Many of
the functions used for deterministic systems are also used for
stochastic systems. Because of the random nature of the inventory
level during a cycle, the meaning of the results are somewhat
different for stochastic systems compared to the deterministic
system. For example the maximum, minimum and average inventory
in a cycle are all random variables for stochastic systems,
so the functions for these results compute estimates of the
expected values of the respective numbers or long run averages
over many cycles. Most of the functions also involve some approximation
as it is very difficult to compute the exact expected values.
When used for stochastic systems, the functions still have two
arguments. The Inv_Level function is as before
**Inv_Level(instance As Range, inventory As
Range)**
Both arguments are ranges. The *instance *is a range
that specifies the variables controlling the inventory. For
a stochastic system, the instance still includes the * lot
size* and *fill rate*, but a third argument is added
for efficient calculation, the *reorder point*. The range
*inventory* holds the parameters of the inventory model
appropriate for the type of system.
The figure below shows computations for a model with the data
name *Inv8*. The *type *shown in cell B3 indicates
that this is a stochastic system with instantaneous replenishments
and that backorders are allowed. The entire data is in the range
B1:B16. We have chosen in this example to analyze the instance
which is the optimal solution. The optimal lot size is computed
in cell B17 with the function expression:
**=Inv_Stoch_EOQ(Inv8)**
This function computes the economic order quantity
for the stochastic system. In cell B18 we have:
**Inv_Stoch_OptFR(B17,Inv8)**
This function computes the optimal fill rate given
the lot size computed in B17. In cell B19 we have:
**Inv_StochRP(Inv8_EOQ,Inv8_OptFR,Inv8)**
This function computes the reorder point for the
lot size in B17 (named *Inv8_EOQ*) and the fill rate
in B18 (named *Inv8_OptFR*). The range B17:B19 has been
named *Inv8_Opt* for subsequent computations. Note that
the three numbers just computed are not independent. The EOQ
computation includes a computation of the optimum fill rate
since both are determined simultaneously. The fill rate depends
depends on the lot size, and the reorder point is function of
the lot size and fill rate.
Subsequent calculations compute various measures
for the system when the optimum parameters are used. The functions
in B20:B29 are the same used for deterministic systems, but
now they compute the expected values of the measures. The computations
performed by the functions depend on the type of system being
analyzed. The results in B30 and B31 are meaningful only for
stochastic systems and are computed with new functions. In B30
we see the probability that a cycle experiences a shortage.
For deterministic systems this probability is 0 when no shortages
are allowed and 1 when shortages are allowed. Here the number
is constrained by the limit in B16 and will generally be some
small value. In B31 we see the safety stock. This is the extra
stock maintained above the expected demand during the lead time
to guard against the risk of shortage. For deterministic systems
the safety stock is always 0 because there is no risk.
Notice that the data for stochastic systems will
always include an indication of the probability distribution
of demand. For the current example the demand is Normal with
a standard deviation of 10 per week. The expected value of demand
for an interval of time is the demand rate multiplied by the
length of the interval. |