Designing efficient processes for accomplishing a particular task is important in our increasing complex society and world. Efficiency can be the difference between a profitable product or company and taking a loss: Everything else can be the same, but one company might be slightly faster or slightly cheaper than another company. On a very large scale, you can think of Amazon as an exercise in systems efficiency. We are going to look at one very small aspect of this: Waste disposal. Every manufacturing process creates waste, and it costs money to safely collect and store this waste until disposal, it costs money for the disposal, and potentially costs money in legal fees if the disposal mechanism damages the environment or harms people. To learn more about systems engineering, you can read the (very detailed) Wikipedia entry on it: https://en.wikipedia.org/wiki/Systems_engineering

The Problem: Innovatron Genetics Corporation is a biotechnology company (no, it doesnt really exist) with one office location. The company produces both organic and non-organic waste. The company has to pay money for the waste to be hauled away. There is a fixed cost (the cost for the truck to show up, a fee to enter a dump site, etc) and a variable cost (how many tons are being disposed of). IG Corp contracts with Jack Dempsey Waste (also non-existent) to haul away their waste and dispose of it. JDW charges $500 per week (one pickup per week) and the variable cost is $235 per ton. Currently IG Corp produces 5 tons of waste per week for a total cost of $1,675 in waste disposal per week, or $87,100 per year.

IG Corp launches a waste reduction program with the goal of reducing the amount that they spend on waste removal. Based on efforts at other companies in the area, IG Corp believes that it can reduce its waste by 3.5% per week. Assuming that they are correct in their ability to reduce waste, determine how much money they will save over the course of one year.

Key Points:

• We will assume that there is one pick up per week and that it occurs at the start of the week before the waste reduction program begins (so, 5 tons of waste are hauled away).
• We will assume that the 3.5% reduction is sustained week over week.
• You will need to create a function that determined how many tons of waste are to be hauled away. Your function should be exponential since the reduction is a percentage and not an absolute.
• This is a course in integral calculus. An integral will show up in a key place and you should use your calculator to compute it.
• The WAMAP homework included several table problems that had velocity (ft/sec) as a function of time (sec). When you integrate velocity you get distance (ft). This will be helpful with the previous point.

A Point to Think About but to do Nothing With: This is, fundamentally, a problem in discrete mathematics rather than continuous mathematics. Since the pick-ups are weekly, we could create a spread sheet showing the waste total per week and avoid calculus altogether. There are a variety of reasons why you generally want to avoid doing this. The company might have more frequent waste collections or want to determine if using a more frequent approach to disposal is in their best interest (more pick-ups means less waste per pick up and a smaller fixed cost). The mathematics for continuously varying variables (like time) is more robust than discrete math (perhaps a bias of mine). And it is easier (in general) to modify the parameters for a continuous model than for a discrete model to determine an optimal arrangement of parameters. On the negative side, for complex problems a discrete approach tends to run more efficiently than a continuous model. For a recent data scienceproject, I worked on the so-called Netflix Challenge. The data set contained 10 million reviews of movies. If I built a continuous model and used calculus to make predictions for individual users, I might get very accurate results, but I could never run the program to compute the model: It would crash any consumer grade computer. Instead, I built a discrete model that would run in less than an hour and get fairly good results.

The Report: You will need to write up a professional looking report for this problem. The report should be self-contained: It should have a description of the problem (in your words), your solution to the problem, and a summary of the results. Your report should be free of spelling mistakes and grammatical errors. All mathematical symbols must be typeset and not handwritten. As an example, you should not write something like, And since the integral of x^2 from x=1 to x=4 is 19 and instead should write

Your goal is to have something that you would be comfortable handing to an employer, employer, or internship as a writing sample (and, yes, all these groups have asked students of mine for such samples in the past). Your report should include at address at least one thing from both of the points below.

• What is one assumption that has been made in the problem (that wasnt already addressed)? Is that assumption valid? How much effect does it have on the problem? How could it be removed?
• How could the problem be extended to a larger context? In our problem were looking at a small company. How could it be extended to a larger company? Or a municipality? What new variables would need to be accounted for?