a common retailer channel: Wholesale price versus revenue-sharing mechanisms. International Journal of Production Economics, 166, 103–118.

    Article  Google Scholar 

  • Chen, J., & Guo, Z. (2015). Strategic sourcing in the presence of uncertain supply and retail competition. Production & Operations Management, 23(10), 1748–1760.

    Article  Google Scholar 

  • Chen, K., & Xiao, T. (2015). Production planning and backup sourcing strategy of a buyer-dominant supply chain with random yield and demand. International Journal of Systems Science, 46(15), 2799–2817.

    Article  Google Scholar 

  • Di, X., & Kewen, P. (2012). Quality control and coordination mechanism in supply chain based on revenue sharing contract. Chinese Journal of Management Science, 40(4), 67–73.

    Google Scholar 

  • Fang, J., Zhao, L., Fransoo, J. C., & Woensel, T. V. (2013). Sourcing strategies in supply risk management: An approximate dynamic programming approach. Computers & Operations Research, 40(5), 1371–1382.

    Article  Google Scholar 

  • Guo, L., Yuchen, K., & Mengqi, L. (2017). Dual-source procurement strategies for manufacturers with supply disruption risks. Journal of Intelligent & Fuzzy Systems, 33(5), 2637–2645.

    Article  Google Scholar 

  • He, B., Huang, H., & Yuan, K. (2016). Managing supply disruption through procurement strategy and price competition. International Journal of Production Research, 54(7), 1980–1999.

    Article  Google Scholar 

  • Huang, H., & Ke, H. (2017). Pricing decision problem for substitutable products based on uncertainty theory. Journal of Intelligent Manufacturing, 28(3), 503–514.

    Article  Google Scholar 

  • Jing, H., Zeng, A. Z., & Li, S. (2016). Backup sourcing with capacity reservation under uncertain disruption risk and minimum order quantity. Computers & Industrial Engineering, 103, 216–226.

    Google Scholar 

  • Jung, K. S., & Hwang, H. (2011). Competition and cooperation in a remanufacturing system with take-back requirement. Journal of Intelligent Manufacturing, 22(3), 427–433.

    Article  Google Scholar 

  • Koulamas, C. (2006). A newsvendor problem with revenue sharing and channel coordination. Decision Sciences, 37(1), 91–100.

    Article  Google Scholar 

  • Ledari, A. M., Pasandideh, S. H. R., & Koupaei, M. N. (2018). A new newsvendor policy model for dual-sourcing supply chains by considering disruption risk and special order. Journal of Intelligent Manufacturing, 29(1), 237–244.

    Article  Google Scholar 

  • Li, X. (2017). Optimal procurement strategies from suppliers with random yield and all-or-nothing risks. Annals of Operations Research, 257(1–2), 167–181.

    Article  Google Scholar 

  • Lu, M., Huang, S., & Shen, Z.-J. M. (2011). Product substitution and dual sourcing under random supply failures. Transportation Research Part B: Methodological, 45(8), 1251–1265.

    Article  Google Scholar 

  • Mansini, R., Savelsbergh, M. W. P., & Tocchella, B. (2012). The supplier selection problem with quantity discounts and truckload shipping. Omega, 40(4), 445–455.

    Article  Google Scholar 

  • Ni, J., Flynn, B. B., & Jacobs, F. R. (2016). The effect of a toy industry product recall announcement on shareholder wealth. International Journal of Production Research, 54(18), 1–12.

    Article  Google Scholar 

  • Ouardighi, F. E. (2014). Supply quality management with optimal wholesale price and revenue sharing contracts: A two-stage game approach. International Journal of Production Economics, 156(5), 260–268.

    Article  Google Scholar 

  • Panab, K., Leung, S. C. H., & Di, X. (2010). Revenue-sharing versus wholesale price mechanisms under different channel power structures. European Journal of Operational Research, 203(2), 532–538.

    Article  Google Scholar 

  • Paul, S. K., Sarker, R., & Essam, D. (2018). A reactive mitigation approach for managing supply disruption in a three-tier supply chain. Journal of Intelligent Manufacturing, 29(7), 1581–1597.

