$10-19$ Triangle Utilities provides electricity for three cit$-$

ies. The company has four electric generators that

are used to provide electricity. The main generator

operates $24$ hours per day, with an occasional shut$-$

down for routine maintenance. Three other genera$-$

tors $,$ and $3)$ are available to provide additional

power when needed. A startup cost is incurred each

time one of these generators is started. The startup

costs are $$6,000$ for $1,$$$5,000$ for $2,$ and $$4,000$ for

$3.$ These generators are used in the following ways:

A generator may be started at $6$:$00$ A$.$M$.$ and run for

either $8$ hours or $16$ hours, or it may be started at

$2$:$00$ P$.$M$.$ and run for $8$ hours $($until $10$:$00$ P$.$M$.).$ All

generators except the main generator are shut down

at $10$:$00$ P$.$M$.$ Forecasts indicate the need for $3,200$

megawatts more than provided by the main genera$-$

tor before $2$:$00$ P$.$M$.,$ and this need goes up to $5,700$

megawatts between $2$:$00$ and $10$:$00$ P$.$M$.$ 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 mega$-$

watt used per eight hour period is $$8$ for $1,$$$9$ for $2,$

and $$7$ for $3.$

$($a$)$ Formulate this problem as an integer program$-$

ming problem to determine the least$-$cost way to

meet the needs of the area.