DOI: 10.1115/1.2205870. Probabilistic Analytical Target Cascading: A Moment Matching Formulation for Multilevel Optimization Under Uncertainty

Probabilistic Analytical Target Cascading: A Moment Matching Formulation for Multilevel Optimization Under Uncertainty

10.1115/1.2205870

Crossref journal-article

ASME International

Journal of Mechanical Design (33)

Abstract

Abstract Analytical target cascading (ATC) is a methodology for hierarchical multilevel system design optimization. In previous work, the deterministic ATC formulation was extended to account for random variables represented by expected values to be matched among subproblems and thus ensure design consistency. In this work, the probabilistic formulation is augmented to allow the introduction and matching of additional probabilistic characteristics. A particular probabilistic analytical target cascading (PATC) formulation is proposed that matches the first two moments of interrelated responses and linking variables. Several implementation issues are addressed, including representation of probabilistic design targets, matching responses and linking variables under uncertainty, and coordination strategies. Analytical and simulation-based optimal design examples are used to illustrate the new formulation. The accuracy of the proposed PATC formulation is demonstrated by comparing PATC results to those obtained using a probabilistic all-in-one formulation.

Bibliography

Liu, H., Chen, W., Kokkolaras, M., Papalambros, P. Y., & Kim, H. M. (2005). Probabilistic Analytical Target Cascading: A Moment Matching Formulation for Multilevel Optimization Under Uncertainty. Journal of Mechanical Design, 128(4), 991â1000.

Authors 5

Huibin Liu (first)
Wei Chen (additional)
Michael Kokkolaras (additional)
Panos Y. Papalambros (additional)
Harrison M. Kim (additional)

