Reorder point

Lead time

Order qty., q

NO

BA

CK

LO

GG

ING

PURCHASE

Inv.

I

PRODUCTION

q

Inv.

II

q

bBA

CK

LO

GG

ING

Inv.

III

b

Inv.

IV

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

Total annual cost = dCCqdq

C 312

Annual usage

7001

4950002 )(EOQ

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

DETERMINISTIC SINGLE ITEM MODEL

t1 t2

tp

t3 t4

t

Imax

0

-b

Rate of fall, dRate of rise p-d

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

SAFETY STOCK DETERMINATION

Chosen reorder level

Distribution of lead time demand

Prob. Of stockout