CH2 Optimum Design Problem Formulation

By Hsiu-Ying Hwang

Problem Formulation1. Project/problem statement2. Data and information collection3. Identification/definition of design variables4. Identification of a criterion to be optimized5. Identification of constraints

Note: Prior to optimization, various preliminary analyses would have been completed and a detailed design of a concept or a sub-problem needs to be carried out

Problem Formulation1. Project/problem statement2. Data and information collection3. Identification/definition of design variables4. Identification of a criterion to be optimized5. Identification of constraints

Key: Knowing overall objectives and requirements

Problem Formulation1. Project/problem statement2. Data and information collection3. Identification/definition of design variables4. Identification of a criterion to be optimized5. Identification of constraints

Key: Analyze trial designs; (Measurable)Analysis procedures and tools should be identified

Problem Formulation1. Project/problem statement2. Data and information collection3. Identification/definition of design variables4. Identification of a criterion to be optimized5. Identification of constraints

Key: The design variables should be independent of each other as far as possible. (If they aredependent, then their values cannot be specifiedindependently.)The number of independent design variablesspecifies the design degrees of freedom for the problem.

Problem Formulation1. Project/problem statement2. Data and information collection3. Identification/definition of design variables4. Identification of a criterion to be optimized5. Identification of constraints

Key: The criterion must be a scalar function; must be function of design variables be minimized (cost function) or maximized

Problem Formulation1. Project/problem statement2. Data and information collection3. Identification/definition of design variables4. Identification of a criterion to be optimized5. Identification of constraints

Note:Linear & Nonlinear ConstraintsFeasible Design (acceptable or workable)Equality and Inequality (unilateral or one-sided) ConstraintsImplicit Constraints

Examples• Design of a Can (Basic)• Insulated Spherical Tank Design

(Intermediate & Design Variables)• Design of a Two-Bar Bracket (Intermediate

& Design Variables)• Design of a Cabinet• Minimum Weight Tubular Column Design

(Integer Programming)• Minimum Cost Cylindrical Tank Design• Design of Coil Springs• Minimum Weigh Design of a Symmetric

Three-Bar Truss

A General Mathematical Model (Standard Design Optimization Model)

Minimize f(x)=f(x1, x2, …., xn)

Subject to hj(x)=hj(x1, x2, …, xn) = 0; j =1 to p

gi(x)=gi(x1, x2, …, xn) <= 0; I =1 to m

Note:Once the problems have been transcribed into mathematical statements using a standard notations, they have the same mathematical form. It’s for any design application.

About Standard Model1. f(x), hj(x), gi(x) must depend explicitly or implicitly on

design variables2. # of independent equality constraints <= number of

design variables (p<=n) 3. The inequality constraints are written as <= 0

There is no restriction on the number of inequality constraintsThe total number of active constraints (satisfied at equality) at the optimum is usually less than or at the most equal to the number of design variables

4. Unconstrained vs. constrained optimization problems5. Linear vs. Nonlinear programming problems6. c* f(x) -- Optimum design does not change but optimum

value changed; similarly for inequality and equalities constraints

Maximization ProblemMaximize F(x)

Is the same as

Minimize f(x)= -F(x)

“>=“ ConstraintsGj(x) >= 0

Is the same as

gj(x)= -Gj(x) <= 0

Discrete & Integer Design Variables

• Discrete Design Variables (Discrete Programming Problems)

• Integer Design Variables (Integer Programming Problems)

Approach• Continue Discrete (Nearest)• Adaptive numerical optimization procedure (Only

variables close to discrete value assigned and held fixed, and then optimized again …)

Feasible Set

• A feasible set for the design problem is a collection of all feasible designs

• “Constraint Set”, “Feasible Design Space”, “Feasible Region”S={x| hj(x)=0, j=1 to p; gi(x)<=0, i=1 to m}

• In general, feasible region usually shrinks when more constraints are added in the design model (f(x) is likely to increase); and expands when some constraints are deleted (f(x) is likely to decrease);

Active/Inactive/Violated Constraints(at a design point x*)

• Active gi(x*)=0hj(x*)=0

(an inequality constraint may or may not be active; equality constraint must be active for all feasible design)

• Tight or Binding gi(x*)=0 (in general for inequality constraints)

• Inactive gi(x*)<0

• Violated gi(x*)>0hj(x*)<>0

(an inequality constraint may or may not be active; equality constraint must be active for all feasible design)

HW1

• Problem 2.19 & 2.20 (page 51)

Note:• Make sure to write your optimization problem in

the Standard Optimization Format• Don’t forget the side constraints

Appendix B