Abstract
In a competitive market, the retailers, in order to encourage the customers to increase their orders, give them the opportunity to pay a fraction of the purchasing cost after delivery of the ordered items (i.e., down-stream partial delayed payment). On the other hand, the suppliers, in order to reduce the risk of cancellations of orders from buyers, may ask the retailers to pay a portion of the purchasing cost before delivery of products (i.e., up-stream partial prepayment). In this paper, an EOQ model with down-stream partial delayed payment and up-stream partial prepayment under three different scenarios (without shortage, with full backordering and with partial backordering) is presented. In order to find the optimal solutions of the models developed for different scenarios, the convexity of the objective functions (i.e., total cost functions) are proved and then closed-form optimal solutions are derived. Also, a solution algorithm is proposed for the model of the third scenario. To demonstrate the applicability of the framework, some numerical examples are presented. Finally, sensitivity analyses are made on several key parameters, in order to gain some managerial insight.
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Acknowledgments
This research was supported by the Iran National Science Foundation (INSF). [grant number INSF-94001671].
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Appendices
Appendix A: Convexity of equation (10) and finding its roots
The first partial derivative of \(\Delta (K,T)\) respect to T is as below;
Setting the above equation equal to zero gives:
Discriminant of \(\gamma (K)=\psi _2 K^{2}-2\psi _3 K+\psi _4 , \Delta =b^{2}-4ac\) is shown in Eq. (63).
Since \(\Delta \) is always negative, \(\gamma (K)\) does not have any root, and so its value is always either positive or negative. On the other hand, regarding \(\gamma (0)=\frac{C_b D}{2}>0\), it is obvious that \(\gamma (K)\) takes always a positive value in the range of [0, 1]. Hence, for any given value ofK, a unique value is obtained for \(T^{*}\), using Eq. (62). If Eq. (62) is substituted into Eq. (26), we will have:
The above Equation gives a unique minimal cost for any given value of K. It should be noticed that \(\Delta (K)\)is continuous on the compact interval [0, 1]. Therefore, it may have more than one local minima, the smallest of which will be the global minimum. In order to find the global minimum solution, the first and the second derivatives of \(\Delta (K)\) with respect to K are taken, as provided in Eqs. (65) and (66), respectively.
For all amounts of K, we have:
Considering that \(\psi _1 ,\psi _2 ,\psi _3 ,\gamma (K)\) are positive and also \((\psi _2 -\psi _3 )=\frac{D(C_h +i_1 C_p )}{2}>0\), function \(\Delta _1 (K)\) is convex and so has a global minimum. In order to obtain this optimal point, the first derivative of \(\mathop \Delta \limits ^\wedge _1 (K)\) with respect to K is set equal to zero:
Since \(\sqrt{\psi _1 }\) and \(\gamma (K)\) are positive, it can be concluded from Eq. (34) that \(\gamma ^{{\prime }}(K)\) is equal to zero, i.e.,
Equation (69) gives:
After Substituting \(K^{*}\) into Eq. (62) and simplifying it, the optimal period length is obtained as follows:
Appendix B: Finding the roots of \(\varphi (K,T)\) with respect to K and T
From Eq. (51), we have:
Taking thefirstderivativesof \(\varphi (K,T)\) with respectto K and T gives:
After some algebra we have:
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Lashgari, M., Taleizadeh, A.A. & Ahmadi, A. Partial up-stream advanced payment and partial down-stream delayed payment in a three-level supply chain. Ann Oper Res 238, 329–354 (2016). https://doi.org/10.1007/s10479-015-2100-5
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DOI: https://doi.org/10.1007/s10479-015-2100-5