Continuous dynamic pricing strategy under competition for deteriorating products in a dual-channel supply chain

Abstract

Dynamic pricing is a crucial revenue management technique when product status changes over time. It can become more prominent if applied in conjunction with an inventory management strategy for deteriorating items. This paper presents a framework for dynamic pricing in a continuous time horizon for a dual-channel supply chain consisting of a manufacturer and a retailer. The manufacturer produces a deteriorating product and sells through direct and indirect retail channels, and both channels compete with each other over retail prices. The freshness of the product varies with time, and thus both channels invest in preservation technology, which further leads to carbon emissions. We derive an analytical solution to the dual-channel pricing model. Furthermore, the case of single-channel pricing has been investigated to study and compare its pricing strategy with the dual-channel pricing strategy. Lastly, a numerical illustration is presented to study the impact of key parameters on optimal decisions through sensitivity analysis. We provide managerial recommendations in various scenarios like an increase in competition, the shift of consumer preference between channels and fluctuation in demand.

Appendices

Appendix A

• The retailer's profit function is jointly concave in both control variable $p_{r}$ and state variable $I_{r}$

Proof

As the ${\text{Hessian of }}H_{r} = \left[ {\begin{array}{*{20}c} { - 2f\left( {\gamma + b_{r} } \right)} & 0 \\ 0 & 0 \\ \end{array} } \right].$

So, the Hamiltonian $H_{r}$ is jointly concave in control and state variables and thus, the retailer's profit function is jointly concave in both control variable $p_{r}$ and state variable $I_{r}$.

• The manufacturer's profit function is jointly concave in both control variables $p_{m}$ and $P$ and state variable $I_{m}$.

Proof

As the ${\text{Hessian of }}H_{m} = \left[ {\begin{array}{*{20}c} { - 2f\gamma } & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right].$

So, the Hamiltonian $H_{m}$ is jointly concave in control and state variables. Thus, the manufacturer's profit function is jointly concave in both control variables $p_{m}$ and $P$ and state variable $I_{m}$.

Appendix B

• The profit function of the retailer is jointly concave in both control variable $p_{rs}$, $P_{s}$ and state variable $I_{rs}$ and $I_{ms}$

Proof

As the ${\text{Hessian of }}H_{s} = \left[ {\begin{array}{*{20}c} { - 2fb_{r} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right].$

So, the Hamiltonian is jointly concave in both control and state variables.