A Dynamic model for requirements
planning with application to supply
chain optimization
by
Stephen Graves, David Kletter and William Hezel (1998)
15.764 The Theory of Operations Management
March 11, 2004
Presenter: Nicolas Miègeville
This summary presentation is based on: Graves, Stephen, D.B. Kletter and W.B. Hetzel.
\"A Dynamic Model for Requirements Planning with Application to Supply Chain Optimization.\"
Operations Research 46, supp. no. 3 (1998): S35-49.

Objective of the paper
Overcome limits of Materials Requirements
Process models
Capture key dynamics in the planning process
Develop a new model for requirements
planning in multi-stage production-inventory
systems
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Quick Review of MRP
At each step of the process, production of
finished good requires components
MRP methodology: iteration of
Multiperiod forecast of demand
Production plan
Requirements forecasts for components (offset
leadtimes and yields)
Assumptions:
accuracy of forecasts
Deterministic parameters
Widely used in discrete parts manufacturing
firms: MRP process is revised periodically
Limits: deviations from the plan due to
incertainty
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Literature Review
Few papers about dynamic modeling of
requirements
No attempt to model a dynamic forecast process,
except:
Graves (1986): two-stage production inventory system
Health and Jackson (1994): MMFE
Conversion forecasts-production plan comes from
previous Graves’ works.
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Methodology and Results
We model:
Forecasts as a stochastic process
Conversion into production plan as a linear system
We create:
A single stage model
production demand
From which we can built general acyclic networks of
multiple stages
We try it on a real case (LFM program, Hetzel)
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Overview
Single-stage model
Extension to multistage systems
Case study of Kodak
Conclusion
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Overview
Single-stage model
Model
Forecasts process
Conversion into production outputs
Measures of interest
Optimality of the model
Extension to multistage systems
Case study of Kodak
Conclusion
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The forecast process model
Demand forecasts
Discrete time
16
At each period t: 14
12
ft(t+i) , i<=H 10
Units
8 d(t+i)
ft(t+i)= Cte , i>H 6
ft(t) is the observed demand
4
2
0
1 3 5 7 9 11 13 15 17
Forecast revision: Week i
∆f t (t + i ) = f t (t + i ) − f t −1 (t + i ) i = 0,1,… H
∆ft is iid random vector E[ ∆ft ] = 0, var[∆ft ] = Σ
ft (t ) − ft −i (t )
i-period forecast error:
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The forecast process properties
Property 1: The i-period forecast is an unbiased estimate of
demand in period t i −1
ft (t ) = ft −i (t ) + ∑ ∆ft + k (t )
k =0
Property 2: Each forecast revision improves the forecast
i −1 i −1
Var[ ft (t ) − ft −i (t )] = Var[∑ ∆f t − k (t )] = ∑ σ k2
k =0 k =0
Property 3: The variance of the forecast error over the
horizon MUST equal the demand variance
H
Var[ ft (t )] = Var[ ft (t ) − ft − H +1 (t )] = ∑ σ k2 = Tr (∑)
k =0
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Production plan: assumptions
At each period t:
Ft(t+i) is the planned production (i=0, actual completed)
It(t+i) is the planned inventory
We introduce the plan revision:
∆Ft (t + i ) = Ft (t + i ) − Ft −1 (t + i )
We SET the production plan Ft(t+i) so that It(t+H)=ss>0
(cumulative revision to the PP=cumulative forecast revision)
From the Fundamental conservation equation:
i
I t (t + i ) = I t (t ) + ∑ ( Ft (t + k ) − f t (t + k ))
k =1
We assume the
H H
∑ ∆F (t + k ) = ∑ ∆f (t + k ) schedule update as
We find: t t
a linear system
k =0 k =0
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Linear system assumption
we model: H
∆Ft (t + i ) = ∑ wij ∆ft (t + j ) i = 0,1,… , H
j =0
H
∑w =1 0 ≤ wij ≤ 1
ij
i =0
wij = proportion of the forecast revision for t+j that is added to the
schedule of production outputs for t+i
Weights express tradeoff between Smooth production and Inventory
wij =1 / (H+1) wii =1 ; wij = 0 (i!=j)
Decisions variables OR parameters
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Smoothing production
4
3
f(t)
units
2 F(t)
I(t)
1
0
1 3 5 7 9 11 13 15 17 19 21 23
periods
F(t) constant ; needed capacity = 1 ; Average inventory = 1,5
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Minimizing inventory cost
4
3
f(t)
units
2 F(t)
I(t)
1
0
1 3 5 7 9 11 13 15 17 19 21 23
periods
F(t) = f(t) ; needed capacity = 4 ; Average inventory = 0
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Measures of interest
∆Ft = W ∆f t
we note:
(H+1) by (H+1) matrix
We obtain : H
Ft = µ + ∑ Bi W ∆ft −i
i =0
We can now measure the smoothness AND the stability of the
production outputs, given by:
outputs
(see section 1.3, “Measures of Interest,” in
the Graves, Kletter, and Hetzel paper)
And if we consider the stock:
stock
(see section 1.3, “Measures of Interest,” in
the Graves, Kletter, and Hetzel paper)
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Global view
smoothness and stability of the production outputs
smoothness and stability of the forecasts for the
upstream stages
var ( ∆ Ft ) = W Σ W ′
var [ ∆ f t ] = Σ
Service level = stock out probability
How to
E ( I t ) > kσ ( I t ) = ss
choose W?
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Optimization problem
Tradeoff between production smoothing and inventory:
= Min(required capacity)
min var[ Ft (t )]
subject to
var[ I t (t )] ≤ K 2 = constraint on amount of ss
H
∑w =1 ∀j
ij
i =0
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Resolution (1/2)
Lagrangian relaxation:
L(λ ) = min var[ Ft (t )] + λ var[ I t (t )] − λ K 2
⎛σ 0 ⎞
We assume:
2
⎜ ⎟
σ12
⎜ ⎟ Forecast revisions
0
∑=⎜ ⎟ are uncorrelated
...
⎜ ⎟
σ H −1
2
⎜ ⎟
0
⎜ σH ⎟
2
⎝ ⎠
We get a decomposition into H+1 subproblems
H H H
L (λ ) = ∑ L j (λ ) − λ K L(λ ) = min ∑ ( wijσ j ) + λ ∑ (bijσ j ) 2
2 2
j =0 i =0 i =0
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Resolution (2/2)
Convex program Karush-Kuhn-Tucker are necessary AND
sufficient H
wij + λ ∑ ( w0 j + … + wkj = ukj ) = γ
k =i
Which can be transformed
wij = Pi (λ ) w0 j i = 0, 2,… , j
wij = Pi (λ ) w0 j − Ri − j (λ ) i = j + 1,… , H
Which defines all elements (with the convexity constraint)
H
∑w =1
ij
j =0
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Solution
The Matrix W is symmetric about the diagonal and the off-
diagonal
Wij >0, increasing and strictly convex over i=1…j
Wij >0, decreasing and strictly convex over i=j…H
The optimal value of each subproblem is
L j (λ ) = w jjσ 2 j = 0,1,… , H
j
we show: k
⎡ (1 − α ) ⎤ λ
w j +k , j = w j −k , j = α ⎢ where α =
(1 + α ) ⎥ (λ + 4)
⎣ ⎦
L(λ ) = min var[ Ft (t )] + λ var[ I t (t )] − λ K 2
And then: λ
− λK 2
≈ tr (Σ)
(λ + 4)
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Computation
⎛ λ +1 −1 ⎞
⎜λ ⎟
λ
⎜ ⎟
λ+2
⎜ −1 −1 ⎟
0
⎜λ ⎟
λ λ
⎜ ⎟
−1
W =⎜ ... ... ... ⎟
⎜ −1 ⎟
λ+2
−1
⎜ ⎟
0
λ λ λ⎟
⎜
λ +1 ⎟
−1
⎜
⎜ ⎟
λ λ⎠
⎝
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interpretation
Wjj 1 when
Lambda increases
low inventory but no
production smoothness
(See Figures 2 and 3 on
pages S43 of the Graves,
Kletter, and Hetzel paper)
Wjj (1/H+1) when
Lambda 0
High inventory
but great production
smoothness
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Overview
Single-stage model
Extension to multistage systems
Case study of Kodak
Conclusion
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Multi stage model assumptions
Acyclic network on n stages, m end-items stages, m<n
1…m End items forecasts independent
Downstream stages decoupled from upstream ones.
Each stage works like the single-one model
fi , i = 1,… , n
We note:
Fi , i = 1,… , n
We find again:
∆Fi = Wi ∆f i for each stage i
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Stages linkage assumptions
At each period, each stage must translate outputs into
production starts
We use a linear system Gi = A i Fi
Ai can model leadtimes, yields, etc.
Push or pull policies
∆fi
We show that each is an iid random vector
All previous assumptions are satisfied
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Overview
Single-stage model
Extension to multistage systems
Case study of Kodak
Conclusion
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Kodak issue
Multistage system: film making supply chain.
Three steps, with
Growth of items
Growth of value
Decrease of leadtimes
How to determine optimal safety stock level at
each stage ?
Use DRP model
Wide data collection
W = I (no production smoothing)
Estimation of forecasts covariance matrix
A captures yields
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Practical results
-20 % recommendation
Inventory pushed upstream
Risk pooling
Lowest value
Shortcomings of DRP model:
Lead time variability
Stationary average demand
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Overview
Single-stage model
Extension to multistage systems
Case study of Kodak
Conclusion
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Conclusion
Model a single inventory production system as
a linear system
Multistage Network:
Material flow
Forecast
of demand
Forecast
of demand
Production plan Production plan
Information flow
Following research topics…
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