Engineering Design . . . Expert Knowledge Is . . . Engineering Design . . . Expert Knowledge Is Needed From Guaranteed . . . for Design under A Designed System . . . Need for Additional . . . Uncertainty: For p-Boxes, Formulation of the . . . Backcalculation is, Title Page in General, NP-Hard ◭◭ ◮◮ ◭ ◮ Vladik Kreinovich Page 1 of 20 Department of Computer Science Go Back University of Texas at El Paso 500 W. University Full Screen El Paso, TX 79968, USA Close Email: vladik@utep.edu Quit
1. Engineering Design Problems Engineering Design . . . Expert Knowledge Is . . . • One of the main objective of engineering design: guar- Engineering Design . . . antee that a quantity c is within a given range [ c, c ]. From Guaranteed . . . • Example: when we design a car engine, we must make A Designed System . . . sure that: Need for Additional . . . Formulation of the . . . – its power is at least as much as needed for the loaded car to climb the steepest mountain roads, Title Page – the concentration c of undesirable substances in the ◭◭ ◮◮ exhaust does not exceed the required threshold. ◭ ◮ • c usually depends on the parameters a of the design and Page 2 of 20 on the parameters b of the environment: c = f ( a, b ). Go Back • Example: the concentration c depends: Full Screen – on the parameter(s) a of the exhaust filters, and Close – on the concentration b of the chemicals in the fuel. Quit
2. Engineering Design Problems and the Notion of Engineering Design . . . Backcalculation: Deterministic Case Expert Knowledge Is . . . Engineering Design . . . • We need to select a design a in such a way that for all From Guaranteed . . . possible values of the environmental parameter(s) b , A Designed System . . . c = f ( a, b ) ∈ [ c, c ] Need for Additional . . . Formulation of the . . . • In this paper, we consider the simplest case when: Title Page – the design of each system is characterized by a sin- ◭◭ ◮◮ gle parameter a , and ◭ ◮ – the environment is also characterized by a single parameter b . Page 3 of 20 • We will show that already in this simple case, the de- Go Back sign problem is computationally difficult (NP-hard). Full Screen Close Quit
3. Expert Knowledge Is Needed Engineering Design . . . Expert Knowledge Is . . . • Reminder: the design problem is computationally dif- Engineering Design . . . ficult (NP-hard). From Guaranteed . . . • Known fact: expert knowledge can help in solving NP- A Designed System . . . hard problems. Need for Additional . . . Formulation of the . . . • Example: the problem of controlling a system is, in general, NP-hard. Title Page • Expert knowledge: human controllers often have exper- ◭◭ ◮◮ tise of controlling the systems. ◭ ◮ • How it can help: intelligent techniques transform this Page 4 of 20 expertise into successful control algorithms. Go Back • Conclusion: to efficiently solve design problem under Full Screen uncertainty, we must use expert knowledge. Close • Relation to fuzzy: fuzzy technique have been invented for using expert knowlegde. Quit
4. Engineering Design Problems and the Notion of Engineering Design . . . Backcalculation: Deterministic Case Expert Knowledge Is . . . Engineering Design . . . • We usually know: the range [ b, b ] of possible values of b . From Guaranteed . . . • Thus, we arrive at the following problem: A Designed System . . . Need for Additional . . . – we know the desired range [ c, c ]; Formulation of the . . . – we know the dependence c = f ( a, b ); Title Page – we know the range [ b, b ] of possible values of b ; ◭◭ ◮◮ – we want to describe the set of all values of a for which f ( a, b ) ∈ [ c, c ] for all b ∈ [ b, b ]. ◭ ◮ • This problem is called backcalculation problem, in con- Page 5 of 20 trast to forward calculation problem, when Go Back – we are given a design a and Full Screen – we want to estimate the value of the desired char- Close acteristic c = f ( a, b ). Quit
5. Linearized Problem Engineering Design . . . Expert Knowledge Is . . . • In many engineering situations, the intervals of possible Engineering Design . . . a , b ≈ � values of a and b are narrow: a ≈ � b . From Guaranteed . . . • In such situations, we can ignore quadratic and higher A Designed System . . . order terms in the Taylor expansion of c = f ( a, b ): Need for Additional . . . Formulation of the . . . c ≈ c 0 + k a · a + k b · b. Title Page • The numerical value of a quantity a depends on the ◭◭ ◮◮ starting point and on the measuring unit. ◭ ◮ • If we re-scale a → c 0 + k a · a and b → k b · b , we get Page 6 of 20 c ≈ a + b. Go Back • We will show that the design problem becomes com- Full Screen putationally difficult (NP-hard) already for c = a + b. Close Quit
