Synchronous approach in interactive multiobjective optimization

We introduce a new approach in the methodology development for interactive multiobjective optimization. The presentation is given in the context of the interactive NIMBUS method, where the solution process is based on the classification of objective functions. The idea is to formulate several scalarizing functions, all using the same preference information of the decision maker. Thus, opposed to fixing one scalarizing function (as is done in most methods), we utilize several scalarizing functions in a synchronous way. This means that we as method developers do not make the choice between different scalarizing functions but calculate the results of different scalarizing functions and leave the final decision to the expert, the decision maker. Simultaneously, (s)he obtains a better view of the solutions corresponding to her/his preferences expressed once during each iteration.In this paper, we describe a synchronous variant of the NIMBUS method. In addition, we introduce a new version of its implementation WWW-NIMBUS operating on the Internet. WWW-NIMBUS is a software system capable of solving even computationally demanding nonlinear problems. The new version of WWW-NIMBUS can handle versatile types of multiobjective optimization problems and includes new desirable features increasing its user-friendliness.     

A review of interactive methods for multiobjective integer and mixed-integer programming

This paper makes a review of interactive methods devoted to multiobjective integer and mixed-integer programming (MOIP/MOMIP) problems. The basic concepts concerning the characterization of the non-dominated solution set are first introduced, followed by a remark about non-interactive methods vs. interactive methods. Then, we focus on interactive MOIP/MOMIP methods, including their characterization according to the type of preference information required from the decision maker, the computing process used to determine non-dominated solutions and the interactive protocol used to communicate with the decision maker. We try to draw out some contrasts and similarities of the different types of methods.     

Interactive multiobjective decision making by the sequential proxy optimization technique: SPOT

In this paper, we propose a new interactive multiobjective decision making technique, which we call the Sequential Proxy Optimization Technique (SPOT), in order to overcome the drawbacks of the conventional multiobjective decision making methods. Our method combines the desirable features of both the Surrogate Worth Trade-off (SWT) method and the Multiattribute Utility Function (MUF) method. We can interactively derive the preferred solution of the decision maker efficiently by assessing his marginal rate of substitution and maximizing sequentially the local proxy preference function. A numerical example illustrates the feasibility and efficiency of the proposed method.     

NAUTILUS method: An interactive technique in multiobjective optimization based on the nadir point

Most interactive methods developed for solving multiobjective optimization problems sequentially generate Pareto optimal or nondominated vectors and the decision maker must always allow impairment in at least one objective function to get a new solution. The NAUTILUS method proposed is based on the assumptions that past experiences affect decision makers’ hopes and that people do not react symmetrically to gains and losses. Therefore, some decision makers may prefer to start from the worst possible objective values and to improve every objective step by step according to their preferences. In NAUTILUS, starting from the nadir point, a solution is obtained at each iteration which dominates the previous one. Although only the last solution will be Pareto optimal, the decision maker never looses sight of the Pareto optimal set, and the search is oriented so that (s)he progressively focusses on the preferred part of the Pareto optimal set. Each new solution is obtained by minimizing an achievement scalarizing function including preferences about desired improvements in objective function values. NAUTILUS is specially suitable for avoiding undesired anchoring effects, for example in negotiation support problems, or just as a means of finding an initial Pareto optimal solution for any interactive procedure. An illustrative example demonstrates how this new method iterates.     

Portfolio selection in the Spanish stock market by interactive multiobjective programming

This study examines new versions of two interactive methods to address multiobjective problems, the aim of which is to enable the decision maker to reach a solution within the range of those considered efficient in a portfolio selection model, in which several objectives are pursued concerning risk and return and given that these are clearly conflicting objectives, the profile of the model proposed is multicriteria. Normally the range of efficient portfolios is fairly extensive thus making the selection of a single one an onerous task. In order to facilitate this process, interactive methods are used aimed at guiding the decision maker towards the optimal solution based on his preferences. Several adaptations were carried out on the original methods in order to facilitate the interactive process, improving the quality of the obtained portfolios, and these were applied to data obtained from the Madrid Stock Market, interaction taking place with two decision makers, one of whom was more aggressive than the other in their selections made.     

