Set Of Noninferior Solutions Research Articles

Managing approximation models in multiobjective optimization

In engineering problems, computationally intensive high-fidelity models or expensive computer simulations hinder the use of standard optimization techniques because they should be invoked repeatedly during optimization, despite the tremendous growth of computer capability. Therefore, these expensive analyses are often replaced with approximation models that can be evaluated at considerably less effort. However, due to their limited accuracy, it is practically impossible to exactly find an actual optimum (or a set of actual noninferior solutions) of the original single (or multi) objective optimization problem. Significant efforts have been made to overcome this limitation. The model management framework is one of such endeavours. The approximation models are sequentially updated during the iterative optimization process in such a way that their capability to accurately model the original functions especially in the region of our interests can be improved. The models are modified and improved by using one or several sample points generated by making a good use of the predictive ability of the approximation models. However, these approaches have been restricted to a single objective optimization problem. It seems that there is no reported management framework that can handle a multi-objective optimization problem. This paper will suggest strategies that can successfully treat not only a single objective but also multiple objectives by extending the concept of sequentially managing approximation models and combining this extended concept with a genetic algorithm which can treat multiple objectives (MOGA). Consequently, the number of exact analyses required to converge to an actual optimum or to generate a sufficiently accurate Pareto set can be reduced considerably. Especially, the approach for multiple objectives will lead to a surprising reduction in number. We will confirm these effects through several illustrative examples.

Multiobjective shape design of an inductor for transverse-flux heating of metal strips

Transverse flux induction heating is a process advantageously applied for heat treatment of thin nonferrous metal strips. In general, the requirements for a good inductor design are in conflict with each other. In the paper, an optimization procedure based on evolutionary strategy is used. In order to obtain a set of noninferior solutions, the Pareto equilibrium is applied. The solutions are obtained by linking the evolutionary strategy with an ad hoc code for the analysis of electromagnetic-thermal problems in three dimensions.

A novel approach to real-time economic emission power dispatch

This paper describes a novel approach that combines an abductive reasoning network (ARN) and a technique for order preference by similarity to an ideal solution (TOPSIS) decision approach to achieve real-time economic emission power dispatch and the best compromise solution. The objectives of fuel cost and the environmental impact of emission are simultaneously considered in this paper. The proposed ARN handles complicated relationships between the load demands (input) and the generation power of each unit (output) using a hierarchical network with several layers of function nodes of simple low-order polynomials to make the computed outputs fit the historical data. Once the ARN is constructed, the desired outputs can be produced as soon as the inputs are given. According to the set of noninferior solutions for a specific load level, the TOPSIS approach is used to provide operators with the best compromise solution. The effectiveness of the proposed approach has been demonstrated by the IEEE 30-bus 6-generator and the practical Taipower 388-bus 27-generator test systems. The test results reveal that the proposed ARN outperforms the artificial neural networks (ANNs) method, in both developing the model and estimating the outputs of the generating units according to the input load demands.

Useful multi-objective hybrid evolutionary approach to optimal power flow

Optimal power flow (OPF) in electric power systems is a static, non-linear, multi-objective optimisation problem of determining the optimal settings of control variables for minimising the cost of generation, emissions, transmission losses and voltage and power flow deviations. OPF is an important problem in power systems operation not only due to operational security considerations but also because even a small saving per hour translates to a large annual saving. The solution of the OPF problem, with a simultaneous and adequate consideration of all its facets within reasonable computing time, is still to be achieved. A multi-objective hybrid evolutionary strategy (MOHES) is presented for the solution of the comprehensive model for OPF. The hybridisation of GA with SA effects a beneficial synergism of both. MOHES concentrates on the ‘better’ areas of the search space. The greater modelling power of the method enables representation of non-linear and discontinuous functions and discrete variables easily without involving approximations, and its enhanced search capabilities lead to better solutions. A complete set of non-inferior solutions representing the trade-off between various objectives is provided in a single run. MOHES has been designed to use the small perturbation analysis to avoid computing a complete load flow in every fitness evaluation. This results in considerable savings in computational expense. Test results provided on standard systems reported in the literature clearly indicate its efficacy.

