A very effective way to study the ideas and methods in this book in a university course environment is through student term projects. Several exercises are aimed at such project work, and the checklists in Chapter 8 should prove useful. Experienced designers will also find this material helpful in their early attempts at applying the ideas of the book in their work.
A term project should be undertaken as early in the semester as possible. After studying the first two chapters, a student should be able to formulate the optimization problem and develop an initial mathematical model. As progress is achieved through the book, the various methods and ideas can be applied to modify, simplify, and eventually solve the optimization problem. Students can use the available optimization codes as “black boxes” to obtain some early results. As their understaning of the algorithms increases they will be able to use the available codes more effectively, perform diagnostics, and interprete the numerical results properly.
Typical project milestones are: a project proposal that contains the description on the design optimization trade-offs and initial mathematical model; a progress report that outlines efforts to analyze and simplify the model based primarily on the material in Chapters 3 through 6; and a final report that contains the final model statement, model reduction, numerical solutions, parametric studies, and interpretation of results based on material throughout the book. Examples of course deliverables for such reports are given below.
Project topics may be assigned by the instructor or chosen by the students. Both approaches have merit and in a typical class a mixture is usually required. For the students who do not have a project idea of their own, topics may be selected from journal articles published in the engineering literature. However, students should be strongly encouraged to take full responsibility in accepting a published model or problem. This usually forces a more than perfunctory study of the problem at hand and a familiarization with the model, its sources, and limitations, which is necessary for an eventually successful optimization study.
The project archive in the present website is a valuable resource for project work. Instructor can obtain access information for their students by contacting the first author.
Modern mathematical software that combine modeling and symbolic and numerical computation capabilities are dramatically increasing the scope and ease of formulating and solving optimal design problems. This book offers many opportunities for the inspired reader to implement or test the ideas and methods presented using such software. These efforts are strongly recommended.
PROJECT PROPOSAL GUIDELINES
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Each team will submit a single proposal about the system to be designed. Each team member will be responsible for an individual subsystem and the team as a whole will study how the subsystem designs must be coordinated to achieve an overall system optimum. Clearly, the team must work together from the beginning but the idea is to assume the viewpoint of individual designers working concurrently on their portion of a larger system.
Each subsystem design problem should have at least 4-5 variables and about twice as many constraints. The overall system should comprise at least two subsystems, which will likely share some common variables.
Each student will be graded separately for their individual subsystem design.
Specific Section Guidelines
The project proposal must be formulated to have the following sections.
This is a 200-word description of the design project, the motivation for performing an optimization study, and the anticipated results.
This section introduces a qualitative statement of the system design project. Describe the system design problem, the anticipated trade-offs that motivate the optimization study, and the previous work that has been done by others. Identify the individual subsystems and explain qualitatively how they are linked. Specifically, explain if you expect that improving the design of each subsystem independently may not lead you to an overall optimal system design.
2. SUBSYSTEM DESIGN
For each subsystem identified in the introduction you must develop the full analytical model, as described in detail below. Each subsystem will be a single section. Within each such section, the individual team member will write the relevant subsections for the individual subsystem.
2.1 Problem Statement
This section contains a more detailed qualitative statement of the subsystem design problem. Building on the introductory description above, you now describe in more detail the anticipated trade-offs that motivate the subsystem optimization study. You also comment on previous work that has been done by others.
Define all symbols used and give units for each quantity. Coordinate with the other subsystem designers so you all use the same symbols for the same quantities and avoid nomenclature inconsistencies.
2.3 Mathematical Model
Describe the objective function in words and then derive its analytical expression in terms of design variables and parameters.
Describe each constraint in words and then derive its analytical expression in terms of design variables and parameters. Try to group the constraints in two categories: physical constraints that express natural laws and engineering specifications, e.g., conservation of mass, energy, strength and deflection requirements, etc; practical constraints that may express limitations of current engineering practice, rules of thumb, etc. — these will often have the form of upper/lower bounds on the design variables.
Design Variables and Parameters
Define and list the design variables and parameters. Give a set of typical values for the parameters that you can use for the particular application. Count and state the number of degrees of freedom. Find a set of values for the design variables that satisfy all of the constraints, i.e., show that there is at least one feasible solution in the model as stated.
At the end of the model development summarize the entire problem in one page, if possible, stating the objective and all the constraints in standard form.
In the derivation of expressions for the objective and constraint functions be as explicit as possible. If you do not have yet an explicit functional form, state it implicitly, e.g., x1 = f(x2, x3), and explain how you will calculate the function f. Examples of that may be curve-fitting from tables or a separate subroutine (e.g., structural analysis). If you have performed curve fitting already, give the details in an Appendix. Throughout the derivations you may cite the references that you used, so that you do not have to re-derive everything in the proposal.
There may be information that you have not obtained yet, for example, appropriate values for all of the parameters. In such cases, state how you expect to get this information.
Occasionally models used for design optimization in this class are created in other courses or in student research work. Acknowledge any such links and the assistance of any individuals that helped you in the preparation of the reported work.
List all references in alphabetical order, complete with author, title, publisher, year, and page number. In the document text give the citation as e.g., (Johnson, 1980).
Suggested Sources for Project Ideas
Ideally, you should choose some problem that is of particular interest to you. The problem may be from any discipline, i.e., you are not limited to mechanical design problems. Projects from previous years and published articles are a good source of ideas. See the archive in project archive.
You may use an existing model developed elsewhere as the basis for your project, but you must take responsibility for it, namely, you must understand all the details of its derivation so you can accept it as your own.
A Cautionary Note
It is easy to begin an optimization project with great expectations and try to use a large, complicated nonlinear model with many parameters and design variables. As your model becomes more concrete, modeling difficulties and numerical shortcomings will become evident. It is a good idea to start with the simplest model that can provide you with meaningful insights into the design tradeoffs. On the other hand you can start with a more complex problem statement and pare it down if you realize that the modeling work will be excessive. For the course, you should aim for subsystem models that have 5-10 design variables and 10-20 constraints, and expand later as needed.
Be prepared for several iterations of your modeling effort, before you perform optimization and after you start your optimization effort. This is normal both in a classroom setting and in a work setting. Often, an optimization study is a way to explore the weakness and deficiencies of your design analysis models.
The final report is a fairly involved document. Keep in mind that the project proposal and the progress report are integral parts of your final report, after appropriate editing. Early planning and execution of the intermediate reporting requirements will make final report production much easier.
PROGRESS REPORT GUIDELINES
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The progress report is an expansion and adaptation of your project proposal. This report will be further expanded and updated to produce your final report.
For each subsystem or subproblem you have identified, you must include the following sections:
1. Problem Statement
An updated version of the material in the proposal.
An updated version of the material in the proposal.
3. Mathematical Models
An updated version of the material in the proposal.
Specifically, you must now have a complete and validated (in terms of feasibility) optimization model statement.
4. Model Analysis
In this section you describe results from monotonicity analysis, as applicable.
Check for well boundedness, model transformations or simplifications you may decide to make. Use monotonicity tables, and activity matrices, where appropriate.
Check constraints for possible redundancy.
Identify active or conditionally active constraints, as appropriate.
5. Numerical Results
You must have coded your model and linked it with one or more of the optimization packages made available. Report initial results, possible local minima etc.
Pay special attention to how exactly you code your model for numerical processing. For example, avoid expressions that have divisions by quantities that can become zero.
Do not include printouts except for 1-2 pages of final results.
For the entire system/problem include a section that identifies possible design conflicts and tradeoffs among the subsystems/subproblems. Resolving these conflicts will be required to complete the project.
FINAL PROJECT REPORT GUIDELINES
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The report must be prepared in a form suitable for electronic transmission. Complicated algebraic manipulations in the appendices may be handwritten and scanned. Figures, tables and equations must be numbered. Figures and tables must each have captions. Printing of text must be at 1 1/2 space with 1 inch margins and the type font size should be similar to the one used in these guidelines (12 point Times).
The report must be submitted electronically at the course website along with one hard copy printed single-sided and loose-leaf. Please do not bind it! Please use the cover page template supplied at the end of these guidelines.
The report must contain the items described below to be acceptable. The sequencing of sections may change after the model development depending on the individual projects. Also, there may be a variation in the length and effort required in particular sections.
1. Cover Page and Abstract
This contains, in a single page, the title of the project, your name and date, and an abstract of approximately 200 words describing your problem and the results obtained. Do not write generalities but be specific about your work.
2. Table of Contents
A list of all sections, subsections, appendices, etc., contained in your report.
3. Problem Statement
As in the proposal/interim report (following the Optimization Checklist in Chapter 8 of your textbook), with any needed modifications
If you are designing a system composed of several subsystems, state the overall system problem and identify the individual subsystems you will first optimize separately, and the rationale for selecting these subsystems. Please note the individual that worked on each subsystem.
Define all symbols that you use, particularly for the mathematical model development, as should have been provided in your proposal/interim report. If you have several subsystems, you should make sure you use a consistent nomenclature and set of symbols. It may also be convenient to divide the symbols list to subsystems.
The following sections 5-9 should be done for each subsystem separately. For some projects, sections 8 and 9 may be more suitable for inclusion after the system integration study in section 10.
5. Mathematical Model
This is the section provided in the proposal/interim report, with any needed corrections. The last part of this section must summarize the model, give list of variables, number of equality and inequality constraints and number of degrees of freedom. Since it is likely that the final model evolved from its original statement, you should describe briefly how the model evolved and what made you change it. If the changes were a result of the optimization study itself, you may include these modeling decisions in the later section on optimization study (Section 7 below). Relegate to appendices any lengthy explanations you feel you need to include, so that the overall report flow is not disrupted.
6. Model Analysis
This section describes any possible bounding agreements, monotonicity properties and tables, constraint activity identification, model transformations and simplifications, scaling, case decomposition and anything else you have done to make the problem easier to solve numerically and/or analytically.
It is suggested that model analysis may be first described for a specific set of parameter values and then generalized to other parameter values to the extent possible.
7. Optimization Study
Identification of the solution and a description of how it was obtained should be presented. Unsuccessful attempts should be reported and documented in an appendix.
The solution should not be given as just a set of numbers. Other issues must be examined and described, e.g., constraint activity, values of multipliers, interior vs. boundary solution, global vs. local results, numerical stability, satisfaction of KKT conditions, different starting points.
Results obtained numerically should motivate attempts for analytical verification. Examine and explain, for example, if monotonicity analysis results agree with numerical results.
8. Parametric Study
The solution should be obtained for different sets of parameter values. Does the optimum change? Can the results be generalized? Are there ranges of parameter values that may dictate the type of solution expected?
9. Discussion of Results
Here, the results of the optimization study are given with an engineering interpretation. What are the design implications? Can you identify a “design rule” for an optimum solution? Do the results make sense? How does the model limit the solution? Are there “practical” constraints active and what would this imply? What would you do next to improve the answers or make the problem more interesting?
In a system design study, you must identify any conflicting requirements stemming from optimizing the subsystems separately. Do the subsystems have common variables, parameters, objectives or constraints? Are some variables in one subsystem parameters in the other? Is there an expected sequence of solving one subsystem before you solve another?
10. System Integration Study
In this section you examine the issues you raised in Section 9 regarding the linking of the subsystems. Can the combined subsystem optima give you the overall system optimum or are there conflicts to be resolved? In the latter case you must attempt the following:
(a) Select a system objective and combine all variables and constraints into a single optimization model. Solve this overall system problem as a single optimization problem. This is what we call the All-in-One (AiO) approach. If you can obtain a solution, compare it with the solutions you obtained from the subsystems. Discuss your results.
(b) The AiO approach may give you a problem that is too complicated and you cannot obtain numerical results, and a decomposition method is applied. Identify the problem partition into subproblems, each with their own local variables. These may be just the subsystems you identified in your earlier individual studies. Further, define a master problem with an appropriate objective that has as design variables the linking variables among the subproblems. Apply a coordination strategy where the master problem is solved wrt the linking variables (local variables fixed) and the subproblems are solved wrt the local variables (linking variables fixed). Examine how the coordination strategy terminates.
(c) Even if you do get results from the AiO strategy, you should perform the study in (b) and compare the results you get from the two approaches.
Complete reference list of any sources that you used to complete your project.
There may be several appendices containing anything that would distract the reader if used in the main text, for example, elaborate algebraic manipulations, proofs of monotonicities, coding of the models, and selected computer runs.
Your report should be a high-quality piece of work similar to technical papers, something you should be proud of. In fact, several student reports have resulted in scientific publications in the past. In any case, you should remember that others must be able to read, understand and duplicate what you have done with only the information contained in your report.