    Article  Google Scholar 

  • Qin, F., Rao, U. S., Gurnani, H., & Bollapragada, R. (2014). Role of random capacity risk and the retailer in decentralized supply chains with competing suppliers. Decision Sciences, 45(2), 255–279.

    Article  Google Scholar 

  • Sawik, T. (2014). Joint supplier selection and scheduling of customer orders under disruption risks: Single vs. dual sourcing. Omega, 43, 83–95.

    Article  Google Scholar 

  • Tang, S. Y., & Kouvelis, P. (2011). Supplier diversification strategies in the presence of yield uncertainty and buyer competition. M&SOM-Manufacturing & Service Operations Management, 13(4), 439–451.

    Article  Google Scholar 

  • Wang, Y., Jiang, L., & Shen, Z.-J. (2004). Channel performance under consignment contract with revenue sharing. Management Science, 50(1), 34–47.

    Article  Google Scholar 

  • Wei, J., & Zhao, J. (2016). Pricing decisions for substitutable products with horizontal and vertical competition in fuzzy environments. Annals of Operations Research, 242(2), 505–528.

    Article  Google Scholar 

  • Wu, H., Han, X., Yang, Q., & Pu, X. (2018). Production and coordination decisions in a closed-loop supply chain with remanufacturing cost disruptions when retailers compete. Journal of Intelligent Manufacturing, 29(1), 227–235.

    Article  Google Scholar 

  • Xu, J., & Liu, N. (2017). Research on closed loop supply chain with reference price effect. Journal of Intelligent Manufacturing, 28(1), 51–64.

    Article  Google Scholar 

  • Yang, H., & Chen, W. (2017). Retailer-driven carbon emission abatement with consumer environmental awareness and carbon tax: revenue-sharing versus cost-sharing. Omega, 78, 179–191.

    Article  Google Scholar 

  • Zhao, J., Tanga, W., & Wei, J. (2012). Pricing decisions for substitutable products with a common retailer in fuzzy environments. European Journal of Operational Research, 216(2), 409–419.

    Article  Google Scholar 

Download references

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yixin Zhang.

Ethics declarations

Conflicts of interest

No conflicts of interest exist in the submission of this manuscript, which has been approved by all authors for publication.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Proof of Lemma 1

According to Eq. (6), it can be proved that \( \varPi_{S1}^{WP} \) is concave functions for \( w_{2} \). Let \( w_{2WP} \) denote the solution of \( \frac{{\partial \varPi_{S2}^{WP} }}{{\partial w_{2} }} = 0 \), we have \( w_{2WP} = \frac{{\mu w_{1} + c_{2} }}{2\mu } \). When \( q_{{_{3} WP}}^{*} > 0 \), then \( w_{2WP} < w_{1} \).

Proof of Lemma 2

If \( w_{1} > \frac{{c_{2} }}{\mu } \), based on Eq. (7) \( \frac{{\partial \varPi_{S1}^{D} }}{{\partial w_{1}^{2} }} = \frac{{ - 4\upsigma^{ 2} - \left( {\uptheta + 2} \right)\upmu^{ 2} }}{{\left( {\uptheta + 2} \right)\upsigma^{ 2} }} < 0 \). Let \( \frac{{\partial \varPi_{S1}^{WP} }}{{\partial w_{1} }} = 0 \), then \( w_{{_{1} A}}^{C} = {{\left( {\left( {c_{1} + w_{2} } \right)\left( {\theta + 2} \right)\mu^{2} + 4\sigma^{2} \left( {a + c_{1} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {c_{1} + w_{2} } \right)\left( {\theta + 2} \right)\mu^{2} + 4\sigma^{2} \left( {a + c_{1} } \right)} \right)} {\left( {\left( {2\theta + 4} \right)\mu^{2} + 8\sigma^{2} } \right)}}} \right. \kern-0pt} {\left( {\left( {2\theta + 4} \right)\mu^{2} + 8\sigma^{2} } \right)}} \). If \( w_{{_{1} A}}^{C} > {{c_{2} } \mathord{\left/ {\vphantom {{c_{2} } \mu }} \right. \kern-0pt} \mu } \), it is easy to prove that \( \varPi_{S1}^{WP} \left( {w_{1A}^{C} } \right) > \varPi_{S1}^{WP} \left( {{{c_{2} } \mathord{\left/ {\vphantom {{c_{2} } \mu }} \right. \kern-0pt} \mu }} \right) \) and the optimal wholesale price of the reliable supplier is \( w_{1WP}^{C} = w_{1A}^{C} \); if \( w_{1A}^{C} \le {{c_{2} } \mathord{\left/ {\vphantom {{c_{2} } \mu }} \right. \kern-0pt} \mu } \), assume \( {{c_{2} } \mathord{\left/ {\vphantom {{c_{2} } \mu }} \right. \kern-0pt} \mu } \le {{\left( {a + c_{1} } \right)} \mathord{\left/ {\vphantom {{\left( {a + c_{1} } \right)} 2}} \right. \kern-0pt} 2} \), it is more advantageous for the reliable supplier to resist the unreliable supplier by adopting low price policy and forming price barrier. So, the optimal wholesale price is \( w_{1WP}^{C} = {{c_{2} } \mathord{\left/ {\vphantom {{c_{2} } \mu }} \right. \kern-0pt} \mu } \).

Proof of Corollary 1

If \( \phi_{1} \le 0 \), the reliable supplier will still monopolize the market. In this case, the profit of manufacturers will be \( C\varPi_{M1}^{WP*} {\text{ = C}}\varPi_{M2}^{WP*} { = }\frac{{\left( {{\text{a}}\upmu - {\text{c}}_{ 2} } \right)^{ 2} }}{{\upmu^{ 2} \left( {\uptheta + 2} \right)^{ 2} }} > \frac{{\left( {{\text{a}} - {\text{c}}_{ 1} } \right)^{ 2} }}{{ 4\left( {\uptheta + 2} \right)^{ 2} }} \). If \( \phi_{1} > 0 \), the reliable supplier will coexist with the unreliable supplier. According to Eq. (8), \( w_{{_{1} WP}}^{C*} < \frac{{a + c_{1} }}{2} < \frac{{3a_{1} - c_{1} }}{2} \) can be easily proved \( C\varPi_{M1}^{WP*} - \varPi_{M1}^{S*} = \frac{{\left( {a - 2w_{1WP}^{C*} + c_{1} } \right)\left( {3a - 2w_{1WP}^{C*} - c_{1} } \right)}}{{4\left( {\theta + 2} \right)^{2} }} \); and \( C\varPi_{M2}^{WP*} - C\varPi_{M1}^{WP*} = \frac{{\left( {w_{1WP}^{C*} - w_{2WP}^{C*} } \right)^{2} \mu^{2} }}{{4\sigma^{2} }} \). Corollary 1 can be proved.

Proof of Corollary 2

\( \phi_{1} - \phi_{2} = \left( {c_{2} - c_{1} \mu } \right)\left( {\theta + 2} \right)\mu^{2} \). (1) If \( c_{2} > c_{1} \mu \),\( \phi_{1} > \phi_{2} \). (1) if \( \phi_{2} \le 0 \) and \( \phi_{1} \le 0 \), the reliable supplier will monopolize the market, and the wholesale price \( w_{1WP}^{C*} = w_{1WP}^{S*} = \frac{{c_{2} }}{\mu } \); (2) if \( \phi_{2} \le 0 \) and \( \phi_{1} > 0 \), \( w_{1WP}^{C*} > w_{1WP}^{S*} = \frac{{c_{2} }}{\mu } \). The unreliable supplier will coexist with the reliable supplier only when they are in an equal competitive position; (3) if \( \phi_{2} > 0 \) and \( \phi_{1} > 0 \), the unreliable supplier can always get an order. And

$$ w_{1WP}^{C*} - w_{1WP}^{S*} = - \frac{{\left( {\left( {c_{2} - c_{1} \mu } \right)\left( {\theta + 2} \right)\mu + 8\sigma^{2} \left( {a - c_{1} } \right)} \right)\left( {\theta + 2} \right)\mu^{2} }}{{2\left( {3\left( {\theta + 2} \right)\mu^{2} + 16\sigma^{2} } \right)\left( {\left( {\theta + 2} \right)\mu^{2} + 8\sigma^{2} } \right)}} < 0 $$
(A.1)

