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MODELING AND OPTIMIZATION OF MULTI-RESPONSE SURFACE PROBLEMS WITH FUZZY APPROACH

Year 2012, Volume: 13 Issue: 1, 65 - 79, 24.08.2012

Abstract

The most widely used approach for solving multi response surface problems is response surface methodology. It is thought to be that the response surface methodology is inadequate for evaluation of unexplained vagueness in real world problems. Therefore in the study, fuzzy approach is proposed as an alternative to solve the multi response surface problems. The main aim of this study is to represent the applicability of the fuzzy approach for solving of the multi-response problems in which the probability distributions of the response variables cannot be determined. At the modeling stage, the fuzzy least squares regression analysis, based on Diamond's distance metric, is used. In the optimization stage, the problem is considered as a fuzzy multi-objective optimization problem. Nondominated Sorting Genetic Algorithm-II (NSGA-II), defined in the literature, is adapted by using centroid index fuzzy ranking approach then called Fuzzy NSGA-II (FNSGA-II). Fuzzy Pareto solution set is obtained by optimizing the problem, which is composed of fuzzy objective functions, with FNSGA-II. The proposed fuzzy solution approaches are applied on a data set defined in the literature. Thus, it is seen that an obtained fuzzy Pareto solution is a set of acceptable different response values for the performed multi-response experiments at the defined levels of input variables.

References

  • Abdullah, L. ve Jamal, N.J. (2010). Centroid- Point of Ranking Fuzzy Numbers and Its Application to Health Related Quality of Life Indicators. International Journal on Computer Science and Engineering (IJCSE) 2(8), 2773-2777.
  • Alvarez, M.J., Ilzarbe, L., Viles, E. ve Tanco, M. (2009). The Use of Genetic Algo- rithms in Response Surface Methodology. Quality Technology and Quantitative Management 6(3), 295-307.
  • Bashiri, M., Kazemzadeh, R.B., Atkinson, A.C. ve Karimi, H. (2011). Metaheuristic Based Multiple Response Process Opti- mization. Journal of Industrial Engineer- ing, Special Issue 13-23.
  • Bashiri, M. ve Hosseininezhad, S.J. (2009). A Fuzzy Programming for Optimizing Multi Response Surface in Robust Designs. Journal of Uncertain Systems 3(3), 163- 173.
  • Bashiri, M. ve Ramezani, M. (2010). An interac- tive fuzzy group decision making ap- proach to multiple response problems considering least significant difference. International Journal of Management Science and Engineering Management 5(4), 243-251.
  • Bera, S. ve Mukherjee, I. (2010). Performance Analysis of Nelder Mead and A Hybrid Simulated Annealing for Multiple Re- sponse Quality Characteristic Optimiza- tion. Proceedings of the International Multi Conference of Engineers and Com- puter Scientist, Vol III, IMECS, 1728- 1732, Hong Kong.
  • Box, G.E.P. and Draper, N.R. (2007). Response Surface Mixtures and Ridge Analysis. John Wiley and Sons, New Jersey.
  • Cheng, C.B., Cheng, C.J. ve Lee, E.S. (2002). Neuro-Fuzzy and Genetic Algorithm in Multiple Response Optimization. Com- puters and Mathematics with Applications 44, 1503-1514.
  • Cheng, C.H. (1998). A new approach for rank- ing by distance method. Fuzzy Sets and Systems 95, 307-317.
  • Chu, T.A. ve Tsao, C. (2002). Ranking fuzzy numbers with an area between the cen- troid point and the original point. Com- puters and Mathematics 43, 111-117.
  • Deb, K., Pratap, A., Agarwal, S. ve Meyarivan, T. (2002). Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Transactions on Evolutionary Computa- tion 6(2), 182-197.
  • Deb, K. (2004). Multi-Objective Optimization Using Evolutionary Algorithms. John- Wiley and Sons, New York.
  • Del Castillo, E., Montgomery, D.C. ve McCar- ville, D.R. (1996). Modified Desirability functions for Multiple Response Optimi- zation. Journal of Quality Technology 28(3), 337-345.
  • Derringer, G. ve Suich, R. (1980). Simultaneous optimization of several response varia- bles. Journal of Quality Technology 12, 214-219.
  • Diamond, P. (1988). Fuzzy least squares. Infor- mation Sciences 46, 141-157.
  • Goel, T., Vaidyanathan, R., Haftka, R.T., Shyy, W., Queipo, N.V. ve Tucker, K. (2007). Response surface approximation of Pareto optimal front in multi-objective optimiza- tion. Computer Methods in Applied Me- chanics and Engineering 196, 879-893.
  • Golestaneh, S.J., Ismail, N., Tangı, S.H., Ar- iffinı, M.K.A.M., Naeini, H.M. ve Maghsoudi, A.A. (2011). A committee machine approach to multiple response optimization. International Journal of the Physical Sciences 6(35), 7935-7949.
  • Jeong, I.J, Kim, K. ve Park, K. (2002). An Inter- active Optimization Approach to Multiple Response Surface Problems. The 2002 IIE Annual Conference (Industrial Engineer- ing Research Conference), Orlando, Flor- ida 1-6.
  • Jeong, I.J. ve Kim, K.J. (2009). An interactive desirability function method to multire- sponse optimization. European Journal of Operational Research 195, 412-426.
  • Kazemzadeh, R.B., Bashiri, M., Atkinson, A.C. ve Noorosana, R. (2008). A general frame work for multi response optimization problems based on goal programming. European Journal of Operational Research 189, 421-429.
  • Khuri, A.I. ve Conlon, M. (1981). Simultaneous Optimization of Multiple Responses Rep- resented by Polinomial Regression Func- tions. Technometrics 23, 363-375.
  • Khuri, A.I. ve Cornell, M. (1996). Response Surfaces, Marcel Dekker Inc., New-York.
  • Kim, K.J. ve Lin, D.K.J. (1998). Dual response surface optimization: A Fuzzy modeling approach. Journal of Quality Technology 30(1), 1-10.
  • Ko, Y.H., Kim, K.J. ve Jun, C.H. (2005). A New Loss Function-Based Method for Multire- sponse Optimization. Journal of Quality Technology 37(1), 50-59.
  • Köksoy, O. ve Doğanaksoy, N. (2003). Joint optimization of mean and standard devia- tion using response surface methods. Journal of Quality Technology 35, 239- 252.
  • Köksoy, O. ve Hocaoğlu, G. (2005). Multi Ob- jective Optimization Solutions To The Taguchi’s Problem. G.U. Journal of Sci- ence 18(4), 613-626.
  • Köksoy, O. (2006). Multiresponse robust de- sign: Mean square error (MSE) criterion. Applied Mathematics and Computation 175, 1716–1729.
  • Köksoy, O. ve Yalçınöz, T. (2006). Mean square error criteria to multiresponse process op- timization by a new genetic algorithm. Applied Mathematics and Computation 175, 1657-1674.
  • Kuhnt, S. ve Erdbrügge, M. (2004). A strategy of robust parameter design for multiple responses. Statistical Modelling 4, 249– 264.
  • Lu, D. ve Antony, J. (2002). Optimization of multiple responses using a fuzzy-rule based inference system. International Journal of Production Research 40(7), 1613-1625.
  • Miettinen, K. (2002). Nonlinear Multiobjective Optimization. Kluwer Academic Publish- ers, USA.
  • Mostafa, J.J., Mohammad, M.A. ve Ehsan, M. (2011). A Hybrid Response Surface Methodology and Simulated Annealing Algorithm: A Case Study on the Optimi- zation of Shrinkage and Warpage of a Fuel Filter. World Applied Sciences Jour- nal 13(10), 2156-2163.
  • Murakami, S., Maeda, S. ve Imamura, S. (1983). Fuzzy decision analysis on the develop- ment of centralized regional energy con- trol system. IFAC Symposium on Fuzzy Information Knowledge Representation and Decision Analysis 363–368.
  • Myers, R.H. ve Montgomery, D.C. (2002). Response Surface Methodology: Process and Designed Experiments. 2nd Ed., John Wiley and Sons, New York. Using
  • Najafi, S., Salmasnia, A. ve Kazemzadeh, R.B. (2011). Optimization of Robust Design for Multiple Response Problem. Australi- an Journal of Basic and Applied Sciences 5(9), 1566-1577.
  • Pasandideh, S.H.R. ve Niaki, S.T.A. (2006). Optimizing Multi-Response Statistical Problems Using a Genetic Algorithm. Scientica Iranica 13(1), 50-59.
  • Park, K.S. ve Kim, K.J. (2005). Optimizing mul- ti-response surface problems: How to use multi-objective optimization techniques. IIE Transactions 37, 523-532.
  • Pignatiello, J.J. (1993). Strategies for Robust Multiresponse Quality Engineering. IIE Transactions 25, 5-15.
  • Rees, L.P., Clayton, E.R. ve Taylor, B.W. (1985). Solving multiple response simula- tion models using modified response sur- face methodology within a lexicographic goal programming framework. IIE Trans- actions 17, 47-57.
  • Prasad, K. ve Nath, N. (2002). Comparison of Sugarcane Juice Based Beverage Optimi- sation Using Response Surface Method- ology with Fuzzy Method. Sugar Tech 4(3-4), 109-115.
  • Sharma, V. (2010). Multi Response Optimiza- tion of Process Parameters Based on Taguchi-Fuzzy Model for Coal Cutting by Water Jet Technology. International Journal on Design and Manufacturing Technologies 4(1), 10-14.
  • Srinivas, N. ve Deb, K. (1996). Multiobjective Optimization Using Nondominated Sort- ing in Genetic Algorithms. Evolutionary Computation 2(3), 221–248.
  • Türkşen, Ö. (2011). Çok Yanıtlı Yüzey Prob- lemlerinin Çözümüne Bulanık ve Sezgisel Yaklaşım. Ankara Üniversitesi Doktora Tezi, 139 s., Ankara.
  • Vining, G. (1998). A compromise approach to multiresponse optimization. Journal of Quality Technology 30(4), 309-314.
  • Wang, J., He, Z., Oh, J. ve Park, S. (2008). Mul- ti-Response Robust Optimization Using Desirability Function. IEEE, (2008).
  • Wang, Y.J. ve Lee, H.S. (2008). The revised method of ranking of fuzzy numbers with an area between the centroid point and original points. Computers and Mathe- matics with Applications 55, 1-9.
  • Xie, H. ve Lee, Y.C. (1994). Process Optimiza- tion Using a Fuzzy Logic Response Sur- face Method. IEEE Transactions on Components, Packaging, and Manufac- turing Technology-Part A, 17(2).
  • Xu, K., Lin, D.K.J., Tang, L.C. ve Xie, M. (2004). Multiresponse systems optimiza- tion using a goal attainment approach. IIE Transactions 36, 433-445.
  • Xu, R. ve Dong, Z. (2006). Fuzzy Modeling in Response Surface Method for Complex Computer Model Based Design Optimiza- tion. Mechatronic and Embedded Systems and Applications Proceedings of the 2nd IEEE/ASME International Conference on 1-6.
  • Yager, R.R. (1980). A procedure for ordering fuzzy subsets of the unit interval. Infor- mation Sciences 24, 143-161.
  • Zadeh, L.A. (1965). Fuzzy Sets. Information and Control 8, 338-353.

BULANIK YAKLAŞIM İLE ÇOK YANITLI YÜZEY PROBLEMLERİNİN MODELLENMESİ VE OPTİMİZASYONU

Year 2012, Volume: 13 Issue: 1, 65 - 79, 24.08.2012

Abstract

Çok yanıtlı yüzey problemlerinin çözümünde en sık kullanılan yaklaşım yanıt yüzey yöntemidir. Gerçek dünya problemlerinde, açıklanamayan, belirsizlik durumlarının varlığı söz konusu olduğunda yanıt yüzey yönteminin yetersiz olduğu düşünülmektedir. Bu nedenle çalışmada, çok yanıtlı bir problemin çözümü için alternatif olarak bulanık yaklaşımın kullanılması önerilmiştir. Bu çalışmanın asıl amacı, yanıt değişkenlerinin olasılık dağılımlarının belirlenemediği durumlarda, çok yanıtlı problemlerin çözümünde bulanık yaklaşımın uygulanabilirliğinin göstermektir. Modelleme aşamasında, Diamond’ın uzaklık metriğine dayalı bulanık en küçük kareler regresyon analizi kullanılmıştır. Optimizasyon aşamasında ise problem, bulanık çok amaçlı optimizasyon problemi biçiminde ele alınmıştır. Literatürde tanımlı Baskın Sıralı Genetik Algoritma-II (BSGA-II) yöntemi, ağırlık merkezi indeksine dayalı bulanık sıralama yaklaşımı ile uyarlanarak, Bulanık BSGA-II (BBSGA-II) olarak adlandırılmıştır. Bulanık yanıtlardan oluşan problemin BBSGA-II ile optimizasyonu sonucu bulanık Pareto kümesine ulaşılmıştır. Önerilen bulanık çözümleme yaklaşımları, literatürde tanımlı çok yanıtlı bir veri setine uygulanmıştır. Böylece, elde edilen bir bulanık Pareto çözümün, belirlenen girdi değişken düzeylerinde yapılan çok yanıtlı deneyler için kabul edilebilir farklı yanıt değerlerinin bir kümesi olduğu görülmüştür.

References

  • Abdullah, L. ve Jamal, N.J. (2010). Centroid- Point of Ranking Fuzzy Numbers and Its Application to Health Related Quality of Life Indicators. International Journal on Computer Science and Engineering (IJCSE) 2(8), 2773-2777.
  • Alvarez, M.J., Ilzarbe, L., Viles, E. ve Tanco, M. (2009). The Use of Genetic Algo- rithms in Response Surface Methodology. Quality Technology and Quantitative Management 6(3), 295-307.
  • Bashiri, M., Kazemzadeh, R.B., Atkinson, A.C. ve Karimi, H. (2011). Metaheuristic Based Multiple Response Process Opti- mization. Journal of Industrial Engineer- ing, Special Issue 13-23.
  • Bashiri, M. ve Hosseininezhad, S.J. (2009). A Fuzzy Programming for Optimizing Multi Response Surface in Robust Designs. Journal of Uncertain Systems 3(3), 163- 173.
  • Bashiri, M. ve Ramezani, M. (2010). An interac- tive fuzzy group decision making ap- proach to multiple response problems considering least significant difference. International Journal of Management Science and Engineering Management 5(4), 243-251.
  • Bera, S. ve Mukherjee, I. (2010). Performance Analysis of Nelder Mead and A Hybrid Simulated Annealing for Multiple Re- sponse Quality Characteristic Optimiza- tion. Proceedings of the International Multi Conference of Engineers and Com- puter Scientist, Vol III, IMECS, 1728- 1732, Hong Kong.
  • Box, G.E.P. and Draper, N.R. (2007). Response Surface Mixtures and Ridge Analysis. John Wiley and Sons, New Jersey.
  • Cheng, C.B., Cheng, C.J. ve Lee, E.S. (2002). Neuro-Fuzzy and Genetic Algorithm in Multiple Response Optimization. Com- puters and Mathematics with Applications 44, 1503-1514.
  • Cheng, C.H. (1998). A new approach for rank- ing by distance method. Fuzzy Sets and Systems 95, 307-317.
  • Chu, T.A. ve Tsao, C. (2002). Ranking fuzzy numbers with an area between the cen- troid point and the original point. Com- puters and Mathematics 43, 111-117.
  • Deb, K., Pratap, A., Agarwal, S. ve Meyarivan, T. (2002). Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Transactions on Evolutionary Computa- tion 6(2), 182-197.
  • Deb, K. (2004). Multi-Objective Optimization Using Evolutionary Algorithms. John- Wiley and Sons, New York.
  • Del Castillo, E., Montgomery, D.C. ve McCar- ville, D.R. (1996). Modified Desirability functions for Multiple Response Optimi- zation. Journal of Quality Technology 28(3), 337-345.
  • Derringer, G. ve Suich, R. (1980). Simultaneous optimization of several response varia- bles. Journal of Quality Technology 12, 214-219.
  • Diamond, P. (1988). Fuzzy least squares. Infor- mation Sciences 46, 141-157.
  • Goel, T., Vaidyanathan, R., Haftka, R.T., Shyy, W., Queipo, N.V. ve Tucker, K. (2007). Response surface approximation of Pareto optimal front in multi-objective optimiza- tion. Computer Methods in Applied Me- chanics and Engineering 196, 879-893.
  • Golestaneh, S.J., Ismail, N., Tangı, S.H., Ar- iffinı, M.K.A.M., Naeini, H.M. ve Maghsoudi, A.A. (2011). A committee machine approach to multiple response optimization. International Journal of the Physical Sciences 6(35), 7935-7949.
  • Jeong, I.J, Kim, K. ve Park, K. (2002). An Inter- active Optimization Approach to Multiple Response Surface Problems. The 2002 IIE Annual Conference (Industrial Engineer- ing Research Conference), Orlando, Flor- ida 1-6.
  • Jeong, I.J. ve Kim, K.J. (2009). An interactive desirability function method to multire- sponse optimization. European Journal of Operational Research 195, 412-426.
  • Kazemzadeh, R.B., Bashiri, M., Atkinson, A.C. ve Noorosana, R. (2008). A general frame work for multi response optimization problems based on goal programming. European Journal of Operational Research 189, 421-429.
  • Khuri, A.I. ve Conlon, M. (1981). Simultaneous Optimization of Multiple Responses Rep- resented by Polinomial Regression Func- tions. Technometrics 23, 363-375.
  • Khuri, A.I. ve Cornell, M. (1996). Response Surfaces, Marcel Dekker Inc., New-York.
  • Kim, K.J. ve Lin, D.K.J. (1998). Dual response surface optimization: A Fuzzy modeling approach. Journal of Quality Technology 30(1), 1-10.
  • Ko, Y.H., Kim, K.J. ve Jun, C.H. (2005). A New Loss Function-Based Method for Multire- sponse Optimization. Journal of Quality Technology 37(1), 50-59.
  • Köksoy, O. ve Doğanaksoy, N. (2003). Joint optimization of mean and standard devia- tion using response surface methods. Journal of Quality Technology 35, 239- 252.
  • Köksoy, O. ve Hocaoğlu, G. (2005). Multi Ob- jective Optimization Solutions To The Taguchi’s Problem. G.U. Journal of Sci- ence 18(4), 613-626.
  • Köksoy, O. (2006). Multiresponse robust de- sign: Mean square error (MSE) criterion. Applied Mathematics and Computation 175, 1716–1729.
  • Köksoy, O. ve Yalçınöz, T. (2006). Mean square error criteria to multiresponse process op- timization by a new genetic algorithm. Applied Mathematics and Computation 175, 1657-1674.
  • Kuhnt, S. ve Erdbrügge, M. (2004). A strategy of robust parameter design for multiple responses. Statistical Modelling 4, 249– 264.
  • Lu, D. ve Antony, J. (2002). Optimization of multiple responses using a fuzzy-rule based inference system. International Journal of Production Research 40(7), 1613-1625.
  • Miettinen, K. (2002). Nonlinear Multiobjective Optimization. Kluwer Academic Publish- ers, USA.
  • Mostafa, J.J., Mohammad, M.A. ve Ehsan, M. (2011). A Hybrid Response Surface Methodology and Simulated Annealing Algorithm: A Case Study on the Optimi- zation of Shrinkage and Warpage of a Fuel Filter. World Applied Sciences Jour- nal 13(10), 2156-2163.
  • Murakami, S., Maeda, S. ve Imamura, S. (1983). Fuzzy decision analysis on the develop- ment of centralized regional energy con- trol system. IFAC Symposium on Fuzzy Information Knowledge Representation and Decision Analysis 363–368.
  • Myers, R.H. ve Montgomery, D.C. (2002). Response Surface Methodology: Process and Designed Experiments. 2nd Ed., John Wiley and Sons, New York. Using
  • Najafi, S., Salmasnia, A. ve Kazemzadeh, R.B. (2011). Optimization of Robust Design for Multiple Response Problem. Australi- an Journal of Basic and Applied Sciences 5(9), 1566-1577.
  • Pasandideh, S.H.R. ve Niaki, S.T.A. (2006). Optimizing Multi-Response Statistical Problems Using a Genetic Algorithm. Scientica Iranica 13(1), 50-59.
  • Park, K.S. ve Kim, K.J. (2005). Optimizing mul- ti-response surface problems: How to use multi-objective optimization techniques. IIE Transactions 37, 523-532.
  • Pignatiello, J.J. (1993). Strategies for Robust Multiresponse Quality Engineering. IIE Transactions 25, 5-15.
  • Rees, L.P., Clayton, E.R. ve Taylor, B.W. (1985). Solving multiple response simula- tion models using modified response sur- face methodology within a lexicographic goal programming framework. IIE Trans- actions 17, 47-57.
  • Prasad, K. ve Nath, N. (2002). Comparison of Sugarcane Juice Based Beverage Optimi- sation Using Response Surface Method- ology with Fuzzy Method. Sugar Tech 4(3-4), 109-115.
  • Sharma, V. (2010). Multi Response Optimiza- tion of Process Parameters Based on Taguchi-Fuzzy Model for Coal Cutting by Water Jet Technology. International Journal on Design and Manufacturing Technologies 4(1), 10-14.
  • Srinivas, N. ve Deb, K. (1996). Multiobjective Optimization Using Nondominated Sort- ing in Genetic Algorithms. Evolutionary Computation 2(3), 221–248.
  • Türkşen, Ö. (2011). Çok Yanıtlı Yüzey Prob- lemlerinin Çözümüne Bulanık ve Sezgisel Yaklaşım. Ankara Üniversitesi Doktora Tezi, 139 s., Ankara.
  • Vining, G. (1998). A compromise approach to multiresponse optimization. Journal of Quality Technology 30(4), 309-314.
  • Wang, J., He, Z., Oh, J. ve Park, S. (2008). Mul- ti-Response Robust Optimization Using Desirability Function. IEEE, (2008).
  • Wang, Y.J. ve Lee, H.S. (2008). The revised method of ranking of fuzzy numbers with an area between the centroid point and original points. Computers and Mathe- matics with Applications 55, 1-9.
  • Xie, H. ve Lee, Y.C. (1994). Process Optimiza- tion Using a Fuzzy Logic Response Sur- face Method. IEEE Transactions on Components, Packaging, and Manufac- turing Technology-Part A, 17(2).
  • Xu, K., Lin, D.K.J., Tang, L.C. ve Xie, M. (2004). Multiresponse systems optimiza- tion using a goal attainment approach. IIE Transactions 36, 433-445.
  • Xu, R. ve Dong, Z. (2006). Fuzzy Modeling in Response Surface Method for Complex Computer Model Based Design Optimiza- tion. Mechatronic and Embedded Systems and Applications Proceedings of the 2nd IEEE/ASME International Conference on 1-6.
  • Yager, R.R. (1980). A procedure for ordering fuzzy subsets of the unit interval. Infor- mation Sciences 24, 143-161.
  • Zadeh, L.A. (1965). Fuzzy Sets. Information and Control 8, 338-353.
There are 51 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Özlem Türkşen

Ayşen Apaydın

Publication Date August 24, 2012
Published in Issue Year 2012 Volume: 13 Issue: 1

Cite

APA Türkşen, Ö., & Apaydın, A. (2012). MODELING AND OPTIMIZATION OF MULTI-RESPONSE SURFACE PROBLEMS WITH FUZZY APPROACH. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering, 13(1), 65-79.
AMA Türkşen Ö, Apaydın A. MODELING AND OPTIMIZATION OF MULTI-RESPONSE SURFACE PROBLEMS WITH FUZZY APPROACH. AUJST-A. August 2012;13(1):65-79.
Chicago Türkşen, Özlem, and Ayşen Apaydın. “MODELING AND OPTIMIZATION OF MULTI-RESPONSE SURFACE PROBLEMS WITH FUZZY APPROACH”. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering 13, no. 1 (August 2012): 65-79.
EndNote Türkşen Ö, Apaydın A (August 1, 2012) MODELING AND OPTIMIZATION OF MULTI-RESPONSE SURFACE PROBLEMS WITH FUZZY APPROACH. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering 13 1 65–79.
IEEE Ö. Türkşen and A. Apaydın, “MODELING AND OPTIMIZATION OF MULTI-RESPONSE SURFACE PROBLEMS WITH FUZZY APPROACH”, AUJST-A, vol. 13, no. 1, pp. 65–79, 2012.
ISNAD Türkşen, Özlem - Apaydın, Ayşen. “MODELING AND OPTIMIZATION OF MULTI-RESPONSE SURFACE PROBLEMS WITH FUZZY APPROACH”. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering 13/1 (August 2012), 65-79.
JAMA Türkşen Ö, Apaydın A. MODELING AND OPTIMIZATION OF MULTI-RESPONSE SURFACE PROBLEMS WITH FUZZY APPROACH. AUJST-A. 2012;13:65–79.
MLA Türkşen, Özlem and Ayşen Apaydın. “MODELING AND OPTIMIZATION OF MULTI-RESPONSE SURFACE PROBLEMS WITH FUZZY APPROACH”. Anadolu University Journal of Science and Technology A - Applied Sciences and Engineering, vol. 13, no. 1, 2012, pp. 65-79.
Vancouver Türkşen Ö, Apaydın A. MODELING AND OPTIMIZATION OF MULTI-RESPONSE SURFACE PROBLEMS WITH FUZZY APPROACH. AUJST-A. 2012;13(1):65-79.