An adaptive penalty function-based maximin desirability multiple-response optimization problems

ORIGINAL ARTICLE

An adaptive penalty function-based maximin desirability index for close tolerance multiple-response optimization problems

Sasadhar Bera &Indrajit Mukherjee

Received:30March 2011/Accepted:13October 2011/Published online:1November 2011#Springer-Verlag London Limited 2011

Abstract Simultaneous optimization of multiple-quality characteristics and determining the process settings is a critical and difficult task for practitioners.Such types of problems are generally referred to as “multiple-response optimization ”problems.To handle high-dimensional multiple-response problems,a popular strategy,using desir-ability functions,is recommended by various researchers.Various types of desirability index functions are recommen-ded to convert multiple scale-free desirability measures to a single composite desirability (or single objective)value.Thus,the objective is then to maximize the single composite desirability for a specific problem.In this paper,a new adaptive penalty function-based “maximin ”desirability index is proposed,which provide superior solution as compared to existing maximin approach,for close (or tight)engineering tolerances of response characteristics.The superiority was proved based on statistical comparison using varied case situations and different swarm intelligent search strategies.Keywords Multiple response .Desirability .Adaptive penalty function .Swarm intelligence

1Introduction

A common interest in product or process quality improve-ment involves determining optimal setting condition for

independent input variables so as to attain the best trade-off solution for dependent bining individual optimal or isolated optimal conditions of each specific response rarely results in overall optima for multiple responses.Interaction between the independent variables and correlation between responses always call for a trade-off solution in real-life situations.Simultaneous optimiza-tion of multiple responses is generally referred to as “multiple-response optimization (MRO)problem ”.The various approaches proposed by researchers to solve MRO are discussed below.

Harington [1]first proposed the concept of desirability function approach to handle MRO problems.Desirability function approach converts a MRO problem into a single-objective optimization problem using suitable desirability transformation ter on,more generalized approaches on desirability function were suggested by Derringer and Suich [2],Catillo et al.[3],and Kim and Lin [4].A major advantage of the desirability function approach is that it allows adjusting the individual desirabil-ity function shape or the relative weights of the responses.Moreover,the desirability values are scale-free and have the same magnitude.The major limitation of desirability approach is that it does not incorporate the dispersion effects of the response variables.A detailed discussion on desirability approach is given in Section 2.

Khuri and Conlon [5]proposed a generalized distance-based optimization scheme for MRO problems.Polynomial regression model(s)are used to build the input –output relationship.The major limitation of this approach is that input –output relationship is based on the same set of input variables and the same order of polynomial regression model.This limitation does not permit it to associate separate design matrix for each response variable.Thus,there is a possibility of model over fitting,due to common

S.Bera :I.Mukherjee (*)

Shailesh J.Mehta School of Management,Indian Institute of Technology,Bombay 400076,India

e-mail:indrajit2005@ S.Bera

e-mail:sasadhar.bera@iitb.ac.in

Int J Adv Manuf Technol (2012)61:379–390DOI 10.1007/s00170-011-3704-9