Inventory Control
IME 451, Lecture 3
Economic Order Quantity
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Harris (1913) developed this basic, widely
used model to find economic lot sizes
Balancing inventory holding costs against
setup (or order) costs
Assumptions
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Production is instantaneous, no capacity constraint
Delivery is immediate, no time lag
Demand is deterministic, no uncertainty
Demand is constant over time
A production run incurs a fixed setup cost
Products can be analyzed individually (single product only
or no interactions such as shared resources or machines)
EOQ Variables
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D = demand rate (in units per year)
c = unit production cost, not counting setup or
inventory cost (in dollars per unit)
A = fixed setup (ordering) cost to produce
(purchase) a lot (in dollars)
h = holding cost (in dollars per unit per year); if
holding costs are only due to interest then h=ic
where i is the annual interest rate
Q = lot size (in units); this is the decision
variable
EOQ Derivation
• Total annual cost Y(Q)
hQ AD
Y (Q) 

 cD
2
Q
• Find the minimum Y(Q) by dY (Q) h AD
  2 0
setting derivative w.r.t. Q
dQ
2 Q
equal to 0
• Check that the second
dY 2 (Q)
AD

2
derivative is positive for any
dQ2
Q3
positive Q (convex function,
so Q* is a min, not a max)
2 AD
*
Q 
• Solve first derivative for Q*
h
Problems with EOQ
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How realistic are assumptions (Instantaneous
production? Deterministic demand?)
Setup costs may be difficult to estimate, especially in
production environments rather than purchasing
systems
D
Average number of lots per year, F
F
Q
Total inventory investment, I
cD
Time between orders, T
I
Q
T
D
2F
2A
T 
hD
*
Sensitivity of EOQ Models
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Holding and setup costs are fairly insensitive to
lot size
Errors caused by restricting lot sizes to powers
of 2 are minimal (no more than 6%)
Powers of 2 ordering can facilitate sharing truck
resources (one week, two weeks, four weeks…)
Extensions involve non-instantaneous
production (economic production lot model),
backorders, major and minor setups, and
quantity discounts
Dynamic Lot Sizing (Wagner
Whitin)
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Notation
t = a time period = 1, 2, …, T where T is the planning horizon
Dt = demand in period t (in units)
ct = unit production cost (in dollars per unit)
At = setup cost to produce a lot in period t (in dollars)
ht = holding cost to carry a unit of inventory from period t to
period t+1 (in dollars per unit per period)
It = inventory (in units) leftover at the end of period t
Qt = lot size for period t (in units); there are T decision variables,
one for each period
Wagner Whitin Procedure
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Qt will be 0 or will be Dt, Dt+Dt+1 , Dt+Dt+1+Dt+2 ...
Produce nothing or produce exactly enough to
cover the current period plus some integer
number of future periods
Produce for the first period in the first period
For each subsequent period, decide whether it
is more economical to produce that period’s
demand in the current period or any previous
period
Follow example in book, pp. 60-61
Models for Uncertain Demand
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Finding statistical reorder points to account for
randomness in demand
News Vendor – single replenishment; vendor
buys paper at start of day and discards any
leftover at end of day
Base Stock – replenish inventory one unit at a
time but carry base stock to cover lag time
(Q,r) model – when inventory reaches or falls
below level r, order a quantity of Q items
News Vendor Model
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Assumptions
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Products are separable, consider 1 at a time
Planning is for a single period only
Demand is random
Deliveries are made in advance of demand
Costs of overage or underage are linear
Notation
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X = demand (in units), a random variable with mean m and
standard deviation s
G(x) = P(X<=x) = c.d.f. of demand
cs = cost (in dollars) per unit of shortage
co = cost (in dollars) per unit of overage
Q = production or order quantity (in units); this is the
decision variable
News Vendor Equations
cs
*
• To balance overage vs
G (Q ) 
co  cs
shortage costs, choose order
quantity Q*
*
• Assume that G is normal,

