CSEL LCCC Workshop on Process Control Sept. 28 - 30, 2016 Control - - PowerPoint PPT Presentation

Control Systems Engineering Laboratory

CSEL

LCCC Workshop on Process Control

• Sept. 28 - 30, 2016

Production-Inventory Systems: Modeling, Forecasting and Control

Daniel E. Rivera

Control Systems Engineering Laboratory School for the Engineering of Matter, Transport and Energy Ira A. Fulton Schools of Engineering Arizona State University

http://csel.asu.edu

Control Systems Engineering Laboratory

CSEL

LCCC Workshop on Process Control

• Sept. 28 - 30, 2016

Production-Inventory Systems: modeling, CONTROL, and Forecasting

Daniel E. Rivera

Control Systems Engineering Laboratory School for the Engineering of Matter, Transport and Energy Ira A. Fulton Schools of Engineering Arizona State University with special acknowledgments to: Jay D. Schwartz, Intel Corp. Naresh N. Nandola, ABB

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Outline

• Dynamical Model of a Production-Inventory System
• Control Strategies:
• IMC-PID and 2DoF Feedback-Only IMC
• 3DoF Combined Feedback/Feedforward IMC
• Model Predictive Control (MPC)
• Improved MPC algorithm / Hybrid MPC
• Control-relevant Demand Modeling / Demand Forecasting
• Summary and Conclusions

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Production-Inventory System

Starts (Manipulated) (Throughput Time) LT LIC Net Stock (Controlled) Demand (Disturbance) Forecast Actual

y(t) u(t) dF (t)

θ K

(Yield)

d(t) = dF (t − θF ) + dU(t)

y(s) = Ke−θs s u(s) − e−θF s s dF (s) − 1 sdU(s)

Integrating System with Delays

4

θd

(Delivery Time)

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Semiconductor Manufacturing Supply Chain Management

Package

• Customer
• X
• I

n c

Fabrication/Sort

• Nonlinear Throughput Time (~Weeks)
• Stochastic output

Assembly/Test

• Linear Throughput Time (~Days)
• Stochastic output

Finish/Pack

• Constant Throughput Time (~Shifts)
• Stochastic Output

Demand Factors

• Stochastic demand
• Inaccurate forecasts

Product Good Not Good Faster Slower Die Good Not Good Faster Slower

I20 LIC Demand Forecast M10 Fab/Test1 M20 Assembly/ Test2 M30 Finish/Pack Actual Demand

LT LT LT

C1 Fab Starts C2 Assembly Starts C3 Finish Starts C4 Shipments

Die/Package Inventory Semi-Finished Inventory Components Warehouse

I10 I20 I30

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Whole Hospital Occupancy

ED Queue Amb. Diverts ED WIP

EA

ED Holding Inpatient WIP Medical Diverts

Surgery Diverts

PACU Holding Surgery WIP PACU WIP

SA Ambulances ED Walk-Ins Medical Direct Admits Surgical Direct Admits ES

ED Queue Amb. Diverts ED WIP

EA

ED Holding Inpatient WIP Medical Diverts

Surgery Diverts

PACU Holding Surgery WIP PACU WIP

SA Ambulances ED Walk-Ins Medical Direct Admits Surgical Direct Admits ES

• Roche, K.T., D.E. Rivera, and J.K. Cochran, “A control engineering framework for managing

whole hospital occupancy,” Mathematical and Computer Modelling, Vol. 55, Issues 3-4,

• pgs. 1401 - 1417, February 2012.

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Global Warming/Climate Change

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• From National Geographic Magazine

(http://ngm.nationalgeographic.com/big-idea/05/carbon-bath)

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Parental function PF(t) is built up by providing an intervention I(t) (frequency of home visits), that is potentially subject to delay, and is depleted by potentially multiple disturbances (adding up to D(t)).

Parental Function-Home Visits Behavioral Intervention as a Production-Inventory Control Problem

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PF(t + 1) = PF(t) + KI I(t − θ) − D(t)

Intervention Dosage (Manipulated Variable) ( Delay Time) Parental Function (Controlled Variable) Exogenous Depletion Effects (Disturbance Variable)

θ

(Gain)

Outflow

Controller/ Decision Rules Measured Parental Function (Feedback Signal) Parental Function Target (Setpoint Signal)

Inflow

PF Goal I(t) PFmeas(t) PF(t) D(t)

KI

• Rivera, D.E., M.D. Pew, and L.M. Collins, “Engineering approaches for the design and analysis of adaptive,

time-varying interventions,” Drug and Alcohol Dependence, Special Issue on Adaptive Treatment Strategies,

• Vol. 88, Supplement 2, pgs. S31-S40, (2007).

