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Supply chain coordination under retail competition using stock dependent price-setting newsvendor framework

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Abstract

Short life-cycle products which are characterized by uncertain demand, short selling season and long lead times have been posing many challenges to supply chain members. Demand of these products depends on several factors such as price, quality, service etc. Apart from these, many business practices have revealed that presence of a larger quantity of goods displayed at retail level also attract customers considerably. This paper captures the stock dependency phenomenon and investigates the role of quantity discounts and returns policies in the coordination of a supply chain. Here, the manufacturer in addition to returns policy provides quantity discounts to two competing retailers who face price-sensitive, stock dependent and uncertain demand. Using the newsvendor framework, a combined contract model is developed and sensitivity analysis is performed to analyze the impact of various parameters on supply chain coordination. The result shows that proposed contract mechanism fails to coordinate when the value of price sensitivity factor approaches the value of cross price sensitivity factor. Further, price-sensitivity and cross-price sensitivity have little effect on coordination benefit at lower values of stock dependency whereas there is a significant impact at higher values of stock dependency.

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Acknowledgments

Authors would like to thank the editor-in-chief and the two anonymous referees for their valuable suggestions and helpful comments.

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Authors and Affiliations

  1. Department of Industrial Engg. and Management, Indian Institute of Technology, Kharagpur, West Bengal, 721302, India

    G. Parthasarathi, S. P. Sarmah & M. Jenamani

Authors
  1. G. Parthasarathi
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  2. S. P. Sarmah
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  3. M. Jenamani
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Corresponding author

Correspondence to G. Parthasarathi.

Appendices

Appendix 1

We know that \( F\left( z \right) = {\frac{p + s - m - sc}{{\left( {1 - c} \right)\left( {p + s} \right)}}} \). Let \( {\frac{p + s - m - sc}{{\left( {1 - c} \right)\left( {p + s} \right)}}} = \rho \) so that \( z = F^{ - 1} \left( \rho \right) \)

$$ \begin{aligned} {\frac{\partial z}{\partial p}} = & {\frac{\partial z}{\partial \rho }}{\frac{\partial \rho }{\partial p}} = {\frac{{\partial F^{ - 1} \left( \rho \right)}}{\partial \rho }}{\frac{\partial \rho }{\partial p}} = {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{\partial \rho }{\partial p}} = {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{\partial }{\partial p}}\left( {{\frac{p + s - m - sc}{{\left( {1 - c} \right)\left( {p + s} \right)}}}} \right) \\ = & {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{{\left( {m + sc} \right)}}{{\left( {1 - c} \right)\left( {p + s} \right)^{2} }}} > 0 \\ \end{aligned} $$
$$ \begin{aligned} {\frac{\partial z}{\partial m}} = & {\frac{\partial z}{\partial \rho }}{\frac{\partial \rho }{\partial m}} = {\frac{{\partial F^{ - 1} \left( \rho \right)}}{\partial \rho }}{\frac{\partial \rho }{\partial m}} = {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{\partial \rho }{\partial m}} = {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{\partial }{\partial m}}\left( {{\frac{p + s - m - sc}{{\left( {1 - c} \right)\left( {p + s} \right)}}}} \right) \\ = & {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{ - 1}{{\left( {1 - c} \right)\left( {p + s} \right)}}} < 0 \\ \end{aligned} $$
$$ \begin{aligned} {\frac{\partial z}{\partial a}} = & {\frac{\partial z}{\partial \rho }}{\frac{\partial \rho }{\partial a}} = {\frac{{\partial F^{ - 1} \left( \rho \right)}}{\partial \rho }}{\frac{\partial \rho }{\partial a}} = {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{\partial \rho }{\partial a}} = {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{\partial }{\partial a}}\left( {{\frac{p + s - m - sc}{{\left( {1 - c} \right)\left( {p + s} \right)}}}} \right) \\ = & {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}\left( 0 \right) = 0 \\ \end{aligned} $$
$$ \begin{aligned} {\frac{\partial z}{\partial b}} = & {\frac{\partial z}{\partial \rho }}{\frac{\partial \rho }{\partial b}} = {\frac{{\partial F^{ - 1} \left( \rho \right)}}{\partial \rho }}{\frac{\partial \rho }{\partial b}} = {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{\partial \rho }{\partial b}} = {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{\partial }{\partial b}}\left( {{\frac{p + s - m - sc}{{\left( {1 - c} \right)\left( {p + s} \right)}}}} \right) \\ = & {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}\left( 0 \right) = 0 \\ \end{aligned} $$
$$ \begin{aligned} {\frac{\partial z}{\partial c}} = & {\frac{\partial z}{\partial \rho }}{\frac{\partial \rho }{\partial c}} = {\frac{{\partial F^{ - 1} \left( \rho \right)}}{\partial \rho }}{\frac{\partial \rho }{\partial c}} = {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{\partial \rho }{\partial c}} = {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{\partial }{\partial c}}\left( {{\frac{p + s - m - sc}{{\left( {1 - c} \right)\left( {p + s} \right)}}}} \right) \\ = & {\frac{1}{{f\left( {F^{ - 1} \left( \rho \right)} \right)}}}{\frac{{\left( {p - m} \right)}}{{\left( {1 - c} \right)\left( {p + s} \right)^{2} }}} > 0 \\ \end{aligned} $$

Appendix 2

$$ \begin{aligned} E\left[ {\left( {z - \varepsilon } \right)^{ + } } \right] = & \int\limits_{A}^{z} {\left( {z - x} \right)} f\left( x \right)dx \\ = & \left[ {\left( {z - x} \right)F\left( x \right) + \int\limits {F\left( x \right)dx} } \right]_{A}^{z} \\ = & \left[ {\left( {z - x} \right)F\left( x \right)} \right]_{A}^{z} + \left[ {\int\limits {F\left( x \right)dx} } \right]_{A}^{z} \\ = & \int\limits_{A}^{z} {F\left( x \right)dx} \\ \end{aligned} $$

For uniform distribution, \( E\left[ {\left( {z - \varepsilon } \right)^{ + } } \right] = \int_{A}^{z} {\left( {{\frac{x - A}{B - A}}} \right)dx} = \left[ {{\frac{{\left( {x - A} \right)^{2} }}{{2\left( {B - A} \right)}}}} \right]_{A}^{z} = {\frac{{\left( {z - A} \right)^{2} }}{{2\left( {B - A} \right)}}} \)

$$ {\frac{\partial }{\partial z}}E\left[ {\left( {z - \varepsilon } \right)^{ + } } \right] = {\frac{\partial }{\partial z}}\left[ {{\frac{{\left( {z - A} \right)^{2} }}{{2\left( {B - A} \right)}}}} \right] = {\frac{z - A}{B - A}} < 1 $$

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Parthasarathi, G., Sarmah, S.P. & Jenamani, M. Supply chain coordination under retail competition using stock dependent price-setting newsvendor framework. Oper Res Int J 11, 259–279 (2011). https://doi.org/10.1007/s12351-010-0077-z

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