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1 Supply Chain Analysis Tools MS&E 262 Supply Chain Management April 14, 2004 Inventory/Service Trade-off Curve Motivation High Inventory Low Poor Service Good Analytical tools help lower the curve! 1

2 Outline EOQ Newsvendor Lot Size Reorder (Q,R) Model Periodic Review (T,S) Model Random Lead Times Risk Pooling/Consolidation Example Multi-echelon Example Other Improvements Economic Order Quantity Model How much to order/produce? Fixed order cost of \$ K Inventory holding cost of \$ h = \$ Ic Shortages prohibited Deterministic (constant) demand rate per year, D Inventory Level Slope = -D Q T (T = Q/D) Time, t 2

3 EOQ Model - Derivation Annual Holding + Setup Cost G(Q) = cd + KD/Q + IcQ/2 Purchasing Setup Holding Total = G(Q)-cD Holding = IcQ/2 Q* Q * = 2KD Ic Setup = KD/Q Q EOQ Model Sensitivities/Shortcomings rule 40 % error in an input parameter results in 20 % error in Q The result is a 2 % increase in the costs, G(Q) Cost function is relatively insensitive to errors in Q Shortcomings of EOQ Model? Zero lead time (easily extended to fixed lead time) Infinite production rate (finite production rate, P, if P > D) No shortages allowed (easily extendable) Constant, deterministic demand rate 3

4 Newsvendor Model How much to order/produce? Underage cost/unit c u Overage cost/unit c o Likelihood f(d) Mean Demand, d Newsvendor Model (cont.) F( Q * ) = c u cu + c o Shortcomings of Newsvendor Model? No consideration of positive lead times One shot model (can be extended to multiple periods) No setup cost for placing orders included 4

5 Lot Size Reorder Point (Q,R) Model We need to decide two things: How much to order each time we place an order (Q)? At which reorder point (R) do we place an order? Inventory Position R s Q τ Safety Stock Time τ= Lead Time (Q,R) Model Notations Average demand rate/year Setup cost Variable cost Holding cost Order quantity Reorder point Lead time Safety stock λ K c h=ic Q R τ s 5

6 (Q,R) Model Unit Shortage Cost p The expected total annual cost is Kλ Q λ G ( Q, R) = + Ic + R λτ + p ( x R) f ( x) dx Q 2 Q R Setup Holding Shortage n Defining n(r) = E [# of units short in a cycle] we obtain 2λ( K + pn( R)) Q = Ic QIc F( R) = 1 pλ (Q,R) Model Unit Shortage Cost p (cont.) For normally distributed demand, define where z = (R-µ L )/σ L. Hence n(r) = σ L L(z) Use the approximate solution: Q = EOQ Obtain z from tables for L(z) R = µ L + zσ L L ( z) = ( t z) φ( t) dt µ L : mean lead time demand; σ L : standard deviation of lead time demand z 6

7 (Q,R) Model Service Level Approaches Type 1: α = Prob(no stock-out in lead time) Type 2: β = Proportion of demand met from on-hand stock Recall n(r) = E[# of units short in cycle] n( R) σ LL( z) = = 1 β Q Q Bad Better (1 β ) Q L( z) = σ L Periodic Review (T,S) Model Review every T units Order up to S units at every review Response Time = T+τ Inventory Lead Time S Τ τ Time τ 7

8 (T,S) Model (cont.) Where T = EOQ/λ S = µ τ+t τ+t + z σ τ+t τ+t µ τ+t = mean demand over τ+t periods σ τ+t = standard deviation of demand over τ+t periods z see (Q,R) model Hence Safety Stock = z σ τ+t Lead Time Demand Variability Expectation of Sum = Sum of Expectations General Variance Formula (Lead time = τ periods, demand in period i = d i ) σ 2 d LT = τ i= 1 σ 2 d i + 2 i< j COV ( d i, d j ), where d LT = τ i= 1 d i Variance of Sum = Sum of Variances (for independent variables) Example: Lead time demand * Mean µ L = τ µ Variance σ L2 = τ σ 2 * assuming independence between periods 8

