ABSTRACT 1 INTRODUCTION

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1 DETERMINING SPC ALLOCATION WITH FUZZY MEMBERSHIP FUNCTION BASED HISTOGRAM EQUALIZATION Katherin Indriawati, Rosalina Witrianti Engineering Physics Department, Faculty of Industrial Technology, Institut Teknologi Sepuluh Nopember Kampus ITS Keputih Sukolilo, Surabaya katherin@ep.its.ac.id ABSTRACT Statistical Process Control (SPC) is a technical tool which can be used to evaluate the performance of process with statistical method. However, because of cost consideration, management need to decide which process should apply SPC. In this paper, the method to determine SPC allocation in the process industry is reported. The proposed method use conditional probability to analyze process failure rate and repair rate. Then, using Markov matrix, probability out-of-control process (PO) is calculated. Finally, the fuzzy membership functions (MFs) are used to analyze PO value. The used fuzzy MFs are the modification of the evenly distributed MFs with the help of a technique termed histogram equalization. To analyze PO, the modified fuzzy MFs are classified into three categories, namely LOW, MEDIUM, and HIGH. Any part of process industry has PO fall into the HIGH category and high degree of membership is prioritized to apply SPC. As an example, the proposed method was applied in the PET continuous process which there is 3 running data from 3 reactor, (reactor R200), 0.5 (reactor R210), and (reactor R220). Later, this running data will be used to determine SPC allocation optimally. Keywords : SPC, modified fuzzy MFs, markov matrix, probability out-of-control. 1 INTRODUCTION The term quality shall mean fitness for use. In the case of an end product, it refers to how well that product meets the requirements of the user or customer. In the case of sequential processing operations, it refers to how well the output of one process operation meets the input requirements of the succeeding operation. Statistical Process Control (SPC) is one of technique that can be used to evaluate performance of the process (Bissel, 1994, Mamzic, 1995, and Montgomery, 1996). This technique give main way in sampling removal, testing as well as evaluation, and information inside the data that is used to control and improve the process. However, because of cost consideration, management need to decide which process should apply SPC. In this case, it is very important to get data of process repair rate and process failure rate. Process failure indicates the process departs from the stable state, it quantifies the frequency of mean shift from the target value, while process repair indicate the process that have shifted from the target, back to stable or normal state (Jang, 1999 and Nembhard, 2001). Since the application of SPC is related to out-ofcontrol condition of the process, Nababan et. al. recommended to use Markov Matrix to get the state probability of the out-of-control process (PO) and the state probability of in-control process (PI) for each part in the production line. In this step, they provide simulation by using computer generated random data, to examine such transition; it is applied to production line that consist of two parts (Nababan et.al. 2003). To make a decision on which parts should be controlled using SPC, there should also a need to consider system sensitivity and manufacturing sensitivity (Jang, 1999). Nababan et. all (2004) conducted research to determine optimal allocation of SPC in a production line based on manufacturing sensitivity and fuzzy membership function. Indriawati (2005) had applied that method in process industry. The flexibility of fuzzy set could overcome the ambiguity of making decision and calibrate the vagueness (El-Shal & Morris (1999); Negnevitsky (2001)). In most fuzzy logic, initial MFs are normally laid evenly all across the universe of discourse. However, for evenly distributed MFs, there exists a potential problem that may adversely affect the performance; that is, if the actual input data is not equally distributed, but instead concentrate within a certain interval that is only part of the entire input area, this will result in two negative effects (Zhuang & Wu, 2001), i.e. the MFs staying in the dense-input area will not be sufficient to react precisely to the inputs and some of the MFs assigned for the sparse-input area are wasted. In this paper we used a mechanism to modify the evenly distributed MFs with the help of a technique termed histogram equalization in according to make a decision on SPC allocation by analyzing PO value. By this way, the result decision about SPC allocation can be better. To prove it, the proposed method was applied in the PET continuous process.

