Lecture 14

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

Example

Service Levels in (Q,R) Systems • •

•

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 • •

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: