A Novel Model Predictive Control Strategy for Tactical Decision-Making in Semiconductor Manufacturing Supply Chain Management

Transcription

1 Paper 42e A Novel Model Predictive Control Strategy for Tactical Decision-Making in Semiconductor Manufacturing Supply Chain Management Wenlin Wang, Daniel E. Rivera, Karl G. Kempf Department of Chemical and Materials Engineering Control Systems Engineering Laboratory, Arizona State University, Tempe, Arizona Decision Technologies, Intel Corporation, 5 W. Chandler Blvd., Chandler, AZ, USA (48) daniel.rivera@asu.edu Prepared for presentation at the Annual AIChE Meeting, Austin, TX November 7 2, 24 Session: [42] Supply Chain Management I Copyright c Control Systems Engineering Laboratory, ASU. November, 24 UNPUBLISHED AIChE shall not be responsible for statements or opinions contained in papers or printed in publications.

2 A Novel Model Predictive Control Strategy for Tactical Decision-Making in Semiconductor Manufacturing Supply Chain Management Wenlin Wang, D. E. Rivera and K. G. Kempf 2 Department of Chemical and Materials Engineering Control Systems Engineering Laboratory, Arizona State University, Tempe, Arizona Decision Technologies, Intel Corporation, 5 W. Chandler Blvd., Chandler, AZ, USA November 5, 24 Abstract A control-oriented approach is presented for supply chain management in semiconductor manufacturing that relies on Model Predictive Control as the tactical decision maker. Short time scale nonlinearity and stochasticity in supply and demand are addressed by Model Predictive Control formulation. Given inventory targets and capacity limits, the Model Predictive Control algorithm makes daily decisions on the starts to track the targets and meet customer demand. Challenges arise from long throughput times that depend nonlinearly on the load of the factory, the presence of unique constraints, and stochasticity in throughput time, yield and customer demand throughout the supply chain. To address these challenges, a novel flexible formulation of MPC is presented in this paper. A multiple degree-of-freedom observer is implemented to achieve robustness and performance in the presence of high stochasticity in both the supply and demand. Based on the expected signal to noise ratio, the tuning parameters in the filter can be chosen to reject disturbances (both measured and unmeasured), suppress noise and track the targets provided by the outer loop. Unique features in semiconductor manufacturing, such capacity limit, are formally addressed by imposing different constraints related to the outputs and inputs of the controller. All these functionalities contrast standard approaches to MPC and make this controller more meaningful as a tactical decision tool for semiconductor manufacturing and similar forms of discrete manufacturing problems. Two representative and challenging problems are examined with this flexible formulation of MPC. Some distinguishing features considered in this problem include multiple products and reconfiguration of high speed devices into lower speed ones. System robustness, performance and high levels of customer service are achieved with proper tuning of MPC, inventory targets and capacity limits setting. To whom all correspondence should be addressed. phone: (48) fax: (48) ; daniel.rivera@asu.edu 2

3 Introduction Supply chain management (SCM) has gained significance as one of the 2st century manufacturing paradigms for improving organizational competitiveness. SCM has been considered as a competitive strategy for integrating supplies and customers with the objective of improving responsiveness and flexibility of manufacturing organizations []. SCM is also vital for discrete parts manufacturing such as semiconductor manufacturing. It integrates factories, warehouses and customers for physical flow of materials, manufacturing planning and control, and physical distribution. The role of integrated decision policies for improving supply chain management in the semiconductor industries is described in some detail in (Kempf, 24)[4]. Model Predictive Control (MPC) is a control strategy that has been widely used in chemical process industries. Recently, the work of Braun et al. [5] applies a partially decentralized MPC structure for inventory control in supply chain. A centralized MPC strategy is successfully implemented for semiconductor manufacturing SCM to track inventory targets and satisfy stochastic customer demands given uncertainty on throughput time and yield in factories [6, 7]. Good system performance and robustness are achieved using judicious choices of move suppression. In this paper, several benchmark supply chain problems in semiconductor manufacturing which include both manufacturing factories and inventory positions are studied using Model Predictive Control. Stochasticity is present on Throughput (TPT) and yield in each factory. In Fab/Test node, Work in Progress (WIP) also influences the distribution and average of TPT, which introduces nonlinearity to these problems. In spite of high stochasticity and nonlinearity in system, a MPC controller is built based on a deterministic linear state space model and tuned to address these features. The advantage of MPC for constraint handling is demonstrated by putting constraints on inventory levels, starts, rate of starts change and capacity of each factory in supply chain. A state estimation based MPC is formulated to handle the short time scale stochasticity in semiconductor manufacturing and make optimal daily decisions for supply chain management. The use of Type I or Type II filter structure address the speed of output responses individually, as opposed to relying on move suppression which acts on all ouputs at the same time. The anticipation of the measured disturbance, i.e. forecasted customer demand, improves system performance by making the system respond to demand changes in a more efficient way. The paper is organized as follows. In Section 2, the state estimation-based Model Predictive Control algorithm is discussed. In Section 3, a basic problem in semiconductor manufacturing is studied. The effects of filter gain tuning and anticipation of customer demand are demonstrated with examples involving customer demand forecast error and periodic demand. In Section 4, a packaging problem is presented. The paper concludes with a discussion of the flexibility and advantages of using MPC in SCM. 2 Model Predictive Control The MPC controller algorithm implemented in this work is explained in this section. 3

4 2. Controller Model The MPC controllers considered in this paper rely on a linear state-space model [2]: x(k) = Ax(k ) + B u u(k ) + B d d(k ) y(k) = Cx(k) + D d d(k) () The measurement vector y m is described as: y m (k) = y(k) + v(k) (2) The inputs are u, d and v representing manipulated inputs, load disturbance and measurement noise respectively. y is the noise free output and y m is the measured inventory levels or controlled variable. The other corresponding variables in supply chain system are the starts of factories (u) and customer demand (d) which is treated as a measured disturbance with anticipation. We assume that d is a stochastic signal described by the following model: x w (k) = A w x w (k ) + B w w(k ) d(k) = C w x w (k) (3) where A w has all the eigenvalues insides unit the disk and w(k) is a vector of integrated white noise. Augmenting () with (3) and taking the difference form gives: X(k) = ΦX(k ) + Γ u u(k ) + Γ w w(k ) ŷ(k) = ΞX(k) + v(k) (4) where X(k) = Φ = Γ u = Γ w = x(k) w (k) y(k) (5) A B d C w A w CA CB d C w + D d C w A w I B u CB u B w D d C w B w (6) (7) (8) Ξ = [ I] (9) Here (k) = (k) (k ). 4

5 2.2 State Estimation and Prediction The states of the process model and the unmeasured disturbance and noise must be estimated using the augmented system model in (4). A Kalman filter can be used for the optimal state estimator as follows [3]. X(k k ) = ΦX(k k ) + Γ u u(k ) () X(k k) = X(k k ) + K f (y m (k) ΞX(k k )) () Here K f is the filter gain usually found by solving the algebric Riccati equation. However, the covariance matrices of load disturbance and measurement noise may not be accurately known. It is more practical to set the filter gain as a tuning parameter based on the signal-to-noise ratio. Since we do not need the prediction of each state in system, it is convenient to lump the effect of all disturbances on the outputs only (i.e. B d =, D d = I). If we assume the load disturbance is double integrated white noise (because of the integrating dynamics in the supply chain) and no information is available on the correlation of disturbances among different outputs (i.e. A w = diag{α, α 2,, α ny }, B w = I, C w = I,α is for Type I filter and 2 for Type II filter) [2], then (4) becomes: x(k) A x(k ) x w (k) = A w x w (k ) y(k) CA A w I y(k ) B u + u(k ) + I w(k ) (2) CB u I where w is white noise with following covariance matrix: E{ w w T } = diag{q,..., q ny } (3) We also assume that the measurement noise is white with following covariance matrix: where n y is the number of outputs. E{vv T } = diag{r,..., r ny } (4) For open-loop stable systems, it can be shown that the Type II optimal filter gain K f for the system is also parameterized in terms of an n y -dimension real vector with each element lying in (,] [2]. where K f = F b F a F b = diag{(f b ),..., (f b ) ny } F a = diag{(f a ),..., (f a ) ny } (5) (f b ) i = (f a ) 2 i + α i α i (f a ) i for i n y (6) 5

