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Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration

  • S.I.: Advances of OR in Commodities and Financial Modelling
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Abstract

In this paper, a deterministic inventory control model with deterioration is developed. Here, the deterioration rate follows stochastic deterioration, especially Weibull distribution deterioration. A time-dependent demand approach is introduced to show the applicability of our proposed model and to be up-to-date with respect to time. The main purpose of the paper is to investigate the optimal retailer’s replenishment decisions for deteriorating items including time-dependent demand for demonstrating more practical circumstances within economic-order quantity frameworks. Keeping in mind the criterion of modern era, we consider that the holding cost is totally dependent on time, and shortages are allowed for this model. Subject to the formulated model, we minimize the total inventory cost. The mathematical model is explored by numerical examples to validate the proposed model. A sensitivity analysis of the optimal solution with regard to important parameters is also carried out to elaborate the quality, e.g., stability, of our result and to possibly modify our model. The paper ends with a conclusion and an outlook to future studies.

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Acknowledgements

The first author is very much thankful to University Grants Commission (UGC) of India for providing financial support to continue this research work under [MANF(UGC)] scheme: Sanctioned letter number [F1-17.1/2012-13/MANF-2012-13-MUS-WES-19170 /(SA-III/Website)] dated 28/02/2013. The authors would like to express their cordial thanks to the Editor-in-Chief and anonymous reviewer for their valuable comments.

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Authors and Affiliations

  1. Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, West Bengal, 721102, India

    Magfura Pervin & Sankar Kumar Roy

  2. Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey

    Gerhard-Wilhelm Weber

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  1. Magfura Pervin
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  2. Sankar Kumar Roy
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  3. Gerhard-Wilhelm Weber
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Correspondence to Sankar Kumar Roy.

Appendix

Appendix

The necessary optimality conditions for finding the optimal values of T and \(t_1\) are calculated by the following derivations:

$$\begin{aligned} \frac{\partial TC}{\partial T}= & {} \frac{-1}{T^2}\left[ A+(h+c)(k-a)\left\{ \frac{t_1^2}{2}-\frac{\alpha \beta t_1^{2+\beta }}{(\beta +1)(2+\beta )}\right\} \right. \\&-\,(h+c)b\left\{ \frac{t_1^3}{6}-\frac{\alpha \beta t_1^{3+\beta }}{2(\beta +2)(3+\beta )}\right\} +\gamma (k-a)\left\{ \frac{t_1^3}{3}-\frac{\alpha \beta t_1^{3+\beta }}{(\beta +1)(3+\beta )}\right\} \\&-\,\gamma b\left\{ \frac{t_1^4}{8}-\frac{\alpha \beta t_1^{4+\beta }}{2(\beta +2)(4+\beta )}\right\} +c\delta a(T-t_1)+\frac{1}{2}c\delta b\left( T^2-t_1^2\right) \\&+\,c\left\{ (k-a)t_1+\frac{bt_1^2}{2}\right\} +Qc_2\left\{ (T-t_1) +(Tt_1^\beta -\frac{T^{\beta +1}}{\beta +1}-\frac{\beta t_1^{\beta +1}}{\beta +1})\right\} \\&+\,ac_2\left[ \frac{2Tt_1+3t_1^2-T^2}{2}+\frac{\alpha }{\beta +1}\left\{ t_1^{1 +\beta }(T-t_1)-\frac{T^{2+\beta }- t_1^{2+\beta }}{2+\beta }\right\} \right. \\&\left. +\,\alpha \left\{ \frac{T^{2+\beta }-t_1^{2+\beta }}{2+\beta }-\frac{t_1T^{1 +\beta }-t_1^{2+\beta }}{\beta +1}\right\} \right] +bc_2\left[ \left\{ \frac{(T-t_1)t_1^2}{2} -\frac{T^3-t_1^3}{6}\right\} \right. \\&+\,\frac{\alpha }{2+\beta }\left\{ t_1^{2+\beta }(T-t_1)-\frac{T^{3+\beta }- t_1^{3+\beta }}{3+\beta }\right\} \\&\left. +\,\frac{\alpha }{2}\left\{ \frac{T^{3+\beta }-t_1^{3+\beta }}{3+\beta } -\frac{t_1^2T^{1+\beta }-t_1^{1+\beta }}{\beta +1}\right\} \right] \\&\left. +\,s(1-\delta )\left\{ a(T-t_1) +\frac{1}{2}b\left( T^2-t_1^2\right) \right\} \right] +\frac{1}{T}\left[ (h+c)(k-a)\left\{ t_1\frac{\partial t_1}{\partial T}\right. \right. \\&\left. -\,\frac{\alpha \beta t_1^{1+\beta }\frac{\partial t_1}{\partial T}}{\beta +1}\right\} -(h+c)b\left\{ \frac{t_1^2\frac{\partial t_1}{\partial T}}{2}-\frac{\alpha \beta t_1^{2+\beta }\frac{\partial t_1}{\partial T}}{2(\beta +2)}\right\} +\gamma (k-a)\left\{ t_1^2\frac{\partial t_1}{\partial T}\right. \\&\left. -\,\frac{\alpha \beta t_1^{2+\beta }\frac{\partial t_1}{\partial T}}{\beta +1}\right\} -\gamma b\left\{ \frac{t_1^3\frac{\partial t_1}{\partial T}}{2}-\frac{\alpha \beta t_1^{3+\beta }\frac{\partial t_1}{\partial T}}{2(\beta +2)}\right\} +c\delta a\left( 1-\frac{\partial t_1}{\partial T}\right) \\&+\,c\delta b\left( T-t_1\frac{\partial t_1}{\partial T}\right) +c\left\{ (k-a)\frac{\partial t_1}{\partial T}+bt_1\frac{\partial t_1}{\partial T}\right\} \\&+\,Qc_2\left\{ \left( 1-\frac{\partial t_1}{\partial T}\right) +t_1^\beta +T\beta t_1^{\beta -1}\frac{\partial t_1}{\partial T}-T^\beta -\beta t_1^\beta \frac{\partial t_1}{\partial T}\right\} \\&+\,ac_2\left[ t_1+T\frac{\partial t_1}{\partial T}+3t_1\frac{\partial t_1}{\partial T}-T+\frac{\alpha }{\beta +1}\left\{ (1+\beta )t_1^\beta \frac{\partial t_1}{\partial T}(T-t_1)\right. \right. \\&\left. +\,t_1^{1+\beta }\left( 1-2\frac{\partial t_1}{\partial T}\right) -T^{1+\beta }\right\} +\alpha \left\{ T^{1+\beta }-t_1^{1+\beta }\frac{\partial t_1}{\partial T}-t_1T^\beta -\frac{T^{1+\beta }\frac{\partial t_1}{\partial T}}{1+\beta }\right. \\&\left. \left. +\,\frac{(2+\beta )t_1^{1+\beta }\frac{\partial t_1}{\partial T}}{1+\beta }\right\} \right] +bc_2\left[ \left\{ \frac{\left( 1 -\frac{\partial t_1}{\partial T}\right) t_1^2}{2}+t_1\frac{\partial t_1}{\partial T}(T-t_1)\right. \right. \\&\left. -\,\frac{T^2}{2}+\frac{t_1^2\frac{\partial t_1}{\partial T}}{2}\right\} +\frac{\alpha }{2+\beta }\left\{ (2+\beta )t_1^{1+\beta }\frac{\partial t_1}{\partial T}(T-t_1)\right. \\&\left. \left. +\,\left( t_1^{2+\beta }-T^{2+\beta }\right) \right\} +\frac{\alpha }{2}\left\{ T^{2+\beta }-t_1^{2+\beta }\frac{\partial t_1}{\partial T}-\frac{2t_1T^{1+\beta }\frac{\partial t_1}{\partial T}}{\beta +1}-t_1^2T^\beta +t_1^\beta \frac{\partial t_1}{\partial T}\right\} \right] \\&\left. +\, s(1-\delta )\left\{ a\left( 1-\frac{\partial t_1}{\partial T}\right) +b\left( T-t_1\frac{\partial t_1}{\partial T}\right) \right\} \right] , \end{aligned}$$
$$\begin{aligned} \frac{\partial TC}{\partial t_1}= & {} (h+c)(k-a)\left\{ t_1-\frac{\alpha \beta t_1^{1+\beta }}{\beta +1}\right\} -(h+c)b\left\{ \frac{t_1^2}{2}-\frac{\alpha \beta t_1^{2+\beta }}{2(\beta +2)}\right\} \\&+\,\gamma (k-a)\left\{ t_1^2-\frac{\alpha \beta t_1^{2+\beta }}{\beta +1}\right\} -\gamma b\left\{ \frac{t_1^3}{2}-\frac{\alpha \beta t_1^{3+\beta }}{2(\beta +2)}\right\} -c\delta (a+bt_1)\\&+c\left\{ (k-a)+bt_1\right\} +Qc_2\left\{ -1+T\beta t_1^{\beta -1}-\beta t_1^\beta \right\} \\&+\,ac_2\left[ T+3t_1+\frac{\alpha }{\beta +1}\left\{ (1+\beta )t_1^\beta (T-t_1)\right\} +\alpha \left\{ \frac{t_1^{1+\beta }}{\beta +1}- \frac{T^{1+\beta }}{\beta +1}\right\} \right] \\&+\,bc_2\left[ t_1(T-t_1)+\alpha t_1^{1+\beta }(T-t_1)+\frac{\alpha }{2}\left\{ t_1^\beta -t_1^{2+\beta }-\frac{2t_1T^{1+\beta }}{1+\beta }\right\} \right] \\&-\,s(1-\delta )(a+bt_1), \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 TC}{\partial t_1^2}= & {} (h+c)(k-a)\left( 1-\alpha \beta t_1^\beta \right) -(h+c)b\left( t_1-\frac{\alpha \beta t_1^{1+\beta }}{2}\right) \\&+\,\gamma (k-a)\left( 2t_1-\frac{\alpha \beta (2+\beta )t_1^{1+\beta }}{\beta +1}\right) -\gamma b\left\{ \frac{3t_1^2}{2}-\frac{\alpha \beta (3+\beta )t_1^{2 +\beta }}{2(\beta +2)}\right\} \\&+\,cb(1-\delta ) +Qc_2\left\{ T\beta (\beta -1)t_1^{\beta -2} -\beta ^2t_1^{\beta -1}\right\} +ac_2\left\{ 3+\alpha \beta t_1^{\beta -1}(T-t_1)\right\} \\&+\,bc_2\left[ (T-2t_1)+\alpha (1+\beta )t_1^\beta (T-t_1) -\alpha t_1^{1+\beta }+\frac{\alpha }{2}\left\{ \beta t_1^{\beta -1}-(2+\beta )t_1^{1+\beta }\right. \right. \\&\left. \left. -\,\frac{2T^{1+\beta }}{1+\beta }\right\} \right] -sb(1-\delta ), \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 TC}{\partial t_1\partial T}= & {} (h+c)(k-a)\left( \frac{\partial t_1}{\partial T}-\alpha \beta t_1^\beta \frac{\partial t_1}{\partial T}\right) -(h+c)b\left( t_1\frac{\partial t_1}{\partial T}-\frac{\alpha \beta t_1^{1+\beta }}{2}\frac{\partial t_1}{\partial T}\right) \\&+\gamma (k-a)\left\{ 2t_1\frac{\partial t_1}{\partial T}-\frac{\alpha \beta (2+\beta ) t_1^{1+\beta }}{(\beta +1)}\frac{\partial t_1}{\partial T}\right\} \\&-\,\gamma b\left\{ \frac{3t_1^2}{2}\frac{\partial t_1}{\partial T}-\frac{\alpha \beta (3+\beta ) t_1^{2+\beta }}{2(\beta +2)}\frac{\partial t_1}{\partial T}\right\} +cb(1-\delta )\frac{\partial t_1}{\partial T}\\&+\,Qc_2\left\{ \beta t_1^{\beta -1}+T\beta (\beta -1) t_1^{\beta -2}\frac{\partial t_1}{\partial T}-\beta ^2t_1^{\beta -1}\frac{\partial t_1}{\partial T}\right\} \\&+\,ac_2\left[ 1+3\frac{\partial t_1}{\partial T}+ \alpha \beta t_1^{\beta -1}(T-t_1)\frac{\partial t_1}{\partial T}+\alpha t_1^\beta \left( 1-\frac{\partial t_1}{\partial T}\right) \right. \\&\left. +\,\alpha \left( \frac{t_1^\beta }{\beta +1} \frac{\partial t_1}{\partial T}-T^\beta \right) \right] +bc_2\left[ (T-2t_1)\frac{\partial t_1}{\partial T}+\alpha (1+\beta )t_1^\beta (T-t_1)\frac{\partial t_1}{\partial T}\right. \\&+\,\alpha t_1^{1+\beta }\left( 1-\frac{\partial t_1}{\partial T}\right) +\frac{\alpha }{2}\left\{ \beta t_1^{\beta -1}\frac{\partial t_1}{\partial T}- (2+\beta )t_1^{1+\beta }\frac{\partial t_1}{\partial T}-\frac{2T^{1+\beta }}{1+\beta }\frac{\partial t_1}{\partial T}\right. \\&\left. \left. -\,2t_1T^\beta \right\} \right] -sb(1-\delta ) \frac{\partial t_1}{\partial T}, \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 TC}{\partial T^2}= & {} \frac{2}{T^3}\left[ A+(h+c)(k-a)\left\{ \frac{t_1^2}{2}-\frac{\alpha \beta t_1^{2+\beta }}{(\beta +1)(2+\beta )}\right\} -(h+c)b\left\{ \frac{t_1^3}{6}\right. \right. \\&\left. -\,\frac{\alpha \beta t_1^{3+\beta }}{2(\beta +2)(3+\beta )}\right\} +\gamma (k-a)\left\{ \frac{t_1^3}{3}-\frac{\alpha \beta t_1^{3+\beta }}{(\beta +1)(3+\beta )}\right\} \\&-\,\gamma b\left\{ \frac{t_1^4}{8}-\frac{\alpha \beta t_1^{4+\beta }}{2(\beta +2)(4+\beta )}\right\} +c\delta a(T-t_1)\\&+\,\frac{1}{2}c\delta b(T^2-t_1^2)+c\left\{ (k-a)t_1+\frac{bt_1^2}{2}\right\} \\&+\,Qc_2\left\{ (T-t_1) +\left( Tt_1^\beta -\frac{T^{\beta +1}}{\beta +1}-\frac{\beta t_1^{\beta +1}}{\beta +1}\right) \right\} \\&+\,ac_2\left[ \frac{2Tt_1+3t_1^2-T^2}{2} +\frac{\alpha }{\beta +1}\left\{ t_1^{1+\beta }(T-t_1)-\frac{T^{2+\beta }- t_1^{2+\beta }}{2+\beta }\right\} \right. \\&\left. +\,\alpha \left\{ \frac{T^{2+\beta } -t_1^{2+\beta }}{2+\beta }-\frac{t_1T^{1+\beta }-t_1^{2+\beta }}{\beta +1}\right\} \right] +bc_2\left[ \left\{ \frac{(T-t_1)t_1^2}{2}-\frac{T^3-t_1^3}{6}\right\} \right. \\&+\,\frac{\alpha }{2+\beta }\left\{ t_1^{2+\beta }(T-t_1)-\frac{T^{3+\beta }- t_1^{3+\beta }}{3+\beta }\right\} +\frac{\alpha }{2}\left\{ \frac{T^{3+\beta } -t_1^{3+\beta }}{3+\beta }\right. \\&\left. \left. \left. -\,\frac{t_1^2T^{1+\beta }-t_1^{1+\beta }}{\beta +1}\right\} \right] +s(1-\delta )\left\{ a(T-t_1)+\frac{1}{2}b\left( T^2-t_1^2\right) \right\} \right] \\&+\,\frac{1}{T}\left[ (h+c)(k-a)\left\{ \left( \frac{\partial t_1}{\partial T}\right) ^2+t_1\frac{\partial ^2 t_1}{\partial T^2}-\alpha \beta t_1^\beta \left( \frac{\partial t_1}{\partial T}\right) ^2-\frac{\alpha \beta t_1^{1+\beta }}{1+\beta }\frac{\partial ^2 t_1}{\partial T^2}\right\} \right. \\&-\,(h+c)b\left\{ t_1\left( \frac{\partial t_1}{\partial T}\right) ^2+\frac{t_1^2}{2}\frac{\partial ^2 t_1}{\partial T^2}-\frac{\alpha \beta t_1^{1+\beta }}{2} \left( \frac{\partial t_1}{\partial T}\right) ^2-\frac{\alpha \beta t_1^{2+\beta }}{2(2+\beta )}\frac{\partial ^2 t_1}{\partial T^2}\right\} \\&+\,\gamma (k-a)\left\{ 2t_1\left( \frac{\partial t_1}{\partial T}\right) ^2+t_1^2\frac{\partial ^2 t_1}{\partial T^2}-\frac{\alpha \beta (2+\beta )t_1^{1+\beta }}{1+\beta } \left( \frac{\partial t_1}{\partial T}\right) ^2-\frac{\alpha \beta t_1^{2+\beta }}{1+\beta }\frac{\partial ^2 t_1}{\partial T^2}\right\} \\&-\,\gamma b\left\{ \frac{3t_1^2}{2} \left( \frac{\partial t_1}{\partial T}\right) ^2+\frac{t_1^3}{2}\frac{\partial ^2 t_1}{\partial T^2}-\frac{\alpha \beta (3+\beta )t_1^{2+\beta }}{2(2+\beta )}\left( \frac{\partial t_1}{\partial T}\right) ^2- \frac{\alpha \beta t_1^{3+\beta }}{2(2+\beta )}\frac{\partial ^2 t_1}{\partial T^2}\right\} \\&-\,c\delta a\frac{\partial ^2 t_1}{\partial T^2}+c\delta b\left\{ 1-t_1\frac{\partial ^2 t_1}{\partial T^2}-\left( \frac{\partial t_1}{\partial T}\right) ^2\right\} +c\left\{ (k-a+bt_1)\frac{\partial ^2 t_1}{\partial T^2} +b\left( \frac{\partial t_1}{\partial T}\right) ^2\right\} \\&+\,Qc_2\left\{ 2\beta t_1^{\beta -1}\frac{\partial t_1}{\partial T}-\frac{\partial ^2 t_1}{\partial T^2}+T\beta (\beta -1) \left( \frac{\partial t_1}{\partial T}\right) ^2+T\beta t_1^{\beta -1}\frac{\partial ^2 t_1}{\partial T^2}\right. \\&\left. -\,\beta T^{\beta -1}-\beta ^2t_1^{\beta -1} \left( \frac{\partial t_1}{\partial T}\right) ^2-\beta t_1^\beta \frac{\partial ^2 t_1}{\partial