# Elite Multi-Criteria Decision Making--Pareto Front Optimization in Multi-Objective Optimization
## 1 Introduction  
Most real-world problems are multi-objective (MO) in nature. In an MO problem, a solution or set of solutions are to be found which are optimal for conflicting objective functions. Pareto optimal solutions represent an optimal trade-off between objectives in most real-world applications. In fact, for conflicting objectives where one Pareto optimal solution is known, in order to find another feasible solution that improves in one objective, one must worsen in one or more other objectives. Resource limiting constraints add to the complexity of finding solutions, and for discrete decision variables, this results in a problem that is classified as NP-Hard, where the time to solve to exact optimality grows exponentially as the problem size increases linearly [1]. The main objective of an optimizer technique is to find a set of solutions which are optimal or near optimal for the conflicting objectives under the given constraints. Algorithms with stochastic components are referred as meta-heuristics [2]. Stochastic optimization techniques are often applied in solving black-box optimization problems [3]. Generally, most real-world problems are classified as black-box optimization problems due to uncertainty in the exact definition of the objective functions, constraints, and other uncertainties. In multi-objective optimization (MOO), the search space needs to be explored to find the best solutions for the objective functions. Exploration and exploitation of the search area both need to be performed. Various techniques such as Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) [4], Non-Dominated Sorting Genetic Algorithm-III (NSGA-III) [5], Particle Swarm Optimization (PSO) [6], and Ant Colony Optimization (ACO) [7] have previously been developed to solve such difficult problems, and the new presented technique will be evaluated against those. Some real-world applications of MOO include finance [8], politics [9], mechanics [10], and spatial optimization [11]. On the contrary, if a set of feasible solutions is already provided, then Multi-Criteria Decision Making (MCDM) is used to find the best solution from the given set of solutions. In MCDM, a single best solution is generally selected from the provided set of solutions depending upon the significance of the objective. Various methods for the selection of a solution have previously been developed [12]. A decision maker (DM) first applies MOO techniques and obtains a set of Pareto front solutions. Next, the DM applies MCDM techniques on the set of Pareto front solutions to pick one best solution. Various researchers have previously implemented this process in [13; 14; 15]. An optimizer combining both of these approaches was previously developed, which was called Multi-Criteria Decision Making-Pareto Front (M-PF) optimizer [16]. The main drawback of the M-PF optimizer is its tendency to converge toward one objective to exploit its search area with low probability of exiting from local minimum fitness. Other researchers have tried to combine the MOO and MCDM techniques [17] in which they use predefined objective weights by decision maker (DM). One of the major issues with predefined objective weights is that this results in a bias toward one of the objective functions. In this paper, an improved version of the M-PF optimizer is developed, which is called the elite Multi-Criteria Decision Making-Pareto Front (eMPF) optimizer. The simultaneous objectives of the new optimizer include both exploitation of the search area and also the exploration of a diverse solution set. The main contribution of this paper is to present a better optimizer than the M-PF optimizer with a lower possibility of being stuck in a local optimal fitness. The other contribution is to reduce the time complexity over the M-PF optimizer. In experiments, the new optimizer has better solution quality over the other evaluated test functions. The structure of the subsequent sections of this paper is as follows.