Conflicting objectives in design Common design objectives: - PowerPoint PPT Presentation

Conflicting objectives in design � Common design objectives: Minimizing mass ( sprint bike; satellite components ) Minimizing volume ( mobile phone; minidisk player ) Objectives Minimizing environmental impact ( packaging, cars ) Maximizing performance ( speed, acceleration of a car ) Minimizing cost ( everything ) � Each defines a performance metric . Example we wish to minimize both mass, m (all constraints being met) cost, C � Conflict : the choice that optimizes one does not optimize the other. � Best choice is a compromise. More info: “ Materials Selection in Mechanical Design ”, Chapters 9 and 10 ME 474-674 Spring 2008 Slides 11 -1

Multiple Constraints and Conflicting Objectives Expensive Solution: a viable choice, meeting constraints, but not necessarily optimum by either criterion. Plot solutions as function of A Dominated � performance metrics. solution Metric 2: Cost C Convention: express objectives to be � minimized B Non-dominated Dominated solution: one that is � solution unambiguously non-optimal (as A ) Non-dominated solution: one that is � Trade-off optimal by one metric (as B: optimal by surface one criterion but not necessarily by both Cheap Trade-off surface: the surface on which the non-dominated solutions lie (also Light Metric 1: Mass m Heavy called the Pareto Front) There are several possible strategies for trading off or compromising among conflicting objectives ME 474-674 Spring 2008 Slides 11 -2

Finding a compromise: Strategy 1 � Make a plot of the conflicting objectives Usually drawn in a manner � Metric 2: Cost C Expensive that both metrics have to be minimized � Sketch the trade off surface � Use intuition to select among the solutions closest to the trade off surface Trade-off surface � But then what is the relative Cheap value of the two variables How important is cost Light Metric 1: Mass m Heavy � compared to mass? Do you pick solutions that � are light but more expensive or those that are heavy, but less expensive? ME 474-674 Spring 2008 Slides 11 -3

Cars: Cost-Performance Trade-off 4-quadrant 0.01 Cheaper but Plot Cost-performance trade-off: cars Reciprocal of performance (1/Top speed) slower Trade-off Slower and Land Rover Defender 8e-3 more expensive surface Smart Fortwo Isuzu Trooper Fiat Punto 1.2 Toyota Land-cruiser Your car 1/Top speed 6e-3 Toyota Yaris Land Rover Range Rover Renault Clio Citroën C2 Peugeot 307 Toyota Corolla 4e-3 Skoda Octavia (98-) Mercedes-Benz C320 SE Porsche Boxster Faster but more Cheaper and Jaguar 4.2 V8 SE expensive faster Mercedes-Benz CL 600 10 20 50 100 200 Pence per mile Cost of ownership (cents/mile) ME 474-674 Spring 2008 Slides 11 -4

Finding a compromise – Strategy 2 Expensive � Reformulate all but one of the objectives as constraints, setting an upper limit for it � Good if budget limit Metric 2: Cost C � Trade-off surface gives the best Best choice within budget choice � Not true optimization -- cost treated as constraint, not Upper limit on C objective. Optimum solution minimising m Trade-off Cheap surface Light Metric 1: Mass m Heavy ME 474-674 Spring 2008 Slides 11 -5

Finding a compromise – Strategy 3 Systematic Methods for multiple constraints � Each constraint gives rise to an equation that must be maximized or minimized � Consider a tie rod Light stiff tie rod Light strong tie rod � F σ = f FL δ = f A AE F ρ = ⎡ ⎤ f FL A = ρ = ρ = 2 σ m LA L L S ⎢ ⎥ δ 1 ⎣ ⎦ E E f ρ ρ = ρ = = m LA LF M σ 2 f 1 E f ρ = M σ 2 � The material parameters to be minimized are different f ME 474-674 Spring 2008 Slides 11 -6

2000 Metals 1800 1600 Thallium, Commercial Purity 1400 Density / Young's modulus Calcium Lead with 0.3% tin, cast 1200 Chemical Lead Corroding Lead 1000 Indium, Commercial Purity, min 99.97% 800 600 400 Lead, Arsenical, F-3 alloy, extruded and air cooled 200 Bismuth Metal, Commercial Purity Antimony metal, Commercial Purity, "Regulus" 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000 Density / Yield strength (elastic limit) ME 474-674 Spring 2008 Slides 11 -7

50 45 40 Density / Young's modulus 35 30 25 Molybdenum, 360 grade, wrought, 150 micron wire 20 15 AlBeMet 162 10 Beryllium Aluminum Casting Alloy, Beralcast 191 5 Beryllium, grade 0-50, hot isostatically pressed 5 10 15 20 25 30 35 40 45 50 55 Density / Yield strength (elastic limit) ME 474-674 Spring 2008 Slides 11 -8

40 38 36 34 Density / Young's modulus 32 30 Carbon steel, AISI 1020 (annealed) Titanium alpha-beta alloy, Ti-6Al-4V, Annealed (generic) 28 26 Wrought aluminum alloy, 6061, T651 24 22 20 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Density / Yield strength (elastic limit) ME 474-674 Spring 2008 Slides 11 -9

Finding a compromise – Strategy 3 � Consider a tie rod with � L= 1m S = 3 x 10 7 N/m and � F f = 10 5 N � � Then the two criteria may give rise to different materials with different values of m 1 and m 2 � For each material, the larger of m 1 and m 2 is the better choice m*, and if all materials are considered, then the one with the smallest of m* is the optimum choice ρ ρ ⎡ ⎤ = = 2 m LF m L S ⎢ ⎥ σ 2 f 1 ⎣ ⎦ E f ρ (kg/m 3 ) σ y (MPa) Material E (GPa) m 1 (kg) m 2 (kg) m* (kg) 1020 steel 7850 205 320 1.15 2.45 2.45 AA 6061 2700 70 120 1.16 2.25 2.25 Ti-6-4 4400 115 950 1.15 0.46 1.15 ME 474-674 Spring 2008 Slides 11 -10

50 45 40 35 Density / Young's modulus Carbon steel, AISI 1020 (as-rolled) 30 Titanium alpha-beta alloy, Ti-6Al-4V, Aged Wrought aluminum alloy, 6061, T652 25 20 15 10 5 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Density / Yield strength (elastic limit) ME 474-674 Spring 2008 Slides 11 -11

Finding a compromise – Strategy 3 � In general, if we can have several equations for minimizing mass = m F G M 1 1 1 1 = m F G M 2 2 2 2 etc � For the problem of the tie rod, where we have two equations for mass that must be minimized. If both equations are satisfied at the same time, then = m m 1 2 = F G M F G M 1 1 1 2 2 2 F G = 1 1 M M 2 1 F G 2 2 = M C M 2 c 1 � C c is called the coupling constant between the two material parameters ME 474-674 Spring 2008 Slides 11 -12

Finding a compromise – Strategy 3 The coupling constant depends upon the particular problem M 2 = C c M 1 Heavier Larger Constraint 2 dominant Large C c Index M 2 Mass m 2 Small C c Best Choice Smaller Lighter Constraint 1 dominant Heavier Lighter Smaller Larger Mass m 1 Index M 1 ME 474-674 Spring 2008 Slides 11 -13

Finding a compromise – Strategy 4 Expensive � If the two conflicting requirements are Z 4 Contours of Z 3 minimum mass m and minimum cost C Z 2 constant Z Z 1 � Define a locally linear penalty function Z Z = α m + C Metric 2: Cost C � Seek material with smallest Z : Either evaluate Z for each solution, � and rank, Decreasing values of Z � Or make a trade-off plot with contours of constant Z Optimum solution, minimising Z C = – α m + Z − α Cheap � Lines of constant Z all have slopes of – α � Read off the solution with the smallest Z . Light Metric 1: Mass m Heavy � This gives the best solution for a given value of α � But what is α ? ME 474-674 Spring 2008 Slides 11 -14

The exchange constant α � The constant α is called the exchange constant since it is a conversion factor between two objective variables, the mass m and the cost C . ∂ ∂ ⎛ ⎞ ⎛ ⎞ C Z = − α ⇒ α = − ⎜ ⎟ = α + ⇒ α = ⎜ ⎟ C Z m Z m C ∂ ∂ ⎝ ⎠ ⎝ ⎠ m m Z C � It is a measure of the value of saving unit mass � The value of a can be obtained: � From historical data � Performing an analysis of the full life cost, or � By interviewing experts ME 474-674 Spring 2008 Slides 11 -15

The exchange constant α � The table below shows the exchange constants for mass saving � The exchange constant for a passenger car was obtained from a full life analysis, i.e., how much fuel cost savings would be obtained if a lighter vehicle were built � Mass becomes very important in space craft, but not for a car α ($ per kg) Transport System: mass saving Family car (based on fuel saving) 1 to 2 Truck (based on payload) 5 to 20 Civil aircraft (based on payload) 100 to 500 Military aircraft (performance payload) 500 to 2000 Space vehicle (based on payload) 1000 to 9000 ME 474-674 Spring 2008 Slides 11 -16

Graphical representation = α + The equation when plotted on a linear plot is straight lines Z m C with a negative slope of – α for different values of Z. On the log scale, these lines become the curves shown. Linear scales Log scales Expensive Expensive Cost, C Cost, C Decreasing Cheap Cheap Decreasing values of Z - α values of Z Lighter mass, m Heavier Lighter mass, m Heavier ME 474-674 Spring 2008 Slides 11 -17