Narrative Information Processing in Electronic Medical Report
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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 = DL
Var(X)1/2 = L = (L)1/2D
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 = 21/2500 = 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
ez 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
ssFS 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 ss1 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)1/2D
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)
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 = (LD2 + D2sL2)1/2
KEY POINT
sL ss
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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 PrY l lD PrY l D l PrY l DL
E X E Z P rY l Var Z l E Z l P rY l
2
2
l
l 1
l 1
2
l σ l D P rY l σ
l 1
2
D
2
2
2
D
l P rY l D
l 1
Lσ 2D D 2 E Y 2 Lσ 2D D 2 VarY E Y
Lσ 2D D 2 sL2 L2
2
2
2
l
P rY l
l 1
VarX E X2 EX Lσ2D D2 sL2
2
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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
i1 FS1 CSL L σi FS1 CSL L i1 σi
k
k
• Centralized: aggregated inventory
FS1 CSL L ik1σ i2 2i j ρij σ i σ j
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Impact of Aggregation on Safety Inventory
Measuring impact (cont’d)
Holding-cost savings on aggregation per unit sold, HCS
HCS
FS1 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
i1 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
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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
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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)/QS + (Q/2 + ss)hC
Q*
2E DS
hC
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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
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