9. Decision-making in Complex Engineering Design. School of Mechanical Engineering Associate Professor Choi, Hae-Jin

Transcription

1 9. Decision-making in Complex Engineering Design School of Mechanical Engineering Associate Professor Choi, Hae-Jin

2 Overview of Lectures Week 1: Decision Theory in Engineering Needs for decision-making in engineering Multiobjective optimization Goal Programming Compromise Decision Support Problem (cdsp) Week 2: Decision-making under Uncertainty (I) Utility theory Utility based selection Decision Support Problem (usdsp)] Week 3: Decision-making under Uncertainty (II) Robust design principles Robust Design Type I : Taguchi method Robust Design Type II: Robust Concept Exploration Method (RCEM) Robust Design Type III: RCEM - Error Margin Indices -2-

3 What is ENGINEERING? As engineers, we frequently think of ourselves as PROBLEM SOLVER. Being taught problem-solving skills as the major element of our education, throughout our lives. (this is important) The problem solving is NOT the principle activity of engineering; rather DECISION-MAKING -3-

4 Science vs Engineering Science (physical science) is the process of rationally and methodically seeking to understand nature, with the principal objective of developing a predictive or problem-solving capability. Engineering involves the manipulation of nature to create systems from the benefit of at least some segment of mankind. -4-

5 Science vs Engineering Load Height Height Maximum Stress Given Height Maximum Stress??? Science Height??? Given Maximum Allowable Stress Engineering -5-

6 Notion of Engineering Definition Man cannot change nature; it is allowed only that man can manipulate nature Something physical is created through engineering process The process of creating something physical requires effective allocation of nature s resources Decision-making -6-

7 System Modeling Why? What? A model is a necessary ingredient to make a decision Why do we need models? What are models? Load Deflection=F(Load, Height) Height F: system model -7-

8 Why do we need models? The ability to make good engineering decisions lies in having large amount of good information. Inexpensive way of trying many different decisions is essential. Without models, we would have to build actual systems in order to gain decision information -> only few tests at most With models, such as computer models, we might be able to test hundreds or even thousands of decision options. -8-

9 What are models? A model is an abstraction of reality. (Hazelrigg) A model is a simplified representation of something real. Three basic classes of models: Iconic models Analog models Symbolic models -9-

10 Iconic Models Scale representations, either large or small, of physical or real things Examples: Wind tunnel models of airplanes, rocket, building and bridges Pilot plants; model trains and automobiles Use when equations that adequately describe the behavior of the systems are not available. -10-

11 Analog Models Models that use one property to represent another. Examples Colors on a relief map Electrical circuit as an analog to heat transfer Voltage Temperature gradient Current Heat flux Resistance Thermal resistivity -11-

12 Symbolic Models Are also called Mathematical Models Use symbols to designate properties of the real system Are more abstract than other models covering a vast range Examples F=m*a; E=m*c 2 Are often transformed to computer simulation Are used to examine system alternatives in a very inexpensive way. -> big benefit for engineering (decision-making) -12-

13 Compromise Decision-Making Multi-Objective Optimization portability cost Compromise Decision Support Problem size performance? ergonomics heat battery life vibration / noise -13-

14 Selection Decision-Making Utility Theory Utility-based Selection Decision Support Problem Multi-criteria Concept Evaluation portability heat size battery life vibration / noise performance cost ergonomics Selected Concept -14-

15 Optimization Optimization is derived from the Latin word optimus, the best. Optimization characterizes the activities involved to find the best. People have been optimizing forever, but the roots for modern day optimization can be traced to the Second World War. -15-

16 Types of Optimization Note: There are MANY different optimization methods/algorithms However, they are can be grouped by fundamental principles of: Model formulation or solution method/algorithm Do not forget: Optimization methods fall in the category of decision support systems/methods Question: What are some other means for decision support? -16-

17 Problem Formulations Different types of optimization model formulations exist: Classical non-linear formulation Linear Programming formulation Baseline model formulation Goal Programming formulation Compromise Decision Support Problem formulation etc. Basic classifications are: Constrained versus unconstrained Linear versus non-linear Single objective versus multi-objective Another classification can be made by variables: continuous/discrete/mixed-integer -17-

18 Single versus Multi-Objective Decision-making is rather multi-objective by nature, so we will look at some multi-objective Some covered are: Baseline model Goal Programming (GP) model Compromise Decision Support Problem model Others exist -18-

