CHAPTER 11 INVENTORY MODELS
11.1 Basic Concepts in Inventory Planning
For many organizations, management of inventories is
of crucial importance.
Our inventory models will concern just a single item
for which the demand per period (year) is D units.
The number of items in stock is depleted over time by
the demand, but also increased from time to time by
instantaneous additions Q called orders, resulting in
sudden jumps in the inventory level.
The time between two consecutive replenishments is
the inventory cycle length tc.
The lead time tL is the time between the placement and
arrival of an order.
Three types of inventory/stock levels:
IO , inventory on hand is stock physically on the shelf,
immediately available to satisfy demand. IO ≥ 0.
IN , net inventory on hand, is inventory on hand minus
backorders (unsatisfied demand). IN = IO 
(backorders) may be negative if backorders exceed
inventory on hand.
IP, inventory position is IP = IN + (outstanding orders).
Order size Q and the lead time tL will usually be the
same for all orders. The cost for carrying one unit of
inventory for one period is called the unit carrying or
holding cost ch. For some inventory models,
backorders are not allowed, otherwise the unit
shortage cost cs is the cost charged to be out of stock by
one unit for one period of time.
Ordering cost co is the cost of placing an order and
having it delivered, considered independent of the
size of the order.
11.2 The Economic Order Quantity (EOQ)
Model
Assumptions of the basic EOQ model are:






Inventory is one single unperishable good,
the demand rate is constant over time,
the same amount is ordered each time,
there are no quantity discounts,
stockouts are not allowed,
the planning horizon is infinite.
The famous sawtooth pattern is shown in the figure
below.
inventory level
Q
tc
time
Ordering costs: if we place N orders, each of size Q, the
total amount ordered is NQ = D, so that N = D/Q is the
number of orders per period and total annual
ordering costs are coN = coD/Q.
For holding cost, we compute the total area under the
sawtooth curve, which can be seen to be ½Q, so that
the total annual holding cost is ch(½Q). The total
inventory costs TC per period are
TC = coD/Q + ½chQ
costs
150
100
50
order quantity Q
The value Q* that minimizes total inventory cost TC, is
the optimum order quantity
Q* 
2 Dco
ch
TC(Q*) = ½ Dco ch  ½ Dco ch  2Dco ch .
Note that holding and ordering costs are equal at
optimum regardless of the value of the parameters D,
co, and ch.
Example: Demand of D = 800 car battery chargers per
year, co = $100 and the holding cost is ch = $4 per
charger per year.
2(800)(100)
 200 chargers and the
4
total cost is TC *  2(800)(100)(4) = $800. The optimal
number of orders per year is N* = D/Q* = 800/200 = 4
orders, and the optimal inventory cycle length is t c* =
Q*/D = 200/800 = ¼ year = 3 months. The total
ordering cost is 4(100) = $400, half of the total cost; the
other half is total holding cost.
Solution: Q* 
11.3 The EOQ with Positive Lead Time
Now a positive lead time tL elapses from the moment
an order is placed and until it has arrived.
The decision when to order will not affect how much to
order; the optimum order quantity Q* still applies.
We determine the order time in terms of the reorder
point R: when IO decreases to the level R, an order is
placed, which will arrive after a delay of tL time units,
just when the inventory level reaches zero. There are
two cases:
Case 1: t L  tc* . Demand during lead time is tLD, so that
if an order of size Q* is placed when the inventory
level reaches R* = t L D , the replenishment arrives
exactly when inventory on hand has been depleted, as
in the figure below.
Case 2: t L  tc* . Demand during the lead time is still
tLD, but since t L  tc* , it follows that tLD > tc* D  Q *, →
the arrival of an order will occur during a subsequent
inventory cycle & not during the cycle in which it was
ordered, as in the figure below, where tc*  t L  2tc* .
In general,
t L 
R*  t L D   *  Q * .
 tc 
which will cover both cases above.
Example: Consider the example above with D = 800, co
= $100, and ch = $4, for which we have obtained an
optimal order quantity of Q* = 200 at an annual cost
of TC* = 800. With a lead time of 2 months (1/6 of a
year), we have tL = 1/6 < ¼ = t c* , so that Case 1 applies
and the optimal reorder point is
R* = tLD = 1/6(800) = 133⅓ units.
On the other hand, with tL = 4 months (⅓ of a year),
we have tL = ⅓ > ¼ = t c* , so that Case 2 applies, and
 13 
R*= ⅓(800) –   (200) = 66⅔ units.
1
 4
A practical way to implement the reordering is the
two-bin system, where the amount Q* is put into two
storage bins, the first with a capacity of Q*  R* units,
and the second R* units. Demand is then satisfied from
the first bin until it is empty. At this time, an order is
placed and demand satisfied from the second bin, will
be depleted exactly when the next shipment arrives.
11.4 The EOQ with Backorders
Now allow backorders, so that IN may become
negative, in the sense that unsatisfied demand is
recorded or “backordered,” and satisfied immediately
upon replenishment of the inventory. Such shortages
incur a shortage cost cs per unit and per period.
The net inventory level will be as in the figure below,
where S denotes the amount of the maximal shortage.
The maximal inventory directly after replenishment is
Q  S, as the stockouts are satisfied first before new
inventory is built up.
Here, t1 denotes the length of time during which the
net inventory IN is nonnegative (there is no stockout);
t2 denotes the length of time during which there is no
stock at hand; t1 + t2 = tc. We find that t1/t2 = (Q − S)/S.
The costs now include ordering and holding costs as
before, and also shortage costs. The annual ordering
costs are coD/Q. For carrying costs, we find that the
average inventory level is obtained by averaging the
inventory level when no stockouts occur, which is
½(Q−S) during the time t1, while the inventory level
during the time stockouts occur is zero for the
duration t2. After some calculations, this leads to
(Q  S ) 2
inventory holding costs of ch
.
2Q
The average annual shortage is ½S during the time t2,
when we have shortages; this leads to total shortage
S2
costs of cs
. The total inventory costs are
2Q
D
(Q  S ) 2
S2
 cs
TC(Q, S) = co  ch
.
Q
2Q
2Q
Using partial derivatives, the total inventory costs are
minimized for
Q* 
2 Dco ch  cs
&
ch
cs
S* 
2 Dco ch
ch
=
Q *.
c s ch  c s
ch  c s
Example: Using the basic EOQ example above with D
= 800, co = $100, and ch = $4 and unit shortage costs of
cs = $6 per unit and year, the optimal order quantity is
2(800)(100) 4  6
Q* =
258.20 units, the optimal
4
6
ch
Q*
shortage is S * 
103.28, and the total
ch  c s
costs are TC(Q*, S*) = 309.84 + 185.90 + 123.94 =
$619.68.
11.5 The EOQ with Quantity Discounts
With the unit purchasing cost p constant and
independent of the order size Q, the sum of ordering,
holding, and purchasing costs is
TC(Q, p) = coD/Q + ½chQ + pD.
In practice, many suppliers offer incentives for
purchases of larger quantities in the form of lower
unit costs. Here we assume that there are three price
levels, the original non-discounted price and two
discount levels.
The unit holding cost ch is redefined as a proportion of
the unit purchasing price p, and the economic order
quantity becomes
Q* 
2 Dco
.
ch p
The given (non-discounted) price level is p0, the price
with the small discount p1, and the price with the large
discount p2, so that p0 > p1 > p2. To obtain the lower
price of p1, we have to purchase at least a quantity Q1
and to get the even cheaper price p2, we must order at
least Q2 units where Q2 > Q1.
Now we have a cost function for each of the price
levels as shown in the figure below.
The cost functions are TC(p0), TC(p1), and TC(p2), and
the dots indicate their optimal points.
For the highest cost curve with no discount we
determine the point of lowest cost with the EOQ and
record the associated cost; this is the optimal solution
given the regular price p0. As Q increases further, the
costs increase as well until we reach Q1, at which point
the actual costs jump down onto the cost curve TC(p1).
The process continues until Q2, when the costs drop to
the third and lowest cost function TC(p2).The
piecewise nonlinear cost function is shown as a bold
line in the figure above. To determine the order
quantity with the overall minimal costs, we have to
examine each cost curve separately.
In the above figure, the EOQ is less than Q1, so that we
increase the order quantity to Q1 and determine the
costs at that point. This is the optimal order quantity
given the price p1.
Example: Annual demand is 10,000 footballs with
purchasing costs of $2 per football, holding cost of 5%
of the price per football and year; cost of placing an
order is $80. There is a ½% discount for orders of at
least 6,000 units, and a 1% discount for orders of at
least 15,000 units.
The parameters of the problem include D = 10,000, ch
=5% of p, and co = $80.
Case 1: No discount, p0 = $2. Then ch = $0.10, and Q* =
4,000 with costs of TC* = 200 + 200 + 20,000 = $20,400.
Case 2: p1 = $1.99. Then ch = $0.0995, and the solution
of the EOQ is Q* = 4,010.038, which does not qualify
for the discount, so that we have to move out of the
optimum to qualify for the discount. That gives Q :=
6,000, with costs of TC(6,000) = 133.33 + 298.50 +
19,900 = $20,331.83.
Case 3: p2 = $1.98. Then ch = $0.099, and the solution
of the EOQ is Q* = 4,020.15, which does not qualify for
the discount, so that we have to move out of the
optimum to qualify for the discount. That gives Q :=
15,000, with costs of TC(15,000) = 53.33 + 742.50 +
19,800 = $20,595.83.
Comparing the three options, Case 2 offers the lowest
total costs, and we order 6,000 footballs, obtain a ½%
discount, and incur total costs of $20,331.83.
11.6 The Production Lot Size Model
In batch or intermittent production a production run
can be considered an order, with the production run
size corresponding to the order size Q, and the
production setup cost corresponding to the ordering
costs co.
Total setup costs are then co(D/Q). For carrying costs,
consider the production phase tr (when production and
demand occur) and the demand phase td (when
demand but no production occurs) separately. In the
production phase, inventory accumulates at the rate of
(r−d), and since the production phase lasts for tr = Q/r,
the maximal level of inventory at the end of each
production run is (r−d)Q/r. During the demand phase,
the inventory starts at (r−d)Q/r and linearly decreases
to zero at a rate of d. The average inventory level
during the entire cycle of duration tc = tr + td is then ½
(r−d)q/r. The total carrying cost per period is
½ch(r−d)Q/r. The total production- and inventoryrelated costs are then
TC = coD/Q + ½ch(r−d)Q/r
inventory on
hand
maximum
inventory level
tr
td
time
We find TC ' = −coD/Q2 + ½ch(r−d)/r and the unique
optimal lot size of
Q* 
2 Dco r
ch r  d
Example: A plant faces an annual demand of 200,000
bottles. It can produce them at a rate of 1,000 bottles
per day during each of the 300 working days in a year.
Setup costs for a production run are $1,000, and each
bottle has a carrying cost of 10¢ per bottle and year.
2(200,000)(1,000)
1,000
0.10
1,000  666.67
109,545 bottles. The corresponding costs are TC(Q*) =
$3,651.50.
We find Q* =
11.7 The EOQ with Stochastic Lead Time
Demand
The demand during the lead time will now be a
random variable which follows a known discrete
probability distribution. This may cause undesired
and unplanned stockouts and surpluses, as in the
figure below. We are only concerned about the
irregularity that occurs between when we have placed
an order, and the time the next shipment arrives.
inventory
level
r
t1
t2
tL
order placed
tL
stockout
t4
time
order arrives
t3
cp is the penalty cost per unit and stockout and
D  E (D) is the expected value of demand per year,
the lead time demand is dL (a random variable), and
the expected value of dL is d L  E (d L ) . The (discrete)
probability distribution of the lead time demand dL is
p(dL), F(dL) is the cumulative probability distribution
of dL. We assume that R  d L ≥ 0, i.e., on average,
there is still a positive inventory level when
replenishment occurs. If this condition were not to be
required, we would, on average, run out of stock at the
end of each cycle.
Therefore, we may regard the quantity R  d L as the
amount of stock that is kept at all times; it is the
expected safety stock or buffer stock.
11.7.1 A Model that Optimizes the Reorder
Point
We minimize the sum of the carrying costs for the
expected safety stock and the expected penalty costs
for stockouts. This sum TC1(R, Q) depends on the
reorder point R as well as on the order quantity Q. To
start, use the order quantity QEOQ, obtained
independently of the reorder point. This can be
justified because of the robustness of the economic
order quantity formula. We then obtain the partial
cost function
TC1(R, QEOQ) =
 D 

(d L  R) p(d L )
= ch( R  d L ) + c p 
 QEOQ 

 dL R

Therefore, we may regard the quantity R  d L as the
amount of stock that is kept at all times. For this
reason, this quantity is usually referred to as the
expected safety stock or buffer stock.
11.7.1 A Model that Optimizes the Reorder
Point
The objective in this section is to minimize the sum of
the carrying costs for the expected safety stock plus
the expected penalty costs for stockouts. This sum will
be denoted by TC1(R, Q), since it depends on the
reorder point R as well as on the order quantity Q. To
start, we will simply use the order quantity QEOQ,
which was obtained independently of the reorder point
by way of the economic order quantity. This can be
justified because of the robustness of the economic
order quantity formula. We then obtain the partial
cost function
TC1(R, QEOQ) =
 D 

(d L  R) p(d L ) ,
= ch( R  d L ) + c p 
 QEOQ 

 dL R

2 D co
, and where the first part of the
ch
relation is the cost for carrying the safety stock. The
summation is taken over all instances, in which
shortages occur, so that we compute the expected
where QEOQ =
shortage level. The optimal reorder point R* must
satisfy
P[d L  R*] 
ch QEOQ
cpD
.
As P[d L  R*] = 1  P[d L  R*] = 1  F( R * ),
F( R * ) = 1 
ch QEOQ
cpD
.
Here, dL is a discrete random variable, so its
cumulative distribution function F is a step function
taking only discrete values in the interval [0, 1]. It is
unlikely that the right-hand side of the above equation
will equal one of these discrete values. Instead, let R*
denote the smallest value that satisfies the inequality
F( R *) ≥ 1 
ch QEOQ
cpD
.
We need only consider the possible values of dL for R*.
Example: Consider again the battery charger example
of Section 10.2 with D = 800, co = $100 , and ch = $4 per
charger per year. The penalty costs are cp = $5 per
charger and stockout; the expected annual demand is
D = 800. The demand during lead time has the
probability distribution:
dL (units)
70
75
80
85
90
p(dL)
.1
.2
.2
.3
.2
QEOQ = 200 units, so that 1 
F(dL)
.1
.3
.5
.8
1.0
ch QEOQ
cpD
=1
4( 200)
=
5(800)
0.8, and since the smallest value of dL with F(dL) ≥ 0.8
equals 85, we have R * = 85. With expected demand
d L  E (d L ) 
xp L ( x) = 81.5, the expected safety
x
stock is R * d L = 85  81.5 = 3.5 units. The carrying
cost for the expected safety stock is then ch ( R * d L ) =
4(85  81.5) = $14, and the expected penalty cost is
 D 
800

cp
(d L  R*) p(d L ) = 5
(90  85)(0.2)
 QEOQ 
200



d L  R*
= $20. Stockouts occur only if dL > R * = 85, which
happens only in case dL = 90, an occurrence that has a
probability of 0.2.
11.7.2 A Stochastic Model with
Simultaneous Computation of Order
Quantity and Reorder Point
Now determine the order quantity Q and the reorder
point R simultaneously. The expected total cost of
ordering, carrying, and penalty is
TC2(Q, R) = co
D
 ch (½Q  R  d L ) +
Q
D
(d L  R) p(d L )
+ cp 
 Q  d R
L

Using partial differentiation we find


2D 

Q* 
c

c
(
d

R
*)
p
(
d
)
o
p
L
L  &

ch 

d L  R*



ch Q
F ( R*)  1 
.
cpD
R *is taken to be the smallest value that satisfies the
inequality. We will use an iterative procedure that
shuttles between these two relations, starting with an
order quantity Q*, uses the second of the two relations
to determine a reorder point R*, then using this
reorder point in the first relation to compute a revised
value of Q*, and so forth.
Example: Take the example in the previous section
with D = 800, co = $100, ch = $4 per charger per year,
cp = $5 per charger and stockout, and the above
probability distribution of the demand.
We obtain Q * = 200, so that R* = 85, just as before.
With the modified economic order quantity, a revised
value of Q* is
Q* 
2(800)
[100  5(5)(0.2)]  204.94 units.
4
Using this revised order quantity in the latter of the
two relations,
( 4)(204.94)
 .795, so that R* = 85
5(800)
again, and thus the procedure terminates.
F (R*) ≥ 1 
Comparing the results for Q* and R* of the simple
model in the previous subsection and the refined
approach in this subsection, in both cases the reorder
point is R* = 85 units, whereas the order quantity is
Q* = 200 units in the simple model which it is not very
different at 204.95  205 units in the refined model.
11.8 Extensions of the Basic Inventory
Models
We consider some inventory policies and define s as
the reorder level (what we have referred to so far as
the reorder point, i.e., the inventory level at which an
order is placed), R as the intervals at which the
inventory level is checked, and S as the inventory level
we have directly after a replenishment.
In a periodic review system, we check the inventory
levels at regular intervals R (e.g., hourly, daily, or
weekly), while in a continuous review system, we
continuously watch the inventory level.
An order-point, order-quantity, or (s, Q) policy involves
continuous review (i.e., R = 0) at which time an order
of a given magnitude Q is placed whenever the
inventory reaches a prespecified reorder level s. An
example of an (s, Q) policy is the two-bin system
described in Section 10.3.
An order-point, order-up-to-level, or (s, S) policy is
another continuous review policy. The inventory level
S is specified by the inventory manager as a level to be
attained directly after a shipment is received. Once the
reorder point s is reached, an order of size S  s is
placed, raising the inventory position level I p to S.
A periodic review, order-up-to-level, replenishment cycle
policy, or (R, S) policy is a periodic review policy. At
intervals of length R time units an order is placed to
raise the inventory position level I p to S.