A state-dependent perishability (s, S) inventory model with random batch demands

Abstract

We consider continuous-review perishable inventory models with random lead times and state-dependent Poisson demand. The paper revises an earlier work of Barron and Baron (IISE Trans 1–52, 2019). While the former studies unit Poisson demands, this paper deals with demand uncertainty and allows for random batch demands. We conduct a comprehensive analysis of two main models that have different lead times and perish times under backorders or lost sales. Thus, our models can be applied to many industries, in situations where the system is subject to random perishability, random lead time, and demand uncertainty. With a probabilistic approach, we derive a long-run average cost function under the (S, s) replenishment policy. Numerical examples are used to demonstrate the impact of changing batch size and other system parameters on the optimal policy. Our numerical study indicates that, although the Markovian policy can be used as a good approximation of the average total cost, it performs better for a general perish time. We further show that the optimal cost may differ for a different average batch size, while the batch variability seems to provide some robustness.

Appendices

Appendix

A. The steady-state probabilities for the general perish time

The steady-state probabilities $\pi _{n},n\le S$ for the general perish time model are given by

$$\begin{aligned} \underset{s\ge 0}{\pi _{n}}=\left\{ \begin{array} [c]{lll} \mathrm{(i)}&{} \frac{1-F^{*}(\lambda _{S})}{\lambda _{S}E(C)}, &{}\quad n=S,\\ \mathrm{(ii)}&{}\sum \limits _{i=1}^{\min (\S ,S-n)}\frac{\pi _{n+i}\lambda _{n+i}p_{_{D}}(i)\left[ 1-R_{n+i}^{*}(\lambda _{n})\right] }{\lambda _{n} }, &{}\quad s<n<S,\\ \mathrm{(iii)}&{}\sum \limits _{i=1}^{\min (\S ,S-n)}\frac{\pi _{n+i}\lambda _{n+i}p_{_{D}}(i)\left[ 1-R_{n+i}^{*}(\lambda _{n}+\xi )\right] }{\lambda _{n}+\xi }, &{}\quad 0<n\le s,\\ \mathrm{(iv)}&{}\frac{\sum \nolimits _{i=1}^{\min (\S ,S)}\pi _{i}\lambda _{i} P_{D}(i)+\frac{1}{E(C)}F^{*}(\lambda _{S})+\sum \nolimits _{i=s+1}^{S-1} \sum \nolimits _{k=1}^{\min (\S ,S-i)}\pi _{i+k}\lambda _{i+k}p_{_{D}}(k)R_{i+k} ^{*}\left( \lambda _{i}\right) }{\lambda _{0}+\xi } &{} \\ &{}\qquad +\frac{\sum \nolimits _{i=1}^{s}\sum \nolimits _{k=1} ^{\min (\S ,S-i)}\pi _{i+k}\lambda _{i+k}p_{_{D}}(k)R_{i+k}^{*}\left( \lambda _{i}+\xi \right) }{\lambda _{0}+\xi }, &{}\quad n=0,\\ \mathrm{(v)}&{} \frac{\sum \nolimits _{i=1}^{\min (\S ,S-n)}\pi _{n+i}\lambda _{n+i}p_{_{D}}(i)}{\lambda _{n}+\xi }, &{}\quad n<0. \end{array} \right. \nonumber \\ \end{aligned}$$

(A.1)

For $s\le 0$, case (iii) is omitted and the numerator of case (iv) becomes

$$\begin{aligned} \sum \limits _{i=1}^{\min (\S ,S)}\pi _{i}\lambda _{i}P_{D}(i)+\frac{1}{E(C)}F^{*}(\lambda _{S})+\sum \limits _{i=s+1}^{S-1}\sum \limits _{k=1} ^{\min (\S ,S-i)}\pi _{i+k}\lambda _{i+k}p_{_{D}}(k)R_{i+k}^{*}\left( \lambda _{i}\right) . \end{aligned}$$

In addition, for $s<0$, the denominator of case (iv) is replaced by $\lambda _{n}$ (instead of $\lambda _{0}+\xi$) and case (v) becomes

$$\begin{aligned} \begin{array} [c]{lll} \mathrm{(v)}&{} \frac{1}{\lambda _{n}}\left( \sum \limits _{i=1}^{\min (\S ,S-n)}\pi _{n+i}\lambda _{n+i}p_{_{D}}(i)\right) , &{} \quad s<n<0,\\ \mathrm{(vi)}&{} \frac{1}{\lambda _{n}+\xi }\left( \sum \limits _{i=1} ^{\min (\S ,S-n)}\pi _{n+i}\lambda _{n+i}p_{_{D}}(i)\right) , &{} \quad n\le s. \end{array} \end{aligned}$$

(A.2)

Note that for $n<0$ the steady-state probability can be directly derived from the analysis of the Markov chain by applying (13) (v).

