Optimization of Critical Systems for Robustness in a Multistate World

ABSTRACT

Critical systems are typically complex systems that are required to perform reliably over a wide range of scenarios, or multistate world. Seldom does a single system exist that performs best for all plausible scenarios. A robust solution, one that performs relatively well over a wide range of scenarios, is often the preferred choice for reduced risk at an acceptable cost. The alternative with the maximum expected utility may possess vulnerabilities that could be exploited. The best strategy is likely to be a hybrid solution. The von Neumann-Morgenstern Expected Utility Theory (EUT) would never select such a solution because, given its linear functional form, the expected utility of a hybrid solution cannot be greater than that of every constituent alternative. The continuity axiom and the independence axiom are assessed to be unrealistic for the problem of interest. Several well-known decision models are analyzed and demonstrated to be potentially misleading. The linear disappointment model modifies EUT by adding a term proportional to downside risk; however, it does not provide a mathematical basis for determining preferred hybrid solutions. The paper proposes a portfolio allocation model with stochastic optimization as a flexible and transparent method for defining choice problems and determining hybrid solutions for critical systems with desirable properties such as diversification and robustness.

Cite this paper

E. Kujawski, "Optimization of Critical Systems for Robustness in a Multistate World,"*American Journal of Operations Research*, Vol. 3 No. 1, 2013, pp. 127-137. doi: 10.4236/ajor.2013.31A012.

E. Kujawski, "Optimization of Critical Systems for Robustness in a Multistate World,"

References

[1] U.S. Office of Homeland Security, “The National Strategy for Homeland Security,” 2002. http://www.ncs.gov/library/policy_docs/nat_strat_hls.pdf

[2] R. D. Luce and H. Raiffa, “Games and Decisions: Introduction and Critical Survey,” Dover Publications, New York, 1957.

[3] J. von Neumann and O. Morgenstern, “Theory of Games and Economic Behavior,” 2nd Edition, Princeton University Press, Princeton, 1947.

[4] L. J. Savage, “The Foundations of Statistics,” 2nd Revised Edition, Dover Publications, New York, 1972.

[5] P. C. Fishburn, “Normative Theories of Decision Making under Risk and Uncertainty,” In: D. Bell, H. Raiffa and A. Tversky, Eds., Decision Making: Descriptive, Normative, and Prescriptive Interactions, Cambridge University Press, Cambridge, 1988, pp. 78-98. doi:10.1017/CBO9780511598951.006

[6] D. Kahneman and A. Tversky, “Prospect Theory: An Analysis of Decisions under Risk,” Econometrica, Vol. 47, No. 3. 1979, pp. 263-291. doi:10.2307/1914185

[7] A. Tversky and D. Kahneman, “Advances in Prospect Theory: Cumulative Representation of Uncertainty,” Journal of Risk and Uncertainty, Vol. 5, No. 4, 1992, pp. 297-323.

[8] D. E. Bell, “Regret in Decision Making under Uncertainty,” Operations Research, Vol. 30, No. 5, 1982, pp. 961-981. doi:10.1287/opre.30.5.961

[9] G. Loomes and R. Sugden, “Regret Theory: An Alternative Theory of Rational Choice under Uncertainty,” Economic Journal, Vol. 92, No. 368, 1982, pp. 805-824. doi:10.2307/2232669

[10] D. E. Bell, “Disappointment in Decision Making under Uncertainty,” Operations Research, Vol. 33, No. 1, 1985, pp. 1-27. doi:10.1287/opre.33.1.1

[11] G. Loomes and R. Sugden, “Disappointment and Dynamic Consistency in Choice under Uncertainty,” Review of Economic Studies, Vol. 53, No. 2, 1986, pp. 271-282. doi:10.2307/2297651

[12] E. Dekel and B. L. Lipman, “How (Not) to Do Decision Theory,” Annual Review of Economics, Vol. 2, No. 1, 2010, pp. 257-282. doi:10.1146/annurev.economics.102308.124328

