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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
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1
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
2
§ 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
3
(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
4
(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
5
• 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
6
*$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
7
$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
8
[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
9
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
10
§ 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
.. ..
..
11
• [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]
12
§.§. 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) ))
13
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]
14
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]
◇◇
15
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
◇◇
16
(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
17
§.§. 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 ))
18
§ 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
19
§ 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.
20
§ 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]
21
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
22
-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
23
[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
24
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
25
§.§. 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
26
§ 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
◇◇
27
§ 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
28
§ 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)
◇◇
29
(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
30
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
31
§ 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
32
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] ◇◇
33
(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
34
(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] ◇◇
35
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)
36
§.§. 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 ...
37
§ 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.)
38
§ 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
*
*
39
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 $800
[Eg 20.1]
1-g-s-61◆
◆2 1-g-s-63◆
40
§.§. I20: I20: Problems & Problems & DiscussionDiscussion
Preparation Time : 25 ~ 30 minutesPreparation Time : 25 ~ 30 minutesDiscussion : 15 ~ 25 minutesDiscussion : 15 ~ 25 minutes
( # C.10 ) ( # C.10 )
41
§ I21: § I21: ∞ periods with starting∞ periods with starting inventory x units.inventory x units.
[
1
22
[c(Q x) L(Q)] cD L(Q)]
E{cost}
[cD L(Q)] ...
Di = demand for period i
2nd period purchases what’s being used in the previous period
Assumptions :
1. Backorders; and assumes (a) that each unit left over at the end of the final period can be salvaged with a return of the initial purchase cost c. (b) if there is a shortage at the end of the final period, this shortage is met by an emergency shipment with the same unit purchase cost c.
2. Demand Distribution & Costs are the same in all periods.
3. α= discount rate =
0.9431.06
1.00 e.g.
value$ Future
value$Present
42
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)] ...
◆
43
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)
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]
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 )
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
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
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
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
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 ))
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
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
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
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 ))
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]
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]
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 )
58
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