5.6 Solving Optimization Problems Homework Answer Key

[Stayla Casillas]

Thefirst part of the course develops some basic mathematical tools of analysiswhich we will use to solve optimization problems. This covers roughly parts II andIII of the text, and may include excerpts from parts VI and VII. The second part(part IV of the textbook) covers classical, calculus-based methods ofoptimization including Lagrange multipliers and the Kuhn-Tucker theorem. Themethods of Lagrange and Kuhn-Tucker have been invaluable in solving manyof the problems you will typically encounter in economics (consumer andproducer choice, social welfare max, etc.). We then cover the solution of difference and differential equations, and their stability properties (part V).If time permits, we will look at dynamic optimization and the Maximum Principle.

Simon and Blume's book is the main text. I plan to coverParts II-IV and VII of Simon and Blume, with some excerpts from Part VI. Time permitting, we will then turn our attention to Part V and dynamic models.

The book by Garrity covers a fair chunk of the math we cover, plus quitea lot we don't use. I found its preface outstanding. The other two are onhow to solve mathematical problems, especially those involving proofs.

Homework will be submitted in person or by emailing it to me. If youemail it, it may be easiest to write it out and then photograph it withyour phone. If so, please combine the pages into a single pdf.I will not be happy if I see 10 separate files for one assignment.

Tentative and subject to change, especially if there are hurricanes. Itis probably a bit over ambitious, but we'll see. There have been yearswhen we've made it all the way to the end. Once the semester is underway,the slides will show what was actuallycovered.

AAE 550 is a fast-paced, graduate-levelcourse that introduces students to the techniques of engineering designoptimization, leading into topics needed for Multidisciplinary DesignOptimization (MDO). The course also presents application of thesetechniques to solve engineering design problems.

To accomplish these two tasks, AAE 550 has two overlapping parts. Thefirst part of the course exposes students to basic concepts about andto implementation of numerical optimization techniques, assuming thatthe student has little or no knowledge of these topics. The second partof the course uses this knowledge as the basis for students toinvestigate approaches for multiobjective and multidisciplinaryoptimization.

My approach to teaching engineering design optimization relies uponhaving students: (1) practice formulating problems, (2) examineimportant features of various optimization algorithms, and (3) usecomputer tools to solve optimization problems. This does notlend itself well to traditional exams, so the grade for AAE 550 willrely upon homework assignments (which essentially take the place ofmid-term exams) and electronic assessments (which take the place oftextbook question-based homework).

Some knowledge of basic statics and strength of materials might helpwith understanding example and homework problems, but this is notrequired. Appropriate equations and formulas for these problems will beprovided.

Students are to complete and submit the electronic assessments and homework assignments by the class session listed in the calendar and in theassignments available on the Brightspace pages;generally the time for this is 11:59pm Eastern. Eight graded electronic assessments and five graded homework assignments are planned for the semester. This course will not have traditional examinations or a final exam.

The Teaching Assistants and I will try to accommodate late submittals of assessments and homework, but this requires advance notice. You must notify the teaching team using the aae5...@ecn.purdue.edu address three business days before the due date if you need extra time to submit your assignment for us to accept your late submittal without penalty. Without the three-business-day advance notice, late assessments or homework assignments may be accepted with a penalty unless there are extenuating circumstances beyond the student's control.

The Office of the Dean of Students (ODOS) student absence policies cover cases of grief/bereavement, military service, jury duty, parenting leave, or emergent medical care (see -policies.html). These are the only cases covered by ODOS.

Because I am concerned about fairness in the class, I do not makethe solutions available until all students have turned in thatassignment, including approved late submittals. If the TAs orI have posted the solutions, we will no longer accept late homeworkassignments.

There are often questions or concerns about how your assignments aregraded. The TAs and I understand this, and we will be willingto entertain questions about your grades and requests forre-grading. However, with a class of this size, we mustreceive these questions and / or re-grade requests no later than oneweek after the graded assignments have been returned.

Copying and plagiarism in the homework is notacceptable. The instructor and teaching assistants will use plagiarism detection software to screen work submitted by thestudents. All code snippets included in your homework and final project must be readable; i.e., cut and paste them into your document rather than using a screen capture.

You may talk with other students, give and receive advice about structuring your Matlab scripts or Excel worksheets, as well as giveand receive advice about using Matlab functions and Excel functions.

Do not share scripts or worksheets that you develop yourself or thatyou have modified from the examples provided by the instructor. Do not simply copy and paste scripts that other students have developed or have modified from the examples provided by the instructor; this includes copying and pasting work from students who have taken the course in previous semesters whether you know them personally or have obtained older work from a commercial website. The work you submit must be demonstrably independent from that of other students, so that the instructor, teaching assistants and/ or graders can reliably judge your mastery of the topics.

As generative Artificial Intelligence (AI) tools continue to develop, these are likely to be both transformative as well as disruptive. The information they provide can sometimes be incorrect or misleading; they can also provide shortcuts that keep you from working to understand content and have a meaningful learning experience. These tools can also generate example MATLAB scripts. Given the discussion above about collaboration, using an AI tool to help you in AAE 550 can provide you with the potential to better understand material in the course; however, you are still responsible for completing your own work. The AAE 550 course has been developed over time and the basic approach of the assessments and homework were set before AI tools became available; there should be no need to use these tools to perform well in the course.

Each assessment and homework will be graded on a (points scored) /(points available) basis. Total points available will vary, but each assessment and homework will have equal weight as other assessments and homework assignments. Grade assignment will use the criterion (straight-scale) approach shown above, but the instructor reserves the right to curve the grades if appropriate. Under no circumstance will the scale be more stringent than the criterion given below (e.g. 93% or above will always earn an A), and the curve will never span more than one grade scale (e.g. the lowest A possible when grades are curved is 93%). A total score of 50% or lower will always fail.

The tutorial explains how to add and where to find Solver in different Excel versions, from 2016 to 2003. Step-by-step examples show how to use Excel Solver to find optimal solutions for linear programming and other kinds of problems.

Everyone knows that Microsoft Excel contains a lot of useful functions and powerful tools that can save you hours of calculations. But did you know that it also has a tool that can help you find optimal solutions for decision problems?

Excel Solver belongs to a special set of commands often referred to as What-if Analysis Tools. It is primarily purposed for simulation and optimization of various business and engineering models.

The Excel Solver add-in is especially useful for solving linear programming problems, aka linear optimization problems, and therefore is sometimes called a linear programming solver. Apart from that, it can handle smooth nonlinear and non-smooth problems. Please see Excel Solver algorithms for more details.

While Solver can't crack every possible problem, it is really helpful when dealing with all kinds of optimization problems where you need to make the best decision. For example, it can help you maximize the return of investment, choose the optimal budget for your advertising campaign, make the best work schedule for your employees, minimize the delivery costs, and so on.

How to add Solver to ExcelThe Solver add-in is included with all versions of Microsoft Excel beginning with 2003, but it is not enabled by default.

Note. If Excel displays a message that the Solver Add-in is not currently installed on your computer, click Yes to install it.Where is Solver in Excel?In the modern versions of Excel, the Solver button appears on the Data tab, in the Analysis group:

Where is Solver in Excel 2003?After the Solver Add-in is loaded to Excel 2003, its command is added to the Tools menu:

Note. The examples discussed in this tutorial use Solver in Excel 2013. If you have another Excel version, the screenshots may not match your version exactly, although the Solver functionality is basically the same.How to use Solver in ExcelBefore running the Excel Solver add-in, formulate the model you want to solve in a worksheet. In this example, let's find a solution for the following simple optimization problem.

Problem. Supposing, you are the owner of a beauty salon and you are planning on providing a new service to your clients. For this, you need to buy a new equipment that costs $40,000, which should be paid by instalments within 12 months.

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