Managing Uncertainty in Supply Chain: Safety Inventory

Spring, 2014

Supply Chain Management:Strategy, Planning, and Operation

Chapter 11

Byung-Hyun Ha

2

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

3

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

order arrival

leadtime

order arrival

leadtime

order arrival

leadtime

4

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

order arrival

leadtime

order arrival

leadtime

order arrival

leadtime

safety inventory

5

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?

leadtime

inventorylevel

Q

ROP

leadtime

Q

OUL

Q

TT

Q'

6

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 availabl

e inventory

Cycle service level (CSL)• Fraction of replenishment cycles that end with all customer demand

met

d1 d2d3 d4

s2

fr = 1 – s2/(d1 + d2 + d3 + d4)CSL = 3/4

time horizon to be considered

7

Determining Level of Safety Inventory

Assumptions No supply uncertainty (deterministic)

• L: constant lead time

Measuring demand uncertainty (general model) Notation

• Xi: demand of period i (random variable)

• X: demand during lead time L; X = X1 + X2 + ... + XL

• Di , 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)

L

ii

L

ii EEE

11

)()( XXX

jijiij

L

ii

L

ii VarVarVarVarVar )()(ρ2)()(

11

XXXXX

)()( XX EVarcv

8

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 t

heorem 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) = DL = DL

Var(X)1/2 = L = (L)1/2D

9

Determining Level of Safety Inventory

Evaluating cycle service level and fill rate Evaluating safety inventory (ss)

• ss = ROP – E(X) = ROP – DL

• Average inventory = Q/2 + ss

order arrival

leadtime

order arrival

leadtime

ROP

ss

E(X) = DL

10

Determining Level of Safety Inventory

Evaluating cycle service level and fill rate Evaluating cycle service level (CSL)

• CSL = Pr(X ROP) = F(ROP) = F(DL + ss)

• where F(x) is the cumulative distribution function of a normally distributed random variable X with mean DL and standard deviation L.

Or• CSL = Pr(X ROP) = Pr((X – DL)/L (ROP – DL)/L)

CSL = Pr(Z ss/L)

CSL = FS(ss/L)

• where Z is a standard normal random variable and FS(z) is the cumulative standard normally distribution function.

11

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 – DL = 1,000, L = 21/2500 = 707

• CSL = FS(ss/L) = FS(1.414) = 92%

Pr(X ROP)= Pr(Z ss/L)

= CSL

Determining Level of Safety Inventory

leadtime

inventorylevel

ROP

DL

PDFof X

ROP =DL + ss

0

DL

ss

DL ss

12

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.

• fS(x) is the standard normal density function.

Observation (KEY POINT)• ss CSL, fr • Q fr

LLL

ROP

ssfssFss

xxfROPxxxfROPxESC

σσσ1

d)()(d)(),0max(

SS

13

Determining Level of Safety Inventory

Evaluating fill rate (cont’d)

LLLssLss

ssLss

ss

z

Lss

z

ss

z

L

ssDL

Dx

L

ROP

ssfssFsszfzFss

zzfzzfss

zzzss

zssz

xssDx

xxfROPxESC

LL

LL

LL

L

L

LL

σσσ1σ

dσd

dπ2

eσd

π2

e

dπ2

eσ

dσπ2

e

d)(

SSσSσS

σ Sσ S

σ

2

σ

2

σ

2

σ2

22

2

22

LL

L

z

xDxz

σ

1

d

d,

σ

14

Determining safety inventory given desired CSL Input

• CSL, L

Determining safety inventory, ss• F(ROP) = F(DL + ss) = CSL

ss = F–1(CSL) – DL

Or• FS(ss/L) = CSL

• ss/L = FS–1(CSL)

ss = FS–1(CSL)L

Determining Level of Safety Inventory

f(x)

Pr(X ROP)= Pr(Z ss/L)

= CSL

DL ROP =DL + ss

0

DL ss

15

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

LLL ssfssFssESCQfr σσσ11 SS

16

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 d

esired product availability (CSL and fr).

Impact of desired product uncertainty on safety inventory ss = FS

–1(CSL)L = FS–1(CSL)(L)1/2D

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 Rate 97.5% 98.0% 98.5% 99.0% 99.5%

Safety Inventory 67 183 321 499 767

17

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 = sL: 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) = DL = DL

Var(X)1/2 = L = (LD2 + D2sL

2)1/2

KEY POINT sL ss

18

Impact of Supply Uncertainty on Safety Inv.

Demand during lead time (cont’d) Let Zl = X1 + X2 + ... + Xl

DLllDllDlEElll

l

111

PrPrPr YYYZX

2222

222222

1

22

1

2

1

222

1

2

1

22

σ

σσ

PrPrσPrσ

PrPr

LsDL

EVarDLEDL

llDlllDll

lEVarlEE

LD

DD

llD

lD

lll

ll

YYY

YYY

YZZYZX

22222 σ LD sDLEEVar XXX

19

Impact of Aggregation on Safety Inventory

Examples HP\Best Buy vs. Dell, Amazon.com vs. Barnes & Noble

Measuring impact Notation

• Di: 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

k

i iS

k

i iS LCSLFLCSLF1

1

1

1 σσ

jiijjiikiS LCSLF σσρσ

221

1

20

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

jiijjiiki

k

i ik

i i

S

D

HLCSLFHCS σσρ2σσ 2

11

1

1

σσσσρσσ kkjiijjiiki

k

i i 2211

21

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

22

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

23

Impact of Replenishment Policy on S. Inv.

Continuous review policy ss = FS

–1(CSL)L

ROP = DL + ss

Q by EOQ formula

Periodic review policy (assuming T is given) ss = FS

–1(CSL)T+L

OUL = DT+L + ss

Optimal T*?

L

Q

OUL

Q

TT

Q'

L

Q'

T

24

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 f

inal customer)• e.g., more retailer safety inventory less required to distributor

Determining stages who carry inventory most• Balancing responsiveness and efficiency!

25

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