Aggie Power Generation supplies electrical power to residential customers for many U$.$S$.$ cities. Its main power generation plants are located in Los Angeles, Tulsa, and Seattle. The following table shows Aggie Power Generation$$s major residential markets, the annual demand in each market $($in megawatts or MW$),$ and the cost to supply electricity to each market from each power generation plant in $$/$MW$)$:

distribution Costs $($$$/$MW$)$

City

Seattle Portland San Francisco Boise Reno Bozeman Laramie Park City Flagstaff Durango

los Angeles

$356\mathrm{.}25356\mathrm{.}25178\mathrm{.}13356\mathrm{.}25237\mathrm{.}50415\mathrm{.}63356\mathrm{.}25356\mathrm{.}25178\mathrm{.}13356\mathrm{.}25$

Tulsa

$593\mathrm{.}75593\mathrm{.}75475\mathrm{.}00475\mathrm{.}00475\mathrm{.}00415\mathrm{.}63415\mathrm{.}63356\mathrm{.}25475\mathrm{.}00296\mathrm{.}88$

Seattle

$59\mathrm{.}38178\mathrm{.}13296\mathrm{.}88296\mathrm{.}88356\mathrm{.}25296\mathrm{.}88356\mathrm{.}25475\mathrm{.}00593\mathrm{.}75593\mathrm{.}75$

demand $($MW$)$

$950\mathrm{.}00$

$831\mathrm{.}252375\mathrm{.}00593\mathrm{.}75950\mathrm{.}00593\mathrm{.}751187\mathrm{.}50712\mathrm{.}501187\mathrm{.}501543\mathrm{.}75$

a$.$ If there are no restrictions on the amount of power that can be supplied by any of the power plants, what is the optimal solution to this problem? Which cities should be sup$-$ plied by which power plants? What is the total annual power distribution cost for this solution?

b$.$ If at most $4000$ MW of power can be supplied by any one of the power plants, what is the optimal solution? What is the annual increase in power distribution cost that results from adding these constraints to the original formulation?