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Math Modeling Research Paper

12/14/2020

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Problem Basics

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Summary (Abstract)

  • Meant to briefly summarize and describe the problem and results
    • Avoid using a “narrative” format
      • Eg. DO NOT DO: “We were stuck for some time because we had some difficulty determining how to characterize the reach of the warehouses of the company. However, after some Googling…”
  • Within 1 page
  • Be sure to save this for last -- keep in mind this provides an overview of your efforts and, thus, should be considered at a later time (despite being at the start of the paper)

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Breaking Down the Summary (Abstract)

  • Things to Include
    • Briefly restate the problem
    • Describe solution methods selected and implemented
    • Final results (avoid variables if possible)
    • Any conclusions drawn

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Restate the Problem�“We were tasked with…”

Briefly Describe the Methods Used

“We created a method to…”

“After implementing this method, we found…”

“We analyzed…”

Describe Final Results Briefly and Conclusion

“Our final solution…”

“We determined…”

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Formal Introduction

  • Restatement of the problem
  • Provide the real-world context to the problem
    • Applications
    • Avoid a history lesson
  • Overview of factors to consider
    • Not the description of weightings, implementation, or quantification of these factors
      • Just a case of “this is what will come up later”
  • Should also be saved for last

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Assumptions & Justifications

  • Sometimes assumptions are necessary to reduce complexity + variables
  • This saves time + ignores specific cases
    • ex. factors that are too difficult to measure, vary too widely, or have little significance
  • Justifications will explain why the variable is less significant or unnecessarily complicated
  • These can be referenced in the Strengths/Weaknesses section

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Example #1

“The number of consumers in each state is proportional to the state’s total population.”

“Although different states may be more inclined to buy the

company’s recreational equipment, for the sake of the metric and

measuring taxes we assume that the state’s population correlates

with the number of consumers from that state to simplify calculations.”

Assumption

Justification

Collecting data on consumers in each state would be too time consuming + not add much to the model.

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Example #2

“The shipping occurs on Monday, November 14, 2016, which is used on the UPS website to find one-day shipping area.”

“We wish to model the situation that occurs most of the time, which

is not during “rush season,” or between Black Friday and

Christmas. Shipping also only occurs during business days, so

sending a package on Friday would be received by Monday. We

chose November 14, 2016, as it is not during “rush season” and is

a Monday, so it is uninterrupted by non-business days.”

Assumption

Justification

This assumption reduces complexity by using data from average cases.

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Actual Math + Data

The model + what it does

Data used to create/test the model

Process + calculations done to create the model + solution

  • Explain how the answers the question
  • Explain how to use the model (ex. what each variable represents, what the numbers given by the model represent)

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  • What data you’re using + why
  • How the data was used
  • Include charts + graphs
  • Explaining your reasoning between steps
  • It’s best to do this in chronological order

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Explaining process + why they set up the model in a certain way.

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Data + Calculations

After finding the mean ZIP code, we allowed a ± 20% range (Table 1), which allowed for freedom in states that are oddly shaped, have inconsistent labelling patterns or have few major roadways, and used the ZIP code within that range that was closest to an interstate or highway.

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Data + Calculations

The closer a ZIP code is to the median amount, the more preferable it was. It was most optimal to find a location near an intersection of highways and interstates. We tested all possible ZIP codes by putting them into the UPS website [1] and compared them to determine which covered the most area.

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Justification

What to do?

  • After listing the data/calculations/methods, provide narrative paragraphs to explain specific reasons for choosing your method over other
  • Questions: is this the most accurate method? Most optimal solution? Most efficient? (think computer run time)
  • Like discussion section
  • Convince others of your method’s effectiveness and credibility

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Example

Example

  • To justify why use zip-code: Using ZIP Code + proximity to highways is the most effective method because UPS has the built-in feature on its website to calculate one-day shipping areas
  • To justify why use middle range of ZIP code of each state: some states are irregular, so can’t just take the middle point; have to standardize with middle range

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Strengths and Weaknesses

List and explain

Most important —> least important

What makes this model accurate/realistic?

Inaccurate/unrealistic?

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Specific &

reason

How final > initial

Walks through reasoning: why is this beneficial?

Explains why another approach could have been better

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Sensitivity Analysis

What to put?

  • How independent variable change in values affect dependent variable change, given a set of assumptions
  • Study how sources of uncertainty in math model would affect the overall model
  • A good model = robust = output are consistently accurate even if input variables change
  • Eg. Even if shipment time changes slightly, can still cover within one-day

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Conclusion and Final Recc, Future Work

Conclude by explaining your solution and how it answers the problem:

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P1: what you determined for 1) min # warehouses 2) where warehouses should be located:

“We determined the minimum number and optimal placement of warehouses depending on their ZIP codes and their proximity to major roadways. The minimum number of warehouses added that would allow the 48 states to receive one-day shipping is 25. The optimal placement of warehouses can be seen in Figure 5… ”

P2: 1) how locations affect customers’ tax liability and 2) any changes/modifications

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Conclusion and Final Recc, Future Work

P3/4: implementation and recommendation:

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“In reality, the warehouses will have to be built one at a time. While the new warehouses could be built east to west, gradually expanding our customer base,

there is a better way.

We propose that the company build new warehouses in the following four cities: Denver, Colorado; Marion, Arkansas; Fresno, California; and Janesville, Wisconsin. Building these four bases ensures that a majority of the United States will have access to two-day shipping from the company. This will allow for a massively increased income for the company, which will allow for faster expansion as they continue to build the 21 remaining warehouses to ensure one-day shipping nationwide.”

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Future work: what else requires further study?

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  • This paper talked about one method for building new warehouses, other methods can also be used.
  • Tax rates: in paper, built model to avoid states w/ high tax rates. Could start by choosing states w/ low tax rates in the future.

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Attendance Form

https://tinyurl.com/ghmm1214

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