Optimisation over the non-dominated set of a multi-objective optimisation problem

In this thesis we are concerned with optimisation over the non-dominated set of a multiobjective optimisation problem. A multi-objective optimisation problem (MOP) involves multiple conflicting objective functions. The non-dominated set of this problem is of interest because it is composed of the “best” trade-off for a decision maker to choose according to his preference. We assume that this selection process can be modelled by maximising a function over the non-dominated set. We present two new algorithms for the optimisation of a linear function over the non-dominated set of a multi-objective linear programme (MOLP). A primal method is developed based on a revised version of Benson’s outer approximation algorithm. A dual method derived from the dual variant of the outer approximation algorithm is proposed. Taking advantage of some special properties of the problem, the new methods are designed to achieve better computational efficiency. We compare the two new algorithms with several algorithms from the literature on a set of randomly generated instances. The results show that the new algorithms are considerably faster than the competitors. We adapt the two new methods for the determination of the nadir point of (MOLP). The nadir point is characterized by the componentwise worst values of the non-dominated points of (MOP). This point is a prerequisite for many multi-criteria decision making (MCDM) procedures. Computational experiments against another exact method for this purpose from the literature reveal that the new methods are faster than the competitor. The last section of the thesis is devoted to optimising a linear function over the non-dominated set of a convex multi-objective problem. A convex multi-objective problem (CMOP) often involves nonlinear objective functions or constraints. We extend the primal and the dual methods to solve this problem. We compare the two algorithms with several existing algorithms from the literature on a set of randomly generated instances. The results reveal that the new methods are much faster than the others

Exact And Representative Algorithms For Multi Objective Optimization

In most real-life problems, the decision alternatives are evaluated with multiple conflicting criteria. The entire set of non-dominated solutions for practical problems is impossible to obtain with reasonable computational effort. Decision maker generally needs only a representative set of solutions from the actual Pareto front. First algorithm we present is for efficiently generating a well dispersed non-dominated solution set representative of the Pareto front which can be used for general multi objective optimization problem. The algorithm first partitions the criteria space into grids to generate reference points and then searches for non-dominated solutions in each grid. This grid-based search utilizes achievement scalarization function and guarantees Pareto optimality. The results of our experimental results demonstrate that the proposed method is very competitive with other algorithms in literature when representativeness quality is considered; and advantageous from the computational efficiency point of view. Although generating the whole Pareto front does not seem very practical for many real life cases, sometimes it is required for verification purposes or where DM wants to run his decision making structures on the full set of Pareto solutions. For this purpose we present another novel algorithm. This algorithm attempts to adapt the standard branch and bound approach to the multi objective context by proposing to branch on solution points on objective space. This algorithm is proposed for multi objective integer optimization type of problems. Various properties of branch and bound concept has been investigated and explained within the multi objective optimization context such as fathoming, node selection, heuristics, as well as some multi objective optimization specific concepts like filtering, non-domination probability, running in parallel. Potential of this approach for being used both as a full Pareto generation or an approximation approach has been shown with experimental studies

An improved version of the augmented epsilon-constraint method (AUGMECON2) for finding the exact Pareto set in Multi-Objective Integer Programming problems

Generation (or a posteriori) methods in Multi-Objective Mathematical Programming (MOMP) is the most computationally demanding category among the MOMP approaches. Due to the dramatic increase in computational speed and the improvement of Mathematical Programming algorithms the generation methods become all the more attractive among today’s decision makers. In the current paper we present the generation method AUGMECON2 which is an improvement of our development, AUGMECON. Although AUGMECON2 is a general purpose method, we will demonstrate that AUGMECON2 is especially suitable for Multi-Objective Integer Programming (MOIP) problems. Specifically, AUGMECON2 is capable of producing the exact Pareto set in MOIP problems by appropriately tuning its running parameters. In this context, we compare the previous and the new version in a series of new and old benchmarks found in the literature. We also compare AUGMECON2’s performance in the generation of the exact Pareto sets with established methods and algorithms based on specific MOIP problems (knapsack, set packing) and on published results. Except from other Mathematical Programming methods, AUGMECON2 is found to be competitive also with Multi-Objective Meta-Heuristics (MOMH) in producing adequate approximations of the Pareto set in Multi-Objective Combinatorial Optimization (MOCO) problems

An improved version of the augmented epsilon-constraint method (AUGMECON2) for finding the exact Pareto set in Multi-Objective Integer Programming problems

Generation (or a posteriori) methods in Multi-Objective Mathematical Programming (MOMP) is the most computationally demanding category among the MOMP approaches. Due to the dramatic increase in computational speed and the improvement of Mathematical Programming algorithms the generation methods become all the more attractive among today’s decision makers. In the current paper we present the generation method AUGMECON2 which is an improvement of our development, AUGMECON. Although AUGMECON2 is a general purpose method, we will demonstrate that AUGMECON2 is especially suitable for Multi-Objective Integer Programming (MOIP) problems. Specifically, AUGMECON2 is capable of producing the exact Pareto set in MOIP problems by appropriately tuning its running parameters. In this context, we compare the previous and the new version in a series of new and old benchmarks found in the literature. We also compare AUGMECON2’s performance in the generation of the exact Pareto sets with established methods and algorithms based on specific MOIP problems (knapsack, set packing) and on published results. Except from other Mathematical Programming methods, AUGMECON2 is found to be competitive also with Multi-Objective Meta-Heuristics (MOMH) in producing adequate approximations of the Pareto set in Multi-Objective Combinatorial Optimization (MOCO) problems

Biobjective Optimization over the Efficient Set Methodology for Pareto Set Reduction in Multiobjective Decision Making: Theory and Application

A large number of available solutions to choose from poses a significant challenge for multiple criteria decision making. This research develops a methodology that reduces the set of efficient solutions under consideration. This dissertation is composed of three major parts: (i) the formalization of a theoretical framework; (ii) the development of a solution approach; and (iii) a case study application of the methodology. In the first part, the problem is posed as a multiobjective optimization over the efficient set and considers secondary robustness criteria when the exact values of decision variables are subjected to uncertainties during implementation. The contributions are centered at the modeling of uncertainty directly affecting decision variables, the use of robustness to provide additional trade-off analysis, the study of theoretical bounds on the measures of robustness, and properties to ensure that fewer solutions are identified. In the second part, the problem is reformulated as a biobjective mixed binary program and the secondary criteria are generalized to any convenient linear functions. A solution approach is devised in which an auxiliary mixed binary program searches for unsupported Pareto outcomes and a novel linear programming filtering excludes any dominated solutions in the space of the secondary criteria. Experiments show that the algorithm tends to run faster than existing approaches for mixed binary programs. The algorithm enables dealing with continuous Pareto sets, avoiding discretization procedures common to the related literature. In the last part, the methodology is applied in a case study regarding the electricity generation capacity expansion problem in Texas. While water and energy are interconnected issues, to the best of our knowledge, this is the first study to consider both water and cost objectives. Experiments illustrate how the methodology can facilitate decision making and be used to answer strategic questions pertaining to the trade-off among different generation technologies, power plant locations, and the effect of uncertainty. A simulation shows that robust solutions tend to maintain feasibility and stability of objective values when power plant design capacity values are perturbed

Methodological review of multicriteria optimization techniques: aplications in water resources

Multi-criteria decision analysis (MCDA) is an umbrella approach that has been applied to a wide range of natural resource management situations. This report has two purposes. First, it aims to provide an overview of advancedmulticriteriaapproaches, methods and tools. The review seeks to layout the nature of the models, their inherent strengths and limitations. Analysis of their applicability in supporting real-life decision-making processes is provided with relation to requirements imposed by organizationally decentralized and economically specific spatial and temporal frameworks. Models are categorized based on different classification schemes and are reviewed by describing their general characteristics, approaches, and fundamental properties. A necessity of careful structuring of decision problems is discussed regarding planning, staging and control aspects within broader agricultural context, and in water management in particular. A special emphasis is given to the importance of manipulating decision elements by means ofhierarchingand clustering. The review goes beyond traditionalMCDAtechniques; it describes new modelling approaches. The second purpose is to describe newMCDAparadigms aimed at addressing the inherent complexity of managing water ecosystems, particularly with respect to multiple criteria integrated with biophysical models,multistakeholders, and lack of information. Comments about, and critical analysis of, the limitations of traditional models are made to point out the need for, and propose a call to, a new way of thinking aboutMCDAas they are applied to water and natural resources management planning. These new perspectives do not undermine the value of traditional methods; rather they point to a shift in emphasis from methods for problem solving to methods for problem structuring. Literature review show successfully integrations of watershed management optimization models to efficiently screen a broad range of technical, economic, and policy management options within a watershed system framework and select the optimal combination of management strategies and associated water allocations for designing a sustainable watershed management plan at least cost. Papers show applications in watershed management model that integrates both natural and human elements of a watershed system including the management of ground and surface water sources, water treatment and distribution systems, human demands,wastewatertreatment and collection systems, water reuse facilities,nonpotablewater distribution infrastructure, aquifer storage and recharge facilities, storm water, and land use

Quality Representation in Multiobjective Programming

In recent years, emphasis has been placed on generating quality representations of the nondominated set of multiobjective programming problems. This manuscript presents two methods for generating discrete representations with equidistant points for multiobjective programs with solution sets determined by convex cones. The Bilevel Controlled Spacing (BCS) method has a bilevel structure with the lower-level generating the nondominated points and the upper-level controlling the spacing. The Constraint Controlled Spacing (CCS) method is based on the epsilon-constraint method with an additional constraint to control the spacing of generated points. Both methods (under certain assumptions) are proven to produce (weakly) nondominated points. Along the way, several interesting results about obtuse, simplicial cones are also proved. Both the BCS and CCS methods are tested and show promise on a variety of problems: linear, convex, nonconvex (CCS only), two-dimensional, and three-dimensional. Sample Matlab code for two of these examples can be found in the appendices as well as tables containing the generated solution points. The manuscript closes with conclusions and ideas for further research in this field

