Analysis of a Forecasting- Production-Inventory System with Stationary Demand L. Beril Toktay and Lawrence W. Wein Presented by Guillaume Roels Operations 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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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