cant freedom and ease for the decision-maker’s self-ex- pression, and requiring a minimal information on his local estimate of the steepest-ascent direction . Zionts and Wallenius (1983) developed a method for interactive multiple objective linear programming by assuming an unknown pseudo concave utility function satisfying cer- tain general properties . Sadagopan and Ravinderan (1986) developed several interactive procedures for solving multiple criteria nonlinear programming prob- lems based on the generalized reduced gradient method for solving single objective nonlinear programming prob- lems . Siegfried (1990) presented an interactive algo- rithm for nonlinear vector optimization problems, after solving only two optimization problems . Kassem (1995) dealt with an interactive stability of multiobjec- tive nonlinear programming problems with fuzzy pa- rameters in the constraints . Aghezzaf and Ouaderh- man (2001) proposed an interactive interior point method for finding the best compromise solution to a multiple objective linear programming problem . Abo-Sinna and Abou-El-Enien (2006) extended the technique of order preference by similarity ideal solution (TOPSIS) for solving large scale multiple objective programming problems involving fuzzyparameters . Luque, Ruiz and Steuer pointed out that many interactive algorithms have two main features: 1) they help a decision maker (DM) learn about a problem while solving it, and 2) they put to work iteratively any new insights gained during the solution process to help the DM navigate to a final solution .
Abstract— Classification techniques are becoming essential in the financial world for reducing risks and possible disasters. Managers are interested in not only high accuracy but also in interpretability and transparency. It is widely accepted now that the comprehension of how inputs and output are related to each other is crucial for taking operative and strategic decisions. Furthermore, inputs are often affected by contextual factors and characterized by a high level of uncertainty. In addition, financial data are usually highly skewed towards the majority class. With the aim of achieving high accuracies, preserving the interpretability and managing uncertain and unbalanced data, the paper presents a novel method to deal with financial data classification by adopting type-2 fuzzy rule-based classifiers (FRBCs) generated from data by a multi-objective evolutionary algorithm (MOEA). The classifiers employ an approach, denoted as scaled dominance, for defining rule weights in such a way to help minority classes to be correctly classified. In particular, we have extended PAES-RCS, an MOEA-based approach to learn concurrently the rule and data bases of FRBCs, for managing both interval type-2 fuzzy sets and unbalanced datasets. To the best of our knowledge, this is the first work that generates type-2 FRBCs by concurrently maximizing accuracy and minimizing the number of rules and the rule length with the objective of producing interpretable models of real-world skewed and incomplete financial datasets. The rule bases are generated by exploiting a rule and condition selection (RCS) approach, which selects a reduced number of rules from a heuristically generated rule base and a reduced number of conditions for each selected rule during the evolutionary process. The weight associated with each rule is scaled by the scaled dominance approach on the fuzzy frequency of the output class, in order to give a higher weight to the minority class. As regards the data base learning, the membership function parameters of the interval type-2 fuzzy sets used in the rules are learned concurrently to the application of RCS. Unbalanced datasets are managed by using, in addition to complexity, selectivity and specificity as objectives of the MOEA rather than only the classification rate. We tested our approach, named IT2-PAES-RCS, on eleven financial datasets and compared our results with the ones obtained by the original
multi-objective optimization of Wire Electrical Discharge Machining parameters, which converts the multi responses into a single fuzzy grade. Based on fuzzy grade, optimal combination of parameters are determined. L27 orthogonal array is used for Design of experiments. Maximum Material removal rate and Minimum surface roughness were chosen as the objectives. In this study Al5052/Sic/Gr Hybrid MMC is considered as the target material for Wire Electrical discharge Machining, because of high corrosion resistance, good mechanical strength and relatively low cost. The process parameters viz., pulse on time, pulse off time, Peak Current and wire feed were optimized with consideration of Grey Relational Grade. The confirmation run, results shows that the better quality is achieved by the optimal combination of process parameters.)
