Multi-objective Combinatorial Problems Research Articles

A Two-Level Fuzzy Multi-Objective Design of ATO Driving Commands for Energy-Efficient Operation of Metropolitan Railway Lines

Policies for reducing CO2 and other GHG emissions have motivated an increase in electrification in metropolitan areas, mandating reductions in energy consumption. Metro systems are keystone contributors to the sustainability of cities; they can reduce the energy consumption of cities through the use of the economic driving parameters in their onboard automatic train operation systems (ATO) and through the strategic design of efficient timetables. This paper proposes a two-level optimization method to design efficient, comfortable, and robust driving commands to be programmed in all the interstations of a metro line. This method aims to increase the sustainability of metro operations by producing efficient timetables with economic driving for each interstation while considering comfort restrictions and train mass uncertainty. First, in the eco-driving level, an optimal Pareto front between every pair of successive stations is obtained using a multi-objective particle swarm optimization algorithm with fuzzy parameters (F-MOPSO). This front contains optimized speed profiles for different running times considering train mass variations. The global problem is stated as a multi-objective combinatorial problem, and a fuzzy greedy randomized adaptive search procedure (F-GRASP) is used to perform an intelligent search for the optimal timetables. Thus, a global front of interstation driving commands is computed for the whole line, showing the minimum energy consumption for different travel times. This method is analyzed in a case study with real data from a Spanish metro line. The results are compared with the minimum running time timetable and a typical timetable design procedure. The proposed algorithms achieve a 24% reduction in energy consumption in comparison to the fastest driving commands timetable, representing a 4% increase in energy savings over the uniform timetable design.

Effects of the Random Forests Hyper-Parameters in Surrogate Models for Multi-Objective Combinatorial Optimization: A Case Study using MOEA/D-RFTS

Surrogate models are techniques to approximate the objective functions of expensive optimization problems. Recently, Random Forests have been studied as a surrogate model technique for combinatorial optimization problems. Nonetheless, Random Forests contain several hyper-parameters that are used to control the prediction process. Despite their importance, research on the effects of these hyper-parameters is scarce. Therefore, this paper performs a systematic investigation of the effects of different combinations of values for the Random Forest hyper-parameters on the approximation of well-known multi-objective combinatorial benchmark problems. The results show that the number of samples to consider when building each tree and the minimum number of samples to be at the leaf node are the two most important hyper-parameters in this context.

Study on Multi-Objective Optimization of Sponge Facilities Combination at Urban Block Level: A Residential Complex Case Study in Nanjing, China

Urban block-scale sponge system design needs address how to specify the optimal approach to combine the number of areas and types of sponge facilities for diverse land conditions and sponge system design objectives, while ensuring sponge performance and economic efficiency. With the gradual application of multi-objective optimization algorithms in the design of sponge cities, multi-objective combinatorial problem solving for sponge facilities based on optimization algorithms is more accurate and efficient than traditional design methods based on the designer’s experience. This study utilizes a residential complex in Nanjing as a practical example, selects six types of typical sponge facilities to construct a multi-objective optimization combination model for sponge facilities, and employs the SPEA-2 algorithm to determine the optimal combination of sponge facility types and quantities. Finally, 186,754 combinations of sponge facilities were calculated. For the three sponge objectives of optimal performance and economy for stormwater infiltration and storage, optimal performance and economy for runoff pollution control, and optimal average overall performance for stormwater infiltration, runoff pollution control, and economy, a number of combinations of sponge types and numbers were obtained.

Objective space division-based hybrid evolutionary algorithm for handing overlapping solutions in combinatorial problems.

Overlapping solutions occur when more than one solution in the space of decisions maps to the same solution in the space of objectives. This situation threatens the exploration capacity of Multi-Objective Evolutionary Algorithms (MOEAs), preventing them from having a good diversity in their population. The influence of overlapping solutions is intensified on multi-objective combinatorial problems with a low number of objectives. This paper presents a hybrid MOEA for handling overlapping solutions that combines the classic NSGA-II with a strategy based on Objective Space Division (OSD). Basically, in each generation of the algorithm, the objective space is divided into several regions using the nadir solution calculated from the current generation solutions. Furthermore, the solutions in each region are classified into non-dominated fronts using different optimization strategies in each of them. This significantly enhances the achieved diversity of the approximate front of non-dominated solutions. The proposed algorithm (called NSGA-II/OSD) is tested on a classic Operations Research problem: the Multi-Objective Knapsack Problem (0-1 MOKP) with two objectives. Classic NSGA-II, MOEA/D and Global WASF-GA are used to compare the performance of NSGA-II/OSD. In the case of MOEA/D two different versions are implemented, each of them with a different strategy for specifying the reference point. These MOEA/D reference point strategies are thoroughly studied and new insights are provided. This paper analyses in depth the impact of overlapping solutions on MOEAs, studying the number of overlapping solutions, the number of solution repairs, the hypervolume metric, the attainment surfaces and the approximation to the real Pareto front, for different sizes of 0-1 MOKPs with two objectives. The proposed method offers very good performance when compared to the classic NSGA-II, MOEA/D and Global WASF-GA algorithms, all of them well-known in the literature.

Dendrogram-directed design space exploration for decision-making of a multi-objective combinatorial design problem

Dendrogram-directed design space exploration for decision-making of a multi-objective combinatorial design problem

Min cost improvement and max gain stability in multicriteria sorting methods on combinatorial domains

AbstractVarious multicriteria sorting methods have been proposed in the literature to assign the feasible alternatives into predefined categories. We consider here problems involving a set of totally ordered categories representing different achievement levels in the satisfaction of criteria. As in many existing methods, the assignment rule of an alternative to a category is based on the comparison of its performance vector to reference profiles defining lower bounds of the categories. Within this standard setting we address a new problem that consists in finding how to modify a given solution, within a combinatorial set of alternatives, to upgrade it in the upper category (or higher) at minimum cost. We also consider the problem of identifying the sequence of solutions that minimize the total cost while satisfying some budget constraint at every step, and the problem of determining how to modify the current solution to save money while staying in the same category. We first propose a general approach based on mixed integer (linear or quadratic) programming to solve these problems. Then, we implement this approach on various multiobjective combinatorial problems, such as multi‐agent assignment problems and multiobjective knapsack problems. Numerical tests are provided to establish the feasibility of the approach on instances of different sizes.

Dendrogram-Based Structural Exploration of Solution Space for Decision-Making of Multi-objective Combinatorial Design Problem

When mathematical optimization is applied to a multi-objective design problem, it is the common practice of decision support to generate Pareto optimal solutions as alternatives and clarify their trade-offs. The decision-maker selects any design solution according to his or her preferences. Although any comparison of alternatives is beneficial for its selection, in the case of combinatorial design problems, a simple comparison study cannot find the significant relationships between alternatives because of their discrete nature. This paper proposes a decision-making support method based on the dendrogram of hierarchical clustering. It is assumed that an optimal solution can be represented as a piling-up of smaller pieces, which correspond to building blocks, in the combinatorial solution space. The distance of alternatives is surveyed using schema representation defined in various levels. The structure of the solution space is reconstructed into a dendrogram through hierarchical clustering. When essential parts of schemas are identified as building blocks, they are organized into a tree-shaped graph. It is expected to help the designer's exploration into the solution space.

Multi-objective communication-aware optimization for virtual machine placement in cloud datacenters

This paper formulates a new multi-objective virtual machine placement (VMP) problem, which is a challenging task in cloud datacenters (DCs). In cloud environment, there are two stakeholders, namely, users and providers. Both sides try to take more benefit whereas a trade-off between conflicting benefits is crucial. From providers’ perspective, power consumption and resource wastage are two objectives to be optimized whereas gaining high quality of service (QoS) is a critical point for users. The unpleasant issue that a user endures in the cloud environment is network delay; this is affected by common bandwidth linkage which is shared between different users’ applications; the reason for considering bandwidth usage optimization as the third objective function in users’ viewpoint. However, inefficient network bandwidth usage has drastic impact on overall performance even makes network links to get saturated and throttles communication-intensive applications. Therefore, VMs with high affinity and traffic dependency must be physically placed as close as possible so less traffic is sent on network layers. To figure out this combinatorial multi-objective problem, we extend a hybrid multi-objective genetic-based optimization solution. To evaluate this work, we conducted extensive scenarios with variable correlation coefficients between resources in requested VMs. The simulation results prove that our proposed hybrid meta-heuristic algorithm outperforms against state-of-the-art ACO-based, well-known heuristic-based FFD algorithms, and random-based approach in terms of total power consumption, resource wastage, the total data transfer rate in network, and number of active servers in use. Also, the simulations in larger search space demonstrated proposed approach has high potential of scalability.

A Decision-Making Tool for a Complex Post Enrollment-Based Course Timetabling

The article describes a new method to construct an enrollment-based course timetable in universities, based on a multi-objective optimization model. The model used mixed-integer and binary variables towards creating a schedule. It satisfies students' preferences for study time, with the number of students in the same class being optimal for training costs while ensuring timetabling business constraints. We use a combination of compromise programming and linear scalarizing to transform many objective functions into single-objective optimization. A scheme of the Genetic Algorithm was developed to solve the proposed model. The proposed method allows approaching several types of multi-objective combinatorial problems. The algorithm was tested by scheduling a study schedule for 3,000 students in the spring semester of 2020 at FPT University, Hanoi, Vietnam. The obtained results show the average students' preference level of 69%. More than 30% of students have a satisfaction level of more than 80% of the timetable after two hours of execution time.

Web service API recommendation for automated mashup creation using multi-objective evolutionary search

Modern software development builds on external Web services reuse as a promising way that allows developers delivering feature-rich software by composing existing Web service Application Programming Interfaces, known as APIs. With the overwhelming number of Web services that are available on the Internet, finding the appropriate Web services for automatic service composition, i.e., mashup creation, has become a time-consuming, difficult, and error-prone task for software designers and developers when done manually. To help developers, a number of approaches and techniques have been proposed to automatically recommend Web services. However, they mostly focus on recommending individual services. Nevertheless, in practice, service APIs are intended to be used together forming a social network between different APIs, thus should be recommended collectively. In this paper, we introduce a novel automated approach, called SerFinder, to recommend service sets for automatic mashup creation. We formulate the service set recommendation as a multi-objective combinatorial problem and use the non-dominated sorting genetic algorithm (NSGA-II) as a search method to extract an optimal set of services to create a given mashup. We aim at guiding the search process towards generating the adequate compromise among three objectives to be optimized (i) maximize services historical co-usage, (ii) maximize services functional matching with the mashup requirements, and (iii) maximize services functional diversity. We perform a large-scale empirical experiment to evaluate SerFinder on a benchmark of real-world mashups and services. The obtained results demonstrate the effectiveness of SerFinder in comparison with recent existing approaches for mashup creation and services recommendation. The statistical analysis results provide an empirical evidence that SerFinder, significantly outperforms four state-of-the-art widely-used multi-objective search-based algorithms as well as random search.

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.

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 multi-population algorithm for multi-objective knapsack problem

Local search algorithms constitute a growing area of interest to approximate the Pareto sets of multi-objective combinatorial problem instances. In this study, we focus on the multi-objective knapsack problem and its optimization thanks to a multi-population based cooperative framework. The proposed approach, Wϵ-CMOLS, uses a multi-objective local search algorithm based on quality indicator (IBMOLS), initially presented by Basseur and Burke in 2007, and integrates it into a cooperative model. The idea is to optimize the overall quality of a Pareto set approximation by evolving several sub-populations in parallel, each population executing a different configuration of IBMOLS. The algorithm uses a weighted version of the epsilon quality indicator by means of different weight vectors. The populations cooperate through sharing a non-dominated archive, which stores the best compromises found during the optimization process, and which is used to re-initialize regularly each sub-population. Wϵ-CMOLS is compared with state-of-the-art algorithms such as IBEA, NSGA-II and SPEA2. Experiments highlight that the use of a cooperative model as well as a weighted indicator to guide the search toward different directions, can lead to interesting results for the multi-objective knapsack problem.

