Info-gap decision theory explained

Info-gap decision theory seeks to optimize robustness to failure under severe uncertainty,[1] [2] in particular applying sensitivity analysis of the stability radius type[3] to perturbations in the value of a given estimate of the parameter of interest. It has some connections with Wald's maximin model; some authors distinguish them, others consider them instances of the same principle.

It was developed by Yakov Ben-Haim,[4] and has found many applications and described as a theory for decision-making under "severe uncertainty". It has been criticized as unsuited for this purpose, and alternatives proposed, including such classical approaches as robust optimization.

Applications

Info-gap theory has generated a lot of literature. Info-gap theory has been studied or applied in a range of applications including engineering, [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] biological conservation,[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] theoretical biology,[30] homeland security,[31] economics,[32] [33] [34] project management[35] [36] [37] and statistics.[38] Foundational issues related to info-gap theory have also been studied.[39] [40] [41] [42] [43] [44]

Engineering

A typical engineering application is the vibration analysis of a cracked beam, where the location, size, shape and orientation of the crack is unknown and greatly influence the vibration dynamics.[8] Very little is usually known about these spatial and geometrical uncertainties. The info-gap analysis allows one to model these uncertainties, and to determine the degree of robustness - to these uncertainties - of properties such as vibration amplitude, natural frequencies, and natural modes of vibration. Another example is the structural design of a building subject to uncertain loads such as from wind or earthquakes.[7] [9] The response of the structure depends strongly on the spatial and temporal distribution of the loads. However, storms and earthquakes are highly idiosyncratic events, and the interaction between the event and the structure involves very site-specific mechanical properties which are rarely known. The info-gap analysis enables the design of the structure to enhance structural immunity against uncertain deviations from design-base or estimated worst-case loads. Another engineering application involves the design of a neural net for detecting faults in a mechanical system, based on real-time measurements. A major difficulty is that faults are highly idiosyncratic, so that training data for the neural net will tend to differ substantially from data obtained from real-time faults after the net has been trained. The info-gap robustness strategy enables one to design the neural net to be robust to the disparity between training data and future real events.[10] [12]

Biology

The conservation biologist faces info-gaps in using biological models. They use info-gap robustness curves to select among management options for spruce-budworm populations in Eastern Canada. Burgman[45] uses the fact that the robustness curves of different alternatives can intersect.

Project management

Project management is another area where info-gap uncertainty is common. The project manager often has very limited information about the duration and cost of some of the tasks in the project, and info-gap robustness can assist in project planning and integration.[36] Financial economics is another area where the future is unpredictable, which may be either pernicious or propitious. Info-gap robustness and opportuneness analyses can assist in portfolio design, credit rationing, and other applications.[32]

Criticism

A general criticism of non-probabilistic decision rules, discussed in detail at decision theory: alternatives to probability theory, is that optimal decision rules (formally, admissible decision rules) can always be derived by probabilistic methods, with a suitable utility function and prior distribution (this is the statement of the complete class theorems), and thus that non-probabilistic methods such as info-gap are unnecessary and do not yield new or better decision rules.

A more general criticism of decision making under uncertainty is the impact of outsized, unexpected events, ones that are not captured by the model. This is discussed particularly in black swan theory, and info-gap, used in isolation, is vulnerable to this, as are a fortiori all decision theories that use a fixed universe of possibilities, notably probabilistic ones.

Sniedovich raises two points to info-gap decision theory, one substantive, one scholarly:

1. the info-gap uncertainty model is flawed and oversold: One should consider the range of possibilities, not its subsets. Sniedovich argues that info-gap decision theory is therefore a "voodoo decision theory."
  • 2. info-gap is maximin: Ben-Haim states (Ben-Haim 1999, pp. 271–2) that "robust reliability is emphatically not a [min-max] worst-case analysis". Note that Ben-Haim compares info-gap to minimax, while Sniedovich considers it a case of maximin.

