## 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: • 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

• 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

Service Levels in (Q,R) Systems

• • • In many circumstances, the penalty cost, 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

p

,

Service Levels in (Q,R) Systems

• • 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 0 0 0 0 45 0 0 10 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 / (p λ)

Imputed (implied) Shortage Cost

Imputed shortage cost vs. service level:

Exchange Curve

Safety stock vs. stockouts: