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Supply Chain Management: Strategy, Planning, and Operation

Seventh Edition

Chapter 12

Managing Uncertainty in a Supply Chain Safety Inventory

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Learning Objectives (1 of 2)

12.1 Understand the role of safety inventory in a supply chain.

12.2 Identify factors that influence the required level of safety inventory.

12.3 Evaluate the appropriate level of safety inventory for a supply chain.

12.4 Discuss the impact of supply uncertainty on safety inventory.

Learning Objectives (2 of 2)

12.5 Understand how aggregation helps reduce the required safety inventory in a supply chain.

12.6 Determine the impact of replenishment policies on safety inventory.

12.7 Improve the management of safety inventory in a multiechelon supply chain.

12.8 Identify managerial levers that lower safety inventory without hurting product availability.

The Role of Safety Inventory (1 of 3)

Safety inventory is required because demand and supply are uncertain

A shortage cold be the result of

Actual demand exceeding forecasted demand, or

Supply comes in later than anticipated

It is carried to satisfy demand that exceeds the amount forecasted

Raising the level of safety inventory increases product availability and thus the margin captured from customer purchases

Raising the level of safety inventory increases inventory holding costs

The Role of Safety Inventory (2 of 3)

Figure 12-1 Inventory Profile with Safety Inventory

Store manager places replenishment order when there are 300 purses remaining in inventory. If demand is higher than 300 purses during replenishment -> lost sales. So, manager decides to place an order when there are 400 purses remaining.

Identify the quantities on the chart

Bloomingdale’s purses:

Lot size = 600

Demand = 100 purses per week

Lead time = 3 weeks

The Role of Safety Inventory (3 of 3)

Three key questions

What is the appropriate level of product availability?

How much safety inventory is needed for the desired level of product availability?

What actions can be taken to reduce safety inventory without hurting product availability?

Summary of Learning Objective 1

Safety inventory helps a supply chain provide customers with a high level of product availability in spite of supply and demand uncertainty. It is carried just in case demand exceeds the amount forecasted or supply arrives later than expected.

Factors Affecting the Level of Safety Inventory

The desired level of product availability

The uncertainty of demand

The uncertainty of supply

Inventory replenishment policies

Factors Affecting the Level of Safety Inventory

The desired level of product availability

The uncertainty of demand

The uncertainty of supply

Flexibility in Inventory replenishment policies (e.g. placing orders anytime in the month)

Safety Inventory

Measuring Product Availability

Product availability refers to the ability to fill a customer’s order with available inventory

Product fill rate (fr)

Fraction of product demand satisfied from product in inventory

Measured over specified amount of demand, not over time

Order fill rate

Fraction of orders filled from available inventory

Measured over specified # of orders, not time (ex. One order for phone and laptop).

Tends to be lower than fill rate

Cycle service level (C S L)

Fraction of replenishment cycles that end with all customer demand being met

Replenishment cycle – the interval between two successive replenishment deliveries

Measuring Demand Uncertainty

Demand uncertainty is the random component of demand

D = Average demand per period

σD = Standard deviation of demand (forecast error) per period

Lead time (L) is the gap between when an order is placed and when it is received -> The company is exposed to uncertainty of demand during lead time

Evaluating Demand Distribution over L Periods

The coefficient of variation

Demand during lead time is normally distributed with a mean D*L and std. dev σL

Measuring Supply Uncertainty

Supply Uncertainty: An order from a supplier arriving late or not arriving complete.

Supply uncertainty is described by uncertainty of the lead time.

Lead time (L) is normally distributed with

L = Average lead time

sL = Standard deviation of lead time

Demand during the lead time is normally distributed with a mean and std dev as follows:

Replenishment Policies

Continuous review

Inventory is continuously tracked

Order for a lot size Q is placed when the inventory declines to the reorder point (R O P)

Periodic review

Inventory status is checked at regular periodic intervals

Order is placed to raise the inventory level to a specified threshold

Summary of Learning Objective 2

Safety inventory is influenced by the desired product availability, demand uncertainty, replenishment lead times, and lead time variability. Product availability is measured using the fill rate or cycle service level. Demand uncertainty is measured by the forecast error. For lead time one measures both the mean and the standard deviation. The required safety inventory is also influenced by the inventory policy implemented. Continuous review policies order a fixed quantity after variable replenishment intervals. Periodic review policies order variable quantities after fixed replenishment intervals.

