# Performance Indicators, Service Levels, Waiting time

**This page provides a summary of problems** **discussed** in the book

In the literature several variations of inventory-related performance measures are discussed which may differ not only with respect to their scope and to the number of considered products but also with respect to the time interval they are related to. These performance measures are the "Key Performance Indicators" (KPI) of an inventory node which must be regularly monitored. If the controlling of the performance of an inventory node is neglected, the decision maker will not be able to optimize the processes within a supply chain.

In the sequel we first discuss several performance measures related to a single
product. After that we consider the situation when the performance of an
inventory node is an aggregate construct related to the simultaneous delivery
of a group of products. Primarily because of their complexity, product
group-specific performance measures are rarely discussed in the literature.
Nevertheless, criteria of this kind are relevant in assemble-to-order
production systems when upon arrival of a customer order the product is
assembled from several components. In addition, in the case when a customer
order includes several lines defining demands for different products, the
customer may require the simultaneous delivery of all products within a single
delivery event. In these situations performance measures based on isolated
products are over-optimistic.

### $\alpha$ service level

The $\alpha$ service level is an event-oriented performance criterion. It measures the probability that all customer orders arriving within a given time interval will be completely delivered from stock on hand, i.e. without delay.

Two versions are discussed in the literature differing with respect to the time interval within which the customers arrive. With reference to a demand period, $\alpha$ denotes the probability that an arbitrarily arriving customer order will be completely served from stock on hand, i.e. without an inventory-related waiting time (period $\alpha_p$ service level):

$\alpha_{\mathrm{p}} = P\{\mathrm{Period demand} \leq \mathrm{Inventory on hand at the beginning of a period}\}

This criterion is also called ready rate. In order to determine the safety stock that guarantees a target $\alpha_{\mathrm{p}}$ service level, the stationary probability distribution of the inventory on hand must be known.

If an order cycle is considered as the standard period of reference, then $\alpha$ denotes the probability of no stock-out within an order cycle which is equal to the proportion of all order cycles with no stock-outs (cycle $\alpha_c$ service level):

$\alpha_{\mathrm{c}} = P\{\mathrm{Demand during replenishment lead time} \leq \mathrm{Inventory on hand at the beginning of the lead time}\}$

The $\alpha_c$ very often does not make much sense as it may be zero although there is a very high probability that a customer is served without delay. This happens in the case of long order cycles (i.e. with large order sizes).

### $\beta$ service level

The $\beta$ service level (fill rate) is a quantity-oriented performance measure describing the proportion of total demand within a reference period which is delivered without delay from stock on hand:

$\beta =1- \frac{{E\left\{ \mathrm{Backorders per period} \right\}}}{E\left\{ {\mathrm{Period
demand}} \right\}}$

This is equal to the probability that an arbitrary demand unit is delivered without delay. Due to the fact that, contrary to the variations of the $\alpha$ service level, the $\beta$ service level does not only reflect the stock-out event but also the amount backordered, it is widely used in industrial practice.

### $\gamma$ service level

The $\gamma$ service level which is a time- and quantity-oriented performance criterion serves to reflect not only the amount of late deliveries but also the waitingtimes of the backordered demands. The $\gamma$ service level is defined as follows:

$\gamma =1- \frac{{E\left\{ \mathrm{Backlog per period}
\right\}}}{E\left\{ {\mathrm{Period demand}} \right\}}$

In contrast to the $\beta$ service level which can be quantified by looking at the size of the backlog immediately before the arrival of a replenishment order, the $\gamma$ service level also reflects the development of the backlog in the preceding periods.

### Duration of a stock-out (probability distribution)

An additional performance criterion that takes the time dimension into consideration is the duration of a stock-out situation $J$ which can be computed as follows. We consider the inventory development in a typical replenishment cycle of
length $\ell$ under a continuous review ($s,q$) inventory policy and we assume a reorder point $s>0$. We approximate the inventory development between the reorder point (at the beginning of a replenishment lead time) and the expected backlog level immediately before the arrival of the replenishment order by a straight line (see Figure in the book). The expected duration of a stock-out situation $J$ is then

$E\{J\} = \frac{{E\left\{ \mathrm{Backorders at the end of
an order cycle} \right\}}}
{\mathrm{Reorder point} +
E\left\{ \mathrm{Backorders at the end of an order cycle}\right\}}
\cdot \ell \quad $(1)

A second measure which is complementary to the duration of a stock-out, $J$, is the coverage $N$. The coverage denotes the number of period demands that can be completely served from a given stock on hand. Obviously, under stochastic conditions both $N$ and $J$ are random variables. From a customer's point of view the duration of a stock-out is important as it defines an upper bound on the actual waiting time a customer order may observe. Particularly when the period demands have large variances, the probability distribution of a stock-out duration will have a large variance, too.

