An Application of Infinite Programming on Desirability Functions

Year 2023, Volume: 6 Issue: 3, 97 - 104, 21.12.2023

Başak Öztürk

https://doi.org/10.33187/jmsm.1194487

Abstract

When assessing the quality of a system or product, it is necessary to take all responses into account and optimize them in a concurrent manner to find the factor levels that satisfy the overall system, process, or product properties to solve the robust design problem. This problem can be solved as a multi-response optimization problem. There are many methods suggested to solve this problem based on different disciplines like multi-objective optimization. In this study, we improve the theory of nondifferentiable desirability functions' optimization for which the so-called gradient-based methods are not useful. In this study, we propose an infinite programming approach for nondifferentiable desirability functions including more than one nondifferentiable point. We employed DNLP model of GAMS/BARON which is a nondifferentiable solver, however, the solution of more than one nondifferentiable point problem is resulted as infeasible. We also tested MATLAB/NOMAD which is a derivative-free solver for MINLP problems, however, MATLAB/NOMAD also did not succeed and could not solve this nondifferentiable problem. Lastly, we use a genetic algorithm that is implemented under MATLAB and it also cannot find a feasible solution. We use an example that is solved by different desirability functions approaches before and show that the desirability functions approach with more than one nondifferentiable point is a good alternative to the ones in the literature. We present the conclusion and future studies at the end of the paper.

Keywords

Desirability functions , MATLAB , Multi-response optimization

References

• [1] G. Taguchi, Introduction to Quality Engineering: Designing Quality into Products and Processes, Kraus, White Plains, NY, 1986.
• [2] G. Taguchi, System of Experimental Design: Engineering Methods to Optimize Quality and Minimize Cost, UNIPUB/Kraus International, White Plains, NY, 1987.
• [3] A. I. Khuri, Multiresponse surface methodology, Handbook of Statist., 13 (1996), 377–406.
• [4] H. H. Chang, Dynamic multi–response experiments by backpropagation networks and desirability functions, J. Chin. Inst. Eng., 23(4) (2010), 280-288.
• [5] I. J. Jeong, K. J. Kim, An interactive desirability function method to multiresponse optimization, European J. Oper. Res., 195(2) (2008), 412-426.
• [6] I. J. Jeong, K.J. Kim, D-STEM: a modified step method with desirability function concept, Comput. Oper. Res., 32 (2005), 3175-3190.
• [7] D.H., Lee, K.J. Kim, M. K¨oksalan, A posterior preference articulation approach to multiresponse surface optimization , European J. Oper. Res., 210 (2011), 301-309.
• [8] D.H. Lee, K.J. Kim, M. Köksalan, An interactive method to multiresponse surface optimization based on pairwise comparisons, IIE Transactions, 44(1) (2012), 13-26.
• [9] B. Akteke Öztürk, G. Köksal, G. W. Weber, Nonconvex optimization desirability functions, Qual. Eng., 30(2) (2017), 293-310.
• [10] B. Akteke Öztürk, G. W. Weber, G. Köksal, Optimization of generalized desirability functions under model uncertainty, Optimization, 66(12) (2017), 2157-2169.
• [11] R. T. Marler, J. S. Arora, Survey of multi-objective optimization methods for engineering, Struct. Multidiscip. Optim., 26(6) (2004), 369-395.
• [12] K. S. Park, K. J. Kim, Optimizing multi-response surface problems: how to use multi-objective optimization techniques, IIE Transactions, 37(6) (2005), 523-532.
• [13] E. C. Jr. Harrington, The desirability function, Ind. Qual. Control, 21 (1965), 494-498.
• [14] G. Derringer, R. Suich, Simultaneous optimization of several responsevariables, J. Qual. Technol., 12 (1980), 214-219.
• [15] G. Derringer, A balancing act, optimizing a products properties, Qual. Prog., 27 (1994), 51-57.
• [16] E. Del Castillo, D. C. Montgomery, D. R. Mc Carville, Modified desirability functions for multiple response optimization, J. Qual. Technol., 28(3) (1996), 337-345.
• [17] B. Akteke Öztürk, G. W. Weber, G. Köksal, Generalized desirability functions: a structural and topological analysis of desirability functions, Optimization, 69(1) (2019), 115-130.
• [18] B. Akteke Öztürk, G. W. Weber, G. Köksal, Desirability functions in multiresponse optimization, Communications in computer and information science, Springer, Cham, (2015), 129–146.
• [19] BARON, https://www.gams.com/latest/docs/S_BARON.html, v. 8.1.5, 2010.
• [20] GAMS, https://www.gams.com/, v. 23.8.2, 2012.
• [21] P. Belotti, C. Kirches, S. Leyffer, J. Linderoth, J. Luedtke, A. Mahajan, Mixed-integer nonlinear optimization. Acta Numer., 22 (2013), 1-131.
• [22] S. L. Digabel, Algorithm 909: NOMAD: Nonlinear optimization with the MADS algorithm, ACM Trans. Math. Software, 37(4) (2011), 1-15.
• [23] S. Sivanandam, S. Deepa, Genetic Algorithm Implementation Using Matlab. In: Introduction to Genetic Algorithms. Springer, Berlin, Heidelberg, 2008.
• [24] R. Porn, K.M. Bjork, T. Westerlund, Global solution of optimization problems with signomial parts, Discrete Optim., 5 (2008), 108-120.
• [25] K. Kim, D. Lin, Simultaneous optimization of multiple responses by maximizing exponential desirability functions, Appl. Stat., 49(C) (2000), 311-325.