Problem $1(20$ points$)$: A company needs to decide how many units of a widget to produce in the next four months. Demands are $150,100,80,$ and $200$ in months $1$ through $4,$ respectively. Producing the widget in any given month incurs a fixed setup cost of $\backslash (\backslash$$ $120\backslash ).$ The table below shows the variable cost of producing one unit of the product in each month, and the inventory cost of carrying one unit from one month to the next. Question: Develop a production plan to satisfy all demands without any stockout at the minimum total cost. You can answer the question by solving a mixed integer linear program $($i$.$e$.,$ a model with both binary and real variables, warning: Excel solver does not work well, you may need to use AMPL$+$CPlex, or other software package$),$ or solving a shortest path problem by hand.