References 43 Referenced 91

10.2514/3.13015 / AIAA J. / Optimization of Coupled Systems: A Critical Overview of Approaches by Balling
Alexandrov, N. M., and Lewis, R. M., 1999, “Comparative Properties of Collaborative Optimization and Other Approaches to MDO,” ICASE Report, No. 99-24, NASA/CR-1999-209354.
10.1137/0804044 / SIAM J. Optim. / Problem Formulation for Multidisciplinary Optimization by Cramer
Kroo, I., Altus, S., Braun, R., Gage, P., and Sobieski, I., 1994, ”Multidisciplinary Optimization Methods for Aircraft Preliminary Design,” 5th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, paper no. AIAA 94-4325, Panama City, Florida. (10.2514/6.1994-4325)
Braun, R. D., Kroo, I. M., and Moore, A. A., 1996, “Use of the Collaborative Optimization Architecture for Launch Vehicle Design,” 6th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, paper no. AIAA 96-4018, Bellevue, WA. (10.2514/6.1996-4018)
{'key': '2022040713072957400_c6', 'first-page': '97', 'volume-title': 'AIAA\n Pap.', 'author': 'Kroo'} / AIAA Pap. by Kroo
{'issue': '3', 'key': '2022040713072957400_c7', 'first-page': '403', 'article-title': 'Multiobjective Collaborative Optimization', 'volume': '119', 'author': 'Renaud', 'journal-title': 'ASME J. Mech. Des.', 'ISSN': 'http://id.crossref.org/issn/1050-0472', 'issn-type': 'print'} / ASME J. Mech. Des. / Multiobjective Collaborative Optimization by Renaud
{'key': '2022040713072957400_c8', 'first-page': '293', 'article-title': 'Target Cascading\n in Optimal System Design', 'volume-title': 'Proceedings of the ASME\n Design Engineering Technical Conferences', 'author': 'Kim'} / Proceedings of the ASME Design Engineering Technical Conferences / Target Cascading in Optimal System Design by Kim
Kim, H. M. , 2001, Target Cascading in Optimal System Design, Ph.D. thesis, University of Michigan, Ann Arbor, Michigan.
{'key': '2022040713072957400_c10', 'first-page': '661', 'article-title': 'Analytical Target Cascading in Automotive Vehicle\n Design', 'volume-title': 'Proceedings of the ASME Design Engineering\n Technical Conferences', 'author': 'Kim'} / Proceedings of the ASME Design Engineering Technical Conferences / Analytical Target Cascading in Automotive Vehicle Design by Kim
10.1504/IJVD.2002.002010 / Int. J. Veh. Des. / Target Cascading in Vehicle Redesign: A Class VI Truck Study by Kim
10.1007/s00158-002-0240-0 / Struct. Multidiscip. Optim. / Extension of the Target Cascading Formulation to the Design of Product Families by Kokkolaras
{'issue': '3,4', 'key': '2022040713072957400_c13', 'first-page': '402', 'article-title': 'Simulation-Based Optimal Design of Heavy Trucks by\n Model-Based Decomposition: An Extensive Analytical Target Cascading Case\n Study', 'volume': '11', 'author': 'Kokkolaras', 'journal-title': 'International Journal of Heavy Vehicle\n Systems'} / International Journal of Heavy Vehicle Systems / Simulation-Based Optimal Design of Heavy Trucks by Model-Based Decomposition: An Extensive Analytical Target Cascading Case Study by Kokkolaras
Wagner, T. C. , 1993, A General Decomposition Methodology for Optimal System Design, Ph.D. thesis, University of Michigan, Ann Arbor, Michigan.
10.1115/DETC2003/DAC-48734 / Proceedings of the ASME Design Engineering Technical Conferences / Analytical Target Setting: An Enterprise Context in Optimal Product Design by Cooper
10.1115/1.1587156 / ASME J. Mech. Des. / An Approach to Decision-Based Design with Discrete Choice Analysis for Demand Modeling by Wassenaar
{'key': '2022040713072957400_c17', 'first-page': '309', 'article-title': 'An Investigation of Multidisciplinary Design Subject to\n Uncertainty', 'volume-title': 'AIAA-1998-4747, 7th\n AIAA/USAF/NASA/ISSMO Multidisciplinary Analysis and Optimization\n Symposium', 'author': 'Gu'} / AIAA-1998-4747, 7th AIAA/USAF/NASA/ISSMO Multidisciplinary Analysis and Optimization Symposium / An Investigation of Multidisciplinary Design Subject to Uncertainty by Gu
10.2514/2.805 / AIAA J. / Robust Design Approach for Achieving Flexibility in Multidisciplinary Design by Chen
10.2514/2.1681 / AIAA J. / Efficient Uncertainty Analysis Methods for Multidisciplinary Robust Design by Du
Batill, S., Renaud, J. E., and Gu, X., 2000, “Modeling and Simulation Uncertainty in Multidisciplinary Design Optimization,” 8th AIAA/NASA/USAF/ISSMO Multidisciplinary Analysis and Optimization Conference and Exhibit, paper no. AIAA 2000-4803, Long Beach, California. (10.2514/6.2000-4803)
10.2514/6.2000-4841 / Proceedings of 8th AIAA/NASA/USAF/ISSMO Multidisciplinary Analysis and Optimization Conference and Exhibit / An Iterative Concurrent Subspace Robust Design Framework by Padmanabhan
10.2514/6.2004-4472 / Proceedings of 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference / An Approach to Robust Decision Making in Multidisciplinary Selection Problems under Uncertainty by Gurnani
Sues, R., Cesare, M., Pageau, S., and Wu, J., 2000, “Reliability Based MDO for Aerospace Systems,” 8th Symposium on Multidisciplinary Analysis and Optimization, AIAA 2000-4804, Long Beach, California.
Chiralaksanakul, A., and Mahadevan, S., 2004, “Multidisciplinary Design Optimization Under Uncertainty,” 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, paper no. AIAA-2004-4311, Albany, NY.
10.1115/1.2168470 / ASME J. Mech. Des. / Design Optimization of Hierarchically Decomposed Multilevel System under Uncertainty by Kokkolaras
{'key': '2022040713072957400_c26', 'volume-title': 'Quality Engineering Using Robust Design', 'author': 'Phadke'} / Quality Engineering Using Robust Design by Phadke
Kim, H., Kumar, D., Chen, W., and Papalambros, P. Y., 2004, “Target Feasibility Achievement in Enterprise-Driven Hierarchical Multidisciplinary Design,” 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, paper no. AIAA-2004-4546, Albany, NY. (10.2514/6.2004-4546)
Allison, J. , 2004, Complex System Optimization: A Comparison of Analytical Target Cascading, Collaborative Optimization, and Other Formulations, MS thesis, University of Michigan, Ann Arbor, Michigan.
10.1115/1.1862674 / ASME J. Mech. Des. / Technical Brief: Weights, Norms, and Notation in Analytical Target Cascading by Michalek
{'key': '2022040713072957400_c30', 'volume-title': 'Probability Concepts in Engineering Planning and Design, Volume\n I–Basic Principles', 'author': 'Ang'} / Probability Concepts in Engineering Planning and Design, Volume I–Basic Principles by Ang
10.1115/1.1290247 / ASME J. Mech. Des. / Towards a Better Understanding of Modeling Feasibility Robustness in Engineering by Du
10.1115/1.1334596 / ASME J. Mech. Des. / A Comprehensive Robust Design Approach for Decision Trade-Offs in Complex Systems Design by Kalsi
10.1115/1.1543978 / ASME J. Mech. Des. / Multidisciplinary Robust Design Optimization of an Internal Combustion Engine by McAllister
10.1115/1.1759358 / ASME J. Mech. Des. / An Integrated Framework for Optimization Using Inverse Reliability Strategy by Du
10.1115/1.1897408 / ASME J. Mech. Des. / Enhancing Discrete Choice Demand Modeling for Decision-Based Design by Wassenaar
10.1115/DETC2004-57487 / An Integrated Latent Variable Choice Modeling Approach to Enhancing Product Demand Modeling by Wassenaar
10.2514/6.2002-5506 / Proceedings of the 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization / Convergence Properties of Analytical Target Cascading by Michelena
10.1023/B:OPTE.0000048534.58121.93 / Opt. Eng. / Multidisciplinary Design Optimization with Quasiseparable Subsystems by Haftka
{'key': '2022040713072957400_c39', 'volume-title': 'Applied Geometric Programming', 'author': 'Beightler'} / Applied Geometric Programming by Beightler
{'key': '2022040713072957400_c40', 'volume-title': 'Statistics and Data Analysis from Elementary to\n Intermediate', 'author': 'Tamhane'} / Statistics and Data Analysis from Elementary to Intermediate by Tamhane
10.1007/s00158-005-0579-0 / Proceedings of the 6th World Conference on Structural and Multidisciplinary Optimization / An Augmented Lagrangian Relaxation for Analytical Target Cascading using the Alternating Directions Method of Multipliers by Tosserams
10.2514/6.2006-7086 / Proceedings of the IFORS / Lagrangian Coordination for Enhancing the Convergence of Analytical Target Cascading by Kim
10.1115/1.1649968 / ASME J. Mech. Des. / Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design by Du

Dates

Type	When
Created	19 years, 2 months ago (June 16, 2006, 6:25 p.m.)
Deposited	3 years, 4 months ago (April 7, 2022, 9:07 a.m.)
Indexed	1 month, 1 week ago (July 19, 2025, 11:24 p.m.)
Issued	19 years, 10 months ago (Oct. 26, 2005)
Published	19 years, 10 months ago (Oct. 26, 2005)
Published Online	19 years, 10 months ago (Oct. 26, 2005)
Published Print	19 years, 1 month ago (July 1, 2006)

Funders 0

None

BibTeX

@article{Liu_2005, title={Probabilistic Analytical Target Cascading: A Moment Matching Formulation for Multilevel Optimization Under Uncertainty}, volume={128}, ISSN={1528-9001}, url={http://dx.doi.org/10.1115/1.2205870}, DOI={10.1115/1.2205870}, number={4}, journal={Journal of Mechanical Design}, publisher={ASME International}, author={Liu, Huibin and Chen, Wei and Kokkolaras, Michael and Papalambros, Panos Y. and Kim, Harrison M.}, year={2005}, month=oct, pages={991–1000} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        7,
        20
      ]
    ],
    "date-time": "2025-07-20T03:24:05Z",
    "timestamp": 1752981845195
  },
  "reference-count": 43,
  "publisher": "ASME International",
  "issue": "4",
  "content-domain": {
    "domain": [
      "asmedigitalcollection.asme.org"
    ],
    "crossmark-restriction": true
  },
  "published-print": {
    "date-parts": [
      [
        2006,
        7,
        1
      ]
    ]
  },
  "abstract": "<jats:title>Abstract</jats:title>\n               <jats:p>Analytical target cascading (ATC) is a methodology for hierarchical multilevel system design optimization. In previous work, the deterministic ATC formulation was extended to account for random variables represented by expected values to be matched among subproblems and thus ensure design consistency. In this work, the probabilistic formulation is augmented to allow the introduction and matching of additional probabilistic characteristics. A particular probabilistic analytical target cascading (PATC) formulation is proposed that matches the first two moments of interrelated responses and linking variables. Several implementation issues are addressed, including representation of probabilistic design targets, matching responses and linking variables under uncertainty, and coordination strategies. Analytical and simulation-based optimal design examples are used to illustrate the new formulation. The accuracy of the proposed PATC formulation is demonstrated by comparing PATC results to those obtained using a probabilistic all-in-one formulation.</jats:p>",
  "DOI": "10.1115/1.2205870",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2006,
        6,
        16
      ]
    ],
    "date-time": "2006-06-16T22:25:35Z",
    "timestamp": 1150496735000
  },
  "page": "991-1000",
  "update-policy": "http://dx.doi.org/10.1115/crossmarkpolicy-asme",
  "source": "Crossref",
  "is-referenced-by-count": 91,
  "title": "Probabilistic Analytical Target Cascading: A Moment Matching\n                    Formulation for Multilevel Optimization Under Uncertainty",
  "prefix": "10.1115",
  "volume": "128",
  "author": [
    {
      "given": "Huibin",
      "family": "Liu",
      "sequence": "first",
      "affiliation": [
        {
          "name": "Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208-3111"
        }
      ]
    },
    {
      "given": "Wei",
      "family": "Chen",
      "sequence": "additional",
      "affiliation": [
        {
          "name": "Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208-3111"
        }
      ]
    },
    {
      "given": "Michael",
      "family": "Kokkolaras",
      "sequence": "additional",
      "affiliation": [
        {
          "name": "Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125"
        }
      ]
    },
    {
      "given": "Panos Y.",
      "family": "Papalambros",
      "sequence": "additional",
      "affiliation": [
        {
          "name": "Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125"
        }
      ]
    },
    {
      "given": "Harrison M.",
      "family": "Kim",
      "sequence": "additional",
      "affiliation": [
        {
          "name": "Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801"
        }
      ]
    }
  ],
  "member": "33",
  "published-online": {
    "date-parts": [
      [
        2005,
        10,
        26
      ]
    ]
  },
  "reference": [
    {
      "issue": "1",
      "key": "2022040713072957400_c1",
      "doi-asserted-by": "crossref",
      "first-page": "6",
      "DOI": "10.2514/3.13015",
      "article-title": "Optimization of\n                        Coupled Systems: A Critical Overview of Approaches",
      "volume": "34",
      "author": "Balling",
      "journal-title": "AIAA J.",
      "ISSN": "http://id.crossref.org/issn/0001-1452",
      "issn-type": "print"
    },
    {
      "key": "2022040713072957400_c2",