6 and (f a ) i as q i /r i, (f a ) i as q i /r i (7) The first term in (), X(k k ), includes all of the deterministic information such as the nominal system delay and measured disturbance anticipation. The second term is the prediction error generated by the stochasticity and uncertainty in the system. As f a approaches zero, the system ignores most of the prediction error and the solution is mainly determined by the deterministic model and the anticipation. The system will compensate all of the prediction error from the stochasticity and uncertainty if f a is one. 2.3 Objective Function As a receding horizon algorithm, at each time instant t, the controller considers the previous information on warehouse inventories, actual customer demands, factory starts and future information on inventory targets, forecasted customer demand to calculate a sequence of future starts by solving the following optimization problem. where the individual terms of J correspond to: min J (8) u(k k)... u(k+m k) J = + + Keep Inventories at Inventory Planning Setpoints {}}{ p Q e (l)(ŷ(k + l k) r(k + l)) 2 l= Penalize Changes in Starts {}}{ m Q u (l)( u(k + l k)) 2 (9) l= Maintain Starts at Strategic Planning Targets {}}{ m Q u (l)(u(k + l k) u target (k + l k)) 2 l= subject to the capacity constraints on the starts, the change rate of starts, the warehouse inventories and factory Work-In-Progress. Here p is the prediction horizon and m is the control horizon. Q u, Q u, Q e are penalty weights on the control signal, move size and control error, respectively. This problem can be solved by standard quadratic program algorithms. 3 The Basic Problem with Backlog In this Section, we present a basic supply chain problem which includes distinguishing features of semiconductor manufacturing. The basic semiconductor manufacturing process is shown in Figure. Wafers are first fabricated and tested in Fab/Test. Transistors are built on a silicon wafer and then interconnected 6

7 Figure : Sequence of steps in semiconductor manufacturing Figure 2: Fluid representation of a three node semiconductor mfg. supply chain 7

8 to form circuits in a fabrication process. In Assembly/Test2 process, the individual die are cut from the wafers and mounted in packages to protect them. The semi-finished goods are configured, packed and shipped to customers in the finishing manufacturing stage. One wafer can be processed and configured to make different products. A fluid representation of a three-node semiconductor manufacturing supply chain (consisting of one Fab/Test, one Assembly/Test2, and one finish node) and its corresponding inventory locations is shown in Figure 2. In a very general sense, the manufacturing nodes are represented as pipes, while the inventory locations are represented as tanks ; material in these pipes and tanks correspond to factory Work-in-Progress (WIP) and warehouse inventory, respectively. A discrete time model is used to describe the dynamics of supply chain. For each inventory position at day k, we can develop the following equation based on material balance. I i (k + ) = I i (k) + Y j C j (k θ j ) C j+ (k) i {, 2, 3} j {, 2, 3} (2) where I i is the inventory stored in position i, C j is the start of factory j and C 4 is the anticipation of future customer demand; Y j and θ j are the yield and throughput time of factory j respectively. These models could be organized in state-space form and used as nominal controller model. Both the throughput time and yield are deterministic in the nominal controller model, while the stochasticity is introduced on these parameters in the simulation model as described in Table. Many requirements of SCM in semiconductor Factory nodes M M2 M3 M4 load Min TPT (days) [%,7%] Ave TPT (days) Max TPT (days) Distribution Unif Unif Unif load Min TPT (days) (7%,9%] Ave TPT (days) Max TPT (days) load Min TPT (days) (9%,%] Ave TPT (days) Max TPT (days) Yield Min % Ave % Max % Distribution Unif Unif Unif Capacity Max Items 4.5E Inventory nodes I I2 I3 UCL(Items).2E4 6E3 3E3 TAR(Items) LCL(Items) Max(Items) 2E4 E4 E4 Table : Manufacturing and inventory nodes data for basic problem with backlog: TPT refers to throughput time; Unif to uniform distribution; UCL to Upper Control Limits, TAR to Target, LCL to Lower Control Limits. manufacturing are more properly expressed as constraints on the process variables. There are three types 8

