ANALYSIS OF INVENTORY MODEL Notes 2 of 2

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1 ANALYSIS OF ANALYSIS OF INVENTORY MODEL INVENTORY MODEL Notes 2 of 2 Notes 2 of 2 By: Prof. Y.P. Chiu By: Prof. Y.P. Chiu 2011 / 09 / 01 2011 / 09 / 01

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ANALYSIS OF INVENTORY MODEL Notes 2 of 2. By: Prof. Y.P. Chiu 2011 / 09 / 01. § I12 : Inventory model: when demand rate λ is not constant. • Periodic review ~ A general model for Production Planning Terms: Periods: 1,2,3,….N - PowerPoint PPT Presentation

Transcript of ANALYSIS OF INVENTORY MODEL Notes 2 of 2

Page 1: ANALYSIS OF  INVENTORY MODEL Notes 2 of  2

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ANALYSIS OF ANALYSIS OF

INVENTORY MODELINVENTORY MODEL

Notes 2 of 2Notes 2 of 2

By: Prof. Y.P. ChiuBy: Prof. Y.P. Chiu

2011 / 09 / 012011 / 09 / 01

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§ I12 : Inventory model:§ I12 : Inventory model: when demand rate λwhen demand rate λ is not constantis not constant

• Periodic review ~ A general model for Production Planning

Terms:

Periods: 1,2,3,….N

i: demand rate in period i

h : holding cost / item / period K : setup cost c : unit cost : cost of producing enough items for period i thru. j at beginning of period i

(j )iC

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(B) Formula

j2i1i

j1ii

ij2h

ck

)(...

...C (j)i

...[Eq.12.1]

)( Ci 1CjjCiMIN

Nji …...[Eq.12.2]

• Lowest cost from period i to N that will satisfies demand

§ I12 : Inventory model:§ I12 : Inventory model: when demand rate λwhen demand rate λ is not constantis not constant

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(C)

2500

3000

2000

3000

4

3

2

02.0$

1.0$

200$

1

h

C

K

Demand

1Q 2Q 3Q 4Q 3000 2000 3000 2500

P1 P2 P3 P4

λ1 λ2 λ3 λ4

X2 X3 X4

[Eg.12.1] ~ When demand rate λ

is not constant ~

§ I12 : Inventory model:§ I12 : Inventory model: when demand rate λwhen demand rate λ is not constantis not constant

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• Use [Eq.12.2]

450

25000.1200h(0))c(λKCC 44(4)

4

*$800

50550200

25000.02

25003000 0.1200

λh) λ λ C(K C

$950

$450030000.1200

$450h(0)])c(λK[ CC MinC

4433

3433

(4)

(3)

[Eg.12.1] ~ When demand rate λ

is not constant ~

§ I12 : Inventory model:§ I12 : Inventory model: when demand rate λwhen demand rate λ is not constantis not constant

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*$1110 160750200

2500230000.02

250030002000 0.1200

)]2λ(λ h )λλC(λ[KC

$1210

$45060]500[200

$45030000.02

30002000 0.1200

C ]λh) λ C(λK[ CC MinC

$1200

$800020000.1200

Ch(0)])c(λK[ CC

43432 2

4332422

3232

(4)

(3)

(2)

[Eg.12.1] ~ When demand rate λ

is not constant ~

§ I12 : Inventory model:§ I12 : Inventory model: when demand rate λwhen demand rate λ is not constantis not constant

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$1,560

7500)

6000(20000.021050200

)]3λ2λ(λ h

)λλλC(λ[KC

$1610

$450

600020000.02800200

C

]2λλh)λ λ C(λK[ CC

*$1,540

80040500200

$800

]λh) λ C(λK[ CC

$16101110300200

$1110h(0)])c(λK[ CC

432

4321 1

4

3232141

22131

121

(4)

(3)

(2)

(1)

C1=Min

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[Answer]

540,1$311)2( CCC

To produce enough items from 1stperiod to 2nd period, then )4(

33 CC

To produce enough items from 3rd

period to 4th period.

In other words , production plan is: “ to produce 5000 items at the beginning of the first period, then to produce 5500 items at the beginning of the 3rd period ”.

0 , 5500 , 0 5000,

PPPP P 4 , 3 , 2 , 1 i

[Eg.12.1] When demand rate λis not constant

§ I12 : Inventory model:§ I12 : Inventory model: when demand rate λis not constantwhen demand rate λis not constant

3)2(

11 CCC )4(33 CC

0 , 5500 , 0 5000,

PPPP P 4 , 3 , 2 , 1 i

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4321

300 200 300 200

K=$20C=$0.1h=$0.02

?CCMin C Find 1j14j1

1(j)

§.§. I12: I12: Problems &Problems &DiscussionDiscussion

Preparation Time : 15 ~ 20 minutesPreparation Time : 15 ~ 20 minutesDiscussion : 10 ~ 20 minutesDiscussion : 10 ~ 20 minutes

#C.4#C.4

#C.5#C.5

3000003.0$ 4000

1$ 300040$ 2000

4

3

2

1

hCK

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§ I13 : Inventory Model: Resource-Constrained Multiple Product System

[Ex.13.1]

Item 1 2 3λj 1850 1150 800Cj $50 $350 $85Kj $100 $150 $50

252185250h

587350250h51250250h

3

2

1jj Cih

.. ..

..

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• [Eq.13.1] Must run under budget →

C EOQjn

000,30$1j

jC

EOQ1= 172 , EOQ2= 63 , EOQ3= 61

$50(172)+$350(63)+$85(61)=$35,835 (over-budget)

• Adjusting Factor

0.837$35,835$30,000

jjj EOQC

Cm scale alProportion

…..Eq.13.2]

51 61(0.837) m )(EOQ *Q52 63(0.837) m )(EOQ *Q144172(0.837) m )(EOQ *Q

33

22

11

$29,735 EOQjn

1j

jC

(Budget)

§ I13 : Resource-Constrained Multiple Product System

……[Eq.13.1]

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§.§. I13: I13: Problems & Problems & Discussion Discussion

Preparation Time : 25 ~ 30 minutesPreparation Time : 25 ~ 30 minutesDiscussion : 15 ~ 25 minutesDiscussion : 15 ~ 25 minutes

( ( # C.6 # C.6 ))

( ( # N4.38(a) # N4.38(a) ))

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18 W 61EOQ

12 W 63EOQ

9 W 172EOQ

33

22

11

feet square 2,000 3402

186112639172WEOQ ii

• Check : wi / hi

w1/ h1 = 9 / 12.5 = 0.72

w2 / h2 =12 / 87.5 = 0.14

w3 / h3 =18 / 21.25 =0.847

Diff.

• Simple solution obtained by a proportional scaling of the EOQ values will not be optimal. • Find Lagrange multiple ?

§ I13 : Inventory Model: Resource-Constrained Multiple Product System

[Ex.13.2]

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1 1 2 2 ...

: /

n n

i

W Q W Q W Q W

where W space consumed unit

" Lagrangean Function "

* 22 i i

i ii

kQ

h w

…..….[Eq.13.3]

1

* n

i i

i

w Q w

θ is a constant to satisfy

…….[Eq.13.4]

§ I13 : Inventory Model: Resource-Constrained Multiple Product System

[Ex.13.2]

◇◇

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0.5879 34022000

w(EOQ)w m

i i

(a) For proportional

1

2

33

172 0.5879

63 0.5879

61

101

37

0.53 8796

Q

Q

Q EOQ m

Not optimal!

§ I13 : Inventory Model: Resource-Constrained Multiple Product System

◇◇

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(b) Find Lagrangean Function

Find

86.632.1

upperLower

w* Qwthat so θ Find i i

1

2

3

Q * 92

Q * 51

75

Q

1

*

.

31

1998* Qw i i

§ I13 : Inventory Model: Resource-Constrained Multiple Product System

◇◇

* 22 i i

i ii

kQ

h w

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§.§. I13.1: Problems & I13.1: Problems & Discussion Discussion

Preparation Time : 15 ~ 20 minutesPreparation Time : 15 ~ 20 minutesDiscussion : 10 ~ 20 minutesDiscussion : 10 ~ 20 minutes

( ( # N4.26 ; N4.28 # N4.26 ; N4.28 ))

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§ I14: The Newsboy Model§ I14: The Newsboy Model

[Eg. 14.1] On consecutive Sundays, Mac, the owner of a local newsstand, purchases a number of copies of The Computer Journal, a popular weekly magazine. He pays 25 cents for each copy and sells each for 75 cents. Copies he has not sold during the week can be returned to his supplier for 10 cents each. The supplier is able to salvage the paper for printing future issues. Mac has kept careful records of the demand each week for the Journal. (This includes the number of copies actually sold plus the number of customer requests that could not be satisfied.) The observed demands during each of the last 52 weeks were

15 19 9 12 9 22 4 7 8 1114 11 6 11 9 18 10 0 14 12 8 9 5 4 4 17 18 14 15 8 6 7 12 15 15 19 9 10 9 16 8 11 11 18 15 17 19 14 14 17

13 12

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§ I14: The Newsboy Model§ I14: The Newsboy Model

There is no discernible pattern to these data, so it is difficult to predict the demand for the Journal in any given week. However, we can represent the demand experience of this item as a frequency histogram, which gives the number of times each weekly demand occurrence was observed during the year. The histogram for this demand pattern appears in the following Figure.

[Eg. 14.1]

Consider the example of Mac’s newsstand. From past experience, we saw that the weekly demand for the Journal is approximately normally distributed with mean μ=11.73 and standard deviationσ= 4.47.

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§ I14: The Newsboy Model§ I14: The Newsboy Model

Co : Overage Cost D : Demand Cu : Underage Cost

Q-D , 0 max C

D-Q , 0 max C D , QG

u

O

.….[Eq.14.1]

D , QG E G(Q) .…..…………..[Eq.14.1.a]

o0

u0

G(Q) C max 0 , Q-x ( )

C max [0, x-Q ] ( )

f x dx

f x dx

…...[Eq.14.1.b]

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uo

u

uuo

uo

uo2

2

uo

QuQ

0o

QuQ

0o

CCC

F(Q*)

0C-F(Q*)CC(Q*)G'

F(Q)-1CQFC0 dQ

dG(Q)

0 Q all for 0f(Q)CC dQG(Q)d

F(Q)-1C-F(Q)C

1- C1 C dQ

dG(Q)

Q-x C x-Q C G(Q)

dxxfdxxf

dxxfdxxf

)()(

)()(

…...[Eq.14.1.b]

…...[Eq.14.1.c]

…(Critical Ratio).[Eq.14.2]

§ I14: The Newsboy Model§ I14: The Newsboy Model

o0

u0

G(Q) C max 0 , Q-x ( )

C max [0, x-Q ] ( )

f x dx

f x dx

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-10 -6 0 100 200 300 400Q→

13

12

11

10

9 8

7 6

5 4

3 2

1

0

Expected Cost Function for Newsboy Model

Fig.14.1

§ I14: The Newsboy Model§ I14: The Newsboy Model

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[Eg. 14.1]

Purchase cost $ 0.25Sell for $ 0.75Salvage $ 0.10Demand has mean = 11.73 (μ)Standard deviation = 4.74 (σ)

• Use [Eq.14.2]

uo

u

CCC

F(Q*)

Cu : Underage Cost = 0.75-0.25 = 0.5

Co : Overage Cost = 0.25-0.10 = 0.15

0.7690.650.50

0.500.150.5

F(Q*)

§ I14: The Newsboy Model§ I14: The Newsboy Model

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f(x)

11.73 Q* X →

76.9%

76.9% → Z = 0.74

15238.1574.4

74.0

0.7411.73

0.74 *Q

-*Q Z

∴ Buy 15 copies every week .

§ I14: The Newsboy Model§ I14: The Newsboy Model

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§.§. I14: I14: Problems & Problems & DiscussionDiscussion

( # N5.8 ; N 5.9 )( # N5.8 ; N 5.9 ) # N 5.2T# N 5.2T # N 5.11# N 5.11

Preparation Time : 15 ~ 20 minutesPreparation Time : 15 ~ 20 minutesDiscussion : 10 ~ 15 minutes Discussion : 10 ~ 15 minutes

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§ I15 : ( R ,Q ) model§ I15 : ( R ,Q ) model

R

J T

S

I

Q

(A) Safety stock(B) Demand in lead time =λ . τ(C)Reorder point = s+ λ . τ = R

◇◇

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§ I16 : ( s , S ) model§ I16 : ( s , S ) model

s

S

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t

• μ : starting inventory in any period• s : reorder point

◆ If μ < s , order S – μ

If μ ≧ s , do not order.

μ1

μ2

μ3

μ4

μ5

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§ I17:§ I17: Stochastic Stochastic Inventory ModelInventory Model (1) : SINGLE PERIOD MODEL WITH No setup Cost

c = unit cost

p = sell price or shortage cost for unsatisfied demand per unit where p > c

h = holding cost or

cost of excess supply per unit

Q = quantities ordered

D= Demand ( A random variable)

d = demand (actual)

Pr(d) Pr(D d)

◇◇

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(A) RISK OF BEING “ SHORT “ ( Shortage cost incurred )

(B) RISK OF HAVING AN “ EXCESS “ ( Wasted unit cost & holding cost ) Recall Q = quantities ordered. D = actual demand , then amount sold:

Q D if QQ D if D

Q , D MIN.

Expected Cost : Discrete R.VExpected Cost : Discrete R.V

(d)P d)-Q , 0 ( hQ)-d , 0 ( pcQ

QD,GEG(Q)

r0d

maxmax

Qd

1Q

0d(d)P d)h(Q (d)PQ)p(d cQ rr

max max

max max

Q

Q

G( D,Q) cQ p ( 0 , D ) h ( 0 , Q D )

Let L(Q) = p ( 0 , D ) h ( 0 , Q D )

G( D,Q) cQ L(Q)

§ I17: § I17: (1) : Single Period Model with(1) : Single Period Model with No setup CostNo setup Cost

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Continuous R.VContinuous R.V

)d( D )]-Q (0,max hQ)-ξ , (0max p[cQ

QD,GEG(Q)

0

( )d

Q

D DQ 0

p( Q) ( )d hcQ

cQ

(Q )

L(Q)

Let L(Q) = Expected [shortage+holding costs] &

function density)(D

QD

0

E G(D, Q)0 ( )

Q( )D

dd

dQ p c

p h

0

d

d2

2Q) (Q) G(D,E

Minimum can be obtained.

累積機率函數 function Prob. Cumulative)Q(D

d

)( d )( ; DPr

DD

)(D

§ I17 § I17 (1) : Single Period Model with(1) : Single Period Model with No setup CostNo setup Cost

u

o u

C Recall : F(Q*)

C C

where Cu = p – c ; Co = c + h

57

◆1-43

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§ I18§ I18 : Another way to look at : Another way to look at SINGLE PERIOD INVENTORYSINGLE PERIOD INVENTORY

Let Q* be the smallest Q for which

0 purchased

*Q ofcost E

purchased 1*Q ofcost

E

Q*

$

Lot size

r r

r r

D D

D D

D

c p

c p h 0

unitP P

Q* 1 is sold * 1 is not sold

P D Q* P D Q*

[1 ( c p h 0

c p p h 0

Q*)] (Q*)

(Q*) (Q*

)

0

p c (Q*)p h

uh

nit

Q

function Prob.

cumulative is (Q*) when D

The Optimal quantity to order Q*, is the smallest integer such that the above function being satisfied.

◆2 投影片 60

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Suppose Demand of a certain single period product

is a random variable and which follows

otherwise 0

250150 100

1)(D

)(D

100

1

150 250

250 1

250150 100

150

150 0

)(D

150 200 250

1

0

)(D

Probability density function:

Cumulative probability function:

[Eg. 18.1] ◇◇

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(A) let c = $100 p = $200 h = $ -25 (salvage value)

250150 when 100

1 )(D

代入 .)(

)( 5710175

100

25200

100200

hp

cpQD

*1500.571

100 Q ( ) 207or

150 250* 207Q

*( )G Q

E G(D,Q)

250

207

2

250 *

07

150

2207150

2 25020

*

7

G(Q* ( )

($100)(207)

)

( 25)(20

($200)(

7 )

207)

414

( )

51.75 ] $207 ] 08

0

d

d

QD D

Q 1

D

D

50

( )

cQ* p(ξ Q*) ( )dξ h(

ξ

Q* )

d

$1849-$406 $20700

$22,143

[Eg. 18.1] ◇◇

~ about COSTS

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(B) To verify minimum G(Q*) = = $22,143

207)G(Q*

FIND

1

2

G(Q 200) ?

G(Q 210) ?

c = $100 p = $200 h = $ -25

1( ) when 150 250

100D

187223132500

508

400

d)h(200dξ )002p(ξcQ

200150

2250200

200

150 1001250

200 1001

1

,$ 20,000

]])($100)(200

)200G(Q

2

1

150224501600

5528

420

d)h(210dξ )102p(ξcQ

210150

2250210

210

150 1001250

210 1001

2

,$ 21,000

].])($100)(210

)210G(Q

2

2

$22,187$22,150$22,143

200 207* 210G(Q)

[Eg. 18.1] ◇◇

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Further discussion Further discussion

2

250

150

207 250D

150 207

207 250207

100 100150 207

207 2501

207200 1050 0 207

E[MIN. Q*,D ] MIN.(Q*, ) ( )

( ) * ( )

E[

] ]

#of items sol

10

d Q*

2

207]

1989 1

D

D

d

d Q d

d d

(a)

250

150

250

2207

200

207

250 250120710007 02 1 0

E[MAX. 0,D-Q* ] MAX. 0,D-Q* ( )

( Q*) (

E[#of shortage Q* 207]

)

( 207) ]

313 518 214 428 9.245

D

D

d

d

d

250150

207150

207 20715015

2

0

[#

[ .(0, ( * ))]

.(0, ( * )) ( )

( * ) ( )

1 (207 ) ]100

428 21

* 207]

207100 200

16.244 311 1 3 51

D

D

E MAX Q D

MAX Q D d

E of overa

Q

e

d

d

g Q

[ $200*9.245=$1849 ]

[ $-25*16.245=-$406 ]

[Eg 18.1]

(b)

(c)

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§.§. I18: I18: Problems & Problems & DiscussionDiscussion

Preparation Time : 25 ~ 30 minutesPreparation Time : 25 ~ 30 minutesDiscussion : 15 ~ 25 minutesDiscussion : 15 ~ 25 minutes

( ( # C.7 # C.7 ) ) ( ( # C.8 # C.8 )) ( ( # C.9 # C.9 ) ) ( ( # C.9.1.c # C.9.1.c ))

Advance TopicsAdvance TopicsFollow ...Follow ...

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§ I19§ I19 : : Single Period with Single Period with “ “ Initial stock x ”Initial stock x ”

Conclusion:Conclusion:

*

*

x

x

D

if Q

if Q

p cwhere Q* satisfies Φ Q*

p h

then order up to Q* (i.e. order Q*-x)

Do not order

◇◇

[Eg 19.1][Eg 19.1]

Let us suppose that in Example 14.1, Mac has received 6 copies of the Journal at the beginning of the week from another supplier.

The optimal police still calls for having 15 copies on hand after ordering, so now he would order the difference 15-6 = 9 copies.

( Set Q*=15 and u=6 to get the order quantity of

Q*-u=9.)

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§ I20 § I20 : Single Period with: Single Period with Ordering ( set-up ) cost “K”Ordering ( set-up ) cost “K”

(s,S) policy :

if on-hand Inv. x < s, order up to S. if on-hand Inv. x s, don’t order.≧

x= s Q*=S

E { G ( D, Q ) }

Lot size

Kc Q + L(Q)

K + c Q + L(Q)

Q

Find s

s order up to S

s do not order

x

x

c s L(s) K cS L(S)

if

if

1◆g-t-60

*

*

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c=20 p=45 h=-9 k=800

ξ10000

DΦ ξ 1 e

E(D) λ 10000

Q 11856 S

c s L(s

s 10674 (how to obtai

) K cS

n th

L(S)

is?)

When Demand Dist.~Exponential

10650120611856ΔSs

1206920

(800)2(10000)Δ

hc

K 2SΔ

s

Suppose ordering cost for the single period product described in Eg. #C.7 is $800

[Eg 20.1]

1-g-s-61◆

◆2 1-g-s-63◆

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§.§. I20: I20: Problems & Problems & DiscussionDiscussion

Preparation Time : 25 ~ 30 minutesPreparation Time : 25 ~ 30 minutesDiscussion : 15 ~ 25 minutesDiscussion : 15 ~ 25 minutes

( # C.10 ) ( # C.10 )

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§ I21: § I21: ∞ periods with starting∞ periods with starting inventory x units.inventory x units.

[

1

22

[c(Q x) L(Q)] cD L(Q)]

E{cost}

[cD L(Q)] ...

Di = demand for period i

2nd period purchases what’s being used in the previous period

Assumptions :

1. Backorders; and assumes (a) that each unit left over at the end of the final period can be salvaged with a return of the initial purchase cost c. (b) if there is a shortage at the end of the final period, this shortage is met by an emergency shipment with the same unit purchase cost c.

2. Demand Distribution & Costs are the same in all periods.

3. α= discount rate =

0.9431.06

1.00 e.g.

value$ Future

value$Present

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2 3

2

cQ cx L(Q) [1 ....]

c E{D} [1 ..

E{cost}

L(Q) c E{D} cQ - cx

1

.]

- 1

α11

Z

1ZZα

1Zαα1α

1Zαα

Zαα1

2

2

2

§ I 21 : § I 21 : ∞periods∞periods (continued)

[

1

22

E{cost} [c(Q x) L(Q)] cD L(Q)]

[cD L(Q)] ...

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D

D

d{E(cost)} 1 dc [L(Q)] 0

dQ 1 α dQ

d c (1 α ) [L(Q)] 0 (1)

dQ

dE{G(D,Q)} dIn single period c [L(Q)] 0

dQ dQ

p cIn single period Φ (Q*

In period

)

p h

From(1)

Φ (Q '

)p c (1 α

h )

p …...[ Eq.21.1 ]

L(Q) c E{D}E{cost} cQ - cx

1 - 1

g-s-30

◆g-b-48

§ I 21 : § I 21 : ∞periods∞periods (continued)

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44

And if p (shortage cost) = $ 15 [ backorder case, p may be less than c, cost just for handling backordering ] c = 35, h = 1 and discount rate α=0.995

800ξ 0 ifξ

ξΦD 800

741Q 800

Q

'

'

0.9266

Suppose Demand of a certain multiple period product is a random variable and it follows

otherwise0

800ξ 0 if ξ 800

1

D

Therefore,

9266.0

1150.995)(1

hp) α(1 cp

3515)(QΦ

)(QΦ period In

'

'

D

D

[Eg 21.1][Eg 21.1]

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45

§.§. I21: I21: Problems & Problems & DiscussionDiscussion

Preparation Time : 25 ~ 30 minutesPreparation Time : 25 ~ 30 minutesDiscussion : 15 ~ 25 minutesDiscussion : 15 ~ 25 minutes

( # C.11 )( # C.11 ) ( # C.12 ) ( # C.12 )

Page 46: ANALYSIS OF  INVENTORY MODEL Notes 2 of  2

46

§I 22 : Interpretation of Co, Cu§I 22 : Interpretation of Co, Cu

Define

period) the of end theat books the on backorders of number theagainst (charged

expense gbookkeepin plus will-good-of-Loss Pperiod the of end theat stock in

remaining inventory ofunit percost holding hitem the ofcost Variable citem the of Price SellingSg

periodSingle

0

0

( ) (

( ) max( ,0) max( ,0) min( ,

) ( ) ( )

)

( )( )Q

Q

Q

Q

G Q

cQ h Q x f x dx p x Q f x dx

Sg Sg f

cQ h Q D p D Q Sg Q

xf

D

dQ xd xx x

( ( ) )

Qu xf x dx

0 0( ) ( ) ( ) ( )

Q

Q Qxf x dx xf x dx xf x dx u xf x dx

'

0

*

( )

G ( )

( ) ( ) ( ) ( ) ( ) ( )

0

F( )

( ) ( )(1 ( )) 0

u

u o

Q

QcQ h Q x f x dx p Sg x Q f x dx SgG Q

Q

p Sg c CQ

p

u

Sg h C C

c hF Q p Sg F Q

…...[ Eq.22.1 ]

◆g-s-66

◆b-t-67

Page 47: ANALYSIS OF  INVENTORY MODEL Notes 2 of  2

47

1

21

2 3

Number of Units Sold in period 1 min(Q,D )

Number of Units Sold in period 2 min(Q, D )max(D Q, 0)

max(D Q,Number of Units Sold in period 3 min(Q,D )

0)

‧ ‧

1 2 1

max(Di Q,0

min(Q,Di) Di)

,

( ) ( )

for i 1,2

( ) m

n n

i

i

for n periods

G Q cQ c Sg E D D D

Sg E

Q D

Q D

‧ ‧

‧ ‧

( 1)

( ) (

in( , ) ( )

( ) ( ) min( , ) (

( ): ( )

min( , ) ( )

)

)

G(Q) (c Sg)u ( )

n

n

n

n

n

c Sg nu c Sg u

n n

Q D nL Q

cQ c Sg

G QAvg G Q

nSg

Sg

E Q

E

DcQ

Q n Q

n

L

Qn

D

L

L Q

letting n

§.I 23 :§.I 23 : ∞ ∞ periods withperiods with backorderedbackordered ◇◇

◆g-s-68

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48

' '

0

'

*

0

'

G ( ) 0 ( ) 0

( ) ( ) ( ) ( ) ( )

( ) ( ) ( 1) ( )

0

( ) ( )

( )

( )

( ) 0

( ) 1 ( )

Q

Q

Q

Q

Q L Q

L Q h Q x f x dx p x Q f x dx

L Q h f x dx p f x dx

h

pF Q

p h

pF Q

Optimal

p F Q

at L Q

p

p

h

hF Q p F Q

◇◇§.I 23 :§.I 23 : ∞ ∞ periods withperiods with backorderedbackordered

…... [ Eq.23.1 ][Eg. 23.1]

Let us return to Mac’s newsstand, described previously. Suppose that Mac is considering how to replenish the inventory of a very popular paperback thesaurus that is ordered monthly. Copies of the thesaurus unsold at the end of a month are still kept on the shelves for future sales.

◆b-t-51

◆g-s-57

◆g-s-43

Page 49: ANALYSIS OF  INVENTORY MODEL Notes 2 of  2

49

Assume that customers who request copies of the thesaurus when they are out of stock will wait until the following month. (Back-ordered allowed)

Mac buys the thesaurus for $1.15 and sells it for $2.75. Mac estimates a loss-of-goodwill cost of 50 cents each time a demand for a thesaurus must be back-ordered. Monthly demand for the book is fairly closely approximated by a Normal distribution with mean 18 and standard deviation 6. Mac uses a 20 percent annual interest rate to determine his holding cost. How many copies of the thesaurus should he purchase at the beginning of each month?

[Eg 23.1][Eg 23.1]

Solution: using [Eq.23.1]

The overage cost in this case is just the cost of holding, which is (1.15)(0.20) / 12 = 0.0192. The underage cost is just the loss-of-goodwill cost, which is assumed to be 50 cents.

Hence, the critical ratio is 0.5/(0.5+0.0192)=.9630. From the Table of Normal Dist., corresponds to a z value of 1.79. The optimal value of the order-up-to

point Q*=σZ+u=(6)(1.79)+18 = 28.74 =29.

( )p

F Qp h

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50

§.§. I23: I23: Problems & Problems & DiscussionDiscussion

Preparation Time : 25 ~ 30 minutesPreparation Time : 25 ~ 30 minutesDiscussion : 15 ~ 25 minutesDiscussion : 15 ~ 25 minutes

( ( # C.13 # C.13 ))

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51

1

2

3

Number of Units Sold in period 1 min(Q,D )

Number of Units Sold in period 2 min(Q,D )

Number of Units Sold in period 3 min(Q,D )

‧ ‧

Q

Q

E min(Q,D) (x Q) f(x) dx

,

( ) ( )( 1)E min(Q,D)

( ) min( , ) (

( ) ( 1) (x Q) f(x) dx

)

n

n

for n periods

G Q

G Q cQ c Sg n

Sg E Q D

cQ n c nSg

nL Q

‧ ‧

( ): ( )

( ) ( ) ( ) ( )

( )

G(Q) (c Sg) ( ) ( ) ( )

n

Q

Q

G QAvg G Q

ncQ c

G Q c Sg x Q f x dx L Qn

n

n

L Q

letting n

x Q f x dx L Q

§ I 24: § I 24: ∞ periods with ∞ periods with Lost Sales Lost Sales

◆b-t-69

◆ b-t-69

◆g-s-68

( ) min( , )

mi

max( ,0)

n( , )

max( ,0)

( )

G Q cQ Sg Qh Q D p D

cQ S

D Q

L Q g Q D

Page 52: ANALYSIS OF  INVENTORY MODEL Notes 2 of  2

52

'

' '

*

( ) ( ) ( ) ( ) 0

( ) 1 ( ) 0

( ) ( ) ( ) 0

( )

( ) 1 ( )

[ ]

( ) ( )

( )

Q

u

G Q c Sg f x dx L Q

c Sg F Q

c Sg c Sg F Q

c Sg h p F Q

L Q

hF Q p F Q

letting C p S

p g c

c

S

g

p Sg cF Qp Sg h c

*

==> ( )

o

u

u o

C h

CF Q

C C

…... [ Eq.24.1 ]

§ I 24: § I 24: ∞ periods with ∞ periods with Lost Sales Lost Sales

[Eg. 24.1] Assume that a local bookstore also stocks the thesaurus and that customers will purchase the thesaurus there, if Mac is out of stock. In this case excess demands are lost rather than back-ordered. ( Lost Sales case )

Determining the order-up-to points (which will be different from that obtained assuming full back-ordering of demand.)

◆g-t-48

Page 53: ANALYSIS OF  INVENTORY MODEL Notes 2 of  2

53

In the lost sales case the underage cost should be interpreted as the loss-of-goodwill cost (i.e. $0.50) plus the lost profit (ie. $2.75-$1.15=$1.60). Therefore, the underage cost is $0.50+$1.60=$2.10. The critical ratio is 2.1/(2.1+0.0192)=0.9909, giving a Z value of 2.36, the optimal value of Q in the lost sales case is

Q*=σZ+u = (6)(2.36)+18 = 32.16 = 32.

Versus backordering allowed Q*= 29

Solution: using [Eq.24.1]

[Eg. 24.1][Eg. 24.1]

* ( )p Sg c

F Qp Sg h c

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54

§.§. I24: I24: Problems & Problems & DiscussionDiscussion

Preparation Time : 25 ~ 30 minutesPreparation Time : 25 ~ 30 minutesDiscussion : 15 ~ 25 minutesDiscussion : 15 ~ 25 minutes

( ( # C.14 # C.14 ))

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55

§ I25: § I25: Markov Model inMarkov Model in Stochastic InventoryStochastic Inventory ManagementManagement

A camera store stocks a particular model camera that can be ordered weekly. Let D1, D2, …, represent the Demands for this camera during the first week, second week, …, respectively. It is assumed that the Dt are independent and identically distributed random variables having a known probability distribution. Let X0 represents the number of cameras on hand at the outset, X1 the number of cameras on hand at the end of week one, X2 the number of cameras on hand at the end of week two, and so on. Assume that X0 = 3. On Saturday night the store places an order that is delivered in time for the opening of the store on Monday.

◇◇[Eg 25.1]

Page 56: ANALYSIS OF  INVENTORY MODEL Notes 2 of  2

56

The store uses (s,S) ordering policy, where (s,S) = (1,3). It is assumed that sales are lost when demand exceeds the inventory on hand. Demand ( i.e. Dt ) has a Poisson distribution with λ= 1.

If the ordering cost K=$10, each camera costs the store $25 to own it and the holding is $0.8 per item per week, while unsatisfied demand is estimated to be $50 per item short per week.

Find the long-run expected total inventory costs per week?

Demonstration follows

Please see C.15C.15.

§ I25: Markov Model (cont’d)§ I25: Markov Model (cont’d)

[Eg 25.1]

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57

§.§. I25: I25: Problems & Problems & DiscussionDiscussion

Preparation Time : 25 ~ 30 minutesPreparation Time : 25 ~ 30 minutesDiscussion : 15 ~ 25 minutes Discussion : 15 ~ 25 minutes

( # C.15 )( # C.15 )

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58

The End The End ofof

Lecture NotesLecture Notes2 of 22 of 2