A method for including a-priori-preferences in Multicriteria Optimization

Genetic algorithms (GAs), which are directed stochastic hill climbing algorithms, are a commonly used optimization technique and are generally applied to single criterion optimization problems with fairly complex solution landscapes. There has been some attempts to apply GA to multicriteria optimization problems. The GA selection mechanism is typically dependent on a single-valued objective function and so no general methods to solve multicriteria optimization problems have been developed so far. In this paper, a new method of transformation of the multiple criteria problem into a single-criterion problem is presented. The problem of transformation brings about the need for the introduction of thePareto set estimation method to perform the multicriteria optimization using GAs. From a given solution set, which is the population of a certain generation of the GA, the Pareto set is found. The fitness of population members in the next GA generation is calculated by a distance metric with a reference to the Pareto set of the previous generation. As we are unable to combine the objectives in some way, we resort to this distance metric in the positive Pareto space of the previous solutions, as the fitness of the current solutions. This new GA-based multicriteria optimization method is proposed here, and it is capable of handling any generally formulated multicriteria optimization problem. The main idea of the method is described in detail in this paper along with a detailed numerical example. Preliminary computer generated results show that our approach produces better, and far more Pareto solutions, than plain stochastic optimization methods.

Regarding the advances in technology and anxieties around high and growing prices of fossil fuels, government incentives increase to produce cleaner and sustainable energy through distributed generations. This makes trends in the using microgrids which consist of electric demands and different distributed generations and energy storage systems. The optimum operation of microgrids with considering demand-side management increases efficiency and reliability and maximize the advantages of using distributed generations. In this paper, the optimal operation scheduling and unit commitment of generation units installed in a microgrid are investigated. The microgrid consists of technologies based on natural gas that are microturbine and phosphoric acid fuel cell and technologies based on renewable energy, including wind turbine and photovoltaic unit along with battery energy storage system and plug-in electric vehicle commercial parking lot. The goal of the paper is to solve a multi-objective problem of maximizing revenues of microgrid operator and minimizing emissions. This paper uses an augmented epsilon constraint method for solving the multi-objective problem in a stochastic framework and also implements a fuzzy-based decision-maker for choosing the suitable optimal solution amid Pareto front solutions. This new model implements the three type of the price-based and incentive-based demand response program. It also considers the generation reserve in order to enhance the flexibility of operations. The presented model is tested on a microgrid and the results demonstrate the efficacy of the proposed model economically and environmentally compared to other methods.

There is an increasing trend in the use of multi-objective evolutionary algorithms (MOEAs) to solve multi-objective optimization problems of the allocation of water resources. However, typically the outcome is a set of Pareto optimal solutions which make up a trade-off surface between the objective functions. For decision makers to choose a satisfactory alternative from a set of Pareto-optimal solutions, this paper suggests a new method based on least squares support vector machine (LSSVM) and k-means clustering for ranking the optimal solutions for the multi-objective allocation of water resources. First, the k-means clustering method was adopted to reduce the large set of solutions to a few representative solutions. Then, to capture and represent the decision maker's preferences as well as to select the most desirable alternative, the LSSVM method was applied to obtain the utility value for each representative solution. According to the magnitude of the utility values, the final priority orders of the representative solutions were determined. Finally, this methodology was applied to rank the Pareto optimal solution set obtained from the multi-objective optimization problems of water resources allocation for the water-receiving areas of the South-to-North Water Transfer Project in Hebei Province, China. Moreover, the comparisons of the proposed method with the information entropy method and the artificial neural network (ANN) model were given. The results of the comparison indicate that the proposed method has the ability to rank the non-dominated solutions of the multi-objective operation optimization model and that it can be employed for decision-making on water allocation and management in a river basin.

Many decision making problems of business and management are formulated in terms of Multiple Attribute Decision Making (MADM): given a set of alternatives evaluated with multiple criteria, find the alternative which according to the Decision Maker (DM), has the most preferred combination of criteria values (attributes), or rank alternatives from the most preferred one to the least preferred one. The MADM methods incorporate mechanisms of building preference models based on information obtained from the DM. In a wide variety of such methods, the DM is supposed to provide information in terms of weights of criteria, usually understood as criteria’s priorities. These weights serve as parameters of the method-pecific preference models. The DM can define weights directly, or by using special weight elicitation techniques such as AHP, MAVT and others. Our concerns are that when using weight-based methods, the DM cannot ensure the correctness of the preference model. First, different weight-based methods use different kinds of preference models, which prioritize criteria based on weights in different manners. Second, interpretation of weights in some MADM methods is far from intuitive. Thus, a situation may occur when an inexperienced DM thinks of weights differently than they actually work in the method, and expresses the preference information incorrectly. In this paper we demonstrate the differences between how weights are interpreted in several methods: simple additive weighting, TOPSIS, VIKOR and PROMETHEE. We do it by comparing rankings produced with methods based on randomly generated data. We demonstrate that differences of interpreting weights significantly contribute to differences in produced rankings. A solution to this problem could be twofold: first, increasing awareness of differences between method-specific weight-based prioritizing mechanisms, and second, providing interpretations of weights for popular methods in the language understandable by the DMs.

Multimodal multiobjective optimization problems (MMOPs) have attracted wide attention in recent years. This kind of problem is very challenging since they need to locate different Pareto-optimal solution sets (PSs) that correspond to the same Pareto front. To resolve it, this article proposes a novel multimodal multiobjective differential evolution (DE) algorithm with species conservation, which develops a new way of locating different PSs. Specifically, the proposed algorithm adopts species conservation to determine different PSs in known areas, while it uses a variant of DE as the basic optimizer to explore new areas. There are three operators in species conservation: 1) species division; 2) seed determination; and 3) seed conservation. Species division mainly partitions the joint population of parents and children into various species in the decision space for retaining different PSs. Seed determination selects superior solutions from each species as seeds that need to be kept in the next generation. Seed conservation is to ensure that all species seeds are retained in the new generation by substituting no promising solutions with them, thereby guarantee not missing some known areas that may contain different PSs. Besides, the DE variant is utilized to produce diverse solutions to find new areas in the decision space where PSs may exist. The comparative experiments with ten state-of-the-art algorithms have been performed on the CEC 2019 MMOPs test set and two real-world problems. The experimental results have verified that the proposed algorithm has a competitive performance for MMOPs.

Objective The aim of this study was to evaluate the role of multicriteria optimization (MCO) in planning of intensity-modulated radiotherapy (IMRT).Methods Twenty IMRT patients (ten with prostate and ten with lung cancers) were randomly selected.For these patients,the treatment plans were designed with direct machine parameter optimization (DMPO).Based on these plans,new plans were designed with MCO,while keeping the setting conditions unchanged.Comparison was made between the two plans including the dose distribution,the dose volume histogram,the time of optimization and number of monitor unit (MU),but were play by pairing-t test.Results The plan designed in both optimizations satisfied all clinical requirements.For the same or better target coverage,rectum,bladder and small bowel were better with MCO compared with DMPO,MCO reduced 58％ of the time for optimization by average while MU increased 32％ by average for prostate cancer.For lung cancer,the whole lung,heart and spinal cord were better with MCO compared with DMPO,MCO reduced 59％ of the time for optimization by average while MU increased 11％ by average.Conclusions In comparison with DMPO,MCO reduces the dose of organs at risk,shorten the time of optimization. Key words: Multicriteria optimization; Direct machine parameter optimization; Intensity-modulated radiotherapy planning

In this thesis we analyze and develop methods based on model order reduction (MOR) for the solution of optimization problems constrained by time-dependent partial differential equations (PDEs). The methods combine a direct solution approach with model reduction via proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). The reduced-order models (ROMs) are used to approximate the high-dimensional dynamic systems originating from a spatial discretization of a PDE. However, when used in an optimization algorithm, conventional POD/DEIM ROMs often lack the ability to give adequate approximations of the gradient. We propose methods for a suitable enhancement of the ROMs for the optimization purpose which are based on the inclusion of derivative information in the POD and DEIM subspaces. We distinguish two types of error between quantities evaluated with the high-dimensional model and its ROM approximation in dependency on the optimization variable q: The reconstruction error which is evaluated with the same q0 which is used constructing the ROM and the prediction error which assesses approximations at q with a ROM constructed at q0 different to q. The novel reconstruction results we present include estimates for solutions of the adjoint equation and the sensitivity equations as well as for the gradient of the objective function. Based on the estimates we explain how the POD and DEIM bases should be extended with either adjoint or sensitivity information. The enhanced ROMs allow control of the reconstruction error for the objective and its gradient up to machine precision. Moreover, we propose a POD prediction estimate for the objective of the optimization problem in a neighborhood of q where the ROM is constructed. In case of sensitivity-extended POD and DEIM bases we give an analogous result for solutions of the states. The derivative-extended ROMs are then used to develop adaptive algorithms for the solution of optimal control and parameter estimation problems which results in great runtime improvements for the optimization while ensuring high approximation quality of the solution of the original problem. For the parameter estimation case a novel a posteriori error estimate is proposed which assesses the quality of suboptimal solutions obtained with the ROM. A further fundamental contribution is a discussion of discretize-then-optimize (DTO) vs. optimize-then-discretize (OTD) approaches in the context of MOR for optimization. We analyze advantages and disadvantages of both approaches and discuss to which extent our methods exhibit properties of either strategy. We also give examples of representative optimization problems in which standard POD/DEIM ROMs show an inacceptable behavior and can be successfully solved by derivative-extended ROMs. We have further implemented the developed methods emphasizing an efficient realization which is important for the investigation of the MOR potential. We showcase the practical performance of the proposed algorithms and the superiority of derivative-extended over conventional ROMs on two academic and one industry-relevant application which exhibit a variety of challenges for the model reduction approach in optimization.

Land-use allocation is of great significance in urban development. This type of allocation is usually considered to be a complex multi-objective spatial optimization problem, whose optimized result is a set of Pareto-optimal solutions (Pareto front) reflecting different tradeoffs in several objectives. However, obtaining a Pareto front is a challenging task, and the Pareto front obtained by state-of-the-art algorithms is still not sufficient. To achieve better Pareto solutions, taking the grid-representative land-use allocation problem with two objectives as an example, an artificial bee colony optimization algorithm for multi-objective land-use allocation (ABC-MOLA) is proposed. In this algorithm, the traditional ABC’s search direction guiding scheme and solution maintaining process are modified. In addition, a knowledge-informed neighborhood search strategy, which utilizes the auxiliary knowledge of natural geography and spatial structures to facilitate the neighborhood spatial search around each solution, is developed to further improve the Pareto front’s quality. A series of comparison experiments (a simulated experiment with small data volume and a real-world data experiment for a large area) shows that all the Pareto fronts obtained by ABC-MOLA totally dominate the Pareto fronts by other algorithms, which demonstrates ABC-MOLA’s effectiveness in achieving Pareto fronts of high quality.

Due to the importance of conserving resources for future generations in the field of energy planning, this paper examines the concept of sustainable development in this area. The energy planning problem is modeled through incorporating environmental and social concepts into a tri-objective mathematical model. The considered objectives are minimizing energy costs, minimizing greenhouse gas emission, and maximizing the rate of social acceptance. In order to show the applicability of the proposed model, it is implemented over a real case study in Iran. Since the considered problem is a multi-objective one, the Pareto optimal set of solutions for the proposed model is obtained using parametric weighting, e-constraint, and augmented e-constraint methods. The results showed that the best Pareto frontiers are obtained using the augmented e-constraint method. Also, four scenarios are defined to help decision makers in selecting the preferred solution under each scenario using the TOPSIS method. The results of scenario evaluation confirm that the socially oriented scenario is more accepted.

Trade-off-based multi-objective optimisation of a simultaneous saccharification and fermentation process

Most of the engineering problems are modeled as evolutionary multiobjective optimization problems, but they always ask for only one best solution, not a set of Pareto optimal solutions. The decision maker′s subjective information plays an important role in choosing the best solution from several Pareto optimal solutions. Generally, the decision‐making processing is implemented after Pareto optimality. In this paper, we attempted to incorporate the decider′s subjective sense with Pareto optimality for chromosomes ranking. A new ranking method based on subjective probability theory was thus proposed in order to explore and comprehend the true nature of the chromosomes on the Pareto optimal front. The properties of the ranking rule were proven, and its transitivity was presented as well. Simulation results compared the performance of the proposed ranking approach with the Pareto‐based ranking method for two multiobjective optimization cases, which demonstrated the effectiveness of the new ranking approach.

