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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)
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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
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)
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 0 2 75 0 3 235 45 4 140 0 5 180 0 6 200 10 7 150 0 8 90 0 9 160 0 10 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) = 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
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:
