Abstract
The purpose of this study is two-fold. The first is to consider supplier’s and retailer’s trade-credit policy for fixed lifetime products and the second is to extend Mahata’s 2012 model with time varying deterioration where Mahata (Expert Syst Appl 39(3):3537–3550, 2012) wrote exponential deterioration but actually he considered constant deterioration. We assume that the suppliers offer full trade-credit to retailers but retailers offer partial trade-credit to their customers. Some numerical examples along with graphical representations are given to illustrate the model.
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Acknowledgments
The authors would like to thank the reviewers for their very helpful comments to improve the paper. This work was supported by the research fund of Hanyang University (HY-2014-N, Project number 201400000002202) for new Faculty members.
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Biswajit Sarkar is in leave on lien from Vidyasagar University.
Appendices
Appendix 1
and
Appendix 2
Case 1 \(M\ge N\)
Case 1.1 \(M\le t_1\) i.e., \(M\le t_M\le T\)
where
To determine the optimal value of \(T\) say \(T^*_1\), we solve the equation \(f_1(T)=0\).
We obtain \(\frac{df_1(T)}{dT}>0 ~~\hbox {if} ~~T>0\). As \(f_1(T)\) is an increasing function on \([0,\infty )\), then \(\frac{dTRC_1(T)}{dT}\) is an increasing function on \([0,\infty )\). Using Lemma, \(TRC_1(T)\) is a convex function on \([0,\infty )\).
In addition, we have as \(\lim T\rightarrow \infty \), then \(f_1(T)\rightarrow \infty \).
Then
By applying the intermediate value theorem, we can state that the optimal solution \(T^*_1\) exists and is unique.
Case 1. (b) \(M\le T\le t_M\)
where
To determine the optimal value of \(T\) say \(T^*_2\), we solve the equation \(f_2(T)=0\).
We obtain \(\frac{df_2(T)}{dT}>0,~~\hbox {if} ~~T>0.\)
As \(f_2(T)\) is an increasing function on \([0,\infty )\), then \(\frac{dTRC_2(T)}{dT}\) is an increasing function on \([0,\infty )\). Using Lemma, \(TRC_2(T)\) is a convex function on \([0,\infty )\).
In addition, we obtain as \(\lim T\rightarrow \infty \), then \(f_2(T)\rightarrow \infty \).
Then
Using intermediate value theorem, we can say that a unique optimal solution \(T^*_2\) exists.
Case 1. (c) \(N\le T\le M\)
where
To determine the optimal value of \(T\) say \(T^*_3\), we solve the equation \(f_3(T)=0.\)
We obtain \(\frac{df_3(T)}{dT}>0 ~~\hbox {if}~~T>0.\)
As \(f_3(T)\) is an increasing function on \([0,\infty )\), then \(\frac{dTRC_3(T)}{dT}\) is an increasing function on \([0,\infty )\). Using the statement of Lemma, \(TRC_3(T)\) is a convex function on \([0,\infty )\).
In addition, we find as \(\lim T\rightarrow \infty \), then \(f_3(T)\rightarrow \infty \).
Now
Then
Again using the intermediate value theorem, we conclude that a unique optimal solution \(T^*_3\) exists.
Case 1. (d) \(0<T\le N\)
where
To determine the optimal value of \(T\) say \(T^*_4\), we solve the equation \(f_4(T)=0\).
We obtain \(\frac{df_4(T)}{dT}>0 ~~\hbox {if}~~ T>0.\)
As \(f_4(T)\) is an increasing function on \([0,\infty )\), then \(\frac{dTRC_4(T)}{dT}\) is an increasing function on \([0,\infty )\). Using the Lemma, \(TRC_4(T)\) is a convex function on \([0,\infty )\).
In addition, we know as \(\lim T\rightarrow \infty \), then \(f_4(T)\rightarrow \infty \).
Then
Using the intermediate value theorem, we can say that a unique optimal solution \(T^*_4\) exists.
Case 2 \(M<N\)
Case 2. (a) \(T\ge t_M\)
where
To determine the optimal value of \(T\) say \(T^*_5\), we solve the equation \(f_5(T)=0\).
We obtain \(\frac{df_5(T)}{dT}>0 ~~\hbox {if} ~~T>0\). As \(f_5(T)\) is an increasing function on \([0,\infty )\), then \(\frac{dTRC_5(T)}{dT}\) is an increasing function on \([0,\infty )\). Using Lemma, \(TRC_5(T)\) is a convex function on \([0,\infty )\).
In addition, we obtain as \(\lim T\rightarrow \infty \), then \(f_5(T)\rightarrow \infty \).
Then
Using the intermediate value theorem, we can state that a unique optimal solution \(T^*_5\) exists.
Case 2. (b) \(M\le T\le t_M\)
where
To determine the optimal value of \(T\) say \(T^*_6\), we solve the equation \(f_6(T)=0\).
We obtain \(\frac{df_6(T)}{dT}>0 ~~\hbox {if} ~~T>0.\)
As \(f_6(T)\) is an increasing function on \([0,\infty )\), then \(\frac{dTRC_6(T)}{dT}\) is an increasing function on \([0,\infty )\). Using Lemma, \(TRC_6(T)\) is a convex function on \([0,\infty )\).
In addition, we find as \(\lim T\rightarrow \infty \), then \(f_6(T)\rightarrow \infty \).
Then
Using the intermediate value theorem, we can state that a unique optimal solution \(T^*_6\) exists.
Case 2. (c) \(0<T\le M\)
where
To determine the optimal value of \(T\) say \(T^*_7\), we solve the equation \(f_7(T)=0.\)
We obtain \(\frac{df_7(T)}{dT}>0, ~~\hbox {if}~~T>0.\)
As \(f_7(T)\) is an increasing function on \([0,\infty )\), then \(\frac{dTRC_7(T)}{dT}\) is an increasing function on \([0,\infty )\). Using the Lemma, \(TRC_7(T)\) is a convex function on \([0,\infty )\).
In addition, as \(\lim T\rightarrow \infty \), then \(f_7(T)\rightarrow \infty \).
Now
Then
Using the intermediate value theorem, we can state that a unique optimal solution \(T^*_7\) exists.
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Sarkar, B., Saren, S. & Cárdenas-Barrón, L.E. An inventory model with trade-credit policy and variable deterioration for fixed lifetime products. Ann Oper Res 229, 677–702 (2015). https://doi.org/10.1007/s10479-014-1745-9
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DOI: https://doi.org/10.1007/s10479-014-1745-9