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Multiobjective optimization (also known as multiobjective programming, vector optimization, multicriteria optimization, multiattribute 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. Multiobjective optimization has been applied in many fields of science, including engineering, economics and logistics (see the section on applications for examples) where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Minimizing weight while maximizing the strength of a particular component and maximizing performance and minimizing fuel consumption and emission of pollutants of a vehicle are examples of multiobjective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives.

For a nontrivial multiobjective optimization problem, there does not exist a single solution 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 nondominated, Pareto optimal, Pareto efficient or noninferior, if none of the objective functions can be improved in value without impairment in some of the other objective values. Without additional preference information, all Pareto optimal solutions can be considered mathematically equally good (as vectors cannot be ordered completely). Researchers study multiobjective optimization problems from different viewpoints and, thus, there exist different solution philosophies and goals when setting and solving them. The goal may be finding a representative set of Pareto optimal solutions, and/or quantifying the trade-offs in satisfying the different objectives, and/or finding a single solution that satisfies the preferences of a human decision maker (DM).

Introduction
A multiobjective optimization problem is an optimization problem that involves multiple objective functions. In mathematical terms, a multiobjective optimization problem can be formulated as

\begin{align} \min &\left(f_1(x), f_2(x),\ldots, f_k(x) \right)^T \\ \text{s.t. } &x\in X, \end{align} $$ where the integer $$k\geq 2$$ is the number of objectives and the set $$X$$ is the feasible set of decision vectors defined by constraint functions. In addition, the vector-valued objective function is often defined as
 * $$f:X\to\mathbb R^k, \ f(x)= (f_1(x),\ldots,f_k(x))^T$$. If some objective function is to be maximized, it is equivalent to minize its negative. The image of $$X$$ is denoted by $$Y \in \mathbb R^k$$

An element $$x^*\in X$$ is called a feasible solution or a feasible decision. A vector $$z^* := f(x^*)\in \mathbb R^k$$ for a feasible solution $$x^*$$ is called an objective vector or an outcome. In multiobjective optimization, there does not typically exist a feasible solution that minimizes all objective functions simultaneously. Therefore, attention is paid to Pareto optimal solutions, i.e., solutions that cannot be improved in any of the objectives without impairment in at least one of the other objectives. In mathematical terms, a feasible solution $$x^1\in X$$ is said to (Pareto) dominate another solution $$x^2\in X$$, if A solution $$x^1\in X$$ (and the corresponding outcome $$f(x^*)$$) is called Pareto optimal, if there does not exist another solution that dominates it. The set of Pareto optimal outcomes is often called the Pareto front.
 * 1) $$f_i(x^2)\leq f_i(x^1)$$ for all indices $$i \in \left\{ {1,2,\dots,k } \right\}$$ and
 * 2) $$f_j(x^2) < f_j(x^1)$$ for at least one index $$j \in \left\{ {1,2,\dots,k } \right\}$$.

The Pareto front of a multiobjective optimization problem is bounded by a so-called nadir objective vector $$ z^{nad} $$ and an ideal objective vector $$ z^{ideal} $$, if these are finite. The nadir objective vector is defined as
 * $$ z^{nad}_i= \max_{x\in X\text{ is Pareto optimal}} f_i(x) \text{ for all } i=1,\ldots,k $$

and the ideal objective vector as
 * $$ z^{ideal}_i=\min_{x\in X}f_i(x)\text{ for all } i=1,\ldots,k.$$

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.

Economics
In economics, many 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 income 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 of indifference curves, representing preferences, and a budget constraint, representing the trade-offs that the consumer is faced with.

Another example involves the production possibilities frontier, which 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 inflation, low unemployment, low 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.

Finance
In finance, a 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 risk, 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 moment) of portfolio return is called a two-moment decision model.

Optimal control
In engineering and economics, many 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, 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. Since these problems typically involve adjusting the controlled variables at various points in time and/or evaluating the objectives at various points in time, intertemporal optimization techniques are employed.

Optimal design
Product and process design can be largely improved using modern modeling, simulation and optimization techniques. The key question in optimal design is the measure of what is good or desirable about a design. Before looking for optimal designs it is important to identify characteristics which contribute the most to the overall value of the design. A good design typically involves multiple criteria/objectives such as capital cost/investment, operating cost, profit, quality and/or recovery of the product, efficiency, process safety, operation time etc. 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.e., achieving the optimal value for one objective requires some compromise on one or more of other objectives.

