Alfredo García Hernández-Díaz

Solving a comprehensive model for multiobjective project portfolio selection

Any organization is routinely faced with the need to make decisions regarding the selection and scheduling of project portfolios from a set of candidate projects. We propose a multiobjective binary programming model that facilitates both obtaining efficient portfolios in line with the set of objectives pursued by the organization, as well as their scheduling regarding the optimum time to launch each project within the portfolio without the need for a priori information on the decision-makers preferences. Resource constraints, the possibility of transferring resources not consumed in a given a period to the following one, and project interdependence have also been taken into account. Given that the complexity of this problem increases as the number of projects and the number of objectives increase, we solve it using a metaheuristic procedure based on Scatter Search that we call SS-PPS (Scatter Search for Project Portfolio Selection). The characteristics and effectiveness of this method are compared with other heuristic approaches (SPEA and a fully random procedure) using computational experiments on randomly generated instances. Statement of scope and purpose: This paper describes a model to aid in the selection and scheduling of project portfolios within an organization. The model was designed assuming strong interdependence between projects, which therefore have to be assessed in groups, while allowing individual projects to start at different times depending on resource availability or any other strategic or political requirements, which involves timing issues. The simultaneous combination of project portfolio selection and scheduling under general conditions involves known drawbacks that we attempt to remedy. Finally, the model takes into account multiple objectives without requiring a priori specifications regarding the decision-makers preferences. The resolution of the problem was approached using a metaheuristic procedure, which showed by computational experiments good performance compared with other heuristics.

g-dominance: Reference point based dominance for multiobjective metaheuristics

One of the main tools for including decision maker (DM) preferences in the multiobjective optimization (MO) literature is the use of reference points and achievement scalarizing functions [A.P. Wierzbicki, The use of reference objectives in multiobjective optimization, in: G. Fandel, T. Gal (Eds.), Multiple-Criteria Decision Making Theory and Application, Springer-Verlag, New York, 1980, pp. 469-486.]. The core idea in these approaches is converting the original MO problem into a single-objective optimization problem through the use of a scalarizing function based on a reference point. As a result, a single efficient point adapted to the DMs preferences is obtained. However, a single solution can be less interesting than an approximation of the efficient set around this area, as stated for example by Deb in [K. Deb, J. Sundar, N. Udaya Bhaskara Rao, S. Chaudhuri, Reference point based multiobjective optimization using evolutionary algorithms, International Journal of Computational Intelligence Research, 2(3) (2006) 273-286]. In this paper, we propose a variation of the concept of Pareto dominance, called g-dominance, which is based on the information included in a reference point and designed to be used with any MO evolutionary method or any MO metaheuristic. This concept will let us approximate the efficient set around the area of the most preferred point without using any scalarizing function. On the other hand, we will show how it can be easily used with any MO evolutionary method or any MO metaheuristic (just changing the dominance concept) and, to exemplify its use, we will show some results with some state-of-the-art-methods and some test problems.

Pareto-adaptive ε-dominance

Efficiency has become one of the main concerns in evolutionary multiobjective optimization during recent years. One of the possible alternatives to achieve a faster convergence is to use a relaxed form of Pareto dominance that allows us to regulate the granularity of the approximation of the Pareto front that we wish to achieve. One such relaxed forms of Pareto dominance that has become popular in the last few years is -dominance, which has been mainly used as an archiving strategy in some multiobjective evolutionary algorithms. Despite its advantages, -dominance has some limitations. In this paper, we propose a mechanism that can be seen as a variant of -dominance, which we call Pareto-adaptive -dominance (pa-dominance). Our proposed approach tries to overcome the main limitation of -dominance: the loss of several nondominated solutions from the hypergrid adopted in the archive because of the way in which solutions are selected within each box.

A new proposal for multi-objective optimization using differential evolution and rough sets theory

This paper presents a new multi-objective evolutionary algorithm (MOEA) based on differential evolution and rough sets theory. The proposed approach adopts an external archive in order to retain the nondominated solutions found during the evolutionary process. Additionally, the approach also incorporates the concept of paε-dominance to get a good distribution of the solutions retained. The main idea of the approach is to use differential evolution (DE) as our main search engine, trying to translate its good convergence properties exhibited in single-objective optimization to the multi-objective case. Rough sets theory is adopted in a second stage of the search in order to improve the spread of the nondominated solutions that have been found so far. Our hybrid approach is validated using standard test functions and metrics commonly adopted in the specialized literature. Our results are compared with respect to the NSGA-II, which is a MOEA representative of the state-of-the-art in the area.

Seeding the initial population of a multi-objective evolutionary algorithm using gradient-based information

In the field of single-objective optimization, hybrid variants of gradient-based methods and evolutionary algorithms have been shown to perform better than an evolutionary method by itself. This same idea has been recently used in Evolutionary Multiobjective Optimization (EMO), obtaining also very promising results. In most cases, gradient information is used along the whole process, which involves a high computational cost, mainly related to the computation of the step lengths required. In contrast, in this paper we propose the use of gradient information only at the beginning of the search process. We will show that this sort of scheme maintains results of good quality while considerably decreasing the computational cost. In our work, we adopt a steepest descent method to generate some nondominated points which are then used to seed the initial population of a multi-objective evolutionary algorithm (MOEA), which will spread them along the Pareto front. The MOEA adopted in our case is the NSGA-II, which is representative of the state-of-the-art in the area. To validate our proposal, we adopt box-constrained continuous problems (the ZDT test suite). The gradients required are approximated using quadratic regressions. Our proposed approach performs a total of 2000 objective function evaluations, which is much lower than the number of evaluations normally adopted with the ZDT test suite in the specialized literature. Our results are compared with respect to the ldquopurerdquo NSGA-II (i.e., without using gradient-based information) so that the potential benefit of these initial solutions fed into the population can be properly assessed.

