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Toward a robust optimal point selection: a multiple-criteria decision-making process applied to multi-objective optimization using response surface methodology

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During the multi-objective optimization process, numerous efficient solutions may be generated to form the Pareto frontier. Due to the complexity of formulating and solving mathematical problems, choosing the best point to be implemented becomes a non-trivial task. Thus, this paper introduces a weighting strategy named robust optimal point selection, based on ratio diversification/error, to choose the most preferred Pareto optimal point in multi-objective optimization problems using response surface methodology. Furthermore, this paper proposes to explore a theoretical gap—the prediction variance behavior related to the weighting. The ratios Shannon’s entropy/error and diversity/error and the unscaled prediction variance are experimentally modeled using mixture design and the optimal weights for the multi-objective optimization process are defined by the maximization of the proposed measures. The study could demonstrate that the weights used in the multi-objective optimization process influence the prediction variance. Furthermore, the use of diversification measures, such as entropy and diversity, associated with measures of error, such as mean absolute percent error, was determined to be useful in mapping regions of minimum variance within the Pareto optimal responses obtained in the optimization process.

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  1. 1.

    Cua KO, Mckone KE, Schroeder RG (2001) Relationships between implementation of TQM, JIT, and TPM and manufacturing performance. J Oper Manag 19(6):675–694

  2. 2.

    Kano M, Nakagawa Y (2008) Data-based process monitoring, process control, and quality improvement: recent developments and applications in steel industry. Comput Chem Eng 32(1–2):12–24

  3. 3.

    Montgomery DC (2009) Design and analysis of experiments, 5th edn. Wiley, New York, p 665

  4. 4.

    Khuri A, Kim HJ, Um Y (1996) Quantile plots of the prediction variance for response surface designs. Comput Stat Data Anal 22(4):395–407

  5. 5.

    Baril C, Yacout S, Clément B (2011) Design for six sigma through collaborative multiobjective optimization. Comput Ind Eng 60(1):43–55

  6. 6.

    Szeląg M, Greco S, Słowiński R (2014) Variable consistency dominance-based rough set approach to preference learning in multicriteria ranking. Inf Sci 277(1):525–552

  7. 7.

    Zeleny M (1974) A concept of compromise solutions and the method of the displaced ideal. Comput Oper Res 1(3–4):479–496

  8. 8.

    Figueira JR, Greco S, Słowiński R (2009) Building a set of additive value functions representing a reference preorder and intensities of preference: GRIP method. Eur J Oper Res 195(2):460–486

  9. 9.

    Tian N, Tang S, Che A, Wu P (2020) Measuring regional transport sustainability using super-efficiency SBM-DEA with weighting preference. J Clean Prod 242:118474

  10. 10.

    Matin A, Zare S, Ghotbi-Ravandi M, Jahani Y (2020) Prioritizing and weighting determinants of workers’ heat stress T control using an analytical network process (ANP) a field study. Urban Clim 31:100587

  11. 11.

    Kamaruzzaman S, Lou E, Wong P, Wood R, Che-Ani A (2018) Developing weighting system for refurbishment building assessment scheme in Malaysia through analytic hierarchy process (AHP) approach. Energy Policy 112:280–290

  12. 12.

    Zhu X, Dapeng N, Wang X, Wang F, Jia M (2019) Comprehensive energy saving evaluation of circulating cooling water system based on combination weighting method. Appl Thermal Eng 157:113735

  13. 13.

    Gaudêncio J, Almeida F, Sabioni RC, Turrioni JB, Paiva AP, Campos PHS (2019) Fuzzy multivariate mean square error in equispaced pareto frontiers considering manufacturing process optimization problems. Eng Comput 35:1213–1236

  14. 14.

    Lakshmi R, Baskar S (2019) Novel term weighting schemes for document representation based on ranking of terms and Fuzzy logic with semantic relationship of terms. Experts Syst Appl 137:493–503

  15. 15.

