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Effect of two-echelon trade credit on pricing-inventory policy of non-instantaneous deteriorating products with probabilistic demand and deterioration functions

  • S.I.: Innovative Supply Chain Optimization
  • Published:
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Abstract

Usually, the profit of companies will increase if they employ trade credit financing policy to encourage customer to purchase more. This paper develops a model for pricing and inventory control of non-instantaneous deteriorating items under two-echelon trade credit in which the vendor provides a credit period to the retailer and the retailer in turn offers a delay in payment to his/her customer. The price-dependent probabilistic demand function and partially backlogged shortages are adopted. Also, deterioration is shown by three different probability distribution function including (1) uniform distribution, (2) triangular distribution, and (3) beta distribution. The theoretical results are designed to determine the optimal selling price and the optimal inventory control variables so that the retailer’s total profit is maximized. Also, the necessary and sufficient conditions to prove the existence and uniqueness of the optimal solution are provided. Moreover, an algorithm is extended to describe the solution procedure. Numerical example, sensitivity analysis, and a simulation approach are presented to illustrate the performance of the algorithm and the theoretical results. Several managerial insights are also driven from computational results. The results indicate that the retailer’s total profit increases by considering the non-instantaneous deteriorating phenomenon and the trade credit policy.

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Authors and Affiliations

  1. Faculty of Industrial Engineering and Management Systems, Amirkabir University of Technology, Tehran, Iran

    Reza Maihami, Behrooz Karimi & Seyyed Mohammad Taghi Fatemi Ghomi

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  1. Reza Maihami
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  2. Behrooz Karimi
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  3. Seyyed Mohammad Taghi Fatemi Ghomi
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Correspondence to Behrooz Karimi.

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Appendices

Appendix 1: Proof of Theorem 1

Proof of part (a): We show that \(F({{t_{1}}})\) is a strictly decreasing function in \({{t_{1}}}{\in }\left[ t_{d},t_{1}^{b} \right) \) and \(\hbox {lim}_{t_{1}\rightarrow t_{1}^{b}}{F\left( {{t_{1}}} \right) {=-\infty }}\). Thus, if \({\Phi }\left( p \right) {\equiv }F\left( t_{d} \right) {\ge 0}\), the intermediate value theorem implies that there exists a unique value of \(\hbox {t}_{1}\) (\(\hbox {t}_{1}^{{*}})\) such that \(F\left( \hbox {t}_{1}^{{*}} \right) =0\). Solving Eq. (24), the unique value \(\hbox {t}_{1}^{{*}}\) is calculated. When the value of \(\hbox {t}_{1}^{{*}}\) is identified, solving Eq. (23), the value of T (\(T^{*})\) is computed.

Proof of part (b): If \({\Phi }\left( p \right) \equiv F\left( t_{d} \right) <0\), then \(F\left( {{t_{1}}} \right) \) is a strictly decreasing function of \({{t_{1}}}\in \left[ t_{d},t_{1}^{b} \right) \). So, for all \({{t_{1}}}\in \left[ t_{d},t_{1}^{b} \right) \), \(F\left( {{t_{1}}} \right) <0\). Therefore, we cannot find a value of \({{t_{1}}}\in \left[ t_{d},t_{1}^{b} \right) \) such that \(F\left( {{t_{1}}} \right) =0\).

Appendix 2: Proof of Theorem 2

Proof of part (a): Let \(\left( {{t_{1}}},T \right) =(t_{1}^{*},T^{*})\) be the solution of Eqs. (21) and (22). We have