9 of constraints that can be imposed in SCM as follows: Manipulated variable constraints: these are hard high and low limits on the starts of factories due to the capacity Cj min C j (k) Cj max j {, 2, 3} (2) where Cj min, Cj max are the high and low limits of the start to factory j. These limits can be constants or time dependent. Manipulated variable rate constraints: these are hard limits on the maximum and minimum move size of the acceptable thrash of factory starts Cj min C j (k) Cj max j {, 2, 3} (22) where Cj min, Cj max are the high and low limits of the start variation to factory j. They can also be constants or vary over a horizon. Capacity constraint: each factory has a capacity limit. Starts shipped to and processed in the factory can not exceed this limit. T P T i C j (k) + C j (k n) Cap i n= C j (k) + C j (k + ) + + C j (k + T P T i 2) + C j (k ) Cap i C j (k) + C j (k + ) + + C j (k + T P T i ) Cap i. (23) C j (k + ) + C j (k + 2) + + C j (k + T P T i ) Cap i C j (k + 2) + C j (k + 3) + + C j (k + T P T i + ) Cap i. (24) C j (k + m T P T i ) + + C j (k + m ) Cap i j {, 2, 3} i {, 2, 3} (25) here Cap i is the capacity limit for each factory, m is the moving horizon. Output variable constraints: inventory levels have specified targets and the objective function minimizes their deviations from the setpoints. Besides the setpoints, they also have high and low limits on the inventories. To avoid infeasible solutions, the constraints on controlled variables are treated as soft constraints in the objective function. Ii min I i (k) Ii max i {, 2, 3} (26) 9

10 The customer demand is treated as a measured disturbance with anticipation. It is kept for one week, and then it is lost. The error between the actual demand and the forecast could be significantly large. Based on different customer demands and anticipation, we develop two scenarios to study the effects of the filter gain and anticipation. 3. Scenario : Stationary Demand and Forecast Error In this case, we demonstrate the effects of Type II filter gain to handle forecast error in customer demand. The actual customer demand and two different demand forecasts are shown in Figure 3. The first forecast is ten day moving average of the actual demand, while the variance of the actual demand is larger than that of the anticipation. If this forecast is used, the results as shown in Figure 4 demonstrate that the starts are similar to the demand forecast and the oscillation on the inventory is mainly caused by the stochastic throughput time and yield in the semiconductor manufacturing process. No backlog is generated and the simulation also shows the inventories are more than enough to meet the customer demand. Compared with the actual customer demand, the starts response for each of the factories is quite smooth. With an erroneous demand forecast as shown in Figure 3, the average of the anticipation signal is units smaller than that of the actual customer demand. If the move suppression is zero, the output weight is for each controlled variable and the filter gain is zero on each output, the controller ignores the average difference between the actual demand and the forecast and tries to follow the anticipation only. Warehouse inventory I3 is depleted and many backorders are accumulated since the controller will not compensate the prediction error, however the starts (C i, i =, 2, 3) are very smooth as shown in Figure 5. If the filter gain f a is increased to.4, the backorders are dramatically reduced while the variance on the starts increases, as shown in Figure 6. The starts begin to increase after the controller measures the actual demand and find the difference between the measurement and the anticipation. The rising speed is determined by the filter gain. The larger the filter gain, the faster the controller increases the starts. If the speed of response is not sufficiently high, the safety stock in inventory I3 is depleted and backorders are generated. As shown in this case, increasing filter gain can make the controller compensate the forecast error. The larger the filter gain, the faster the controller can compensate the error. However, if the filter gain is too large, the responses will be very noisy since the controller tries to chase the noise from the customer demand and stochastic manufacturing process and this also generates backorders. As shown in Figure 7, the backorders reach minimum aound f a =.5 and then they increase again with increasing filter gain. The reason that larger filter gain creates more backorders is the increasing variance on starts as shown in Figure 8. There is always a tradeoff between the response speed and robustness, either of which will influence the total cost of inventory holding, revenue and backlog. 3.2 Scenario 2: Periodic Customer Demand In practice, customer demand may change dramatically from day to day, while operations may want the starts of the factories to stay as constant as possible. In this case, we apply a sinusoidal customer demand to test the effects of anticipation in the demand forecast and to evaluate the flexibility of this MPC algorithm.

