Stock Available Demand Occurs (units withdrawn) Determine Stock Position (on hand + on order - backorders) Is Stock Position ≤ Reorder Point ? Issue Replenishment Order stock receipt FIXED ORDER SIZE SYSTEM no yes
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Stock Available
Demand Occurs(units withdrawn)
Determine Stock Position(on hand + on order - backorders)
Is Stock Position ≤ Reorder Point ?
Issue Replenishment Order
stock receipt
FIXED ORDER SIZE SYSTEM
no
yes
TYPICAL INVENTORY CYCLE
TIME
QU
AN
TIT
Y
CLASSICAL INVENTORY MODEL
a b c d e f
TIME
Q = lot size , B = reorder point, ab = cd = ef = lead time
INV
EN
TO
RY
B
Q
TOTAL ANNUAL COST = PURCHASE + ORDER + HOLDING
02
H
Q
CRdQ
)Q(dTC2
=+-=
2
HQ
Q
CRPR(Q) =
TC ++
PFCR2
HCR2
*Q == = EOQ
ANNUAL INVENTORY COST
ORDER QUANTITY (Q)
Q*
CO
ST
TC(Q)
HQ/2
PR
CR/Q
2
*HQ
2
*HQPR ++=
2
*HQ
H/CR2
*CRQPR ++=
2
*HQ
*Q*Q
*CRQPR ++=
2HQ
QCR
PR)Q(TC ++=
2
*HQ
*Q
CRPR*)Q(TC ++=
If Q = Q*, then :
OPTIMUM TOTAL ANNUAL COST
* HQPR +=*)Q(TC
INVENTORY STOCK POSITION (B > Q)Q
UA
NT
ITY
TIME
Q -
B -
L
T
stock on-hand
stock on-hand + on-order
Q SENSITIVITY OF C, R, & H
H
CR2*Q
*Q
* QX
XX*Q
*Q
*QQ H
RC −=−
X
XX*Q
X
XX
H
CR2Q
H
RC
H
RC ==
=
= order quantity error fraction
1 − =H
RC
X
XX
TVC(Q) SENSITIVITY OF C, R, AND H
*HQPR*)Q(TC2
HQ
Q
CRPR)Q(TC
2
*)Q(TVC
2
*HQ
*Q
CR ==
+=++=
*HQ*)Q(TVC2
HQ
Q
CR)Q(TVC =+=
HRCHH
RC XXXX
*Q
X
XX
H
CR2Q =
=
H
CR2*Q =
*)Q(TVC
*)Q(TVCXXX*)Q(TVC
*)Q(TVC
*)Q(TVC)Q(TVC HRC −=−
HRC XXX*)Q(TVC=
HRC XXX2
*HQ
*Q
CR
+=
2
HQ
Q
CR)Q(TVC +=
H
HRCH
HRC
HRC
X2
XXX*QHX
XXX*Q
XXCRX +=
fraction errorTVC1XXX HRC =−=
TVC(Q) SENSITIVITY OF Q
=
+
=
+=
Q
2Q
QQ X
1X
2
*)Q(TVCX
X
1
2
*)Q(TVC
+=+= Q
Q 2
X*HQ
X*Q
CR
2
HQ
Q
CR)Q(TVC
= =Q factorerrorEOQEOQactual
EOQestimatedX
==2
*)Q(TVC
2
*HQ
*Q
CR
*HQ*)Q(TVC
*)Q(TVC
*)Q(TVCX
1X
2
*)Q(TVC
*)Q(TVC
*)Q(TVC)Q(TVC Q
2Q
X2
1X2X
Q
Q2
Q + −=
1X2
1X
Q
2Q − +
=
−
+
=−
fractionerrorTVCX2
)1X(
Q
2Q =
−=
EOQ SENSITIVITY
Order quantity
Q - Q* XC, XR, XH XCXR
Q* XH
Total variable cost
TVC(Q) – TVC(Q*) XC, XR, XH √ XCXR XH - 1
TVC(Q*)
Total variable cost
TVC(Q) – TVC(Q*) XQ (XQ – 1)2
TVC(Q*) 2 XQ
Error Fraction Error Factor Formula
- 1
BACKORDERING INVENTORY MODELIN
VE
NT
OR
Y L
EV
EL
TIME
Q
t2
t3
t1 J0
TC(Q, J) = Purchase + Order + Holding + Backordering
R
Jtand
R
JQt
KJttJQHCPQ
Q
R
22
)(
21
21
Q
KJ
Q
HJHJ
HQ
Q
CRPR
222
22
++−++=
Q
KJ
Q
JJQQH
Q
CRPR
22
)2( 222
++−++=
Q
KJ
Q
JQH
Q
CRPR
22
)( 22
+−++=
=−
=
+−++=
R
KJ
R
JQHCPQ
Q
R22
)( 22
+−++=
Q
KJ
Q
HJH
Q
CRQ
JQTC −−+−=∂
∂
KH
HQJ
+=*
JJQTC =
∂∂
QJKH
HQKJ
QHJ
H = ++−=++− 0)(),(
222
),(
H
JKH
H
CRQ
++= )(2*
22
2
2
2
2
2
JKHCR
Q
H =
+−−= 02
)(1
2
2
2
K
KH
H
CRQ
+= 2* B = RL - J*
Q* - J* = KQ*
H + K
PROOF: TC(Q*, J*) = PR + KJ*
2
22
***
**
2*
22
)(),(
KH
KQJQ
KH
HQJ
K
KH
H
CRQ
Q
kJ
Q
JQH
Q
CRPRJQTC
2
2
2
2
)(2
*
)(2
*)(2
*
KH
KQH
KH
QHK
K
KH
H
CRCRQ
PR+
++
++
+=
22*
*2
*
*2*
*
**)*,(
KH
HQ
Q
K
KH
KQ
Q
H
Q
Q
Q
CRPRJQTC
++
++
+=
+=−
+ =
+=
+−++=
TC(Q*, J*) *)(2)(2 2
22
QKH
KHHK
KH
HKPR
*
KH
HKQPR
++=
)(
*)(*
)( 22
2
KH
QKHHKPRQ
KH
HKPR
+++=
+++=
++ +
++=
*)(2 2
2222
QKH
KHHKHKKHPR
+
++++=
*KJPR +=TC(Q*, J*)
**
KH
HQJ
+ =
H K 2
Lower unit cost Higher holding costs
Lower ordering costs Larger inventory investment
Fewer stockouts Older stock
Price increase hedge Slow inventory turnover
QUANTITY DISCOUNTS
Advantages Disadvantages
ALL - UNITS QUANTITY DISCOUNTS
j
j
jjj
i
PPP
UUU
where
UQUforP
UQUforP
UQUforP
P
>>
<<
<≤
<≤
<≤
=
+
10
21
1
211
100
:
breakpoints =
unit purchase cost =
ORDER QUANTITY (Q)
CO
ST
INVENTORY COSTS: ALL-UNITS QUANTITY DISCOUNTS
U1 U2
TC(Q)PFQ 2
PRCR Q
ALL-UNITS QUANTITY DISCOUNTS
CASE 1 CASE 2 CASE 3
U1 U1 U1
QUANTITY
TO
TA
L C
OS
T
Decision Rule: Order the quantity with the lowest cost.Case 1: Order > U1
Case 2: Order = U1
Case 3: Order < U1
ALL-UNITS QUANTITY DISCOUNT LOGIC
Calculate the EOQ for the lowest unit cost
Is the EOQ valid ?
Calculate the total cost for the valid EOQ and all larger price-break quantities
Calculate the EOQfor the next higher unit cost
Is the EOQ valid ?
Select the EOQ as the order quantity
Select the order quantity with the lowest total cost
Start
No
No
Yes
Yes
KNOWN PRICE INCREASE
q
Q̂
Q*Q*
ta t1 t2 t3
TIME
QU
AN
TIT
Y
a
Special Order
No Special Order
KNOWN PRICE INCREASE
l. SPECIAL ORDER OF Q FOR TIME PERIOD (Q+q)/R
Total Cost = Purchase + Holding + Order
CR2
PFq
R2 Q ˆ PF
R Q ˆ PFq
Q̂PTC 2 2
s ++++=
ˆ ˆ
CRq
PF2q
RQ̂
PF2Q̂
R
qPFQ̂Q̂PTCs ++++=
holding cost of
Q from t2 to t3
holding cost of
q from t1 to t2
holding cost of
Q from t1 to t2ˆ ˆ
II. NO SPECIAL ORDER
( )( )
RPFq
RQ̂*FQkP
Q̂kPTC2a
n ++
++=
( ) ( )*QQ̂C
R
qPF
2
q
R
Q̂FkP
2
*QQ̂kPTC
a
an +++++=
( )( )
*Q
Q̂C
R2
PFq
R2
Q̂*FQkPQ̂kP
a
2a
Q
+++
++=
holding cost of q from t1 to t2
holding cost of Q* from t2 to t3a
C
( ) ( )
( )
R2
Q̂a*FQkP
FkPCR/2
a*QQ̂2
a*Q
a*Q̂C
a*Q
Q̂C +=
+==
KNOWN PRICE INCREASE=−= SavingsCostTCTCg
*=−−
+ += 0
ˆ)(
ˆ RQPF
RPFq
R
FQkPk
Qd
dg a
−−
−
++=
2
ˆˆ
*)( 2C
RQPF
QR
PFq
R
FQkPkg a
−−−22
ˆ 22C
RPFq
RQPF
ˆ−−+
+++=
ˆˆ
2
*)(ˆ)(
2
RQPFq
QPR
PFq
R
QFQkPQkP
a
sn
−++=*)(
*ˆ qP
QkP
PFkR
Q a = optimum special order quantity
1−
=
**ˆ
*
2
Cg
= optimum cost savings of special order
KNOWN PRICE INCREASE: DERIVATION OF g*
CRQPF
QR
PFq
R
FQkPkg
RQPF
R
PFq
R
FQkPkq
P
QkP
PF
kRQ
a
aa
−−
−
++=
=−++−++=
2
ˆˆ)(
*ˆ)()(*ˆ
2*
**
=*)(2 2Q
C
RPF
−
= 1
*
*ˆ*
2
Cg
−=*)(
*)ˆ(2
2C
Q
QC
CR
QPF
R
QPFg −−=2
*)ˆ(*)ˆ(*22
CR
QPF−=
2
*)ˆ( 2
−
−
−−+
++
+
1*
*ˆ1
*
*ˆ*
)()()(
*ˆ
22
**
Q
QC
Q
QCg
BqP
QkP
PFkR
P
QkP
PFkR
Q aa
KNOWN PRICE INCREASE OPTIMIZATION
During Regular Replenishment Before Regular Replenishment
PRODUCTION ORDER QUANTITY
Q1
Q
B
0 tp L
t1
TIME
QU
AN
TIT
Y (
Q)
- r
p
p - r
EPQ - SINGLE ITEM
prpQ
rptp
)(
)(
prp
Q
CRdQ
QdTC0
2
)H()(2
=−
+−=
pH rpQ
QCR
PR2
)( −++=
HOLDINGSETUPPRODUCTIONQTC )( ++=
−=
−=MAX. INV. tp = Q/p
rpH
CRpQ
)(2
*−
= = EPQ
pHQ rp
PRQTC* )(
*)(−
+=
EPQ WITH BACKORDERING
)(
2*
K
KH
rpH
CRpQ
*JN
RLB −=
)()(*
*pKH
rpHQJ
+−=
+−
=
)()(*
*)*,(pKH
rpKHQPRJQTC
+−+=
*KJPR +=
MAKE / BUY INFLUENCING FACTORS● Idle plant capacity
● In-house capabilities
— Personnel
— Equipment
— Future capabilities
● Economic Advantage
— Incremental cost
— Allocation of overhead
● Reliability of Supply
● Trade Relations (reciprocity)
● Employment stabilization
● Alternative resource uses
EPQ - MULTIPLE ITEMS
Qi = Ri/m* = lot size for item i
Ri = annual demand for item i
m* = optimum number of runs (cycles) per year
N = operating days per year
n = number of products or itemsQi / pi = run time in days per cycle for item i
pi = production rate for item i in units per day
N/m* = run (cycle) time in days
Nm*
1 2 3 4 5 1Q1
p1
Q2
p2
Q3 Q4 Q5
p3 p4 p5slack time
n∑=i ii pQ
1/
slack time ≥ 0
≥
HoldingSetupoductionPr)m(TC ++=
EPQ - MULTIPLE ITEMSm/Ri
tpipi mpi / Ri tpi ===Qi
m pi
)ri(Ri)ripi(tpi
-=-=
MAX INVfor item i
pi
∑
∑
=
==
-
= n
1i
n
1iyearperrunsof.nooptimal
Ci2
pi
)ripi(RiHi
m*
pi
)ripi(RiHi
m21CimRiPi
n
1i
n
1i
n
1i
-++= ∑∑∑
===
0pi
)ripi(RiHi
m 221Cidm
)m(dTC n
1i
n
1i=
--= ∑∑
==
= ii
∑=
−=n
1iii p/Q
*mNCycle/TimeSlack
∑=
≥
≥
n
1i i
ipR
N
TimeDemandTimeSupply
∑∑==
+=n
1ii
n
1iii C*m2RP*)m(TC*m/RQ*
FIXED ORDER INTERVAL SYSTEM
Backorder/Lost Sale
Issue Replenishment Order
Determine Stock Position(on hand + on order - backorders)
Compute Order Quantity(max. stock - stock position)
Stock Available
Demand Occurs
Has Review Period Arrived ?
Is Stock > Demand? no
yes
yes
no
no
FIXED ORDER INTERVAL SYSTEMQ
UA
NT
ITY
TIME
E
0T
L
T
L
TLL
EOI - SINGLE ITEM
TOTAL ANNUAL COST = PURCHASE + ORDER + HOLDING
)( LTRE +=
2)(
RFPTTC
PRTTC ++=
2
)(2
RFP
T
CdT
TdTC +−= = 0
2*RFP
CT = Q* = RT* = R 2C
PFR= 2CR
PF
TC(T*) = PR + HRT*
ANNUAL INVENTORY COSTS
ORDER INTERVAL (T)
T*
CO
ST
TC(T)
PFRT
PR
C
2
T
EOI - MULTIPLE ITEMS
TOTAL ANNUAL COST = PURCHASE + ORDER + HOLDING
)( LTRE ii +=
02
)()(
12
PRF
T
ncCdT
TdTC n
iii =++−= ∑
=
2)()(
11PRTF
TncC
PRTTCn
iii
n
iii +++= ∑∑
==
∑=
n
1iiiRPTC(T*) = ( 1 + FT*)
)(2*
1PRF
ncCT n
iii
+=
∑=
= EOI
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