A Robust Posterior Preference Decision Making Approach to Multiple Response Process Design

Setting of process variables to meet required specification of quality characteristics is one of the important problems in quality control process. In general, most industrial and production systems are dealing with several different responses and
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     Int. J. Applied Decision Sciences, Vol. X, No. Y, xxxx 1  Copyright © 200x Inderscience Enterprises Ltd. A robust posterior preference decision-makingapproach to multiple response process design  Ali Salmasnia* Department of Industrial Engineering,   Faculty of Engineering,Tarbiat Modares University,Tehran, IranE-mail: *Corresponding author   Asghar Moeini School of Computer Science, Engineering and Mathematics,Flinders University,AustraliaE-mail: Hadi Mokhtari Department of Industrial Engineering,   Faculty of Engineering,Tarbiat Modares University,Tehran, IranE-mail:  Cyrus Mohebbi Department of Information, Operation, and Management Science,    New York University, NY, USAE-mail: Abstract: Setting of process variables to meet required specification of qualitycharacteristics is one of the important problems in quality control process. Ingeneral, most industrial and production systems are dealing with severaldifferent responses and the problem is to simultaneously optimise theseresponses. To obtain the most satisfactory solution, a decision-maker (DM)’s preference on the trade-offs among the quality characteristics should beincorporated into the optimisation procedure. This study suggests a robust posterior preference articulation approach based on a non-dominated sortinggenetic algorithm (NSGA-II) to optimise multiple responses. In order tominimise the variation in deviation of responses from targets, maximum andsum of deviations are taken into consideration. To investigate the performanceof the suggested approach, a computational analysis on a real world chemicalengineering example is performed. Results show the superiority of the proposed approach compared to the existing techniques. Comment [t1]: Author: Please provide fullmailing address of Ali Salmasnia Comment [t2]: Author: Please provide fullmailing address of Asghar Moeini. Comment [t3]: Author: Please provide fullmailing address of Hadi Mokhtari. Comment [t4]: Author: Please provide fullmailing address of Cyrus Mohebbi.   2  A. Salmasniaet al. Keywords: non-dominated sorting genetic algorithm; NSGA-II; multipleresponse optimisation; posterior preference approach; VIKOR method. Reference to this paper should be made as follows: Salmasnia, A., Moeini, A.,Mokhtari, H. and Mohebbi, C. (xxxx) ‘A robust posterior preference decision-making approach to multiple response process design’,  Int. J. Applied DecisionSciences , Vol. X, No. Y, pp.000–000. Biographical notes: Ali Salmasnia is a Doctoral student in the Department of Industrial Engineering at the Tarbiat Modares University in Tehran, Iran. Hereceived his BS degree from the Mazandaran University of Science andTechnology and his MS degrees in Industrial Engineering from the ShahedUniversity in Iran. His current research interests include multiple responseoptimisation, reliability and project management. He has published in  IEEE Transactions on Engineering Management  ,  International Journal of Advanced  Manufacturing Technology ,  International Journal of Applied DecisionSciences ,  International Journal of Industrial Engineering & Production Management  ,  Neurocomputing  , and  Applied Soft Computing  .Asghar Moeini is currently a Doctoral student and Research Assistant at theSchool of Computer Science, Engineering and Mathematics of the FlindersUniversity in Australia. He holds MSc and BSc in Industrial Engineering fromShahed University and Islamic Azad University in Iran, respectively. Hiscurrent research interests are in multi criteria decision-making under uncertainty, Markov decision process, Monte Carlo simulation and stochasticmodelling. Furthermore, he works on Flinders Hamiltonian Cycle Project(FHCP) at the present time. In this project, numerous approaches to analysingand solving the Hamiltonian cycle problem have been developed incollaboration with FHCP team members.Hadi Mokhtari received his BSc and MSc degrees in Industrial Engineeringfrom K.N. Toosi University of Technology in Iran. He is currently workingtowards his PhD degree in Industrial Engineering from Tarbiat ModaresUniversity. His current research interests include the applications of operationsresearch and artificial intelligence techniques to the areas of project scheduling, production scheduling, and manufacturing supply chains. He also publishedmore than ten ISI-indexed papers in international journals such as InternationalJournal of Production Research, Neurocomputing, International Journal of Advanced Manufacturing Technology, IEEE Transactions on EngineeringManagement, and Expert Systems with Applications . Cyrus Mohebbi is a Managing Director and Head of Quantitative Strategists for Global Wealth Management at Morgan Stanley. Before joining MorganStanley, he was the Head of the ABS Structuring and Analytics Group atHSBC. He began his career on Wall Street in 1987 at Prudential Securities Inc.,where he eventually became the Head of the Structured Finance/QuantitativeGroup. Prior to joining Prudential Securities, he taught Statistics, Marketingand Finance at the Wharton School of the University of Pennsylvania. He is anAdjunct Professor at the Stern School of Business at New York University andat the Graduate School of Business at Columbia University. He holds an MBAin Management and Accounting from LaSalle University, an MS in BusinessStatistics from Temple University, and an MA in Marketing, an MA, a PhD inStatistics, and Postdoctoral in Management from the Wharton School at theUniversity of Pennsylvania.     