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How Design Quality Improves with Increasing Computational Abilities: General Formulas and Case Study of Aircraft Fuel Efficiency

Joe Lorkowski1, Olga Kosheleva1, Vladik Kreinovich1, and Sergei Soloviev2

1University of Texas at El Paso, El Paso, TX 79968, USA

lorkowski@computer.org, olgak@utep.edu, vladik@utep.edu

2Institut de Recherche en Informatique de Toulouse (IRIT)

Toulouse, France, sergei.soloviev@irit.fr

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1. Outline

• It is known that the problems of optimal design are

NP-hard.

• This means that, in general, a feasible algorithm can
• nly produce close-to-optimal designs.
• The more computations we perform, the better design

we can produce.

• In this paper, we theoretically derive the dependence
• f design quality on computation time.
• We then empirically confirm this dependence on the

example of aircraft fuel efficiency.

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2. Formulation of the Problem

• Since 1980s, computer-aided design (CAD) has become

ubiquitous in engineering; example: Boeing 777.

• The main objective of CAD is to find a design which
• ptimizes the corresponding objective function.
• Example: we optimize fuel efficiency of an aircraft.
• The corresponding optimization problems are non-linear,

and such problems are, in general, NP-hard.

• So – unless P = NP – a feasible algorithm cannot al-

ways find the exact optimum, only an approximate one.

• The more computations we perform, the better the de-

sign.

• It is desirable to quantitatively describe how increasing

computational abilities improve the design quality.

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3. Because of NP-Hardness, More Computations Simply Means More Test Cases

• In principle, each design optimization problem can be

solved by exhaustive search.

• Let d denote the number of parameters.
• Let C denote the average number of possible values of

a parameter.

• Then, we need to analyze Cd test cases.
• For large systems (e.g., for an aircraft), we can only

test some combinations.

• NP-hardness means that optimization algorithms to be

significantly faster than exponential time Cd.

• This means that, in effect, all possible optimization al-

gorithms boil down to trying many possible test cases.

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4. Enter Randomness

• Increasing computational abilities mean that we can

test more cases.

• Thus, by increasing the scope of our search, we will

hopefully find a better design.

• Since we cannot do significantly better than with a

simple search, – we cannot meaningfully predict whether the next test case will be better or worse, – because if we could, we would be able to signifi- cantly decrease the search time.

• The quality of the next test case cannot be predicted

and is, in this sense, a random variable.

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5. Which Random Variable?

• Many different factors affect the quality of each indi-

vidual design.

• Usually, the distribution of the resulting effect of sev-

eral independent random factors is close to Gaussian.

• This fact is known as the Central Limit Theorem.
• Thus, the quality of a (randomly selected) individual

design is normally distributed, with some µ and σ.

• After we test n designs, the quality of the best-so-far

design is x = max(x1, . . . , xn).

• We can reduce the case of yi with µ = 0 and σ = 1:

namely, xi = µ + σ · yi hence x = µ + σ · y, where y

def

= max(y1, . . . , yn).

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6. Let Us Use Max-Central Limit Theorem

• For large n, y’s cdf is F(y) ≈ FEV

y − µn σn

• , where:
• FEV (y)

def

= exp(− exp(−y)) (Gumbel distribution),

• µn

def

= Φ−1

• 1 − 1

n

• , where Φ(y) is cdf of N(0, 1),
• σn

def

= Φ−1

• 1 − 1

n · e−1

• − Φ−1
• 1 − 1

n

• .
• Thus, y = µn + σn · ξ, where ξ is distributed according

to the Gumbel distribution.

• The mean of ξ is the Euler’s constant γ ≈ 0.5772.
• Thus, the mean value mn of y is equal to µn + γ · σn.
• For large n, we get asymptotically mn ∼ γ ·
• 2 ln(n).
• Hence the mean value en of x = µ + σ · y is asymptot-

ically equal to en ∼ µ + σ · γ ·

• 2 ln(n).

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7. Resulting Formula: Let Us Test It

• Situation: we test n different cases to find the optimal

design.

• Conclusion: the quality en of the resulting design in-

creases with n as en ∼ µ + σ · γ ·

• 2 ln(n).
• We test this formula: on the example of the average

fuel efficiency E of commercial aircraft.

• Empirical fact: E changes with time T as

E = exp(a + b · ln(T)) = C · T b, for b ≈ 0.5.

• Question: can our formula en ∼ µ + σ · γ ·
• 2 ln(n)

explain this empirical dependence?

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8. How to Apply Our Theoretical Formula to This Case?

• The formula q ∼ µ + σ · γ ·
• 2 ln(n) describes how the

quality changes with the # of computational steps n.

• In the case study, we know how it changes with time T.
• According to Moore’s law, the computational speed

grows exponentially with time T: n ≈ exp(c · T).

• Crudely speaking, the computational speed doubles ev-

ery two years.

• When n ≈ exp(c · T), we have ln(n) ∼ T; thus,

q ≈ a + b · √ T.

• This is exactly the empirical dependence that we actu-

ally observe.

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9. Caution

• Idea: cars also improve their fuel efficiency.
• Fact: the dependence of their fuel efficiency on time is

piece-wise constant.

• Explanation: for cars, changes are driven mostly by

federal and state regulations.

• Result: these changes have little to do with efficiency
• f Computer-Aided design.

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10. Acknowledgments This work was supported in part:

• by the National Science Foundation grants:

– HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and – DUE-0926721,

• by Grants 1 T36 GM078000-01 and 1R43TR000173-01

from the National Institutes of Health, and

• by grant N62909-12-1-7039 from the Office of Naval

Research.

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11. Bibliography: CAD

• D. A. Madsen and D. P. Madsen, Engineering Drawing

and Design, Delmar, Cengage Learning, Clifton Park, New York, 2012.

• K. Sabbagh, Twenty-First-Century Jet: The Making

and Marketing of the Boeing 777, Scribner, New York, 1996.

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12. Bibliography: Statistics of Extremes

• J. Beirlant, Y. Goegevuer, J. Teugels, and J. Segers,

Statistics of Extremes: Theory and Applications, Wi- ley, Chichester, 2004.

• L. de Haan and A. Ferreira, Extreme Value Theory: An

Introduction, Springer Verlag, Berlin, Hiedelberg, New York, 2006.

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uppelberg, and T. Mikosch, Mod- elling Extremal Events for Insurance and Finance, Springer Verlag, Berlin, Heidelberg, New York, 2012.

• E. J. Gumbel, Statistics of Extremes, Dover Publ., New

York, 2004.

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13. Bibliography: Aircraft Fuel Efficiency

• J. J. Lee, S. P. Lukachko, I. A. Waitz, and A. Schafer,

“Historical and future trends in aircraft performance, cost and emissions”, Annual Review of Energy Envi- ronment, 2001, Vol. 26, pp. 167–200.

• P. M. Peeters, J. Middel, and A. Hoolhorst, Fuel Effi-

ciency of Commercial Aircraft: An Overview of Histor- ical and Future Trends, Netherlands National Aerospace Laboratory NLR, Technical Report NLR-CR-2005-669, 2005.

• J. E. Penner, D. H. Lister, D. J. Griggs, D. J. Dokken,

and M. McFarland, Aviation and the Global Atmo- sphere: A Special Report of Intergovernmental Panel

• n Climate Change (IPCC) Working Groups I and III,

Cambridge University Press, Cambridge, UK, 1999.

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14. Bibliography: Moore’s Law

• G. E. Moore, “Cramming More Components onto Inte-

grated Circuits”, Electronics, April 19, 1965, pp. 114– 117.

• J. L. Hennessy and D. A. Patterson, Computer Archi-

tecture: A Quantitative Approach, Morgan Kaufmann, Waltham, Massachusetts, 2012.