Math and Time Optimization

Since this course is about optimization and problem solving, you will need to write a maximum two-page paper with a one-page maximum mathematical modelon a problem that you have encountered and explain how you solved it.  Any papers over the page allotments will receive a 20 point deduction.  This paper is meant to be a thought paper that will link what you are learning in class to your everyday life.  Please make sure that your writing style is formal.  There are five major components you should discuss/have in the paper.  The first component of the paper (first paragraph) will have you elaborate on what optimization problem you tried to solve, e.g., maximizing your score for a game, minimizing the amount of reading you did for a class, maximizing the number of events you participated in during the quarter, etc.  Make sure in this paragraph you motivate why this optimization problem is important to you.  The next part of the paper (second paragraph) will have you discuss what the decision variables are for the problem you solved, and how they are related to your objective you tried to optimize.  A decision variable can be defined as a choice that you get to make that either directly or indirectly affects your objective function.  You should categorize the decision variables into constrained and unconstrained.  A constrained decision variable is one that shows up in your constraints, while an unconstrained decision variable is one that does not.  You should note that most if not all decision variables should be constrained.  If any of these decision variables are directly in your objective function, then you need to explain whether they positively or negatively affect the objective function.  For any decision variable you do not do this for, you will lose 2 points.  To have a positive effect on an objective function means that when you increase the decision variable, it causes a positive effect on the objective function, e.g., more studying leads to higher GPA.  For a decision variable to have a negative effect on the objective function means that increasing the decision variable will negatively affect the objective function, e.g., increasing partying has a negative effect on GPA.  For any decision variables that are unconstrained, you need to explain why they are unconstrained. The third part of the paper (paragraph 3) will explain what constraints you encountered when trying to solve the problem.  A constraint can be defined as something that holds back a decision variable or multiple decision variables from the optimal choice of it/them.  This portion of the paper should explain which constraints are affecting which decision variables.  You should explain how the constraint is affecting the decision variable from being optimal, e.g., a time constraint will not allow you to study more.  You should note that if you have a constraint that has only one decision variable and is met with equality, then the decision variable is truly not a decision variable, i.e., if your constraint is sleep and you say that you must get exactly eight hours of sleep, then you would not consider sleep a decision variable because it is predetermined and already set to a value.  The fourth component of the paper (paragraph 4) will explain how you went about solving the problem.  No mathematical discussion is needed in this paragraph.  You should write about the thought process that you undertook or will undertake to solve the problem.  Any discussion about solving the problem using a Lagrangian or any other mathematical means will get you a 10 point deduction.  The first four components will be worth 75 points. The last component of the paper, which will be on its own page, is a mathematical model of your problem.  Your grading of this component will be based on how well you are able to capture in mathematical form what you have written in paragraphs 1 through 3.  This model should use generalized notation for any types of functions (e.g., objective functions, constraint functions, etc.).  A function, which is designated with parenthesis, could be a single letter like F(·) or multiple meaningful letters like GPA(·) where you will put the decision variables separated by a comma where the dot is.  You need to define all the variables with letters or combination of letters as well as the functions (e.g., Ts = time studying, Tw = time working, Q = quantity eaten, ft(Ts, Tw) is a the time constraint, etc.).  You will receive up to a 10-point deduction for not defining your variables or functions.  You should make sure that each constraint is set to some value using an equality or inequality constraint (e.g., ft(Ts, Tw) = 24 or ft(Ts, Tw) ≤ 24).  You are allowed to write each function out (e.g., Ts + Tw = 24), but you must also have the functional notation (e.g., f(Ts, Tw) = Ts + Tw = 24) or 5 points will be deducted.  You will receive a 5-point deduction if your model and what is written in your paper do not align with each other (e.g., talking about a decision variable in the paper, but not having it in the mathematical model or vice versa).  If you do not put the mathematical model in the paper you will receive a 25-point deduction.  You must type out the mathematical model.  If you do not type out the model, you will receive a 10-point deduction.  Trying to solve the mathematical model will get you a 5-point deduction.  I will discuss more about this modelling in class as we get closer to the due date of the paper.  I would strongly encourage meeting with me to discuss this model if you need assistance. The paper should be well written and is worth 100 points.  It should be uploaded to Canvas by 9:00 pm on the due date given above.   A paper submitted late will receive up to a 20-point deduction.  For every spelling and grammatical error that is found written in the paper, up to 5 points will be deducted for each error up to 30 points.  If you have more than six grammatical errors, you will need to resubmit the paper corrected for all the mistakes and you will receive at most the lowest score given to the other students.  I would encourage you to write it well the first time.  The second part of the grading will examine how well you discuss the four major components that are asked for in the paper, as well as development of the mathematical model.  Make sure you thoroughly discuss how your decision variables are incorporated into your constraints and objective function.  You should be able to easily make a mathematical model from what you have written in your four paragraph discussion.  If you decide to write the paper on any aspect of minimizing the amount of effort/time/space allocated to the paper, you will lose 25 points.  I will discuss more about the paper in a class video.  I encourage you to attend my online office hours to discuss your paper.  Early papers will be joyously accepted.  Any paper that is submitted by a student that is similar to one that has been submitted in past quarters will be considered plagiarism and will receive a zero.