Proposing a discount policy for two-level supply chain coordination with periodic review replenishment and promotional efforts decisions

Abstract

This study develops a coordination scheme based upon discount contract to coordinate promotional efforts and periodic review replenishment decisions. We consider a supplier-retailer chain with stochastic demand dependent on promotional efforts. The retailer uses a periodic review order-up-to-level inventory system and makes promotional efforts as a means of enhancing market demand. In traditional trading market, the retailer individually determines the replenishment and promotional efforts decisions based upon his/her own profitability. However, these decisions affect the sales volume of the whole system and consequently influence the supplier’s profitability. Therefore, it is worth devising a coordination plan that not only enhances the entire system’s profitability but also brings great benefits to both the retailer and the supplier. To this end, we first propose a decentralized model where each member’s profit function is optimized regardless of the others’ profitability. Then, a centralized model is investigated to achieve the best values of replenishment and promotional efforts decisions from the whole supply chain viewpoint. Afterwards, an incentive plan based upon a discount factor is developed to guarantee the members’ participation in the coordination plan. Upper and lower limits of the discount factor are derived based upon the members’ conditions for taking part in the plan. Moreover, we propose an industrial case study and a comprehensive sensitivity analysis. The outcomes of the model indicate that the proposed coordination scheme is effective and desirable even when the level of demand uncertainty increases.

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Acknowledgements

The authors are grateful to the editor and two anonymous reviewers for their valuable comments, which have greatly improved this paper.

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Correspondence to Seyyed-Mahdi Hosseini-Motlagh.

Appendices

Appendix 1. Proving the concavity of the supplier’s profit function

To prove concavity of the retailer’s profit function over \(k\) and \(a\), it suffices to prove that the Hessian matrix represented below is negative definite.

$$H\left( { {\text{TP}}_{\text{r}} \left( {{\text{T}},{\text{k}},{\text{a}}} \right)} \right) = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} {\text{TP}}_{\text{r}} \left( {{\text{T}},{\text{k}},{\text{a}}} \right)}}{{\partial k^{2} }}} & {\frac{{\partial^{2} {\text{TP}}_{\text{r}} \left( {{\text{T}},{\text{k}},{\text{a}}} \right)}}{\partial k\partial a}} \\ {\frac{{\partial^{2} {\text{TP}}_{\text{r}} \left( {{\text{T}},{\text{k}},{\text{a}}} \right)}}{\partial a\partial k}} & {\frac{{\partial^{2} {\text{TP}}_{\text{r}} \left( {{\text{T}},{\text{k}},{\text{a}}} \right)}}{{\partial a^{2} }}} \\ \end{array} } \right]$$
(33)

where,

$$H_{11} = \frac{{\partial^{2} {\text{TP}}_{\text{r}} \left( {{\text{T}},{\text{k}},{\text{a}}} \right)}}{{\partial k^{2} }} = - \left[ {{\text{h}}_{\text{r}} \alpha + \frac{{{{\uppi }} + {{\upalpha }}\left( {{\text{r}} - {\text{w}}} \right)}}{\text{T}}} \right]\sigma \left( {T + L} \right)^{{\frac{1}{2}}} \varphi_{Z} \left( k \right) < 0$$
(34)
$$\begin{aligned} H_{22} & = \frac{{\partial^{2} {\text{TP}}_{\text{r}} \left( {{\text{T}},{\text{k}},{\text{a}}} \right)}}{{\partial k^{2} }}\frac{{\partial^{2} {\text{TP}}_{\text{r}} \left( {{\text{T}},{\text{k}},{\text{a}}} \right)}}{{\partial a^{2} }} - \frac{{\partial^{2} {\text{TP}}_{\text{r}} \left( {{\text{T}},{\text{k}},{\text{a}}} \right)}}{\partial k\partial a}\frac{{\partial^{2} {\text{TP}}_{\text{r}} \left( {{\text{T}},{\text{k}},{\text{a}}} \right)}}{\partial a\partial k} \\ & = \left( { - \left[ {{\text{h}}_{\text{r}} \alpha + \frac{{{{\uppi }} + {{\upalpha }}\left( {{\text{r}} - {\text{w}}} \right)}}{\text{T}}} \right]\sigma \left( {T + L} \right)^{{\frac{1}{2}}} \varphi_{Z} \left( k \right)} \right)\left( { - \frac{\eta }{T}} \right) > 0 \\ \end{aligned}$$
(35)

\(H_{11}\) is always less than zero over \(k\) and \(a\), for any T and \(H_{22}\) is always greater than zero. Thus, the Hessian matrix \(H\left( { {\text{TP}}_{\text{r}} \left( {{\text{T}},{\text{k}},{\text{a}}} \right)} \right)\) is proved to be negative definite and therefore the concavity of the retailer’s profit function over \(k\) and \(a\) is guaranteed.

Appendix 2. Proving the concavity of supplier’s profit function

To prove concavity of \(TP_{s} \left( n \right)\) over n, the second derivative of \(TP_{s} \left( n \right)\) with respect to n is calculated and is shown that it is negative.