Determining the Appropriate Level of Safety Inventory (1 of 8)

We focus on continuous review policy

Continuous review policy consists of a lot size Q ordered when inventory on hand declines to the ROP.

Weekly demand is normally distributed.

Lead time is constant and fixed = L weeks

Evaluating Safety Inventory Given a Reorder Point

Expected demand during lead time = D × L

Safety inventory, ss = R O P − D × L

Example 12-1

Determining the Appropriate Level of Safety Inventory (2 of 8)

Example 12.1 (page 317)

Average demand per week, D = 2,500

Standard deviation of weekly demand, sD = 500

Average lead time for replenishment, L = 2 weeks

Reorder point, R O P = 6,000

Average lot size, Q = 10,000

Determining the Appropriate Level of Safety Inventory (3 of 8)

Average inventory = cycle inventory + safety inventory

= 5,000 + 1,000 = 6,000

Safety inventory, ss = R O P −D L = 6,000 −5,000 = 1,000

Determining the Appropriate Level of Safety Inventory (4 of 8)

Evaluating Cycle Service Level Given a Reorder Point

(ddlt = demand during lead time)

C S L = F(R O P, DL, σL) = NORMDIST(R O P, DL, σL, 1)

Probability of not stocking out in a replenishment cycle

Stockout occurs if demand during lead time is larger than the ROP.

CSL is the inverse

The relationship between CSL and ROP

Probability Distribution of demand during lead time (ddlt)

ROP

Example 12-2

Q = 10,000, R O P = 6,000, L = 2 weeks

D = 2,500/week, σD = 500

DL = 5000

Determining the Appropriate Level of Safety Inventory (5 of 8)

Q = 10,000, R O P = 6,000, L = 2 weeks

D = 2,500/week, σD = 500

C S L = F(R O P, DL, σL) = NORMDIST(ROP, DL, σL, 1)

= NORMDIST(6,000, 5,000, 707, 1) = 0.92

Example 12.2 (page 318)

Determining the Appropriate Level of Safety Inventory (6 of 8)

Evaluating Required Safety Inventory Given a Desired Cycle Service Level

Desired cycle service level = C S L

Mean demand during lead time = DL

Standard deviation of demand during lead time = σL

Probability(demand during lead time

Identify safety inventory ss so that

F(DL + ss, DL, sL) = CSL

ROP

Determining the Appropriate Level of Safety Inventory (7 of 8)

Example 12.3

Desired

Determining the Appropriate Level of Safety Inventory (8 of 8)

Example 12.3 (page 320)

1.28

Problem 12.2 (page 346)

3 Key factors that impact safety inventory

Desired Product Availability

Lead Time

Demand Uncertainty

Impact of Desired Product Availability, Lead Time, and Demand Uncertainty

As desired product availability goes up the required safety inventory increases

Table 12- Required Safety Inventory for Different Values of Fill Rate

Fill Rate	Safety Inventory
97.5%	67
98.0%	183
98.5%	321
99.0%	499
99.5%	767

From example 12-5

Exponential growth

Impact of Desired Product Availability and Uncertainty

Goal is to reduce the level of safety inventory required in a way that does not adversely affect product availability

Reduce the supplier lead time L

Reduce the underlying uncertainty of demand (represented by σD )

Safety inventory grows linearly with underlying uncertainty of demand

Safety inventory is proportional to the square root of lead time, therefore it grows at a lower proportion (not linear)

