MER 160, Prof. Bruno1 Optimization The idea behind “optimization” is to find the “best”...

MER 160, Prof. Bruno 1

OptimizationOptimization

• 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.

MER 160, Prof. Bruno 2

Optimization in Design

Need Identified

Problem Definition

Concept Generation

Modeling/Simulation

Workable Design

Optimization/Optimal Design

Concept Selection

MER 160, Prof. Bruno 3

OptimizationOptimization

x1

x2

*

Set of all “workable”or “functional designs”(Allowed by physics,orange boarder)

Set of all “acceptable”designs. (allowed by constraints, yellow boarder)

Optimal DesignU(x1,x2) = Umax

H2 : X2> c2

H1 : X1> c1

MER 160, Prof. Bruno 4

LingoLingo

• Objective Function: represents the quantity (U) which is to be optimized (the “objective”) as a function of one or more independent variables (x1, x2, x3…)

– The best form to put the objective function in depends on the optimization technique to be employed.

U = U(x1, x2, x3…)→Uopt

MER 160, Prof. Bruno 5

LingoLingo

• Design Variables: The independent variables (x1, x2, x3…) 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.

MER 160, Prof. Bruno 6

Lingo Cont.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.

MER 160, Prof. Bruno 7

Mathematical FormulationMathematical Formulation• Objective Function of n independent design

variables:For U( x1, x2, x3…xn) Find Uopt

• Equality Constraints: Gi( x1, x2, x3…)=0 i=1,2,…,m

• Inequality Constraints:Hj(x1, x2, x3…) < or > Cj j=1,2,…l

If n>m → An Optimization problem resultsIf n=m → A unique solution exists…just solve all

equations simultaneouslyIf n<m → The problem is “overconstrained” no solution

which satisfies all of the constraints is possible

MER 160, Prof. Bruno 8

OptimizationOptimization

x1

x2

*

Set of all “workable”or “functional designs”(Allowed by physics,orange boarder)

Set of all “acceptable”designs. (allowed by constraints, yellow boarder)

Optimal DesignU(x1,x2) = Umax

H2 : X2> c2

H1 : X1> c1

MER 160, Prof. Bruno 9

Classification of Optimization TechniquesClassification 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

MER 160, Prof. Bruno 10

Techniques Cont.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

MER 160, Prof. Bruno 11

Techniques Cont.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.

MER 160, Prof. Bruno 12

Exhaustive SearchExhaustive Search

x1

x2

*

Optimal DesignU(x1,x2) = Umax

H2 : X2> c2

H1 : X1> c1

Note: None of the searchpoints exactly hits the optimum. The spacebetween search points isknown as the “interval of uncertainty”

The aptly named

MER 160, Prof. Bruno 13

*

*

*

*

Lattice SearchLattice Search1

2

3

4

MER 160, Prof. Bruno 14

““Programming” MethodsProgramming” 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.

MER 160, Prof. Bruno 15

ExampleExample

• Set up an optimization problem formulation for a water Chilling system to minimize first cost.

MER 160, Prof. Bruno 16

Homework?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.