Multiobjective Optimization (also known as multiobjective programming, vector optimization, multi-criteria optimization, multi-attribute optimization, or Pareto optimization) is a field of multi-criteria decision-making that deals with mathematical optimization problems involving more than one objective function that can 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 made in the presence of trade-offs between two or more conflicting objectives. Cost minimization while maximizing car purchase comfort and performance maximization while minimizing vehicle fuel consumption and pollutant emissions are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there may be more than three objectives.
The optimal value can be found through the optimization process. Multiobjective Optimization problems involve finding a maximum or minimum value or using a single objective or multiple objectives. Problems that have more than one objective are referred to as multi-objective optimization (MOO). These types of problems are found in everyday life such as mathematics, engineering, economics, social studies, agriculture, aviation, automotive, and many others.
The application of MOO in the economic field can be used to optimize the bioeconomic model of fisheries (Mardle, Pascoe, & Tamiz, 1998). This model can be used as an optimal tool for estimating resource utilization and management plan effectiveness. The basis of the bioeconomic model of fisheries is derived from the open access or public goods theory of economics, which is based on a logistic model of population growth. The case the author studies develops a model for North Sea fisheries with four objectives to consider: maximizing profits, maintaining relatively historical quota shares between countries, maintaining jobs in the industry, and minimizing waste.
In the field of finance (Horn, Nafpliotis, and Goldberg, 1994; Ruspini & Zwir, 1999; Tapia & Coello, 2007; Zwir & Ruspini, 1999) to identify significant patterns of technical analysis in financial time series, the so-called Pareto Genetic Algorithm (NPGA) is used. Two objectives are considered, which are the quality of the matches (a measure of the extent of the financial time series, whether it is ascending, descending, or head-and-shoulders) and area (the size, through a linear function, the length of the described interval). NPGA is used to determine the appropriate sharp interval for downtrends, uptrends, and heads and shoulders.
An application of MOO in politics (Gunasekara, Mehrotra, & Mohan, 2014) is to identify key players who profit from political campaigns. The problems studied were the average eigenvector centrality and the distance between key players and two network models, the Dolphin Network and the Prisoners Network. The selection of the best key player is based on the assumption that the selected key player is needed to perform well together or as a whole.
In the field of mechanics (Jena, 2013; Deb & Datta, 2012), there are shell and tube heat exchangers. It is used to achieve the minimization of total equipment cost including capital investment and annual energy expenditure including pumping using Genetic Algorithm (GA) software (Matlab). At the same time, it is aimed at minimizing the length of the heat exchanger. Multi-objective algorithms search for optimal values of design variables such as pipe OD, OD, and baffle spacing.
Different types of MOO issues are also followed by multiple settlement methods. The global criteria method (Miettinen, 1999; Zeleny, 1973) is used to transform multi-problem optimization into single-problem optimization by minimizing the distance between multiple reference points and viable target regions. The reference point is the ideal solution. By the weighted sum method (Cohon, 1983; Das & Dennis, 1997; Kim & de Weck, 2005; Messac, Sukam, & Melachrinoudis, 2000a; Messac, Sundararaj, Tappet, & Renaud, 2000b; Odu & Charles-Owaba, 2013; Triantaphyllou, Shu, Sanchez and Ray, 1998), all problems are combined into one problem using a weight vector. The numbers of weights are usually normalized to one. Although the weighted sum method is simple and easy to use, there are two fundamental problems. First, it is difficult to choose weights for problems that have different sizes. Therefore, there will be a bias in finding a compromise solution. Second, a problem arises if the plural being optimized is not convex.
The ε-constraint method is used to overcome the difficulties in manifold problems that are not convex. The ε-constraint method (Haimes, Lasdon, & Wismer, 1971) optimizes only one problem while the other problems are transformed into constraints. The vector Îµ is determined and uses a limit (upper limit in case of minimization) for all problems. For certain Îµ vectors, this method finds an optimal solution by optimizing all problems. By changing Îµ we are able to obtain many optimal solutions. The disadvantage of this method is that there is no feasible solution for certain Îµ vectors.
In the lexicographic method (Fishburn, 1974), decision-makers are asked to regulate objective functions by applying their absolute interests. The optimization process is carried out individually for each objective according to the order of importance. If only one solution is returned after optimizing the most important objective (the first objective), then the solution is the optimal solution. Conversely, the optimization will continue with the second objective and with new constraints on the solution obtained from the first objective. This cycle continues until the final destination.
In goal programming (Chang, 2007; Charnes, Clower, & Kortanek, 1967; Charnes & Cooper, 1961; Charnes, Cooper, & Ferguson, 1955; Hokey & James, 1991; Ignizio, 1974; Steuer, 1986), the decision-maker determines the aspiration level of the objective function. The optimization of the objective function with the aspiration level is considered the goal to be achieved.
Multi-objective evolutionary algorithm (MOEA) introduced by Lam & Sameer, 2008, is a stochastic optimization technique. Similar to other optimization algorithms, MOEAs are used to find Pareto optimal solutions for specific problems, but they differ from population-based approaches. Most of the existing MOEAs use the concept of dominance in their actions and some do not. Therefore, we focus here on the MOEA class, which is based on dominance. The optimization mechanism of MOEA is very similar to evolutionary algorithms, except that the dominance relation is used. In detail, at each iteration, an objective value for each individual is calculated and then used to determine the dominance relationship in the population in order to choose a potentially better solution for producing the hereditary population. This population can be combined with parent populations to create populations for the next generation. In addition, the existence of an objective space can provide MOEA with the flexibility to apply some conventional support techniques such as niche.