Multidisciplinary optimisation of an Unmanned Aerial Vehicle with a fuel cell powered energy system

ALF/ENGAER 139425-J Bernardo Miguel Teixeira Alves. Examination Committee: Chairperson: COR/ENGAER Luís António Monteiro Pessanha; Supervisors: Prof. André Calado Marta, MAJ/ENGAER Luís Filipe da Silva Félix; Member of the Committee: Prof. Pedro Vieira GamboaPara explorar a utilização de células de combustível a hidrogénio como alternativa viável aos combustíveis nocivos em veículos aéreos não-tripulados, um conceito de UAV de classe I foi desenvolvido no Centro de Investigação da Força Aérea (CIAFA). Este trabalho foca-se nos estudos trade-off realizados durante a sua conceção e na subsequente otimização. Primeiro, uma abordagem de otimização multi-objetivo foi utilizada com o auxílio do algoritmo genético NSGA-II para balancear dois objetivos em conflito: peso reduzido; e elevada autonomia. Conclui-se que é possível voar mais de três horas com um peso máximo à descolagem de 21,6 kg, uma célula de hidrogénio de 800 W e 148 g de hidrogénio. Uma configuração mais pesada com maior potência nominal e mais combustível foi descartada devido a um constragimento na envergadura. Posteriormente, com um conceito que satisfaz os requisitos impostos, uma abordagem multi-disciplinar (MDO) foi utilizada para maximizar a autonomia. O software utilizado foi o OpenAeroStruct, método dos elementos finitos (FEM) e o método da malha de vórtices (VLM) para modelar superfícies sustentadoras. Inicialmente, uma condição de cruzeiro e de carga foram utilizadas com torção geométrica da asa como variável de projeto. Posteriormente, maior complexidade foi introduzida atrav´es da utilização de afilamento, corda e envergadura. Finalmente, uma terceira condição de voo foi introduzida com o intuito de garantir o requisito de perda. Com a utilização de MDO foi possível aumentar a autonomia em 21% satisfazendo todos os requisitos. Este trabalho marca um passo importante no desenvolvimento de um futuro protótipo no Centro de Investigação.To explore the use of hydrogen fuel cells as a feasible alternative to pollutant fuels on Unmanned Aerial Vehicles (UAVs), a class I concept was designed at the Portuguese Air Force Research Centre. This work focuses on the trade-off studies performed during its design and on the optimisation that followed. First, a multi-objective optimisation approach was used with the aid of the Algorithm NSGAII to balance between two conflicting objectives: low weight and high endurance. It was found that it is possible to fly for more than 3 hours with a Maximum Take-off Weight of 21.6 kg, an 800 W fuel cell and 148 g of hydrogen. A heavier configuration with more power and fuel was discarded due to a wingspan constraint. Later, after the concept satisfied the project requirements, Multi-Disciplinary Design Optimisation (MDO) was performed to achieve the maximum endurance possible. The software used was OpenAeroStruct, low fidelity Finite Element Analysis (FEA) and Vortex Lattice Method (VLM) to model lifting surfaces. Initially, a cruise and a load flight point were used with wing geometric twist only as design variable. After, more complexity was added by introducing taper, wing chord and span. Finally, a third flight point was introduced to ensure the stall requirements were satisfied. The use of MDO allowed a 21% increase in endurance with a smaller wing area. Other improvements could not be achieved without violation of the constraints. This work marks an important milestone in the development of a future prototype at the Research Centre.N/

User-Oriented Methodology and Techniques of Decision Analysis and Support

This volume contains 26 papers selected from Workshop presentations. The book is divided into two sections; the first is devoted to the methodology of decision analysis and support and related theoretical developments, and the second reports on the development of tools -- algorithms, software packages -- for decision support as well as on their applications. Several major contributions on constructing user interfaces, on organizing intelligent DSS, on modifying theory and tools in response to user needs -- are included in this volume

Multiple Criteria Decision Support; Proceedings of an International Workshop, Helsinki, Finland, August 7-11, 1989

Multiple Criteria Decision Making has been an important and active research area for some 20 years. In the 1970's, research focused on the theory of multiple objective mathematical programming and on procedures for solving multiple objective mathematical programming problems. During the 1980's, a shift in emphasis towards multiple criteria decision support was observed. Accordingly, much research has focused on the user interface, the behavioral foundations of decision making, and on supporting the entire decision-making process from problem structuring to solution implementation. Because of the shift in research emphasis the authors decided to make "Multiple Criteria Decision Support" the theme for the International Workshop, which was held at Suomen Saeaestoepankkiopisto in Espoo, Finland. The Workshop was organized by the Helsinki School of Economics, and sponsored by the Helsinki School of Economics and IIASA, Austria. This volume provides an up-to-date coverage of the theory and practice of multiple criteria decision support. The authors trust that it will serve the research community as well as the previously published Conference Proceedings based on IIASA Workshops