A manufacturing company purchases a new machine at the beginning of year one. The cost of buying this machine at the beginning of years 1,2,3 and 4 (at the start of each subsequent year) is given in Table 1.Since the company wants to use up-to-date machines, the company can replace the used machine with a new machine at any time after one year of use. For example, if the company switched to a new machine at the beginning of year two and used it for two years, the total cost would be $ 365 (new machine price + 2 years use = $350 + $ 15). The cost of maintaining this type of machine, depending on its period of use, is also given in Table 2. For example, if the machine is used for three years, the total of three years of maintenance cost would be $20. Assume that there is no trade-in value when a machine is replaced. The goal of this company is to minimize the total cost (which includes the purchase and maintenance costs) of owning and using this type of machine for the next four years (e.g., the end time is the beginning of year 5). In particular, the company wants to determine the years in which a new machine should be purchased.

Table 1: New machine price at the beginning of the year.

Year	1	2	3	4
Purchase Price	$300	$350	$400	$450

Table 2: Total machine maintenance cost depending on the period of use

Year	1-year use	2 years use	3 years use	4 years use
Maintenance Cost	$10	$15	$20	$30

1. Define the decision variables. (1 points)
2. Define the objective function. (3 points)
3. Define the constraints (3 points)
4. Determine an optimal solution (3 points)