## (PDF) Evolutionary Algorithms for Solving Multi-Objective Problems | Vahid FARYAD - stscholapr.ga

Evolutionary Algorithms for Solving Multi-Objective Problems Second Edition. This is a major motivation to use these algorithms for solving multi-objective optimization problems . Evolutionary Algorithms for Solving Multi-Objective Problems (Genetic and Evolutionary Computation) Authors: Carlos A. Coello Coello: Inés M. Galván, Optimizing the DFCN Broadcast Protocol with a Parallel Cooperative Strategy of Multi-Objective Evolutionary Algorithms, Proceedings of the 5th International Conference on Evolutionary Multi Cited by: Multi-objective evolutionary algorithms (MOEAs) are receiving increasing and unprecedented attention. Researchers and practitioners are finding an irresistible match be­ tween the popUlation available in most genetic and evolutionary algorithms and the need in multi-objective problems to approximate the Pareto trade-off curve or surface.

## Evolutionary algorithm - Wikipedia

Multi-objective optimization also known as multi-objective programmingvector optimizationmulticriteria optimizationmultiattribute optimization or Pareto optimization is an area of multiple criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously.

Multi-objective optimization has been applied in many fields of science, including engineering, economics and logistics where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives.

Minimizing cost while maximizing comfort while buying a car, and maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives. For a nontrivial multi-objective optimization problem, no single solution exists that simultaneously optimizes each objective.

In that case, the objective functions are said to be conflicting, and there exists a possibly infinite number of Pareto optimal solutions. A solution is called nondominatedevolutionary algorithms for solving multi-objective problems, Pareto optimal, Pareto efficient or noninferior, if none of the objective functions can be improved in value without degrading some of the other objective values.

Without additional subjective preference information, all Pareto optimal solutions are considered equally good as vectors cannot be ordered completely. Researchers study multi-objective optimization problems from different viewpoints and, thus, there exist different solution philosophies and goals when setting and solving them.

A multi-objective optimization problem is an optimization problem that involves multiple objective functions. The feasible set is typically defined by some constraint functions, evolutionary algorithms for solving multi-objective problems. In addition, evolutionary algorithms for solving multi-objective problems, the vector-valued objective function is often defined as.

In multi-objective optimization, there does not typically exist a feasible solution that minimizes all objective functions simultaneously.

Therefore, attention is paid to Pareto optimal solutions; that is, solutions that cannot be improved in any of the objectives without degrading at least one of the other objectives.

The set of Pareto optimal outcomes is often called the Pareto frontPareto frontier, or Pareto boundary, evolutionary algorithms for solving multi-objective problems. The nadir objective vector is defined as. In other words, the components of a nadir and an ideal objective vector define upper and lower bounds for the objective function values of Pareto optimal solutions, respectively. In practice, the nadir objective vector can only be approximated as, typically, the whole Pareto optimal set is unknown.

In economicsmany problems involve multiple objectives along with constraints on what combinations of those objectives are attainable. For example, consumer's demand for various goods is determined by the process of maximization of the utilities derived from those goods, subject to a constraint based on how much evolutionary algorithms for solving multi-objective problems is available to spend on those goods and on the prices of those goods.

This constraint allows more of one good to be purchased only at the sacrifice of consuming less of another good; therefore, the various objectives more consumption of each good is preferred are in conflict with each other.

A common method for analyzing such a problem is to use a graph evolutionary algorithms for solving multi-objective problems indifference curvesrepresenting preferences, and a budget constraint, representing the trade-offs that the consumer is faced with. Another example involves the production possibilities frontierwhich specifies what combinations of various types of goods can be produced by a society with certain amounts of various resources.

The frontier specifies the trade-offs that the society is faced with — if the society is fully utilizing its resources, more of one good can be produced only at the expense of producing less of another good. A society must then use some process to choose among the possibilities on the frontier. Macroeconomic policy -making is a context requiring multi-objective optimization.

Typically a central bank must choose a stance for monetary policy that balances competing objectives — low inflationlow unemploymentlow balance of trade deficit, etc. To do this, the central bank uses a model of the economy that quantitatively describes the various causal linkages in the economy; it simulates the model repeatedly under various possible stances of monetary policy, in order to obtain a menu of possible predicted outcomes for the various variables of interest.

Then in principle it can use an aggregate objective function to rate the alternative sets of predicted outcomes, although in practice central banks use a non-quantitative, judgement-based, process for ranking the alternatives and making the policy choice.

In financea common problem is to choose a portfolio when there are two conflicting objectives — the desire to have the expected value of portfolio returns be as high as possible, and the desire to have riskoften measured by the standard deviation of portfolio returns, be as low as possible.

This problem is often represented by a graph in which the efficient frontier shows the best combinations of risk and expected return that are available, and in which indifference curves show the investor's preferences for various risk-expected return combinations. The problem of optimizing a function of the expected value first moment and the standard deviation square root of the second central moment of portfolio return is called a two-moment decision model.

In engineering and economicsmany problems involve multiple objectives which are not describable as the-more-the-better or the-less-the-better; instead, there is an ideal target value for each objective, and the desire is to get as close as possible to the desired value of each objective. For example, energy systems typically have a trade-off between performance and cost   or one might want to adjust a rocket's fuel usage and orientation so that it arrives both at a specified place and at a specified time; or one might want to conduct open market operations so that both the inflation rate and the unemployment rate are as close as possible to their desired values.

Often such problems are subject to linear equality constraints that prevent all objectives from being simultaneously perfectly met, especially when the number of controllable variables is less than the number of objectives and when the presence of random shocks generates uncertainty. Commonly a multi-objective quadratic objective function is used, with the cost associated with an objective rising quadratically with the distance of the objective from its ideal value.

Product and process design can be largely improved using modern modeling, simulation and optimization techniques. Before looking for optimal designs it is important to identify characteristics which contribute the most to the overall value of the design.

Therefore, in practical applications, the performance of process and product design is often measured with respect to multiple objectives. These objectives typically are conflicting, i. For example, when designing a paper mill, one can seek to decrease the amount of capital invested in a paper mill and enhance the quality of paper simultaneously. If the design of a paper mill is defined by large storage volumes and paper quality is defined by quality parameters, evolutionary algorithms for solving multi-objective problems the problem of optimal design of a paper mill can include objectives such as: i minimization of expected variation of those quality parameter from their nominal values, ii minimization of expected time of breaks and iii minimization of investment cost of storage volumes.

Here, maximum volume of towers are design variables. This example of optimal design of a paper mill is a simplification of the model used in.

Multi-objective optimization has been increasingly employed in chemical engineering and manufacturing. InFiandaca and Fraga used the multi-objective genetic algorithm MOGA to optimize the pressure swing adsorption process cyclic separation process.

The design problem involved the dual maximization of nitrogen recovery and nitrogen purity. The results provided a good approximation of the Pareto frontier with acceptable trade-offs between the objectives. They tackled two case studies bi-objective and triple objective problems with nonlinear dynamic models and used a hybrid approach consisting of the weighted Tchebycheff and the Normal Boundary Intersection approach.

The novel hybrid approach was able to construct a Pareto optimal set for the thermal processing of foods. InGanesan et al. The objective functions were methane conversion, carbon monoxide selectivity and hydrogen to carbon monoxide ratio. InAbakarov et al proposed an alternative technique to solve multi-objective optimization problems arising evolutionary algorithms for solving multi-objective problems food engineering.

The Analytic Hierarchy Process and Tabular Method were used simultaneously for choosing the best alternative among the computed subset of non-dominated solutions for osmotic dehydration processes. InPearce et al. Their approach used a Mixed-Integer Linear Program to solve the optimization problem for a weighted sum of the two objectives to calculate a set of Pareto optimal solutions.

The application of the approach to several manufacturing tasks showed improvements in at least one objective in most tasks and in both objectives in some of the processes. The purpose of radio resource management is to satisfy the data rates that are requested by the users of a cellular network, evolutionary algorithms for solving multi-objective problems. Each user has its own objective function that, for example, can represent some combination of the data rate, latency, and energy efficiency.

These objectives are conflicting since the evolutionary algorithms for solving multi-objective problems resources are very scarce, thus there is a need for tight spatial frequency reuse which causes immense inter-user interference if not properly controlled. Multi-user MIMO techniques are nowadays used to reduce the interference by adaptive precoding. The network operator would like to both bring great coverage and high data rates, thus the operator would like to find a Pareto optimal solution that balance the total network data throughput and the user fairness in an appropriate subjective manner.

Radio resource management is often solved by scalarization; that is, selection of evolutionary algorithms for solving multi-objective problems network utility function that tries to balance throughput and user fairness. The choice of utility function has a large impact on the computational complexity of the resulting single-objective optimization problem. Reconfiguration, by exchanging the functional links between the elements of the system, represents one of the most important measures which can improve the operational performance of a distribution system.

The problem of optimization through the reconfiguration of a power distribution system, evolutionary algorithms for solving multi-objective problems, in terms of its definition, is a historical single objective problem with constraints. Sincewhen Merlin and Back  introduced the idea of distribution system reconfiguration for active power loss reduction, until nowadays, a lot of researchers have proposed diverse methods and algorithms to solve the reconfiguration problem as a single objective problem.

Some authors have proposed Pareto optimality based approaches including active power losses and reliability indices as objectives. For this purpose, different artificial intelligence based methods have been used: microgenetic, evolutionary algorithms for solving multi-objective problems,  branch exchange, evolutionary algorithms for solving multi-objective problems,  particle swarm optimization  and non-dominated sorting genetic algorithm.

Autonomous inspection of infrastructure has the potential to reduce costs, risks and environmental impacts, as well as ensuring better periodic maintenance of inspected assets. Typically, planning such missions has been viewed as evolutionary algorithms for solving multi-objective problems single-objective optimization problem, where one aims to minimize the energy or time spent in inspecting an entire target structure .

A recent study has indicated that multiobjective inspection planning indeed has the potential to outperform traditional methods on complex structures . As there usually evolutionary algorithms for solving multi-objective problems multiple Pareto optimal solutions for multi-objective optimization problems, what it means to solve such a problem is not as straightforward as it is for a conventional single-objective optimization problem.

Therefore, different researchers have defined evolutionary algorithms for solving multi-objective problems term "solving a multi-objective optimization problem" in various ways.

This section summarizes some of them and the contexts in which they are used. Many methods convert the original problem with multiple objectives into a single-objective optimization problem. This is called a scalarized problem. If scalarization is done neatly, [ clarification needed ] Pareto optimality of the solutions obtained can be guaranteed. Solving a multi-objective optimization problem is sometimes understood as approximating or computing all or a representative set of Pareto optimal solutions.

Here, a human decision maker DM plays an important role. The DM is expected to be an expert in the problem domain. The most preferred results can be found using different philosophies. Multi-objective optimization methods can be divided into evolutionary algorithms for solving multi-objective problems classes.

In a priori methods, preference information is first asked from the DM and then a solution best satisfying these preferences is found. In a posteriori methods, a representative set of Pareto optimal solutions is first found and then the DM must choose one of them.

In interactive methods, the decision maker is allowed to iteratively search for the most preferred solution. In each iteration of the interactive method, the DM is shown Pareto optimal solution s and describes how the solution s could be improved. The information given by the decision maker is then taken into account while generating new Pareto optimal solution s for the DM to study in the next iteration.

More information and examples of different methods in the four classes are given in the following sections. Scalarizing a multi-objective optimization problem is an a priori method, which means evolutionary algorithms for solving multi-objective problems a single-objective optimization problem such that optimal solutions to the single-objective optimization problem are Pareto optimal solutions to the multi-objective optimization problem.

A general formulation for a scalarization of a multiobjective optimization is thus. For example, portfolio optimization is often conducted in terms of mean-variance analysis. When a decision maker does not explicitly articulate any preference information the multi-objective optimization method can be classified as no-preference method. A priori methods require that sufficient preference information is expressed before the solution process.

In the utility function method, it is assumed that the decision maker's utility function is available. The utility function specifies an ordering of the decision vectors recall that vectors can be ordered in many different ways. The lexicographic method assumes that the objectives can be ranked in the order of importance. The lexicographic method consists of solving a sequence of single-objective optimization problems of the form.

Note that a goal or target value is not specified for any objective here, which makes it different from the Lexicographic Goal Programming method. 