a dynamic model for requirements planning with application to supply chain optimization - forecast...

A Dynamic model for requirements planning with application to supply chain optimization

byStephen Graves, David Kletter and William Hezel (1998)

15.764 The Theory of Operations ManagementMarch 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

7/6/2004 15.764 The Theory of Operations Management 2

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

Quick Review of MRP


At each step of the process, production of finished good requires components

MRP methodology: iteration ofMultiperiod forecast of demandProduction planRequirements forecasts for components (offset leadtimes and yields)

Assumptions: accuracy of forecastsDeterministic parameters

Widely used in discrete parts manufacturing firms: MRP process is revised periodically

Limits: deviations from the plan due to incertainty

Literature Review


Few papers about dynamic modeling of requirements

No attempt to model a dynamic forecast process, except:

Graves (1986): two-stage production inventory systemHealth and Jackson (1994): MMFE

Conversion forecasts-production plan comes from previous Graves’ works.

Methodology and Results


We model:Forecasts as a stochastic processConversion into production plan as a linear system

We create:A single stage model

From which we can built general acyclic networks of multiple stages

We try it on a real case (LFM program, Hetzel)

demandproduction

Overview


Single-stage model

Extension to multistage systems

Case study of Kodak

Conclusion

Overview


Single-stage modelModel

Forecasts processConversion into production outputsMeasures of interest

Optimality of the model


Case study of Kodak

Conclusion

The forecast process model


Discrete timeAt each period t:

ft(t+i) , i<=Hft(t+i)= Cte , i>Hft(t) is the observed demand

Forecast revision:

i-period forecast error:

Demand forecasts

02468

10121416

1 3 5 7 9 11 13 15 17

Week i

Uni

ts

d(t+i)

1( ) ( ) ( ) 0,1,t t tf t i f t i f t i i H−∆ + = + − + = …

( ) ( )t t if t f t−−

is iid random vector [ ] 0, var[ ]t t tf E f f∆ ∆ = ∆ = Σ

The forecast process properties


Property 1: The i-period forecast is an unbiased estimate of demand in period t

Property 2: Each forecast revision improves the forecast

Property 3: The variance of the forecast error over the horizon MUST equal the demand variance

1

0( ) ( ) ( )

i

t t i t kk

f t f t f t−

− +=

= + ∆∑

1 12

0 0[ ( ) ( )] [ ( )]

i i

t t i t k kk k

Var f t f t Var f t σ− −

− −= =

− = ∆ =∑ ∑

21

0[ ( )] [ ( ) ( )] ( )

H

t t t H kk

Var f t Var f t f t Trσ− +=

= − = = ∑∑

Production plan: assumptions


At each period t:Ft(t+i) is the planned production (i=0, actual completed)It(t+i) is the planned inventoryWe introduce the plan revision:

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:

We find:

1( ) ( ) ( ( ) ( ))

i

t t t tk

I t i I t F t k f t k=

+ = + + − +∑

0 0( ) ( )

H H

t tk k

F t k f t k= =

∆ + = ∆ +∑ ∑We assume the

schedule update as a linear system

1( ) ( ) ( )t t tF t i F t i F t i−∆ + = + − +

Linear system assumption


we model:

wij = proportion of the forecast revision for t+j that is added to the schedule of production outputs for t+i

Weights express tradeofftradeoff between Smooth productionSmooth production and InventoryInventory

Decisions variables OR parameters

wij =1 / (H+1) wii =1 ; wij = 0 (i!=j)

0( ) ( ) 0,1, ,

H

t ij tj

F t i w f t j i H=

∆ + = ∆ + =∑ …

01 0 1

H

ij iji

w w=

= ≤ ≤∑

Smoothing production


F(t) constant ; needed capacity = 1 ; Average inventory = 1,5

0

1

2

3

4

1 3 5 7 9 11 13 15 17 19 21 23

periods

units

f(t)F(t)I(t)

Minimizing inventory cost


0

1

2

3

4

1 3 5 7 9 11 13 15 17 19 21 23

periods

units

f(t)F(t)I(t)

F(t) = f(t) ; needed capacity = 4 ; Average inventory = 0

Measures of interest


we note:

We obtain :

We can now measure the smoothnesssmoothness AND the stability of the stability of the production outputsproduction outputs, given by:

And if we consider the stockthe stock:

t tF f∆ = ∆W(H+1) by (H+1) matrix

0

Hi

t t ii

F fµ −=

= + ∆∑B W

(see section 1.3, “Measures of Interest,” in the Graves, Kletter, and Hetzel paper)

(see section 1.3, “Measures of Interest,” in the Graves, Kletter, and Hetzel paper)

Global view


smoothness and stability of the production outputs

smoothness and stability of the forecasts for the upstream stages

var ( )tF ′∆ = ΣW Wvar [ ]tf∆ = Σ

( ) ( )t tE I k Iσ>

Service level = stock out probability

= ss How to choose W?

Optimization problem


Tradeoff between production smoothing and inventory:

min var[ ( )]tF t capacity)= Min(required

subject to

2var[ ( )]tI t K≤ on amount of ss= constraint

01

H

iji

w j=

= ∀∑

Resolution (1/2)


Lagrangian relaxation:

We assume:

We get a decomposition into H+1 subproblems

20

21

21

2

0...

0 H

H

σσ

σσ

−

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟∑ =⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

Forecast revisionsare uncorrelated

2( ) min var[ ( )] var[ ( )]t tL F t I t Kλ λ λ= + −

2 2 2

0 0 0( ) ( ) ( ) min ( ) ( )

H H H

j ij j ij jj i i

L L K L w bλ λ λ λ σ λ σ= = =

= − = +∑ ∑ ∑

Resolution (2/2)


Convex program Karush-Kuhn-Tucker are necessary AND sufficient

Which can be transformed

Which defines all elements (with the convexity constraint)

0( )H

ij j kj kjk i

w w w uλ γ=

+ + + = =∑ …

0

0

( ) 0, 2, ,

( ) ( ) 1, ,ij i j

ij i j i j

w P w i j

w P w R i j H

λ

λ λ−

= =

= − = +

……

01

H

ijj

w=

=∑

Solution


The Matrix W is symmetric about the diagonal and the off-diagonal

Wij >0, increasing and strictly convex over i=1…jWij >0, decreasing and strictly convex over i=j…H

The optimal value of each subproblem is

we show:

And then:

2( ) 0,1, ,j jj jL w j Hλ σ= = …

, ,(1 )(1 ) ( 4)

k

j k j j k jw w whereα λα αα λ+ −

⎡ ⎤−= = =⎢ ⎥+ +⎣ ⎦

2

2

( ) min var[ ( )] var[ ( )]

( )( 4)

t tL F t I t K

tr K

λ λ λ

λ λλ

= + −

≈ Σ −+

Computation


1

1 1

1 2 1 0

... ... ...1 2 10

1 1

W

λλ λ

λλ λ λ

λλ λ λ

λλ λ

−

+ −⎛ ⎞⎜ ⎟⎜ ⎟− + −⎜ ⎟

⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟− + −⎜ ⎟⎜ ⎟

− +⎜ ⎟⎜ ⎟⎝ ⎠

interpretation


Wjj 1 when Lambda increases

low inventory but noproduction 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 productionsmoothness

Overview


Single-stage model


Case study of Kodak

Conclusion

Multi stage model assumptions


Acyclic network on n stages, m end-items stages, m<n1…m End items forecasts independentDownstream stages decoupled from upstream ones.Each stage works like the single-one model

We note:

We find again:

, 1, ,

, 1, ,i

i

f i n

F i n

=

=

…

…

for each stagei i iF f i∆ = ∆W

Stages linkage assumptions


At each period, each stage must translate outputs into production startsWe use a linear system

Ai can model leadtimes, yields, etc.Push or pull policies

We show that each is an iid random vector

All previous assumptions are satisfied

i i iG F= A

if∆

Overview


Single-stage model


Case study of Kodak

Conclusion

Kodak issue


Multistage system: film making supply chain. Three steps, with

Growth of itemsGrowth of valueDecrease of leadtimes

How to determine optimal safety stock level at each stage ?Use DRP model

Wide data collectionW = I (no production smoothing)Estimation of forecasts covariance matrixA captures yields

Practical results


-20 % recommendation

Inventory pushed upstreamRisk poolingLowest value

Shortcomings of DRP model:Lead time variabilityStationary average demand

Overview


Single-stage model


Case study of Kodak

Conclusion

Conclusion


Model a single inventory production system as a linear systemMultistage Network:

Following research topics…

Forecast of demand

Production planForecast of demandProduction plan

Material flow

Information flow

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