6. From Guaranteed Bounds to p-Boxes Engineering Design . . . Expert Knowledge Is . . . • Ideally: it is desirable to provide a 100% guarantee that Engineering Design . . . the quantity c never exceeds the threshold c . From Guaranteed . . . • In practice: however, too many unpredictable factors A Designed System . . . affect the performance of a system. Need for Additional . . . Formulation of the . . . • What we can realistically guarantee: the probability of exceeding c is small enough: Prob( c ≤ c ) ≥ 1 − ε c . Title Page def • Such constraints bound the cdf F c ( x ) = Prob( c ≤ x ): ◭◭ ◮◮ F c ( x ) ≤ F c ( x ) ≤ F c ( x ) , ◭ ◮ Page 7 of 20 where: Go Back – F c ( x ) is the largest of the lower bounds, and – F c ( x ) is the smallest of the upper bounds. Full Screen Close • The interval [ F c ( x ) , F c ( x )] is called a probability box ( p-box ). Quit
7. From Guaranteed Bounds to p-Boxes (cont-d) Engineering Design . . . Expert Knowledge Is . . . • Similarly: for the environmental parameter b , we rarely Engineering Design . . . know guaranteed bounds b and b . From Guaranteed . . . • Example: we know that for a given bound b , the prob- A Designed System . . . ability of exceeding this bound is small. Need for Additional . . . Formulation of the . . . • In precise terms: we know that Prob( b ≤ b ) ≥ 1 − ε b for some small ε b . Title Page • Conlusion: here too, instead of a single bound, in ef- ◭◭ ◮◮ fect, we have a p-box [ F b ( x ) , F b ( x )] . ◭ ◮ • In manufacturing: it is not possible to guarantee that Page 8 of 20 the value a is within the given interval. Go Back • At best, we can guarantee that, e.g., Full Screen Prob( a ≤ a ) ≥ 1 − ε a . Close • In other words, the design restriction on a can also be formulated in terms of p-boxes. Quit
8. Backcalculation Problem for p-Boxes Engineering Design . . . Expert Knowledge Is . . . • Given: Engineering Design . . . – the desired p-box [ F c ( x ) , F c ( x )] for c ; From Guaranteed . . . A Designed System . . . – the dependence c = f ( a, b ); and Need for Additional . . . – the p-box [ F b ( x ) , F b ( x )] describing b . Formulation of the . . . • Objective: find a p-box [ F a ( x ) , F a ( x )] for which: Title Page – for every probability distribution F a ( x ) ∈ [ F a ( x ) , F a ( x )] , ◭◭ ◮◮ – for every probability distribution F b ( x ) ∈ [ F b ( x ) , F b ( x )], ◭ ◮ and – for all possible correlations between a and b , Page 9 of 20 Go Back the distribution of c = f ( a, b ) is within the given p-box Full Screen [ F c ( x ) , F c ( x )] . Close Quit
9. Reminder: Forward Calculation for p-Boxes Engineering Design . . . Expert Knowledge Is . . . • Our objective: back calculation problem for p-boxes. Engineering Design . . . • Let us first recall: forward calculation problem. From Guaranteed . . . A Designed System . . . • Given: Need for Additional . . . – the p-box [ F a ( x ) , F a ( x )] for a ; and Formulation of the . . . – the p-box [ F b ( x ) , F b ( x )] describing b . Title Page • We want: to find the range [ F c ( x ) , F c ( x )] of possible ◭◭ ◮◮ values of F c ( x ) for c = a + b . ◭ ◮ • Solution: best formulated in terms of bounds c i and c i Page 10 of 20 on quantiles c i , values for which F c ( c i ) = i n : Go Back c i = max j ( a j + b i − j ); c i = min j ( a j − i + b n − j ) . Full Screen Close Quit
10. Quantile Reformulation of the Problem Engineering Design . . . Expert Knowledge Is . . . • Forward problem (reminder): Engineering Design . . . From Guaranteed . . . c i = max j ( a j + b i − j ); c i = min j ( a j − i + b n − j ) . A Designed System . . . • In terms of quantile bounds: the backcalculation prob- Need for Additional . . . lem takes the following form. Formulation of the . . . • Given: Title Page – the quantile intervals [ b i , b i ] corresponding to the ◭◭ ◮◮ environmental variable b ; ◭ ◮ � � c i , � – the intervals that should contain the quan- � c i Page 11 of 20 tiles for c = a + b . Go Back • Objective: find the bounds a i and a i for which Full Screen � � c i , � [ c i , c i ] ⊆ � c i . Close Quit
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