Transcript Narrative Information Processing in Electronic Medical Report

Managing Uncertainty in Supply
Chain: Safety Inventory
Fall, 2014
Supply Chain Management:
Strategy, Planning, and Operation
Chapter 11
Byung-Hyun Ha
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
1
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
lead
time
lead
time
order arrival
order arrival
lead
time
order
arrival
2
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
lead
time
lead
time
lead
time
safety inventory
order arrival
order arrival
order
arrival
3
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?
inventory
level
OUL
Q'
Q
Q
Q
ROP
T
lead
time
T
lead
time
4
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
fr = 1 – s2/(d1 + d2 + d3 + d4)
CSL = 3/4
d1
d3
d2
d4
s2
time horizon to be considered
5
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 of demand of period i
ij: correlation coefficient of demand between periods i and j
 L
 L
E (X)  E  Xi    E (Xi )
 i 1  i 1
 Standard deviation and coefficient of variation (cv)
 L
 L
Var(X)  Var  Xi   Var(Xi )  2 ρij Var(Xi )Var(X j )
i j
 i 1  i 1
cv  Var(X) E(X)
6
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 without addition cost by stock out (no lost sales)
 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 = DL
 Var(X)1/2 = L = (L)1/2D
7
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
lead
time
lead
time
ROP
E(X) = DL
ss
order arrival
order arrival
8
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.
9
Determining Level of Safety Inventory
 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 = 21/2500 = 707
• CSL = FS(ss/L) = FS(1.414) = 92%
inventory
level
PDF
of X
DL
ss
ROP
Pr(X  ROP)
= Pr(Z  ss/L)
= CSL
DL
ss
lead
time
0
DL
ROP =
DL + ss
10
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
• (Appendix 11C)



ROP
ESC   max(0, x  ROP) f ( x)dx  
( x  ROP) f ( x)dx
 ss 1  FS ss σ L   σ L fS ss σ L 
• 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 
11
Determining Level of Safety Inventory
 Evaluating fill rate (cont’d)
ESC  

ROP
x  ROP f ( x)dx
e   x  DL  2 σ L
x  DL  ss 

dx
DL  ss
2 π σL
2

2
z
ez 2
   z σ L  ss 
dz
ss σ L
2π
2

  ss 

  ss 

ss σ L
ss σ L
e
z2 2
2π
dz  σ L 

z
ss σ L
f S  z dz  σ L 

ss σ L
e
x  DL dx 1
,

σL
dz σ L
z2 2
2π
dz
f S z dz
  ssFS  z ss σ L  σ L  f S  z ss σ L   ss 1  FS ss σ L   σ L f S ss σ L 


12
Determining Level of Safety Inventory
 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
f(x)
DL
ss
Pr(X  ROP)
= Pr(Z  ss/L)
= CSL
0
DL
ROP =
DL + ss
13
Determining Level of Safety Inventory
 Determining safety inventory given desired fr
 Input
• fr, Q, L
 Determining safety inventory, ss
• fr = 1 – ESC/Q
1  fr  Q  ESC  ss1 FS ss σL   σL fS ss σL 
 No analytical solution
• ESC is a decreasing function with regard to ss.
• Using line search, e.g., Goal Seek in Excel
14
Determining Level of Safety Inventory
 Impact of desired product availability on safety inventory
Fill Rate
Safety Inventory
97.5%
98.0%
98.5%
99.0%
99.5%
67
183
321
499
767
 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 = FS–1(CSL)L = FS–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)
15
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 = (LD2 + D2sL2)1/2
 KEY POINT
 sL   ss 
16
Impact of Supply Uncertainty on Safety Inv.
 Demand during lead time (cont’d)
 Let Zl = X1 + X2 + ... + Xl



l 1
l 1
l 1
E X   E Zl   PrY  l    lD  PrY  l   D l  PrY  l   DL
 
 




E X   E Z  P rY  l    Var Z l   E Z l   P rY  l 
2
2
l
l 1


l 1
2

  l  σ  l D  P rY  l   σ
l 1
2
D
2
2
 


2
D
 l  P rY  l   D
l 1
 Lσ 2D  D 2 E Y 2  Lσ 2D  D 2 VarY   E Y 

 Lσ 2D  D 2 sL2  L2
 

2


2
2
l
  P rY  l 
l 1
VarX  E X2  EX  Lσ2D  D2 sL2
2
17
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
i1 FS1 CSL  L  σi  FS1 CSL  L  i1 σi
k
k
• Centralized: aggregated inventory
FS1 CSL  L  ik1σ i2  2i  j ρij σ i σ j
18
Impact of Aggregation on Safety Inventory
 Measuring impact (cont’d)
 Holding-cost savings on aggregation per unit sold, HCS
HCS 
FS1 CSL  L  H

k
i 1
Di


k
k
2
σ


σ
i 1 i  2 i  j ρ ij σ i σ j
i 1 i

• 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 = .
k
2
σ


σ
i1 i
i 1 i  2 i  j ρ ij σ i σ j  kσ  k σ
k
 Disadvantage of aggregating inventories
 Increase in response time to customer order
 Increase in transportation cost to customer
19
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
20
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
21
Impact of Replenishment Policy on S. Inv.
 Continuous review policy
 ss = FS–1(CSL)L
 ROP = DL + ss
 Q by EOQ formula
• Suppose
• total cost TC (random variable)
• annual demand D = X1 + X2 + X3 + ...
• E(TC)  E(D)/QS + (Q/2 + ss)hC
 Q* 
2E DS
hC
22
Impact of Replenishment Policy on S. Inv.
 Periodic review policy
 ss = FS–1(CSL)T+L
 OUL = DT+L + ss
 T by EOQ-like formula
OUL
Q'
Q
Q
L
T
Q'
L
T
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
 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
25