isye3104chapter2-inventory+control+Q%2Cr

ISyE 3104 Fall 2014 © Georgia Tech, 2014 1

Inventory Control PART 3: STOCHASTIC DEMAND: Q ,r MODEL


The Single Product (Q,r) Model

Consider a central distribution facility which orders from a manufacturer and delivers to retailers. The distributor periodically places orders to replenish its inventory:

Random demand

Fixed lead time

Fixed setup/order cost

Why hold inventory?


The Single Product (Q,r) Model

Motivation: Either

1. Fixed cost associated with replenishment orders and cost per backorder.

2. Constraint on number of replenishment orders per year and service constraint.

Objective: Under (1) costbackorder cost holdingcost setup fixed min

Q,r

As in EOQ, this makes

batch production attractive.

The relevant costs are: Set-up each time an order is placed Holding at per unit held per unit time ( i. e., per year) Penalty per unit of unsatisfied/backordered demand


Inventory Profile of (Q, r) Policy

Time

Inve

nto

ry L

evel

Q+r

r

0

Lead Time Lead Time

Inventory Position

Q Q

On-hand Inventory

Q

s


The (Q, r) Policy

Policy: If the inventory level is at or below r, order Q

r is the reorder point, and Q is the order quantity ◦ r is chosen to protect against uncertainty of demand during the lead time

◦ Q is chosen to balance the holding and set-up costs


Summary of (Q,r) Model Assumptions

1. Demand is uncertain, but stationary over time and distribution is known.

2. Continuous review of inventory level.

3. Fixed replenishment lead time l

4. Constant replenishment batch sizes (Q)

5. Stockouts are backordered.


(Q,r) Model Derivation


(Q,r) Notation

cost backorder unit annual

stockout per cost

cost holding unit annual

item an of cost unit

order per cost fixed

time lead during demand of cdf ) ()(

time lead during demand of pmf

time lead entreplenishm during demand of deviation standard

time lead entreplenishm during demand expected ][

time lead entreplenishm during demand (random)

constant) (assumed time lead entreplenishm

year per demand expected

b

k

h

c

A

xXPxG

xXPg(x)

XE

X

D

)(


(Q,r) Notation (cont.)

levelinventory average),(

levelbackorder average ),(

rate) (fill level service average ),(

frequencyorder average )(

by impliedstock safety

pointreorder

quantityorder

rQI

rQB

rQS

QF

rrs

r

Q

Decision Variables:

Performance Measures:


Costs in (Q,r) Model with Backordered Demand

),(),(),( rQhIrQbBAQ

DrQY


Inventory Profile of (Q, r) Policy

Time

Inve

nto

ry L

evel

Q+r

r

0

Lead Time Lead Time

Inventory Position

Q Q

On-hand Inventory

Q

s

Expected Inventory Position


Inventory Costs in (Q,r) Model

Recall: Inventory Position = On hand Inventory + On order – Backorders - Committed

On average: expected inventory position declines from Q +r to r+1

(Q+ r)+ (r+1)

2= I(Q, r)+q -B(Q, r)

I(Q, r) =Q+1

2+ r -q +B(Q, r)


Backorder Costs

Backorder cost = bB(Q,r)

)(),(

)]()([1

),(

)(1

),(

rBrQB

QrBrBQ

rQB

dxxBQ

rQB

Qr

r

Averaging the backorders over all ranges of Q

Could be approximated by


Backorder Approximation

/)(),()](1)[()(

)()1()(1

rzwherezzrrB

dxxgrxrBr

:to simplified is this then ddistribute normally is demand If

)](1)[()())(1()()()(1

0

rGrrgxGxgrxrBr

xrx

simpler version for

spreadsheet

computing, but

only works for

Poisson demand

Recall from Base Stock model: If g(x) is continuous:

If g(x) is discrete:


(Q,r) Model with Backorder Cost

Objective Function:

),(),(),( rQhIrQbBAQ

DrQY

))(2

1()(),(

~),( rBr

QhrbBA

Q

DrQYrQY


Finding Q* and r*


Two Solution Procedures to find Q* and r*:

Based on minimizing the expected costs

Based on achieving a predetermined service level: Type 1 service: not stocking out in a cycle

Type 2 service: proportion of demand met from stock


Solution Approach 1: Minimizing expected costs


Results of Approximate Optimization

Assumptions: ◦ Q,r can be treated as continuous variables

◦ G(x) is a continuous cdf

Results:

zrbh

brG

h

ADQ

**)(

2*

if G is normal(,),

where (z)=b/(h+b)

Note: this is just the EOQ formula

Note: this is just the

base stock formula


Service Level Approximations

Type I (base stock):

Type II:

)(),( rGrQS

Q

rBrQS

)(1),(

Note: computes number

of stockouts per cycle,

underestimates S(Q,r)

Note: neglects B(r,Q)

term, underestimates S(Q,r)


(Q,r) Example

Stocking Repair Parts:

D = 14 units per year

c = $150 per unit

h = 0.1 × 150 + 10 = $25 per unit

l = 45 days

ϴ = (14 × 45)/365 = 1.726 units during replenishment lead time

A = $10

b = $40

Demand during lead time is Poisson


Values for Poisson(q) Distribution

r g(r) G(r) B(r)

0 0.178 0.178 1.726

1 0.307 0.485 0.904

2 0.265 0.750 0.389

3 0.153 0.903 0.140

4 0.066 0.969 0.042

5 0.023 0.991 0.011

6 0.007 0.998 0.003

7 0.002 1.000 0.001

8 0.000 1.000 0.000

9 0.000 1.000 0.000

10 0.000 1.000 0.000

)](1)[()( rGrrg


Calculations for Example

615.04025

40

43.415

)14)(10(22*

bh

b

h

ADQ

From the table r* = 2, Or, using the normal approximation:

2107.2)314.1(29.0726.1*

29.0615.0)29.0(

zr

z so ,


Performance Measures for Example

823.2049.0726.122

14*)*,(*

2

1**)*,(

049.0]003.0011.0042.0140.0[4

1

)]6()5()4()3([1

)(*

1*)*,(

904.0]003.0389.0[4

11

)]42()2([1

1*)]*(*)([*

11**

5.34

14

**)(

**

1*

rQBrQ

rQI

BBBBQ

xBQ

rQB

BBQ

QrBrBQ

),rS(Q

Q

DQF

Qr

rx


Observations on Example

•Orders placed at rate of 3.5 per year

•Fill rate fairly high (90.4%)

•Very few outstanding backorders (0.049 on average)

•Average on-hand inventory just below 3 (2.823)


Varying the Example

Change: suppose we order twice as often so F=7 per year, then Q=2 and:

which may be too low, so increase r from 2 to 3:

This is better. For this policy (Q=2, r=4) we can compute B(2,3)=0.026, I(Q,r)=2.80.

Conclusion: this has higher service and lower inventory than the original policy (Q=4, r=2). But the cost of achieving this is an extra 3.5 replenishment orders per year.

826.0]042.0389.0[2

11)]()([

11),( QrBrB

QrQS

936.0]011.0140.0[2

11)]()([

11),( QrBrB

QrQS


Solution Approach 2: Meeting Predetermined Service Levels


Finding Q* and r* based on Service Levels in (Q,r) Systems

In many circumstances, the backorder costs (b) are difficult to estimate.

For this reason, it is common business practice to set inventory levels to meet a specified service objective instead. The two most common service objectives are:

1) Type 1 service: Choose r* so that the probability of not stocking out in the lead time is equal to a specified value.

2) Type 2 service (fill rate). Choose both Q* and r* so that the proportion of demands satisfied from stock equals a specified value.


Service Level Approximations

Type I:

Type II:

)(),( rGrQS

Q

rBrQS

)(1),(


Finding Q* and r* based on Service Levels in (Q,r) Systems

For type 1 service, if the desired service level is α then one finds r* from G(r*)= α and Q*=EOQ.

For type 2 service, if the desired service level is β then ◦ set Q*=EOQ

◦ find r* to satisfy

*)(*

11 rB

Q


(Q,r) Example

Stocking Repair Parts:

D = 14 units per year

c = $150 per unit

h = 0.1 × 150 + 10 = $25 per unit

l = 45 days

ϴ = (14 × 45)/365 = 1.726 units during replenishment lead time

A = $10

Demand during lead time is Poisson


Values for Poisson(q) Distribution

r p(r) G(r) B(r)

0 0.178 0.178 1.726

1 0.307 0.485 0.904

2 0.265 0.750 0.389

3 0.153 0.903 0.140

4 0.066 0.969 0.042

5 0.023 0.991 0.011

6 0.007 0.998 0.003

7 0.002 1.000 0.001

8 0.000 1.000 0.000

9 0.000 1.000 0.000

10 0.000 1.000 0.000


Calculations for Example

43.415

)14)(10(22*

h

ADQ

If the desired type 1 service level is 90% => G(r*) = 0.9 =>

341.3)314.1(28.1726.1*

28.19.0)(

zr

zz so ,

If the desired type 2 service level is 90%

4.0*)(9.0*)(*

11 rBrB

Q

2*

)()](1)[()(

r

zzrrB

using


Performance Measures for Example

*)*,(*2

1**)*,(

)(*

1*)*,(

*)]*(*)([*

11**

**)(

**

1*

rQBrQ

rQI

xBQ

rQB

QrBrBQ

),rS(Q

Q

DQF

Qr

rx


Single Product (Q,r) Insights

Increasing D tends to increase optimal order quantity Q.

Increasing ϴ increase the reorder point.

Increasing the variability of the demand process tends to increase the optimal reorder point (provided z > 0).

Increasing the holding cost tends to decrease the optimal order quantity and reorder point.

isye3104chapter2-inventory+control+Q%2Cr

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