11 95 Items 9 Actual customer demand Demand forecast Demand forecast Figure 3: Stationary customer demand and forecast error in scenario : dashed: actual customer demand, star: forecast with the same average of actual demand, dot: forecast with units error to the average of actual demand..5 x 4 5 I M I M I M Backorders Shipment&Demand Figure 4: System responses using a forecast with the same average as demand: f a =, Scenario

12 5 I 8 C I2 I3 Backorders C2 C3 Shipment&Demand Figure 5: System responses using an erroneous forecast with f a =, Scenario 5 I 5 C I C I3 2 C3 9 Backorders Shipment&Demand Figure 6: System responses using erroneous forecast with f a =.4, Scenario 2

13 3 x Backorders Filter gain Figure 7: Relationship between backorders and filter gain 2 x 5 8 Variance Filter gain Figure 8: Relationship between Fab/Test starts and filter gain The customer demand is a stochastic sinusoidal signal with the mean of 95 units, amplitude of units and frequency of. rad/sec. The first demand forecast is a deterministic sinusoidal signal with the same mean, amplitude and frequency as the stochastic demand and the second demand forecast is the average of the stochastic demand. In both cases, the move suppression and output weight parameters are zero and one respectively. The filter gain is set to be. to achieve the robustness. As shown in Figure 9, for this first demand forecast, there is no backorder, and the inventory has low variance and no offset with the target. However, the starts change sinusoidally just like the actual demand; this may not be desirable to operations. If the stationary demand forecast is applied, the results are shown in Figure. No backorders are generated and the starts display similar variability to that of the forecast, while the closest inventory I3 varies periodically corresponding to the customer demand and absorbs the variance in this variable. 3

14 2 I 5 C I2 4 2 C I3 4 2 C2 9 Backorders Shipment&Demand Figure 9: System responses with periodic demand forecast, Scenario 2: f a =. 2 I 5 C I2 I C2 C Backorders Shipment&Demand Figure : System responses with stationary demand forecast, Scenario 2: f a =. 4

15 4 The Assembly/Test2 stochastic split problem A representative semiconductor manufacturing supply chain for a problem involving the manufacture and demand of two products with different speeds is considered. The fluid analogy corresponding to this problem is shown in Figure. It contains one Fab/Test node M, one Assembly/Test2 node M2, two Finish/Pack nodes M 3, one Assembly-Die Inventory (ADI) I, one Semi-Finished Goods Inventory (SFGI) for high speed devices I2, one SFGI for low speed devices I2, one Components Warehouse (CW) for high speed devices I3 and one CW I3 for low speed devices. Two M4 nodes represent shipments with one day delay and no uncertainty. The valves from C35 to C39 are the starts for factories. C4 and C4 are where customer demands enter. Items coming into M2 through C36 will be split into two bins, one is made up of high speed components, while the other one has low speed devices. The number of items in each bin is determined by a split factor in A/T2 which is stochastic and can have different average and variance values. Besides meeting the fast device demand D, fast devices in I2 can also be used to make slow devices to meet the demand E. Devices in I2 can only be used to meet the customer demand E. In other words, if there are more than enough products available to meet customer demand D while not enough to meet E, through C38 some fast devices will be transferred from I2 to I3 to meet the demand E. The opposite direction is not allowed. Excess devices in I2 will be discarded through C9 if the inventory level reaches a maximum. Figure : The Assembly/Test2 stochastic split problem The controlled variables are the inventories, I, I2, I2, I3 and I3. The associated variables are 5

