Problem $4(20$ points$)$

A company knows that demand for a product for the next $4$ months is:

At the beginning of each month the company decides how many units to produce. During a month in which any units are produced there is a setup cost of $\backslash (\backslash$$ $3\backslash ).$ In addition, there is a variable cost of $\backslash (\backslash$$ $1\backslash )$ per unit produced. At the end of each month there is a holding cost of $\backslash (\backslash$$ $0\mathrm{.}5\backslash )$ per unit. Determine the optimal production schedule.