(2) If \( c_{2} \le c_{1} \mu \), \( \phi_{2} \ge \phi_{1} > 0 \). Similarly, the unreliable supplier has access to the supply market. And \( w_{1WP}^{C*} < w_{1WP}^{S*} \), if \( \left( {c_{1} \mu - c_{2} } \right)\mu < \frac{{8\sigma^{2} \left( {a - c_{1} } \right)}}{{\left( {\theta + 2} \right)}} \); otherwise \( w_{1WP}^{C*} \ge w_{1WP}^{S*} \).

Proof of Corollary 3

If \( \phi_{i} \le 0,i = 1,2 \), \( w_{1WP}^{*} { = }w_{2WP}^{*} { = }\frac{{c_{2} }}{\mu } \). From proposition1, the production quantity, sales price, and profit are equal. Otherwise, \( w_{1WP}^{*} \ge w_{2WP}^{*} { > }\frac{{c_{2} }}{\mu } \). \( E\left( {q_{M2}^{WP} } \right) = q_{2WP}^{*} + q_{3WP}^{*} \mu = \frac{{a - w_{1WP}^{*} }}{\theta + 2} \), \( \varPi_{M2}^{WP} { = }\frac{{\left( {w_{1WP}^{*} - w_{2WP}^{*} } \right)^{2} \mu^{2} }}{{4\sigma^{2} }} + \frac{{\left( {a - w_{1WP}^{*} } \right)^{2} }}{{\left( {\theta + 2} \right)^{2} }} \).

Proof of Proposition 5

According to Eq. (11), we can get the Hessian matrix of the profit function \( \Pi _{\text{M2}}^{RS} \) as \( H = \left[ {\begin{array}{*{20}c} { - 2} & {\left( { -\uplambda - 1} \right)\mu } \\ {\left( { -\uplambda - 1} \right)\mu } & { - 2\uplambda\left( {\upmu^{ 2} +\upsigma^{ 2} } \right)} \\ \end{array} } \right] \). When \( 4\uplambda \upsigma ^{ 2} > \left( {\uplambda - 1} \right)^{ 2}\upmu^{ 2} \), the matrix will be negative definite. Taking the derivative of \( \Pi _{\text{M2}}^{RS} \) with respect to the variable \( \left( {q_{2} ,q_{3} } \right) \), then we have:

$$ \left\{ \begin{aligned} & q_{2} = \frac{{\mu^{2} \left( {a - \theta q_{1} } \right)\lambda^{2} + \left( {\left( {\theta q_{1} - a + 2w_{1} - w_{2} } \right)\mu^{2} - 2\sigma^{2} \left( {a - \theta q_{1} - w_{1} } \right)} \right)\lambda - \mu^{2} w_{2} }}{{\left( {\lambda - 1} \right)^{2} \mu^{2} - 4\lambda \sigma^{2} }} \hfill \\ & q_{3} = \frac{{2\left( {w_{1} - w_{2} } \right)\mu - \left( {1 - \lambda } \right)\left( {a - \theta q_{1} + w_{1} } \right)\mu }}{{4\lambda \sigma^{2} - \left( {\lambda - 1} \right)^{2} \mu^{2} }} \hfill \\ \end{aligned} \right. $$
(A.2)

Taking the derivative of \( \Pi _{\text{M1}}^{RS} \) with respect to the variable \( q_{1} \), then we have:

$$ q_{1} = - \frac{1}{2}\mu \theta q_{3} - \frac{1}{2}\theta q_{2} + \frac{1}{2}a - \frac{1}{2}w_{1} $$
(A.3)

Simultaneous Eqs. (A.2) and (A.3), the quantity that the manufacturer ordered from each supplier when the reliable supplier coexists with the reliable supplier can get the order, as shown in Eq. (12).

