Proposing a delay in payment contract for coordinating a two-echelon periodic review supply chain with stochastic promotional effort dependent demand

Abstract

In a decentralized supply chain (SC), each member individually makes decisions according to its personal objectives while these decisions impact on the other SC actors. The promotional effort made by retailers is one of the main factors that largely impacts on the market demand of a commodity, which in turn boosts the profitability of the whole SC members. In this paper, we investigate coordination of promotional effort and replenishment decisions in a two-echelon SC including single supplier and single retailer. The investigated SC faces a stochastic demand influenced by the retailer’s promotional effort. To replenish items, the retailer uses a periodic review inventory system and decides on the review period, order-up-to-level and promotional effort. On the other side, the supplier employs a periodic review lot-for-lot strategy and determines its replenishment cycle multiplier. Firstly, we model the SC under the decentralized and centralized decision-making models. Exact solution procedures are presented using mathematical and concavity analysis to obtain the decentralized and centralized optimal solutions. Afterwards, a coordination model based on delay in payment contract is proposed to motivate the retailer to participate in the joint decision-making model. To create a more realistic model, in the proposed models we assume that the SC members’ rates of return on investment are different. The minimum and maximum length of credit period which are acceptable to both members are determined. Finally, numerical examples and sensitivity analysis are conducted to investigate the performance and applicability of the developed models. The results show that the proposed coordination scheme considerably improves the profitability of both SC members and the whole SC in comparison with the decentralized setting.

Appendix

1.1 Appendix A. Proof of proposition 1

To prove concavity with respect to $k$ and $a$ for a given $T$, the Hessian matrix of the retailer’s profit function for a given $T$ is calculated:

$$H\left( {{{{{\Pi}}}_R}} \right)=\left[ {\begin{array}{*{20}{c}} {\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial {a^2}}}}&{\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial a\partial k}}} \\ {\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial k\partial a}}}&{\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial {k^2}}}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} { - \frac{\eta }{T}}&0 \\ 0&{\sigma \sqrt {T+L} \left( {{h_R}\alpha +\frac{{\pi +\alpha \left( {r - w} \right)}}{T}} \right)\varphi \left( k \right)} \end{array}} \right]$$

$$\frac{{\partial {{{{\Pi}}}_R}}}{{\partial a}}=\left( {r - w} \right)\mu - {h_R}\frac{{T\mu }}{2} - \frac{{\eta a}}{T}$$

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$$\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial {a^2}}}= - \frac{\eta }{T}<0$$

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where,

$${H_{11}}=\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial {a^2}}}= - \frac{\eta }{T}<0$$

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$$\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial a\partial k}}=0$$

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$$\frac{{\partial {{{{\Pi}}}_R}}}{{\partial k}}= - {h_R}\sigma \sqrt {T+L} - \sigma \sqrt {T+L} \left( {{h_R}\alpha +\frac{{\pi +\alpha \left( {r - w} \right)}}{T}} \right)\left( {\Phi \left(k \right) - 1} \right)$$

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$$\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial {k^2}}}= - \sigma \sqrt {T+L} \left( {{h_R}\alpha +\frac{{\pi +\alpha \left( {r - w} \right)}}{T}} \right)\varphi \left( k \right)<0$$

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$$\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial k\partial a}}=0$$

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$${H_{22}}=\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial {a^2}}}*\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial {k^2}}}=( - \frac{\eta }{T})*( - \sigma \sqrt {T+L} \left( {{h_R}\alpha +\frac{{\pi +\alpha \left( {r - w} \right)}}{T}} \right)\varphi \left( k \right))>0$$

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The first principal minor of the above Hessian matrix (${H_{11}}$) has a negative value. Also, the second principal minor (${H_{22}}$) is always positive. Therefore, the Hessian matrix is negative definite and therefore the retailer’s profit function is concave with respect to $k$ and $~a$ for a given $T$. By optimizing the retailer’s profit function with respect to $k$ and $a$, the optimal values of $k$ and $a~$ for a given $T$ will be:

$$1 - \Phi \left( {{k^*}} \right)=\frac{{{h_R}T}}{{{h_R}\alpha T+\pi +\alpha \left( {r - w} \right)}}~~~~$$

