lot size/reorder level (q,r) models - isye.gatech.edupinar/teaching/isye3104-fall2013/...3 demand...

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013

Lot size/Reorder level (Q,R) Models

ISYE 3104 – Fall 2013

Recap: Basic EOQ

T 2T 3T 4T time

Inventory I(t)

d

QT

Q

Place an order when the inventory level is R. The order arrives after time periods Q was the only decision variable R could be computed easily because D was deterministic

R=d

Lead time

-d

2

Uncertain demand

Both Q and R are decision variables Cycle time is no longer constant!

Inventory

RQ

Q

s1s2

Q

s3

T1 T2 T3

time

(Q,R) Decisions

We choose R to meet the demand during lead time Service levels: Protect against uncertainties in

demand (or lead time) Balance the costs: stock-outs and inventory

Tradeoff in Q: Fixed cost versus holding cost

Objective: Minimize

fixed cost + holding cost + stockout (backorder) cost

3

Demand during lead time

Inventory

Time

Reorderlevel

What will happen if demandfollows one of these patterns?

Stockout

Excess inventoryR

Demand during lead time

Inventory

Time

Often the probability distributionof demand during lead time follows a Normal pattern

R

Expected demandduring lead time

R

P(D>R)=Probability of stockout

4

(Q,R) Model Assumptions Continuous review Demand is random and stationary. Expected demand is d per

unit time. Lead time is Costs

K: Setup cost per order h: Holding cost per unit per unit time c: Purchase price (cost) per unit p: Stockout (backorder) cost per unit

Demand during lead time is a continuous random variable Dwith pdf (density function) f(x) and cdf (distribution function) F(x) Mean= and standard deviation=

(Q,R) Model – Expected total cost per unit time

)()()()()()(0)(

0 shortage

shortage

cycleper shortage Expected )(

leadtime) during demand (expected level)(Reorder

arrivesorder an before levelinventory Average

:Recap)(

2)(

0

cost Shortagecost Fixedcost Holding

zLdxxfRxdxxfRxdxxfRn

RD

D-RRD

Rn

R-

s

d

QT

T

Rnp

T

KQshQC

R

R

R

Standard lossfunction

5


)()()()()()(0)(

0 shortage

shortage




:Recap)(

2)(

0



RD

D-RRD

Rn

R-

s

d

QT

T

Rnp

T

KQshQC

R

R

R



)()()()()()(0)(

0 shortage

shortage




:Recap)(

2)(

0



RD

D-RRD

Rn

R-

s

d

QT

T

Rnp

T

KQshQC

R

R

R


Same expression as the “expected number of stockouts” in the newsvendor model (Q replaced by R)

6


pd

QhRF

Q

RFpdh

R

Gh

RpnKdQ

Q

RpnKdh

Q

Rpdn

Q

Kdh

Q

G

Q

Rndp

Q

dKdR

Qh

QG

)(1))(1(

)]([2

)]([

20

)(

2

)(

2

cost Shortage cost Fixed cost Holding )(

222


pd

QhRF

h

RnpKdQ

Q

Rndp

Q

dKdR

Qh

QC

1)()]([2

:solution Optimal

)(

2

cost Shortage cost Fixed cost Holding )(

1 2

How do we pull Q and R from these equations? Solve iteratively!!

7

Solving for optimal Q and R

Start with a Q0 value and iterate until the Q values converge

Q0=EOQ

R1Q1

R2Q2

Q3 R3

R4

2

1

2

1

Remember: To find Q, you need n(R) = L(z)Lookup for z in the Normaltables

Example – Rainbow Colors

Rainbow Colors paint store uses a (Q,R) inventory system to control its stock levels. For a popular eggshell latex paint, historical data show that the distribution of monthly demandis approximately Normal, with mean 28 and standard deviation 8. Replenishment lead time for this paint is about 14 weeks. Each can of paint costs the store $6. Although excess demands are backordered, each unit of stockoutcosts about $10 due to bookkeeping and loss-of-goodwill. Fixed cost of replenishment is $15 per order and holding costs are based on a 30% annual interest rate. What is the optimal lot size (order quantity) and reorder level? What is the expected inventory level (safety stock) just before

an order arrives?

8

Example – Rainbow Colors Input

Monthly demand Normal mean=28 std.dev.=8 =14 weeks c=$6, p=$10, K=$15 h=ic=(0.3)(6)=$1.8/unit/year

Computed input d= ? (Expected annual demand) Expected demand during lead time

Variance of demand during lead time

?2

?



Computed input d=(28)(12)=336 units/year (Expected annual demand) Expected demand during lead time


38.1477.20652

14768demand timelead of Variance

768)8)(12( varianceAnnual 2

units90weeks)14(ar weeks/ye52

units/year336

9





38.1477.20652




units/year)12)(28(





38.1477.20652




units/year)12)(28(

As the lead time increases, so does the mean and varianceof demand during lead time

Shorter lead times Less variability of demand during lead time

10


Iteration 0: Compute EOQ

758.1

)336)(15)(2(20

h

KdQ


Iteration 1: Compute R1 (given Q0) and then compute Q1 (given R1)

11590)75.1)(38.14(

)()()()( :Recap

75.1)(96.0)336)(10(

)8.1)(75(11)(

1

01

RzRR

z

zzZPRD

PRDPRF

zzpd

hQRF

Z z Standard Normal

From standardNormal table

2

Q0=75

R1=115Q1

2

1

Safety Stock

Expected Demand during Lead time

11


Iteration 1 (continued): Compute Q1 (given R1)

808.1

)]233.0)(10(15)[336)(2(

233.0)0162.0)(38.14()()(

)]([2

1

1

Q

zLRnh

RnpKdQ

1

Q0=75

R2

2

1

2

Q1=80 R1=115

Q0 and Q1 not close, continue iterations



11590)72.1)(38.14(

72.1)(957.0)336)(10(

)8.1)(80(11)(

2

12

RzR

zzpd

hQRF

Q0=75

Q1=80 R1=115

R2=115STOP! R values converged, optimal (Q,R)=(80,115)

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(Q,R)=(80,115) Reorder level is larger than expected demand during

lead time. Why? Optimal order quantity is larger than EOQ. Why?

Safety stock s=R-=115-90=25 units


Weekly demand~5.64 Avg cycle time T=Q/d=80/6.64=12.38 weeks. Lead time=14 weeks. Cycle time shorter than lead time

Impact of R on the costs and Q

R inventory , therefore Q

R

Total cost

Holding cost

Fixed cost

Shortage cost

115

Q

115

As R Q

13

Optimal R as a function of Q

As the order quantity increases, the reorder level decreases

Q holding cost and setup cost , therefore R so that we can bring the holding cost , although a lower R means shortage cost

pd

QhRF 1)(

The Impact of Holding Cost on the Optimal (Q,R)

i Q R0.2 97 1160.3 81 1150.4 71 1140.5 64 1130.6 59 1120.7 55 111

As h goes up, both Q and R go down, but in this exampleQ drops at a faster rate!

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The Impact of Stockout Cost on the Optimal (Q,R)

As p goes up, Q goes ??? and R goes ???

The Impact of Stockout Cost on the Optimal (Q,R)

p Q R2 84 1016 81 11110 81 11514 81 11718 80 11822 80 120

As p goes up, Q goes down and R goes up!

15

Summary: (Q,R) Models

Balance between holding cost, setup/fixed cost, and shortage cost To save on the shortage cost, we want large R To save on the holding cost, we want small Q and

small R To save on the fixed cost, we want large Q

Choose Q and R to strike a good balance among these three costs!!!

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Service Levels in (Q,R) Models

ISYE 3104 – Fall 2013

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Service objectives

Type I service level () The proportion of cycles in which no stockouts occur Example: 90% Type I service level There are no

stockouts in 9 out of 10 cycles (on average)

Type II service level (fill rate, ) Fraction of demand satisfied on time

Service objectives - Example

Order cycle Demand Stock-outs1 180 02 75 03 235 454 140 05 180 06 200 107 150 08 90 09 160 010 40 0

TOTAL: 1450 55

Fraction of periodswith no stock-outs = 8/10Type I service = 80%

= 0.8

Fraction of demand satisfied on time = (1450-55)/1450=0.96 Type II service = 96%

= 0.96

In general, is it easier to achieve an x% Type I service or Type II service level?

17

Type I service level,

: Long-run average proportion of cycles with no stock-outs : Probability of having no stock-outs in a cycle : Probability of having no stock-outs during lead time : Probability that demand during lead time is less than R !!!

)()()(:Recap

)(

zzZPRD

PRDP

RDP

Set Q=EOQ Find z that satisfies (z)= Set R=z+ (safety stock + expected demand during lead time)



)()()(:Recap

)(

zzZPRD

PRDP

RDP


18



)()()(:Recap

)(

zzZPRD

PRDP

RDP


Why is Q=EOQ optimal in this case?


Rainbow Colors paint store uses a (Q,R) inventory system to control its stock levels. For a popular eggshell latex paint, historical data show that the distribution of monthly demandis approximately Normal, with mean 28 and standard deviation 8. Replenishment lead time for this paint is about 14 weeks. Each can of paint costs the store $6. Although excess demands are backordered, each unit of stockoutcosts about $10 due to bookkeeping and loss-of-goodwill. Fixed cost of replenishment is $15 per order and holding costs are based on a 30% annual interest rate. What is the optimal lot size (order quantity) and reorder level? What is the expected inventory level (safety stock) just before

an order arrives?

19


Rainbow Colors is not sure whether the $10 estimate for the shortage cost is accurate. Hence, they decided to use a service level approach.What are the optimal (Q,R) values if they want to

achieve no stockouts in 90% of the order cycles? satisfy 90% of the demand on time?


Monthly demand Normal mean=28 std.dev.=8 =14 weeks, c=$6, K=$15 h=Ic=(0.3)(6)=$1.8/unit/year = 0.9 or = 0.9



38.1477.20652



units90weeks)14(year / weeks52

year / units)12)(28(

20

Rainbow Colors – Type I service

Find (Q,R) to have 90% Type I service level Q=EOQ=75 (z)= = 0.9 z=1.28 R= z+ R=(14.38)(1.28)+90=108 For 90% Type I service level (Q,R)=(75,108)

Remember: With unit penalty cost of $10, we found (Q,R)=(80,115). What is the Type I service level that corresponds to (Q,R)=(80,115)?

R= z+ 115=(14.38)z+90 z=1.7385 (1.7385)=0.96 96% Type I service level when (Q,R)=(80,115)

Type II service level

: Fraction of demand met on time 1- : Fraction of demand not met on time (stock-outs)

Recap:

Q

Rn

Q

Rn

d

QT

Q

Rnd

T

Rn

)(1

)(

unit timeper demand Expected

unit timeper stockouts of # Expected1

since)()(

unit timeper stockouts of # Expected

With this information, for a given (Q,R), we can compute .

4

21

Rainbow Colors

For 90% Type I service level we found (Q,R)=(75,108)

What is the Type II service level which corresponds to this policy?

99.00097.075

7276.0)(1

7276.0)0506.0)(38.14()25.1()()(

25.138.14

90108

Q

Rn

LzLRn

zR

The same policy results in 90% Type I service and 99% Type II service!!

Finding the optimal (Q,R) for a desired Type II service level

pd

QhRF

h

RnpKdQ

p

1)()]([2

:cost out -stock

have hen wesolution w Optimal:Remember

1 2

22


QRn

RF

Rn

h

Kd

RF

RnQ

p

RFd

Qhp

pd

QhRF

h

RnpKdQ

p

)1()( usly with simultaneo solved be To

)(1

)(2

)(1

)(

: 1 into Substitute

)(1: 2 From

1)()]([2

:cost out -stock have hen wesolution w Optimal

2

1 2

3

4

5 Imputed shortage cost

Impact of service level on R

For a given Q As n(R)=(1- )Q i.e., R

As the service level , the reorder level as well

23


QRnRF

Rn

h

Kd

RF

RnQ )1()(

)(1

)(2

)(1

)(2

3 4

Q0=EOQ

R1Q1

R2Q2

Q3 R3

R4

4

4

4

4

3

3

3

Start with a Q0 value and iterate until the Q values (or the R values) converge


Iteration 0: Compute EOQ

758.1

)336)(15)(2(20

h

KdQ

24



Q0=75

R1=87Q1=89

4

3

89

5871.0)22.0()22.0(1

table.Normal at theLook ).(1 need we find To

8783.8690)22.0)(38.14(

22.05216.0)(

)(5.7)75)(9.01()1()(

1

11

1

01

Q

FF

R-FQ

zR

zzL

zLQRn

4

3



25



90

648.0)38.0()38.0(1

table.Normal at theLook ).(1 need we find To

855.8490)38.0)(38.14(

38.0619.0)(

)(9.8)89)(9.01()1()(

2

11

2

12

Q

FF

R-FQ

zR

zzL

zLQRn

4

3

Q0=75

R1=87Q1=89

4

3

R2=85Q2=904

3



851.8590)34.0)(38.14(

34.0591.0)(

)(5.8)85)(9.01()1()(

3

23

zR

zzL

zLQRn4

Q0=75

R1=87Q1=89

4

3

R2=85Q2=904

3

R3=854STOP! R values converged, optimal (Q,R)=(90,85)

lot size/reorder level (q,r) models - isye.gatech.edupinar/teaching/isye3104-fall2013/...3 demand...

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