Identifying preferred solutions to Multi-Objective Binary Optimisation problems,with an application to the Multi-Objective Knapsack Problem

In this paper we present a new framework for identifying preferred solutions to multi-objective binary optimisation problems. We develop the necessary theory which leads to new formulations that integrate the decision space with the space of criterion weights. The advantage of this is that it allows for incorporating preferences directly within a unique binary optimisation problem which identifies efficient solutions and associated weights simultaneously. We discuss how preferences can be incorporated within the formulations and also describe how to accommodate the selection of weights when the identification of a unique solution is required. Our results can be used for designing interactive procedures for the solution of multi-objective binary optimisation problems. We describe one such procedure for the multi-objective multi-dimensional binary knapsack formulation of the portfolio selection problem.     

An algorithm for systems welfare interactive goal programming modelling

Every human system is faced with the problem of choosing between alternative options, and methods of interactive programming have been suggested as the best way to lead decision makers reach decisions that are consistent with their preferences. However, even though a large number of interactive algorithms have been proposed for multiobjective decision making (MODM), there is yet no truly interactive goal programming (GP) algorithm, despite the preference of GP over other MODM methodologies. The current paper presents an algorithm for interactive GP modelling called SWIGP (systems welfare interactive GP) which ensures that the overall welfare of the system under consideration is adequately taken into account in the interactive process. To achieve this, this paper distinguishes between technical, allocative and economic efficiencies and combines an economic efficiency index with interactive GP process. Besides being of wide applicability, the algorithm exerts little cognitive burden on the decision maker (DM). Indeed, even if the DM is assumed to operate under conditions of complete ignorance, SWIGP provides the direction for searching the “best” compromise solution. Moreover, the algorithm converges very fast because of the economic efficiency index that complements the interactive process in aiding the DM arrive at a most preferred solution.     

Robust optimization for interactive multiobjective programming with imprecise information applied to R&D project portfolio selection

A multiobjective binary integer programming model for R&D project portfolio selection with competing objectives is developed when problem coefficients in both objective functions and constraints are uncertain. Robust optimization is used in dealing with uncertainty while an interactive procedure is used in making tradeoffs among the multiple objectives. Robust nondominated solutions are generated by solving the linearized counterpart of the robust augmented weighted Tchebycheff programs. A decision maker’s most preferred solution is identified in the interactive robust weighted Tchebycheff procedure by progressively eliciting and incorporating the decision maker’s preference information into the solution process. An example is presented to illustrate the solution approach and performance. The developed approach can also be applied to general multiobjective mixed integer programming problems.     

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.     

An interactive fuzzy satisficing method for multiobjective nonconvex programming problems through floating point genetic algorithms

In this paper, we focus on multiobjective nonconvex nonlinear programming problems and present an interactive fuzzy satisficing method through floating point genetic algorithms. After determining the fuzzy goals of the decision maker, if the decision maker specifies the reference membership values, the corresponding Pareto optimal solution can be obtained by solving the augmented minimax problems for which the floating point genetic algorithm, called GENOCOP III, is applicable. In order to overcome the drawbacks of GENOCOP III, we propose the revised GENOCOP III by introducing a method for generating an initial feasible point and a bisection method for generating a new feasible point efficiently. Then an interactive fuzzy satisficing method for deriving a satisficing solution for the decision maker efficiently from a Pareto optimal solution set is presented together with an illustrative numerical example.     

Multiobjective stochastic programming for feed formulation

The minimum cost linear programming model used traditionally for feed formulation does not take account of variability of nutrients in feed ingredients. Therefore, it may be that the nutrient requirements of the animal are not adequately met. In this paper, we show how a multiobjective stochastic model that permits confronting the cost of the ration with the probabilities of meeting the nutrient requirements of the animal can enhance the process of animal diet formulation. The model presented here does not require any a priori information from the decision maker, eliciting his preferences through an interactive process. This is the main advantage in relation to other models found in the literature for treating the problem of nutrient variability, which introduce stochastic constraints in the single objective minimum cost model requiring fixing the level of probability desired for each one of the nutrients in advance.     

