Improving the computational efficiency in a global formulation (GLIDE) for interactive multiobjective optimization

In this paper, we present a new general formulation for multiobjective optimization that can accommodate several interactive methods of different types (regarding various types of preference information required from the decision maker). This formulation provides a comfortable implementation framework for a general interactive system and allows the decision maker to conveniently apply several interactive methods in one solution process. In other words, the decision maker can at each iteration of the solution process choose how to give preference information to direct the interactive solution process, and the formulation enables changing the type of preferences, that is, the method used, whenever desired. The first general formulation, GLIDE, included eight interactive methods utilizing four types of preferences. Here we present an improved version where we pay special attention to the computational efficiency (especially significant for large and complex problems), by eliminating some constraints and parameters of the original formulation. To be more specific, we propose two new formulations, depending on whether the multiobjective optimization problem to be considered is differentiable or not. Some computational tests are reported showing improvements in all cases. The generality of the new improved formulations is supported by the fact that they can accommodate six interactive methods more, that is, a total of fourteen interactive methods, just by adjusting parameter values.     

An interactive approach for multiobjective decision making

We develop an interactive approach for multiobjective decision-making problems, where the solution space is defined by a set of constraints. We first reduce the solution space by eliminating some undesirable regions. We generate solutions (partition ideals) that dominate portions of the efficient frontier and the decision maker (DM) compares these with feasible solutions. Whenever the decision maker prefers a feasible solution, we eliminate the region dominated by the partition ideal. We then employ an interactive search method on the reduced solution space to help the DM further converge toward a highly preferred solution. We demonstrate our approach and discuss some variations.     

An interactive method for multiple-objective mathematical programming problems

An interactive method is developed for solving the general nonlinear multiple objective mathematical programming problems. The method asks the decision maker to provide partial information (local tradeoff ratios) about his utility (preference) function at each iteration. Using the information, the method generates an efficient solution and presents it to the decision maker. In so doing, the best compromise solution is sought in a finite number of iterations. This method differs from the existing feasible direction methods in that (i) it allows the decision maker to consider only efficient solutions throughout, (ii) the requirement of line search is optional, and (iii) it solves the problems with linear objective functions and linear utility function in one iteration. Using various problems selected from the literature, five line search variations of the method are tested and compared to one another. The nonexisting decision maker is simulated using three different recognition levels, and their impact on the method is also investigated.     

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.     

An interactive evolutionary multi-objective optimization algorithm with a limited number of decision maker calls

This paper presents a preference-based method to handle optimization problems with multiple objectives. With an increase in the number of objectives the computational cost in solving a multi-objective optimization problem rises exponentially, and it becomes increasingly difficult for evolutionary multi-objective techniques to produce the entire Pareto-optimal front. In this paper, an evolutionary multi-objective procedure is combined with preference information from the decision maker during the intermediate stages of the algorithm leading to the most preferred point. The proposed approach is different from the existing approaches, as it tries to find the most preferred point with a limited budget of decision maker calls. In this paper, we incorporate the idea into a progressively interactive technique based on polyhedral cones. The idea is also tested on another progressively interactive approach based on value functions. Results are provided on two to five-objective unconstrained as well as constrained test problems.     

Interactive multiple objective programming subject to flexible constraints

An interactive decision support system is introduced which aids in solving multiple objective programming problems subject to strict and flexible constraints. Integral part is an extension of a well-known fuzzy sets approach evaluating possible solutions by their degrees of membership to objectives and constraints. This approach is linked to classical multiple objective programming models. If the decision maker cannot determine membership functions a priori the system suggests functions dependent on the given information and interactive modifications are allowed.     

The tolerance approach in multiobjective linear fractional programming

When solving a multiobjective programming problem by the weighted sum approach, weights represent the relative importance associated to the objectives. As these values are usually imprecise, it is important to analyze the sensitivity of the solution under possible deviations on the estimated values. In this sense, the tolerance approach provides a direct measure of how weights may vary simultaneously and independently from their estimated values while still retaining the same efficient solution. This paper provides an explicit expression to the maximum tolerance on weights in a multiobjective linear fractional programming problem when all the denominators are equal. An application is also presented to illustrate how the results may help the decision maker to choose a most satisfactory solution in a production problem.     

The specification of a reference direction using the analytic hierarchy process

In this paper we deal with the use of the Analytic Hierarchy Process (AHP) for specifying a reference direction, which is used to find a search direction in the visual interactive method developed by Korhonen and Laakso for multiple criteria problems. The reference direction describes how the decision maker would like to improve the values of multiple objectives and we show that the AHP is a convenient way to structure requisite preference information.     

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.     

ISTMO: An interval reference point-based method for stochastic multiobjective programming problems

In this paper, we present an interactive algorithm (ISTMO) for stochastic multiobjective problems with continuous random variables. This method combines the concept of probability efficiency for stochastic problems with the reference point philosophy for deterministic multiobjective problems. The decision maker expresses her/his references by dividing the variation range of each objective into intervals, and by setting the desired probability for each objective to achieve values belonging to each interval. These intervals may also be redefined during the process. This interactive procedure helps the decision maker to understand the stochastic nature of the problem, to discover the risk level (s)he is willing to assume for each objective, and to learn about the trade-offs among the objectives.     