For example, in paper industry 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, then 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.

Solving a multiobjective optimization problem
As there usually exist multiple Pareto optimal solutions for multiobjective optimization problems, what it means to solve such a problem is not as straightforward as it is for a single objective optimization problem. Therefore, different researchers have defined the term "solving a multiobjective 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 carefully, Pareto optimality of the solutions obtained can be guaranteed.

Solving a multiobjective optimization problem is sometimes understood as approximating or computing all or a representative set of Pareto optimal solutions. This is done, e.g., in and.

When decision making is emphasized, the objective of solving a multiobjective optimization problem is referred to supporting a decision maker in finding the most preferred Pareto optimal solution according to his/her preferences. This is followed e.g., in and. The underlying assumption is that one solution to the problem must be identified to be implemented in practice. Here, a human decision maker (DM) plays an important role. (S)he is expected to be an expert in the problem domain.

The most preferred solution can be found using different philosophies. In, multiobjective optimization methods are divided into four classes. In so-called no preference methods, no decision maker is expected to be available, but a neutral compromise solution is identified without preference information. The other classes are so-called a priori, a posteriori and interactive methods and they all involve preference information from the decision maker in different ways.

In a priori methods, preference information is first asked from the decision maker 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 decision maker 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 decision maker is shown Pareto optimal solution(s) and (s)he can tell 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 decision maker to study in the next iteration. In this way, the decision maker learns about the feasibility of his/her wishes and can concentrate on solutions that are interesting to him/her. The decision maker may stop the search whenever he/she wants to. More information and examples of different methods in the four classes are given in the following sections.

Scalarizing multiobjective optimization problems
Scalarizing a multiobjective optimization problem means formulating a single-objective optimization problems such that optimal solutions to the single-objective optimization problem are Pareto optimal solutions to the multiobjective optimization problem. With different parameters for the scalarization, different Pareto optimal solutions are produced. A well-known example is the so-called linear scalarization (see, e.g., )

\min_{x\in X} \sum_{i=1}^k w_if_i(x), $$ where the weights of the objectives $$w_i>0$$ are the parameters of the scalarization.

No-preference methods
Multiobjective optimization methods that do not require any preference information to be explicitly articulated by a decision maker can be classified as no-preference methods. A well-known example is the method of global criterion, in which a scalarized problem of the form

\begin{align} \min&\|f(x)-z^{ideal}\|\\ \text{s.t. }&x\in X \end{align} $$ is solved. In the above problem, $$\|\cdot\|$$ can be any $L_p$ norm, with common choices including $$L_1$$, $$L_2$$ and $$L_\infty$$. The method of global criterion is sensitive to the scaling of the objective functions, and thus, it is recommended that the objectives are normalized into a uniform, dimensionless scale,.

A priori methods
A priori methods require that sufficient preference information is expressed before the solution process. Well-known examples of a priori methods include the utility function method, lexicographic method, and goal programming. In the utility function method, it is assumed that the decision maker's utility function is available. A mapping $$ u\colon Y\rightarrow\mathbb{R}$$ is a utility function if for all $$\mathbf{y}^1,\mathbf{y}^2\in Y$$ it holds that $$u(\mathbf{y}^1)>u(\mathbf{y}^2)$$ if the decision maker prefers $$\mathbf{y}^1$$ to $$\mathbf{y}^2$$, and $$u(\mathbf{y}^1)=u(\mathbf{y}^2)$$ if the decision maker is indifferent between $$\mathbf{y}^1$$ and $$\mathbf{y}^2$$. Once $$u$$ is obtained, it suffices to solve
 * $$   \max\;u(\mathbf{f}(\mathbf{x}))\text{ subject to }\mathbf{x}\in X,$$

but in practice it is very difficult to construct a utility function that would accurately represent the decision maker's preferences.

Lexicographic method assumes that the objectives can be ranked in the order of importance. We can assume, without loss of generality, that the objective functions are in the order of importance so that $$f_1$$ is the most important and $$f_k$$ the least important to the decision maker. The lexicographic method consists of solving a sequence of single objective optimization problems of the form

\begin{align} \min&f_l(\mathbf{x})\\ \text{s.t. }&f_j(\mathbf{x})\leq\mathbf{y}^*_j,\;j=1,\dotsc,l-1,\\ &\mathbf{x}\in X, \end{align} $$ where $$\mathbf{y}^*_j$$ is the optimal value of the above problem with $$l=j$$. Thus, $$\mathbf{y}^*_1:=\min\{f_1(\mathbf{x})\mid\mathbf{x}\in X\}$$ and each new problem of the form in the above problem in the sequence adds one new constraint as $$l$$ goes from $$1$$ to $$k$$.

A posteriori methods
A posteriori methods aim at producing all the Pareto optimal solutions or a representative subset of the Pareto optimal solutions. Well-known examples are the Normal Boundary Intersection (NBI) and see also this report, Modified Normal Boundary Intersection (NBIm), Normal Constraint (NC) , Successive Pareto Optimization (SPO) methods that solve the multi-objective optimization problem by constructing several scalarizations. The solution to each scalarization yields a Pareto otpimal solution, whether locally or globally. The scalarizations of the NBI, NBIm, NC methods are constructed with the target of obtaining evenly distributed Pareto points that give a good evenly distributed approximation of the real set of Pareto points.

Evolutionary algorithms are popular approaches to generating Pareto optimal solutions to a multiobjective optimization problem. Currently, most evolutionary multiobjective optimization (EMO) algorithms apply Pareto-based ranking schemes. Evolutionary algorithms such as the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) and Strength Pareto Evolutionary Algorithm 2 (SPEA-2) have become standard approaches, although some schemes based on particle swarm optimization and simulated annealing are significant. The main advantage of evolutionary algorithms, when applied to solve multiobjective optimization problems, is the fact that they typically generate sets of solutions, allowing computation of an approximation of the entire Pareto front. The main disadvantage of evolutionary algorithms is their lower speed and the Pareto optimality of the solutions cannot be guaranteed. It is only known that none of the generated solutions dominates the others.

Other a posteriori methods are:
 * PGEN (Pareto surface generation for convex multiobjective instances)
 * IOSO (Indirect Optimization on the basis of Self-Organization)
 * SMS-EMOA (S-metric selection evolutionary multiobjective algorithm)
 * Reactive Search Optimization (using machine learning for adapting strategies and objectives), implemented in LIONsolver
 * Benson's algorithm for linear vector optimization problems
 * Multiobjective particle swarm optimization

Interactive methods
In interactive methods, the solution process is iterative and the decision maker continuously interacts with the method when searching for the most preferred solution (see e.g.,, ). In other words, the decision maker is expected to express preferences at each iteration in order to get Pareto optimal solutions that are of interest to him/her and learn what kind of solutions are attainable. The following steps are commonly present in interactive methods :


 * 1) initialize (e.g., calculate ideal and approximated nadir objective vectors and show them to the decision maker)
 * 2) generate a Pareto optimal starting point (by using e.g. some no-preference method or solution given by the decision maker)
 * 3) ask for preference information from the decision maker (e.g., aspiration levels or number of new solutions to be generated)
 * 4) generate new Pareto optimal solution(s) according to the preferences and show it/them and possibly some other information about the problem to the decision maker
 * 5) if several solutions were generated, ask the decision maker to select the best solution so far
 * 6) stop, if the decision maker wants to; otherwise, go to step 3).

Above, aspiration levels refer to desirable objective function values forming a reference point. Instead of mathematical convergence that is often used as a stopping criterion in mathematical optimization methods, a psychological convergence is emphasized in interactive methods. Generally speaking, a method is terminated when the decision maker is confident that (s)he has found the most preferred solution available. Different interactive methods involve different types of preference information. For example, three types can be identified: methods based on 1) trade-off information, 2) reference points and 3) classification of objective functions . On the other hand, a fourth type of generating a small sample of solutions is included in and . An example of interactive method utilizing trade-off information is the Zionts-Wallenius method, where the decision maker is shown several objective trade-offs at each iteration, and (s)he is expected to say whether (s)he likes, dislikes or is indifferent with respect to each trade-off. In reference point based methods (see e.g., , ), the decision maker is expected at each iteration to specify a reference point consisting of desired values for each objective and a corresponding Pareto optimal solution(s) is then computed and shown to him/her for analysis. In classification based interactive methods, the decision maker is assumed to give preferences in the form of classifying objectives at the current Pareto optimal solution into different classes indicating how the values of the objectives should be changed to get a more preferred solution. Then, the classification information given is taken into account when new (more preferred) Pareto optimal solution(s) are computed. In the satisficing trade-off method (STOM) three classes are used: objectives whose values 1) should be improved, 2) can be relaxed, and 3) are acceptable as such. In the NIMBUS method, , two additional classes are also used: objectives whose values 4) should be improved until a given bound and 5) can be relaxed until a given bound.

