Inventory Control Subject to
Quantity decisions made together with decision
When to order?
One of the major decisions in management of the inventory systems.
Impacts: inventory levels, inventory costs, level of service provided
One time decisions
Continuous review systems
Periodic review systems
Structure of timing decisions
(Q, R) System
(S, T) System
(s, S) System
Situation is common to retail and manufacturing environment
Consider seasonal goods, which are in demand during short period only.
Product losses its value at the end of the season. The lead time can be
longer than the selling season if demand is higher than the original
order, can not rush order for additional products.
Christmas ornament retailer
finished good inventory
“Christmas tree” model
Trivial problem if demand is known (deterministic case), in practical
situations demand is described as random variable (stochastic case).
Example: One-Time Decision
Mrs. Kandell has been in the Christmas tree business for years. She
keeps track of sales volume each year and has made a table of the
demand for the Christmas trees and its probability (frequency
Q – order quantity; Q* - optimal
D – demand: random variable with
probability density function f(D)
F(D) – cumulative probability function:
F(D) = Pr (demand ≤ D)
co – cost per unit of positive inventory
cu – cost per unit of unsatisfied demand
Economics marginal analysis: overage
and underage costs are balanced
Example: One-Time Decision (cont)
Shortages = lost profit + lost of goodwill
Overage = unit cost + cost of disposal of the overage
Either ignore the purchase cost, because it does not impact the optimal
solution or implicitly consider it in the overage and underage costs.
Expected overage cost of the order Q* is
Expected shortage cost is
For order Q* those two costs are equal:
F(Q*)co = (1-F(Q*))cu
- probability of satisfying demand during the
period, also is known as critical ratio
To calculate Q* we must use cumulative probability distribution.
Example: One-Time Decision (cont.)
Probability Cum Probability
cu co 40 40
Mrs. Kandell estimates that
if she buys more trees than
she can sell, it costs about
$40 for the tree and its
disposal. If demand is
higher than the number of
trees she orders, she looses
a profit of $40 per tree.
The Nature of Uncertainty
Suppose that we represent demand as
D = Ddeterministic + Drandom
If the random component is small compared to the deterministic
component, the models used in chapter 4 will be accurate. If not,
randomness must be explicitly accounted for in the model.
In chapter 5, assume that demand is a random variable with
cumulative probability distribution F(D) and probability density
D - continuous random variable, N(μ, σ)
estimated from history of demand
seems to model many demands accurately
Objective: minimize the expected costs – law of large numbers
The Newsboy Model
The critical ration can also be derived mathematically.
At the start of each day, a newsboy must decide on the
number of papers to purchase. Daily sales cannot be
predicted exactly, and are represented by the random
variable D with normal distribution N(μ, σ), where
μ = 11.73 and σ = 4.74
It can be shown that the optimal number of papers to
purchase is the fractile of the demand distribution given
by F(Q*) = cu / (cu + co). See Figure 5-4 when demand
is normal with μ = 11.73 and σ = 4.74, and the critical
fractile is 0.77.
Determination of the Optimal
Order Quantity for Newsboy Example
Q z 15.24
Lot Size Reorder Point Systems (Q, R)
levels are reviewed continuously
(the level of on-hand inventory is known at all times)
is random but the mean and variance of demand are
constant (stationary demand)
is a positive lead time, τ.
Set-up each time an order is placed at $K per order
Unit order cost at $c for each unit ordered
Holding at $h per unit held per unit time ( i.e. per year)
Penalty cost of $p per unit of unsatisfied demand
For EOQ model there was a single decision variable Q.
The value of the reorder level, R, was determined by Q:
R = λτ, if τ < T
R = λ*MOD(τ/T), if τ > T
In the stochastic demand case, we treat Q and R as
independent decision variables
R is chosen to protect against uncertainty of demand
during the lead time
Q is chosen to balance the holding and set-up costs
Changes in Inventory Over Time
for Continuous-Review (Q, R) System
Order Q whenever inventory is at level R
The Expected Number of Stockouts
The Cost Function
The average annual cost is given by:
G (Q, R ) h(Q / 2 R ) K / Q p n( R ) / Q.
Interpret n(R) as the expected number of stockouts per cycle
calculated using the standardized loss function L(z):
The standardized loss integral values appear in Table A-4.
The optimal values of (Q,R) that minimizes G(Q,R) can be
found by iterating between equations: Q 2 ( K pn( R))
1 F ( R ) Qh / p
Service Levels in (Q,R) Systems
In many circumstances, the penalty cost, p, is difficult
to estimate. For this reason, it is common business
practice to set inventory levels to meet a specified
service objective instead. The two most common
service objectives are:
Type 1 service: Choose R so that the probability of not
stocking out in the lead time is equal to a specified
Type 2 service. Choose both Q and R so that the
proportion of demands satisfied from stock equals a
For type 1 service, if the desired service level is α then one
finds R from F(R)= α and sets Q=EOQ
Type 2 service requires a complex iterative solution
procedure to find the best Q and R.
See Example 5.5 on page 256.
Type 1 finds fraction of periods in which there is no stockout (no matter one item short or 1000).
Type 2 measures the percentage of all filled orders in all
periods (95% or 98% service objective).
Comparison of Type 1 and Type 2 Services
For a type 1 service objective there are two cycles out of ten in which a
stock-out occurs, so the type 1 service level is 80%. For type 2 service,
there are a total of 1,450 units demand and 55 stockouts (which means
that 1,395 demand are satisfied). This translates to a 96% fill rate.
Other Continues Review System:
Order-Up-To-Level (R, S) System
Periodic Review System:
Order-Up-To-Level (s, S) vs (s, Q) System
(s, S) Policies
The (Q,R) policy is appropriate when inventory levels
are reviewed continuously. In the case of periodic
review, a slight alteration of this policy is required.
Define two levels, s < S, and let u be the starting
inventory at the beginning of a period. Then
If u s, order S u.
If u s, don't order.
In general, computing the optimal values of s and S is
much more difficult than computing Q and R.
Read Ch. 5 (5.1 – 5.8)
5.3, 5.6 – 5.8, 5.12, 5.15, 5.19, 5.25 – 5.27