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Internal Model Control (IMC) Design Procedure

• Step 1 (Nominal Performance): Obtain an H2 (ISE)-optimal q(s)
• An external input form is specified (e.g., step or ramp)
• Closed-form solution for q(s) is obtained
• Resulting controller is stable and causal
• Step 2 (Robust Stability and Performance)
• Augment the IMC controller from Step 1 with a filter, f(s).
• Proper choice and tuning of the filter ensures that:

the final controller q(s) is proper. the control system achieves stability and performance under uncertainty.

p(s) ˜ p(s) q(s) c(s) r(s) u(s) y(s)

+ + +

• p(s)

u(s) r(s) y(s)

⇐ ⇒

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IMC-PID Tuning Rules

c(s) = Kc

• 1 + 1

τIs + τDs

• 1

(τF s + 1) Kc = 3θ + 4λ K(θ2 + 4θλ + 2λ2), τI = 3 2θ + 2λ τD = θ2 + 2θλ 3θ + 4λ , τF = θλ2 θ2 + 4θλ + 2λ2 D.E. Rivera, M. Morari, and S. Skogestad. “Internal Model Control 4: PID Controller Design”. Ind. Eng. Chem. Process Des. Dev. 25, 252-265, 1986.

r(s) u(s) y(s) p

+ - + -

Inventory Target Factory Starts Inventory

d(s)

pd

c

Representing the delay with a first-order Padé approximation and applying the IMC design procedure leads to the PID with filter controller.

˜ p(s) = K(− θ

2s + 1)

s( θ

2s + 1)

p(s) = Ke−θs s q(s) = s K(λs + 1)

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IMC-PID Controller Response

50 100 150 500 1000 1500 2000

Net Stock

50 100 150 −100 100 200 300 400 500 600

Factory Starts

50 100 150 50 100 150 200

Customer Demand Time (Days)

Inventory Setpoint Change Forecasted Demand Change Unforecasted Demand Change

K = 1 θ = 5 θd = 0 λ = 5

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Two Degree-of-Freedom (2DoF) Feedback-Only IMC p(s) = Ke−θs s ˜ p(s) = Ke−θs s

No approximation is applied to the plant delay.

r(s) u(s) y(s) qr p ˜ p qd

• +
• +

Inventory Target Demand Factory Starts Inventory

pd

d(s)

+ +

12

qr(s) = s K 1 (λrs + 1)nr qd(s) = s(θs + 1) K (ndλds + 1) (λds + 1)nd

J.D. Schwartz and D.E. Rivera. “A process control approach to tactical inventory management in production-inventory systems,” International Journal of Production Economics, Volume 125, Issue 1, Pages 111-124, 2010.

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2DoF Feedback-Only IMC

50 100 150 500 1000 1500 2000

Net Stock qr qd

50 100 150 −100 100 200 300 400 500 600

Factory Starts

50 100 150 50 100 150 200

Customer Demand Time (Days)

K = 1 θ = 5 θd = 0 λr = 1 nr = 2 λd = 2 nd = 3

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3DoF Combined Feedback/Feedforward IMC Control

pd dU(s) dF (s) r(s) u(s) y(s) qF qr p ˜ p ˜ pd qd pd2 pd1

+ - ++ + -

• +
• +

+ +

Inventory Target Demand Forecast Unforecasted Demand Factory Starts Inventory

p(s) = Ke−θs s ˜ p(s) = Ke−θs s pd(s) = e−θF s s ˜ pd(s) = e−θF s s

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J.D. Schwartz and D.E. Rivera. “A process control approach to tactical inventory management in production-inventory systems,” International Journal of Production Economics, Volume 125, Issue 1, Pages 111-124, 2010.

qr(s) = s K 1 (λrs + 1)nr qd(s) = s(θs + 1) K (ndλds + 1) (λds + 1)nd qF (s) = e−(θF −θd−θ)s(nF λF s + 1) K (λF s + 1)nF , θF ≥ θ + θd

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3DoF Combined Feedback/Feedforward IMC Control

50 100 150 500 1000 1500 2000

Net Stock qr qF qd

50 100 150 −100 100 200 300 400 500 600

Factory Starts

50 100 150 50 100 150 200

Time (Days) Customer Demand Actual Demand θF−day ahead Forecast

λr = 1 nr = 2 λF = 1 nF = 3 λd = 2 nd = 3 K = 1 θ = 5 θd = 0 θF = 10

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Model Predictive Control (MPC)

Keep Inventories at Planning Setpoints Penalize Changes in Factory Starts

min

∆u(k|k)...∆u(k+M−1|k)

z }| {

P

X

`=1

Qe(`)(ˆ y(k + `|k) − r(k + `))2 + z }| {

M

X

`=1

Q∆u(`)(∆u(k + ` − 1|k))2

Predicted Inventory y(t+k) Future Starts u(t+k) Actual Demand Previous Starts

t t+1 t+M t+P

Move Horizon Prediction Horizon Umax Umin

Forecasted Demand dF(t+k) Inventory Setpoint r(t+k)

Disturbance Output Input

Past Future

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IMC/MPC Comparison

50 100 150 50 100 150 200 250 300

Net Stock MPC IMC

50 100 150 50 100 150 200 250 300

Factory Starts MPC IMC

50 100 150 50 100 150 200

Time (Days) Customer Demand Actual Demand θF−day ahead Forecast

θ = 5, P = 20, M = 10, Qe = 1, Q∆u = 10

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Constrained MPC (with Stpt Anticipation)

20 40 60 80 100 120 140 160 180 200 200 400 600

Net Stock MPC Setpoint

20 40 60 80 100 120 140 160 180 200 50 100 150 200

Factory Starts

20 40 60 80 100 120 140 160 180 200 50 100 150 200

Time (Days) Customer Demand Demand Forecast

Simulation under conditions of active constraints in net stock and factory starts.

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Some Observations

• Feedback-only control strategies (even if multi-degree-of-freedom)

are unsatisfactory (in general).

• Combined feedback-feedforward strategies that rely on the

availability of a demand forecast signal are necessary for good, comprehensive control.

• Model predictive control can provide useful functionality (e.g.,

constraint handling, anticipation) but the traditional move suppression/single-degree-of-freedom formulation can be lacking.

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Motivation for an Improved MPC Formulation

• Integrating dynamics (i.e., ramp responses and disturbances)
• Need to take advantage of anticipated future system inputs (i.e.,

forecasted demand)

• Multiple degrees-of-freedom (forecasted + unforecasted demand +

inventory setpoint tracking) with ease of tuning

• Ability to incorporate problem-specific constraints and possibly

hybrid dynamics

• Robustness in the presence of stochasticity and nonlinearity

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Nandola, N. and D. E. Rivera, “An Improved Formulation of Hybrid Model Predictive Control with Application to Production-Inventory Systems,” IEEE Trans. Control Systems Technology, Vol. 21, No. 1, pgs. 121-135, 2013.

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Block Diagram for 3 DoF MPC Controller

Optimization Filter II Filter I Plant Prediction and Estimation

MPC Controller

Measurement Noise ++ u y Forecasted Demand Actual Demand Error Projection Inventory Targets

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Three Degree-of-Freedom (3-DoF) MPC Tuning

• 1. Filter I for inventory target setpoint tracking (Type I /asymptotically

step signals)

• 2. Filter II for forecasted demand satisfaction (Type II /asymptotically

ramp signals)

fj(z) =

• (1 − αIIj) + 3

5αIIj

• − 1

5αIIjz−1 − 2 5αIIjz−2

1 − αIIjz−1 , j = 1, . . . , n

fi(z) = (1 − αIi)z z − αIi , i = 1, . . . , n

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Three-degree-of-freedom (3-DoF) MPC tuning (cont.)

• State estimation and unmeasured disturbance rejection (J.H. Lee and

Yu, Computers and Chemical Engineering,

• Vol. 18, No. 1, pgs. 15-37, 1994)

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• is focused on each output ; constrained to
• Speed of dist. rejection is proportional to the tuning parameter

(fa)j j 0 ≤ (fa)j ≤ 1 (fa)j

Kf = [0 Fb Fa]T Fa = diag{(fa)1, · · · , (fa)ny} Fb = diag{(fb)1, · · · , (fb)ny} (fb)j = (fa)j

2

1 + αj − αj(fa)j , 0 ≤ (fa)j ≤ 1, 1 ≤ j ≤ ny Step-B1: Xflt(k|k − 1): one step ahead prediction using filtered measured disturbance (dflt) Step-B2: Xflt(k|k) = Xflt(k|k − 1) + Kf(y(k) − CX(k|k − 1)) Step-A1: X(k|k − 1): one step ahead prediction using actual measured disturbance (d) Step-A2: X(k|k) = X(k|k − 1) + Kf(y(k) − CX(k|k − 1))

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3-DoF MPC for Continuous Input

20 40 60 80 100 200 400 y(k) 20 40 60 80 100 50 100 u(k) 20 40 60 80 100 20 40 60 Time (day) d(k)

fa=0.3;αr=0.9;αd=0.9 fa=1;αr=0;αd=0

independent controller adjustment without the need for move suppression!

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Controller Model (includes hybrid dynamics)

Disturbance Model

xw(k + 1) = Awxw(k) + Bww(k) d0(k + 1) = Cwxw(k + 1) Aw = diag{α1, α1, · · · , αny}, Bw = Cw = I

Integrated white noise

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: Unmeasured disturbance

d0

: Measured disturbance

d

Plant Model Mixed Logical Dynamical (MLD) Framework

x(k + 1) = Ax(k) + B1u(k) + B2δ(k) + B3z(k) + Bdd(k) y(k + 1) = Cx(k + 1) + d0(k + 1) + ν(k + 1) E5 ≥ E2δ(k) + E3z(k) − E4y(k) − E1u(k) + Edd(k)

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MPC Objective Function

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min

{[u(k+i)]m−1

i=0 , [δ(k+i)]p−1 i=0 , [z(k+i)]p−1 i=0 }

J

4

=

p

X

i=1

k(y(k + i) yr)k2

Qy + m1

X

i=0

k(∆u(k + i))k2

Q∆u

+

m1

X

i=0

k(u(k + i) ur)k2

Qu + p1

X

i=0

k(δ(k + i) δr)k2

Qd + p1

X

i=0

k(z(k + i) zr)k2

Qz

Subject to

ymin ≤ y(k + i) ≤ ymax, 1 ≤ i ≤ p umin ≤ u(k + i) ≤ umax, 0 ≤ i ≤ m − 1 ∆umin ≤ ∆u(k + i) ≤ ∆umax, 0 ≤ i ≤ m − 1 E5 ≥ E2δ(k + i) + E3z(k + i) − E4y(k + i) − E1u(k) + Edd(k + i), 0 ≤ i ≤ p − 1

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Hybrid 3 DoF Model Predictive Control, Production-Inventory System

(Throughput Time)

LT LIC

Net Stock (Controlled) Demand Actual (Disturbance) (Yield) (Delivery Time) Forecast Horizon Forecast

u(k)

Starts (Manipulated)

θ K θF dF (k) y(k) θd d(k)

u(k) ∈ {0, 33.33, 66.66, 100}

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y(k + 1) = y(k) + Ku(k − (θ − 1)) − d(k) d(k) = df(k) | {z }

forecasted

+ du(k) | {z }

unforecasted

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20 40 60 80 100 200 400 y(k) 20 40 60 80 100 50 100 u(k) 20 40 60 80 100 20 40 60 Time (day) d(k)

fa=0.3;αr=0.9;αd=0.9 fa=1;αr=0;αd=0

20 40 60 80 100 200 400 y(k) 20 40 60 80 100 50 100 u(k) 20 40 60 80 100 20 40 60 Time (day) d(k)

fa=0.3;αr=0.9;αd=0.9 fa=1;αr=0;αd=0

df d

Continuous u(t) Discrete-level u(t)

u(k) ∈ {0, 33.33, 66.66, 100}

Hybrid vs Continuous 3 DoF MPC Production-Inventory System

Solution involves solving a Mixed Integer Quadratic Program (MIQP) to address continuous error but discrete-level inputs (i.e., a hybrid problem).

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Production-Inventory System in the Presence of Forecast Error

Integrating System with Delays

Starts (Manipulated) (Throughput Time) LT LIC Net Stock (Controlled) Demand Forecast Actual

y(t) u(t)

(Disturbance)

dF (t)

θ K θd

(Yield) (Delivery Time)

eF (t)

Forecast Error

Σ

d(t) = dF (t − θF ) + dU(t)

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System Response to Forecast Error

10

−2

10

−1

10 10

−2

10

−1

10 10

1

Frequency, ω (rad/sec) Amplitude Ratio Inventory Response to Forecast Error Starts Response to Forecast Error 10 20 30 40 50 −2 2 ∆Inventory 10 20 30 40 50 −1 1 ∆Starts 10 20 30 40 50 −1 1 ∆Forecast Error Time (Days)

Impulse Response Frequency Response The closed-loop system response to a unit pulse in forecast error provides a basis for understanding modeling requirements for control-relevant demand models.

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J.D. Schwartz and D.E. Rivera. “A control-relevant approach to demand modeling for supply chain management,” Computers and Chemical Engineering, 70:78-90, 2014.

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Understanding C-L Response to Forecast Error

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The effect of forecast error on closed-loop performance is most significant in an intermediate frequency range.

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Response to Forecast Error (MPC, changing move suppression)

ω rad sec

• AR

10−3 102 10−3 100

Inventory response to forecast error Starts response to forecast error

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Control-Relevant Estimation

d(t) = ˜ pd(z)ud(t) + ˜ pee(t) Parseval’s theorem allows for frequency domain analysis of the problem. True demand is defined by a demand transfer function pd(z) and a stochastic component H(z)a(t). The control-relevant estimation step consists of minimizing the one-step-ahead prediction error, where is the prefilter. L(z) d(t) = pd(z)ud(t) + H(z)a(t) The estimated demand is defined by and a noise model . ˜ pd(z) ˜ pe(z) lim

N→∞

1 N

N

• t=1

e2

L(t) = 1

2π π

−π

• L(ejω)

˜ pe(ejω)

• 2

pd(ejω) − ˜ pd(ejω)

• 2 Φud(ω)+
• H(ejω)
• 2 Φa(ω)
• dω

min

˜ pd,˜ pe V = min ˜ pd,˜ pe

1 N

N

• t=1

[L(z)e(t)]2 = min

˜ pd,˜ pe

1 N

N

• t=1

e2

L(t)

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Multi-Objective Formulation

It is desirable to minimize a weighted combination of inventory and factory starts variance.

|L(ejω)|2 |˜ pe(ejω)|2 ΦeF (ω) = (1 − γ)|Lec(ejω)|2ΦeF (ω) + γλ|L∆u(ejω)|2ΦeF (ω) By assuming an output error model structure, L(z) can be reduced to the following form. |L(ejω)|2 = (1 − γ)|Lec(ejω)|2 + γλ|L∆u(ejω)|2 A curve fitting procedure is then used to obtain an Infinite Impulse Response filter that matches the amplitude ratio of the control-relevant prefilter.

The control-relevant prefilter then takes the following form.

min

˜ pd,˜ pe

∞

• t=0

(1 − γ)e2

c(t) + λ ∞

• t=0

γ∆u2(t)

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Multi-Objective Formulation (Cont.)

|L(ejω)|2 = (1 − γ)|Lec(ejω)|2 + γλ|L∆u(ejω)|2 γ = 0 : Inventory Variance Optimal γ = 1 : Starts Change Variance Optimal γ : Weighted Combination

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Final Observations

• Production-inventory systems are iconic dynamical systems that describe

interesting problems in the process industries (and beyond).

• Combined feedback-feedforward strategies relying on demand forecast signals

are necessary to adequately control these systems. Improved formulations of MPC can be developed in this regard.

• Demand modeling is a problem of significant importance in production-

inventory systems; analysis of closed-loop decision policies show that these are most responsive to forecast error in an intermediate frequency bandwidth.

• Prefiltering can be used to apply the proper emphasis in control-relevant

demand modeling.

• Multivariable extensions exist for both the control and demand modeling /

demand forecasting segments of this presentation.

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Primary References

• Schwartz, J.D., W. Wang, and D.E. Rivera, “Optimal tuning of process control policies for

inventory management in supply chains,” Automatica, 42, pgs. 1311- 1320, 2006.

• Wang, W. and D.E. Rivera, "Model predictive control for tactical decision-making in

semiconductor manufacturing supply chain management," IEEE Transactions on Control Systems Technology, Vol. 16, No. 5, pgs. 841 - 855, 2008.

• Schwartz, J.D., M.R. Arahal, D.E. Rivera, and K.D. Smith, "Control-relevant demand

forecasting for tactical decision-making in semiconductor manufacturing supply chain management," IEEE Trans. on Semiconductor Mfg, Vol. 22, No. 1, pgs. 154 - 163, 2009.

• Schwartz, J.D. and D.E. Rivera, "A process control approach to tactical inventory

management in production-inventory systems," International Journal of Production Economics, Volume 125, Issue 1, Pages 111-124, 2010.

• Nandola, N. and D. E. Rivera, “An Improved Formulation of Hybrid Model Predictive

Control with Application to Production-Inventory Systems,” IEEE Trans. Control Systems Technology, Vol. 21, No. 1, pgs. 121-135, 2013.

• Schwartz, J.D. and D.E. Rivera, “A control-relevant approach to demand modeling for

supply chain management,” Computers and Chemical Engineering, 70:78-90, 2014.

Additional references in http://csel.asu.edu/SCMpapers

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Acknowledgements

• Intel Research Council
• National Science Foundation (DMI-0432439)
• National Institutes of Health (K25 DA021173 and R21 DA024266).
• Jay D. Schwartz, Naresh N. Nandola, Martin W. Braun, Wenlin Wang, Manuel Arahal,

and Kirk D. Smith

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