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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
][
][][
][)1(][]1[
][][][1
1
tekeadtDt
k
][][][ tetFtD
Expansion of inventory
][)1(])[][(]0[
][][]0[
]1[][]1[][
1
1
LtFLLtDtDI
kqkDI
LtqtDtItIt
k
Lt
Lk
])[(])[]2[]1[(
])[]1[][(]0[
])[][(])[][(]0[
LteLtaetaete
LtaetaeteI
LtFLtDLtFtDI
L
k
kakteItI0
)1]([]0[][
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
k
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
)111
11]([1)111(]0[2
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)