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).

Forecasting-Production-
Inventory
Demand
Make-to-Stock environment
forecast
Ct ~ N ( µ , σ )
2 update
Rt C
Order
Dt
release
It
Qt −1
Pt = min{Qt −1 , Ct }

Objective
Minimize steady-state
Inventory holding costs h
Shortage penalty costs b

Recap: MMFE Model
Rolling horizon H
Forecast Dt ,t +i i = 0,… , H
Forecast Update
ε t ,t +i = Dt ,t +i − Dt −1,t +i
Assumptions
Stationary Demand with rate λ
Unbiased Forecasts
Uncorrelated Forecast Updates

Production-Inventory Model
MRP-Type Release Policy:
H −1
Rt = ∑ ε t ,t +i + Dt ,t + H = eT ε t + λ
i =0
Inventory Policy
H
Qt + I t − ∑ Dt ,t +i = sH
i =1
It

Production Policy
Forecast-corrected base-stock policy
⎧Ct if sH > I t −1 + Ct ⎫
⎪ ⎪
P ( I t −1 ) = ⎨
*
⎬
⎩ sH − I t −1 if sH ≤ I t −1 + Ct ⎭
⎪ ⎪
State-dependent Optimal Policy
( Dt −1,t , Dt −1,t +1 ,… , Dt −1,t + H −1 , λ )

Benchmark: Myopic Policy
Do not use available forecast
information
Rt = Dt
Constant Inventory
Qt + I t = sm

Outline
Model
Steady-State Distribution of WIP
Base-Stock Levels
Discussion
Conclusion

In Heavy Traffic, the WIP has
an exponential distribution
Average Excess Capacity
2( µ − λ )
v= T
e Σe + σ C 2
Variance of the forecasts Variance of the production

WIP at time n n
X n = ∑ ( Rt − Ct )
units
t =1
Qn = X n − inf1≥t ≥ n X t
R2 − C2
time

Heavy traffic analysis 1
Consider a sequence of systems k s.t.:
λ →λ
(k )
µ →µ
(k )
k (λ −µ )→c<0
(k ) (k )

Heavy traffic analysis 2
− inf1≥t ≥ n X
(k ) (k ) (k )
Q X
= +
n n t
k k k
−m ⎢⎥
(k ) (k ) (k )
⎢ kt ⎥
⎣ kt ⎦
(k )
X X ⎣⎦
m
⎢ kt ⎥ ⎢ kt ⎥
= +
⎣⎦ ⎣⎦
k k k
where m = λ − µ
(k ) (k ) (k )

Heavy traffic analysis 2
σ B(t ) ct
−m ⎢ kt ⎥
(k ) (k ) (k )
⎢ kt ⎥
⎣⎦
(k )
X X ⎣⎦
m
⎢ kt ⎥ ⎢ kt ⎥
⎣⎦ ⎣⎦
= +
k k k
BM (c, σ ) 2

Heavy traffic analysis 3
− inf1≥t ≥ n X
(k ) (k ) (k )
Q X
= +
n n t
k k k
RBM (c, σ )2
Reflected Brownian Motion
on the nonnegative halfline
Estimated by an exponential
random variable

Steady-state WIP distribution
− vβ
P(Q∞ = 0) = 1 − e impulse
(x + β )
− vx
P(Q∞ > x) = e
β is a correction term, coming from
Random Walks

Outline
Model
Steady-State Distribution of WIP
Base-Stock Levels
Discussion
Conclusion

Determination of the Base-
Stock levels 1
Myopic Policy
⎛b⎞
−1
sm = F ⎜ ⎟
⎝b+h⎠
Q∞
Newsboy quantity…
… considering the distr. of the WIP!
1 ⎛ b⎞
sm = ln ⎜1 + ⎟ − β
v ⎝ h⎠

Determination of the Base-
Stock levels 2
MRP-type Policy
⎛b⎞
−1
sH = F ⎜ ⎟
⎝b+h⎠
W
W = max{Q∞ + Y0 , max1≤ k ≤ H Yk }
Y0 is the difference between:
Total Forecast Error over the horizon H and
Total Capacity

Determination of the Base-
Stock levels 3
MRP-type policy: asymptotic
b h
⎛1⎞ 2
s = s + µY 0 + ⎜ ⎟ σ Y0 v
a *
H m
⎝2⎠
Proportional to the variance!
Good approximation when
b / h large
High utilization rate

Outline
Model
Steady-State Distribution of WIP
Base-Stock Levels
Discussion
Conclusion

Capacity – Stock Trade-Off
No advance information
s = s − Hλ
a *
H m
Full advance information
⎛σ ⎞
2
s = s − H λ − H (µ − λ ) ⎜ 2
a *
2⎟
D
⎝ σ D +σC ⎠
H m
Interchangeability Capacity/Safety Stock
Demand variability Capacity

Discussion
Correlation ↑→ v ↓→ sm ↑
*
σ Y0 is the system variability over H not
2
resolved at the beginning of the horizon
Preference for accurate early forecasts
Optimize over all planning horizons
H −1
∑ε
Rt = + Dt , t + H
t , t +1
i=0
Greater costs due to
misspecification of the forecast model than
misuse of the information in production

Outline
Model
Steady-State Distribution of WIP
Base-Stock Levels
Discussion
Conclusion

Conclusion
Integrated view
Forecast
Production
Inventory
Lots of improvement for current MRP
systems
Is heavy traffic of practical value?

Thank you
Questions?