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 at least one 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

Design of a Can

The purpose of this project is to design a can to hold at least 400 ml ofliquid, as well as to meet other design requirements (1 ml = 1cm3). Thecans will be produced in the billions so it is desirable to minimizemanufacturing costs. Since cost can be directly related to the surfacearea of the sheet metal, it is reasonable to minimize the amount ofsheet metal required to fabricate the can. Fabrication, handling,aesthetics, and shipping considerations impose the followingrestrictions on the size of the can: the diameter should be no more than8 cm and no less than 3.5 cm, whereas the height should be no morethan 18 cm and no less than 8 cm.

Design of a Can (cont.)

Design of a Can (cont.)

Examp: Design of a CanProblem Formulation

1. Project/problem statement

2. Data and information collection

3. Identification/definition of design variables

4. Identification of a criterion to be optimized

5. Identification of constraints

Design of a Can

1. Vol>= 400 ml; minimize costs(sheet metal)

2. Data Collection 3. Identification/Definition

of Design Variables

4. Cost Function

5. Constraints

Examp: Insulated Spherical Tank Design

Problem Formulation

1. Project/problem statement

2. Data and information collection

3. Identification/definition of design variables

4. Identification of a criterion to be optimized

5. Identification of constraints

Insulated Spherical Tank

1. minimize the life-cycle cooling cost with r (m) as its radius.

2. Data Collection 3. Identification/Definition

of Design Variables

4. Cost Function

5. Constraints

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)

Formulation with Intermediate Variables

Formulation with Design Variables Only

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

design variables2. # of independent equality 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)

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 RegionS={x| hj(x)=0, j=1 to p; gi(x)

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*)0hj(x*)0

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

HW

Problem 2.19 & 2.20 (page 51)

Note: Make sure to write your optimization problem in

the Standard Optimization Format Dont forget the side constraints

Appendix B

Lab Exercise

5.0 ,5.0 valueinitialLet

520 520

0211

..),(min

:, :

=

=

Step 1 View the base input template in HyperGraph

1. Start HyperGraph.2. Using the window mode selector, change to TextView

Step 1 View the base input template in HyperGraph(cont.)

3. Click the Open Document toolbar button

4. Use the file browser to open the file Simple.tpl from your working directory.

The following text is displayed:

{parameter(DVAR1,"Area 1",.5,0.2,5)}

{parameter(DVAR2,"Area 2",.5,0.2,5)}

{RES = DVAR1 + DVAR2}

{CON = 1/DVAR1 + 1/DVAR2 - 2}

{RES}

{CON}

{DVAR1}

{DVAR2}

Step 1 View the base input template in HyperGraph(cont.)

5. Click the Edit Text icon on the tool bar The Templex statements are evaluated, and the

parameters are replaced with their initial values.

The following numbers are displayed:

1.0

2.0

0.5

0.5

Step 2 Begin the study

From Applications pull-down on the menu bar in HyperGraph, select HyperStudy

Step 3 Perform the study setup1. Click Add study and select New.

The Add Study dialog is displayed.2. Click OK to accept the default Label: and Variable:.3. Select your working directory as Study directory: using the file browser

in the upper right of the window.4. Click Next to continue on to the Create models dialog.5. Click Add model and set Model type: to Template.6. Click OK.

A new model is added to the list.7. Select the Template file using the file browser in the upper right.8. Open the file Simple.tpl from your working directory.9. Click Next to continue to the Create design variables dialog.10. Check the settings of lower and upper bounds.11. Click Next to continue to the Do nominal run dialog.12. In the Solver input file field enter res.

This is the name of the input file created by HyperStudy for the solver.

Step 3 Perform the study setup (cont.)13. Choose Templex from the Solver execution script pull-down menu.

The input to Templex will be res. You do not need to edit the Solver input arguments field.

Step 4 Perform the base run1. Click Write and then Execute, or Write/Execute.

This creates a nom_run/m_1 directory inside the study directory. The res file is the result of the nominal run and is created in the nom_run/m_1 directory.

2. Click NextThe Create responses dialog is displayed.

Step 5 Define responses1. Click Add Response.2. Click OK.3. Repeat steps 1 and 2 to create a second response.4. Click Response_1.

Step 5 Define responses (cont.)

Step 6 Run an optimization study

Step 7 View the iteration history of an optimization study

Results

Results (cont.)

Results (cont.)

Results (cont.)

CH2 Optimum Design Problem FormulationProblem FormulationProblem FormulationProblem FormulationProblem FormulationProblem FormulationProblem FormulationExamplesDesign of a CanDesign of a Can (cont.)Design of a Can (cont.)Examp: Design of a CanExamp: Insulated Spherical Tank DesignA General Mathematical Model (Standard Design Optimization Model)Formulation with Intermediate VariablesFormulation with Design Variables OnlyAbout Standard ModelMaximization Problem>= ConstraintsDiscrete & Integer Design VariablesFeasible SetActive/Inactive/Violated Constraints(at a design point x*)HWAppendix BLab ExerciseStep 1 View the base input template in HyperGraphStep 1 View the base input template in HyperGraph(cont.)Step 1 View the base input template in HyperGraph(cont.)Step 2 Begin the studyStep 3 Perform the study setupStep 3 Perform the study setup (cont.)Step 4 Perform the base runStep 5 Define responsesStep 5 Define responses (cont.)Step 6 Run an optimization studyStep 7 View the iteration history of an optimization studyResultsResults (cont.)Results (cont.)Results (cont.)