1 7. Managing Flow Variability: Safety Inventory Chapter 7 Managing Flow Variability: Safety...

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7. Managing Flow Variability: Safety Inventory

Chapter 7

Managing Flow Variability: Safety Inventory

7.1 Demand Forecasts and Forecast Errors

7.2 Safety Inventory and Service Level

7.3 Optimal Service Level – The Newsvendor Problem

7.4 Lead Time Demand Variability

7.5 Pooling Efficiency through Aggregation

7.6 Shortening the Forecast Horizon

7.7 Levers for Reducing Safety Inventory

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7.1 Demand Forecast and Forecast Errors

In review, we have 3 stages of a process:

1. Input (e.g. raw materials)

2. Process

3. Output (finished goods)

This is important to forecasting because it will allow us to

more closely match outputs to inputs and vice versa.

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We have previously assumed demand is knownand is constant.

Demand varies in predictable and unpredictable ways.

Unpredictable, random factors affecting demand is referred to as “noise”.

“As a process of predicting future demand, forecasting is, among other things, an effort to deal with NOISE.”

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Why do we forecast?

We forecast so that we can make decisions about the future.

We need to make rational decisions about process inventory.

* How to spend money and how not to spend money.

* When to buys more widgets.

* When to hire more workers.

* How to avoid stockouts (upset customers =

business losses)

* How to avoid holding excess inventory (= $ lost)

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Forecasting methods

Subjective – Based on judgement and experience• Surveys and expert judgements

Objective – Based on data analysis• Causal models - Forecast methods that assume that in

addition to data, there are other factors that influence demand (eg. Consumer prices.)

• Time series analyses - Methods that rely solely on past data.

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4 Characteristics of Forecasts

Forecasts are usually wrong.Because of random noise – forecasts are inaccurate.

Forecasts should be accompanied by a measure of forecast error.

A measure of forecast error quantifies the manager’s degree of confidence in the forecast.

Aggregate forecasts are more accurate than individual forecasts.Aggregate forecasts reduce the amount of variability – relative to the

aggregate mean demand.

Long-range forecasts are less accurate than short-range forecasts.

Precise forecasting of events far out in the future are much more difficult to predict than something that will occur in a matter

of moments from now.

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Forecasts should incorporate hard quantitaive data as well as qualitative factors such as managerial judgement, intuition, and market savvy.

Forecasting is as much art as science.

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Safety Inventory cushions the process against supply disruptions or surges in demand.

Having adequate Safety Inventory reduces the uncertainty in supply and demand.

Ensuring reliable suppliers and stable demand eliminates the need for Safety Inventory.

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Objective:

Review of common terms and a discussion of Service

Level

Where: SL = f (Safety Inventory, I safety)

And some math using Excel…

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SL: Service Level

I safety: Safety Inventory (or Safety Stock)

I cycle: Cycle Inventory

LTD: Lead Time Demand

ROP: Re-order Point

L: Replenishment Lead Time

Q: Order Size

NORMDIST: Standard Normal Tables

NORMSINV: Standard Normal Tables

NORMINV: Standard Normal Tables

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Inventory, I (t)

ROP

0

Safety Inventory

(I safety)

LTD, # of Units used during lead time

ORDER ORDER

L LTime, t

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I safety = ROP – LTD

0

ROP

LTD, # of Units used during lead time

Safety Inventory

(I safety)

Inventory, I (t)

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An Inventory with an Order Size = Q

Average Inventory = Q/2

I cycle = Q/2

I = I cycle + I safety = Q/2 + I safety

Average Flow Rate = R

Average Flow Time as expressed by Little’s Law

T = I /R = (Q/2 + I safety )

R

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The Service Level for a given ROP is given by:

SL = Prob (LTD < ROP)

To calculate SL, recall first that if LTD is normally distributed

with mean LTD and standard deviation LTD then

I safety = z x LTD , where z is a multiple of LTD

Or the number of standard deviations

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Example:

At GE Lighting’s Paris warehouse,

LTD (average Lead Time Demand) = 20,000 lamps

Actual Demand varies daily and LTD = 5,000

The warehouse re-orders whenever ROP = 24,000

Therefore, I safety = ROP – LTD = 24,000 – 20,000 = 4,000

And: z = I safety / LTD = 4,000 / 5,000 = 0.8

And: SL= Prob (Z< 0.8) from Appendix II

SL= 0.7881

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A B C D E F

1 z = 0.802 SL (z<0.8)3 SL = 0.788144

5

678

EXCEL

Service Level

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EXCEL

Safety Inventory

A B C D E F

1 SL = 0.788142 z = 0.803

4

5

678

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EXCEL

Reorder Point

A B C D E F

1 SL = 0.788142 LTD = 20,000

3

LTD = 5,0004 ROP 24,0005

678

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7.3 Optimal Service Level: The Newsvendor Problem

So Far…

Safety inventory has been defined for a desired level of customer service.

But…

How do we choose what level of service a firm should offer?

Examples:

• Newspapers / Magazines• Perishables (fish, produce, bread, milk, etc.)• Seasonal Items (Summer & Winter Apparel)

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Cost of Holding Extra Inventory

Improved Service

Optimal Service Level??

The Newsvendor ProblemDecision making under uncertainty whereby the decision maker balances the expected costs of ordering too much with the expected costs of ordering too little to determine the optimal order quantity.

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Cost:

Price:

Salvage:

$1,800

$2,500

$1,700

Demand ProbabilityCumulative Probability

Complementary Cumulative Probability

r Prob(R = r) Prob(R ≤ r) Prob(R > r)

100 0.02 0.02 0.98

110 0.05 0.07 0.93

120 0.08 0.15 0.85

130 0.09 0.24 0.76

140 0.11 0.35 0.65

150 0.16 0.51 0.49

160 0.2 0.71 0.29

170 0.15 0.86 0.14

180 0.08 0.94 0.06

190 0.05 0.99 0.01

200 0.01 1 0

Predicted Demand for HDTV’s

Profit:

Loss:

$700

$100

Mean:

Std. Dev:

151.6

22.44

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100 110 120 130 140 150 160 170 180 190 200

100 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000

110 $69,000 $77,000 $77,000 $77,000 $77,000 $77,000 $77,000 $77,000 $77,000 $77,000 $77,000

120 $68,000 $76,000 $84,000 $84,000 $84,000 $84,000 $84,000 $84,000 $84,000 $84,000 $84,000

130 $67,000 $75,000 $83,000 $91,000 $91,000 $91,000 $91,000 $91,000 $91,000 $91,000 $91,000

140 $66,000 $74,000 $82,000 $90,000 $98,000 $98,000 $98,000 $98,000 $98,000 $98,000 $98,000

150 $65,000 $73,000 $81,000 $89,000 $97,000 $105,000 $105,000 $105,000 $105,000 $105,000 $105,000

160 $64,000 $72,000 $80,000 $88,000 $96,000 $104,000 $112,000 $112,000 $112,000 $112,000 $112,000

170 $63,000 $71,000 $79,000 $87,000 $95,000 $103,000 $111,000 $119,000 $119,000 $119,000 $119,000

180 $62,000 $70,000 $78,000 $86,000 $94,000 $102,000 $110,000 $118,000 $126,000 $126,000 $126,000

190 $61,000 $69,000 $77,000 $85,000 $93,000 $101,000 $109,000 $117,000 $125,000 $133,000 $133,000

200 $60,000 $68,000 $76,000 $84,000 $92,000 $100,000 $108,000 $116,000 $124,000 $132,000 $140,000

=IF($A3>B$1,B$1*700-($A3-B$1)*100,$A3*700)

Demand

Qua

ntity

Ord

ered

100 x $700 – (110-100) x $100 = $69,000

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100 110 120 130 140 150 160 170 180 190 200

100 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000 $70,000

110 $69,000 $77,000 $77,000 $77,000 $77,000 $77,000 $77,000 $77,000 $77,000 $77,000 $77,000

120 $68,000 $76,000 $84,000 $84,000 $84,000 $84,000 $84,000 $84,000 $84,000 $84,000 $84,000

130 $67,000 $75,000 $83,000 $91,000 $91,000 $91,000 $91,000 $91,000 $91,000 $91,000 $91,000

140 $66,000 $74,000 $82,000 $90,000 $98,000 $98,000 $98,000 $98,000 $98,000 $98,000 $98,000

150 $65,000 $73,000 $81,000 $89,000 $97,000 $105,000 $105,000 $105,000 $105,000 $105,000 $105,000

160 $64,000 $72,000 $80,000 $88,000 $96,000 $104,000 $112,000 $112,000 $112,000 $112,000 $112,000

170 $63,000 $71,000 $79,000 $87,000 $95,000 $103,000 $111,000 $119,000 $119,000 $119,000 $119,000

180 $62,000 $70,000 $78,000 $86,000 $94,000 $102,000 $110,000 $118,000 $126,000 $126,000 $126,000

190 $61,000 $69,000 $77,000 $85,000 $93,000 $101,000 $109,000 $117,000 $125,000 $133,000 $133,000

200 $60,000 $68,000 $76,000 $84,000 $92,000 $100,000 $108,000 $116,000 $124,000 $132,000 $140,000

$69,000(0.02) + $77,000(0.05) + …+ $77,000(0.01) = $76,840

Demand

Qua

ntity

Ord

ered

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Order Quantity (Q) Expected Profit

100 $70,000

110 $76,840

120 $83,280

130 $89,080

140 $94,160

150 $98,360

160 $101,280

170 $102,600

180 $102,720

190 $102,200

200 $101,280

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Net Marginal Benefit:

Net Marginal Cost:

MB = p – c

MC = c - v

MB = $2,500 - $1,800 = $700

MC = $1,800 - $1,700 = $100

We receive Marginal Benefit when R > Q, therefore at any order quantity Q,

Expected MB = MB x Prob(R > Q)

We receive Marginal Cost when R ≤ Q, therefore at any order quantity Q,

Expected MC = MC x Prob(R ≤ Q)

MC x Prob(R ≤ Q*) ≥ MB x Prob(R > Q*)

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MB

MB MC

Time for Algebra…

MC x Prob(R ≤ Q*) ≥ MB x Prob(R > Q*)

Since, Prob(R > Q) = 1 – Prob(R ≤ Q)

We can write, MC x Prob(R ≤ Q*) ≥ MB x [1 – Prob(R ≤ Q*)]

After rearranging, Prob(R ≤ Q*) ≥

Newsvendor formula: SL* = Prob(R ≤ Q*) = MB

MB MC

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$700* 0.875

$700 $100

MBSL

MB MC

Going back to the example…

So what quantity corresponds to this service level ?

If we assume demand is normally distributed then,

* RQ R z

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-4 -3 -2 -1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4Probability Less than Upper Bound is 0.87493

Den

sity

Critical Value (z)

z = 1.15

* 151.6 1.15 22.44 177.41RQ R z

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7.4 Lead Time Demand Variability

LTD L R 2 2LTD RL

2 2 2LTD LR

Average Lead Time Demand:

Variability in Periodic Demand:

Variability in Lead Time:

Variability in Demand and Lead Time:

2 2 2LTD R LL R

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7.5: Pooling Efficiency through Aggregation

Third characteristic of forecasts

Aggregation: pooling demand for several similar

products

Aggregate sales

Safety Inventory: Uncertain demand

Assume Decentralized: Warehouses operates

independently

Imbalance of inventory - Customer demand not satisfied

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Physical Centralization: that the firm can consolidate all its stock in

one location from which is can serve all its customers.

ELIMINATES stock imbalance

BETTER customer service

SAME total inventory

LESS inventory

Location 1 Location 2

Lead times demands: LTD1 LTD2

Mean of LTD Standard Deviation LTD

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LTD1 and LTD2: statistically identically distributed

To provide desired level of service, SL each location must carry

Safety Inventory:

Z determined by the desired service level

Each facility: Identical demand and service levels

Total safety inventory decentralized system:

LTDsafetyI

LTDdsafetyI 2

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Independent DemandsCentralizing the two locations in one location when

lead time demands at the two locations are independent.

LTD = LTD1 + LTD2 Centralized PoolThe mean of total lead time demand is:

LTD + LTD = 2 LTDVariance is:

LTDLTDLTD222 2

LTD2Standard Deviation is:

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Comparing safety inventories of decentralized

and centralized systems.

Safety Inventory in Centralized system is in a 2

location decentralized system by a factor of

)( dsafetyI

)( csafetyI

2

1

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Centralization of N locations:

Safety Inventory needed is

Centralization will reduce inventory by factor of

N

LTDcsafety NI

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Example

GE lighting operating 7 warehouses

Consolidated in to one centralized

warehouse

Replenishment lead time remain at 10 days

What will be the impact of accepting

the task force recommendations?

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A warehouse with average lead time demand of 20,000 units

with a standard deviation of 5,000 units needs to carry a safety

inventory to provide a 95% service level.

Total safety inventory across 7 warehouses:

Task force accepted, single central warehouse will face total lead

time demand with mean and standard deviation of:

722,57246,87 dsafetyI

246,8safetyI

80.228,13000,57

000,140000,207

LTD

LTD

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95% service level, the central warehouse must carry a safety

inventory:

Safety inventory with the single central warehouses is 35,894

less than that required under the current decentralized network

of 7 warehouses.

Decrease in safety inventory by a factor of

828,2180.228,1365.165.1 LTDcsafetyI

65.27

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Square Root Law

States that the total safety inventory required to provide

a specific level of service increases by the square root of

the number of locations in which it is held.

Previous example

Correlated Demands

Does centralization offer similar benefits when demands

in multiple locations are correlated?

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LTD1 and LTD2 are statistically identically distributed

but correlated.

Correlation between two locations with coefficient

Mean of total lead time: LTD + LTD = 2 LTD

Variance is:

Total safety in centralized system is:

LTDLTDLTDLTD2222 )1(22

LTDcsafetyI )1(2

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The total safety inventory in the decentralized system:

The safety inventory in the two-location decentralized

system is larger than in the centralized system by a factor

of

If demand is positively correlated

centralization offers no benefits in the reduction of safety

inventory

)1(

2

LTDdsafetyI 2

)1.,.( ei

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Advantages

1. Centralized systems as the demand on the two locations

become negatively correlated.

2. Centralized systems diminishes as the demand in the two locations

become positively correlated

Disadvantages of Centralization

1. Response time to Customers

2. Shipping Cost

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7.5.2: Principle of Aggregation and Pooling Inventory

Statistical Principle

Principle Aggregation: the standard deviation of the sum

of random variables is less than the sum of the

individual standard deviations.

Pooling inventory: available inventory is shared among

various sources of demand

Pooling inventory applied in other ways other than physical

centralization

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Virtual Centralization

Specialization

Component Commonality

Product Substitute

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Virtual Centralization

Distribution System

Location A Location B

Exceeds Available stock Available

1. Information about product demand and availability must

be available at both locations

2. Shipping the product from one location to a customer at

another location must be fast and cost effective

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Correlation is less than one – Pooling is Effective

Inventory Decentralized instead of physically

consolidated

Virtual Centralization: is a system in which inventory

pooling in a network of locations is facilitated using

information regarding availability of goods and

subsequent transshipment of goods between locations to

satisfy demand.

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Specialization

Each product only one specialized warehouse

EXAMPLE

Location A Location B

P1 P2

Safety Inventory is reduced because each inventory is now

centralized at one location

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Component Commonality

Aggregating demand across various products.

Computer companies with models that vary.

Make-to-stock: produce in anticipation of product

demand

Make-to-Order: Produce in response to customer

orders

Reduce inventory investment maintaining the same level

of service and product variety

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Disadvantage Make-to-Order Strategy

Customer must wait for firm to produce product

Advantage Make-to-Stock Strategy

Available for immediate consumption

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7.6: Shortening the Forecast Horizon through Postponement

Postponement (or Delayed Differentiation): More Effective

Short-Range forecast more accurate

Two Alternative processes (both two weeks)

Process A: Coloring the fabric, assembling

Process B: Assembling T-shirts, coloring

Does one have the advantage over the other?

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By Reversing: assembling and dyeing process

Process B postponed the color difference until one

week closer to the time of sale

Postponement: the practice of delaying part of a process in order to reduce the need for safety inventory

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Process B has the advantage

Aggregation Reduces Variability1. Aggregates demands by color in the first phase2. Requires shorter-range forecasts of individual T-shirts

needed by color in the second phase.

Less Demand Variability Less Total Safety Inventory

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7.7: Levers for Reducing Safety Inventory

Levers for Reducing Flow Variability and the Required

Safety Inventory

1. Reduce demand variability through improved

forecasting

2. Reduce replenishment lead time

3. Reduce variability in replenishment lead time

4. Pool safety inventory for multiple locations or products

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7.7: Levers for Reducing Safety Inventory

5. Exploit product substitution

6. Use common components

7. Postpone product-differentiation processing until

closer to the point of actual demand

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