    • Sniedovich has challenged the validity of info-gap theory for making decisions under severe uncertainty. Sniedovich notes that the info-gap robustness function is "local" to the region around

    \displaystyle\tilde{u}

    , where

    \displaystyle\tilde{u}

    is likely to be substantially in error.

    Maximin

    Symbolically, max

    \alpha

    assuming min (worst-case) outcome, or maximin.

    In other words, while it is not a maximin analysis of outcome over the universe of uncertainty, it is a maximin analysis over a properly construed decision space.

    Ben-Haim argues that info-gap's robustness model is not min-max/maximin analysis because it is not worst-case analysis of outcomes; it is a satisficing model, not an optimization model – a (straightforward) maximin analysis would consider worst-case outcomes over the entire space which, since uncertainty is often potentially unbounded, would yield an unbounded bad worst case.

    Stability radius

    Sniedovich has shown that info-gap's robustness model is a simple stability radius model, namely a local stability model of the generic form

    \hat{\rho}(\tilde{p}):=max \{\rho\ge0:p\inP(s),\forallp\inB(\rho,\tilde{p})\}

    where

    B(\rho,\tilde{p})

    denotes a ball of radius

    \rho

    centered at

    \tilde{p}

    and

    P(s)

    denotes the set of values of

    p

    that satisfy pre-determined stability conditions.

    In other words, info-gap's robustness model is a stability radius model characterized by a stability requirement of the form

    rc\leR(q,p)

    . Since stability radius models are designed for the analysis of small perturbations in a given nominal value of a parameter, Sniedovich argues that info-gap's robustness model is unsuitable for the treatment of severe uncertainty characterized by a poor estimate and a vast uncertainty space.

    Discussion

    Satisficing and bounded rationality

    It is correct that the info-gap robustness function is local, and has restricted quantitative value in some cases. However, a major purpose of decision analysis is to provide focus for subjective judgments. That is, regardless of the formal analysis, a framework for discussion is provided. Without entering into any particular framework, or characteristics of frameworks in general, discussion follows about proposals for such frameworks.

    Simon [46] introduced the idea of bounded rationality. Limitations on knowledge, understanding, and computational capability constrain the ability of decision makers to identify optimal choices. Simon advocated satisficing rather than optimizing: seeking adequate (rather than optimal) outcomes given available resources. Schwartz,[47] Conlisk[48] and others discuss extensive evidence for the phenomenon of bounded rationality among human decision makers, as well as for the advantages of satisficing when knowledge and understanding are deficient. The info-gap robustness function provides a means of implementing a satisficing strategy under bounded rationality. For instance, in discussing bounded rationality and satisficing in conservation and environmental management, Burgman notes that "Info-gap theory ... can function sensibly when there are 'severe' knowledge gaps." The info-gap robustness and opportuneness functions provide "a formal framework to explore the kinds of speculations that occur intuitively when examining decision options."[49] Burgman then proceeds to develop an info-gap robust-satisficing strategy for protecting the endangered orange-bellied parrot. Similarly, Vinot, Cogan and Cipolla [50] discuss engineering design and note that "the downside of a model-based analysis lies in the knowledge that the model behavior is only an approximation to the real system behavior. Hence the question of the honest designer: how sensitive is my measure of design success to uncertainties in my system representation? ... It is evident that if model-based analysis is to be used with any level of confidence then ... [one must] attempt to satisfy an acceptable sub-optimal level of performance while remaining maximally robust to the system uncertainties." They proceed to develop an info-gap robust-satisficing design procedure for an aerospace application.

    Alternatives

    Of course, decision in the face of uncertainty is nothing new, and attempts to deal with it have a long history. A number of authors have noted and discussed similarities and differences between info-gap robustness and minimax or worst-case methods[6] [34] [36] [51] .[52] Sniedovich [53] has demonstrated formally that the info-gap robustness function can be represented as a maximin optimization, and is thus related to Wald's minimax theory. Sniedovich [53] has claimed that info-gap's robustness analysis is conducted in the neighborhood of an estimate that is likely to be substantially wrong, concluding that the resulting robustness function is equally likely to be substantially wrong. On the other hand, the estimate is the best one has, so it is useful to know if it can err greatly and still yield an acceptable outcome. This critical question clearly raises the issue of whether robustness (as defined by info-gap theory) is qualified to judge whether confidence is warranted,[54] [55] [56] and how it compares to methods used to inform decisions under uncertainty using considerations not limited to the neighborhood of a bad initial guess. Answers to these questions vary with the particular problem at hand. Some general comments follow.

