Alpha$-$Beta, Inc., needs to schedule the weekly production of widgets and gadgets for the next $4$ weeks. The production of widgets and gadgets requires machining a single raw material. Each widget requires $2$ pounds of raw material and $24$ minutes of machine time; each gadget requires $2\mathrm{.}4$ pounds of raw material and $18$ minutes of machine time. Each pound of raw material currently costs $$2\mathrm{.}10,$ but the price is expected to rise $$0\mathrm{.}20$ per pound each week. There is no storage area available for raw material and Alpha$-$Beta must purchase the raw material used in a given week during that week. The total machine time available during each week is $120$ hours at a cost of $$18$ per hour. In addition, $40$ hours of overtime are available each week at a cost of $$30$ per hour. However, the cost of a machine$-$hour is expected to rise $10\%$ during the last week. The projected weekly demands for widgets are $100,120,200,150$ units, and the weekly demands for Gadgets are $80,90,115,160$ units. A maximum of $40$ units of excess production can be stored from one week to the next at a cost of $$3$ per unit. Assuming that beginning inventory levels are zero, develop a linear programming model to determine the optimal production and inventory schedules.