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A Two-Echelon Inventory Model for Ameliorating/Deteriorating Items with Single Vendor and Multi-buyers

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Abstract

This research work introduces an inventory model for ameliorating items as livestock (say fishes, chickens, ducklings), where the demand rate and deterioration rate are assumed as constant. Delivery times for buyers are introduced in the proposed model. The accumulated inventory contains amelioration rate of livestock with deterioration rate due to death of ameliorating items. The proposed model reduces the integrated total cost of inventory. In addition, the demand and deterioration rates are constant, and the amelioration rate is assumed to adhere to the Weibull distribution. The inventory model is discussed for single manufacturer, who produces the ameliorating items and sells the finished goods to the multiple buyers. The connections between framework parameters and solution procedure illustrate the optimal solutions. Numerical examples are provided to illustrate the theoretical results. Finally, the sensitivity analysis of the framework parameters is discussed.

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Author information

Author notes

  1. Vandana

    Present address: Department of Management Studies, Indian Institute of Technology, Chennai, Tamil Nadu, 600036, India

  2. Shib Sankar Sana

    Present address: Kishore Bharati Bhagini Nivedita College, Ramkrishna Palli, Behala, Kolkata, 700060, India

Authors and Affiliations

  1. School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, C.G., 492010, India

    Vandana

  2. Department of Mathematics, Bhangar Mahavidyalaya, South 24 Parganas, India

    Shib Sankar Sana

Authors
  1. Vandana
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  2. Shib Sankar Sana
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Corresponding author

Correspondence to Shib Sankar Sana.

Appendix

Appendix

To prove that Eq. (19) has one global optimum solution, we differentiate \({\text {JTRC}}(n_i, T_1, T_2)\) with respect to \(T_1\) and \(T_2\) such that

