The Markov model for base-stock control of an inventory system with Poisson demand, non-crossing lead times and lost sales

Highlights

• Extension of a previous Markov model for base-stock control with non-crossing Erlangian lead times (LTs).

• Our extended Erlangian LTs are specified only by their mean and standard deviation (SD).

• Our Markov model and its Gauss–Seidel algorithm suffer from the curse of dimensionality.

• We present a reasonable and well-performing base-stock policy, which is easy to compute.

• The average cost goes up as SD goes up, which is in sharp contrast to independence of SD when the LTs are i.i.d.

Abstract

We study base-stock control of a continuous review single-item inventory system with Poisson demand and lost sales. The item is supplied by an exogenous and sequential system with stochastic lead times (LTs) specified by their mean and standard deviation (SD). Define $R$ as the square of mean/SD and define $r$ as the smallest integer which is at least $R$. When $R<r$, we introduce a novel tractable LT distribution called Extended Erlangian. When $R=r$ then Extended Erlangian is the same as pure Erlangian and the simple Markov model presented by Johansen (2005) makes it easy to compute and minimize the long-run average cost per unit time. We present an algorithm to compute the steady state distribution of a new Markov model for base-stock control with Extended Erlangian LTs. For fixed base-stock $S$, this algorithm can be applied to compute the average cost and it is straightforward (but burdensome for large $r$) to compute the optimal $S$ and the minimum average cost. We also suggest a reasonable $S$ computed in closed form from simple models with pure Erlangian LTs. The reasonable $S$ is easy to compute and it performs well. Our numerical study illustrates that the average cost goes up as SD goes up, which is in sharp contrast to independence of SD when the LTs are i.i.d.

Introduction

We consider a single-item inventory system with Poisson demand, lost sales, negligible set-up costs and continuous review. Karush (1957) and Zipkin (2000, Section 7.2.3) show for this system that Erlang’s loss formula, see the end of Appendix B, can be applied to compute the optimal base-stock when the lead times are independent and identically distributed (i.i.d.). Then it is also straightforward to allow compound Poisson demand (Kouki et al., 2019). This reference shows how to compute and minimize the average cost per unit time exactly in the complete rejection case (loss of customer orders which cannot be satisfied completely) and approximately in the partial rejection case (loss of the part of a customer order which cannot be satisfied immediately). However, rather than assuming i.i.d. lead times and allowing compound Poisson demand, we assume that the lead times are so that replenishment orders are delivered in the same sequence as they were issued. Such non-crossing lead times are most common in practice (Axsäter, 2015 page 99) and we let them be generated by an Exogenous and Sequential Supply System (ESSS) as suggested by Zipkin, 1986, Zipkin, 2000.

The evolution of the ESSS is independent of our demand and orders because their contribution to the supplier’s overall workload is assumed to be negligible, which is often realistic. We can illustrate this and lost sales by small service departments (or companies) who provide maintenance service of failed equipment with one or more brand-specific components having a single supplier, who behaves like an ESSS. If such components are out of stock when needed to replace a broken one, then they are changed to a different available component, which is applicable and more expensive. Karush (1957) describes an OR study of the distribution of automobile replacement parts through a nation-wide system of warehouses. The eventual solution for this lost-sales system was obtained through simulation because it turned out to be necessary to correlate the random lead times of successive replenishment orders.

We represent the ESSS by a continuous-time Markov chain with $r$ phases numbered from 1 to $r$. This chain is illustrated for $r=4$ in Fig. 1. We assume that

• the time spent in Phase $j$ is exponential with mean denoted $m\left(j\right)$

• the phase times are mutually independent

• the phases are visited in rotation.

For a replenishment order issued when the actual phase of the ESSS is Phase $j$, the lead time is the sum of the remaining time (being exponential with mean $m\left(j\right)$) in this phase and the times spent in the other $r-1$ phases before the next visit into Phase $j$. Hence, the replenishment orders are delivered in the same sequence as they were issued and all lead times have the same distribution. It is Generalized Erlangian of order $r$ ($G{E}_r$), see Neuts (1981, page 46) and Bladt and Nielsen (2017, page 128). It is also known as the hypo-exponential distribution (Lipsky, 2009 page 152). Observe that Tijms (2003, pages 209 and 444) defines a Generalized Erlangian distribution differently. According to him it is a mixture of Erlangian distributions with the same scale parameters. His definition excludes that more than one exponential phase can be applied to describe an ESSS.

Our generic variable for non-crossing lead times is $L$. When $L$ is $G{E}_r$, the mean is

$$\bar{L}=\sum _{j=1}^{r}m\left(j\right),$$

the variance is

$${\sigma }^2=\sum _{j=1}^{r}m{\left(j\right)}^2$$

and $\sigma \le \bar{L}$. For given values of $\bar{L}$ and $\sigma \le \bar{L}$, we define $R={\bar{L}}^2∕{\sigma }^2$ and let $\hat{r}$ be the smallest integer which is at least $R$. As a special case of $G{E}_{\hat{r}}$ with mean $\bar{L}$ and standard deviation $\sigma$, we introduce the novel Extended Erlangian of order $\hat{r}$ ($E{E}_{\hat{r}}$) specified by

$$m\left(j\right)=\left\{\begin{array}{cc}{\hat{m}}_{\bar{L},\sigma },\hfill & j=1,2,\dots ,\hat{r}-1\hfill \\ \bar{L}-(\hat{r}-1){\hat{m}}_{\bar{L},\sigma },\hfill & j=\hat{r},\hfill \end{array}\right.$$

where

$${\hat{m}}_{\bar{L},\sigma }=\frac{\bar{L}+\sqrt{\frac{\hat{r}{\sigma }^2-{\bar{L}}^2}{\hat{r}-1}}}{\hat{r}}.$$

