Dr. RAVI SHANKARProfessor

Department of Management Studies

Indian Institute of Technology DelhiHauz Khas, New Delhi 110 016, India

Phone: +91-11-26596421 (O); 2659-1991(H); (0)-+91-9811033937 (m)Fax: (+91)-(11) 26862620

Email: r.s.research@gmail.comhttp://web.iitd.ac.in/~ravi1

SESSION#3: TUTORIAL ON RISK POOLING (CFVG: 2012)

A TUTORIAL ON RISK POOLING

RISK POOLINGRisk pooling is an important concept in supply chain management. The idea of risk pooling is executed by a centralized distribution system which caters to the requirements of all the markets in a given region instead of separate warehouse allocated for different markets.

Market Two

Risk Pooling

• Consider these two systems:

Supplier

Warehouse One

Warehouse Two

Market One

Market Two

Supplier Warehouse

Market One

Supplier

Warehouse

Retailers

Centralized Systems

Decentralized System

Supplier

Warehouses

Retailers

Demand Forecasts

• The three principles of all forecasting techniques:

– Forecasting is always wrong

– The longer the forecast horizon the worst is the

forecast

– Aggregate forecasts are more accurate

The Effect of

Demand Uncertainty• Most companies treat the world as if it were predictable:

– Production and inventory planning are based on forecasts of demand made far in advance of the selling season

– Companies are aware of demand uncertainty when they create a forecast, but they design their planning process as if the forecast truly represents reality

• Recent technological advances have increased the level

of demand uncertainty:

– Short product life cycles

– Increasing product variety

Market one

Market two

Factory

Central

warehouse

Warehouse 1

Warehouse 2

Factory

Decentralized Warehouses

Market one

Market two

Factory

Centralised

warehouse at

Ayutthaya

Market Two

ABC Chiang Pai

Market One

Market Two

ABC Chiang Pai

Market One

Prachin Buri Warehouse

Pathumthani Warehouse

Central

warehouse:

Ayutthaya

Market Pathumthani

Market Prachin Buri

Factory: ABC

Central

warehouse

Market Two

ABC company

Market One

Market Two

ABC company

Market One

Prachin Buri Warehouse

Pathumthani Warehouse

Central

warehouse

(Ayutthaya)

Market one

Market two

Market one

Market two

WEEK 1 2 3 4 5 6 7 8

Pathumthani 68(-17) 37(+14) 45(+6) 58(-7) 16(+35) 32(+19) 72(-21) 80(-29)

Prachin Buri 87(-27) 62(-3) 55(+4) 67(-8) 12(+47) 42(+17) 69(-10) 81(-22)

TOTAL 155(-45) 99(+11) 100(+10) 125(-15) 28(+82) 74(+36) 141(-31) 161(-51)

PRODUCT A

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8

WEEK

AV

ER

AG

E W

EE

KL

Y D

EM

AN

D

DEMAND Pathumthani

DEMAND Prachin Buri

HISTORICAL DEMAND DATA

51

59

110

Average

Theoretical Approach

• Consider two markets

– Risk Polling by Aggregating Demand by

Centralized procurement, centralized

warehousing, centralized distribution like

super stores etc

– Risk Polling by Aggregating time horizon by

combining orders as discussed in previous

slide

A Detail Analysis of

RISK POOLING Case

The Basic EOQ Model

We assumed that, we will only keep half the inventory over a year then

The total carry cost/yr = Cc x (Q/2). Total order cost = Co x (D/Q)

Then , Total cost = 2Q

CQDCTC co +=

Finding optimal Q*

Cost Relationships for Basic EOQ(Constant Demand, No Shortages)

TC

–A

nn

ual

Co

st

Total Cost

CarryingCost

OrderingCost

EOQ balances carryingcosts and ordering costs in this model.

Q* Order Quantity (how much)

The Basic EOQ Model

• EOQ occurs where total cost curve is at minimum value and carrying cost equals

ordering cost:

•Where is Q* located in our model?

c

o

co

CDCQ

QCQDCTC

2

2

*

min

=

+=

(How to obtain this?)Then, *c

o

co

CDCQ

QCQDCTC

2

2

*

min

=

+=

A Revision of model discussed in Sesion-3:

Model with “re-order points”• The reorder point is the inventory level at which a new order is placed.

• Order must be made while there is enough stock in place to cover demand during lead time.

• Formulation: R = dL, where d = demand rate per time period, L = lead time

Then R = dL = (10,000/311)(10) = 321.54

Working days/yr

Reorder Point• Inventory level might be depleted at slower or faster rate during lead time.

• When demand is uncertain, safety stock is added as a hedge against stockout.

Two possible scenarios

Safety stock!

No Safety

stocks!

We should then ensure

Safety stock is secured!

Determining Safety Stocks Using Service Levels

• We apply the Z test to secure its safety level,

)( LZLdR dσ+=

Reorder point

Safety stock

Average sample demand

How these values are represented in the diagram of normal distribution?

Reorder Point with Variable Demand

stocksafety

yprobabilit level service toingcorrespond deviations standard ofnumber

demanddaily ofdeviation standard the

timelead

demanddaily average

pointreorder

where

=

=

=

=

=

=

+=

LZ

Z

L

d

R

LZLdR

d

d

d

σ

σ

σ

Reorder Point with Variable Demand

Example

Example: determine reorder point and safety stock for service level of 95%.

26.1. : formulapoint reorder in termsecond isstock Safety

yd 1.3261.26300)10)(5)(65.1()10(30

1.65 Zlevel, service 95%For

dayper yd 5 days, 10 L day,per yd 30 d

=+=+=+=

=

===

LZLdR

d

dσ

σ

A detail treatment of

this case study

TERMINOLOGY

• AVG: Average daily demand faced by the distributor.

• STD: standard deviation of the daily demand faced by the distributor.

• L: Replenishment lead time from the supplier to the distributor in days

• K: Fixed cost (set up cost) incurred every time the warehouse places an order, it includes transportation cost.

• h: Cost of holding one unit of the product in the inventory for one day at the warehouse.

• α: Service level -the probability of not stocking out during lead time.

• Average demand during lead time=L×AVG. This ensures that if a distributor places an order the system has enough inventory to cover expected demand during lead time.

• Safety stock= z×STD× this is the amount of inventory distributor needs to keep to meet deviations from average demand during lead time.

• z: Safety factor which is chosen from statistical table to ensure that probability of stock out is exactly 1-α

• Reorder level (s) = average demand during lead time + safety stock

=L×AVG + z×STD×Whenever the inventory level drops below reorder

level the distributor should place new order to raise itsinventory.

L

L

• . Order quantity (Q): It is the number of items ordered each time places an order that minimizes the average total cost per unit of time distributor.

Q=

• Order-up-to level (S): Since there is variability in demand the distributor places an order for Q items whenever inventory is below reorder level (s).

S= Q + s

2K AVG

h

×

• Average inventory = Q/2 + z STD

• Coefficient of variation =

×× L

STD

AVG L×

A View of (s, S) Policy

Time

Inven

tory

Lev

el

S

s

0

Lead

Time

Lead

Time

Inventory Position

EXAMPLE OF RISK

POOLINGLet us illustrate this with an example of a Chiang Paibased company ABC that produces certain type of products and distributes them in the South Thailand region .The current distribution system partitions S-Thailand region into two markets each of which has a warehouse.

1. One warehouse is located in Prachin Buri

2. Another one located in Pathumthani.

alternative strategy of centralized distribution system replaces two warehouses by a single warehouse located between the two cities in Ayutthaya that will serve all customer orders in both markets

Market Two

Consider these two systems:

ABC company

Pathumthani Warehouse

Prachin Buri. Warehouse

Market One

Market Two

ABC companyCentral

warehouse

Market OneMarket one

Market two

Market two

Market one

Chiang Rai

Chiang Rai

ASSUMPTIONS

• Manufacturing facility has sufficient capacity to

satisfy any warehouse demand

• Lead time for delivery to each warehouse is

about one week and is assumed to be constant.

• Delivery time does not change significantly if we

adopt a centralized distribution system.

• Service level of 95% that is the probability of

stocking out is 5% is maintained.

DATA ANALYSIS

Now with analysis of weekly demand for two different products, product A and product B produced by ABC company for last 8 weeks in both market zones we will be able to decide which distribution strategy will be more efficient and cost effective.

WEEK 1 2 3 4 5 6 7 8

Pathum 68 37 45 58 16 32 72 80

Prachine 87 62 55 67 12 42 69 81

TOTAL 155 99 100 125 28 74 141 161

PRODUCT A

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8

WEEK

AV

ER

AG

E W

EE

KL

Y D

EM

AN

D

DEMAND Pathum

DEMAND Prachine

HISTORICAL DEMAND DATA FOR PRODUCT A

WEEK 1 2 3 4 5 6 7 8

Pathum 0 0 1 3 2 4 0 1

Prachine 1 0 2 0 0 3 1 1

TOTAL 1 0 3 3 2 7 1 2

PRODUCT B

00.5

1

1.52

2.5

33.5

4

4.5

1 2 3 4 5 6 7 8

WEEK

AV

ER

AG

E D

EM

AN

D

DEMAND Pathum DEMAND Prachine

HISTORICAL DEMAND DATA FOR PRODUCT B

ANALYSIS OF HISTORICAL DATA

PRODUCT AVERAGEDEMAND

STANDARDDEVIATION

COEFFICIENT OF

VARIATION

Pathum A 51 20.70 0.41

Prachin B 1.38 1.41 1.02

Pathum A 59.38 22.23 0.32

Prachin B 1 1 1

CENTRAL A 110.38 39.14 0.35

CENTRAL B 2.38 1.99 0.84

SAMPLE CALCULATIONS

FOR PRODUCT A IN Pathumthani WAREHOUSE

1. Average demand = (68+37+45+58+16+32+72+80)/8=51

2. Standard deviation of demand =

= 20.7

3. Coefficient of variation = 20.7/51 = 0.41

2 2 2(68 51) (51 37) .............. (80 51)

8

− + − + −

GENERALIZATIONS

• average demand for product A is much higher than product B which is a slow moving product.

• Both standard deviation (absolute) and coefficient of variation (relative to average demand) are measure of variability of demand but we find that STD for product A is higher but coefficient of variation of product B is higher.

• For centralized distribution average demand is simply the sum of the demand faced by each of existing warehouse

• However the variability of demand as measured by STD or COV faced by central warehouse is lower than that faced by the two existing ones.

NUMERICAL VALUES

• Safety factor (Z) =1.65

• Fixed cost for both the products (Co) = Rs 3500

• Inventory holding cost (Cc) = Rs 18.5 per unit per week.

• Cost of transportation from warehouse to a customer – Current distribution system = Rs 50 per product

– Centralized distribution system = Rs 60 per product.

INVENTORY LEVELS

PRODUCT AVERAGE DEMAND DURINGLEAD TIME

SAFETY STOCK

(SS)

REORDERPOINT

(s)

ORDERQUANTITY

(Q)

ORDERUPTOLEVEL(S)

AVERAGE

INVENTORY

Pathum A 51 34.16 85 139 224 104

Prachine B 1.38 2.33 4 23 27 14

Pathum A 59.38 36.68 96 150 246 112

Prachine B 1 1.65 3 19 22 11

CENTRAL A 110.38 64.58 175 204 379 167

CENTRAL B 2.38 3.28 6 30 36 18

4. Safety stock =1.65 20.7 = 34.16

5. Reorder point = 51 + 34.16 = 85.16

6. Order quantity = = 139

7. Order up to level = 139 +85 = 224

8. Average inventory = 139/2 +34.16 = 103.66

× × 1

2 3500 51

18.5

× ×

SAMPLE CALCULATIONSFOR PRODUCT A IN Pathumthani WAREHOUSE

% REDUCTION IN

INVENTORY

REDUCTION IN AVERAGE INVENTORY

PRODUCT A = = 22.7%

PRODUCT B = = 28%

(104 112 167)100

(104 112)

+ −×

+

(14 11 18)100

(14 11)

+ −×

+

NORMAL DISTRIBUTIONAverage mean = 0

Standard deviation = 1

X axis- safety factor

Shaded area under curve= service level

Z=1.65P(z)=.95

Z=0

Demand Variability: Example 1

Product Demand

150

75

225

100

150

50

125

6148 53

104

45

0

50

100

150

200

250

Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar

Month

Demand

(000's)

Reminder:

The Normal Distribution

0 10 20 30 40 50 60Average = 30

Standard Deviation = 5

Standard Deviation = 10

ANALYSIS AT DIFFERENT

SERVICE LEVELS

When average inventory for different level of service is calculated corresponding to varying value of z it was found that there exists a trade-off between service level and reduction in inventory through risk pooling.

SERVICE LEVEL (%)

90 91 92 93 94 95 96 97 98 99 99.9

Z 1.29 1.34 1.41 1.48 1.56 1.65 1.75 1.88 2.05 2.33 3.08

PERCENTAGE REDUCTION IN AVERAGE INVENTORY VS

SERVICE LEVEL

0

5

10

15

20

25

30

90 93 96 99

SERVICE LEVEL

% R

ED

UC

TIO

N I

N A

VG

INV

EN

TO

RY

PRODUCTAPRODUCTB

SERVICE

LEVEL (%)90 91 92 93 94 95 96 97 98 99 99.9

PRODUCT A

24 23.7 23.4 23.1 23 22.7 22.3 21.8 21.7 21.2 19.5

PRODUCT B

27.12 27.07 27.0 26.94 26.89 26.82 26.72 26.59 26.44 26.2 25.65

% REDUCTION IN AVERAGE INVENTORY

Following generalizations are made

• If a company goes for higher level of service it has to compromise with the % of reduction in the inventory level and vice versa.

• To provide high service level company has to maintain high inventory too.

• % reduction in inventory decreases with increase in service level.

IDEAL SITUATION

This works best for

– High coefficient of variation, which reduces required

safety stock.

– Negatively correlated demand as in such a case the

high demand from one customer will be offset by low demand from another