Desirability function approach: A review and performance evaluation in adverse conditions

doi:10.1016/j.chemolab.2011.04.004

Chemometrics and Intelligent Laboratory Systems

Volume 107, Issue 2, July 2011, Pages 234-244

https://doi.org/10.1016/j.chemolab.2011.04.004 Get rights and content

Abstract

Adverse conditions in terms of quality of predictions and robustness are simulated to evaluate the ability of desirability-based methods for yielding compromise solutions with desired response's properties. The method's solutions are assessed at optimal variable settings with respect to bias, quality of predictions and robustness through optimization measures, and the usefulness of those measures to select the compromise solution is evaluated. Three examples with different features in terms of responses variance are used and the performance of various analysis methods is compared. Results show that a less sophisticated desirability-based method can compete with other methods designed to perform well under adverse conditions and that the optimization measures justify its use in real life problems.

Introduction

A widely used methodology for developing, improving and optimizing systems (process and product), the so-called response surface methodology (RSM), consists of the following general phases: 1) screening: experiments are designed with the purpose of discovering the vital few control factors that cause statistically significant effects of practical importance for the goal of the study; 2) modeling: experiments are designed with the purpose of modeling the quality characteristic of interest (response) as a function of control factors; 3) optimization: response model is analyzed to determine the variable settings at which optimum conditions of system property are achieved.

This methodology has inherent sequential experimentation strategy that, if the technical and non-technical issues in the experimental phases are properly managed, leads to a high level of system knowledge. For an understanding of the assumptions and conditions necessary to successfully apply RSM the reader is referred to Refs. [1], [2], [3].

This article focuses on the optimization phase and, in particular, on the optimization of multiple responses. These problems are usual in various fields of science and often involve incommensurate and conflicting responses that must be in some sense optimized simultaneously, because their separate analysis may result in incompatible solutions. This has been a much researched subject and a strategy widely used in the RSM framework consists of converting the multiple responses into a single one by combining the individual responses into a composite function followed by its optimization. Although the desirability function and loss function approaches are popular among practitioners, other approaches have been used for optimizing multiple responses. Among them are those based on Compromise Programming [4], Goal Programming [5], [6], Inspection of contour plots [7], Physical Programming [8], [9], Probability-based [10], Performance Index [11], [12], Neural Networks [13], [14], and Vectorial Optimization [15]. All these approaches have their own merits. However, the lack of recommendations for proper use, unavailability of the algorithms employed, and (mathematical/statistical) complexity of some approaches and methods are major reasons by which they are of little practical use or appealing to practitioners, in particular to non-statisticians.

The less sophisticated desirability-based methods are easy to understand, easy to use, and flexible for incorporating the decision-maker's preferences (weights or priorities assigned to responses). Moreover, the most popular desirability-based method, the Derringer and Suich's method [16] or modifications of it [17] are available in many data analysis software packages. This method has been extensively used in practice, namely in chemometrics, such as illustrated in Refs. [18], [19], [20], [21], [22]. However, the analyst needs to specify values to four types of parameters (shape factors and weights) to use the so-called Derringer and Suich's method. This is not a simple task and impacts on the optimal variable settings. So, other less known and used methods that require less subjective information from the user and can yield effective compromise solutions are particularly welcome, namely to non-statistician practitioners who are using statistics more than ever before [23].

This article provides a review on existing desirability-based methods and aims at evaluating the ability of two less sophisticated ones along with appropriate optimization performance measures to give desirable response's properties at optimal variable settings, namely low bias (responses deviation from target) and low variance. This is illustrated with three case studies where adverse conditions in terms of responses variance are simulated. In the first simulated scenario the response's models differ in terms of the quality of predictions (variance due to uncertainty in the regression coefficients). In the second simulated scenario the response's models are characterized by unequal sensitivity to uncontrollable variables (robustness) and high quality of predictions. In the third one, both the quality of predictions and robustness are low.

The remainder of the article is organized as follows: Section 2 provides a review on desirability-based methods; optimization criteria and measures to evaluate the compromise solutions in terms of response's bias, quality of predictions and robustness are introduced in 3 Proposed approach, 4 Examples characterizes the simulated scenarios and includes three examples to illustrate the feasibility of the proposed approach; Section 5 includes the discussion of results and Section 6 presents the final conclusions.

Section snippets

Literature review

Desirability function-based approach consists of converting the estimated response models ( $\hat{y}$ ), which usually are second order models, into individual desirability functions (d) that are then aggregated into a composite function (D). This function is usually a geometric or an arithmetic mean, which will be maximized or minimized, respectively.

Next subsections provide a chronological review on the existing desirability-based methods, which are separated into two groups of different

Proposed approach

Progress in research on desirability functions has been focused on flexibility, minimization of the impact in decision-making of certain types of dependencies caused by data limitations, consideration of response's variance, and use of genetic algorithms. Approaches that require minimum subjective information from the user and are easy to understand and implement are especially appealing to practitioners and useful if they provide compromise solutions with desired response's properties.

Examples

Myers and Montgomery [42] reported an experimental study where the objective was to maximize the conversion of a polymer and minimize the thermal activity by setting the variables reaction time(x₁), reaction temperature (x₂), and amount of catalyst (x₃) at appropriate levels. The ranges for the conversion (y₁) and thermal activity (y₂) responses are [80, 100] and [55, 60], respectively. Assuming that y₁ is a LTB-type response, its target value is set equal to 100; y₂ is a NTB-type response and

Discussion

To date, there is no effective procedure to guide the analyst in the selection of weights or priorities to responses. The optimization results can vary significantly if the weights change, and very little is usually known about how to choose their values. In practice, it is difficult, if at all possible, to determine beforehand the modifications required in the weights so as to produce a solution of interest, that is, so as to know which response(s) will change and which is the direction and

Conclusions

This article provides a review on desirability-based methods and evaluates, under adverse variance conditions, two of the less sophisticated ones, namely the methods proposed by Ch'ng et al. [27] and Wu [28]. To assess the properties of the compromise solutions at variables setting with respect to bias, quality of predictions and robustness, optimization measures were used. These measures can be used along with any method of practitioner's interest, and have proved to be useful for selecting a

Acknowledgment

The authors are grateful to reviewers for their comments and suggestions. The Fundação para a Ciência e Tecnologia (FCT) through the PIDDAC Program funds (INESC-ID multiannual funding) supported the second author.

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View full text

Desirability function approach: A review and performance evaluation in adverse conditions

Abstract

Introduction

Section snippets

Literature review

Proposed approach

Examples

Discussion

Conclusions

Acknowledgment

Chemom. Intell. Lab.

Chemom. Intell. Lab.

Chemom. Intell. Lab.

Chemom. Intell. Lab.

Chemom. Intell. Lab.

Appl. Math. Comput.

Guidelines to help practitioners of design of experiments

Total Qual. Manage. Mag.

Response Surface Methodology: Process and Product Optimization Using Designed Experiments

Implementation of design of experiments projects in industry

Appl. Stochastic Models Data Anal.

Robust design modeling and optimization with correlated quality characteristics using a multicriteria decision framework

Int. J. Adv. Manuf. Technol.

Multiresponse systems optimization using a goal attainment approach

IIE Trans.

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Eng. Optim.

From dubious construction of objective functions to the application of physical programming

AIAA J.

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Int. J. Adv. Manuf. Technol.

A Bayesian reliability approach to multiple response optimization with seemingly unrelated regression models

Qual. Technol. Quant. Manage.

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Qual. Eng.

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