In the following, the probability distribution of the duration of a stock-out for an $(s,q)$ policy in discrete time is computed. Assume that the inventory on hand at the beginning of a replenishment lead time of length $\ell$ is exactly $s$. Assume further that the order size is large enough to clear all outstanding backorders at the time of replenishment. Under the given assumptions the maximum stock-out duration, $J$, is $\ell$ periods. This happens if the demand arriving in the first period of a replenishment lead time is larger than the reorder point. This is equivalent to a coverage of $N=0$. The probability for this situation is:

$P\{N=0\}=P\{D > s\}=1 - P\{D \leq s\}$

The coverage of the quantity $s$ is exactly $n$ periods, if the cumulated demand of $n$ periods is less than or equal to $s$ and the cumulated demand of $n + 1$ periods is greater than $s$. Let $Y^{(n)}$ denote the sum of the first $n$ period demands of the replenishment lead time. Then the probability that the coverage of the reorder point $s$ is exactly $n$, can be written as follows:

$P\{N=n\}=P\{Y^{(n)} \leq s\} - P\{Y^{{(n+1)}} \leq s \} \qquad n=1,2,\ldots$

The probabilities of the cumulated demand of $n$ periods, $Y^{(n)}$, can be computed through the $n$-fold convolution of the one-period demand distribution. In case of normally distributed demands we obtain

$P\{N=0\} = 1 - \Phi_N \left( \frac{s - \mu_{D}}{\sigma_{D}} \right)$

$P\{N=n\} = \Phi_N \left( \frac{s - \mu_{D}\cdot n}{\sigma_{D}\cdot\sqrt{n}} \right) - \Phi_N \left( \frac{s - \mu_{D}\cdot(n+1)}{\sigma_{D}\cdot\sqrt{n+1}} \right)\qquad n=1,2,\ldots\quad$ (2)

where $\Phi_N(\cdot)$ denotes the cumulative probability function of the standard normal distribution. Consider an example. The period demands are independent, identically distributed (i.i.d.) ormal random variables with mean $\mu_D=100$ and standard deviation $\sigma_D=30$. The lead time is deterministic $\ell = 5$. For a given reorder point $s=400$ we obtain

$P\{N=0\}=P\{J=5\} = 1 - \Phi_N \left( \frac{400 - 100}{30} \right) \approx 0$

where $\Phi_N$ denotes the cumulated distribution function of the standardized normal distribution. The following Table shows the remaining results with $v_1$ ($v_2$) denoting the first (second) term in brackets on the right side of equation (2).

$n$ |
$j=\ell-n$ |
$v_1$ |
$\Phi_N$($v_1$) |
$v_2$ |
$\Phi_N$($v_2$) |
$P\{N=n\}=P\{J=j\}$ |

1 |
4 |
10.00 |
0.9999 |
4.71 |
0.9999 |
0.0000 |

2 |
3 |
4.71 |
0.9999 |
1.92 |
0.9728 |
0.0271 |

3 |
2 |
1.92 |
0.9728 |
0.00 |
0.5000 |
0.4728 |

4 |
1 |
0.00 |
0.5000 |
-1.49 |
0.0681 |
0.4319 |

The probability of no stock-out and, consequently, no backorders occurring in an order cycle is $P\{Y^{(\ell)}<s\}$. In the considered example with $s=400$ we have $P\{Y^{(\ell)}<s\}=0.0681$. If we assume an order quantity $q=1000$ and a target $\beta$ service level of 90\% ($E\{\mathrm{Backorders per order cycle}\}=100$), then from equation (1) we obtain

$E\{J\} = \frac{100}{400+100}\cdot 5=1.0$

By contrast, the expected value of $J$ computed from the probabilities given in the
Table is $E\{J\}=1.4588$.

It is also possible to compute the probability distribution of the stock-out duration for an $(r,S)$ policy. For this inventory policy, the risk period, i.e. the sum of the review interval $r$ and the replenishment lead time $\ell$, may be significantly longer than the risk period for an $(s,q)$ policy. Therefore, the maximum duration of a stock-out situation may also be longer. This has negative consequences for the customer order waiting times.

The probability distribution of the stock-out duration may be used for the computation of the probability distribution of the waiting time that a customer order observes.

### Order reliability window

Further information are available in the book.

### Customer Order Waiting time

From the point of view of a customer order, the logistic performance of the inventory node in a supply network is mainly about whether the order is immediately fulfilled or whether it has to wait and if so, how long.

The fact that a customer order possibly has to wait is covered by the service levels discussed above which might also be called "supplier focused". For the customer, however, it is not only important whether he has to wait but also how long this waiting time (replenishment lead time from the customer's point of view) will be. If the customer is a retailer who himself keeps an inventory, the probability distribution of the waiting time may also be of interest to him as this influences the demand during the replenishment lead time which is important for his safety inventory.

None of the so far discussed performance criteria, neither the $\gamma$ service level nor the duration of a stock-out, provide adequate information on this. Here, the inventory related \textbf{customer order waiting time} lends itself as a customer-focused criterion. It is closely related to the duration of a stock-out.

Further information are available in the book.