A robust posterior preference decision-making approach 3  1 Introduction An industrial process optimisation problem includes setting design variables to meet therequired specification of responses. To this end, among global approximation approaches,the response surface methodology (RSM) has recently gained the most attention since ithas performed well in comparison with other optimum seeking approaches (Garcia-Diazet al., 1981; Greenwood et al., 1993; Smith et al., 1973). RSM consists of a collection of mathematical and statistical techniques used in investigating the relation between theresponse and process variables. The detailed descriptions on various RSM techniques are presented by Box and Draper (1987), Khuri and Cornell (1996) and Myers andMontgomery (2002). Most of these techniques have focused on cases with just oneresponse, whereas in practice there is typically more than one response. These types of  problems are usually called multi-response optimisation (MRO) (Hwang et al., 1979) inwhich selection of optimum process variables would require the simultaneousconsideration of all the responses.According to the definition of MRO, it can generally be considered as amulti-objective optimisation (MOO) problem and the approaches developed in context of MOO can be successfully utilised to deal with MRO problems. In literature, the MOOapproaches can be classified into three major categories:1 prior preference articulation2 progressive preference articulation3 posterior preference articulation methods (Korhonen et al., 1992; Steuer, 1986).In the prior preference articulation approach, the DM expresses all the preferenceinformation before beginning the problem solving process. If the DM can explicitly pre-specify his/her preference information without error, prior methods will ensure themost satisfactory solution. However in many situations, the DM has difficulties inspecifying all the required preference information in advance. Progressive methodsreferred as the interactive approaches and often require that the DM’s preferenceinformation is inputted into the model during the solving process. In comparison with the prior method, the progressive approaches have some advantages. They allow an active participation of the DM during the solving procedure and thus the DM can direct thesearch and concentrate on more desirable solutions. The major disadvantage of interactive approaches is that they often require a considerable amount of time on the partof the DM and may not be particularity efficient for large size problems.Posterior methods do not require any substantial articulation of the DM’s preference before or during the solution process. After all (or most) of the non-dominated solutionsare generated, the DM manually selects the best one. The posterior approach has a limited practical applicability because a large number of solutions are produced in this methodand hence it becomes very difficult to select the best solution.Most of the existing methods in MRO can be categorised into the prior preferencearticulation. Some typical examples of these techniques are described by Lin and Tu(1995), Lu and Antony (2002), Ozler et al. (2008), Wu (2005), Bernard et al. (2009),Tatjana et al. (2012), Koach and Cho (2008), and Hejazi et al. (in press). Moreover several approaches have been proposed to solve MRO problems by using progressivemethods such as Montgomery and Bettencourt (1977), Mollaghasemi and Evans (1994),   Comment [t5]: Author: Please provide fullreference or delete from the text if not required. Comment [t6]: Author: Please provide year of  publication.   4  A. Salmasniaet al. Koksalan and Plante (2003), Jeong and Kim (2003, 2005, 2009), Park and Kim (2005)and Koksoy (2006). To the best of our knowledge there is only one piece of researchemploying the posterior method to solve MRO problems (Lee et al., 2010).Main contributions of this paper could be summarised as follows:1 A robust posterior preference articulation approach based on a non-dominatedsorting genetic algorithm (NSGA-II) is presented to optimise multiple responses.Unlike existing approaches, this new method does not require any information aboutthe DM’s preference before or during solving process.2 The proposed method employs the VIKOR tool to ensure that the variation of boththe individual response deviations and the overall average deviation is small.Ignoring this issue may lead to an optimisation in which deviations from the targetassociated with a number of responses are very small while deviation associated withothers are very large. Such an optimisation with a large variation in responsedeviations is usually unacceptable to engineers, even when the overall averagedeviation is small.The rest of paper is organised as follows. A review of related literature is given inSection 2. In Section 3 the proposed approach is presented and Section 4 illustrates theapplicability of the suggested method using a numerical example from the literature. Acomparative study between the suggested methodology and other existing approaches is performed in Section 5. Finally, concluding remarks and some suggestions for futureresearch are reported in Section 6. 2 Multiple response optimisation techniques As mentioned earlier, according to the role of the DM, the MRO approaches can becategorised into three groups of prior, progressive and posterior methods. In the sequel,the related literature of aforementioned three classes is presented. 2.1 Prior preference articulation methods Most of existing prior approaches aggregate the multiple responses into a single functionand solve it as a single objective optimisation problem. The most popular approaches inaggregating multiple responses are based on a desirability function (Derringer andSuich, 1980; Plante, 1999; Kim and Lin, 2000; Pan et al., 2009), a loss function(Pignatiello, 1993; Vining, 1998; Tsui, 1999; Wu and Chyu, 2004), a distance function(Khuri and Conlon, 1981), a proportion of conformance (Chiao and Hamada, 2001) or a principal component analysis (PCA) (Salmasnia et al., 2012a, 2012b). Some other approaches reduce multiple responses to one objective by employing multiple criteriadecision-making (MCDM) techniques such as goal programming (Kazemzadeh et al.,2008; Ramezani et al., 2011; Hejazi et al., 2012), goal attainment (Xu et al., 2004), andTOPSIS (Tong and Su, 1997).The desirability function approach was initially proposed by Harrington (1965) andthen modified by Derringer and Suich (1980). This approach transforms an estimatedresponseˆ() i  yx into a scaled free value of  d  i (  x ) systematically. It assigns a value from0 to 1 to the value of each response, in which a number closer to 1 indicates the more     A robust posterior preference decision-making approach 5 desirable value of response. To aggregate several desirability functions, Harrington(1965) introduces the overall desirability  D  by taking the geometric mean of theindividual desirability values d  i (  x ) as follows: ( ) 112 ()()() r r   Ddxdxdx = × × × … (1)where r  denotes the number of responses. Later, Derringer (1994) suggested a weightedgeometric mean to aggregate the individual desirability functions as below: ( ) 2 112 ()()() r i wwwr   Ddxdxdx = × × × ∑ … (2)where w i   is the relative importance of  i th response. The most important feature of thisapproach is that the obtained solution does not violate the acceptable limits. Kim and Lin(2000, 2006) suggested a minimum operator to combine the individual desirabilityfunctions in order to maximise the minimum degree of satisfaction within theexperimental region. In another work, Plante (1999) considered the deviation of responses from targets. The major difference between the approaches of Kim and Lin(2006) and Plante (1999) is that while the former applies a unique minimum operator, thelatter suggests a different minimum operator for each response.As it can be seen, the majority of the above desirability function methods is based onthe location effect and ignores the dispersion of responses. On the contrary, the lossfunction based methods focus on the both mean and variance-covariance effects. Thereare a few standard loss functions used for optimising multiple response problems. The primal idea of the loss functions is based on the squared error loss function as follows: ( ) ( ) 2 ,  Ly τ   y τ  = − (3)where  y and τ   are the actual process response and the target value, respectively. Inaddition Taguchi (1986) suggested that the expected value of the squared error loss to beminimised to reach a robust process. Afterwards Pignatiello (1993) extended Taguchi’ssingle response loss function to a multi-response case as below: ( ) [ ] ( ) [ ] ( ) () (),()() T  yx  ELyx τ   Eyx τ  CEyx τ  traceC  ⎡ ⎤⎡ ⎤= − − +⎣ ⎦⎢ ⎥⎣ ⎦ ∑ (4)where  y (  x ) is a r  × 1 vector and represents response  y at parameter setting  x , τ   is a r  × 1vector and shows the target values and ()  yx ∑ indicates a r  × r  variance-covariancematrix for   y (  x ). Since the mean vector and variance-covariance are usually unknown,Pignatiello (2009) suggests using the sample mean vector and the sample variance-covariance matrix. Vining (1998) introduced a new loss function by substituting  y (  x ) byˆ()  yx in the expected multi-response squared error loss proposed by Pignatiello (1993).Khuri and Conlon (1981) introduced a distance function based on Mahalanobis distance between estimated and idealised response vectors so as to set the controllable variables.Moreover, Ko et al. (2005) defined another loss function with properties of small bias,high robustness, and high quality of predictions.
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