$$\begin{aligned}&{\frac{\partial ^{2}{\textit{TP}}_{1}\left( p,{{t_{1}}},T\right) }{\partial t_{1}^{2}}|}_{\left( t_{1}^{*},T^{*}\right) }\\&\quad =\frac{\left( \mu _{1}R\left( p \right) +\mu _{2} \right) }{T^{*}}\left\{ \frac{{-\updelta }\left( {\hbox {p}}-{\hbox {c}}+{\hbox {M}} \right) }{\left[ {1+\updelta }\left( T^{*}-t_{1}^{*} \right) \right] ^{2}}-\hbox {N}\hbox {e}^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( t_{1}^{*}-t_{d} \right) }-ce^{\left( {t_{1}^{*}}-Y\right) \left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }I_{p} \right\} <O\\&{\frac{\partial ^{2}{\textit{TP}}_{1}\left( p,{{t_{1}}},T\right) }{\partial T^{2}}|}_{\left( t_{1}^{*},T^{*}\right) }=\frac{\left( \mu _{1}R\left( p \right) +\mu _{2} \right) }{T^{*}}\nonumber \\&\quad \times \left( \frac{{-\updelta }\left( {\hbox {p}}-{\hbox {c}}+{\hbox {M}} \right) }{\left[ {1+\updelta }\left( T^{*}-t_{1}^{*} \right) \right] ^{2}}-\frac{2\left( -\frac{1}{2}\hbox {I}_{e}p\left( Y^{2}-Z^{2} \right) +\frac{c\hbox {I}_{p}\left( -t_{1}^{*}+Y-\frac{1-e^{\left( t_{1}^{*}-Y\right) \left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\right) }{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\right) }{T^{{*}^{2}}}\right) <0\\&{\frac{\partial ^{2}{\textit{TP}}_{1}\left( p,{{t_{1}}},T\right) }{\partial T\partial {{t_{1}}}}|}_{\left( t_{1}^{*},T^{{*}}\right) }\\&\quad =\frac{\left( \mu _{1}R\left( p \right) +\mu _{2} \right) }{T^{{*}}}\left\{ \frac{{\updelta }\left( {\hbox {p}}-{\hbox {c}}+{\hbox {M}} \right) }{\left[ {1+\updelta }\left( T^{{*}}-t_{1}^{*} \right) \right] ^{2}}+\frac{c\left( -1+e^{\left( t_{1}^{*}-Y\right) \left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\right) \hbox {I}_{p}}{T^{{*}}\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) } \right\} \end{aligned}$$

Therefore, the determinant of the Hessian matrix is

$$\begin{aligned}&{\frac{\partial ^{2}TP\left( p,{{t_{1}}},T\right) }{\partial t_{1}^{2}}|}_{\left( t_{1}^{*},T^{*}\right) }\times {\frac{\partial ^{2}{\textit{TP}}_{1}\left( t_{1,}T,p \right) }{\partial T^{2}}|}_{\left( t_{1}^{*},T^{*}\right) }-{\left[ {\frac{\partial ^{2}{\textit{TP}}_{1}\left( t_{1,}T,p \right) }{\partial T\partial t_{1,}}|}_{\left( t_{1}^{*},T^{*}\right) }\right] }^{2}\\&\quad =\left( \frac{\left( \mu _{1}R\left( p \right) +\mu _{2} \right) }{T^{*}} \right) ^{2}\left( \left( {\left( \frac{\left( -c+M+p\right) \delta }{{\left( 1+T^{*}\delta -t_{1}^{*}\delta \right) }^{2}}+\frac{c\left( -1+e^{\left( t_{1}^{*}-Y\right) \left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\right) \hbox {I}_{p}}{T^{*}\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\right) }^{2}\right. \right. \\&\quad \quad +\left( -ce^{\left( t_{1}^{*}-Y\right) \left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\hbox {I}_{p}-e^{\left( t_{1}^{*}-\hbox {td}\right) \left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }N-\frac{\left( -c+M+p\right) \delta }{{\left( 1+T^{*}\delta -t_{1}^{*}\delta \right) }^{2}}\right) \left( -\frac{\left( -c+M+p\right) \delta }{{\left( 1+T^{*}\delta -t_{1}^{*}\delta \right) }^{2}}\right. \\&\left. \left. \left. \quad +\frac{\hbox {I}_{e}p\left( Y^{2}-Z^{2}\right) \left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) ^{2}-2c\hbox {I}_{p}\left( -1+e^{\left( t_{1}^{*}-Y\right) \left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }-t_{1}^{*}\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) +Y\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \right) }{T^{{*}^{2}}\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) ^{2}}\right) \right) \right) >0 \end{aligned}$$

