It is now the end of December and as manager of a local rese

It is now the end of December, and as manager of a local reservoir you must devise a reservoir management plan for the next year. This plan must satisfy several conditions:

(a) In each month, a fixed amount of water must be released to meet municipal water supply irrigation needs.

(b) The amount of water in storage at the end of each month must not exceed the reservoir’s capacity. Also, the storage at the end of month 12 must not be less than the initial storage.

(c) To ensure adequate hydropower generation, the amount of water in storage at the end of each month must be at least one-third of the reservoir’s capacity.

(d) In each month, some amount of water will be released for recreation and for the ecological integrity of the river downstream from the dam. A monetary value has been placed on these releases: $3 per m3 for releases made in June, July and August; $2.00 per m3 in May and September, and is $1 per m3 in all other months.

Factors that determine the amount of water in storage at the end of each month include: the amount of water in storage at the end of the previous month (St-1 , see below); the total amount of water withdrawn during the month (Dt plus Xt); and the total amount of water flowing into the reservoir from upstream during the month (It).

You have the following information (parameters whose values are known).

S0 = amount of water (in m3 ) currently in the reservoir (initial storage at end of month t = 0)

C = the capacity of the reservoir (in m3 )

It = estimated inflow of water (in m3 ) to reservoir from upstream during month t (t = 1,…,12)

Dt = amount of water (in m3 ) released during month t for water supply, irrigation (t = 1,…,12)

The following unknowns (decision variables) have been defined.

St = the amount of water in storage (in m3 ) in the reservoir at the end of month t (t = 1,…,12);

Xt = the amount of water to release (in m3 ) for recreation and ecological integrity during month t (t = 1,…,12).

Using the above notation, formulate a general linear program that, if solved, would suggest an optimal management plan. (Of course, the LP might instead show that the given set of requirements has no feasible solution). The objective is to maximize the total monetary value associated with releases for recreation and ecological integrity for the year.

Solution

Using the above notation, formulate a general linear program that, if solved, would suggest an optimal management plan. (Of course, the LP might instead show that the given set of requirements has no feasible solution). The objective is to maximize the total monetary value associated with releases for recreation and ecological integrity for the year.