MECH L# 10 Conflicting Objectives 1/30 MECH , Lecture 10 Objectives in Conflict: Trade-off Methods and Penalty Functions Textbook Chapters 9 & 10 Tutorial 5 (2 exercises, two afternoons, due Oct 13) Technical Papers: P. Sirisalee, M. F. Ashby, G. T. Parks and P. J. Clarkson, "Multi-Criteria Material Selection in Engineering Design", Adv. Engng. Mater., 2004, 6, (Simple, readable) C. H. Cáceres, "Economical and environmental factors in light alloys automotive applications", Metall. Mater. Trans. A, 2007, 38, (Automotive applications) M. F. Ashby, "Multi-objective optimization in material design and selection", Acta Materialia, 2000, 48, (Advanced reading)

MECH L# 10 Conflicting Objectives 2/30 Examples of Conflicting Objectives in design Some objectives may mass, m conflict with another cost, c We wish to minimize both (all constraints being met) Common design objectives: Minimising mass (sprint bike; satellite components) Minimising volume (mobile phone; minidisk player) Minimising environmental impact (packaging, cars) Minimising cost (everything) Objectives Conflict : the choice that optimises one does not optimise the other. Best choice is a compromise. Each defines a performance metric

MECH L# 10 Conflicting Objectives 3/30 Light Metric 1: Mass m Heavy Cheap Metric 2: Cost C Expensive Multi-objective optimisation: The terminology Trade-off surface: the surface on which the non-dominated solutions lie (also called the Pareto Front) (after Pareto, 1898) Solution: a viable choice, meeting constraints, but not necessarily optimum by either criterion. Trade-off surface Plot all viable solutions as function of performance metrics. (Convention: express objectives to be minimised) Dominated solution: one that is unambiguously non-optimal (as A) (there are better ones) A Dominated solution Non-dominated solution: one that is optimal by one metric (as B: optimal by one criterion but not necessarily by both) B Non-dominated solution

MECH L# 10 Conflicting Objectives 4/30 Example of Conflicting Objectives in Pushbikes Price vs. mass of bicycles: a matter of perception? Price $ Mass (kg) The price we are prepared to pay for a light bike does not relate to the actual cost of the materials it is made of. Then, how do we decide what is the “best” material ? Three strategies for finding the best compromise (next 4 frames)

MECH L# 10 Conflicting Objectives 5/30 Strategy 1: compromise by intuition and experience Make trade-off plot and Sketch trade-off surface Use intuition to select a solution on the trade-off surface “Solutions” on or near the surface offer the best compromise between mass and cost The choice depends on how highly you value a light weight, -- a question of relative values Light Metric 1: Mass m Heavy Cheap Metric 2: Cost C Expensive Trade-off surface select current material

MECH L# 10 Conflicting Objectives 6/30 Finding a compromise: Strategy 2 Reformulate all but one of the objectives as constraints, setting an upper limit for it Optimum solution minimising m Light Metric 1: Mass m Heavy Cheap Metric 2: Cost C Expensive Trade-off surface Mass and price of bicycles: Good if you have budget limit Trade-off surface leads you to the best choice within budget But not a true optimisation -- mass has been treated as a constraint, not an objective. Optimum solution minimising c Constraint: mass = 11 kg Upper limit for cost: $200.

MECH L# 10 Conflicting Objectives 7/30 Light Metric 1: Mass m Heavy Cheap Metric 2: Cost C Expensive Strategy 3: Penalty functions and exchange constants Optimum solution, minimising Z (lowers both m and c) Z1Z1 Z2Z2 Z3Z3 Z4Z4 Contours of constant Z Decreasing values of Z Seek material with smallest Z: Either evaluate Z for each solution, and rank, Or make trade-off plot But what is the meaning of  ? plot on it contours of Z -- lines of constant Z have slope -  Read off solution with lowest Z Define locally linear Penalty function Z Z = y-intcpt (in this example)

MECH L# 10 Conflicting Objectives 8/30 Light Metric 1: Mass m Heavy Cheap Metric 2: Cost C Expensive Z = penalty, value or utility function. Z1Z1 Along the line Z = cost +  mass = constant cost mass Z is the combined “value” of (cost &  mass)

MECH L# 10 Conflicting Objectives 9/30 The exchange constant  The quantity  is called an “exchange constant” -- it measures the value of performance, here the value of saving 1 kg of mass ($/kg). How get  …? Effect of metric on Z market survey (perceived value) full life cost (engineering criteria)  = drop in Z per unit mass, at constant cost Metric P1: Mass m Metric P2: Cost C Exchange Constant: quantifies the effect of a material substitution on the total value, or the (value) penalty involved in the substitution.

