scholarly journals Multiobjective Optimization, Scalarization, and Maximal Elements of Preorders

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Paolo Bevilacqua ◽  
Gianni Bosi ◽  
Magalì Zuanon
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problem ◽  
Design Space ◽  
Pareto Optimal ◽  
Multiobjective Optimization Problem ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Maximal Elements ◽  
Compact Design

We characterize the existence of (weak) Pareto optimal solutions to the classical multiobjective optimization problem by referring to the naturally associated preorders and their finite (Richter-Peleg) multiutility representation. The case of a compact design space is appropriately considered by using results concerning the existence of maximal elements of preorders. The possibility of reformulating the multiobjective optimization problem for determining the weak Pareto optimal solutions by means of a scalarization procedure is finally characterized.

Multiobjective optimization of the flaxseed mucilage extraction process using normal-boundary intersection approach

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mariana Souza Rocha ◽  
Luiz Célio Souza Rocha ◽  
Marcia Barreto da Silva Feijó ◽  
Paula Luiza Limongi dos Santos Marotta ◽  
Samanta Cardozo Mourão
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Method ◽  
Extraction Process ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Fiber Concentration ◽  
Trade Off ◽  
Content Type ◽  
Normal Boundary

PurposeThe mucilage of the Linum usitatissimum L. seed (Linseed) is one of the natural mucilages that presents a great potential to provide a food hydrocolloid with potential applications in both food and pharmaceutical industries. To increase the yield and quality of linseed oil during its production process, it is necessary to previously extract its polysaccharides. Because of this, flax mucilage production can be made viable as a byproduct of oil extraction process, which is already a product of high commercial value consolidated in the market. Thus, the purpose of this work is to optimize the mucilage extraction process of L. usitatissimum L. using the normal-boundary intersection (NBI) multiobjective optimization method.Design/methodology/approachCurrently, the variables of the process of polysaccharide extraction from different sources are optimized using the response surface methodology. However, when the optimal points of the responses are conflicting it is necessary to study the best conditions to achieve a balance between these conflicting objectives (trade-offs) and to explore the available options it is necessary to formulate an optimization problem with multiple objectives. The multiobjective optimization method used in this work was the NBI developed to find uniformly distributed and continuous Pareto optimal solutions for a nonlinear multiobjective problem.FindingsThe optimum extraction point to obtain the maximum fiber concentration in the extracted material was pH 3.81, temperature of 46°C, time of 13.46 h. The maximum extraction yield of flaxseed was pH 6.45, temperature of 65°C, time of 14.41 h. This result confirms the trade-off relationship between the objectives. NBI approach was able to find uniformly distributed Pareto optimal solutions, which allows to analyze the behavior of the trade-off relationship. Thus, the decision-maker can set extraction conditions to achieve desired characteristics in mucilage.Originality/valueThe novelty of this paper is to confirm the existence of a trade-off relationship between the productivity parameter (yield) and the quality parameter (fiber concentration in the extracted material) during the flaxseed mucilage extraction process. The NBI approach was able to find uniformly distributed Pareto optimal solutions, which allows us to analyze the behavior of the trade-off relationship. This allows the decision-making to the extraction conditions according to the desired characteristics of the final product, thus being able to direct the extraction for the best applicability of the mucilage.

scholarly journals Evaluation Method of Multiobjective Functions’ Combination and Its Application in Hydrological Model Evaluation

2020 ◽  
Vol 2020 ◽  
pp. 1-23 ◽  
Author(s):  
Jiuyuan Huo ◽  
Liqun Liu
Keyword(s):  
Objective Function ◽  
Parameter Optimization ◽  
Optimization Problem ◽  
Hydrological Model ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Hydrological Forecasting ◽  
Model Parameter Optimization

Parameter optimization of a hydrological model is intrinsically a high dimensional, nonlinear, multivariable, combinatorial optimization problem which involves a set of different objectives. Currently, the assessment of optimization results for the hydrological model is usually made through calculations and comparisons of objective function values of simulated and observed variables. Thus, the proper selection of objective functions’ combination for model parameter optimization has an important impact on the hydrological forecasting. There exist various objective functions, and how to analyze and evaluate the objective function combinations for selecting the optimal parameters has not been studied in depth. Therefore, to select the proper objective function combination which can balance the trade-off among various design objectives and achieve the overall best benefit, a simple and convenient framework for the comparison of the influence of different objective function combinations on the optimization results is urgently needed. In this paper, various objective functions related to parameters optimization of hydrological models were collected from the literature and constructed to nine combinations. Then, a selection and evaluation framework of objective functions is proposed for hydrological model parameter optimization, in which a multiobjective artificial bee colony algorithm named RMOABC is employed to optimize the hydrological model and obtain the Pareto optimal solutions. The parameter optimization problem of the Xinanjiang hydrological model was taken as the application case for long-term runoff prediction in the Heihe River basin. Finally, the technique for order preference by similarity to ideal solution (TOPSIS) based on the entropy theory is adapted to sort the Pareto optimal solutions to compare these combinations of objective functions and obtain the comprehensive optimal objective functions’ combination. The experiments results demonstrate that the combination 2 of objective functions can provide more comprehensive and reliable dominant options (i.e., parameter sets) for practical hydrological forecasting in the study area. The entropy-based method has been proved that it is effective to analyze and evaluate the performance of different combinations of objective functions and can provide more comprehensive and impersonal decision support for hydrological forecasting.

scholarly journals A Quantum Adiabatic Algorithm for Multiobjective Combinatorial Optimization

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 32 ◽  
Author(s):  
Benjamín Barán ◽  
Marcos Villagra
Keyword(s):  
Combinatorial Optimization ◽  
Multiobjective Optimization ◽  
Theorem Proving ◽  
Optimization Problems ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Quantum Annealing ◽  
Multiobjective Optimization Problems ◽  
First Time

In this work we show how to use a quantum adiabatic algorithm to solve multiobjective optimization problems. For the first time, we demonstrate a theorem proving that the quantum adiabatic algorithm can find Pareto-optimal solutions in finite-time, provided some restrictions to the problem are met. A numerical example illustrates an application of the theorem to a well-known problem in multiobjective optimization. This result opens the door to solve multiobjective optimization problems using current technology based on quantum annealing.

New Dynamic Multi-Objective Constrained Optimization Evolutionary Algorithm

2015 ◽  
Vol 32 (05) ◽  
pp. 1550036 ◽  
Author(s):  
Chun-An Liu ◽  
Yuping Wang ◽  
Aihong Ren
Keyword(s):  
Constrained Optimization ◽  
Evolutionary Algorithm ◽  
Optimization Problem ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Constrained Optimization Problem ◽  
Multi Objective ◽  
Time Period ◽  
Pareto Fronts

For dynamic multi-objective constrained optimization problem (DMCOP), it is important to find a sufficient number of uniformly distributed and representative dynamic Pareto optimal solutions. In this paper, the time period of the DMCOP is first divided into several random subperiods. In each random subperiod, the DMCOP is approximately regarded as a static optimization problem by taking the time subperiod fixed. Then, in order to decrease the amount of computation and improve the effectiveness of the algorithm, the dynamic multi-objective constrained optimization problem is further transformed into a dynamic bi-objective constrained optimization problem based on the dynamic mean rank variance and dynamic mean density variance of the evolution population. The evolution operators and a self-check operator which can automatically checkout the change of time parameter are introduced to solve the optimization problem efficiently. And finally, a dynamic multi-objective constrained optimization evolutionary algorithm is proposed. Also, the convergence analysis for the proposed algorithm is given. The computer simulations are made on four dynamic multi-objective optimization test functions and the results demonstrate that the proposed algorithm can effectively track and find the varying Pareto optimal solutions or the varying Pareto fronts with the change of time.