19 Goal Programming (GP) Multiobjective mathematical programming technique is Goal Programming (GP) The term "goal programming" is used by its developers to indicate the search for an "optimal" program (i.e., a set of policies to be implemented) for a mathematical model that is composed solely of goals. Developers argue that any mathematical programming model may find an equivalent representation in GP. GP provides an alternative representation that often is more effective in capturing the nature of real world problems. -19-

20 Difference between Objectives and Goals In Goal Programming a distinction is made between an objective and a goal: Objective: In mathematical programming, an objective is a function that we seek to optimize, via changes in the problem variables. The most common forms of objectives are those in which we seek to maximize or minimize. For example, Minimize Z = A(X) Goal: In short, a goal is an objective with a right hand side. This right hand side (T) is the target value or aspiration level associated with the goal. For example, A(X) T -20-

21 Solving Multi-objective Models Solving multi-objective models is NOT standard practice (yet). Often, a first step in solving these models is a model transformation into a model that CAN be solved using an existing algorithm/solver. -21-

22 Transforming into a GP model Step 1 : Transform all objectives into goals by establishing associated aspiration levels based on the belief that a real world decision maker can usually cite (initial) estimates of his or her aspiration levels. Hence, maximize A r (X) becomes A r (X) T r for all r minimize A s (X) becomes A s (X) T s for all s. where T r and T s are the respective aspiration levels (targets). Step 2 : Rank-order each goal according to its perceived importance. Hence, the set of hard goals (i.e., constraints in traditional math programming) is always assigned the top priority or rank. Step 3 : All the goals must be converted into equations through the addition of deviation variables -22-

23 Deviation Variables - Distance to target In Goal Programming and other approaches (like compromise Decision Support Problem) deviation variables are used to convert inequalities to equalities. The deviation variable d is (then) defined as: d = Ti -Ai(X) Note: Mathematically, the deviation variable d can be negative, positive, or zero. From a reality point of view, a deviation variable represents the distance (deviation) between the aspiration level (target) and the actual attainment of the goal. -23-

24 Two Deviation Variables instead of One The deviation variable d can be replaced by two variables: d = d i- -d i + where d i- d i+ = 0 and d i-, d i + 0 Why? Many optimization algorithms do not like negative numbers and the preceding ensures that the deviation variables never take on negative values. The product constraint ensures that one of the deviation variables will always be zero. The goal formulation (now) becomes: A i (X) + d i- -d i+ = T i ; i = 1,2,..., m subject to d i- d i+ = 0 and d i-, d + i 0-24-

25 Values of Deviation Variables Note that a goal is always expressed as an equality: A i (X) + d i- -d i+ =T i ; i=1,2,...,m And when considering this equality, the following will be true: if A i (X) < T i is true, then (d i- > 0 AND d i + = 0) must be true; if A i (X) > T i is true, then (d i- = 0 AND d i + > 0) must be true; if A i (X) = T i is true, then (d i- = 0 AND d i + = 0) must be true. When in doubt, just use a numerical example. -25-

26 Desired Values of Deviation Variables Again, note that a goal is always expressed as an equality. A i (X) + d i- -d i+ = T i ; i = 1,2,..., m To achieve a goal (i.e., reach the target), 3 situations are possible: 1. To satisfy A i (X) T i, we must ensure that the deviation variable d i+ is zero. - The deviation variable d i- is a measure of how far the performance of the actual design is from the goal. 2. To satisfy A i (X) T i, the deviation variable d i- must be made equal to zero. - In this case, the degree of overachievement is indicated by the positive deviation variable d i+. 3. To satisfy A i (X) = T i, both deviation variables, d i- and d i+ must be zero. Question: How would this change if we only had a single d i that can be positive or negative? Thus, to achieve a target, we must minimize the unwanted deviation(s)! -26-

27 Minimizing deviations Consider the preceding three situations again. To achieve a goal (i.e., reach the target), 3 situations are possible: 1. To achieve A i (X) T i, we must minimize ( d i+ ) 2. To achieve A i (X) T i, we must minimize ( d i- ) 3. To achieve A i (X) = T i, we must minimize (d i- + d i+ ). (How would this change if we only had a single d i that can be positive or negative?_ Big Question: What if you have more than one goal? That is, how do you minimize multiple deviation variables? -27-

28 Two Approaches to Prioritizing Objectives Goals are not equally important to a decision maker. How do we represent our preferences? Two approaches are: Assign weights and calculate the sum of the deviation variables ( distance to target ) multiplied by their individual weights. Rank order goal deviations in priority levels, often referred to as a preemptive formulation. The preemptive formulation does not exclude the assignment of weights. -28-

29 Weighted Sum Approach Assigning weights, or weighted sum approach, is one of the most common ways of converting multi-objective/multi-goal problems into a single objective problem. Min z = (w 1 d 1- + w 2 d 2+ +.) = (w i d i- + w k d k+ ) The weights (w) can be any value, in principle. In case the sum of the weights equals 1, then we speak of an archimedean formulation. However, assigning weights without thought can cause problems. -29-

30 Rank Ordering In Rank Ordering, you prioritize one goal/objective above each other without giving explicit mathematical weights. Basically, in words, Goal A has to be achieved before Goal B. I should not even think about Goal B yet if Goal A has not been achieved yet. One mathematical construct that is used in rank ordered formulations is the Lexicographic Minimum. The concept of a lexicographic minimum is used in several multi-objective formulations Goal Programming Compromise DSP -30-

31 Compromise Decision Support Problem Traditional Single-Objective Optimization Multi-Objective Decision Support: Compromise DSP Given n, number of decision variables p, number of equality constraints q, number of inequality constraints f(x), an objective function g i (x), constraint functions Find x Subject to g(x)=0 i=1,...,p g(x)<0 i=p+1,...,p+q Optimize f(x) Constraints from Math. Programming Goals and Deviation Variables from Goal Programming Given Find n, number of decision variables p, number of equality constraints q, number of inequality constraints m, number of system goals g i (x), constraint functions x (system variables) d i-,d i+ (deviation variables) Satisfy System constraints: g(x)=0 i=1,...,p g(x)<0 i=p+1,...,p+q System goals: A i (x)/g i + d i- -d i+ = 1 Bounds: X min i <X i <Xmax i d i-,d i+ >0 and d -. i d i+ = 0 Minimize Z = [f 1 (d i-,d i+ ),, f k (d i-,d i+ )] preemptive Z = W i (d i- + d i+ ) Archimedean

32 The Effect of Selecting a Formulation It is important to note that differences in formulation CAN cause differences in results. The most influential factors are the choices of: Objectives versus goals Goal Priorities Constraints versus goals (constraints are higher priority) Goal targets -32-

33 Pareto Optimality The typical role of a design engineer is to resolve conflicting objectives and arrive at a design that represents an acceptable or desired balance of all objectives. (Mattson & Messac 2002) Classical examples of conflicting objectives: Truss Design: Weight versus Strength Flywheel design: Kinetic Energy stored versus Weight Finite Element Meshes: Aspect Ratio versus Distortion Parameter Standard problem definition (Textbook s notation): Minimize f = [ f 1 (x), f 2 (x),, f m (x) ], where each f i is an objective function Subject to x Ω (constraints on space of design variables) -33-

34 Methods for Trading Off Across Objectives 1. Weighting of objectives (Archimedean) minimize f = w 1 f 1 (x) + w 2 f 2 (x)+ ; subject to x Ω; where w i > 0 and Σ w i = Lexicographic minimum: preemptive ranking of objectives 3. A slight twist: Picking one objective as primary, transforming remaining objectives into constraints minimize f 1 (x); subject to f 2 (x) c 2, f 3 (x) c 3, and f m (x) c m where c i is a limit x Ω These all provide point solutions (x*) based on an assignment of preferences among objectives. -34-

35 The Need Globally Viewing Tradeoffs in Optimality Thus far in class, preferences, weights, & limits were all chosen by engineering judgment trial and error, experience, etc. Varying weights & preferences to explore goal tradeoffs is manually intensive. How can we visualize a global picture of the tradeoffs in optimum solutions over a wide range of weights? Answer: Transform graphical solutions from design (variable) space to criterion space (also called objective space ). x 2 f 1 design space f 2 criterion space f 2 Ω Ω' x 1 f 1-35-

36 The Pareto Optimality Curve In criterion space, we can identify a special trade-off curve on the boundary where: Changing the weights in an Archimedean (weighted) objective function traces out the curve s path. No point is better than any other point on the line with respect to both objectives. No improvements can be made in any objective without trading off (worsening) the other. f 2 f 2 This part of the boundary is called the Pareto Curve (or Pareto Frontier) Or, the functionally efficient solution set There are Pareto curves in both the design variable space and the criterion space. Pareto curves contain Pareto points (solutions) Bold lines in the pictures (right) represent Pareto curves when maximizing objectives. f 2 f 2 f 1 f 1 Pareto f 1 f 1 Maximization Problem -36-

37 Rotating Disk (Flywheel) Example w(r) ( L + U )/2.0 w 1 (r) L Experimental evidence suggests that at high speeds the stresses are high near the hub of the rotating disk. For this reason, to get thestresseswithinsafelimits,itisadvisabletohavemoremass near the hub. The design criterion is to locate the points, P2 to P4 such that the kinetic energy is maximized and the mass of the rotating disk is minimized. w 2 (r) w 3 (r) w 4 (r) P 2 P 3 P = 0.10 P 1 5 = 1.0 P 4 U r Z -37-

38 Baseline Model for Flywheel Given The relevant information for the disk: Angular velocity of the disk = user input (rad/sec) Lower limit of thickness L = 0.01 (m) Upper limit of thickness U = 0.10 (m) Location of the hub P1 = 0.05 (m) Location of the rime P5 = 0.5 (m) Slope of the linear portion = 0.9 Density of the material of disk =7830 (kg/m 3 ) Yield stress of the material of disk YS =1.48E9 (N/m 2 ) Relevant equations for the physics of the problem. Find System variables They determine the profile of the rotating disk, P2, P3, and P4. Satisfy System constraints The stress constraints, R (r) y, T (r) y, where R, T, and y are the radial stress, tangential stress and yield stress respectively. The constraints on the geometry of the rotating disk, P1 P2, P 2 P3, P 3 P4, P 4 P5. Maximize The kinetic energy (K) of the rotating disk is to be maximized. Minimize The weight (M) of the rotating disk is to be minimized. -38-

39 Different Design Scenarios The traditional single-objective model is exercised in three ways, one with kinetic energy as objective function and mass of the disk as constraint, the other with mass of the disk as objective function and kinetic energy as constraint, and as a weighted sum of the two objectives. The compromise DSP template is exercised in three ways, The deviation function is modeled in the preemptive form with the achievement of the kinetic energy goal as first priority. the deviation function is modeled in the preemptive form with the achievement of the weight goal as first priority. the deviation function is formulated for the Archimedean form giving a weight of 0.6 for the achievement of the aspiration level of the kinetic energy of the disk and a weight of 0.4 for the weight of the disk. -39-

40 Different single objective functions Minimize The kinetic energy (K) of the rotating disk is to be maximized, -K(P 2, P 3, P 4 ) = -K where f is the objective function. Minimize The weight (M) of the rotating disk is to be minimized, W(P 2, P 3, P 4 ) = M where f is the objective function. Weighted sum approach: Minimize The kinetic energy of the disk is to be maximized and its weight (M) is to be minimized f ( P 2, P 3, P 4 ) = 0.6(-K )+ 0.4M, where f is the objective function. -40-

41 Compromise Decision Support Problem Satisfy (continued from the previous baseline model) System Goals K(P 2, P 3, P 4 ) /G kinetic_energy + d 1- -d 1+ = 1 W(P 2, P 3, P 4 ) /G weight + d 2- -d 2+ = 1 d i-, d i+ >0 and d i- d i+ = 0 where i=1,2 Minimize Z = [g 1 (d 1- ), g 2 ( d 2+ )] Z = W 1 d 1- + W 2 d 2 + Preemptive Archimedean -41-

42 Differences in Results By way of illustrating the "power" of a preemptive formulation, a comparison of the results obtained is made for Scenario I: First priority: maximize the kinetic energy of the disk Second priority: minimize the mass. The aspiration levels for these objectives are set at 1000 MJ and 800 kgs respectively KINETIC ENERGY (MJ) ASPIRED KINETIC ENERGY = 1000 MJ ASPIRED WEIGHT = 800 KGS Compromise DSP Single-objective approach ROTATION SPEED (rad/s) Note that compromise DSP solution "sticks" to 1000 MJ while trying to minimize weight. Question: At what speed do you expect a minimum weight? -42-

43 References Decision-based Design Theory: Kemper E. Lewis, W. C., and Linda C. Schmidt, 2006, Decision Making in Engineering Design, ASME, New York, NY. Hazelrigg, G. A., 1996, Systems Engineering: An Approach to Information-based Design, Prentice-Hall, Upper Saddle River, NJ. The Decision Support Problems: Mistree, F., Hughes, O. F. and Bras, B. A., "The Compromise Decision Support Problem and the Adaptive Linear Programming Algorithm" in Structural Optimization: Status and Promise, pages , (M. P. Kamat, Ed.), Washington, D.C.: AIAA, (1993). -43-