Inclusion of these constraints ensures that the solution obtained would be physically feasible. The optimization problem can be formulated as,

Maximize

23.5

Subject to

	23.6
	23.7
	23.8
	23.9
	23.10
	23.11
	23.12
	23.13
	23.14
	23.15
	23.16
	23.17
	23.18

The objective function will try to maximize the head value of the system while satisfying all the constraints.

The constraint (23.16) ensures that the minimum withdrawn is N_min and the constraints (23.17) and (23.18) are the non negativity constraints. Here the decision variables are N₁ to N₁₀ and h₁ to h₁₀ are the state variables. If we look at the optimization problem, the objective function and the constraints are linear in nature. As such the problem is a linear problem (LP) and can be solved using any LP solving algorithm like Simplex method. We will not discuss about any optimization algorithms here. However, students are requested to go through the NPTEL course developed by Prof. Nagesh Kumar on “Optimization Method” for details about the optimization algorithms.

The solution of the optimization problem will give the spatial distribution of pumping pattern where total pumping from all the pumping wells is equal to N_min.