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$${a^*}=\frac{{T(\left( {r - w} \right)\mu - {h_R}\frac{{T\mu }}{2})}}{\eta }$$

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1.2 Appendix B. Proof of proposition2

To prove concavity with respect to $n$, it is sufficient to indicate that second-order derivative of ${\Pi _S}(n)$ with respect to $n$ is negative. The first order and second order derivatives of ${\Pi _S}(n)$ with respect to $n$ can be calculated as follows:

$$\frac{{\partial {{{{\Pi}}}_S}}}{{\partial n}}=\frac{{{A_S}}}{{{n^2}T}} - \frac{{{h_S}}}{2}(\left( {D+\mu a} \right)T - \alpha \sigma \sqrt {T+L} G\left( k \right))$$

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$$\frac{{{\partial ^2}{{{{\Pi}}}_S}}}{{\partial {n^2}}}= - \frac{{2{A_S}}}{{{n^3}T}}<0$$

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The second order derivative of ${\Pi _S}(n)$ has negative value. Hence, the supplier’s profit function is concave with respect to $n$. By setting Eq. [30] equal to zero, the optimal value of $n$ can be calculated as:

$${n^*}=\sqrt {\frac{{2{A_S}}}{{T{h_S}(\left( {D+\mu a} \right)T - \alpha \sigma \sqrt {T+L} G\left( k \right))}}}$$

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1.3 Appendix C. Proof of proposition 3

To prove concavity with respect to $k,~a,~n$ for a given $~T$, the Hessian matrix is calculated as follows:

$$H\left( {{{{{\Pi}}}_{SC}}} \right)=\left[ {\begin{array}{*{20}{c}} {\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial {a^2}}}}&{\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial a\partial k}}}&{\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial a\partial n}}} \\ {\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial k\partial a}}}&{\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial {k^2}}}}&{\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial k\partial n}}} \\ {\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial n\partial a}}}&{\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial n\partial k}}}&{\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial {n^2}}}} \end{array}} \right]$$

$$\frac{{\partial {{{{\Pi}}}_{SC}}}}{{\partial a}}=\left( {r - m} \right)\mu - \frac{{T\mu }}{2}\left( {\left( {n - 1} \right){h_S}+{h_R}} \right) - \frac{{\eta a}}{T}$$

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$$\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial {a^2}}}= - \frac{\eta }{T}<0$$

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$${H_{11}}=\frac{{{\partial ^2}{{{{\Pi}}}_R}}}{{\partial {a^2}}}= - \frac{\eta }{T}<0$$

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$$\frac{{\partial { \Pi _{SC}}}}{{\partial k}}= - \frac{1}{T}\left( {\pi +\alpha \left( {r - m} \right) - \frac{{{h_S}\alpha \left( {n - 1} \right)}}{2}T} \right)\sigma \sqrt {T+L} \left[ {\Phi \left( k \right) - 1} \right] - {h_R}[\sigma \sqrt {T+L} +\alpha \sigma \sqrt {T+L} \left( {\Phi \left( k \right) - 1} \right)]$$

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$$\frac{{{\partial ^2}{ \Pi _{SC}}}}{{\partial {k^2}}}= - \frac{1}{T}\left[ {\pi +\alpha \left( {r - m} \right)+\alpha T{h_R} - \frac{{{h_S}\alpha \left( {n - 1} \right)T}}{2}} \right]\sigma \sqrt {T+L} \varphi \left(k \right)$$

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$$\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial a\partial k}}=\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial k\partial a}}=0$$

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$${H_{22}}=\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial {a^2}}}*\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial {k^2}}}=( - \frac{\eta }{T})*( - \frac{1}{T}\left[ {\pi +\alpha \left( {r - m} \right)+\alpha T{h_R} - \frac{{{h_S}\alpha \left( {n - 1} \right)T}}{2}} \right]\sigma \sqrt {T+L} \varphi \left( k \right))>0$$

