In this section we
consider an isolated inventory in which external demanders remove
items from the inventory and external suppliers replenish the
inventory. Rather than demand occurring in a random and uncertain
manner, we assume that items are withdrawn from the inventory
at a continuous rate. Replenishments to the inventory are of
a fixed size *q*, called the lot size. The time between
when a replenishment is requested and when the amount enters
the inventory is called the lead-time. We assume that the lead-time
is zero or a constant.
Six different models are developed in this section. They vary
in whether shortages are allowed or not and whether the replenishment
rate is infinite or finite. Two customer responses to shortages
are considered: the backorder case when the customer will wait
for delivery and the lost sales case when the customer will
not wait. For the backorder case, the cost associated with a
backorder is either proportional to the waiting time or it is
independent of the waiting time. The cases are illustrated by
the graphs of inventory position versus time shown below. Links
adjacent to the figures and in the navigation bar on the left
lead to pages providing detailed development of the optimization
and analysis formulas. The formulas are implemented in the Inventory
add-in.
Practitioners sometimes criticize of the results of inventory
theory because the simple models described in this section do
not often mirror reality. We justify their consideration on
several grounds. First because of the simplicity we are able
to find algebraic formulas that give the optimum solutions for
the models. In some cases the computed optimum values may prove
useful. The familiar EOQ formula is perhaps one of the most
used formulas in practice. Perhaps more valuable are the insights
into the effects of the several parameters on inventory cost
and optimum operating policies. Further, the results for the
deterministic system are used as approximation of the formulas
describing stochastic systems. Further, we use the deterministic
models for individual stations in models that involve systems
of inventories. Finally, the process of creating a cost model
and finding its optimum using calculus is a good example for
the application of mathematics to the solution of other practical
problems.
Care must be taken that the models are not used inappropriately.
Just because a formula has been derived does not justify its
use without a careful analysis of its validity in a particular
situation. We hope that the presentation of inventory theory
provided here and in many excellent textbooks provides both
the basis and cautionary limitations for application. |