Non-inferior Solutions Research Articles

Application of an Interior-Point Algorithm For Optimization of a Large-Scale Reservoir System

Optimization of a multi-reservoir system operation is challenging due to the non-linearity, stochasticity, and dimensionality involved in such a problem. In this research, a long-term planning model is presented for optimizing the operation of Iranian Karoon-Dez reservoir system using an interior-point algorithm. The system is the largest multi-purpose reservoir system in Iran with hydropower generation, water supply, and environmental objectives. The focus is on resolving the dimensionality of this problem while considering hydropower generation and water supply objectives. The weighting and constraints methods of multi-objective programming are used to assess the trade-off between water supply and hydropower objectives so as to find noninferior solutions. The computational efficiency of the proposed approach is demonstrated using historical data taken from Karoon-Dez reservoir system.

Stochastic Multi-Objective Generation Dispatch

This article explores the use of genetic algorithm to search for the assigned weightage pattern of objectives to obtain the best compromised thermal power generation schedule in the multi-objective framework. The multi-objective problem is formulated considering non-commensurable objectives viz. operating cost, NOx emission, and variance of real, as well as reactive, power generation mismatch with explicit recognition of statistical uncertainties in the thermal power generation cost curves, emission curves, and power demands, which are considered random variables. The solution set of such formulated problems is non-inferior due to contradictions among the objectives taken. The generation of non-inferior solutions requires an enormous amount of computation time when the objectives are more than two. In this article, the weighting method is used to convert the multi-objective optimization problem into a scalar optimization problem. Scalar optimization problem is solved many times for a different set of weight pattern to generate non-inferior solutions. The weighting patterns are either presumed on the basis of the decision maker's intuition or simulated with suitable step size. Such a simulation procedure may fail to provide the decision maker with the non-inferior solution that actually corresponds to the best compromised solution by virtue of the step size chosen. Thus, scalar weights are searched by genetic algorithms in the non-inferior domain. Among the generated population of scalar weights that generate non-inferior solutions, the system operator chooses the population of weighting pattern that provides maximum satisfaction level from the membership function of participating objectives and is termed as fitness function. The goals/objectives of fuzzy nature can be quantified by defining their membership functions. The validity of the proposed method has been demonstrated on an 11 node IEEE system comprising of five generators. The result of the proposed method is compared with the interactive method in which weighting patterns are simulated by giving suitable variation to weights in a specific manner.

A goal-seeking approach to capital budgeting

This paper articulates how a goal-seeking model addresses a variation of the capital-budgeting problem. Focused toward technology modernization in the public sector, this multi-criteria optimization model explicitly considers the diverse functions of the organization. In particular, the synergism amongst the functions is modeled as a multiplicative value function. The model is solved by the “constraint reduced feasible-region method”, resulting in a non-convex mathematical program that produces numerical intricacies. Linearization of the criterion (objective) functions reduces such intricacy. An Army-modernization acquisition-study was used to illustrate the proposed model, showing that its non-inferior solutions are remarkably stable. Comparison is also made with other approaches, typically formulated as goal-setting programs. The model highlights how technology acquisitions are affected as the priorities of each organizational function changes.

An Interactive Fuzzy Satisfying-Based Simulated Annealing Technique for Economic Emission Load Dispatch with Nonsmooth Fuel Cost and Emission Level Functions

The economic emission load dispatch problem treats economic and emission as competing objectives for optimal dispatch, which requires some form of conflict resolution to arrive at a solution. Assuming that the decision maker (DM) has imprecise or fuzzy goals for each of the objective functions, the multiobjective problem is transformed into a minimax problem, which is then handled by the simulated annealing (SA) technique. The solution methodology can offer noninferior optimal solution for the DM. Numerical results for a sample test system have been presented to demonstrate the performance and applicability of the proposed method.

Optimal management of waste loading into a river system with nonpoint source pollutants

An optimization model, fit for practical use, to allocate COD (Chemical Oxygen Demand)-based wasteload into a river system among outfalls is developed within the framework of robust optimization (RO). Nonpoint source COD loading, estimated based on the unit loading factor to be assumed known, is treated as uncontrollable one. The total amount of expected allowable COD load from point sources is then, under all possible scenarios of uncertain input information, maximized while satisfying the constraints on in-stream COD and DO (Dissolved Oxygen) transport, effluent standards and river water quality standards. Advantage of the e-RO model using the ∈-constraint method for optimization practice is brought to light from theoretical and practical aspects, in comparison with the conventional RO model resorting to the Lagrangian method. Solving a simple hypothetical example problem, it is demonstrated that the model developed is competent for successfully generating noninferior and robust solutions on optimal COD load allocation.

Noninferior Nash Strategies for Multi-Team Systems

This paper is concerned with the optimization of systems that are controlled by several teams of decision makers. The decision makers within each team cooperate for the benefit of their team. On the other hand, the teams compete among themselves in order to achieve an objective that relates to the overall performance of the system. An approach that merges concepts from team theory and game theory for dealing with such systems and a solution called the noninferior Nash strategy are introduced. This multi-team solution provides a new framework for analyzing hierarchically controlled systems so as to address complicated coordination problems among the decision makers. The properties of the noninferior Nash solution in static multi-team systems are investigated and necessary conditions for its existence are derived. Analytical expressions for the noninferior Nash strategies are derived for a class of linear-quadratic static multi-team games. In order to deal with the issue of nonuniqueness of the solution, the concept of a noninferior Nash strategy with a team leader is introduced. Several examples are presented to illustrate the results.

Multiobjective Optimization for Safety-related Decision Making in Chemical Processes

This paper proposes a novel decision making procedure based on multiobjective optimization programming (MOOP) to find the investment priority to reduce plant accidents. The new method rank the accident scenarios in order among many possible accident scenarios in a plant to decide the invest priority as considering safety and cost simultaneously. The method is explained by taking an example, which includes 30 accident scenario data (Kim et al., 2001) consisting of accident consequence, accident frequency, safety activity cost, and non-operating time. In this example, four goals are considered: (1) minimization of total safety activity cost, (2) minimization of total accident consequence, (3) minimization of the number of accident scenarios for unreasonable frequency, and (41 minimization of non-operating time. To analyze this optimization problem in terms of process safety and cost, this study obtained the noninferior solution curve (Pareto curve) by using the Goal programming (GP) methods. We found the ideal compromise solution set based on the Pareto curve. This result assists the business decision maker to select the best compromise in improving process safety as well as reducing investment cost.

An interactive fuzzy satisfying method based on evolutionary programming technique for multiobjective short-term hydrothermal scheduling

Abstract This paper presents an interactive fuzzy satisfying method based on evolutionary programming technique for short-term multiobjective hydrothermal scheduling. The multiobjective problem is formulated considering two objectives: (i) cost and (ii) emission. Assuming that the decision maker (DM) has fuzzy goals for each of the objective functions, evolutionary programming technique based fuzzy satisfying method is applied for generating a corresponding optimal noninferior solution for the DM’s goals. Then, by considering the current solution, the DM acts on this solution by updating the reference membership values until the satisfying solution for the DM can be obtained. A multi-reservoir cascaded hydroelectric system with a nonlinear relationship between water discharge rate, net head and power generation is considered. The water transport delay between connected reservoirs is taken into account. Thermal plants with nonsmooth fuel cost and emission level function are also taken into consideration. Results of the application of the proposed method are presented.

Fuzzy satisfying stochastic multi-objective generation scheduling by weightage pattern search methods

In the multi-objective framework, an interactive fuzzy satisfying method is presented to decide the generation schedule with explicit recognition of statistical uncertainties in system production cost data, NO x pollutant emission data and system load demand. In deciding the optimal schedule, four objectives viz. operating cost, NO x emission and risk due to variance of active and of reactive power generation mismatch are simultaneously minimized. Specific technique is put forth to convert the stochastic models into their respective deterministic equivalents. The weighting method is used to simulate the trade-off relationship between the conflicting objectives in the non-inferior domain. Generally, the weights are either simulated or searched in the non-inferior domain. In the paper, Hooke–Jeeves and evolutionary search techniques are implemented to search the ‘preferred’ weightage pattern in the non-inferior domain, which corresponds to the ‘best’ optimal solution. Fuzzy set theory has been exploited to decide the ‘preferred’ optimal operating point by interacting with the decision maker. The non-inferior solution, which attains maximum satisfaction level from the membership functions of the participating objectives, has been adjudged the ‘best’ solution. The goal/objectives being of fuzzy nature can be quantified by defining their membership functions. The validity of proposed method has been demonstrated on an 11-node IEEE system having five generators. The results obtained by searching weightage pattern using Hooke–Jeeves and evolutionary search techniques, are compared with the interactive method in which weightage patterns are simulated by giving suitable variation to weights in a specific manner.

