Chapter 16, Problem 21P

The Splash River has a flow rate of $2\times {10}^6{\text{m}}^{\text{3}}\text{/d}$, of which up to 70% can be diverted into two channels where it flows through Splish County. These channels are used for transportation, irrigation, and electric power generation, with the latter two being sources of revenue. The transportation use requires a minimum diverted flow rate of $0.3\times {10}^6{\text{m}}^{\text{3}}\text{/d for Channel }1\text{ and }0.2\times {10}^6{\text{m}}^{\text{3}}\text{/d}$ for Channel 2. For political reasons it has been decided that the absolute difference between the flow rates in the two channels can-not exceed 40% of the total flow diverted into the channels. The Splish County Water Management Board has also limited maintenance costs for the channel system to be no more than $\$1.8\text{x}{10}^6$ per year. Annual maintenance costs are estimated based on the daily flow rate. Channel 1 costs per year are estimated by multiplying $1.1 times the ${\text{m}}^{\text{3}}\text{/d}$ of flow; while for Channel 2 the multiplication factor is $\$1.4\text{ per }{m}^3/d$. Electric power production revenue is also estimated based on daily flow rate. For Channel 1 this is $\$4.0{\text{ per m}}^{\text{3}}\text{/d}$, while for Channel 2 it is $\$3.0{\text{ per m}}^{\text{3}}\text{/d}$. Annual revenue from irrigation is also estimated based on daily flow rate, but the flow rates must first be corrected for water loss in the channels previous to delivery for irrigation. This loss is 30% in Channel 1 and 20% in Channel 2. In both channels the revenue is $\$3.2{\text{ per m}}^{\text{3}}\text{/d}$. Determine the flows in the channels that maximize profit.