Non-inferior Solutions Research Articles

The optimal group consensus models for 2-tuple linguistic preference relations

We establish in this paper the optimization model of group consensus of 2-tuple linguistic preferential relations (LPRGCO Model), put forward three kinds of solutions to this model, and discover in it the convergence of group consensus. To detect the LPRGCO Model, we first build two kinds of optimal matrices as standards to measure the group consensus of 2-tuple linguistic preference relations (LPRs). And to analyze consensus deviations, we then adopt three types of measures, namely, the individual degree of consistency regarding alternative decision pairs, the deviational degree of the group consensus regarding alternative decision pairs, and the degree of group consensus regarding the original 2-tuple LPRs. On the basis of the previous analysis we not only construct an optimization model to probe into the deviation of the group consensus of 2-tuple LPRs by minimizing the weighted arithmetic average of deviation degrees of individual consistency, but also point out three feasible solutions to this optimization model: the optimal solution, satisfactory solutions, and non-inferior solutions. Accordingly, we discover different conditions in terms of the three solutions. And hence, we can from the aforementioned discussion draw a conclusion that the deviation of group consensus either decreases or stays invariant as the number of decision makers (DM) increases. To expatiate on the practical value of the model proposed, we will display in this paper numerical examples.

An Advanced Multi-Objective Genetic Algorithm Based on Borda Number

When applied to Multi-Objective Decision Making (MODM), genetic algorithm is plagued with two problems: how to appreciate non-inferior solutions and how to store them. Using Borda number as the fitness of a chromosome, an advanced Multi-Objective Genetic Algorithm ( MOGA) is provided in this paper, which can solve these problems in an easier way; and with the characteristic of Genetic Algorithm (GA) of producing a number of feasible solutions, this approach is able to obtain the set of non-inferior solutions without information of the decision-maker’s preferences. Finally, a simulated example to apply this algorithm to a multipurpose reservoir’s operation is provided, indicating the feasibility and effectiveness of this advanced MOGA.

Bidding based generator maintenance scheduling with triple-objective optimization

A new bidding based generating unit maintenance scheduling method under the electricity market environment is presented in this paper. The method can balance overall reliability and economy, and ensure the fairness to all generation companies (GENCOs). The unit maintenance scheduling is modeled as a triple-objective non-convex optimization problem with discrete variables and is solved using an improved version of the Non-dominated Sorting Genetic Algorithm (NSGA-II). A non-inferior solution set is directly produced from huge combinations. A few candidate solutions are selected from the non-inferior set by independent system operator (ISO) based on the overall reliability and economy, whereas the final maintenance schedule is determined by voting of GENCOs. The shares of vote are calculated from the bidding information of GENCOs and the right of each GENCO is fairly guaranteed. A case study of 50 units demonstrates the effectiveness of the proposed method.

A bi-objective modeling approach applied to an urban semi-desirable facility location problem

This paper introduces a mixed-integer, bi-objective programming approach to identify the locations and capacities of semi-desirable (or semi-obnoxious) facilities. The first objective minimizes the total investment cost; the second one minimizes the dissatisfaction by incorporating together in the same function “pull” and “push” characteristics of the decision problem (individuals do not want to live too close, but they do not want to be too far, from facilities). The model determines the number of facilities to be opened, the respective capacities, their locations, their respective shares of the total demand, and the population that is assigned to each candidate site opened. The proposed approach was tested with a case study for a particular urban planning problem: the location of sorted waste containers. The complete set of (supported or unsupported) non-inferior solutions, consisting of combinations of multi-compartment containers for the disposal of four types of sorted waste in nineteen candidate sites, and population assignments, was generated. The results obtained for part of the historical center of an old European city (Coimbra, Portugal) show that this approach can be applied to a real-world planning scenario.

The optimal group consensus deviation measure for multiplicative preference relations

Consistency and consensus measures are two important procedures employed in group decision making with multiplicative preference relations (MPRs). In this paper, we first establish the concept of individual consistency deviation degree between the original MPR and its optimal estimation. Then we develop a group consensus deviation degree optimization model (GCO Model) by minimizing the weighted arithmetic average of individual consistency deviation degrees. Our established theorems enlist the conditions for the existence of the optimal solution, the satisfactory solution, and non-inferior solution to the GCO Model. These results also provide an existence condition of redundant MPR in group decision making. Additionally, we show that the optimal value function of the GCO Model converges to 0, which implies that the consensus degree of DMs converges as the number of the decision makers increases indefinitely.