Proof of Lemma 3

When \( q_{3} > 0 \), substitute \( q_{3RS}^{*} \) into equal (13), then we can get

$$ \frac{{\partial \varPi_{S2}^{RS} }}{{\partial w_{2}^{2} }} = \frac{{ - 2\left( {\theta^{4} - 4} \right)^{2} \left( {\lambda + 1} \right)\sigma^{2} \mu^{2} }}{{\left( {\left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right)\mu^{2} + 2\sigma^{2} \left( {4 - \theta^{2} } \right)\lambda } \right)^{2} }} < 0 $$
(A.4)

Let \( \frac{{\partial_{S2}^{RS} }}{{\partial w_{2}^{2} }} = 0 \) and \( {\text{w}}_{{ 2 {\text{B}}}} \) denote the optimal price, then

$$ {\text{w}}_{{ 2 {\text{B}}}} = \frac{{\left( {{\text{w}}_{ 1} \mu - {\text{c}}_{ 2} } \right)\left( {\uplambda - 1} \right)^{ 2} \left( {\uptheta^{ 2} - 2} \right)\upmu^{ 2} }}{{ 2\sigma^{ 2}\upmu\left( {\theta^{ 2} - 4} \right)\left( {\uplambda + 1} \right)}} + \frac{{\left( {\uptheta{\text{w}}_{ 1} + {\text{a}} + {\text{w}}_{ 1} } \right)\uplambda^{ 2} - {\text{a}} + {\text{w}}_{ 1} }}{{\left( {\theta { + 2}} \right)\left( {\uplambda + 1} \right)}}{ + }\frac{{\uplambda{\text{c}}_{ 2} }}{{\left( {\uplambda + 1} \right)\upmu}} $$
(A.5)

When \( w_{2B} < \bar{w}_{2} \), \( q_{3RS}^{*} > 0 \) where \( \bar{w}_{2} { = }\frac{{\left( {\theta w_{1} + a + w_{1} } \right)\lambda - a + w_{1} }}{\theta + 2} \).

(1) If \( w_{1} \le \frac{a - a\lambda }{\theta \lambda + \lambda + 1},\bar{w}_{2} \le 0 \), The unreliable supplier needs to subsidize manufacturer 2 in order to receive an order. (2) If \( w_{1} > \frac{a - a\lambda }{\theta \lambda + \lambda + 1},\;\bar{w}_{2} { > 0} \) and \( w_{2B} - \bar{w}_{2} = \frac{{\left( {\left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right)\mu^{2} + 2\sigma^{2} \lambda \left( {4 - \theta^{2} } \right)} \right)\left( {\mu w_{1} - c_{2} } \right)}}{{2\left( {\theta^{2} - 4} \right)\mu \left( {\lambda + 1} \right)\sigma^{2} }} \). Note \( 4\uplambda \upsigma ^{ 2} > \left( {\uplambda - 1} \right)^{ 2}\upmu^{ 2} \), we have \( \left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right)\mu^{2} + 2\lambda \sigma^{2} \left( {4 - \theta^{2} } \right) > 4\lambda \sigma^{2} - \left( {\lambda - 1} \right)^{2} \mu^{2} > 0 \). Let \( \psi \left( \theta \right){ = }\left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right)\mu^{2} + 2\lambda \sigma^{2} \left( {4 - \theta^{2} } \right) - \left( {4\lambda \sigma^{2} - \left( {\lambda - 1} \right)^{2} \mu^{2} } \right) \). Obviously, \( \psi \left( \theta \right) \) is a monotone function when \( \theta \in \left[ {0,1} \right] \), it is easy to verify that \( \psi \left( \theta \right) > 0 \). Therefore, if \( \mu w_{1} > c_{2} \), \( {\text{w}}_{{ 2 {\text{B}}}} < \overline{{w_{2} }} \) and \( q_{3} > 0 \); if \( \mu w_{1} \le c_{2} \), \( {\text{w}}_{{ 2 {\text{B}}}} \ge \overline{{w_{2} }} \) and \( q_{3} { = }0 \).

