Discrete Multi-objective Optimization Problem Research Articles

Analysis of the weighted Tchebycheff weight set decomposition for multiobjective discrete optimization problems

Scalarization is a common technique to transform a multiobjective optimization problem into a scalar-valued optimization problem. This article deals with the weighted Tchebycheff scalarization applied to multiobjective discrete optimization problems. This scalarization consists of minimizing the weighted maximum distance of the image of a feasible solution to some desirable reference point. By choosing a suitable weight, any Pareto optimal image can be obtained. In this article, we provide a comprehensive theory of this set of eligible weights. In particular, we analyze the polyhedral and combinatorial structure of the set of all weights yielding the same Pareto optimal solution as well as the decomposition of the weight set as a whole. The structural insights are linked to properties of the set of Pareto optimal solutions, thus providing a profound understanding of the weighted Tchebycheff scalarization method and, as a consequence, also of all methods for multiobjective optimization problems using this scalarization as a building block.

A NSGA-II based approach for multi-objective optimization of a reconfigurable manufacturing transfer line supported by Digital Twin: A case study

In response to the wide range of customer demands, the concept of reconfigurable manufacturing systems (RMS) was introduced in the industrial sector. RMS enables producers to meet varying volumes of demand over varying time periods by swiftly adjusting its production capacity and functionality within a part family in response to abrupt market changes. In these circumstances, RMS are made to swiftly reconfigure their Reconfigurable Machine Tools (RMTs). RMTs are designed to have a variety of configurations that may be conditionally chosen and reconfigured in accordance with specific performance goals. However, the reconfiguration process is not an easy process, which entails optimization of several objectives and many of which are inherently conflictual. As a result, it necessitates real-time monitoring of the RMS's condition, which may be achieved by digital twinning, or the real-time capture of system data. The concept of using a digital replica of a physical system to provide real-time optimization is known as digital twin. This work considered a case study of discrete parts manufacturing on a reconfigurable single manufacturing transfer line (SMTL). Six manufacturing operations are required to be performed on the parts at six production stages. This work uses the Digital Twin (DT) based approach to assist a discrete multi-objective optimization problem for a reconfigurable manufacturing transfer line. This multi-objective optimization problem consists of four objective functions which is illustrated by using DT-based Non-dominated Sorting Genetic Algorithm-II (NSGA-II). The innovative aspect of the current study is the use of a DT-based framework for RMS reconfiguration to produce the best optimum solutions. The produced real-time solutions will be of great assistance to the decision maker in selecting the appropriate real-time optimal solutions for reconfigurable manufacturing transfer lines.

Local-Diversity Evaluation Assignment Strategy for Decomposition-Based Multiobjective Evolutionary Algorithm

Decomposition-based multiobjective evolutionary algorithms (MOEAs) transform a multiobjective optimization problem (MOP) into a set of subproblems and then optimize them collaboratively. When two adjacent objective vectors are similar but the difference in the corresponding decision vectors is large, it is difficult to transfer from a solution of one subproblem to solutions of its neighboring subproblems. In this article, we argue that the key to overcoming this difficulty is the evaluation assignment for different subproblems based on local density and global utility. We propose a local-density measurement model to estimate the solution density around each subproblem. Based on the model, a local-diversity evaluation assignment strategy for the decomposition-based MOEA is designed to assign fitness evaluations among different subproblems. Two discrete MOPs are selected as test instances for experiments because they satisfy the premise that the difference in the corresponding decision vectors of adjacent objective vectors is large. Experimental results indicate that the proposed strategy enhances the population diversity. Besides, the combination of the proposed strategy and the evaluation assignment for high global utility subproblems can overcome the difficulty in transferring from a solution of one subproblem to solutions of its neighboring subproblems. The results also verify that the proposed algorithm significantly outperforms the state-of-the-art evaluation assignment MOEAs regarding the inverted generational distance and the hypervolume metrics.

FERN: Fair Team Formation for Mutually Beneficial Collaborative Learning

Automated <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">team formation</i> is becoming increasingly important for a plethora of applications in open-source community projects, remote working platforms, as well as online educational systems. The latter case, in particular, poses significant challenges that are specific to the educational domain. Indeed, teaming students aims to accomplish far more than the successful completion of a specific task. It needs to ensure that all the members in the team benefit from the collaborative work, while also ensuring that the participants are not discriminated against with respect to their protected attributes, such as race and gender. Toward achieving these goals, this article introduces <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FERN</monospace> , a fair team formation approach that promotes mutually beneficial peer learning, dictated by protected group fairness as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">equality of opportunity</i> in collaborative learning. We formulate the problem as a multi-objective discrete optimization problem. We show this problem to be NP-hard and propose a heuristic hill-climbing algorithm. Extensive experiments on both the synthetic and real-world datasets against well-known team formation techniques show the effectiveness of the proposed method.

Integrating Energy Smart Grid’s ontologies through multi-objective particle swarm optimization algorithm with competitive mechanism

Smart city seeks safer, resource-oriented, environment friendly, and cost-effective energy solutions, and due to the distributed and independent nature of energy management systems, Energy Smart Grid (ESG) quickly evolves from a widely accepted concept to an industrial reality. To monitor the energy consumption across various domains and assist efficient energy distribution and storage, it is necessary to integrate various ESG agents’ ontologies. To this end, this work models ESG’s Ontology Matching Problem (ESG-OMP) as a discrete Multi-Objective Optimization Problem (MOOP), which simultaneously optimizes the alignment’s completeness and correctness through determining an optimal entity corresponding set. After that, a Multi-Objective Particle Swarm Optimization Algorithm With Competitive Mechanism (MOPSO-CM) is proposed to address this issue by trading off the alignment’s completeness and correctness, which introduces the CM to update the particles with the pairwise competition that performed in the swarm of current generation. Moreover, an instance-based hybrid similarity measure is used to distinguish the heterogeneous ontology entities. The experiment uses OAEI’s testing datasets, 13 smart grid’s ontology matching tasks and 5 energy smart grid’s ontology matching tasks to test MOPSO-CM’s performance. The experimental results show that MOPSO-CM can effectively address various heterogeneous ontology matching problems and determine high-quality ESG’s ontology alignments.

Thermal performance revival of composite PCM for hybrid BTMSs by architecture and formula integrated optimization

Composite phase change material (CPCM) cooling is an essential method to ensure better performance of battery thermal management system (BTMS) due to high latent heat. However, long recovery time of CPCM makes it unaffordable especially during continuous high rate discharge–charge cycles. In order to reduce the energy consumption while enhancing the latent heat recovery efficiency, a novel hybrid BTMS architecture coupling with CPCM and liquid cooling is proposed. Liquid cooling is employed to recover the latent heat of PCM. To enhance the heat transfer in BTMS, the structural layout of liquid cooling arrangement is investigated. Meanwhile, mass fraction of expanded graphite (EG) and bulk density of CPCM are considered. Specially, the experiments are conducted to verify the effectiveness of present hybrid BTMS. In solidification, temperature performance especially temperature uniformity is not ideal. Therefore, a novel successive multiple attribute decision making (MADM) algorithm is employed to achieve optimized temperature difference, maximum temperature, solidification rate and mass of CPCM. In MADM algorithm, the order preference by similarity to ideal solution (TOPSIS) is coupled with the analytic hierarchy process (AHP) to solve the multi-objective discrete optimization problem. Results indicate that the optimal solution can make the maximum temperature and temperature difference at the lowest level of 317.51 K and 3.45 K at discharge rate of 4C. Meanwhile, the latent heat of CPCM is improved by about 27.5% at charge rate of 2C. The solution can improve the long-term reliability for the BTMS and the safety and service life for battery module.

Network Models for Multiobjective Discrete Optimization

This paper provides a novel framework for solving multiobjective discrete optimization problems with an arbitrary number of objectives. Our framework represents these problems as network models, in that enumerating the Pareto frontier amounts to solving a multicriteria shortest-path problem in an auxiliary network. We design techniques for exploiting network models in order to accelerate the identification of the Pareto frontier, most notably a number of operations to simplify the network by removing nodes and arcs while preserving the set of nondominated solutions. We show that the proposed framework yields orders-of-magnitude performance improvements over existing state-of-the-art algorithms on five problem classes containing both linear and nonlinear objective functions. Summary of Contribution: Multiobjective optimization has a long history of research with applications in several domains. Our paper provides an alternative modeling and solution approach for multiobjective discrete optimization problems by leveraging graphical structures. Specifically, we encode the decision space of a problem as a layered network and propose graph reduction operators to preserve only solutions whose image are part of the Pareto frontier. The nondominated solutions can then be extracted through shortest-path algorithms on such a network. Numerical results comparing our method with state-of-the-art approaches on several problem classes, including the knapsack, set covering, and the traveling salesperson problem (TSP), suggest orders-of-magnitude runtime speed-ups for exactly enumerating the Pareto frontier, especially when the number of objective functions grows.

Development and application of hybrid teaching-learning genetic algorithm in fuel reloading optimization

The problems of fuel reloading optimization is a complex multi-objective discrete optimization problem with multiple local optimums. In recent years, great process has been made in solving these problems by using meta-heuristic optimization algorithms. Teaching-learning based optimization algorithm (TLBO) is one kind of novel meta-heuristic optimization algorithms. However, it is seldom used to solve the problems of fuel reloading optimization, because its original purpose is to solve continuous optimization problems. In this paper, a hybrid teaching-learning genetic algorithm (HTLGA) is developed, which could be directly applied to solve the problems of fuel reloading optimization. This hybrid algorithm takes TLBO as main part, combines three operators of genetic algorithms (GA) which are coding, crossover and mutation. The optimization solutions which are represented as students in TLBO are further divided into top students, ordinary students and poor students in HTLGA. The calculation phases “Teacher phase” and “Learner phase” in TLBO are improved into “Teacher phase”, “Discussion phase” and “Self-study phase” in HTLGA. For testing the optimization ability of HTLGA, it is applied to solve the problems of fuel reloading optimization for the 1/6 core of thorium-based block-type HTGRs. The results showed that the developed HTLGA has more powerful optimization ability than TLBO and GA.

Surrogate-assisted evolutionary algorithm for expensive constrained multi-objective discrete optimization problems

Surrogate-assisted optimization has attracted much attention due to its superiority in solving expensive optimization problems. However, relatively little work has been dedicated to addressing expensive constrained multi-objective discrete optimization problems although there are many such problems in the real world. Hence, a surrogate-assisted evolutionary algorithm is proposed in this paper for this kind of problem. Specifically, random forest models are embedded in the framework of the evolutionary algorithm as surrogates to improve approximate accuracy for discrete optimization problems. To enhance the optimization efficiency, an improved stochastic ranking strategy based on the fitness mechanism and adaptive probability operator is presented, which also takes into account both convergence and diversity to advance the quality of candidate solutions. To validate the proposed algorithm, it is comprehensively compared with several well-known optimization algorithms on several benchmark problems. Numerical experiments are demonstrated that the proposed algorithm is very promising for the expensive constrained multi-objective discrete optimization problems.

Explainable Interactive Evolutionary Multiobjective Optimization

We present a new interactive procedure for evolutionary multiobjective optimization involving the decision rule preference model in the search for the best compromise solution. As usual in the interactive multiobjective optimization, preference elicitation phases alternate with optimization phases, however, in this evolutionary procedure, one preference elicitation phase falls into many iterations of the evolutionary multiobjective optimization. In the preference elicitation phase, the Decision Maker (DM) is asked to select those solutions she considers relatively good in the current population. Using the Dominance-based Rough Set Approach (DRSA) this information is represented in terms of ``if ..., then ... decision rules which represent DM’s preferences. They are used in the next phases of the evolutionary multiobjective optimization to iteratively converge towards the part of the Pareto front containing the best compromise solution. Besides guiding the search process, the decision rules can be read as arguments explaining the DM's choices. In this way, the proposed method implements the postulate of transparency and explainability expected from interactive procedures, because the DM has a chance to understand how her answers given in the preference elicitation phase are translated into guidelines for the algorithm in the optimization phase, i.e., what is the impact of the answers on newly proposed compromise solutions. This can be considered the distinctive aspect of what we define explainable multiobjective optimization evolutionary approach (XIMEA). In particular, the method proposed in this paper is called XIMEA-DRSA. From the decision psychology point of view, in line with the main results of behavioral sciences, the decision rules support the DM to construct and learn her preferences during the decision process. For its explanatory feature resulting from the conjunction of evolutionary multiobjective optimization with DRSA, the method has been called XIMEA-DRSA. The computational effectiveness of the method has been tested on both continuous and discrete multiobjective optimization problems. In the latter case, the XIMEA-DRSA method has been applied to the multiobjective knapsack problem.

Enumeration of the Nondominated Set of Multiobjective Discrete Optimization Problems

In this paper, we propose a generic algorithm to compute exactly the set of nondominated points for multiobjective discrete optimization problems. Our algorithm extends the ε-constraint method, originally designed for the biobjective case only, to solve problems with two or more objectives. For this purpose, our algorithm splits the search space into zones that can be investigated separately by solving an integer program. We also propose refinements, which provide extra information on several zones, allowing us to detect, and discard, empty parts of the search space without checking them by solving the associated integer programs. This results in a limited number of calls to the integer solver. Moreover, we can provide a feasible starting solution before solving every program, which significantly reduces the time spent for each resolution. The resulting algorithm is fast and simple to implement. It is compared with previous state-of-the-art algorithms and is seen to outperform them significantly on the experimented problem instances.

Generalized Equivalence Set Method for Solving Multiobjective Optimization Problems

Various aspects of solving multiobjective discrete optimization problems are considered. Advantages of the equivalence set method are shown compared to other methods often used to solve multiobjective problems such as the method of successive concessions and the method of seeking the set of Pareto-optimal solutions. Theorems reflecting the main properties of the equivalence set method and showing the relationship and interrelation between the set of Pareto-optimal solutions and the equivalence set are formulated and proved.

Multi-Objective Approach for Power Quality Monitor Allocation With Symmetry in Short-Duration Voltage Variations

In this paper, we present a new approach for solving the problem of power quality monitors (PQMs) optimal allocation, for the monitoring of short-duration voltage variations caused by a fault condition in a power grid system. The problem is treated in a multiobjective perspective with two optimal criteria: minimization of the number of PQMs and maximization of the number of identified faults. Non-identification of an event can occur as a result of symmetry conditions in the network, i.e., in cases where two or more faults generate the same signals in some buses, which leads to ambiguity in the monitoring results. Symmetry increases the complexity of both the problem formulation and solution. The problem is described as a multiobjective discrete optimization problem and is solved by the algorithm for bicriteria discrete optimization within reasonable computational time. That approach was tested in power grids of different characteristics and sizes. The results demonstrate the proposed methodology applicability for solving the problem in real-size networks.

Multi-period price optimization problem for omnichannel retailers accounting for customer heterogeneity

In recent times many brick and mortar retailers have considered adding online channels to compete with e-tailers. These retailers have further enabled omnichannel features into their businesses considering customer's channel switching behavior. This article presents a price optimization problem for such a retailer who already had many offline stores but then added an online channel to improve its digital presence. The retailer has a distribution center that replenishes these stores periodically. Furthermore, the retailer uses either the distribution center or any of the offline stores to fulfill online orders. The paper takes an integrated approach to price optimization, inventory control, and e-fulfillment problem and developed a decision support model for such a retailer having multiple objectives of profit maximization and lost sales minimization. The lost sales minimization is achieved either by decreasing the probability of no-purchase option (by reducing the posted price) or by fulfilling online demands at relatively higher costs (it is not optimal to meet all the demand). The paper tests the efficacy of conventional Non-dominated sorting genetic algorithm-II (NSGA-II) and recently proposed direct Zigzag search in order to solve the resulting NP-hard discrete multi-objective optimization problem. We offer a new algorithm that exploits the neighborhood search feature of the Zigzag method to extend the NSGA-II front further. Furthermore, using realistic and real-world inspired data, we generated some useful insights for omnichannel retailers.

Adaptive wireless network multi-objective optimization algorithm based on image synthesis

Multi-objective optimization problems can be divided into continuous multi-objective optimization problems and discrete multi-objective optimization problems, and discrete multi-objective optimization is not universal. In practical applications, there are many discrete multi-objective optimization problems. The solution of different problems needs to design different evolutionary multi-objective algorithms according to their specific conditions. The threshold in the traditional multi-objective optimization of wireless network is a preset constant. It has relatively poor performance in the synthesis of images with different noise levels. An adaptive wireless network multi-objective optimization algorithm based on image synthesis is proposed. Based on the maximum inter class variance and maximum peak signal to noise ratio (SNR), an adaptive wireless network multi-objective optimization algorithm is established. The accuracy and noise immunity of image synthesis are also considered. In order to avoid the effect of threshold increase on algorithm efficiency, multi-objective optimization algorithm is introduced into the algorithm. Experiments show that the method proposed in this paper is accurate and robust and has good universality for the synthesis of different noise images.

Solving discrete multi-objective optimization problems using modified augmented weighted Tchebychev scalarizations

In this paper we present the modified augmented weighted Tchebychev norm, which can be used to generate a complete efficient set of solutions to a discrete multi-objective optimization problem. We contribute a generating algorithm that will, without supervision, generate the entire non-dominated set for any number of objectives. To our knowledge, this is the first generating method for general discrete multi-objective problems that uses a variant of the Tchebychev norm. In a computational study, our algorithm’s running times are comparable to previously proposed algorithms.

Pareto front generation with knee-point based pruning for mixed discrete multi-objective optimization

This note proposes an algorithm to generate the Pareto front of a mixed discrete multi-objective optimization problem based on the pruning of irrelevant subproblems. The knee point is introduced as a new reference point for pruning decision. The point can overcome the drawback of the existing reference point – over-pruning, and be naturally defined and used in the context of multi-objective optimization. The validity of the proposed procedure is demonstrated through case studies.

A Novel Multi-Objective Discrete Particle Swarm Optimization with Elitist Perturbation for Reconfiguration of Ship Power System

Abstract A novel multi-objective discrete particle swarm optimization with elitist perturbation strategy (EPSMODPSO) is proposed and applied to solve the reconfiguration problem of shipboard power system(SPS). The new algorithm uses the velocity to decide each particle to move one step toward positive or negative direction to update the position. An elitist perturbation strategy is proposed to improve the local search ability of the algorithm. Reconfiguration model of SPS is established with multiple objectives, and an inherent homogeneity index is adopted as the auxiliary estimating index. Test results of examples show that the proposed EPSMODPSO performs excellent in terms of diversity and convergence of the obtained Pareto optimal front. It is competent to solve network reconfiguration of shipboard power system and other multi-objective discrete optimization problems.

A cooperative swarm intelligence algorithm for multi-objective discrete optimization with application to the knapsack problem

We propose a novel cooperative swarm intelligence algorithm to solve multi-objective discrete optimization problems (MODP). Our algorithm combines a firefly algorithm (FA) and a particle swarm optimization (PSO). Basically, we address three main points: the effect of FA and PSO cooperation on the exploration of the search space, the discretization of the two algorithms using a transfer function, and finally, the use of the epsilon dominance relation to manage the size of the external archive and to guarantee the convergence and the diversity of Pareto optimal solutions.We compared the results of our algorithm with the results of five well-known meta-heuristics on nine multi-objective knapsack problem benchmarks. The experiments show clearly the ability of our algorithm to provide a better spread of solutions with a better convergence behavior.

Multiobjective optimization in a pseudometric objective space as applied to a general model of business activities

It is shown that finding the equivalence set for solving multiobjective discrete optimization problems is advantageous over finding the set of Pareto optimal decisions. An example of a set of key parameters characterizing the economic efficiency of a commercial firm is proposed, and a mathematical model of its activities is constructed. In contrast to the classical problem of finding the maximum profit for any business, this study deals with a multiobjective optimization problem. A method for solving inverse multiobjective problems in a multidimensional pseudometric space is proposed for finding the best project of firm’s activities. The solution of a particular problem of this type is presented.