Multi-objective Combinatorial Optimization Research Articles

A decomposition-based coevolutionary multiobjective local search for combinatorial multiobjective optimization

Abstract Multiobjective evolutionary algorithm based on decomposition (MOEA/D) divides a multiobjective optimization problem into a number of single-objective subproblems and solves them in a collaborative way. MOEA/D can be naturally extended by the common, intensification oriented method of local search for solving combinatorial multiobjective optimization problems (CMOPs). However, the performance of MOEA/D strongly depends on the distribution of direction vectors and the decomposition method it adopts. In this paper, an efficient coevolutionary multiobjective local search based on decomposition (CoMOLS/D) is proposed. In CoMOLS/D, two sets of direction vectors and two populations with different decomposition methods are adopted to coevolve with each other. Among them, one population aims to achieve fast convergence while the other one puts more effort for maintaining the complementarily diverse solutions based on the convergence population. In the experimental studies, CoMOLS/D is compared with four decomposition-based local search heuristics, i.e., MOEA/D-LS (WS, TCH, PBI and iPBI); a dominance-based local search, i.e., e-MOEA-LS; an indicator-based local search, i.e., IBEA-LS; and a state-of-the-art local search with dual populations, i.e., ND/DPP-LS; on two well-known CMOPs. The experimental results show that CoMOLS/D significantly outperforms the compared algorithms on most of the test instances.

Identifying preferred solutions in multiobjective combinatorial optimizationproblems

We develop an evolutionary algorithm for multiobjective combinatorial optimization problems. The algorithm aims at converging the preferred solutions of a decision-maker. We test the performance of the algorithm on the multiobjective knapsack and multiobjective spanning tree problems. We generate the true nondominated solutions using an exact algorithm and compare the results with those of the evolutionary algorithm. We observe that the evolutionary algorithm works well in approximating the solutions in the preferred regions.

Evaluation method for Green jack-up drilling platform design scheme based on improved grey correlation analysis

This paper presents an evaluation method for green jack-up drilling platform(GJDP) design scheme based on the improved grey correlation analysis(IGCA). Under the situation of the low-carbon economy it is important and complicated for jack-up drilling platform design to select the optimum design scheme from the feasible schemes. In this paper, the evaluation index system for GJDP has been proposed which includes 3 factors: advanced technology, economic feasibility and environmental coordination, as well as 13 sub-indices. At the same time, a multi-objective combinatorial optimization model is established by using IGCA, in which the analysis hierarchy process (AHP) is used to obtain the weight values of the indices. The IGCA method is fully considered the connection of the indices reflecting the objective true nature of the things, and the subjective interest of the platform owner and the needs of the designer. This research can not only conduct to green design of jack-up drilling platform, but also can contribute to the development, popularization and application of low-carbon technology on offshore platform. Finally, the case study results demonstrate the feasibility and efficiency of the proposed evaluation method.

Multi-objective unconstrained combinatorial optimization: a polynomial bound on the number of extreme supported solutions

The multi-objective unconstrained combinatorial optimization problem (MUCO) can be considered as an archetype of a discrete linear multi-objective optimization problem. It can be interpreted as a specific relaxation of any multi-objective combinatorial optimization problem with linear sum objective function. While its single criteria analogon is analytically solvable, MUCO shares the computational complexity issues of most multi-objective combinatorial optimization problems: intractability and NP-hardness of the $$\varepsilon $$ -constraint scalarizations. In this article interrelations between the supported points of a MUCO problem, arrangements of hyperplanes and a weight space decomposition, and zonotopes are presented. Based on these interrelations and a result by Zaslavsky on the number of faces in an arrangement of hyperplanes, a polynomial bound on the number of extreme supported solutions can be derived, leading to an exact polynomial time algorithm to find all extreme supported solutions. It is shown how this algorithm can be incorporated into a solution approach for multi-objective knapsack problems.

Sustainability Assessment of Constructive Solutions for Urban Spain: A Multi-Objective Combinatorial Optimization Problem

Industrial areas are set up on plots of roads and associated infrastructure. These use materials and machinery that have environmental impacts, and thus require constructive solutions throughout their lifecycles. In turn, these solutions and their components cause environmental impacts that can be measured by sustainability indicators. The concept of sustainability is closely tied to sustainable development, which is defined as “development that meets the needs of the present, without compromising the ability of future generations to meet their own needs”. The large number of possible and available solutions means that identifying the best one for a given road section must employ a set of heuristic techniques, which conceptualize the issue as a combinatorial optimization problem that is purely discrete and non-differential. The system chosen can be based on a genetic algorithm method that differentiates individuals based on three sustainability indicators: CO2 emissions, embedded energy (also known as embodied energy, defined as the energy expended to manufacture a product), and economic cost. In this paper, we supplement traditional cost analyses using a three-objective multi-objective genetic algorithm that considers the aforementioned criteria, thus addressing sustainability in aggregate planning. The procedure is applied to three objective functions—CO2 emissions, economic cost and embedded energy—for each possible solution. We used the non-dominated sorting genetic algorithm (NSGA-II) to implement multi-objective optimization in MATLAB. Additional results for a random walk and multi-objective search algorithm are shown. This study involved 26 design variables, with different ranks of variation, and the application of the algorithm generates results for the defined Pareto fronts. Our method shows that the optimal approach effectively solves a real-world multi-objective project planning problem, as our solution is one of the Pareto-optimal solutions generated by the NSGA-II.

Optimizing a linear function over the nondominated set of multiobjective integer programs

AbstractIn this paper, we develop two algorithms to optimize a linear function over the nondominated set of multiobjective integer programs. The algorithms iteratively generate nondominated points and converge to the optimal solution reducing the feasible set. The first algorithm proposes improvements to an existing algorithm employing a decomposition and search procedure in finding a new point. Differently, the second algorithm maximizes one of the criteria throughout the algorithm and generates new points by setting bounds on the linear function value. The decomposition and search procedure in the algorithms is accompanied by problem‐specific mechanisms in order to explore the objective function space efficiently. The algorithms are designed to produce solutions that meet a prespecified accuracy level. We conduct experiments on multiobjective combinatorial optimization problems and show that the algorithms work well.

Optimization of workcell layout for hybrid medical device fabrication

Manufacturers often overlook the importance of optimizing workcell layout. Optimizing manufacturing workcell layouts can help increase production efficiency by reducing unnecessary actions and movements. However, the application of conventional optimization methods to modern production lines is not only becoming too complex, but also becoming inaccurate and unreliable. In this paper, we proposed an accurate and practical methodology that is readily implementable. The methodology uses multi-objective combinatorial optimization (MOCO) concepts to optimize manufacturing workcell layouts, specifically for the production of hybrid medical devices. The proposed model, also known as the Two-Step MOCO model, considers objectives and constraints that are selected by the manufacturer. The first step determines the optimal number of equipment and operators required given a fixed budget, while the second step determines the optimal positions for every equipment, given a fixed space and the required safety distance among each equipment, after considering the results of the first step. We illustrated the implementation of the proposed model on the fabrication of a hybrid medical device. Virtual manufacturing workcells were developed and utilized to evaluate the proposed methodology. The evaluation results proved the feasibility and effectiveness of the proposed model, and its ability to provide flexibility to manufacturers. This paper highlights the potential of our proposed optimization model in improving production efficiency in medical device and other manufacturing industries.

Multi-objective evolutionary algorithm on simplified bi-objective minimum weight minimum label spanning tree problems

As general purpose optimisation methods, evolutionary algorithms have been efficiently used to solve multi-objective combinatorial optimisation problems. However, few theoretical investigations have been conducted to understand the efficiency of evolutionary algorithms on such problems, and even fewer theoretical investigations have been conducted on multi-objective combinatorial optimisation problems coming from real world. In this paper, we analyse the performance of a simple multi-objective evolutionary algorithm on two simplified instances of the bi-objective minimum weight minimum label spanning tree problem, which comes from real world. Though these two instances are similar, the analysis results show that the simple multi-objective evolutionary algorithm is efficient for one instance, but it may be inefficient for the other. According to the analysis on the second instance, we think that the restart strategy may be useful in making the multi-objective evolutionary algorithm more efficient for the bi-objective problem.