MECH L# 10 Conflicting Objectives 10/30 Materials substitution and exchange constants Engineering definition of  Cost of substituting D for A ($/kg) Cost of substituting B for A Upper bound to  C. H. Cáceres, "Economical and environmental factors in light alloys automotive applications", Metall. Mater. Trans. A, 2007, 38,

MECH L# 10 Conflicting Objectives 11/30 Family car (based on fuel saving) Truck (based on payload) Civil aircraft (based on payload) Military aircraft (performance payload) Bicycle frame (perceived value) Space vehicle (based on payload) Transport System: mass saving  ($US per kg) 0.5 ~ 6 5 to to to to ( Upper bounds to ) Exchange constants for mass saving in transport systems Finding  : engineering criteria. Example of upper bounds to exchange constants for transport systems The is how much you can afford to expend in a material substitution. If the substitution costs you more than the upper bound, you won’t get your $ back. Savings over 2x10 5 km C. H. Cáceres, "Economical and environmental factors in light alloys automotive applications", Metall. Mater. Trans. A, 2007, 38, M. F. Ashby, "Multy-objective optimization in material design and selection", Acta Materialia, 2000, 48,

MECH L# 10 Conflicting Objectives 12/30 Penalty function on log scales Log scales Lighter mass, m Heavier Cheap Cost, C Expensive Decreasing values of Z A linear relation, on log scales, plots as a curve Linear scales Lighter mass, m Heavier Cheap Cost, C Expensive Decreasing values of Z --

MECH L# 10 Conflicting Objectives 13/30 Penalty function in transport systems. Mass of a beam vs. cost for given stiffness P 2 = Cost for given stiffness P 1 =Mass for given stiffness Exchange constant  = 1 $/kg Exchange constant  = 50 $/kg Exchange constant  = 5 $/kg Exchange constant  = 500 $/kg Trade-off surface  c/E 1/2  /E 1/2 Family car Truck Civil aircraft Military aircraft Bicycle frame Space vehicle System  ($US per kg) 0.5~6 5 to to to to Engineering definition of  Penalty Function & Exchange Constants: Powerful and Unambiguous Strategy for Material Substitutions under Conflicting Objectives

MECH L# 10 Conflicting Objectives 14/30 Case study: casing for electronic equipment Electronic equipment -- portable computers, players, mobile phones, cameras – are miniaturised; many less than 12 mm thick Minidisk player: An ABS or Polycarbonate casing has to be > 1mm thick to be stiff enough to protect; casing takes 20% of the volume stiff, light, thin casing bending stiffness EI at least that of existing case minimise casing thickness minimise casing mass choice of material casing thickness, t Constraints Objectives Function Free variables The thinnest may not be the lightest … need to explore trade-off

MECH L# 10 Conflicting Objectives 15/30 Performance metrics for the casing: t and m Function Stiff casing t w L F Metric 1 Objective 2 Minimise mass m Metric 2 m = mass w = width L = length  = density t = thickness S = required stiffness I = second moment of area E = Youngs Modulus Objective 1 Minimise thickness t Constraints Adequate toughness, G 1c > 1kJ/m 2 Stiffness, S with Unit 5, Frame 5.10 Materials Index to minimise the thickness Materials Index to minimise the mass

MECH L# 10 Conflicting Objectives 16/30 Relative performance metrics The thickness of a casing made from an alternative material M, differs (for the same stiffness) from one made of M o by the factor The mass differs by the factor Explore the trade-off between and We are interested here in substitution. Suppose the casing is currently made of a material M o, elastic modulus E o, density  o. Define a relative penalty function, Z* (  now dimensionless) Relative mass = ratio of Materials Indices (mass) Relative thickness = ratio of Materials Indices (t)

MECH L# 10 Conflicting Objectives 17/30 Plotting the relative penalty function, Z* Penalty lines for casing Assume mass and thickness are equally important:  * = 1 Thickness relative to ABS Mass relative to ABS 1 10 Low alloy steel Al-alloys Mg-alloys GFRP CFRP Al-SiC Composites Ti-alloys ABSNi-alloys Thickness relative to ABS Mass relative to ABS Z* 1 Z* 2 Z* 3 Polymers are all dominated solutions Materials on trade- off surface are metals and high performance composites Explains the use of Mg alloys in mobile phones and laptop computer casings, cameras Penalty functions of gradient -  * = -1  * = ??? Current casing Decreasing values of Z* at constant  *

MECH L# 10 Conflicting Objectives 18/30 Thickness relative to ABS, t/t o Mass relative to ABS, m/m o Trade-off surface Conclusion: Four-sector trade-off plot for minidisk player Q: Is material cost relevant? Not a lot -- the case only weighs a few grams. Volume and weight are much more valuable. The four sectors of a trade-off plot for substitution A. Better by both metrics C. Lighter but thicker D. Worse by both metrics B. Thinner but heavier win-win sector win-lose sectors: worth exploring win-lose sector: worth exploring sometimes Don’t bother Current casing

MECH L# 10 Conflicting Objectives 19/30 Tute 5: E 7.4. Compressed air cylinders for trucks Design goal: lighter, cheap air cylinders for trucks Compressed air tank

MECH L# 10 Conflicting Objectives 20/30 Design requirements for the air cylinder Pressure vessel Minimise mass Minimise cost Dimensions L, R, pressure p, given Must not corrode in water or oil Working temperature -50 to C Safety: must not fail by yielding Adequate toughness: K 1c > 15 MPa.m 1/2 Wall thickness, t; Choice of material Specification Function Objectives Constraints Free variables R = radius L = length  = density p = pressure t = wall thickness L 2R Pressure p t

MECH L# 10 Conflicting Objectives 21/30 Light and Cheap Air Cylinder Metric 1: mass Eliminate t to give: L 2R Pressure p t Constraint (no yielding) Objective 2 Vol of material in cylinder wall Aspect ratio Q Objective 1 Metric 2: cost R = radius L = length  = density p = pressure t = wall thickness = yield strength S = safety factor Q = aspect ratio 2R/L Materials Index to minimise the mass Materials Index to minimise the cost

MECH L# 10 Conflicting Objectives 22/30 Conflicting Objectives: Relative mass and cost This is a problem of material substitution. The tank is currently made of a plain carbon steel. The mass m and cost C of a tank made from an alternative material M, differs (for the same strength) from one made of M o by the factors Explore the trade-off between and Relative mass = ratio of Materials Indices (mass) Relative cost = ratio of Materials Indices (cost)

MECH L# 10 Conflicting Objectives 23/30 Four sectors trade-off plot for air tank Trade-off surface Additional constraints: K 1c >15 MPa.m 1/2 T max > 373 K T min < 223 K Water: good + Organics: good + Anything in this corner is slightly better (cheaper and lighter) Current tank. Axes normalised to locate current tank material at origin (1,1) win-win sector win-lose sector: Anything in this corner is a trade-off (lighter but more $): eg, Ti, Mg, GFRP or CFRP Al alloys; stronger steels For 2009: Explain how the normalising is done by reading the bubble’s coordinates on CES, and then dividing the axes scales by those values)

MECH L# 10 Conflicting Objectives 24/30 The trade-off plot: Conclusions Aluminium alloys and low alloy steels offer modest reductions in mass and material cost. Need a strategy to explore the win-lose (trade-off) sectors as well: Penalty functions and Exchange constants Win-win sector: Safe options, but kind of boring.

MECH L# 10 Conflicting Objectives 25/30 Cost relative to plain carbon steel, C/C o Mass relative to plain carbon steel, m/m o Penalty Functions and Exchange Constants Z*=1  * -1 = 0.05 (trucks,  = 20$/kg) To the left: OK; to the right: too expensive Z*=1  * -1 = 0.01  = 100$/kg Ti, expensive! Z*= 2  = 1 (current, cheap and heavy) Z*= 0.6  = 1 (cheaper and lighter) (safe bet, boring) GFRP: border line for 20$/kg CFRP is a cheaper option

MECH L# 10 Conflicting Objectives 26/30 The main points Real design problems involve conflicting objectives -- often technical or environmental performance vs. economic performance (cost). Trade-off plots reveal the options for material selection or material substitutions that solve the conflict, and (when combined with the other constraints of the design) frequently point to a sensible final choice. If the relative value of the two metrics of performance (measured by an exchange constant) is known, a penalty function allows an unambiguous selection: the exchange constants allow exploring the chart's win-lose (trade-off) sectors as well as the win-win sector. Engineering definition of  P 1, P 2 = performance metrics (mass, cost)

MECH L# 10 Conflicting Objectives 27/30 Tute 5, E 7.5. Refrigerated truck: Solution 1: CES chart for and 1/E Use foamed materials data base (level 3) Grapher version =-3*x+.7 =-0.001*x+.035 For 2009: Explain here that using a high alpha means that you value thermal properties more than stiffness. A low alpha puts stiffness ahead of thermal behaviour.

MECH L# 10 Conflicting Objectives 28/30 xy=-3*x+.7y=-0.001*x =-3*x+.7 =-0.001*x+.035 Tute 5, E 7.5. Refrigerated truck: Solution 1: CES chart for and 1/E Use foamed materials data base (level 3) Excel version

MECH L# 10 Conflicting Objectives 29/30 Refrigerated Truck Penalty Function Lines =-3*x+0.7 makes stiffness very important. As Ceramic foams are very stiff, they are selected but the thermal losses may be high, and the toughness may be low. =-0.001*x Medium density polymeric foams ( ) are good if thermal losses are more important than having a high stiffness.

MECH L# 10 Conflicting Objectives 30/30 The End For 2009: This lecture is too messy and complicated. The penalty functions Z are not well explained. Use the minidisk case as an illustration of how to reduce Z at constant alpha, and the truck tank as an example of changing alpha at constant Z. Cut the maths a bit.