Statistical Inventory
control models I
(Q, r) model
Learning objective

After this class the students should be able to:
•
Apply optimization technique to inventory model
• calculate the appropriate order quantity in the
•
•
face of uncertain demand.
Derive the (Q, r) model from the integration of the
EOQ model and the EPL model.
analyze the implication of the (Q, r) model
Time management

The expected time to deliver this module
is 50 minutes. 30 minutes are reserved
for team practices and exercises and 20
minutes for lecture.
The Base Stock Model

Consider the situation facing an appliance store that sells a
particular model of refrigerator. Because space is limited and
because the manufacturer makes frequent deliveries of other
appliances, the store finds it practical to order replacement
refrigerators each time one is sold. In fact, they have a system
that places purchase orders automatically whenever a sale is
made. But, because the manufacturer is slow to fill
replenishment orders, the store must carry some stock in order
to meet customer demands promptly. Under these conditions,
the key question is how much stock to carry.
Assumptions
1.
2.
Demands occur one at a time;
Any demand not filled from stock is backordered;
3.
Replenishment lead times are fixed and known;
4.
There is no setup cost associated with placing an order; and
5.
There is no constraint on the number of orders that can be
placed per year.

Last two assumptions imply that there is no incentive to
replenish stock in anything other than one-at-a-time fashion.
Notation
l  Replenishm ent lead - time (in years)
X  Demand during replenishm ent (in units), a random variable
G  x   P X  x  cumulative distributi on function of demand
during replenishm ent lead - time;
  E  X  Demand (in units) during lead - time l ;
h  Cost (in dollar per unit per year) to carry one unit of inventory for one year;
b  Cost (in dollar per unit per year) to carry one unit backorder for one year
r  reorder point (in units), which represents the inventory level that trig ger
a replenshme nt order; this is the decision v ariable;
R  r  1, base stock level (in units)
s  r   , base stock level (in units)
The question

We place an order when there are r units in stock and
expect to incur demand for θ units while we wait for the
replenishment order to arrive. Hence, r - θ is the amount
of inventory we expect to have on hand when the order
arrives. If s = r - θ > 0, then we call this the safety stock
for this system, since it represents inventory that protects
it against stockouts due to fluctuations in either demand
or deliveries. Since finding r - θ is equivalent to finding r
(because θ is a constant), we can view the problem either
as finding the optimal base stock level (R = r -1), reorder
point (r), or safety stock level (s = r - θ).
Solution

Since an order is placed every time a demand
occurs, the relationship “on-hand + purchase
orders - backorders = R” holds at all times.

Because lead times are constant, we know that all of
the other R-1 items in inventory and on order will be
available to fill new demand before the order under
consideration arrives.

Therefore, the only way the order can arrive after the
demand for it has occurred is if demand during the
replenishment lead time is greater than or equal to R
(i.e., X ≥ R).
Solution

Hence, the probability that the order arrives before
its demand (i.e., does not result in a backorder) is
given by P(X < R). If demand has a continuous
distribution then
P(X < R) = P(X ≤ R-1) = G(R).

However, if demand has a discrete distribution (i.e.,
X can take on only integer values and has a
probability mass function instead of a density
function), then
P(X < R) = P(X ≤ R-1) = G(R-1)=G(R).
Solution

Since all orders are alike with regard to this
calculation, the fraction of demands that are
filled from stock is equal to the probability that
an order arrives before the demand for it has
occurred, or
G ( R) if demand is continous
P X  R   
Gr  if demand is discrete

Hence, G(R) or G(r) represents the fraction of
demands that will be filled from stock (i.e., the fill
rate).
The (Q, r) model

Consider the situation where inventory is
monitored continuously and demands occur
randomly, possibly in batches. When the
inventory level reaches (or goes below) r, an
order of size Q is placed. After a lead-time of ℓ,
during which a stockout might occur, the order
is received. The problem is to determine
appropriate values of Q and r. The model we
use to address this problem is known as the
(Q, r) model
Example

The manager of a maintenance department must stock spare parts to
facilitate equipment repairs.

Demand for parts is a function of machine breakdowns, therefore
random.

Unlike the base stock model, the costs incurred in placing a purchase
order (for parts obtained from an outside supplier) or the costs
associated with setting up the production facility (for parts produced
internally) are significant enough to make one-at-a-time replenishment
impractical.

Thus, the maintenance manager must determine not only how much
stock to carry (as in the base stock model), but also how many to
produce/order at a time (as in the EOQ and newsboy models).
Assumptions

From a modeling perspective, the (Q, r) model is identical
to the base stock model, except that we will assume that
either
•
•

There is a fixed cost associated with a replenishment order.
There is a constraint on the number of replenishment orders
per year.
Therefore, replenishment quantities greater than one may
make sense.
Basic mechanics of the (Q, r) model
Demands occur randomly, possibly in batches. When the inventory level
reaches (or goes below) the reorder point r, a replenishment order for
quantity Q is placed. After a (constant) lead time of ℓ, during which a
stockout might occur, the order is received. The problem is to determine
appropriate values of Q and r.
Total cost
In some sense, the (Q, r) model represents the integration of the EOQ
model and the base stock model two models. Then, the total cost is
D
D
Q

Y Q, r   A  h  r     b n(r )
Q
Q
2

Where n(r) is the expected number of backorders that
will be placed during a cycle
Optimal replenishment quantity

The optimal
replenishment
quantity Q*, and
reorder point r*,
can be found by
simultaneously
solving the
following
equations:
2 D( A  bn(r )
Q
h
hQ
G ( R)  1 
bD
Reflections
Each team is invited to analyze the
following insights, based on the
statistical model (10) minutes):
1.
2.
“Cycle stock increase as replenishment
frequency decrease”
“Safety stock provide a buffer against
stockout”
Reflections

Suppose you are stocking parts
purchased from vendors in a warehouse.
How could you use a (Q, r) model to
determine whether a vendor of a part
with a higher price but a shorter lead
time is offering a good deal? What other
factors should you consider in deciding
to change vendors. (10 minutes)
Reflections

What is the key difference between the
EOQ model and the (Q, r) model?
Between the base stock model and the
(Q, r) model?
Reference

Factory Physics. Hopp & Spearmen,
Irwin, 1996. Chapter 2, p.72-100