    Rocha LCS, de Paiva AP, Junior PR, Balestrassi PP, da Silva Campos PH, Davim JP (2017) Robust weighting applied to optimization of AISI H13 hardened-steel turning process with ceramic wiper tool: a diversity-based approach. Precis Eng 50:235–247

  16. 16.

    Davoudabadi R, Mousavi SM, Sharifi E (2020) An integrated weighting and ranking model based on entropy, DEA and PCA considering two aggregation approaches for resilient supplier selection problem. J Comput Sci 40:101074

  17. 17.

    Aquila G, Rocha LCS, Pamplona E, Queiroz A, Rotela Junior P, Balestrassi P, Fonseca M (2018) Proposed method for contracting of wind-photovoltaic projects connected to T the Brazilian electric system using multiobjective programming. Renew Sustain Energy Rev 97:377–389

  18. 18.

    Ibáñes-Forés V, Bovea MD, Pérez-Belis V (2014) A holistic review of applied methodologies for assessing and selecting the optimal technological alternative from a sustainability perspective. J Clean Prod 70(1):259–281

  19. 19.

    Taboada HA, Baheranwala F, Coit DW, Wattanapongsakorn N (2007) Practical solutions for multi-objective optimization: an application to system reliability design problems. Reliab Eng Syst Saf 92(3):314–322

  20. 20.

    Gaudreault C, Samson R, Stuart P (2009) Implications of choices and interpretation in LCA for multi-criteria process design: de-inked pulp capacity and cogeneration at a paper mill case study. J Clean Prod 17(17):1535–1546

  21. 21.

    Pilavachi PA, Stephanidis SD, Pappas VA, Afgan NH (2009) Multi-criteria evaluation of hydrogen and natural gas fuelled power plant technologies. Appl Therm Eng 29(11–12):2228–2234

  22. 22.

    Zeleny M (1975) The theory of the displaced ideal. In: Zeleny M (ed) Lecture notes in economics and mathematical systems, no 123: multiple criteria decision making—Kyoto. Springer, Berlin

  23. 23.

    Melachrinoudis E (1985) Determining an optimum location for an undesirable facility in a workroom environment. Appl Math Model 9(5):365–369

  24. 24.

    Saaty TL (1990) How to make a decision: the analytic hierarchy process. Eur J Oper Res 48(1):9–26

  25. 25.

    Grzybowski AZ (2012) Note on a new optimization based approach for estimating priority weights and related consistency index. Expert Syst Appl 39(14):11699–11708

  26. 26.

    Promentilla MAB, Furuichi T, Ishii K, Tanikawa N (2008) A fuzzy analytic network process for multi-criteria evaluation of contaminated site remedial countermeasures. J Environ Manag 88(3):479–495

  27. 27.

    Tran NH, Tran K (2007) Combination of fuzzy ranking and simulated annealing to improve discrete fracture inversion. Math Comput Model 45(7–8):1010–1020

  28. 28.

    Narang N, Dhillon JS, Kothari DP (2014) Weight pattern evaluation for multiobjective hydrothermal generation scheduling using hybrid search technique. Int J Electr Power Energy Syst 62:665–678

  29. 29.

    Wan S-P, Dong J-Y (2015) Power geometric operators of trapezoidal intuitionistic fuzzy numbers and application to multiattribute group decision making. Appl Soft Comput 29:153–168

  30. 30.

    Wang Z-J (2015) Consistency analysis and priority derivation of triangular fuzzy preference relations based on modal value and geometric mean. Inf Sci 314:169–183

  31. 31.

    Gomes JHF, Paiva AP, Costa SC, Balestrassi PP, Paiva EJ (2013) Weighted multivariate mean square error for processes optimization: a case study on flux-cored arc welding for stainless steel claddings. Eur J Oper Res 226(3):522–535

  32. 32.

    Savier JS, Das D (2011) Loss allocation to consumers before and after reconfiguration of radial distribution networks. Int J Electr Power Energy Syst 33(3):540–549

  33. 33.

    Huang HZ, Gu YK, Du X (2006) An interactive fuzzy multi-objective optimization method for engineering design. Eng Appl Artif Intell 19(5):451–460

  34. 34.

    Rubio L, De La Sen M, Longstaff AP, Fletcher S (2013) Model-based expert system to automatically adapt milling forces in Pareto optimal multi-objective working points. Expert Syst Appl 40(6):2312–2322

  35. 35.

    Luo D, Wang X (2012) The multi-attribute grey target decision method for attribute value within three-parameter interval grey number. Appl Math Model 36(5):1957–1963

  36. 36.

    Zhu J, Hipel KW (2012) Multiple stages grey target decision making method with incomplete weight based on multigranularity linguistic label. Inf Sci 212:15–32

  37. 37.

    Luo D (2009) Decision-making methods with three-parameter interval grey number. Syst Eng Theory Pract 29(1):124–130

  38. 38.

    Monghasemi S, Nikoo MR, Khaksar Fasaee MA, Adamowski J (2015) A novel multi criteria decision making model for optimizing time-cost-quality trade-off problems in construction projects. Expert Syst Appl 42(6):3089–3104

  39. 39.

    Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423

  40. 40.

    Rocha LCS, Paiva AP, Balestrassi PP, Severino G, Rotela Junior P (2015a) Entropy-based weighting for multiobjective optimization: an application on vertical turning. Math Probl Eng Article ID 608325

  41. 41.

    Rocha LCS, Paiva AP, Balestrassi PP, Severino G, Rotela Junior P (2015) Entropy-based weighting applied to normal boundary intersection approach: the vertical turning of martensitic gray cast iron piston rings case. Acta Sci Technol 37(4):361–371

  42. 42.

    Wang JJ, Jing YY, Zhang CF, Zhao JH (2009) Review on multi-criteria decision analysis aid in sustainable energy decision-making. Renew Sustain Energy Rev 13(9):2263–2278

  43. 43.

    Wuwongse V, Kobayashi S, Iwai S-I, Ichikawa A (1983) Optimal design of linear control systems by an interactive optimization method. Comput Ind 4(4):381–394

  44. 44.

    Bonano EJ, Apostolakis GE, Salter PF, Ghassemi A, Jennings S (2000) Application of risk assessment and decision analysis to the evaluation, ranking and selection of environmental remediation alternatives. J Hazard Mater 71(1–3):35–57

  45. 45.

    Dijkmans R (2000) Methodology for selection of best available techniques (BAT) at the sector level. J Clean Prod 8(1):11–21

  46. 46.

    Geldermann J, Rentz O (2004) The reference installation approach for the techno-economic assessment of emission abatement options and the determination of BAT according to the IPPC-directive. J Clean Prod 12(4):389–402

  47. 47.

    Halog A, Shultmann F, Rentz O (2001) Using quality function deployment for technique selection for optimum environmental performance improvement. J Clean Prod 9:387–394

  48. 48.

    Prabhu TR, Vizayakumar K (2001) Technology choice using FHDM: a case of iron-making technology. IEEE Trans Eng Manag 48(2):209–222

  49. 49.

    Vignes RP (2001) Use limited life-cycle analysis for environmental decision-making. Chem Eng Prog 97(2):40–54

  50. 50.

    Zhang W, Yang H (2001) A study of the weighting method for a certain type of multicriteria optimization problem. Comput Struct 79(31):2741–2749

  51. 51.

    Derden A, Vercaemst P, Dijkmans R (2002) Best available techniques (BAT) for the fruit and vegetable processing industry. Resour Conserv Recycl 34(4):261–271

  52. 52.

    Beccali M, Cellura M, Mistretta M (2003) Decision-making in energy planning. Application of the Electre method at regional level for the diffusion of renewable energy technology. Renew Energy 28(13):2063–2087

  53. 53.

    Afgan NH, Carvalho MG (2004) Sustainability assessment of hydrogen energy systems. Int J Hydrogen Energy 29(13):1327–1342

  54. 54.

