Using aspiration levels in an interactive interior multiobjective linear programming algorithm

In this paper, we propose the use of an interior-point linear programming algorithm for multiple objective linear programming (MOLP) problems. At each iteration, a Decision Maker (DM) is asked to specify aspiration levels for the various objectives, and an achievement scalarizing function is applied to project aspiration levels onto the nondominated set. The interior-point algorithm is used to find an interior solution path from a starting solution to a nondominated solution corresponding to the optimum of the achievement scalarizing function. The proposed approach allows the DM to re-specify aspiration levels during the solution process and thus steer the interior solution path toward different areas in objective space. We illustrate the use of the approach with a numerical example.     

The optimality conditions for optimization problems with convex constraints and multiple fuzzy-valued objective functions

The optimality conditions for multiobjective programming problems with fuzzy-valued objective functions are derived in this paper. The solution concepts for these kinds of problems will follow the concept of nondominated solution adopted in the multiobjective programming problems. In order to consider the differentiation of fuzzy-valued functions, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the optimality conditions for obtaining the (strongly, weakly) Pareto optimal solutions are elicited naturally by introducing the Lagrange multipliers.     

The Karush-Kuhn-Tucker optimality conditions for multi-objective programming problems with fuzzy-valued objective functions

The KKT optimality conditions for multiobjective programming problems with fuzzy-valued objective functions are derived in this paper. The solution concepts are proposed by defining an ordering relation on the class of all fuzzy numbers. Owing to this ordering relation being a partial ordering, the solution concepts proposed in this paper will follow from the similar solution concept, called Pareto optimal solution, in the conventional multiobjective programming problems. In order to consider the differentiation of fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the KKT optimality conditions are elicited naturally by introducing the Lagrange function multipliers.     

The aspiration level interactive method (AIM) reconsidered: Robustness of solutions

In this paper we explore theory and practice for the aspiration-level interactive model (AIM), a useful decision tool that takes advantage of the concepts of satisficing as well as other concepts of multiple criteria decision making (MCDM). We examine the relationship between aspiration levels and their mapped-to solutions in the MCDM context using AIM.We extend the concept of robustness in decision making, by defining a solution to be robust if many (suitably defined) aspiration levels map to it. We use simulation to help explore robustness, by generating three groups of random test problems. Each problem has twenty alternatives. For each group of problems, the top 5 (most-mapped-to) alternatives out of 20 are mapped to by at least 50% of the aspiration levels. We also relate the concept of robustness in decision making to the ideas of simply ranking alternatives using equal weights. There is a strong correlation between the robustness ranking and the equal-weights ranking. Based on our analyses, we then randomly generate additional problems to explore certain other factors. We also discuss practical aspects of robustness.     

The stochastic bottleneck linear programming problem

In this paper we consider some stochastic bottleneck linear programming problems. We overview the solution methods in the literature. In the case when the coefficients of the objective functions are simple randomized, the minimum-risk approach will be used for solving these problems. We prove that, under some positivity conditions, these stochastic problems are reduced to certain deterministic bottleneck linear problems. An application of these problems to bottleneck spanning tree problems is given. Two simple numerical examples are presented. This paper was written when I.M. Stancu-Minasian was visiting the Instituto Complutense de Análisis Económico, in the Universidad Complutensen de Madrid, from October 1, 1997 to November 15, 1997 and from October 24, 1998 to November, 9, 1998, as invited researcher. He is grateful to the Institution.     

Analysis and comparisons of some solution concepts for stochastic programming problems

The aim of this study is to analyse the resolution of Stochastic Programming Problems in which the objective function depends on parameters which are continuous random variables with a known distribution probability. In the literature on these questions different solution concepts have been defined for problems of these characteristics. These concepts are obtained by applying a transformation criterion to the stochastic objective which contains a statistical feature of the objective, implying that for the same stochastic problem there are different optimal solutions available which, in principle, are not comparable. Our study analyses and establishes some relations between these solution concepts. The work of these authors was supported byMinisterio de Ciencia y Tecnología andConsejería de Educación y Ciencia, Junta de Andalucía.     

Finding an efficient solution to linear bilevel programming problem: An effective approach

Multilevel programming is developed to solve the decentralized problem in which decision makers (DMs) are often arranged within a hierarchical administrative structure. The linear bilevel programming (BLP) problem, i.e., a special case of multilevel programming problems with a two level structure, is a set of nested linear optimization problems over polyhedral set of constraints. Two DMs are located at the different hierarchical levels, both controlling one set of decision variables independently, with different and perhaps conflicting objective functions. One of the interesting features of the linear BLP problem is that its solution may not be Paretooptimal. There may exist a feasible solution where one or both levels may increase their objective values without decreasing the objective value of any level. The result from such a system may be economically inadmissible. If the decision makers of the two levels are willing to find an efficient compromise solution, we propose a solution procedure which can generate effcient solutions, without finding the optimal solution in advance. When the near-optimal solution of the BLP problem is used as the reference point for finding the efficient solution, the result can be easily found during the decision process.     