Proof of Lemma 4

If \( w_{1} > \underline{{w_{1} }} \), \( \frac{{\partial \varPi_{S1}^{RS} }}{{\partial w_{1}^{2} }} = \frac{{\left( {\theta - 2} \right)\left( {\theta \mu^{2} \lambda^{2} + \left( {4\mu^{2} + 4\sigma^{2} - 2\sigma^{2} \theta } \right)\lambda + \mu^{2} \theta } \right)}}{{\left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right) \cdot \mu^{2} + 2\sigma^{2} \left( {4 - \theta^{2} } \right)\lambda }} - 1 \). Note \( \left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right)\mu^{2} > 2\lambda \sigma^{2} \left( {4 - \theta^{2} } \right) \), then \( \frac{{\partial \varPi_{S1}^{RS} }}{{\partial w_{1}^{2} }} < 0 \) 。Let \( \frac{{\partial \varPi_{S1}^{RS} }}{{\partial w_{1}^{2} }} = 0 \), we have

$$ w_{1B}^{C} = \frac{{\left( {\theta - 2} \right)\left( {\left( {\theta + \lambda + 1} \right)w_{2} - a\left( {\lambda^{2} - \lambda } \right) + \left( {\theta + 2} \right)c_{1} \lambda } \right)\mu^{2} + \left( {a + c_{1} } \right)\left( {\lambda - 1} \right)^{2} \left( {1 - \theta } \right)\mu^{2} + 4\sigma^{2} \lambda \left( {\theta - 2} \right)\left( {a + c_{1} } \right)}}{{2\left( {1 - \theta } \right)\left( {\lambda - 1} \right)^{2} \mu^{2} + 2\lambda \left( {\theta - 2} \right)\left( {\left( {\theta + 2} \right)\mu^{2} + 4\sigma^{2} } \right)}} $$

(1) If \( \underline{{w_{1} }} < \frac{{a + c_{1} }}{2} \) and \( w_{1B}^{C} > \underline{{w_{1} }} \), supplier1 has two different pricing options: set price barriers for the unreliable supplier (\( w_{1RS}^{C} { = }\underline{{w_{1} }} \)) or coexist with the unreliable supplier (\( w_{1RS}^{C} { = }w_{1B}^{C} \)). It is easy to prove that \( C\varPi_{S1}^{RS} \left( {w_{1} } \right) \) is a continuous function and increases with \( w_{1} \) when \( w_{1} \in \left( {\underline{{w_{1} }} ,w_{1B}^{C} } \right] \). It means \( C\varPi_{S1}^{RS} \left( {w_{1B}^{S} } \right) > C\varPi_{S1}^{RS} \left( \underline{{w_{1} }} \right) \) and the optimal strategy for the reliable supplier coexists with the unreliable supplier.

(2) If \( \underline{{w_{1} }} < \frac{{a + c_{1} }}{2} \) and \( w_{1B}^{C} \le \underline{{w_{1} }} \), the optimal wholesale price is \( w_{1RS}^{C} { = }\underline{{w_{1} }} \) and the unreliable supplier will not enter the market.

(3) If \( \underline{{w_{1} }} > \frac{{a + c_{1} }}{2} \), the appearance of the unreliable supplier will not affect the original monopoly position of The reliable supplier. Thus, \( w_{1RS}^{C} { = }\frac{{a{ + }c_{1} }}{2} \).

Proof of Proposition 7

If \( w_{2B} \ge 0 \),\( w_{1} \ge \widehat{{w_{1} }} \), where \( \widehat{{w_{1} }} = \frac{{c_{2} \mu^{2} \left( {\lambda - 1} \right)^{2} \left( {2 - \theta^{2} } \right) + 2\sigma^{2} \left( {\theta - 2} \right)\left( {a\mu \left( {\lambda^{2} - 1} \right) + \lambda c_{2} \left( {\theta + 2} \right)} \right)}}{{\mu^{3} \left( {\lambda - 1} \right)^{2} \left( {2 - \theta^{2} } \right) + \left( {4 - 2\theta } \right)\mu \sigma^{2} \left( {\left( {\theta + 1} \right)\lambda^{2} + 1} \right)}} \).

Then,\( \widehat{{w_{1} }} - \frac{{c_{2} }}{\mu } = \frac{{2\left( {\lambda + 1} \right)\left( {\theta - 2} \right)\left( {\left( {a\mu + \theta c_{2} + c_{2} } \right)\lambda - a\mu + c_{2} } \right)\sigma^{2} }}{{\mu \left( {\mu^{2} \left( {\lambda - 1} \right)^{2} \left( {2 - \theta^{2} } \right) + 2\left( {2 - \theta } \right)\sigma^{2} \left( {\left( {\theta + 1} \right)\lambda^{2} + 1} \right)} \right)}} \)

$$ \widehat{{w_{1} }} - \frac{a - a\lambda }{\theta \lambda + \lambda + 1} = \frac{{\left( {\left( {\uplambda - 1} \right)^{ 2} \left( {\uptheta^{ 2} - 2} \right)\upmu^{ 2} + 2\uplambda \upsigma ^{ 2} \left( { 4-\uptheta^{ 2} } \right)} \right)\left( {\left( {{\text{a}}\upmu + {\text{c}}_{ 2}\uptheta + {\text{c}}_{ 2} } \right)\uplambda - {\text{a}}\upmu + {\text{c}}_{ 2} } \right)}}{{\mu \left( { 1+ \left( {\uptheta + 1} \right)\lambda } \right) \cdot \left( {\left( {\uplambda - 1} \right)^{ 2} \left( {\uptheta^{ 2} - 2} \right)\mu^{ 2} + 2\left( { 1+ \left( {\uptheta + 1} \right)\uplambda^{ 2} } \right)\left( {\uptheta - 2} \right)\upsigma^{ 2} } \right)}}. $$

Note \( \frac{{c_{2} }}{\mu } < \frac{{a + c_{1} }}{2} \), if \( w_{1} > \frac{a - a\lambda }{\theta \lambda + \lambda + 1} \) and \( w_{1} > \frac{{c_{2} }}{\mu } \), substitute \( w_{2RS} = w_{2B} \), \( w_{2RS} = 0 \) into the profit of the reliable supplier, respectively. We have

$$ {\text{w}}_{1B}^{S} = \frac{{\left( {{\text{c}}_{ 1}\upmu + {\text{c}}_{ 2} } \right)\left( {\uptheta +\uplambda + 1} \right)\upmu + 4\upsigma^{ 2} \left( {\uplambda + 1} \right)\left( {{\text{a}} + {\text{c}}_{ 1} } \right)}}{{ 2\left( {\uptheta +\uplambda + 1} \right)\upmu^{ 2} + 8\upsigma^{ 2} \left( {\uplambda + 1} \right)}},\quad w_{2RS} = w_{2B} $$
(A.6)
$$ w_{1C}^{S} = \frac{{\left( {a + c_{1} } \right)\left( {\theta - 1} \right)\left( {4 \sigma^{2} \lambda - \left( {\lambda - 1} \right)^{2} \mu^{2} } \right) - 4 \sigma^{2} \left( {a + c_{1} } \right)\lambda + \left( {\theta^{2} - 4} \right)c_{1} \lambda \mu^{2} + a\left( {\theta - 2} \right)\left( {\lambda - \lambda^{2} } \right)\mu^{2} }}{{2 \cdot \left( {\mu^{2} \left( {\theta - 1} \right)\left( {\lambda - 1} \right)^{2} + \left( {2 - \theta } \right)\lambda \left( {\left( {2 + \theta } \right)\mu^{2} + 4 \sigma^{2} } \right)} \right)}},w_{2RS} = 0 $$
(A.7)

(1) If \( \lambda \le \frac{{a\mu - c_{2} }}{{\left( {a\mu + \theta c_{2} + c_{2} } \right)}} \),\( \widehat{{w_{1} }} > \frac{a - a\lambda }{\theta \lambda + \lambda + 1} > \frac{{c_{2} }}{\mu } \). The pricing of the reliable supplier affects the price and order of the unreliable supplier. The reliable supplier has three pricing options \( {\text{w}}_{1RS}^{S} \).

