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Optimal pricing and lot-sizing policy for a two-warehouse supply chain system with perishable items under partial trade credit financing

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Abstract

Usually, in real life business, supplier is willing to provide a retailer a trade credit period if the retailer orders a large quantity. So the retailer purchases more goods than that can be stored in his own warehouse and these excess quantities are stored in a rented warehouse. In practice, we observe that the demand depends on the selling price and the retailer offers the partial trade credit option to his customers. In this paper, we develop an economic-order-quantity-based model with perishable items and two-storage facility as a profit maximization problem under retailer’s partial trade credit policy and price dependent demand. Mathematical theorems are developed to determine optimal inventory policy for the retailer and numerical examples are given to illustrate the theory. We also obtain a lot of managerial phenomena.

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Acknowledgements

We sincerely express our gratitude to the anonymous referees for their valuable comments and suggestions. This research work is fully supported by Senior Research Fellowship (Grant No: 09/715(0002)/2006 EMR‐I) under Council of Scientific and Industrial Research (CSIR) ‐ India. We also thank University Grants Commision, India for providing Special Assistance Program (UGC ‐ SAP) in the department of Mathematics, Gandhigram Rural University.

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Authors and Affiliations

  1. Department of Mathematics, Gandhigram Rural University, Gandhigram, Dindigul, TamilNadu, 624302, India

    A. Thangam & R. Uthayakumar

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  1. A. Thangam
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  2. R. Uthayakumar
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Correspondence to A. Thangam.

Appendices

Appendix A

(a). From Eq. (21), we could not obtain an explicit closed-form solution for T*. For simplicity, we use Taylor’s series approximation of eθ T to obtain the closed-form solution. Utilizing the fact that

$$ \hbox{e}^{\theta T} \approx 1+\theta T+(\theta T)^2/2 \hbox { as } \theta T \hbox { is small} $$

we obtain

$$ \theta T \hbox{e}^{\theta T} - \hbox{e}\hbox{e}^{\theta T} +1 \approx (\theta T)^2/2 $$
(56)

Substituting Eq. (56) in Eq. (21), we get

$$ (h_1+c\theta+sI_{\rm e} \alpha) \lambda(s) T^2 \approx 2A $$

Consequently we obtain the optimal value of T 1*(s) as

$$ T_1^*(s)=\sqrt{\frac{2A}{\lambda(s)(h_1+c\theta+sI_{\rm e} \alpha)}} $$
(57)

To ensure T 1*(s) ≤ t 1, substituting Eq. (57) in to this inequality we have

$$ 2A \leq t_1^2 \lambda(s)(h_1+c\theta+sI_{\rm e} \alpha) $$

and so δ11(s) ≥ 2A implies that T* = T 1*(s).

(b). Following the similar approach given above, from Eq. (23) we get

$$ T_2^*(s)=\sqrt{\frac{2A+\lambda(s)(h_2-h_1)t_1^2}{\lambda(s)(h_2+c\theta+sI_{\rm e} \alpha)}} $$
(58)

To ensure that t 1 < T < M, substituting Eq. (58) in to this inequality, we have

$$ \lambda(s)(h_1+c\theta+sI_{\rm e} \alpha)t_1^2 < 2A < \lambda(s)[(h_2+c\theta+sI_{\rm e} \alpha)M^2-(h_2-h_1) t_1^2] $$

This implies that if δ11(s) < 2A < δ12(s) then T* = T 2*(s).

(c). By Taylor’s series expansion, \(\theta T e^{\theta(T-M)}-e^{\theta(T-M)}+1 \approx \frac{(\theta T)^2}{2}- \frac{(\theta M)^2}{2}+\theta M\) as θT is small (for details, see Chang et al. (2003))

Therefore, the Eq. (25) gives that

$$ \lambda(s)(h_2+c\theta+cI_k)T^2 \approx 2A+\lambda(s)[(h_2-h_1)t_1^2+(cI_k-sI_{\rm e} \alpha)M^2] $$

It implies that

$$ T_3^*(s)=\sqrt{\frac{2A+\lambda(s)[(h_2-h_1)t_1^2+(cI_k-sI_{\rm e} \alpha)M^2]}{\lambda(s)(h_2+c\theta+cI_k)}} $$
(59)

To ensure that T ≥ M, substituting Eq. (59) in to this inequality, we have

$$ 2A \geq \lambda(s)[(h_2+c\theta+sI_{\rm e} \alpha)M^2-(h_2-h_1) t_1^2] $$

It implies that if δ12(s) ≤ 2A, then T* = T 3*(s).

Appendix B

Solving Eq. (26) by using Taylor’s series approximations, we get

$$ T_4^*(s)=\sqrt{\frac{2A+(cI_k-sI_{\rm e}\alpha) \lambda(s) M^2} {\lambda(s)(h_1+c \theta+cI_k)}} $$
(60)

Following the similar approach as in Appendix A, we prove the Theorem 2 by using the inequalities 0 < T 1*(s) ≤ M, M < T 4*(s) < t 1 and T 3*(s) ≥ t 1.

Appendix C

Here, we use the Taylor’s series approximation as in Appendix A. Solving the Eqs. (27), (29), (31) and (33), we get

$$ \begin{aligned} T_5^*(s)=&\sqrt{\frac{2A}{\lambda(s)(h_1+c\theta+sI_{\rm e} \alpha)}}\\ T_6^*(s)=&\sqrt{\frac{2A+\lambda(s)(h_2-h_1)t_1^2}{\lambda(s)(h_2+c\theta+sI_{\rm e} \alpha)}}\\ T_7^*(s)=&\sqrt{\frac{2A+\lambda(s)[(h_2-h_1)t_1^2+sI_{\rm e}N^2(1-\alpha)]}{\lambda(s)(h_2+c\theta+sI_{\rm e})}}\\ T_8^*(s)=&\sqrt{\frac{2A+\lambda(s)[(h_2-h_1)t_1^2+cI_kM^2-sI_{\rm e}(M^2-N^2(1-\alpha))]}{\lambda(s)(h_2+c\theta+cI_k)}}\\ \end{aligned} $$

Using the above solutions and the inequalities 0 < T 5*(s) ≤ t 1, t 1 < T 6*(s) < N, N ≤ T 7*(s) < M and T 8*(s) ≥ M, we easily get the required results in Theorem 3.

Appendix D

Solving the Eq. (34), we get

$$ T_9^*(s)= \sqrt{\frac{2A+sI_{\rm e} \lambda(s) N^2 (1-\alpha)} {\lambda(s)(h_1+c\theta + sI_{\rm e})}} $$

Following the approach given in Appendix A and using the inequalities 0 < T 5*(s) ≤ N, N < T 9*(s) < t 1, t 1 ≤ T 7*(s) <  M and T 8*(s) ≥ M, we easily derive the theorem 4.

Appendix E

Solving the Eq. (35), we get

$$ T_{10}^*(s)= \sqrt{\frac{2A+ \lambda(s)[cI_kM^2-sI_{\rm e}(M^2- N^2 (1-\alpha))]}{\lambda(s)(h_1+c\theta + cI_k)}} $$

Following the approach given in Appendix A and using the inequalities 0 < T 5*(s) ≤ N, N < T 9*(s) < M, M ≤ T 10*(s) <  t 1 and T 8*(s) ≥ t 1, we easily derive the theorem 5.

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Thangam, A., Uthayakumar, R. Optimal pricing and lot-sizing policy for a two-warehouse supply chain system with perishable items under partial trade credit financing. Oper Res Int J 10, 133–161 (2010). https://doi.org/10.1007/s12351-009-0066-2

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