Robust multi‐response surface optimization: a posterior preference approach

• Author: Seyed Taghi Akhavan Niaki,Mahdi Bashiri,Amir Moslemi
• Date: 01 May 2020
• DOI: http://doi.org/10.1111/itor.12450
• Published date: 01 May 2020

Intl. Trans. in Op. Res. 27 (2020) 1751–1770

DOI: 10.1111/itor.12450

INTERNATIONAL

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IN OPERATIONAL

RESEARCH

Robust multi-response surface optimization: a posterior

preference approach

Mahdi Bashiria, Amir Moslemiband Seyed Taghi Akhavan Niakic

aDepartment of Industrial Engineering, Faculty of Engineering, Shahed University, Tehran,Iran

bDepartment of Industrial Engineering, West Tehran Branch,Islamic Azad University, Tehran, Iran

cDepartment of Industrial Engineering, Sharif University of Technology, Tehran, Iran

E-mail: bashiri@shahed.ac.ir [Bashiri]; moslemi.amir@wtiau.ac.ir [Moslemi]; niaki@sharif.edu[Akhavan Niaki]

Received 18 May2016; received in revised form 1 April 2017; accepted 24 July 2017

Abstract

This paper discusses the use of multi-response surface optimization (MRSO) to select the preferredsolutions

from among various non-dominated solutions (NDS). Since MSRO often involves conflicting responses,

the decision-maker’s (DM) preference information should be included in the model in order to choose the

preferred solutions.In some approaches this information is added to the model after the problem is solved. In

contrast, this paper proposesa three-stage method for solving the problem. In the first stage,a robust approach

is used to construct a regression model. In the second phase, non-dominated solutions are generated by the

ε-constraint approach. The robust solutions obtained in the third phase are NDS that are more likely to

be Pareto solutions during consecutive iterations. A simulation study is then presented to show the effective

performance of the proposed approach. Finally, a numerical example from the literature is brought in to

demonstrate the efficiency and applicability of the proposed methodology.

Keywords:robust; multi-response; posterior; optimization

1. Introduction

The response surface methodology involves relationships between differentvariables, such as exper-

imental inputs as controllable factors, and one or more response variables that are uncontrollable,

called nuisances (Khuri, 1996). The main purpose of this approach is to find the optimum setting

of controllable factors, using design and analysis of experiments that will maximize or minimize

the responses based on their types. A regression model is applied, considering these factors and

responses variables, to illustrate the relationships. Both single responses and multiple responses are

common in this approach; to some extent, multiple response surfaces are more suitable for real-

world cases, where different responses are considered simultaneously. When several responses are

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2017 The Authors.

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2017 International Federation ofOperational Research Societies

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1752 M. Bashiri et al. / Intl. Trans. in Op.Res. 27 (2020) 1751–1770

to be analyzed simultaneously, a multi-response optimization problem arises, whose main goal is

to find the settings of the input variables that achieve an optimal compromise among the response

variables. In this paper, an optimization method is proposed to draw conclusions about the input

variables as wellas the responses in order to determine the optimum level of each variable. However,

as conflicting responses are involved in many cases, a Pareto solution is first extracted from a set

of feasible solutions. Then, a decision maker can select the preferred solution based on his/her

preferences. In this case, the decision-maker’s (DM) preference information is very important to

make a reliable decision.

In most multi-response surface optimization (MRSO) methods, the DM’s preferences are incor-

porated using three approaches. The first is the very common approach of prior preference, where

it is very difficult to incorporate the DM’s preference information precisely, and in many cases,

it cannot be determined easily. The second is the progressive approach, which incorporates the

DM’s preference during the solution procedure. The thirdone is the posterior preference approach,

in which a set of non-dominated solutions (NDS) is presented and the preferred one is selected

according to the DM’s preference.

Estimating regression models using design and analysis of experiments is the first step in existing

approaches. In order to obtain a reliable solution that does not have inordinate effects on the

conclusion, these models should not be sensitive to outliers and trends obtained by experiments

(Maronna et al., 2006). In other words, models should be as robust as possible to outliers. After

designing and performing experiments and finding a robust model, the next step is generally a

statistical approach to finally select the controllable factor levels. As the optimization stage is based

on the models obtained using regression, estimating a robust regression model is very important

that may affect the optimization steps.

A review of the literature reveals that while there are a few papers on the use of posterior

approaches in multi-response surface optimization problems, a combination of robust regression

and robust response surface methods is new. In other words, this study is the first to use a posterior

approach in which the model construction procedure is modified iteratively by robust weighting

functions to estimate the regression coefficients with sufficient robustness. Then, a set of robust

Pareto (non-dominated) solutions (NDS) is obtained by the ε-constraint method in each iteration.

Finally, among these NDS, a novel robust preferred solution is presented that is more likely to be

the NDS during consecutive iterations. A novel approach involving two steps, model construction

and robust solution selection, is proposed.

Often, it is assumed in linear regression models that residuals are normally distributed. How-

ever, sometimes practical applications violate this assumption. The presence of outliers and non-

normality of error terms can have a large misleading influence on the performance of classical

statistical methods, which are ideal for non-contaminated data under the assumption of normality.

A robust approach to statistical modeling and data analysis aims to derive methods that produce

reliable parameter estimation in cases where the data follow a given non-normal distribution and

there are outliers.

In fact, since the ordinary least squares (OLS) method is very sensitive to outliers, reliable

and precise response surface estimates cannot be obtained in the presence of contaminated data.

Hence, subsequent calculations in the optimization phase depend on the accuracy of estimated

response functions and can be affected by estimation errors that occur in that phase. Moreover,

epsilon constraints and robust selection methods improve preferred non-dominated solutions in the

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2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation ofOperational Research Societies