A novel elitist multiobjective optimization algorithm: Multiobjective extremal optimization

Recently, a general-purpose local-search heuristic method called extremal optimization (EO) has been successfully applied to some NP-hard combinatorial optimization problems. This paper presents an investigation on EO with its application in numerical multiobjective optimization and proposes a new novel elitist (1 + λ) multiobjective algorithm, called multiobjective extremal optimization (MOEO). In order to extend EO to solve the multiobjective optimization problems, the Pareto dominance strategy is introduced to the fitness assignment of the proposed approach. We also present a new hybrid mutation operator that enhances the exploratory capabilities of our algorithm. The proposed approach is validated using five popular benchmark functions. The simulation results indicate that the proposed approach is highly competitive with the state-of-the-art multiobjective evolutionary algorithms. Thus MOEO can be considered a good alternative to solve numerical multiobjective optimization problems.     

Evaluation of the multiobjective ant colony algorithm performances on biobjective quadratic assignment problems

The difficulty of resolving the multiobjective combinatorial optimization problems with traditional methods has directed researchers to investigate new approaches which perform better. In recent years some algorithms based on ant colony optimization (ACO) metaheuristic have been suggested to solve these multiobjective problems. In this study these algorithms have been reported and programmed both to solve the biobjective quadratic assignment problem (BiQAP) instances and to evaluate the performances of these algorithms. The robust parameter sets for each 12 multiobjective ant colony optimization (MOACO) algorithms have been calculated and BiQAP instances in the literature have been solved within these parameter sets. The performances of the algorithms have been evaluated by comparing the Pareto fronts obtained from these algorithms. In the evaluation step, a multi significance test is used in a non hierarchical structure, and a performance metric (P metric) essential for this test is introduced. Through this study, decision makers will be able to put in the biobjective algorithms in an order according to the priority values calculated from the algorithms’ Pareto fronts. Moreover, this is the first time that MOACO algorithms have been compared by solving BiQAPs.     

A multiobjective based approach for mathematical programs with linear flexible constraints

In this paper a mathematical problem with linear flexible constraints is considered. In order to solve the problem an approach is proposed based on multiobjective linear programming. Indeed, allowing violations for the constraints, and using multiobjective linear programming to minimize these violations, a subset of solution set which has less violations, namely efficiently feasible set, is obtained. Then, the corresponding objective function is optimized over efficiently feasible set in order to obtain an optimal solution. An application of the proposed approach in pattern classification is introduced.     

A new multiobjective evolutionary algorithm

The Pareto-based approaches have shown some success in designing multiobjective evolutionary algorithms (MEAs). Their methods of fitness assignment are mainly from the information of dominated and nondominated individuals. On the top of the hierarchy of MEAs, the strength Pareto evolutionary algorithm (SPEA) has been elaborately designed with this principle in mind. In this paper, we propose a (μ+λ) multiobjective evolutionary algorithm ((μ+λ) MEA), which discards the dominated individuals in each generation. The comparisons of the experimental results demonstrate that the (μ+λ) MEA outperforms SPEA on five benchmark functions with less computational efforts.     

The influence of the fitness evaluation method on the performance of multiobjective search algorithms

In this paper we are concerned with finding the Pareto optimal front or a good approximation to it. Since non-dominated solutions represent the goal in multiobjective optimisation, the dominance relation is frequently used to establish preference between solutions during the search. Recently, relaxed forms of the dominance relation have been proposed in the literature for improving the performance of multiobjective search methods. This paper investigates the influence of different fitness evaluation methods on the performance of two multiobjective methodologies when applied to a highly constrained two-objective optimisation problem. The two algorithms are: the Pareto archive evolutionary strategy and a population-based annealing algorithm. We demonstrate here, on a highly constrained problem, that the method used to evaluate the fitness of candidate solutions during the search affects the performance of both algorithms and it appears that the dominance relation is not always the best method to use.     

On the transformation of lexicographic nonlinear multiobjective programs to single objective programs

This paper deals with multiobjective optimization programs in which the objective functions are ordered by their degree of priority. A number of approaches have been proposed (and several implemented) for the solution of lexicographic (preemptive priority) multiobjective optimization programs. These approaches may be divided into two classes. The first encompasses the development of algorithms specifically designed to deal directly with the initial model. Considered only for linear multiobjective programs and multiobjective programs with a finite discrete feasible region, the second one attempts to transform, efficiently, the lexicographic multiobjective model into an equvivalent model, i.e. a single objective programming problem. In this paper, we deal with the second approach for lexicographic nonlinear multiobjective programs.     

A branch-and-bound algorithm with fuzzy inference for a permutation flowshop scheduling problem

This paper considers an m-machine permutation flowshop scheduling problem of minimizing the makespan. This classical scheduling problem is still important in modern manufacturing systems, and is well known to be intractable (i.e., NP-hard). In fact branch-and-bound algorithms developed so far for this problem have not come to solve large scale problem instances with over a hundred jobs. In order to improve the performance of branch-and-bound algorithms this paper proposes a new dominance relation by which the search load could be reduced, and notices that it is based on a sufficient precondition. This suggests that the dominance relation holds with high possibility even if the precondition approximately holds, thus being more realistic. The branch-and-bound algorithm proposed here takes advantage of this possibility for obtaining an optimal solution as early as possible in the branch-and-bound search. For this purpose this paper utilizes membership functions in the context of the fuzzy inference. Extensive numerical experiments that were executed through Monte Carlo simulations and benchmark tests show that the developed branch-and-bound algorithm can solve 3-machine problem instances with up to 1000 jobs with probability of over 99%, and 4-machine ones with up to 900 jobs with over 97%.     

Minimizing makespan in a two-machine flowshop scheduling with batching and release time

This paper studies two-machine flowshop scheduling with batching and release time, whose objective is to minimize the makespan. We formulate the scheduling problem as a mixed integer programming model and show that it is a strongly NP-hard problem. We derive a lower bound and develop dynamic programming-based heuristic algorithms to solve the scheduling problem. Computational experiments are carried out to evaluate the performance of the heuristic algorithms. The numerical results show that some of the heuristic algorithms can indeed find effective solutions for the scheduling problem.     

Solving multiobjective,multiconstraint knapsack problems using mathematical programming and evolutionary algorithms

In this paper, we solve instances of the multiobjective multiconstraint (or multidimensional) knapsack problem (MOMCKP) from the literature, with three objective functions and three constraints. We use exact as well as approximate algorithms. The exact algorithm is a properly modified version of the multicriteria branch and bound (MCBB) algorithm, which is further customized by suitable heuristics. Three branching heuristics and a more general purpose composite branching and construction heuristic are devised. Comparison is made to the published results from another exact algorithm, the adaptive ε-constraint method [Laumanns, M., Thiele, L., Zitzler, E., 2006. An efficient, adaptive parameter variation scheme for Metaheuristics based on the epsilon-constraint method. European Journal of Operational Research 169, 932–942], using the same data sets. Furthermore, the same problems are solved using standard multiobjective evolutionary algorithms (MOEA), namely, the SPEA2 and the NSGAII. The results from the exact case show that the branching heuristics greatly improve the performance of the MCBB algorithm, which becomes faster than the adaptive ε -constraint. Regarding the performance of the MOEA algorithms in the specific problems, SPEA2 outperforms NSGAII in the degree of approximation of the Pareto front, as measured by the coverage metric (especially for the largest instance).     

Polynomial reduction of time-space scheduling to time scheduling

We study the University Course Timetabling Problem (UCTP). In particular we deal with the following question: is it possible to decompose UCTP into two problems, namely, (i) a time scheduling, and (ii) a space scheduling. We have arguments that it is not possible. Therefore we study UCTP with the assumption that each room belongs to exactly one type of room. A type of room is a set of rooms, which have similar properties. We prove that in this case UCTP is polynomially reducible to time scheduling. Hence we solve UCTP with the following method: at first we solve time scheduling and subsequently we solve space scheduling with a polynomial O(n³) algorithm. In this way we obtain a radical (exponential) speed-up of algorithms for UCTP. The method was applied at P.J. Šafárik University.     

A neural network model for scheduling problems

Artificial neural networks (ANNs) have been successfully applied to solve a variety of problems. This paper proposes a new neural network approach to solve the single machine mean tardiness scheduling problem and the minimum makespan job shop scheduling problem. The proposed network combines the characteristics of neural networks and algorithmic approaches. The performance of the network is compared with the existing scheduling algorithms under various experimental conditions. A comprehensive bibliography is also provided in the paper.     

An experimental analysis of evolutionary heuristics for the biobjective traveling purchaser problem

Given a set of markets and a set of products to be purchased on those markets, the Biobjective Traveling Purchaser Problem (2TPP) consists in determining a route through a subset of markets to collect all products, minimizing the travel distance and the purchasing cost simultaneously. As its single objective version, the 2TPP is an NP-hard Combinatorial Optimization problem. Only one exact algorithm exists that can solve instances up to 100 markets and 200 products and one heuristic approach that can solve instances up to 500 markets and 200 products. Since the Transgenetic Algorithms (TAs) approach has shown to be very effective for the single objective version of the investigated problem, this paper examines the application of these algorithms to the 2TPP. TAs are evolutionary algorithms based on the endosymbiotic evolution and other interactions of the intracellular flow interactions. This paper has three main purposes: the first is the investigation of the viability of Multiobjective TAs to deal with the 2TPP, the second is to determine which characteristics are important for the hybridization between TAs and multiobjective evolutionary frameworks and the last is to compare the ability of multiobjective algorithms based only on Pareto dominance with those based on both decomposition and Pareto dominance to deal with the 2TPP. Two novel Transgenetic Multiobjective Algorithms are proposed. One is derived from the NSGA-II framework, named NSTA, and the other is derived from the MOEA/D framework, named MOTA/D. To analyze the performance of the proposed algorithms, they are compared with their classical counterparts. It is also the first time that NSGA-II and MOEA/D are applied to solve the 2TPP. The methods are validated in 365 uncapacitated instances of the TPPLib benchmark. The results demonstrate the superiority of MOTA/D and encourage further researches in the hybridization of Transgenetic Algorithms and Multiobjective Evolutionary Algorithms specially the ones based on decomposition.     

Comparative analysis of multiobjective evolutionary algorithms for random and correlated instances of multiobjective d-dimensional knapsack problems

This study analyzes multiobjective d-dimensional knapsack problems (MOd-KP) within a comparative analysis of three multiobjective evolutionary algorithms (MOEAs): the ε-nondominated sorted genetic algorithm II (ε-NSGAII), the strength Pareto evolutionary algorithm 2 (SPEA2) and the ε-nondominated hierarchical Bayesian optimization algorithm (ε-hBOA). This study contributes new insights into the challenges posed by correlated instances of the MOd-KP that better capture the decision interdependencies often present in real world applications. A statistical performance analysis of the algorithms uses the unary ε-indicator, the hypervolume indicator and success rate plots to demonstrate their relative effectiveness, efficiency, and reliability for the MOd-KP instances analyzed. Our results indicate that the ε-hBOA achieves superior performance relative to ε-NSGAII and SPEA2 with increasing number of objectives, number of decisions, and correlative linkages between the two. Performance of the ε-hBOA suggests that probabilistic model building evolutionary algorithms have significant promise for expanding the size and scope of challenging multiobjective problems that can be explored.     

Opportunity Losses Due to Uncertainty in Multiobjective Decision Problems

A multiobjective maximization problem is considered in which at least one objective function f_i(x, C) depends on a random parameter C. If a single-valued measure, such as weighting or an l_p distance, is used to determine the preferred solution among the nondominated solutions, then standard decision-theoretic methods can be used to determine the expected opportunity loss (EOL). By an example hydrologic problem, it is shown that EOL is highly dependent on the single-valued measure selected to solve the multiobjective problem. The expected multiobjective opportunity loss (EMOL) is developed as a vector-valued measure of the effect of uncertainty on the problem which is independent of the technique. Finding the decision point with minimum EOL or EMOL is a possible way of selecting the preferred point. Problems pertaining to a multiobjective formulation of the EOL concept are examined.     

A novel Lagrangian relaxation approach for a hybrid flowshop scheduling problem in the steelmaking-continuous casting process

One of the largest bottlenecks in iron and steel production is the steelmaking-continuous casting (SCC) process, which consists of steel-making, refining and continuous casting. The SCC scheduling is a complex hybrid flowshop (HFS) scheduling problem with the following features: job grouping and precedence constraints, no idle time within the same group of jobs and setup time constraints on the casters. This paper first models the scheduling problem as a mixed-integer programming (MIP) problem with the objective of minimizing the total weighted earliness/tardiness penalties and job waiting. Next, a Lagrangian relaxation (LR) approach relaxing the machine capacity constraints is presented to solve the MIP problem, which decomposes the relaxed problem into two tractable subproblems by separating the continuous variables from the integer ones. Additionally, two methods, i.e., the boundedness detection method and time horizon method, are explored to handle the unboundedness of the decomposed subproblems in iterations. Furthermore, an improved subgradient level algorithm with global convergence is developed to solve the Lagrangian dual (LD) problem. The computational results and comparisons demonstrate that the proposed LR approach outperforms the conventional LR approaches in terms of solution quality, with a significantly shorter running time being observed.     

New solution methods for single machine bicriteria scheduling problem: Minimization of average flowtime and number of tardy jobs

We consider the bicriteria scheduling problem of minimizing the number of tardy jobs and average flowtime on a single machine. This problem, which is known to be NP-hard, is important in practice, as the former criterion conveys the customer’s position, and the latter reflects the manufacturer’s perspective in the supply chain. We propose four new heuristics to solve this multiobjective scheduling problem. Two of these heuristics are constructive algorithms based on beam search methodology. The other two are metaheuristic approaches using a genetic algorithm and tabu-search. Our computational experiments indicate that the proposed beam search heuristics find efficient schedules optimally in most cases and perform better than the existing heuristics in the literature.     

An interactive dynamic approach based on hybrid swarm optimization for solving multiobjective programming problem with fuzzy parameters

Real engineering design problems are generally characterized by the presence of many often conflicting and incommensurable objectives. Naturally, these objectives involve many parameters whose possible values may be assigned by the experts. The aim of this paper is to introduce a hybrid approach combining three optimization techniques, dynamic programming (DP), genetic algorithms and particle swarm optimization (PSO). Our approach integrates the merits of both DP and artificial optimization techniques and it has two characteristic features. Firstly, the proposed algorithm converts fuzzy multiobjective optimization problem to a sequence of a crisp nonlinear programming problems. Secondly, the proposed algorithm uses H-SOA for solving nonlinear programming problem. In which, any complex problem under certain structure can be solved and there is no need for the existence of some properties rather than traditional methods that need some features of the problem such as differentiability and continuity. Finally, with different degree of α we get different α-Pareto optimal solution of the problem. A numerical example is given to illustrate the results developed in this paper.     

On solving multiobjective bin packing problems using evolutionary particle swarm optimization

The bin packing problem is widely found in applications such as loading of tractor trailer trucks, cargo airplanes and ships, where a balanced load provides better fuel efficiency and safer ride. In these applications, there are often conflicting criteria to be satisfied, i.e., to minimize the bins used and to balance the load of each bin, subject to a number of practical constraints. Unlike existing studies that only consider the issue of minimum bins, a multiobjective two-dimensional mathematical model for bin packing problems with multiple constraints (MOBPP-2D) is formulated in this paper. To solve MOBPP-2D problems, a multiobjective evolutionary particle swarm optimization algorithm (MOEPSO) is proposed. Without the need of combining both objectives into a composite scalar weighting function, MOEPSO incorporates the concept of Pareto’s optimality to evolve a family of solutions along the trade-off surface. Extensive numerical investigations are performed on various test instances, and their performances are compared both quantitatively and statistically with other optimization methods to illustrate the effectiveness and efficiency of MOEPSO in solving multiobjective bin packing problems.     

On the convergence of multiobjective evolutionary algorithms

We consider the usage of evolutionary algorithms for multiobjective programming (MOP), i.e. for decision problems with alternatives taken from a real-valued vector space and evaluated according to a vector-valued objective function. Selection mechanisms, possibilities of temporary fitness deterioration, and problems of unreachable alternatives for such multiobjective evolutionary algorithms (MOEAs) are studied. Theoretical properties of MOEAs such as stochastic convergence with probability 1 are analyzed.