Simultaneous Optimization of Incomplete Multi-Response Experiments
ABSTRACT
This article attempts to develop a simultaneous optimization procedure of several response variables from incomplete multi-response experiments. In incomplete multi-response experiments all the responses (p) are not recorded from all the experimental units (n). Two situations of multi-response experiments considered are (i) on
units all the responses are recorded while on 
units a subset of 
responses is recorded and (ii) on 
units all the responses (p) are recorded, on 
units a subset of 
responses is recorded and on 
units the remaining subset of 
responses is recorded. The procedure of estimation of parameters from linear multi-response models for incomplete multi-response experiments has been developed for both the situations. It has been shown that the parameter estimates are consistent and asymptotically unbiased. Using these parameter estimates, simultaneous optimization of incomplete multi-response experiments is attempted following the generalized distance criterion [1]. For the implementation of these procedures, SAS codes have been developed for both complete (k ≤ 5, p = 5) and incomplete (k ≤ 5, p1 = 2, 3 and p2 = 2, 3, where k is the number of factors) multi-response experiments. The procedure developed is illustrated with the help of a real data set.
This article attempts to develop a simultaneous optimization procedure of several response variables from incomplete multi-response experiments. In incomplete multi-response experiments all the responses (p) are not recorded from all the experimental units (n). Two situations of multi-response experiments considered are (i) on
units all the responses are recorded while on units a subset of responses is recorded and (ii) on units all the responses (p) are recorded, on units a subset of responses is recorded and on units the remaining subset of responses is recorded. The procedure of estimation of parameters from linear multi-response models for incomplete multi-response experiments has been developed for both the situations. It has been shown that the parameter estimates are consistent and asymptotically unbiased. Using these parameter estimates, simultaneous optimization of incomplete multi-response experiments is attempted following the generalized distance criterion [1]. For the implementation of these procedures, SAS codes have been developed for both complete (k ≤ 5, p = 5) and incomplete (k ≤ 5, p1 = 2, 3 and p2 = 2, 3, where k is the number of factors) multi-response experiments. The procedure developed is illustrated with the help of a real data set.
KEYWORDS
Incomplete Multi-Response Experiments, Simultaneous Optimization, Response Surface Designs, Ridge Analysis
Incomplete Multi-Response Experiments, Simultaneous Optimization, Response Surface Designs, Ridge Analysis
Cite this paper
Nandi, P. , Parsad, R. and Gupta, V. (2015) Simultaneous Optimization of Incomplete Multi-Response Experiments. Open Journal of Statistics, 5, 430-444. doi: 10.4236/ojs.2015.55045.
Nandi, P. , Parsad, R. and Gupta, V. (2015) Simultaneous Optimization of Incomplete Multi-Response Experiments. Open Journal of Statistics, 5, 430-444. doi: 10.4236/ojs.2015.55045.
References
[1] Khuri, A.I. and Conlon, M. (1981) Simultaneous Optimization of Multiple Responses Represented by Polynomial Regression Functions. Technometrics, 23, 363-375.
http://dx.doi.org/10.1080/00401706.1981.10487681 [2] Khuri, A.I. and Cornell, J.A. (1996) Response Surfaces: Designs and Analyses. 2nd Edition, Marcel Dekker, New York. [3] Harrington, E.C. (1965) The Desirability Function. Industrial Quality Control, 21, 494-498. [4] Myers, R.H. and Carter Jr., W.H. (1973) Response Surface Techniques for Dual Response Systems. Technometrics, 15, 301-317.
http://dx.doi.org/10.1080/00401706.1973.10489044 [5] Biles, W.E. (1975) A Response Surface Method for Experimental Optimization of Multi-Response Processes. Industrial and Engineering Chemistry Process Design and Development, 14, 152-158.