An interior multiobjective primal-dual linear programming algorithm using approximated gradients and sequential generation of anchor points

An algorithm for addressing multiple objective linear programming (MOLP) problems is presented. The algorithm modifies the path-following primal-dual algorithm to MOLP problems by using the single objective algorithm to generate interior search directions and later combine them to derive a single direction along which to step to the next iterate. Combining the different interior search directions is done by interacting with a Decision Maker (DM) to obtain locally-relevant preference information for the value vectors along these directions. This preference information is then used to derive an approximation to the gradient of an implicity-known utility function, and using a projection of this gradient provides a direction gradient of an implicitly-known utility function, and using a projection of this gradient provides a direction vector along which we step to the next iterate. At each iteration the algorithm also generates boundary points that aid in deriving the combined search direction. We refer to these boundary points, generated sequentially during the process, as anchor points that serve as candidate solutions at which to terminate the iterative process.     

Interactive fuzzy programming approach to Bi-level quadratic fractional programming problems

In this paper we propose an interactive fuzzy programming method for obtaining a satisfactory solution to a “bi-level quadratic fractional programming problem” with two decision makers (DMs) interacting with their optimal solutions. After determining the fuzzy goals of the DMs at both levels, a satisfactory solution is efficiently derived by updating the satisfactory level of the DM at the upper level with consideration of overall satisfactory balance between both levels. Optimal solutions to the formulated programming problems are obtained by combined use of some of the proper methods. Theoretical results are illustrated with the help of a numerical example.     

A unified approach for different concepts of robustness and stochastic programming via non-linear scalarizing functionals

Abstract

We show that many different concepts of robustness and of stochastic programming can be described as special cases of a general non-linear scalarization method by choosing the involved parameters and sets appropriately. This leads to a unifying concept which can be used to handle robust and stochastic optimization problems. Furthermore, we introduce multiple objective (deterministic) counterparts for uncertain optimization problems and discuss their relations to well-known scalar robust optimization problems by using the non-linear scalarization concept. Finally, we mention some relations between robustness and coherent risk measures.     

Probability maximization models for portfolio selection under ambiguity

This paper considers several probability maximization models for multi-scenario portfolio selection problems in the case that future returns in possible scenarios are multi-dimensional random variables. In order to consider occurrence probabilities and decision makers’ predictions with respect to all scenarios, a portfolio selection problem setting a weight with flexibility to each scenario is proposed. Furthermore, by introducing aspiration levels to occurrence probabilities or future target profit and maximizing the minimum aspiration level, a robust portfolio selection problem is considered. Since these problems are formulated as stochastic programming problems due to the inclusion of random variables, they are transformed into deterministic equivalent problems introducing chance constraints based on the stochastic programming approach. Then, using a relation between the variance and absolute deviation of random variables, our proposed models are transformed into linear programming problems and efficient solution methods are developed to obtain the global optimal solution. Furthermore, a numerical example of a portfolio selection problem is provided to compare our proposed models with the basic model.     

Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming

Decomposition has proved to be one of the more effective tools for the solution of large-scale problems, especially those arising in stochastic programming. A decomposition method with wide applicability is Benders' decomposition, which has been applied to both stochastic programming as well as integer programming problems. However, this method of decomposition relies on convexity of the value function of linear programming subproblems. This paper is devoted to a class of problems in which the second-stage subproblem(s) may impose integer restrictions on some variables. The value function of such integer subproblem(s) is not convex, and new approaches must be designed. In this paper, we discuss alternative decomposition methods in which the second-stage integer subproblems are solved using branch-and-cut methods. One of the main advantages of our decomposition scheme is that Stochastic Mixed-Integer Programming (SMIP) problems can be solved by dividing a large problem into smaller MIP subproblems that can be solved in parallel. This paper lays the foundation for such decomposition methods for two-stage stochastic mixed-integer programs.     

Linear programming with fuzzy parameters: An interactive method resolution

This paper proposes a method for solving linear programming problems where all the coefficients are, in general, fuzzy numbers. We use a fuzzy ranking method to rank the fuzzy objective values and to deal with the inequality relation on constraints. It allows us to work with the concept of feasibility degree. The bigger the feasibility degree is, the worst the objective value will be. We offer the decision-maker (DM) the optimal solution for several different degrees of feasibility. With this information the DM is able to establish a fuzzy goal. We build a fuzzy subset in the decision space whose membership function represents the balance between feasibility degree of constraints and satisfaction degree of the goal. A reasonable solution is the one that has the biggest membership degree to this fuzzy subset. Finally, to illustrate our method, we solve a numerical example.     

