Patricia runs a clothing store in Maine that is open every year from April through October. Today is January $1$ and Patricia has no inventory. She both produces and sells her own pieces. She is currently planning her production for the upcoming season $($until October$).$ Each piece retails for $$100$ and costs $$50$ to make. She can produce up to $90$ pieces per month in the winter months $($January through March$),$ but only $60$ pieces per month after the store opens in April. She can store at most $300$ pieces at any time and her holding cost is $$2$ per month. Her demand during the early season $($April to June$)$ is of $60$ pieces a month, during the peak season $($July and August$)$ is of $150$ pieces a month and during the late season $($September and October$)$ is of $100$ pieces a month. Assume for simplicity that all demand occurs on the last day of the month. Formulate Patricia$$s linear optimization problem to help her figure out how many units she should produce on each month.