1. MECH4301 2008 L# 10 Conflicting Objectives 1/30 MECH4301 2008, 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, 84-92. (Simple, readable) C. H. Cáceres, "Economical and environmental factors in light alloys automotive applications", Metall. Mater. Trans. A, 2007, 38, 1649-1662. (Automotive applications) M. F. Ashby, "Multi-objective optimization in material design and selection", Acta Materialia, 2000, 48, 359-369. (Advanced reading)

2. MECH4301 2008 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

3. MECH4301 2008 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

4. MECH4301 2008 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)

5. MECH4301 2008 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

6. MECH4301 2008 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.

7. MECH4301 2008 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)

8. MECH4301 2008 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)

9. MECH4301 2008 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.

10. MECH4301 2008 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, 1649-1662.

11. MECH4301 2008 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 20 100 to 500 500 to 1000 20-4000 3000 to 10000 ( 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, 1649-1662. M. F. Ashby, "Multy-objective optimization in material design and selection", Acta Materialia, 2000, 48, 359-369.

12. MECH4301 2008 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 --

13. MECH4301 2008 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 20 100 to 500 500 to 1000 20-4000 3000 to 10000 Engineering definition of  Penalty Function & Exchange Constants: Powerful and Unambiguous Strategy for Material Substitutions under Conflicting Objectives

14. MECH4301 2008 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

15. MECH4301 2008 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

16. MECH4301 2008 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)

17. MECH4301 2008 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 0.1110 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  *

18. MECH4301 2008 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

19. MECH4301 2008 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

20. MECH4301 2008 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 +100 0 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

21. MECH4301 2008 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

22. MECH4301 2008 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)

23. MECH4301 2008 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)

24. MECH4301 2008 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.

25. MECH4301 2008 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

26. MECH4301 2008 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)

27. MECH4301 2008 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.

28. MECH4301 2008 L# 10 Conflicting Objectives 28/30 xy=-3*x+.7y=-0.001*x+.035 0.0010.6970.034999 0.0020.6940.034998 0.0030.6910.034997 0.0050.6850.034995 0.010.670.03499 0.020.640.03498 0.030.610.03497 0.050.550.03495 0.10.40.0349 0.150.250.03485 0.170.190.03483 0.20.10.0348 0.210.070.03479 0.220.040.03478 0.230.010.03477 0.3 0.0347 0.50.0345 10.034 20.033 30.032 50.03 100.025 200.015 240.011 =-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

29. MECH4301 2008 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+0.035 Medium density polymeric foams (0.08-0.16) are good if thermal losses are more important than having a high stiffness.

30. MECH4301 2008 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.