Abstract— Multiobjective evolutionary method is a way to overcome the limitation of the classical methods, by finding multiple solutions within a single run of the solution procedure. The aim of having a solution method for multiobjectiveoptimization problem is to help the decision maker in getting the best solution. Usually the decision maker is not interested in a diverse set of Pareto optimal points. So, it is necessary to incorporate the decision maker’s preference so that the algorithm gives out alternative solutions around the decision maker’s preference. The problem in incorporating the decision maker’s preference is that the decision maker may not have a solid guide line in comparing tradeoffs of objectives. However, it is easy for the decision maker to compare in a fuzzy way. This paper discusses on incorporating a fuzzy tradeoffs in the evolutionary algorithm to zoom out the region where the decision maker’s preference lies. By using test functions it has shown that it is possible to give points in the region on the Pareto front where the decision maker’s interest lies.
The game defined by (2) is called a two-person zero-sum game with fuzzy payoffs. When each of the players chooses a strategy, a payoff for each of them is represented as a fuzzy number, but an outcome of the game has a zero-sum structure such that, when one player receives a gain the order player suffers an equal loss. Assuming that each of the two players has r objectives, the following multiple fuzzy payoff matrices represent a multiobjective two-person zero-sum game with fuzzy payoffs :
The above procedure (3.2) can be used to develop ranking for hexagonal fuzzy numbers. Based on this ranking procedure, a ranking algorithm is developed for a hexagonal fuzzy numbers. Moreover, it is applied to MOLPP under constraints with fuzzy coefficients.
corresponding solution concept called potential solutions. In this paper, we especially focus on hierarchical fuzzymultiobjective linear programming problems where mul- tiple decision makers in a hierarchical organization have fuzzy goals for their own multiple objective linear func- tions together with common linear constraints. In section 2, hierarchical fuzzymultiobjective linear programming problems are formulated and the corresponding solution concept called a generalized Λ-extreme point is intro- duced. In section 3, an interactive algorithm is proposed to obtain the satisfactory solution from among a general- ized Λ-extreme point set, where the corresponding hyper- plane problem [6, 10] is solved. In section 4, interactive processes of the proposed method are demonstrated by means of an illustrative numerical example.
A multiobjective fractional optimization problem (MFP), which consists of more than two fractional objective functions with convex numerator functions and convex denominator functions, finitely many convex constraint functions, and a geometric constraint set, is considered. Using parametric approach, we transform the problem (MFP) into the non-fractional multiobjective convex optimization problem (NMCP) v with parametric v Î ℝ p , and then give the equivalent relation between (weakly) ε - efficient solution of (MFP) and (weakly) ε ¯ -efficient solution of (NMCP) ¯ v . Using the
The water level control system of the steam generator in a pressurized water reactor and its control problem are analysed. In this work, a stable control strategy particularly during low power operation based on the fuzzy control method has studied. The Control strategy employs substitutional information using the bypass valve opening instead of incorrectly measured signal at the low flow rate as the fuzzy variable of the flow rate during low power operation, and includes the flexible scale adjusting method for fast response at a large transient. A self- tuning algorithm based on the control performance and the descent method is also suggested for tuning the membership function scale. It gives a practical way to tune the controller under real operation. Simulation was carried out on the Compact Nuclear Simulator set up at Korea Atomic Energy Research Institute and its results showed the good performance of the controller and effectiveness of its tuning.
It has been well recognized that convergence and diversity are two main but hard-to- balance goals in designing MOEAs. Any bias toward one goal will inevitably aggra- vate the other. In many-objective optimization, the balance between them is still of great importance. However, when handling MaOPs, most MOEAs inherit elitist preservation from their counterparts of multiobjectiveoptimization that emphasizes nondominated so- lutions in the population, resulting in very little room left for diversity maintenance. Even if these MOEAs did not intentionally emphasize convergence, they could not elude the fact that an increasingly large fraction of population becomes nondominated with an in- crease in the number of objectives. In other words, they perform environmental selection in a convergence-first-and-diversity-second manner. As a result, when the MOEAs are ap- plied to many-objective optimization, there will be a large number of nondominated indi- viduals after the convergence-first selection, and diversity preservation will be performed only on the nondominated individuals. Correspondingly, some regions occupied by domi- nated individuals will be scarcely explored, and diversity preservation becomes of limited use in this case. In contrast, SPEA/R adopts a DFCS strategy to perform environmental selection, at an attempt to maximize population diversity and strengthen exploitation in less-converged regions during the search. Our experiments have shown its promise for both multiobjective and many-objective optimization.
Inspiration by nature has been a creative force for dealing with optimization algorithm design. Apart from biological evolution, many other natural phenomena have been con- sidered. While many of these algorithmic ideas have so far remained in a somewhat experimental and immature state, some non-evolutionary bio-inspired optimization algo- rithms have gained maturation and competitive perfor- mance. Among others, this seems to hold for particle swarm optimization (Reyes-Sierra and Coello Coello 2006), ant colony optimization (Bara´n and Schaerer 2003), and artificial immune systems Coello Coello and Corte´s (2005). As with evolutionary algorithms, also these algo- rithms have first been developed for single-objective opti- mization, and subsequently, they have been generalized to multiobjectiveoptimization. Moreover, there is some recent research on bio-inspired techniques that are specif- ically developed for multiobjectiveoptimization. An example of such a development is the Predator-Prey Evo- lutionary Algorithm, where different objectives are repre- sented by different types of predators to which the prey (solutions) have to adapt (Laumanns et al. 1998; Grimme and Schmitt 2006).
The no-preference methods assume no information about the importance of objec- tives, but a heuristic is used to find a single optimal solution. It is important to note that although no preference information is used, these methods do not make any at- tempt to find multiple Pareto-optimal solutions. Posteriori methods do use preference information on each objective and iteratively generate a set of Pareto-optimal solutions. The classical method of generating Pareto optimal solutions requires some knowledge of the algorithmic parameters that will guarantee the finding of a Pareto-optimal solution. On the other hand, A priori methods use more information about the preferences of objectives and usually find one preferred Pareto-optimal solution. Interactive meth- ods use the preference information progressively during the optimization process as the decision-maker interacts with the optimization program during the optimization process. Typically the system provides an updated set of solutions and lets the decision-maker consider whether or not to change the weighting of individual objective functions.
Contrarily to the majority of classical methods, evolutionary algorithms are population- based optimization techniques. This feature makes them particularly suitable for solving multiobjectiveoptimization problems. The main aspects to be taken into consideration when implementing MOEAs are: (i) fitness assignment, (ii) diversity preservation, and (iii) elitism. Although the discussed algorithms are organized on the basis of their variation operators, one can trace how each of these aspects was included into MOEAs since the first pioneering studies appeared in the mid 1980s. Regarding the fitness assignment, one can distinguish the following most popular techniques: dominance-based, scalarizing-based, and indicator-based fitness assignment. Furthermore, diversity of solutions within the current Pareto set approximation is maintained by incorporating density information into the selection process of EMO algorithms. Some widely used diversity preserving methods are: (i) fitness sharing, (ii) hypergrid, (iii) clustering, (iv) nearest neighbor, (v) crowding distance. Recently, the use of quality indicators as the diversity estimator has become very popular. Elitism is addressed to the issue of maintaining good solutions found during the optimization process. The elite preserving operator is usually included in MOEAs to make them better convergent to the Pareto front. One way to deal with this problem is to combine the parent and offspring populations and to apply selection procedure to select a new population. Alternatively, the external archive can be maintained where the promising solutions are stored during the search.
The type of algorithm and implementation clearly affects the optimization process. Among the algorithms, and of particular interest, is the Non-dominated Sorting Genetic Algorithm-2 (NSGA-2), as shown in (Deb, 2000, Deb, 2001, Deb, 2002). The differences between NSGA-2 and other MOGAs include methods of working with dominated solutions and consequent crossover between solutions. NSGA-2 preserves a number of individuals with the highest fitness onto the next iterations, making it an elitist MOGA. It uses sorting to compare performances between multiple dimensions of the individuals in the population. Its complexity in sorting through the solutions is given as ( ), where n is the number of objectives and z is the size of the population (Deb, 2002). Furthermore, it also includes niching techniques that preserve design diversity (ibid).
