$10-19$ Triangle Utilities provides electricity for three cities. The company has four electric generators that are used to provide electricity. The main generator operates $24$hours per day, with an occasional shutdown for routine maintenance. Three other generators $(1,2,$ and $3)$ are available to provide additional power when needed. A start$-$up cost is incurred each time one of these generators is started. The start$-$up costs are $$6,000$ for $1,$ $$5,000$ for $2,$ and $$4,000$ for $3.$ These generators are used in one of the following ways: a generator may be started at $6$:$00$ AM and run for either $8$hours or $16$hours$,$ or it may be started at $2$:$00$ PM and run for $8$hours $($until $10$:$00$ PM$).$ All generators except the main generator are shut down at $10$:$00$ PM Forecasts indicate the need for $3,200$megawatts more than is provided by the main generator before $2$:$00$ PM$,$ and this need goes up to $5,700$megawatts$$ between $2$:$00$ and $10$:$00$ PM$.$ Generator $1$ may provide up to $2,400$megawatts$,$ generator $2$ may provide up to $2,100$megawatts$,$ and generator $3$ may provide up to $3,300$megawatts$.$ The cost per megawatt used per $8-$hour period is $$8$ for $1,$ $$9$ for $2,$ and $$7$ for $3.$ QX

$($a$)$ Formulate this problem as an integer programming problem to determine the least$-$cost way to meet the needs of the area.

$($b$)$ Solve using computer software.