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ISEN 315 Spring 2011 Dr. Gary Gaukler

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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

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The Inventory Control Policy Keep track of inventory position (IP) IP = net inventory + on order When IP reaches R, place order of size Q

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Inventory Levels

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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.

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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)

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Example

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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

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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.

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Comparison Order Cycle Demand Stock-Outs 11800 2 750 3235 45 41400 51800 6200 10 71500 8 900 91600 10 400 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.

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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

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Type II Service Level a.k.a. Fill rate Fraction of all demands filled without backordering Fill rate = 1 – unfilled rate

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Type II Service Level

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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.

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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!

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Imputed (implied) Shortage Cost Calculate Q, R using service level formulas Then, 1 - F(R) = Qh / (pλ)

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Imputed (implied) Shortage Cost Imputed shortage cost vs. service level:

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Exchange Curve Safety stock vs. stockouts:

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5-1 ISE 315 – Production Planning, Design and Control Chapter 5 – Inventory Control Subject to Unknown Demand McGraw-Hill/Irwin Copyright © 2005 by The.

5-1 ISE 315 – Production Planning, Design and Control Chapter 5 – Inventory Control Subject to Unknown Demand McGraw-Hill/Irwin Copyright © 2005 by The.

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