MME445: Lecture 12 Multiple Constraints and Conflicting Objectives Part 1: Selection with multiple constraints Learning Objectives Knowledge & Understanding Knowledge on analytical and graphical selection method in design with multiple constraints Skills & Abilities Ability to select systematically when there are multiple constraints Values & Attitudes Understand the fact that “the most restrictive constraint wins” in selection methods with multiple constraints Resources • M F Ashby, Materials Selection in Mechanical Design, 4th Ed., Ch. 07 1 Outline of today’s lecture Introduction and synopsis Selection with multiple constraints Summary and conclusions Introduction & Synopsis Most decisions in life involve trade-offs. Sometimes the trade-off is to cope with conflicting constraints: I must pay this bill but I must also pay that one — you pay the one that is most pressing At other times the trade-off is to balance divergent objectives: I want to be rich but I also want to be happy — and resolving this is more difficult since you must balance the two, and wealth is rarely measured in the same units as happiness. 2 Solution of the problem having one design objective with many non-conflicting constraints is straightforward: Apply the constraints in sequence, rejecting at each step the materials that fail to meet them. The survivors are viable candidates. Rank them by their ability to meet the single objective and then explore documentation for the top-ranked candidates. The selection of materials and processes often must satisfy several conflicting constraints. In the design of an aircraft wing spar, weight must be minimized, with constraints on stiffness, fatigue strength, toughness and geometry. In the design of a disposable hot-drink cup, cost is what matters; it must be minimized subject to constraints on stiffness, strength, and thermal conductivity. A second class of problem involves more than one objective, and here the conflict is more severe. Nature being what it is, the choice of materials that best meets one objective will not usually be that which best meets others. The designer charged with selecting a material for a wing spar that must be both light and cheap faces an obvious difficulty: The lightest materials are not always the least expensive, and vice versa. To make any progress, the designer needs a way of trading weight against cost — a problem we have not encountered until now. We shall deal these type of selection problems in the next class. 3 Selection with Multiple Constraints Nearly all material selection problems are over-constrained, “Should not deflect more than something, must not fail by yielding, by fatigue, by fast-fracture …” There are more constraints than free variables !! Recapitulating, we identify the constraints and the objective imposed by the design requirements, and apply the following steps. Screen, using each constraint in turn. Rank, using the performance metric describing the objective (often mass, volume, or cost) or simply by the value of the material index that appears in the equation for the metric. Seek documentation for the top-ranked candidates and use this to make the final choice. Design with Multiple Constraints and Multiple Objectives Function One Objective: Multiple Objectives: the performance metric several performance metrics One Constraint Rank by performance metric Many Constraints Rank by most restrictive performance metric One Constraint Many Constraints Penalty function method Combination of methods 4 Simplest case: Design with one objective, meeting a single constraint Tie rod Function Minimise mass Carry force F without yielding; Length given One Objective: Multiple Objectives: one performance metric several performance metrics One Constraint Rank by performance metric Many Constraints Rank by most restrictive performance metric One Constraint Many Constraints Penalty function method Combination of methods Or several non-conflicting constraints, such as melting point, corrosion resistance, etc. Selection with multiple non-conflicting constraints (a) and a single objective (b). Screen using the constraints and rank using the objective. 5 The previous example is a special case of a single objective that can be limited by more than one non-conflicting constraints. But what if the requirements for a tie-rod of minimum mass might specify both stiffness and strength, leading to two independent equations for the mass? One notch up in complexity: Single objective / Conflicting Constraints Tie rod Function Minimise mass No yield & Given deflection No corrosion Tmax > 100 °C One Objective: Multiple Objectives: one performance metric several performance metrics One Constraint Rank by performance metric Many Constraints Rank by most restrictive performance metric One Constraint Many Constraints Penalty function method Combination of methods The most restrictive constraint determines the performance metric (mass) 6 material for a stiff, light tie rod E r sy r material for a strong, light tie rod A= area, L*= length, ρ = density, S*= stiffness, E = Young’s modulus, F*f = collapse load, σy= yield strength or elastic limit Conflicting Constraints: Strong /Stiff Light Tie Rod Evaluate competing constraints and performance metrics Requires stiffer material Max. deflection Must not yield Stiffness constraint Strength constraint s 1%; 0.1% L E sy = yield strength σ σy m1 = deflection E = elastic modulus F r L2 E r m2 FL s y Competing performance metrics Rank by the more restrictive of the two, meaning…? 7 If stiffness is the dominant constraint, the mass of the rod is m1; if it is strength, the mass is m2. If the tie is to meet the requirements on both, its mass has to be the greater of m1 and m2. Writing ~ = max (m1, m2) m (7.1) ~ we search for the material that offers the smallest value of m. This is an example of a “min–max” problem, not uncommon in the world of optimization. We seek the smallest value (min) of a metric that is the larger (max) of two or more alternatives. Can be solved either analytically or graphically The analytical method Analytical solution in 3 steps: Rank by the more restrictive of the constraints 1. evaluate both m1 and m2 for each member of the population, 2. assign the larger of the two to each member, and then 3. rank the members by the assigned value, seeking a minimum. 8 1. Calculate m1 and m2 for given L and F m1 r m2 FL s y r L2 E F = 0.5% L=1m F = 10 kN r, Mg/m3 E, GPa sy, MPa m1 (E), g Epoxy-Aramid Fibre 1.38 80 1400 34.5 9.9 Glass-Epoxy UD Composite 1.95 45 1100 86.7 17.7 Low alloy steel 7.90 217 2445 72.8 32.3 Epoxy-Carbon Fibre (SMC) 1.70 150 345 22.7 49.3 Al 6061-T6 2.73 74 320 73.8 85.3 Titanium 4.45 117 1138 76.1 39.1 m2 (TS), g ~ 2. Find the largest of every pair of m’s (which is m) 3. Find the smallest of the larger ones The most restrictive constraint requires a larger mass and thus becomes the controlling or active constraint. Advantages and disadvantages of analytical method When there are 3,000 rather than 3 materials to choose from, simple computer codes can be used to sort and rank them. But this numerical approach 1. yields a solution that is not general (as it depends on F and L) 2. lacks visual immediacy and the stimulus for creative thinking that a more graphical method allows. 9 The graphical method Graphical solution using indices and bubble charts: • More general/powerful. • Allows for a visual while physically based selection. • Involves all available materials. • Incorporates geometrical constraints through coupling factors. • For a population of materials, m1 is plotted against m2. • Each bubble represents a material. • All the variables in both equations for m1 and m2 are specified except the material so the only difference between one bubble and another is the material FIGURE 7.4 The graphical approach to min–max problems. (a) Coupled selection using performance metrics (here, mass m). 10 To minimize mass, the best choices lie somewhere near the bottom left. But where, exactly? If stiffness is paramount and strength is unimportant the choice must surely differ from that if the opposite were true. FIGURE 7.4 The graphical approach to min–max problems. (a) Coupled selection using performance metrics (here, mass m). The line m1 = m2 separates the chart into two regions. • In one, m1 > m2 and constraint 1 (stiffness) is dominant. • In the other, m2 > m1 and constraint 2 (strength) dominates In region 1 our objective is to minimize m1, since it is the larger of the two; in region 2 the opposite is true. • This defines a box-shaped selection envelope with its corner on the m1 = m2 line. The nearer the box is pulled to ~ the bottom left, the smaller is m. • The best choice is the last material left in the box. FIGURE 7.4 The graphical approach to min–max problems. (a) Coupled selection using performance metrics (here, mass m). 11 - We need a new chart. Index M2 • What will happen if these values change? Larger • Earlier graph (Fig. 7.4(a)) was plotted with m1 and m2 as axes, using specific single values of L*,S*; and Ff* Smaller Now, the data is re-plotted against materials indices M1 = E/r and M2 = sy/r. Each bubble still represents a material, but now its position depends only on material properties, not on the values of L*,S*; and Ff* Smaller Index M1 Larger FIGURE 7.4 (b) A more general approach: coupled selection using material indices M and a coupling constant Cc. m1 = L*2 S* r M1 = E r m2 = L* F *f s y M2 = r E r sy For the minimisation problem, the RHS factor is not altered !! m1 = m2 L*2 S* r E r = L* F *f s y L*2 S* M1 = L* F *f M2 M2 = L* S* F *f Coupling constant, CC M1 log M2 = log M1 + log L* S* F *f Coupling line 12 Larger Coupling lines • This describes a line of slope 1, in a position that depends on the value of L*S*/Ff*. Smaller Index M2 M2 = (CC) M1 Smaller Larger Index M1 A box, with its corner on the coupling line, is pulled down toward the bottom left. Large CC Constraint 2 dominant Coupling lines M2 = CC M1 Index M2 Now, the selection strategy remains the same: Larger FIGURE 7.4 (b) A more general approach: coupled selection using material indices M and a coupling constant Cc. Small CC Changing either one of these, or the geometry of the component (here described by L*) moves the coupling line and changes the selections. Smaller But the chart is now more general, covering all values of L*, S* and Ff*. Decreasing ~ m Smaller Constraint 1 dominant Index M1 Larger FIGURE 7.4 (b) A more general approach: coupled selection using material indices M and a coupling constant Cc. 13 Summary & Conclusions Real designs are over-constrained and many have multiple objectives Method of maximum restrictiveness copes with conflicting multiple constraints Analytical method useful but depends on the particular conditions set and lacks the visual power of the graphical method Graphical method produces a more general solution Next Class MME445: Lecture 24 Multiple Constraints and Conflicting Objectives Case studies with multiple constraints 14