In particular when $R$ is an integer (which is equal to $\hat{r}$), then all phase-means equal $\bar{L}∕\hat{r}$ and $E{E}_{\hat{r}}$ is the same as pure Erlangian of order $\hat{r}$ ($E_{\hat{r}}$). We introduce $E{E}_{\hat{r}}$ as a tractable distribution of $L$ when $R$ is not an integer. Observe for $\sigma =\bar{L}$ that $\hat{r}=1$ and that $E_1$, $E{E}_1$ and $G{E}_1$ identify the same exponential distribution. If $\bar{L}∕\sqrt{2}\le \sigma <\bar{L}$ then $\hat{r}=2$, and $E{E}_2$ and $G{E}_2$ identify the same distribution. If $0<\sigma <\bar{L}∕\sqrt{2}$ then $E{E}_{\hat{r}}$ is unique and it identifies the same distribution as $E_{\hat{r}}$ and $G{E}_{\hat{r}}$ if $R$ is an integer. If $R$ is not an integer then $G{E}_{\hat{r}}$ is not unique.

We present a new continuous-time Markov Model (MM) for base-stock control of the considered inventory system when $L$ is $G{E}_r$, but in this paper we make use of this model only in the special case where $L$ is $E{E}_{\hat{r}}$. As a function of time the MM describes the actual phase of the ESSS and how many single-unit orders we have issued when the ESSS latest was in each of the $r$ phases. Observe that the actual phase of the ESSS can be observed only if $r=1$ (implying that $L$ is exponential). However, missing state observation for $r>1$ does not exclude that the steady state distribution of the MM can be computed numerically by a Gauss–Seidel algorithm, which can be applied to compute the long-run average cost per unit time. Because the average cost is a convex function of the base-stock, it is straightforward (but very burdensome for large $r$) to compute the optimal base-stock and the minimum average cost.

Johansen (2005) has studied the special setting where $L$ is $E_{\hat{r}}$. The continuous-time Markov model (described in Appendix B) for this setting is much simpler than the new MM. His model can be used for any base-stock to obtain a closed-form for the equilibrium distribution of the number of single-unit replenishment orders outstanding. This expression makes it easy to compute and minimize the long-run average cost per unit time. His numerical study illustrates for fixed $\bar{L}$ that the minimum average cost is a clearly increasing function of the allowed values of $\sigma$. The same result is obtained in this paper where any $\sigma$ from 0 to $\bar{L}$ is allowed. The result is in sharp contrast to complete independence of $\sigma$ when the lead times are i.i.d.

Our MM and the Gauss–Seidel algorithm suffer from the curse of dimensionality as the number $r$ of phases and the base-stock increase. Therefore, when $R={\bar{L}}^2∕{\sigma }^2$ is large because $\sigma$ is small but positive, then the curse of dimensionality might kick in. It does not happen if $R$ is an integer because then $\hat{r}=R$ and it is easy to compute the optimal base-stock for $E_{\hat{r}}$ lead times. However, it happens if a large $R$ is not an integer, implying that $\hat{r}-1<R<\hat{r}$. Then we suggest a reasonable base-stock computed from the two simple models where $L$ for unchanged $\bar{L}$ is changed to be $E_{\hat{r}-1}$ and $E_{\hat{r}}$. Our reasonable base-stock is easy to compute, it performs well and it facilitate exact computation of the optimal base-stock. When $\sigma =0$ then $L$ equals $\bar{L}$ and Erlang’s loss formula can be applied to compute the optimal base-stock.

Hill (1999) and Johansen, 2001, Johansen, 2013 explain for constant lead times that the global minimum average cost cannot be found in the class of policies specified by a base-stock for the inventory position (stock on hand + stock on order). However, such policies are attractive in practice because they are inexpensive to implement and, as noted by Kouki et al. (2019), they allow a tractable analysis of more complex systems, such as multi-echelon and multi-item systems.

Riezebos (2006) concludes that a necessary but not sufficient condition for order crossovers to occur is the existence of lead time variability. Our MM is designed to cope with continuous review, lost sales and variability originating from one ESSS, whereas Kouki et al. (2019) let i.i.d. lead times cause order crossovers. Bischak et al. (2014), Disney et al. (2016) and many of their references assume periodic review, back-ordering and order crossings. Disney et al. present a case where almost 40% of orders cross between a supplier in Colorado, USA and a customer in Shenzhen, China. However, their paper measures and illustrates graphically crossings as change in queue positions. We doubt that the measure applied in the paper provides useful information for decision making because the time dimension is neglected.

The paper is organized as follows. Our MM with $G{E}_r$ lead times and our Gauss–Seidel algorithm designed to compute its equilibrium probabilities are described in Section 2. Here we also derive expressions for the average cost and the cost-minimal base-stock ${\hat{S}}_{\sigma }$. Based on the similar expressions for the simple model with $E_r$ lead times, we present the reasonable base-stock ${\tilde{S}}_{\sigma }$. Our numerical study is presented in Section 3. Section 4 has concluding remarks. Appendix A specifies various mappings used in Section 2.1 and Appendix B presents the equilibrium distribution of the Markov model with $E_r$ lead times.