Path Optimization of Hazardous Chemicals Transportation Based on Improved NSGA-II Algorithm

Different stakeholders have different concerns about the transportation of hazardous chemicals. Multi-objective optimization comprehensively considering transportation risks and transportation interests is the primary demand for the transportation of hazardous chemicals. For this, this paper studies the multi-objective combinatorial optimization problem of multiple hazardous chemicals in the same distribution range of transportation network. Firstly, according to the transportation characteristics of hazardous chemicals, the calculation methods of distribution paths in terms of physical dissimilarity and spatial dissimilarity were proposed. Then, a multi-objective optimization model of hazardous chemicals transportation was established with transportation risk, cost and time as optimization objectives, and the model was solved by the improved NSGA-II algorithm. Finally, the feasibility of the optimization method was verified in practical examples. The research results show that the proper increase of transportation risk value can greatly reduce the transportation cost and transportation time. When the calculation results only emphasize the optimal risk, there exist a large number of shared road sections in different transportation planning paths, so that the transportation risks are concentrated; however, after setting the threshold and increasing the constraint conditions, the phenomenon of the shared road section is significantly reduced, and the distribution of the hazardous chemicals transportation risks is relatively dispersed, to further control the potential hazards. This shall provide a theoretical reference for the risk regulation of hazardous chemicals transportation and the cost control of carriers.

PPLS/D: Parallel Pareto Local Search Based on Decomposition.

Pareto local search (PLS) is a basic building block in many metaheuristics for a multiobjective combinatorial optimization problem. In this paper, an enhanced PLS variant called parallel PLS based on decomposition (PPLS/D) is proposed. PPLS/D improves the efficiency of PLS using the techniques of parallel computation and problem decomposition. It decomposes the original search space into L subregions and executes L parallel processes searching in these subregions simultaneously. Inside each subregion, the PPLS/D process is guided by a unique scalar objective function. PPLS/D differs from the well-known two phase PLS in that it uses the scalar objective function to guide every move of the PLS procedure in a fine-grained manner. In the experimental studies, PPLS/D is compared against the basic PLS and a recently proposed PLS variant on the multiobjective unconstrained binary quadratic programming problems and the multiobjective traveling salesman problems with, at most, four objectives. The experimental results show that regardless of whether the initial solutions are randomly generated or generated by heuristic methods, PPLS/D always performs significantly better than the other two PLS variants.

Experimental analysis of design elements of scalarizing function-based multiobjective evolutionary algorithms

In this paper, we systematically study the influence of the main design elements of scalarizing function-based multiobjective evolutionary algorithms (MOEAs) on the performance of these algorithms. Such algorithms proved to be very successful in multiple computational experiments and practical applications. Well-known examples of this class of MOEAs are Jaszkiewicz’s multiobjecitve genetic local search and multiobjective evolutionary algorithm based on decomposition (MOEA/D). The two algorithms share the same common structure and differ in two aspects, i.e., the selection of parents for recombination and the selection of weight vectors of scalarizing functions. Using three different multiobjective combinatorial optimization problems, i.e., the multiobjective symmetric traveling salesperson problem, the traveling salesperson problem with profits, and the multiobjective set covering problem, we show that the design element with the highest influence on the performance is the choice of a mechanism for parents selection, while the selection of weight vectors, either random or evenly distributed, has practically negligible influence if the number of evenly distributed weight vectors is sufficiently large.

Estimating the number of basins of attraction of multi-objective combinatorial problems

The efficiency of local search is proportional to the number and the distribution of basins of attraction. Often combinatorial optimisation problems have a large number of local optima, uncountable with available computational resources. Approximating the number of basins of attraction and the minimal number of samples for visiting all basins at least once are complex problems for which we assume specific distributions of basins of attraction. We define two types of basins of attraction of multi-objective combinatorial optimisation problems with complementary properties. Acknowledging that each local search run generates a Pareto front of solutions, either each Pareto local solution corresponds to a basin of attraction, or a Pareto basin matches an entire Pareto local front. Simulations compare parametric and non-parametric estimators on bi-objective quadratic assignment problem instances.