(36) shows the first derivative of \(TP_{s} \left( n \right)\) with respect to n;

$$\frac{{\partial TP_{s} \left( n \right)}}{\partial n} = \frac{{A_{s} }}{{n^{2} T}} - \frac{{h_{s} }}{2}\left( {\left( {{\text{D}}_{0} + {{\upmu a}}} \right)T - \alpha \sigma \left( {T + L} \right)^{{\frac{1}{2}}} G\left( k \right)} \right)$$
(36)

And (37) shows the second derivative of \(TP_{s} \left( n \right)\) with respect to n;

$$\frac{{\partial^{2} {\text{TP}}_{\text{s}} \left( {\text{n}} \right)}}{{\partial n^{2} }} = - 2\frac{{A_{s} }}{{n^{3} T}}$$
(37)

As can be seen, the second derivative of \({\text{TP}}_{\text{s}} \left( n \right)\) over n is always negative. Therefore, the supplier’s profit function is concave over \(n\).

Appendix 3. Proving the concavity of supply chain’s profit function

To prove concavity of \({\text{TP}}_{\text{sc}} \left( {{\text{T}},{\text{k}},{\text{a}},{\text{n}}} \right)\) with respect to \(k\), \(a\) and n simultaneously, it suffices to show that the Hessian matrix of \({\text{TP}}_{\text{sc}} \left( {{\text{T}},{\text{k}},{\text{a}},{\text{n}}} \right)\) represented by (38), is negative definite for any \(T\).

The Hessian matrix of \({\text{TP}}_{\text{sc}} \left( {{\text{T}},{\text{k}},{\text{a}},{\text{n}}} \right)\) for any \(T\) is:

$$\left[ { \begin{array}{*{20}c} {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{{\partial a^{2} }}} & {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial a\partial k}} & {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial a\partial n}} \\ {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial k\partial a}} & {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{{\partial k^{2} }}} & {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial k\partial n}} \\ {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial n\partial a}} & {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial n\partial k}} & {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{{\partial n^{2} }}} \\ \end{array} } \right]$$
(38)

where,

$$H_{11} = \frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{{\partial a^{2} }} = - \frac{\eta }{T} < 0$$
(39)
$$\begin{aligned} H_{22} & = \frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{{\partial a^{2} }}\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{{\partial k^{2} }} - \frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial a\partial k}\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial k\partial a} \\ & = \left( { - \frac{\eta }{T}} \right)\left( { - \frac{1}{T}\left[ {\alpha \left( {r - p} \right) + \pi + h_{r} \alpha T - h_{s} \frac{{\alpha \left( {n - 1} \right)T}}{2}} \right]\sigma \left( {T + L} \right)^{{\frac{1}{2}}} \varphi_{z} \left( k \right) } \right) > 0 \\ \end{aligned}$$
(40)
$$\begin{aligned} H_{33} & = \left( { - 1} \right)^{2} \frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{{\partial a^{2} }}\left( {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{{\partial k^{2} }}\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{{\partial n^{2} }} - \frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial k\partial n}\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial n\partial k}} \right) \\ & \quad+ \left( { - 1} \right)^{3} \frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial a\partial k}\left( {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial k\partial a}\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{{\partial n^{2} }} - \frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial k\partial n}\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial n\partial a}} \right) \\ & \quad+ \left( { - 1} \right)^{4} \frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial a\partial n}\left( {\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial k\partial a}\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial n\partial k} - \frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{{\partial k^{2} }}\frac{{\partial^{2} TP_{sc} \left( {T,k,a,n} \right)}}{\partial n\partial a}} \right) \\ & = \left[ {\left( {\frac{{h_{s} \mu T}}{2}} \right)^{2} - \frac{{2\eta A_{s} }}{{n^{3} T^{2} }}} \right]\left[ {\alpha \left( {r - p} \right) + \pi + h_{r} \alpha T - h_{s} \frac{{\alpha \left( {n - 1} \right)T}}{2}} \right]\varphi_{z} \left( k \right) \\ & \quad+ \left[ {\frac{{\eta h_{s} }}{2}\alpha \sigma \left( {T + L} \right)^{{\frac{1}{2}}} \left( {1 -\upphi_{z} \left( k \right)} \right)^{2} } \right] < 0 \\ \end{aligned}$$
(41)

A Hessian matrix is called to be negative definite when \(k^{TH}\) principal minor of the matrix has the sign of \(\left( { - 1} \right)^{k}\). Thus, for the above matrix to be negative definite, the first minor (\(H_{11}\)), the second minor (\(H_{22}\)) and the third minor (\(H_{33}\)) must be negative, positive and negative, respectively. According to (39), \(H_{11}\) is always negative. (40) shows that \(H_{22}\) is positive when the condition represented by (10) is satisfied and (41) indicates that \(H_{33}\) will be negative, if the condition represented by (11) is satisfied.

Appendix 4. Proof of Theorem 2

Finding the upper limit on \(T\). under the centralized model.