    Sensitivity analysis

    See main article: Sensitivity analysis.

    Sensitivity analysis – how sensitive conclusions are to input assumptions – can be performed independently of a model of uncertainty: most simply, one may take two different assumed values for an input and compares the conclusions. From this perspective, info-gap can be seen as a technique of sensitivity analysis, though by no means the only.

    Robust optimization

    See main article: Robust optimization. The robust optimization literature [57] [58] [59] [60] [61] [62] provides methods and techniques that take a global approach to robustness analysis. These methods directly address decision under severe uncertainty, and have been used for this purpose for more than thirty years now. Wald's Maximin model is the main instrument used by these methods. The principal difference between the Maximin model employed by info-gap and the various Maximin models employed by robust optimization methods is in the manner in which the total region of uncertainty is incorporated in the robustness model. Info-gap takes a local approach that concentrates on the immediate neighborhood of the estimate. In sharp contrast, robust optimization methods set out to incorporate in the analysis the entire region of uncertainty, or at least an adequate representation thereof. In fact, some of these methods do not even use an estimate.

    Comparative analysis

    Classical decision theory,[63] [64] offers two approaches to decision-making under severe uncertainty, namely maximin and Laplaces' principle of insufficient reason (assume all outcomes equally likely); these may be considered alternative solutions to the problem info-gap addresses.

    Further, as discussed at decision theory: alternatives to probability theory, probabilists, particularly Bayesians probabilists, argue that optimal decision rules (formally, admissible decision rules) can always be derived by probabilistic methods (this is the statement of the complete class theorems), and thus that non-probabilistic methods such as info-gap are unnecessary and do not yield new or better decision rules.

    Maximin

    As attested by the rich literature on robust optimization, maximin provides a wide range of methods for decision making in the face of severe uncertainty.

    Indeed, as discussed in criticism of info-gap decision theory, info-gap's robustness model can be interpreted as an instance of the general maximin model.

    Bayesian analysis

    As for Laplaces' principle of insufficient reason, in this context it is convenient to view it as an instance of Bayesian analysis.

    The essence of the Bayesian analysis is applying probabilities for different possible realizations of the uncertain parameters. In the case of Knightian (non-probabilistic) uncertainty, these probabilities represent the decision maker's "degree of belief" in a specific realization.

    In our example, suppose there are only five possible realizations of the uncertain revenue to allocation function. The decision maker believes that the estimated function is the most likely, and that the likelihood decreases as the difference from the estimate increases. Figure 11 exemplifies such a probability distribution.

    Now, for any allocation, one can construct a probability distribution of the revenue, based on his prior beliefs. The decision maker can then choose the allocation with the highest expected revenue, with the lowest probability for an unacceptable revenue, etc.

    The most problematic step of this analysis is the choice of the realizations probabilities. When there is an extensive and relevant past experience, an expert may use this experience to construct a probability distribution. But even with extensive past experience, when some parameters change, the expert may only be able to estimate that

    A

    is more likely than

    B

    , but will not be able to reliably quantify this difference. Furthermore, when conditions change drastically, or when there is no past experience at all, it may prove to be difficult even estimating whether

    A

    is more likely than

    B

    .

    Nevertheless, methodologically speaking, this difficulty is not as problematic as basing the analysis of a problem subject to severe uncertainty on a single point estimate and its immediate neighborhood, as done by info-gap. And what is more, contrary to info-gap, this approach is global, rather than local.

    Still, it must be stressed that Bayesian analysis does not expressly concern itself with the question of robustness.

