### Chapter 7 Variability Safety Inventory 1-16

```Managing Business Process Flows:
Supply Chain Management Module
• Managing the Supply Chain
• Economies of Scale (Chapter 6)
• Managing Flow Variability: Safety Inventory (Chapter 7)
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Characteristics of Forecasts
Continuous Review System (Reorder Point Policy)
Inventory Pooling
Accurate Response (Newsvendor model)
Postponement / Delayed Differentiation
1
Demand uncertainty and
forecasting
• Forecasts depend on
– historical data
– “market intelligence”
• Forecasts are usually (always?) wrong.
• A good forecast has at least 2 numbers (includes a
measure of forecast error, e.g., standard deviation).
• Aggregate forecasts tend to be more accurate.
• The longer the forecast horizon, the less accurate the
forecast.
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7.2 Safety Inventory and Service
Level
Example 7.1
Throughput rate
Order Quantity
Reorder point
Definitions:
Cycle service level (SL)
Fill rate
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Reorder Point and Cycle
Service Level
desired cycle service level
1.0-(desired cycle service level)
SL = Prob (LTD <= ROP)
MeanDemand
Reorder Point
Reorder Point (ROP)
= Mean Demand over Leadtime + Safety Stock
= LTD + Isafety
I safety = Z * sLTD
Examples 7.3 & 7.4
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The standard normal distribution
F(z)
• Transform X = N(m,s) to z =
N(0,1)
z = (X - m) / s.
F(z) = Prob( N(0,1) < z)
F(z)
0
z
• Transform back, knowing z*:
X* = m + z*s.
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z
0 .0 0
0 .0 1
0 .0 2
0 .0 3
0 .0 4
0 .0 5
0 .0 6
0 .0 7
0 .0 8
0 .0 9
0 .0
0 .5 0 0 0
0 .5 0 4 0
0 .5 0 8 0
0 .5 1 2 0
0 .5 1 6 0
0 .5 1 9 9
0 .5 2 3 9
0 .5 2 7 9
0 .5 3 1 9
0 .5 3 5 9
0 .1
0 .5 3 9 8
0 .5 4 3 8
0 .5 4 7 8
0 .5 5 1 7
0 .5 5 5 7
0 .5 5 9 6
0 .5 6 3 6
0 .5 6 7 5
0 .5 7 1 4
0 .5 7 5 3
0 .2
0 .5 7 9 3
0 .5 8 3 2
0 .5 8 7 1
0 .5 9 1 0
0 .5 9 4 8
0 .5 9 8 7
0 .6 0 2 6
0 .6 0 6 4
0 .6 1 0 3
0 .6 1 4 1
0 .3
0 .6 1 7 9
0 .6 2 1 7
0 .6 2 5 5
0 .6 2 9 3
0 .6 3 3 1
0 .6 3 6 8
0 .6 4 0 6
0 .6 4 4 3
0 .6 4 8 0
0 .6 5 1 7
0 .4
0 .6 5 5 4
0 .6 5 9 1
0 .6 6 2 8
0 .6 6 6 4
0 .6 7 0 0
0 .6 7 3 6
0 .6 7 7 2
0 .6 8 0 8
0 .6 8 4 4
0 .6 8 7 9
0 .5
0 .6 9 1 5
0 .6 9 5 0
0 .6 9 8 5
0 .7 0 1 9
0 .7 0 5 4
0 .7 0 8 8
0 .7 1 2 3
0 .7 1 5 7
0 .7 1 9 0
0 .7 2 2 4
0 .6
0 .7 2 5 7
0 .7 2 9 1
0 .7 3 2 4
0 .7 3 5 7
0 .7 3 8 9
0 .7 4 2 2
0 .7 4 5 4
0 .7 4 8 6
0 .7 5 1 7
0 .7 5 4 9
0 .7
0 .7 5 8 0
0 .7 6 1 1
0 .7 6 4 2
0 .7 6 7 3
0 .7 7 0 4
0 .7 7 3 4
0 .7 7 6 4
0 .7 7 9 4
0 .7 8 2 3
0 .7 8 5 2
0 .8
0 .7 8 8 1
0 .7 9 1 0
0 .7 9 3 9
0 .7 9 6 7
0 .7 9 9 5
0 .8 0 2 3
0 .8 0 5 1
0 .8 0 7 8
0 .8 1 0 6
0 .8 1 3 3
0 .9
0 .8 1 5 9
0 .8 1 8 6
0 .8 2 1 2
0 .8 2 3 8
0 .8 2 6 4
0 .8 2 8 9
0 .8 3 1 5
0 .8 3 4 0
0 .8 3 6 5
0 .8 3 8 9
1 .0
0 .8 4 1 3
0 .8 4 3 8
0 .8 4 6 1
0 .8 4 8 5
0 .8 5 0 8
0 .8 5 3 1
0 .8 5 5 4
0 .8 5 7 7
0 .8 5 9 9
0 .8 6 2 1
1 .1
0 .8 6 4 3
0 .8 6 6 5
0 .8 6 8 6
0 .8 7 0 8
0 .8 7 2 9
0 .8 7 4 9
0 .8 7 7 0
0 .8 7 9 0
0 .8 8 1 0
0 .8 8 3 0
1 .2
0 .8 8 4 9
0 .8 8 6 9
0 .8 8 8 8
0 .8 9 0 7
0 .8 9 2 5
0 .8 9 4 4
0 .8 9 6 2
0 .8 9 8 0
0 .8 9 9 7
0 .9 0 1 5
1 .3
0 .9 0 3 2
0 .9 0 4 9
0 .9 0 6 6
0 .9 0 8 2
0 .9 0 9 9
0 .9 1 1 5
0 .9 1 3 1
0 .9 1 4 7
0 .9 1 6 2
0 .9 1 7 7
1 .4
0 .9 1 9 2
0 .9 2 0 7
0 .9 2 2 2
0 .9 2 3 6
0 .9 2 5 1
0 .9 2 6 5
0 .9 2 7 9
0 .9 2 9 2
0 .9 3 0 6
0 .9 3 1 9
1 .5
0 .9 3 3 2
0 .9 3 4 5
0 .9 3 5 7
0 .9 3 7 0
0 .9 3 8 2
0 .9 3 9 4
0 .9 4 0 6
0 .9 4 1 8
0 .9 4 2 9
0 .9 4 4 1
1 .6
0 .9 4 5 2
0 .9 4 6 3
0 .9 4 7 4
0 .9 4 8 4
0 .9 4 9 5
0 .9 5 0 5
0 .9 5 1 5
0 .9 5 2 5
0 .9 5 3 5
0 .9 5 4 5
1 .7
0 .9 5 5 4
0 .9 5 6 4
0 .9 5 7 3
0 .9 5 8 2
0 .9 5 9 1
0 .9 5 9 9
0 .9 6 0 8
0 .9 6 1 6
0 .9 6 2 5
0 .9 6 3 3
1 .8
0 .9 6 4 1
0 .9 6 4 9
0 .9 6 5 6
0 .9 6 6 4
0 .9 6 7 1
0 .9 6 7 8
0 .9 6 8 6
0 .9 6 9 3
0 .9 6 9 9
0 .9 7 0 6
1 .9
0 .9 7 1 3
0 .9 7 1 9
0 .9 7 2 6
0 .9 7 3 2
0 .9 7 3 8
0 .9 7 4 4
0 .9 7 5 0
0 .9 7 5 6
0 .9 7 6 1
0 .9 7 6 7
2 .0
0 .9 7 7 2
0 .9 7 7 8
0 .9 7 8 3
0 .9 7 8 8
0 .9 7 9 3
0 .9 7 9 8
0 .9 8 0 3
0 .9 8 0 8
0 .9 8 1 2
0 .9 8 1 7
2 .1
0 .9 8 2 1
0 .9 8 2 6
0 .9 8 3 0
0 .9 8 3 4
0 .9 8 3 8
0 .9 8 4 2
0 .9 8 4 6
0 .9 8 5 0
0 .9 8 5 4
0 .9 8 5 7
2 .2
0 .9 8 6 1
0 .9 8 6 4
0 .9 8 6 8
0 .9 8 7 1
0 .9 8 7 5
0 .9 8 7 8
0 .9 8 8 1
0 .9 8 8 4
0 .9 8 8 7
0 .9 8 9 0
2 .3
0 .9 8 9 3
0 .9 8 9 6
0 .9 8 9 8
0 .9 9 0 1
0 .9 9 0 4
0 .9 9 0 6
0 .9 9 0 9
0 .9 9 1 1
0 .9 9 1 3
0 .9 9 1 6
2 .4
0 .9 9 1 8
0 .9 9 2 0
0 .9 9 2 2
0 .9 9 2 5
0 .9 9 2 7
0 .9 9 2 9
0 .9 9 3 1
0 .9 9 3 2
0 .9 9 3 4
0 .9 9 3 6
2 .5
0 .9 9 3 8
0 .9 9 4 0
0 .9 9 4 1
0 .9 9 4 3
0 .9 9 4 5
0 .9 9 4 6
0 .9 9 4 8
0 .9 9 4 9
0 .9 9 5 1
0 .9 9 5 2
2 .6
0 .9 9 5 3
0 .9 9 5 5
0 .9 9 5 6
0 .9 9 5 7
0 .9 9 5 9
0 .9 9 6 0
0 .9 9 6 1
0 .9 9 6 2
0 .9 9 6 3
0 .9 9 6 4
2 .7
0 .9 9 6 5
0 .9 9 6 6
0 .9 9 6 7
0 .9 9 6 8
0 .9 9 6 9
0 .9 9 7 0
0 .9 9 7 1
0 .9 9 7 2
0 .9 9 7 3
0 .9 9 7 4
2 .8
0 .9 9 7 4
0 .9 9 7 5
0 .9 9 7 6
0 .9 9 7 7
0 .9 9 7 7
0 .9 9 7 8
0 .9 9 7 9
0 .9 9 7 9
0 .9 9 8 0
0 .9 9 8 1
2 .9
0 .9 9 8 1
0 .9 9 8 2
0 .9 9 8 2
0 .9 9 8 3
0 .9 9 8 4
0 .9 9 8 4
0 .9 9 8 5
0 .9 9 8 5
0 .9 9 8 6
0 .9 9 8 6
3 .0
0 .9 9 8 7
0 .9 9 8 7
0 .9 9 8 7
0 .9 9 8 8
0 .9 9 8 8
0 .9 9 8 9
0 .9 9 8 9
0 .9 9 8 9
0 .9 9 9 0
0 .9 9 9 0
3 .1
0 .9 9 9 0
0 .9 9 9 1
0 .9 9 9 1
0 .9 9 9 1
0 .9 9 9 2
0 .9 9 9 2
0 .9 9 9 2
0 .9 9 9 2
0 .9 9 9 3
0 .9 9 9 3
3 .2
0 .9 9 9 3
0 .9 9 9 3
0 .9 9 9 4
0 .9 9 9 4
0 .9 9 9 4
0 .9 9 9 4
0 .9 9 9 4
0 .9 9 9 5
0 .9 9 9 5
0 .9 9 9 5
3 .3
0 .9 9 9 5
0 .9 9 9 5
0 .9 9 9 5
0 .9 9 9 6
0 .9 9 9 6
0 .9 9 9 6
0 .9 9 9 6
0 .9 9 9 6
0 .9 9 9 6
0 .9 9 9 7
5
• L
• R=N(R, sR ) =Demand per unit time is normally distributed
with mean R and standard deviation sR ,
• Cycle service level = P(no stock out)
= P(demand during lead time < ROP)
= F(z*)
[use tables to find z*]
Safety stock
Reorder point
I safety  z  s
L
ROP = L x R + Isafety
Example 7.8 (see data from 7.1)
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Total variability in lead time demand =
Ls
2
R
 R s
Flow rate random
2
2
L
Flow rate constant
Example 7.9
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Learning Objectives: safety
stocks
Safety stock increases (decreases) with an increase
(decrease) in:
• demand variability or forecast error, z *s
L
• delivery lead time for the same level of service,
• delivery lead time variability for the same level of service.
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7.5 The Effect of Centralization
I
N * Z * s
LTD
N s
LTD
d
safety

c
 Z *
I safety
Example 7.10
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Concept of Centralization
•
•
•
•
•
Physical Centralization
Information Centralization
Specialization
Commonality
Postponement
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Learning Objectives:
Centralization/pooling
 Centralization reduces safety stocks (pooling) and cycle stocks
(economies of scale)
Can offer better service for the same inventory investment or
same service with smaller inventory investment.
 Different methods to achieve pooling efficiencies:
– Physical centralization,Information centralization, Specialization,
Commonality, Postponement/late customization.
 Cost savings are proportional to square root of # of locations
pooled.
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7.3 Newsvendor Problem
• Marginal benefit of stocking an additional unit = MB (e.g., retail price
- purchase price)
• Marginal cost of stocking an additional unit = MC (e.g., purchase
price - salvage price)
Given an order quantity Q, increase it by one unit if and only if the
expected benefit of being able to sell it exceeds the expected cost of
having that unit left over.
MB
 At optimal Q, P rob (D em and  Q ) 
MB  MC
Q* = R + ZsR
Data from example 7.5
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