Genetic based fuzzy goal programming for multiobjective chance constrained programming problems with continuous random variables

Solution procedure consisting of fuzzy goal programming and stochastic simulation-based genetic algorithm is presented, in this article, to solve multiobjective chance constrained programming problems with continuous random variables in the objective functions and in chance constraints. The fuzzy goal programming formulation of the problem is developed first using the stochastic simulation-based genetic algorithm. Without deriving the deterministic equivalent, chance constraints are used within the genetic process and their feasibilities are checked by the stochastic simulation technique. The problem is then reduced to an ordinary chance constrained programming problem. Again using the stochastic simulation-based genetic algorithm, the highest membership value of each of the membership goal is achieved and thereby the most satisfactory solution is obtained. The proposed procedure is illustrated by a numerical example.     

Fuzzy random programming models for location-allocation problem with applications

Three types of fuzzy random programming models based on the mean chance for the capacitated location-allocation problem with fuzzy random demands are proposed according to different criteria, including the expected cost minimization model, the α-cost minimization model, and the chance maximization model. In order to solve the proposed models, some hybrid intelligent algorithms are designed by integrating the network simplex algorithm, fuzzy random simulation, and genetic algorithm. Finally, some numerical examples about a container freight station problem are given to illustrate the effectiveness of the devised algorithms.     

Interactive multiobjective fuzzy random programming through the level set-based probability model

This paper considers a multiobjective linear programming problem involving fuzzy random variable coefficients. A new fuzzy random programming model is proposed by extending the ideas of level set-based optimality and a stochastic programming model. The original problem involving fuzzy random variables is transformed into a deterministic equivalent problem through the proposed model. An interactive algorithm is provided to obtain a satisficing solution for a decision maker from among a set of newly defined Pareto optimal solutions. It is shown that an optimal solution of the problem to be solved iteratively in the interactive algorithm is analytically obtained by a combination of the bisection method and the simplex method.     

Taylor series approach to fuzzy multiobjective linear fractional programming

This paper presents the use of a Taylor series for fuzzy multiobjective linear fractional programming problems (FMOLFP). The Taylor series is a series expansion that a representation of a function. In the proposed approach, membership functions associated with each objective of fuzzy multiobjective linear fractional programming problem transformed by using a Taylor series are unified. Thus, the problem is reduced to a single objective. Practical applications and numerical examples are used in order to show the efficiency and superiority of the proposed approach.     

On solving a multiobjective fixed charge problem with imprecise fractional objectives

The fixed charge problem is a special type of nonlinear programming problem which forms the basis of many industry problems wherein a charge is associated with performing an activity. In real world situations, the information provided by the decision maker regarding the coefficients of the objective functions may not be of a precise nature. This paper aims to describe a solution algorithm for solving such a fixed charge problem having multiple fractional objective functions which are all of a fuzzy nature. The enumerative technique developed not only finds the set of efficient solutions but also a corresponding fuzzy solution, enabling the decision maker to operate in the range obtained. A real life numerical example in the context of the ship routing problem is presented to illustrate the proposed method.     

Expected model for portfolio selection with random fuzzy returns

This paper researches portfolio selection problem in combined uncertain environment of randomness and fuzziness. Due to the complexity of the security market, expected values of the security returns may not be predicted accurately. In the paper, expected returns of securities are assumed to be given by fuzzy variables. Security returns are regarded as random fuzzy variables, i.e. random returns with fuzzy expected values. Following Markowitz's idea of quantifying investment return by the expected value of the portfolio and risk by the variance, a new type of mean–variance model is proposed. In addition, a hybrid intelligent algorithm is provided to solve the new model problem. A numeral example is also presented to illustrate the optimization idea and the effectiveness of the proposed algorithm.     

A new perspective for optimal portfolio selection with random fuzzy returns

The aim of this paper is to solve the portfolio selection problem when security returns contain both randomness and fuzziness. Utilizing a different perspective, this paper gives a new definition of risk for random fuzzy portfolio selection. A new optimal portfolio selection model is proposed based on this new definition of risk. A new hybrid intelligent algorithm is designed for solving the new optimization problem. In the proposed new algorithm, neural networks are employed to calculate the expected value and the chance value. These greatly reduce the computational work and speed up the process of solution as compared with the random fuzzy simulation used in our previous algorithm. A numerical example is also presented to illustrate the new modelling idea and the proposed new algorithm.     

Commentary on “On solving a multiobjective fixed charge problem with imprecise fractional objectives”

Upmanyu and Saxena (Applied Soft Computing 40 (2016) 64–69) proposed a method for solving a multiobjective fixed charge problem having multiple fractional objective functions which are all of a fuzzy nature. The aim of this note is to aware the researchers that the method, proposed by Upmanyu and Saxena, is not valid and hence, to propose a method for solving this type of fixed charge problem is still an open challenging research problem.     

Asset portfolio optimization using fuzzy mathematical programming

By morphing mean-variance optimization (MVO) portfolio model into semi-absolute deviation (SAD) model, we apply multi criteria decision making (MCDM) via fuzzy mathematical programming to develop comprehensive models of asset portfolio optimization (APO) for the investors’ pursuing either of the aggressive or conservative strategies.     

A hybrid metaheuristic for multiobjective unconstrained binary quadratic programming

The conventional unconstrained binary quadratic programming (UBQP) problem is known to be a unified modeling and solution framework for many combinatorial optimization problems. This paper extends the single-objective UBQP to the multiobjective case (mUBQP) where multiple objectives are to be optimized simultaneously. We propose a hybrid metaheuristic which combines an elitist evolutionary multiobjective optimization algorithm and a state-of-the-art single-objective tabu search procedure by using an achievement scalarizing function. Finally, we define a formal model to generate mUBQP instances and validate the performance of the proposed approach in obtaining competitive results on large-size mUBQP instances with two and three objectives.     

The quadratic programming problem with fuzzy relation inequality constraints

The quadratic programming has been widely applied to solve real world problems. The quadratic functions are often applied in the inventory management, portfolio selection, engineering design, molecular study, and economics, etc. Fuzzy relation inequalities (FRI) are important elements of fuzzy mathematics, and they have recently been widely applied in the fuzzy comprehensive evaluation and cybernetics. In view of the importance of quadratic functions and FRI, we present a fuzzy relation quadratic programming model with a quadratic objective function subject to the max-product fuzzy relation inequality constraints. Some sufficient conditions are presented to determine its optimal solution in terms of the maximum solution or the minimal solutions of its feasible domain. Also, some simplification operations have been given to accelerate the resolution of the problem by removing the components having no effect on the solution process. The simplified problem can be converted into a traditional quadratic programming problem. An algorithm is also proposed to solve it. Finally, some numerical examples are given to illustrate the steps of the algorithm.     

Risk assessment of serious crime with fuzzy random theory

This paper presents a novel approach for assessing the potential risk of serious crime events (e.g. terrorist attack). The modelling and assessment of such risk is carried out under uncertain circumstances because of both the randomness and fuzziness inherent in crime data. The approach is based on fuzzy random theory that complements probability theory, with an additional dimension of imprecision. This allows for potential loss caused by a crime to be expressed as a fuzzy random variable. Crime risk is therefore estimated as the mean chance of a fuzzy random event, where the resulting loss reaches a given confidence level. The concept of the average loss per unit of time is also introduced, in order to calculate the rate at which the loss may increase due to possible crime events. The work is compared with typical existing approaches and supported with examples throughout that illustrate its utility.     

Stability and regularization of the lexicographic vector problem of quadratic discrete programming

The paper studies the bounds of variation of input parameters for a vector quadratic discrete optimization problem, which do not expand the set of lexicographic optima. A stability criterion is described and a regularization method is presented, which makes it possible to pass from a possibly unstable problem to a series of perturbed stable problems with a previous set of lexicographic optima. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 54–62, March–April, 2000.     

Duality theory in fuzzy mathematical programming problems with fuzzy coefficients

In this paper, the notions of subgradient, subdifferential, and differential with respect to convex fuzzy mappings are investigated, which provides the basis for the fuzzy extremum problem theory. We consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and/or sufficient optimality conditions. Furthermore, the concept of saddle-points and minimax theorems under fuzzy environment is discussed. The results obtained are used to formulate the Lagrangian dual of fuzzy programming. Under certain fuzzy convexity assumptions, KKT conditions for fuzzy programming are derived, and the “perturbed” convex fuzzy programming is considered. Finally, these results are applied to fuzzy linear programming and fuzzy quadratic programming.     

Fuzzy random programming with equilibrium chance constraints

To model fuzzy random decision systems, this paper first defines three kinds of equilibrium chances via fuzzy integrals in the sense of Sugeno. Then a new class of fuzzy random programming problems is presented based on equilibrium chances. Also, some convex theorems about fuzzy random linear programming problems are proved, the results provide us methods to convert primal fuzzy random programming problems to their equivalent stochastic convex programming ones so that both the primal problems and their equivalent problems have the same optimal solutions and the techniques developed for stochastic convex programming can apply. After that, a solution approach, which integrates simulations, neural network and genetic algorithm, is suggested to solve general fuzzy random programming problems. At the end of this paper, three numerical examples are provided. Since the equivalent stochastic programming problems of the three examples are very complex and nonconvex, the techniques of stochastic programming cannot apply. In this paper, we solve them by the proposed hybrid intelligent algorithm. The results show that the algorithm is feasible and effectiveness.     

Ensemble of multi-objective metaheuristic algorithms for multi-objective unconstrained binary quadratic programming problem

Metaheuristics have been widely utilized for solving NP-hard optimization problems. However, these algorithms usually perform differently from one problem to another, i.e., one may be effective on a problem but performs badly on another problem. Therefore, it is difficult to choose the best algorithm in advance for a given problem. In contrast to selecting the best algorithm for a problem, selection hyper-heuristics aim at performing well on a set of problems (instances). This paper proposes a selection hyper-heuristic based algorithm for multi-objective optimization problems. In the proposed algorithm, multiple metaheuristics exhibiting different search behaviors are managed and controlled as low-level metaheuristics in an algorithm pool, and the most appropriate metaheuristic is selected by means of a performance indicator at each search stage. To assess the performance of the proposed algorithm, an implementation of the algorithm containing four metaheuristics is proposed and tested for solving multi-objective unconstrained binary quadratic programming problem. Experimental results on 50 benchmark instances show that the proposed algorithm can provide better overall performance than single metaheuristics, which demonstrates the effectiveness of the proposed algorithm.     

Hedging an option portfolio with minimum transaction lots: A fuzzy goal programming problem

Options are designed to hedge against risks to their underlying assets such as stocks. One method of forming option-hedging portfolios is using stochastic programming models. Stochastic programming models depend heavily on scenario generation, a challenging task. Another method is neutralizing the Greek risks derived from the Black–Scholes formula for pricing options. The formula expresses the option price as a function of the stock price, strike price, volatility, risk-free interest rate, and time to maturity. Greek risks are the derivatives of the option price with respect to these variables. Hedging Greek risks requires no human intervention for generating scenarios. Linear programming models have been proposed for constructing option portfolios with neutralized risks and maximized investment profit. However, problems with these models exist. First, feasible solutions that can perfectly neutralize the Greek risks might not exist. Second, models that involve multiple assets and their derivatives were incorrectly formulated. Finally, these models lack practicability because they consider no minimum transaction lots. Considering minimum transaction lots can exacerbate the infeasibility problem. These problems must be resolved before option hedging models can be applied further. This study presents a revised linear programming model for option portfolios with multiple underlying assets, and extends the model by incorporating it with a fuzzy goal programming method for considering minimum transaction lots. Numerical examples show that current models failed to obtain feasible solutions when minimum transaction lots were considered. By contrast, while the proposed model solved the problems efficiently.     

A new algorithm for resolution of the quadratic programming problem with fuzzy relation inequality constraints

The minimization problem of a quadratic objective function with the max-product fuzzy relation inequality constraints is studied in this paper. In this problem, its objective function is not necessarily convex. Hence, its Hessian matrix is not necessarily positive semi-definite. Therefore, we cannot apply the modified simplex method to solve this problem, in a general case. In this paper, we firstly study the structure of its feasible domain. We then use some properties of n × n real symmetric indefinite matrices, Cholesky’s decomposition, and the least square technique, and convert the problem to a separable programming problem. Furthermore, a relation in terms of a closed form is presented to solve it. Finally, an algorithm is proposed to solve the original problem. An application example in the economic area is given to illustrate the problem. Of course, there are other application examples in the area of digital data service and reliability engineering.     

Stability analysis of efficient solutions in multiobjective integer programming: A case study in load management

This paper presents a multiobjective linear integer programming model for supporting the choice of remote load control strategies in electric distribution network management. The model takes into account the main concerns in load management, considering three objective functions: minimization of the peak demand as perceived by the distribution network dispatch center, maximization of the utility profit associated with the energy services delivered by the controlled loads and minimization of the discomfort caused to consumers. The problem was analyzed using an interactive reference point method for multiobjective integer (and mixed-integer) linear programming. This approach exploits the use of the branch-and-bound algorithm for solving the reference point scalarizing programs through which efficient solutions are computed. Post-optimality techniques enable a stability analysis of the efficient solutions by means of computing and displaying graphically sets of reference points that correspond to the same solution.     

An integrated multi-objective Markowitz–DEA cross-efficiency model with fuzzy returns for portfolio selection problem

In this paper, a novel multi objective model is proposed for portfolio selection. The proposed model incorporates the DEA cross-efficiency into Markowitz mean–variance model and considers return, risk and efficiency of the portfolio. Also, in order to take uncertainty in proposed model, the asset returns are considered as trapezoidal fuzzy numbers. Due to the computational complication of the proposed model, the second version of non-dominated sorting genetic algorithm (NSGA-II) is applied. To illustrate the performance of our model, the model is implemented for 52 firms listed in stock exchange market of Iran and the results are analyzed. The results show that the proposed model is suitable in compared with Markowitz and DEA models due to considering return, risk and efficiency, simultaneously.