    Bayesian analysis raises the issue of learning from experience and adjusting probabilities accordingly. In other words, decision is not a one-stop process, but profits from a sequence of decisions and observations.

    See also

    External links

    What is Info-Gap Theory? informal introduction

    Making Responsible Decisions (When it Seems that You Can't): Engineering Design and Strategic Planning Under Severe Uncertainty

    How Did Info-Gap Theory Start? How Does it Grow?

    Frequently Asked Questions about info-gap theory

    Frequently Asked Questions about Info-Gap Decision Theory (PDF)

    Notes and References

    1. Yakov Ben-Haim, Information-Gap Theory: Decisions Under Severe Uncertainty, Academic Press, London, 2001.
    2. Yakov Ben-Haim, Info-Gap Theory: Decisions Under Severe Uncertainty, 2nd edition, Academic Press, London, 2006.
    3. 10.1108/15265941011043648 . Sniedovich . M. . 2010 . A bird's view of info-gap decision theory . Journal of Risk Finance . 11 . 3. 268–283 .
    4. Web site: How Did Info-Gap Theory Start? How Does it Grow? . 2009-03-18 . https://web.archive.org/web/20091128095051/http://www.technion.ac.il/yakov/IGT/start-grow02.html . 2009-11-28 . dead .
    5. 10.1109/5326.798765 . Hipel . Keith W. . Ben-Haim . Yakov . 1999 . Decision making in an uncertain world: Information-gap modelling in water resources management . IEEE Transactions on Systems, Man, and Cybernetics - Part C: Applications and Reviews. 29 . 4. 506–517 . 14135581 .
    6. Yakov Ben-Haim, 2005, Info-gap Decision Theory For Engineering Design. Or: Why `Good' is Preferable to `Best', appearing as chapter 11 in Engineering Design Reliability Handbook, Edited by Efstratios Nikolaidis, Dan M.Ghiocel and Surendra Singhal, CRC Press, Boca Raton.
    7. 10.1016/j.ijsolstr.2005.06.088 . Kanno . Y. . Takewaki . I. . 2006 . Robustness analysis of trusses with separable load and structural uncertainties . International Journal of Solids and Structures . 43 . 9. 2646–2669 .
    8. Kaihong Wang, 2005, Vibration Analysis of Cracked Composite Bending-torsion Beams for Damage Diagnosis, PhD thesis, Virginia Politechnic Institute, Blacksburg, Virginia.
    9. 10.1007/s10957-006-9102-z . Kanno . Y. . Takewaki . I. . 2006 . Sequential semidefinite program for maximum robustness design of structures under load uncertainty . Journal of Optimization Theory and Applications . 130 . 2. 265–287 . 16514524 .
    10. 10.1016/j.jsv.2005.09.029 . Pierce . S.G. . Worden . K. . Manson . G. . 2006 . A novel information-gap technique to assess reliability of neural network-based damage detection . Journal of Sound and Vibration . 293 . 1–2. 96–111 . 2006JSV...293...96P .
    11. 10.1109/TNN.2006.880363 . Pierce . Gareth . Ben-Haim . Yakov . Worden . Keith . Manson . Graeme . 2006 . Evaluation of neural network robust reliability using information-gap theory . IEEE Transactions on Neural Networks . 17 . 6. 1349–1361 . 17131652 . 13019088 .
    12. Chetwynd . D. . Worden . K. . Manson . G. . 2006 . An application of interval-valued neural networks to a regression problem . 10.1098/rspa.2006.1717 . . 462 . 2074. 3097–3114 . 2006RSPSA.462.3097C . 122820264 .
    13. 10.1007/s10710-006-9013-7 . Lim . D. . Ong . Y. S. . Jin . Y. . Sendhoff . B. . Lee . B. S. . 2006 . Inverse Multi-objective Robust Evolutionary Design . Genetic Programming and Evolvable Machines . 7 . 4. 383–404 . 9244713 .