16 Work-In-Progress (WIP) for M, M2 and M3. The manipulated variables are the starts for all the factories, C35, C36, C37, C38 and C39. Customer demands for high speed devices D and for low speed devices E are treated as measured disturbances coming into system through C4 and C4 respectively. Although we do not know the exact customer demand in future, a reasonable forecast can be achieved. This forecast is used as an anticipation for future measured disturbance in MPC. There are errors between the actual demand and the forecast. However, the approximate anticipation can still improve the system performance. The nominal controller model used for inventories in this problem is based on a material balance. The representative equations for I2 and I2 which are related to the split factor can be written as follows I 2 (k + ) = I 2 (k) + Y 2 C 36 (k θ 2 ) α C 37 (k) C 38 (k) I 2 (k + ) = I 2 (k) + Y 2 C 36 (k θ 2 ) ( α) C 39 (k) (27) Here Y 2 stands for the yield of M2, θ 2 is the average throughput time in M2 and α is the split factor which is assumed in simulation as a random number with a uniform distribution. The representative model for the WIP of the manufacturing node M3 for low speed devices can be described in the following equation. W IP low 3 (k + ) = W IP low 3 (k) + C 38 (k) + C 39 (k) C 38 (k θ 3 ) C 39 (k θ 3 ) (28) where θ 3 is the throughput time for M3. In this formulation, the nominal model for the controller is linear in nature with fixed throughput time and yield. In the simulation model, the throughput time of M is also nonlinearly dependent on the load or WIP. The prediction horizon p is 7 days and control horizon m is 6 days.the manufacturing nodes data implemented in this problem are listed in Table 2. The inventory targets and constraints are presented in Table 3. The two customer demands for fast and slow devices are stochastic with different means but similar variances. Usually the demand for the slow device is higher than that for the fast device; the average of the slow device demand is larger than that of the fast device demand, as shown in Figure 2. The constraints are enforced to keep high and low limits on inventory levels, WIPs, starts and the change of starts. I min I (k + i k) I max (29) W IP min W IP (k + i k) W IP max (3) i =, 2,..., p C (k + j k) C max (3) C min C (k + j k) C max (32) j =, 2,..., m (33) 6

17 M M2 M3 M4 load=% TO 7%Max TPT Min days TPT Ave days TPT Max days Distribution Unif Unif Unif load=7% TO 9%Max TPT Min days TPT Ave days TPT Max days Distribution Unif Unif Unif load=9% TO %Max TPT Min days TPT Ave days TPT Max days Distribution Unif Unif Unif Applied at Output Yield Min % Yield Ave % Yield Max % Distribution Unif Unif Unif Capacity Load Max Items Load Initial Items Table 2: Manufacturing nodes data in the test2 stochastic split problem: TPT-throughput time; Unifuniform distribution I I2 I3 I3 I3 Inventory UCL Item TAR Item LCL Item Level Max Level Initial Table 3: Inventory setting in test2 stochastic split problem 7