A multi-start algorithm for a balanced real-world Open Vehicle Routing Problem

The aim of this paper is to solve a real-world problem proposed by an international company operating in Spain and modeled as a variant of the Open Vehicle Routing Problem in which the makespan, i.e., the maximum time spent on the vehicle by one person, must be minimized. A competitive multi-start algorithm, able to obtain high quality solutions within reasonable computing time is proposed. The effectiveness of the algorithm is analyzed through computational testing on a set of 19 school-bus routing benchmark problems from the literature, and on 9 hard real-world problem instances.

A new proposal for multiobjective optimization using particle swarm optimization and rough sets theory

This paper presents a new multi-objective evolutionary algorithm which consists of a hybrid between a particle swarm optimization approach and some concepts from rough sets theory. The main idea of the approach is to combine the high convergence rate of the particle swarm optimization algorithm with a local search approach based on rough sets that is able to spread the nondominated solutions found, so that a good distribution along the Pareto front is achieved. Our proposed approach is able to converge in several test functions of 10 to 30 decision variables with only 4,000 fitness function evaluations. This is a very low number of evaluations if compared with todays standards in the specialized literature. Our proposed approach was validated using nine standard test functions commonly adopted in the specialized literature. Our results were compared with respect to a multi-objective evolutionary algorithm that is representative of the state-of-the-art in the area: the NSGA-II.

Improving the vector generation strategy of Differential Evolution for large-scale optimization

Several potential weaknesses of DE deteriorate its performance specially when dealing with large-scale problems.Controlling the diversity of trial vectors and exploration capabilities of DE is very important when designing new DE approaches for high-dimensional problems.Two new schemes based on diversifying the trial vectors substantially improve the capabilities of DE for dealing with high-dimensional problems.The new proposals are an alternative way to control the balance between exploration and exploitation in DE.Several state of the art non-hybrid DE schemes are improved by incorporating our proposals. Differential Evolution is an efficient metaheuristic for continuous optimization that suffers from the curse of dimensionality. A large amount of experimentation has allowed researchers to find several potential weaknesses in Differential Evolution. Some of these weaknesses do not significantly affect its performance when dealing with low-dimensional problems, so the research community has not paid much attention to them. The aim of this paper is to provide a better insight into the reasons of the curse of dimensionality and to propose techniques to alleviate this problem. Two different weaknesses are revisited and schemes for dealing with them are devised. The schemes increase the diversity of trial vectors and improve on the exploration capabilities of Differential Evolution. Some important mathematical properties induced by our proposals are studied and compared against those of related schemes. Experimentation with a set of problems with up to 1000 dimensions and with several variants of Differential Evolution shows that the weaknesses analyzed significantly affect the performance of Differential Evolution when used on high-dimensional optimization problems. The gains of the proposals appear when highly exploitative schemes are used. Our proposals allow for high-quality solutions with small populations, meaning that the most significant advantages emerge when dealing with large-scale optimization problems, where the benefits of using small populations have previously been shown.

The challenge of optimizing expensive black boxes: a scatter search/rough set theory approach

There is renewed interest in the development of effective and efficient methods for optimizing models of which the optimizer has no structural knowledge. This is what in the literature is referred to as optimization of black boxes. In particular, we address the challenge of optimizing expensive black boxes, that is, those that require a significant computational effort to be evaluated. We describe the use of rough set theory within a scatter search framework, with the goal of identifying high-quality solutions with a limited number of objective function evaluations. The rough set strategies that we developed take advantage of the information provided by the best and diverse solutions found during the search, in order to define areas of the solution space that are promising for search intensification. We test our procedure on a set of 92 nonlinear multimodal functions of varied complexity and size and compare the results with a state-of-the-art procedure based on particle swarm optimization.

Using Box Indices in Supporting Comparison in Multiobjective Optimization

Because of the conflicting nature of criteria or objectives, solving a multiobjective optimization problem typically requires interaction with a decision maker who can specify preference information related to the objectives in the problem in question. Due to the difficulties of dealing with multiple objectives, the way information is presented plays a very important role. Questions posed to the decision maker must be simple enough and information shown must be easy to understand. For this purpose, visualization and graphical representations can be useful and constitute one of the main tools used in the literature. In this paper, we propose to use box indices to represent information related to different solution alternatives of multiobjective optimization problems involving at least three objectives. Box indices are an intelligible and easy to handle way to represent data. They are based on evaluating the solutions in a natural and rough enough scale in order to let the decision maker easily recognize the main characteristics of a solution at a glance and to facilitate comparison of two or more solutions in an easily understandable way.