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

Transcription

1 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): S A Dynamic model for requirements planning with application to supply chain optimization by Stephen Graves, David Kletter and William Hezel (1998) The Theory of Operations Management March 11, 2004 Presenter: Nicolas Miègeville

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/ 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/ 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/ 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/ The Theory of Operations Management 5

6 Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/ 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/ The Theory of Operations Management 7

8 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 Demand forecasts 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/ 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 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/ The Theory of Operations Management 9

10 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/ The Theory of Operations Management 10

11 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/ The Theory of Operations Management 11

12 Smoothing production 4 3 units 2 1 f(t) F(t) I(t) periods F(t) constant ; needed capacity = 1 ; Average inventory = 1,5 7/6/ The Theory of Operations Management 12

13 Minimizing inventory cost 4 3 units 2 1 f(t) F(t) I(t) periods F(t) = f(t) ; needed capacity = 4 ; Average inventory = 0 7/6/ The Theory of Operations Management 13

14 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/ 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 ( F ) = WΣW t var[ f t ] =Σ Service level = stock out probability EI ( ) > kσ ( I) t t = ss How to choose W? 7/6/ The Theory of Operations Management 15

16 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/ The Theory of Operations Management 16

17 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 = σ H 1 2 σ H Forecast revisions are uncorrelated We get a decomposition into H+1 subproblems H H H 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/ The Theory of Operations Management 17

18 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/ The Theory of Operations Management 18

19 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/ The Theory of Operations Management 19

20 Computation W 1 λ λ λ 1 λ λ λ λ = λ λ λ λ 1 λ + 1 λ λ 7/6/ The Theory of Operations Management 20

21 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/ The Theory of Operations Management 21

22 Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/ 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 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/ 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 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/ The Theory of Operations Management 24

25 Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/ 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/ 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/ The Theory of Operations Management 27

28 Overview Single-stage model Extension to multistage systems Case study of Kodak Conclusion 7/6/ The Theory of Operations Management 28

29 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/ The Theory of Operations Management 29