9 Random Lead Times If lead time is random, with mean τ and variance s 2 And demand in time t has mean µt and variance σ 2 t Then the demand during (random) lead time has * Mean µ L = τµ Variance σ L2 = µ 2 s 2 +τσ 2 * Assuming orders do not cross and successive lead times are independent Lead Time Example Supplier s Production Time = 3 weeks Transportation Time from Supplier = 4 weeks End Product Demand, per week ~ N(µ,σ 2 ) Case 1: No variability in transportation time ~ N(4,0) weeks τ = 7 weeks µ L = 7µ, σ L2 = 7σ 2 Case 2: Transportation Time from Supplier ~ N(4,0.81) weeks τ = 7 weeks µ L = 7µ, σ L2 = µ 2 (0.81) + 7σ 2 9

10 Uniform vs. Non-uniform Service Levels Risk Pooling/Consolidation Multi-Echelon Analysis Postponement Lead-time Reduction Review Period Reduction Variable Lead-time Risk Pooling/Consolidation What is meant by Risk Pooling? Example Laser Printer Supply Chain 10

11 Laser Printer: Finished Goods Logistics Penang, Malaysia Long Beach CA, USA Memphis TN, USA Represents a DC location for distributor D1 UPS Ground Map for Memphis, TN 11

12 Laser Printer s Distributor Network Assume the following distributor network: 5 Independent Distributors (D1, D2, D3, D4, D5) Each distributor operates 8 DCs across the US Who are the distributors customers? Who owns the printer inventory? Relevant inventory metrics for a DC? Laser Printer s Distributor Network Opportunity for Risk Pooling For any particular distributor? For any particular location (e.g., Memphis, TN)? For the original equipment mfg (OEM)? 12

13 Assume that: Laser Printer s Distributor Network Demands at the multiple DCs are statistically independent. The means and standard deviations of demand for the multiple product DCs are identical. The leadtimes for the multiple DCs are identical/constant. The review periods at the DCs are identical. The safety factors for the DCs are identical. All DCs have the same inventory value. Laser Printer s Distributor Network Let: σ i = standard deviation of demand per period at DC i; L i = lead time for DC i; T i = review period for DC i; z i = safety factor for DC i; n = number of DCs in a region (e.g., DCs in Memphis, TN). 13

14 Laser Printer s Distributor Network Safety Stock at DC i = z i a i T i + L i n Total System Safety Stock = > i=1 z i a i T i + L i Safety Stock, Unpooled Total = nza T + L Safety Stock, Pooled Total = za Pooled T + L a Pooled = > i a i2 = a n Laser Printer s Distributor Network Reduction Effect through Pooling: SafetyStock,Unpooled Total?SafetyStock,Pooled Total SafetyStock,Unpooled Total 1? za Pooled T + L nza T + L = 1? a n na = 1? 1 n Reduction Effect in Memphis (n = 5): 55.3% Recall, the Distributors operate 8 separate DCs across the US 14

15 Laser Printer s Distributor Network What Happened? Benefit to the OEM in a particular region? Cost Savings attributed to Risk Pooling? Benefits, other than logistics? Multi-Echelon Supply Chain Analysis Concept Interactions of various levels in supply chain Levels are referred to as echelons Example Beer Game 15

16 Multi-Echelon Beer Game O I O I O I ORDER FLOW BEER! Retailer Wholesaler Distributor Factory D D D D D D D D Inventory Inventory Inventory Inventory PRODUCT FLOW Multi-Echelon Beer Game What Happened During Play? Communication? Lead Time? Holding vs. Backlogging Costs? Time Horizon/Duration? 16

17 Multi-Echelon Beer Game Applying Inventory Theory: Holding vs. Backlogging Costs? (p = \$1/wk, h = \$0.50/wk) Lead Time Effects? Information Sharing? What policy could you play? Postponement/Delayed Differentiation Concept Delay product differentiation until as late as possible in the production process Examples Advantages? Trade-offs? 17

19 Summary Avoid black box approach Understand underlying assumptions Perform sensitivity analysis on different parameters At worst, simulate the system! Analytical tools can help significantly improve supply chain performance! 19

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