The operational research project was a lesson in graphing systems of linear inequalities, and applying the skills learned in the real world. We learned on the job as the class worked to solve a problem called "The Really BIG Question." An anonymous student was attempting to sell cupcakes in order to raise funds for a trip to Thailand. She had two types of cupcakes that she could make: plain and iced. She wanted to maximize her profit, but she had four constraints: the time available to make cupcakes, the amount of batter, the amount of icing, and oven space. We also had information on the profit that she could get from each type of cupcake. We made a graph to find the feasible region (the polygon that satisfies all of the inequalities), then made another clearer graph (above, top) and found the coordinates of the point that would produce the most profit. The solution turned out to be 75 plain and 50 iced cupcakes, for a profit of $212.50.
To close out our operational research project, students at a local college made problems - one per table group in each class - and brought them in for us to solve. My table group received a problem titled, "The Artist's Dilemma." An artist could create two types of art for a local art fair - pastel and watercolor - and he wanted to maximize his profit. He had constraints on time to make the art, and on art supplies available. We received information on the profit that he could get from each type of art, and were let loose on the problem. Unfortunately, due to bad timing on an FRC robotics competition, I could not attend school on the day that my class received the problem. So, my group solved the problem, and I solved it separately on the following Tuesday.
The first thing to do when solving a problem such as the Artist's Dilemma is determine what the constraints are. In this case, the constraints are time and supplies. The pastels and watercolors will take the same amount of time to make, and he has 16 hours total to paint. Also, each pastel will cost $5 and each watercolor will cost $15 for materials, and he has $180 to spend. Next, to make inequalities for these constraints, one determines which variable will be x and which will be y - I chose to call x the number of pastel paintings, and y the number of watercolor paintings. For the constraint on time, the inequality is x+y≤16. The ≤ sign is used because the artist can use 16 hours or less, but he cannot use more. The constraint on supplies is 5x+15y≤180, because it costs $5 per each x and $15 per each y. If the y-intercept is also in dollars, the units work out. We also have non-negative constraints; the artist cannot make negative paintings. These are simply x≥0 and y≥0. These inequalities can be graphed in a graphing computer program called GeoGebra - which I have used on previous projects - and the result is shown in blue above. The darkest region is called the feasible region; only the points inside that region or on its borders will satisfy all of the constraints. Now comes a very important part of information: since (0,0) will result in the lowest possible profit, the solution must be the farthest point on the feasible region from the origin. For any given feasible region, the closest and farthest points to the origin will always be vertices.
Now we need to figure out which vertex will result in the greatest profit. The thick black line represents the total profit. We know that each pastel will yield a profit of $40 and each watercolor will yield a profit of $100. So, the equation becomes 40x+100y=profit. For the black line, I substituted a random number - 1,000 - for profit. As one can see, as the line is translated farther and farther from the origin, the profit will increase, and the final point on the feasible region that it will intersect will be the vertex (6,10). To test this conjecture, one can plot a new parallel profit line: 40x+100y = 40(6)+100(10), which comes out to 40x+10y=1,240. This line is represented by the thick green line on the graph. It is apparent that this line intersects the point (6,10) and no other on the feasible region. So, 6 pastels and 10 watercolors will provide the greatest profit: $1,240.
In our groups, there were four roles, one per person: facilitator (Erik Lennox), documenter (Nidhal Dawood), spokesperson (Alejandra Sandoval), and GeoGebra guru (myself). The job of the facilitator was to make sure that the group worked efficiently, and that we all stayed on task. The documenter's job was to take notes of everything, so that the clients and group members could reference them if/when necessary. The spokesperson was the only group member who could speak to the client, to ask questions or to provide the answer once found. My job as GeoGebra guru was to plot the constraints in the form of inequalities in GeoGebra in order to create the feasible region. I was also in charge of making the graphs appear presentable, as needed.
Since I was unable to be there on the day of the Artist's Dilemma, I will do my group reflections on a practice problem that my group did prior to the Artist's Dilemma: the Breakfast Dilemma, in which the client was a cereal company working to create rations for the military.
a) Erik was a little overconfident with his position of being in charge at times, but it was obviously all in good fun, and he definitely added an element of humor to the group. That might sound trivial, but it can actually be useful in keeping morale up, and having a good time in general. When he was doing his job correctly, he did a good job of making sure that everyone else did their jobs correctly. His strongest habit of a mathematician was, "Be Systematic," because he did well at maintaining the various steps we were supposed to take to obtain our final answer. Nidhal took awesome and really useful notes; being the person to put the inequalities into GeoGebra, I referenced her notes on multiple occasions. Her strongest habit was the "Stay organized" habit. Alejandra didn't offer much while we were doing the Breakfast Dilemma, but she didn't have much of a job then. I'm sure that she did her job well during the Artist's Dilemma, and when I can say is that when I did the problem on Tuesday, she was eager to learn where the group got off-track during the problem, and what they needed to do to solve the problem. They were actually extremely close - they just mixed up the ≤ signs with ≥ signs - so kudos to them for knowing how to solve a problem that, frankly, could have been worded more clearly. Her strongest habit was, "Collaborate and Listen," because she paid close attention and asked questions when she needed to. For myself, I wasn't there, but when I got back, I solved the problem fairly quickly, and I did go over what I did with Alejandra. My strongest habit was, "Describe and Articulate," because I did a lot of that, when describing my answer for why the solution had to be a vertex and when talking to Alejandra after the problem.
b) During the breakfast problem, my group was extremely focused once Erik calmed down a bit. I mostly led the group in doing the solving, but I'm not sure if everyone was completely awake. Everyone certainly knew how to do their jobs, so while I was gone, I'm sure the group functioned extremely well. Also, the group paid attention during the Breakfast Dilemma problem, so I know that they knew how to solve the Artist's Dilemma. They did, if fact, know how to solve it; it was just the signs that mixed them up, so I could picture them having created their graph in no more than 30 minutes, then having become confused and spent the rest of the period trying to figure out what they did incorrectly. I didn't consult anyone on this, but just based off of these logical assumptions, I would give the rest of my group at least a 9/10 for teamwork. They probably did better, but like I said, I can't give a definite grade since I wasn't there.
c) I generally pick up on software such as GeoGebra fairly quickly, so I was proficient in the use of GeoGebra during the Breakfast Dilemma. I also used it well when I was solving the Artist's Dilemma on my own. The GeoGebra guru role is fairly straightforward; I am naturally inquisitive, so I had learned how to use all of the GeoGebra tools I needed by the time the Breakfast Dilemma and Artist's Dilemma were introduced. I feel like, although I did well in the Operational Research project, there's always more that I can learn with GeoGebra, and I look forward to doing this.
To close out our operational research project, students at a local college made problems - one per table group in each class - and brought them in for us to solve. My table group received a problem titled, "The Artist's Dilemma." An artist could create two types of art for a local art fair - pastel and watercolor - and he wanted to maximize his profit. He had constraints on time to make the art, and on art supplies available. We received information on the profit that he could get from each type of art, and were let loose on the problem. Unfortunately, due to bad timing on an FRC robotics competition, I could not attend school on the day that my class received the problem. So, my group solved the problem, and I solved it separately on the following Tuesday.
The first thing to do when solving a problem such as the Artist's Dilemma is determine what the constraints are. In this case, the constraints are time and supplies. The pastels and watercolors will take the same amount of time to make, and he has 16 hours total to paint. Also, each pastel will cost $5 and each watercolor will cost $15 for materials, and he has $180 to spend. Next, to make inequalities for these constraints, one determines which variable will be x and which will be y - I chose to call x the number of pastel paintings, and y the number of watercolor paintings. For the constraint on time, the inequality is x+y≤16. The ≤ sign is used because the artist can use 16 hours or less, but he cannot use more. The constraint on supplies is 5x+15y≤180, because it costs $5 per each x and $15 per each y. If the y-intercept is also in dollars, the units work out. We also have non-negative constraints; the artist cannot make negative paintings. These are simply x≥0 and y≥0. These inequalities can be graphed in a graphing computer program called GeoGebra - which I have used on previous projects - and the result is shown in blue above. The darkest region is called the feasible region; only the points inside that region or on its borders will satisfy all of the constraints. Now comes a very important part of information: since (0,0) will result in the lowest possible profit, the solution must be the farthest point on the feasible region from the origin. For any given feasible region, the closest and farthest points to the origin will always be vertices.
Now we need to figure out which vertex will result in the greatest profit. The thick black line represents the total profit. We know that each pastel will yield a profit of $40 and each watercolor will yield a profit of $100. So, the equation becomes 40x+100y=profit. For the black line, I substituted a random number - 1,000 - for profit. As one can see, as the line is translated farther and farther from the origin, the profit will increase, and the final point on the feasible region that it will intersect will be the vertex (6,10). To test this conjecture, one can plot a new parallel profit line: 40x+100y = 40(6)+100(10), which comes out to 40x+10y=1,240. This line is represented by the thick green line on the graph. It is apparent that this line intersects the point (6,10) and no other on the feasible region. So, 6 pastels and 10 watercolors will provide the greatest profit: $1,240.
In our groups, there were four roles, one per person: facilitator (Erik Lennox), documenter (Nidhal Dawood), spokesperson (Alejandra Sandoval), and GeoGebra guru (myself). The job of the facilitator was to make sure that the group worked efficiently, and that we all stayed on task. The documenter's job was to take notes of everything, so that the clients and group members could reference them if/when necessary. The spokesperson was the only group member who could speak to the client, to ask questions or to provide the answer once found. My job as GeoGebra guru was to plot the constraints in the form of inequalities in GeoGebra in order to create the feasible region. I was also in charge of making the graphs appear presentable, as needed.
Since I was unable to be there on the day of the Artist's Dilemma, I will do my group reflections on a practice problem that my group did prior to the Artist's Dilemma: the Breakfast Dilemma, in which the client was a cereal company working to create rations for the military.
a) Erik was a little overconfident with his position of being in charge at times, but it was obviously all in good fun, and he definitely added an element of humor to the group. That might sound trivial, but it can actually be useful in keeping morale up, and having a good time in general. When he was doing his job correctly, he did a good job of making sure that everyone else did their jobs correctly. His strongest habit of a mathematician was, "Be Systematic," because he did well at maintaining the various steps we were supposed to take to obtain our final answer. Nidhal took awesome and really useful notes; being the person to put the inequalities into GeoGebra, I referenced her notes on multiple occasions. Her strongest habit was the "Stay organized" habit. Alejandra didn't offer much while we were doing the Breakfast Dilemma, but she didn't have much of a job then. I'm sure that she did her job well during the Artist's Dilemma, and when I can say is that when I did the problem on Tuesday, she was eager to learn where the group got off-track during the problem, and what they needed to do to solve the problem. They were actually extremely close - they just mixed up the ≤ signs with ≥ signs - so kudos to them for knowing how to solve a problem that, frankly, could have been worded more clearly. Her strongest habit was, "Collaborate and Listen," because she paid close attention and asked questions when she needed to. For myself, I wasn't there, but when I got back, I solved the problem fairly quickly, and I did go over what I did with Alejandra. My strongest habit was, "Describe and Articulate," because I did a lot of that, when describing my answer for why the solution had to be a vertex and when talking to Alejandra after the problem.
b) During the breakfast problem, my group was extremely focused once Erik calmed down a bit. I mostly led the group in doing the solving, but I'm not sure if everyone was completely awake. Everyone certainly knew how to do their jobs, so while I was gone, I'm sure the group functioned extremely well. Also, the group paid attention during the Breakfast Dilemma problem, so I know that they knew how to solve the Artist's Dilemma. They did, if fact, know how to solve it; it was just the signs that mixed them up, so I could picture them having created their graph in no more than 30 minutes, then having become confused and spent the rest of the period trying to figure out what they did incorrectly. I didn't consult anyone on this, but just based off of these logical assumptions, I would give the rest of my group at least a 9/10 for teamwork. They probably did better, but like I said, I can't give a definite grade since I wasn't there.
c) I generally pick up on software such as GeoGebra fairly quickly, so I was proficient in the use of GeoGebra during the Breakfast Dilemma. I also used it well when I was solving the Artist's Dilemma on my own. The GeoGebra guru role is fairly straightforward; I am naturally inquisitive, so I had learned how to use all of the GeoGebra tools I needed by the time the Breakfast Dilemma and Artist's Dilemma were introduced. I feel like, although I did well in the Operational Research project, there's always more that I can learn with GeoGebra, and I look forward to doing this.