1. MER 160, Prof. Bruno1 Optimization The idea behind “optimization” is to find the “best” solution from a domain of “possible” solutions. Optimization methods provide mathematical tools that allow the search for this “best” solution to carried out in a rational and efficient way. Before these tools can be applied the design problem needs to be recast in an appropriate form.

2. MER 160, Prof. Bruno2 Optimization in Design Need Identified Problem Definition Concept Generation Modeling/Simulation Workable Design Optimization/ Optimal Design Concept Selection

3. MER 160, Prof. Bruno3 Optimization x1x1 x2x2 * Set of all “workable” or “functional designs” (Allowed by physics, orange boarder) Set of all “acceptable” designs. (allowed by constraints, yellow boarder) Optimal Design U(x 1,x 2 ) = Umax H 2 : X 2 > c 2 H1 : X 1 > c 1

4. MER 160, Prof. Bruno4Lingo Objective Function: represents the quantity (U) which is to be optimized (the “objective”) as a function of one or more independent variables (x 1, x 2, x 3 …) –The best form to put the objective function in depends on the optimization technique to be employed. U = U(x 1, x 2, x 3 …)→U opt

5. MER 160, Prof. Bruno5 Lingo Design Variables: The independent variables (x 1, x 2, x 3 …) that the objective function depends on. –It is generally best to minimize the number of design variables … the more variables the tougher the optimization will be.

6. MER 160, Prof. Bruno6 Lingo Cont. Constraints: Relations which limit the possible (physical limitations) or the permissible (external constraints) solutions to the objective function. Constraints come in two mathematical “flavors.” –Equality Constraints: Often come from fundamental physics considerations (e.g. cons. of mass) –Inequality Constraints: Often from safety, cost, space, material strength limits etc. Generally Equality Constraints are easier to deal with than inequality constraints. Generally it is desirable to reduce the number of constraints.

7. MER 160, Prof. Bruno7 Mathematical Formulation Objective Function of n independent design variables: For U( x 1, x 2, x 3 …x n ) Find U opt Equality Constraints: G i ( x 1, x 2, x 3 …)=0i=1,2,…,m Inequality Constraints: H j (x 1, x 2, x 3 …) C j j=1,2,…l If n>m → An Optimization problem results If n=m → A unique solution exists…just solve all equations simultaneously If n<m → The problem is “overconstrained” no solution which satisfies all of the constraints is possible

8. MER 160, Prof. Bruno8 Optimization x1x1 x2x2 * Set of all “workable” or “functional designs” (Allowed by physics, orange boarder) Set of all “acceptable” designs. (allowed by constraints, yellow boarder) Optimal Design U(x 1,x 2 ) = Umax H 2 : X 2 > c 2 H1 : X 1 > c 1

9. MER 160, Prof. Bruno9 Classification of Optimization Techniques Calculus based Techniques –Lagrange Multipliers Search Methods –Elimination Methods Exhaustive Fibonacci golden section search –“Hill Climbing” techniques Lattice Search Steepest ascent “Programming” methods –Linear Programming –Geometric Programming

10. MER 160, Prof. Bruno10 Techniques Cont. Calculus Methods: Related in principle to the simple “minimization” and “maximization” techniques that you used in Calculus. –Require all equations to be differentiable (a.k.a. Continuous, “well behaved” and explicit) –This puts severe limitations on the usefulness of this technique

11. MER 160, Prof. Bruno11 Techniques Cont. Search Methods: Are exactly what they sound like … you keep trying different solutions searching for the “best.” –These techniques are Very flexible, making them applicable to a broad array of problems. –The “techniques” provide algorithms which help to find the optimum in the minimum possible number of searches. –Very useful when a design variable can only take on certain discrete values. –These methods do result in a finite amount of uncertainty about the optimum.

12. MER 160, Prof. Bruno12 Exhaustive Search x1x1 x2x2 * Optimal Design U(x 1,x 2 ) = Umax H 2 : X 2 > c 2 H1 : X 1 > c 1 Note: None of the search points exactly hits the optimum. The space between search points is known as the “interval of uncertainty” The aptly named

13. MER 160, Prof. Bruno13 * * * * Lattice Search 1 2 3 4

14. MER 160, Prof. Bruno14 “Programming” Methods These methods have nothing to do with “Programming” in the sense that you usually think of it! Linear Programming: Very powerful, but very limited! –Applies only when the objective function and all constraints can be expressed as Linear Functions … Not often the case in thermal/fluid systems Dynamic Programming: Related to optimizing a “process” –Lets you find the “best” order to do steps in –Very useful in Project Management and Assembly optimization Geometric Programming: Similar to Linear Programming, but now all functions must be polynomials. –Reduces the restrictions on linear programming quite a bit –Very useful where empirical correlations for system behavior are known –More difficult and computationally intensive than Linear Prog.

15. MER 160, Prof. Bruno15 Example Set up an optimization problem formulation for a water Chilling system to minimize first cost.

16. MER 160, Prof. Bruno16 Homework? What exactly are you trying to optimize? → “What is your Objective Function?” What is the Absolute maximum that one would be willing to pay? → “Is there a cost inequality constraint that we can use to help limit our design domain?” What are your “design variables” ? What is the nature of your functions? (continuous / discrete) (linear/non-linear) etc.