Tchebycheff Approach Research Articles

An Improved MOEA/D with Optimal DE Schemes for Many-Objective Optimization Problems

MOEA/D is a promising multi-objective evolutionary algorithm based on decomposition, and it has been used to solve many multi-objective optimization problems very well. However, there is a class of multi-objective problems, called many-objective optimization problems, but the original MOEA/D cannot solve them well. In this paper, an improved MOEA/D with optimal differential evolution (oDE) schemes is proposed, called MOEA/D-oDE, aiming to solve many-objective optimization problems. Compared with MOEA/D, MOEA/D-oDE has two distinguishing points. On the one hand, MOEA/D-oDE adopts a newly-introduced decomposition approach to decompose the many-objective optimization problems, which combines the advantages of the weighted sum approach and the Tchebycheff approach. On the other hand, a kind of combination mechanism for DE operators is designed for finding the best child solution so as to do the a posteriori computing. In our experimental study, six continuous test instances with 4–6 objectives comparing NSGA-II (nondominated sorting genetic algorithm II) and MOEA/D as accompanying experiments are applied. Additionally, the final results indicate that MOEA/D-oDE outperforms NSGA-II and MOEA/D in almost all cases, particularly in those problems that have complicated Pareto shapes and higher dimensional objectives, where its advantages are more obvious.

Robust Multi-Objective Beamforming Design for Power Efficient and Secure Communication in MU-MISO Networks

In this paper, we consider transmission optimization in a multiple-input-single-output downlink network, in which each user is wiretapped by a specific eavesdropper. Particularly, we aim to minimize the total transmit power and maximize the sum secrecy rate (SSR) of the system simultaneously, under the assumption that the channel state information (CSI) of the eavesdroppers is not perfectly known at the transmitter. Considering the conflict between two objectives, a multi-objective optimization (MOO) framework based on the weighted Tchebycheff approach is proposed. The formulated MOO problem is non-convex and intractable. To tackle it, several auxiliary variables are introduced and the corresponding Taylor series expansion is employed to linearize the term related to each auxiliary variable. Then, the non-convexity resulting from the CSI errors is recast as a convex one with the aid of ${S}$ -procedure and Cauchy–Schwarz inequality. Based on above treatments, a robust iterative algorithm is proposed to solve the original problem. Simulation results not only demonstrate the effectiveness of the proposed design, but also unveil the tradeoff between the total transmit power and the SSR.

A Novel Memetic Algorithm Based on Decomposition for Multiobjective Flexible Job Shop Scheduling Problem

A novel multiobjective memetic algorithm based on decomposition (MOMAD) is proposed to solve multiobjective flexible job shop scheduling problem (MOFJSP), which simultaneously minimizes makespan, total workload, and critical workload. Firstly, a population is initialized by employing an integration of different machine assignment and operation sequencing strategies. Secondly, multiobjective memetic algorithm based on decomposition is presented by introducing a local search to MOEA/D. The Tchebycheff approach of MOEA/D converts the three-objective optimization problem to several single-objective optimization subproblems, and the weight vectors are grouped by K-means clustering. Some good individuals corresponding to different weight vectors are selected by the tournament mechanism of a local search. In the experiments, the influence of three different aggregation functions is first studied. Moreover, the effect of the proposed local search is investigated. Finally, MOMAD is compared with eight state-of-the-art algorithms on a series of well-known benchmark instances and the experimental results show that the proposed algorithm outperforms or at least has comparative performance to the other algorithms.

A lexicographic weighted Tchebycheff approach for multi-constrained multi-objective optimization of the surface grinding process

ABSTRACTIn this article a multi-objective mathematical model is developed to minimize total time and cost while maximizing the production rate and surface finish quality in the grinding process. The model aims to determine optimal values of the decision variables considering process constraints. A lexicographic weighted Tchebycheff approach is developed to obtain efficient Pareto-optimal solutions of the problem in both rough and finished conditions. Utilizing a polyhedral branch-and-cut algorithm, the lexicographic weighted Tchebycheff model of the proposed multi-objective model is solved using GAMS software. The Pareto-optimal solutions provide a proper trade-off between conflicting objective functions which helps the decision maker to select the best values for the decision variables. Sensitivity analyses are performed to determine the effect of change in the grain size, grinding ratio, feed rate, labour cost per hour, length of workpiece, wheel diameter and downfeed of grinding parameters on each value of the objective function.

Multiobjective Resource Allocation for Secure Communication in Cognitive Radio Networks With Wireless Information and Power Transfer

In this paper, we study resource allocation for multiuser multiple-input–single-output secondary communication systems with multiple system design objectives. We consider cognitive radio (CR) networks where the secondary receivers are able to harvest energy from the radio frequency when they are idle. The secondary system provides simultaneous wireless power and secure information transfer to the secondary receivers. We propose a multiobjective optimization framework for the design of a Pareto-optimal resource allocation algorithm based on the weighted Tchebycheff approach. In particular, the algorithm design incorporates three important system design objectives: total transmit power minimization, energy harvesting efficiency maximization, and interference-power-leakage-to-transmit-power ratio minimization. The proposed framework takes into account a quality-of-service (QoS) requirement regarding communication secrecy in the secondary system and the imperfection of the channel state information (CSI) of potential eavesdroppers (idle secondary receivers and primary receivers) at the secondary transmitter. The proposed framework includes total harvested power maximization and interference power leakage minimization as special cases. The adopted multiobjective optimization problem is nonconvex and is recast as a convex optimization problem via semidefinite programming (SDP) relaxation. It is shown that the global optimal solution of the original problem can be constructed by exploiting both the primal and the dual optimal solutions of the SDP-relaxed problem. Moreover, two suboptimal resource allocation schemes for the case when the solution of the dual problem is unavailable for constructing the optimal solution are proposed. Numerical results not only demonstrate the close-to-optimal performance of the proposed suboptimal schemes but unveil an interesting tradeoff among the considered conflicting system design objectives as well.

A REGION DECOMPOSITION-BASED MULTI-OBJECTIVE PARTICLE SWARM OPTIMIZATION ALGORITHM

In this paper, a novel multi-objective particle swarm optimization algorithm based on MOEA/D-M2M decomposition strategy (MOPSO-M2M) is proposed. MOPSO-M2M can decompose the objective space into a number of subregions and then search all the subregions using respective sub-swarms simultaneously. The M2M decomposition strategy has two very desirable properties with regard to MOPSO. First, it facilitates the determination of the global best (gbest) for each sub-swarm. A new global attraction strategy based on M2M decomposition framework is proposed to guide the flight of particles by setting an archive set which is used to store the historical best solutions found by the swarm. When we determine the gbest for each particle, the archive set is decomposed and associated with each sub-swarm. Therefore, every sub-swarm has its own archive subset and the gbest of the particle in a sub-swarm is selected randomly in its archive subset. The new global attraction strategy yields a more reasonable gbest selection mechanism, which can be more effective to guide the particles to the Pareto Front (PF). This strategy can ensure that each sub-swarm searches its own subregion so as to improve the search efficiency. Second, it has a good ability to maintain the diversity of the population which is desirable in multi-objective optimization. Additionally, MOPSO-M2M applies the Tchebycheff approach to determine the personal best position (pbest) and no additional clustering or niching technique is needed in this algorithm. In order to demonstrate the performance of the proposed algorithm, we compare it with two other algorithms: MOPSO and DMS-MO-PSO. The experimental results indicate the validity of this method.

A multiobjective evolutionary algorithm based on decomposition with normal boundary intersection for traffic grooming in optical networks

This paper presents a multiobjective optimization algorithm for solving tri-objective optimization problems with objectives of very different scales. As a version of the Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D), it adopts the Normal Boundary Intersection (NBI) based Tchebycheff approach to decompose a multiobjective optimization problem into a number of single objective subproblems. Particular attention has been paid to set the weight vectors of these subproblems for handling disparately scaled objectives. We have applied our proposed algorithm to a traffic grooming problem in the telecommunication fields. We also propose to use an indirect encoding approach to handle this very complicated problem. Our experimental studies show that our proposed method is able to produce very promising results for this real-world telecommunication problem.

Multiobjective evolutionary algorithm based on decomposition for 3-objective optimization problems with objectives in different scales

In Multiobjective Optimization problems the objective functions may have different scales, which leads to a neglecting of one or more objective functions. The most common-used way in the literature to solve this drawback is to normalize the objective space; however, a set of uniformly distributed solutions in the normalized objective space may not be uniformly distributed in the original objective space with more than two objective functions. In this work, we present an improved version of the Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D) which incorporates a new aggregation technique based on the Normal Boundary Intersection approach and the Tchebycheff approach (MOEA/D-NBI) for solving 3-objective optimization problems with different scales of objectives.

Multi-objective ant colony optimization based on decomposition for bi-objective traveling salesman problems

This paper proposes a framework named multi-objective ant colony optimization based on decomposition (MoACO/D) to solve bi-objective traveling salesman problems (bTSPs). In the framework, a bTSP is first decomposed into a number of scalar optimization subproblems using Tchebycheff approach. To suit for decomposition, an ant colony is divided into many subcolonies in an overlapped manner, each of which is for one subproblem. Then each subcolony independently optimizes its corresponding subproblem using single-objective ant colony optimization algorithm and all subcolonies simultaneously work. During the iteration, each subproblem maintains an aggregated pheromone trail and an aggregated heuristic matrix. Each subcolony uses the information to solve its corresponding subproblem. After an iteration, a pheromone trail share procedure is evoked to realize the information share of those subproblems solved by common ants. Three MoACO algorithms designed by, respectively, combining MoACO/D with AS, MMAS and ACS are presented. Extensive experiments conducted on ten bTSPs with various complexities manifest that MoACO/D is both efficient and effective for solving bTSPs and the ACS version of MoACO/D outperforms three well-known MoACO algorithms on large bTSPs according to several performance measures and median attainment surfaces.