Water Cycle Algorithm for Small Scale Electrical Economic Power Dispatch Problem

Efficient and optimal planning of economic electrical power dispatch problem is an integral part of economic electrical energy generation planning and it is the need of time for the electrical engineers to browse this area in multi-scale planning scenarios. Intelligent and Optimization techniques based on evolutionary computing, metaheuristic,biological base,nature inspired, search method establish their applications in the area of electrical economic power dispatch planning(EEPDP) to reach global optimal solution for this multi scale, multi-decision, multi-objective combinatorial problem subjected to different constraints. In this paper, water cycle algorithm (WCA) has been proposed to solve electrical economic power dispatch problem for three and six unit system.This is based on how the streams and rivers flow downhill towards the sea and recycle in nature The suggested technique is tested on small scale system of three and six unit system of EEPDP considering various equality and inequality constraints . Test results are compared with other techniques grey wolf optimization(GWO),cuckoo search(CS),artificial bee colony(ABC),firefly algorithm(FA),particle swarm optimization(PSO),shuffled frog leap (SFL) ,bacteria foraging algorithm(BFO),harmony search(HS) applied in literature.Convergence of solution with iteration is presented for both cases.Simulation results proved that the WCA technique is better as compared to other nature inspired,heuristic,metaheuristic techniques to find global minima and maintain the solution quality in terms of low fuel cost.

Ant Lion Optimizer based Approach for Optimal Scheduling of Thermal Units for Small Scale Electrical Economic Power Dispatch Problem

A novel nature inspired algorithm ant lion optimizer (ALO) is recently developed which is motivated from the hunting mechanism of ant lions .Inherit steps of hunting prey such as the random walk of ants, building traps, entrapment of ants in traps, catching preys, and re-building are simulated to find the optimal solution of real life problems . Intelligent and Optimization techniques based on evolutionary computing, metaheuristic,biological base,nature inspired, search method establish their applications in the area of electrical economic power dispatch planning(EEPDP) to reach global optimal solution for this multi scale, multi-decision, multi-objective combinatorial problem subjected to different constraints.An application of ALO to solve non linear electric economic power dispatch problem(EEPDP) is proposed in this paper. Efficient and optimal planning of economic electrical power dispatch problem is an integral part of economic electrical energy generation planning and it is the need of time for the electrical engineers to browse this area in multi-scale planning scenarios.. The performance of s ant lion optimizer (ALO) to solve electrical economic power dispatch problem is tested on three and six unit system.Test results are compared with other techniques grey wolf optimization(GWO),cuckoo search(CS),artificial bee colony(ABC),firefly algorithm(FA),particle swarm optimization(PSO),shuffled frog leap (SFL) ,bacteria foraging algorithm(BFO),harmony search(HS) applied in literature. Simulation results proved that the ALO technique is better as compared to other nature inspired,heuristic,metaheuristic techniques to find global minima and maintain the solution quality in terms of low fuel cost.

Solution to Economic Power Dispatch Planning Problem considering Generator Constraints using Artificial Bee Colony Algorithm

Optimum active and reactive power dispatch is an inherent part of power system generation planning and it is the need of an hour for the electrical utilities and power engineers to dig this area in short and long term planning scenarios. Load demand requirements subjected to economic feasible solutions matching voltage profile, power demand, minimization of losses, voltage stability and improve the capacity of the system is the need of the hour. Optimization techniques based on evolutionary computing, artificial intelligence, search method finds their applications in the area of economic load dispatch planning to reach global optimal solution for this multi-decision, multi-objective combinatorial problem subjected to different constraints. In this paper, ant bee colony based algorithm has been proposed to solve economic dispatch problem. Unlike other heuristic algorithms, ABC utilizes search space as multi-step decision process. It possesses a flexible and well-balanced operator to enhance and adapt the global and fine tune local search to follow the minimum cost path in the search boundary. The suggested technique is tested on IEEE 25 bus system. Test results are compared with other techniques presented in literature.

A multi-objective military system of systems architecting problem with inflexible and flexible systems: formulation and solution methods

System of systems (SoS) architecting is the process of bringing together and connecting a set of systems so that the collection of the systems, i.e., the SoS is equipped with a set of required capabilities. A system is defined as inflexible in case it contributes to the SoS with all of the capabilities it can provide. On the other hand, a flexible system can collaborate with the SoS architect in the capabilities it will provide. In this study, we formulate and analyze a SoS architecting problem representing a military mission planning problem with inflexible and flexible systems as a multi-objective mixed-integer-linear optimization model. We discuss applications of an exact and an evolutionary method for generating and approximating the Pareto front of this model, respectively. Furthermore, we propose a decomposition approach, which decomposes the problem into smaller sub-problems by adding equality constraints, to improve both the exact and the evolutionary methods. Results from a set of numerical studies suggest that the proposed decomposition approach reduces the computational time for generating the exact Pareto front as well as it reduces the computational time for approximating the Pareto front while not resulting in a worse approximated Pareto front. The proposed decomposition approach can be easily used for different problems with different exact and heuristic methods and thus is a promising tool to improve the computational time of solving multi-objective combinatorial problems. Furthermore, a sample scenario is presented to illustrate the effects of system flexibility.

Real-Time Multifault Rush Repairing Strategy Based on Utility Theory and Multiagent System in Distribution Networks

The problem of multifault rush repair in distribution networks (DNs) is a multiobjective dynamic combinatorial problem with topology constraints. The problem consists of archiving an optimal faults’ allocation strategy to squads and an admissible multifault rush repairing strategy with coordinating switch operations. In this article, the utility theory is introduced to solve the first problem and a new discrete bacterial colony chemotaxis (DBCC) algorithm is proposed for the second problem to determine the optimal sequence for each squad to repair faults and the corresponding switch operations. The above solution is called the two-stage approach. Additionally, a double mathematical optimization model based on the fault level is proposed in the second stage to minimize the outage loss and total repairing time. The real-time adjustment multiagent system (RA-MAS) is proposed to provide facility to achieve online multifault rush repairing strategy in DNs when there are emergencies after natural disasters. The two-stage approach is illustrated with an example from a real urban distribution network and the simulation results show the effectiveness of the two-stage approach.

Hybridization of decomposition and local search for multiobjective optimization.

Combining ideas from evolutionary algorithms, decomposition approaches, and Pareto local search, this paper suggests a simple yet efficient memetic algorithm for combinatorial multiobjective optimization problems: memetic algorithm based on decomposition (MOMAD). It decomposes a combinatorial multiobjective problem into a number of single objective optimization problems using an aggregation method. MOMAD evolves three populations: 1) population P(L) for recording the current solution to each subproblem; 2) population P(P) for storing starting solutions for Pareto local search; and 3) an external population P(E) for maintaining all the nondominated solutions found so far during the search. A problem-specific single objective heuristic can be applied to these subproblems to initialize the three populations. At each generation, a Pareto local search method is first applied to search a neighborhood of each solution in P(P) to update P(L) and P(E). Then a single objective local search is applied to each perturbed solution in P(L) for improving P(L) and P(E), and reinitializing P(P). The procedure is repeated until a stopping condition is met. MOMAD provides a generic hybrid multiobjective algorithmic framework in which problem specific knowledge, well developed single objective local search and heuristics and Pareto local search methods can be hybridized. It is a population based iterative method and thus an anytime algorithm. Extensive experiments have been conducted in this paper to study MOMAD and compare it with some other state-of-the-art algorithms on the multiobjective traveling salesman problem and the multiobjective knapsack problem. The experimental results show that our proposed algorithm outperforms or performs similarly to the best so far heuristics on these two problems.

Vulnerability Assessment of Water Distribution Networks under Seismic Actions

AbstractA methodology to analyze the vulnerability of water distribution networks (WDNs) to earthquakes by means of risk assessment is presented. The consequences of multiple pipe failures due to earthquakes are investigated in terms of unsupplied demand to customers. To this aim the steady-state WDN analysis is performed considering system topology changes due to closing isolation valves in order to separate the network segments where failures occur. The pipe failure probability is calculated using fragility curves from the American Lifelines Association (ALA). The identification of the worst pipe failure scenarios as trade-offs between unsupplied demand and probability of occurring is formulated as a multiobjective combinatorial problem and solved using a multiobjective genetic algorithm as optimization strategy. The methodology is applied to the Exnet network. Results show that WDN seismic vulnerability depends also on network segmentation due to the existing isolation valve system. The methodology all...