In this paper, we utilize the concept of multiobjectiveoptimization and provide a general framework for a specific class of applications in WSNs. It is well known that the TCP/IP reference model, one of the most popular layered architectures, divides the whole architecture into five layers (modules), and each layer only communicates with the layers next to it, while recent work on the NUM approach divides the architectures according to applications. We first list all the constraints and objectives which the applications may have. Then the network architecture is divided into n modules, each of which has several interface variables with other modules. In this way, we can inherit the advantages of both the layered architectures (as we have fixed modules) and the NUM approach (as each module can communicate with other modules). We illustrate this in Figure 1, in which λ and μ are the interface variables (see Section 3 for detailed definition of λ and μ ), and O i is the objective vector function
On the other hand, the equilibrium problem provides a general mathematical model for a wide range of practical problems, such as optimization problems, Nash equilibria prob- lems, ﬁxed point problems, variational inequality problems, and complementarity prob- lems, and has been investigated intensively. For more details, we refer to [12–15]. As a particular case of the vector equilibrium problem, multiobjectiveoptimization problems arise in a large number of applications such as transportation, ﬁnance, communication, etc. Naturally, the issue of uncertain data aﬀects single objective optimization problems in the same way as it aﬀects these multiobjective ones. The essential problem in multiob- jective optimization is to ﬁnd the Pareto eﬃcient solutions, meaning the feasible solutions
We study first- and second-order necessary and suﬃcient optimality conditions for approximate weakly, properly eﬃcient solutions of multiobjectiveoptimization problems. Here, tangent cone, -normal cone, cones of feasible directions, second-order tangent set, asymptotic second-order cone, and Hadamard upper lower directional derivatives are used in the characterizations. The results are first presented in convex cases and then generalized to nonconvex cases by employing local concepts.
In this paper, an exact method for generating all eﬃcient solutions for multiple objective integer linear fractional programming problems is presented The method does not require any nonlinear optimization. A linear fractional program is solved using the Cambini and Martein’s algorithm in the original format and then by using the well-known concept of branching in integer linear programming, integer solutions are generated. The proposed eﬃcient cut exploits all the criteria in the simplex table, and only the parts of the feasible solutions domain containing eﬃcient solutions are explored. Also it is easy to implement the proposed cut since to obtain integer solution x k 1 from x k , one has just to append the cut in the simplex
A multi-objective evolutionary algorithm approach for determination of optimum pole shape of SRM is proposed in this paper , considering average torque, torque ripple and copper loss as objectives. The application of elitist Non- dominated Sorting Genetic Algorithm version II (NSGAII) to determine optimum pole shape design for performance enhancement of Switched Reluctance Machine (SRM) is presented. In SRM, torque output and torque ripple are sensitive to stator and rotor pole arcs and their selection is a vital part of SRM design process. The problem of determining optimal pole arc is formulated as a multi- objective optimization problem and the Finite Element Method (FEM) is used to determine the performance of the machine. NSGA-II is used in the search for Pareto solutions and the proposed optimization technique is applied to determine optimal pole shape of an 8/6, four-phase, 5 HP, 1500 rpm SRM. The Pareto fronts obtained using the proposed approach is in close agreement with the fronts obtained using weighted sum method. The results indicate that the optimization algorithm has yielded new motor designs improving the three objectives considered. Analyzing the performance of the machine using FEA confirm the application of NSGA-II to determine various viable pole shape designs for performance enhancement of SRM. The results show the effectiveness of the proposed approach and confirm the application of NSGA-II as a promising tool for solving SRM design problems. The results obtained by NSGA-II are compared and validated with classical multi- objective approach based on weighted sum method using Differential Evolution (DE) algorithm.
The rotor geometry is defined in Fig. 1 for an example motor with two pole-pairs and 5 layers. The set of parameters used to describe the rotor geometry has a critical role in the design optimization. The barrier shape types in the literature are various and full of parameters and a strong simplification is needed here to keep the optimization process as simple as possible. Each variable should have a reasonable impact on at least one of the performance indexes of the optimization. It was demonstrated in  that circular barriers, with two input variables per layer can get close to the performance of more complex geometries despite their very basic set of parameters. For circular layers the geometric inputs are: