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. 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
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
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
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
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
Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/2004 15.764 The Theory of Operations Management 6
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
The forecast process model Discrete time At each period t: f t (t+i), i<=h f t (t+i)= Cte, i>h f t (t) is the observed demand Forecast revision: t t t 1 Units 16 14 12 10 8 6 4 2 0 Demand forecasts 1 3 5 7 9 11 13 15 17 Week i f ( t+ i) = f ( t+ i) f ( t+ i) i = 0,1, H f is iid random vector E[ f ] = 0, var[ f ] =Σ t t t d(t+i) i-period forecast error: f () t f () t t t i 7/6/2004 15.764 The Theory of Operations Management 8
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 i 1 f () t = f () t + f () t t t i t+ k k = 0 i 1 i 1 2 t t i = t k = k k= 0 k= 0 Var[ f ( t) f ( t)] Var[ f ( t)] σ Property 3: The variance of the forecast error over the horizon MUST equal the demand variance H [ 2 t( )] = [ t( ) t H+ 1( )] = σ k = ( ) k = 0 Var f t Var f t f t Tr 7/6/2004 15.764 The Theory of Operations Management 9
Production plan: assumptions At each period t: F t (t+i) is the planned production (i=0, actual completed) I t (t+i) is the planned inventory We introduce the plan revision: F( t+ i) = F( t+ i) F ( t+ i) t t t 1 We SET the production plan F t (t+i) so that I t (t+h)=ss>0 (cumulative revision to the PP=cumulative forecast revision) From the Fundamental conservation equation: I ( t+ i) = I ( t) + ( F( t+ k) f ( t+ k)) t t t t k = 1 H H We find: i F( t+ k) = f ( t+ k) t k= 0 k= 0 t We assume the schedule update as a linear system 7/6/2004 15.764 The Theory of Operations Management 10
Linear system assumption we model: H F ( t+ i) = w f ( t+ j) i= 0,1,, H t ij t j= 0 H i= 0 w ij = 1 0 w 1 ij w ij = 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
Smoothing production 4 3 units 2 1 f(t) F(t) I(t) 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
Minimizing inventory cost 4 3 units 2 1 f(t) F(t) I(t) 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
Measures of interest we note: F = W f t t (H+1) by (H+1) matrix We obtain : F t H i µ B W ft i i= 0 = + We can now measure the smoothness AND the stability of the production outputs, given by: (see section 1.3, Measures of Interest, in the Graves, Kletter, and Hetzel paper) And if we consider the 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
Global view smoothness and stability of the production outputs smoothness and stability of the forecasts for the upstream stages var ( F ) = WΣW t var[ f t ] =Σ Service level = stock out probability EI ( ) > kσ ( I) t t = ss How to choose W? 7/6/2004 15.764 The Theory of Operations Management 15
Optimization problem Tradeoff between production smoothing and inventory: min var[ Ft ( t)] subject to = Min(required capacity) var[ It ( t)] K 2 = constraint on amount of ss H i= 0 w ij = 1 j 7/6/2004 15.764 The Theory of Operations Management 16
Resolution (1/2) Lagrangian relaxation: L( λ) = min var[ F( t)] + λvar[ I ( t)] λk t t 2 We assume: 2 σ 0 2 σ1 0 =... 2 0 σ H 1 2 σ H Forecast revisions are uncorrelated We get a decomposition into H+1 subproblems H H H 2 2 2 j ij j ij j j= 0 i= 0 i= 0 L( λ) = L ( λ) λk L( λ) = min ( w σ ) + λ ( b σ ) 7/6/2004 15.764 The Theory of Operations Management 17
Resolution (2/2) Convex program Karush-Kuhn-Tucker are necessary AND sufficient w + λ ( w + + w = u ) = γ ij 0 j kj kj k= i Which can be transformed H w = P( λ) w i = 0, 2,, j ij i 0 j w = P( λ) w R ( λ) i = j + 1,, H ij i 0 j i j Which defines all elements (with the convexity constraint) H j= 0 w ij = 1 7/6/2004 15.764 The Theory of Operations Management 18
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 w j H 2 j( λ) = jjσ j = 0,1,, we show: k (1 α) λ wj+ k, j= wj k, j= α whereα (1 α) = + ( λ+ 4) And then: L( λ) = minvar[ F( t)] + λvar[ I ( t)] λk λ tr( Σ) λk ( λ + 4) t 2 t 2 7/6/2004 15.764 The Theory of Operations Management 19
Computation W 1 λ + 1 1 λ λ 1 λ + 2 1 0 λ λ λ =......... 1 λ + 2 1 0 λ λ λ 1 λ + 1 λ λ 7/6/2004 15.764 The Theory of Operations Management 20
interpretation Wjj1 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
Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/2004 15.764 The Theory of Operations Management 22
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 We note: f, i = 1,, n i F, i = 1,, n i We find again: F = W f for each stagei i i i 7/6/2004 15.764 The Theory of Operations Management 23
Stages linkage assumptions At each period, each stage must translate outputs into production starts We use a linear system G = A F i i i Ai can model leadtimes, yields, etc. Push or pull policies f i 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
Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/2004 15.764 The Theory of Operations Management 25
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
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
Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/2004 15.764 The Theory of Operations Management 28
Conclusion Model a single inventory production system as a linear system Multistage Network: Production plan Material flow Forecast of demand Production plan Forecast of demand Information flow Following research topics 7/6/2004 15.764 The Theory of Operations Management 29