This repository consists of notebooks using which, we formulate an optimization problem to identify the optimal placement of pressure sensors in a Water Distribution Network (WDN). The WDN is represented on a constrained graph G(V,E), where V is the set of vertices and E is the set of edges.
The solution to this graph problem is explored using different reformulations of the problem.
First, we formulate the graph optimization problem as a Mixed Integer Program (MIP) (Speziali et al. (2021)):
min_{x} sum_{i in V} c_{i}x_{i} + sum_{(i, j) in E} w_{ij}(1 - x_{j} - x_{i} + x_{i}x_{j})
s.t. sum_{i in V} x_{i} = s
x_{i} in {0,1}^{n}
Here, c_{i} represents the cost of the i-th node, w_{ij} represents the weight corresponding to the edge between nodes i and j, and x_{i} in {0, 1}^{n} is a binary decision variable that indicates whether a sensor is placed at the i-th node. s is the predefined total number of sensors.
Next, the Quadratic Unconstrained Binary Optimization (QUBO) formulation of the graph problem is explored. Consider the MIP model from before:
min_{x} sum_{i in V} c_{i}x_{i} + sum_{(i, j) in E} w_{ij}(1 - x_{j} - x_{i} + x_{i}x_{j})
s.t. sum_{i in V} x_{i} = s
x_{i} in {0,1}^{n}
To implement the problem as a QUBO, the constraint should be lifted up into the objective function such that the problem becomes unconstrained. This is performed as follows:
min_{x} sum_{i in V} c_{i}x_{i} + sum_{(i, j) in E} (w_{ij}(1 - x_{j} - x_{i} + x_{i}x_{j})) + rho(sum_{i in V} x_{i} - s)^2
where rho is a scalar penalty term. Further simplification leads to
min_{x} sum_{i in V} c_{i}x_{i} + sum_{(i, j) in E} w_{ij} - sum_{(i, j) in E} w_{ij}x_{j} - sum_{(i, j) in E} w_{ij}x_{i} + sum_{(i, j) in E} w_{ij}x_{i}x_{j} + rho sum_{i in V} x_{i}^2 - 2rho sum_{i in V} x_{i}s + rho s^2
Since x_{i} is a binary variable, we can write x_{i} as x_{i}^2. Taking this into account, the problem now becomes:
min_{x} sum_{i in V} c_{i}x_{i}^2 + sum_{(i, j) in E}w_{ij} - sum_{(i, j) in E} w_{ij}x_{j}^2 - sum_{(i, j) in E} w_{ij}x_{i}^2 + sum_{(i, j) in E} w_{ij}x_{i}x_{j} + rho sum_{i in V} x_{i}^2 - 2rho sum_{i in V} x_{i}^2s + rho s^2
Performing some algebraic manipulations, we have our QUBO problem:
min_{x} ( sum_{i in V} (c_i + rho - 2rho s - w_{ij})x_i^2 + sum_{(i,j) in E} w_{ij}x_i x_j + rho s^2 + sum_{(i,j) in E} w_{ij} )
where s is the total number of sensors that is predefined.
The optimal sensor placement problem is solved using five different solvers: Gurobi, dwave-tabu Tabu Search Sampler, dwave-neal Simulated Annealing Sampler, D-Wave Leap Hybrid Quantum-Classical solver and D-Wave Advantage 4.1 Quantum Annealer. Additionally, we leverage Networkx for network models and graphs.