Exercise:- (Wagner) Ft. Loudoun and Watts Bar are two large hydroelectric dams, the former upstream of the latter. The level of Watts Bar Lake must be kept within limits for recreational purposes, and the problem is to plan releases from Ft. Loudoun to do so. In reality this problem is solved simultaneously for numerous dams covering an entire watershed, but we focus on a single reservoir. There are also sophisticated models for predicting runoff into the reservoirs, but we will suppose that runoff is negligible. Thus any water entering Watts Bar Lake must be released through Ft. Loudoun Dam.
The planning period is 20 months. During month t, let Xt be the average water level of Watts Bar Lake before augmentation by water from Ft. Loudoun; x1 = 25. Let Yt be the number of feet added to the average level in month t from Ft. Loudoun. Lt and Ut are the lower and upper bounds on the lake level in month t (more restrictive in summer). To model seepage, evaporation and hydroelectric release through Watts Bar Dam we suppose that Watts Bar Lake begins month t+1 at a level equal to .75 times the average level of the previous month (including the augmentation from Ft. Loudoun). The cost of water from Ft. Loudoun Lake is ct for every foot added to the level of Watts Bar. Formulate the appropriate LP model. (In reality the model is a huge nonlinear program.)