  1. (1)

    if \( {\text{w}}_{1RS}^{S} > \widehat{{w_{1} }} \), the reliable supplier coexists with the unreliable supplier and \( w_{2RS} > 0 \). The optimal price \( {\text{w}}_{1RS}^{S} {\text{ = max}}\left\{ {{\text{w}}_{1B}^{S} ,\widehat{{w_{1} }}} \right\} \).

  2. (2)

    if \( \frac{a - a\lambda }{\theta \lambda + \lambda + 1}{\text{ < w}}_{1RS}^{S} < \widehat{{w_{1} }} \), the reliable supplier coexists with the unreliable supplier and \( w_{2RS} = 0 \). The optimal price of supplier1 is \( {\text{w}}_{1RS}^{S} {\text{ = max}}\left\{ {\frac{a - a\lambda }{\theta \lambda + \lambda + 1},{ \hbox{min} }\left\{ {{\text{w}}_{1C}^{S} ,\widehat{{w_{1} }}} \right\}} \right\} \)

  3. (3)

    if \( {\text{w}}_{1RS}^{S} < \frac{a - a\lambda }{\theta \lambda + \lambda + 1} \), the reliable supplier monopolizes the supply market and. \( w_{2RS} = 0 \)\( {\text{w}}_{1RS}^{S} {\text{ = min}}\left\{ {\frac{a - a\lambda }{\theta \lambda + \lambda + 1},\frac{{a + c_{1} }}{2}} \right\} \).

(2) If \( \lambda > \frac{{a\mu - c_{2} }}{{\left( {a\mu + \theta c_{2} + c_{2} } \right)}} \), \( \widehat{{w_{1} }} < \frac{a - a\lambda }{\theta \lambda + \lambda + 1} < \frac{{c_{2} }}{\mu } \). The reliable supplier has two pricing options.

  1. (1)

    If \( {\text{w}}_{1B}^{S} > \frac{{c_{2} }}{\mu } \), the reliable supplier coexists with the unreliable supplier and \( w_{2RS} > 0 \). \( {\text{w}}_{1RS}^{S} {\text{ = w}}_{1B}^{S} \);

  2. (2)

    If \( {\text{w}}_{1B}^{S} \le \frac{{c_{2} }}{\mu } \) the reliable supplier monopolizes the supply market, \( w_{1RS}^{S*} = \frac{{c_{2} }}{\mu } \).

Let \( w_{1D}^{S} = \mathop {\arg \hbox{max} }\limits_{{w_{1} \in Z}} S\varPi_{S1}^{RS} \left( {w_{1} } \right) \), \( Z = \Big\{ { \hbox{max} }\left\{ {{\text{w}}_{1B}^{S} ,\hat{w}_{1} } \right\}, {\hbox{max} }\left\{ {\frac{a - a\lambda }{\theta \lambda + \lambda + 1} , {\text{min}}\left\{ {{\text{w}}_{1C}^{S} ,\hat{w}_{1} } \right\}} \right\},{ \hbox{min} }\left\{ {\frac{a - a\lambda }{\theta \lambda + \lambda + 1},\frac{{a + c_{1} }}{2}} \right\} \Big\} \).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Wang, X. Optimal pricing and sourcing strategies in the presence of supply uncertainty and competition. J Intell Manuf 32, 61–76 (2021). https://doi.org/10.1007/s10845-020-01557-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10845-020-01557-2

Keywords

Navigation