A Layered Approach Based on Objectives to Multi-Objective Optimization

A new multi-objective optimization approach based on layered objective is proposed. Two improvements on the non-dominated sorting genetic algorithm are given which are new layering method and the new selection mech- anism. On the basis of the traditional non-dominated sorting genetic algorithm, a new objective layer approach is adapted and cycle copy through the first half layers. Another one is making a restriction mechanism before crossover on the classical non-dominated sorting genetic algorithm with elitist strategy. Through these operations, it can ef- fectively improve the convergence rate of the optimal solution set. At last, the corresponding non-inferior solutions will be maintained. Several classic test functions commonly adopted in evolutionary multi-objective optimization and the network design problem are used. The results indicate that the non-dominated sorting genetic algorithm based on layered objective can effectively speed up the convergence rate.

Multi-Objective Optimization of Pilots’ FFS Recurrent Training Problem

Two multi-objective programming models are built to describe Pilots’ full flight simulator (FFS) recurrent training (PFRT) problem. There are two objectives for them. One is the best matching of captains and copilots in the same aircraft type. The other is that pilots could attend his training courses at proper month. Usually the two objectives are conflicting because there are copilots who will promote to captains or transfer to other aircraft type and new trainees will enter the company every year. The main theme in the research is to find the final non-inferior solutions of PFRT problem. Graph models are built to help to analyze the problem and we convert the original problem into a longest-route problem with weighted paths. An algorithm is designed with which we can obtain all the non-inferior solutions by a graphic method. A case study is present to demonstrate the effectiveness of the algorithm as well.

Suspension Parameters Optimize Based on Competition-Cooperation Game Model

Game theory is used to solve engineering problems, according to behavioral expressions, derived a number of algorithms from simulation of human behavior. The paper, adopts competition-cooperation game method to optimize passive suspension. The calculation results show that the competition-cooperation game method could be used to solve muti- objective optimization problems. During the game, all game partiers abide by some constraint protocol based on collective rationality; by negotiating, game partiers make some “concession” mutually, and reach trade-off equilibrium solution. Maybe the results is not the best optimal results to the game party, it is a relatively acceptable solution, which means the co-operative game result for multi-objective is a non-inferior solution.

梯级水库防洪库容分配策略研究

针对梯级水库防洪任务和边界条件发生的变化，建立了一种确定梯级水库防洪库容分配的数学模型，耦合水库电力调度优化、防洪效益优化等多种目标，把梯级水库预留的防洪库容最小作为约束条件，采用逐次优化算法求解出使梯级水库发电效益最大的非劣解集。以清江干流梯级水库为例，确定水布垭和隔河岩水库防洪库容的最优分配策略。在水布垭预留5亿m3和隔河岩预留不小于1.5亿m3防洪库容的前提下，若发生100年一遇的洪水时，隔河岩水库的最高调洪水位不会超过200 m，可以满足防洪安全。该研究为梯级水库防洪库容分配提供了一种新的探讨思路。 Due to the changes of flood control task and boundary condition of cascade reservoirs, a mathematical model was established to allocate cascade reservoir flood prevention storage. Coupled hydropower generation with flood control operation objections, progressive optimality algorithm was used to obtain Pareto non-inferior solutions of maximum hydropower generation under the minimum flood prevention storage constrains for cascade reservoirs. The Qingjiang basin was selected as case study, the optimal storage allocation strategy for the Shuibuya and Geheyan reservoirs was given. If the Shuibuya reservoir keeps 500 million m3 and Geheyan reservoir remains not less than 150 million m3 flood prevention storages, the maximum regulation water level of the Geheyan reservoir will not over 200mwhen 100-year flood occurs, which can assurance flood control safety. The proposed method might provide a new approach for cascade reservoir storage allocation.

A Simple Algorithm for Multi-Objective, Short-Term Hydrothermal Scheduling

This investigation proposes a new algorithm for the solution of multi-objective, short-term hydrothermal scheduling problems. In the common algorithm using weighting method, a set of weight vectors are fed infor each weight vector objective function values in the optimisation interval, and the corresponding fuzzy membership functions are computed and the best compromise solution is identified from the set of non-inferior solutions obtained. In the proposed algorithm, a set of weight vectors are genetically generated and the best solution is identified as described already. Further, the weight vectors are genetically modified and the process is repeated to identify the best solution using the modified weight vectors. The process is continued till change in the fitness value of the best solution is marginal. The investigation further proposes three methods to determine the objective function values in the optimisation interval, which is a major sub-process in the above algorithm. These are the Newton-Raphson method, a genetic search method and a random search method. A comparison is made among the performance data of the three methods, and the random search method is projected as the simple algorithm for multi-objective, short-term hydrothermal scheduling problem. A coal-constrained thermal unit is introduced in the investigation to make the scheduling problem more universal.

Intelligent Multi-Objective Nonlinear Model Predictive Control (iMO-NMPC): Towards the ‘on-line’ optimization of highly complex control problems

The benefits of using the Nonlinear Model Predictive Control (NMPC) for the response optimization of highly complex controlled plants are well known. Nevertheless the complexity and associated high computational cost make its implementation and reliability the focus of the discussion. This paper proposes an Intelligent and Multi-Objective NMPC (iMO-NMPC) scheme where several, and often conflicting, control objectives can be competing simultaneously in the control problem. In the iMO-NMPC, the combination of a Neural Network, a Multi-Objective Genetic Algorithm and a Fuzzy Inference System, help us in the nonlinear search for near-optimal control actions, aiming to fulfil all the specified control objectives simultaneously. The proposed scheme adds an expert stage that can adaptively change the degree of importance (weight) of each control objective according to the state of the plant. Therefore, once the nonlinear multi-objective optimization problem is solved at each sampling time and the non-inferior control solutions belonging to the set of Pareto are obtained, the most appropriate one is selected by using the control objectives weights inferred from the expert stage. Some experimental results showing the iMO-NMPC effectiveness and the details about its implementation over control systems with low sampling times are also presented and discussed in this paper.

Processing Parameters Optimization Based on PSO Algorithm

Particle swarm optimization (PSO) is classified into swarm intelligence algorithm. Focusing on the NC processing parameters optimization, this paper has made several improvements on PSO: dividing solutions into feasible and infeasible, then shrinking the infeasible solutions; dividing feasible solutions into inferior and non-inferior, then choosing the non-inferior solution with most domination solutions as the global best position for inferior solutions. It has shown a significant achievement in actual processing.

Multi-objective Optimization of Distributed Generation Planning Using Impact Indices and Trade-off Technique

This article presents a multi-objective formulation for determining the best location and size of distributed generation. This multi-objective formulation includes reliability of service, system operational efficiency, cost of purchased energy, power quality, and system security as objective functions that are the primary concern of system planners. These objectives contradict each other and have trade-off relations. Conventional approaches for optimizing a single objective yield an uncompromised solution for such multi-objective problems. The multi-objective formulation is solved using an interactive trade-off algorithm to obtain compromised or most satisfactory non-inferior solutions. The system planner has a choice to include his/her preference on each objective through interactive steps. The practical situations, such as voltage rise phenomenon and voltage dependency of loads, are addressed, incorporating certain voltage constraints and appropriate load models. The test system is an existing Indian rural distribution network.

Regional allocation of irrigation water in a rice paddy area with water-saving practices

A multiobjective optimization model has been presented for allocating irrigation waters in a rice paddy area. Water-saving practices conducted on the field-plot basis are considered and expressed in the optimal water allocation problem in the regional scale. Irrigation water allocated to the blocks of paddy fields is divided into two components of a basic amount and a safety margin in order to mitigate the effect of hydrological and hydraulic uncertainties on the study area. Four competing management objectives with regard to total rice yield, total water-saving cost, equity of water allocation, and safety of water supply are defined in the linear programming formulation so that noninferior solutions can be procured which are informative and persuasive in decision-making. Applicability of the optimization model is examined using water allocation problems for a hypothetical irrigation system. It is demonstrated that the optimization model can provide satisficing solutions where the four objectives are harmonized under substantial variations of total water supply to the irrigation system.