Since \(\upphi_{z} \left( k \right)\). is the normal distribution function, it always satisfies \(\upphi_{z} \left( k \right) \ge 0\). Accordingly, the non-equality \(1 - \upphi_{z} \left( k \right) \le 1\). is always satisfied. Besides, Eq. (12) can be rewritten as \(1 - \upphi_{z} \left( {k^{**} } \right) = \frac{{Th_{r} }}{{h_{r} \alpha T + \left( {\alpha \left( {r - p} \right) + \pi - h_{s} \frac{{\alpha \left( {n - 1} \right)T}}{2}} \right)}}\). Due to the fact that \(1 - \upphi_{z} \left( {k^{**} } \right) \le 1\), we have \(\frac{{Th_{r} }}{{h_{r} \alpha T + \left( {\alpha \left( {r - p} \right) + \pi - h_{s} \frac{{\alpha \left( {n - 1} \right)T}}{2}} \right)}} \le 1\). Making some simplifications will result in the upper bound represented by (15).

Appendix 5. Proving the concavity of supply chain’s profit function under the simplified model

To prove the concavity of \({\text{TP}}_{\text{sc}} \left( {{\text{T}},{\text{k}},{\text{n}}} \right)\). with respect to \(k\). and n simultaneously, it suffices to show that the Hessian matrix of \({\text{TP}}_{\text{sc}} \left( {{\text{T}},{\text{k}},{\text{n}}} \right)\) represented by (42), is negativeefinite for any \(T\).

The Hessian matrix of \({\text{TP}}_{\text{sc}} \left( {{\text{T}},{\text{k}},{\text{n}}} \right)\) for any \(T\) is:

$$\left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} TP_{sc} \left( {T,k,n} \right)}}{{\partial k^{2} }}} & {\frac{{\partial^{2} TP_{sc} \left( {T,k,n} \right)}}{\partial k\partial n}} \\ {\frac{{\partial^{2} TP_{sc} \left( {T,k,n} \right)}}{\partial n\partial k}} & {\frac{{\partial^{2} TP_{sc} \left( {T,k,n} \right)}}{{\partial n^{2} }}} \\ \end{array} } \right]$$
(42)

where,

$$\begin{aligned} H_{11} & = \frac{{\partial^{2} {\text{TP}}_{\text{sc}} \left( {{\text{T}},{\text{k}},{\text{n}}} \right)}}{{\partial k^{2} }} \\ & = - \frac{1}{T}\left[ {\alpha \left( {r - p} \right) + \pi + h_{r} \alpha T - h_{s} \frac{{\alpha \left( {n - 1} \right)T}}{2}} \right]\sigma \left( {T + L} \right)^{{\frac{1}{2}}} \varphi_{z} \left( k \right) \\ \end{aligned}$$
(43)
$$\begin{aligned} H_{22} & = \frac{{\partial^{2} TP_{sc} \left( {T,k,n} \right)}}{{\partial k^{2} }}\frac{{\partial^{2} TP_{sc} \left( {T,k,n} \right)}}{{\partial n^{2} }} - \frac{{\partial^{2} TP_{sc} \left( {T,k,n} \right)}}{\partial k\partial n}\frac{{\partial^{2} TP_{sc} \left( {T,k,n} \right)}}{\partial n\partial k} \\ & = \frac{{2A_{s} }}{{n^{3} T^{2} }}\left[ {\alpha \left( {r - p} \right) + \pi + h_{r} \alpha T - h_{s} \frac{{\alpha \left( {n - 1} \right)T}}{2}} \right]\sigma \left( {T + L} \right)^{{\frac{1}{2}}} \varphi_{z} \left( k \right) \\ & \quad- \left( {\frac{{h_{s} }}{2}\alpha \sigma \left( {T + L} \right)^{{\frac{1}{2}}} \left( {1 -\upphi_{z} \left( k \right)} \right)} \right)^{2} \\ \end{aligned}$$
(44)

for the above matrix to be negative definite, the first minor (\(H_{11}\)) and the second minor (\(H_{22}\)) must be negative and positive, respectively. According to (43), \(H_{11}\) is negative when the condition \(\left( {r - p} \right) + \pi + h_{r} \alpha T - h_{s} \frac{{\alpha \left( {n - 1} \right)T}}{2} > 0\) is satisfied. (44) shows that \(H_{22}\) is positive when the condition \(\frac{{2A_{s} \varphi_{z} \left( k \right)}}{{n^{3} T^{2} }}[ {\alpha \left( {r - p} \right) + \pi + h_{r} \alpha T - h_{s} \frac{{\alpha \left( {n - 1} \right)T}}{2}} ] > \sigma \left( {T + L} \right)^{{\frac{1}{2}}} ( {\frac{{h_{s} \alpha }}{2}\left( {1 - \varphi_{z} \left( k \right)} \right)} )^{2}\) is satisfied.

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Nouri, M., Hosseini-Motlagh, SM. & Nematollahi, M. Proposing a discount policy for two-level supply chain coordination with periodic review replenishment and promotional efforts decisions. Oper Res Int J 21, 365–398 (2021). https://doi.org/10.1007/s12351-018-0434-x

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Keywords

  • Supply chain coordination
  • Promotional efforts
  • Periodic review inventory system
  • Discount scheme
  • Profit allocation policy