[13] G. Chichilnisky, “An Axiomatic Approach to Choice under Uncertainty with Catastrophic Risks,” Resource and Energy Economics, Vol. 22, No. 3, 2000, pp. 221-231. doi:10.2139/ssrn.1522307

[14] G. G. Brown and L. A. Cox, Jr., “Making Terrorism Risk Analysis Less Harmful and More Useful: Another Try,” Risk Analysis, Vol. 31, No. 2, 2011, pp. 193-195. doi:10.1111/j.1539-6924.2010.01563.x

[15] P. K. Davis, R. D. Shaver and J. Beck, “Portfolio-Analysis Methods for Assessing Capability Options,” RAND, Santa Monica, 2008.

[16] S. Savage, “The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty,” John Wiley & Sons, Hoboken, 2009.

[17] Y. Y. Haimes, “Risk Modeling, Assessment, and Management,” 3rd Edition, John Wiley & Sons, Hoboken, 2009.

[18] H. M. Markowitz, “Portfolio Selection: Efficient Diversification of Investments,” 2nd Edition, Blackwell, Cambridge, 1997.

[19] Office of the Deputy under Secretary of Defense for Acquisition and Technology, Systems and Software Engineering. “Systems Engineering Guide for Systems of Systems,” Version 1.0, ODUSD(A&T)SSE, Washington, 2008.

[20] D. H. Wagner, W. C. Mylander and T. J. Sanders, “Naval Operations Analysis,” 3rd Edition, Naval Institute Press, Annapolis, 1999.

[21] A. Washburn and M. Kress, “Combat Modeling,” Springer, New York, 2009. doi:10.1007/978-1-4419-0790-5

[22] 2011 Naval Surface Warfare Center Crane Cohort, “System Engineering Approach to Improving Arizona Border Patrol C4ISR Mission Operations,” MSSE Capstone Project, Naval Postgraduate School, Monterey, 2011.

[23] R. Hastie and R. M. Dawes, “Rational Choice in an Uncertain World: The Psychology of Judgment and Decision Making,” Sage Publications, Thousand Oaks, 2001.

[24] A. Einstein, “Ideas and Opinions,” Bonanza Books, New York, 1954.

[25] E. Malinvaud, “Note on von Neumann-Morgenstern’s Strong Independence Axiom,” Econometrica, Vol. 20, No. 4, 1952, p. 679. doi:10.2307/1907650

[26] R. de Neuville, “Applied Systems Analysis: Engineering Planning and Technology Management,” McGraw-Hill, New York, 1999.

[27] E. Kujawski, M. Alvaro and W. Edwards, “Incorporating Psychological Influences in Probabilistic Cost Analysis,” Systems Engineering, Vol. 7, No. 3, 2004, pp. 195-216. doi:10.1002/sys.20004

[28] S. H. Chew, “Axiomatic Utility Theories with the Betweenness Property,” Annals of Operations Research, Vol. 19, No. 1, 1989, pp. 273-298. doi:10.1007/BF02283525

[29] M. Allais, “Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l’Ecole Américaine,” Econometrica, Vol. 21, No. 4, 1953, pp. 503-546. doi:10.2307/1907921

[30] P. Samuelson, “Risk and Uncertainty: A Fallacy of Large Numbers,” Scientia, Vol. 98, No. 4, 1963, pp. 108-113.

[31] M. Rabin and R. H. Thaler, “Anomalies: Risk Aversion,” The Journal of Economic Perspectives, Vol. 15, No. 1, 2001, pp. 219-232. doi:10.1257/jep.15.1.219

[32] E. Dekel, “Asset Demands without the Independence Axiom,” Econometrica, Vol. 57, No.1, 1989, pp. 163-169. doi:10.2307/1912577

[33] R. J. Lempert, D. G. Groves, S. W. Popper and S. C. Bankes, “A General, Analytic Method for Generating Robust Strategies and Narrative Scenarios,” Management Science, Vol. 52, No. 4, 2006, pp. 51-528. doi:10.1287/mnsc.1050.0472

[34] P. Krokhmal, R. Murphey, P. Pardalos, S. Uryasev and G. Zrazhevsky, “Robust Decision Making: Addressing Uncertainties in Distributions,” In: S. Butenko, et al., Eds., Cooperative Control: Models, Applications and Algorithms, Kluwer Academic Publishers, Dordrecht, 2003, pp. 165-185.

[35] D. G. Ullman, “Making Robust Decisions: Decision Management for Technical, Business, & Service,” Trafford Publishing, Victoria, 2006.

[36] G. Friedman, “The Intransitivity of Pairwise Comparisons Even with a Single Rational Decision Maker or: Homomorphisms from Allegedly Paradoxical Dice to Decision-Making in the Military, Business and Sports World,” Presentation to the NSF Decision-Based Design Workshop, Long Beach, 1999. http://dbd.eng.buffalo.edu/papers/friedman.html

[37] P. Delquié and A. Cillo, “Disappointment without Prior Expectation: A Unifying Perspective on Decision under Risk,” Journal of Risk and Uncertainty, Vol. 33, No. 3, pp. 197-215. doi:10.1007/s11166-006-0499-4

[38] D. Nawrocki, “A Brief History of Downside Risk Measures,” Journal of Investing, Vol. 8, No. 3, 1999, pp. 9-25. doi:10.3905/joi.1999.319365

[39] E. Kujawski and G. A. Miller, “Quantitative Risk-Based Analysis for Military Counterterrorism Systems,” Systems Engineering, Vol. 10, No. 4, 2007, pp. 273-289. doi:10.1002/sys.20075

[40] D Bertsimas, D. B. Brown and C. Caramanis, “The Theory and Applications of Robust Optimization,” SIAM Review, Vol. 53, No. 3, 2011, pp. 464-501. doi:10.1137/080734510

[41] M. Levy, “Almost Stochastic Dominance and Efficient Investment Sets,” American Journal of Operations Research, Vol. 2, No. 3, 2012, pp. 313-321. doi:10.4236/ajor.2012.23038

[42] E. Kujawski, I. M. Jacobs and A. M. Smith, “An Evaluation of the Use of Signal Validation Techniques as a Defense against Common-Cause Failures,” Electric Power Research Institute, Palo Alto, 1987.

[43] F. Glover, M. Laguna and R. Martí, “Scatter Search,” In: A. Ghosh and S. Tsutsui, Eds., Advances in Evolutionary Computation: Theory and Applications, Springer-Verlag, New York, 2003, pp. 519-537.

[44] 2008 Cohort from the Naval Surface Warfare Center Dahlgren Division and the Naval Undersea Warfare Center Division Newport, “Enhanced Vessel Awareness Capability,” MSSE Capstone Project, Naval Postgraduate School, Monterey, 2011.

[1] U.S. Office of Homeland Security, “The National Strategy for Homeland Security,” 2002. http://www.ncs.gov/library/policy_docs/nat_strat_hls.pdf

[2] R. D. Luce and H. Raiffa, “Games and Decisions: Introduction and Critical Survey,” Dover Publications, New York, 1957.

[3] J. von Neumann and O. Morgenstern, “Theory of Games and Economic Behavior,” 2nd Edition, Princeton University Press, Princeton, 1947.

[4] L. J. Savage, “The Foundations of Statistics,” 2nd Revised Edition, Dover Publications, New York, 1972.

[5] P. C. Fishburn, “Normative Theories of Decision Making under Risk and Uncertainty,” In: D. Bell, H. Raiffa and A. Tversky, Eds., Decision Making: Descriptive, Normative, and Prescriptive Interactions, Cambridge University Press, Cambridge, 1988, pp. 78-98. doi:10.1017/CBO9780511598951.006

[6] D. Kahneman and A. Tversky, “Prospect Theory: An Analysis of Decisions under Risk,” Econometrica, Vol. 47, No. 3. 1979, pp. 263-291. doi:10.2307/1914185

[7] A. Tversky and D. Kahneman, “Advances in Prospect Theory: Cumulative Representation of Uncertainty,” Journal of Risk and Uncertainty, Vol. 5, No. 4, 1992, pp. 297-323.

