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Case b: MTBF > T s
In this case the defnition of the average number
of parts storing during the supplying time T has to
S

take into account both contributions, in terms of
average stock and , of and . LS is
Fig. 1 - Time scheme for The value is calculated using the following: connected to , and the minimum real value of N
model based on binomial is therefore .
distribution
The annual stock cost C can be evaluated by
(1)
(5)
Then the second term is estimated in the time
that can be used in an iterative process in order
period in order to minimize the total cost as to fnd the optimum level N according to the
indicated in fgure 1. minimization of previous cost C:
During the time T production losses could occur
s
if the number of failures exceeds the number N
of supplied parts which are available after the
consignment at the time zero, affecting the service (6)
level LS. The corresponding cumulative probability
can be calculated by formula:

(7)
(2)
If d and CM represent respectively the customer
annual part consumption and the cost for a 3.2. Optimal policy with fexible
production lack, the total cost C due to stockout delay (Teunter et al, 2012)
1
is:
In this policy, as for the standard (s, S) one, an
order is placed if the inventory level goes to or
(3) below s and the order quantity allows to restore
stock levels to S. In this case, the order is delayed
The storage cost C requires the defnition of by time units or until next demand occurs,
2
the average number of parts storing during the where q is the order size and u is the undershoot.
supplying time T . The authors (Teunter et al, 2012) derived the
S
Two different situations are possible related to optimality conditions using marginal analysis,
spare part MTBF and T : differing the normal situation with a perturbed one,
S
where the delay is increased. The marginal cost
function has been introduced and the optimal delay
Case a: MTBF > T s is found as unique solution of this formula:
In this case

because
(8)
If the warehouse contains N parts at time zero, the
probability P of N failures in T can be pointed out where h is the holding cost, b is the backorder
N
S
as done in Persona et al. 2006. Let R indicate the cost, P t is the probability that the total number
cost of each spare part, and t the annual stoking of demanded items in the analyzed period is lower
cost index; the global annual storage cost C can be than as follows:
evaluated by the following:

(9)
(4)
where N t are the probabilities that k demands

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