2 2 DETERMINING SPC ALLOCATION The methodology to determine the SPC allocation consists of four stages: modelling the production line, analyzing and calculating PI/PO, constructing membership function, and deciding the SPC allocation. 2.1 Modelling the production line of PET Polyethylene terephtalate or PET is a thermoplastic polyester resin. Such resins may be classified as lowviscosity or high-viscosity. Low-viscosity PET typically has an intrinsic viscosity of less than 0.75 are used in a wide variety of product such as apparel fiber, bottles, and photographic film, while high-viscosity PET typically has an intrinsic viscosity of 0.9 or higher are used in tire cord, seat belts, and the like. In the PET continuous process (see figure 1), the production line is assumed consisted of 3 main parts and is modelled as shown in figure 2. Each part represents one dimension of one unit of process production that have a mean work to decide the quality of product. Part 1: first step polycondensation reaction vessel (R200), part 2: second step polycondensation reaction vessel (R210), part 3: third step polycondensation reaction vessel (R220). Since many variables to be measured, the variables which have more dominant error from 3 reactors are determined by using multivariable control chart. According to the result, the determined variables are: TI 720 pv for R200, TC 722 mv for R210, and TC 724 pv for R220. Figure 1. Modelling from 3 parts of production line (Nababan, et.al., 2004) 2.2 Analyzing and calculating PI/PO Because of this research is focused on the application of SPC, repair rate and failure rate become the main observation. These two states will be analyzed by using transition probability and Markov matrix to get the state probability of out-of-control (PO) and in-control process (PI). For this purpose, control chart is used as analyze tool for the observation data. By using transition probability of the process, we can find the probability in next state. Transition probability of failure rate is notated with P f while P r as transition probability of repair rate. Both states are parameters of a geometric distribution 1 1 p f = and pr = (1) x f x r If the system is assumed as the experience of multiplies trials then a transition probability can be defined as a set of transition and can be suggest as Markov process. For the calculation PI and PO from Pf and Pr can be done by using Markov matrix. The formula of the matrix to calculate PI and PO from each part of production line is: Pf Pr PI 0 = (2) 1 1 PO 1 Pr PI P f + Pr (3) = PO Pf Pf + Pr In this calculation, the used data from each unit of models is divided into n groups which each group contain k samples data. Then the next step is finding a condition of out-of-control and in-control process from each group so that the value of Xf and Xr could be determined and soon the value from the average of Xf and Xr from each group also could be determined. These values from the average is later will be used to calculate value of Pf and Pr so that the value of PI and PO can be found. 2.3 Constructing membership function Fuzzy membership function can be applied to classify the condition of out-of-control process by using linguistic value, such as low, medium, and high which is defined on a 0 to 100 percent unit interval of PO probability. By this mean, the part which is optimal to apply SPC can be determined by looking at the linguistic value of each part, beside the numerical value of the PO, just like Nababan and friends have done before (Nababan, et. all, 2004). Since every part of production line has different sensitivity degree to the final product, then every part of a production line has different membership function (MF). By using its MF, the process that is considered HIGH is decided to apply SPC. The MF is constructed from a set of observation data. From the result MF, the exact class of the actual PO can be determined. For this purpose, the observation data is divided into two groups: the historical data and the running data. The historical data is used to construct the MF, while the running data is used to evaluate PO by applying the constructed MF.

3 The steps to build the membership function are: a. Divide the observation data into n groups with each group contain k samples. b. Calculate the PO from each group i (i = 1,2,3,...n) to be arranged in one dimensional matrix, P = [PO1,PO2,PO3,...,POn]. c. Construct the membership function of PO from the set of P with the method of histogram equalization: Construct a histogram with PO as an input Find and plot the running sums of PO values Divide evenly the vertical axis by the number of MFs and obtain the new positions of the MFs Move MFs to new input location. Some proper stretching and shrinking on the resulting membership function are needed to make sure that membership function cover the entire range while not overlapping each other too much. 2.4 Deciding SPC Allocation After obtaining the degree of membership of PO from each part in production line, it is continued by deciding which part should apply SPC based on the high value of PO and the highest degree of membership. By symbolizing PO in part 1, 2, and 3 in a series with x, y, and z, the determination of the SPC allocation is based on the if-then rules as follow: If x HIGH, y HIGH, z HIGH and x > y, and x > z, then apply SPC at part 1 If x HIGH, y HIGH, z HIGH and x > y, and x < z, then apply SPC at part 3 If x HIGH, y HIGH, z HIGH and x < y, and y > z, then apply SPC at part 2 If x HIGH, y HIGH, z HIGH and x < y, and y < z, then apply SPC at part 3 If x HIGH, y HIGH, z MEDIUM and x > y, then apply SPC at part 1 If x HIGH, y HIGH, z MEDIUM and x < y, then apply SPC at part 2 If x HIGH, y MEDIUM, z HIGH and x > z, then apply SPC at part 1 If x HIGH, y MEDIUM, z HIGH and x < z, then apply SPC at part 3 If x MEDIUM, y HIGH, z HIGH and y > z, then apply SPC at part 2 If x MEDIUM, y HIGH, z HIGH and y < z, then apply SPC at part 3 If x HIGH, y MEDIUM, z MEDIUM, then apply SPC at part 1 If x MEDIUM, y HIGH, z MEDIUM, then apply SPC at part 2 If x MEDIUM, y MEDIUM, z HIGH, then apply SPC at part 3 If x HIGH, y MEDIUM, z LOW, then apply SPC at part 1 If x MEDIUM, y HIGH, z LOW, then apply SPC at part 2 If x LOW, y MEDIUM, z HIGH, then apply SPC at part 3 If x HIGH, y HIGH, z LOW and x > y, then apply SPC at part 1 If x HIGH, y HIGH, z LOW and x < y, then apply SPC at part 2 3 RESULT Based on the formula to calculate PO and by using training data (12050 samples) in 2 months (May to July 2006), value of PO is obtained as followed in table 1. To know the distribution of PO value from each part, the histogram curve is made for each as shown in figure 2, 3, and 4. Figure 2. Histogram of PO Value for R200 Figure 3. Histogram of PO Value for R210 Figure 4 Histogram of PO Value for R220

4 TI 720 pv Table 1. Value of PO TC 722 mv TC 724 pv Using histogram equalization method to make MFs, the curve of MF for each observation part is shown in figure 5, 6, and 7. After the MFs was obtained from historical data, then using the rules that have been stated in 2.4 and using running data in final week operation in July, the PO value and membership degree in each category of each part are obtained as shown in table 2. According to the result on table 2, it is shown that the temperature (TI 720 pv) of reactor R200 has PO value , and because PO is located between two category of membership function i.e. medium and high, so that membership degree is calculated by using intersection fuzzy formula: f ( A B)( x) = min( µ A( x), µ B( x) ) (4) The calculation result of PO of reactor R200 is This value is in high category. Figure 5. MF of PO Value for R200 Figure 6. MF of PO Value for R210 Figure 7. MF of PO Value for R220 Table 2. PO value and membership degree Variabel PO low medium high TI 720 pv x TC 722 mv y TC 724 pv z Next, the temperature (TC 722 mv) of reactor R210 has PO value 0.5 and degree of membership value 1 in medium category. Finally, the temperature (TC 724 pv) of reactor R220 has PO value and degree of membership value located in high category. According to the three conditions above and based on if-then rules in section 2.4, so the running data has a rule like this: If PO value in reactor R200 (x) is HIGH, PO value in reactor R210 (y) is MEDIUM, and PO value in reactor R220 (z) is HIGH, and x < z then apply SPC in part R220 (z). So that, the decision of which part to be applied SPC is in reactor R220.

5 4 CONCLUSION The conclusions of this research are: The biggest PO value in the historical data is (from TI 720 pv or reactor R200), but it did not indicate that SPC allocation is applied on that part. By using running data and fuzzy membership function, then the temperature part of R220 is the most appropriate place to apply SPC There are three parts in production line to observe in this research, but not all of plant in industry has three production lines. For next research, it can determine SPC allocation with more than three production lines. REFERENCE Bissel, D. (1994), Statistical Method For Spc And Tqm, Chapman & Hall, London, UK. El-Shal, S. M., Morris, A.S. (1999), A Fuzzy Rule-Based Algorithms to Improve The Performance of SPC in Quality Systems, Proceedings of IEEE International Conference on Systems, Man., and Cybernetics. Tokyo, Japan: Indriawati, K. (2005), Penentuan Alokasi Spc Pada Jalur Produksi Kimia Dengan Menggunakan Fungsi Keanggotaan Fuzzy. Laporan Penelitan, Teknik Fisika, Fakultas Teknologi Industri, ITS. Jang, Y.J. (1999). Mathematical Model for Optimal Allocation Statistical Process Control. Masters Thesis. Department of Mechanical Engineering. Massachusetts Institute of Technology, Cambridge, MA. Mamzic, C.L. (1995), Introduction To Statistical Process Control. dalam STATISTICAL PROCESS CONTROL, Bab 1, Mamzic, C.L, Editor, Instrument Society of America Montgomery, D.C. (1996). Introduction To Statistical Quality Control, 3 rd ed., John Wiley & Sons, New York, NY. Nababan, E.B, Hamdan, A.R, Hasan, M.K. dan Mohamed, H. (2004), FUZZY MEMBERSHIP FUNCTION In DETERMINING SPC ALLOCATION. Nababan, E.B, Hamdan, A.R, Hasan, M.K. & Mohamed, H. (2003), Transition Probability in Allocating SPC, Proceeding on International Simposium on Information Technology (ITSim2003). Kuala Lumpur Negnevitsky, M. (2002), Artificial Intelligent: A Guide to Intelligent Systems, Essex: Addison Wesley. Zhuang, H., Wu, Xiaomin. (2001), Membership Function Modification Of Fuzzy Logic Controllers With Histogram Equalization. Departement of Electrical Engineering, Florida Atlantic University, Boca Raton, USA.