We have modeled power-demand routing as multiobjective network optimization problem. Hence, we can not apply the solution concept used in a single-objective optimization problem, i.e. a feasible solution is optimal for a multiobjective optimization problem, if it minimizes all objective functions simultaneously. In general such an optimal, ideal, solution does not usually exist, when the objectives conflict with each other. This leads to a concept of Pareto optimality.

In our case, the above problem has the form \(\min_{\mathbf{x}\in \mathbb{X}}(C_E(\mathbf{x}),C_N(\mathbf{x}),C_P(\mathbf{x}))^T\), where the set of feasible flows is described by the constraints C1-C6. We now ready to define the Pareto optimality.

Clearly, each ideal solution is a Pareto optimal solution and a Pareto optimal solution is a weak Pareto optimal solution. Thus the weak Pareto optimality concept is weaker than the Pareto optimality concept.

The set of Pareto optimal solutions is called Pareto frontier. Enumeration solutions in a Pareto frontier, moving a long this frontier, may decrease the value of some objective functions for these solutions, but it may increase the values other ones. It is worth to pointed out that a Pareto frontier may contain a large number of solutions and thus computing the all solution in the Pareto frontier may be a hard task in practice.

Several methods for computing a (weak) Pareto optimal solution have been proposed in the literature (see, e.g.,   : 1 ). For instance, scalarizing methods, nonscalarizing methods, etc.

From now on, we assume that all the objective function values in multiobjective optimization problems are in a common unit (e.g. in a monetary unit).

A natural approach for solving multiobjective optimization problems according to the Pareto concept of optimality is the scalarization, which reduces a multiobjective optimization problem to a single-objective optimization problem. The reduction consists in replacing objective functions with a real-valued scalarizing function, which is a function of these objectives functions. There are many the scalarization methods, for instance, the weighted sum method, the weighted minimax method, the constraint method, etc. (see, e.g.,  : 1 ).

The one of the simplest methods is the weighted sum one, that consists in solving a single-objective optimization problem whose objective is the weighted sum of all the objective functions of the multiobjective optimization problem under consideration. Thus the weighted sum problem can be formulated as follows: \[ \min_{\mathbf{x}\in \mathbb{X}} \sum_{k=1}^K \lambda_kC_k(\mathbf{x}), \] where \(\lambda=(\lambda_1,\ldots,\lambda_K)\geq 0\) is a nonnegative vector of weights.
Our network problem after scalarization has the form: \[ \min_{\mathbf{x}\in \mathbb{X}} \lambda_E C_E(\mathbf{x})+ \lambda_N C_N(\mathbf{x})+ \lambda_P C_P(\mathbf{x}), \] where \(\lambda_E, \lambda_N, \lambda_P\geq 0\). It turns out that an optimal solution \(\mathbf{x}^*\in \mathbb{X}\) of a weighted problem for some weights \(\lambda=(\lambda_1,\ldots,\lambda_K)> 0\), is a Pareto optimal solution of the underlying multiobjective optimization problem, which justifies the weighted sum method and makes it useful for our network problem.

The second approach is the constraint method, which consists in solving a single-objective constrained problem formed by taking one objective cost function as the objective function of the constrained problem and extending the set of constraints of the underlying multiobjective optimization problem by including all the others objective cost functions as inequality constraints. Thus, a constrained problem related to a general multiobjective optimization problem has the form:
\[ \begin{aligned} &\min C_j(\mathbf{x})&\\ \text{subject to}&\\ &C_k(\mathbf{x})\leq \alpha_k&\text{for } k=1,\ldots,K, j\not=k\\ &\mathbf{x}\in \mathbb{X}& \end{aligned} \] The above single-objective optimization problem has useful property. Namely, an optimal solution \(\mathbf{x}^*\in \mathbb{X}\) of the constrained problem for some \(\alpha_k\), \(k=1,\ldots,K, j\not=k\), is a weak Pareto optimal solution of the underlying multiobjective optimization problem (if \(\mathbf{x}^*\in \mathbb{X}\) is a unique solution, then it is Pareto optimal solution), which justifies the method.

Computational efficiency of algorithms for solving multiobjective optimization problems, in consequence for solving multiobjective network optimization problem, and quality of solutions computed (local, global optima) heavy relay on properties of constraints determining a set of feasible solution and properties of cost functions.

If the components forming cost functions \(C_E\), \(C_N\) and \(C_P\), in our multiobjective network optimization problem, are linear functions of request flow \(\mathbf{x}\), then after applying a scalarization technique one obtains a linear programming problem, i.e. the problem with a linear objective function and linear constraints (see, e.g.,   : 2 ). Algorithms for solving a linear programming problem are highly efficient and output globally optimal solutions - even very large linear problems. It is worth pointing out that practical linear programming problems can be solved by some standard off-the-shelf linear programming solvers (see, e.g., Glpk: GNU Linear Programming Kit   : 3 ).

In practice, cost functions are nonlinear or piecewise linear or nonconvex functions of \(\mathbf{x}\). Moreover, functions may have discontinuities or may be nondifferentiable at some points, etc.(see,   : 4  for examples nonlinear network, performance and energy cost functions and nonconvex cost functions). In this case solving the multiobjective network optimization problem under consideration can be a hard task.

If cost functions are nonlinear convex functions then computationally efficient methods exist that can output globally optimal solutions. In some cases linear programming methods are used to solve convex nonlinear optimization problems. Otherwise, there are less efficient algorithms which can return only a locally solution (see, e.g.   : 5 ).