$$\begin{aligned}&\frac{\delta \hbox {JTRC}}{\delta T_1} = \biggl [-\frac{O_{\mathrm{v}}}{({}{T_1}+{}{T_2})^2}-\frac{c_{\mathrm{m}}}{({}{T_1}+{}{T_2})^2} \biggl (\int _{0}^{{}{T_2}} {}{{t_2}}^{-1+y} x y \sum _{i=1}^{n} \biggl (1+{}{{t_2}}^y x-{}{{t_2}} \theta \biggl ) \\&\quad \biggl (-{}{{t_2}}+{}{T_2}-\frac{\biggl (-{}{{t_2}}^{1+y}+{}{T_2}^{1+y}\biggl ) x}{1+y} \\&\quad +\frac{1}{2}\biggl (-{}{{t_2}}^2+{}{T_2}^2\biggl ) \theta \biggl ) d_i \, \hbox {d}{}{{t_2}} \\&\quad +\frac{{}{T_1}^y x \biggl (2 (1+y)+{}{T_1}^y x (1+y)-2 {}{T_1} y \theta \biggl ) \sum _{i=1}^n \biggl (1-{}{T_1}^y x+{}{T_1} \theta \biggl ) \biggl ({}{T_2}-\frac{{}{T_2}^{1+y} x}{1+y}+\frac{{}{T_2}^2 \theta }{2}\biggl ) d_i}{2 (1+y)}\biggl ) \\&\quad + \frac{c_{\mathrm{m}}}{({}{T_1}+{}{T_2})^2} \biggl [ \frac{{}{T_1}^y x \biggl (2 (1+y)+{}{T_1}^y x (1+y)-2 {}{T_1} y \theta \biggl )\sum _{i=1}^{n} \biggl (-{}{T_1}^{-1+y} x y+\theta \biggl ) \biggl ({}{T_2}-\frac{{}{T_2}^{1+y} x}{1+y}+\frac{{}{T_2}^2 \theta }{2}\biggl ) d_i}{2 (1+y)} \\&\quad +\frac{{}{T_1}^y x \biggl ({}{T_1}^{-1+y} x y (1+y)-2 y \theta \biggl )\sum _{i=1}^{n} \biggl (1-{}{T_1}^y x+{}{T_1} \theta \biggl ) \biggl ({}{T_2}-\frac{{}{T_2}^{1+y}x}{1+y}+\frac{{}{T_2}^2 \theta }{2}\biggl ) d_i}{2 (1+y)} \\&\quad +\frac{{}{T_1}^{-1+y} x y \biggl (2 (1+y)+{}{T_1}^y x (1+y)-2 {}{T_1} y \theta \biggl )\sum _{i=1}^{n} \biggl (1-{}{T_1}^y x+{}{T_1} \theta \biggl ) \biggl ({}{T_2}-\frac{{}{T_2}^{1+y} x}{1+y}+\frac{{}{T_2}^2 \theta }{2}\biggl ) d_i}{2 (1+y)}\biggl ] \\&\quad +\sum _{i=1}^{n} -\frac{n_i O_{\mathrm{b}}}{({}{T_1}+{}{T_2})^2} -\frac{1}{({}{T_1}+{}{T_2})^2}{}{h_{\mathrm{v}}} \biggl (\int _0^{{}{T_2}} \biggl (\sum _{i=1}^{n} \biggl (1+{}{{t_2}}^y x-{}{{t_2}} \theta \biggl ) \\&\quad \biggl (-{}{{t_2}}+{}{T_2}-\frac{\biggl (-{}{{t_2}}^{1+y}+{}{T_2}^{1+y}\biggl ) x}{1+y} \\&\quad +\frac{1}{2} \biggl (-{}{{t_2}}^2+{}{T_2}^2\biggl ) \theta \biggl ) d_i\biggl ) \, \hbox {d}{}{{t_2}} +\biggl ({}{T_1}+\frac{{}{T_1}^{1+y} x}{1+y}-\frac{{}{T_1}^2 \theta }{2}\biggl ) \\&\quad \sum _{i=1}^{n} \biggl (1-{}{T_1}^y x+{}{T_1} \theta \biggl ) \biggl ({}{T_2}-\frac{{}{T_2}^{1+y} x}{1+y}+\frac{{}{T_2}^2 \theta }{2}\biggl ) d_i \\&\quad -\sum _{i=1}^{n} n_i \sum _{i=1}^{n} \int _0^{\frac{{}{T_1}+{}{T_2}}{n_i}} \biggl (\sum _{i=1}^{n} \frac{d_i}{\theta }\biggl ) \sum _{i=1}^{n} (1-t \theta ) \\&\quad \biggl (-t \theta +\frac{({}{T_1}+{}{T_2}) \theta }{n_i}\biggl ) \, \hbox {d}t\biggl )-\frac{h_{\mathrm{b}} p_{\mathrm{b}}}{({}{T_1}+{}{T_2})^2} \sum _{i=1}^{n} n_i \sum _{i=1}^{n} \int _0^{\frac{{}{T_1}+{}{T_2}}{n_i}} \biggl (\sum _{i=1}^n \frac{d_i}{\theta }\biggl ) \\&\quad \sum _{i=1}^n (1-t \theta ) \biggl (-t \theta +\frac{({}{T_1}+{}{T_2}) \theta }{n_i}\biggl ) \, \hbox {d}t \end{aligned}$$
$$\begin{aligned}&+\frac{h_{\mathrm{v}}}{{{}{T_1}+{}{T_2}}} \biggl [\biggl ({}{T_1}+\frac{{}{T_1}^{1+y} x}{1+y}-\frac{{}{T_1}^2 \theta }{2}\biggl ) \sum _{i=1}^{n} \biggl (-{}{T_1}^{-1+y} x y+\theta \biggl ) \nonumber \\&\quad \biggl ({}{T_2}-\frac{{}{T_2}^{1+y} x}{1+y}+\frac{{}{T_2}^2 \theta }{2}\biggl ) d_i + \biggl (1+{}{T_1}^y x-{}{T_1} \theta \biggl ) \nonumber \\&\quad \sum _{i=1}^{n} \biggl (1-{}{T_1}^y x+{}{T_1} \theta \biggl ) \biggl ({}{T_2}-\frac{{}{T_2}^{1+y} x}{1+y}+\frac{{}{T_2}^2 \theta }{2}\biggl ) d_i \nonumber \\&\quad -\sum _{i=1}^{n} n_i \sum _{i=1}^{n} \int _0^{\frac{{}{T_1}+{}{T_2}}{n_i}} \biggl (\sum _{i=1}^n \frac{d_i}{\theta }\biggl )\sum _{i=1}^{n}\frac{\theta (1-t \theta )}{n_i} \, \hbox {d}t\biggl ] \nonumber \\&+ \frac{{h_{\mathrm{b}}} {}{p_{\mathrm{b}}}}{{{}{T_1}+{}{T_2}}} \sum _{i=1}^n n_i \sum _{i=1}^{n} \int _0^{\frac{{}{T_1}+{}{T_2}}{n_i}} \biggl (\sum _{i=1}^{n} \frac{d_i}{\theta }\biggl ) \sum _{i=1}^{n} \frac{\theta (1-t \theta )}{n_i} \, \hbox {d}t +\sum _{i=1}^{n} \nonumber \\&\quad \biggl (\frac{{}{p_{\mathrm{v}}} \sum _{i=1}^{n} \biggl (-{}{T_1}^{-1+y} x y+\theta \biggl ) \biggl ({}{T_2}-\frac{{}{T_2}^{1+y} x}{1+y}+\frac{{}{T_2}^2 \theta }{2}\biggl ) d_i}{{}{T_1}+{}{T_2}}\nonumber \\&\quad -\frac{{}{p_{\mathrm{v}}} \sum _{i=1}^{n} \biggl (1-{}{T_1}^y x+{}{T_1} \theta \biggl ) \biggl ({}{T_2}-\frac{{}{T_2}^{1+y} x}{1+y}+\frac{{}{T_2}^2 \theta }{2}\biggl ) d_i}{({}{T_1}+{}{T_2})^2}\biggl ) \nonumber \\&\quad +\sum _{i=1}^n \biggl (\frac{n_i p_{\mathrm{b}} \biggl (-\frac{d_i}{n_i}+\sum _{i=1}^{n} \frac{d_i}{n_i}\biggl )}{{}{T_1}+{}{T_2}} \nonumber \\&\quad -\frac{n_i p_{\mathrm{b}} \biggl (-\frac{({}{T_1}+{}{T_2}) d_i}{n_i}+\sum _{i=1}^n \frac{({}{T_1}+{}{T_2}) d_i}{n_i}\biggl )}{({}{T_1}+{}{T_2})^2}\biggl ), {}{Re}[y] > 0, \biggl ]. \end{aligned}$$
(19)
$$\begin{aligned}&\quad +\frac{{}{h_{\mathrm{v}}}}{{{}{T_1}+{}{T_2}}} \biggl (\int _0^{{}{T_2}} \biggl (\sum _{i=1}^{n} \biggl (1+{}{{t_2}}^y x-{}{{t_2}} \theta \biggl ) \nonumber \\&\quad \biggl (1-{}{T_2}^y x+{}{T_2} \theta \biggl ) d_i\biggl ) \, \hbox {d}{}{{t_2}}\nonumber \\&\quad +\biggl ({}{T_1}+\frac{{}{T_1}^{1+y} x}{1+y}-\frac{{}{T_1}^2 \theta }{2}\biggl ) \nonumber \\&\quad \sum _{i=1}^{n} \biggl (1-{}{T_1}^y x+{}{T_1} \theta \biggl ) \biggl (1-{}{T_2}^y x+{}{T_2} \theta \biggl ) d_i \nonumber \\&\quad -\sum _{i=1}^{n} n_i \sum _{i=1}^{n} \int _{0}^{\frac{{}{T_1}+{}{T_2}}{n_i}} \biggl (\sum _{i=1}^{n} \frac{d_i}{\theta }\biggl ) \sum _{i=1}^{n} \frac{\theta (1-t \theta )}{n_i} \, \hbox {d}t\biggl ) \nonumber \\&\quad +\frac{{}{h_{\mathrm{b}}} {}{p_{\mathrm{b}}}}{{{}{T_1}+{}{T_2}}} \sum _{i=1}^{n} n_i \sum _{i=1}^{n} \int _{0}^{\frac{{}{T_1}+{}{T_2}}{n_i}} \biggl (\sum _{i=1}^{n} \frac{d_i}{\theta }\biggl ) \sum _{i=1}^{n} \frac{\theta (1-t \theta )}{n_i} \, \hbox {d}t \nonumber \\&\quad +\sum _{i=1}^{n} \biggl (\frac{{}{p_{\mathrm{v}}} \sum _{i=1}^{n} \biggl (1-{}{T_1}^y x+{}{T_1} \theta \biggl ) \biggl (1-{}{T_2}^y x+{}{T_2} \theta \biggl ) d_i}{{}{T_1}+{}{T_2}}\nonumber \\&\quad -\frac{{}{p_{\mathrm{v}}}}{{({}{T_1}+{}{T_2})^2}} \sum _{i=1}^{n} \biggl (1-{}{T_1}^y x+{}{T_1} \theta \biggl ) \nonumber \\&\quad \biggl ({}{T_2}-\frac{{}{T_2}^{1+y}x}{1+y}+\frac{{}{T_2}^2 \theta }{2}\biggl ) d_i\biggl ) +\sum _{i=1}^{n} \nonumber \\&\quad \biggl (\frac{n_i p_{\mathrm{b}} \biggl (-\frac{d_i}{n_i}+\sum _{i=1}^{n} \frac{d_i}{n_i}\biggl )}{{}{T_1}+{}{T_2}} \nonumber \\&\quad -\frac{n_i p_{\mathrm{b}} \biggl (-\frac{({}{T_1}+{}{T_2}) d_i}{n_i}+\sum _{i=1}^n \frac{({}{T_1}+{}{T_2}) d_i}{n_i}\biggl )}{({}{T_1}+{}{T_2})^2}\biggl ), {}{Re}[y]>0\biggl ]. \end{aligned}$$
(20)

Now, set \(\frac{\delta {\text {JTRC}}}{\delta T_1} = \frac{\delta {\text {JTRC}}}{\delta T_2} = 0\), and find the optimum value of \(T_1\) and \(T_2\), say \(T_{1}^{*}\) and \(T_{2}^{*}\). The sufficient condition for the joint relevant total cost \( \hbox {JRTC} (n_i, T_1, T_2) \) for global optimum solution is

$$\left\vert\begin{array}{ll}\frac{{\delta}^2{\text{JTRC}}}{\delta T_1^2}& \frac{{\delta}^2{\text{JTRC}}}{\delta T_1 \delta T_2}\\\frac{{\delta}^2{\text{JTRC}}}{\delta T_2 \delta T_1} &\frac{{\delta}^2{\text{JTRC}}}{\delta T_2^2}\\ \end{array}\right\vert >0$$

Hence, the expression of two derivative is nonlinear. Therefore, we are not writing the whole expression.

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Vandana, Sana, S.S. A Two-Echelon Inventory Model for Ameliorating/Deteriorating Items with Single Vendor and Multi-buyers. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 90, 601–614 (2020). https://doi.org/10.1007/s40010-018-0568-5

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