Example 12.6

D = 2,500/week σD, CSL = 0.95

Benefits of Reducing Lead Time and Demand Uncertainty

D = 2,500/week σD, CSL = 0.95

If lead time is reduced to one week

If standard deviation is reduced to 400

Example 12.6 (page 325)

1.65

Adjusting Safety Inventory for Demand Lumpiness and Seasonality

Orders typically in large lots

Demand at various stages in the supply chain tends to be lumpy

Raise safety inventory by half the average size of a customer order

Demand is often seasonal

Fixing a R O P may lead to stockouts

Keep R O P constant in terms of days of demand (flow time)

Summary of Learning Objective 3 (1 of 2)

Given a desired cycle service level C S L, a lead time L, and a standard deviation of periodic demand σD , the required safety inventory ss for a continuous review policy is given by

Given a reorder point R O P, a lead time L, a standard deviation of periodic demand σD, and periodic demand D, the resulting cycle service level is given by

Summary of Learning Objective 3 (2 of 2)

Given a level of safety inventory, one can evaluate the resulting fill rate. Given a desired fill rate, one can evaluate the required safety inventory. The required safety inventory increases with an increase in desired product availability, lead time, and uncertainty of periodic demand. In practice, it is best to evaluate safety inventory in terms of days of demand to account for seasonality of demand.

Impact of Supply Uncertainty on Safety Inventory

We incorporate supply uncertainty by assuming that lead time is uncertain

D : Average demand per period

σL : Standard deviation of demand during lead time

L : Average lead time for replenishment

sL : Standard deviation of lead time

Demand during lead time

Std deviation of demand

Example 12.7

Average demand per period, D = 2,500

Standard deviation of demand per period, σD = 500

Average lead time for replenishment, L = 7 days

Standard deviation of lead time, sL = 7 days

Mean ddlt, DL = DL = 2,500 × 7 = 17,500

Impact of Lead Time Uncertainty on Safety Inventory (1 of 3)

Average demand per period, D = 2,500

Standard deviation of demand per period, σD = 500

Average lead time for replenishment, L = 7 days

Standard deviation of lead time, sL = 7 days

Mean ddlt, DL = DL = 2,500 × 7 = 17,500

Example 12.7 (page 327)

Impact of Lead Time Uncertainty on Safety Inventory (2 of 3)

Required safety inventory

22,491 ÷ 2,500 = 9.0 therefore, 22,491 tablets is equivalent to 9 days of demand

Impact of Lead Time Uncertainty on Safety Inventory (3 of 3)

Table 12-2 Required Safety Inventory as a Function of Lead Time Uncertainty

sL	σL	ss (units)	ss (days)
6	15,058	19,298	7.72
5	12,570	16,109	6.44
4	10,087	12,927	5.17
3	7,616	9,760	3.90
2	5,172	6,628	2.65
1	2,828	3,625	1.45
0	1,323	1,695	0.68

Problem 12.11(page 347)

Summary of Learning Objective 4

An increase in supply uncertainty significantly increases the amount of safety inventory required for a given level of product availability. Lead time uncertainty has a more significant impact on the required safety inventory than lead time itself. A reduction in supply uncertainty can help to dramatically reduce the required safety inventory without hurting product availability.

Impact of Aggregation on Safety Inventory (1 of 5)

How does aggregation affect forecast accuracy and safety inventories

Di: Mean periodic demand in region i, i = 1, …, k

σi : Standard deviation of periodic demand in region i, i =, …, k

ρij: Correlation of periodic demand for regions i, j,

Customers in the k regions can be served via local inventories in each region OR via a centralized facility with aggregate inventories

Impact of Aggregation on Safety Inventory (2 of 5)

Total safety inventory in decentralized option

Simplified to

The aggregated demand:

Impact of Aggregation on Safety Inventory (3 of 5)

Require safety inventory on aggregation

Holding – cost savings on aggregation per unit sold

Impact of Aggregation on Safety Inventory (4 of 5)

The safety inventory savings on aggregation increase with the desired cycle service level C S L

The safety inventory savings on aggregation increase with the replenishment lead time L

The safety inventory savings on aggregation increase with the holding cost H

The safety inventory savings on aggregation increase with the coefficient of variation of demand

The safety inventory savings on aggregation decrease as the correlation coefficients increase

Example 12.8

Standard deviation of weekly demand, σD =5

Replenishment, L = 2 weeks; Decentralized C S L = 0.9

Correlation Coefficient =0

4 Retail Stores

Geographic Area Demand

1 Large Outlet

Geographic Area Demand

Impact of Correlation on Value of Aggregation (1 of 3)

Standard deviation of weekly demand, σD =5

Replenishment, L = 2 weeks; Decentralized C S L = 0.9

Example 12.8 (page 330)

Total required safety inventory,

DECENTRALIZED

EACH OUTLET CARRIES 36/4 = 9 CARS

Impact of Correlation on Value of Aggregation (1 of 3)

Aggregate ρ = 0

Standard deviation of weekly demand at central outlet,

Example 12.8 (page 330)

cars

Impact of Correlation on Value of Aggregation (2 of 3)

Table 12-3 Safety Inventory in the Disaggregate and Aggregate Options

Rho	Disaggregate Safety Inventory	Aggregate Safety Inventory
0	36.25	18.12
0.2	36.25	22.93
0.4	36.25	26.88
0.6	36.25	30.33
0.8	36.25	33.42
1.0	36.25	36.25

Impact of Aggregation on Safety Inventory (5 of 5)

Figure 12-4 Square-Root Law

Square Root Law: If the # of independent stocking locations decrease by a factor of n, the average safety stock is expected to decrease by a factor of √n

Impact of Correlation on Value of Aggregation (3 of 3)

Two possible disadvantages to aggregation

Increase in response time to customer order

Increase in transportation cost to customer

Example 12.9

Use four regional or one national distribution center

4 DCs

Regional Demand

1 DC

Regional Demand

Trade-Offs of Physical Centralization (1 of 2)

Use four regional or one national distribution center

Four regional centers

Total required safety inventory,

Example 12.9 (page 332)

Trade-Offs of Physical Centralization (2 of 2)

One national distribution center,  = 0

Standard deviation of weekly demand,

Decrease in holding costs

Decrease in facility costs

Increase in transportation

Example 12.9 (page 332)

Problem 12.12

Information Centralization

Online systems that allow customers or stores to locate stock

Improves product availability without adding to inventories

Reduces the amount of safety inventory

Impact of Coefficient of Variation on Value of Aggregation (1 of 2)

Table 12-4 Value of Aggregation at W.W. Grainger

Blank	Motors	Cleaner
Inventory is stocked in each store	Blank	Blank
Mean weekly demand per store	20	1,000
Standard deviation	40	100
Coefficient of variation	2.0	0.1
Safety inventory per store	132	329
Total safety inventory	211,200	526,400
Value of safety inventory	$105,600,000	$15,792,000
Inventory is aggregated at the DC	Blank	Blank
Mean weekly aggregate demand	32,000	1,600,000
Standard deviation of aggregate demand	1,600	4,000
Coefficient of variation	0.05	0.0025
Aggregate safety inventory	5,264	13,159
Value of safety inventory	$2,632,000	$394,770

Example 12.10 (page 334)

Impact of Coefficient of Variation on Value of Aggregation (2 of 2)

Table 12-4 [Continued]

Blank	Motors	Cleaner
Savings	Blank	Blank
Total inventory saving on aggregation	$102,968,000	$15,397,230
Total holding cost saving on aggregation	$25,742,000	$3,849,308
Holding cost saving per unit sold	$15.47	$0.046
Savings as a percentage of product cost	3.09%	0.15%

Specialization (1 of 2)

Supply Chain provides a wide variety of products to customers

Inventory is carried at multiple locations

Key decision for Supply Chain Managers:

Should all products should be stocked at every location?

Required level of safety inventory

Affected by coefficient of variation of demand

Low demand, slow-moving items, typically have a high coefficient of variation

High demand, fast-moving items, typically have a low coefficient of variation

The reduction in safety inventory due to aggregation is heavily influenced by the demand’s CoV

Specialization (2 of 2)

Item Type	Centralized Inventories	Decentralized Inventories
Fast Moving Predictable {Low Value}	Customer willing to pay premium?	Low cost
Slow Moving Unpredictable {High Value}	Low cost	Customer willing to pay premium?

Figure 12-5 Specialization of Inventory Based on Product Type

Specialization of distribution network:

Product Substitution

The use of one product to satisfy demand for a different product

Manufacturer-driven substitution

Allows aggregation of demand

Reduce safety inventories

Influenced by the cost differential, correlation of demand

Customer-driven substitution

Allows aggregation of safety inventory

Component Commonality

High product variety drives high # of components

Using common components exploits the benefits of aggregation and reduces component inventory

Without common components

Uncertainty of demand for a component is the same as for the finished product

Results in high levels of safety inventory

With common components

Demand for a component is an aggregation of the demand for the finished products

Component demand is more predictable

Component inventories are reduced

Example 12.11

27 servers, 3 components, 3 × 27 = 81 distinct components

Monthly demand = 5,000

Standard deviation = 3,000

Replenishment lead time = 1 month

C S L = 0.95

Value of Component Commonality (1 of 3)

27 servers, 3 components, 3 × 27 = 81 distinct components

Monthly demand = 5,000

Standard deviation = 3,000

Replenishment lead time = 1 month

C S L = 0.95

Total safety inventory required

Safety inventory per common component

Example 12.11 (page 337)

Value of Component Commonality (2 of 3)

With component commonality

Nine distinct components

Total safety inventory required

Value of Component Commonality (3 of 3)

Table 12-5 Marginal Benefit of Component Commonality

Number of Finished Products per Component	Safety Inventory	Marginal Reduction in Safety Inventory	Total Reduction in Safety Inventory
1	399,699	Blank	Blank
2	282,630	117,069	117,069
3	230,766	51,864	168,933
4	199,849	30,917	199,850
5	178,751	21,098	220,948
6	163,176	15,575	236,523
7	151,072	12,104	248,627
8	141,315	9,757	258,384
9	133,233	8,082	266,466

Note: Marginal benefit decreases as a component is used in more and more finished products

Postponement (1 of 2)

Delay product differentiation or customization until closer to the time the product is sold

Have common components in the supply chain for most of the push phase

Move product differentiation as close to the pull phase of the supply chain as possible

Inventories in the supply chain are mostly aggregate

Examples:

Final mixing of painting at the store

Benetton – postpones the color customization of knit garments. When thread is purchased, only aggregate demand across all colors needs to be forecasted

Postponement (2 of 2)

Figure 12-6 Supply Chain Flows without and with Postponement

Postponement delays product differentiation. Most of the inventories in the SC are aggregate

Example 12.12

100 different paint colors, D = 30/week, D = 10, L = 2 weeks, CSL = 0.95

Value of Postponement

100 different paint colors, D = 30/week, D = 10, L = 2 weeks, CSL = 0.95

Total required safety inventory,

Standard deviation of base paint weekly demand,

Example 12.12 (page 340)

Summary of Learning Objective 5

Aggregation reduces the required safety inventory as long as the demand across the aggregated regions is not perfectly, positively correlated. The safety inventory savings on aggregation increase with the desired C S L, the replenishment lead time, the product holding cost, and the coefficient of variation of demand. The safety inventory savings on aggregation decrease as the correlation of demand across regions increases. Firms can aggregate inventories through physical aggregation, information centralization, product substitution, component commonality, and postponement.

Optional Topics

Evaluating Fill Rate Given a Reorder Point (1 of 4)

Expected shortage per replenishment cycle (E S C) is the average units of demand that are not satisfied from inventory in stock per replenishment cycle

Product fill rate

Evaluating Fill Rate Given a Reorder Point (2 of 4)