ProblemSet 1 -- Optimization

1. An optimization question in widget manufacturing

A manufacturing firm makes a profit of $1200 per unit on the sale of a product known as a widget. The firm hopes to increase widget sales by offering a rebate; after some number-crunching, it is estimated that for every $100 of rebate, the number of widgets sold in a given month will increase by 15%.

1. What amount of rebate will maximize the manufacturers profit for the month? Model the question as a single-variable optimization problem.

2. Compute the sensitivity of your answer to the 15% assumption. Consider both the amount of rebate and the resulting profit.

3. Suppose that rebates actually generate only a 10% increase in sales per $100. What is the effect? What if the response is somewhere between 10% and 15% per $100 of rebate?

4. Under what circumstances would an offer of a rebate cause a reduction in profit?

2. Computing yields with multi-variate optimization

A chemist is synthesizing a compound. In the last step, she must dissolve her reagents in a solution with a particular pH level \(H\), for \(1.2 ≤ H ≤ 2.7\), and heated to a temperature \(T\) (in degrees Celsius), for \(66 ≤ T ≤ 98\). Her goal is to maximize her percent yield as a percentage of the initial mass of the reagents.

The equation determining the percentage \(F (H,T)\) is

\[F(H,T) = −0.038\cdot T^2 − 0.223 \cdot T\cdot H − 10.982 \cdot H^2 + 7.112 \cdot T + 60.912 \cdot H − 328.898.\]

1. Find the optimal temperature and pH level in the allowed range.

2. Use matplotlib to produce a graph and a contour plot of the percentage of the powder function \(F (H, T )\).

(You should consult this week’s jupyter notebooks to see some examples. To get a usable copy of your image, you can proceed in a few ways:

• if you produce the graph in colab you can right-click on the image and Save As a file on your file system.

• if you work in Python on your computer, you can save the image via a command like

  > g.savefig(”myGraphImage.png”)