An interactive weight space reduction procedure for nonlinear multiple objective mathematical programming

To make a decision that is defined by multiple, conflicting objectives it is necessary to know the relative importance of the different objectives. In this paper we present an interactive method and the underlying theory for solving multiple objective mathematical programming problems defined by a convex feasible region and concave, continuously differentiable objective functions. The relative importance of the different objectives for a decision maker is elicited by using binary comparisons of objective function vectors. The method is cognitively easy to use and in test problems has rapidly converged to an optimal solution.     

Interactive procedure for a multiobjective stochastic discrete dynamic problem

Multiple objectives and dynamics characterize many sequential decision problems. In the paper we consider returns in partially ordered criteria space as a way of generalization of single criterion dynamic programming models to multiobjective case. In our problem evaluations of alternatives with respect to criteria are represented by distribution functions. Thus, the overall comparison of two alternatives is equivalent to the comparison of two vectors of probability distributions. We assume that the decision maker tries to find a solution preferred to all other solutions (the most preferred solution). In the paper a new interactive procedure for stochastic, dynamic multiple criteria decision making problem is proposed. The procedure consists of two steps. First, the Bellman principle is used to identify the set of efficient solutions. Next interactive approach is employed to find the most preferred solution. A numerical example and a real-world application are presented to illustrate the applicability of the proposed technique.     

Lexicographic improvement of the target values in convex goal programming

The aim of this paper is to carry out an exhaustive post optimization analysis in a Convex Goal Programming problem, so as to study the possible existence of satisfying solutions for different levels of the target values. To this end, an interactive algorithm is proposed, which allows us to improve the values of the objective functions, after obtaining a satisfying solution, if such a solution exists, in such a way that a Pareto optimal solution is finally reached, through a successive actualization of such target values. This way, the target values are lexicographically improved, according to the priority order previously given by the decision maker, in an attempt to harmonize the concepts of satisfying and efficient solutions, which have traditionally been in conflict.     

An interactive satisficing method based on alternative tolerance for fuzzy multiple objective optimization

An interactive satisficing method based on alternative tolerance is proposed for fuzzy multiple objective optimization. The new tolerances of the dissatisficing objectives are generated using an auxiliary programming problem. According to the alternative tolerant limits, either the membership functions are changed, or the objective constraints are added. The lexicographic two-phase programming is implemented to find the final solution. The results of the dissatisficing objectives are iteratively improved. The presented method not only acquires the efficient or weak efficient solution of all the objectives, but also satisfies the progressive preference of decision maker. Numerical examples show its power.     

Pruned Pareto-optimal sets for the system redundancy allocation problem based on multiple prioritized objectives

In this paper, a new methodology is presented to solve different versions of multi-objective system redundancy allocation problems with prioritized objectives. Multi-objective problems are often solved by modifying them into equivalent single objective problems using pre-defined weights or utility functions. Then, a multi-objective problem is solved similar to a single objective problem returning a single solution. These methods can be problematic because assigning appropriate numerical values (i.e., weights) to an objective function can be challenging for many practitioners. On the other hand, methods such as genetic algorithms and tabu search often yield numerous non-dominated Pareto optimal solutions, which makes the selection of one single best solution very difficult. In this research, a tabu search meta-heuristic approach is used to initially find the entire Pareto-optimal front, and then, Monte-Carlo simulation provides a decision maker with a pruned and prioritized set of Pareto-optimal solutions based on user-defined objective function preferences. The purpose of this study is to create a bridge between Pareto optimality and single solution approaches.     

An Interactive Method for 0-1 Multiobjective Problems Using Simulated Annealing and Tabu Search

This paper presents an interactive method for solving general 0-1 multiobjective linear programs using Simulated Annealing and Tabu Search. The interactive protocol with the decision maker is based on the specification of reservation levels for the objective function values. These reservation levels narrow the scope of the search in each interaction in order to identify regions of major interest to the decision maker. Metaheuristic approaches are used to generate potentially nondominated solutions in the computational phases. Generic versions of Simulated Annealing and Tabu Search for 0-1 single objective linear problems were developed which include a general routine for repairing unfeasible solutions. This routine improves significantly the results of single objective problems and, consequently, the quality of the potentially nondominated solutions generated for the multiobjective problems. Computational results and examples are presented.     

An interactive desirability function method to multiresponse optimization

Multiresponse optimization problems often involve incommensurate and conflicting responses. To obtain a satisfactory compromise in such a case, a decision maker (DM)’s preference information on the tradeoffs among the responses should be incorporated into the problem. This paper proposes an interactive method based on the desirability function approach to facilitate the preference articulation process. The proposed method allows the DM to adjust any of the preference parameters, namely, the shape, bound, and target of a desirability function in a single, integrated framework. The proposed method would be highly effective in generating a compromise solution that is faithful to the DM’s preference structure.     

A concept of a robust solution of a multicriterial linear programming problem

A new concept of a robust solution of a multicriterial linear programming problem is proposed. The robust solution is understood here as the best starting point, prepared while the preferences of the decision maker with respect to the criteria are still unknown, for the adaptation of the solution to the preferences of the decision maker, once they are finally known. The objective is the total cost of the initial preparation and of the later potential adaptation of the solution. In the starting robust solution the decision variables may have interval values. The problem can be solved by means of the simplex algorithm. A numerical example illustrates the approach.     

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.