Multi-objective Combinatorial Optimization Research Articles

Core-guided method for constraint-based multi-objective combinatorial optimization

Multi-Objective Combinatorial Optimization(MOCO), which consists of several conflicting objectives to be optimized, finds an ever-increasing number of uses in many real-world applications. In past years, the research of MOCO mainly focuses on evolutionary algorithms. Recently, constraint-based methods come into the view and have been proved to be effective on MOCO problems. This paper builds on the previous works of constraint-based algorithm MCSEnumPD(AAAI-18) using path diversification method. Due to the inadequacy that the original method fails to prune the redundant search space effectively, this paper proposes the definition of infeasible path and develops a novel algorithm that exploits the properties of unsatisfiable cores, referred as CgPDMCS. The approach extends MCSEnumPD algorithm with a core-guided path diversification method, which avoids solving infeasible paths representing the supersets of the unsatisfiable cores. Experimental results show that the novel approach provides further performance gains over the previous constraint-based algorithms, especially for the instances tightly constrained.

A New Scatter Search Design for Multiobjective Combinatorial Optimization with an Application to Facility Location

Metaheuristic optimization is at the heart of the intersection between computer science and operations research. The INFORMS Journal of Computing has been fundamental in advancing the ideas behind metaheuristic methodologies. Fred Glover’s “Tabu Search—Part I” was published more than 30 years ago in the first volume of the then ORSA Journal on Computing. This article, one of the most cited in the area of heuristic optimization, paved the way for many contributions to the methodology and practice of operations research. As a continuation of this stream of research, we describe a new scatter search design for multiobjective optimization. The design includes a short-term memory tabu search and a path relinking combination method. We show how the strategies and mechanisms within scatter search and tabu search can be combined to produce a highly effective approach to multiobjective optimization.

Analysis of multiobjective evolutionary algorithms on the biobjective traveling salesman problem (1,2)

Multiobjective evolutionary algorithms have been successfully used to deal with multiobjective combinatorial optimization problems for more than two decades. However, we know little about the performance of multiobjective evolutionary algorithms on multiobjective combinatorial optimization problems in theory so far, especially on NP-hard ones from real-world, since Pareto curves are often of exponential size, meanwhile, evolutionary algorithms rely heavily on the use of randomness and are hard to understand from a theoretical point of view. In this paper, we theoretically investigate the performance of two simple multiobjective evolutionary algorithms with different population diversity mechanisms on the biobjective traveling salesman problem (1,2). It is found that one of them can efficiently find a $\frac {3}{2}$ -approximation Pareto curve for the problem, the best result so far. At the same time, these two multiobjective evolutionary algorithms are proved to be superior to a multiobjective local search algorithm, and the multiobjective local search algorithm is also proven to outperform these two multiobjective evolutionary algorithms as well. Finally, the population diversity is proved to be helpful in reducing the expected runtime of multiobjective evolutionary algorithm.

Industry choice for an airport economic zone by multi-objective optimization

Airport Economic Zone (AEZ) is a suburban area whose infrastructure, land-use, and economy are centered on an airport. More than 100 AEZs are under planning and development in China. Industry choice is the critical issue for all the AEZs. To ensure that all the industries located in AEZs are closely related to airports and air transport, first, we create a pool of optional industries for AEZs based on the industrial input-output model. Second, we construct a multi-objective combinatorial optimization model subject to the constraints of land area of an AEZ, targeting the maximum number of employees, the largest GDP, the highest degree of aviation correlation, the fullest utilization of basic resources, and the greatest policy support. Third, to verify the feasibility of this model in practice, we choose the Zhengzhou Airport Economic Comprehensive Experimental Zone for case study. We divide these five objectives into two categories: economic objectives and non-economic objectives, and put different weights on them to describe different political preferences, and then discuss the calculation results in three scenarios. Finally, we come to the conclusion that comprehensive consideration of every characteristic of each industry can reasonably determine the scale of industrial land use and increase industrial economic benefit in AEZs. Our research provides a new quantitative and reliable method for industry choice and land use planning of an AEZ.

Multi‐objective hybrid decomposition and local dominance based meter placement for distribution system state estimation

The study proposes a new hybrid multi-objective evolutionary optimisation algorithm based on decomposition and local dominance for meter placement in distribution system state estimation. The evenly distributed qualitative and diverse solutions on the Pareto front are required for a decision-maker for selecting a final optimal solution. Such a Pareto front can be achieved by obtaining the balance between convergence and diversity of multi-objective optimisation algorithm. Therefore, the proposed method combined dominance and decomposition techniques, modelled meter placement as a constrained combinatorial multi-objective optimisation. The meter placement is designed as a trade-off between three objectives that are minimising the cost of the meters, average relative percentage error (ARPE) of voltage magnitude and ARPE of voltage angle. As the meter placement problem is a combinatorial optimisation, the binomial distribution-based Monte Carlo method is utilised to initialise the population, which aims to improve the diversity, as a consequence it improves the convergence, which is a by-product of this method. The results of the proposed method are compared with multi-objective evolutionary algorithm based on decomposition, non-dominated sorting genetic algorithm-II and with multi-objective hybrid particle swarm optimisation-krill herd algorithm, multi-objective hybrid estimation of distribution algorithm-interior point method and demonstrated on PG&E 69-bus distribution system and Practical Indian 85-bus distribution system.

A bi-objective heuristic approach for green identical parallel machine scheduling

Sustainability in manufacturing has become a fundamental topic in the scientific literature due to the preeminent role of manufacturing industry in total world energy consumption and carbon emission. This paper tackles the multi-objective combinatorial optimization problem of scheduling jobs on multiple parallel machines, while minimizing both the makespan and the total energy consumption. The electricity prices vary according to a time-of-use policy, as in many cases of practical interest. In order to face this problem, an ad-hoc heuristic method is developed. The first part of the method, called Split-Greedy heuristic, consists in an improved and refined version of the constructive heuristic (CH) proposed in Wang, Wang, Yu, Ma and Liu (2018). The second part, called Exchange Search, is a novel local search procedure aimed at improving the quality of the Pareto optimal solutions. The experimental results prove the effectiveness of the proposed method with respect to three competitors: CH, NSGA-III, and MOEA/D.

Trainingless multi-objective evolutionary computing-based nonintrusive load monitoring: Part of smart-home energy management for demand-side management

Electricity is the most widely used form of energy in modern society. One method of satisfying the continuously increasing industrial, commercial, and residential electrical-energy demands of consumers in smart grids is to use an Internet-of-things (IoT) service-oriented electrical-energy management system (EMS) to intrusively monitor and manage electrical loads, which can effectively react to demand-response schemes for demand-side management (DSM). Nonintrusive load monitoring (NILM), a viable cost-effective load disaggregation technique, has recently gained considerable attention as a nonintrusive alternative to EMS in the research field of smart grids. This paper presents a smart IoT-oriented home EMS founded on trainingless multi-objective evolutionary computing-based NILM for DSM in a smart grid. Evolutionary computing-based NILM is considered and addressed as a multi-objective combinatorial optimization problem. The proposed NILM technique can determine the electrical appliances based on their individual electrical characteristics extracted from composite electrical-load consumption with no intrusive deployment of smart plugs or power meters. A fully nonintrusive NILM alternative is considered and proposed. In addition, this alternative is different from conventional NILM because conventional NILM considers artificial intelligence including artificial neural networks (NNs) and deep NN as load classifiers of NILM where training and retraining stages and a hyperparameter tuning procedure are required. The proposed smart IoT-oriented home EMS was experimentally investigated with the trainingless multi-objective evolutionary computing-based NILM in a real house environment. The experimental results confirm that the proposed methodology is feasible.

A model of anytime algorithm performance for bi-objective optimization

Anytime algorithms allow a practitioner to trade-off runtime for solution quality. This is of particular interest in multi-objective combinatorial optimization since it can be infeasible to identify all efficient solutions in a reasonable amount of time. We present a theoretical model that, under some mild assumptions, characterizes the “optimal” trade-off between runtime and solution quality, measured in terms of relative hypervolume, of anytime algorithms for bi-objective optimization. In particular, we assume that efficient solutions are collected sequentially such that the collected solution at each iteration maximizes the hypervolume indicator, and that the non-dominated set can be well approximated by a quadrant of a superellipse. We validate our model against an “optimal” model that has complete knowledge of the non-dominated set. The empirical results suggest that our theoretical model approximates the behavior of this optimal model quite well. We also analyze the anytime behavior of an $$\varepsilon $$ -constraint algorithm, and show that our model can be used to guide the algorithm and improve its anytime behavior.

The multi-objective supplier selection problem with fuzzy parameters and solving the order allocation problem with coverage

PurposeThis study aims to deal with supplier selection problem. The supplier selection problem has significantly become attractive to researchers and practitioners in recent years. Many real-world supply chain problems are assumed as multiple objectives combinatorial optimization problems.Design/methodology/approachIn this paper, the authors propose a multi-objective model with fuzzy parameters to select suppliers and allocate orders considering multiple periods, multiple resources, multiple products and two-echelon supply chain. The objective functions consist of total purchase costs, transportation, order and on-time delivery, coverage and the weights of suppliers. Distance-based partial and general coverage of suppliers makes the number of orders of products more realistic. In this model, the weights of suppliers are determined by fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method, as a multi-criteria decision analysis method, in the objective function. Also, the authors consider the parameters related to delays as triangular fuzzy numbers.FindingsA small-sized numerical example is provided to clearly show the proposed model. The exact epsilon constraint method is used to solve this given multi-objective combinatorial optimization problem. Subsequently, the sensitivity analysis is conducted to testify the proposed model. The obtained results demonstrate the validity of the proposed multiple objectives mixed integer mathematical programming model and the efficiency of the solution approach.Originality/valueIn real-life situations, supplier selection parameters are uncertain and incomplete. Hence, the fuzzy set theory is used to tackle uncertainty. In this paper, a multi-objective supplier selection problem is formulated taking into consideration the coverage of suppliers and suppliers’ weights. Integrating coverage of suppliers to select and allocate the order to them can be mentioned as the main contribution of this study. The proposed model considers the delay from suppliers as fuzzy parameters.

An Interactive Regret-Based Genetic Algorithm for Solving Multi-Objective Combinatorial Optimization Problems

We propose a new approach consisting in combining genetic algorithms and regret-based incremental preference elicitation for solving multi-objective combinatorial optimization problems with unknown preferences. For the purpose of elicitation, we assume that the decision maker's preferences can be represented by a parameterized scalarizing function but the parameters are initially not known. Instead, the parameter imprecision is progressively reduced by asking preference queries to the decision maker during the search to help identify the best solutions within a population. Our algorithm, called RIGA, can be applied to any multi-objective combinatorial optimization problem provided that the scalarizing function is linear in its parameters and that a (near-)optimal solution can be efficiently determined when preferences are known. Moreover, RIGA runs in polynomial time while asking no more than a polynomial number of queries. For the multi-objective traveling salesman problem, we provide numerical results showing its practical efficiency in terms of number of queries, computation time and gap to optimality.

Half-open polyblock for the representation of the search region in multiobjective optimization problems: its application and computational aspects

The search region in multiobjective optimization problems is a part of the objective space where nondominated points could lie. It plays an important role in the generation of the nondominated set of multiobjective combinatorial optimization (MOCO) problems. In this paper, we establish the representation of the search region by half-open polyblocks (a variant concept of “polyblock” in monotonic optimization) and propose a new procedure for updating the search region. We also study the impact of stack policies to the new procedure and the existing methods that update the search region. Stack policies are then analyzed, pointing out their performance effectiveness by means of the results of rich computational experiments on finding the whole set of nondominated points of MOCO problems.

Allocation of coal de-capacity quota among provinces in China: A bi-level multi-objective combinatorial optimization approach

Coal de-capacity, or capacity cut, is an important part of China's energy transition. Formulating a quota allocation scheme for coal de-capacity is the key to realizing smooth exit of coal overcapacity. This study proposes a novel method of allocation of coal de-capacity quota among provinces, based on bi-level multi-objective combinatorial optimization. In this bi-level optimal allocation scheme (BOAS), the upper level is the central government and the lower level is the provincial governments. The results indicate that, because of the different costs of coal de-capacity in each province, the execution rate of each province for tasks assigned by the central government is quite different. Compared with the government allocation scheme (GAS) and the single-level optimal allocation scheme (SOAS), the growth rate of total factor productivity of the BOAS increases by 2.14% and 0.60%, respectively; the total de-capacity cost of BOAS has reduced 64 billion yuan and 19 billion yuan, respectively; and the environmental benefits of BOAS has increased 73 billion yuan and 71 billion yuan, respectively; the Gini coefficient of BOAS calculated by various indexes is less than 0.3, placing the scheme within the category of considerable or absolute fairness. In addition, the proposed allocation model truly reflects the complex dynamics of the game process of China's coal overcapacity governance system, and can provide a more effective decision-making reference for the Chinese government in formulating the allocation scheme of coal de-capacity. • A novel allocation model based on bi-level multi-objective optimization is constructed. • The environmental benefit is incorporated into government's de-capacity target appeal. • Optimal scheme is more efficient, environment-friendly, and cost-effective than the others. • The execution rate of each province for assigned de-capacity task is quite different. • The optimal allocation ratio of coal de-capacity in different scenarios is proposed.

Solving Combinatorial Multi-Objective Bi-Level Optimization Problems Using Multiple Populations and Migration Schemes

Many decision making situations are characterized by a hierarchical structure where a lower-level (follower) optimization problem appears as a constraint of the upper-level (leader) one. Such kind of situations is usually modeled as a BLOP (Bi-Level Optimization Problem). The resolution of the latter usually has a heavy computational cost because the evaluation of a single upper-level solution requires finding its corresponding (near) optimal lower-level one. When several objectives are optimized in each level, the BLOP becomes a multi-objective task and more computationally costly as the optimum corresponds to a whole non-dominated solution set, called the PF (Pareto Front). Despite the considerable number of recent works in multi-objective evolutionary bi-level optimization, the number of methods that could be applied to the combinatorial (discrete) case is much reduced. Motivated by this observation, we propose in this paper an Indicator-Based version of our recently proposed Co-Evolutionary Migration-Based Algorithm (CEMBA), that we name IB-CEMBA, to solve combinatorial multi-objective BLOPs. The indicator-based search choice is justified by two arguments. On the one hand, it allows selecting the solution having the maximal marginal contribution in terms of the performance indicator from the lower-level PF. On the other hand, it encourages both convergence and diversity at the upper-level. The comparative experimental study reveals the outperformance of IB-CEMBA on a multi-objective bi-level production-distribution problem. From the effectiveness viewpoint, the upper-level hyper-volume values and inverted generational distance ones vary in the intervals [0.8500, 0.9710] and [0.0072, 0.2420], respectively. From the efficiency viewpoint, IB-CEMBA has a good reduction rate of the Number of Function Evaluations (NFEs), lying in the interval [30.13%, 54.09%]. To further show the versatility of our algorithm, we have developed a case study in machine learning, and more specifically we have addressed the bi-level multi-objective feature construction problem.

A Multi-Objective Optimized Service Level Agreement Approach Applied on a Cloud Computing Ecosystem

The cloud ecosystem provides transformative advantages that allow elastically offering on-demand services. However, it is not always possible to provide adequate services to all customers and thus to fulfill service level agreements (SLA). To enable compliance with these agreements, service providers leave the customer responsible for determining the service settings and expect that the client knows what to do. Some studies address SLA compliance, but the existing works do not adequately address the problem of resource allocation according to clients' needs since they consider a limited set of objectives to be analyzed and fulfilled. In previous work, we have already addressed the problem considering a single-objective approach. In that work, we identified that the problem has a multi-objective characteristic since several attributes simultaneously influence the SLA agreement, which can lead to conflicts. This paper proposes a multi-objective combinatorial optimization approach for computational resources provisioning, seeking to optimize the efficient use of the infrastructure and provide the client with greater flexibility in contract closure.

Bi-level multi-objective combinatorial optimization using reference approximation of the lower level reaction

Abstract Bi-level optimization has gained a lot of interest during the last decade. This framework is suitable to model several real-life situations. Bi-level optimization problems refer to two related optimization tasks, each one is assigned to a decision level (i.e., upper and lower levels). In this way, the evaluation of an upper level solution requires the evaluation of the lower level. This hierarchical decision making necessitates the execution of a significant number of Function Evaluations (FEs). When dealing with a multi-objective optimization context, new complexities are added and imposed by the conflicting objectives and their evaluation techniques. In this paper, we aim to reduce the induced complexity using approximation techniques in order to obtain the lower level optimality. To this end, ideas from multi-objective optimization have been extracted, improved, and hybridized with evolutionary methods to build an efficient approach for Multi-objective Bi-Level Optimization Problems (MBLOPs). In this work, three techniques are suggested: (1) a complete lower level approximation Pareto front procedure, (2) a reference-based approximation selection procedure, and (3) a sub-set reference-based approximation selection one. The proposed variants are applied to a new multi-objective formulation of a well-known combinatorial problem integrating two systems in the supply chain management, namely, the Bi-level Multi Depot Vehicle Routing Problem (Bi-MDVRP). The statistical analysis demonstrates the efficiency of each algorithm according to a set of metrics. Indeed, a large number of savings are detected which confirms the efficiency of our proposals for solving combinatorial optimization problems.

Topology control for constructive interference-based data dissemination in WSN

Data dissemination is a fundamental service in wireless sensor networks. Traditional protocols often suffer severe medium contention problem or incur significant overhead in contention resolution. Constructive interference (CI) enables concurrent transmissions to interfere constructively, so as to enhance network performance. It has been proved that leveraging CI can achieve near-optimal network flooding latency. However, recent studies show that redundant nodes transmitting simultaneously may degrade the dissemination performance and also consume extra energy. Considering the limited energy of sensor nodes, how to choose the appropriate nodes for simultaneous transmission is important for CI-based data dissemination schemes. In this paper, we formulate this problem as a multi-objective combinatorial optimisation problem theoretically. Considering the complexity of the problem, we design a distributed greedy node selection algorithm to select suitable nodes which forward simultaneously based on CI in each time slot. The main idea of the algorithm is that, by maintaining RecSet and unRecSet in the two-hop neighbour, each node selects suitable nodes from RecSet to transmit simultaneously, such that the maximum number of nodes in unRecSet can receive the packet. Numerical simulations show the efficiency of our proposed algorithm.

Managing False Data Injection Attacks During Contingency of Secured Meters

One major cyber-threat to energy management system is false data injection attack, where an attacker corrupts a set of meters maliciously in such a way that the conventional methods cannot detect it. For an $n$ bus power system, a method to prevent such threats is by securing a set of $(n-1)$ meters, called a basic measurement set (BMS). We prove, in this paper, that the failure in the protection system of a single meter in a BMS is sufficient to create the most vulnerable form of attacks, if the BMS is not chosen appropriately. Selecting the appropriate BMS (called optimal BMS) is formulated as a multi-objective combinatorial optimization problem. First, a greedy algorithm is developed to find the optimal BMS. Second, if all the necessary meters cannot be secured, a greedy algorithm is developed for securing less than $(n-1)$ meters in an optimal BMS, in such a way that the vulnerability of attack is minimum. Our simulation results confirm that there is a significant improvement in security when the optimal BMS is chosen.

Design and Analysis of Multi - Heuristic Based Solution for Task Graph Scheduling Problem

Heuristic based strategies have always been of interest for researchers to achieve sub-optimal solutions for various NP-Complete problems. Human evolution based methods have been an inspiration for research since ages. One of the many evolutionary strategies based on the principle of genetic algorithm have been able to provide much sought after sub-optimal solutions for various NP-Complete problems. One of the most sought after NP-Complete problem is Task graph scheduling i.e. optimally execute the schedule of tasks on available parallel and distributed environment so as to achieve efficient utilization of available resources. Task scheduling is a multi-objective combinatorial optimization problem, with key parameters being reduced completion time and effective load balance on the available resources. Various algorithms have been proposed by various authors to achieve the above mentioned goal with the help of various heuristics like list scheduling, task duplication and critical path based. The algorithms proposed by various authors like Highest Level First Execution Time (HLFET), Modified Critical Path (MCP), Duplication Scheduling Heuristic (DSH), Linear Clustering (LC) and Dynamic Critical Path (DCP), belonging to each heuristic mentioned before will be taken under study. Previously these algorithms have been individually reported to be efficient in some certain restricted environment parameters with certain limitations; offering very preliminary improvement on the state of art of one single type of environment. Designers face difficulty in choosing the optimal algorithm for the generalized environment. This paper will identify the gaps in existing literature that forms the base of every research focusing in the direction of improvement of task graph scheduling algorithms. Further, this paper will propose a hybrid meta-heuristic i.e. Genetic Algorithm based task graph scheduling solution and perform a comparative study of aforementioned algorithms.

Random walk’s correlation function for multi-objective NK landscapes and quadratic assignment problem

The random walk’ correlation matrix of multi-objective combinatorial optimization problems utilizes both local structure and general statistics of the objective functions. Reckoning time of correlation, or the random walk of lag 0, is quadratic in problem size L and number of objectives D. The computational complexity of the correlation coefficients of mNK is $$O(D^2 K^2 L)$$ , and of mQAP is $$O(D^2 L^2)$$ , where K is the number of interacting bits. To compute the random walk of a lag larger than 0, we employ a weighted graph Laplacian that associates a mutation operator with the difference in the objective function. We calculate the expected objective vector of a neighbourhood function and the eigenvalues of the corresponding transition matrix. The computational complexity of random walk’s correlation coefficients is polynomial with the problem size L and the number of objectives D. The computational effort of the random walks correlation coefficients of mNK is $$O(2^K L D^2)$$ , whereas of mQAP is $$O(L^6 D^2)$$ . Numerical examples demonstrate the utilization of these techniques.

The Collaborative Local Search Based on Dynamic-Constrained Decomposition With Grids for Combinatorial Multiobjective Optimization.

The decomposition-based algorithms [e.g., multiobjective evolutionary algorithm based on decomposition (MOEA/D)] transform a multiobjective optimization problem (MOP) into a number of single-objective optimization subproblems and solve them in a collaborative manner. It is a natural framework for using single-objective local search (LS) to solve combinatorial MOPs. However, commonly used decomposition methods, such as weighted sum (WS), Tchebycheff (TCH), and penalty-based boundary intersection (PBI) may not be good at maintaining the population diversity while providing diverse initial solutions for different LS procedures in a collaborative way. Based on our previous work on the constrained decomposition with grids (CDG), this article proposes a dynamic CDG (DCDG) framework used to design a multiobjective memetic algorithm (DCDG-MOMA). DCDG uses grids for maintaining diversity, supporting the collaborative LS. In addition, DCDG dynamically increases the number of grids for obtaining more nondominated solutions as well as the better collaborative search among them. DCDG-MOMA has been compared with several classical and state-of-the-art algorithms on multiobjective traveling salesman problem (MOTSP), multiobjective quadratic assignment problem (MOQAP), and multiobjective capacitated arc routing problem (MOCARP).