A Novel Approach to Real-Tine Economic Eimission Power Dispatch

This paper describes a novel approach that combines abductive reasoning networks (ARN) and the technique for order preference by similarity to ideal solution (TOPSIS) decision approach to achieve real-time economic emission power dispatch and the best compromise solution. The objectives of fuel cost and environmental impact of emission are considered simultaneously. The proposed ARN handles complicated relationships between the load demands (input) and the generation power of each unit (output) using a hierarchical network with several layers of function nodes of simple low-order polynomials to make the computed outputs fit the historical data. Once the ARN is constructed, the desired outputs can be produced as soon as the inputs are given. According to the set of noninferior solutions for a specific load level, the TOPSIS approach is used to provide operators with the best compromise solution. The effectiveness of the proposed approach has been demonstrated by the IEEE 30-bus 6-generator and the practical Taipower 388-bus 27-generator test systems. The test results reveal that the proposed ARN outperforms the artificial neural network (ANN) method in both developing the model and estimating the outputs of the generating units according to the input load demands.

Combined wavelet-based networks and game-theoretical decision approach for real-time power dispatch

This paper proposes a novel approach combining wavelet-based networks and game-theoretical decision approach to reach the terms real-time power dispatch and the best compromise solution. The goals considered are both fuel cost and environment impact of NO/sub x/ emission. The wavelet-based networks, evolved by an evolutionary computing algorithm, are composed of 3-layer structures, which contain the wavelet, weighting, and summing nodes. The parameters of translation and dilation in the wavelet nodes and the weighting factors in the weighting nodes are tuned to make the computed outputs fit the historical data. Once the networks are trained properly, the desired outputs can be produced as soon as the inputs are given. Based on the set of noninferior solutions for a certain load level, a game-theoretical approach is relied on to provide operators the best compromise solution. The effectiveness of the proposed approach has been demonstrated by the IEEE 30-bus 6-generator test system. Comparisons of learning performances are made to the existing artificial neural networks (ANNs) method.

Combined Wavelet-Based Networks and Game-Theoretical Decision Approach for Real-Time Power Dispatch

This paper proposes a novel approach combining wavelet-based networks and the game-theoretical decision approach to reach the terms of real-time power dispatch and the best compromise solution. Both fuel cost and environmental impact of NO emission are considered. The wavelet-based networks, evolved by an evolutionary computing algorithm, are composed of three-layer structures that contain the wavelet weighting and summing nodes. The parameters of translation and dilation in the wavelet nodes and the weighting factors in the weighting nodes are tuned to make the computed outputs fit the historical data. Once the networks are trained properly, the desired outputs can be produced as soon as the inputs are given. Based on the set of noninferior solutions for a certain load level, a game-theoretical approach is relied on to provide operators with the best compromise solution. The effectiveness of the proposed approach has been demonstrated by the IEEE 30-bus six-generator test system. Comparisons of learning performances are made to the existing artificial neural network method.

Multiobjective optimal load flow based on ideal distance minimization in 3D space

In this paper, three objectives of Multiobjective Optimal Load Flow (MOLF) problem — cost of generation, system transmission loss and pollution — are considered. The MOLF problem is formulated as a multiobjective optimization problem using weighting method and a number of noninferior solutions are generated in 3D space. The optimal power system operation is attained by Ideal Distance Minimization method. This method employs the concept of an ‘Ideal Point’ (IP) to scalarize the problems having multiple objectives and it minimizes the Euclidean distance between IP and the set of noninferior solutions. This method has been applied to three IEEE standard systems.

Multiobjective optimal design of interior permanent magnet synchronous motors considering improved core loss formula

This paper describes the optimal design of interior permanent magnet synchronous motors for which two objective functions regarding motor efficiency and weight are used. Multiobjective optimization technique is applied to finding the optimal solution in this case. An optimal design method that determines both the noninferior solution set and the best compromise solution employing a modified genetic algorithm is proposed. The proposed algorithm adopts the structure of the conventional genetic algorithm, but fitness value and convergence criterion are redefined and some major parameters of the algorithm are adjusted to the multiobjective optimization. In order to predict the motor performance more accurately, a core loss formula is derived considering the flux variation due to the stator currents as well as that due to the magnet.

An integrated logical and physical design flow for deep submicron circuits

This paper presents a set of techniques and a new design flow to be used in the synthesis of high-performance deep-submicron logic circuits. The design flow consists of circuit partitioning into tree like clusters, floorplanning, global routing, and timing analysis/budgeting steps, followed by simultaneous technology mapping and linear placement of each cluster. The strength of this approach lies in the dynamic programming-based algorithms used in performing simultaneous technology mapping and linear placement of the logic clusters. The two algorithms we propose, one for exact total (gate plus routing) area minimization and the other for total (gate plus routing) delay minimization, generate a set of noninferior solutions for each cluster enabling designers to perform tradeoffs between total-area and total-delay. Experimental results on large benchmarks prove the effectiveness of the proposed flow.

