Inventory Managementand Risk Pooling

Tokyo University of

Marine Science and Technology

Mikio Kubo

Why the inventory management is so important?

• GM reduced their inventory and transportation costs by 26% using a decision support tool that optimizes their fright shipment schedule.

• In 1994, IBM struggled with shortages in the ThinkPad line due to ineffective inventory management

• The average inventory levels of Japanese super markets are 2 weeks in food, 4 weeks in non-food products.

Two basic laws in inventory management

• The first lawThe demand forecast is always wrong.

• The second lawThe aggregation of inventories reduces the total amount of inventories

Other reasons for holding inventories are: • Economy of scale in production and/or transportation (lot-sizing inventories)• Uncertainty of the lead time • To catch up with the seasonal demand (seasonal inventories)

Economic ordering quantity modelInventory of Beers

　　　 Demand ratio per day =10 cans. Inventory holding cost is 10 yen per day per can. Ordering cost is 300 yen. Stock out is prohibited. What is the best ordering policy of beers?

Economic order quantity (EOQ) model

在 庫

需 要

発 注 量

サ イ ク ル 時 間 時 間

OrderingQuantity

Demand

Inventory

Cycle timeTime

EOQ formula

• Fixed ordering cost ＝３００ yen

• Demand ＝１０ cans/day

• Inventory ＝１０ yen/can ・ day

5.2410

103002quantity ordering optimal

costinventory

demandcost fixed2quantity ordering optimal

Economic Ordering Quantity (EOQ) model

Economic Ordering Quantity (EOQ) model

• d (units/day): Demand per day.• Q (units): Ordering quantity （ variable ）• K ($): Fixed ordering cost• h ( $/(day ・ unit) ） : Inventory holding costObjective ：

Find the minimum cost ordering policy• Constraints ：

– Backorder is not allowed ．– The lead time, the time that elapses between the placement of an order

and its receipt, is zero.– Initial inventory is zero.– The planning horizon is long (infinite).

Time

Inventorylevel

Cycle time (T days) = [ ]

Q

d ： demand speed

Total cost over T days = Ordering cost +Inventory Cost =f(Q)= Cost per day =

h×Area

EOQ formulaEOQ formula

• minimize f(Q)– ∂f(Q)/∂Q =– ∂2f(Q)/∂Q2 =– f(Q) is [ ] function.

• Q* =

• f(Q* )=

Swimsuit Production using Excel

• Fixed production cost 100000 $• Variable production cost 80$/unit• Selling price 125 $/unit • Salvage value 20$/unit

=125*MIN($D$1,$B2)=20*MAX($D$1-B2,0)

=D2+E2-F2-G2

Demand Probability 9000 Salvage value Variable Cost Fixed Cost Profit Expectation8000 0.11 1000000 20000 720000 100000 200000 22000

10000 0.11 1125000 0 720000 100000 305000 3355012000 0.27 1125000 0 720000 100000 305000 8235014000 0.22 1125000 0 720000 100000 305000 6710016000 0.19 1125000 0 720000 100000 305000 5795018000 0.1 1125000 0 720000 100000 305000 30500

1 Profit 293450

When the company produced 9000 units, the expected profit is 293450$.

Swimsuit Production (Continued)

0

50000

100000

150000

200000

250000

300000

350000

400000

0 5000 10000 15000 20000 25000

確率 8000 9000 10000 11000 12000 13000 14000 15000 16000 17000 180000.11 28600 22000 15400 8800 2200 - 4400 - 11000 - 17600 - 24200 - 30800 - 374000.11 28600 33550 38500 31900 25300 18700 12100 5500 - 1100 - 7700 - 143000.27 70200 82350 94500 106650 118800 102600 86400 70200 54000 37800 216000.22 57200 67100 77000 86900 96800 106700 116600 103400 90200 77000 638000.19 49400 57950 66500 75050 83600 92150 100700 109250 117800 106400 950000.1 26000 30500 35000 39500 44000 48500 53000 57500 62000 66500 71000

1 260000 293450 326900 348800 370700 364250 357800 328250 298700 249200 199700

Effect of initial inventory

• If initial inventory is 5000 units.

• Do not produce:225000+5000×80(pink line)

• Produce up to 12000 units: 370700+5000×80

(blue line)

販売（需要）量 確率 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 150008000 0.11 13750 18700 23650 28600 22000 15400 8800 2200 - 4400 - 11000 - 17600

10000 0.11 13750 18700 23650 28600 33550 38500 31900 25300 18700 12100 550012000 0.27 33750 45900 58050 70200 82350 94500 106650 118800 102600 86400 7020014000 0.22 27500 37400 47300 57200 67100 77000 86900 96800 106700 116600 10340016000 0.19 23750 32300 40850 49400 57950 66500 75050 83600 92150 100700 10925018000 0.1 12500 17000 21500 26000 30500 35000 39500 44000 48500 53000 57500

生産する場合 1 125000 170000 215000 260000 293450 326900 348800 370700 364250 357800 328250生産しない場合 225000 270000 315000 360000 393450 426900 448800 470700 464250 457800 428250

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

500000

0 5000 10000 15000 20000

Truncated Normal distribution with mean100 and standard deviation 100

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0 100 200 300 400 500 600 700

需 要 量

確 率 密 度

Demand

Probability density function

Service Level and Critical Ratio

• Service Level ： The probability with which stock-out does not happen.

Optimal service level=Critical ratio

99.01100

100

costinventory costbackorder

costbackorder ratio critical

Service level and density function

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0 100 200 300 400 500 600 700

需 要 量

確 率 密 度

３３３

Critical ratio=0.99The area (probability) that rhe demand is below 333 isSet to 0.99.

Inverse of cumulative distribution function

0

0.2

0.4

0.6

0.8

1

1.2

1 24 47 70 93 116

139

162

185

208

231

254

277

300

323

346

369

392

415

438

461

484

507

530

553

576

599

622

０．９９

３３３

Excel NORMINV(0.99,100,100)

Service level and safety stock ratio

0

0.5

1

1.5

2

2.5

60 65 70 75 80 85 90 95 100

サ ー ビ ス レ ベ ル

安 全 在 庫 係 数

NORMINV(service level,0,1)

Service level

Safety stock ratio

Base stock level• Base stock level ： target level of inventory

position

backorderinventorytransit -ininventory hand-inpositioninventory

timeleaddeviation standardratiostock safety

timeleaddemand averagelevelstock base

　 　 　 　 　 　 　

333=100×1+2.33×100×SQRT(1)

TV set example

月 9 10 11 12 1 2 3 4 5 6 7 8 平均 標準偏差販売量 200 152 100 221 287 176 151 198 246 309 98 156 191.1667 66.53479

=AVERAGE(B2:M2)

=STDEV(B2:M2)

• Lead time =2 weeks • Service level =97%-> safety stock ratio =[ ]• Average demand in a week (note that 1 month =4.4 week) ＝ [

]• Standard deviation in a week ＝ [ ]• Base stock level ＝ [ ] ；

Week of Supply ？ =[ ]

(s,S) Policy

• Fixed cost of an order （ K ）-> determine the ordering quantity Q using EOQ model

• (s,S) policy ： If the inventory position is below a re-ordering point s, order the amount so that it becomes an order-up-to level S

•サプライ・チェインの設計と管理 p.58 事例　秋葉原無線

TV set example (Continued)

• Fixed cost of ordering （ K ） =4500$

• Price =250$ ， interest rate = 18 % /year （ 1 year = 52 weeks ）->Inventory holding cost/week =[ ]

• Q is determined by EQO formula　　 Q= [ formula ] = [ ]

• inventory position （ S ） =[ ]

When the lead time L is a random variable

• Lead time L: Normal distribution with mean （ AVGL ） and standard deviation （ STDL ）

• Remark that the assumption that L follows a normal distribution is not realistic.

222 STDLAVGSTDAVGLzAVGLAVGs

Non-stationary demand case

CustomerRetailer

Demand D[t]Inventory I[t]

For each period t=1,2…,

Ordering quantity q[t]

Derive a formula for determining the safety stock levelWhen the demand is NOT stationary.

Discrete time model(Periodic ordering system)

Lead time L Items ordered at the end of period t will arrive at the beginning of period t+L+1.

2)Demand

D[t]occurs

t t+1 t+2 t+3 t+4

3) Forecast demand F[t+1]4) Order q[t]

１） Arrive the items ordered in period t-L-1

Arrive the itemsin period t+L+1 （ L=3)

Demand process

• Mean d• A parameter that represents the un-stability of demand process a (0<a<1)• Forecast error e[t], t=1,2,…

D[1]= d+e[1] D[t]= D[t-1] -(1-a) e[t-1] +e[t], t=2,3,…

Exceld=100,a=0.9,e[t]=[-10,10] (uniform r.v.)

D[t]=D[t- 1]- (1- a)e[t- 1]+e[t]需要量 e[t] a92.50207999 - 7.497920006 0.997.95128683 - 1.298921166 0.998.18900674 - 0.93130914 0.9108.4135819 9.386396971 0.999.5439356 - 0.421889065 0.9

92.32458517 - 7.599050589 0.9

１２３４５６７

A B C

=100+B2

=A2- （１－ C2 ） *B2+B3=RAND()*(-20)+10 =C3

Demand processa=0.9

D[t]=D[t-1]- (1-a)e[t-1]+e[t]需要量

0

20

40

60

80

100

120

1 10 19 28 37 46 55 64 73 82 91

D[t]=D[t-1]-需要量(1-a)e[t-1]+e[t]

Demand process a=0.5

D[t]=D[t-1]- (1-a)e[t-1]+e[t]需要量

0

20

40

60

80

100

120

1 10 19 28 37 46 55 64 73 82 91

D[t]=D[t-1]-需要量(1-a)e[t-1]+e[t]

Demand processa=0.1

D[t]=D[t-1]- (1-a)e[t-1]+e[t]需要量

0

20

40

60

80

100

120

1 10 19 28 37 46 55 64 73 82 91

D[t]=D[t-1]-需要量(1-a)e[t-1]+e[t]

Ordering quantity q[t]• Forecast future demands

(exponential smoothing method ）

F[1]=dF[t]=a D[t-1] + (1-a) F[t-1], t=2,3,…

• Ordering quantity: At the end of period t, order the amount

q[t]=D[t]+(L+1) (F[t+1]-F[t]) ,t=1,2,… where q[t]=d, t<=0.

Forecast and ordering amounta=0.5

940

960

980

1000

1020

1040

1060

1080

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101

D[t]=D[t- 1]- (1- a)e[t- 1]+e[t]需要量

F[t]

q[t]

Inventory I[t]

• Inventory flow conservation equation:Final inventory (period t)=Final inventory (period t-1)-Demand ＋ Arrival Volume

I[0]=A Safety Stock LevelI[t] =I[t-1] –D[t] +q[t-L-1],t=1,2,…

Example using Excel

需要量D[t]=D[t- 1]-

(1- a ) e[ t -1] + e[ t ] e[t] a F[t] L q[t] I[t]

1000100010001000 100

993.1913284 - 6.8086716 0.5 1000 3 982.9783209 106.80871004.405119 7.80945476 0.5 996.5956642 3 1016.119301 102.4036992.9290505 - 7.5713411 0.5 1000.500392 3 981.5720389 109.4745

=C7*A6+(1-C7)*D6=A6+(E6+1)*(D7-D6)

=G5-A6+F2

12345678

A B C D E F G

Inventory process: a=0.5I[t]

0

20

40

60

80

100

120

140

160

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101

I[t]

Relationship between demand and forecast

]1[]1[]1[ eFD

])1[]1[]1[)(1(][][][ tetDtFatetDtF

,...3,2,1],[][][ ttetFtD

Expansion of demand and forecast

t

k

kead

taetF

tFataDtF

1

][

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][)1(][]1[

][][][1

1

tekeadtDt

k

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Expansion of inventory

][)1(])[][(]0[

][][]0[

]1[][]1[][

1

1

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k

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LtFLtDLtFtDI

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Derived formula

• e[t] ： mean = ０， S.D. =σ, normal distribution

• Expected value of inventory

• Standard deviation

L

k

katISTD0

2)1(])[(

]0[])[( ItIE

6

)12(11

2

LLaaLL

Safety stock

• z ： Safety stock ratio

• When a=0 （ stationary ） :

• When a=1 （ random walk ） :

6

)12(11]0[

2

LLaaLLzI

1]0[ LzI

6

)32)(2)(1(]0[

LLLzI

Echelon Inventory

Supplier Warehouse

RelailerEchelon lead time （２ weeks ）

Echelon inventoryof warehouse

Echelon inventory position os warehouse

Multi echelon model

CustomerRetailer

Demand D1[t]Inventory I1[t]

For each period t=1,2…

Demand in the second level D2[t]＝ ordering quantity of the retailer q1[t]= Demand+Lead time × （ Forecast Error ）= D1[t]+(L1+1) (F1[t+1]-F1[t])

Warehouse(or Supplier)

Inventory I2[t]Order q2[t]

Lead time L1Lead time L2

Expansion of 2nd level demand (1)

D2[t]=D1[t]+(L1+1) (F1[t+1]-F1[t])

][1][11][11

1

tekeadtDt

k

t

k

keadtF1

][11]1[1

][1)111(][11][21

1

teaLkeadtDt

k

Expansion of 2nd level demand (2)

][1)111(][11][21

1

teaLkeadtDt

k

111

12

aL

aa

][1)111(][2 teaLte

][2][22][21

1

tekeadtDt

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Same as the first level demand ！

Inventory in the 2nd level

2

0

2)1)1(1(])[2(L

k

akLtISTD

2

0

)21]([2]0[2][2L

k

kakteItI

2

0

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L

k aL

akkteaLI

2

0

)1)1(1]([1]0[2L

k

akLkteI

]0[2])[2( ItIE

When the inventory is controlled by the warehouse (supplier)

• Warehouse (or supplier) controls the echelon inventory are controlled EI[t]

• Echelon lead time Ｌ１＋Ｌ２ (=EL)

CustomerRetailerWarehouse

(or Supplier)

Echelon lead time L1+L2

EL

k

katEISTD0

2)11(])[(

When the inventories are controlled by th

e retailer and the warehouse separately

1

0

2)11(])[1(L

k

katISTD

2

0

2)1)1(1(])[2(L

k

akLtISTD

Retailer

Warehouse (or Supplier)