In the following a typical Inventory Management Test is considered. Elementay questions regarding EOQ, Reorder Point, Fill Level, Cycle Service Level are considered.

Also a chart showing the Inventory Time curve of a Periodic Review, order Point Policy (R,s,S) is shown.




A paint company follows an inventory  policy Continuous Review (s, Q) to control the level of inventories.

For a particular paint, historical data show a normal distribution of monthly demand, with average 28 pieces / month and a standard deviation of 8 pieces / month.

The lead time to supply the warehouse is approximately 14 weeks.

Each can of paint costs 6 €.

The unmet demand goes in back-order with an estimated  cost of 10 € /can.

Setup  costs A are 15 € / order and the carry costs of replenishment are 30% per year.

A) Determine the optimal lot size and reorder point of the paint.

B) Determine the optimal level of safety stock.

C) What is the reorder point if you want to avoid stock-out in 90% of cases

D) Determine the fill rate reached following the Inventory policy in a) and c).

E) Answer the points A), B), C), D) when orders of paints are lauched each month.

F) Suppose you have the following monthly demand:


Month demand

Month Demand (cans)
January 37
February 33
March 26
April 31
May 12
June 40


If the Inventory level at the beginning of January is equal to 26 cans of paint, determine the number of cans of paint ordered in 6 months, following the policy (R, s, S), where R and S are those of point E and s is the one found in A).

 Solution to point A)


First we e approximate the solution by finding the EOQ and the reorder point:


, k=1.75


If we want to find the exact solution we have to satisfy both the following:


Starting from the approximate solution EOQ and s we find after 5 iterations the following results.


Qexact=84.72 (Qexact - EOQ =9.28, 11%)

sexact=123.3 (sexact-s=-0.0071, 0%)


Solution to point b)

The answer to this point is in the Equation (1) above so ss=26.2

Solution to point c)

In order to have a Cycle Service Level of 90% the safety factor k=1.28

so considering that demand during lead time is D⋅LT=28 items/months ⋅ (14 weeks/4 weeks/month)=98 items,

and that σLT=(14/4).5⋅8 items=14.96 items

then s=98+1.28⋅14.96=117.16


Solution to point d)

Fill Rate in case of BackLog  with k=1.28:

Fill Rate in case of Backlog with k=1.75:


Solution to point e)

In the case a) and b) we have:



in case c)

While for case D) the FIll Rates are:


Solution to point f)

 The value of  s, S and R are :




R=1 month


The period under observation start from January, 1st up to June, 30th.

If the starting level of stocks is 26, on the hypothesis of not order outstanding, at the begining of January a order is launched up to S level.

In this manner the Inventory Position of Stocks go to S. Afterwards the level of stocks is shown in the following table and the following figure.

It is noteworthy that even if Stock on OH reduce you do not launch other orders until Inventory Position is above the Reorder point.

Month 0 1 2 3 4 5 6 Total
Demand   37 33 26 31 14 40 181
Inventory OH begin 26 26 -11 -44 -70 35.82 21.82 -15.36
Inventory OH after 26 -11 -44 -70 35.82 21.82 51.82 10.46
Inventory Position begin 26 125.82 92.82 136.82 105.8 148.82 108.82 744.92
Order 136.82 0 70 0 57 0 54 317.82
Inventory Positon after 162.82 125.82 162.8 136.82 162.8 148.82 162.82 1062.74