[8] D. E. Bell, “Regret in Decision Making under Uncertainty,” Operations Research, Vol. 30, No. 5, 1982, pp. 961-981. doi:10.1287/opre.30.5.961

[9] G. Loomes and R. Sugden, “Regret Theory: An Alternative Theory of Rational Choice under Uncertainty,” Economic Journal, Vol. 92, No. 368, 1982, pp. 805-824. doi:10.2307/2232669

[10] D. E. Bell, “Disappointment in Decision Making under Uncertainty,” Operations Research, Vol. 33, No. 1, 1985, pp. 1-27. doi:10.1287/opre.33.1.1

[11] G. Loomes and R. Sugden, “Disappointment and Dynamic Consistency in Choice under Uncertainty,” Review of Economic Studies, Vol. 53, No. 2, 1986, pp. 271-282. doi:10.2307/2297651

[12] E. Dekel and B. L. Lipman, “How (Not) to Do Decision Theory,” Annual Review of Economics, Vol. 2, No. 1, 2010, pp. 257-282. doi:10.1146/annurev.economics.102308.124328

[13] G. Chichilnisky, “An Axiomatic Approach to Choice under Uncertainty with Catastrophic Risks,” Resource and Energy Economics, Vol. 22, No. 3, 2000, pp. 221-231. doi:10.2139/ssrn.1522307

[14] G. G. Brown and L. A. Cox, Jr., “Making Terrorism Risk Analysis Less Harmful and More Useful: Another Try,” Risk Analysis, Vol. 31, No. 2, 2011, pp. 193-195. doi:10.1111/j.1539-6924.2010.01563.x

[15] P. K. Davis, R. D. Shaver and J. Beck, “Portfolio-Analysis Methods for Assessing Capability Options,” RAND, Santa Monica, 2008.

[16] S. Savage, “The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty,” John Wiley & Sons, Hoboken, 2009.

[17] Y. Y. Haimes, “Risk Modeling, Assessment, and Management,” 3rd Edition, John Wiley & Sons, Hoboken, 2009.

[18] H. M. Markowitz, “Portfolio Selection: Efficient Diversification of Investments,” 2nd Edition, Blackwell, Cambridge, 1997.

[19] Office of the Deputy under Secretary of Defense for Acquisition and Technology, Systems and Software Engineering. “Systems Engineering Guide for Systems of Systems,” Version 1.0, ODUSD(A&T)SSE, Washington, 2008.

[20] D. H. Wagner, W. C. Mylander and T. J. Sanders, “Naval Operations Analysis,” 3rd Edition, Naval Institute Press, Annapolis, 1999.

[21] A. Washburn and M. Kress, “Combat Modeling,” Springer, New York, 2009. doi:10.1007/978-1-4419-0790-5

[22] 2011 Naval Surface Warfare Center Crane Cohort, “System Engineering Approach to Improving Arizona Border Patrol C4ISR Mission Operations,” MSSE Capstone Project, Naval Postgraduate School, Monterey, 2011.

[23] R. Hastie and R. M. Dawes, “Rational Choice in an Uncertain World: The Psychology of Judgment and Decision Making,” Sage Publications, Thousand Oaks, 2001.

[24] A. Einstein, “Ideas and Opinions,” Bonanza Books, New York, 1954.

[25] E. Malinvaud, “Note on von Neumann-Morgenstern’s Strong Independence Axiom,” Econometrica, Vol. 20, No. 4, 1952, p. 679. doi:10.2307/1907650

[26] R. de Neuville, “Applied Systems Analysis: Engineering Planning and Technology Management,” McGraw-Hill, New York, 1999.

[27] E. Kujawski, M. Alvaro and W. Edwards, “Incorporating Psychological Influences in Probabilistic Cost Analysis,” Systems Engineering, Vol. 7, No. 3, 2004, pp. 195-216. doi:10.1002/sys.20004

[28] S. H. Chew, “Axiomatic Utility Theories with the Betweenness Property,” Annals of Operations Research, Vol. 19, No. 1, 1989, pp. 273-298. doi:10.1007/BF02283525

[29] M. Allais, “Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l’Ecole Américaine,” Econometrica, Vol. 21, No. 4, 1953, pp. 503-546. doi:10.2307/1907921

[30] P. Samuelson, “Risk and Uncertainty: A Fallacy of Large Numbers,” Scientia, Vol. 98, No. 4, 1963, pp. 108-113.

[31] M. Rabin and R. H. Thaler, “Anomalies: Risk Aversion,” The Journal of Economic Perspectives, Vol. 15, No. 1, 2001, pp. 219-232. doi:10.1257/jep.15.1.219

[32] E. Dekel, “Asset Demands without the Independence Axiom,” Econometrica, Vol. 57, No.1, 1989, pp. 163-169. doi:10.2307/1912577

[33] R. J. Lempert, D. G. Groves, S. W. Popper and S. C. Bankes, “A General, Analytic Method for Generating Robust Strategies and Narrative Scenarios,” Management Science, Vol. 52, No. 4, 2006, pp. 51-528. doi:10.1287/mnsc.1050.0472

[34] P. Krokhmal, R. Murphey, P. Pardalos, S. Uryasev and G. Zrazhevsky, “Robust Decision Making: Addressing Uncertainties in Distributions,” In: S. Butenko, et al., Eds., Cooperative Control: Models, Applications and Algorithms, Kluwer Academic Publishers, Dordrecht, 2003, pp. 165-185.

[35] D. G. Ullman, “Making Robust Decisions: Decision Management for Technical, Business, & Service,” Trafford Publishing, Victoria, 2006.

[36] G. Friedman, “The Intransitivity of Pairwise Comparisons Even with a Single Rational Decision Maker or: Homomorphisms from Allegedly Paradoxical Dice to Decision-Making in the Military, Business and Sports World,” Presentation to the NSF Decision-Based Design Workshop, Long Beach, 1999. http://dbd.eng.buffalo.edu/papers/friedman.html

[37] P. Delquié and A. Cillo, “Disappointment without Prior Expectation: A Unifying Perspective on Decision under Risk,” Journal of Risk and Uncertainty, Vol. 33, No. 3, pp. 197-215. doi:10.1007/s11166-006-0499-4

[38] D. Nawrocki, “A Brief History of Downside Risk Measures,” Journal of Investing, Vol. 8, No. 3, 1999, pp. 9-25. doi:10.3905/joi.1999.319365

[39] E. Kujawski and G. A. Miller, “Quantitative Risk-Based Analysis for Military Counterterrorism Systems,” Systems Engineering, Vol. 10, No. 4, 2007, pp. 273-289. doi:10.1002/sys.20075

[40] D Bertsimas, D. B. Brown and C. Caramanis, “The Theory and Applications of Robust Optimization,” SIAM Review, Vol. 53, No. 3, 2011, pp. 464-501. doi:10.1137/080734510

[41] M. Levy, “Almost Stochastic Dominance and Efficient Investment Sets,” American Journal of Operations Research, Vol. 2, No. 3, 2012, pp. 313-321. doi:10.4236/ajor.2012.23038

[42] E. Kujawski, I. M. Jacobs and A. M. Smith, “An Evaluation of the Use of Signal Validation Techniques as a Defense against Common-Cause Failures,” Electric Power Research Institute, Palo Alto, 1987.

[43] F. Glover, M. Laguna and R. Martí, “Scatter Search,” In: A. Ghosh and S. Tsutsui, Eds., Advances in Evolutionary Computation: Theory and Applications, Springer-Verlag, New York, 2003, pp. 519-537.

[44] 2008 Cohort from the Naval Surface Warfare Center Dahlgren Division and the Naval Undersea Warfare Center Division Newport, “Enhanced Vessel Awareness Capability,” MSSE Capstone Project, Naval Postgraduate School, Monterey, 2011.