The role of the waiting time distribution in a multilevel inventory system
In the following we consider a twolevel supplychain, which is known in the literature as the OneWarehouseNRetailer system. Such a system can be found very often in industrial practice. We will show, that the probability distribution of the waiting time of a retailer order at the warehouse has a significant influence on the inventory that is required at the retailer level.
In the example, we make the following assumptions:
 a single warehouse
 using an $(r,s,q)$ inventory policy in discrete time with $r=1$
 deterministic replenishment lead time $L=10$
 holding costs $h=0.024$
 fixed ordering costs $s=120$
 replenishment order size is fixed $q=1000$
 daily review
 10 identical retailers
 with normallydistributed period (daily) demands ($\mu_D=20$, $\sigma_D=5$)
 using a basestock policy with daily review $(1,S)$
 with holding costs $h=0.024$
 with target service level $\beta=0.9
The following table has been constructed by varying the reorder point in the warehouse $s$. For each reorder point the resulting probability distribution has been computed analytically. With this given probability distribution (which is the replenishment lead time distribution seen by a retailer) for each retailer the optimum basestock level w.r.t. to the target $\beta$ servicelevel is computed. Finally the average costs on both stages of the supply chain are calculated.
reorder point $s$ 
200 
0 
200 
400 
600 
800 
900 
1000 
1050 
1100 
Waiting time

$P\{W=w\}$ 
$P\{W=w\}$ 
$P\{W=w\}$ 
$P\{W=w\}$ 
$P\{W=w\}$ 
$P\{W=w\}$ 
$P\{W=w\}$ 
$P\{W=w\}$ 
$P\{W=w\}$ 
$P\{W=w\}$ 
0

0 
0.0032 
0.1494 
0.3494 
0.5494 
0.7494 
0.8494 
0.9462 
0.9812 
0.9966 
1

0 
0.0492 
0.1 
0.1 
0.1 
0.1 
0.0971 
0.0508 
0.0186 
0.0034 
2

0.0026 
0.0969 
0.1 
0.1 
0.1 
0.0974 
0.0508 

3

0.0492 
0.1 
0.1 
0.1 
0.1 
0.0508 

4

0.0975 
0.1 
0.1 
0.1 
0.0979 

5

0.1 
0.1 
0.1 
0.1 
0.0508 

6

0.1 
0.1 
0.1 
0.0985 

7

0.1 
0.1 
0.1 
0.0508 

8

0.1 
0.1 
0.099 

9

0.1 
0.1 
0.0508 

10

0.1 
0.0994 

11

0.1012 
0.0506 

12

0.0992 

13

0.0492 

warehouse inventory 
0 
0 
5.69 
45.68 
125.69 
245.68 
320.69 
405.69 
451.91 
500.34 
warehouse costs 
0 
0 
0.14 
1.1 
3.02 
5.9 
7.7 
9.74 
10.85 
12.01 
expected leadtime 
8.99 
6.99 
5.05 
3.45 
2.24 
1.44 
1.2 
1.05 
1.02 
1 
basestock level $S$ 
287 
246 
205 
165 
123 
83 
63 
48 
45 
45 
retailer inventory 
882.52 
873.67 
852 
765.73 
590.76 
350.34 
193.47 
80.17 
56.73 
57.19 
retailer costs 
21.18 
20.97 
20.45 
18.38 
14.18 
8.41 
4.64 
1.92 
1.36 
1.37 
total costs 
21.18 
20.97 
20.58 
19.47 
17.19 
14.3 
12.34 
11.66 
12.21 
13.38 
Note that the minimum lead time is one period, caused by the discrete review. The results show that in the current setting it is optimal to use a warehouse reorder point $s=1000$ and a retailer basestock level $S=48$. The development of the total cost curve is shown in the following graph.
The procedure used to evaluate the performance of an $(r,s,q)$ policy which is the basis for finding the optimum distribution of the inventory over all nodes in the supply chain is applicable without any change to the case of nonidentical retailers. Being able to compute the complete waiting time distribution is the key for the solution of the problem. The remaining calculations are more or less standard.
Further information are available in the book.