AN INTELLIGENT DECISION SUPPORT BASED ON A SUBTRACTIVE CLUSTERING AND FUZZY INFERENCE SYSTEM FOR MULTIOBJECTIVE OPTIMIZATION PROBLEM IN SERIOUS GAME

2011 ◽  
Vol 10 (05) ◽  
pp. 793-810 ◽  
Author(s):  
I. G. P. ASTO BUDITJAHJANTO ◽  
HAJIME MIYAUCHI
Keyword(s):  
Decision Support ◽  
Multiobjective Optimization ◽  
Optimization Problem ◽  
Fuzzy Inference ◽  
Subtractive Clustering ◽  
Multiobjective Optimization Problem ◽  
Optimal Solutions ◽  
Inference System ◽  
Intelligent Decision ◽  
Intelligent Decision Support

Learning decision making through playing a game is an interesting activity for the decision maker or player. In this paper, a multiobjective optimization problem for economic and emission dispatch in which the player can learn about the tradeoff between fuel cost (economic) and emission problems to achieve optimal decisions is considered. A nonplayer character (NPC) is an entity that is built to provide intelligent decision support for the player. The proposed approach is carried out in two stages for the NPC module: the first stage uses the nondominated sorting genetic algorithm II method to solve the multiobjective optimization problem; this stage produces some optimal solutions. The next stage uses subtractive clustering to cluster optimal solutions; furthermore, these clusters are used to build a fuzzy inference system based on the Mamdani type. In this stage, players can select the best decision offered by the NPC.

A Newton method for capturing Pareto optimal solutions of fuzzy multiobjective optimization problems

2019 ◽  
Vol 53 (3) ◽  
pp. 867-886
Author(s):  
Mehrdad Ghaznavi ◽  
Narges Hoseinpoor ◽  
Fatemeh Soleimani
Keyword(s):  
Multiobjective Optimization ◽  
Newton Method ◽  
Optimization Problems ◽  
Order Relation ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Multiobjective Optimization Problems ◽  
Generalized Hukuhara Differentiability

In this study, a Newton method is developed to obtain (weak) Pareto optimal solutions of an unconstrained multiobjective optimization problem (MOP) with fuzzy objective functions. For this purpose, the generalized Hukuhara differentiability of fuzzy vector functions and fuzzy max-order relation on the set of fuzzy vectors are employed. It is assumed that the objective functions of the fuzzy MOP are twice continuously generalized Hukuhara differentiable. Under this assumption, the relationship between weakly Pareto optimal solutions of a fuzzy MOP and critical points of the related crisp problem is discussed. Numerical examples are provided to demonstrate the efficiency of the proposed methodology. Finally, the convergence analysis of the method under investigation is discussed.

scholarly journals MODIFICATION OF FIREWORKS METHOD FOR MULTIOBJECTIVE OPTIMIZATION BASED ON NON-DOMINATED SORTING

2019 ◽  
Vol 22 (3) ◽  
pp. 67-78
Author(s):  
A. V. Panteleev ◽  
A. U. Krychkov
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Complex Structure ◽  
Optimal Solution ◽  
Future Research ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Equal Importance ◽  
Approximate Result

The article suggests a modification for numerical fireworks method of the single-objective optimization for solving the problem of multiobjective optimization. The method is metaheuristic. It does not guarantee finding the exact solution, but can give a good approximate result. Multiobjective optimization problem is considered with numerical criteria of equal importance. A possible solution to the problem is a vector of real numbers. Each component of the vector of a possible solution belongs to a certain segment. The optimal solution of the problem is considered a Pareto optimal solution. Because the set of Pareto optimal solutions can be infinite; we consider a method for finding an approximation consisting of a finite number of Pareto optimal solutions. The modification is based on the procedure of non-dominated sorting. It is the main procedure for solutions search. Non-dominated sorting is the ranking of decisions based on the values of the numerical vector obtained using the criteria. Solutions are divided into disjoint subsets. The first subset is the Pareto optimal solutions, the second subset is the Pareto optimal solutions if the first subset is not taken into account, and the last subset is the Pareto optimal solutions if the rest subsets are not taken into account. After such a partition, the decision is made to create new solutions. The method was tested on well-known bi-objective optimization problems: ZDT2, LZ01. Structure of the location of Pareto optimal solutions differs for the problems. LZ01 have complex structure of Pareto optimal solutions. In conclusion, the question of future research and the issue of modifying the method for problems with general constraints are discussed.

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