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When Using the Continuous Review System Stock Outs Can Occur

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Abstract

This paper examines a continuous review inventory model for perishable items with two demand classes. Demands for both classes occur according to Poisson process. The items in inventory are perishable products and have exponential lifetimes. The time after placing an order is an exponential random variable. When the on-hand inventory drops to pre-specified level s, only the priority customer demands are met whereas the demands from ordinary customers are lost. And also, the demand occurring stock-out periods are lost. The inventory system is characterized by continuous-time Markov process and steady-state probabilities are derived. The expected cost function is formulated and a numerical study is provided for optimization.

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Acknowledgements

We would like to thank the anonymous referees who have improved our work with their invaluable comments.

Author information

Authors and Affiliations

  1. Department of Statistics, Faculty of Science, Dokuz Eylul University, Kaynaklar Campus Tinaztepe-Buca, Izmir, Turkey

    Umay Uzunoglu Kocer & Bahar Yalcin

Corresponding author

Correspondence to Umay Uzunoglu Kocer.

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Appendix

Appendix

The pseudo-convex functions have many properties of convex functions, particularly about derivative properties and finding local minima. If the total cost function \( TC(s+1,Q)-TC(s,Q)\) is non-positive, then the total cost function values are non-increasing of s; that is the total cost function is pseudo-convex.

Proof of Theorem 1

For a fixed Q, let the first difference \(TC(s+1,Q)-TC(s,Q)\) be \(\frac{N(s)}{D(s)}\).

$$\begin{aligned} N(s)=\left( h+g_{1} \gamma \right) N_{1}(s)+(K+cQ) N_{2}(s)+ N_{3}(s)+N_{4}(s) \end{aligned}$$

where

$$\begin{aligned} N_{1}(s)&= \mu ^{2} \left[ ( \lambda _{2}+Q \gamma ) VY+X( \lambda _{1}+QV- \lambda _{2} W) \right] +\mu Y \\ &\quad+\, C \mu \left\{ V(s+1)+WQ \left[ \lambda _{1}+(s+1)\gamma \right] +A_{s+2} \left[ (s+1)X-Y \right] \right\} \\ N_{2}(s)&= \mu \left[ 1+\mu V(\lambda _{2}+Q \gamma )-C A_{s+2}\right] \\ N_{3}(s)&= -\,g_{2} \lambda _{1}( \mu X+C) \\ N_{4}(s)&= g_{3} \lambda _{2}\left[ 1+\mu V(\lambda _{2}+Q \gamma )+\mu A_{s+2}X-(1+\mu Z)(\mu X +C)\right] \end{aligned}$$

and

$$\begin{aligned} X&= \sum _{n=s+2}^{Q+s+1} \frac{1}{\lambda _{1}+\lambda _{2}+n \gamma }\\ Y&= \sum _{n=s+2}^{Q+s+1} \frac{n}{\lambda _{1}+\lambda _{2}+n \gamma }\\ V&= \sum _{n=1}^{s+1} \frac{1}{(\lambda _{1}+n \gamma )\left[ \lambda _{1}+\lambda _{2}+(Q+n) \gamma \right] } A_{n}\\ W&= \sum _{n=1}^{s+1} \frac{n}{(\lambda _{1}+n \gamma )\left[ \lambda _{1}+\lambda _{2}+(Q+n) \gamma \right] } A_{n}\\ Z&= \sum _{n=1}^{s+1} \frac{1}{(\lambda _{1}+n \gamma )} A_{n}\\ C&= \frac{\lambda _{2}}{\lambda _{1}+\lambda _{2}+(s+1) \gamma } \end{aligned}$$

The dominator D(s) is always positive. To prove that TC(s,Q) is pseudo-convex, it sufficies to show that N(s) is an increasing function. Let \(\bigtriangleup N(s)=N(s+1)-N(s)\). The difference is also given in four parts.

Let \(\bigtriangleup N_{1}(s)=N_{1}(s+1)-N_{1}(s)\); and it is given as

$$\begin{aligned}&\bigtriangleup N_{1}(s)=\mu A_{s+2}X\lambda _{2}Q \gamma (\lambda _{1}+\lambda _{2}+\mu (s+1)) \\&\quad +\frac{\mu A_{s+2}X}{\lambda _{1}+(s+2)\gamma }\left\{ \lambda _{2}Q \mu \gamma (\lambda _{1}+\lambda _{2}+\lambda _{2}(s+1))+\lambda _{2}(\lambda _{1}+\lambda _{2})M4+\lambda _{1}Q \mu M3 \right\} \\&\quad +\mu \gamma A_{s+2}Y \left\{ Q \mu \left[ \lambda _{1}+\lambda _{2}+(s+1)\gamma \right] +\lambda _{2}\left[ \lambda _{1}+\lambda _{2}+(Q+s+2)\gamma \right] \right\} \\&\quad +\mu V\left\{ \frac{Q \mu (\lambda _{2}+Q \gamma )(\lambda _{1}+\lambda _{2})}{M1} +\frac{\lambda _{2}^{2}(\lambda _{1}+\lambda _{2}+Q \gamma )}{M3} \right\} \\&\mu \lambda _{2} \gamma W\left( \frac{Q \mu }{M1}+\frac{\lambda _{2}}{M3}\right) +(\lambda _{1}+\lambda _{2})\left( \frac{Q}{M1}+\frac{\lambda _{2}}{M3}\right) \\&\bigtriangleup N_{2}(s)=\gamma A_{s+2}\left[ \frac{Q \mu +\lambda _{2}\left[ \lambda _{1}+\lambda _{2}+(Q+s+2)\gamma \right] }{M1} \right] \\&\bigtriangleup N_{3}(s)= \frac{Q \mu \gamma }{M1}+\frac{\gamma }{M3} \\&\bigtriangleup N_{4}(s)= \mu \gamma \left[ \frac{Q \mu }{M1}+\frac{1}{M3} \right] Z +\frac{Q \mu \gamma (A_{s+2}+1)}{M1}+\frac{\gamma }{M3} \end{aligned}$$

where

$$\begin{aligned} M1&= \left[ \lambda _{1}+\lambda _{2}+(s+2)\gamma \right] \left[ \lambda _{1}+\lambda _{2}+(Q+s+2)\gamma \right] \\ M2&= \left[ \lambda _{1}+(s+2)\gamma \right] \left[ \lambda _{1}+\lambda _{2}+(Q+s+2)\gamma \right] \\ M3&= \left[ \lambda _{1}+\lambda _{2}+(s+1)\gamma \right] \left[ \lambda _{1}+\lambda _{2}+(s+2)\gamma \right] \\ M4&= \left[ \lambda _{1}+(s+2)\gamma \right] \left[ \lambda _{1}+\lambda _{2}+(s+2)\gamma \right] \end{aligned}$$

Then

$$\begin{aligned} \bigtriangleup N(s)=(h+g_{1})\bigtriangleup N_{1}(s)+(K+cQ)\mu \bigtriangleup N_{2}(s) +g_{2}\lambda _{1} \bigtriangleup N_{3}(s)+g_{3}\lambda _{2} \bigtriangleup N_{4}(s) \end{aligned}$$

Since \(\bigtriangleup N(s) > 0\), N(s) is an increasing function of s. Therefore TC(s,Q) is pseudo-convex. \(\square \)

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Uzunoglu Kocer, U., Yalcin, B. Continuous review (s, Q) inventory system with random lifetime and two demand classes. OPSEARCH 57, 104–118 (2020). https://doi.org/10.1007/s12597-019-00393-0

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Keywords

  • Markov process
  • Perishable inventory
  • Two demand classes
  • Random lifetime
  • Random lead time

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