Managing Uncertainty in Supply Chain: Safety Inventory Spring, 2014 Supply Chain Management: Strategy, Planning, and Operation Chapter 11 Byung-Hyun Ha

1 Contents  Introduction  Determining the appropriate level of safety inventory  Impact of supply uncertainty on safety inventory  Impact of aggregation on safety inventory  Impact of replenishment policies on safety inventory  Managing safety inventory in a multi-echelon supply chain  Estimating and managing safety inventory in practice

2 Introduction  Uncertainty in demand  Forecasts are rarely completely accurate.  If you kept only enough inventory in stock to satisfy average demand, half the time you would run out.  Safety inventory  Inventory carried for the purpose of satisfying demand that exceeds the amount forecasted in a given period  Average inventory = cycle inventory + safety inventory orderarrival lead time orderarrival lead time orderarrival lead time

3 Introduction  Tradeoff in raising safety inventory  Higher levels of product availability and customer service  Increasing holding costs, risk in obsolescence  Factors to determine appropriate level of safety inventory  Uncertainty of both demand and supply  Desired level of product availability orderarrival lead time orderarrival lead time orderarrival lead time safety inventory

4 Introduction  Replenishment policies (very basic)  Continuous review Inventory is continuously monitored and an order of size Q is placed when the inventory level reaches the reorder point ROP  Periodic review Inventory is checked at regular (periodic) intervals T and an order is placed to raise the inventory to the order-up-to level OUL  Decision variables? lead time inventory level Q ROP lead time Q OUL Q TT Q'

5 Introduction  Measuring product availability  Product fill rate (fr) Fraction of demand that is satisfied from product in inventory  Order fill rate Fraction of orders (i.e., multiple products) that are filled from available inventory  Cycle service level (CSL) Fraction of replenishment cycles that end with all customer demand met d1d1 d2d2 d3d3 d4d4 s2s2 fr = 1 – s 2 /(d 1 + d 2 + d 3 + d 4 ) CSL = 3/4 time horizon to be considered

6 Determining Level of Safety Inventory  Assumptions  No supply uncertainty (deterministic) L: constant lead time  Measuring demand uncertainty (general model)  Notation X i : demand of period i (random variable) X: demand during lead time L; X = X 1 + X X L D i,  i : mean and standard deviation demand of period i  ij : correlation coefficient of demand between periods i and j  Standard deviation and coefficient of variation (cv)

7 Determining Level of Safety Inventory  Further assumptions  Demand of each of L periods is independent.  Demand for each period is normally distributed, or, central limit theorem can be effectively applied (with sufficiently large L).  Taking continuous review policy  Back-order (not lost sales) by stock out  Demand statistics  D: average demand of each period   D : standard deviation of demand of each period  Demand during lead time, X  X is normally distributed.  E(X) = D L = DL  Var(X) 1/2 =  L = (L) 1/2  D

8 Determining Level of Safety Inventory  Evaluating cycle service level and fill rate  Evaluating safety inventory (ss) ss = ROP – E(X) = ROP – D L Average inventory = Q/2 + ss orderarrival lead time orderarrival lead time ROP ss E(X) = D L

9 Determining Level of Safety Inventory  Evaluating cycle service level and fill rate  Evaluating cycle service level (CSL) CSL = Pr(X  ROP) = F(ROP) = F(D L + ss) where F(x) is the cumulative distribution function of a normally distributed random variable X with mean D L and standard deviation  L.  Or CSL = Pr(X  ROP) = Pr((X – D L )/  L  (ROP – D L )/  L ) CSL = Pr(Z  ss/  L ) CSL = F S (ss/  L ) where Z is a standard normal random variable and F S (z) is the cumulative standard normally distribution function.

10  Evaluating cycle service level and fill rate (cont’d)  Example 11-2 Input Q = 10,000, ROP = 6,000, L = 2 periods D = 2,500/period,  D = 500 Cycle service level ss = ROP – D L = 1,000,  L = 2 1/2  500 = 707 CSL = F S (ss/  L ) = F S (1.414) = 92% Pr(X  ROP) = Pr(Z  ss/  L ) = CSL Determining Level of Safety Inventory lead time inventory level ROP DLDL PDF of X ROP = D L + ss 0 DLDL ss DLDL

11 Determining Level of Safety Inventory  Evaluating fill rate (fr)  Fill rate, fr = (Q – ESC)/Q = 1 – ESC/Q where ESC is expected shortage per replenishment cycle  Expected shortage per replenishment cycle (Appendix 11C) where f(x) is the probability density function of X. f S (x) is the standard normal density function.  Observation (KEY POINT) ss   CSL, fr  Q   fr 

12 Determining Level of Safety Inventory  Evaluating fill rate (cont’d)

13  Determining safety inventory given desired CSL  Input CSL,  L  Determining safety inventory, ss F(ROP) = F(D L + ss) = CSL  ss = F –1 (CSL) – D L  Or F S (ss/  L ) = CSL ss/  L = F S –1 (CSL)  ss = F S –1 (CSL)  L Determining Level of Safety Inventory f(x)f(x) Pr(X  ROP) = Pr(Z  ss/  L ) = CSL DLDL ROP = D L + ss 0 DLDL ss

14 Determining Level of Safety Inventory  Determining safety inventory given desired fr  Input fr, Q,  L  Determining safety inventory, ss fr = 1 – ESC/Q  No analytical solution ESC is a decreasing function with regard to ss. Using line search, e.g., Goal Seek in Excel

15 Determining Level of Safety Inventory  Impact of desired product availability on safety inventory  KEY POINT The required safety inventory grows rapidly with an increase in the desired product availability (CSL and fr).  Impact of desired product uncertainty on safety inventory  ss = F S –1 (CSL)  L = F S –1 (CSL)  (L) 1/2  D  KEY POINT The required safety inventory increases with an increase in the lead time and the standard deviation of periodic demand.  Reducing safety inventory without decreasing product availability Reduce supplier lead time, L (e.g., Wal-Mart) Reduce uncertainty in demand,  L (e.g., Seven-Eleven Japan) Fill Rate97.5%98.0%98.5%99.0%99.5% Safety Inventory

16 Impact of Supply Uncertainty on Safety Inv.  Assumptions  Uncertain supply Y: lead time for replenishment (random variable) E(Y) = L: average lead time Var(Y) 1/2 = s L : standard deviation of lead time  D: average demand of each period   D : standard deviation of demand of each period  Demand during lead time, X  E(X) = D L = DL  Var(X) 1/2 =  L = (L  D 2 + D 2  s L 2 ) 1/2  KEY POINT  s L   ss 

17 Impact of Supply Uncertainty on Safety Inv.  Demand during lead time (cont’d)  Let Z l = X 1 + X X l

18 Impact of Aggregation on Safety Inventory  Examples  HP\Best Buy vs. Dell, Amazon.com vs. Barnes & Noble  Measuring impact  Notation D i : mean weekly demand in region i, i = 1,..., k  i : standard deviation of weekly demand in region i, i = 1,..., k  ij : correlation of weekly demand for regions i and j L: lead time in weeks CSL: desired cycle service level  Required safety inventory Decentralized: local inventory in each region Centralized: aggregated inventory

19 Impact of Aggregation on Safety Inventory  Measuring impact (cont’d)  Holding-cost savings on aggregation per unit sold, HCS where H is the holding cost per unit.  Observations HCS  0 CSL   HCS , L   HCS , H   HCS ,  ij   HCS   Square-root law Suppose  ij = 0 and  i = .  Disadvantage of aggregating inventories  Increase in response time to customer order  Increase in transportation cost to customer

20 Impact of Aggregation on Safety Inventory  Exploiting benefits from aggregation  Information centralization Virtual aggregation of inventories e.g., McMaster-Carr, Gap, Wal-Mart  Specialization Items with high cv  centralization (usually slow-moving) Items with low cv  decentralization (usually fast-moving) e.g., Barnes & Nobles + barnesandnoble.com  Product substitution Manufacturer-driven substitution Substituting a high-value product for lower-value product that is not in inventory No lost sales & savings from aggregation vs. substitution cost Customer-driven substitution Suggesting a different product instead of out-of-inventory one

21 Impact of Aggregation on Safety Inventory  Exploiting benefits from aggregation (cont’d)  Component commonality Using common components in a variety of different products Safety inventory savings vs. component cost increasing by flexibility  Postponement Differentiation  disaggregated inventories Inventory cost savings by delayed differentiation (usually with component commonality) Examples Dell, Benetton

22 Impact of Replenishment Policy on S. Inv.  Continuous review policy  ss = F S –1 (CSL)  L  ROP = D L + ss  Q by EOQ formula  Periodic review policy (assuming T is given)  ss = F S –1 (CSL)  T+L  OUL = D T+L + ss  Optimal T*? L Q OUL Q TT Q' L T

23 Managing Safety Inv. in Multiechelon SC  Two-stage case  Inventory relationship Supplier’s safety inventory   short lead time to retailer  retailer’s safety inventory can be reduced And vice versa.  Implications Safety inventories of all stages in multiechelon SC should be related.  Inventory management decision  Considering echelon inventory (all inventory between a stage to final customer) e.g., more retailer safety inventory  less required to distributor  Determining stages who carry inventory most Balancing responsiveness and efficiency!

24 Further Discussion  Role of IT in inventory management  Appendix 11D (SKIP)  Estimating and managing safety inventory in practice  Account for the fact that supply chain demand is lumpy  Adjust inventory policies if demand is seasonal  Use simulation to test inventory policies  Start with a pilot  Monitor service levels  Focus on reducing safety inventories