Hybrid methods
Different hybrid methods exist, but here we consider hybridizing MCDM and EMO. A hybrid algorithm in the context of multiobjective optimization is a combination of algorithms/approaches from these two fields (see e.g., ). Hybrid algorithms of EMO and MCDM are mainly used to overcome shorcomings by utilizing strengths. Several types of hybrid algorithms have been proposed in the literature, e.g., incorporating MCDM approaches into EMO algorithms as a local search operator and to lead a DM to the most preferred solution(s) etc. A local search operator is mainly used to enhance the rate of convergence of EMO algorithms.

The roots for hybrid multiobjective optimization can be traced to the first Dagstuhl seminar organized in November 2004 (see, here). Here some of the best minds in EMO (Professor Kalyanmoy Deb, Professor Jürgen Branke etc.) and MCDM (Professor Kaisa Miettinen, Professor Ralph E. Steuer etc.) realized the potential in combining ideas and approaches of MCDM and EMO fields to prepare hybrids of them. Subsequently many more Dagstuhl seminars have been arranged to foster collaboration. Recently, hybrid multiobjective optimization has become an important theme in several international conferences in the area of EMO and MCDM (see e.g., and.

Visualization of the Pareto frontier
Visualization of the Pareto frontier is one of the a posteriori preference techniques of multi-objective optimization. The a posteriori preference techniques (see, for example, ) provide an important class of multi-objective optimization techniques. Usually the a posteriori preference techniques include four steps: (1) computer approximates the Pareto frontier, i.e. the Pareto optimal set in the objective space; (2) the decision maker studies the Pareto frontier approximation; (3) the decision maker identifies the preferred point at the Pareto frontier; (4) computer provides the Pareto optimal decision, which output coincides with the objective point identified by the decision maker. From the point of view of the decision maker, the second step of the a posteriori preference techniques is the most complicated one. There are two main approaches to informing the decision maker. First, a number of points of the Pareto frontier can be provided in the form of a list (interesting discussion and references are given in ). Alternative idea consists in visualizing the Pareto frontier.

Visualization in bi-objective problems: tradeoff curve
In the case of bi-objective problems, informing the decision maker concerning the Pareto frontier is usually carried out by its visualization: the Pareto frontier, often named the tradeoff curve in this case, can be drawn at the objective plane. The tradeoff curve gives full information on objective values and on objective tradeoffs, which inform how improving one objective is related to deteriorating the second one while moving along the tradeoff curve. The decision maker takes this information into account while specifying the preferred Pareto optimal objective point. The idea to approximate and visualize the Pareto frontier was introduced for linear bi-objective decision problems by S.Gass and T.Saaty. This idea was developed and applied in environmental problems by J.L. Cohon. A review of methods for approximating the Pareto frontier for various decision problems with a small number of objectives (mainly, two) is provided in.

Visualization in high-order multi-objective optimization problems
There are two generic ideas how to visualize the Pareto frontier in high-order multi-objective decision problems (problems with more than two objectives). One of them, which is applicable in the case of a relatively small number of objective points that represent the Pareto frontier, is based on using the visualization techniques developed in statistics (various diagrams, etc – see the corresponding subsection below). The second idea proposes the display of bi-objective cross-sections (slices) of the Pareto frontier. It was introduced by W.S. Meisel in 1973 who argued that such slices inform the decision maker on objective tradeoffs. The figures that display a series of bi-objective slices of the Pareto frontier for three-objective problems are known as the decision maps. They give a clear picture of tradeoffs between three criteria. Disadvantages of such an approach are related to two following facts. First, the computational procedures for constructing the bi-objective slices of the Pareto frontier are not stable since the Pareto frontier is usually not stable. Secondly, it is applicable in the case of only three objectives. In the 1980s, the idea W.S. Meisel of implemented in a different form – in the form of the Interactive Decision Maps (IDM) technique.