B. The steady-state probabilities for the general lead time

The steady-state probabilities $\pi _{n},n\le S$ for the general lead time model are given by

$$\begin{aligned} \underset{s>0}{\pi _{n}}=\left\{ \begin{array} [c]{lll} \mathrm{(i)}&{}\frac{1}{\lambda _{S}+\mu }\frac{1}{E(C)}, &{} \quad n=S,\\ \mathrm{(ii)}&{}\frac{\sum \nolimits _{i=1}^{\min (\S ,S-n)}\lambda _{n+i}\pi _{n+i}p_{_{D}}(i)}{\lambda _{n}+\mu }, &{} \quad s<n<S,\\ \mathrm{(iii)}&{} \frac{\sum \nolimits _{i=1}^{\min (\S ,S-n)}\lambda _{n+i}\pi _{n+i}p_{_{D}}(i)\left( 1-R_{n+i}^{*}(\lambda _{n}+\mu )\right) }{\lambda _{n}+\mu }, &{} \quad 0<n\le s,\\ \mathrm{(iv)}&{} \frac{\sum \nolimits _{i=1}^{\min (\S ,S)}\lambda _{i}\pi _{i}p_{_{D} }(i)\left( 1-R_{i}^{*}(\lambda _{0})\right) +\sum _{i=1}^{s}\pi _{i} \mu (1-R_{i}^{*}(\lambda _{0}))+\sum _{i=s+1}^{S}\pi _{i}\mu (1-G^{*} (\lambda _{0}))}{\lambda _{0}} &{} \\ &{}\qquad =\frac{1-\overline{R}^{*}(\lambda _{0})}{\lambda _{0} }\left( \sum \limits _{i=1}^{\min (\S ,S)}\pi _{i}\lambda _{i}p_{_{D}}(i)+ {\textstyle \sum \nolimits _{i=1}^{S}} \mu \pi _{i}\right) , &{} \quad n=0,\\ \mathrm{(v)}&{}\frac{\sum \nolimits _{i=1}^{\min (\S ,S-n)}\lambda _{n+i}\pi _{n+i}p_{_{D}}(i)\left( 1-R_{n+i}^{*}(\lambda _{n})\right) }{\lambda _{n} }. &{} \quad n<0, \end{array} \right. \nonumber \\ \end{aligned}$$

(B.1)

For $s=0$, case (iii) is omitted, and case (iv) becomes

$$\begin{aligned} \frac{\sum \nolimits _{i=1}^{\min (\S ,S)}\lambda _{i}\pi _{i}p_{_{D}}(i)\left( 1-R_{i}^{*}(\lambda _{0})\right) +\sum _{i=1}^{S}\pi _{i}\mu (1-G^{*}(\lambda _{0}))}{\lambda _{0}}. \end{aligned}$$

For $s<0$, we have similar equations for cases (i) and (ii), case (iii) is omitted, the region $s<n\le 0$ is derived from the Markov chain analysis, and case (v) holds for $n\le s$.

C. Selected studies on continuous review (S, s) inventory policy with perishability

In the following table, we summarize the most important literature studies concerning the (S, s) policy with perishability.

Authors	Demand dist.	Lead time dist.	Life time dist.	Policy
Weiss (1980)	Poisson	Zero	Fixed	Backorder, lost sales
Kalpakam and Sapna (1994)	Poisson	Exp	Exp	Lost sales
Kalpakam and Sapna (1996)	Renewal process	Exp	Exp	$(S,S-1)$ policy, lost sales
Liu and Lian (1999)	Renewal process	Zero	Fixed	Backorder
Liu and Yang (1999)	Poisson	Exp	Exp	Backorder
Lian and Liu (2001)	Renewal process general batch size	Zero	Fixed	Backorder
Gürler and Özkaya (2008)	Renewal process	Zero	General	Backorder
Baron et al. (2010)	Compound poisson	Zero	Exp	Policy, lost sales
Kouki et al. (2016)	Poisson	Zero	General	Backorder
Baron et al. (2017)	State-dependent Compound Poisson	Exp	Exp	Lost sales
Barron and Baron (2019)	State-dependent Poisson	General L.T. and exp life time or exp lead time and general life time		Backorder, key steps for lost sales