#### Transcript 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, 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: