Super Size

1
Super Size
e
An Optimization Problem

• MME 2007

2
Background

• In 2003, documentary film-maker Morgan Spurlock
  set out to expose the fast food industrys role
  in Americas obesity epidemic.

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Background

• For thirty consecutive days, Spurlock ate nothing
  but McDonalds food anything offered on the menu
  was fair game.

4
Background

• He had three rules for his experiment

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Background

• He had three rules for his experiment

• He would eat exclusively from the McDonalds menu

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Background

• He had three rules for his experiment

• He would eat exclusively from the McDonalds menu

2. He would super size his meal whenever an
employee asked
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Background

• He had three rules for his experiment

• He would eat exclusively from the McDonalds menu

2. He would super size his meal whenever an
employee asked
3. He would try everything on the menu at least
once.
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Background

• The results were not pretty

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Background

• The results were not pretty
• Spurlock gained about 25 pounds over the
  month-long experiment,

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Background

• The results were not pretty
• Spurlock gained about 25 pounds over the
  month-long experiment,
• And he suffered from various health problems such
  as chest pains, shortness of breath, and liver
  trouble.

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What does this have to do with math?

• We are interested in whether or not Spurlock
  could have designed his experiment more
  intelligently.
• Simply put, could Spurlock go on an
  all-McDonalds diet for thirty days without
  causing such harm to his health?

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What does this have to do with math?

• This is where optimization comes into play
• We want to design an optimal menu for Spurlock.
• But how should we define optimal?

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McStrategy

• In this case, we will define an optimal menu as
  one that meets these criteria

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McStrategy

• In this case, we will define an optimal menu as
  one that meets these criteria

• The menu consists only of food served at
  McDonalds restaurants.

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McStrategy

• In this case, we will define an optimal menu as
  one that meets these criteria

2. Spurlock super sizes his meal when asked.
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McStrategy

• In this case, we will define an optimal menu as
  one that meets these criteria

3. Specific nutritional requirements are met,
namely the RDI of vitamins A and C, calcium,
iron, protein, carbohydrates, fat, and fiber.
For our purposes, these values will be based on
Spurlocks needs, taking into account his sex,
age, height, and weight.
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McStrategy

• In this case, we will define an optimal menu as
  one that meets these criteria

4. Spurlocks daily caloric intake is minimized.
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McStrategy

• We also need some simplifying assumptions
• We will only consider one day in the experiment,
  as each days nutritional and caloric
  requirements are independent of the others. We
  can later scale this one-day simulation to
  approximate a 30-day experiment.

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McStrategy

• We also need some simplifying assumptions
• We will consider only non-negative integral
  values for the amount of each menu item eaten.

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McStrategy

• We also need some simplifying assumptions
• We will assume that Spurlock will not mind eating
  the same thing every day.

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McStrategy

• We also need some simplifying assumptions
• We will exclude Spurlocks rule that each item
  available on the menu must be eaten at least once
  over the course of the month. (Remember, we are
  only looking at one day of the experiment.)

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A First Helping The Basic Model

• The first model we consider is an integer
  programming model.

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A First Helping The Basic Model

• We have 136 decision variables (the number of
  items on McDonalds menu).

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A First Helping The Basic Model

• If we let xi represent the number of item i eaten
  (1 i 136), then our objective function is

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A First Helping The Basic Model

• If we let xi represent the number of item i eaten
  (1 i 136), then our objective function is
• Minimize
• Where ci is the number of calories of item i.

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A First Helping The Basic Model

• Our objective function is subject to our
  pre-specified nutritional constraints

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A First Helping The Basic Model

• Nutritional Constraints

(Vitamin A, Vitamin C, Calcium, Iron, protein,
fat, carbohydrates, and dietary fiber.)
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A First Helping The Basic Model

• We also specify the implicit constraint that the
  amount of each menu item ordered (and eaten) be a
  non-negative integer
• xi 0, xi integer

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A First Helping The Basic Model

• We can set up this integer programming model in a
  spreadsheet program (were using gnumeric) and
  use the built-in Solver function to specify our
  constraints.

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A First Helping The Basic Model

• We can set up this integer programming model in a
  spreadsheet program (were using gnumeric) and
  use the built-in Solver function to specify our
  constraints.
• When we run this model, we obtain a menu in which
  Spurlock consumes a total of 1795 calories each
  day.
• Spurlocks nutritionist advised him to eat 2500
  calories a day, so we would actually expect
  Spurlock to lose weight (about 6 pounds) with
  this menu!

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A First Helping The Basic Model

• A six-pound weight loss is a good thing, right?
  Welllets look at the actual menu.

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A First Helping The Basic Model

• This model generated the following daily menu for
  Spurlock

1 Ketchup packet 2 Caesar salads with grilled
chicken 45 side salads 46 half-and-half creamer
packets
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Going Back for Seconds Binary Model
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Going Back for Seconds Binary Model

• Due to the unrealistic nature of our previous
  results, we adopt a new constraint for our model.

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Going Back for Seconds Binary Model

• Due to the unrealistic nature of our previous
  results, we adopt a new constraint for our model.
• We will limit Spurlock to eating no more than 1
  of each menu item during the day.
• So each xi0 or xi1.

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Going Back for Seconds Binary Model

• This new constraint will hopefully yield a more
  realistic menu as well as add some variety to
  Spurlocks menu.

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Going Back for Seconds Binary Model

• When we run the revised model, we see that
  Spurlocks daily caloric intake has increased to
  2820 calories also, his daily fat intake is at
  118.5 grams.

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Going Back for Seconds Binary Model

• Spurlocks new menu

1 Quarter Pounder 1 California Cobb salad 1
Small French Fries 1 side salad 1 Medium French
Fries 1 English muffin 1 Large French Fries 1
order of hash browns 1 Hot Mustard Sauce 1
packet of peanuts 1 Tangy Honey Mustard Sauce 1
oatmeal raisin cookie 1 Bacon Ranch salad 1
childs Coca Cola 1 Caesar salad w/ grilled
chicken
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Going Back for Seconds Binary Model

• Initial conclusions
• Higher caloric intake will only cause him to gain
  about 3 pounds over the month.
• Increased fat intake cant really say for sure
• Menu is more realistic not only is Spurlock able
  to eat typical McDonalds fare, but he also
  eats from the breakfast menu (which was actually
  an implicit requirement in the original
  experiment).

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Ready, Set, SUPER SIZE!

• Our next goal is to incorporate Spurlocks super
  sizing condition into our model. We can do this
  using Monte Carlo simulation.

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Ready, Set, SUPER SIZE!

• During Spurlocks original experiment, he was
  asked to super size his meal a total of nine
  times.

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Ready, Set, SUPER SIZE!

• During Spurlocks original experiment, he was
  asked to super size his meal a total of nine
  times.
• We can use this figure in our simulation to
  determine best- and worst-case scenarios.

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Ready, Set, SUPER SIZE!

• Our simulation yields that Spurlock will super
  size a minimum of zero times, an average of nine
  times (as expected), and a maximum of 25 times
  during the month.

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Ready, Set, SUPER SIZE!

• If Spurlock is asked to super size zero times, we
  will not see a change in the results from our
  binary model.

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Ready, Set, SUPER SIZE!

• If Spurlock is asked to super size nine times,
  his total weight gain for the month is 4.4 pounds.

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Ready, Set, SUPER SIZE!

• If Spurlock is asked to super size twenty-five
  times, his total weight gain for the month is 7
  pounds.

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Ready, Set, SUPER SIZE!
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Conclusions

• The basic integer programming model with which we
  initially began is easy to implement and adapt.
• However, the bizarre combination of foods that
  Spurlock would have to eat with this model was
  highly unrealistic.
• Thus, the basic model, as it stands, is
  inadequate for this type of situation.

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Conclusions

• The binary model is also easy to implement and
  adapt.
• The binary constraint provided more variety in
  Spurlocks menu, thus proving to be more like his
  original experiment.
• The higher caloric intake did not drastically
  affect Spurlocks weight gain, although the high
  fat intake is a bit troubling.

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Conclusions

• The super size model, even under the worst-case
  scenario, yielded a menu that would cause
  moderate weight gain for Spurlock.
• Despite our promising results, there are still
  ways to improve upon our model.
• Can you think of what you might do differently?

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Questions?