    14. 10.1016/j.jsv.2005.07.007 . Vinot . P. . Cogan . S. . Cipolla . V. . 2005 . A robust model-based test planning procedure . Journal of Sound and Vibration . 288 . 3. 571–585 . 2005JSV...288..571V . 122895551 .
    15. 10.1016/j.jsv.2005.07.005 . Takewaki . Izuru . Ben-Haim . Yakov . 2005 . Info-gap robust design with load and model uncertainties . Journal of Sound and Vibration . 288 . 3. 551–570 . 2005JSV...288..551T . 53466062 .
    16. Izuru Takewaki and Yakov Ben-Haim, 2007, Info-gap robust design of passively controlled structures with load and model uncertainties, Structural Design Optimization Considering Uncertainties, Yiannis Tsompanakis, Nikkos D. Lagaros and Manolis Papadrakakis, editors, Taylor and Francis Publishers.
    17. 10.1016/j.ymssp.2004.03.001 . Hemez . Francois M. . Ben-Haim . Yakov . 2004 . Info-gap robustness for the correlation of tests and simulations of a nonlinear transient . Mechanical Systems and Signal Processing . 18 . 6. 1443–1467 . 2004MSSP...18.1443H .
    18. 10.1016/S0304-3800(00)00226-X . Levy . Jason K. . Hipel . Keith W. . Kilgour . Marc . 2000 . Using environmental indicators to quantify the robustness of policy alternatives to uncertainty . Ecological Modelling . 130 . 1–3. 79–86 . 2000EcMod.130...79L .
    19. 10.1016/j.biocon.2005.11.006 . Moilanen . A. . Wintle . B.A. . 2006 . Uncertainty analysis favours selection of spatially aggregated reserve structures . Biological Conservation . 129 . 3. 427–434 .
    20. 10.1111/j.1461-0248.2005.00827.x . Halpern . Benjamin S. . Regan . Helen M. . Possingham . Hugh P. . McCarthy . Michael A. . 2006 . Accounting for uncertainty in marine reserve design . Ecology Letters . 9 . 1. 2–11 . 16958861 . 2006EcolL...9....2H .
    21. 10.1890/03-5419 . Regan . Helen M. . Ben-Haim . Yakov . Langford . Bill . Wilson . Will G. . Lundberg . Per . Andelman . Sandy J. . Burgman . Mark A. . 2005 . Robust decision making under severe uncertainty for conservation management . Ecological Applications . 15 . 4. 1471–1477 . 2005EcoAp..15.1471R .
    22. 10.1007/s00267-006-0022-3 . McCarthy . M.A. . Lindenmayer . D.B. . 2007 . Info-gap decision theory for assessing the management of catchments for timber production and urban water supply . Environmental Management . 39 . 4. 553–562 . 17318697 . 2007EnMan..39..553M . 45674554 . 1885/28692 . free .
    23. 10.1016/j.biocon.2007.06.007 . Crone . Elizabeth E. . Pickering . Debbie . Schultz . Cheryl B. . 2007 . Can captive rearing promote recovery of endangered butterflies? An assessment in the face of uncertainty . Biological Conservation . 139 . 1–2. 103–112 . 2007BCons.139..103C .
    24. L. Joe Moffitt, John K. Stranlund and Craig D. Osteen, 2007, Robust detection protocols for uncertain introductions of invasive species, Journal of Environmental Management, In Press, Corrected Proof, Available online 27 August 2007.
    25. 10.1890/04-0906 . Burgman . M. A. . Lindenmayer . D.B. . Elith . J. . 2005 . Managing landscapes for conservation under uncertainty . Ecology . 86 . 8. 2007–2017 . 2005Ecol...86.2007B . 10.1.1.477.4238 .
    26. 10.1111/j.1523-1739.2006.00560.x . Moilanen . A. . Elith . J. . Burgman . M. . Burgman . 2006 . M . Uncertainty analysis for regional-scale reserve selection . Conservation Biology . 20 . 6. 1688–1697 . 17181804 . 2006ConBi..20.1688M . 28613050 .
