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
„
„
„
†
Assumptions:
„
„
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Multiperiod forecast of demand
Production plan
Requirements forecasts for components (offset
leadtimes and yields)
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
„
„
„
ft(t+i) , i<=H
ft(t+i)= Cte , i>H
ft(t) is the observed demand
Units
† Discrete time
† At each period t:
16
14
12
10
8
6
4
2
0
d(t+i)
1
3
5
7
† Forecast revision:
† i-period forecast error:
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11
13
15
17
Week i
∆f t (t + i ) = f t (t + i ) − f t −1 (t + i )
∆ft is iid random vector
9
i = 0,1,… H
E[ ∆ft ] = 0, var[∆ft ] = Σ
ft (t ) − ft −i (t )
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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
k =0
k =0
Var[ ft (t ) − ft −i (t )] = Var[∑ ∆f t − k (t )] = ∑ σ k2
† 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
H
†
We find:
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H
∑ ∆F (t + k ) = ∑ ∆f (t + k )
k =0
t
k =0
t
We assume the
schedule update as
a linear system
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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
i =0
ij
=1
0 ≤ wij ≤ 1
†
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
units
3
f(t)
F(t)
I(t)
2
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
units
3
f(t)
F(t)
I(t)
2
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:
†
We obtain :
(H+1) by (H+1) matrix
H
Ft = µ + ∑ Bi W ∆ft −i
i =0
†
We can now measure the smoothness AND the stability of the
production outputs,
outputs given by:
(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
E ( I t ) > kσ ( I t ) = ss
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How to
choose W?
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Optimization problem
† Tradeoff between production smoothing and inventory:
min var[ Ft (t )]
= Min(required capacity)
subject to
var[ I t (t )] ≤ K 2
H
∑w
i =0
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ij
=1
= constraint on amount of ss
∀j
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Resolution (1/2)
† Lagrangian relaxation:
L(λ ) = min var[ Ft (t )] + λ var[ I t (t )] − λ K 2
† We assume:
⎛ σ 02
⎞
⎜
⎟
2
σ
0
1
⎜
⎟ Forecast revisions
⎟ are uncorrelated
∑=⎜
...
⎜
⎟
2
σ H −1
0
⎜
⎟
2 ⎟
⎜
σ
H ⎠
⎝
† We get a decomposition into H+1 subproblems
H
L (λ ) = ∑ L j (λ ) − λ K 2
j =0
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H
H
i =0
i =0
L(λ ) = min ∑ ( wijσ j ) 2 + λ ∑ (bijσ j ) 2
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Resolution (2/2)
† Convex program Æ 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
j =0
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ij
=1
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Solution
† The Matrix W is symmetric about the diagonal and the offdiagonal
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 2j
† we show:
⎡ (1 − α ) ⎤
w j +k , j = w j −k , j = α ⎢
⎥
(1
α
)
+
⎣
⎦
† And then:
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j = 0,1,… , H
k
where α =
λ
(λ + 4)
L(λ ) = min var[ Ft (t )] + λ var[ I t (t )] − λ K 2
≈ tr (Σ)
λ
(λ + 4)
− λK 2
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Computation
⎛ λ +1
⎜ λ
⎜
⎜ −1
⎜ λ
⎜
W −1 = ⎜
⎜
⎜
⎜
⎜
⎜
⎝
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−1
λ
λ+2
λ
−1
λ
0
...
...
−1
...
λ+2
λ
λ
0
−1
λ
⎞
⎟
⎟
⎟
⎟
⎟
⎟
−1 ⎟
⎟
λ ⎟
λ +1 ⎟
⎟
λ ⎠
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interpretation
†WjjÆ1 when
Lambda increases
†Ælow inventory but no
production smoothness
(See Figures 2 and 3 on
pages S43 of the Graves,
Kletter, and Hetzel paper)
†WjjÆ(1/H+1) when
Lambda Æ 0
†ÆHigh 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
† We note:
fi , i = 1,… , n
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
† We show that each
∆fi
is an iid random vector
Æ 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
Production plan
Forecast
of demand
Forecast
of demand
Production plan
Information flow
† Following research topics…
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