http://dx.doi.org/10.1021/i260054a010 [6] Derringer, G. and Suich, R. (1980) Simultaneous Optimization of Several Response Variables. Journal of Quality Technology, 12, 214-219. [7] Conlon, M. and Khuri, A.I. (1990) MR: Multiple Response Optimization. Report No. 322, Department of Statistics, University of Florida, Gainesville, FL. [8] Kar, A., Chandra, P., Parsad, R., Samuel, D.V.K. and Khurdiya, D.S. (2001) Osmotic Dehydration of Banana (Dwarf Cavendish) Slices. Indian Journal of Agricultural Engineering, 38, 12-21. [9] Schimdt, P. (1977) Estimation of Seemingly Unrelated Regressions with Unequal Number of Observations. Journal of Econometrics, 5, 365-377.
http://dx.doi.org/10.1016/0304-4076(77)90045-8 [10] Kakwani, N.C. (1967) The Unbiasedness of Zellner’s Seemingly Unrelated Regression Equations Estimator. Journal of the American Statistical Association, 62, 141-142.
http://dx.doi.org/10.1080/01621459.1967.10482895 [11] Sharma, V.K. (1993) Estimation of Seemingly Unrelated Regressions with Unequal Numbers of Observation. Sankhya, B 55, 135-138. [12] Price, W.L. (1977) A Controlled Random Search Procedure for Global Optimization. Computer Journal, 20, 367-370.
http://dx.doi.org/10.1093/comjnl/20.4.367 [13] SAS Institute Inc. (1989) SAS/STAT User’s Guide. Vol. 2, Version 6, 4th Edition, Cary, NC.
[1] Khuri, A.I. and Conlon, M. (1981) Simultaneous Optimization of Multiple Responses Represented by Polynomial Regression Functions. Technometrics, 23, 363-375.
http://dx.doi.org/10.1080/00401706.1981.10487681 [2] Khuri, A.I. and Cornell, J.A. (1996) Response Surfaces: Designs and Analyses. 2nd Edition, Marcel Dekker, New York. [3] Harrington, E.C. (1965) The Desirability Function. Industrial Quality Control, 21, 494-498. [4] Myers, R.H. and Carter Jr., W.H. (1973) Response Surface Techniques for Dual Response Systems. Technometrics, 15, 301-317.
http://dx.doi.org/10.1080/00401706.1973.10489044 [5] Biles, W.E. (1975) A Response Surface Method for Experimental Optimization of Multi-Response Processes. Industrial and Engineering Chemistry Process Design and Development, 14, 152-158.
http://dx.doi.org/10.1021/i260054a010 [6] Derringer, G. and Suich, R. (1980) Simultaneous Optimization of Several Response Variables. Journal of Quality Technology, 12, 214-219. [7] Conlon, M. and Khuri, A.I. (1990) MR: Multiple Response Optimization. Report No. 322, Department of Statistics, University of Florida, Gainesville, FL. [8] Kar, A., Chandra, P., Parsad, R., Samuel, D.V.K. and Khurdiya, D.S. (2001) Osmotic Dehydration of Banana (Dwarf Cavendish) Slices. Indian Journal of Agricultural Engineering, 38, 12-21. [9] Schimdt, P. (1977) Estimation of Seemingly Unrelated Regressions with Unequal Number of Observations. Journal of Econometrics, 5, 365-377.
http://dx.doi.org/10.1016/0304-4076(77)90045-8 [10] Kakwani, N.C. (1967) The Unbiasedness of Zellner’s Seemingly Unrelated Regression Equations Estimator. Journal of the American Statistical Association, 62, 141-142.
http://dx.doi.org/10.1080/01621459.1967.10482895 [11] Sharma, V.K. (1993) Estimation of Seemingly Unrelated Regressions with Unequal Numbers of Observation. Sankhya, B 55, 135-138. [12] Price, W.L. (1977) A Controlled Random Search Procedure for Global Optimization. Computer Journal, 20, 367-370.
http://dx.doi.org/10.1093/comjnl/20.4.367 [13] SAS Institute Inc. (1989) SAS/STAT User’s Guide. Vol. 2, Version 6, 4th Edition, Cary, NC.