ISEN 315
Spring 2011
Dr. Gary Gaukler
Lot Size Reorder Point Systems
Assumptions
– Inventory levels are reviewed continuously (the
level of on-hand inventory is known at all times)
– Demand is random but the mean and variance of
demand are constant. (stationary demand)
– There is a positive leadtime, τ. This is the time that
elapses from the time an order is placed until it
arrives.
– The costs are:
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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
The Inventory Control Policy
• Keep track of inventory position (IP)
• IP = net inventory + on order
• When IP reaches R, place order of size Q
Inventory Levels
Solution Procedure
• The optimal solution procedure requires iterating
between the two equations for Q and R until
convergence occurs (which is generally quite fast).
• A cost effective approximation is to set Q=EOQ and
find R from the second equation.
• In this class, we will use the approximation.
Example
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Selling mustard jars
Jars cost $10, replenishment lead time 6 months
Holding cost 20% per year
Loss-of-goodwill cost $25 per jar
Order setup $50
Lead time demand N(100, 25)
Example
Example
Service Levels in (Q,R) Systems
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In many circumstances, the penalty cost, p,
is difficult to estimate
Common business practice is to set
inventory levels to meet a specified service
objective instead
Service objectives: Type 1 and Type 2
Service Levels in (Q,R) Systems
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Type 1 service: Choose R so that the
probability of not stocking out in the lead
time is equal to a specified value.
Type 2 service. Choose both Q and R so
that the proportion of demands satisfied
from stock equals a specified value.
Comparison
Order Cycle
1
2
3
4
5
6
7
8
9
10
Demand
180
75
235
140
180
200
150
90
160
40
Stock-Outs
0
0
45
0
0
10
0
0
0
0
For a type 1 service objective there are two cycles out of ten
in which a stockout 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.
Type I Service Level
Determine R from F(R) = a
Q=EOQ
E.g., if a = 0.95:
“Fill all demands in 95% of the order
cycles”
Type II Service Level
a.k.a. “Fill rate”
Fraction of all demands filled without
backordering
Fill rate = 1 – unfilled rate
Type II Service Level
Summary of Computations
• For type 1 service, if the desired service level is α,
then one finds R from F(R)= α and Q=EOQ.
• For Type 2 service, set Q=EOQ and find R to satisfy
n(R) = (1-β)Q.
Imputed (implied) Shortage Cost
Why did we want to use service levels
instead of shortage costs?
Each choice of service level implies a
shortage cost!
Imputed (implied) Shortage Cost
Calculate Q, R using service level formulas
Then, 1 - F(R) = Qh / (pλ)
Imputed (implied) Shortage Cost
Imputed shortage cost vs. service level:
Exchange Curve
Safety stock vs. stockouts: