Course SummaryProspective Students
Course Summary
(3 credits) Theory/practice of linear programming will be developed, including the determination of the optimum mix of products, levels of staffing, blending, network analysis, and multi-period planning.
*Prerequisite: SYSEN 520 or instructor’s permission
Overview
Engineers face problems involving making decisions. They have to pick the “best” from a list of alternatives. This requires a clear definition of the objective and the ability to clearly express it in terms of the decision variables. A thorough understanding of the problem and a detailed description are necessary to define the performance parameters that are needed to specify the objective function. The design variables can be either controllable or uncontrollable. We set the uncontrollable variables to meaningful values and write the objective function in terms of controllable variables. Lastly, we need to define the constraints. They could be the equality or the inequality types. Having clearly defined the optimization task, we solve the problem and determine the design variables that will yield the optimum solution.
Finding the best solution to a problem is a basic human desire. In this process, one may or may not encounter problems with restrictions or constraints which makes them either constrained or unconstrained sets, respectively. There are two basic approaches scientists use to solve these problems: optimality criteria methods and search methods. In the optimality criteria approach, a set of conditions has to be satisfied at the optimal point. In the search technique, a solution set, say a design, is hypothesized and improved upon until an optimum set is found that satisfies any optimality criteria. In this course, we will address both of these methodologies.
When the number of variables is two, graphical methods are possible to obtain the optimum design variables. In practice, engineers face situations that involve hundreds of design variables, and the objective function and the constraint equations are generally nonlinear in nature. With the advent of powerful computers and sophisticated computer programs like Matlab and Excel, it is possible to solve practical problems without making a lot of simplifying assumptions. The optimization toolbox available in Matlab provides a very user-friendly graphical user interface (GUI) and makes the solution process much easier.
We will start with unconstrained optimization in which there are no constraints and extend the concepts to constrained optimization. The optimality conditions assume that we are at an optimum point and at that point these conditions are to be satisfied. Some of the conditions are necessary and some of the conditions are sufficient. We will discuss situations when they are necessary and sufficient.
Linear Programming problems are a class of problems in which the objective function and all the constraints are linear. This is a very popular and practical approach that is applicable to many real-world problems. The Simplex algorithm developed by George Dantzig in 1947 is a very popular and powerful algorithm for linear programming problems. We will learn the detailed procedure for both the one-phase and the two-phase Simplex algorithm.
The optimization toolbox is available in Matlab and will be used extensively throughout the course. We will also use the solver capabilities in Excel.
*Note: All students are expected to have a working knowledge of Matlab and Excel and an understanding of mathematics at the undergraduate engineering level. The mathematics needed for the course will be quite basic and will be reviewed as needed. You will also be expected to use our online discussion forums, chat rooms, and electronic mail on a regular basis. The most important prerequisite of all, however, is your interest in the course, motivation, and commitment to learning.
Course Objectives
After completing the course, a student will be able to:
- Understand the theoretical and conceptual basis of setting up and solving optimization problems.
- Understand the mathematics of linear and nonlinear programming.
- Appreciate the use of software like Matlab and Excel to be able to solve optimization problems.
- Acquire skills to analyze practical problems and the ability to convey the results.
Course Materials
Required Textbook
- Introduction to Optimum Design by Jasbir S. Arora, Academic Press 4th Edition, 2017. (Free E-book option available). ISBN 978-0-12-800806-5L
- E-Book Option: An online version of your text is available at no cost as a Penn State Library E-Book. You can access the E-Book through the Library Resources link on the course navigation. You may choose to use the E-Book as an alternative to purchasing a physical copy of the text. For questions or issues, contact the University Libraries Reserve Help (UL-RESERVESHELP@LISTS.PSU.EDU).
- You may choose to use the E-Book as an alternative to purchasing a physical copy of the text. For questions or issues, you can contact the University Libraries Reserve Help (UL-RESERVESHELP@LISTS.PSU.EDU).
Optional Texts
-
- An Introduction to Optimization by E.K. P. Chang and S. H. Zak, Second Edition, Wiley-InterScience, John Wiley & Sons, Inc, 2001 ISBN: 0-471-39126-3
- Optimization Concepts and Applications in Engineering by A. D. Belegundu and T. R. Chandrupatla, Second Edition, Cambridge University Press, 2011.
- Applied Optimization with Matlab Programming by P. Venkatraman, Second Edition, John Wiley & Sons, Inc, 2009 ISBN: 978-0-470-08488-5
Required Software
1. MATLAB
In this course, we will use MATLAB, which is available for free via Penn State WebLabs. Instructions for accessing MATLAB can be found in the course modules.
- In order to access MATLAB via WebLabs, you will first need to be connected through GlobalProtect VPN. View the instructions on how to connect to the GlobalProtect VPN.
- Once you are connected to the VPN, you can access PSU WebLabs and launch Mathworks MATLAB R2022a.
***If you have any issues accessing GlobalProtect or PSU WebLabs, please contact the IT Service Desk for assistance.
A variety of MATLAB and Simulink Tutorials are available at MathWorks.com
2. Excel Software
All Penn State students have free access to Microsoft Office 365, which includes Excel.
Proctored Exams
Proctored Exams – None
Grading and Examinations
Note: A grade is given solely on the basis of the instructor’s judgment as to the student’s scholarly attainment (see the Penn State Graduate Degree Programs Bulletin).
The following grading system applies to graduate students:
- “A” (Excellent) indicates exceptional achievement;
- “B” (Good) indicates substantial achievement;
- “C” (Satisfactory) indicates acceptable but substandard achievement; and
- “D” (Poor) indicates inadequate achievement and is a failing grade for a graduate student.
Students are reminded that simply meeting the minimal requirements of any assignment (both in terms of content and presentation) will result in a letter grade of “B” or less for that assignment.
Grades will be based on the following scale:
A = 95-100, A- = 90-94, B+ = 87-89, B = 84-86, B- = 80-83, C+ = 77-79, C = 70-76, D = 65-69, F = 64 and Below
| Assignment Category | Quantity | % of Final Grade |
| Weekly Homework Sets (Individual) | 6 | 20% |
| Midterm Exam (Individual) | 1 | 40% |
| Final Exam (Individual) | 1 | 40% |
Class Participation (Discussion Forums)
There will be discussion forums for students to discuss among themselves different aspects of the course, and the instructor will participate in the discussions when it is appropriate. You are encouraged to work together on homework assignments. Use these discussion forums to post your questions and to read the responses from your classmates.
Homework Assignments
You will be assigned homework problems every week. It is very important that you complete and submit all the homework assigned in order to master the materials and keep up with the course schedule.
All assignments (homework, exams) for each week must be submitted to the appropriate assignment in Canvas before 11:59 PM EST on the following Sunday (see Course Schedule below for exact dates) for the submission to be considered on time.
- You may collaborate on homework assignments through the discussion forums, but the midterm and final exams are individual assignments.
- Homework may require Excel, Matlab, Word, handwritten, or graphical solutions. Word, handwritten, or graphical solutions should be converted to PDF files.
Exams
There will be two exams (one midterm exam and one final). Exams are on an individual basis. Any questions on exams should be directed to the instructor, who will add information to the exams if many students are experiencing particular problems.
Course Topics
- Optimization – The Basics
- Graphical Optimization Excel & Matlab
- Optimality Conditions
- Optimum Design with Excel Solver
- Optimum Design with Matlab
- Linear Programming Methods
- Numerical Methods for Optimum Design
Prospective Students
For more information on this program, check out the Master of Engineering Management website!