    27. 10.1016/j.ecolmodel.2006.07.004 . Moilanen . Atte . Runge . Michael C. . Elith . Jane . Tyre . Andrew . Carmel . Yohay . Fegraus . Eric . Wintle . Brendan . Burgman . Mark . Benhaim . 2006 . Y . Planning for robust reserve networks using uncertainty analysis . Ecological Modelling . 199 . 1. 115–124 . 2006EcMod.199..115M . 2406867 .
    28. 10.1890/1051-0761(2007)017[0251:MCDUUF]2.0.CO;2 . Nicholson . Emily . Possingham . Hugh P. . 2007. Making conservation decisions under uncertainty for the persistence of multiple species . Ecological Applications . 17 . 1. 251–265 . 17479849 .
    29. Burgman, Mark, 2005, Risks and Decisions for Conservation and Environmental Management, Cambridge University Press, Cambridge.
    30. 10.1086/491691 . Carmel . Yohay . Ben-Haim . Yakov . 2005 . Info-gap robust-satisficing model of foraging behavior: Do foragers optimize or satisfice? . American Naturalist . 166 . 5. 633–641 . 16224728 . 20509139 .
    31. 10.2202/1547-7355.1134 . Moffitt . Joe . Stranlund . John K. . Field . Barry C. . 2005 . Inspections to Avert Terrorism: Robustness Under Severe Uncertainty . Journal of Homeland Security and Emergency Management . 2 . 3 . 3 . 55708128 . 2006-04-21 . https://web.archive.org/web/20060323110629/http://www.bepress.com/jhsem/vol2/iss3/3/ . 2006-03-23 . dead .
    32. 10.1108/15265940710721055 . Beresford-Smith . Bryan . Thompson . Colin J. . 2007 . Managing credit risk with info-gap uncertainty . The Journal of Risk Finance . 8 . 1. 24–34 .
    33. John K. Stranlund and Yakov Ben-Haim, (2007), Price-based vs. quantity-based environmental regulation under Knightian uncertainty: An info-gap robust satisficing perspective, Journal of Environmental Management, In Press, Corrected Proof, Available online 28 March 2007.
    34. 10.1108/15265940510633460 . Ben-Haim . Yakov . 2005 . Value at risk with Info-gap uncertainty . Journal of Risk Finance . 6 . 5. 388–403 . 154808813 .
    35. 10.1061/(ASCE)0733-9364(1998)124:2(125) . Ben-Haim . Yakov . Alexander Laufer . Laufer . Alexander . 1998 . Robust reliability of projects with activity-duration uncertainty . Journal of Construction Engineering and Management. 124 . 2. 125–132 .
    36. 10.1002/sys.20021 . Tahan . Meir . Ben-Asher . Joseph Z. . 2005 . Modeling and analysis of integration processes for engineering systems . Systems Engineering . 8 . 1. 62–77 . 3178866 .
    37. Regev . Sary . Shtub . Avraham . Ben-Haim . Yakov . 2006 . Managing project risks as knowledge gaps . Project Management Journal . 37 . 5. 17–25 . 10.1177/875697280603700503 . 110857106 .
    38. 10.1002/env.811 . Fox . D.R. . Ben-Haim . Y. . Hayes . K.R. . McCarthy . M. . Wintle . B. . Dunstan . P. . 2007 . An Info-Gap Approach to Power and Sample-size calculations . Environmetrics . 18 . 2. 189–203 . 2007Envir..18..189F . 53609269 .
    39. Ben-Haim . Yakov . 1994 . Convex models of uncertainty: Applications and Implications . Erkenntnis . 41 . 2. 139–156 . 10.1007/BF01128824 . 121067986 .
    40. 10.1016/S0016-0032(99)00024-1 . Ben-Haim . Yakov . 1999 . Set-models of information-gap uncertainty: Axioms and an inference scheme . Journal of the Franklin Institute . 336 . 7. 1093–1117 .
    41. 10.1016/S0016-0032(00)00016-8 . Ben-Haim . Yakov . 2000 . Robust rationality and decisions under severe uncertainty . Journal of the Franklin Institute . 337 . 2–3. 171–199 .
    42. 10.1016/j.ress.2004.03.015 . Ben-Haim . Yakov . 2004 . Uncertainty, probability and information-gaps . Reliability Engineering and System Safety . 85 . 1–3. 249–266 .
    43. George J. Klir, 2006, Uncertainty and Information: Foundations of Generalized Information Theory, Wiley Publishers.
    44. Yakov Ben-Haim, 2007, Peirce, Haack and Info-gaps, in Susan Haack, A Lady of Distinctions: The Philosopher Responds to Her Critics, edited by Cornelis de Waal, Prometheus Books.
    45. Burgman, Mark, 2005, Risks and Decisions for Conservation and Environmental Management, Cambridge University Press, Cambridge, pp.399.
    46. Simon . Herbert A. . 1959 . Theories of decision making in economics and behavioral science . American Economic Review . 49 . 253–283 .
    47. Schwartz, Barry, 2004, Paradox of Choice: Why More Is Less, Harper Perennial.
    48. Conlisk . John . 1996 . Why bounded rationality? . Journal of Economic Literature . XXXIV . 669–700 .
    49. Burgman, Mark, 2005, Risks and Decisions for Conservation and Environmental Management, Cambridge University Press, Cambridge, pp.391, 394.
    50. Vinot . P. . Cogan . S. . Cipolla . V. . 2005 . A robust model-based test planning procedure . Journal of Sound and Vibration . 288 . 3. 572 . 10.1016/j.jsv.2005.07.007. 2005JSV...288..571V . 122895551 .
    51. Z. Ben-Haim and Y. C. Eldar, Maximum set estimators with bounded estimation error, IEEE Trans. Signal Process., vol. 53, no. 8, August 2005, pp. 3172-3182.
    52. Babuška, I., F. Nobile and R. Tempone, 2005, Worst case scenario analysis for elliptic problems with uncertainty, Numerische Mathematik (in English) vol.101 pp.185–219.
    53. Sniedovich . M. . 2007 . The art and science of modeling decision-making under severe uncertainty . Decision Making in Manufacturing and Services . 1 . 1–2. 109–134 . 10.7494/dmms.2007.1.2.111 . free .
    54. Yakov Ben-Haim, Robust Reliability in the Mechanical Science, Springer, Berlin,1996.
    55. 10.1006/mssp.1996.0137 . Ben-Haim . Yakov . Cogan . Scott . Sanseigne . Laetitia . 1998 . Usability of Mathematical Models in Mechanical Decision Processes . Mechanical Systems and Signal Processing . 12 . 1. 121–134 . 1998MSSP...12..121B .
    56. (See also chapter 4 in Yakov Ben-Haim, Ref. 2.)
    57. 10.1057/jors.1972.72 . Rosenhead . M.J. . Elton . M. . Gupta . S.K. . 1972 . Robustness and Optimality as Criteria for Strategic Decisions . Operational Research Quarterly . 23 . 4. 413–430 .
    58. 10.1080/00207548708919855 . Rosenblatt . M.J. . Lee . H.L. . 1987 . A robustness approach to facilities design . International Journal of Production Research . 25 . 4. 479–486 .
    59. P. Kouvelis and G. Yu, 1997, Robust Discrete Optimization and Its Applications, Kluwer.
    60. B. Rustem and M. Howe, 2002, Algorithms for Worst-case Design and Applications to Risk Management, Princeton University Press.
    61. R.J. Lempert, S.W. Popper, and S.C. Bankes, 2003, Shaping the Next One Hundred Years: New Methods for Quantitative, Long-Term Policy Analysis, The Rand Corporation.
    62. A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, 2006, Mathematical Programming, Special issue on Robust Optimization, Volume 107(1-2).
    63. Resnik, M.D., Choices: an Introduction to Decision Theory, Universityof Minnesota Press, Minneapolis, MN, 1987.
    64. French, S.D., Decision Theory, Ellis Horwood, 1988.

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