In the following a new performance in considered in the Safety Stock management: the waiting time of backordered clients.

In particular the percentage of client waiting more than a time limit (1 month) is limited. For the rest dimensioning Safety Stock following different policy is required


 Consider the following items:

item v € / piece LT months D piece / month sD

piece / month

1 80 4 950 200

2 200 6 60 15

3 20 8 500 100

item v [€/item] LT [month] D [items/month] σD [items/month]
1 80 4 950 200
2 200 6 60 15
3 20 8 500 100

 Where v is the cost of the item, D the average monthly demand, sD the standard deviation of demand, LT Lead Time of delivery.

Considering the following:

  • a setup fee cost A=100 €/order;
  • a carrying cost rate r=25% [€/€-year];
  •  stock out backordered;
  • distribution of demand is gaussian.

Answer the following questions:

a) safety stock should ensure that in the average no more than 10% of customers wait more than 1 month. Hint use formula ESPRC  considering  LT-1 Months, whereas 10% = ESPRC (LT-1) / Q) :


item SS [item] (only 10% of customers expect more than 1 month)

Compile the table at the bottom answering the following:

b) It allocates a value of € 50,000 for safety stock so that all 3 items have the same safety coefficient k (k Same);

c) As in the previous but by imposing the same B1 [€ / stock-out incident] for all items (hint it starts from B1 around 1000) (Same B1);

d) As in previous but by imposing the same B2 [% / stock-out incident] for all items (hint we start from B2 = 0.08) (Same B2);

e) Same Cycle Service Level P1, percentage of cycles in which you go to stock out (Same P1);

f) Same Time Between Stock Out (Same TBS);

g) Same Fill Rate P2, percentage of satisfied customers, assuming back order (hint try P2 = 93%) (same P2);


Quantity  b) the same k  c) the same B1 d) the same B2 e) the same P1 f) the same TBS g) the same P2
Expected Total Stock Out Per Year            
Expected Total Shortage in Value  Per Year            
SS  item 1  in value €            
SS  item 2  in value €            
SS  item 3  in value €            
Global TBS            



As preliminar Operation we calculate σLT and EOQ for each items:

400 338
37 54
283 490


Solution to point a)

In order to solve point a we consider that the average number of clients satisfied in one Replenishment cycle as Q=EOQ.

As known P2, the Fill rate,  represents the percentage of clients satisfied respect to the total during LT (or during a Replenishment Cycle).

We can then introduce a P2(LT-1), a Fill Rate considering the time interval LT-1. If P2(LT-1) is 90% it means that only the 10% of clients goes in backorder more than one month before next Replenishment. 

This is the results we would like to obtain in point a). So we have the following (where ESPRC(LT-1) is the Expected Shortage per Replenishment Cycle referring to the time interval LT-1 and the same for σLT-1):


 then we can easily obtain:


σLT-1 Gu(k) SS SS (€)
346,41 0,08441 0,993 343,99 27518,82
33,54 0,146059 0,687 23,04 4608,536
264,58 0,173205 0,5835 154,38 3087,592
      Total 35214,95





 Solution to points from b) to g):

In order to solve the other problem we could use classical methods and considering that same B1 is equivalent to minimize ETSOPY,   same B2 is equivalent to  both same TBS and minimizing ETSVPY, same k to same P1 and that global TBS is 1/ETSOPY we have:


  same k same b1 same b2 same P1 same TBS same P2
ETSOPY [events/year] 7,92 7,45 9,88 7,92 9,88 8,74
ETSVPY  [€/year] 83815,73 101692,72 75393,31 83815,73 75393,31 77596,35
share item 1 [€] 35551,35 30841,14 41459,66 35551,35 41459,66 38496,00
share item 2 [€] 8164,00 10458,46 5060,89 8164,00 5060,89 6797,33
share item 3 [€] 6284,65 8702,13 3484,78 6284,65 3484,78 4695,19
global TBS [year/events] 0,13 0,13 0,10 0,13 0,10 0,11


 you can find calculations here and the txt in Italian here and other documents here.