Interactive decision maps technique
The IDM technique is based on approximating the Edgeworth-Pareto Hull (EPH) of the feasible objective set, that is, the feasible objective set broadened by the objective points dominated by it. Alternatively, this set is known as Free Disposal Hull. It is important that the EPH has the same Pareto frontier as the feasible objective set, but the bi-objective slices of the EPH look much simpler. The frontiers of bi-objective slices of the EPH contain the slices of the Pareto frontier. It is important that, in contrast to the Pareto frontier itself, the EPH is usually stable in respect to disturbances of data. The IDM technique applies fast on-line display of bi-objective slices of the EPH approximated in advance.

Since the bi-objective slices of the EPH for two selected objectives are extending (or shrinking) monotonically, while the value of one of the other objectives (the “third” objective) changes monotonically, the frontiers of the slices of the EPH, for which the values only of the “third” objective changes, do not intersect. This is why a figure with superimposed bi-objective slices of the EPH looks like an ordinary topographical map and is named the decision map, too. To study influence of the other (fourth, fifth, etc.) objectives, one can use animation of the decision maps. Such animation is possible due to the preliminary approximating the EPH. Alternatively, one can study various collections of snap-shots of the animation. Computers can visualize the Pareto frontier in the form of decision maps for linear and nonlinear decision problems for three to about eight objectives. Computer networks are able to bring, for example, Java applets that display graphs of the Pareto frontiers on request. Real-life applications of the IDM technique are described in.

Illustration of the IDM technique
The above figure represents a gray scale copy of a color computer display for a real-life water quality problem involving five objectives. The decision map consists of four superimposed bi-objective differently colored slices. A palette shows the relation between the values of the “third” objective and colors. Two scroll-bars are related to the values of the fourth and the fifth objectives.

A movement of a scroll-bar results in a change of the decision map. One can move the slider manually. However, the most effective form of displaying information to the DM is based on an automatic movement of the slider, that is, on a gradual increment (or decrement) in the constraint imposed on the value of an objective. A fast replacement of the decision maps offers the effect of animation. Because any reasonable number of scroll-bars can be located on the display, one can explore the influence of the fourth, the fifth (and maybe even the sixth and the seventh etc.) objectives on the decision map.

Approximating the EPH
The EPH must be approximated in the IDM technique before the decision maps are displayed. Methods for approximating the EPH depend on the convexity properties of the EPH. Approximation methods are typically based either on approximation of the EPH by a convex polyhedral set or on approximation of the EPH by a large but finite number of domination cones in objective space with vertices that are close to the Pareto frontier. The first form can be applied only in the convex problems, while the second form is universal and can be used in general nonlinear problems

Approximation and visualization in the case of convex EPH
The EPH approximated by a polyhedral set is described by a system of a finite number of linear inequalities, which must be constructed by the approximation technique. Mathematical theory of optimal polyhedral approximation of convex bodies was developed during recently, and its results can be applied for developing the effective methods for approximating the EPH (see details in ). A large number of bi-objective slices of such approximations can be computed and displayed in the form of a decision map in several seconds.

Point-wise approximation of the Pareto frontier and its visualization
An EPH approximation by a large but finite number of domination cones can be constructed on the basis of any point-wise approximation of the Pareto frontier, which can be found by using a broad range of techniques from classic single-objective optimization methods up to modern evolutionary methods Hybrid methods for approximating the EPH based on combination of classic and evolutionary methods can be used, too. The bi-objective slices of such approximation can be computed very fast as well. Application of these methods results in decision maps that look fairly understandable if the number of approximating points is sufficiently large.

Search for the preferred decision
In the IDM technique, search for the preferred decision is based on identification of a preferred Pareto optimal objective point (feasible goal). Decision maps help the user to identify the goal directly at a tradeoff curve drawn at the computer display. Then, a Pareto optimal decision associated with the goal is found automatically.

Detailed discussion of the Pareto frontier visualization problems is provided in.

Multiobjective optimization software
The International MCDM society keeps up a list of software related to MCDM. Weierstroff et al. have written a book chapter on multiobjective optimization software.