      "unstructured": "Alexandrov, N.\n                                M., and Lewis, R.\n                                M., 1999,\n                        \u201cComparative Properties of Collaborative Optimization and\n                        Other Approaches to MDO,\u201d ICASE Report, No. 99-24,\n                    NASA/CR-1999-209354."
    },
    {
      "issue": "4",
      "key": "2022040713072957400_c3",
      "doi-asserted-by": "publisher",
      "first-page": "754",
      "DOI": "10.1137/0804044",
      "article-title": "Problem Formulation for Multidisciplinary\n                        Optimization",
      "volume": "4",
      "author": "Cramer",
      "journal-title": "SIAM J. Optim.",
      "ISSN": "http://id.crossref.org/issn/1052-6234",
      "issn-type": "print"
    },
    {
      "key": "2022040713072957400_c4",
      "doi-asserted-by": "crossref",
      "unstructured": "Kroo,\n                            I.,\n                                Altus,\n                            S.,\n                                Braun,\n                            R.,\n                                Gage,\n                            P., and\n                                Sobieski,\n                                I.,\n                        1994, \u201dMultidisciplinary\n                        Optimization Methods for Aircraft Preliminary Design,\u201d 5th\n                    AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization,\n                    paper no. AIAA 94-4325, Panama City, Florida.",
      "DOI": "10.2514/6.1994-4325"
    },
    {
      "key": "2022040713072957400_c5",
      "doi-asserted-by": "crossref",
      "unstructured": "Braun, R.\n                                D., Kroo, I.\n                                M., and Moore, A.\n                                A., 1996,\n                        \u201cUse of the Collaborative Optimization Architecture for\n                        Launch Vehicle Design,\u201d 6th AIAA/USAF/NASA/ISSMO Symposium\n                    on Multidisciplinary Analysis and Optimization, paper no. AIAA 96-4018,\n                    Bellevue, WA.",
      "DOI": "10.2514/6.1996-4018"
    },
    {
      "key": "2022040713072957400_c6",
      "first-page": "97",
      "volume-title": "AIAA\n                        Pap.",
      "author": "Kroo"
    },
    {
      "issue": "3",
      "key": "2022040713072957400_c7",
      "first-page": "403",
      "article-title": "Multiobjective Collaborative Optimization",
      "volume": "119",
      "author": "Renaud",
      "journal-title": "ASME J. Mech. Des.",
      "ISSN": "http://id.crossref.org/issn/1050-0472",
      "issn-type": "print"
    },
    {
      "key": "2022040713072957400_c8",
      "first-page": "293",
      "article-title": "Target Cascading\n                        in Optimal System Design",
      "volume-title": "Proceedings of the ASME\n                        Design Engineering Technical Conferences",
      "author": "Kim"
    },
    {
      "key": "2022040713072957400_c9",
      "unstructured": "Kim, H.\n                                M.\n          , 2001,\n                    Target Cascading in Optimal System Design, Ph.D. thesis, University of Michigan,\n                    Ann Arbor, Michigan."
    },
    {
      "key": "2022040713072957400_c10",
      "first-page": "661",
      "article-title": "Analytical Target Cascading in Automotive Vehicle\n                        Design",
      "volume-title": "Proceedings of the ASME Design Engineering\n                        Technical Conferences",
      "author": "Kim"
    },
    {
      "issue": "3",
      "key": "2022040713072957400_c11",
      "doi-asserted-by": "publisher",
      "first-page": "199",
      "DOI": "10.1504/IJVD.2002.002010",
      "article-title": "Target Cascading in Vehicle Redesign: A Class VI Truck\n                        Study",
      "volume": "29",
      "author": "Kim",
      "journal-title": "Int. J. Veh. Des.",
      "ISSN": "http://id.crossref.org/issn/0143-3369",
      "issn-type": "print"
    },
    {
      "issue": "4",
      "key": "2022040713072957400_c12",
      "doi-asserted-by": "crossref",
      "first-page": "293",
      "DOI": "10.1007/s00158-002-0240-0",
      "article-title": "Extension of the Target Cascading Formulation to the Design\n                        of Product Families",
      "volume": "24",
      "author": "Kokkolaras",
      "journal-title": "Struct. Multidiscip.\n                        Optim.",
      "ISSN": "http://id.crossref.org/issn/1615-147X",
      "issn-type": "print"
    },
    {
      "issue": "3,4",
      "key": "2022040713072957400_c13",
      "first-page": "402",
      "article-title": "Simulation-Based Optimal Design of Heavy Trucks by\n                        Model-Based Decomposition: An Extensive Analytical Target Cascading Case\n                        Study",
      "volume": "11",
      "author": "Kokkolaras",
      "journal-title": "International Journal of Heavy Vehicle\n                        Systems"
    },
    {
      "key": "2022040713072957400_c14",
      "unstructured": "Wagner, T.\n                                C.\n          , 1993, A\n                    General Decomposition Methodology for Optimal System Design, Ph.D. thesis,\n                    University of Michigan, Ann Arbor, Michigan."
    },
    {
      "key": "2022040713072957400_c15",
      "doi-asserted-by": "crossref",
      "DOI": "10.1115/DETC2003/DAC-48734",
      "article-title": "Analytical Target Setting: An Enterprise Context in Optimal\n                        Product Design",
      "volume-title": "Proceedings of the ASME Design\n                        Engineering Technical Conferences",
      "author": "Cooper"
    },
    {
      "issue": "3",
      "key": "2022040713072957400_c16",
      "doi-asserted-by": "publisher",
      "first-page": "490",
      "DOI": "10.1115/1.1587156",
      "article-title": "An Approach to\n                        Decision-Based Design with Discrete Choice Analysis for Demand\n                        Modeling",
      "volume": "125",
      "author": "Wassenaar",
      "journal-title": "ASME J. Mech. Des.",
      "ISSN": "http://id.crossref.org/issn/1050-0472",
      "issn-type": "print"
    },
    {
      "key": "2022040713072957400_c17",
      "first-page": "309",
      "article-title": "An Investigation of Multidisciplinary Design Subject to\n                        Uncertainty",
      "volume-title": "AIAA-1998-4747, 7th\n                        AIAA/USAF/NASA/ISSMO Multidisciplinary Analysis and Optimization\n                        Symposium",
      "author": "Gu"
    },
    {
      "issue": "8",
      "key": "2022040713072957400_c18",
      "doi-asserted-by": "crossref",
      "first-page": "982",
      "DOI": "10.2514/2.805",
      "article-title": "Robust Design\n                        Approach for Achieving Flexibility in Multidisciplinary\n                        Design",
      "volume": "37",
      "author": "Chen",
      "journal-title": "AIAA J.",
      "ISSN": "http://id.crossref.org/issn/0001-1452",
      "issn-type": "print"
    },
    {
      "issue": "3",
      "key": "2022040713072957400_c19",
      "doi-asserted-by": "crossref",
      "first-page": "545",
      "DOI": "10.2514/2.1681",
      "article-title": "Efficient\n                        Uncertainty Analysis Methods for Multidisciplinary Robust\n                        Design",
      "volume": "40",
      "author": "Du",
      "journal-title": "AIAA J.",
      "ISSN": "http://id.crossref.org/issn/0001-1452",
      "issn-type": "print"
    },
    {
      "key": "2022040713072957400_c20",
      "doi-asserted-by": "crossref",
      "unstructured": "Batill,\n                                S.,\n                                Renaud, J.\n                                E., and Gu,\n                                X.,\n                        2000, \u201cModeling and\n                        Simulation Uncertainty in Multidisciplinary Design\n                        Optimization,\u201d 8th AIAA/NASA/USAF/ISSMO Multidisciplinary\n                    Analysis and Optimization Conference and Exhibit, paper no. AIAA 2000-4803, Long\n                    Beach, California.",
      "DOI": "10.2514/6.2000-4803"
    },
    {
      "key": "2022040713072957400_c21",
      "doi-asserted-by": "crossref",
      "DOI": "10.2514/6.2000-4841",
      "article-title": "An Iterative Concurrent Subspace Robust Design\n                        Framework",
      "volume-title": "Proceedings of 8th AIAA/NASA/USAF/ISSMO\n                        Multidisciplinary Analysis and Optimization Conference and Exhibit",
      "author": "Padmanabhan"
    },
    {
      "key": "2022040713072957400_c22",
      "doi-asserted-by": "crossref",
      "DOI": "10.2514/6.2004-4472",
      "article-title": "An Approach to\n                        Robust Decision Making in Multidisciplinary Selection Problems under\n                        Uncertainty",
      "volume-title": "Proceedings of 10th AIAA/ISSMO\n                        Multidisciplinary Analysis and Optimization Conference",
      "author": "Gurnani"
    },
    {
      "key": "2022040713072957400_c23",
      "unstructured": "Sues,\n                            R.,\n                                Cesare,\n                                M.,\n                                Pageau,\n                                S., and\n                                Wu,\n                            J.,\n                        2000, \u201cReliability Based\n                        MDO for Aerospace Systems,\u201d 8th Symposium on\n                    Multidisciplinary Analysis and Optimization, AIAA 2000-4804, Long Beach,\n                    California."
    },
    {
      "key": "2022040713072957400_c24",
      "unstructured": "Chiralaksanakul,\n                                A., and\n                                Mahadevan,\n                                S.,\n                        2004, \u201cMultidisciplinary\n                        Design Optimization Under Uncertainty,\u201d 10th AIAA/ISSMO\n                    Multidisciplinary Analysis and Optimization Conference, paper no.\n                    AIAA-2004-4311, Albany, NY."
    },
    {
      "issue": "2",
      "key": "2022040713072957400_c25",
      "doi-asserted-by": "publisher",
      "first-page": "503",
      "DOI": "10.1115/1.2168470",
      "article-title": "Design Optimization of Hierarchically Decomposed Multilevel\n                        System under Uncertainty",
      "volume": "128",
      "author": "Kokkolaras",
      "journal-title": "ASME J. Mech.\n                        Des.",
      "ISSN": "http://id.crossref.org/issn/1050-0472",
      "issn-type": "print"
    },
    {
      "key": "2022040713072957400_c26",
      "volume-title": "Quality Engineering Using Robust Design",
      "author": "Phadke"
    },
    {
      "key": "2022040713072957400_c27",
      "doi-asserted-by": "crossref",
      "unstructured": "Kim,\n                            H.,\n                                Kumar,\n                            D.,\n                                Chen,\n                            W., and\n                                Papalambros, P.\n                                Y., 2004,\n                        \u201cTarget Feasibility Achievement in Enterprise-Driven\n                        Hierarchical Multidisciplinary Design,\u201d 10th AIAA/ISSMO\n                    Multidisciplinary Analysis and Optimization Conference, paper no.\n                    AIAA-2004-4546, Albany, NY.",
      "DOI": "10.2514/6.2004-4546"
    },
    {
      "key": "2022040713072957400_c28",
      "unstructured": "Allison,\n                                J.\n          ,\n                        2004, Complex System Optimization: A\n                    Comparison of Analytical Target Cascading, Collaborative Optimization, and Other\n                    Formulations, MS thesis, University of Michigan, Ann Arbor,\n                    Michigan."
    },
    {
      "issue": "3",
      "key": "2022040713072957400_c29",
      "doi-asserted-by": "publisher",
      "first-page": "499",
      "DOI": "10.1115/1.1862674",
      "article-title": "Technical Brief: Weights, Norms, and Notation in Analytical\n                        Target Cascading",
      "volume": "127",
      "author": "Michalek",
      "journal-title": "ASME J. Mech. Des.",
      "ISSN": "http://id.crossref.org/issn/1050-0472",
      "issn-type": "print"
    },
    {
      "key": "2022040713072957400_c30",
      "volume-title": "Probability Concepts in Engineering Planning and Design, Volume\n                        I\u2013Basic Principles",
      "author": "Ang"
    },
    {
      "issue": "4",
      "key": "2022040713072957400_c31",
      "doi-asserted-by": "publisher",
      "first-page": "385",
      "DOI": "10.1115/1.1290247",
      "article-title": "Towards a Better\n                        Understanding of Modeling Feasibility Robustness in\n                        Engineering",
      "volume": "122",
      "author": "Du",
      "journal-title": "ASME J. Mech. Des.",
      "ISSN": "http://id.crossref.org/issn/1050-0472",
      "issn-type": "print"
    },
    {
      "key": "2022040713072957400_c32",
      "doi-asserted-by": "publisher",
      "first-page": "1",
      "DOI": "10.1115/1.1334596",
      "article-title": "A Comprehensive\n                        Robust Design Approach for Decision Trade-Offs in Complex Systems\n                        Design",
      "volume": "123",
      "author": "Kalsi",
      "journal-title": "ASME J. Mech. Des.",
      "ISSN": "http://id.crossref.org/issn/1050-0472",
      "issn-type": "print"
    },
    {
      "key": "2022040713072957400_c33",
      "doi-asserted-by": "publisher",
      "first-page": "124",
      "DOI": "10.1115/1.1543978",
      "article-title": "Multidisciplinary Robust Design Optimization of an Internal\n                        Combustion Engine",
      "volume": "125",
      "author": "McAllister",
      "journal-title": "ASME J. Mech. Des.",
      "ISSN": "http://id.crossref.org/issn/1050-0472",
      "issn-type": "print"
    },
    {
      "issue": "4",
      "key": "2022040713072957400_c34",
      "doi-asserted-by": "publisher",
      "first-page": "562",
      "DOI": "10.1115/1.1759358",
      "article-title": "An Integrated\n                        Framework for Optimization Using Inverse Reliability\n                        Strategy",
