Chapter 7: Stochastic Inventory Model

Proportional Cost Models:

x: initial inventory,

y: inventory position (on hand + on order-backorder),

: random demand, () , (),

(y- )+: ending inventory position, N.B.L,

(y- ) : ending inventory position, B.L,

=1/(1+r) : discount factor,

ordering cost : c(y-x),

holding cost : h (y- )+

penalty cost : p( -y)+

salvage cost : - s(y- )+

Minimum cost f(x) satisfies:

L(y) convex, L’() < -c (otherwise never order) L′ eventually becomes positive

)2()()(min

)()(

)()()()(min)(0

yLxyc

dy

dyshxycxf

xy

y

pcp

y

xy

SxIfxSotherwisexq

Sxxy

cSL

,,0

*

*

)(

},max{)(

PolicyStockBase

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0))(1)(()()(

bcc

c

shccp

cpyLB

acc

c

shccp

cpyLBN

yyshc

ou

u

ou

u

pcp

Example

c=$1, h=1¢ per month, =0.99, p=$2(NBL), p=$0.25(BL),

s=50 ¢, c+h- s=51.5 ¢,

NBL: p-c = 100 ¢, BL: p-c(1- )=24 ¢,

)32.0(,32.05.5124

24)()(

)66.0(,66.05.51100

100)()(

1

1

yyii

yyi

Set up cost K

L(x) if we order nothing

K+c(S-x)+L(S) if we order upto S If we order, L’(S)+c=0.

Use the cheaper of alternatives L(x) and K+ c(S-x)+L(S)

)()()(min)( yLxycxyKxfxy

S x

cost

L(x)

KK

s S x

L(x)+cx

c

s

K+c(S-x)+L(S)

Two-bin or (s,S) policy

order S-x if x ≤ s

order nothing if x > s

Multiperiod models

Infinite Horizon (f1000 & f1001 cannot be different)

dyfyLxycxfxy

)()()()(min)(0

12

)9()()()()(min)(0

dyfyLxycxf

xy

Taking derivative of {}

)10()()(')('00

dSfSLc

If f convex, find S the base stock level, then for x ≤ S

)11()()()()()(0

dSfSLxScxf

We see from (11) thatf’(x)=-c for x ≤ S . (12)

)18(0))(1()('

N.B.Lfor Similarly

(13) B.L0)1()('

toreduces)(10

ScSL

cSL

Proportional costs:

So that

dypdyhyLy

y)()()()()(

0

policyS)(s,stilld,complicatemorecostSetuptime,Lead

:Remark

c-by replaced s- : (5) with cf.

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)1()(:.

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)(:..

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Example 4:

1800

5

4

205

20

)(

5 ,20

:Solution

loaves. ofnumber daily optimal theFind loaf.per

cents 5 of loss aat sellsoutlet store a otherwise, on time; sold if loafper

20cents ofprofit Makes ].[1000,2000 Uniform~Demand Bread

S

cc

cS

cc

ou

u

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Example 5:

5.73.05.1882

)(25

1

2

3)(

25

1

10

3

)()()()()(

:Solution

0,2

3p(z):costshortage

0,z ,10

3)(:costHolding

,1,15,25

1)(:ondistributiDemand

25

252

25

0

0

25

2

ye

deydey

dypdyhyL

zz

zzh

cKe

y

y

y

y

y

Intuition: The current period would be a separate one

period if we know what the next period would be willing

to pay for our leftover inventory. Assuming we are not

“overstocked”, every unit leftover will mean the next period will

order one less, thus saving c. So the next period should be

willing to pay c per unit in salvage for one leftover inventory.

5.805.1010

2525

25

25

q

:policy optimal The

5.80

:ionapproximatSuccesive

5.73.05.18825.101155.73.05.1882

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5.101

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3.025.75)(

xifxotherwise

Ss

S

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y

s

Seses

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S

edy

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edy

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Multiperiod models: No Setup Cost Begin with two periods

Demand D1, D2, i.i.d

Density: ()

L(y) = expected one period holding+ shortage penalty cost;

strictly convex with linear cost and () >0,

c purchase cost /unit

c1(x1) optimal cost with 1 period to go;

c+L’(S1)=0

while S1 is the optimal base stock level.

gotoperiods2withlevelbasestock

convexiswhich)]([)()(min)(

)()]()([)()(

)()())((

)()(

)(

2

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22

12

12

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dSLyScdyL

dycxcE

Dycxc

xc

xy

Sy

Sy

SDyifDyLSDyifSLDySc

SxifxLSxifSLxSc

Example: c=10, h=10, p=15 the demand density is

Solution:

100

10

1

0)(

if

otherwise

2

0

10

11

1

1

)4/5(1575

10

)(10

10

)(15)(

2,10

)(

5

1

1015

1015)(

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dz

dz

zL

SS

SSince

hp

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z

z

3/3592/194/)(24/)(

)4/5(1575)(10min)(

3/3592/194/)(24/)(

10

1)](1070[

10

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10

1]2)4/5(2*1575)2(10[

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22

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2

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:policy optimal The

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{}

zerotoequalitsetting,ytorespectwithderivativeTake

xifxotherwise

xifxotherwise

q

q

S

SS

S

SSdy

d

Multi-Period Dynamic Inventory Model with no Setup Cost

Cn(xn): n periods to go,

: discount factor.

DP equations:

SS

xcxc

dycyLxycxc

S

SSSSSS

xc

DycEyLxycxc

nn

nn

xy

nn

nnnnnnxy

nnnn

lim)3

)(lim)(bysatisfied

)()()()(min)(2)

optimal)horizon (Infinite 0;)-c(1)(L'

where,...................... 1)

:Properties

0)(

])([)()(min)(

0

1321

00

1

Multi-Period Dynamic Inventory Model with Setup Cost

nnnn

nn

nn

nn

nn

SxifxSSxifn

xyifKxyifnn

nnnnnnnxy

nn

q

xyK

dycyLxycxyKxc

0

nn

n

0

01

:)policyS,(soptimalThe

.Sfindthenconvex,isL(y)If

)(

)()()()()(min)(

Multi-Period Dynamic Inventory Model with Lead TimesLead time:

0)1()('

horizoninfinite

)()()(

:follows as timelead 0 toormCan transf

periods) 1-next in arrive order toon hand(on position inventoryu

)()()()()()(min)(

0

n

01

0

cS

dyLy

dyfdyLuycuyKuf nnnnnnnuy

nnnn

L

L