The equality relations in scientific computing

The equality relation (more generally, the ordering relations) in floating point arithmetic is the exact translation of the mathematical equality relation. Because of the propagation of round-off errors, the floating point arithmetic is not the exact representation of the theoretical arithmetic which is continuous on the real numbers.This leads to some incoherence when the equality concept is used in floating point arithmetic. A well known example is the detection of a zero element in the pivoting column and equation when applying Gaussian elimination, which is almost impossible in floating point arithmetic.We shall begin by showing the inadequacy of the equality relation used in floating point arithmetic (we will call it floating point equality), and then introduce two new concepts: stochastic numbers and the equality relation between such numbers which will be called the stochastic equality. We will show how these concepts allow to recover the coherence between the arithmetic operators and the ordering relations that was missing in floating point computations.     

A new adaptive Runge–Kutta method for stochastic differential equations

In this paper, we will present a new adaptive time stepping algorithm for strong approximation of stochastic ordinary differential equations. We will employ two different error estimation criteria for drift and diffusion terms of the equation, both of them based on forward and backward moves along the same time step. We will use step size selection mechanisms suitable for each of the two main regimes in the solution behavior, which correspond to domination of the drift-based local error estimator or diffusion-based one. Numerical experiments will show the effectiveness of this approach in the pathwise approximation of several standard test problems.     

The C 3 Theorem and a D 2 Algorithm for Large Scale Stochastic Mixed-Integer Programming: Set Convexification

This paper considers the two-stage stochastic integer programming problem, with an emphasis on instances in which integer variables appear in the second stage. Drawing heavily on the theory of disjunctive programming, we characterize convexifications of the second stage problem and develop a decomposition-based algorithm for the solution of such problems. In particular, we verify that problems with fixed recourse are characterized by scenario-dependent second stage convexifications that have a great deal in common. We refer to this characterization as the C³ (Common Cut Coefficients) Theorem. Based on the C³ Theorem, we develop a decomposition algorithm which we refer to as Disjunctive Decomposition (D²). In this new class of algorithms, we work with master and subproblems that result from convexifications of two coupled disjunctive programs. We show that when the second stage consists of 0-1 MILP problems, we can obtain accurate second stage objective function estimates after finitely many steps. This result implies the convergence of the D² algorithm.This research was funded by NSF grants DMII 9978780 and CISE 9975050.     

A visual interactive approach for scenario-based stochastic multi-objective problems and an application

In many practical applications of stochastic programming, discretization of continuous random variables in the form of a scenario tree is required. In this paper, we deal with the randomness in scenario generation and present a visual interactive method for scenario-based stochastic multi-objective problems. The method relies on multi-variate statistical analysis of solutions obtained from a multi-objective stochastic problem to construct joint confidence regions for the objective function values. The decision maker (DM) explores desirable parts of the efficient frontier using a visual representation that depicts the trajectories of the objective function values within confidence bands. In this way, we communicate the effects of randomness inherent in the problem to the DM to help her understand the trade-offs and the levels of risk associated with each objective.     

Efficient Solution Concepts and Their Relations in Stochastic Multiobjective Programming

In this work, different concepts of efficient solutions to problems of stochastic multiple-objective programming are analyzed. We center our interest on problems in which some of the objective functions depend on random parameters. The existence of different concepts of efficiency for one single stochastic problem, such as expected-value efficiency, minimum-risk efficiency, etc., raises the question of their quality. Starting from this idea, we establish some relationships between the different concepts. Our study enables us to determine what type of efficient solutions are obtained by each of these concepts.     

Relations between stochastic and parametric programming for decision problems with a random objective function

In this paper we study the relation between the general concept for an optimal solution for stochastic programming problems with a random objective function-the concept of an £-efficient solution-and the associated parametric problem, We show that it is possible under certain assumptions to obtain some or even all £-efficient solutions of the stochastic problem by solving the parametric problem with respect to a certain parameter set.     

Evolutionary multiobjective optimization in noisy problem environments

This paper presents a multiobjective evolutionary algorithm (MOEA) capable of handling stochastic objective functions. We extend a previously developed approach to solve multiple objective optimization problems in deterministic environments by incorporating a stochastic nondomination-based solution ranking procedure. In this study, concepts of stochastic dominance and significant dominance are introduced in order to better discriminate among competing solutions. The MOEA is applied to a number of published test problems to assess its robustness and to evaluate its performance relative to NSGA-II. Moreover, a new stopping criterion is proposed, which is based on the convergence velocity of any MOEA to the true Pareto optimal front, even if the exact location of the true front is unknown. This stopping criterion is especially useful in real-world problems, where finding an appropriate point to terminate the search is crucial.