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- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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