Chance-constrained multi-objective programming for optimal multi-layer aquifer remediation design

Groundwater remediation works are expensive, complex in nature, and generally involve uncertain design parameters and multiple conflicting objectives. Reduction in remediation cost is, normally, the primary objective of optimal remediation designs. Remediation time has significant influence on remediation cost. The main contributions of this study are: development of a chance-constrained multi-objective programming (CCMOP) model, exploration of the significance of flexible remediation time on optimal remediation designs, and exploration of some basic characteristics of multi-objective optimization algorithms. The developed methodology excludes inferior yet optimal solutions from the set of possible solutions, identifies the ranges in which non-inferior solutions lie, and prevents one from making inferior decisions. It provides insightful information about the problem and simplifies the decision making process considerably. The results suggest that by avoiding the use of faulty remediation times, highly significant amounts of remediation cost could be saved. The CCMOP is an effective uncertainty and reliability prediction technique.

Study on Multiobjective Programming Problem under Bifuzzy Environment

In this paper, based on bifuzzy theory, we have studied the multiobjective programming problem under bifuzzy environment, and presented the expected-value model which is a deterministic multiobjective problem. To the expected value model, the concepts of non-inferior solution are defined, and their relations are also discussed. According to practical decision-making process, a solution method, called the method of main objective function, has been studied, whose results can facilitate us to design algorithms to solve the bifuzzy multiobjective programming problem.

The Expected Value Model of Multiobjective Programming and its Solution Method Based on Bifuzzy Environment

In this paper, based on bifuzzy theory, we study the multiobjective programming problem under bifuzzy environment and present the expected value model to the problem. Furthermore, to the proposed model, the concepts of non-inferior solution are defined, and their relations are also discussed. According to practical decision-making process, a solution method, called the method of main objective function, has been studied, whose results can facilitate us to design algorithms to solve the bifuzzy multiobjective programming problem. Finally, a numerical example is given to explain the proposed method.

QoS-driven Global Optimization Approach for Large-scale Web Services Composition

One of the aims of SOA is to compose atomic web services into a powerful composite service. QoS based selection approaches are used to choose the best solution among candidate services with the same functionality. Due to the increasing scale of the candidate services and demands for real-time in some specific application domains, the rapid convergent algorithm for large-scale web service composition is especially important, but rare work has been done to solve the problem. This paper describes the Web services composition model and constructs the web service selection mathematical model. According to these models, service composition problem can be considered as Single-Objective Multi-Constraints optimization problem. We propose a new algorithm named GAELS (Genetic Algorithm Embedded Local Searching), which uses the strategies of enhanced initial population and mutation with local searching, to speed up the convergence. Finally, the in-depth experimental results show that the GAELS algorithm can get the non-inferior solution more quickly and more adaptively than simple genetic algorithm in large-scale web service composition.

Approximate Greatest Descent Methods for Optimization with Equality Constraints

In an optimization problem with equality constraints the optimal value function divides the state space into two parts. At a point where the objective function is less than the optimal value, a good iteration must increase the value of the objective function. Thus, a good iteration must be a balance between increasing or decreasing the objective function and decreasing a constraint violation function. This implies that at a point where the constraint violation function is large, we should construct noninferior solutions relative to points in a local search region. By definition, an accessory function is a linear combination of the objective function and a constraint violation function. We show that a way to construct an acceptable iteration, at a point where the constraint violation function is large, is to minimize an accessory function. We develop a two-phases method. In Phase I some constraints may not be approximately satisfied or the current point is not close to the solution. Iterations are generated by minimizing an accessory function. Once all the constraints are approximately satisfied, the initial values of the Lagrange multipliers are defined. A test with a merit function is used to determine whether or not the current point and the Lagrange multipliers are both close to the optimal solution. If not, Phase I is continued. If otherwise, Phase II is activated and the Newton method is used to compute the optimal solution and fast convergence is achieved.

Optimal capacitors appointment though immune multi-objective algorithm

This work proposes a two-stage immune algorithm that embeds the compromise programming to perform multi-objective optimal compensator placement. A new problem formulation model that involves fuzzy sets to reflect the imprecise nature of objectives and incorporates multiple planning requirements is presented. The proposed approach finds a set of non-inferior (Pareto) solutions rather than any single aggregated optimal solution. Additionally, this developed approach eliminates the need for any user-definedweight factor to aggregate all objectives. Comparative studies are conducted on an actual system with encouraging results, demonstrating the effectiveness of the proposed approach.