Protein Bars Ranking Project

Helynna Lin, Tommy Macneil

Summary

        

Companies that sell protein bars claim that their product is the best, but do not specify why. We made a ranking function in order to specify which protein bars are the best. We took into account the amount of protein, the amount of calories, the amount of carbohydrates, the amount of fat, the amount of sugar, the price, and the taste of each protein bar. Through taste tests and research we were able to design the function: Score = amt. of protein / amt. of carbs * (amt. of protein + amt. of carbs + (amt. of calories/10) - amt. of sugar - amt. of fat) - cost + taste.  

Introduction and Analysis of the problem

Ideally this function is for someone who is looking to gain muscle mass by eating protein bars. Based on this goal, we identify our variables. A list of our variables, their units, and the reason why we included them are listed as below:

Variables

units

Reason why we included this variable

Domains (estimated from data table)

Amount of protein

grams

In order to gain muscle mass, which is why one would eat a protein bar, a lot of protein, calories and carbohydrates will help accomplish that goal.

0-25

Amount of carbs

grams

10-25

Amount of calories

calories

180-400

Amount of sugar

grams

To gain muscle mass, minimizing fats and sugars is important.

1-30

Amount of fat

grams

1-20

Cost

dollars

We assumed that taste and cost were the least important values. We assumed this because if one wants to get fit and strong, they must sacrifice the extra dollar, and possibly a not as great tasting bar.

1-3

Taste

A single-digit number between 1-10

1-10

Taste tests are conducted by having a group of five people taste the same protein bar and give a score from 1-10, 1 being the worst taste and 10 being the best taste. Then, we take the average of the five scores and get the taste score for the protein bar.

Assumptions and Hypothesis

Based on our research, we made the following assumptions:

I. We assumed that the most important constants in the equation is the protein/carbohydrate ratio.

II. We assumed out of the nutrition facts that protein, carbohydrates, calories, sugar, and fats are equally as important.

III. We assumed that taste and cost were the least important values.

IV. We assumed this because if one wants to get fit and strong, they must sacrifice the extra dollar, and possibly a not as tasty bar.  

V. We assumed that when tasting the bars, everyone was honest and not biased towards any bars.

Overall, we hypothesized that the bar with the largest P/C will have the greatest protein bar score.

Design of model and Justification

Score = amt. of protein / amt. of carbs * (amt. of protein + amt. of carbs + (amt. of calories/10) - amt. of sugar - amt. of fat) - cost + taste

I. Explanation of the function

Function families

Reason

amt. of protein / amt. of carbs

This is the most important factor because if the ratio is too small, it is just like eating a bowl of your everyday cereal.  Because of this assumption, we used this as a multiplier to all the other nutrition facts.  If the P/C is too small, than the score should be lower because it is basically dividing the nutrition facts.  If it is higher, than it is used as a multiplier to raise the score.

(amt. of protein + amt. of carbs + (amt. of calories/10) - amt. of sugar - amt. of fat)

All of these are factors of nutritional values, so we put them in parenthesis and multiply them by P/C.

Amts. of protein, carbs and calories have a positive effect (the bigger the number, the better), so they are followed by a plus sign; amts. of sugar and fat have a negative effect (the smaller the number, the better), so they are followed by a negative sign.

amt. of calories/10

Since calories are equally as important and their numbers have a range that is bigger than most of the other numbers, we divided the value of most calories by 10 in order to get a number similar to carbohydrates and proteins.

Taste and cost

Taste was a positive factor, so it was followed by a positive sign; cost was a negative factor, so it was followed by a negative sign.

II. Normal and extreme cases

        We applied normal and extreme cases for our function that we created.  For example, we looked at a Quaker Oats Chewy granola bar which had only 1g of protein.  Since we are modeling protein bars, we expected this to be a low ranking due to its lack of protein. The value of the bar was 7.7, which was much smaller than all the other values we received from real protein bars.  We were unable to look for a really extreme value with a large protein number.  We thought the Met-RX protein bar was an extreme value as it contains 30g of protein, which is much more than the normal 20g.  As expected, the value of the bar was 74.2, which was the largest of the data we collected.  We then used other hypothetic values making possible variables larger or smaller. For instance, we doubled the amount of fat for the quest bar and found that the value was only 9.4 less than before.  However, when we doubled the protein we found that the value was 96.6 points more.  We then doubled the carb amount and found that the new score is 36.6 points, which is 15.7 points less than the original quest bar.  Finally we doubled the calorie amount and found that the score was 72.4 points, 21.1 points more than the original quest bar.  We did this to see which factors have a larger effect on the score.  As shown above, when the protein increases, the score will increase a lot.  This makes sense because when the protein/carbohydrates value is high, the score will be high.  These variables worked the way that we hoped they would in our model.  

III. Explanation of asymptotes and intercepts

        There is no asymptote in our graph. The reason is that all of the variables (amount of protein, amount of fat and final score) are dependent on the production processes of protein bars, and none is approaching a certain value.

        The intercepts in our graph and the corresponding analysis are below:

1) Intercept on the x-axis: (0.018,0,0). When there are 0.018 grams of protein and 0 gram of fat in a Quest protein bar, we will get a score of 0. This information does not yield much significance. There is little possibility that a protein bar contains only 0.018 grams of protein and no fat at all; meanwhile, we are not concerned about when the score will be a 0.

2) Intercept on the y-axis: none. It is impossible that there is no protein in a protein bar.

3) Intercept on the z-axis: none. Similarly, it is impossible that there is no protein in a protein bar.

Graphs

Looking back at our function:

Score = amt. of protein / amt. of carbs * (amt. of protein + amt. of carbs + (amt. of calories / 10) - amt. of sugar - amt. of fat) - cost + taste

        We make “amt. of protein” x, and “amt. of fat” y. The amt. of protein is directly related to the (amt. of protein / amt. of carbs) ratio, which has most significant effect on the nutritional value of a protein bar; we don’t make the (amt. of protein / amt. of carbs) ratio our x because it is dependent on two other variables, and it makes more sense to use an independent variable. Since amt. of protein has positive effect on the score, we choose a y variable that has a negative effect - amt. of fat. We make the score our z. Thus, the function turns into the following:

z = x / amt. of carbs * (x + amt. of carbs + amt. of calories - amt. of sugar - y) - cost + taste

We choose a random brand of protein bars to plug in the constants into the rest of the variables. For a Quest bar, we have the following values:

Protein Bar

Protein(g)

Carbs(g)

Calories

Protein/Carbs

Sugar(g)

Fat(g)

Cost ($)

Taste (1-10)

Quest

21

20

190

1.05

1

9

2.39

2.20

Therefore, our function turns into:

z = (x / 20) * (x + 20 + 190/10 - 1 - y) - 2.39 + 2.20 = (x / 20) * (x - y + 38) - 0.19

We make a graph of this function:

geogebra-export.png

In our graph, the red axis is the x axis, the green axis is the y axis, and the blue axis is the z axis. The red area indicates the graph of our function. The justifications of the graph are shown below:

I. When x increases, z increases.

This makes sense, for the amt. of protein has a positive effect on the final score. The more the amt. of protein, the better the protein bar.

Protein Bar

Protein(g) (x)

Carbs(g)

Calories

Protein/Carbs

Sugar(g)

Fat(g) (y)

Cost ($)

Taste (1-10)

Score (z)

Quest (x=21, y=9)

21

20

190

1.05

1

9

2.39

2.2

52.3

Quest (x=42, y=9)

42

20

190

2.10

1

9

2.39

2.2

148.9

        As shown in the chart, doubling the amt. of protein boosts the final score.

II. When y increases, z decreases.

        This makes sense, for the amt. of fat has a negative effect on the final score. The more the amt. of fat, the worse the protein bar.

Protein Bar

Protein(g) (x)

Carbs(g)

Calories

Protein/Carbs

Sugar(g)

Fat(g) (y)

Cost ($)

Taste (1-10)

Score (z)

Quest (x=21, y=9)

21

20

190

1.05

1

9

2.39

2.2

52.3

Quest (x=21, y=18)

21

20

190

1.05

1

18

2.39

2.2

42.9

        

        As shown in the chart, doubling the amt. of fat makes the final score go down.

III. Changing a constant does not dramatically affect the graph.

We change the constant of carbs from 20 g to 40 g (doubling the value). That gives us this function:

        

z = (x / 40) * (x + 40 + 190/10 - 1 - y) - 2.39 + 2.20 = (x / 40) * (x - y + 58) - 0.19

We graph it as shown below:

geogebra-export2.png

In comparison to our original function:

geogebra-export1.png

As shown in the graphs, there isn’t a drastic change happening to its shape and range.

Similarly, we change the constant of calories from 190 to 380. That gives us this function:

        

z = (x / 20) * (x + 40 + 380/10 - 1 - y) - 2.39 + 2.20 = (x / 20) * (x - y + 77) - 0.19

We graph it as below:

geogebra-export3.png

In comparison to our original graph:

geogebra-export1.png

As shown in the graphs, there isn’t a drastic change happening to its shape and range.

        Therefore, our function is proper.

Explanation of units and constraints

        Due to our model containing the ratio of proteins/carbs, if a protein bar did not contain any carbohydrates we would not be able to give a score because we would get an undefined value.  Ideally, this model works best for protein bars because of the variables we use to rank, but this model could rank any food.  This is a good system for anyone who is trying to gain muscle mass because it takes into account the nutrition facts that are important to getting bigger and stronger.  Protein, carbohydrates, sugar, and fat are all measured in grams, and calories are divided by 10 to get a value similar to those.  The price was measured in dollars, and taste was measured from 1-10.  The score was a combination of these, and its units in points.  The higher the points, the better the value of the protein bar.

Solution

        Tommy went to the store and bought the 7 different bars available that are sold at Cumberland Farms in Lancaster for a taste test.  Allowing each student in the class to test, we ranked the taste from 1-10 with 1 being the worst thing you have ever eaten, and 10 being the best thing you have ever eaten.  We then looked at the nutrition facts and plugged in the values accordingly to our equation.  The table below shows the nutrition facts and the score each protein bar received.

Protein Bar

Protein(g)

Carbs(g)

Calories

Protein/Carbs

Sugar(g)

Fat(g)

Cost ($)

Taste (1-10)

Score

Quest

21

20

190

1.05

1

9

2.39

2.2

52.3

Gatorade

20

41

360

0.49

29

12

2.18

9

34.1

Clif

20

29

280

0.69

21

10

1.99

5.6

35.3

Met-Rx

30

34

300

0.88

2

10

3.19

5

74.2

Nature Valley

10

14

190

0.71

6

12

1.49

6.8

23.2

Protein Plus

20

25

210

0.80

12

6

2.33

6.5

42.6

Quaker Chewy

1

17

100

0.06

7

3.5

0.31

7

7.7


Analysis of Model

        Although this model is a great representation of the variables that are important when buying a protein bar, we did leave some out.  We did not include sodium, potassium, and other nutrition facts.  This was on purpose because we assumed that the other variables were more important for a diet of someone who is trying to gain muscle mass.  This model is very strong as it takes into account the protein/carbs ratio.  Many protein bars advertise that they have a high protein count, yet there are still more carbohydrates.  Although carbohydrates are good to gain muscle mass, you cannot have much more than protein or else it becomes less healthy and efficient.  Our model is good because it gives a high score if there is a lot of protein and carbohydrates, but the ratio is what makes the score move to a higher value or a lower value.  This model is also strong because it can be used for any type of food, not only protein bars and still will work. As stated earlier, it is important to note that the model is unable to take into account a food that does not contain carbohydrates because the score will be undefined.        

Group Contributions/sources

Helynna and Tommy worked well together on finding the model.  Tommy used his knowledge on protein bars for finding the importance of each variable.  Helynna used her skills in math to work on the model and helped find the equation and the graphs.  Overall the the work was split equally and we both were able to utilize their skills to find the model.