Third Coast Chemicals (TCC) is planning to invest in manufacturing capacity to produce a new non-toxic chemical product. The company's initial estimate is that the demand for the product will be 50,000 in the current year (year 0), and will grow at a rate of 6% per year for the first 7 years, and then decrease at a rate of 2% per year. Of Course, TCC may choose to only partially meet the demand if it is not economical to meet all of it.

Building a plant with capacity of C per year requires an investment now (year 0) of $16*C, the plant's lifetime (before it needs to be shut down) is 10 years (year 0 to year 9). The plant will be ready for operation (production) in time to support this year's sales. The cost for operating this plant is $0.40 per year per unit of installed capacity (regardless of actual usage of this capacity). The variable production cost is $0.30 per unit produced. Assume that the company does not hold products in inventory from year to year.

The product's selling price is $3 per unit; this price is expected to increase at a rate of 2% per year for the first 4 years and then remain steady.

The firm's discount is 8%.

TCC wants to determine how much manufacturing capacity to install now for this product in order to maximize the net present of the project.

a. Provide a mathematical formulation for the problem. Clearly define the decision variable(s) and any parameters you use, and specify/explain any intermediate expressions and the final function that needs to be optimized.