Control-Oriented Approaches to Inventory Management in Semiconductor Manufacturing Supply Chains

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1 Control-Oriented Approaches to Inventory Management in Semiconductor Manufacturing Supply Chains Daniel E. Rivera, Wenlin Wang, Martin W. Braun* Control Systems Engineering Laboratory Department of Chemical and Materials Engineering Arizona State University Karl G. Kempf Decision Technologies Intel Corporation * currently with Texas Instruments, Dallas, TX Motivation Billions of dollars in potential savings in the semiconductor industry alone by eliminating supply chain inefficiencies (PricewaterhouseCoopers, 2000) Increasing realization that inventory management is a control problem, and not just an optimization/or problem. Previous work (Braun et al., (2001, 2002)) presented a partially decentralized MPC-based approach for a three echelon, six-node problem involving assembly/test, distribution, and retailing. Need to reflect on issues of controller design and structure. The IMC design procedure is used to build insight, while MPC represents an appropriate implementation environment AIChE Annual Meeting, Indianapolis 1

2 Motivation Presentation Outline Internal Model Control (IMC)-based decision policies for a Single Node Inventory System Pull Control Structure Push Control Structure IMC-based Analysis of a Two Node Problem Decentralized (with Pull ) Decentralized (with Push ) Centralized (with Push ) Application of Model Predictive Control to the Two Node Problem Extensions, Summary and Conclusions Single Node Inventory Control Problem - Pull Structure θ (production time) LIC Forecast (known θ f days beforehand) θ d (delivery time) Meet demand (with forecast given θ f days beforehand) for a node with θ day production (or order fulfillment) time and θ d delivery time AIChE Annual Meeting, Indianapolis 2

3 Single Node Equations (Laplace Domain) y 1 (s) = e θs u(s) 1 s s d(s) y 2 (s) =e θds d(s) = e θ fs θ ˆd(s)e d s n(s) d(s) =e (θ f θ d )s ˆd(s)n(s) y 1 (s) Inventory (Net Stock) y 2 (s) Supply to downstream node (received by customer) u(s) Starts (or Orders) ˆd(s) Forecast d(s) Actual (ordered by customer) n(s) Forecast Error θ Production (or Order Fulfillment) Time θ d Delivery Time θ f Forecast Time Horizon Internal Model Control (IMC) The IMC (or Q-parametrization) structure is an alternate yet equivalent means of representing a classical feedback-feedforward structure. The IMC design procedure is a convenient two-step procedure for designing Q-parametrized control systems AIChE Annual Meeting, Indianapolis 3

4 Three-Degree-of -Freedom FB/FF IMC structure ^ d ( Forecast) r (Net Stock Setpoint) e q - - r d q F (Orders /Starts) u p p d p d1 d p d2 n (Forecast Error) p d3 y 2 (Supply Received by Customer) y 1 (Net Stock) p y - q d Internal Model Control Design Procedure Step 1 (Nominal Performance): Obtain H 2 optimal q s Must specify a form for the external input (e.g, step, ramp). Closed-form solutions can be obtained The controllers generated per step 1 are stable and causal. Step 2 (Robust Stability and Performance): Augment the IMC controller(s) from step 1 with a filter(s) (f(s)); the filter(s) are specified to insure that the q s are proper and that the control system demonstrates stability and performance in the face of uncertainty AIChE Annual Meeting, Indianapolis 4

5 1. Design for nominal optimal performance: q r (s), q d (s), and q F (s) are designed for H 2 -optimal setpoint tracking, unmeasured disturbance rejection, and measured disturbance rejection, respectively. min (1 p q r ) r 2 qr min q d (1 p q d ) p d2 n) 2 min ( p d p q F )p d1 p d2 ˆd 2 q F subject to the requirement that q r (s), q d (s) and q F (s) be stable and causal. 2. Design for robust stability and performance: In this step q r (s), q d (s) and q F (s) are augmented with low-pass filters which can be tuned to detune the nominal performance (e.g., reduce aggressive manipulated variable action associated with the optimal controller per Step 1) or to satisfy a robust performance objective. The final controllers obtained from this step are q r (s) = q r (s)f r (s) q d (s) = q d (s)f d (s) q F (s) = q F (s)f F (s) IMC Controllers, Single Node Inventory Problem p(s) = e θs p d1 = e (θ f θ d )s s 1. Setpoint Tracking. s q r (s) = (λ r s 1) n r 2. Unmeasured Disturbance Rejection. ( q d (s) = s(θs 1) (n ) dλ d s 1) (λ d s 1) n d 3. Measured Disturbance Rejection. p d2 = 1 s λ r 0 n r 1 ˆd, r, n = 1 s λ d 0 n d 3 { e q F (s) = (θ f θ d θ)s if θ f (θ θ d ) (θ θ d θ f )s 1 ifθ f < (θ θ d ) Filtering could be used with the measured disturbance IMC controller, but is not needed for physical realizability when θ f (θ θ d ) AIChE Annual Meeting, Indianapolis 5

6 Three Degree of Freedom IMC Pull Results Feedback-only Combined FB/FF θ f = 20, θ = 10, θ d = 2, λ f = 1, λ r = 1, λ d = 1 Single Node Inventory Control Problem Push θ (production time) LIC Forecast (known θ f days beforehand) θ d (delivery time) Track demand (with forecast given θ f days beforehand) for a node with θ day production (or order fulfillment) time and θ d delivery time. Controller now manipulates both the starts/orders and the stream out release to customer AIChE Annual Meeting, Indianapolis 6

7 IMC Design: Decoupled Deadtime Compensation Theorem. The diagonal P decoupled matrix such that the multivariable IMC controller Q(s) =P 1 (s)p decoupled (s) is realizable has the form P decoupled = diag(r ii, r jj, r nn ) where r jj = e s(max i max(0,(ˆq ij ˆp ij )) and ˆp ij is the minimum delay in the numerator of element ij of P 1,ˆq ij is the minimum delay in the denominator of element ij of P 1. Holt, B.R. and M. Morari, Design of resilient process plants: the effect of deadtime on dynamic resilience, Chem. Eng. Science, 40, 1229, (1985) Single Node Decoupled Deadtime Compensation [ ] [ y1 (s) e θs ][ ] = s 1 s u(s) y 2 (s) 0 e θ ds d(s) y(s) =P (s) u (s) P d (s) n(s) [ ] P decoupled e θs 0 (s) = Q(s) 0 e (θθ = d)s Q(s) = Q(s)F (s) = [ ] y1 (s) y 2 (s) = [ e θs λ 1 s1 0 0 e (θθ d )s λ 2 s1 [ s λ 1 s1 [ ] 1 s e θ n(s) ds 1 λ 2 s1 e 0 θs λ 2 s1 ] [ r1 (s) r 2 (s) [ ] s 1 0 e θs r 1 and r 2 are the net stock and demand setpoints, respectively. ] ] 2002 AIChE Annual Meeting, Indianapolis 7

8 Decoupled IMC implementation (two degrees of freedom but 4 adjustable parameters) Net Stock and Setpoints 1 In1 2 In2 In1 In3 Out1 Out3 Q(s) Mux Mux3 Sum2 2 Out2 Orders/ Starts and Stream Out Flow In1 In2 Out1 Out2 Mux Mux2 Mux 1 Out1 Net Stock and 3 Out3 p(s) Mux Sum3 In1 In2 Out1 Out2 Mux p(s)1 Mux1 Sum1 Out1 Out3 In1 In3 Demux Q(s)1 Demux1 Decoupled IMC Push Results θ = 5, θ d = 2, λ 1 = 1, λ 2 = AIChE Annual Meeting, Indianapolis 8

9 Two Node Example F/S ADI A/T SFGI Center =Inventory Holding =Mfg Node =Transport Link F/S: Fabrication/Sort Facility A/T: Assembly/Test Facility ADI: Assembly-die Inventory SFGI: Semi-finished goods inventory Fluid Analogy to the Two Node Network F/S starts θ 1 (8 weeks) ADI A/T starts θ 2 (2 weeks) SFGI Shipments θ 3 (1 week) 2002 AIChE Annual Meeting, Indianapolis 9

10 Two Node Inventory Control Problem Decentralized Pull Structure LIC Order Forecast LIC Forecast Two Node Inventory Control Problem Decentralized Push Structure LIC LIC 2002 AIChE Annual Meeting, Indianapolis 10

11 Decentralized Push Results λ 1 = 1, λ 2 = 0 for each node Two Node Inventory Control Problem Centralized Push Structure LIC 2002 AIChE Annual Meeting, Indianapolis 11

12 Centralized Push Results MPC Appeal for Dynamic Inventory Management in Supply Chains As an optimizer, an MPC-based algorithm can minimize or maximize an objective function that represents a suitable measure for supply chain performance. As a controller, an MPC algorithm can be tuned to achieve stability, robustness, and performance in the presence of plant/model mismatch, failures and disturbances which affect the system AIChE Annual Meeting, Indianapolis 12

13 The MPC optimization problem can be written min J u(k k)... u(km 1 k) s.t. J = p Q e (l)(ŷ(k l k) r(k l)) 2 <- Satisfy demand l=1 m Q u (l)( u(k l 1 k)) 2 l=1 m Q u (l)(u(k l 1 k) u target (k l 1 k)) 2 l=1 u min u(k l 1 k) u max, u min u(k l 1 k) u min, y min y(k l 1 k) y max, Penalizes changes in <- order quantities (i.e. move suppression) 2002 AIChE Annual Meeting, Indianapolis 13

14 Centralized Pull Structure F/S Information flow A/T (MD) Forecast P S F/S MPC I F/S I A/T D Measured disturbance: D Controlled Variables: I F/S, I A/T Manipulated Variables: S F/S, P Centralized Pull Structure with Autoregressive Variability Output Weights=[1 1]; Move Suppression=[1 1] 2002 AIChE Annual Meeting, Indianapolis 14

15 Centralized Pull Structure with Autoregressive Variability Output Weights=[1 100]; Move Suppression =[1 1] Centralized Pull Structure with Autoregressive Variability Output Weights=[100 1]; Move Suppression =[1 1] 2002 AIChE Annual Meeting, Indianapolis 15

16 RMS Error Comparison Output Weights Centralized:[1 1] Centralized:[1 100] Centralized:[100 1] ADI SFGI Output weights can be used to shift variability between inventories, as desired by the enterprise! =Materials Supply =Inventory Holding =Manufacturing =Transport 1.1 F/S 2.1 A/T 2.2 Center 1 Wafers Packages 1.2 F/S F/S=Fab/Sort Facility A/T=Assembly/Test Facility A/T Center AIChE Annual Meeting, Indianapolis 16

17 Summary and Conclusions Supply chains (demand networks, value chains) are dynamical systems whose efficient operation merits a control-oriented approach The IMC design procedure provides important insights into controller structure and tuning. MPC offers a powerful implementation environment that has the potential for good performance in uncertain, noisy environments. Acknowledgements NSF Scalable Enterprise Systems Phase I Award: Designing and Managing Dynamic Supply Chains Using Model-on- Predictive Control Intel Research Council: A Modular, Scalable Approach to Modeling and Analysis of Semiconductor Supply Networks. Additional students: Melvin E. Flores, Mark Szwast 2002 AIChE Annual Meeting, Indianapolis 17