In this paper, we discusses the problem that involving the conflict fuzzy multi-objective in transportation planning which is one of special case in multi-objective linear programming. To find the simultaneously optimal solution of the problem, we use the Interactive Fuzzy Multi-Objective Linear Programming (IFMOLP) method. This method can reduce the fuzzy multi-objective linear programming problem into deterministic single objective linear programming which can be solved using simplex method. Beside that , with IFMOLP method, decision maker (DM) can establish interactively the goal from the objective function to produce the pareto optimal solution so that.

A computationally-efficient method for multi-criteria optimisation is developed for performance-based seismic design of friction energy dissipation dampers in RC structures. The proposed method is based on the concept of Uniform Distribution of Deformation (UDD), where the slip-load distribution along the height of the structure is gradually modified to satisfy multiple performance targets while minimising the additional loads imposed on existing structural elements and foundation. The efficiency of the method is demonstrated through optimisation of 3, 5, 10, 15 and 20-storey RC frames with friction wall dampers subjected to design representative earthquakes using single and multi-criteria optimisation scenarios. The optimum design solutions are obtained in only a few steps, while they are shown to be independent of the selected initial slip loads and convergence factor. Optimum frames satisfy all predefined design targets and exhibit up to 48% lower imposed loads compared to designs using a previously proposed slip-load distribution. It is also shown that dampers designed with optimum slip load patterns based on a set of spectrum-compatible synthetic earthquakes, on average, provide acceptable design solutions under multiple natural seismic excitations representing the design spectrum.

The allocation of limited capital to a variety of assets available is one of the important problems in financial management. It is actually a constrained multi-objective optimization problem called portfolio selection. Most studies have been made to solve the problem with single-objective optimization techniques by aggregating conflicting and incommensurate objectives into a single one. The solutions obtained may be unsatisfactory because their nondominance is not guaranteed, especially when some practical constraints, such as cardinality constraints, are incorporated into the portfolio optimization models. This paper presents an evolutionary multi-objective optimization technique, which called Knapsack Multi-Objective Particle Swarm Optimization (K-MOPSO) to generate the efficient portfolios in terms of expected return and variance under short sales and/or cardinality constraints. Finally, a Multi-Attribute Decision Making (MADM) method named Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is employed to outrank the efficient portfolio that decision makers satisfy most.

Multicriterial design methodology with safety as one of the design objectives is presented. The aim of the paper is to analyze the influence of safety based design objectives on generated nondominated designs on the Pareto frontier. Possible improvements in nondominated designs are investigated by comparison to ones obtained with the standard design procedure when safety criteria are used as design constraints only. It is assumed that safety based objectives and targets act as attractors, driving nondominated designs along the constant cost/weight contours in design space towards its safer regions. Global safety objectives (for hogging/sagging modes), are based on the maximization of ultimate longitudinal strength in vertical bending calculated via the extended IACS incremental-iterative method. Applied compound safety measures for gross-panel (stiffened panel with associated girders) are based upon 34 failure modes, belonging to serviceability/collapse subsets. Objectives based on the maximization of safety measures are applied together with standard design objectives such as minimization of initial cost and weight. The following problems were solved with different sets of objectives: (a) minimize cost and weight objectives subject to safety constraints (used for reference), (b) only the maximization of local safety measures is added to (a) as additional objectives, (c) only the maximization of global safety measures is added to (a) as additional objectives, (d) maximization of safety measures ad (b) and (c) are added to (a) as additional objectives. For each of the problems (a–d) the developed design procedure is executed. It contains two basic tasks for structural design of realistic (non-academic) problems: (1) multicriterial optimization with topology / geometry design variables; (2) multicriterial optimization of gross-panels with scantling / material design variables. Design procedure steps are executed using a fast and balanced collection of analysis and synthesis modules/methods of the OCTOPUS design system: • Determination of design load sets; • MOGA / MOPSO based generation of nondominated designs for the selected ship structure; • For each design the following analysis blocks are executed: – calculation of ship’s primary and racking response fields, – calculation of ship’s ultimate longitudinal strength, – calculation of serviceability and collapse safety criteria on the gross-panel level. Comparisons of results, based on generated Pareto hyper-surfaces and on subset of preferred designs, are given for problems (a–d). Insights into the results of optimization process, using 5-D graphics for design and attribute spaces, are also presented. Design problems of modern RoPax and SWATH structures are used in case studies.