ο€  Advances in Technology Innovation, vol. 2, no. 1, 2017, pp. 25 - 28 25 EWMA Controller with Concurrent Adjustment for a High-Mixed Production Process Shui-Pin Lee Department of Industrial Management, Chien Hsin University of Science and Technology, Taoyuan, Taiwan. Received 22 February 2016; received in revised form 15 April 2016; accepted 17 April 2016 Abstract The exponentially weighted moving average (EWMA) feedback controller is a very popular run-to-run (RtR) process control scheme in the semiconductor industry. Traditionally, the manufacturing environment was simplified as a single product and single tool process. In the feedback control, the adjustment of the recipe for the next run is related to the deviation of the current output against the desired target. How- ever, in a commercial foundry, every tool always works for many products. It is called a multiple products and single tool (MPST) process. The challenge of this process is how to adjust the recipes among different products in a production row. In this study, a modified threaded EWMA feedback controller, the EWMA with concurrent adjustment, is proposed to deal with the issue of multiple products in a tool. When the process disturbances follow an IMA(1,1) time series model, the stability of the proposed method will be proven. The optimum discount factors of the proposed EWMA controller will be investigated by several simulations in terms of the number of multiple products, their distribution and the scheduling of the process. Moreover, according to results of the performance comparisons with threaded EWMA, the proposed controller is advantage in the large number of multiple products and in low-frequency products. Keywords: EWMA, run-to-run, feedback con- troller, high-mixed production pro- cess, concurrent adjustment 1. Introduction Run-to-Run (R2R) process controller has been a conversional quality control technique in semiconductor manufacturing process. It was purposed by integrating statistical process con- trol (SPC) and engineering process control (EPC) for overcoming the shift or gradually drift in the complex manufacturing process [1-2]. In most semiconductor manufacturing process, the run-to-run process controller for adjusting the input recipe is based on a known prediction model. Assume that the I/O relationship of the SISO process is linear, the process outputs ( 𝑦𝑑 ) can be expressed as follows: 0 0 1t t ty x  ο€­ο€½   (1) where π‘₯π‘‘βˆ’1 denotes the process input at run t that has been adjusted after its previous output π‘¦π‘‘βˆ’1 obtained, 𝛼0 and 𝛽0 are the intercept and slope parameters, respectively, and πœ–π‘‘ denotes the process disturbances. The next process input at run 𝑑 + 1 will be adjusted by: t t a x b  ο€­ ο€½ (2) where Ο„ is the process target, 𝑏 is the estimate of the slope 𝛽 and π‘Žπ‘‘ is the new estimate of 𝛼0 by using the following EWMA formula:    1 11t t t ta y bx a  ο€­ο€½ ο€­  ο€­ (3) Typically, the researches related to the topic of how to enhance the performances of R2R controller are very restricted in a single product and single tool (SPST) production environment. However, in many real commercial production processes, a tool will produce several different products. The same type of products might not be produced in-a-row. The kind of manufacturing * Corresponding author, Email: shuipin@uch.edu.tw Advances in Technology Innovation, vol. 2, no. 1, 2017, pp. 25 - 28 26 Copyright Β© TAETI mode is often called a multiple-product- multi- ple-tool (MPMT) or a high-mixed production process. The foundries in Taiwan are the typical examples of the high-mixed manufacturing mode. The manufacturing environment consists of mul- tiple products passing through a sequence of batch processing steps being that are out by multiple parallel tools. Recently, R2R control implementation in a MPMT environment has been discussed by many authors [3-5]. Most of these works have attempted to identify parame- ters that can characterize the product and tool states. However, information is never shared between products or tools in such an approach. Thus, EPC or SPC are ineffective for low fre- quency products. Lee et al. [6] proposed a cu- mulative sum-type statistical process control for a MPMT process to detect substantial changes in tool and product effects. The estimate of gain parameter will be updated after signals. The recipes of inactive products will be adjusted if SPC emits a signal. For statistical significance, the signal is related to the number of different products. In this short paper, the EWMA with concurrent adjustment algorithm for a multiple products and single tool (MPST) process is proposed for inactive products. Such that, when those products become active, their corresponding recipes can reflect the effect of the tool change. The organization of the paper is as follows. The proposed EWMA with concurrent adjust- ment algorithm is introduced in section 2. In section 3, the stability condition of the proposed method and the evaluation of its performance will be shown. Conclusions were then drawn in section 4. 2. EWMA Controller with Con- current Adjustment If we assume the initial estimates of param- eters Ξ± and Ξ² of (1) are correct, the input recipe will be set οΏ½ΜƒοΏ½ = πœβˆ’π›Ό 𝛽0 for all runs to make the ex- pected output meet the process target. Without loss generality, we can simplify (1) by the fol- lowing expression: 0 1t t ty x ο€­ο€½  (4) Assume that there are J products will be produced in a tool. Let π‘₯𝑗,π‘‘βˆ’1and 𝑦𝑗,𝑑denote the process input and output at run 𝑑. Since a tool just can produce one product at each run, the product index 𝑗 is a function of 𝑑. The product 𝑗 is active at run 𝑑, but other 𝐽 βˆ’ 1 products are inactive. In this paper, the I/O relationship of the MPST environment is , , 1 ,j t j j t j ty x ο€­ο€½  (5) where 𝛽𝑗 denotes the slope parameter with re- spect to product 𝑗 and πœ–π‘—,𝑑 denotes the process disturbance. Denote 𝛽𝑗 = 𝛽0𝑓𝐽 , where 𝛽0 de- notes the nominal tool effect parameter and 𝑓𝑗 denotes the nominal product effect for product 𝑗 about the reference product. Moreover, we take the contrast 𝑓1 = 1 for identifiability. De- note 𝑏𝑗,0 as the initial estimate of 𝛽𝑗 , 𝑗 = 1, β‹― , J then 𝑏1,0 is also the estimate of 𝛽0 and the esti- mates of the nominal product effects are 𝑓𝑗 = 𝑏𝑗,0 𝑏1,0 . Hence, the initial prediction model can be expressed as follows:  , , 1 0 , 1Λ†|j t j t j j tE y x b f xο€­ ο€­ο€½ (6) For the current active product 𝑗, its recipe can be adjusted by , , 0 , Λ† t j t j t j a x b f  ο€­ ο€½ (7) where π‘Žπ‘—,𝑑 = π‘Žπ‘—,π‘‘βˆ’1 + πœ” (𝑦𝑗,𝑑 βˆ’ πœπ‘‘ ) is the formal EWMA formula, but for the inactive products 𝑗′ β‰  𝑗. We propose the concurrent adjustment ', ', 1j t j t ta a c  (8) where 𝑐𝑑 = πœ” (𝑦𝑗,𝑑 βˆ’ πœπ‘‘ ) and πœ— denotes the con- current factor. 3. Results and Discussion The offset of the process output can be ex- pressed by       1 1 0,2 , , 1, 2 0, 0, , , 0, , 1 1 1 j j l l j l j l j l j j t j j lj t h h j j h j l j t j h j t h y y C b C B b B        ο€­ ο€­ ο€­ο€­ ο€­ ο€­ ο€­ ο€­ ο€­ ο€­ ο€­ ο€­  ο€­ ο€½  ο€­ (9) where 𝑙 denotes the series index of product 𝑗 , Advances in Technology Innovation, vol. 2, no. 1, 2017, pp. 25 - 28 27 Copyright Β© TAETI βˆ…π‘— = (1 βˆ’ πœ”π›½0,𝑗 𝑏0,𝑗 ) denote the stable factors, 𝐢𝑗,𝑙 = βˆ‘ 𝑐𝑖 π‘‘βˆ’1 𝑖=π‘‘βˆ’β„Ž 𝑙 𝑗 denotes the cumulative devia- tion from target and 𝐡 denotes the backward operator. When the process disturbances is an IMA (1, 1), or a ARMA (p, q), the variance of the process output is bounded if |βˆ…π‘— | is smaller than 1 for all 𝑗. According to several simulations in different conditions of the production envi- ronment, πœ” = 0.2 and πœ— = 1.0 are suggested. The performance of the proposed method was evaluated based on the criteria of the relative efficiency, the ratio of mean square errors of the proposed method to that of the threaded EWMA controller under the same conditions. Fig. 1 shows the results of the relative efficiency comparisons No matter what product scheduling (random or concentration) is, what the number of product types (2 or 4) is and what the mag- nitude of slope shift is, Fig. 1 explicitly shows the proposed method is better than the threaded EWMA controller. Fig. 1 The relative efficiency of the EWMA controller with concurrent adjustment with respect to the threaded EWMA controller 4. Conclusions In this paper, the EWMA with concurrent adjustment algorithm control was proposed for a multiple products process. The exact expression of the process output of the proposed control algorithm is derived. Hence, its stability condi- tions can be obtained. Based on several numer- ical simulations,πœ” = 0.2 and πœ— = 1.0 are sug- gested for the discount factor of EWMA con- troller and concurrent factor when the process disturbances are a white noise series. Compare to the threaded EWMA controller in different production scheduling, the advantage of the EWMA with concurrent adjustment is increas- ing in terms of the magnitude of the slope shift. Moreover, the larger the number of product types, the bigger the advantage. Acknowledgement The support of the Minister of Science and Technology (Taiwan), under Grant Most 104-2221-E-231-005 is gratefully acknowledged. References [1] A. Ingolfsson A and E. Sachs, β€œStability and sensitivity of an EWMA controller,” Jour- nal of Quality Technology, vol. 25, pp. 271-287, 1993. [2] E. Sachs, A. Hu, and E. Ingolfsson, β€œRun by run process control: combining SPC and feedback control,” IEEE Transactions Sem- iconductor Manufacturing, vol. 8, pp. 26-43, 1995. [3] A. V. Prabhu and T. F. 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