ISEN 315Spring 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 Demand Stock-Outs

1 180 02 75 03 235 454 140 05 180 06 200 107 150 08 90 09 160 010 40 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) = aQ=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: