reorder point lead time order qty., q no backlogging purchase inv. i
TRANSCRIPT
WITHOUT BACKLOGGING
02
1
3 *b,
CdC
*q
C = unit cost (Rs/piece)C1 = i X C = carrying cost (Rs/unit/time)C2 = shortage/backlogging cost (Rs/unit/time)C3 = order cost (Rs/order)
C = unit cost (Rs/piece)C1 = i X C = carrying cost (Rs/unit/time)C2 = shortage/backlogging cost (Rs/unit/time)C3 = order cost (Rs/order)
012
1
3
*b,
)p/d(CdC
*q
C = unit cost (Rs/piece)C1 = i X C = carrying cost (Rs/unit/time)C2 = shortage/backlogging cost (Rs/unit/time)C3 = order cost (Rs/order)
b
WITH BACKLOGGING
2
21
1
32C
CCC
dC*q
21
1
CC*qC
*b
C = unit cost (Rs/piece)C1 = i X C = carrying cost (Rs/unit/time)C2 = shortage/backlogging cost (Rs/unit/time)C3 = order cost (Rs/order)
b
2
21
1
3
12
CCC
)p/d(CdC
*q
21
1 1CC
)p/d(*qC*b
SENSITIVITY STUDIES ON CLASSICAL LOT-SIZE MODEL
q
AN
NU
AL
CO
ST
S
LOT SIZE qq1
q*
q2
TC
TCmin
Total cost
Carrying cost
Order cost
221 iCqqC
qdC3
Inv.Level
Ave
rage
In
vent
ory
= q
/2
qdCqC
)q(K31
2
1
32CdC
*q 312 CdCTCmin
Sensitivity Q = bq*, b > 0
b 0.5 0.8 0.9 1.0 1.1 1.2 1.5 2.0
TC/TCmin 1.250 1.025 1.006 1.000 1.005 1.017 1.083 1.29
bb
)b(TC
TCbmin 2
121 2
1
OPERATING AT A LOT-SIZE of 1000 rather than EOQ of 700 is WARRANTED HERE
ANNUALCOSTS
q700 2500
25,700
TAC(Rs 5)
1000
TAC(Rs 4.85)
TAC(Rs 4.75)
25,098
24,995
79220
525022 ==
)(.XX
)Rs(EOQ 112120
525021 ==
)(.
XX)Rs(EOQ
IN THIS CASE A LOT SIZE OF 112 RESULTS IN MINIMUM COST
ANNUALCOSTS
qEOQ =79 112100
COSTS/CYCLE
Notice that
)dp(dp
bddp
b)tt(
11
41
)dp(dp
Iddp
I)tt( axmaxm
11
32
bpd
qI& axm
1
t1 t2 t3 t4
AVERAGE ANNUAL COSTS K (q, b)
K (b, q) = t
Ctb
'Ct
ttbC
tttI
Caxm 3
241
232
122
Substituting for t, (t1 + t4), (t2 + t3) & Imax in terms of q, b we obtain
K (b, q) =
qdC
qbd'C
)p/d(qbC
)p/d(qb)p/d(qC 32
221
12121 2
OPTIMAL RESULTS
Annual cost is K (b, q)
The solution of these simultaneous equations yields the optimum values q* and b* as follows:
0
bK
qK
2
21
2111
3
)()/1(
2*
)'( 2
2
C
CC
CCCpdC
dCq
dC
21
21 1CC
)p/d)(d'C*qC(*b
and
U3U2U1
Amt. of inventory on hand
LT2
Amt. of inventory on hand
Amt. of inventory on hand
Amt. of inventory on hand
Reorder level, R
Safety stock (s s)
Avg. lead time usage (U)
Amt. used during Lead time
Amt. of inventory on hand
Amt. of inventory on hand
Amt. of inventory on hand
Q
LT1 LT3
QOrder qty, Q
Time
COMPUTATIONS FOR R
LT__
d_
U_
2)d_
)(VarLT(LT__
)Vard()U(Var
)U(Varu
uZUR
)u,U(N 2
Z Probability of stock out
Similar data on demands for last six months yield
EXAMPLE (p305, ch. 10)
d = 40 units/day
Var (d) = 30 (units/day)2
LT = 7 + 12 +25 + 16 + 14 + 156
= 14.83 days
Var (LT) = (7 – 14.83)2 + (12 – 14.83)2 + …
6 -1= 34.97 (day)2
(contd.)
EXAMPLE (contd.)
3975640 29734831430 ,)().().)(()U(Var
08304330
..
nyr/SO
3593831440 .).)((U Units demanded per lead time
523739756 .,u Units per lead time
Desired SO/yr = 0.33 (as stated earlier)
P = desired probability of stockout per order cycle
From tables Z = 1.39
SS = 1.39 (237.5) = 330.1
R = 593.3 + 330.1 = 923.4
Z
0.083
Order cycles/yr = 4 (given) = n
IMPROVING RELIABILITY OF LEAD TIME
Safety stock = 1.39 (21.09) = 29 ( compared to 330 earlier)
R = 622 (compared to 923 earlier)
Inventory lowered by 301 units
Annul savings = Rs 1 X 301 = Rs 301
Thus it is worthwhile to improve reliability of lead time
If var (LT) = 0
Then var (U) = 30 X 14.83 = 444.9
u = 444.9 = 21.09 units per lead time
(compared to the original 237.5)
Lead time
Reorder point
Avg.Lead time consumption
Reserve stock
Safety stock
Demand uncertainties
Lead time uncertainties
Sto
ck
Avg. demand for maximum delay
Probability of delay
a) Avg. demand during avg. lead time (buffer)
b) Variations in demand during avg. lead time, depending on service level (reserve stock)
c) Avg. demand during delivery delays (safety stock)
)LXD(
)kx( L D
Q system
units 1414x).(
x,x qty Order
1020
100000202
Safety Reserve Buffer point reorder
units 1540 4 x 52
20,000 time lead during demand avg. Buffer
time lead during demand of deviation std. x k stock Reserve
units 164 50) x 4( 1.64 delay of yprobabilit x delay max during demand Avg. stock Safety
units .x.x)3 x 20,000( 358310115431052
units 2062 358 164 1540 point Reorder
time
S2062
Stock level
Q = 1414
Lead time = 4 weeks1.64
Mean,
0.95
P system
wks.3.7 weeks52 x,
years,Demand
EOQ periodReview
000201414
000201414
Can be rounded off to either 3 or 4 weeks depending on cost consideration3 weeks
1,154 Rs .xxx17.3320,000 cost carrying inventory Annual 20010
21
Total inv. Cost = 1733 + 1154 = Rs 2887
1733 Rs 100 Rs x3
52 cost ordering Annual
4 weeks
1300 Rs 100 x4
52 cost ordering Annual
1,154 Rs .xxx13
20,000 cost carrying inventory Annual 20010
21
Total inv. Cost = 1300 + 1154 = Rs 2840
Review period is 4 weeks
Desired inventory level = Buffer + Reserve + Safety = 3668
units 3080 8 x52
20,000 Buffer
units 230 1.64 x 50 x 8 Reserve
units 350 0.31 x 3 x )(20,000/52 Safety
P system