Skip to main content
Log in

Quantifying tradeoffs to reduce the dimensionality of complex design optimization problems and expedite trade space exploration

  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript


Multi-objective optimization is increasingly being employed to solve complex design problems. However, multi-objective optimization problems might be formulated with extraneous objective functions that could be eliminated without affecting the final solution. Furthermore, as the number of objectives increases, the effort required to visualize and explore the resulting solution set, herein referred to as the trade space, increases. Although visual analytic techniques exist to facilitate analytical reasoning for exploring a high-dimensional trade space through graphical interfaces, existing techniques often rely upon exhaustive two-dimensional representations to identify all tradeoffs. Yet, the knowledge of the tradeoffs among competing objectives is important for decision-making because it fosters learning from the trade space and hence aids preference formation. In this paper, an index to quantify the tradeoff between any two objectives from multi-objective optimization is presented and incorporated into a visual analytic technique that can be used as a tool for reducing the dimensionality of the problem formulation. The tradeoff index reveals the conflicting and correlated objectives; hence, it can be used to expedite trade space exploration by focusing cognitive effort only on those objectives that have tradeoff. The utility and efficiency of the proposed technique is illustrated through application to a Pareto approximate solution set from a benchmark optimization problem with eight objectives. The results of this exercise are compared to the solutions obtained using other existing visual analytic techniques found in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others


  • Agrawal G, Parashar S, Bloebaum CL (2006) Intuitive visualization of hyperspace pareto frontier for robustness in multi-attribute decision-making. 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Portsmouth, VA, AIAA, AIAA-2006-6962

  • Bader J, Zitzler E (2010) A hypervolume-based optimizer for high-dimensional objective spaces. New developments in multiple objective and goal programming. Springer, Berlin Heidelberg, pp 35–54

    Book  Google Scholar 

  • Blasco X, Herrero JM, Sanchis J, Martínez M (2008) A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization. Inf Sci 178(20):3908–3924

    Article  MATH  Google Scholar 

  • Chambers JM, Cleveland WS, Kleiner B, Tukey PA (1983) Graphical methods for data analysis. Chapman and Hall, New York

    MATH  Google Scholar 

  • Chipperfield AJ, Bica B, Fleming PJ (2002) Fuzzy scheduling control of a gas turbine aero-engine: a multiobjective approach. IEEE Trans Ind Electron 49(3):536–548

    Article  Google Scholar 

  • Chiu P-W, Bloebaum, CL (2008) Hyper-radial visualization (HRV) with weighted preferences for multi-objective decision making. 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Victoria, British Columbia, Canada, AIAA, AIAA-2008-5986

  • Coello Coello CA, Lamont GB, Van Veldhuizen DA (2007) Evolutionary algorithms for solving multi-objective problems, Springer

  • Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast elitist multi-objective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197

    Article  Google Scholar 

  • Fleming PJ, Purshouse RC, Lygoe RJ (2005) Many-objective optimization: an engineering design perspective. Evolutionary multi-criterion optimization. Springer, Berlin, Heidelberg, pp 14–32

    Book  MATH  Google Scholar 

  • Fu G, Kapelan Z, Kasprzyk JR, Reed P (2012) Optimal design of water distribution systems using many-objective visual analytics. J Water Resour Plan Manag 139(6):624–633

    Article  Google Scholar 

  • Inselberg A (1997) Multidimensional detective. IEEE Symposium on Information Visualization, pp 100–107

  • Kohonen T (2001) Self-organizing maps. Springer

  • Kollat JB, Reed P (2007) A framework for visually interactive decision-making and design using evolutionary multi-objective optimization (VIDEO). Environ Model Softw 22(12):1691–1704

    Article  Google Scholar 

  • Kollat JB, Reed PM, Maxwell RM (2011) Many‐objective groundwater monitoring network design using bias‐aware ensemble Kalman filtering, evolutionary optimization, and visual analytics. Water Resour Res 47(2):W02529

    Article  Google Scholar 

  • Laumanns M, Laumanns N (2005) Evolutionary multiobjective design in automotive development. Appl Intell 23(1):55–70

    Article  MATH  Google Scholar 

  • Matlab. Matlab version 7.14 R2012a. The Mathworks Inc 2012. Matick, M.A.: Natick, MA. Accessed 8 Feb 2006

  • Pareto V (1971) Manual of political economy. A. M. Kelley, New York

    Google Scholar 

  • Purshouse RC, Fleming PJ (2003) Conflict, harmony, and independence: relationships in evolutionary multi-criterion optimisation. Evolutionary multi-criterion optimization. Springer, Berlin Heidelberg, pp 16–30

    MATH  Google Scholar 

  • Richardson T, Nekolny B, Holub J, Winer EH (2014) Visualizing design spaces using two-dimensional contextual self-organizing maps. AIAA J 52(4):725–738

    Article  Google Scholar 

  • Shah R, Reed PM, Simpson TW (2011) Many-objective evolutionary optimization and visual analytics for product family design. In: Wang L, Ng A, Deb K (eds) Multi-objective evolutionary optimisation for product design and manufacturing. Springer, London, pp 137–159

    Chapter  Google Scholar 

  • Stump GM, Yukish M, Simpson TW, Bennett L (2002) Multidimensional visualization and its application to a design by shopping paradigm. 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, AIAA-2002-5622

  • Stump GM, Simpson TW, Donndelinger JA, Lego S, Yukish M (2009) Visual steering commands for trade space exploration: user-guided sampling with example. ASME J Comput Inf Sci Eng 9(4):044501

    Article  Google Scholar 

  • Thomas J, Wong PC (2004) Visual analytics. IEEE Comput Graph Appl 24(5):0020–0021

    Article  Google Scholar 

  • Unal M, Warn G, Simpson TW (2015) Introduction of a tradeoff index for efficient trade space exploration. ASME International Design Engineering Technical Conferences, Boston, Massachusetts, IDETC/CIE 2015

  • Winer EH, Bloebaum CL (2001) Visual design steering for optimization solution improvement. Struct Multidiscip Optim 22(3):219–229

    Article  Google Scholar 

  • Winer EH, Bloebaum CL (2002) Development of visual design steering as an aid in large-scale multidisciplinary design optimization. Part I: method development. Struct Multidiscip Optim 23(6):412–424

    Article  Google Scholar 

  • Woodruff MJ, Reed PM, Simpson TW (2013) Many objective visual analytics: rethinking the design of complex engineered systems. Struct Multidiscip Optim 48:201–219

    Article  Google Scholar 

  • Zitzler E, Thiele L, Laumanns M, Fonseca CM, Da Fonseca VG (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(2):117–132

    Article  Google Scholar 

Download references


The authors acknowledge support from the National Science Foundation (NSF) under NSF Grants CMMI-1436236 and CMMI-1455444. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Mehmet Unal.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Unal, M., Warn, G.P. & Simpson, T.W. Quantifying tradeoffs to reduce the dimensionality of complex design optimization problems and expedite trade space exploration. Struct Multidisc Optim 54, 233–248 (2016).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: