Cost minimising order schedules for a capacitated inventory system

Abstract

In this paper, we study an inventory system with multiple retailers under periodic review and stochastic demand. The demand is modelled as a discrete random variable. Linear holding and backorder costs as well as fixed order costs are assumed. Orders to replenish inventories can be placed at a manufacturer with a limited capacity according to a cyclic order schedule. A fixed portion of the total available capacity in a period is allocated to each retailer, who follows a modified base-stock policy to determine the order quantities. Thus, the order policy consists of four policy parameters for each retailer: the length of the review period, the first order point within a planning horizon, the individual capacity limit, and the modified base-stock level. We present an algorithm to compute the exact optimal policy parameters and two heuristics. In a numerical study, we compare the results of these approaches and derive insights into the performance of the heuristics. In addition, we introduce three different schedule types and identify the situations, in which they perform best.

Appendices

Appendix I: Derivation of the cost functions

1.1 The exact cost function

In order to be able to compute the average stock on hand or backorders, we need information about the distribution of the inventory position, which can be obtained using a Markov chain approach. The state space of the Markov chain is given by the undershoot $U_t$, which is defined as the difference between the modified base-stock level $S_m$ and the inventory position $IP_m$ at the beginning of a period $t$ after an order has been placed. The dynamic of the system is described by the following recursive equation for the undershoot:

$$\begin{aligned} U_{t} = \left\{ \begin{array}{ll} 0 &{} \quad U_{t-R_m} +D_{m,R_m} \le CP_m\\ U_{t-R_m} +D_{m,R_m} - CP_m &{}\quad U_{t-R_m} +D_{m,R_m} > CP_m\\ \end{array} \right. \end{aligned}$$

(21)

The equilibrium distribution $\pi$ defined as

$$\begin{aligned} \lim _{t \rightarrow \infty } P(U_{t}=u) =: \pi _u \; \forall \; u \in \mathbb {N}_0 \end{aligned}$$

(22)

can be computed by solving the system of equations

$$\begin{aligned} \begin{array}{l} \pi _u = \sum \nolimits _{\check{u} = 0,1,2, \ldots } \pi _{\check{u}} \cdot p_{\check{u},u} \; \forall \; u \in \mathbb {N}_0\\ \sum \nolimits _{u=0,1, \ldots } \pi _u = 1\\ \end{array} \end{aligned}$$

(23)

where $P$ denotes the transition matrix given as

$$\begin{aligned} \begin{array}{l} P = (p_{i,j}) \text{ with } p_{i,j} = P(U_{t} = j| U_{t-R_m} = i)\\ \end{array} \end{aligned}$$

(24)

With Eq. (2), this results in the following expression for the average holding and backorder cost:

$$\begin{aligned} C_m(CP_m,R_m,S_m)&= \frac{1}{R_m} \sum _{u=0,1, \ldots } \pi _u \sum ^{R_m}_{r=1} \Big (h \cdot \sum ^{S_m-u}_{i=0} (S_m-u-i)f_{m,L+r}(i) \nonumber \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad +\,\,b \cdot \sum ^{\infty }_{i=S_m-u} (i-(S_m-u))f_{m,L+r}(i)\Big )\qquad \end{aligned}$$

(25)

For the given values of $CP_m, R_m$, the cost function depends on the modified base-stock level $S_m$ only. The optimal modified base-stock level $S^*_m$ then has to satisfy the following Newsboy like condition:

$$\begin{aligned} S^*_m = \arg \min _{S_m} \left\{ \frac{b_m}{h+b_m} \le \sum _u \frac{\pi _u}{R_m} \sum ^{R_m}_{r=1} F_{m,L+r}(S_m-u)\right\} \end{aligned}$$

(26)

1.2 The approximated cost function

During the optimisation, many combinations of policy parameters have to be evaluated. Since for each $(R_m,CP_m)$ combination a system of equations has to be solved to determine $S_m$, this may result in long computation times. Therefore, in some parts of the heuristic approaches, we use an approximated cost function where the Eq. (21) for the undershoot is replaced by

$$\begin{aligned} U_{t} = (D_{m,R_m} - CP_m)^+ \end{aligned}$$

(27)

and the convolution of the resulting probability mass function with the probability mass function for the period demand is replaced by a two-moment fit of these. Both assumptions correspond to those found in Kleintje-Ell and Kiesmüller (2013). With the fitted probability mass function $f_{fit}$, the cost function results in:

$$\begin{aligned} \widetilde{C_m}(CP_m,R_m,S_m)&= \frac{1}{R_m} \sum ^{R_m}_{r=1}\left\{ \sum ^{S_m}_{x=0}(S_m-x) f_{fit}(x)\right. \nonumber \\&\left. +\,\, \sum ^{\infty }_{x=S_m}(x-S_m)f_{fit}(x)\right\} \end{aligned}$$

(28)

Appendix II Upper bound for $R_m$

In the following section, we show why $(E[D_i]+1)\cdot R_m$ is the lowest capacity for the other retailers that needs to be taken into account of all schedules $X$ with any length $T$ containing the order interval $R_m$ for retailer $m$. The following distinction of cases for the possible order intervals of the retailers $i \ne m$ demonstrates this:

$R_i < R_m$ :

We determine the highest possible number of orders of retailer $i$ for an interval $R_m: count:= \max _{\bar{t}<T} \sum ^{\bar{t}+R_m}_{t=\bar{t}} x_{i,t}$. In this case, it holds that: $R_m \cdot (E[D_i]+1)\le count \cdot R_i \cdot (E[D_i]+1)$ as $count \ge \frac{R_m}{R_i}$.

$R_i = R_m$ :

We assume the correct minimum capacity with $(E[D_i]+1)\cdot R_m$.

$R_m<R_i$ :

If there exists an order interval of length $R_m$, in which both retailer $m$ and retailer $i$ place an order, the capacity limit of a retailer is the same at each of his orders. In this case, it holds that $R_m \cdot (E[D_i]+1) < R_i \cdot (E[D_i]+1)$ and we even underestimate the amount required by retailer $i$.

$R_i,R_j \ge 2 \cdot R_m\; , \; R_i = R_j$ :

There may be certain combinations of order intervals in these cases, such that the three retailers $m,i,j$ never order all together in an interval of length $R_m$, but rather in intervals in which $i$ orders, the distance to the order points of $m$ are the same as for $j$ in the corresponding intervals. If we assume w.l.o.g. $E[D_i]\le E[D_j]$, it holds that $(E[D_i]+1)\cdot R_m + (E[D_j]+1)\cdot R_m \le (E[D_j]+1)\cdot 2 \cdot R_m \le (E[D_j]+1)\cdot R_j$ and in these cases as well, the assumed capacity does not overestimate the amount needed by other retailers in an interval of length $R_m$.