CONTAINER PORT LOCATION STRATEGY BASED ON DOMESTIC PORT CHOICE MODELING AND OPTIMAL LINER ROUTING APPROACH

The focal point of the paper is the introduction and application of logit model in port choice modeling and double integer formulation in optimal liner routing approach. In port choice modeling, the model specification without port specific intercepts and six effective variables provides the suitable results (parameter estimation). The choice probability results (observed and predicted) to determined ports selection potential and demarcation of hinterland were both validated. In liner routing modeling, the resultant route choice and the optimum cost for all different vessel sizes were determined. The model also proves beneficial from the standpoint of attracting foreign vessels by providing non-inferior solution sets (relationship between carrier's cost and shipper's cost-trade off). The paper also proposed a new port planning framework concept base on location strategy by integrating both models. Transportation cost, frequency and volume, are the port operators control variables.

Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method

The convexification of a noninferior frontier can be achieved in an appropriate equivalent objective space for general nonconvex multiobjective optimization problems. Specifically, this paper proves that taking the exponentials of the objective functions can act as a convexification scheme. This convexification scheme further leads to the exponential generating method that guarantees the identification of the entire set of noninferior solutions.

A Parametric Mixed-Integer Optimization Algorithm for Multiobjective Engineering Problems Involving Discrete Decisions

An iterative, decomposition-based algorithm is proposed in this paper, for the solution of convex parametric MINLP problems and, in extension, the identification of the noninferior solution set in multiobjective problems involving continuous and discrete decisions. The parametric optimal solution is constructed via an upper- and lower-bound procedure. Parametric upper bounds are identified with the solution of parametric NLP problems, while the parametric lower bound is updated in each iteration via the solution of a parametric MILP Master problem, which involves only the binary variables of the initial problem. Convergence properties and computational requirements are discussed in example problems from process synthesis under uncertainty and simultaneous product/process design.

A multi-modal approach to the location of a rapid transit line

The location of a rapid transit line (RTL) represents a very complex decision problem because of the large number of decision makers, unquantifiable criteria and uncertain data. In this context Operational Research can help in the design process by providing tools to generate and assess alternative solutions. For this purpose two bicriterion mathematical programming models — the Maximum Coverage Shortest Path model and the Median Shortest Path model — have been developed in the past. In this paper a new bicriterion model, which can evaluate in a more realistic way the attractivity of an RTL is introduced. To calculate an estimation of the non-inferior solution set of the problem, a procedure based on a k-shortest path algorithm was developed. This approach was applied to a well-known sample problem and the results are discussed and compared with those obtained using a Median Shortest Path model.

Bi-objective dynamic programming for trading off hydropower and irrigation

Reservoir management is examined from the bicriterion viewpoints of hydropower generation and irrigation. Specifically, a decision support system is developed which includes constructing the set of non-inferior solutions, using the constraint, penalty and weighting methods. The reservoir operation model, as well as the optimization and simulation software, were initially provided by the R & D Division of the French Electrical Utility (Electricité de France); the method is based on a Stochastic Dynamic Programming—SDP—model with daily time steps, using 50 annual values of daily historical inflows and marginal electricity production costs. The water requirement for irrigation is taken either as preset using a demand curve, or as a soil reservoir model based on SDP and running simultaneously with the reservoir model. The model yields trade-off curves of electrical production losses versus, respectively, irrigated surface area, crop yield, and agricultural benefits. A numerical application illustrates the methodology in the French case of the Bort reservoir, in the Dordogne valley. The model reveals that releasing water for irrigation could present economic advantages within certain constraints on maximum irrigated surface area and minimum water price. Consequently, the current economic justification of single purpose reservoir management for hydropower ought to be changed. Negotiation should be initiated for the purpose of selecting one of the non-dominated reservoir management policies, hereby minimizing the distance of the “most preferred” solution point from the ideal one or by using the Nash cooperative game solution. Multicriterion decision-making seems to be a promising operational management tool for hydrosystems.

Multiobjective Zone TP Reduction Analyses for an Off-Stream Reservoir

Multiobjective linear programming (MOLP) models are developed to evaluate total phosphorus (TP) load reduction strategies for an off-stream reservoir. Cost, water quality, and equity are the three major objectives considered. Watershed-based, subwatershed-based, and equal-transport-path-length-based strategies are the three zone control strategies analyzed and compared. The agricultural nonpoint-source pollution (AGNSP) model is used to simulate the non–point-source pollution loads on the reservoir. A two step modeling procedure is applied to verify sediment mass balance and water quality of storm runoff. MOLP models are established based on the simulation results for the three control strategies and are solved using the ADBASE package to generate the noninferior solution sets of the models. The subwatershed-based control strategy is observed to be the most cost-effective one based on results obtained for TP concentration between 13 and 20 ppb. With respect to equity, however, the watershed-based control strategy is superior to the others. The proposed multiobjective analyses are intended to improve the comparison and evaluation of nonpoint source pollution control strategies.

Application of vector optimization employing modified genetic algorithm to permanent magnet motor design

This paper presents a method to solve the vector optimization problem that determines both the noninferior solution set and the best compromise solution employing a modified genetic algorithm. The algorithm differs from the conventional one in the definition of fitness value and convergence criterion. Some parameters of the algorithm are adjusted to the vector optimization. The algorithm also contains the additional routine for searching the best compromise solution. This method is applied to the optimal design of the permanent magnet synchronous motor for which two objective functions regarding motor efficiency and weight are used.

Bi-objective power dispatch using fuzzy satisfaction-maximizing decision approach

A fuzzy satisfaction-maximizing decision approach is presented to solve the bi-objective power dispatch problem regarding both minimum fuel cost and minimum environmental impact of NO/sub x/ emission. Based on the fuzzy utility functions of system operators and a fuzzy satisfaction-maximizing scheme, a set of noninferior solutions are obtained. For each noninferior solution achieved, the associated marginal rate of substitution (MRS) is then calculated to help the operators more rationally determine the weight factors of the objectives considered and hence the final best compromise solution. Numerical results of the IEEE 30-bus 6-generator test systems are presented to demonstrate the proposed fuzzy satisfaction-maximizing decision approach.

Heuristic search strategies for multiobjective state space search

The multiobjective search model is a framework for solving multicriteria optimization problems using heuristic search techniques, where the different dimensions of a multiobjective search problem are mapped into a vector valued cost structure and partial order search is employed to determine the set of non-inferior solutions. This new framework for solving multicriteria optimization problems has been introduced by Stewart and White, who presented a generalization of the well known algorithmA* in this model. This paper presents several results on multiobjective state space search which helps in refining the scheme proposed by them. In particular, the following results have been presented.• The concept ofpathmax has been generalized to the multiobjective framework. It has been established that unlike in the conventional model, multidimensionalpathmax (in the multiobjective model) is useful for nonpathological tree search instances as well.We investigate the utility of an induced total order on the partial order search mechanism. The results presented are as follows: - If an induced total order is used in the selection process, then in general it is not necessary to compute the entire set of heuristic vectors at a node. - In memory-bounded search, a multiobjective search strategy that uses an induced total order for selection can back up a single cost vector while backtracking and yet guarantee admissibility though multiple noninferior candidate back-up vectors may be present in the space pruned while backtracking. • In this paper we study multiobjective state space search using inadmissible heuristics. We show that if heuristics are allowed to overestimate, then no algorithm is guaranteed to find all non-inferior solutions unless it expands dominated nodes also.The paper also addresses the task of multiobjective search under memory bounds, which is important in order to make the search scheme viable for practical purposes. The paper presents a linear space multiobjective search strategy called MOR*0 and suggests several variants of the strategy which may be useful under different situations.

The hierarchical and social decision making procedure for bi-objective optimization problems by using the simple majority decision rule

Multi-objective optimization has been successfully applied to problems of industrial design, problems of quality control and production management, and problems of finance. The theme of these applications is how to choose the best solution for the decision makers out of a set of non-inferior solutions to a multi-objective optimization problem. For this purpose, an optimization model with hierarchical structure, whose lower problem is a multi-objective optimization problem and the upper problem is a preference optimization problem on a set of non-inferior solutions, must be constructed. This kind of hierarchical problems have been previously analyzed only with regard to linear programming problems by Benson[6]. In this paper, an algorithm is derived that provides a solution as a social choice, obtained by aggregating plural decision-makers' preferences. In the case of the simple majority rule, the bi-objective problem is transformed into an ɛ-parameter choice problem, and the golden section method is applied. The availability of the approach is demonstrated with the means of an illustrative example.