18 7 Fast Demand Slow Demand Figure 2: Demand sets in Assembly/Test2 stochastic split problem represents the index for different inventories, WIPs and starts. The constraints are forced over the time horizon. Depending on the split in Assembly/Test2, we will have different amounts of products to meet different customer demands. Based on the split factor in A/T2 node, we developed three cases for study, which are:. balanced demand, 2. reconfiguring high speed device, 3. discard low speed device. The average and variance of the split are shown in Table 4. These three cases are studied by using different tuning parameters to achieve robustness and better customer service. In each case, the results are compared based on the variance of each variable and the backorders created by the process. 4. Case (Balanced) The average of the split in Case is the same as the proportion of the two customer demands D E (balanced), which means the split will make the amount of the fast devices and the slow devices the same as the demands for the fast and the slow devices if no uncertainty occurs. The move suppression is still zero for all the manipulated variables. Output weights coresponding to Q e in equation 2 are [,, 8,, 8] for I, I2, I2, I3 and I3 respectively. These output weights puts more penalty on inventory levels of I2 and I3 and force them closer to the targets than I2 and I3. The filter gain is.5 since the forecast and actual demand are at the same average. The results are shown in Figure 3. The WIP is within the capacity limit for each factory. The starts are smooth and a small amount of high speed devices are reconfigured to low speed devices throught C38 due to the stochasticity. Each inventory is close to the target and only 8

19 few backorders are generated for high speed device demand. 4.2 Case 2 (Reconfigure High) In this case, the split average is larger than the average of D E. So there are more products to meet the demand for D than to meet the demand for E at steady state. Since the split in M2 does not make enough products to meet the demand for E, some of the fast devices in I2 are used to make the slow devices through M2 to meet the demand for E. Move suppression is set to be zero. Output weights are [,, 5,, 5] and filter gain is.5. The results are shown in Figure 5. The inventories are well kept around the targets and the WIPs are lower than the capacity limits. However compared to the results in Case, since not enough low speed devices are generated by the split, lots of high speed devices have to be reconfigured to meet low speed devices through C38. Only few of backorders are observed for high speed devices in this case. 4.3 Case 3 (Discard Low) In Case 3, the average split is smaller than the average of D E. At steady state, there will be not enough items to meet the demand for D if demand for E is met without any excessive products left in I2 or I3. Even in a deterministic setting (not shown), in order to meet the demand for D, the total amount of items processed in M and M2 must be increased. While enough items will be shipped to I2 and I3 to meet the demand for D, but there will be more than enough products to satisfy the demand for E. The inventory will be held until it reaches the high limit of capacity and excessive products will be scrapped due to the limited capacity of I2. Although no backorder is generated, the loads of F/T and A/T2 exceed 95%. When stochasticity is introduced in the throughput times, yields of the manufacturing nodes and two cutomer demands, we first try to test the original capacity which is 45, 75, 25 for M, M2 and M 3 respectively. The total amount of products starting from C35 have to increase to produce enough high speed devices and excess low speed devices. Consequently, the inventory I2 needs to store the excess slow devices and scrap when it reaches the limit. Therefore, the output weight for this variable is set to be zero. The output weight is [5] and filter gain is.5. The results are shown in Figure 7. The inventory level of I2 stays at the capacity limit all the time and many slow devices are scrapped though C9. The loads of M and M2 exceed the capacity most of the time and not enough products can be produced on time with these capacity settings. Many backorders are observed as shown in Figure 8 becuase the inventories of I3 and I3 are depleted. This is a result of the limited capacity and can not be overcome simply by controller tuning. The only way to solve this problem is to allocate capacity for each manufacturing node which is sufficient to meet the demands. If we increase the capacities of M and M3 by 2% and M2 by 3% and apply the same filter gain and output weight, the performance is significantly improved as shown Figure 9. One can find here the loads of M, M 2 and M 3 are all below %. We have enough capacity to process items as many as they are needed. Only a few backorders are 9

20 I2 Hi spd I2 Low spd Bin Splits Max Average Min Max Average Min Case Case Case Table 4: Three different split factor distribution in Assembly/Test2 stochastic split problem Load of M Load of M2 Load of M C35 C36 C I I2 I Bleed C38 5 I C39 56 I Figure 3: Assembly/Test2 stochastic split problem: Invetory, starts and WIP in Case 5 5 Fast Device Shipment Fast Device Backorder Slow Device Shipment Slow Device Backorder Figure 4: Assembly/Test2 stochastic split problem: Backorders in Case 2

21 Load of M Load of M2 Load of M C35 C36 C I I2 I Bleed C38 I C39 5 I Figure 5: Assembly/Test2 stochastic split problem: Invetory, starts and WIP in Case Fast Device Shipment Fast Device Backorder Slow Device Shipment Slow Device Backorder Figure 6: Assembly/Test2 stochastic split problem: Backorders in Case 2 2

22 Load of M C I 2 x Load of M C I Load of M x 5 C I Bleed C38 I C39 I Figure 7: Assembly/Test2 stochastic split problem: Invetory, starts and WIP in Case 3 (limited capacity) Fast Device Shipment Fast Device Backorder Slow Device Shipment 5 5 Slow Device Backorder Figure 8: Assembly/Test2 stochastic split problem: backorders in Case 3 (limited capacity) 22

23 created for high speed device demand and no backorders for slow device demand. This simulation study shows it necessary to have an external decision policy that allocates capacity based on longer-term demand information and economic considerations. 5 Conclusions A novel Model Predictive Control formulation is presented for supply chain management in semiconductor manufacturing. The selection of proper tuning parameter helps the system to achieve both robustness and performance with the linear deterministic nominal model despite the stochasticity and nonlinearity in the plant. In most cases, the choice of a small filter gain improves robustness. However, if there is a large error between the average of customer demand and the forecast, a larger filter gain has to be used to make the controller compensate for the error sufficiently fast. There is always a tradeoff to pick the proper filter gain. The anticipation of the measured disturbance is critical for achieving better performance. Proper anticipation can help the system to meet different requirements with no backlog cost. Simulation results demonstrate the effects of both filter gain and anticipation. From these we see that MPC serves as a flexible and powerful tool for making daily decisions in SCM in semiconductor manufacturing, in the presence of high stochasticity and nonlinearity. 6 Acknowledgement Support for this research from the Intel Research Council and the National Science Foundation (grant DMI ) is gratefully acknowledged. References [] Editorial. Supply chain management: Theory and application European Journal of Operational Research, 59 (24) [2] Lee, J and Z. H. Yu. Tuning of model predictive controllers for robust performancecomputers chem. Engng, Vol. 8, No., pp. 5-37, 994. [3] Åström, K. J. and B. Wittenmark, Computer Controlled Systems: Theory and Design,Prentice-Hall, Englewood Cliffs, NJ, 984. [4] K. G. Kempf. Control-Oriented Approaches to Supply Chain Management in Semiconductor Manufacturing, Proc. American Control Conference, Boston, MA, June 2-July 2 24, pp [5] M. W. Braun, D. E. Rivera, M. E. Flores, W. M. Carlyle and K. G. Kempf. A Model Predictive Control framework for robust management of multi-product, multi-echelon demand networks, Annual Reviews in Control, Special Issue on Enterprise Integration and Networking, Vol.27, Issue 2, pp , 23. [6] W. Wang, D. E. Rivera and K. G. Kempf. Centralized Model Predictive Control Strategies for Inventory Management in Semiconductor manufacturing Supply Chains, Proc. American Control Conference, Denver, CO, June 23, pp

24 [7] W. Wang, D. E. Rivera, K. G. Kempf and K. D. Smith. A Model Predictive Control Strategy for Supply Chain Management in Semiconductor Manufacturing under Uncertainty, Proc. American Control Conference, Boston, MA, June 2-July 2 24, pp

25 Load of M C I Load of M C I Load of M x 5 C I Bleed 2 C38 I C39 I Figure 9: Assembly/Test2 stochastic split problem: Invetory, starts and WIP in Case 3 (increased capacity) Fast Device Shipment Fast Device Backorder Slow Device Shipment Slow Device Backorder Figure 2: Assembly/Test2 stochastic split problem: backorders in Case 3 (increased capacity) 25