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An integrated inventory model with warranty dependent credit period under two policies of a manufacturer

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Abstract

This article considers an integrated inventory model that deals with one manufacturer and one retailer. The manufacturer provides warranty period for the finished product to ensure product reliability. Also, the manufacturer gives the opportunity to take the facility of delay payment to the retailer. The main objective of this article is to analyze the effect of the credit period offered by the manufacturer which is functionally depended on warranty period of the product. Again, the retailer also provides trade credit to the end customers. Thus, because of presence of product warranty and customers’ credit, demand rate of customers has been assumed depending on warranty period and customers’ credit period. In this paper, two policies have been discussed according to the manufacturer’s policy on warranty period such as (1) free repairing for the defective products within the warranty period and (2) no warranty period to be offered. The purpose of this article is to maximize the integrated model optimizing product warranty, customers’ credit period and cycle length. Also, solution algorithms have been developed to solve the derived model. Finally, numerical examples have been carried out to illustrate the feasibility of the model.

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Acknowledgements

The authors are heartily thankful to the anonymous referees and the Editor of the Journal for their contractive comments and suggestions to improve the quality of the paper. Also, first author is highly thankful to the University Grant Commission (UGC) of India for financial support under F1-17.1/2014-15/MANF-2014-15-MUS-WES-35645.

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Authors and Affiliations

  1. Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, 721 102, W.B., India

    Ateka Banu & Shyamal Kumar Mondal

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  1. Ateka Banu
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  2. Shyamal Kumar Mondal
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Correspondence to Ateka Banu.

Appendix A

Appendix A

Taking the first order partial derivative of the average profit \(TP_{2}(N,T)\) w.r.t N, we have

$$\begin{aligned} \frac{\partial }{\partial N}(TP_{2}(N,T))= &\, {} \frac{D_0(1+\beta _2 N)}{T}\left[ -\beta _{3}(T-M_0+N)(s-c-pI_{p}M_{0})+\frac{h\beta _3}{2}(T^2-(M_0-N)^2)\right. \\&\left. +\,\frac{pI_{c}\beta _3}{2}(T-M_0)^2+\frac{sI_e}{2}\left[ 4N-2M_{0}-\beta _{3}(T-M_{0}+N)^2\right] \right] \\&+\,\frac{D_0\beta _2}{T}\left[ (s-c-pI_pM_0)\left[ T-\frac{\beta _{3}}{2}(T-M_0+N)^2\right] \right. \\&-\,\frac{h}{2}\left[ T^2-\frac{\beta _3}{3}(T-M_0+N)^2(2T+M_0-N)\right] \\&-\,\frac{pI_c(T-M_0)^2}{2}\left[ 1-\beta _3\left[ \frac{2}{3}(T-M_0)+N\right] \right] \\&\left. +\,\frac{sI_e}{2}\left[ T^2-2N(M_0-N)-\frac{\beta _3}{3}(T-M_0+N)^3\right]\right] \\=\, & {} G_1(N,T), say \end{aligned}$$

Taking the first order partial derivative of the average profit \(TP_{2}(N,T)\) w.r.t T, we have

$$\begin{aligned} \frac{\partial }{\partial T}(TP_{2}(N,T))=\, & {} D_0(1+\beta _2N)\left[ (s-c-pI_pM_0)\left\{ \frac{1}{T}(1-\beta _3(T-M_0+N))-\frac{1}{T^2}\left[ T-\frac{\beta _{3}}{2}(T-M_0+N)^2\right] \right\} \right. \\&-\,\frac{h}{2}\left\{ \frac{2}{T}[T-\beta _3(T^2-T(M_0-N))]-\frac{1}{T^2}\left[ T^2-\frac{\beta _3}{3}(T-M_0+N)^2(2T+M_0-N)\right] \right\} \\&-\,\frac{pI_c}{2}\left\{ \frac{2(T-M_0)}{T}[1-\beta _3(T-M_0+N)]-\frac{(T-M_0)^2}{T^2}\left[ 1-\beta _3\left[ \frac{2}{3}(T-M_0)+N\right] \right] \right\} \\&\left. +\,\frac{sI_e}{2}\left\{ \frac{1}{T}[2T-\beta _3(T-M_0+N)^2]-\frac{1}{T^2}[T^2-2N(M_0-N)-\frac{\beta _3}{3}(T-M_0+N)^3]\right\} \right] +\frac{A+F}{T^2}\\=\, & {} G_2(N,T), say \end{aligned}$$

Taking the second order partial derivative of the average profit \(TP_{2}(N,T)\) w.r.t N, we have

$$\begin{aligned} \frac{\partial ^2}{\partial ^2 N}(TP_{2}(N,T))= & {} \frac{D_0(1+\beta _2 N)}{T}\left[ -\beta _3(s-c-pI_pM_0)+h\beta _3(M_0-N)+sI_e(2-\beta _3(T-M_0+N))\right] \\&+\,\frac{2D_0\beta _2 }{T}\left[ -\beta _{3}(T-M_0+N)(s-c-pI_{p}M_{0})+\frac{h\beta _3}{2}(T^2-(M_0-N)^2)\right. \\&\left. +\,\frac{pI_{c}\beta _3}{2}(T-M_0)^2+\frac{sI_e}{2}[4N-2M_{0}-\beta _{3}(T-M_{0}+N)^2]\right] \end{aligned}$$

Taking the second order partial derivative of the average profit \(TP_{2}(N,T)\) w.r.t T, we have

$$\begin{aligned} \frac{\partial ^2}{\partial ^2 T}(TP_{2}(N,T))= & {} D_0(1+\beta _2N)\left[ (s-c-pI_pM_0)\left\{ -\frac{\beta _3}{T}-\frac{2}{T^2}[1-\beta _3(T-M_0+N)]\right. \right. \\&\left. +\,\frac{2}{T^3}\left[ T-\frac{\beta _3}{2}(T-M_0+N)^2\right] \right\} -h\left\{ \frac{1}{T}[1-\beta _3(2T-M_0+N)]-\frac{2}{T^2}[T-\beta _3(T^2-T(M_0-N))]\right. \\&+\left. \,\frac{1}{T^3}\left[ T^2-\frac{\beta _3}{3}(T-M_0+N)^2(2T+M_0-N)\right] \right\} -pI_c\left\{ \frac{1}{T}[1-\beta _3(T-M_0+N)]-\frac{2}{T^2}(T-M_0)[1-\beta _3(T-M_0+N)]\right. \\&\left. +\,\frac{2}{T^3}\left[ 1-\beta _3\left[ \frac{2}{3}(T-M_0)+N\right] \right] \right\} +sI_e\left\{ \frac{1}{T}[1-\beta _3(T-M_0+N)]-\frac{1}{T^2}[2T-\beta _3(T-M_0+N)^2]\right. \\&\left. +\,\frac{1}{T^3}\left[ T^2-2N(M_0-N)-\frac{\beta _3}{3}(T-M_0+N)^3]\right\} \right] -\frac{2}{T^3}(A+F)\\ \frac{\partial ^2}{\partial T \partial N}(TP_{2}(N,T))= & {} \frac{D_0(1+\beta _2N)}{T}\bigg [-\beta _3(s-c-pI_pM_0)+h\beta _3T+\beta _3pI_c(T-M_0)-sI_e\beta _3(T-M_0+N)\bigg ]\\&-\,\frac{D_0(1+\beta _2N)}{T^2}\left[ -\beta _{3}(T-M_0+N)(s-c-pI_{p}M_{0})+\frac{h\beta _3}{2}(T^2-(M_0-N)^2)\right. \\&+\,\frac{pI_{c}\beta _3}{2}(T-M_0)^2+\frac{sI_e}{2}[4N-2M_{0}-\beta _{3}(T-M_{0}+N)^2]\bigg ]\\&+\,\frac{D_0\beta _2}{T}\bigg[(s-c-pI_pM_0)[1-\beta _3(T-M_0+N)]-h[T-\beta _3(T^2-T(M_0-N))]-pI_c(T-M_0)[1-\beta _3(T-M_0+N)]\\&\left. +\,\frac{sI_e}{2}\left[ 2T-\beta _3(T-M_0+N)^2\right] \right] -\frac{D_0\beta _2}{T^2}\left[ (s-c-pI_pM_0)\left[ T-\frac{\beta _{3}}{2}(T-M_0+N)^2\right] \right. \\&-\,\frac{h}{2}\left[ T^2-\frac{\beta _3}{3}(T-M_0+N)^2(2T+M_0-N)\right] -\frac{pI_c(T-M_0)^2}{2}\left[ 1-\beta _3\left[ \frac{2}{3}(T-M_0)+N\right] \right] \\&\left. +\,\frac{sI_e}{2}\left[ T^2-2N(M_0-N)-\frac{\beta _3}{3}(T-M_0+N)^3\right] \right] \end{aligned}$$

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Banu, A., Mondal, S.K. An integrated inventory model with warranty dependent credit period under two policies of a manufacturer. OPSEARCH 55, 677–702 (2018). https://doi.org/10.1007/s12597-018-0345-x

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