An integration design optimization framework of robust design, axiomatic design, and reliability-based design

Robust design, axiomatic design, and reliability-based design provide effective approaches to deal with quality problems, and their integration will achieve better quality improvement. An integration design optimization framework of robust design, axiomatic design, and reliability-based design is proposed in this paper. First, the fitted response model of each quality characteristic is obtained by response surface methodology and the mean square error (MSE) estimation is given by a second-order Taylor series approximation expansion. Then the multiple quality characteristics robust design model is developed by the MSE criteria. Finally, the independence axiom constraints for decoupling and reliability constraints are integrated into the multiple quality characteristics robust design model, and the integration design optimization framework is formulated, where the weighted Tchebycheff approach is adopted to solve the multiple objective programming. An illustrative example is presented at the end, and the results show that the proposed approach can obtain better trade-offs among conflicting quality characteristics, variability, coupling degree and reliability requirements. Copyright © 2011 John Wiley & Sons, Ltd.

Hybrid immune algorithm with Lamarckian local search for multi-objective optimization

Lamarckian learning has been introduced into evolutionary computation as local search mechanism. The relevant research topic, memetic computation, has received significant amount of interests. In this study, a novel Lamarckian learning strategy is designed for improving the Nondominated Neighbor Immune Algorithm, a novel hybrid multi-objective optimization algorithm, Multi-objective Lamarckian Immune Algorithm (MLIA), is proposed. The Lamarckian learning performs a greedy search which proceeds towards the goal along the direction obtained by Tchebycheff approach and generates the improved progenies or improved decision vectors, so single individual will be optimized locally and the newcomers yield an enhanced exploitation around the nondominated individuals in less-crowded regions of the current trade-off front. Simulation results based on twelve benchmark problems show that MLIA outperforms the original immune algorithm and NSGA-II in approximating Pareto-optimal front in most of the test problems. When compared with the state of the art algorithm MOEA/D, MLIA shows better performance in terms of the coverage of two sets metric, although it is laggard in the hypervolume metric.

Studies on a biobjective robust design optimization problem

The vast majority of the multiobjective robust design research reported in the literature has been performed under the assumption of multiple quality characteristics. This paper differs in that the process mean and variance are considered to be a biobjective problem since the primary goal of robust design is to determine the optimal robust design factor settings by minimizing performance variability and deviation from a target value of a product. A more comprehensive set of solutions is developed using a lexicographic weighted Tchebycheff approach to the biobjective robust design model rather than the approaches traditionally used in the dual-response approach to obtain efficient solutions. Numerical examples show that the proposed model is far more effective than the traditional weighted sum approach. [Supplementary materials are available for this article. Go to the publisher's online edition of IIE Transactions for the following free supplemental resource: Appendix with mathematical proof, figures, and tables.]

An Interactive Multistage ε-Inequality Constraint Method For Multiple Objectives Decision Making

Multiple objectives decision making (MODM) in engineering design refers to obtaining a preferred optimal solution in the context of conflicting design objectives. Problems with multiple objectives do not have a unique optimal solution but a set of Pareto optimal solutions. This paper presents a new interactive multistage MODM method which captures a decision maker’s preference structure in order to obtain a preferred Pareto solution even for non-convex problems. Representative subsets of an entire Pareto optimal set are generated and expanded based on the decision maker’s preference. The ε-constraint method is used to constrain the multiple objectives problem based on the decision maker’s feedback. In addition, ideas from an interactive weighted Tchebycheff approach are applied to reduce the feasible region at each stage, ensuring that the process eventually converges to a preferred solution. The method is demonstrated with two examples: (i) a simple two-bar truss design, and (ii) a more complicated problem in power electronic module design.

MULTIOBJECTIVE DECISION MAKING TECHNIQUES FOR RESERVOIR OPERATION1

ABSTRACT: Six applications of multiobjective decision making techniques for finding optimal or satisfying operating rules for reservoir systems are presented. The examples include situations with hydropower vs. water supply (for irrigation), flood control vs. low flow augmentation, selection of an operating rule, low‐flow vs. reliability, and low flow and recreation vs. water quality. The techniques applied include the constraint method, compromise programming, goal programming, Tchebycheff approach (max‐mm), Consensus, and ELECTRE I and II.

Multicriteria analysis of water allocation in a river basin: The Tchebycheff Approach

A new interactive multiple objective methodology is applied to the seasonal water allocation problem. Linear optimization and filtering procedures are combined within a multiobjective frame‐work in order to identify nondominated solutions associated with randomly sampled criteria vectors. The procedure samples from the entire nondominated set, not just the set of nondominated extreme points, by computing the nondominated criteria vector that is closest to an ideal criteria vector according to a weighted Tchebycheff metric. Unlike many other multicriteria methods, the Tchebycheff procedure is capable of finding solutions of highest utility which may not correspond to extreme point solutions. The procedure is applied to the problem of water allocation in a river basin where a large number of conflicting objectives, including environmental objectives, compete for the available supply.