It means that the Hessian matrix H at point \(\left( t_{1}^{*},T^{*} \right) \) is negative definite. As a result, we find that the point \(\left( t_{1}^{*},T^{*} \right) \) is a global maximum.

Proof of part (b): For any given p, if \({\Phi }\left( p \right) <0\), then \(F\left( {{t_{1}}} \right) <0\) for all \({{t_{1}}}\in \left[ t_{d},t_{1}^{b} \right) \). Therefore,

$$\begin{aligned}&\frac{d{TP}_{1}\left( p,{{t_{1}}},T\right) }{dT}=\frac{\left( \mu _{1}R\left( p \right) +\mu _{2} \right) }{T^{2}}\\&\quad \left( \left( \hbox {p}-\hbox {c}+\hbox {M} \right) \left( \frac{\hbox {t}_{1}+\frac{\hbox {N}\left[ e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( {{t_{1}}}-t_{d} \right) }-1 \right] +ht_{d}+\frac{cI_{p}}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\left( e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( \hbox {t}_{1}-Y \right) }-1 \right) }{{\updelta }\left\{ \hbox {p}-\hbox {c}+\hbox {M}-\hbox {N}\left[ e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( {{t_{1}}}-t_{d} \right) }-1 \right] -ht_{d+}\frac{cI_{p}}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\left( e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( \hbox {t}_{1}-Y \right) }-1 \right) \right\} }}{1+\updelta \left( \hbox {t}_{1}+\frac{\hbox {N}\left[ e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( {{t_{1}}}-t_{d} \right) }-1 \right] +ht_{d}+\frac{cI_{p}}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\left( e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( \hbox {t}_{1}-Y \right) }-1 \right) }{{\updelta }\left\{ \hbox {p}-\hbox {c}+\hbox {M}-\hbox {N}\left[ e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( {{t_{1}}}-t_{d} \right) }-1 \right] -ht_{d+}\frac{cI_{p}}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\left( e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( \hbox {t}_{1}-Y \right) }-1 \right) \right\} }-\hbox {t}_{1}\right) }-\hbox {t}_{1}\right. \right. \\&\quad \left. -\frac{\hbox {ln}[1+\updelta \left( \hbox {t}_{1}+\frac{\hbox {N}\left[ e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( {{t_{1}}}-t_{d} \right) }-1 \right] +ht_{d}+\frac{cI_{p}}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\left( e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( \hbox {t}_{1}-Y \right) }-1 \right) }{{\updelta }\left\{ \hbox {p}-\hbox {c}+\hbox {M}-\hbox {N}\left[ e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( {{t_{1}}}-t_{d} \right) }-1 \right] -ht_{d+}\frac{cI_{p}}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\left( e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( \hbox {t}_{1}-Y \right) }-1 \right) \right\} }-\hbox {t}_{1}\right) }{{\updelta }}\right) \\&\quad \quad +\frac{\hbox {N}}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\times \left[ \hbox {e}^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( \hbox {t}_{1}-\hbox {t}_{\mathrm{d}} \right) }-\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( \hbox {t}_{1}-\hbox {t}_{\mathrm{d}} \right) -1 \right] +\hbox {h}\hbox {t}_{\mathrm{d}}\hbox {t}_{1}-\frac{\hbox {h}\hbox {t}_{\mathrm{d}}^{2}}{2}+\frac{\hbox {A}}{\left( \mu _{1}R\left( p \right) +\mu _{2} \right) }\\&\quad \quad \left. -\frac{1}{2}I_{e}p\left( \mu _{1}R\left( p \right) +\mu _{2} \right) \left( Y^{2}-Z^{2}\right) +\frac{cI_{p}\left( \mu _{1}R\left( p \right) +\mu _{2}\right) \left( -\hbox {t1}+Y-\frac{1-e^{\left( \hbox {t1}-Y\right) \left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\right) }{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\right) \\&\quad =\frac{\left( \mu _{1}R\left( p \right) +\mu _{2} \right) F\left( {{t_{1}}} \right) }{T^{2}} \end{aligned}$$

It is clear that \(\frac{\left( \mu _{1}R\left( p \right) +\mu _{2} \right) }{T^{2}}{>}0\) and we showed that \(F\left( {{t_{1}}} \right) {<}0\). Therefore, \(\frac{\left( \mu _{1}R\left( p \right) +\mu _{2} \right) F\left( {{t_{1}}} \right) }{T^{2}}{<}0\), which means that \({\textit{TP}}_{1}(p,{{t_{1}}},T)\) is a strictly decreasing function of T. Thus, when T is minimum, \({\textit{TP}}_{1}(p,{{t_{1}}},T)\) attains its maximum value. From Eq. (23), the minimum value of T occurs at

$$\begin{aligned} \hbox {T}= t_{d}+\frac{ht_{d}+\frac{cI_{p}}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\left( e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( t_{d}-Y \right) }-\hbox {1} \right) }{{\updelta }\left\{ {\hbox {p}}-{\hbox {c}}+{\hbox {M}}-ht_{d+}\frac{cI_{p}}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\left( e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( t_{d}-Y \right) }-\hbox {1} \right) \right\} } \end{aligned}$$

As \({{t_{1}}}=t_{d}\), \({\textit{TP}}_{1}(p,{{t_{1}}},T)\) has a maximum value at point \((t_{1}^{*},T^{{*}})\), where \(t_{1}^{*}=t_{d}\) and

$$\begin{aligned} T^{{*}}=t_{d}+\frac{ht_{d}+\frac{cI_{p}}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\left( e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( t_{d}-Y \right) }-\hbox {1} \right) }{{\updelta }\left\{ {\hbox {p}}-{\hbox {c}}+{\hbox {M}}-ht_{d+}\frac{cI_{p}}{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) }\left( e^{\left( \frac{\alpha _{1}+\alpha _{2}}{2} \right) \left( t_{d}-Y \right) }-\hbox {1} \right) \right\} } \end{aligned}$$

Appendix 3: Proof of Theorem 3

Using step 1, the initial total profit function with staring value of \(\hbox {t}_{1}^{0}, \hbox {T}^{0},{\hbox {and}\,\hbox {p}}_{{0}}\) is computed. For notational convenience, let

$$\begin{aligned} {\textit{TP}}_{1}\left( \hbox {t}_{1}^{{0}}\hbox {, }\hbox {T}^{{0}}, \hbox {p}_{{0}} \right) = {\Pi }^{\hbox {(o)}} \end{aligned}$$

Using step 2, \(\hbox {p}_{{0}}\) is fixed and the value of \(\hbox {t}_{1}^{1}\hbox { and }\hbox {T}^{1}\) is obtained. Therefore, the new total profit function is resulted.

$$\begin{aligned} {\textit{TP}}_{1}\left( \hbox {t}_{1}^{1}, \hbox {T}^{1}, \hbox {p}_{0} \right) ={\Pi }^{(1)} \end{aligned}$$

Applying Theorem 1, we determined that \({\textit{TP}}_{1}\left( \hbox {t}_{1}^{1}, \hbox {T}^{1}, \hbox {p}_{{0}} \right) \) is concave and takes its global solution at \({\hbox {(t}}_{1}^{1}, \hbox {T}^{1})\). So, \({\textit{TP}}_{1}\left( \hbox {t}_{1}^{1}, \hbox {T}^{1}, \hbox {p}_{0} \right) \ge {\textit{TP}}_{1}\left( \hbox {t}_{1}^{0},\hbox {T}^{0},\hbox {p}_{0}\right) \).

If \({\textit{TP}}_{1}\left( \hbox {t}_{1}^{1},\hbox {T}^{1}, \hbox {p}_{{0}} \right) ={\textit{TP}}_{1}\left( \hbox {t}_{1}^{{0}},\hbox {T}^{{0}},\hbox {p}_{{0}} \right) \), then the algorithm is convergent. Else,

$$\begin{aligned} {\textit{TP}}_{1}\left( \hbox {t}_{1}^{1},\hbox {T}^{1},\hbox {p}_{{0}} \right) {>}{\textit{TP}}_{1}\left( \hbox {t}_{1}^{{0}},\hbox {T}^{{0}},\hbox {p}_{{0}} \right) \Longrightarrow {\Pi }^{(1)}{>}{\Pi }^{\hbox {(o)}} \end{aligned}$$

Now, by fixing \(\hbox {t}_{1}^{1}{\hbox { and T}}^{1}\), we solve Eq. (27) and the new selling price \(\hbox {p}_{1}\) is obtained. Thus, the new total profit function is found.

$$\begin{aligned} {\textit{TP}}_{1}\left( \hbox {t}_{1}^{1},\hbox {T}^{1},\hbox {p}_{1} \right) ={\Pi }^{(2)} \end{aligned}$$

Using Theorem 2, we proved that \({\textit{TP}}_{1}\left( \hbox {t}_{1}^{1},\hbox {T}^{1},\hbox {p}_{1} \right) \) is concave and obtained its global solution at \(\hbox {p}_{1}\). Therefore,

$$\begin{aligned} {\textit{TP}}_{1}\left( \hbox {t}_{1}^{1},\hbox {T}^{1},\hbox {p}_{1} \right) \ge {\textit{TP}}_{1}\left( \hbox {t}_{1}^{1},\hbox {T}^{1},\hbox {p}_{{0}} \right) \end{aligned}$$

If \({\textit{TP}}_{1}\left( \hbox {t}_{1}^{1},\hbox {T}^{1},\hbox {p}_{1} \right) ={\textit{TP}}_{1}\left( \hbox {t}_{1}^{1},\hbox {T}^{1},\hbox {p}_{{0}} \right) \), then the algorithm is convergent. Else,

$$\begin{aligned} {\textit{TP}}_{1}\left( \hbox {t}_{1}^{1},\hbox {T}^{1},\hbox {p}_{1} \right) {>}{\textit{TP}}_{1}\left( \hbox {t}_{1}^{1},\hbox {T}^{1},\hbox {p}_{{0}} \right) \Longrightarrow {\Pi }^{(2)}{>}{\Pi }^{(1)} \end{aligned}$$

Using the above approach, we obtain the strictly increasing sequence of \({\textit{TP}}_{1}\left( {{t_{1}}}\hbox {,T,p} \right) \)as follows:

$$\begin{aligned} {\Pi }^{(n)}{>}{\Pi }^{({\hbox {n}}-1)}{>}\cdots >{\Pi ^{(1)}}{>}{\Pi }^{\hbox {(o)}} \end{aligned}$$

We assume that the retailer’s total profit is finite; i.e. the retailer’s total profit has an upper bound. It implies that the obtained strictly increasing sequence of \({\textit{TP}}_{1}\left( {{t_{1}}}\hbox {,T,p} \right) \) has an upper bound. On the other hand, a strictly increasing sequence with an upper bound is convergent.

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Maihami, R., Karimi, B. & Fatemi Ghomi, S.M.T. Effect of two-echelon trade credit on pricing-inventory policy of non-instantaneous deteriorating products with probabilistic demand and deterioration functions. Ann Oper Res 257, 237–273 (2017). https://doi.org/10.1007/s10479-016-2195-3

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