Chapter 8, Problem 18P

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 MWs), and the cost to supply electricity to each market from each power generation plant (prices are in $/MW).

1. 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 supplied by which power plants? What is the total annual power distribution cost for this solution?
2. b. If at most 4,000 MWs 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?

To determine

Find the optimal solution and total annual power distribution plan.

Explanation of Solution

The linear programming model can be formulated as:

$Min\left(\begin{array}{l}356.26x_{L1}+356.25x_{L2}+178.13x_{L3}+356.25x_{L4}+237.50x_{L5}\\ +415.63x_{L6}+356.25x_{L7}+356.25x_{L8}+178.13x_{L9}+356.25x_{L10}\\ +593.75x_{T1}+593.75x_{T2}+475.00x_{T3}+475.00x_{T4}+475.00x_{T5}\\ +415.60x_{T6}+415.63x_{T7}+356.25x_{T8}+475.00x_{T9}+296.88x_{T10}\\ +59.38x_{S1}+178.13x_{S2}+296.88x_{S3}+296.88x_{S4}+356.25x_{S5}\\ +296.88x_{S6}+356.28x_{S7}+475.00x_{S8}+593.75x_{S9}+593.75x_{S10}\end{array}\right)$

subject to

$\begin{array}{c}{x}_{L1}+x_{r1}+x_{s1}\ge 950\\ x_{L2}+x_{r2}+x_{s2}\ge 831.25\\ x_{L3}+x_{r3}+x_{s3}\ge 2375\\ x_{L4}+x_{r4}+x_{s4}\ge 593.75\\ x_{L5}+x_{r5}+x_{s5}\ge 950\\ x_{L6}+x_{r6}+x_{s6}\ge 593.75\\ x_{L7}+x_{r7}+x_{s7}\ge 1187.50\\ x_{L8}+x_{r8}+x_{s8}\ge 712.50\\ x_{L9}+x_{r9}+x_{s9}\ge 1187.50\\ x_{L10}+x_{r10}+x_{s10}\ge 1543.75\\ x_{ij}\ge 0,i=L,T,S,j=1,\mathrm{2...},10\end{array}$

Formula:

Solver:

Output:

The total distribution cost of this solution is $2,552,382.81

To determine

Find the optimal solution if at most 4,000 MWs of power is supplied.

Explanation of Solution

Other constraints that should be added in the model are:

$\begin{array}{c}{x}_{L1}+x_{L2}+x_{L3}+x_{L4}+x_{L5}+x_{L6}+x_{L7}+x_{L8}+x_{L9}+x_{L10}\le 4000\\ x_{r1}+x_{r2}+x_{r3}+x_{r4}+x_{r5}+x_{r6}+x_{r7}+x_{r8}+x_{r9}+x_{r10}\le 4000\\ x_{s1}+x_{s2}+x_{s3}+x_{s4}+x_{s5}+x_{s6}+x_{s7}+x_{s8}+x_{s9}+x_{s10}\le 4000\end{array}$

Formula:

Solver:

Output:

The optimal solution is shown in the above output.