Analysis of a Forecasting-Production-Inventory System with Stationary Demand

L. Beril Toktay and Lawrence W. WeinPresented by Guillaume RoelsOperations Research Center

This summary presentation is based on: Toktay, L. Beryl, and Lawrence M. Wein. "Analysis of a Forecasting-Production-Inventory System with Stationary Demand." Management Science 47, no. 9 (2001).

Forecasting-Production-Inventory

Make-to-Stock environment Demand forecast update

Order release

2~ ( , )t CC N µ σtR

tD

tI

1tQ −

1min{ , }t t tP Q C−=

Objective

Minimize steady-stateInventory holding costs hShortage penalty costs b

Recap: MMFE Model

Rolling horizon ForecastForecast Update

AssumptionsStationary Demand with rateUnbiased ForecastsUncorrelated Forecast Updates

H, 0, ,t t iD i H+ = …

, , 1,t t i t t i t t iD Dε + + − += −

λ

Production-Inventory Model

MRP-Type Release Policy:

Inventory Policy

1

, ,0

HT

t t t i t t H ti

R D eε ε λ−

+ +=

= + = +∑

,1

H

t t t t i Hi

Q I D s+=

+ − =∑

tI

Production Policy

Forecast-corrected base-stock policy

State-dependent Optimal Policy

1*1

1 1

( )tt H t

tt tH H t

C if s I CP I

s I if s I C

−−

− −

⎧ ⎫> +⎪ ⎪= ⎨ ⎬− ≤ +⎪ ⎪⎩ ⎭

1, 1, 1 1, 1( , , , , )t t t t t t HD D D λ− − + − + −…

Benchmark: Myopic Policy

Do not use available forecast information

Constant Inventory

t tR D=

t t mQ I s+ =

Outline

ModelSteady-State Distribution of WIPBase-Stock LevelsDiscussionConclusion

In Heavy Traffic, the WIP has an exponential distribution

Average Excess Capacity

2

2( )T

C

ve e

µ λσ

−=

Σ +

Variance of the forecasts Variance of the production

WIP at time n

time

units1( )

n

n t tt

X R C=

= −∑

1infn n t n tQ X X≥ ≥= −

2 2R C−

Heavy traffic analysis 1

Consider a sequence of systems s.t.:

( )

( )

( ) ( )( ) 0

k

k

k kk c

λ λ

µ µ

λ µ

→

→

− → <

k

Heavy traffic analysis 2( ) ( ) ( )

1infk k kn n t n tQ X Xk k k

≥ ≥−= +

( ) ( ) ( ) ( )

( ) ( ) ( )where

k k k kkt kt

k k k

X X m kt m ktk k k

m λ µ

⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦− ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦= +

= −

Heavy traffic analysis 2

( ) ( ) ( ) ( )k k k kkt ktX X m kt m ktk k k⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

− ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦= +

2( , )BM c σ

( )B tσ ct

Heavy traffic analysis 3( ) ( ) ( )

1infk k kn n t n tQ X Xk k k

≥ ≥−= +

2( , )RBM c σReflected Brownian Motion on the nonnegative halfline

Estimated by an exponential random variable

Steady-state WIP distribution

is a correction term, coming from Random Walks

impulse( 0) 1

( ) ( )

v

vx

P Q e

P Q x e x

β

β

−∞

−∞

= = −

> = +

β

Outline

ModelSteady-State Distribution of WIPBase-Stock LevelsDiscussionConclusion

Determination of the Base-Stock levels 1

Myopic Policy

Newsboy quantity…… considering the distr. of the WIP!

1m Q

bs Fb h∞

− ⎛ ⎞= ⎜ ⎟+⎝ ⎠

1 ln 1mbs

v hβ⎛ ⎞= + −⎜ ⎟

⎝ ⎠

Determination of the Base-Stock levels 2

MRP-type Policy

is the difference between:Total Forecast Error over the horizon H andTotal Capacity

1H W

bs Fb h

− ⎛ ⎞= ⎜ ⎟+⎝ ⎠

0 1max{ ,max }k H kW Q Y Y∞ ≤ ≤= +

0Y

Determination of the Base-Stock levels 3

MRP-type policy: asymptotic

Proportional to the variance!Good approximation when

largeHigh utilization rate

0 0

* 212

aH m Y Y

b h

s s vµ σ⎛ ⎞= + + ⎜ ⎟⎝ ⎠

/b h

Outline

ModelSteady-State Distribution of WIPBase-Stock LevelsDiscussionConclusion

Capacity – Stock Trade-Off

No advance information

Full advance information

Interchangeability Capacity/Safety StockDemand variability Capacity

*aH ms s Hλ= −

2*

2 2( )a DH m

D C

s s H H σλ µ λσ σ⎛ ⎞

= − − − ⎜ ⎟+⎝ ⎠

Discussion

Correlationis the system variability over H not

resolved at the beginning of the horizon Preference for accurate early forecasts

Optimize over all planning horizons

Greater costs due to misspecification of the forecast model than misuse of the information in production

*mv s↑→ ↓→ ↑

0

2Yσ

1

, 1 ,0

H

t t t t t Hi

R Dε−

+ +=

= +∑

Outline

ModelSteady-State Distribution of WIPBase-Stock LevelsDiscussionConclusion

Conclusion

Integrated viewForecastProductionInventory

Lots of improvement for current MRP systemsIs heavy traffic of practical value?

Thank you

Questions?