      "volume": "126",
      "author": "Du",
      "journal-title": "ASME J. Mech. Des.",
      "ISSN": "http://id.crossref.org/issn/1050-0472",
      "issn-type": "print"
    },
    {
      "issue": "4",
      "key": "2022040713072957400_c35",
      "doi-asserted-by": "publisher",
      "first-page": "514",
      "DOI": "10.1115/1.1897408",
      "article-title": "Enhancing\n                        Discrete Choice Demand Modeling for Decision-Based Design",
      "volume": "127",
      "author": "Wassenaar",
      "journal-title": "ASME J. Mech. Des.",
      "ISSN": "http://id.crossref.org/issn/1050-0472",
      "issn-type": "print"
    },
    {
      "key": "2022040713072957400_c36",
      "doi-asserted-by": "crossref",
      "article-title": "An Integrated\n                        Latent Variable Choice Modeling Approach to Enhancing Product Demand\n                        Modeling",
      "author": "Wassenaar",
      "DOI": "10.1115/DETC2004-57487"
    },
    {
      "key": "2022040713072957400_c37",
      "doi-asserted-by": "crossref",
      "DOI": "10.2514/6.2002-5506",
      "article-title": "Convergence\n                        Properties of Analytical Target Cascading",
      "volume-title": "Proceedings of the 9th AIAA/ISSMO Symposium on Multidisciplinary\n                        Analysis and Optimization",
      "author": "Michelena"
    },
    {
      "key": "2022040713072957400_c38",
      "doi-asserted-by": "crossref",
      "first-page": "9",
      "DOI": "10.1023/B:OPTE.0000048534.58121.93",
      "article-title": "Multidisciplinary Design Optimization with Quasiseparable\n                        Subsystems",
      "volume": "6",
      "author": "Haftka",
      "journal-title": "Opt. Eng.",
      "ISSN": "http://id.crossref.org/issn/0091-3286",
      "issn-type": "print"
    },
    {
      "key": "2022040713072957400_c39",
      "volume-title": "Applied Geometric Programming",
      "author": "Beightler"
    },
    {
      "key": "2022040713072957400_c40",
      "volume-title": "Statistics and Data Analysis from Elementary to\n                        Intermediate",
      "author": "Tamhane"
    },
    {
      "key": "2022040713072957400_c41",
      "doi-asserted-by": "crossref",
      "DOI": "10.1007/s00158-005-0579-0",
      "article-title": "An Augmented Lagrangian Relaxation for Analytical Target\n                        Cascading using the Alternating Directions Method of\n                        Multipliers",
      "volume-title": "Proceedings of the 6th World\n                        Conference on Structural and Multidisciplinary Optimization",
      "author": "Tosserams"
    },
    {
      "key": "2022040713072957400_c42",
      "doi-asserted-by": "crossref",
      "DOI": "10.2514/6.2006-7086",
      "article-title": "Lagrangian\n                        Coordination for Enhancing the Convergence of Analytical Target\n                        Cascading",
      "volume-title": "Proceedings of the IFORS",
      "author": "Kim"
    },
    {
      "issue": "2",
      "key": "2022040713072957400_c43",
      "doi-asserted-by": "publisher",
      "first-page": "225",
      "DOI": "10.1115/1.1649968",
      "article-title": "Sequential\n                        Optimization and Reliability Assessment Method for Efficient Probabilistic\n                        Design",
      "volume": "126",
      "author": "Du",
      "journal-title": "ASME J. Mech. Des.",
      "ISSN": "http://id.crossref.org/issn/1050-0472",
      "issn-type": "print"
    }
  ],
  "container-title": "Journal of Mechanical Design",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://asmedigitalcollection.asme.org/mechanicaldesign/article-pdf/128/4/991/6870818/991_1.pdf",
      "content-type": "application/pdf",
      "content-version": "vor",
      "intended-application": "syndication"
    },
    {
      "URL": "https://asmedigitalcollection.asme.org/mechanicaldesign/article-pdf/128/4/991/6870818/991_1.pdf",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2022,
        4,
        7
      ]
    ],
    "date-time": "2022-04-07T13:07:54Z",
    "timestamp": 1649336874000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://asmedigitalcollection.asme.org/mechanicaldesign/article/128/4/991/471952/Probabilistic-Analytical-Target-Cascading-A-Moment"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        2005,
        10,
        26
      ]
    ]
  },
  "references-count": 43,
  "journal-issue": {
    "issue": "4",
    "published-print": {
      "date-parts": [
        [
          2006,
          7,
          1
        ]
      ]
    }
  },
  "URL": "http://dx.doi.org/10.1115/1.2205870",
  "relation": {},
  "ISSN": [
    "1050-0472",
    "1528-9001"
  ],
  "subject": [],
  "published": {
    "date-parts": [
      [
        2005,
        10,
        26
      ]
    ]
  }
}