Problem $1$: $(25$ points$)$ ABC Manufacturing Company committed to deliver Type A windows over the next $6$ months. The demands fo each month are $100,250,190,140,220,$ and $110$ windows respectively. Production costs per windows varies from month to month depending on the cost of labor, material, and utilities. The estimates of the production cost per window over the next $6$ months are $$50,$$$45,$$$55,$$$48,$$$52,$ and $$50,$ respectively. To take advantage of the fluctuations in manufacturing costs, ABC is willing to produce more than needed in a given month and hold the excess windows for future delivery. For each window produced and not delivered in the same month will storage cost of $$8$ per month will apply. Develop a linear program to determine the optimum production schedule.