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

     by
     Stephen Graves, ...
Objective of the paper


    Overcome limits of Materials Requirements
    Process models

    Capture key dynamics in the...
Quick Review of MRP
                At each step of the process, production of
                finished good requires comp...
Literature Review

    Few papers about dynamic modeling of
    requirements

    No attempt to model a dynamic forecast p...
Methodology and Results
    We model:
           Forecasts as a stochastic process
           Conversion into production p...
Overview


  Single-stage model

  Extension to multistage systems

  Case study of Kodak

  Conclusion




7/6/2004      ...
Overview

  Single-stage model
       Model
           Forecasts process
           Conversion into production outputs
   ...
The forecast process model
                                                                               Demand forecasts...
The forecast process properties
    Property 1: The i-period forecast is an unbiased estimate of
    demand in period t   ...
Production plan: assumptions
    At each period t:
           Ft(t+i) is the planned production (i=0, actual completed)
  ...
Linear system assumption
    we model:                      H
                    ∆Ft (t + i ) = ∑ wij ∆ft (t + j )       ...
Smoothing production

                  4


                  3

                                                         ...
Minimizing inventory cost
              4


              3

                                                             ...
Measures of interest

               ∆Ft = W ∆f t
    we note:
                                      (H+1) by (H+1) matrix...
Global view
    smoothness and stability of the production outputs


    smoothness and stability of the forecasts for the...
Optimization problem

    Tradeoff between production smoothing and inventory:

                                          ...
Resolution (1/2)
    Lagrangian relaxation:
                  L(λ ) = min var[ Ft (t )] + λ var[ I t (t )] − λ K 2
       ...
Resolution (2/2)
    Convex program                 Karush-Kuhn-Tucker are necessary AND
    sufficient              H
   ...
Solution
    The Matrix W is symmetric about the diagonal and the off-
    diagonal
           Wij >0, increasing and stri...
Computation

         ⎛ λ +1     −1                                  ⎞
         ⎜λ                                        ...
interpretation


  Wjj 1 when
Lambda increases

   low inventory but no
production smoothness
                            ...
Overview


  Single-stage model

  Extension to multistage systems

  Case study of Kodak

  Conclusion




7/6/2004      ...
Multi stage model assumptions
    Acyclic network on n stages, m end-items stages, m<n
    1…m End items forecasts indepen...
Stages linkage assumptions
    At each period, each stage must translate outputs into
    production starts
    We use a l...
Overview


  Single-stage model

  Extension to multistage systems

  Case study of Kodak

  Conclusion




7/6/2004      ...
Kodak issue

  Multistage system: film making supply chain.
  Three steps, with
       Growth of items
       Growth of va...
Practical results

  -20 % recommendation

  Inventory pushed upstream
       Risk pooling
       Lowest value


  Shortco...
Overview


  Single-stage model

  Extension to multistage systems

  Case study of Kodak

  Conclusion




7/6/2004      ...
Conclusion

  Model a single inventory production system as
  a linear system
  Multistage Network:
                      ...
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A Dynamic Model For Requirements Planning With Application To Supply Chain Optimization - Forecast Evolution Models - Nicolas Miegeville

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A Dynamic Model For Requirements Planning With Application To Supply Chain Optimization - Forecast Evolution Models - Nicolas Miegeville

  1. 1. 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. quot;A Dynamic Model for Requirements Planning with Application to Supply Chain Optimization.quot; Operations Research 46, supp. no. 3 (1998): S35-49.
  2. 2. 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 7/6/2004 15.764 The Theory of Operations Management 2
  3. 3. 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 7/6/2004 15.764 The Theory of Operations Management 3
  4. 4. 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. 7/6/2004 15.764 The Theory of Operations Management 4
  5. 5. 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) 7/6/2004 15.764 The Theory of Operations Management 5
  6. 6. Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/2004 15.764 The Theory of Operations Management 6
  7. 7. 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 7/6/2004 15.764 The Theory of Operations Management 7
  8. 8. 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: 7/6/2004 15.764 The Theory of Operations Management 8
  9. 9. 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 7/6/2004 15.764 The Theory of Operations Management 9
  10. 10. 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 7/6/2004 15.764 The Theory of Operations Management 10
  11. 11. 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 7/6/2004 15.764 The Theory of Operations Management 11
  12. 12. 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 7/6/2004 15.764 The Theory of Operations Management 12
  13. 13. 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 7/6/2004 15.764 The Theory of Operations Management 13
  14. 14. 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) 7/6/2004 15.764 The Theory of Operations Management 14
  15. 15. 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? 7/6/2004 15.764 The Theory of Operations Management 15
  16. 16. 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 7/6/2004 15.764 The Theory of Operations Management 16
  17. 17. 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 7/6/2004 15.764 The Theory of Operations Management 17
  18. 18. 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 7/6/2004 15.764 The Theory of Operations Management 18
  19. 19. 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) 7/6/2004 15.764 The Theory of Operations Management 19
  20. 20. 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⎠ ⎝ 7/6/2004 15.764 The Theory of Operations Management 20
  21. 21. 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 7/6/2004 15.764 The Theory of Operations Management 21
  22. 22. Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/2004 15.764 The Theory of Operations Management 22
  23. 23. 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 7/6/2004 15.764 The Theory of Operations Management 23
  24. 24. 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 7/6/2004 15.764 The Theory of Operations Management 24
  25. 25. Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/2004 15.764 The Theory of Operations Management 25
  26. 26. 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 7/6/2004 15.764 The Theory of Operations Management 26
  27. 27. Practical results -20 % recommendation Inventory pushed upstream Risk pooling Lowest value Shortcomings of DRP model: Lead time variability Stationary average demand 7/6/2004 15.764 The Theory of Operations Management 27
  28. 28. Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/2004 15.764 The Theory of Operations Management 28
  29. 29. 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… 7/6/2004 15.764 The Theory of Operations Management 29

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