University:
Massachusetts Institute of Technology
Course:
15.992 | S-Lab: Laboratory for Sustainable Business
Academic year:

2023
Views:

135

Pages:

29
Author:

acertantb9

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