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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).
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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−=
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Objective
Minimize steady-stateInventory holding costs hShortage penalty costs b
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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
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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
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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− − + − + −…
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Benchmark: Myopic Policy
Do not use available forecast information
Constant Inventory
t tR D=
t t mQ I s+ =
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Outline
ModelSteady-State Distribution of WIPBase-Stock LevelsDiscussionConclusion
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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
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WIP at time n
time
units1( )
n
n t tt
X R C=
= −∑
1infn n t n tQ X X≥ ≥= −
2 2R C−
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Heavy traffic analysis 1
Consider a sequence of systems s.t.:
( )
( )
( ) ( )( ) 0
k
k
k kk c
λ λ
µ µ
λ µ
→
→
− → <
k
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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
⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦− ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦= +
= −
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Heavy traffic analysis 2
( ) ( ) ( ) ( )k k k kkt ktX X m kt m ktk k k⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
− ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦= +
2( , )BM c σ
( )B tσ ct
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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
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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
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Outline
ModelSteady-State Distribution of WIPBase-Stock LevelsDiscussionConclusion
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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⎛ ⎞= + −⎜ ⎟
⎝ ⎠
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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
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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
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Outline
ModelSteady-State Distribution of WIPBase-Stock LevelsDiscussionConclusion
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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⎛ ⎞
= − − − ⎜ ⎟+⎝ ⎠
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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−
+ +=
= +∑
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Outline
ModelSteady-State Distribution of WIPBase-Stock LevelsDiscussionConclusion
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Conclusion
Integrated viewForecastProductionInventory
Lots of improvement for current MRP systemsIs heavy traffic of practical value?
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Thank you
Questions?