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$$\frac{{\partial {{{{\Pi}}}_{SC}}}}{{\partial n}}=\frac{1}{T}(\frac{{{A_S}}}{{{n^2}}}) - \frac{{{h_S}}}{2}(\left( {D+\mu a} \right)T - \alpha \sigma \sqrt {T+L} G\left( k \right))$$

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$$\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial {n^2}}}= - \frac{1}{T}\left( {\frac{{2n{A_S}}}{{{n^4}}}} \right)<0$$

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$$\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial n\partial a}}=\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial a\partial n}}= - \frac{{{h_S}}}{2}T{{{\upmu}}}$$

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$$\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial k\partial n}}=\frac{{{\partial ^2}{{{{\Pi}}}_{SC}}}}{{\partial n\partial k}}=\frac{{{h_S}}}{2}\alpha \sigma \sqrt {T+L} \left( {\Phi \left( k \right) - 1} \right)$$

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$$\begin{aligned} {H_{33}}= & {( - 1)^2}\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial {a^2}}} \times \left[ {\left( {\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial {k^2}}} \times \frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial {n^2}}}} \right) - \left( {\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial n\partial k}} \times \frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial k\partial n}}} \right)} \right]+{\left( { - 1} \right)^3}\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial a\partial k}} \\ & \quad \times \left[ {\left( {\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial k\partial a}} \times \frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial {n^2}}}} \right) - \left( {\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial n\partial a}} \times \frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial k\partial n}}} \right)} \right]+{\left( { - 1} \right)^4}\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial a\partial n}} \\ & \quad \times \left[ {\left( {\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial k\partial a}} \times \frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial n\partial k}}} \right) - \left( {\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial n\partial a}} \times \frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial {k^2}}}} \right)} \right] \\ & =\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial {k^2}}} \times \left[ {\left( {\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial {a^2}}} \times \frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial {n^2}}}} \right)+\left( {\frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial a\partial n}} \times \frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial n\partial a}}} \right)} \right] - \frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial {a^2}}} \times \frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial n\partial k}} \\ & \quad \times \frac{{{\partial ^2}{\Pi _{SC}}}}{{\partial k\partial n}} \\ & = - \frac{1}{T}\left[ {\pi +\alpha \left( {r - m} \right)+\alpha T{h_R} - \frac{{{h_S}\alpha \left( {n - 1} \right)T}}{2}} \right]\sigma \sqrt {T+L} \varphi \left( k \right) \\ & \quad \times \left[ {\left( {\frac{\eta }{{{T^2}}}\left( {\frac{{2{A_S}}}{{{n^3}}}} \right)} \right)+{{\left( { - \frac{{{h_S}}}{2}T\mu } \right)}^2}} \right]+\left( {\frac{\eta }{T}{\alpha ^2}{\sigma ^2}\left( {T+L} \right){{\left( {\Phi \left( k \right) - 1} \right)}^2}\frac{{{h_S}}}{2}} \right)<0 \\ \end{aligned}$$

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The first principal minor of the above Hessian matrix (${H_{11}}$) has a negative value. Also, the second principal minor (${H_{22}}$) is always positive and the third principal minor (${H_{33}}$) is negative under the condition. Therefore, the Hessian matrix is negative definite, thus the SC’s profit function is concave with respect to $k,~a,~n$ for a given $T$. By optimizing the SC’s profit function with respect to $k,~a,~n$, the optimal values of $k,~a,~n$ for a given $T$ will be:

$${a^{**}}=\frac{{T\left( {\left( {r - m} \right)\mu - \frac{{T\mu }}{2}\left( {\left( {n - 1} \right){h_S}+{h_R}} \right)} \right)}}{\eta }$$

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$${n^{**}}=\sqrt {\frac{{2{A_S}}}{{T{h_S}\left( {\left( {D+\mu a} \right)T - \alpha \sigma \sqrt {T+L} G\left( k \right)} \right)}}}$$

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$$1 - \Phi \left( {{k^{**}}} \right)=\frac{{{h_R}T}}{{{h_R}\alpha T+[\pi +\alpha \left( {r - m} \right) - \frac{{{h_S}\alpha \left( {n - 1} \right)T}}{2}]}}$$

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