T^2}\right\} +ac_2\left[ 2\frac{\partial t_1}{\partial T} +T\frac{\partial ^2t_1}{\partial T^2}\right. \\&+\,3\left( \frac{\partial t_1}{\partial T}\right) ^2+3t_1\frac{\partial ^2 t_1}{\partial T^2}-1 +\frac{\alpha }{\beta +1}\left\{ (1+\beta )\beta t_1^{\beta -1}(T-t_1)\left( \frac{\partial t_1}{\partial T}\right) ^2\right. \\&+\,(1+\beta )t_1^\beta \left( 2-\frac{\partial t_1}{\partial T}\right) \frac{\partial t_1}{\partial T}+(1+\beta )t_1^\beta \frac{\partial ^2 t_1}{\partial T^2}(T-t_1)+(1+\beta )t_1^\beta \frac{\partial t_1}{\partial T}\\&\left. -\,2t_1^{1+\beta }\frac{\partial ^2 t_1}{\partial T^2}\right\} +\alpha \beta T^\beta +\alpha \left\{ t_1^\beta \left( \frac{\partial t_1}{\partial T}\right) ^2-t_1^{1+\beta }\frac{\partial ^2 t_1}{\partial T^2}-2T^\beta \frac{\partial t_1}{\partial T}-t_1\beta T^{\beta -1}\right. \\&\left. \left. -\,\frac{T^{\beta +1}}{\beta +1}\frac{\partial ^2 t_1}{\partial T^2}+\frac{(2+\beta )t_1^{\beta +1}}{\beta +1}\frac{\partial ^2 t_1}{\partial T^2}\right\} \right] +bc_2\left[ \left\{ \left( \frac{\partial t_1}{\partial T}\right) ^2(T-t_1)\right. \right. \\&\left. \left. +\,t_1(T-t_1)\frac{\partial ^2 t_1}{\partial T^2}+t_1 \frac{\partial t_1}{\partial T}\left( 1-\frac{\partial t_1}{\partial T}\right) -T\right\} + \alpha (1+\beta )t_1^\beta (T-t_1)\left( \frac{\partial t_1}{\partial T}\right) ^2\right. \\&+\,\alpha t_1^{1+\beta }(T-t_1)\frac{\partial ^2 t_1}{\partial T^2}- \alpha t_1^{1+\beta }\left( \frac{\partial t_1}{\partial T}\right) ^2+2\alpha t_1^{1+\beta }\frac{\partial t_1}{\partial T}-\alpha T^{1+\beta }\\&+\frac{\alpha }{2}\left\{ (2+\beta )T^{1+\beta }-t_1^{2+\beta }\frac{\partial ^2 t_1}{\partial T^2}-(2+\beta )t_1^{1+\beta }\frac{\partial ^2 t_1}{\partial T^2} -2t_1 \frac{\partial t_1}{\partial T}-\frac{2T^{1+\beta }}{1+\beta }\frac{\partial ^2 t_1}{\partial T^2}\right. \\&\left. \left. -\,\frac{2t_1T^{1+\beta }}{1+\beta } \left( \frac{\partial t_1}{\partial T}\right) ^2-2t_1T^\beta \frac{\partial t_1}{\partial T}-t_1^2 \beta T^{\beta -1}+\beta t_1^{\beta -1}\left( \frac{\partial t_1}{\partial T}\right) ^2 +t_1^\beta \frac{\partial ^2 t_1}{\partial T^2}\right\} \right] \\&\left. +\,s(1-\delta )\left\{ b\left( 1-\left( \frac{\partial t_1}{\partial T}\right) ^2-t_1 \frac{\partial ^2 t_1}{\partial T^2}\right) -a\frac{\partial ^2 t_1}{\partial T^2}\right\} \right] . \end{aligned}$$

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Pervin, M., Roy, S.K. & Weber, GW. Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration. Ann Oper Res 260, 437–460 (2018). https://doi.org/10.1007/s10479-016-2355-5

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