A Dynamic Model for Requirements Planning With Application to Supply Chain Optimization

A Dynamic model for requirements planning with application to supply chain optimization 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 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 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. 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) Overview † Single-stage model † Extension to multistage systems † Case study of Kodak † Conclusion 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 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 † Forecast revision: † i-period forecast error: 5 7 9 11 Week i ∆f t (t + i ) = f t (t + i ) − f t −1 (t + i ) ∆ft is iid random vector 3 i = 0,1,… H E[ ∆ft ] = 0, var[∆ft ] = Σ ft (t ) − ft −i (t ) 13 15 17 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[ f t (t ) − ft − H +1 (t )] = ∑ σ k2 = Tr (∑) k =0 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: H ∑ ∆F (t + k ) = ∑ ∆f (t + k ) k =0 t k =0 t We assume the schedule update as a linear system 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) † Decisions variables OR parameters wii =1 ; wij = 0 (i!=j) 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 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 Measures of interest ∆Ft = W ∆f t † we note: † We obtain : H (H+1) by (H+1) matrix Ft = µ + ∑ Bi W ∆ft −i i =0 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 How to choose W? 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 ij =1 ∀j = constraint on amount of ss Resolution (1/2) † Lagrangian relaxation: L(λ ) = min var[ Ft (t )] + λ var[ I t (t )] − λ K 2 † We assume: ⎛ σ 02 ⎞ ⎜ ⎟ 2 0 σ 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 H H i =0 i =0 L(λ ) = min ∑ ( wijσ j ) 2 + λ ∑ (bijσ j ) 2 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 ij =1 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: j = 0,1,… , H k where α = L(λ ) = min var[ Ft (t )] + λ var[ I t (t )] − λ K 2 ≈ tr (Σ) λ (λ + 4) − λK 2 λ (λ + 4) Computation ⎛ λ +1 ⎜ λ ⎜ ⎜ −1 ⎜ λ ⎜ W −1 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −1 λ λ+2 λ −1 λ 0 ... ... −1 ... λ+2 λ λ 0 −1 λ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ −1 ⎟ ⎟ λ ⎟ λ +1 ⎟ ⎟ λ ⎠ interpretation †WjjÆ1 when Lambda increases †Ælow inventory but no production smoothness †WjjÆ(1/H+1) when Lambda Æ 0 †ÆHigh inventory but great production smoothness Overview † Single-stage model † Extension to multistage systems † Case study of Kodak † Conclusion Multi stage model assumptions † † † † Acyclic network on n stages, m end-items stages, m