7.2 Multicriteria Optimization Applied to MANET Routing
7.2.1 Characterization of an Optimization Problem
Optimization is the process of minimizing or maximizing an objective single- or multivariable function, which can be subject to a number of equality or inequality restrictions. Common com-ponents of any optimization problem are listed below [9]:
◾ Decision or design variable(s): These are the unknown parameters of the real-world prob-lem that can be modified to obtain different results. Depending on the number of decision variables involved, the problem can be single variable or multivariable. In the case of routing protocols, these variables are related to the design parameters of the algorithms that those protocols implement.
◾ Objective functions: These functions are a representation of the decision maker that must be optimized. When there is only one objective function, the problem is named single-criterion or single-objective. On the contrary, a problem with several objective functions is known as a multicriteria or multiobjective problem (i.e., considering either a single-metric or multiple-metric routing protocol).
◾ Restrictions to decision or design space: The decision variables of the problem can be restricted or nonrestricted, depending on the existence of equality or inequality restrictions.
In most routing scenarios, these constraints will be derived from bandwidth/delay/number of hops requirements.
For simplicity, from now on, the optimization problem will be analyzed first from a theoreti-cal point of view and applied later (in Section 7.2.2) to the specific routing use case. Furthermore, for simplicity purposes it will be considered a minimization problem, since this is the most usual situation, keeping in mind that any conclusion could be extensible to the maximization problem.
7.2.1.1 Single-Variable and Single-Criterion Optimization Problem
The final aim of any optimization problem is to obtain a local or global minimum of the objec-tive function f(x) (Figure 7.4). A single-variable objecobjec-tive function f(x) has a global minimum in x = x* when
f x( )* ≤ f x( *+h) ∀h (7.1)
A single-variable objective function f(x) has a local minimum in x = x* when
f x( )* ≤ f x( *+h) (7.2)
for values of h sufficiently close to 0. Therefore, the single-variable, single-criterion optimization problem provides a way of selecting the “best” possible value of the input parameter (design or decision variable) to optimize the single criterion or objective function f(x).
The minimization problem can also be expressed as
min{ ( )f x =z} (7.3)
7.2.1.2 Multivariable Single-Criterion Optimization Problem
In this case, the value of the single criterion analyzed depends on several factors or input param-eters and, as a result, the multivariable single-criterion optimization problem provides a way of selecting the “best” possible combination of these input parameters to optimize the single criterion considered.
Assuming x is any combination of input parameters, the multivariable single-criterion optimi-zation problem has the following definition:
min{ ( )f x =z}withx S∈ (7.4) where S is the feasible region (often also called feasible design space, feasible decision space, or restricted set).
The concept of S leads to the classification of optimization problems according to another of their defining features: the format of the problem. Some problems do not need the solution to meet any prerequisite. In these cases, S ≡ Rn, and the optimization is generally called “restriction-less.” However, usually x will be subject to a number of constraints and, hence, S will be a subset of S ≡ Rn defined by a group of equality and/or inequality functions that members of S must satisfy.
This kind of problem is known as “restricted” and is mathematically formulated as
min{ ( )f x =z} (7.5)
with g xi( )≤ ∀ =0 i 1 2, ,....,m with h xj( )≤ ∀ =0 j 1 2, ,....,l
Therefore, the single-criterion optimization problem can be defined in short as “the search for x in S so that f(x) = z is maximum.”
f(x)
Local minimums
Global minimum
x Figure 7.4 Local and global minimums.
7.2.1.3 Multicriteria Optimization Problem
Not only routing but many real-world problems are often based on the existence of several oppo-site objectives or criteria, which has led to the multicriteria or multiobjective problem approach.
The basic aim of this approach is to minimize all the criteria simultaneously and meet the equality and inequality constraints of the feasible space S.
The multiobjective problem can thus be defined as an extension of the single-criterion
where fi is the i-th objective function.
The simplest solution to this kind of problem lies in finding the input vector x*, which satisfies
∃ ∈x* Smin
{
f xi( *)=zi*}
∀ =i 1 2, ,...,k (7.7) Figure 7.5 depicts an optimization problem with two variables or parameters to optimize (x1, x2) and two criteria (f1, f2) for a space S constrained by two inequality functions (g1, g2). The feasible region S in the design space is mapped into Y in the criteria space, which is constrained by q1, q2, which correspond to g1, g2 in this space. The optimal solution f * = f(x*) that minimizes all the criteria simultaneously is seldom achievable. In this case, it is known as a utopian solution [10].The solution to the problem should then be as close as possible to such a utopian solution, resulting in the Pareto-optimum front ϕ.
Figure 7.5 Graphic representation of a bivariable and bicriteria optimization problem in design and criteria spaces.
In most cases, there is no x* that simultaneously minimizes all the criteria. Thus, it is neces-sary to redefine the optimization problem to attain, from within the whole feasible set of possible solutions, the solution that optimizes the global result, for example, that which is closer to the utopian solution. As shown in Figure 7.5, this solution often is not unique, and the Pareto front is composed of a set of possible solutions. The final aim of a high-quality multicriteria optimization problem is to find a sufficiently representative solution set, homogeneously distributed along that Pareto front.
7.2.1.4 Taxonomy of a Multicriteria Optimization Problem
Optimization problems can be classified according to a wide variety of perspectives, resulting in a general taxonomy of the multicriteria optimization problems composed from the works in [9,11–15]:
◾ Derivative vs. nonderivative: This classification is according to whether the objective func-tions to be optimized can be derived using the resolution algorithm. This point settles the possibility for the optimization problem to explore the feasible region in a simple way by the use of a gradient.
◾ Qualitative vs. quantitative criteria: This classification is according to the way of measuring the analyzed criteria. When the criteria that model the preferences of the decision maker are quantitative, they are represented by a numeric value, indicating the preference degree of one preference against the others. When the criteria are qualitative, preference lacks an exact numeric value and a descriptive value is assigned.
◾ Preference articulation: The moment the preferences of the decision maker are given classi-fies optimization problems into another four categories:
− A priori preference articulation: These preferences add additional restrictions to the problem (weighted sum and lexicographic methods).
− A posteriori preference articulation: The final solution is selected from the set of opti-mal solutions that result from the optimization process (i.e., in evolutive and genetic methods).
− Progressive preference articulation: The decision maker’s preferences are gradually incor-porated in an interactive way during the optimization process.
− Without preference articulation: When the decision maker cannot (or does not want to) define the preference for the solution set (i.e., max–min formulation, global criterion method).
◾ Continuous vs. discrete: This classification refers to the variable type to work with. Some optimization problems are restricted to the manipulation of discrete variables, such as inte-gers, binary values, or other abstract objects. The goal of the discrete optimization is to attain the optimum point from a finite, but usually huge, set. Continuous optimization problems, on the contrary, operate with infinite variable values. Continuous problems are usually easier to solve due to their predictability, for the solution can be achieved with an approximative iterative process.
◾ Constrained vs. nonconstrained: In some cases, the purpose of multiobjective optimization problems is not only to find a solution that optimizes a set of criteria, but also to must meet a series of requirements expressed through (in)equality equations. Nonconstrained methods are also used to solve constrained methods, substituting restrictions for penalizations on objective functions to prevent possible constraint violations.
◾ Global vs. local: Global optimization methods search for the solution all over the feasible space.
But some applications just need to find a locally optimum solution near a certain point. A global solution is also locally optimal, but the contrary is not always true. However, many global opti-mization methods make use of local methods to locate a first approach to the solution.
◾ Stochastic vs. deterministic: In some cases, the model to be optimized is not completely described because some specifications remain unknown at the analysis of the problem. In these situations, and in contrast with deterministic methods, stochastic optimization problems use uncertainty quantifications to achieve an optimum solution for the expected model.
This classification does not result into disjoint categories, but a multicriteria optimization prob-lem can fall into one or several of the categories listed above. This is the main difficulty in finding a definitive taxonomy of multicriteria optimization problems or methods in the existing literature.
7.2.1.5 Preference or Criteria Modeling
Before beginning with the process of decision-making analysis of a multicriteria optimization problem, there is a fundamental first step: the characterization of the criteria to be optimized, i.e., the preferences of the decision maker about the adequacy of the selected solution.
A key factor in the analysis for decision making is the fact that the functions that model the decision maker’s preferences (criteria or objective functions) are not usually known a priori. In this sense, several methodologies have been proposed within the field of multicriteria mathematical programming in order to develop a successful interaction with the decision maker that makes it possible to provide relevant information about local solutions to the decision maker and to get useful information about the decision maker’s preferences at the same time.
It must be kept in mind that the final goal is to establish an ordered ranking as a natural foun-dation to solve decision or selection problems.
7.2.1.6 Valuation Scale
A preference model establishes a formal representation of the comparison between alternatives that express both the structure of the described situation and the variety of manipulations that can be made on it [16]. Logical language proves to be appropriate for these descriptions. Classic logic, however, can turn out to be too rigid to sufficiently define expressive models. Hence, other formalisms must be considered to provide the model with the required flexibility.
The preference or criteria representation format plays an important role since it defines the nature and structure of the information on which the decision maker bases his inclinations toward the dif-ferent alternatives. Experts from each field of knowledge use the representation format that better applies to their area of expertise. In some cases, preferences will be expressed through numerical values, and in other cases, more natural expressions, such as words or linguistic terms, are used.
The final objective is the comparison of potential actions in order to make a decision, so it becomes mandatory to determine a scale for every considered criterion. The elements of the scale are denoted degrees, levels, or ranks. When comparing the behavior of the different options, it is important to analyze the specific meaning in terms of preference associated with each criterion.
This leads to the differentiation of two main types of scales [17]:
◾ Pure ordinal or qualitative scale: In these cases, the gap between two levels or degrees has no clear meaning regarding preference disparity. This has no relation to the levels expressed in a verbal or numerical way. For example, talking about temperatures, 20ºC does not mean
twice as hot as 10ºC, and the increase from 5 to 8ºC does not feel like the increase from 15 to 18ºC, even when the numerical difference in both gaps remains the same. In this case, the aggregation of criteria using an arithmetical procedure makes no sense due to the lack of real numerical meaning of the graduation.
◾ Pure cardinal or quantitative scale: It is a numerical scale with definite meaning regard-ing the relative difference of preference between two levels, regardless of the levels beregard-ing considered.
Between these two opposite cases, there is a wide variety of possibilities, especially in the case of interval scales.
Once the scale is determined, the comparison between two alternatives boils down to one of the following scenarios [16]:
◾ To determine whether option A is preferable to option B.
◾ To check whether option A is close to option B in the sense of being considered indifferent from the decision maker’s point of view.
7.2.1.7 Objectivity and Uncertainty
To build a model from reality is always an abstraction drill. Even in the case of a highly objective decision maker, this objectivity is limited, basically due to the following factors [18]:
◾ The boundary between feasible and nonfeasible solutions is often fuzzy and varies as the analysis of the scenario gets deeper.
◾ Even for a well-defined decision maker, the preferences are seldom perfectly characterized.
Uncertainty zones, conflicts, or contradictions almost always exist and constitute a source of ambiguity or arbitrariness.
◾ Available information is often imprecise, uncertain, or is poorly defined, leading to misinterpretation.
◾ Optimization processes are objective by definition, avoiding subjective aspects associated with organizational, cultural, or social fairness considerations.
The goal is to compose a preference model based on incomplete or biased information and to know to what extent uncertainty or ambiguity is propagated along a model built in those circumstances.
In this case, there are two possibilities:
◾ When there is a well-known preference relation, but there is not enough available informa-tion to build the model, probability distribuinforma-tions are used to associate an uncertainty level to each statement.
◾ When the concept of preference itself is poorly defined, independently from the available information, multivalued logic is used to associate each statement on preference with a value representing its “veracity intensity.”