In industrial practice we often find statements from logistic managers with respect to the target logistic performance of an inventory policy, such as "*90% of all orders must be delivered from inventory on hand without delay (i.e. inventory related waiting time=0), 95% after one day at the latest (i.e. inventory related waiting time=1), and all other orders after two days at the latest.*" This statement demonstrates the significance of the aspect of time as a competitive weapon. At the same time it reveals ignorance of the fact that an objective function which includes the shape of the probability distribution of the inventory related waiting time may not be systematically achieved by determining the parameters of an inventory policy, as the shape of the distribution of the inventory related waiting time cannot be influenced.

The shape of the probability distribution of the customer order waiting time depends on the probability distribution of the period demands and the length of the risk period. To clarify the influence of the risk period on the waiting time, we take a look at four hypothetical suppliers who supply the same product and use an $(s,q)$ inventory policy. The period demand is assumed to be normally distributed with the parameters $\mu_D=100$ and $\sigma_D=30$. All suppliers offer their customers a $\beta$ service level of 90%, respectively. If the suppliers are also equally good regarding all other criteria (quality, price, after sales service, etc.), a potential customer has no clue as to which supplier to select if he only looks at the $\beta$ service level.

If we now assume that the suppliers have different replenishment lead times, also different probability distributions of the inventory related waiting times result. In the following Table the waiting time distributions are shown for the different suppliers depending on their replenishment lead times $\ell$. For the calculation of these probability distributions -- which is not shown here -- an order quantity of $q=500$ was assumed.

Waiting time | ||||||

$\ell$ |
0 |
1 |
2 |
3 |
4 |
5 |

5 |
0.9000 |
0.0828 |
0.0168 |
0.0004 |
-- |
-- |

10 |
0.9000 |
0.0759 |
0.0216 |
0.0024 |
0.0001 |
-- |

15 |
0.9000 |
0.0708 |
0.0244 |
0.0044 |
0.0004 |
-- |

30 |
0.9000 |
0.0613 |
0.0275 |
0.0089 |
0.0020 |
0.0003 |

If the suppliers (or rather their inventory policies) are assessed solely with the help of the $\beta$ service level, all suppliers are equally good. The Table reveals, however, that the variance of the waiting time distribution increases with increasing replenishment lead time. From a customer's point of view this information may be of great importance as will be illustrated later in Section B.3.5. Thus, under otherwise equal conditions, he will normally choose the supplier whose waiting time shows the lowest variation.

Let us now assume that the customer is a retailer who himself applies an inventory policy. For the supplier with $\ell=5$, he would require a reorder point of $s=228$ to achieve a service level of $\beta=95%$ with respect to his own customers. If he instead chooses the supplier with $\ell=30$, however, the reorder point would be $s=250$. Even worse, if the retailer wants to provide a service level of $\beta=99%$, then for $\ell=5$ a reorder point of $s=312$ will be required and for $\ell=30$ the reorder point will be $s=383$. From the retailer's point of view there is a difference as to the inventory costs between the best supplier and the poorest supplier which can be quantified with sufficient precision only with available information on the probability distribution of the supplier's customer order waiting time.

If a supplier uses the waiting time as a criterion for evaluating his inventory performance, he can apply its probability distribution as a competitive argument. In addition, the waiting time as a unifying dimension for the description of logistic processes also provides the opportunity to consider the integrated optimization of several consecutive processes within a supply chain. Note that it is not possible to influence the exact shape of the waiting time distribution as this is mainly determined by the probability distribution of the period demands. However, it is possible to adjust the parameters of an inventory policy in such a way that the scale of the distribution is modified. In this case it is possible, for example, with a constant total waiting time of the customers, to determine the optimal allocation of this total waiting time to the different logistic sub-processes. Also, an objective function related to the cumulative probability like "*98% of the customer orders must be delivered after two days at the latest!*" may be achieved. An example for the optimal coordination between make-to-stock\index{make-to-stock} and assemble-to-order production is given by Tempelmeier(2000). In this paper, also a method for the approximation of the waiting time distribution for a $(r,S)$ policy in discrete time is described. Below, in Section C.2.4 we will show the derivation of the exact probability distribution of the waiting time for a base-stock policy or rather a $(r=1,S)$ policy in discrete time.

- Examples for waiting time distributions under an ($r,s,q$) Policy in discrete time
- The role of the waiting time distribution in a multi-level inventory system

### Performance measures related to a group of products

Further information are available in the book.

### Customer-class specific performance measures

Further information are available in the book.

### The effect of the inventory policy on the transportation system

Further information are available in the book.

### The Effect of Inventory Performance on Subsequent Logistical Processes

Compute the optimum transportation capacity for a given inventory policy. Further information are available in the book.

### The Key Performance indicator "Inventory Run-Out time"

In industrial practice, the performance of a company's logistics system is often measured with the indicator "inventory run-out-time". This is also a popular measure used by financial analysts. The run-out-time is equal to the average inventory on hand divided by the average demand per period. This criterion suggests that a company with a long run-out-time performs poorer than a company with a short run-out-time. In a benchmark study, a company with a short run-out-time would thus be evaluated as superior.

For the analysis of the logistical performance, the number of average period demands is a misleading criterion. In order to improve performance by reducing the holding costs, it is necessary to look at the causes of inventory. In a hierarchical planning system, inventory may be built up as a consequence of decisions made on several planning levels.