Gradient projection and local region search for multiobjective optimisation

This paper presents a new method for multiobjective optimisation based on gradient projection and local region search. The gradient projection is conducted through the identification of normal vectors of an efficient frontier. The projection of the gradient of a nonlinear utility function onto the tangent plane of the efficient frontier at a given efficient solution leads to the definition of a feasible local region in a neighbourhood of the solution. Within this local region, a better efficient solution may be sought. To implement such a gradient-based local region search scheme, a new auxiliary problem is developed. If the utility function is given explicitly, this search scheme results in an iterative optimisation algorithm capable of general nonseparable multiobjective optimisation. Otherwise, an interactive decision making algorithm is developed where the decision maker (DM) is expected to provide local preference information in order to determine trade-off directions and step sizes. Optimality conditions for the algorithms are established and the convergence of the algorithms is proven. A multiobjective linear programming (MOLP) problem is taken for example to demonstrate this method both graphically and analytically. A nonlinear multiobjective water quality management problem is finally examined to show the potential application of the method to real world decision problems.     

An interactive fuzzy satisficing method for structured multiobjective linear fractional programs with fuzzy numbers

In this paper, by considering the experts' vague or fuzzy understanding of the nature of the parameters in the problem formulation process, multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated. Using the a-level sets of fuzzy numbers, the corresponding nonfuzzy a-multiobjective linear fractional programming problem is introduced. The fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones. Through the introduction of extended Pareto optimality concepts, if the decision maker specifies the degree a and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method and Ritter's partitioning procedure are applicable. Then a linear programming-based interactive fuzzy satisficing method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an extended Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

Interactive Polyhedral Outer Approximation (IPOA) strategy for general multiobjective optimization problems

We propose an interactive polyhedral outer approximation (IPOA) method to solve a broad class of multiobjective optimization problems (MOP) with, possibly, nonlinear and nondifferentiable objective and constraint functions, and with continuous or discrete decision variables. During the interactive optimization phase, the method progressively constructs a polyhedral approximation of the decision-maker’s (DM’s) unknown preference structure and a polyhedral outer-approximation of the feasible set of MOP. The piecewise linear approximation of the DM’s preferences also provides a mechanism for testing the consistency of the DM’s assessments and removing inconsistencies; it also allows post-optimality analysis. All the feasible trial solutions are non-dominated (efficient, or Pareto-optimal) so preference assessments are made in the context of non-dominated alternatives only. Upper and lower bounds on the yet unknown optimal value are produced at every iteration, allowing terminating the search prematurely at a good-enough solution and providing information about the closeness of this solution to the optimal solution. The IPOA method includes a preliminary phase in which a limited probe of the efficient set is conducted in order to find a good initial trial solution for the interactive phase. The computational requirements of the algorithm are relatively simple. The results of an extensive computational study are reported.     

A multiobjective interactive approach to determine the optimal electricity mix in Andalucía (Spain)

It seems clear that energy production is one of the key aspects of global sustainability. Economic, social and environmental aspects must be taken into account in order to design appropriate policies and thus, multicriteria analysis becomes a very adequate tool to deal with real problems of this kind. This study was directed by the Regional Ministry of Environment of Andalucía, who wanted to know the impact on the cost and on the environmental damage of a potential mix, more focused on renewable sources. Some authorities of the Ministry acted as decision maker in the interactive process. As a result, we have built a linear multiobjective model, in order to determine the optimal electrical mix for the Spanish region of Andalucía. Namely, we determine how much electricity power should be installed and produced, by each of the eight generation systems considered (lignite, other coals, oil, natural gas, nuclear, photovoltaic, wind and mini-hydro). Apart from the economic criterion (yearly cost), we have considered the vulnerability (in terms of percentage of imported fuel) as a strategic criterion, and 12 environmental criteria, which have been derived using the Life Cycle Analysis method on the different production systems. The interactive system PROMOIN was used to solve the multiobjective problem. PROMOIN allows the decision maker to choose how to give preference information to the system, and enables changing it anytime during the solution process, which gives more flexibility to the decision maker and increases the confidence of the decision maker in the final solution.     

An interactive fuzzy satisficing method for multiobjective stochastic linear programming problems through an expectation model

For decision making problems involving uncertainty, both stochastic programming as an optimization method based on the theory of probability and fuzzy programming representing the ambiguity by fuzzy concept have been developing in various ways. In this paper, we focus on multiobjective linear programming problems with random variable coefficients in objective functions and/or constraints. For such problems, as a fusion of these two approaches, after incorporating fuzzy goals of the decision maker for the objective functions, we propose an interactive fuzzy satisficing method for the expectation model to derive a satisficing solution for the decision maker. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

Preference modelling by estimating local utility functions for multiobjective optimization

This paper is intended to design goal programming models for capturing the decision maker's (DM's) preference information and for supporting the search for the best compromise solutions in multiobjective optimization. At first, a linear goal programming model is built to estimate piecewise linear local utility functions based on pairwise comparisons of efficient solutions as well as objectives. The interactive step trade-off method (ISTM) is employed to generate a typical subset of efficient solutions of a multiobjective problem. Another general goal programming model is then constructed to embed the estimated utility functions in the original multiobjective problem for utility optimization using ordinary nonlinear programming algorithms. This technique, consisting of the ISTM method and the newly investigated search process, facilitates the identification and elimination of possible inconsistent information which may exist in the DM's preferences. It also provides various ways to carry out post-optimality analysis to test the robustness of the obtained best solutions. A modified nonlinear multiobjective management problem is taken as example to demonstrate the technique.     

Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems

As indicated by the most widely accepted classification, the Multi-Objective Mathematical Programming (MOMP) methods can be classified as a priori, interactive and a posteriori, according to the decision stage in which the decision maker expresses his/her preferences. Although the a priori methods are the most popular, the interactive and the a posteriori methods convey much more information to the decision maker. Especially, the a posteriori (or generation) methods give the whole picture (i.e. the Pareto set) to the decision maker, before his/her final choice, reinforcing thus, his/her confidence to the final decision. However, the generation methods are the less popular due to their computational effort and the lack of widely available software. The present work is an effort to effectively implement the ε-constraint method for producing the Pareto optimal solutions in a MOMP. We propose a novel version of the method (augmented ε-constraint method - AUGMECON) that avoids the production of weakly Pareto optimal solutions and accelerates the whole process by avoiding redundant iterations. The method AUGMECON has been implemented in GAMS, a widely used modelling language, and has already been used in some applications. Finally, an interactive approach that is based on AUGMECON and eventually results in the most preferred Pareto optimal solution is also proposed in the paper.     

Fuzzy programming for multiobjective 0–1 programming problems through revised genetic algorithms

Recently, genetic algorithms (GAs), a new learning paradigm that models a natural evolution mechanism, have received a great deal of attention regarding their potential as optimization techniques for solving combinatorial optimization problems. In this paper, we focus on multiobjective 0–1 programming problems as a generalization of the traditional single objective ones. By considering the imprecise nature of human judgements, we assume that the decision maker may have fuzzy goal for each of the objective functions. After eliciting the linear membership functions through the interaction with the decision maker, we adopt the fuzzy decision of Bellman and Zadeh or minimum-operator for combining them. In order to investigate the applicability of the conventional GAs for the solution of the formulated problems, a lot of numerical simulations are performed by assuming several genetic operators. Then, instead of using the penalty function for treating the constraints, we propose three types of revised GAs which generate only feasible solutions. Illustrative numerical examples demonstrate both feasibility and efficiency of the proposed methods.     

Tradeoff-based decomposition and decision-making in multiobjective programming

To facilitate the evaluation of tradeoffs and the articulation of preferences in multiple criteria decision-making, a multiobjective decomposition scheme is proposed that restructures the original problem as a collection of smaller-sized subproblems with only subsets of the original criteria. A priori preferences on objective tradeoffs are integrated into this process by modifying the ordinary Pareto order by more general domination cones, and decision makers are supported by an interactive decision-making procedure to coordinate any remaining tradeoffs using concepts of approximate efficiency. A theoretical foundation for this method is provided, and an illustrative application to multiobjective portfolio optimization is described in detail.