A Random Forest-Assisted Evolutionary Algorithm for Data-Driven Constrained Multiobjective Combinatorial Optimization of Trauma Systems.

Many real-world optimization problems can be solved by using the data-driven approach only, simply because no analytic objective functions are available for evaluating candidate solutions. In this paper, we address a class of expensive data-driven constrained multiobjective combinatorial optimization problems, where the objectives and constraints can be calculated only on the basis of a large amount of data. To solve this class of problems, we propose using random forests (RFs) and radial basis function networks as surrogates to approximate both objective and constraint functions. In addition, logistic regression models are introduced to rectify the surrogate-assisted fitness evaluations and a stochastic ranking selection is adopted to further reduce the influences of the approximated constraint functions. Three variants of the proposed algorithm are empirically evaluated on multiobjective knapsack benchmark problems and two real-world trauma system design problems. Experimental results demonstrate that the variant using RF models as the surrogates is effective and efficient in solving data-driven constrained multiobjective combinatorial optimization problems.

CEA: Combination-based gene set functional enrichment analysis

Functional enrichment analysis is a fundamental and challenging task in bioinformatics. Most of the current enrichment analysis approaches individually evaluate functional terms and often output a list of enriched terms with high similarity and redundancy, which makes it difficult for downstream studies to extract the underlying biological interpretation. In this paper, we proposed a novel framework to assess the performance of combination-based enrichment analysis. Using this framework, we formulated the enrichment analysis as a multi-objective combinatorial optimization problem and developed the CEA (Combination-based Enrichment Analysis) method. CEA provides the whole landscape of term combinations; therefore, it is a good benchmark for evaluating the current state-of-the-art combination-based functional enrichment methods in a comprehensive manner. We tested the effectiveness of CEA on four published microarray datasets. Enriched functional terms identified by CEA not only involve crucial biological processes of related diseases, but also have much less redundancy and can serve as a preferable representation for the enriched terms found by traditional single-term-based methods. CEA has been implemented in the R package CopTea and is available at http://github.com/wulingyun/CopTea/.

Parallel–Series Multiobjective Genetic Algorithm for Optimal Tests Selection With Multiple Constraints

It is known that optimal tests selection is an important issue in design for testability field. The selection is subjected to constraints of testability metrics. At the same time, test time and economic costs need to be minimized. The tests selection is a combinatorial multiobjective optimization problem. According to the schema theory, the more the amount of testability constraints, the more the constraints are violated. Therefore, the canonical genetic algorithm (GA) evolves slowly, and the Pareto-optimal solutions are less likely to be found. Based on these considerations, a parallel-series multiobjective GA (PSMOGA) is proposed. First, each test procedure is handled by a submultiobjective GA (MOGA) independently. The chromosome length of the $i$ th sub-MOGA is equal to the available tests number of the $i$ th test procedure. The sub-MOGAs are executed in parallel. The Pareto-optimal solutions to every procedure are saved for further process. Second, the MOGA is used to handle the optimal test selection problem for the whole product. The length of the chromosome is equal to the amount of the test procedures. The $i$ th gene can vary between one and $k_{i}$ , where $k_{i}$ is the solution amount of the $i$ th procedure. The genetic material from the subproblem will not be changed. Hence, the subconstraints will never be violated in the MOGA. The effectiveness and efficiency of the proposed method are verified by statistical experiments.

A Grid Weighted Sum Pareto Local Search for Combinatorial Multi and Many-Objective Optimization.

Combinatorial multiobjective optimization problems (CMOPs) are very popular due to their widespread applications in the real world. One common method for CMOPs is Pareto local search (PLS), a natural extension of single-objective local search (LS). However, classical PLS tends to reserve all of the nondominated solutions for LS, which causes the inefficient LS, as well as unbearable computational and space cost. Due to the aforementioned reasons, most PLS approaches can only handle CMOPs with no more than two objectives. In this paper, by combining the Pareto dominance and weighted sum (WS) approach in a grid system, the grid weighted sum dominance (gws-dominance) is proposed and integrated into PLS for CMOPs with multiple objectives. In the grid system, at most one representative solution is maintained in each grid for more efficient LS, thus largely reducing the computational and space complexity. The grid-based WS approach can further guide the LS in different grids for maintaining more widely and uniformly distributed Pareto front approximations. In the experimental studies, the grid WS PLS is compared with the classical PLS, three decomposition-based LS approaches [multiobjective evolutionary algorithm based on decomposition-LS (WS, Tchebycheff, and penalty-based boundary intersection)], a grid-based algorithm ( ϵ -MOEA), and a state-of-the-art hybrid approach (multiobjective memetic algorithm based on decomposition) on two sets of benchmark CMOPs. The experimental results show that the grid weighted sum Pareto local search significantly outperforms the compared algorithms and remains effective and efficient on combinatorial multiobjective and even many-objective optimization problems.

Research on Receiving Routes and Departure Routes Assignment Problem in High-speed Railway Station Based on SFLA

The assignment problem of receiving routes and departure routes in high-speed railway stations is a complex multi-objective combinatorial optimization problem. According to the characteristics of receiving routes and departure routes, assignment model of receiving routes and departure routes in High-speed railway stations is set up. Shuffled Frog Leaping Algorithm (short for SFLA) is adopted to solve that model. As using SFLA, it could reduce the amount of calculation effectively, improve the execution speed of the algorithm and increase the diversity of solutions. And with its unique mechanism of exchanging and sharing information, the solving process is not easy to fall into local optimum, which is beneficial to search the global optimal solution. Using SFLA offers a new method to set up the assignment model of receiving routes and departure routes in High-speed railway stations.

Full-Duplex Relaying for D2D Communication in Millimeter Wave-based 5G Networks

Device-to-device (D2D) communication, which can offload data from base stations by direct transmission between mobile devices, is a promising technology for fifth generation (5G) wireless networks. However, the limited battery capacity of mobile devices is a barrier to fully exploiting the benefits of D2D communication. Meanwhile, high data rate D2D communication is required to support the increasing traffic demand of emerging applications. In this paper, we study relay-assisted D2D communication in millimeter wave (mmWave)-based 5G networks to address these issues. Multiple D2D user pairs are assisted by full-duplex relays that are equipped with directional antennas. To design an efficient relay selection and power allocation scheme, we formulate a multi-objective combinatorial optimization problem, which balances the trade-off between total transmit power and system throughput. The problem is transformed into a weighted bipartite matching problem. We then propose a centralized relay selection and power allocation algorithm and prove that it can achieve a Pareto optimal solution in polynomial time. We further propose a distributed algorithm based on stable matching. Simulation results show that our proposed algorithms substantially reduce the total transmit power and improve the system throughput compared with two existing algorithms in the literature.

A Simulated Annealing-Based Multiobjective Optimization Algorithm for Political Districting

Redistricting consists of partitioning a set of basic units into a given number of larger groups for electoral purposes. These groups should be designed to fulfill federal and state requirements such as contiguity, population equality and compactness to promote democratic fairness. Political districting can be modeled as a multiobjective combinatorial optimization problem where the criteria are often difficult to optimize. In fact, it has been proven to be a computationally intractable (NP-hard) problem. Due to these reasons the use of heuristics to provide good approximations in a reasonable amount of time is justified. In the literature, most approaches manage the different objectives as mono-objective through weighted aggregation functions and the use of Pareto search techniques is limited. In this work, a multi-objective meta-heuristic algorithm based on simulated annealing that uses Pareto dominance during the search process is proposed and applied to a bi-objective model to solve the problem. The current strategy applied by the Mexican national redistricting authorities is used to compare the performance of the algorithm on 12 real data sets. We conclude that the proposed algorithm can generate better quality solutions than its counterpart.

Survey and unification of local search techniques in metaheuristics for multi-objective combinatorial optimisation

Metaheuristics are algorithms that have proven their efficiency on multi-objective combinatorial optimisation problems. They often use local search techniques, either at their core or as intensification mechanisms, to obtain a well-converged and diversified final result. This paper surveys the use of local search techniques in multi-objective metaheuristics and proposes a general structure to describe and unify their underlying components. This structure can instantiate most of the multi-objective local search techniques and algorithms in literature.