Takashi Hasuike

Portfolio selection problems with random fuzzy variable returns

This paper considers several portfolio selection problems including probabilistic future returns with ambiguous expected returns assumed as random fuzzy variables. Random fuzzy portfolio selection problems are formulated as nonlinear programming problems based on both stochastic and fuzzy programming approaches Since there is no efficient solution method to solve these problems directly, main problems are transformed into equivalent deterministic quadratic programming problems using probabilistic chance constraints, possibility measure and fuzzy goals, and their efficient solution methods to find a global optimal solution of each problem is constructed. Furthermore, numerical examples of portfolio selection problems are provided to illustrate our proposed models and solution methods compared with several previous basic models and to show that our proposed model is a versatile model to be applicable to various unexpected conditions.

Product mix problems considering several probabilistic conditions and flexibility of constraints

This paper considers product mix problems including randomness of future returns, ambiguity of coefficients and flexibility of upper value with respect to each constraint such as budget, human resource, time and several costs. Particularly, the flexibility is assumed to be a fuzzy goal. Then, several models based on maximizing total future profits under a level of satisfaction to each fuzzy goal are proposed. Furthermore, the model considering preference ranking to each fuzzy goal of constraints is proposed. Since these problems are basically formulated as nonlinear programming problems, the transformations into deterministic equivalent problems are introduced and the efficient solution methods are developed. A numerical example for product mix problem is given to illustrate our proposed models.

Probability maximization model of 0–1 knapsack problem with random fuzzy variables

This paper considers a new model of 0-1 knapsack problem including probabilistic coefficients with ambiguous expected returns assumed as random fuzzy variables. Since the random fuzzy 0-1 knapsack problem is not well-defined integer programming problem due to involve random fuzzy variables, it is hard to construct the efficient solution method to solve this problem directly. In this paper, using chance constraints, possibility measure and fuzzy goal based on both stochastic and fuzzy programming approaches, the main problem is transformed into a deterministic equivalent quadratic integer programming. Then, the efficient solution method to find a strict optimal solution based on dynamic programming is constructed.

A random fuzzy minimum spanning tree problem through a possibility-based value at risk model

This paper considers a minimum spanning tree problem under the situation where costs for constructing edges in a network include both fuzziness and randomness. In particular, this article focuses on the case that the edge costs are expressed by random fuzzy variables. A new decision making model based on a possibility measure and a value at risk measure is proposed in order to find a solution which fully reflects random and fuzzy information. It is shown that an optimal solution of the proposed model is obtained by a polynomial-time algorithm.

Robust shortest path problem based on a confidence interval in fuzzy bicriteria decision making

This paper proposes a model for a robust shortest path problem with random edge costs. The proposed model is formulated as a bicriteria optimization problem to minimize the total cost of the path from a start node to an end node as well as to maximize the range of the confidence interval satisfying the condition that the total cost of any path in the region is less than a target value in terms of robustness. It is not always the case that a solution that simultaneously optimizes all of the objectives exists when the objectives conflict with each other. In this paper, fuzzy goals for both objectives and the aggregation function are introduced in a solution approach that considers the decision makers preference and the subjectivity of her or his judgment in multicriteria decision making. The fuzzy bicriteria problem is transformed into a constrained shortest path problem. Furthermore, an exact algorithm based on Dijkstras algorithm and a heuristic algorithm based on the Lagrange relaxation-based algorithm are developed for the constrained shortest path problem.

Robust Expectation Optimization Model Using the Possibility Measure for the Fuzzy Random Programming Problem

This paper considers an expectation optimization model using a possibility measure to the objective function in the fuzzy random programming problem, based on possibilistic programming and stochastic programming. The main fuzzy random programming problem is not a well-defined problem due to including random variables and fuzzy numbers. Therefore, in order to solve it analytically, a criterion for goal of objective function is set and the chance constraint are introduced. Then, considering decision maker’s subjectivity and flexibility of the original plan, a fuzzy goal for each objective function is introduced. Furthermore, this paper considers that the occurrence probability of each scenario has ambiguity, and is represented as an interval value. Considering this interval of probability, a robust expectation optimization problem is proposed. Main problem is transformed into the deterministic equivalent linear programming problem, and so the analytical solution method extending previous solution approaches is constructed.

Tour route planning problem for sightseeing with the multiroute under several uncertain conditions

This paper proposes a route planning problem for sightseeing with fuzzy random variables based on constraints of required traveling times and satisfactions of the total sightseeing activity. In general, traveling times among sightseeing places and the satisfactions of activities depend on weather and climate conditions. Tourists will like to do their favorable route planning without drastically changing the tour route of usual condition such as fine even if the weather condition changes for the worse. Therefore, the fuzzy random variables for traveling times and satisfactions of activities derived from uncertainty conditions are introduced, and a new route planning problem is proposed to obtain the favorable route similar to the optimal route under the usual condition. As a basic case of fuzzy numbers, interval values and the order relation are introduced. From the order relation, the proposed model is transformed into an extended model of network optimization problems. A numerical example is provided to compare the proposed model with standard route planning problems for sightseeing.

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.

A random fuzzy programming models based on possibilistic programming

This paper considers linear programming problems where each coefficient of the objective function is expressed by a random fuzzy variable. New decision making models are proposed based on stochastic and possibilistic programming in order to maximize both of possibility and probability with respect to the objective function value. It is shown that each of the proposed models is transformed into a deterministic equivalent one. Solution algorithms using convex programming techniques and/or the bisection method are provided for obtaining an optimal solution of each model.

A Constructing Algorithm for Appropriate Piecewise Linear Membership Function based on Statistics and Information Theory

Abstract This paper proposes a constructing algorithm for an appropriate membership function to integrate the fuzzy Shannon entropy with a piecewise linear function into subjective intervals estimation by the heuristic method based on the human cognitive behavior and subjectivity under a given probability density function. It is important to set a membership function appropriately in real-world decision making. The main parts of our proposed approach are to give membership values a decision maker confidently set, and to obtain the others by solving a nonlinear mathematical programming problem objectively. It is difficult to solve the initial mathematical programming problem efficiently using previous constructing approaches. In this paper, introducing some natural assumptions in the real-world and performing deterministic equivalent transformations to the initial problem using nonlinear programming, an efficient algorithm to obtain the optimal condition of each appropriate membership value is developed.