A multiobjective model for non-point source pollution control for an off-stream reservoir catchment

Phosphorus loads from agricultural non-point source pollution (NPSP) significantly degrade reservoir water quality, making adequate control of agricultural NPSP necessary for improving the water quality. Controlling NPSP is generally accomplished using various Best Management Practices (BMPs). The present study applies the Agricultural Non-Point Source Pollution (AGNPS) model to simulate NPSP loading and BMP efficiencies and establishes an enhanced multiobjective mixed-integer programming model for NPSP control strategy analyses based on these results. Cost, phosphorus load, sediment load and equity are the four major objectives considered. A case study for the Posan reservoir is presented. Four commonly proposed and applicable BMPs are chosen. Non-inferior solutions obtained using the constraint method and trade-off relationships among different control objectives are described and discussed. Compared with a previously proposed fertilizer control model, results show that the model established herein is more cost-effective and achieves better phosphorus and sediment loading reduction and equity goals. Furthermore, the current model is expected to facilitate decision-making analysis for development of an appropriate cost-sharing program to encourage adoption of appropriate BMPs by farmers.

Design for Operability: A Singular-Value Optimization Approach within a Multiple-Objective Framework

Design-for-operability is a field of active research in process systems engineering because of the high economic impact of design on operability. Dynamic operability has been widely studied by means of the so-called (open-loop) controllability and resiliency indices, which are mostly based on linearized (Laplace/frequency domains) models of the dynamic systems. In particular, the minimum singular value of the transfer function matrix is a fair measure of resilience to disturbances. Among the plethora of approaches to design for dynamic operability, a multiple-objective formulation between cost and controllability naturally arises because of the conflicting characters of the two objectives. In this contribution the cost/minimum-singular-value multiple-objective design problem is solved for the meaningful reactor-separator-recycle system and two different control strategies. The generation of the noninferior solution set is performed here within the framework of an eigenvalue optimization approach.

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 class of linear interval programming problems and its application to portfolio selection

This paper discusses a class of linear programming problems with interval coefficients in both the objective functions and constraints. The noninferior solutions to such problems are defined based on two order relations between intervals, and can be found by solving a parametric linear programming problem. Considering the uncertain returns of assets in capital markets as intervals, we propose a model for portfolio selection based on the semiabsolute deviation measure of risk, which can be transformed to a linear interval programming model studied in the paper. The method is illustrated by solving a simplified portfolio selection problem.

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 load dispatch by fuzzy logic based searching weightage pattern

Abstract A multiobjective thermal power dispatch problem minimizes number of objectives viz cost and emission together while allocating the electricity demand among the committed generating units subject to physical and technological constraints. Such problems are solved to generate non-inferior solutions using weighting method or e-constraint method. Afterwards the decision maker is provided with a set of simple but effective tools to choose the best alternative among non-inferior solutions. The generation of non-inferior solution requires an enormous amount of computation time when the number of objectives is more than two. In the paper, the multiobjective problem has been solved a using weighted technique. The Evolutionary optimization technique has been employed in which the ‘preferred’ weightage pattern has been searched to get the ‘best’ optimal solution in non-inferior domain. Decision making theories attempt to deal with the vagueness or fuzziness inherent in subjective or impressive determination of goals. So fuzzy set theory has been exploited to decide the ‘preferred’ optimal operating point. The non-inferior solution that attains maximum satisfaction level from the membership functions of the participating objectives has been adjudged the ‘best’ solution. The proposed method requires few search moves to get the optimal operating point in the non-inferior domain for any number of goals. The validity of the proposed method has been demonstrated on a 25 nodes IEEE system comprising five generators.