cs
Q
m 
*


G(Q )  F


where F is the cdf of the
 s  co  cs
standard normal function
• Find z in a standard normal
Q*  m
z
table (Table 1 at end of text)
s
• Solve for Q*
Q*  m  zs
Base Stock Model
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Assumptions
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Products can be analyzed individually
Demands occur one at a time
Unfilled demand is backordered
Replenishment leadtimes are fixed and known
Replenishments are ordered one at a time
Notation
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l = replenishment leadtime (in days), assumed constant
X = demand during replenishment leadtime (in units)
G(x) = P(X<=x) = c.d.f. probability demand during
replenishment leadtime is less than or equal to x
q = E[X] = mean demand (in units) during leadtime l
s = E[X] = standard deviation of demand (in units) during
leadtime l
Base Stock Model
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Notation (continued)
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h = cost to carry one unit of inventory for one year
b = cost to carry one unit of backorder for one year
r = reorder point (in units)
R = r + 1 = base stock level (in units)
s = r - q = safety stock level (in units)
S( R ) = fill rate or service level, fraction of orders filled from
stock as a function of R
B( R ) = average outstanding backorders
I( R ) = average on-hand inventory
Inventory position = on-hand inventory –
backorders + orders
Base Stock Equations
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For this model, at all times inventory position = R
Service level, S( R ) = G (R – 1)
R
Backorders are 0 if x < R
B( R)  q  [1  G( x)]
Backorders are x-R if x>=R
x 0
Expected backorder level B( R )
I ( R ) = R – q + B( R )
Use table 2.5 to find fill rates
b
*
G(R ) 
R*  q  zs
bh
(Q,r) Model
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Assumptions
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Similar to base stock except
There is a fixed cost for each replenishment
order, OR
There is a constraint on the number of orders per
year
Decide how much safety stock to carry to
cover leadtimes AND what quantity to
order
(Q,r) Notation
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D = expected demand per year (in units)
l = replenishment leadtime (in days)
X = demand during replenishment leadtime (in units), random variable
q = E[X] = Dl/365 = expected demand (in units) during leadtime l
s = E[X] = standard deviation of demand (in units) during leadtime l
G(x) = P(X<=x) = c.d.f. probability demand during replenishment leadtime is less than or equal
to x
A = setup cost per replenishment (in dollars)
c = unit production cost (in dollars per unit)
h = cost to carry one unit of inventory for one year
k = cost per stockout
b = cost to carry one unit of backorder for one year
r = reorder point (in units)
Q = replenishment quantity
s = r - q = safety stock level (in units)
F (Q,r) = order frequency, as a fuction of Q and r
S( Q,r ) = fill rate or service level, fraction of orders filled from stock as a function of Q and r
B( Q,r ) = average outstanding backorders
I( Q,r ) = average on-hand inventory
(Q,r) Equations
• Replenishment quantity Q affects cycle stock,
inventory that is held to avoid excessive
replenishment costs (like EOQ)
• Reorder point r affects safety stock, inventory
held to avoid stockouts (like Base Stock)
• Either minimize:
min fixedsetup cos t  backorder cos t  holding cos t
Q ,r
or
min fixedsetup cos t  stockout cos t  holding cos t
Q,r
Fixed Setup Cost & Backorder Cost
• Fixed setup (or order) cost
• First, set number of orders per year
• Then, find annual fixed order costs
D
F (Q, r ) 
Q
 D
F (Q, r ) A    A
Q
• Backorder Cost
• Inventory position is uniformly distributed between r+1 and r+Q
• Averaging the backorder level over the range r+1 to r+Q
1
B(Q, r )  B(r  1)  ...  B(r  Q)
Q
Stockout Cost in (Q,r) Model
• Penalizes poor customer service
• Charge a cost each time a stockout occurs
• Charge a penalty that is proportional to the time a customer
waits to have their order filled
1
S (Q, r )  1  B(r )  B(r  Q)
Q
• Approximations
• Type I (base stock) – computes # of stockouts per cycle,
underestimates S(Q,r)
S (Q, r )  G(r )
• Type II – neglects B(Q,r) term, also underestimates S(Q,r)
S (Q, r )  1 
B(r )
Q
Holding Costs in (Q,r) Model
• Inventory holding cost = hI(Q,r)
• First equation approximates and underestimates
average inventory, since demand is variable and
thus backorders sometimes occur
I (Q, r ) 
(Q  s)  ( s  1) Q  1
Q 1

s
 r q
2
2
2
• Exact formulation:
I (Q, r ) 
Q 1
 r  q  B(Q, r )
2
Backorder Cost Approach
• Compute Q and r values that minimize:
• approximate, since B(r) replaces B(Q,r)
D
Q 1

Y (Q, r )  A  bB(r )  h 
 r  q  B( r ) 
Q
 2

• Optimal Q* and r*
2 AD
Q* 
h
b
G (r*) 
bh
• Then, assume normal distribution for lead
time as in base stock model
r*  q  zs
Stockout Cost Approach
• Compute Q and r values that minimize:
• approximate, since B(r) replaces B(Q,r)
Y (Q, r ) 
D
B(r )
Q 1

A  kD
 h
 r  q  B( r ) 
Q
Q
 2

• Optimal Q* and r*
2 AD
Q* 
h
G (r*) 
• Assume normal distribution for lead time
kD
kD  hQ
r*  q  zs
• Note: larger Q results in smaller r* because a smaller
reorder point is needed to achieve the same fill rate
(Q,r) Insights
• Cycle stock increases as replenishment frequency
decreases
• Safety stock provides a buffer against stockouts
• The base stock model is a (Q,r) model where Q=1
• Increasing annual demand (D) increases Q
• Increasing demand during leadtime increases r (thus,
high annual demand or long leadtimes require
inventory)
• Increased demand variability increases r (for most
parts where high fill rates are desired)
• Increased holding costs (h) decreases both Q and r
(if it is expensive to hold inventory, avoid doing so!)