    Cziner K, Tuomaala M, Hurme M (2005) Multicriteria decision making in process integration. J Clean Prod 13(5):475–483

  55. 55.

    Sadiq R, Khan FI, Veitch B (2005) Evaluating offshore technologies for produced water management using GreenPro-I: a risk-based life cycle analysis for green and clean process selection and design. Comput Chem Eng 29(5):1023–1039

  56. 56.

    Chowdhury S, Husain T (2006) Evaluation of drinking water treatment technology: an entropy-based fuzzy application. J Environ Eng 132(10):1264–1271

  57. 57.

    Critto A, Cantarella L, Carlon C, Giove S, Petruzzelli G, Marcomini A (2006) Decision support-oriented selection of remediation technologies to rehabilitate contaminated sites. Integr Environ Assess Manag 2(3):273–285

  58. 58.

    Doukas H, Patlitzianas KD, Psarras J (2006) Supporting sustainable electricity technologies in Greece using MCDM. Resour Policy 31(2):129–136

  59. 59.

    Khelifi O, Dalla Giovanna F, Vranes S, Lodolo A, Miertus S (2006) Decision support tool for used oil regeneration technologies assessment and selection. J Hazard Mater 137(1):437–442

  60. 60.

    Pilavachi PA, Roumpeas CP, Minett S, Afgan NH (2006) Multi-criteria evaluation for CHP system options. Energy Convers Manag 47(20):3519–3529

  61. 61.

    Shehabuddeen N, Probert D, Phaal R (2006) From theory to practice: challenges in operationalising a technology selection framework. Technovation 26(3):324–335

  62. 62.

    Begić F, Afgan NH (2007) Sustainability assessment tool for the decision making in selection of energy system-Bosnian case. Energy 32(10):1979–1985

  63. 63.

    Fijal T (2007) An environmental assessment method for cleaner production technologies. J Clean Prod 15(10):914–919

  64. 64.

    Grandinetti L, Guerriero F, Lepera G, Mancini M (2007) A niched genetic algorithm to solve a pollutant emission reduction problem in the manufacturing industry: a case study. Comput Oper Res 34(7):2191–2214

  65. 65.

    Krajnc D, Mele M, Glavič P (2007) Fuzzy logic model for the performance benchmarking of sugar plants by considering best available techniques. Resour Conserv Recycl 52(2):314–330

  66. 66.

    Mavrotas G, Georgopoulou E, Mirasgedis S, Sarafidis Y, Lalas D, Hontou V, Gakis N (2007) An integrated approach for the selection of best available techniques (BAT) for the industries in the greater Athens area using multiobjective combinatorial optimization. Energy Econ 29(4):953–973

  67. 67.

    Zeng G, Jiang R, Huang G, Xu M, Li J (2007) Optimization of wastewater treatment alternative selection by hierarchy grey relational analysis. J Environ Manag 82(2):250–259

  68. 68.

    Bollinger D, Pictet J (2008) Multiple criteria decision analysis of treatment and land-filling technologies for waste incineration residues. Omega 36(3):418–428

  69. 69.

    Georgopoulou E, Hontou V, Gakis N, Sarafidis Y, Mirasgedis S, Lalas DP, Loukatos A, Gargoulas N, Mentzis A, Economidis D, Triantafilopoulos T, Korizi K (2008) BEAsT: a decision-support tool for assessing the environmental benefits and the economic attractiveness of best available techniques in industry. J Clean Prod 16(3):359–373

  70. 70.

    Schollenberger H, Treitz M, Geldermann J (2008) Adapting the European approach of best available techniques: case studies from Chile and China. J Clean Prod 16(17):1856–1864

  71. 71.

    Bréchet T, Tulkens H (2009) Beyond BAT: selecting optimal combinations of available techniques, with an example from the limestone industry. J Environ Manag 90(5):1790–1801

  72. 72.

    Cavallaro F (2009) Multi-criteria decision aid to assess concentrated solar thermal technologies. Renew Energy 34(7):1678–1685

  73. 73.

    Daim T, Intarode N (2009) A framework for technology assessment: case of a Thai building material manufacturer. Energy Sustain Dev 13(4):280–286

  74. 74.

    Gómez-López MD, Bayo J, García-Cascales MS, Angosto JM (2009) Decision support in disinfection technologies for treated wastewater reuse. J Clean Prod 17(16):1504–1511

  75. 75.

    Karagiannidis A, Perkoulidis G (2009) A multi-criteria ranking of different technologies for the anaerobic digestion for energy recovery of the organic fraction of municipal solid wastes. Bioresour Technol 100(8):2355–2360

  76. 76.

    Karavanas A, Chaloulakou A, Spyrellis N (2009) Evaluation of the implementation of best available techniques in IPPC context: an environmental performance indicators approach. J Clean Prod 17(4):480–486

  77. 77.

    Paiva AP, Paiva EJ, Ferreira JF, Balestrassi PP, Costa SC (2009) A multivariate mean square error optimization of AISI 52100 hardened steel turning. Int J Adv Manuf Technol 43(7):631–643

  78. 78.

    Yang QZ, Chua BH, Song B (2009) A matrix evaluation model for sustainability assessment of manufacturing technologies. World Acad Sci Eng Technol Int J Mech Aerosp Ind Mech Manuf Eng 3(8):953–958

  79. 79.

    Kazagić A, Smajević I, Duić N (2010) Selection of sustainable technologies for combustion of Bosnian coals. Thermal Sci 14(3):715–727

  80. 80.

    Lin GTR, Shen YC (2010) A collaborative model for technology evaluation and decision-making. J Sci Ind Res 69(2):94–100

  81. 81.

    Bottero M, Comino E, Riggio V (2011) Application of the analytic hierarchy process and the analytic network process for the assessment of different wastewater treatment systems. Environ Model Softw 26(10):1211–1224

  82. 82.

    García N, Caballero JA (2011) Economic and environmental assessment of alternatives to the extraction of acetic acid from water. Ind Eng Chem Res 50(18):10717–10729

  83. 83.

    Inoue Y, Katayama A (2011) Two-scale evaluation of remediation technologies for a contaminated site by applying economic input-output life cycle assessment: risk-cost, risk-energy consumption and risk-CO2 emission. J Hazard Mater 192(3):1234–1242

  84. 84.

    San Cristóbal JR (2011) A multi criteria data envelopment analysis model to evaluate the efficiency of the renewable energy technologies. Renew Energy 36(10):2742–2746

  85. 85.

    Cristóbal J, Guillén-Gosálbez G, Jiménez L, Irabien A (2012) Optimization of global and local pollution control in electricity production from coal burning. Appl Energy 92:369–378

  86. 86.

    De Lange WJ, Stafford WHL, Forsyth GG, Le Maitre DC (2012) Incorporating stakeholder preferences in the selection of technologies for using invasive alien plants as a bio-energy feedstock: applying the analytical hierarchy process. J Environ Manag 99(30):76–83

  87. 87.

    Giner-Santonja G, Aragonés-Beltrán P, Niclós-Ferragut J (2012) The application of the analytic network process to the assessment of best available techniques. J Clean Prod 25:86–95

  88. 88.

    Liu X, Wen Z (2012) Best available techniques and pollution control: a case study on China’s thermal power industry. J Clean Prod 23(1):113–121

  89. 89.

    Liu F, Zhang W-G, Wang ZX (2012) A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making. Eur J Oper Res 218(3):747–754

  90. 90.

    Severino G, Paiva EJ, Ferreira JR, Balestrassi PP, Paiva AP (2012) Development of a special geometry carbide tool for the optimization of vertical turning of martensitic gray cast iron piston rings. Int J Adv Manuf Technol 63(5–8):523–534

  91. 91.

    Yu OY, Guikema SD, Briaud JL, Burnett D (2012) Sensitivity analysis for multiattribute system selection problems in onshore environmentally friendly drilling (EFD). Syst Eng 15(2):153–171

  92. 92.

    Khorasani G, Mirmohammadi F, Motamed H, Fereidoon M, Tatari A, Maleki Verki MR, Khorasani M, Fazelpour S (2013) Application of multi criteria decision making tools in road safety performance indicators and determine appropriate method with average concept. Int J Innov Technol Explor Eng 3(5):173–177

  93. 93.

    Shahraki AF, Noorossana R (2014) Reliability-based robust design optimization: a general methodology using genetic algorithm. Comput Ind Eng 74:199–207

  94. 94.

    Hein N, Kroenke A, Rodrigues Junior MM (2015) Professor assessment using multi-criteria decision analysis. Proc Comput Sci 55:539–548

  95. 95.

    Shahhosseini H, Farsi M, Eini S (2016) Mult-objective optimization of industrial membrane SMR to produce syngas for Fischer-Tropsch production using NSGA-II and decision makins. J Nat Gas Sci Eng 32:222–238

  96. 96.

    Howard E, Kamper M (2016) Weighted Factor multiobjective design optimization of a reluctance synchronous machine. IEEE Trans Ind Appl 52:3

  97. 97.

    Prakash C, Barua M (2016) Robust multi-criteria decision making framework for evaluation of the airport service quality enablers for ranking the airports. J Qual Assur Hosp Tour 17:3

  98. 98.

    Rocha LCS, de Paiva AP, Junior PR, Balestrassi PP, da Silva Campos PH (2017) Robust multiple criteria decision making applied to optimization of AISI H13 hardened steel turning with PCBN wiper tool. Int J Adv Manuf Technol 89(5–8):2251–2268

  99. 99.

    Zhou R, Cai R, Tong G (2013) Applications of entropy in finance: a review. Entropy 15(11):4909–4931

  100. 100.

    Fang S-C, Rajasekera JR, Tsao H-SJ (1997) Entropy optimization and mathematical programming. Kluwer Academic Publishers, Boston

  101. 101.

    Hickey EA, Carlson JL, Loomis D (2010) Issues in the determination of the optimal portfolio of electricity supply options. Energy Policy 38:2198–2207

  102. 102.

    Stirling A (1994) Diversity and ignorance in electricity supply investment: addressing the solution rather than the problem. Energy Policy 22(3):195–216

  103. 103.

    Stirling A (2007) A general framework for analysing diversity in science, technology and society. J R Soc Interface 4(15):707–719

  104. 104.

    Das I, Dennis JE (1998) Normal boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8(3):631–657

  105. 105.

    Rocha LCS, Paiva AP, Paiva EJ, Balestrassi PP (2015) Comparing DEA and principal component analysis in the multiobjective optimization of P-GMAW process. J Braz Soc Mech Sci Eng.

  106. 106.

    Montgomery DC, Jennings CL, Kulahci M (2008) Introduction to time series analysis and forecasting. Wiley, New York, p 445

  107. 107.

    Zahran A, Anderson-Cook CM, Myers RH (2003) Fraction of design space to assess prediction capability of response surface designs. J Qual Technol 35(4):377–386

  108. 108.

    Myers RH, Montgomery DC, Anderson-Cook CM (2009) Response surface methodology: process and product optimization using designed experiments, 3rd edn. Wiley, New York, p 680

  109. 109.

    Campos PHS (2015) (Doctoral dissertation) Metodologia DEA-OTS: Uma contribuição para a seleção ótima de ferramentas no Torneamento do Aço ABNT H13 Endurecido. Universidade Federal de Itajubá, Itajubá

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The authors would like to express their gratitude to CAPES, CNPq and FAPEMIG for their financial support and research incentive.

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Rocha, L.C.S., Rotela Junior, P., Aquila, G. et al. Toward a robust optimal point selection: a multiple-criteria decision-making process applied to multi-objective optimization using response surface methodology . Engineering with Computers (2020).

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  • Multi-objective programming
  • Multi-criteria analysis
  • Robust optimal point selection (ROPS)
  • Diversification measures
  • Error measures