Tri-objective Optimization Problem Research Articles

A method to identify a representation of the set of non-dominated points for discrete tri-objective optimization problems

A method to identify a representation of the set of non-dominated points for discrete tri-objective optimization problems

Multi-objective optimization of forest ecosystem services under uncertainty

Forest ecosystems deliver multiple services crucial for human wellbeing and welfare, with complex relationships among forest ecosystem services (FESs). Implementing sustainable forest management (SFM) policy requires insights into trade-offs, conflicts and synergies among key FESs, necessitating decision-support tools for multiple objectives. However, uncertainty in key parameters adds complexity. We formulated a tri-objective optimization problem concentrating on three vital ecosystem services in Nordic boreal forests: timber supply, climate change mitigation, and biodiversity conservation. Furthermore, future timber price uncertainty was integrated into the optimization model. Utilizing the TreeSim individual-tree growth and yield simulator, we simulated forest growth and yields, carbon stocks, and biodiversity values. Pareto-optimal solutions were found for four distinct management strategies using the epsilon-constrain method to solve the optimization problem. These management strategies differed in spatial restrictions on harvest location and timing, including considerations like harvest adjacency, green-up and maximum harvest opening area (MOA). The analysis revealed trade-offs among all three objectives, with variations influenced by management intensity. Furthermore, within the Pareto solutions for each strategy, a compromise solution was identified based on the knee-point method. The study demonstrated nuanced impacts of green-up constraints related to harvest practices, showing minor effects on harvesting, carbon sequestration, and biodiversity conservation when MOA was restricted to 35 hectares or less. However, more substantial impacts were observed with a MOA limited to 60 hectares or in a strategy without spatial restrictions. Our results underscore the importance of considering multiple objectives in forest management and policy under uncertain wood prices, preventing undesired effects from a singular or deterministic approach. This work provides insights into trade-offs and synergies, contributing to strategic planning and policy design.

Density Function-Based Trust Region Algorithm for Approximating Pareto Front of Black-Box Multiobjective Optimization Problems

In this paper, we consider a black-box multiobjective optimization problem, whose objective functions are computationally expensive. We propose a density function-based trust region algorithm for approximating the Pareto front of this problem. At every iteration, we determine a trust region and then in this trust region, select several sample points, at which are evaluated objective function values. In order to obtain non-dominated solutions in the trust region, we convert given objective functions into one function: scalarization. Then, we construct quadratic models of this function and the objective functions. In current trust region, we find optimal solutions of all single-objective optimization problems with these models as objectives. After that, we remove dominated points from the set of obtained solutions. In order to estimate the distribution of non-dominated solutions, we introduce a density function. By using this density function, we obtain the most “isolated” point among the non-dominated points. Then, we construct a new trust region around this point and repeat the algorithm. We prove convergence of proposed algorithm under the several assumptions. Numerical results show that even in case of tri-objective optimization problems, the points generated by proposed algorithm are uniformly distributed over the Pareto front.

Efficient, economical and energy-saving multi-workflow scheduling in hybrid cloud

Benefiting from the flexible, scalable and secure environment, hybrid cloud can overcome the shortage of limited resources in private cloud to simultaneously execute large-scale scientific workflows. In hybrid cloud, privacy-sensitive tasks are not allowed to be executed on public resources, while non-sensitive tasks are unrestricted. As an NP-Complete problem, it is extraordinarily challenging to schedule multiple workflows efficiently, economically and energy-savingly under quality-of-service constraints. This paper models the hybrid-cloud-based privacy-aware multi-workflow scheduling as a tri-objective optimization problem that optimizes workflow-oriented total tardiness, private-cloud-oriented total energy consumption, and public-cloud-oriented total monetary cost. To the best of authors’ knowledge, few studies have been conducted on the tri-objective privacy-aware multi-workflow scheduling in hybrid cloud (PMWS-HC). To solve this problem, we dissect various factors involved during task scheduling and devise a novel Heuristic Scheduling Algorithm based on 9 Factors (HSA9Fs), which dynamically selects the workflows and tasks to be scheduled, and the corresponding VMs to execute them. To optimize the three conflicting objectives simultaneously, we propose a nested algorithm called MSIA, which first employs a Multi-objective Salp swarm algorithm to explore for the Pareto solutions, and then uses an Iterative greedy Algorithm to perform a refined search on individuals to obtain high-quality solutions. Extensive Medium-Small-Scale and Large-Scale simulation experiments show that both HSA9Fs and MSIA outperform state-of-the-art scheduling algorithms in several multi-objective performance metrics.

A multi-objective model for optimizing hydrogen injected-high pressure natural gas pipeline networks

The paper develops a statistical model for optimizing the Hydrogen-injected Natural gas (H-NG) high-pressure pipeline network. Gas hydrodynamic principles are utilized to construct the pipeline and compressor station model. The model developed is implemented on a pipeline grid that is supposed to carry Hydrogen as an energy carrier in a natural gas-carrying pipeline. The paper aims to optimize different objectives using ant colony optimization (ACO). The first objective includes a single objective optimization problem that evaluates the maximum permissible hydrogen amounts blended with natural gas (NG) for a set of pipeline constraints. We also evaluated the variations in operational variables on injecting Hydrogen into the natural gas pipeline networks at varying fractions. The study further develops a multi-objective optimization model that includes bi-objective and tri-objective problems and is optimized using ACO. Traditional studies have focused on single-objective optimization with minimal bi-objective issues. In addition, none of the earlier research has shown the effect of introducing Hydrogen to the NG network using tri-objective function evaluations. The bi-objective and tri-objective functions help evaluate the effect of injecting Hydrogen on different operational parameters. The study further attempts to fill the gap by detailing the modelling equations implemented through a bi-objective and tri-objective function for the H-NG pipeline network and optimized through ACO. Pareto fronts that show the tradeoff between the different objectives for the multi-objective problem have been generated. The primary objective of the bi-objective and tri-objective optimization problems is maximizing hydrogen mole percent in natural gas. The other objective chosen is minimizing compressor fuel consumption and maximizing delivery pressure, throughput, and power delivered at the delivery station. The findings will serve as a roadmap for pipeline operators interested in repurposing natural gas pipeline networks to transport hydrogen and natural gas blend (H-NG) and seeking to reduce carbon intensity per unit of energy-delivered fuel.

Multi-criteria analysis of cell-recycle based continuous lactic acid production process

ABSTRACT This study focuses on the optimization of the cell-recycle based continuous lactic acid (LA) fermentation production process with the estimation of improved kinetic parameters. The non-dominated sorting genetic algorithm II (NSGA-II) is used considering seven variables and five imposed constraints for multi-objective optimization (MOO) study. Three objectives namely LA productivity, substrate conversion, and yield are formed for the maximization purpose. Three different cases are constructed by using a combination of two bi-objectives and one tri-objective optimization problem, and the modified differential evolution (MDE) algorithm is used to individually optimize each case. Pareto ranking is carried out for MOO, and the best optimal solution is determined using the technique for order preference by similarity to the ideal solution (TOPSIS) method. The highest value of volumetric productivity of LA (184.363 kg/m3.h) is reported in case-4a. The highest yield (1.176) is reported in case-2, while the highest substrate conversion (0.963) is reported in case-1.

A comparative study of reverse flow chromatographic reactor and fixed bed reactor: A multi-objective optimization approach

In this study, we have carried out a comparative analysis of a fixed bed reactor (FBR) and reverse flow chromatographic reactor (RFCR) for a series reaction. This comparison has been carried out using mathematical model based single and multi-objective optimization. In addition, since the model merely approximates the actual process, there is an inherent presence of uncertainty in it. To account for the uncertainty, optimization is done considering the uncertainty in the key model parameters. Three objective functions are used in this work for the comparative study, namely yield, selectivity, and conversion. The decision variables used in the optimization problem are the inlet concentration, Damkohler number, and dimensionless switching time. The multi-objective optimization results have been analysed in terms of optimal Pareto fronts for three bi-objective and one tri-objective optimization problems. A detailed analysis of the Pareto fronts for both the reactors is presented for five distinct representative Pareto points. The results of the work indicated the superiority of RFCR over FBR. In particular, a representative Pareto point solution of 3-objective problem in RFCR corresponds to 51.36% higher yield, 27.06% higher selectivity and 19.15% higher conversion as compared to FBR.

Multi-objective optimization of fly ash-slag based geopolymer considering strength, cost and CO2 emission: A new framework based on tree-based ensemble models and NSGA-II

Fly ash-slag based geopolymer has excellent mechanical performance with low carbon footprints, which has emerged as a promising alternative to Portland cement. The optimization of geopolymers requires trade-offs between multiple objectives (strength, cost, and CO2 emission) while considering a large number of highly nonlinear variables. To solve this multi-objective optimization (MOO) problem, this study developed a MOO model combining the tree-based ensemble learning algorithm and non-dominated sorting genetic algorithm (NSGA-II). The MOO model was trained on a database collected from published literature with 676 mixture proportions. The chemical components as well as their contents and the curing conditions were selected as the input variables, while the uniaxial compressive strength (UCS) was the output variable. The UCS of fly ash-slag based geopolymers were modeled using the random forest regressor, extra trees regressor, gradient boosted regressor and extreme gradient boosting regressor. The results show that the gradient boosted regressor has the highest prediction accuracy with R (0.966) and RMSE (5.295 MPa) on the testing set. The analysis of the variable importance by the developed model indicates that the curing age, contents of slag and sodium silicate are more important to the strength development of the binary blended geopolymer. The MOO model developed based on the gradient-boosted regressor and NSGA-II successfully found the cost-UCS and CO2 emission-UCS Pareto fronts of the bi-objective optimization problem, as well as the cost-UCS-CO2 emission Pareto front of the tri-objective optimization problem. This developed framework improves the efficiency of geopolymer design and can be applied to the mixture optimization of other construction materials.

Economic Dispatch with High Penetration of Wind Power Using Extreme Learning Machine Assisted Group Search Optimizer with Multiple Producers Considering Upside Potential and Downside Risk

The power system with high penetration of wind power is gradually formed, and it would be difficult to determine the optimal economic dispatch (ED) solution in such an environment with significant uncertainties. This paper proposes a multi-objective ED (MuOED) model, in which the expected generation cost (EGC), upside potential (USP), and downside risk (DSR) are simultaneously considered. The heterogeneous indices of upside potential and downside risk mean the potential economic gains and losses brought by high penetration of wind power, respectively. Then, the MuOED model is formulated as a tri-objective optimization problem, which is related to uncertain multi-criteria decision-making against uncertainties. After-wards, the tri-objective optimization problem is solved by an extreme learning machine (ELM) assisted group search optimizer with multiple producers (GSOMP). Pareto solutions are obtained to reflect the trade-off among the expected generation cost, the upside potential, and the downside risk. And a fuzzy decision-making method is used to choose the final ED solution. Case studies based on the Midwestern US power system verify the effectiveness of the proposed MuOED model and the developed optimization algorithm.

Planning sustainable routes: Economic, environmental and welfare concerns

We introduce a problem called the Sustainable Vehicle Routing Problem (SVRP) in which the sustainability notion is considered in terms of economic, environmental and social impacts. Inspired by real-world problems that large cargo companies face for their delivery decisions, we introduce a new facet to the classical vehicle routing problem by considering the welfare of all three stakeholders of the problem: an environmentally conscious company, the drivers, and the indistinguishable customers, as our setting assumes that all customers belong to the same delivery class. Thus, the proposed problem consists of three objective functions. The first one is to minimize the total fuel consumption and emission to represent the companies’ main economic and environmental concerns. The second one is to maximize total welfare of the drivers through a function that encourages equitable payment across drivers while encouraging low total driver cost and the third one is to maximize total welfare of the customers through a function that encourages fairness in terms of delivery times. The last two objectives are measured using slots for tour lengths and delivery times. We implement an efficient solution approach based on the ϵ-constraint scalarization to find the nondominated solutions of our triobjective optimization problem and present computational analysis that provide insights on the trade-off between the objectives. Our experiments demonstrate the potential of the suggested framework under the customer anonymity assumption to help decision makers make effective plans that all parties involved would give consent to.

Multi-objective algorithm based on tissue P system for solving tri-objective optimization problems

Multi-objective algorithm based on tissue P system for solving tri-objective optimization problems

An improved dynamic multi-objective optimization approach for nonlinear equation systems

Solving nonlinear equation systems using evolutionary algorithms involves solving two key problems. One problem is how to efficiently optimize the nonlinear equations derived from the physical features, and the other problems is how to locate more than one optimal solution in a single trial. To address these two problems, an improved dynamic tri-objective differential evolution method is proposed in this paper. First, we transform a given system with any type and number of nonlinear equations into a dynamic tri-objective optimization problem, which targets the first problem. Second, we develop a self-adaptive ranking multi-objective differential evolution, which targets the second problem. In addition, a probability distribution-based local search is introduced, which aims to identify the optimal solutions with a high level of accuracy. Based on previous studies of numerical optimizations with multiple solutions, each component is elaborately proposed and developed, so it is more suitable for a nonlinear equation system. Experiments were conducted on 30 numerical examples collected from real-world applications. The statistical results are encouraging, showing that the performance of the proposed approach is better than that of eight state-of-the-art evolutionary algorithms, with respect to root ratio and success rate metrics.

Resetting Weight Vectors in MOEA/D for Multiobjective Optimization Problems With Discontinuous Pareto Front.

When a multiobjective evolutionary algorithm based on decomposition (MOEA/D) is applied to solve problems with discontinuous Pareto front (PF), a set of evenly distributed weight vectors may lead to many solutions assembling in boundaries of the discontinuous PF. To overcome this limitation, this article proposes a mechanism of resetting weight vectors (RWVs) for MOEA/D. When the RWV mechanism is triggered, a classic data clustering algorithm DBSCAN is used to categorize current solutions into several parts. A classic statistical method called principal component analysis (PCA) is used to determine the ideal number of solutions in each part of PF. Thereafter, PCA is used again for each part of PF separately and virtual targeted solutions are generated by linear interpolation methods. Then, the new weight vectors are reset according to the interrelationship between the optimal solutions and the weight vectors under the Tchebycheff decomposition framework. Finally, taking advantage of the current obtained solutions, the new solutions in the decision space are updated via a linear interpolation method. Numerical experiments show that the proposed MOEA/D-RWV can achieve good results for bi-objective and tri-objective optimization problems with discontinuous PF. In addition, the test on a recently proposed MaF benchmark suite demonstrates that MOEA/D-RWV also works for some problems with other complicated characteristics.

Density-Enhanced Multiobjective Evolutionary Approach for Power Economic Dispatch Problems

Economic dispatching of generating units in a power system can significantly reduce the energy cost of the system. However, the economic dispatch (ED) problem is highly constrained, and often has disconnected feasible regions because of various physical features. Enhancing population diversity is critical for the evolutionary approach to fully explore and exploit the feasible regions. In this article, we propose a density-enhanced multiobjective evolutionary approach to solve ED problem. An ED problem is first transformed into a tri-objective optimization problem, and then multiobjective optimization techniques are employed to fully optimize the constraints and cost function simultaneously. The first two objectives are derived from the original ED problem, while the third one is a novel density objective constructed by niching methods to enhance population diversity. These three objectives are optimized simultaneously by a dynamic dominance relation, which can make a good balance among feasibility, diversity, and convergence. To evaluate the performance of this proposed approach, 22 benchmark problems and seven real-world ED problems with different features are tested in this article. The experimental results show that our approach performs better than or at least competitive to the state-of-the-art algorithms, especially on large-scale ED problems.

An Automatic and Optimal MPA Design Method.

Raw polarimetric images are captured by a focal plane polarimeter which is covered by a micro-polarizer array (MPA). The design of the MPA plays a crucial role in polarimetric imaging. MPAs are predominantly designed according to expert engineering experience and rules of thumb. Typically, only one optimization criterion, maximizing bandwidth, is used to design the MPA. To select a design, an exhaustive search is usually performed on a very limited set of available polarizing patterns, which must be constrained in order to make the search tractable. In contrast, this paper proposes a fully automated and optimal MPA design method (AO-MPA) which generates significantly improved MPAs. Instead of the single criterion of bandwidth, we propose six design principles, and show how they can be utilized to mutually optimize the MPA design by formulating a tri-objective optimization problem with multiple constraints. A much larger set of possible MPA patterns is rapidly and automatically searched by applying advanced multi-objective optimization techniques. We have tested AO-MPA using two groups of experiments, in which AO-MPA is compared against several other leading MPA design methods, and the patterns generated by AO-MPA are compared against state-of-the-art patterns from the literature. The results, obtained using a public benchmark dataset, show that the AO-MPA method is very computationally efficient, and can find all optimal MPA patterns for all array sizes. Moreover, for each size, AO-MPA obtains all optimal layouts simultaneously. AO-MPA generates designs which require fewer polarization orientations, while also yielding better performance in estimating intensity measurements, Stokes vector and the degree of linear polarization. This results in MPAs which are easier to manufacture while also being more robust to noise.

Risk-constrained self-scheduling of a hybrid power plant considering interval-based intraday demand response exchange market prices

Hybrid power plants (HPPs) integrating dispatchable and non-dispatchable generation are gaining attention by generation companies due to their increased flexibility in the operation of the power system. In this paper, an offering and bidding framework for an HPP consisting wind, photovoltaic (PV), compressed air energy storage (CAES), battery energy storage (BES), and thermal units in day-ahead (DA) and intraday markets is presented. Moreover, the interaction between the HPP and demand response providers (DRPs) through the intraday demand response exchange (IDREX) market is incorporated into the proposed model. The existing uncertainties such as DA, intraday, imbalance prices, along with renewable energy and IDREX market, are tackled via a hybrid stochastic-interval approach. The suggested structure is not only capable of handling both stochastic and interval uncertainties but also can manage the risk associated with both uncertainty characterization methods. To this end, the proposed risk-constrained offering and bidding model turns into a tri-objective optimization problem in which the normal boundary intersection (NBI) procedure is applied for its solution. The numerical results demonstrate that the proposed framework is well capable of simultaneously reaching risk-taker and risk-averse strategies.

Rotary unmanned aerial vehicles path planning in rough terrain based on multi-objective particle swarm optimization

This paper presents a path planning approach for rotary unmanned aerial vehicles (R-UAVs) in a known static rough terrain environment. This approach aims to find collision-free and feasible paths with minimum altitude, length and angle variable rate. First, a three-dimensional (3D) modeling method is proposed to reduce the computation burden of the dynamic models of R-UAVs. Considering the length, height and tuning angle of a path, the path planning of R-UAVs is described as a tri-objective optimization problem. Then, an improved multi-objective particle swarm optimization algorithm is developed. To render the algorithm more effective in dealing with this problem, a vibration function is introduced into the collided solutions to improve the algorithm efficiency. Meanwhile, the selection of the global best position is taken into account by the reference point method. Finally, the experimental environment is built with the help of the Google map and the 3D terrain generator World Machine. Experimental results under two different rough terrains from Guilin and Lanzhou of China demonstrate the capabilities of the proposed algorithm in finding Pareto optimal paths.

Equidistant representations: Connecting coverage and uniformity in discrete biobjective optimization

The nondominated frontier of a multiobjective optimization problem can be overwhelming to a decision maker, as it is often either very large or infinite in size. Instead, a discrete representation of this set in the form of a small sample of points is often preferred. In this paper we consider the Discrete Representation Problem (DRP), which is itself a triobjective optimization problem. The three objectives comprise three standard quality measures for discrete representations, namely coverage, uniformity and the cardinality of the set. We introduce the notion of complete equidistant representations, and prove that such a representation provides a nondominated solution to the DRP. In addition, we show through the help of complete equidistant representations that coverage and uniformity can be seen as dual problems given a fixed cardinality, and therefore that optimality gaps for coverage and uniformity can be obtained given any representation. Moreover, even though the definition of the coverage error requires the full nondominated set, we show how the coverage error for a given representation can be calculated by generating a much smaller set. Finally, we present a new method for finding discrete representations of a desired cardinality that outperforms existing methods w.r.t. coverage and uniformity on a set of mixed-integer programming benchmark instances.

Tri-objective optimal scheduling of smart energy hub system with schedulable loads

Many researches have investigated the optimal energy scheduling to meet the demand regarding the sustainable development issues and the economic and environmental indices. The energy hub system (EHS) is introduced as an appropriate framework in which the demands are supplied by multi carrier energy (MCE) such as electricity, natural gas, and thermal energy. In this paper, the optimal scheduling problem of an EHS is modeled as a tri-objective optimization problem in which the operation cost, the emission pollution, and the deviation of the electrical load profile from its desired value is minimized. In the proposed model, the third objective function as a Demand-side Management (DSM) strategy is considered for the optimal shifting of the electrical deferrable loads (EDLs) based on the day-ahead electrical energy prices. Moreover, the reserve scheduling is proposed by the interruptible loads (ILs) of the thermal and the electrical demand. The proposed model is solved using the augmented ε-constraint method in the General Algebraic Modeling System (GAMS) optimization software environment. The generated solutions by the augmented ε-constraint method have diverse non-dominated solutions in which the best solution is selected by the fuzzy approach. In order to validate the proposed approach, three Case studies are investigated and their results are compared with each other. The operation cost and the emission pollution of the EHS in the first case is 402130.12 $ and 36162.26 kg. With the participation of EDLs in the second case, the operation cost and the emission reduces 0.4% and 3.02%, respectively. Finally, with the simultaneous participation of EDLs and ILs in the third case, the operation cost and the emission pollution reduces 0.38% and 2.53% in comparison with the second case. • Proposing tri-objective functions in smart EHS with optimal shifting of EDLs. • Implementation of optimal shifting and strategic conversion of DSM strategies. • Proposing ILs as reserve in smart EHS. • Utilization of augmented ɛ-constraint method to solve objective functions. • Finding best solution by fuzzy approach.

INFERRING PARAMETERS OF A RELATIONAL SYSTEM OF PREFERENCES FROM ASSIGNMENT EXAMPLES USING AN EVOLUTIONARY ALGORITHM

Most evolutionary multi-objective algorithms perform poorly in many objective problems. They normally do not make selective pressure towards the Region of Interest (RoI), the privileged zone in the Pareto frontier that contains solutions important to a DM. Several works have proved that a priori incorporation of preferences improves convergence towards the RoI. The work of (E. Fernandez, E. Lopez, F. Lopez & C.A. Coello Coello, 2011) uses a binary fuzzy outranking relational system to map many-objective problems into a tri-objective optimization problem that searches the RoI; however, it requires the elicitation of many preference parameters, a very hard task. The use of an indirect elicitation approach overcomes such situation by allowing the parameter inference from a battery of examples. Even though the relational system of Fernandez et al. (2011) is based on binary relations, it is more convenient to elicit its parameters from assignment examples. In this sense, this paper proposes an evolutionary-based indirect parameter elicitation method that uses preference information embedded in assignment examples, and it offers an analysis of their impact in a priori incorporation of DM’s preferences. Results show, through an extensive computer experiment over random test sets, that the method estimates properly the model parameter’s values.